Datasets:

JohnYang88

/

lean-dojo-mathlib4

like 0

https://github.com/leanprover-community/mathlib4

3ce43c18f614b76e161f911b75a3e1ef641620ff

Mathlib/Analysis/Normed/Group/Basic.lean

nndist_nnnorm_nnnorm_le'

[]

https://github.com/leanprover-community/mathlib4

3ce43c18f614b76e161f911b75a3e1ef641620ff

Mathlib/Topology/Homeomorph.lean

Homeomorph.quotientMap

[{"tactic": "simp only [self_comp_symm, QuotientMap.id]", "annotated_tactic": ["simp only [<a>self_comp_symm</a>, <a>QuotientMap.id</a>]", [{"full_name": "Homeomorph.self_comp_symm", "def_path": "Mathlib/Topology/Homeomorph.lean", "def_pos": [201, 9], "def_end_pos": [201, 23]}, {"full_name": "QuotientMap.id", "def_path": "Mathlib/Topology/Maps.lean", "def_pos": [293, 19], "def_end_pos": [293, 21]}]], "state_before": "X : Type u_1\nY : Type u_2\nZ : Type u_3\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : TopologicalSpace Y\ninst\u271d\u00b2 : TopologicalSpace Z\nX' : Type u_4\nY' : Type u_5\ninst\u271d\u00b9 : TopologicalSpace X'\ninst\u271d : TopologicalSpace Y'\nh : X \u2243\u209c Y\n\u22a2 QuotientMap (\u2191h \u2218 \u2191(Homeomorph.symm h))", "state_after": "no goals"}]

https://github.com/leanprover-community/mathlib4

3ce43c18f614b76e161f911b75a3e1ef641620ff

Mathlib/NumberTheory/Bertrand.lean

Bertrand.real_main_inequality

[{"tactic": "let f : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x", "annotated_tactic": ["let f : \u211d \u2192 \u211d := fun x => <a>log</a> x + <a>sqrt</a> (2 * x) * <a>log</a> (2 * x) - <a>log</a> 4 / 3 * x", [{"full_name": "Real.log", "def_path": "Mathlib/Analysis/SpecialFunctions/Log/Basic.lean", "def_pos": [43, 19], "def_end_pos": [43, 22]}, {"full_name": "Real.sqrt", "def_path": "Mathlib/Data/Real/Sqrt.lean", "def_pos": [166, 19], "def_end_pos": [166, 23]}, {"full_name": "Real.log", "def_path": "Mathlib/Analysis/SpecialFunctions/Log/Basic.lean", "def_pos": [43, 19], "def_end_pos": [43, 22]}, {"full_name": "Real.log", "def_path": "Mathlib/Analysis/SpecialFunctions/Log/Basic.lean", "def_pos": [43, 19], "def_end_pos": [43, 22]}]], "state_before": "x : \u211d\nn_large : 512 \u2264 x\n\u22a2 x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) \u2264 4 ^ x", "state_after": "x : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\n\u22a2 x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) \u2264 4 ^ x"}, {"tactic": "have hf' : \u2200 x, 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h =>\n div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _)", "annotated_tactic": ["have hf' : \u2200 x, 0 < x \u2192 0 < x * (2 * x) ^ <a>sqrt</a> (2 * x) / 4 ^ (x / 3) := fun x h =>\n <a>div_pos</a> (<a>mul_pos</a> h (<a>rpow_pos_of_pos</a> (<a>mul_pos</a> <a>two_pos</a> h) _)) (<a>rpow_pos_of_pos</a> <a>four_pos</a> _)", [{"full_name": "Real.sqrt", "def_path": "Mathlib/Data/Real/Sqrt.lean", "def_pos": [166, 19], "def_end_pos": [166, 23]}, {"full_name": "div_pos", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [89, 9], "def_end_pos": [89, 16]}, {"full_name": "mul_pos", "def_path": "Mathlib/Algebra/Order/Ring/Lemmas.lean", "def_pos": [345, 7], "def_end_pos": [345, 14]}, {"full_name": "Real.rpow_pos_of_pos", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/Real.lean", "def_pos": [92, 9], "def_end_pos": [92, 24]}, {"full_name": "mul_pos", "def_path": "Mathlib/Algebra/Order/Ring/Lemmas.lean", "def_pos": [345, 7], "def_end_pos": [345, 14]}, {"full_name": "two_pos", "def_path": "Mathlib/Algebra/Order/Monoid/NatCast.lean", "def_pos": [113, 7], "def_end_pos": [113, 14]}, {"full_name": "Real.rpow_pos_of_pos", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/Real.lean", "def_pos": [92, 9], "def_end_pos": [92, 24]}, {"full_name": "four_pos", "def_path": "Mathlib/Algebra/Order/Monoid/NatCast.lean", "def_pos": [119, 7], "def_end_pos": [119, 15]}]], "state_before": "x : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\n\u22a2 x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) \u2264 4 ^ x", "state_after": "x : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\n\u22a2 x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) \u2264 4 ^ x"}, {"tactic": "have hf : \u2200 x, 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by\n intro x h5\n have h6 := mul_pos (zero_lt_two' \u211d) h5\n have h7 := rpow_pos_of_pos h6 (sqrt (2 * x))\n rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne',\n log_rpow h6, log_rpow zero_lt_four, \u2190 mul_div_right_comm, \u2190 mul_div, mul_comm x]", "annotated_tactic": ["have hf : \u2200 x, 0 < x \u2192 f x = <a>log</a> (x * (2 * x) ^ <a>sqrt</a> (2 * x) / 4 ^ (x / 3)) := by\n intro x h5\n have h6 := <a>mul_pos</a> (<a>zero_lt_two'</a> \u211d) h5\n have h7 := <a>rpow_pos_of_pos</a> h6 (<a>sqrt</a> (2 * x))\n rw [<a>log_div</a> (<a>mul_pos</a> h5 h7).<a>ne'</a> (<a>rpow_pos_of_pos</a> <a>four_pos</a> _).<a>ne'</a>, <a>log_mul</a> h5.ne' h7.ne',\n <a>log_rpow</a> h6, <a>log_rpow</a> <a>zero_lt_four</a>, \u2190 <a>mul_div_right_comm</a>, \u2190 <a>mul_div</a>, <a>mul_comm</a> x]", [{"full_name": "Real.log", "def_path": "Mathlib/Analysis/SpecialFunctions/Log/Basic.lean", "def_pos": [43, 19], "def_end_pos": [43, 22]}, {"full_name": "Real.sqrt", "def_path": "Mathlib/Data/Real/Sqrt.lean", "def_pos": [166, 19], "def_end_pos": [166, 23]}, {"full_name": "mul_pos", "def_path": "Mathlib/Algebra/Order/Ring/Lemmas.lean", "def_pos": [345, 7], "def_end_pos": [345, 14]}, {"full_name": "zero_lt_two'", "def_path": "Mathlib/Algebra/Order/Monoid/NatCast.lean", "def_pos": [89, 7], "def_end_pos": [89, 19]}, {"full_name": "Real.rpow_pos_of_pos", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/Real.lean", "def_pos": [92, 9], "def_end_pos": [92, 24]}, {"full_name": "Real.sqrt", "def_path": "Mathlib/Data/Real/Sqrt.lean", "def_pos": [166, 19], "def_end_pos": [166, 23]}, {"full_name": "Real.log_div", "def_path": "Mathlib/Analysis/SpecialFunctions/Log/Basic.lean", "def_pos": [133, 9], "def_end_pos": [133, 16]}, {"full_name": "mul_pos", "def_path": "Mathlib/Algebra/Order/Ring/Lemmas.lean", "def_pos": [345, 7], "def_end_pos": [345, 14]}, {"full_name": "LT.lt.ne'", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [328, 9], "def_end_pos": [328, 12]}, {"full_name": "Real.rpow_pos_of_pos", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/Real.lean", "def_pos": [92, 9], "def_end_pos": [92, 24]}, {"full_name": "four_pos", "def_path": "Mathlib/Algebra/Order/Monoid/NatCast.lean", "def_pos": [119, 7], "def_end_pos": [119, 15]}, {"full_name": "LT.lt.ne'", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [328, 9], "def_end_pos": [328, 12]}, {"full_name": "Real.log_mul", "def_path": "Mathlib/Analysis/SpecialFunctions/Log/Basic.lean", "def_pos": [128, 9], "def_end_pos": [128, 16]}, {"full_name": "Real.log_rpow", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/Real.lean", "def_pos": [401, 9], "def_end_pos": [401, 17]}, {"full_name": "Real.log_rpow", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/Real.lean", "def_pos": [401, 9], "def_end_pos": [401, 17]}, {"full_name": "zero_lt_four", "def_path": "Mathlib/Algebra/Order/Monoid/NatCast.lean", "def_pos": [81, 15], "def_end_pos": [81, 27]}, {"full_name": "mul_div_right_comm", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [542, 9], "def_end_pos": [542, 27]}, {"full_name": "mul_div", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [324, 9], "def_end_pos": [324, 16]}, {"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [302, 9], "def_end_pos": [302, 17]}]], "state_before": "x : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\n\u22a2 x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) \u2264 4 ^ x", "state_after": "x : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\n\u22a2 x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) \u2264 4 ^ x"}, {"tactic": "have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large", "annotated_tactic": ["have h5 : 0 < x := <a>lt_of_lt_of_le</a> (by norm_num1) n_large", [{"full_name": "lt_of_lt_of_le", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [115, 9], "def_end_pos": [115, 23]}]], "state_before": "x : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\n\u22a2 x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) \u2264 4 ^ x", "state_after": "x : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\n\u22a2 x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) \u2264 4 ^ x"}, {"tactic": "rw [\u2190 div_le_one (rpow_pos_of_pos four_pos x), \u2190 div_div_eq_mul_div, \u2190 rpow_sub four_pos, \u2190\n mul_div 2 x, mul_div_left_comm, \u2190 mul_one_sub, (by norm_num1 : (1 : \u211d) - 2 / 3 = 1 / 3),\n mul_one_div, \u2190 log_nonpos_iff (hf' x h5), \u2190 hf x h5]", "annotated_tactic": ["rw [\u2190 <a>div_le_one</a> (<a>rpow_pos_of_pos</a> <a>four_pos</a> x), \u2190 <a>div_div_eq_mul_div</a>, \u2190 <a>rpow_sub</a> <a>four_pos</a>, \u2190\n <a>mul_div</a> 2 x, <a>mul_div_left_comm</a>, \u2190 <a>mul_one_sub</a>, (by norm_num1 : (1 : \u211d) - 2 / 3 = 1 / 3),\n <a>mul_one_div</a>, \u2190 <a>log_nonpos_iff</a> (hf' x h5), \u2190 hf x h5]", [{"full_name": "div_le_one", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [425, 9], "def_end_pos": [425, 19]}, {"full_name": "Real.rpow_pos_of_pos", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/Real.lean", "def_pos": [92, 9], "def_end_pos": [92, 24]}, {"full_name": "four_pos", "def_path": "Mathlib/Algebra/Order/Monoid/NatCast.lean", "def_pos": [119, 7], "def_end_pos": [119, 15]}, {"full_name": "div_div_eq_mul_div", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [449, 9], "def_end_pos": [449, 27]}, {"full_name": "Real.rpow_sub", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/Real.lean", "def_pos": [225, 9], "def_end_pos": [225, 17]}, {"full_name": "four_pos", "def_path": "Mathlib/Algebra/Order/Monoid/NatCast.lean", "def_pos": [119, 7], "def_end_pos": [119, 15]}, {"full_name": "mul_div", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [324, 9], "def_end_pos": [324, 16]}, {"full_name": "mul_div_left_comm", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [537, 9], "def_end_pos": [537, 26]}, {"full_name": "mul_one_sub", "def_path": "Mathlib/Algebra/Ring/Defs.lean", "def_pos": [393, 9], "def_end_pos": [393, 20]}, {"full_name": "mul_one_div", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [300, 9], "def_end_pos": [300, 20]}, {"full_name": "Real.log_nonpos_iff", "def_path": "Mathlib/Analysis/SpecialFunctions/Log/Basic.lean", "def_pos": [213, 9], "def_end_pos": [213, 23]}]], "state_before": "x : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\n\u22a2 x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) \u2264 4 ^ x", "state_after": "x : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\n\u22a2 f x \u2264 0"}, {"tactic": "suffices \u2203 x1 x2, 0.5 < x1 \u2227 x1 < x2 \u2227 x2 \u2264 x \u2227 0 \u2264 f x1 \u2227 f x2 \u2264 0 by\n obtain \u27e8x1, x2, h1, h2, h0, h3, h4\u27e9 := this\n exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4", "annotated_tactic": ["suffices \u2203 x1 x2, 0.5 < x1 \u2227 x1 < x2 \u2227 x2 \u2264 x \u2227 0 \u2264 f x1 \u2227 f x2 \u2264 0 by\n obtain \u27e8x1, x2, h1, h2, h0, h3, h4\u27e9 := this\n exact (h.right_le_of_le_left'' h1 ((h1.trans h2).<a>trans_le</a> h0) h2 h0 (h4.trans h3)).<a>trans</a> h4", [{"full_name": "LT.lt.trans_le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [148, 7], "def_end_pos": [148, 21]}, {"full_name": "LE.le.trans", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [120, 7], "def_end_pos": [120, 18]}]], "state_before": "x : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\n\u22a2 f x \u2264 0", "state_after": "x : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\n\u22a2 \u2203 x1 x2, 0.5 < x1 \u2227 x1 < x2 \u2227 x2 \u2264 x \u2227 0 \u2264 f x1 \u2227 f x2 \u2264 0"}, {"tactic": "refine' \u27e818, 512, by norm_num1, by norm_num1, n_large, _, _\u27e9", "annotated_tactic": ["refine' \u27e818, 512, by norm_num1, by norm_num1, n_large, _, _\u27e9", []], "state_before": "x : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\n\u22a2 \u2203 x1 x2, 0.5 < x1 \u2227 x1 < x2 \u2227 x2 \u2264 x \u2227 0 \u2264 f x1 \u2227 f x2 \u2264 0", "state_after": "case refine'_1\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\n\u22a2 0 \u2264 f 18\n\ncase refine'_2\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\n\u22a2 f 512 \u2264 0"}, {"tactic": "intro x h5", "annotated_tactic": ["intro x h5", []], "state_before": "x : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\n\u22a2 \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))", "state_after": "x\u271d : \u211d\nn_large : 512 \u2264 x\u271d\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nx : \u211d\nh5 : 0 < x\n\u22a2 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))"}, {"tactic": "have h6 := mul_pos (zero_lt_two' \u211d) h5", "annotated_tactic": ["have h6 := <a>mul_pos</a> (<a>zero_lt_two'</a> \u211d) h5", [{"full_name": "mul_pos", "def_path": "Mathlib/Algebra/Order/Ring/Lemmas.lean", "def_pos": [345, 7], "def_end_pos": [345, 14]}, {"full_name": "zero_lt_two'", "def_path": "Mathlib/Algebra/Order/Monoid/NatCast.lean", "def_pos": [89, 7], "def_end_pos": [89, 19]}]], "state_before": "x\u271d : \u211d\nn_large : 512 \u2264 x\u271d\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nx : \u211d\nh5 : 0 < x\n\u22a2 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))", "state_after": "x\u271d : \u211d\nn_large : 512 \u2264 x\u271d\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nx : \u211d\nh5 : 0 < x\nh6 : 0 < 2 * x\n\u22a2 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))"}, {"tactic": "have h7 := rpow_pos_of_pos h6 (sqrt (2 * x))", "annotated_tactic": ["have h7 := <a>rpow_pos_of_pos</a> h6 (<a>sqrt</a> (2 * x))", [{"full_name": "Real.rpow_pos_of_pos", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/Real.lean", "def_pos": [92, 9], "def_end_pos": [92, 24]}, {"full_name": "Real.sqrt", "def_path": "Mathlib/Data/Real/Sqrt.lean", "def_pos": [166, 19], "def_end_pos": [166, 23]}]], "state_before": "x\u271d : \u211d\nn_large : 512 \u2264 x\u271d\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nx : \u211d\nh5 : 0 < x\nh6 : 0 < 2 * x\n\u22a2 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))", "state_after": "x\u271d : \u211d\nn_large : 512 \u2264 x\u271d\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nx : \u211d\nh5 : 0 < x\nh6 : 0 < 2 * x\nh7 : 0 < (2 * x) ^ sqrt (2 * x)\n\u22a2 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))"}, {"tactic": "rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne',\n log_rpow h6, log_rpow zero_lt_four, \u2190 mul_div_right_comm, \u2190 mul_div, mul_comm x]", "annotated_tactic": ["rw [<a>log_div</a> (<a>mul_pos</a> h5 h7).<a>ne'</a> (<a>rpow_pos_of_pos</a> <a>four_pos</a> _).<a>ne'</a>, <a>log_mul</a> h5.ne' h7.ne',\n <a>log_rpow</a> h6, <a>log_rpow</a> <a>zero_lt_four</a>, \u2190 <a>mul_div_right_comm</a>, \u2190 <a>mul_div</a>, <a>mul_comm</a> x]", [{"full_name": "Real.log_div", "def_path": "Mathlib/Analysis/SpecialFunctions/Log/Basic.lean", "def_pos": [133, 9], "def_end_pos": [133, 16]}, {"full_name": "mul_pos", "def_path": "Mathlib/Algebra/Order/Ring/Lemmas.lean", "def_pos": [345, 7], "def_end_pos": [345, 14]}, {"full_name": "LT.lt.ne'", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [328, 9], "def_end_pos": [328, 12]}, {"full_name": "Real.rpow_pos_of_pos", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/Real.lean", "def_pos": [92, 9], "def_end_pos": [92, 24]}, {"full_name": "four_pos", "def_path": "Mathlib/Algebra/Order/Monoid/NatCast.lean", "def_pos": [119, 7], "def_end_pos": [119, 15]}, {"full_name": "LT.lt.ne'", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [328, 9], "def_end_pos": [328, 12]}, {"full_name": "Real.log_mul", "def_path": "Mathlib/Analysis/SpecialFunctions/Log/Basic.lean", "def_pos": [128, 9], "def_end_pos": [128, 16]}, {"full_name": "Real.log_rpow", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/Real.lean", "def_pos": [401, 9], "def_end_pos": [401, 17]}, {"full_name": "Real.log_rpow", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/Real.lean", "def_pos": [401, 9], "def_end_pos": [401, 17]}, {"full_name": "zero_lt_four", "def_path": "Mathlib/Algebra/Order/Monoid/NatCast.lean", "def_pos": [81, 15], "def_end_pos": [81, 27]}, {"full_name": "mul_div_right_comm", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [542, 9], "def_end_pos": [542, 27]}, {"full_name": "mul_div", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [324, 9], "def_end_pos": [324, 16]}, {"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [302, 9], "def_end_pos": [302, 17]}]], "state_before": "x\u271d : \u211d\nn_large : 512 \u2264 x\u271d\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nx : \u211d\nh5 : 0 < x\nh6 : 0 < 2 * x\nh7 : 0 < (2 * x) ^ sqrt (2 * x)\n\u22a2 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))", "state_after": "no goals"}, {"tactic": "norm_num1", "annotated_tactic": ["norm_num1", []], "state_before": "x : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\n\u22a2 0 < 512", "state_after": "no goals"}, {"tactic": "norm_num1", "annotated_tactic": ["norm_num1", []], "state_before": "x : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\n\u22a2 1 - 2 / 3 = 1 / 3", "state_after": "no goals"}, {"tactic": "apply ConcaveOn.sub", "annotated_tactic": ["apply <a>ConcaveOn.sub</a>", [{"full_name": "ConcaveOn.sub", "def_path": "Mathlib/Analysis/Convex/Function.lean", "def_pos": [888, 9], "def_end_pos": [888, 22]}]], "state_before": "x : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\n\u22a2 ConcaveOn \u211d (Set.Ioi 0.5) f", "state_after": "case hf\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\n\u22a2 ConcaveOn \u211d (Set.Ioi 0.5) fun x => log x + sqrt (2 * x) * log (2 * x)\n\ncase hg\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\n\u22a2 ConvexOn \u211d (Set.Ioi 0.5) fun x => log 4 / 3 * x"}, {"tactic": "apply ConcaveOn.add", "annotated_tactic": ["apply <a>ConcaveOn.add</a>", [{"full_name": "ConcaveOn.add", "def_path": "Mathlib/Analysis/Convex/Function.lean", "def_pos": [196, 9], "def_end_pos": [196, 22]}]], "state_before": "case hf\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\n\u22a2 ConcaveOn \u211d (Set.Ioi 0.5) fun x => log x + sqrt (2 * x) * log (2 * x)\n\ncase hg\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\n\u22a2 ConvexOn \u211d (Set.Ioi 0.5) fun x => log 4 / 3 * x", "state_after": "case hf.hf\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\n\u22a2 ConcaveOn \u211d (Set.Ioi 0.5) fun x => log x\n\ncase hf.hg\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\n\u22a2 ConcaveOn \u211d (Set.Ioi 0.5) fun x => sqrt (2 * x) * log (2 * x)\n\ncase hg\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\n\u22a2 ConvexOn \u211d (Set.Ioi 0.5) fun x => log 4 / 3 * x"}, {"tactic": "exact strictConcaveOn_log_Ioi.concaveOn.subset\n (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5)", "annotated_tactic": ["exact strictConcaveOn_log_Ioi.concaveOn.subset\n (<a>Set.Ioi_subset_Ioi</a> (by norm_num)) (<a>convex_Ioi</a> 0.5)", [{"full_name": "Set.Ioi_subset_Ioi", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [598, 9], "def_end_pos": [598, 23]}, {"full_name": "convex_Ioi", "def_path": "Mathlib/Analysis/Convex/Basic.lean", "def_pos": [325, 9], "def_end_pos": [325, 19]}]], "state_before": "case hf.hf\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\n\u22a2 ConcaveOn \u211d (Set.Ioi 0.5) fun x => log x\n\ncase hf.hg\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\n\u22a2 ConcaveOn \u211d (Set.Ioi 0.5) fun x => sqrt (2 * x) * log (2 * x)\n\ncase hg\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\n\u22a2 ConvexOn \u211d (Set.Ioi 0.5) fun x => log 4 / 3 * x", "state_after": "case hf.hg\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\n\u22a2 ConcaveOn \u211d (Set.Ioi 0.5) fun x => sqrt (2 * x) * log (2 * x)\n\ncase hg\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\n\u22a2 ConvexOn \u211d (Set.Ioi 0.5) fun x => log 4 / 3 * x"}, {"tactic": "convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap\n ((2 : \u211d) \u2022 LinearMap.id))) using 1", "annotated_tactic": ["convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap\n ((2 : \u211d) \u2022 <a>LinearMap.id</a>))) using 1", [{"full_name": "LinearMap.id", "def_path": "Mathlib/Algebra/Module/LinearMap.lean", "def_pos": [263, 5], "def_end_pos": [263, 7]}]], "state_before": "case hf.hg\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\n\u22a2 ConcaveOn \u211d (Set.Ioi 0.5) fun x => sqrt (2 * x) * log (2 * x)\n\ncase hg\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\n\u22a2 ConvexOn \u211d (Set.Ioi 0.5) fun x => log 4 / 3 * x", "state_after": "case h.e'_9\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\n\u22a2 Set.Ioi 0.5 = \u2191(2 \u2022 LinearMap.id) \u207b\u00b9' Set.Ioi 1\n\ncase hg\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\n\u22a2 ConvexOn \u211d (Set.Ioi 0.5) fun x => log 4 / 3 * x"}, {"tactic": "apply ConvexOn.smul", "annotated_tactic": ["apply <a>ConvexOn.smul</a>", [{"full_name": "ConvexOn.smul", "def_path": "Mathlib/Analysis/Convex/Function.lean", "def_pos": [975, 9], "def_end_pos": [975, 22]}]], "state_before": "case hg\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\n\u22a2 ConvexOn \u211d (Set.Ioi 0.5) fun x => log 4 / 3 * x", "state_after": "case hg.hc\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\n\u22a2 0 \u2264 log 4 / 3\n\ncase hg.hf\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\n\u22a2 ConvexOn \u211d (Set.Ioi 0.5) fun x => x"}, {"tactic": "refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1)", "annotated_tactic": ["refine <a>div_nonneg</a> (<a>log_nonneg</a> (by norm_num1)) (by norm_num1)", [{"full_name": "div_nonneg", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [94, 9], "def_end_pos": [94, 19]}, {"full_name": "Real.log_nonneg", "def_path": "Mathlib/Analysis/SpecialFunctions/Log/Basic.lean", "def_pos": [209, 9], "def_end_pos": [209, 19]}]], "state_before": "case hg.hc\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\n\u22a2 0 \u2264 log 4 / 3\n\ncase hg.hf\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\n\u22a2 ConvexOn \u211d (Set.Ioi 0.5) fun x => x", "state_after": "case hg.hf\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\n\u22a2 ConvexOn \u211d (Set.Ioi 0.5) fun x => x"}, {"tactic": "exact convexOn_id (convex_Ioi (0.5 : \u211d))", "annotated_tactic": ["exact <a>convexOn_id</a> (<a>convex_Ioi</a> (0.5 : \u211d))", [{"full_name": "convexOn_id", "def_path": "Mathlib/Analysis/Convex/Function.lean", "def_pos": [91, 9], "def_end_pos": [91, 20]}, {"full_name": "convex_Ioi", "def_path": "Mathlib/Analysis/Convex/Basic.lean", "def_pos": [325, 9], "def_end_pos": [325, 19]}]], "state_before": "case hg.hf\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\n\u22a2 ConvexOn \u211d (Set.Ioi 0.5) fun x => x", "state_after": "no goals"}, {"tactic": "norm_num", "annotated_tactic": ["norm_num", []], "state_before": "x : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\n\u22a2 0 \u2264 0.5", "state_after": "no goals"}, {"tactic": "ext x", "annotated_tactic": ["ext x", []], "state_before": "case h.e'_9\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\n\u22a2 Set.Ioi 0.5 = \u2191(2 \u2022 LinearMap.id) \u207b\u00b9' Set.Ioi 1", "state_after": "case h.e'_9.h\nx\u271d : \u211d\nn_large : 512 \u2264 x\u271d\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\u271d\nx : \u211d\n\u22a2 x \u2208 Set.Ioi 0.5 \u2194 x \u2208 \u2191(2 \u2022 LinearMap.id) \u207b\u00b9' Set.Ioi 1"}, {"tactic": "simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply,\n LinearMap.id_coe, id_eq, smul_eq_mul]", "annotated_tactic": ["simp only [<a>Set.mem_Ioi</a>, <a>Set.mem_preimage</a>, <a>LinearMap.smul_apply</a>,\n <a>LinearMap.id_coe</a>, <a>id_eq</a>, <a>smul_eq_mul</a>]", [{"full_name": "Set.mem_Ioi", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [151, 9], "def_end_pos": [151, 16]}, {"full_name": "Set.mem_preimage", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [64, 9], "def_end_pos": [64, 21]}, {"full_name": "LinearMap.smul_apply", "def_path": "Mathlib/Algebra/Module/LinearMap.lean", "def_pos": [841, 9], "def_end_pos": [841, 19]}, {"full_name": "LinearMap.id_coe", "def_path": "Mathlib/Algebra/Module/LinearMap.lean", "def_pos": [272, 9], "def_end_pos": [272, 15]}, {"full_name": "id_eq", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [284, 17], "def_end_pos": [284, 22]}, {"full_name": "smul_eq_mul", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [93, 9], "def_end_pos": [93, 20]}]], "state_before": "case h.e'_9.h\nx\u271d : \u211d\nn_large : 512 \u2264 x\u271d\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\u271d\nx : \u211d\n\u22a2 x \u2208 Set.Ioi 0.5 \u2194 x \u2208 \u2191(2 \u2022 LinearMap.id) \u207b\u00b9' Set.Ioi 1", "state_after": "case h.e'_9.h\nx\u271d : \u211d\nn_large : 512 \u2264 x\u271d\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\u271d\nx : \u211d\n\u22a2 OfScientific.ofScientific 5 true 1 < x \u2194 1 < 2 * x"}, {"tactic": "rw [\u2190 mul_lt_mul_left (two_pos)]", "annotated_tactic": ["rw [\u2190 <a>mul_lt_mul_left</a> (<a>two_pos</a>)]", [{"full_name": "mul_lt_mul_left", "def_path": "Mathlib/Algebra/Order/Ring/Lemmas.lean", "def_pos": [197, 9], "def_end_pos": [197, 24]}, {"full_name": "two_pos", "def_path": "Mathlib/Algebra/Order/Monoid/NatCast.lean", "def_pos": [113, 7], "def_end_pos": [113, 14]}]], "state_before": "case h.e'_9.h\nx\u271d : \u211d\nn_large : 512 \u2264 x\u271d\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\u271d\nx : \u211d\n\u22a2 OfScientific.ofScientific 5 true 1 < x \u2194 1 < 2 * x", "state_after": "case h.e'_9.h\nx\u271d : \u211d\nn_large : 512 \u2264 x\u271d\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\u271d\nx : \u211d\n\u22a2 2 * OfScientific.ofScientific 5 true 1 < 2 * x \u2194 1 < 2 * x"}, {"tactic": "norm_num1", "annotated_tactic": ["norm_num1", []], "state_before": "case h.e'_9.h\nx\u271d : \u211d\nn_large : 512 \u2264 x\u271d\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\u271d\nx : \u211d\n\u22a2 2 * OfScientific.ofScientific 5 true 1 < 2 * x \u2194 1 < 2 * x", "state_after": "case h.e'_9.h\nx\u271d : \u211d\nn_large : 512 \u2264 x\u271d\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\u271d\nx : \u211d\n\u22a2 1 < 2 * x \u2194 1 < 2 * x"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case h.e'_9.h\nx\u271d : \u211d\nn_large : 512 \u2264 x\u271d\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\u271d\nx : \u211d\n\u22a2 1 < 2 * x \u2194 1 < 2 * x", "state_after": "no goals"}, {"tactic": "norm_num1", "annotated_tactic": ["norm_num1", []], "state_before": "x : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\n\u22a2 1 \u2264 4", "state_after": "no goals"}, {"tactic": "norm_num1", "annotated_tactic": ["norm_num1", []], "state_before": "x : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\n\u22a2 0 \u2264 3", "state_after": "no goals"}, {"tactic": "obtain \u27e8x1, x2, h1, h2, h0, h3, h4\u27e9 := this", "annotated_tactic": ["obtain \u27e8x1, x2, h1, h2, h0, h3, h4\u27e9 := this", []], "state_before": "x : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis : \u2203 x1 x2, 0.5 < x1 \u2227 x1 < x2 \u2227 x2 \u2264 x \u2227 0 \u2264 f x1 \u2227 f x2 \u2264 0\n\u22a2 f x \u2264 0", "state_after": "case intro.intro.intro.intro.intro.intro\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nx1 x2 : \u211d\nh1 : 0.5 < x1\nh2 : x1 < x2\nh0 : x2 \u2264 x\nh3 : 0 \u2264 f x1\nh4 : f x2 \u2264 0\n\u22a2 f x \u2264 0"}, {"tactic": "exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4", "annotated_tactic": ["exact (h.right_le_of_le_left'' h1 ((h1.trans h2).<a>trans_le</a> h0) h2 h0 (h4.trans h3)).<a>trans</a> h4", [{"full_name": "LT.lt.trans_le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [148, 7], "def_end_pos": [148, 21]}, {"full_name": "LE.le.trans", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [120, 7], "def_end_pos": [120, 18]}]], "state_before": "case intro.intro.intro.intro.intro.intro\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nx1 x2 : \u211d\nh1 : 0.5 < x1\nh2 : x1 < x2\nh0 : x2 \u2264 x\nh3 : 0 \u2264 f x1\nh4 : f x2 \u2264 0\n\u22a2 f x \u2264 0", "state_after": "no goals"}, {"tactic": "norm_num1", "annotated_tactic": ["norm_num1", []], "state_before": "x : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\n\u22a2 0.5 < 18", "state_after": "no goals"}, {"tactic": "norm_num1", "annotated_tactic": ["norm_num1", []], "state_before": "x : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\n\u22a2 18 < 512", "state_after": "no goals"}, {"tactic": "have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1)", "annotated_tactic": ["have : <a>sqrt</a> (2 * 18) = 6 := (<a>sqrt_eq_iff_mul_self_eq_of_pos</a> (by norm_num1)).<a>mpr</a> (by norm_num1)", [{"full_name": "Real.sqrt", "def_path": "Mathlib/Data/Real/Sqrt.lean", "def_pos": [166, 19], "def_end_pos": [166, 23]}, {"full_name": "Real.sqrt_eq_iff_mul_self_eq_of_pos", "def_path": "Mathlib/Data/Real/Sqrt.lean", "def_pos": [222, 9], "def_end_pos": [222, 39]}, {"full_name": "Iff.mpr", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [92, 3], "def_end_pos": [92, 6]}]], "state_before": "case refine'_1\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\n\u22a2 0 \u2264 f 18", "state_after": "case refine'_1\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis : sqrt (2 * 18) = 6\n\u22a2 0 \u2264 f 18"}, {"tactic": "rw [hf, log_nonneg_iff, this]", "annotated_tactic": ["rw [hf, <a>log_nonneg_iff</a>, this]", [{"full_name": "Real.log_nonneg_iff", "def_path": "Mathlib/Analysis/SpecialFunctions/Log/Basic.lean", "def_pos": [206, 9], "def_end_pos": [206, 23]}]], "state_before": "case refine'_1\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis : sqrt (2 * 18) = 6\n\u22a2 0 \u2264 f 18", "state_after": "case refine'_1\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis : sqrt (2 * 18) = 6\n\u22a2 1 \u2264 18 * (2 * 18) ^ 6 / 4 ^ (18 / 3)\n\ncase refine'_1\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis : sqrt (2 * 18) = 6\n\u22a2 0 < 18 * (2 * 18) ^ sqrt (2 * 18) / 4 ^ (18 / 3)\n\ncase refine'_1.a\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis : sqrt (2 * 18) = 6\n\u22a2 0 < 18"}, {"tactic": "rw [one_le_div] <;> norm_num1", "annotated_tactic": ["rw [<a>one_le_div</a>] <;> norm_num1", [{"full_name": "one_le_div", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [422, 9], "def_end_pos": [422, 19]}]], "state_before": "case refine'_1\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis : sqrt (2 * 18) = 6\n\u22a2 1 \u2264 18 * (2 * 18) ^ 6 / 4 ^ (18 / 3)\n\ncase refine'_1\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis : sqrt (2 * 18) = 6\n\u22a2 0 < 18 * (2 * 18) ^ sqrt (2 * 18) / 4 ^ (18 / 3)\n\ncase refine'_1.a\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis : sqrt (2 * 18) = 6\n\u22a2 0 < 18", "state_after": "case refine'_1\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis : sqrt (2 * 18) = 6\n\u22a2 4 ^ 6 \u2264 18 * 36 ^ 6\n\ncase refine'_1\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis : sqrt (2 * 18) = 6\n\u22a2 0 < 4 ^ 6\n\ncase refine'_1\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis : sqrt (2 * 18) = 6\n\u22a2 0 < 18 * (2 * 18) ^ sqrt (2 * 18) / 4 ^ (18 / 3)\n\ncase refine'_1.a\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis : sqrt (2 * 18) = 6\n\u22a2 0 < 18"}, {"tactic": "apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1", "annotated_tactic": ["apply <a>le_trans</a> _ (<a>le_mul_of_one_le_left</a> _ _) <;> norm_num1", [{"full_name": "le_trans", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [56, 9], "def_end_pos": [56, 17]}, {"full_name": "le_mul_of_one_le_left", "def_path": "Mathlib/Algebra/Order/Ring/Lemmas.lean", "def_pos": [670, 9], "def_end_pos": [670, 30]}]], "state_before": "case refine'_1\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis : sqrt (2 * 18) = 6\n\u22a2 4 ^ 6 \u2264 18 * 36 ^ 6\n\ncase refine'_1\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis : sqrt (2 * 18) = 6\n\u22a2 0 < 4 ^ 6\n\ncase refine'_1\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis : sqrt (2 * 18) = 6\n\u22a2 0 < 18 * (2 * 18) ^ sqrt (2 * 18) / 4 ^ (18 / 3)\n\ncase refine'_1.a\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis : sqrt (2 * 18) = 6\n\u22a2 0 < 18", "state_after": "x : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis : sqrt (2 * 18) = 6\n\u22a2 4 ^ 6 \u2264 36 ^ 6\n\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis : sqrt (2 * 18) = 6\n\u22a2 0 \u2264 36 ^ 6\n\ncase refine'_1\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis : sqrt (2 * 18) = 6\n\u22a2 0 < 4 ^ 6\n\ncase refine'_1\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis : sqrt (2 * 18) = 6\n\u22a2 0 < 18 * (2 * 18) ^ sqrt (2 * 18) / 4 ^ (18 / 3)\n\ncase refine'_1.a\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis : sqrt (2 * 18) = 6\n\u22a2 0 < 18"}, {"tactic": "apply Real.rpow_le_rpow <;> norm_num1", "annotated_tactic": ["apply <a>Real.rpow_le_rpow</a> <;> norm_num1", [{"full_name": "Real.rpow_le_rpow", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/Real.lean", "def_pos": [433, 9], "def_end_pos": [433, 21]}]], "state_before": "x : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis : sqrt (2 * 18) = 6\n\u22a2 4 ^ 6 \u2264 36 ^ 6\n\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis : sqrt (2 * 18) = 6\n\u22a2 0 \u2264 36 ^ 6\n\ncase refine'_1\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis : sqrt (2 * 18) = 6\n\u22a2 0 < 4 ^ 6\n\ncase refine'_1\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis : sqrt (2 * 18) = 6\n\u22a2 0 < 18 * (2 * 18) ^ sqrt (2 * 18) / 4 ^ (18 / 3)\n\ncase refine'_1.a\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis : sqrt (2 * 18) = 6\n\u22a2 0 < 18", "state_after": "x : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis : sqrt (2 * 18) = 6\n\u22a2 0 \u2264 36 ^ 6\n\ncase refine'_1\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis : sqrt (2 * 18) = 6\n\u22a2 0 < 4 ^ 6\n\ncase refine'_1\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis : sqrt (2 * 18) = 6\n\u22a2 0 < 18 * (2 * 18) ^ sqrt (2 * 18) / 4 ^ (18 / 3)\n\ncase refine'_1.a\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis : sqrt (2 * 18) = 6\n\u22a2 0 < 18"}, {"tactic": "apply rpow_nonneg_of_nonneg", "annotated_tactic": ["apply <a>rpow_nonneg_of_nonneg</a>", [{"full_name": "Real.rpow_nonneg_of_nonneg", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/Real.lean", "def_pos": [141, 9], "def_end_pos": [141, 30]}]], "state_before": "x : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis : sqrt (2 * 18) = 6\n\u22a2 0 \u2264 36 ^ 6\n\ncase refine'_1\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis : sqrt (2 * 18) = 6\n\u22a2 0 < 4 ^ 6\n\ncase refine'_1\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis : sqrt (2 * 18) = 6\n\u22a2 0 < 18 * (2 * 18) ^ sqrt (2 * 18) / 4 ^ (18 / 3)\n\ncase refine'_1.a\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis : sqrt (2 * 18) = 6\n\u22a2 0 < 18", "state_after": "case hx\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis : sqrt (2 * 18) = 6\n\u22a2 0 \u2264 36\n\ncase refine'_1\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis : sqrt (2 * 18) = 6\n\u22a2 0 < 4 ^ 6\n\ncase refine'_1\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis : sqrt (2 * 18) = 6\n\u22a2 0 < 18 * (2 * 18) ^ sqrt (2 * 18) / 4 ^ (18 / 3)\n\ncase refine'_1.a\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis : sqrt (2 * 18) = 6\n\u22a2 0 < 18"}, {"tactic": "norm_num1", "annotated_tactic": ["norm_num1", []], "state_before": "case hx\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis : sqrt (2 * 18) = 6\n\u22a2 0 \u2264 36\n\ncase refine'_1\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis : sqrt (2 * 18) = 6\n\u22a2 0 < 4 ^ 6\n\ncase refine'_1\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis : sqrt (2 * 18) = 6\n\u22a2 0 < 18 * (2 * 18) ^ sqrt (2 * 18) / 4 ^ (18 / 3)\n\ncase refine'_1.a\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis : sqrt (2 * 18) = 6\n\u22a2 0 < 18", "state_after": "case refine'_1\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis : sqrt (2 * 18) = 6\n\u22a2 0 < 4 ^ 6\n\ncase refine'_1\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis : sqrt (2 * 18) = 6\n\u22a2 0 < 18 * (2 * 18) ^ sqrt (2 * 18) / 4 ^ (18 / 3)\n\ncase refine'_1.a\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis : sqrt (2 * 18) = 6\n\u22a2 0 < 18"}, {"tactic": "apply rpow_pos_of_pos", "annotated_tactic": ["apply <a>rpow_pos_of_pos</a>", [{"full_name": "Real.rpow_pos_of_pos", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/Real.lean", "def_pos": [92, 9], "def_end_pos": [92, 24]}]], "state_before": "case refine'_1\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis : sqrt (2 * 18) = 6\n\u22a2 0 < 4 ^ 6\n\ncase refine'_1\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis : sqrt (2 * 18) = 6\n\u22a2 0 < 18 * (2 * 18) ^ sqrt (2 * 18) / 4 ^ (18 / 3)\n\ncase refine'_1.a\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis : sqrt (2 * 18) = 6\n\u22a2 0 < 18", "state_after": "case refine'_1.hx\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis : sqrt (2 * 18) = 6\n\u22a2 0 < 4\n\ncase refine'_1\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis : sqrt (2 * 18) = 6\n\u22a2 0 < 18 * (2 * 18) ^ sqrt (2 * 18) / 4 ^ (18 / 3)\n\ncase refine'_1.a\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis : sqrt (2 * 18) = 6\n\u22a2 0 < 18"}, {"tactic": "norm_num1", "annotated_tactic": ["norm_num1", []], "state_before": "case refine'_1.hx\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis : sqrt (2 * 18) = 6\n\u22a2 0 < 4\n\ncase refine'_1\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis : sqrt (2 * 18) = 6\n\u22a2 0 < 18 * (2 * 18) ^ sqrt (2 * 18) / 4 ^ (18 / 3)\n\ncase refine'_1.a\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis : sqrt (2 * 18) = 6\n\u22a2 0 < 18", "state_after": "case refine'_1\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis : sqrt (2 * 18) = 6\n\u22a2 0 < 18 * (2 * 18) ^ sqrt (2 * 18) / 4 ^ (18 / 3)\n\ncase refine'_1.a\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis : sqrt (2 * 18) = 6\n\u22a2 0 < 18"}, {"tactic": "apply hf' 18", "annotated_tactic": ["apply hf' 18", []], "state_before": "case refine'_1\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis : sqrt (2 * 18) = 6\n\u22a2 0 < 18 * (2 * 18) ^ sqrt (2 * 18) / 4 ^ (18 / 3)\n\ncase refine'_1.a\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis : sqrt (2 * 18) = 6\n\u22a2 0 < 18", "state_after": "case refine'_1\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis : sqrt (2 * 18) = 6\n\u22a2 0 < 18\n\ncase refine'_1.a\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis : sqrt (2 * 18) = 6\n\u22a2 0 < 18"}, {"tactic": "norm_num1", "annotated_tactic": ["norm_num1", []], "state_before": "case refine'_1\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis : sqrt (2 * 18) = 6\n\u22a2 0 < 18\n\ncase refine'_1.a\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis : sqrt (2 * 18) = 6\n\u22a2 0 < 18", "state_after": "case refine'_1.a\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis : sqrt (2 * 18) = 6\n\u22a2 0 < 18"}, {"tactic": "norm_num1", "annotated_tactic": ["norm_num1", []], "state_before": "case refine'_1.a\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis : sqrt (2 * 18) = 6\n\u22a2 0 < 18", "state_after": "no goals"}, {"tactic": "norm_num1", "annotated_tactic": ["norm_num1", []], "state_before": "x : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\n\u22a2 0 < 6", "state_after": "no goals"}, {"tactic": "norm_num1", "annotated_tactic": ["norm_num1", []], "state_before": "x : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\n\u22a2 6 * 6 = 2 * 18", "state_after": "no goals"}, {"tactic": "have : sqrt (2 * 512) = 32 :=\n (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1)", "annotated_tactic": ["have : <a>sqrt</a> (2 * 512) = 32 :=\n (<a>sqrt_eq_iff_mul_self_eq_of_pos</a> (by norm_num1)).<a>mpr</a> (by norm_num1)", [{"full_name": "Real.sqrt", "def_path": "Mathlib/Data/Real/Sqrt.lean", "def_pos": [166, 19], "def_end_pos": [166, 23]}, {"full_name": "Real.sqrt_eq_iff_mul_self_eq_of_pos", "def_path": "Mathlib/Data/Real/Sqrt.lean", "def_pos": [222, 9], "def_end_pos": [222, 39]}, {"full_name": "Iff.mpr", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [92, 3], "def_end_pos": [92, 6]}]], "state_before": "case refine'_2\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\n\u22a2 f 512 \u2264 0", "state_after": "case refine'_2\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis : sqrt (2 * 512) = 32\n\u22a2 f 512 \u2264 0"}, {"tactic": "rw [hf, log_nonpos_iff (hf' _ _), this, div_le_one] <;> norm_num1", "annotated_tactic": ["rw [hf, <a>log_nonpos_iff</a> (hf' _ _), this, <a>div_le_one</a>] <;> norm_num1", [{"full_name": "Real.log_nonpos_iff", "def_path": "Mathlib/Analysis/SpecialFunctions/Log/Basic.lean", "def_pos": [213, 9], "def_end_pos": [213, 23]}, {"full_name": "div_le_one", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [425, 9], "def_end_pos": [425, 19]}]], "state_before": "case refine'_2\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis : sqrt (2 * 512) = 32\n\u22a2 f 512 \u2264 0", "state_after": "case refine'_2\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis : sqrt (2 * 512) = 32\n\u22a2 512 * 1024 ^ 32 \u2264 4 ^ (512 / 3)\n\ncase refine'_2\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis : sqrt (2 * 512) = 32\n\u22a2 0 < 4 ^ (512 / 3)"}, {"tactic": "have : (512 : \u211d) = 2 ^ (9 : \u2115)", "annotated_tactic": ["have : (512 : \u211d) = 2 ^ (9 : \u2115)", []], "state_before": "case refine'_2\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis : sqrt (2 * 512) = 32\n\u22a2 512 * 1024 ^ 32 \u2264 4 ^ (512 / 3)\n\ncase refine'_2\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis : sqrt (2 * 512) = 32\n\u22a2 0 < 4 ^ (512 / 3)", "state_after": "case this\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis : sqrt (2 * 512) = 32\n\u22a2 512 = 2 ^ \u21919\n\ncase refine'_2\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis\u271d : sqrt (2 * 512) = 32\nthis : 512 = 2 ^ \u21919\n\u22a2 512 * 1024 ^ 32 \u2264 4 ^ (512 / 3)\n\ncase refine'_2\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis : sqrt (2 * 512) = 32\n\u22a2 0 < 4 ^ (512 / 3)"}, {"tactic": "conv_lhs => rw [this]", "annotated_tactic": ["conv_lhs => rw [this]", []], "state_before": "case refine'_2\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis\u271d : sqrt (2 * 512) = 32\nthis : 512 = 2 ^ \u21919\n\u22a2 512 * 1024 ^ 32 \u2264 4 ^ (512 / 3)\n\ncase refine'_2\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis : sqrt (2 * 512) = 32\n\u22a2 0 < 4 ^ (512 / 3)", "state_after": "case refine'_2\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis\u271d : sqrt (2 * 512) = 32\nthis : 512 = 2 ^ \u21919\n\u22a2 2 ^ \u21919 * 1024 ^ 32 \u2264 4 ^ (512 / 3)\n\ncase refine'_2\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis : sqrt (2 * 512) = 32\n\u22a2 0 < 4 ^ (512 / 3)"}, {"tactic": "have : (1024 : \u211d) = 2 ^ (10 : \u2115)", "annotated_tactic": ["have : (1024 : \u211d) = 2 ^ (10 : \u2115)", []], "state_before": "case refine'_2\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis\u271d : sqrt (2 * 512) = 32\nthis : 512 = 2 ^ \u21919\n\u22a2 2 ^ \u21919 * 1024 ^ 32 \u2264 4 ^ (512 / 3)\n\ncase refine'_2\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis : sqrt (2 * 512) = 32\n\u22a2 0 < 4 ^ (512 / 3)", "state_after": "case this\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis\u271d : sqrt (2 * 512) = 32\nthis : 512 = 2 ^ \u21919\n\u22a2 1024 = 2 ^ \u219110\n\ncase refine'_2\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis\u271d\u00b9 : sqrt (2 * 512) = 32\nthis\u271d : 512 = 2 ^ \u21919\nthis : 1024 = 2 ^ \u219110\n\u22a2 2 ^ \u21919 * 1024 ^ 32 \u2264 4 ^ (512 / 3)\n\ncase refine'_2\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis : sqrt (2 * 512) = 32\n\u22a2 0 < 4 ^ (512 / 3)"}, {"tactic": "rw [this, \u2190 rpow_mul, \u2190 rpow_add] <;> norm_num1", "annotated_tactic": ["rw [this, \u2190 <a>rpow_mul</a>, \u2190 <a>rpow_add</a>] <;> norm_num1", [{"full_name": "Real.rpow_mul", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/Real.lean", "def_pos": [317, 9], "def_end_pos": [317, 17]}, {"full_name": "Real.rpow_add", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/Real.lean", "def_pos": [173, 9], "def_end_pos": [173, 17]}]], "state_before": "case refine'_2\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis\u271d\u00b9 : sqrt (2 * 512) = 32\nthis\u271d : 512 = 2 ^ \u21919\nthis : 1024 = 2 ^ \u219110\n\u22a2 2 ^ \u21919 * 1024 ^ 32 \u2264 4 ^ (512 / 3)\n\ncase refine'_2\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis : sqrt (2 * 512) = 32\n\u22a2 0 < 4 ^ (512 / 3)", "state_after": "case refine'_2\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis\u271d\u00b9 : sqrt (2 * 512) = 32\nthis\u271d : 512 = 2 ^ \u21919\nthis : 1024 = 2 ^ \u219110\n\u22a2 2 ^ 329 \u2264 4 ^ (512 / 3)\n\ncase refine'_2\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis : sqrt (2 * 512) = 32\n\u22a2 0 < 4 ^ (512 / 3)"}, {"tactic": "have : (4 : \u211d) = 2 ^ (2 : \u2115)", "annotated_tactic": ["have : (4 : \u211d) = 2 ^ (2 : \u2115)", []], "state_before": "case refine'_2\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis\u271d\u00b9 : sqrt (2 * 512) = 32\nthis\u271d : 512 = 2 ^ \u21919\nthis : 1024 = 2 ^ \u219110\n\u22a2 2 ^ 329 \u2264 4 ^ (512 / 3)\n\ncase refine'_2\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis : sqrt (2 * 512) = 32\n\u22a2 0 < 4 ^ (512 / 3)", "state_after": "case this\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis\u271d\u00b9 : sqrt (2 * 512) = 32\nthis\u271d : 512 = 2 ^ \u21919\nthis : 1024 = 2 ^ \u219110\n\u22a2 4 = 2 ^ \u21912\n\ncase refine'_2\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis\u271d\u00b2 : sqrt (2 * 512) = 32\nthis\u271d\u00b9 : 512 = 2 ^ \u21919\nthis\u271d : 1024 = 2 ^ \u219110\nthis : 4 = 2 ^ \u21912\n\u22a2 2 ^ 329 \u2264 4 ^ (512 / 3)\n\ncase refine'_2\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis : sqrt (2 * 512) = 32\n\u22a2 0 < 4 ^ (512 / 3)"}, {"tactic": "rw [this, \u2190 rpow_mul] <;> norm_num1", "annotated_tactic": ["rw [this, \u2190 <a>rpow_mul</a>] <;> norm_num1", [{"full_name": "Real.rpow_mul", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/Real.lean", "def_pos": [317, 9], "def_end_pos": [317, 17]}]], "state_before": "case refine'_2\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis\u271d\u00b2 : sqrt (2 * 512) = 32\nthis\u271d\u00b9 : 512 = 2 ^ \u21919\nthis\u271d : 1024 = 2 ^ \u219110\nthis : 4 = 2 ^ \u21912\n\u22a2 2 ^ 329 \u2264 4 ^ (512 / 3)\n\ncase refine'_2\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis : sqrt (2 * 512) = 32\n\u22a2 0 < 4 ^ (512 / 3)", "state_after": "case refine'_2\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis\u271d\u00b2 : sqrt (2 * 512) = 32\nthis\u271d\u00b9 : 512 = 2 ^ \u21919\nthis\u271d : 1024 = 2 ^ \u219110\nthis : 4 = 2 ^ \u21912\n\u22a2 2 ^ 329 \u2264 2 ^ (1024 / 3)\n\ncase refine'_2\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis : sqrt (2 * 512) = 32\n\u22a2 0 < 4 ^ (512 / 3)"}, {"tactic": "apply rpow_le_rpow_of_exponent_le <;> norm_num1", "annotated_tactic": ["apply <a>rpow_le_rpow_of_exponent_le</a> <;> norm_num1", [{"full_name": "Real.rpow_le_rpow_of_exponent_le", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/Real.lean", "def_pos": [491, 9], "def_end_pos": [491, 36]}]], "state_before": "case refine'_2\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis\u271d\u00b2 : sqrt (2 * 512) = 32\nthis\u271d\u00b9 : 512 = 2 ^ \u21919\nthis\u271d : 1024 = 2 ^ \u219110\nthis : 4 = 2 ^ \u21912\n\u22a2 2 ^ 329 \u2264 2 ^ (1024 / 3)\n\ncase refine'_2\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis : sqrt (2 * 512) = 32\n\u22a2 0 < 4 ^ (512 / 3)", "state_after": "case refine'_2\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis : sqrt (2 * 512) = 32\n\u22a2 0 < 4 ^ (512 / 3)"}, {"tactic": "apply rpow_pos_of_pos four_pos", "annotated_tactic": ["apply <a>rpow_pos_of_pos</a> <a>four_pos</a>", [{"full_name": "Real.rpow_pos_of_pos", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/Real.lean", "def_pos": [92, 9], "def_end_pos": [92, 24]}, {"full_name": "four_pos", "def_path": "Mathlib/Algebra/Order/Monoid/NatCast.lean", "def_pos": [119, 7], "def_end_pos": [119, 15]}]], "state_before": "case refine'_2\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis : sqrt (2 * 512) = 32\n\u22a2 0 < 4 ^ (512 / 3)", "state_after": "no goals"}, {"tactic": "norm_num1", "annotated_tactic": ["norm_num1", []], "state_before": "x : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\n\u22a2 0 < 32", "state_after": "no goals"}, {"tactic": "norm_num1", "annotated_tactic": ["norm_num1", []], "state_before": "x : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\n\u22a2 32 * 32 = 2 * 512", "state_after": "no goals"}, {"tactic": "rw [rpow_nat_cast 2 9]", "annotated_tactic": ["rw [<a>rpow_nat_cast</a> 2 9]", [{"full_name": "Real.rpow_nat_cast", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/Real.lean", "def_pos": [357, 9], "def_end_pos": [357, 22]}]], "state_before": "case this\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis : sqrt (2 * 512) = 32\n\u22a2 512 = 2 ^ \u21919", "state_after": "case this\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis : sqrt (2 * 512) = 32\n\u22a2 512 = 2 ^ 9"}, {"tactic": "norm_num1", "annotated_tactic": ["norm_num1", []], "state_before": "case this\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis : sqrt (2 * 512) = 32\n\u22a2 512 = 2 ^ 9", "state_after": "no goals"}, {"tactic": "rw [rpow_nat_cast 2 10]", "annotated_tactic": ["rw [<a>rpow_nat_cast</a> 2 10]", [{"full_name": "Real.rpow_nat_cast", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/Real.lean", "def_pos": [357, 9], "def_end_pos": [357, 22]}]], "state_before": "case this\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis\u271d : sqrt (2 * 512) = 32\nthis : 512 = 2 ^ \u21919\n\u22a2 1024 = 2 ^ \u219110", "state_after": "case this\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis\u271d : sqrt (2 * 512) = 32\nthis : 512 = 2 ^ \u21919\n\u22a2 1024 = 2 ^ 10"}, {"tactic": "norm_num1", "annotated_tactic": ["norm_num1", []], "state_before": "case this\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis\u271d : sqrt (2 * 512) = 32\nthis : 512 = 2 ^ \u21919\n\u22a2 1024 = 2 ^ 10", "state_after": "no goals"}, {"tactic": "rw [rpow_nat_cast 2 2]", "annotated_tactic": ["rw [<a>rpow_nat_cast</a> 2 2]", [{"full_name": "Real.rpow_nat_cast", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/Real.lean", "def_pos": [357, 9], "def_end_pos": [357, 22]}]], "state_before": "case this\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis\u271d\u00b9 : sqrt (2 * 512) = 32\nthis\u271d : 512 = 2 ^ \u21919\nthis : 1024 = 2 ^ \u219110\n\u22a2 4 = 2 ^ \u21912", "state_after": "case this\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis\u271d\u00b9 : sqrt (2 * 512) = 32\nthis\u271d : 512 = 2 ^ \u21919\nthis : 1024 = 2 ^ \u219110\n\u22a2 4 = 2 ^ 2"}, {"tactic": "norm_num1", "annotated_tactic": ["norm_num1", []], "state_before": "case this\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis\u271d\u00b9 : sqrt (2 * 512) = 32\nthis\u271d : 512 = 2 ^ \u21919\nthis : 1024 = 2 ^ \u219110\n\u22a2 4 = 2 ^ 2", "state_after": "no goals"}]

https://github.com/leanprover-community/mathlib4

3ce43c18f614b76e161f911b75a3e1ef641620ff

Mathlib/Algebra/ContinuedFractions/Computation/Approximations.lean

GeneralizedContinuedFraction.abs_sub_convergents_le'

[{"tactic": "have not_terminated_at_n : \u00ac(of v).TerminatedAt n := by\n simp [terminatedAt_iff_part_denom_none, nth_part_denom_eq]", "annotated_tactic": ["have not_terminated_at_n : \u00ac(<a>of</a> v).<a>TerminatedAt</a> n := by\n simp [<a>terminatedAt_iff_part_denom_none</a>, nth_part_denom_eq]", [{"full_name": "GeneralizedContinuedFraction.of", "def_path": "Mathlib/Algebra/ContinuedFractions/Computation/Basic.lean", "def_pos": [195, 15], "def_end_pos": [195, 17]}, {"full_name": "GeneralizedContinuedFraction.TerminatedAt", "def_path": "Mathlib/Algebra/ContinuedFractions/Basic.lean", "def_pos": [145, 5], "def_end_pos": [145, 17]}, {"full_name": "GeneralizedContinuedFraction.terminatedAt_iff_part_denom_none", "def_path": "Mathlib/Algebra/ContinuedFractions/Translations.lean", "def_pos": [53, 9], "def_end_pos": [53, 41]}]], "state_before": "K : Type u_1\nv : K\nn : \u2115\ninst\u271d\u00b9 : LinearOrderedField K\ninst\u271d : FloorRing K\nb : K\nnth_part_denom_eq : Stream'.Seq.get? (partialDenominators (of v)) n = some b\n\u22a2 |v - convergents (of v) n| \u2264 1 / (b * denominators (of v) n * denominators (of v) n)", "state_after": "K : Type u_1\nv : K\nn : \u2115\ninst\u271d\u00b9 : LinearOrderedField K\ninst\u271d : FloorRing K\nb : K\nnth_part_denom_eq : Stream'.Seq.get? (partialDenominators (of v)) n = some b\nnot_terminated_at_n : \u00acTerminatedAt (of v) n\n\u22a2 |v - convergents (of v) n| \u2264 1 / (b * denominators (of v) n * denominators (of v) n)"}, {"tactic": "refine' (abs_sub_convergents_le not_terminated_at_n).trans _", "annotated_tactic": ["refine' (<a>abs_sub_convergents_le</a> not_terminated_at_n).<a>trans</a> _", [{"full_name": "GeneralizedContinuedFraction.abs_sub_convergents_le", "def_path": "Mathlib/Algebra/ContinuedFractions/Computation/Approximations.lean", "def_pos": [454, 9], "def_end_pos": [454, 31]}, {"full_name": "LE.le.trans", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [120, 7], "def_end_pos": [120, 18]}]], "state_before": "K : Type u_1\nv : K\nn : \u2115\ninst\u271d\u00b9 : LinearOrderedField K\ninst\u271d : FloorRing K\nb : K\nnth_part_denom_eq : Stream'.Seq.get? (partialDenominators (of v)) n = some b\nnot_terminated_at_n : \u00acTerminatedAt (of v) n\n\u22a2 |v - convergents (of v) n| \u2264 1 / (b * denominators (of v) n * denominators (of v) n)", "state_after": "K : Type u_1\nv : K\nn : \u2115\ninst\u271d\u00b9 : LinearOrderedField K\ninst\u271d : FloorRing K\nb : K\nnth_part_denom_eq : Stream'.Seq.get? (partialDenominators (of v)) n = some b\nnot_terminated_at_n : \u00acTerminatedAt (of v) n\n\u22a2 1 / (denominators (of v) n * denominators (of v) (n + 1)) \u2264 1 / (b * denominators (of v) n * denominators (of v) n)"}, {"tactic": "rcases (zero_le_of_denom (K := K)).eq_or_gt with\n ((hB : (GeneralizedContinuedFraction.of v).denominators n = 0) | hB)", "annotated_tactic": ["rcases (<a>zero_le_of_denom</a> (K := K)).<a>eq_or_gt</a> with\n ((hB : (<a>GeneralizedContinuedFraction.of</a> v).<a>denominators</a> n = 0) | hB)", [{"full_name": "GeneralizedContinuedFraction.zero_le_of_denom", "def_path": "Mathlib/Algebra/ContinuedFractions/Computation/Approximations.lean", "def_pos": [269, 9], "def_end_pos": [269, 25]}, {"full_name": "LE.le.eq_or_gt", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [430, 7], "def_end_pos": [430, 21]}, {"full_name": "GeneralizedContinuedFraction.of", "def_path": "Mathlib/Algebra/ContinuedFractions/Computation/Basic.lean", "def_pos": [195, 15], "def_end_pos": [195, 17]}, {"full_name": "GeneralizedContinuedFraction.denominators", "def_path": "Mathlib/Algebra/ContinuedFractions/Basic.lean", "def_pos": [374, 5], "def_end_pos": [374, 17]}]], "state_before": "K : Type u_1\nv : K\nn : \u2115\ninst\u271d\u00b9 : LinearOrderedField K\ninst\u271d : FloorRing K\nb : K\nnth_part_denom_eq : Stream'.Seq.get? (partialDenominators (of v)) n = some b\nnot_terminated_at_n : \u00acTerminatedAt (of v) n\n\u22a2 1 / (denominators (of v) n * denominators (of v) (n + 1)) \u2264 1 / (b * denominators (of v) n * denominators (of v) n)", "state_after": "case inl\nK : Type u_1\nv : K\nn : \u2115\ninst\u271d\u00b9 : LinearOrderedField K\ninst\u271d : FloorRing K\nb : K\nnth_part_denom_eq : Stream'.Seq.get? (partialDenominators (of v)) n = some b\nnot_terminated_at_n : \u00acTerminatedAt (of v) n\nhB : denominators (of v) n = 0\n\u22a2 1 / (denominators (of v) n * denominators (of v) (n + 1)) \u2264 1 / (b * denominators (of v) n * denominators (of v) n)\n\ncase inr\nK : Type u_1\nv : K\nn : \u2115\ninst\u271d\u00b9 : LinearOrderedField K\ninst\u271d : FloorRing K\nb : K\nnth_part_denom_eq : Stream'.Seq.get? (partialDenominators (of v)) n = some b\nnot_terminated_at_n : \u00acTerminatedAt (of v) n\nhB : 0 < denominators (of v) n\n\u22a2 1 / (denominators (of v) n * denominators (of v) (n + 1)) \u2264 1 / (b * denominators (of v) n * denominators (of v) n)"}, {"tactic": "simp [terminatedAt_iff_part_denom_none, nth_part_denom_eq]", "annotated_tactic": ["simp [<a>terminatedAt_iff_part_denom_none</a>, nth_part_denom_eq]", [{"full_name": "GeneralizedContinuedFraction.terminatedAt_iff_part_denom_none", "def_path": "Mathlib/Algebra/ContinuedFractions/Translations.lean", "def_pos": [53, 9], "def_end_pos": [53, 41]}]], "state_before": "K : Type u_1\nv : K\nn : \u2115\ninst\u271d\u00b9 : LinearOrderedField K\ninst\u271d : FloorRing K\nb : K\nnth_part_denom_eq : Stream'.Seq.get? (partialDenominators (of v)) n = some b\n\u22a2 \u00acTerminatedAt (of v) n", "state_after": "no goals"}, {"tactic": "simp only [hB, mul_zero, zero_mul, div_zero, le_refl]", "annotated_tactic": ["simp only [hB, <a>mul_zero</a>, <a>zero_mul</a>, <a>div_zero</a>, <a>le_refl</a>]", [{"full_name": "MulZeroClass.mul_zero", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [38, 3], "def_end_pos": [38, 11]}, {"full_name": "MulZeroClass.zero_mul", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [36, 3], "def_end_pos": [36, 11]}, {"full_name": "div_zero", "def_path": "Mathlib/Algebra/GroupWithZero/Basic.lean", "def_pos": [295, 9], "def_end_pos": [295, 17]}, {"full_name": "le_refl", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [50, 9], "def_end_pos": [50, 16]}]], "state_before": "case inl\nK : Type u_1\nv : K\nn : \u2115\ninst\u271d\u00b9 : LinearOrderedField K\ninst\u271d : FloorRing K\nb : K\nnth_part_denom_eq : Stream'.Seq.get? (partialDenominators (of v)) n = some b\nnot_terminated_at_n : \u00acTerminatedAt (of v) n\nhB : denominators (of v) n = 0\n\u22a2 1 / (denominators (of v) n * denominators (of v) (n + 1)) \u2264 1 / (b * denominators (of v) n * denominators (of v) n)", "state_after": "no goals"}, {"tactic": "apply one_div_le_one_div_of_le", "annotated_tactic": ["apply <a>one_div_le_one_div_of_le</a>", [{"full_name": "one_div_le_one_div_of_le", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [451, 9], "def_end_pos": [451, 33]}]], "state_before": "case inr\nK : Type u_1\nv : K\nn : \u2115\ninst\u271d\u00b9 : LinearOrderedField K\ninst\u271d : FloorRing K\nb : K\nnth_part_denom_eq : Stream'.Seq.get? (partialDenominators (of v)) n = some b\nnot_terminated_at_n : \u00acTerminatedAt (of v) n\nhB : 0 < denominators (of v) n\n\u22a2 1 / (denominators (of v) n * denominators (of v) (n + 1)) \u2264 1 / (b * denominators (of v) n * denominators (of v) n)", "state_after": "case inr.ha\nK : Type u_1\nv : K\nn : \u2115\ninst\u271d\u00b9 : LinearOrderedField K\ninst\u271d : FloorRing K\nb : K\nnth_part_denom_eq : Stream'.Seq.get? (partialDenominators (of v)) n = some b\nnot_terminated_at_n : \u00acTerminatedAt (of v) n\nhB : 0 < denominators (of v) n\n\u22a2 0 < b * denominators (of v) n * denominators (of v) n\n\ncase inr.h\nK : Type u_1\nv : K\nn : \u2115\ninst\u271d\u00b9 : LinearOrderedField K\ninst\u271d : FloorRing K\nb : K\nnth_part_denom_eq : Stream'.Seq.get? (partialDenominators (of v)) n = some b\nnot_terminated_at_n : \u00acTerminatedAt (of v) n\nhB : 0 < denominators (of v) n\n\u22a2 b * denominators (of v) n * denominators (of v) n \u2264 denominators (of v) n * denominators (of v) (n + 1)"}, {"tactic": "have : 0 < b := zero_lt_one.trans_le (of_one_le_get?_part_denom nth_part_denom_eq)", "annotated_tactic": ["have : 0 < b := zero_lt_one.trans_le (<a>of_one_le_get?_part_denom</a> nth_part_denom_eq)", [{"full_name": "GeneralizedContinuedFraction.of_one_le_get?_part_denom", "def_path": "Mathlib/Algebra/ContinuedFractions/Computation/Approximations.lean", "def_pos": [138, 9], "def_end_pos": [138, 34]}]], "state_before": "case inr.ha\nK : Type u_1\nv : K\nn : \u2115\ninst\u271d\u00b9 : LinearOrderedField K\ninst\u271d : FloorRing K\nb : K\nnth_part_denom_eq : Stream'.Seq.get? (partialDenominators (of v)) n = some b\nnot_terminated_at_n : \u00acTerminatedAt (of v) n\nhB : 0 < denominators (of v) n\n\u22a2 0 < b * denominators (of v) n * denominators (of v) n", "state_after": "case inr.ha\nK : Type u_1\nv : K\nn : \u2115\ninst\u271d\u00b9 : LinearOrderedField K\ninst\u271d : FloorRing K\nb : K\nnth_part_denom_eq : Stream'.Seq.get? (partialDenominators (of v)) n = some b\nnot_terminated_at_n : \u00acTerminatedAt (of v) n\nhB : 0 < denominators (of v) n\nthis : 0 < b\n\u22a2 0 < b * denominators (of v) n * denominators (of v) n"}, {"tactic": "apply_rules [mul_pos]", "annotated_tactic": ["apply_rules [<a>mul_pos</a>]", [{"full_name": "mul_pos", "def_path": "Mathlib/Algebra/Order/Ring/Lemmas.lean", "def_pos": [345, 7], "def_end_pos": [345, 14]}]], "state_before": "case inr.ha\nK : Type u_1\nv : K\nn : \u2115\ninst\u271d\u00b9 : LinearOrderedField K\ninst\u271d : FloorRing K\nb : K\nnth_part_denom_eq : Stream'.Seq.get? (partialDenominators (of v)) n = some b\nnot_terminated_at_n : \u00acTerminatedAt (of v) n\nhB : 0 < denominators (of v) n\nthis : 0 < b\n\u22a2 0 < b * denominators (of v) n * denominators (of v) n", "state_after": "no goals"}, {"tactic": "conv_rhs => rw [mul_comm]", "annotated_tactic": ["conv_rhs => rw [<a>mul_comm</a>]", [{"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [302, 9], "def_end_pos": [302, 17]}]], "state_before": "case inr.h\nK : Type u_1\nv : K\nn : \u2115\ninst\u271d\u00b9 : LinearOrderedField K\ninst\u271d : FloorRing K\nb : K\nnth_part_denom_eq : Stream'.Seq.get? (partialDenominators (of v)) n = some b\nnot_terminated_at_n : \u00acTerminatedAt (of v) n\nhB : 0 < denominators (of v) n\n\u22a2 b * denominators (of v) n * denominators (of v) n \u2264 denominators (of v) n * denominators (of v) (n + 1)", "state_after": "case inr.h\nK : Type u_1\nv : K\nn : \u2115\ninst\u271d\u00b9 : LinearOrderedField K\ninst\u271d : FloorRing K\nb : K\nnth_part_denom_eq : Stream'.Seq.get? (partialDenominators (of v)) n = some b\nnot_terminated_at_n : \u00acTerminatedAt (of v) n\nhB : 0 < denominators (of v) n\n\u22a2 b * denominators (of v) n * denominators (of v) n \u2264 denominators (of v) (n + 1) * denominators (of v) n"}, {"tactic": "exact mul_le_mul_of_nonneg_right (le_of_succ_get?_denom nth_part_denom_eq) hB.le", "annotated_tactic": ["exact <a>mul_le_mul_of_nonneg_right</a> (<a>le_of_succ_get?_denom</a> nth_part_denom_eq) hB.le", [{"full_name": "mul_le_mul_of_nonneg_right", "def_path": "Mathlib/Algebra/Order/Ring/Lemmas.lean", "def_pos": [156, 9], "def_end_pos": [156, 35]}, {"full_name": "GeneralizedContinuedFraction.le_of_succ_get?_denom", "def_path": "Mathlib/Algebra/ContinuedFractions/Computation/Approximations.lean", "def_pos": [284, 9], "def_end_pos": [284, 30]}]], "state_before": "case inr.h\nK : Type u_1\nv : K\nn : \u2115\ninst\u271d\u00b9 : LinearOrderedField K\ninst\u271d : FloorRing K\nb : K\nnth_part_denom_eq : Stream'.Seq.get? (partialDenominators (of v)) n = some b\nnot_terminated_at_n : \u00acTerminatedAt (of v) n\nhB : 0 < denominators (of v) n\n\u22a2 b * denominators (of v) n * denominators (of v) n \u2264 denominators (of v) (n + 1) * denominators (of v) n", "state_after": "no goals"}]

https://github.com/leanprover-community/mathlib4

3ce43c18f614b76e161f911b75a3e1ef641620ff

Mathlib/CategoryTheory/Sites/Grothendieck.lean

CategoryTheory.GrothendieckTopology.Cover.Arrow.middle_spec

[]

https://github.com/leanprover-community/mathlib4

3ce43c18f614b76e161f911b75a3e1ef641620ff

Mathlib/CategoryTheory/Preadditive/AdditiveFunctor.lean

CategoryTheory.Functor.map_nsmul

[]

https://github.com/leanprover-community/mathlib4

3ce43c18f614b76e161f911b75a3e1ef641620ff

Mathlib/Analysis/SumIntegralComparisons.lean

MonotoneOn.integral_le_sum

[{"tactic": "rw [\u2190 neg_le_neg_iff, \u2190 Finset.sum_neg_distrib, \u2190 intervalIntegral.integral_neg]", "annotated_tactic": ["rw [\u2190 <a>neg_le_neg_iff</a>, \u2190 <a>Finset.sum_neg_distrib</a>, \u2190 <a>intervalIntegral.integral_neg</a>]", [{"full_name": "neg_le_neg_iff", "def_path": "Mathlib/Algebra/Order/Group/Defs.lean", "def_pos": [342, 3], "def_end_pos": [342, 14]}, {"full_name": "Finset.sum_neg_distrib", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [1814, 3], "def_end_pos": [1814, 14]}, {"full_name": "intervalIntegral.integral_neg", "def_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "def_pos": [591, 16], "def_end_pos": [591, 28]}]], "state_before": "x\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhf : MonotoneOn f (Icc x\u2080 (x\u2080 + \u2191a))\n\u22a2 \u222b (x : \u211d) in x\u2080..x\u2080 + \u2191a, f x \u2264 \u2211 i in Finset.range a, f (x\u2080 + \u2191(i + 1))", "state_after": "x\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhf : MonotoneOn f (Icc x\u2080 (x\u2080 + \u2191a))\n\u22a2 \u2211 x in Finset.range a, -f (x\u2080 + \u2191(x + 1)) \u2264 \u222b (x : \u211d) in x\u2080..x\u2080 + \u2191a, -f x"}, {"tactic": "exact hf.neg.sum_le_integral", "annotated_tactic": ["exact hf.neg.sum_le_integral", []], "state_before": "x\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhf : MonotoneOn f (Icc x\u2080 (x\u2080 + \u2191a))\n\u22a2 \u2211 x in Finset.range a, -f (x\u2080 + \u2191(x + 1)) \u2264 \u222b (x : \u211d) in x\u2080..x\u2080 + \u2191a, -f x", "state_after": "no goals"}]

https://github.com/leanprover-community/mathlib4

3ce43c18f614b76e161f911b75a3e1ef641620ff

Mathlib/Algebra/GCDMonoid/Basic.lean

normUnit_one

[]

https://github.com/leanprover-community/mathlib4

3ce43c18f614b76e161f911b75a3e1ef641620ff

Mathlib/Data/Polynomial/Degree/Definitions.lean

Polynomial.coeff_mul_X_sub_C

[{"tactic": "simp [mul_sub]", "annotated_tactic": ["simp [<a>mul_sub</a>]", [{"full_name": "mul_sub", "def_path": "Mathlib/Algebra/Ring/Defs.lean", "def_pos": [365, 7], "def_end_pos": [365, 14]}]], "state_before": "R : Type u\nS : Type v\na\u271d b c d : R\nn m : \u2115\ninst\u271d : Ring R\np : R[X]\nr : R\na : \u2115\n\u22a2 coeff (p * (X - \u2191C r)) (a + 1) = coeff p a - coeff p (a + 1) * r", "state_after": "no goals"}]

https://github.com/leanprover-community/mathlib4

3ce43c18f614b76e161f911b75a3e1ef641620ff

Mathlib/Algebra/Periodic.lean

Function.Periodic.const_inv_mul

[]

https://github.com/leanprover-community/mathlib4

3ce43c18f614b76e161f911b75a3e1ef641620ff

Mathlib/Data/Polynomial/Expand.lean

Polynomial.expand_eq_zero

[]

https://github.com/leanprover-community/mathlib4

3ce43c18f614b76e161f911b75a3e1ef641620ff

Mathlib/NumberTheory/LegendreSymbol/JacobiSymbol.lean

jacobiSym.value_at

[{"tactic": "conv_rhs => rw [\u2190 prod_factors hb.pos.ne', cast_list_prod, \u03c7.map_list_prod]", "annotated_tactic": ["conv_rhs => rw [\u2190 <a>prod_factors</a> hb.pos.ne', <a>cast_list_prod</a>, \u03c7.map_list_prod]", [{"full_name": "Nat.prod_factors", "def_path": "Mathlib/Data/Nat/Factors.lean", "def_pos": [67, 9], "def_end_pos": [67, 21]}, {"full_name": "Nat.cast_list_prod", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [2202, 9], "def_end_pos": [2202, 23]}]], "state_before": "a : \u2124\nR : Type u_1\ninst\u271d : CommSemiring R\n\u03c7 : R \u2192* \u2124\nhp : \u2200 (p : \u2115) (pp : Nat.Prime p), p \u2260 2 \u2192 legendreSym p a = \u2191\u03c7 \u2191p\nb : \u2115\nhb : Odd b\n\u22a2 J(a | b) = \u2191\u03c7 \u2191b", "state_after": "a : \u2124\nR : Type u_1\ninst\u271d : CommSemiring R\n\u03c7 : R \u2192* \u2124\nhp : \u2200 (p : \u2115) (pp : Nat.Prime p), p \u2260 2 \u2192 legendreSym p a = \u2191\u03c7 \u2191p\nb : \u2115\nhb : Odd b\n\u22a2 J(a | b) = List.prod (List.map (\u2191\u03c7) (List.map Nat.cast (factors b)))"}, {"tactic": "rw [jacobiSym, List.map_map, \u2190 List.pmap_eq_map Nat.Prime _ _ fun _ => prime_of_mem_factors]", "annotated_tactic": ["rw [<a>jacobiSym</a>, <a>List.map_map</a>, \u2190 <a>List.pmap_eq_map</a> <a>Nat.Prime</a> _ _ fun _ => <a>prime_of_mem_factors</a>]", [{"full_name": "jacobiSym", "def_path": "Mathlib/NumberTheory/LegendreSymbol/JacobiSymbol.lean", "def_pos": [80, 5], "def_end_pos": [80, 14]}, {"full_name": "List.map_map", "def_path": "lake-packages/std/Std/Data/List/Init/Lemmas.lean", "def_pos": [96, 17], "def_end_pos": [96, 24]}, {"full_name": "List.pmap_eq_map", "def_path": "Mathlib/Data/List/Basic.lean", "def_pos": [3001, 9], "def_end_pos": [3001, 20]}, {"full_name": "Nat.Prime", "def_path": "Mathlib/Data/Nat/Prime.lean", "def_pos": [46, 5], "def_end_pos": [46, 10]}, {"full_name": "Nat.prime_of_mem_factors", "def_path": "Mathlib/Data/Nat/Factors.lean", "def_pos": [50, 9], "def_end_pos": [50, 29]}]], "state_before": "a : \u2124\nR : Type u_1\ninst\u271d : CommSemiring R\n\u03c7 : R \u2192* \u2124\nhp : \u2200 (p : \u2115) (pp : Nat.Prime p), p \u2260 2 \u2192 legendreSym p a = \u2191\u03c7 \u2191p\nb : \u2115\nhb : Odd b\n\u22a2 J(a | b) = List.prod (List.map (\u2191\u03c7) (List.map Nat.cast (factors b)))", "state_after": "a : \u2124\nR : Type u_1\ninst\u271d : CommSemiring R\n\u03c7 : R \u2192* \u2124\nhp : \u2200 (p : \u2115) (pp : Nat.Prime p), p \u2260 2 \u2192 legendreSym p a = \u2191\u03c7 \u2191p\nb : \u2115\nhb : Odd b\n\u22a2 List.prod (List.pmap (fun p pp => legendreSym p a) (factors b) (_ : \u2200 (x : \u2115), x \u2208 factors b \u2192 Nat.Prime x)) =\n List.prod (List.pmap (fun a x => (\u2191\u03c7 \u2218 Nat.cast) a) (factors b) (_ : \u2200 (x : \u2115), x \u2208 factors b \u2192 Nat.Prime x))"}, {"tactic": "congr 1", "annotated_tactic": ["congr 1", []], "state_before": "a : \u2124\nR : Type u_1\ninst\u271d : CommSemiring R\n\u03c7 : R \u2192* \u2124\nhp : \u2200 (p : \u2115) (pp : Nat.Prime p), p \u2260 2 \u2192 legendreSym p a = \u2191\u03c7 \u2191p\nb : \u2115\nhb : Odd b\n\u22a2 List.prod (List.pmap (fun p pp => legendreSym p a) (factors b) (_ : \u2200 (x : \u2115), x \u2208 factors b \u2192 Nat.Prime x)) =\n List.prod (List.pmap (fun a x => (\u2191\u03c7 \u2218 Nat.cast) a) (factors b) (_ : \u2200 (x : \u2115), x \u2208 factors b \u2192 Nat.Prime x))", "state_after": "case e_a\na : \u2124\nR : Type u_1\ninst\u271d : CommSemiring R\n\u03c7 : R \u2192* \u2124\nhp : \u2200 (p : \u2115) (pp : Nat.Prime p), p \u2260 2 \u2192 legendreSym p a = \u2191\u03c7 \u2191p\nb : \u2115\nhb : Odd b\n\u22a2 List.pmap (fun p pp => legendreSym p a) (factors b) (_ : \u2200 (x : \u2115), x \u2208 factors b \u2192 Nat.Prime x) =\n List.pmap (fun a x => (\u2191\u03c7 \u2218 Nat.cast) a) (factors b) (_ : \u2200 (x : \u2115), x \u2208 factors b \u2192 Nat.Prime x)"}, {"tactic": "apply List.pmap_congr", "annotated_tactic": ["apply <a>List.pmap_congr</a>", [{"full_name": "List.pmap_congr", "def_path": "Mathlib/Data/List/Basic.lean", "def_pos": [3006, 9], "def_end_pos": [3006, 19]}]], "state_before": "case e_a\na : \u2124\nR : Type u_1\ninst\u271d : CommSemiring R\n\u03c7 : R \u2192* \u2124\nhp : \u2200 (p : \u2115) (pp : Nat.Prime p), p \u2260 2 \u2192 legendreSym p a = \u2191\u03c7 \u2191p\nb : \u2115\nhb : Odd b\n\u22a2 List.pmap (fun p pp => legendreSym p a) (factors b) (_ : \u2200 (x : \u2115), x \u2208 factors b \u2192 Nat.Prime x) =\n List.pmap (fun a x => (\u2191\u03c7 \u2218 Nat.cast) a) (factors b) (_ : \u2200 (x : \u2115), x \u2208 factors b \u2192 Nat.Prime x)", "state_after": "case e_a.h\na : \u2124\nR : Type u_1\ninst\u271d : CommSemiring R\n\u03c7 : R \u2192* \u2124\nhp : \u2200 (p : \u2115) (pp : Nat.Prime p), p \u2260 2 \u2192 legendreSym p a = \u2191\u03c7 \u2191p\nb : \u2115\nhb : Odd b\n\u22a2 \u2200 (a_1 : \u2115), a_1 \u2208 factors b \u2192 \u2200 (h\u2081 : Nat.Prime a_1), Nat.Prime a_1 \u2192 legendreSym a_1 a = (\u2191\u03c7 \u2218 Nat.cast) a_1"}, {"tactic": "exact fun p h pp _ => hp p pp (hb.ne_two_of_dvd_nat <| dvd_of_mem_factors h)", "annotated_tactic": ["exact fun p h pp _ => hp p pp (hb.ne_two_of_dvd_nat <| <a>dvd_of_mem_factors</a> h)", [{"full_name": "Nat.dvd_of_mem_factors", "def_path": "Mathlib/Data/Nat/Factors.lean", "def_pos": [146, 9], "def_end_pos": [146, 27]}]], "state_before": "case e_a.h\na : \u2124\nR : Type u_1\ninst\u271d : CommSemiring R\n\u03c7 : R \u2192* \u2124\nhp : \u2200 (p : \u2115) (pp : Nat.Prime p), p \u2260 2 \u2192 legendreSym p a = \u2191\u03c7 \u2191p\nb : \u2115\nhb : Odd b\n\u22a2 \u2200 (a_1 : \u2115), a_1 \u2208 factors b \u2192 \u2200 (h\u2081 : Nat.Prime a_1), Nat.Prime a_1 \u2192 legendreSym a_1 a = (\u2191\u03c7 \u2218 Nat.cast) a_1", "state_after": "no goals"}]

https://github.com/leanprover-community/mathlib4

3ce43c18f614b76e161f911b75a3e1ef641620ff

Mathlib/Data/Set/Intervals/Basic.lean

Set.Ico_subset_Ico_union_Ico

[]

https://github.com/leanprover-community/mathlib4

3ce43c18f614b76e161f911b75a3e1ef641620ff

Mathlib/RingTheory/RingHomProperties.lean

RingHom.RespectsIso.cancel_left_isIso

[{"tactic": "convert hP.2 (f \u226b g) (asIso f).symm.commRingCatIsoToRingEquiv H", "annotated_tactic": ["convert hP.2 (f \u226b g) (<a>asIso</a> f).symm.commRingCatIsoToRingEquiv H", [{"full_name": "CategoryTheory.asIso", "def_path": "Mathlib/CategoryTheory/Iso.lean", "def_pos": [307, 19], "def_end_pos": [307, 24]}]], "state_before": "P : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\nhP : RespectsIso P\nR S T : CommRingCat\nf : R \u27f6 S\ng : S \u27f6 T\ninst\u271d : IsIso f\nH : P (f \u226b g)\n\u22a2 P g", "state_after": "case h.e'_5\nP : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\nhP : RespectsIso P\nR S T : CommRingCat\nf : R \u27f6 S\ng : S \u27f6 T\ninst\u271d : IsIso f\nH : P (f \u226b g)\n\u22a2 g = comp (f \u226b g) (RingEquiv.toRingHom (Iso.commRingCatIsoToRingEquiv (asIso f).symm))"}, {"tactic": "exact (IsIso.inv_hom_id_assoc _ _).symm", "annotated_tactic": ["exact (<a>IsIso.inv_hom_id_assoc</a> _ _).<a>symm</a>", [{"full_name": "CategoryTheory.IsIso.inv_hom_id_assoc", "def_path": "Mathlib/CategoryTheory/Iso.lean", "def_pos": [298, 9], "def_end_pos": [298, 25]}, {"full_name": "Eq.symm", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [310, 9], "def_end_pos": [310, 16]}]], "state_before": "case h.e'_5\nP : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\nhP : RespectsIso P\nR S T : CommRingCat\nf : R \u27f6 S\ng : S \u27f6 T\ninst\u271d : IsIso f\nH : P (f \u226b g)\n\u22a2 g = comp (f \u226b g) (RingEquiv.toRingHom (Iso.commRingCatIsoToRingEquiv (asIso f).symm))", "state_after": "no goals"}]

https://github.com/leanprover-community/mathlib4

3ce43c18f614b76e161f911b75a3e1ef641620ff

Mathlib/MeasureTheory/Measure/MeasureSpace.lean

MeasureTheory.ae_restrict_biUnion_iff

[{"tactic": "simp_rw [Filter.Eventually, ae_restrict_biUnion_eq s ht, mem_iSup]", "annotated_tactic": ["simp_rw [<a>Filter.Eventually</a>, <a>ae_restrict_biUnion_eq</a> s ht, <a>mem_iSup</a>]", [{"full_name": "Filter.Eventually", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1072, 15], "def_end_pos": [1072, 25]}, {"full_name": "MeasureTheory.ae_restrict_biUnion_eq", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2503, 9], "def_end_pos": [2503, 31]}, {"full_name": "Filter.mem_iSup", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [582, 9], "def_end_pos": [582, 17]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d : Set \u03b1\ns : \u03b9 \u2192 Set \u03b1\nt : Set \u03b9\nht : Set.Countable t\np : \u03b1 \u2192 Prop\n\u22a2 (\u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc (\u22c3 i \u2208 t, s i), p x) \u2194 \u2200 (i : \u03b9), i \u2208 t \u2192 \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc (s i), p x", "state_after": "no goals"}]

https://github.com/leanprover-community/mathlib4

3ce43c18f614b76e161f911b75a3e1ef641620ff

Mathlib/Data/Set/Basic.lean

Set.ite_univ

[{"tactic": "simp [Set.ite]", "annotated_tactic": ["simp [<a>Set.ite</a>]", [{"full_name": "Set.ite", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [2265, 15], "def_end_pos": [2265, 18]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\na b : \u03b1\ns\u271d s\u2081 s\u2082 t t\u2081 t\u2082 u s s' : Set \u03b1\n\u22a2 Set.ite univ s s' = s", "state_after": "no goals"}]

https://github.com/leanprover-community/mathlib4

3ce43c18f614b76e161f911b75a3e1ef641620ff

Mathlib/MeasureTheory/Measure/OuterMeasure.lean

MeasureTheory.OuterMeasure.sInf_eq_boundedBy_sInfGen

[{"tactic": "refine' le_antisymm _ _", "annotated_tactic": ["refine' <a>le_antisymm</a> _ _", [{"full_name": "le_antisymm", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [188, 9], "def_end_pos": [188, 20]}]], "state_before": "\u03b1 : Type u_1\nm : Set (OuterMeasure \u03b1)\n\u22a2 sInf m = boundedBy (sInfGen m)", "state_after": "case refine'_1\n\u03b1 : Type u_1\nm : Set (OuterMeasure \u03b1)\n\u22a2 sInf m \u2264 boundedBy (sInfGen m)\n\ncase refine'_2\n\u03b1 : Type u_1\nm : Set (OuterMeasure \u03b1)\n\u22a2 boundedBy (sInfGen m) \u2264 sInf m"}, {"tactic": "refine' le_boundedBy.2 fun s => le_iInf\u2082 fun \u03bc h\u03bc => _", "annotated_tactic": ["refine' <a>le_boundedBy</a>.2 fun s => <a>le_iInf\u2082</a> fun \u03bc h\u03bc => _", [{"full_name": "MeasureTheory.OuterMeasure.le_boundedBy", "def_path": "Mathlib/MeasureTheory/Measure/OuterMeasure.lean", "def_pos": [866, 9], "def_end_pos": [866, 21]}, {"full_name": "le_iInf\u2082", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [887, 9], "def_end_pos": [887, 17]}]], "state_before": "case refine'_1\n\u03b1 : Type u_1\nm : Set (OuterMeasure \u03b1)\n\u22a2 sInf m \u2264 boundedBy (sInfGen m)", "state_after": "case refine'_1\n\u03b1 : Type u_1\nm : Set (OuterMeasure \u03b1)\ns : Set \u03b1\n\u03bc : OuterMeasure \u03b1\nh\u03bc : \u03bc \u2208 m\n\u22a2 \u2191(sInf m) s \u2264 \u2191\u03bc s"}, {"tactic": "apply sInf_le h\u03bc", "annotated_tactic": ["apply <a>sInf_le</a> h\u03bc", [{"full_name": "sInf_le", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [265, 9], "def_end_pos": [265, 16]}]], "state_before": "case refine'_1\n\u03b1 : Type u_1\nm : Set (OuterMeasure \u03b1)\ns : Set \u03b1\n\u03bc : OuterMeasure \u03b1\nh\u03bc : \u03bc \u2208 m\n\u22a2 \u2191(sInf m) s \u2264 \u2191\u03bc s", "state_after": "no goals"}, {"tactic": "refine' le_sInf _", "annotated_tactic": ["refine' <a>le_sInf</a> _", [{"full_name": "le_sInf", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [269, 9], "def_end_pos": [269, 16]}]], "state_before": "case refine'_2\n\u03b1 : Type u_1\nm : Set (OuterMeasure \u03b1)\n\u22a2 boundedBy (sInfGen m) \u2264 sInf m", "state_after": "case refine'_2\n\u03b1 : Type u_1\nm : Set (OuterMeasure \u03b1)\n\u22a2 \u2200 (b : OuterMeasure \u03b1), b \u2208 m \u2192 boundedBy (sInfGen m) \u2264 b"}, {"tactic": "intro \u03bc h\u03bc t", "annotated_tactic": ["intro \u03bc h\u03bc t", []], "state_before": "case refine'_2\n\u03b1 : Type u_1\nm : Set (OuterMeasure \u03b1)\n\u22a2 \u2200 (b : OuterMeasure \u03b1), b \u2208 m \u2192 boundedBy (sInfGen m) \u2264 b", "state_after": "case refine'_2\n\u03b1 : Type u_1\nm : Set (OuterMeasure \u03b1)\n\u03bc : OuterMeasure \u03b1\nh\u03bc : \u03bc \u2208 m\nt : Set \u03b1\n\u22a2 \u2191(boundedBy (sInfGen m)) t \u2264 \u2191\u03bc t"}, {"tactic": "refine' le_trans (boundedBy_le t) (iInf\u2082_le \u03bc h\u03bc)", "annotated_tactic": ["refine' <a>le_trans</a> (<a>boundedBy_le</a> t) (<a>iInf\u2082_le</a> \u03bc h\u03bc)", [{"full_name": "le_trans", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [56, 9], "def_end_pos": [56, 17]}, {"full_name": "MeasureTheory.OuterMeasure.boundedBy_le", "def_path": "Mathlib/MeasureTheory/Measure/OuterMeasure.lean", "def_pos": [838, 9], "def_end_pos": [838, 21]}, {"full_name": "iInf\u2082_le", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [861, 9], "def_end_pos": [861, 17]}]], "state_before": "case refine'_2\n\u03b1 : Type u_1\nm : Set (OuterMeasure \u03b1)\n\u03bc : OuterMeasure \u03b1\nh\u03bc : \u03bc \u2208 m\nt : Set \u03b1\n\u22a2 \u2191(boundedBy (sInfGen m)) t \u2264 \u2191\u03bc t", "state_after": "no goals"}]

https://github.com/leanprover-community/mathlib4

3ce43c18f614b76e161f911b75a3e1ef641620ff

Mathlib/Data/Set/Intervals/Basic.lean

Set.Ioo_insert_left

[{"tactic": "rw [insert_eq, union_comm, Ioo_union_left h]", "annotated_tactic": ["rw [<a>insert_eq</a>, <a>union_comm</a>, <a>Ioo_union_left</a> h]", [{"full_name": "Set.insert_eq", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1310, 9], "def_end_pos": [1310, 18]}, {"full_name": "Set.union_comm", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [786, 9], "def_end_pos": [786, 19]}, {"full_name": "Set.Ioo_union_left", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [865, 9], "def_end_pos": [865, 23]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : PartialOrder \u03b1\na b c : \u03b1\nh : a < b\n\u22a2 insert a (Ioo a b) = Ico a b", "state_after": "no goals"}]

https://github.com/leanprover-community/mathlib4

3ce43c18f614b76e161f911b75a3e1ef641620ff

Mathlib/LinearAlgebra/CliffordAlgebra/Grading.lean

CliffordAlgebra.even_induction

[{"tactic": "refine' evenOdd_induction Q 0 (fun rx => _) (@hadd) h\u03b9\u03b9_mul x hx", "annotated_tactic": ["refine' <a>evenOdd_induction</a> Q 0 (fun rx => _) (@hadd) h\u03b9\u03b9_mul x hx", [{"full_name": "CliffordAlgebra.evenOdd_induction", "def_path": "Mathlib/LinearAlgebra/CliffordAlgebra/Grading.lean", "def_pos": [163, 9], "def_end_pos": [163, 26]}]], "state_before": "R : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ : QuadraticForm R M\nP : (x : CliffordAlgebra Q) \u2192 x \u2208 evenOdd Q 0 \u2192 Prop\nhr : \u2200 (r : R), P (\u2191(algebraMap R (CliffordAlgebra Q)) r) (_ : \u2191(algebraMap R (CliffordAlgebra Q)) r \u2208 evenOdd Q 0)\nhadd :\n \u2200 {x y : CliffordAlgebra Q} {hx : x \u2208 evenOdd Q 0} {hy : y \u2208 evenOdd Q 0},\n P x hx \u2192 P y hy \u2192 P (x + y) (_ : x + y \u2208 evenOdd Q 0)\nh\u03b9\u03b9_mul :\n \u2200 (m\u2081 m\u2082 : M) {x : CliffordAlgebra Q} {hx : x \u2208 evenOdd Q 0},\n P x hx \u2192 P (\u2191(\u03b9 Q) m\u2081 * \u2191(\u03b9 Q) m\u2082 * x) (_ : \u2191(\u03b9 Q) m\u2081 * \u2191(\u03b9 Q) m\u2082 * x \u2208 evenOdd Q 0)\nx : CliffordAlgebra Q\nhx : x \u2208 evenOdd Q 0\n\u22a2 P x hx", "state_after": "R : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ : QuadraticForm R M\nP : (x : CliffordAlgebra Q) \u2192 x \u2208 evenOdd Q 0 \u2192 Prop\nhr : \u2200 (r : R), P (\u2191(algebraMap R (CliffordAlgebra Q)) r) (_ : \u2191(algebraMap R (CliffordAlgebra Q)) r \u2208 evenOdd Q 0)\nhadd :\n \u2200 {x y : CliffordAlgebra Q} {hx : x \u2208 evenOdd Q 0} {hy : y \u2208 evenOdd Q 0},\n P x hx \u2192 P y hy \u2192 P (x + y) (_ : x + y \u2208 evenOdd Q 0)\nh\u03b9\u03b9_mul :\n \u2200 (m\u2081 m\u2082 : M) {x : CliffordAlgebra Q} {hx : x \u2208 evenOdd Q 0},\n P x hx \u2192 P (\u2191(\u03b9 Q) m\u2081 * \u2191(\u03b9 Q) m\u2082 * x) (_ : \u2191(\u03b9 Q) m\u2081 * \u2191(\u03b9 Q) m\u2082 * x \u2208 evenOdd Q 0)\nx : CliffordAlgebra Q\nhx : x \u2208 evenOdd Q 0\nrx : CliffordAlgebra Q\n\u22a2 \u2200 (h : rx \u2208 LinearMap.range (\u03b9 Q) ^ ZMod.val 0), P rx (_ : rx \u2208 \u2a06 i, LinearMap.range (\u03b9 Q) ^ \u2191i)"}, {"tactic": "rintro \u27e8r, rfl\u27e9", "annotated_tactic": ["rintro \u27e8r, rfl\u27e9", []], "state_before": "R : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ : QuadraticForm R M\nP : (x : CliffordAlgebra Q) \u2192 x \u2208 evenOdd Q 0 \u2192 Prop\nhr : \u2200 (r : R), P (\u2191(algebraMap R (CliffordAlgebra Q)) r) (_ : \u2191(algebraMap R (CliffordAlgebra Q)) r \u2208 evenOdd Q 0)\nhadd :\n \u2200 {x y : CliffordAlgebra Q} {hx : x \u2208 evenOdd Q 0} {hy : y \u2208 evenOdd Q 0},\n P x hx \u2192 P y hy \u2192 P (x + y) (_ : x + y \u2208 evenOdd Q 0)\nh\u03b9\u03b9_mul :\n \u2200 (m\u2081 m\u2082 : M) {x : CliffordAlgebra Q} {hx : x \u2208 evenOdd Q 0},\n P x hx \u2192 P (\u2191(\u03b9 Q) m\u2081 * \u2191(\u03b9 Q) m\u2082 * x) (_ : \u2191(\u03b9 Q) m\u2081 * \u2191(\u03b9 Q) m\u2082 * x \u2208 evenOdd Q 0)\nx : CliffordAlgebra Q\nhx : x \u2208 evenOdd Q 0\nrx : CliffordAlgebra Q\n\u22a2 \u2200 (h : rx \u2208 LinearMap.range (\u03b9 Q) ^ ZMod.val 0), P rx (_ : rx \u2208 \u2a06 i, LinearMap.range (\u03b9 Q) ^ \u2191i)", "state_after": "case intro\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ : QuadraticForm R M\nP : (x : CliffordAlgebra Q) \u2192 x \u2208 evenOdd Q 0 \u2192 Prop\nhr : \u2200 (r : R), P (\u2191(algebraMap R (CliffordAlgebra Q)) r) (_ : \u2191(algebraMap R (CliffordAlgebra Q)) r \u2208 evenOdd Q 0)\nhadd :\n \u2200 {x y : CliffordAlgebra Q} {hx : x \u2208 evenOdd Q 0} {hy : y \u2208 evenOdd Q 0},\n P x hx \u2192 P y hy \u2192 P (x + y) (_ : x + y \u2208 evenOdd Q 0)\nh\u03b9\u03b9_mul :\n \u2200 (m\u2081 m\u2082 : M) {x : CliffordAlgebra Q} {hx : x \u2208 evenOdd Q 0},\n P x hx \u2192 P (\u2191(\u03b9 Q) m\u2081 * \u2191(\u03b9 Q) m\u2082 * x) (_ : \u2191(\u03b9 Q) m\u2081 * \u2191(\u03b9 Q) m\u2082 * x \u2208 evenOdd Q 0)\nx : CliffordAlgebra Q\nhx : x \u2208 evenOdd Q 0\nr : R\n\u22a2 P (\u2191(Algebra.linearMap R (CliffordAlgebra Q)) r)\n (_ : \u2191(Algebra.linearMap R (CliffordAlgebra Q)) r \u2208 \u2a06 i, LinearMap.range (\u03b9 Q) ^ \u2191i)"}, {"tactic": "exact hr r", "annotated_tactic": ["exact hr r", []], "state_before": "case intro\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ : QuadraticForm R M\nP : (x : CliffordAlgebra Q) \u2192 x \u2208 evenOdd Q 0 \u2192 Prop\nhr : \u2200 (r : R), P (\u2191(algebraMap R (CliffordAlgebra Q)) r) (_ : \u2191(algebraMap R (CliffordAlgebra Q)) r \u2208 evenOdd Q 0)\nhadd :\n \u2200 {x y : CliffordAlgebra Q} {hx : x \u2208 evenOdd Q 0} {hy : y \u2208 evenOdd Q 0},\n P x hx \u2192 P y hy \u2192 P (x + y) (_ : x + y \u2208 evenOdd Q 0)\nh\u03b9\u03b9_mul :\n \u2200 (m\u2081 m\u2082 : M) {x : CliffordAlgebra Q} {hx : x \u2208 evenOdd Q 0},\n P x hx \u2192 P (\u2191(\u03b9 Q) m\u2081 * \u2191(\u03b9 Q) m\u2082 * x) (_ : \u2191(\u03b9 Q) m\u2081 * \u2191(\u03b9 Q) m\u2082 * x \u2208 evenOdd Q 0)\nx : CliffordAlgebra Q\nhx : x \u2208 evenOdd Q 0\nr : R\n\u22a2 P (\u2191(Algebra.linearMap R (CliffordAlgebra Q)) r)\n (_ : \u2191(Algebra.linearMap R (CliffordAlgebra Q)) r \u2208 \u2a06 i, LinearMap.range (\u03b9 Q) ^ \u2191i)", "state_after": "no goals"}]

https://github.com/leanprover-community/mathlib4

3ce43c18f614b76e161f911b75a3e1ef641620ff

Mathlib/Topology/Basic.lean

clusterPt_principal_iff

[]

https://github.com/leanprover-community/mathlib4

3ce43c18f614b76e161f911b75a3e1ef641620ff

Mathlib/Data/Matrix/PEquiv.lean

PEquiv.toMatrix_injective

[{"tactic": "intro f g", "annotated_tactic": ["intro f g", []], "state_before": "k : Type u_1\nl : Type u_2\nm : Type u_3\nn : Type u_4\n\u03b1 : Type v\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : MonoidWithZero \u03b1\ninst\u271d : Nontrivial \u03b1\n\u22a2 Function.Injective toMatrix", "state_after": "k : Type u_1\nl : Type u_2\nm : Type u_3\nn : Type u_4\n\u03b1 : Type v\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : MonoidWithZero \u03b1\ninst\u271d : Nontrivial \u03b1\nf g : m \u2243. n\n\u22a2 toMatrix f = toMatrix g \u2192 f = g"}, {"tactic": "refine' not_imp_not.1 _", "annotated_tactic": ["refine' <a>not_imp_not</a>.1 _", [{"full_name": "not_imp_not", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [383, 9], "def_end_pos": [383, 20]}]], "state_before": "k : Type u_1\nl : Type u_2\nm : Type u_3\nn : Type u_4\n\u03b1 : Type v\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : MonoidWithZero \u03b1\ninst\u271d : Nontrivial \u03b1\nf g : m \u2243. n\n\u22a2 toMatrix f = toMatrix g \u2192 f = g", "state_after": "k : Type u_1\nl : Type u_2\nm : Type u_3\nn : Type u_4\n\u03b1 : Type v\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : MonoidWithZero \u03b1\ninst\u271d : Nontrivial \u03b1\nf g : m \u2243. n\n\u22a2 \u00acf = g \u2192 \u00actoMatrix f = toMatrix g"}, {"tactic": "simp only [Matrix.ext_iff.symm, toMatrix_apply, PEquiv.ext_iff, not_forall, exists_imp]", "annotated_tactic": ["simp only [Matrix.ext_iff.symm, <a>toMatrix_apply</a>, <a>PEquiv.ext_iff</a>, <a>not_forall</a>, <a>exists_imp</a>]", [{"full_name": "PEquiv.toMatrix_apply", "def_path": "Mathlib/Data/Matrix/PEquiv.lean", "def_pos": [58, 9], "def_end_pos": [58, 23]}, {"full_name": "PEquiv.ext_iff", "def_path": "Mathlib/Data/PEquiv.lean", "def_pos": [87, 9], "def_end_pos": [87, 16]}, {"full_name": "Classical.not_forall", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [686, 9], "def_end_pos": [686, 19]}, {"full_name": "exists_imp", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [367, 9], "def_end_pos": [367, 19]}]], "state_before": "k : Type u_1\nl : Type u_2\nm : Type u_3\nn : Type u_4\n\u03b1 : Type v\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : MonoidWithZero \u03b1\ninst\u271d : Nontrivial \u03b1\nf g : m \u2243. n\n\u22a2 \u00acf = g \u2192 \u00actoMatrix f = toMatrix g", "state_after": "k : Type u_1\nl : Type u_2\nm : Type u_3\nn : Type u_4\n\u03b1 : Type v\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : MonoidWithZero \u03b1\ninst\u271d : Nontrivial \u03b1\nf g : m \u2243. n\n\u22a2 \u2200 (x : m), \u00ac\u2191f x = \u2191g x \u2192 \u2203 x x_1, \u00ac(if x_1 \u2208 \u2191f x then 1 else 0) = if x_1 \u2208 \u2191g x then 1 else 0"}, {"tactic": "intro i hi", "annotated_tactic": ["intro i hi", []], "state_before": "k : Type u_1\nl : Type u_2\nm : Type u_3\nn : Type u_4\n\u03b1 : Type v\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : MonoidWithZero \u03b1\ninst\u271d : Nontrivial \u03b1\nf g : m \u2243. n\n\u22a2 \u2200 (x : m), \u00ac\u2191f x = \u2191g x \u2192 \u2203 x x_1, \u00ac(if x_1 \u2208 \u2191f x then 1 else 0) = if x_1 \u2208 \u2191g x then 1 else 0", "state_after": "k : Type u_1\nl : Type u_2\nm : Type u_3\nn : Type u_4\n\u03b1 : Type v\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : MonoidWithZero \u03b1\ninst\u271d : Nontrivial \u03b1\nf g : m \u2243. n\ni : m\nhi : \u00ac\u2191f i = \u2191g i\n\u22a2 \u2203 x x_1, \u00ac(if x_1 \u2208 \u2191f x then 1 else 0) = if x_1 \u2208 \u2191g x then 1 else 0"}, {"tactic": "use i", "annotated_tactic": ["use i", []], "state_before": "k : Type u_1\nl : Type u_2\nm : Type u_3\nn : Type u_4\n\u03b1 : Type v\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : MonoidWithZero \u03b1\ninst\u271d : Nontrivial \u03b1\nf g : m \u2243. n\ni : m\nhi : \u00ac\u2191f i = \u2191g i\n\u22a2 \u2203 x x_1, \u00ac(if x_1 \u2208 \u2191f x then 1 else 0) = if x_1 \u2208 \u2191g x then 1 else 0", "state_after": "case h\nk : Type u_1\nl : Type u_2\nm : Type u_3\nn : Type u_4\n\u03b1 : Type v\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : MonoidWithZero \u03b1\ninst\u271d : Nontrivial \u03b1\nf g : m \u2243. n\ni : m\nhi : \u00ac\u2191f i = \u2191g i\n\u22a2 \u2203 x, \u00ac(if x \u2208 \u2191f i then 1 else 0) = if x \u2208 \u2191g i then 1 else 0"}, {"tactic": "cases' hf : f i with fi", "annotated_tactic": ["cases' hf : f i with fi", []], "state_before": "case h\nk : Type u_1\nl : Type u_2\nm : Type u_3\nn : Type u_4\n\u03b1 : Type v\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : MonoidWithZero \u03b1\ninst\u271d : Nontrivial \u03b1\nf g : m \u2243. n\ni : m\nhi : \u00ac\u2191f i = \u2191g i\n\u22a2 \u2203 x, \u00ac(if x \u2208 \u2191f i then 1 else 0) = if x \u2208 \u2191g i then 1 else 0", "state_after": "case h.none\nk : Type u_1\nl : Type u_2\nm : Type u_3\nn : Type u_4\n\u03b1 : Type v\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : MonoidWithZero \u03b1\ninst\u271d : Nontrivial \u03b1\nf g : m \u2243. n\ni : m\nhi : \u00ac\u2191f i = \u2191g i\nhf : \u2191f i = none\n\u22a2 \u2203 x, \u00ac(if x \u2208 none then 1 else 0) = if x \u2208 \u2191g i then 1 else 0\n\ncase h.some\nk : Type u_1\nl : Type u_2\nm : Type u_3\nn : Type u_4\n\u03b1 : Type v\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : MonoidWithZero \u03b1\ninst\u271d : Nontrivial \u03b1\nf g : m \u2243. n\ni : m\nhi : \u00ac\u2191f i = \u2191g i\nfi : n\nhf : \u2191f i = some fi\n\u22a2 \u2203 x, \u00ac(if x \u2208 some fi then 1 else 0) = if x \u2208 \u2191g i then 1 else 0"}, {"tactic": "cases' hg : g i with gi", "annotated_tactic": ["cases' hg : g i with gi", []], "state_before": "case h.none\nk : Type u_1\nl : Type u_2\nm : Type u_3\nn : Type u_4\n\u03b1 : Type v\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : MonoidWithZero \u03b1\ninst\u271d : Nontrivial \u03b1\nf g : m \u2243. n\ni : m\nhi : \u00ac\u2191f i = \u2191g i\nhf : \u2191f i = none\n\u22a2 \u2203 x, \u00ac(if x \u2208 none then 1 else 0) = if x \u2208 \u2191g i then 1 else 0", "state_after": "case h.none.none\nk : Type u_1\nl : Type u_2\nm : Type u_3\nn : Type u_4\n\u03b1 : Type v\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : MonoidWithZero \u03b1\ninst\u271d : Nontrivial \u03b1\nf g : m \u2243. n\ni : m\nhi : \u00ac\u2191f i = \u2191g i\nhf : \u2191f i = none\nhg : \u2191g i = none\n\u22a2 \u2203 x, \u00ac(if x \u2208 none then 1 else 0) = if x \u2208 none then 1 else 0\n\ncase h.none.some\nk : Type u_1\nl : Type u_2\nm : Type u_3\nn : Type u_4\n\u03b1 : Type v\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : MonoidWithZero \u03b1\ninst\u271d : Nontrivial \u03b1\nf g : m \u2243. n\ni : m\nhi : \u00ac\u2191f i = \u2191g i\nhf : \u2191f i = none\ngi : n\nhg : \u2191g i = some gi\n\u22a2 \u2203 x, \u00ac(if x \u2208 none then 1 else 0) = if x \u2208 some gi then 1 else 0"}, {"tactic": "rw [hf, hg] at hi", "annotated_tactic": ["rw [hf, hg] at hi", []], "state_before": "case h.none.none\nk : Type u_1\nl : Type u_2\nm : Type u_3\nn : Type u_4\n\u03b1 : Type v\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : MonoidWithZero \u03b1\ninst\u271d : Nontrivial \u03b1\nf g : m \u2243. n\ni : m\nhi : \u00ac\u2191f i = \u2191g i\nhf : \u2191f i = none\nhg : \u2191g i = none\n\u22a2 \u2203 x, \u00ac(if x \u2208 none then 1 else 0) = if x \u2208 none then 1 else 0", "state_after": "case h.none.none\nk : Type u_1\nl : Type u_2\nm : Type u_3\nn : Type u_4\n\u03b1 : Type v\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : MonoidWithZero \u03b1\ninst\u271d : Nontrivial \u03b1\nf g : m \u2243. n\ni : m\nhi : \u00acnone = none\nhf : \u2191f i = none\nhg : \u2191g i = none\n\u22a2 \u2203 x, \u00ac(if x \u2208 none then 1 else 0) = if x \u2208 none then 1 else 0"}, {"tactic": "exact (hi rfl).elim", "annotated_tactic": ["exact (hi <a>rfl</a>).<a>elim</a>", [{"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}, {"full_name": "False.elim", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [223, 21], "def_end_pos": [223, 31]}]], "state_before": "case h.none.none\nk : Type u_1\nl : Type u_2\nm : Type u_3\nn : Type u_4\n\u03b1 : Type v\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : MonoidWithZero \u03b1\ninst\u271d : Nontrivial \u03b1\nf g : m \u2243. n\ni : m\nhi : \u00acnone = none\nhf : \u2191f i = none\nhg : \u2191g i = none\n\u22a2 \u2203 x, \u00ac(if x \u2208 none then 1 else 0) = if x \u2208 none then 1 else 0", "state_after": "no goals"}, {"tactic": "use gi", "annotated_tactic": ["use gi", []], "state_before": "case h.none.some\nk : Type u_1\nl : Type u_2\nm : Type u_3\nn : Type u_4\n\u03b1 : Type v\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : MonoidWithZero \u03b1\ninst\u271d : Nontrivial \u03b1\nf g : m \u2243. n\ni : m\nhi : \u00ac\u2191f i = \u2191g i\nhf : \u2191f i = none\ngi : n\nhg : \u2191g i = some gi\n\u22a2 \u2203 x, \u00ac(if x \u2208 none then 1 else 0) = if x \u2208 some gi then 1 else 0", "state_after": "case h\nk : Type u_1\nl : Type u_2\nm : Type u_3\nn : Type u_4\n\u03b1 : Type v\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : MonoidWithZero \u03b1\ninst\u271d : Nontrivial \u03b1\nf g : m \u2243. n\ni : m\nhi : \u00ac\u2191f i = \u2191g i\nhf : \u2191f i = none\ngi : n\nhg : \u2191g i = some gi\n\u22a2 \u00ac(if gi \u2208 none then 1 else 0) = if gi \u2208 some gi then 1 else 0"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case h\nk : Type u_1\nl : Type u_2\nm : Type u_3\nn : Type u_4\n\u03b1 : Type v\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : MonoidWithZero \u03b1\ninst\u271d : Nontrivial \u03b1\nf g : m \u2243. n\ni : m\nhi : \u00ac\u2191f i = \u2191g i\nhf : \u2191f i = none\ngi : n\nhg : \u2191g i = some gi\n\u22a2 \u00ac(if gi \u2208 none then 1 else 0) = if gi \u2208 some gi then 1 else 0", "state_after": "no goals"}, {"tactic": "use fi", "annotated_tactic": ["use fi", []], "state_before": "case h.some\nk : Type u_1\nl : Type u_2\nm : Type u_3\nn : Type u_4\n\u03b1 : Type v\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : MonoidWithZero \u03b1\ninst\u271d : Nontrivial \u03b1\nf g : m \u2243. n\ni : m\nhi : \u00ac\u2191f i = \u2191g i\nfi : n\nhf : \u2191f i = some fi\n\u22a2 \u2203 x, \u00ac(if x \u2208 some fi then 1 else 0) = if x \u2208 \u2191g i then 1 else 0", "state_after": "case h\nk : Type u_1\nl : Type u_2\nm : Type u_3\nn : Type u_4\n\u03b1 : Type v\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : MonoidWithZero \u03b1\ninst\u271d : Nontrivial \u03b1\nf g : m \u2243. n\ni : m\nhi : \u00ac\u2191f i = \u2191g i\nfi : n\nhf : \u2191f i = some fi\n\u22a2 \u00ac(if fi \u2208 some fi then 1 else 0) = if fi \u2208 \u2191g i then 1 else 0"}, {"tactic": "simp [hf.symm, Ne.symm hi]", "annotated_tactic": ["simp [hf.symm, <a>Ne.symm</a> hi]", [{"full_name": "Ne.symm", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [575, 9], "def_end_pos": [575, 16]}]], "state_before": "case h\nk : Type u_1\nl : Type u_2\nm : Type u_3\nn : Type u_4\n\u03b1 : Type v\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : MonoidWithZero \u03b1\ninst\u271d : Nontrivial \u03b1\nf g : m \u2243. n\ni : m\nhi : \u00ac\u2191f i = \u2191g i\nfi : n\nhf : \u2191f i = some fi\n\u22a2 \u00ac(if fi \u2208 some fi then 1 else 0) = if fi \u2208 \u2191g i then 1 else 0", "state_after": "no goals"}]

https://github.com/leanprover/std4

869c615eb10130c0637a7bc038e2b80253559913

lake-packages/std/Std/Data/Nat/Lemmas.lean

Nat.pos_iff_ne_zero

[]

https://github.com/leanprover-community/mathlib4

3ce43c18f614b76e161f911b75a3e1ef641620ff

Mathlib/ModelTheory/Substructures.lean

FirstOrder.Language.Substructure.closure_mono

[]

https://github.com/leanprover-community/mathlib4

3ce43c18f614b76e161f911b75a3e1ef641620ff

Mathlib/Algebra/Order/Ring/Defs.lean

nonneg_of_mul_nonneg_left

[]

https://github.com/leanprover-community/mathlib4

3ce43c18f614b76e161f911b75a3e1ef641620ff

Mathlib/Data/Sigma/Basic.lean

Sigma.mk.inj_iff

[{"tactic": "cases h", "annotated_tactic": ["cases h", []], "state_before": "\u03b1 : Type u_1\n\u03b1\u2081 : Type u_2\n\u03b1\u2082 : Type u_3\n\u03b2 : \u03b1 \u2192 Type u_4\n\u03b2\u2081 : \u03b1\u2081 \u2192 Type u_5\n\u03b2\u2082 : \u03b1\u2082 \u2192 Type u_6\na\u2081 a\u2082 : \u03b1\nb\u2081 : \u03b2 a\u2081\nb\u2082 : \u03b2 a\u2082\nh : { fst := a\u2081, snd := b\u2081 } = { fst := a\u2082, snd := b\u2082 }\n\u22a2 a\u2081 = a\u2082 \u2227 HEq b\u2081 b\u2082", "state_after": "case refl\n\u03b1 : Type u_1\n\u03b1\u2081 : Type u_2\n\u03b1\u2082 : Type u_3\n\u03b2 : \u03b1 \u2192 Type u_4\n\u03b2\u2081 : \u03b1\u2081 \u2192 Type u_5\n\u03b2\u2082 : \u03b1\u2082 \u2192 Type u_6\na\u2081 : \u03b1\nb\u2081 : \u03b2 a\u2081\n\u22a2 a\u2081 = a\u2081 \u2227 HEq b\u2081 b\u2081"}, {"tactic": "exact \u27e8rfl, heq_of_eq rfl\u27e9", "annotated_tactic": ["exact \u27e8<a>rfl</a>, <a>heq_of_eq</a> <a>rfl</a>\u27e9", [{"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}, {"full_name": "heq_of_eq", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [627, 9], "def_end_pos": [627, 18]}, {"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "case refl\n\u03b1 : Type u_1\n\u03b1\u2081 : Type u_2\n\u03b1\u2082 : Type u_3\n\u03b2 : \u03b1 \u2192 Type u_4\n\u03b2\u2081 : \u03b1\u2081 \u2192 Type u_5\n\u03b2\u2082 : \u03b1\u2082 \u2192 Type u_6\na\u2081 : \u03b1\nb\u2081 : \u03b2 a\u2081\n\u22a2 a\u2081 = a\u2081 \u2227 HEq b\u2081 b\u2081", "state_after": "no goals"}, {"tactic": "subst h\u2081", "annotated_tactic": ["subst h\u2081", []], "state_before": "\u03b1 : Type u_1\n\u03b1\u2081 : Type u_2\n\u03b1\u2082 : Type u_3\n\u03b2 : \u03b1 \u2192 Type u_4\n\u03b2\u2081 : \u03b1\u2081 \u2192 Type u_5\n\u03b2\u2082 : \u03b1\u2082 \u2192 Type u_6\na\u2081 a\u2082 : \u03b1\nb\u2081 : \u03b2 a\u2081\nb\u2082 : \u03b2 a\u2082\nx\u271d : a\u2081 = a\u2082 \u2227 HEq b\u2081 b\u2082\nh\u2081 : a\u2081 = a\u2082\nh\u2082 : HEq b\u2081 b\u2082\n\u22a2 { fst := a\u2081, snd := b\u2081 } = { fst := a\u2082, snd := b\u2082 }", "state_after": "\u03b1 : Type u_1\n\u03b1\u2081 : Type u_2\n\u03b1\u2082 : Type u_3\n\u03b2 : \u03b1 \u2192 Type u_4\n\u03b2\u2081 : \u03b1\u2081 \u2192 Type u_5\n\u03b2\u2082 : \u03b1\u2082 \u2192 Type u_6\na\u2081 : \u03b1\nb\u2081 b\u2082 : \u03b2 a\u2081\nx\u271d : a\u2081 = a\u2081 \u2227 HEq b\u2081 b\u2082\nh\u2082 : HEq b\u2081 b\u2082\n\u22a2 { fst := a\u2081, snd := b\u2081 } = { fst := a\u2081, snd := b\u2082 }"}, {"tactic": "rw [eq_of_heq h\u2082]", "annotated_tactic": ["rw [<a>eq_of_heq</a> h\u2082]", [{"full_name": "eq_of_heq", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [451, 9], "def_end_pos": [451, 18]}]], "state_before": "\u03b1 : Type u_1\n\u03b1\u2081 : Type u_2\n\u03b1\u2082 : Type u_3\n\u03b2 : \u03b1 \u2192 Type u_4\n\u03b2\u2081 : \u03b1\u2081 \u2192 Type u_5\n\u03b2\u2082 : \u03b1\u2082 \u2192 Type u_6\na\u2081 : \u03b1\nb\u2081 b\u2082 : \u03b2 a\u2081\nx\u271d : a\u2081 = a\u2081 \u2227 HEq b\u2081 b\u2082\nh\u2082 : HEq b\u2081 b\u2082\n\u22a2 { fst := a\u2081, snd := b\u2081 } = { fst := a\u2081, snd := b\u2082 }", "state_after": "no goals"}]

https://github.com/leanprover-community/mathlib4

3ce43c18f614b76e161f911b75a3e1ef641620ff

Mathlib/Data/Multiset/Basic.lean

Multiset.le_zero

[]

https://github.com/leanprover-community/mathlib4

3ce43c18f614b76e161f911b75a3e1ef641620ff

Mathlib/Order/Filter/AtTopBot.lean

Filter.Ici_mem_atTop

[]

https://github.com/leanprover-community/mathlib4

3ce43c18f614b76e161f911b75a3e1ef641620ff

Mathlib/CategoryTheory/Limits/Shapes/ZeroMorphisms.lean

CategoryTheory.Limits.zero_comp

[]

https://github.com/leanprover-community/mathlib4

3ce43c18f614b76e161f911b75a3e1ef641620ff

Mathlib/Data/Set/Pointwise/Interval.lean

Set.preimage_neg_uIcc

[{"tactic": "simp only [\u2190 Icc_min_max, preimage_neg_Icc, min_neg_neg, max_neg_neg]", "annotated_tactic": ["simp only [\u2190 <a>Icc_min_max</a>, <a>preimage_neg_Icc</a>, <a>min_neg_neg</a>, <a>max_neg_neg</a>]", [{"full_name": "Set.Icc_min_max", "def_path": "Mathlib/Data/Set/Intervals/UnorderedInterval.lean", "def_pos": [220, 9], "def_end_pos": [220, 20]}, {"full_name": "Set.preimage_neg_Icc", "def_path": "Mathlib/Data/Set/Pointwise/Interval.lean", "def_pos": [150, 9], "def_end_pos": [150, 25]}, {"full_name": "min_neg_neg", "def_path": "Mathlib/Algebra/Order/Group/MinMax.lean", "def_pos": [35, 15], "def_end_pos": [35, 26]}, {"full_name": "max_neg_neg", "def_path": "Mathlib/Algebra/Order/Group/MinMax.lean", "def_pos": [43, 15], "def_end_pos": [43, 26]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : LinearOrderedAddCommGroup \u03b1\na b c d : \u03b1\n\u22a2 -[[a, b]] = [[-a, -b]]", "state_after": "no goals"}]

https://github.com/leanprover-community/mathlib4

3ce43c18f614b76e161f911b75a3e1ef641620ff

Mathlib/NumberTheory/LucasLehmer.lean

succ_mersenne

[{"tactic": "rw [mersenne, tsub_add_cancel_of_le]", "annotated_tactic": ["rw [<a>mersenne</a>, <a>tsub_add_cancel_of_le</a>]", [{"full_name": "mersenne", "def_path": "Mathlib/NumberTheory/LucasLehmer.lean", "def_pos": [42, 5], "def_end_pos": [42, 13]}, {"full_name": "tsub_add_cancel_of_le", "def_path": "Mathlib/Algebra/Order/Sub/Canonical.lean", "def_pos": [30, 9], "def_end_pos": [30, 30]}]], "state_before": "k : \u2115\n\u22a2 mersenne k + 1 = 2 ^ k", "state_after": "k : \u2115\n\u22a2 1 \u2264 2 ^ k"}, {"tactic": "exact one_le_pow_of_one_le (by norm_num) k", "annotated_tactic": ["exact <a>one_le_pow_of_one_le</a> (by norm_num) k", [{"full_name": "one_le_pow_of_one_le", "def_path": "Mathlib/Algebra/GroupPower/Order.lean", "def_pos": [423, 9], "def_end_pos": [423, 29]}]], "state_before": "k : \u2115\n\u22a2 1 \u2264 2 ^ k", "state_after": "no goals"}, {"tactic": "norm_num", "annotated_tactic": ["norm_num", []], "state_before": "k : \u2115\n\u22a2 1 \u2264 2", "state_after": "no goals"}]

https://github.com/leanprover-community/mathlib4

3ce43c18f614b76e161f911b75a3e1ef641620ff

Mathlib/RingTheory/Polynomial/Bernstein.lean

bernsteinPolynomial.iterate_derivative_at_1_eq_zero_of_lt

[{"tactic": "intro w", "annotated_tactic": ["intro w", []], "state_before": "R : Type u_1\ninst\u271d : CommRing R\nn \u03bd k : \u2115\n\u22a2 k < n - \u03bd \u2192 eval 1 ((\u2191derivative)^[k] (bernsteinPolynomial R n \u03bd)) = 0", "state_after": "R : Type u_1\ninst\u271d : CommRing R\nn \u03bd k : \u2115\nw : k < n - \u03bd\n\u22a2 eval 1 ((\u2191derivative)^[k] (bernsteinPolynomial R n \u03bd)) = 0"}, {"tactic": "rw [flip' _ _ _ (tsub_pos_iff_lt.mp (pos_of_gt w)).le]", "annotated_tactic": ["rw [<a>flip'</a> _ _ _ (tsub_pos_iff_lt.mp (<a>pos_of_gt</a> w)).<a>le</a>]", [{"full_name": "bernsteinPolynomial.flip'", "def_path": "Mathlib/RingTheory/Polynomial/Bernstein.lean", "def_pos": [81, 9], "def_end_pos": [81, 14]}, {"full_name": "pos_of_gt", "def_path": "Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean", "def_pos": [305, 9], "def_end_pos": [305, 18]}, {"full_name": "LT.lt.le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [142, 7], "def_end_pos": [142, 15]}]], "state_before": "R : Type u_1\ninst\u271d : CommRing R\nn \u03bd k : \u2115\nw : k < n - \u03bd\n\u22a2 eval 1 ((\u2191derivative)^[k] (bernsteinPolynomial R n \u03bd)) = 0", "state_after": "R : Type u_1\ninst\u271d : CommRing R\nn \u03bd k : \u2115\nw : k < n - \u03bd\n\u22a2 eval 1 ((\u2191derivative)^[k] (comp (bernsteinPolynomial R n (n - \u03bd)) (1 - X))) = 0"}, {"tactic": "simp [Polynomial.eval_comp, iterate_derivative_at_0_eq_zero_of_lt R n w]", "annotated_tactic": ["simp [<a>Polynomial.eval_comp</a>, <a>iterate_derivative_at_0_eq_zero_of_lt</a> R n w]", [{"full_name": "Polynomial.eval_comp", "def_path": "Mathlib/Data/Polynomial/Eval.lean", "def_pos": [1098, 9], "def_end_pos": [1098, 18]}, {"full_name": "bernsteinPolynomial.iterate_derivative_at_0_eq_zero_of_lt", "def_path": "Mathlib/RingTheory/Polynomial/Bernstein.lean", "def_pos": [146, 9], "def_end_pos": [146, 46]}]], "state_before": "R : Type u_1\ninst\u271d : CommRing R\nn \u03bd k : \u2115\nw : k < n - \u03bd\n\u22a2 eval 1 ((\u2191derivative)^[k] (comp (bernsteinPolynomial R n (n - \u03bd)) (1 - X))) = 0", "state_after": "no goals"}]

https://github.com/leanprover-community/mathlib4

3ce43c18f614b76e161f911b75a3e1ef641620ff

Mathlib/Data/Set/Pointwise/Interval.lean

Set.preimage_mul_const_Iic

[]

https://github.com/leanprover-community/mathlib4

3ce43c18f614b76e161f911b75a3e1ef641620ff

Mathlib/Algebra/Group/Units.lean

Units.mk_val

[]

https://github.com/leanprover-community/mathlib4

3ce43c18f614b76e161f911b75a3e1ef641620ff

Mathlib/Data/Set/Sups.lean

Set.infs_inter_subset_right

[]

https://github.com/leanprover-community/mathlib4

3ce43c18f614b76e161f911b75a3e1ef641620ff

Mathlib/RingTheory/DiscreteValuationRing/Basic.lean

DiscreteValuationRing.addVal_uniformizer

[{"tactic": "simpa only [one_mul, eq_self_iff_true, Units.val_one, pow_one, forall_true_left, Nat.cast_one]\n using addVal_def \u03d6 1 h\u03d6 1", "annotated_tactic": ["simpa only [<a>one_mul</a>, <a>eq_self_iff_true</a>, <a>Units.val_one</a>, <a>pow_one</a>, <a>forall_true_left</a>, <a>Nat.cast_one</a>]\n using <a>addVal_def</a> \u03d6 1 h\u03d6 1", [{"full_name": "one_mul", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [464, 9], "def_end_pos": [464, 16]}, {"full_name": "eq_self_iff_true", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [86, 9], "def_end_pos": [86, 25]}, {"full_name": "Units.val_one", "def_path": "Mathlib/Algebra/Group/Units.lean", "def_pos": [235, 9], "def_end_pos": [235, 16]}, {"full_name": "pow_one", "def_path": "Mathlib/Algebra/GroupPower/Basic.lean", "def_pos": [97, 9], "def_end_pos": [97, 16]}, {"full_name": "forall_true_left", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [931, 17], "def_end_pos": [931, 33]}, {"full_name": "Nat.cast_one", "def_path": "Mathlib/Data/Nat/Cast/Defs.lean", "def_pos": [141, 9], "def_end_pos": [141, 17]}, {"full_name": "DiscreteValuationRing.addVal_def", "def_path": "Mathlib/RingTheory/DiscreteValuationRing/Basic.lean", "def_pos": [404, 9], "def_end_pos": [404, 19]}]], "state_before": "R : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : DiscreteValuationRing R\n\u03d6 : R\nh\u03d6 : Irreducible \u03d6\n\u22a2 \u2191(addVal R) \u03d6 = 1", "state_after": "no goals"}]

https://github.com/leanprover-community/mathlib4

3ce43c18f614b76e161f911b75a3e1ef641620ff

Mathlib/Analysis/NormedSpace/Units.lean

Ideal.closure_ne_top

[{"tactic": "have h := closure_minimal (coe_subset_nonunits hI) nonunits.isClosed", "annotated_tactic": ["have h := <a>closure_minimal</a> (<a>coe_subset_nonunits</a> hI) <a>nonunits.isClosed</a>", [{"full_name": "closure_minimal", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [443, 9], "def_end_pos": [443, 24]}, {"full_name": "coe_subset_nonunits", "def_path": "Mathlib/RingTheory/Ideal/Basic.lean", "def_pos": [850, 9], "def_end_pos": [850, 28]}, {"full_name": "nonunits.isClosed", "def_path": "Mathlib/Analysis/NormedSpace/Units.lean", "def_pos": [101, 19], "def_end_pos": [101, 27]}]], "state_before": "R : Type u_1\ninst\u271d\u00b9 : NormedRing R\ninst\u271d : CompleteSpace R\nI : Ideal R\nhI : I \u2260 \u22a4\n\u22a2 Ideal.closure I \u2260 \u22a4", "state_after": "R : Type u_1\ninst\u271d\u00b9 : NormedRing R\ninst\u271d : CompleteSpace R\nI : Ideal R\nhI : I \u2260 \u22a4\nh : closure \u2191I \u2286 nonunits R\n\u22a2 Ideal.closure I \u2260 \u22a4"}, {"tactic": "simpa only [I.closure.eq_top_iff_one, Ne.def] using mt (@h 1) one_not_mem_nonunits", "annotated_tactic": ["simpa only [I.closure.eq_top_iff_one, <a>Ne.def</a>] using <a>mt</a> (@h 1) <a>one_not_mem_nonunits</a>", [{"full_name": "Ne.def", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [59, 9], "def_end_pos": [59, 15]}, {"full_name": "mt", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [516, 9], "def_end_pos": [516, 11]}, {"full_name": "one_not_mem_nonunits", "def_path": "Mathlib/RingTheory/Ideal/Basic.lean", "def_pos": [846, 9], "def_end_pos": [846, 29]}]], "state_before": "R : Type u_1\ninst\u271d\u00b9 : NormedRing R\ninst\u271d : CompleteSpace R\nI : Ideal R\nhI : I \u2260 \u22a4\nh : closure \u2191I \u2286 nonunits R\n\u22a2 Ideal.closure I \u2260 \u22a4", "state_after": "no goals"}]

https://github.com/leanprover-community/mathlib4

3ce43c18f614b76e161f911b75a3e1ef641620ff

Mathlib/MeasureTheory/Function/SimpleFunc.lean

MeasureTheory.SimpleFunc.apply_mk

[]

https://github.com/leanprover-community/mathlib4

3ce43c18f614b76e161f911b75a3e1ef641620ff

Mathlib/MeasureTheory/MeasurableSpace/Defs.lean

MeasurableSet.empty

[]

https://github.com/leanprover-community/mathlib4

3ce43c18f614b76e161f911b75a3e1ef641620ff

Mathlib/Analysis/Convex/Between.lean

wbtw_or_wbtw_smul_vadd_of_nonneg

[{"tactic": "rcases le_total r\u2081 r\u2082 with (h | h)", "annotated_tactic": ["rcases <a>le_total</a> r\u2081 r\u2082 with (h | h)", [{"full_name": "le_total", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [295, 9], "def_end_pos": [295, 17]}]], "state_before": "R : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u00b3 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx : P\nv : V\nr\u2081 r\u2082 : R\nhr\u2081 : 0 \u2264 r\u2081\nhr\u2082 : 0 \u2264 r\u2082\n\u22a2 Wbtw R x (r\u2081 \u2022 v +\u1d65 x) (r\u2082 \u2022 v +\u1d65 x) \u2228 Wbtw R x (r\u2082 \u2022 v +\u1d65 x) (r\u2081 \u2022 v +\u1d65 x)", "state_after": "case inl\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u00b3 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx : P\nv : V\nr\u2081 r\u2082 : R\nhr\u2081 : 0 \u2264 r\u2081\nhr\u2082 : 0 \u2264 r\u2082\nh : r\u2081 \u2264 r\u2082\n\u22a2 Wbtw R x (r\u2081 \u2022 v +\u1d65 x) (r\u2082 \u2022 v +\u1d65 x) \u2228 Wbtw R x (r\u2082 \u2022 v +\u1d65 x) (r\u2081 \u2022 v +\u1d65 x)\n\ncase inr\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u00b3 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx : P\nv : V\nr\u2081 r\u2082 : R\nhr\u2081 : 0 \u2264 r\u2081\nhr\u2082 : 0 \u2264 r\u2082\nh : r\u2082 \u2264 r\u2081\n\u22a2 Wbtw R x (r\u2081 \u2022 v +\u1d65 x) (r\u2082 \u2022 v +\u1d65 x) \u2228 Wbtw R x (r\u2082 \u2022 v +\u1d65 x) (r\u2081 \u2022 v +\u1d65 x)"}, {"tactic": "exact Or.inl (wbtw_smul_vadd_smul_vadd_of_nonneg_of_le x v hr\u2081 h)", "annotated_tactic": ["exact <a>Or.inl</a> (<a>wbtw_smul_vadd_smul_vadd_of_nonneg_of_le</a> x v hr\u2081 h)", [{"full_name": "Or.inl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [517, 5], "def_end_pos": [517, 8]}, {"full_name": "wbtw_smul_vadd_smul_vadd_of_nonneg_of_le", "def_path": "Mathlib/Analysis/Convex/Between.lean", "def_pos": [763, 9], "def_end_pos": [763, 49]}]], "state_before": "case inl\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u00b3 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx : P\nv : V\nr\u2081 r\u2082 : R\nhr\u2081 : 0 \u2264 r\u2081\nhr\u2082 : 0 \u2264 r\u2082\nh : r\u2081 \u2264 r\u2082\n\u22a2 Wbtw R x (r\u2081 \u2022 v +\u1d65 x) (r\u2082 \u2022 v +\u1d65 x) \u2228 Wbtw R x (r\u2082 \u2022 v +\u1d65 x) (r\u2081 \u2022 v +\u1d65 x)", "state_after": "no goals"}, {"tactic": "exact Or.inr (wbtw_smul_vadd_smul_vadd_of_nonneg_of_le x v hr\u2082 h)", "annotated_tactic": ["exact <a>Or.inr</a> (<a>wbtw_smul_vadd_smul_vadd_of_nonneg_of_le</a> x v hr\u2082 h)", [{"full_name": "Or.inr", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [519, 5], "def_end_pos": [519, 8]}, {"full_name": "wbtw_smul_vadd_smul_vadd_of_nonneg_of_le", "def_path": "Mathlib/Analysis/Convex/Between.lean", "def_pos": [763, 9], "def_end_pos": [763, 49]}]], "state_before": "case inr\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u00b3 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx : P\nv : V\nr\u2081 r\u2082 : R\nhr\u2081 : 0 \u2264 r\u2081\nhr\u2082 : 0 \u2264 r\u2082\nh : r\u2082 \u2264 r\u2081\n\u22a2 Wbtw R x (r\u2081 \u2022 v +\u1d65 x) (r\u2082 \u2022 v +\u1d65 x) \u2228 Wbtw R x (r\u2082 \u2022 v +\u1d65 x) (r\u2081 \u2022 v +\u1d65 x)", "state_after": "no goals"}]

https://github.com/leanprover-community/mathlib4

3ce43c18f614b76e161f911b75a3e1ef641620ff

Mathlib/Data/Finset/NAry.lean

Finset.image₂_image₂_image₂_comm

[{"tactic": "push_cast", "annotated_tactic": ["push_cast", []], "state_before": "\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3\u271d : Type u_5\n\u03b3' : Type u_6\n\u03b4\u271d : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\ninst\u271d\u00b9\u2070 : DecidableEq \u03b1'\ninst\u271d\u2079 : DecidableEq \u03b2'\ninst\u271d\u2078 : DecidableEq \u03b3\u271d\ninst\u271d\u2077 : DecidableEq \u03b3'\ninst\u271d\u2076 : DecidableEq \u03b4\u271d\ninst\u271d\u2075 : DecidableEq \u03b4'\ninst\u271d\u2074 : DecidableEq \u03b5\ninst\u271d\u00b3 : DecidableEq \u03b5'\nf\u271d f'\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\u271d\ng\u271d g'\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\u271d \u2192 \u03b4\u271d\ns s' : Finset \u03b1\nt t' : Finset \u03b2\nu\u271d u' : Finset \u03b3\u271d\na a' : \u03b1\nb b' : \u03b2\nc : \u03b3\u271d\n\u03b3 : Type u_14\n\u03b4 : Type u_15\nu : Finset \u03b3\nv : Finset \u03b4\ninst\u271d\u00b2 : DecidableEq \u03b6\ninst\u271d\u00b9 : DecidableEq \u03b6'\ninst\u271d : DecidableEq \u03bd\nf : \u03b5 \u2192 \u03b6 \u2192 \u03bd\ng : \u03b1 \u2192 \u03b2 \u2192 \u03b5\nh : \u03b3 \u2192 \u03b4 \u2192 \u03b6\nf' : \u03b5' \u2192 \u03b6' \u2192 \u03bd\ng' : \u03b1 \u2192 \u03b3 \u2192 \u03b5'\nh' : \u03b2 \u2192 \u03b4 \u2192 \u03b6'\nh_comm : \u2200 (a : \u03b1) (b : \u03b2) (c : \u03b3) (d : \u03b4), f (g a b) (h c d) = f' (g' a c) (h' b d)\n\u22a2 \u2191(image\u2082 f (image\u2082 g s t) (image\u2082 h u v)) = \u2191(image\u2082 f' (image\u2082 g' s u) (image\u2082 h' t v))", "state_after": "\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3\u271d : Type u_5\n\u03b3' : Type u_6\n\u03b4\u271d : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\ninst\u271d\u00b9\u2070 : DecidableEq \u03b1'\ninst\u271d\u2079 : DecidableEq \u03b2'\ninst\u271d\u2078 : DecidableEq \u03b3\u271d\ninst\u271d\u2077 : DecidableEq \u03b3'\ninst\u271d\u2076 : DecidableEq \u03b4\u271d\ninst\u271d\u2075 : DecidableEq \u03b4'\ninst\u271d\u2074 : DecidableEq \u03b5\ninst\u271d\u00b3 : DecidableEq \u03b5'\nf\u271d f'\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\u271d\ng\u271d g'\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\u271d \u2192 \u03b4\u271d\ns s' : Finset \u03b1\nt t' : Finset \u03b2\nu\u271d u' : Finset \u03b3\u271d\na a' : \u03b1\nb b' : \u03b2\nc : \u03b3\u271d\n\u03b3 : Type u_14\n\u03b4 : Type u_15\nu : Finset \u03b3\nv : Finset \u03b4\ninst\u271d\u00b2 : DecidableEq \u03b6\ninst\u271d\u00b9 : DecidableEq \u03b6'\ninst\u271d : DecidableEq \u03bd\nf : \u03b5 \u2192 \u03b6 \u2192 \u03bd\ng : \u03b1 \u2192 \u03b2 \u2192 \u03b5\nh : \u03b3 \u2192 \u03b4 \u2192 \u03b6\nf' : \u03b5' \u2192 \u03b6' \u2192 \u03bd\ng' : \u03b1 \u2192 \u03b3 \u2192 \u03b5'\nh' : \u03b2 \u2192 \u03b4 \u2192 \u03b6'\nh_comm : \u2200 (a : \u03b1) (b : \u03b2) (c : \u03b3) (d : \u03b4), f (g a b) (h c d) = f' (g' a c) (h' b d)\n\u22a2 image2 f (image2 g \u2191s \u2191t) (image2 h \u2191u \u2191v) = image2 f' (image2 g' \u2191s \u2191u) (image2 h' \u2191t \u2191v)"}, {"tactic": "exact image2_image2_image2_comm h_comm", "annotated_tactic": ["exact <a>image2_image2_image2_comm</a> h_comm", [{"full_name": "Set.image2_image2_image2_comm", "def_path": "Mathlib/Data/Set/NAry.lean", "def_pos": [338, 9], "def_end_pos": [338, 34]}]], "state_before": "\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3\u271d : Type u_5\n\u03b3' : Type u_6\n\u03b4\u271d : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\ninst\u271d\u00b9\u2070 : DecidableEq \u03b1'\ninst\u271d\u2079 : DecidableEq \u03b2'\ninst\u271d\u2078 : DecidableEq \u03b3\u271d\ninst\u271d\u2077 : DecidableEq \u03b3'\ninst\u271d\u2076 : DecidableEq \u03b4\u271d\ninst\u271d\u2075 : DecidableEq \u03b4'\ninst\u271d\u2074 : DecidableEq \u03b5\ninst\u271d\u00b3 : DecidableEq \u03b5'\nf\u271d f'\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\u271d\ng\u271d g'\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\u271d \u2192 \u03b4\u271d\ns s' : Finset \u03b1\nt t' : Finset \u03b2\nu\u271d u' : Finset \u03b3\u271d\na a' : \u03b1\nb b' : \u03b2\nc : \u03b3\u271d\n\u03b3 : Type u_14\n\u03b4 : Type u_15\nu : Finset \u03b3\nv : Finset \u03b4\ninst\u271d\u00b2 : DecidableEq \u03b6\ninst\u271d\u00b9 : DecidableEq \u03b6'\ninst\u271d : DecidableEq \u03bd\nf : \u03b5 \u2192 \u03b6 \u2192 \u03bd\ng : \u03b1 \u2192 \u03b2 \u2192 \u03b5\nh : \u03b3 \u2192 \u03b4 \u2192 \u03b6\nf' : \u03b5' \u2192 \u03b6' \u2192 \u03bd\ng' : \u03b1 \u2192 \u03b3 \u2192 \u03b5'\nh' : \u03b2 \u2192 \u03b4 \u2192 \u03b6'\nh_comm : \u2200 (a : \u03b1) (b : \u03b2) (c : \u03b3) (d : \u03b4), f (g a b) (h c d) = f' (g' a c) (h' b d)\n\u22a2 image2 f (image2 g \u2191s \u2191t) (image2 h \u2191u \u2191v) = image2 f' (image2 g' \u2191s \u2191u) (image2 h' \u2191t \u2191v)", "state_after": "no goals"}]

https://github.com/leanprover-community/mathlib4

3ce43c18f614b76e161f911b75a3e1ef641620ff

Mathlib/LinearAlgebra/TensorProduct.lean

LinearMap.rTensor_smul

[]

https://github.com/leanprover-community/mathlib4

3ce43c18f614b76e161f911b75a3e1ef641620ff

Mathlib/Algebra/Order/Group/Defs.lean

mul_inv_lt_iff_lt_mul

[{"tactic": "rw [\u2190 mul_lt_mul_iff_right b, inv_mul_cancel_right]", "annotated_tactic": ["rw [\u2190 <a>mul_lt_mul_iff_right</a> b, <a>inv_mul_cancel_right</a>]", [{"full_name": "mul_lt_mul_iff_right", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [113, 9], "def_end_pos": [113, 29]}, {"full_name": "inv_mul_cancel_right", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [1165, 9], "def_end_pos": [1165, 29]}]], "state_before": "\u03b1 : Type u\ninst\u271d\u00b2 : Group \u03b1\ninst\u271d\u00b9 : LT \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x < x_1\na b c : \u03b1\n\u22a2 a * b\u207b\u00b9 < c \u2194 a < c * b", "state_after": "no goals"}]

https://github.com/leanprover-community/mathlib4

3ce43c18f614b76e161f911b75a3e1ef641620ff

Mathlib/Analysis/Calculus/BumpFunction/Basic.lean

ContDiffWithinAt.contDiffBump

[{"tactic": "change ContDiffWithinAt \u211d n (uncurry (someContDiffBumpBase E).toFun \u2218 fun x : X =>\n ((f x).rOut / (f x).rIn, (f x).rIn\u207b\u00b9 \u2022 (g x - c x))) s x", "annotated_tactic": ["change <a>ContDiffWithinAt</a> \u211d n (<a>uncurry</a> (<a>someContDiffBumpBase</a> E).<a>toFun</a> \u2218 fun x : X =>\n ((f x).<a>rOut</a> / (f x).<a>rIn</a>, (f x).<a>rIn</a>\u207b\u00b9 \u2022 (g x - c x))) s x", [{"full_name": "ContDiffWithinAt", "def_path": "Mathlib/Analysis/Calculus/ContDiffDef.lean", "def_pos": [417, 5], "def_end_pos": [417, 21]}, {"full_name": "Function.uncurry", "def_path": "Mathlib/Init/Function.lean", "def_pos": [217, 5], "def_end_pos": [217, 12]}, {"full_name": "someContDiffBumpBase", "def_path": "Mathlib/Analysis/Calculus/BumpFunction/Basic.lean", "def_pos": [104, 5], "def_end_pos": [104, 25]}, {"full_name": "ContDiffBumpBase.toFun", "def_path": "Mathlib/Analysis/Calculus/BumpFunction/Basic.lean", "def_pos": [87, 3], "def_end_pos": [87, 8]}, {"full_name": "ContDiffBump.rOut", "def_path": "Mathlib/Analysis/Calculus/BumpFunction/Basic.lean", "def_pos": [69, 8], "def_end_pos": [69, 12]}, {"full_name": "ContDiffBump.rIn", "def_path": "Mathlib/Analysis/Calculus/BumpFunction/Basic.lean", "def_pos": [69, 4], "def_end_pos": [69, 7]}, {"full_name": "ContDiffBump.rIn", "def_path": "Mathlib/Analysis/Calculus/BumpFunction/Basic.lean", "def_pos": [69, 4], "def_end_pos": [69, 7]}]], "state_before": "E : Type u_1\nX : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup X\ninst\u271d\u00b9 : NormedSpace \u211d X\ninst\u271d : HasContDiffBump E\nc\u271d : E\nf\u271d : ContDiffBump c\u271d\nx\u271d : E\nn : \u2115\u221e\nc g : X \u2192 E\ns : Set X\nf : (x : X) \u2192 ContDiffBump (c x)\nx : X\nhc : ContDiffWithinAt \u211d n c s x\nhr : ContDiffWithinAt \u211d n (fun x => (f x).rIn) s x\nhR : ContDiffWithinAt \u211d n (fun x => (f x).rOut) s x\nhg : ContDiffWithinAt \u211d n g s x\n\u22a2 ContDiffWithinAt \u211d n (fun x => \u2191(f x) (g x)) s x", "state_after": "E : Type u_1\nX : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup X\ninst\u271d\u00b9 : NormedSpace \u211d X\ninst\u271d : HasContDiffBump E\nc\u271d : E\nf\u271d : ContDiffBump c\u271d\nx\u271d : E\nn : \u2115\u221e\nc g : X \u2192 E\ns : Set X\nf : (x : X) \u2192 ContDiffBump (c x)\nx : X\nhc : ContDiffWithinAt \u211d n c s x\nhr : ContDiffWithinAt \u211d n (fun x => (f x).rIn) s x\nhR : ContDiffWithinAt \u211d n (fun x => (f x).rOut) s x\nhg : ContDiffWithinAt \u211d n g s x\n\u22a2 ContDiffWithinAt \u211d n\n (uncurry (someContDiffBumpBase E).toFun \u2218 fun x => ((f x).rOut / (f x).rIn, (f x).rIn\u207b\u00b9 \u2022 (g x - c x))) s x"}, {"tactic": "refine (((someContDiffBumpBase E).smooth.contDiffAt ?_).of_le le_top).comp_contDiffWithinAt x ?_", "annotated_tactic": ["refine (((<a>someContDiffBumpBase</a> E).smooth.contDiffAt ?_).<a>of_le</a> <a>le_top</a>).<a>comp_contDiffWithinAt</a> x ?_", [{"full_name": "someContDiffBumpBase", "def_path": "Mathlib/Analysis/Calculus/BumpFunction/Basic.lean", "def_pos": [104, 5], "def_end_pos": [104, 25]}, {"full_name": "ContDiffAt.of_le", "def_path": "Mathlib/Analysis/Calculus/ContDiffDef.lean", "def_pos": [1361, 9], "def_end_pos": [1361, 25]}, {"full_name": "le_top", "def_path": "Mathlib/Order/BoundedOrder.lean", "def_pos": [98, 9], "def_end_pos": [98, 15]}, {"full_name": "ContDiffAt.comp_contDiffWithinAt", "def_path": "Mathlib/Analysis/Calculus/ContDiff.lean", "def_pos": [712, 9], "def_end_pos": [712, 41]}]], "state_before": "E : Type u_1\nX : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup X\ninst\u271d\u00b9 : NormedSpace \u211d X\ninst\u271d : HasContDiffBump E\nc\u271d : E\nf\u271d : ContDiffBump c\u271d\nx\u271d : E\nn : \u2115\u221e\nc g : X \u2192 E\ns : Set X\nf : (x : X) \u2192 ContDiffBump (c x)\nx : X\nhc : ContDiffWithinAt \u211d n c s x\nhr : ContDiffWithinAt \u211d n (fun x => (f x).rIn) s x\nhR : ContDiffWithinAt \u211d n (fun x => (f x).rOut) s x\nhg : ContDiffWithinAt \u211d n g s x\n\u22a2 ContDiffWithinAt \u211d n\n (uncurry (someContDiffBumpBase E).toFun \u2218 fun x => ((f x).rOut / (f x).rIn, (f x).rIn\u207b\u00b9 \u2022 (g x - c x))) s x", "state_after": "case refine_1\nE : Type u_1\nX : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup X\ninst\u271d\u00b9 : NormedSpace \u211d X\ninst\u271d : HasContDiffBump E\nc\u271d : E\nf\u271d : ContDiffBump c\u271d\nx\u271d : E\nn : \u2115\u221e\nc g : X \u2192 E\ns : Set X\nf : (x : X) \u2192 ContDiffBump (c x)\nx : X\nhc : ContDiffWithinAt \u211d n c s x\nhr : ContDiffWithinAt \u211d n (fun x => (f x).rIn) s x\nhR : ContDiffWithinAt \u211d n (fun x => (f x).rOut) s x\nhg : ContDiffWithinAt \u211d n g s x\n\u22a2 Ioi 1 \u00d7\u02e2 univ \u2208 \ud835\udcdd ((f x).rOut / (f x).rIn, (f x).rIn\u207b\u00b9 \u2022 (g x - c x))\n\ncase refine_2\nE : Type u_1\nX : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup X\ninst\u271d\u00b9 : NormedSpace \u211d X\ninst\u271d : HasContDiffBump E\nc\u271d : E\nf\u271d : ContDiffBump c\u271d\nx\u271d : E\nn : \u2115\u221e\nc g : X \u2192 E\ns : Set X\nf : (x : X) \u2192 ContDiffBump (c x)\nx : X\nhc : ContDiffWithinAt \u211d n c s x\nhr : ContDiffWithinAt \u211d n (fun x => (f x).rIn) s x\nhR : ContDiffWithinAt \u211d n (fun x => (f x).rOut) s x\nhg : ContDiffWithinAt \u211d n g s x\n\u22a2 ContDiffWithinAt \u211d n (fun x => ((f x).rOut / (f x).rIn, (f x).rIn\u207b\u00b9 \u2022 (g x - c x))) s x"}, {"tactic": "exact prod_mem_nhds (Ioi_mem_nhds (f x).one_lt_rOut_div_rIn) univ_mem", "annotated_tactic": ["exact <a>prod_mem_nhds</a> (<a>Ioi_mem_nhds</a> (f x).<a>one_lt_rOut_div_rIn</a>) <a>univ_mem</a>", [{"full_name": "prod_mem_nhds", "def_path": "Mathlib/Topology/Constructions.lean", "def_pos": [568, 9], "def_end_pos": [568, 22]}, {"full_name": "Ioi_mem_nhds", "def_path": "Mathlib/Topology/Order/Basic.lean", "def_pos": [363, 9], "def_end_pos": [363, 21]}, {"full_name": "ContDiffBump.one_lt_rOut_div_rIn", "def_path": "Mathlib/Analysis/Calculus/BumpFunction/Basic.lean", "def_pos": [116, 9], "def_end_pos": [116, 28]}, {"full_name": "Filter.univ_mem", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [148, 9], "def_end_pos": [148, 17]}]], "state_before": "case refine_1\nE : Type u_1\nX : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup X\ninst\u271d\u00b9 : NormedSpace \u211d X\ninst\u271d : HasContDiffBump E\nc\u271d : E\nf\u271d : ContDiffBump c\u271d\nx\u271d : E\nn : \u2115\u221e\nc g : X \u2192 E\ns : Set X\nf : (x : X) \u2192 ContDiffBump (c x)\nx : X\nhc : ContDiffWithinAt \u211d n c s x\nhr : ContDiffWithinAt \u211d n (fun x => (f x).rIn) s x\nhR : ContDiffWithinAt \u211d n (fun x => (f x).rOut) s x\nhg : ContDiffWithinAt \u211d n g s x\n\u22a2 Ioi 1 \u00d7\u02e2 univ \u2208 \ud835\udcdd ((f x).rOut / (f x).rIn, (f x).rIn\u207b\u00b9 \u2022 (g x - c x))", "state_after": "no goals"}, {"tactic": "exact (hR.div hr (f x).rIn_pos.ne').prod ((hr.inv (f x).rIn_pos.ne').smul (hg.sub hc))", "annotated_tactic": ["exact (hR.div hr (f x).rIn_pos.ne').<a>prod</a> ((hr.inv (f x).rIn_pos.ne').<a>smul</a> (hg.sub hc))", [{"full_name": "ContDiffWithinAt.prod", "def_path": "Mathlib/Analysis/Calculus/ContDiff.lean", "def_pos": [517, 9], "def_end_pos": [517, 30]}, {"full_name": "ContDiffWithinAt.smul", "def_path": "Mathlib/Analysis/Calculus/ContDiff.lean", "def_pos": [1529, 9], "def_end_pos": [1529, 30]}]], "state_before": "case refine_2\nE : Type u_1\nX : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup X\ninst\u271d\u00b9 : NormedSpace \u211d X\ninst\u271d : HasContDiffBump E\nc\u271d : E\nf\u271d : ContDiffBump c\u271d\nx\u271d : E\nn : \u2115\u221e\nc g : X \u2192 E\ns : Set X\nf : (x : X) \u2192 ContDiffBump (c x)\nx : X\nhc : ContDiffWithinAt \u211d n c s x\nhr : ContDiffWithinAt \u211d n (fun x => (f x).rIn) s x\nhR : ContDiffWithinAt \u211d n (fun x => (f x).rOut) s x\nhg : ContDiffWithinAt \u211d n g s x\n\u22a2 ContDiffWithinAt \u211d n (fun x => ((f x).rOut / (f x).rIn, (f x).rIn\u207b\u00b9 \u2022 (g x - c x))) s x", "state_after": "no goals"}]

https://github.com/leanprover-community/mathlib4

3ce43c18f614b76e161f911b75a3e1ef641620ff

Mathlib/Data/Real/CauSeq.lean

CauSeq.const_neg

[]

https://github.com/leanprover-community/mathlib4

3ce43c18f614b76e161f911b75a3e1ef641620ff

Mathlib/Topology/Order.lean

DiscreteTopology.of_continuous_injective

[]

https://github.com/leanprover-community/mathlib4

3ce43c18f614b76e161f911b75a3e1ef641620ff

Mathlib/GroupTheory/Sylow.lean

Sylow.coe_comapOfInjective

[]

https://github.com/leanprover-community/mathlib4

3ce43c18f614b76e161f911b75a3e1ef641620ff

Mathlib/Data/List/Indexes.lean

List.mapIdx_eq_enum_map

[{"tactic": "rw [List.new_def_eq_old_def]", "annotated_tactic": ["rw [<a>List.new_def_eq_old_def</a>]", [{"full_name": "List.new_def_eq_old_def", "def_path": "Mathlib/Data/List/Indexes.lean", "def_pos": [137, 19], "def_end_pos": [137, 37]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\nl : List \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\n\u22a2 mapIdx f l = map (uncurry f) (enum l)", "state_after": "\u03b1 : Type u\n\u03b2 : Type v\nl : List \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\n\u22a2 List.oldMapIdx f l = map (uncurry f) (enum l)"}, {"tactic": "induction' l with hd tl hl generalizing f", "annotated_tactic": ["induction' l with hd tl hl generalizing f", []], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\nl : List \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\n\u22a2 List.oldMapIdx f l = map (uncurry f) (enum l)", "state_after": "case nil\n\u03b1 : Type u\n\u03b2 : Type v\nf\u271d f : \u2115 \u2192 \u03b1 \u2192 \u03b2\n\u22a2 List.oldMapIdx f [] = map (uncurry f) (enum [])\n\ncase cons\n\u03b1 : Type u\n\u03b2 : Type v\nf\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2\nhd : \u03b1\ntl : List \u03b1\nhl : \u2200 (f : \u2115 \u2192 \u03b1 \u2192 \u03b2), List.oldMapIdx f tl = map (uncurry f) (enum tl)\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\n\u22a2 List.oldMapIdx f (hd :: tl) = map (uncurry f) (enum (hd :: tl))"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case nil\n\u03b1 : Type u\n\u03b2 : Type v\nf\u271d f : \u2115 \u2192 \u03b1 \u2192 \u03b2\n\u22a2 List.oldMapIdx f [] = map (uncurry f) (enum [])", "state_after": "no goals"}, {"tactic": "rw [List.oldMapIdx, List.oldMapIdxCore, List.oldMapIdxCore_eq, hl]", "annotated_tactic": ["rw [<a>List.oldMapIdx</a>, <a>List.oldMapIdxCore</a>, <a>List.oldMapIdxCore_eq</a>, hl]", [{"full_name": "List.oldMapIdx", "def_path": "Mathlib/Data/List/Indexes.lean", "def_pos": [37, 15], "def_end_pos": [37, 24]}, {"full_name": "List.oldMapIdxCore", "def_path": "Mathlib/Data/List/Indexes.lean", "def_pos": [31, 15], "def_end_pos": [31, 28]}, {"full_name": "List.oldMapIdxCore_eq", "def_path": "Mathlib/Data/List/Indexes.lean", "def_pos": [46, 19], "def_end_pos": [46, 35]}]], "state_before": "case cons\n\u03b1 : Type u\n\u03b2 : Type v\nf\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2\nhd : \u03b1\ntl : List \u03b1\nhl : \u2200 (f : \u2115 \u2192 \u03b1 \u2192 \u03b2), List.oldMapIdx f tl = map (uncurry f) (enum tl)\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\n\u22a2 List.oldMapIdx f (hd :: tl) = map (uncurry f) (enum (hd :: tl))", "state_after": "case cons\n\u03b1 : Type u\n\u03b2 : Type v\nf\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2\nhd : \u03b1\ntl : List \u03b1\nhl : \u2200 (f : \u2115 \u2192 \u03b1 \u2192 \u03b2), List.oldMapIdx f tl = map (uncurry f) (enum tl)\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\n\u22a2 f 0 hd :: map (uncurry fun i a => f (i + (0 + 1)) a) (enum tl) = map (uncurry f) (enum (hd :: tl))"}, {"tactic": "simp [map, enum_eq_zip_range, map_uncurry_zip_eq_zipWith]", "annotated_tactic": ["simp [<a>map</a>, <a>enum_eq_zip_range</a>, <a>map_uncurry_zip_eq_zipWith</a>]", [{"full_name": "List.map", "def_path": "lake-packages/lean4/src/lean/Init/Data/List/Basic.lean", "def_pos": [151, 19], "def_end_pos": [151, 22]}, {"full_name": "List.enum_eq_zip_range", "def_path": "Mathlib/Data/List/Range.lean", "def_pos": [187, 9], "def_end_pos": [187, 26]}, {"full_name": "List.map_uncurry_zip_eq_zipWith", "def_path": "Mathlib/Data/List/Zip.lean", "def_pos": [389, 9], "def_end_pos": [389, 35]}]], "state_before": "case cons\n\u03b1 : Type u\n\u03b2 : Type v\nf\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2\nhd : \u03b1\ntl : List \u03b1\nhl : \u2200 (f : \u2115 \u2192 \u03b1 \u2192 \u03b2), List.oldMapIdx f tl = map (uncurry f) (enum tl)\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\n\u22a2 f 0 hd :: map (uncurry fun i a => f (i + (0 + 1)) a) (enum tl) = map (uncurry f) (enum (hd :: tl))", "state_after": "no goals"}]

https://github.com/leanprover-community/mathlib4

3ce43c18f614b76e161f911b75a3e1ef641620ff

Mathlib/FieldTheory/PolynomialGaloisGroup.lean

Polynomial.Gal.mul_splits_in_splittingField_of_mul

[{"tactic": "apply splits_mul", "annotated_tactic": ["apply <a>splits_mul</a>", [{"full_name": "Polynomial.splits_mul", "def_path": "Mathlib/Data/Polynomial/Splits.lean", "def_pos": [112, 9], "def_end_pos": [112, 19]}]], "state_before": "F : Type u_1\ninst\u271d\u00b2 : Field F\np q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np\u2081 q\u2081 p\u2082 q\u2082 : F[X]\nhq\u2081 : q\u2081 \u2260 0\nhq\u2082 : q\u2082 \u2260 0\nh\u2081 : Splits (algebraMap F (SplittingField q\u2081)) p\u2081\nh\u2082 : Splits (algebraMap F (SplittingField q\u2082)) p\u2082\n\u22a2 Splits (algebraMap F (SplittingField (q\u2081 * q\u2082))) (p\u2081 * p\u2082)", "state_after": "case hf\nF : Type u_1\ninst\u271d\u00b2 : Field F\np q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np\u2081 q\u2081 p\u2082 q\u2082 : F[X]\nhq\u2081 : q\u2081 \u2260 0\nhq\u2082 : q\u2082 \u2260 0\nh\u2081 : Splits (algebraMap F (SplittingField q\u2081)) p\u2081\nh\u2082 : Splits (algebraMap F (SplittingField q\u2082)) p\u2082\n\u22a2 Splits (algebraMap F (SplittingField (q\u2081 * q\u2082))) p\u2081\n\ncase hg\nF : Type u_1\ninst\u271d\u00b2 : Field F\np q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np\u2081 q\u2081 p\u2082 q\u2082 : F[X]\nhq\u2081 : q\u2081 \u2260 0\nhq\u2082 : q\u2082 \u2260 0\nh\u2081 : Splits (algebraMap F (SplittingField q\u2081)) p\u2081\nh\u2082 : Splits (algebraMap F (SplittingField q\u2082)) p\u2082\n\u22a2 Splits (algebraMap F (SplittingField (q\u2081 * q\u2082))) p\u2082"}, {"tactic": "rw [\u2190\n (SplittingField.lift q\u2081\n (splits_of_splits_of_dvd (algebraMap F (q\u2081 * q\u2082).SplittingField) (mul_ne_zero hq\u2081 hq\u2082)\n (SplittingField.splits _) (dvd_mul_right q\u2081 q\u2082))).comp_algebraMap]", "annotated_tactic": ["rw [\u2190\n (<a>SplittingField.lift</a> q\u2081\n (<a>splits_of_splits_of_dvd</a> (<a>algebraMap</a> F (q\u2081 * q\u2082).<a>SplittingField</a>) (<a>mul_ne_zero</a> hq\u2081 hq\u2082)\n (<a>SplittingField.splits</a> _) (<a>dvd_mul_right</a> q\u2081 q\u2082))).<a>comp_algebraMap</a>]", [{"full_name": "Polynomial.SplittingField.lift", "def_path": "Mathlib/FieldTheory/SplittingField/Construction.lean", "def_pos": [333, 5], "def_end_pos": [333, 9]}, {"full_name": "Polynomial.splits_of_splits_of_dvd", "def_path": "Mathlib/Data/Polynomial/Splits.lean", "def_pos": [254, 9], "def_end_pos": [254, 32]}, {"full_name": "algebraMap", "def_path": "Mathlib/Algebra/Algebra/Basic.lean", "def_pos": [125, 5], "def_end_pos": [125, 15]}, {"full_name": "Polynomial.SplittingField", "def_path": "Mathlib/FieldTheory/SplittingField/Construction.lean", "def_pos": [230, 5], "def_end_pos": [230, 19]}, {"full_name": "mul_ne_zero", "def_path": "Mathlib/Algebra/GroupWithZero/Basic.lean", "def_pos": [88, 9], "def_end_pos": [88, 20]}, {"full_name": "Polynomial.SplittingField.splits", "def_path": "Mathlib/FieldTheory/SplittingField/Construction.lean", "def_pos": [326, 19], "def_end_pos": [326, 25]}, {"full_name": "dvd_mul_right", "def_path": "Mathlib/Algebra/Divisibility/Basic.lean", "def_pos": [82, 9], "def_end_pos": [82, 22]}, {"full_name": "AlgHom.comp_algebraMap", "def_path": "Mathlib/Algebra/Algebra/Hom.lean", "def_pos": [240, 9], "def_end_pos": [240, 24]}]], "state_before": "case hf\nF : Type u_1\ninst\u271d\u00b2 : Field F\np q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np\u2081 q\u2081 p\u2082 q\u2082 : F[X]\nhq\u2081 : q\u2081 \u2260 0\nhq\u2082 : q\u2082 \u2260 0\nh\u2081 : Splits (algebraMap F (SplittingField q\u2081)) p\u2081\nh\u2082 : Splits (algebraMap F (SplittingField q\u2082)) p\u2082\n\u22a2 Splits (algebraMap F (SplittingField (q\u2081 * q\u2082))) p\u2081", "state_after": "case hf\nF : Type u_1\ninst\u271d\u00b2 : Field F\np q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np\u2081 q\u2081 p\u2082 q\u2082 : F[X]\nhq\u2081 : q\u2081 \u2260 0\nhq\u2082 : q\u2082 \u2260 0\nh\u2081 : Splits (algebraMap F (SplittingField q\u2081)) p\u2081\nh\u2082 : Splits (algebraMap F (SplittingField q\u2082)) p\u2082\n\u22a2 Splits\n (RingHom.comp (\u2191(SplittingField.lift q\u2081 (_ : Splits (algebraMap F (SplittingField (q\u2081 * q\u2082))) q\u2081)))\n (algebraMap F (SplittingField q\u2081)))\n p\u2081"}, {"tactic": "exact splits_comp_of_splits _ _ h\u2081", "annotated_tactic": ["exact <a>splits_comp_of_splits</a> _ _ h\u2081", [{"full_name": "Polynomial.splits_comp_of_splits", "def_path": "Mathlib/Data/Polynomial/Splits.lean", "def_pos": [433, 9], "def_end_pos": [433, 30]}]], "state_before": "case hf\nF : Type u_1\ninst\u271d\u00b2 : Field F\np q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np\u2081 q\u2081 p\u2082 q\u2082 : F[X]\nhq\u2081 : q\u2081 \u2260 0\nhq\u2082 : q\u2082 \u2260 0\nh\u2081 : Splits (algebraMap F (SplittingField q\u2081)) p\u2081\nh\u2082 : Splits (algebraMap F (SplittingField q\u2082)) p\u2082\n\u22a2 Splits\n (RingHom.comp (\u2191(SplittingField.lift q\u2081 (_ : Splits (algebraMap F (SplittingField (q\u2081 * q\u2082))) q\u2081)))\n (algebraMap F (SplittingField q\u2081)))\n p\u2081", "state_after": "no goals"}, {"tactic": "rw [\u2190\n (SplittingField.lift q\u2082\n (splits_of_splits_of_dvd (algebraMap F (q\u2081 * q\u2082).SplittingField) (mul_ne_zero hq\u2081 hq\u2082)\n (SplittingField.splits _) (dvd_mul_left q\u2082 q\u2081))).comp_algebraMap]", "annotated_tactic": ["rw [\u2190\n (<a>SplittingField.lift</a> q\u2082\n (<a>splits_of_splits_of_dvd</a> (<a>algebraMap</a> F (q\u2081 * q\u2082).<a>SplittingField</a>) (<a>mul_ne_zero</a> hq\u2081 hq\u2082)\n (<a>SplittingField.splits</a> _) (<a>dvd_mul_left</a> q\u2082 q\u2081))).<a>comp_algebraMap</a>]", [{"full_name": "Polynomial.SplittingField.lift", "def_path": "Mathlib/FieldTheory/SplittingField/Construction.lean", "def_pos": [333, 5], "def_end_pos": [333, 9]}, {"full_name": "Polynomial.splits_of_splits_of_dvd", "def_path": "Mathlib/Data/Polynomial/Splits.lean", "def_pos": [254, 9], "def_end_pos": [254, 32]}, {"full_name": "algebraMap", "def_path": "Mathlib/Algebra/Algebra/Basic.lean", "def_pos": [125, 5], "def_end_pos": [125, 15]}, {"full_name": "Polynomial.SplittingField", "def_path": "Mathlib/FieldTheory/SplittingField/Construction.lean", "def_pos": [230, 5], "def_end_pos": [230, 19]}, {"full_name": "mul_ne_zero", "def_path": "Mathlib/Algebra/GroupWithZero/Basic.lean", "def_pos": [88, 9], "def_end_pos": [88, 20]}, {"full_name": "Polynomial.SplittingField.splits", "def_path": "Mathlib/FieldTheory/SplittingField/Construction.lean", "def_pos": [326, 19], "def_end_pos": [326, 25]}, {"full_name": "dvd_mul_left", "def_path": "Mathlib/Algebra/Divisibility/Basic.lean", "def_pos": [169, 9], "def_end_pos": [169, 21]}, {"full_name": "AlgHom.comp_algebraMap", "def_path": "Mathlib/Algebra/Algebra/Hom.lean", "def_pos": [240, 9], "def_end_pos": [240, 24]}]], "state_before": "case hg\nF : Type u_1\ninst\u271d\u00b2 : Field F\np q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np\u2081 q\u2081 p\u2082 q\u2082 : F[X]\nhq\u2081 : q\u2081 \u2260 0\nhq\u2082 : q\u2082 \u2260 0\nh\u2081 : Splits (algebraMap F (SplittingField q\u2081)) p\u2081\nh\u2082 : Splits (algebraMap F (SplittingField q\u2082)) p\u2082\n\u22a2 Splits (algebraMap F (SplittingField (q\u2081 * q\u2082))) p\u2082", "state_after": "case hg\nF : Type u_1\ninst\u271d\u00b2 : Field F\np q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np\u2081 q\u2081 p\u2082 q\u2082 : F[X]\nhq\u2081 : q\u2081 \u2260 0\nhq\u2082 : q\u2082 \u2260 0\nh\u2081 : Splits (algebraMap F (SplittingField q\u2081)) p\u2081\nh\u2082 : Splits (algebraMap F (SplittingField q\u2082)) p\u2082\n\u22a2 Splits\n (RingHom.comp (\u2191(SplittingField.lift q\u2082 (_ : Splits (algebraMap F (SplittingField (q\u2081 * q\u2082))) q\u2082)))\n (algebraMap F (SplittingField q\u2082)))\n p\u2082"}, {"tactic": "exact splits_comp_of_splits _ _ h\u2082", "annotated_tactic": ["exact <a>splits_comp_of_splits</a> _ _ h\u2082", [{"full_name": "Polynomial.splits_comp_of_splits", "def_path": "Mathlib/Data/Polynomial/Splits.lean", "def_pos": [433, 9], "def_end_pos": [433, 30]}]], "state_before": "case hg\nF : Type u_1\ninst\u271d\u00b2 : Field F\np q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np\u2081 q\u2081 p\u2082 q\u2082 : F[X]\nhq\u2081 : q\u2081 \u2260 0\nhq\u2082 : q\u2082 \u2260 0\nh\u2081 : Splits (algebraMap F (SplittingField q\u2081)) p\u2081\nh\u2082 : Splits (algebraMap F (SplittingField q\u2082)) p\u2082\n\u22a2 Splits\n (RingHom.comp (\u2191(SplittingField.lift q\u2082 (_ : Splits (algebraMap F (SplittingField (q\u2081 * q\u2082))) q\u2082)))\n (algebraMap F (SplittingField q\u2082)))\n p\u2082", "state_after": "no goals"}]

https://github.com/leanprover-community/mathlib4

3ce43c18f614b76e161f911b75a3e1ef641620ff

Mathlib/Order/Monotone/Union.lean

MonotoneOn.union_right

[{"tactic": "intro x hx y hy hxy", "annotated_tactic": ["intro x hx y hy hxy", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : Preorder \u03b2\na : \u03b1\nf : \u03b1 \u2192 \u03b2\ns t : Set \u03b1\nc : \u03b1\nh\u2081 : MonotoneOn f s\nh\u2082 : MonotoneOn f t\nhs : IsGreatest s c\nht : IsLeast t c\nA : \u2200 (x : \u03b1), x \u2208 s \u222a t \u2192 x \u2264 c \u2192 x \u2208 s\nB : \u2200 (x : \u03b1), x \u2208 s \u222a t \u2192 c \u2264 x \u2192 x \u2208 t\n\u22a2 MonotoneOn f (s \u222a t)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : Preorder \u03b2\na : \u03b1\nf : \u03b1 \u2192 \u03b2\ns t : Set \u03b1\nc : \u03b1\nh\u2081 : MonotoneOn f s\nh\u2082 : MonotoneOn f t\nhs : IsGreatest s c\nht : IsLeast t c\nA : \u2200 (x : \u03b1), x \u2208 s \u222a t \u2192 x \u2264 c \u2192 x \u2208 s\nB : \u2200 (x : \u03b1), x \u2208 s \u222a t \u2192 c \u2264 x \u2192 x \u2208 t\nx : \u03b1\nhx : x \u2208 s \u222a t\ny : \u03b1\nhy : y \u2208 s \u222a t\nhxy : x \u2264 y\n\u22a2 f x \u2264 f y"}, {"tactic": "rcases lt_or_le x c with (hxc | hcx)", "annotated_tactic": ["rcases <a>lt_or_le</a> x c with (hxc | hcx)", [{"full_name": "lt_or_le", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [336, 9], "def_end_pos": [336, 17]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : Preorder \u03b2\na : \u03b1\nf : \u03b1 \u2192 \u03b2\ns t : Set \u03b1\nc : \u03b1\nh\u2081 : MonotoneOn f s\nh\u2082 : MonotoneOn f t\nhs : IsGreatest s c\nht : IsLeast t c\nA : \u2200 (x : \u03b1), x \u2208 s \u222a t \u2192 x \u2264 c \u2192 x \u2208 s\nB : \u2200 (x : \u03b1), x \u2208 s \u222a t \u2192 c \u2264 x \u2192 x \u2208 t\nx : \u03b1\nhx : x \u2208 s \u222a t\ny : \u03b1\nhy : y \u2208 s \u222a t\nhxy : x \u2264 y\n\u22a2 f x \u2264 f y", "state_after": "case inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : Preorder \u03b2\na : \u03b1\nf : \u03b1 \u2192 \u03b2\ns t : Set \u03b1\nc : \u03b1\nh\u2081 : MonotoneOn f s\nh\u2082 : MonotoneOn f t\nhs : IsGreatest s c\nht : IsLeast t c\nA : \u2200 (x : \u03b1), x \u2208 s \u222a t \u2192 x \u2264 c \u2192 x \u2208 s\nB : \u2200 (x : \u03b1), x \u2208 s \u222a t \u2192 c \u2264 x \u2192 x \u2208 t\nx : \u03b1\nhx : x \u2208 s \u222a t\ny : \u03b1\nhy : y \u2208 s \u222a t\nhxy : x \u2264 y\nhxc : x < c\n\u22a2 f x \u2264 f y\n\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : Preorder \u03b2\na : \u03b1\nf : \u03b1 \u2192 \u03b2\ns t : Set \u03b1\nc : \u03b1\nh\u2081 : MonotoneOn f s\nh\u2082 : MonotoneOn f t\nhs : IsGreatest s c\nht : IsLeast t c\nA : \u2200 (x : \u03b1), x \u2208 s \u222a t \u2192 x \u2264 c \u2192 x \u2208 s\nB : \u2200 (x : \u03b1), x \u2208 s \u222a t \u2192 c \u2264 x \u2192 x \u2208 t\nx : \u03b1\nhx : x \u2208 s \u222a t\ny : \u03b1\nhy : y \u2208 s \u222a t\nhxy : x \u2264 y\nhcx : c \u2264 x\n\u22a2 f x \u2264 f y"}, {"tactic": "intro x hx hxc", "annotated_tactic": ["intro x hx hxc", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : Preorder \u03b2\na : \u03b1\nf : \u03b1 \u2192 \u03b2\ns t : Set \u03b1\nc : \u03b1\nh\u2081 : MonotoneOn f s\nh\u2082 : MonotoneOn f t\nhs : IsGreatest s c\nht : IsLeast t c\n\u22a2 \u2200 (x : \u03b1), x \u2208 s \u222a t \u2192 x \u2264 c \u2192 x \u2208 s", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : Preorder \u03b2\na : \u03b1\nf : \u03b1 \u2192 \u03b2\ns t : Set \u03b1\nc : \u03b1\nh\u2081 : MonotoneOn f s\nh\u2082 : MonotoneOn f t\nhs : IsGreatest s c\nht : IsLeast t c\nx : \u03b1\nhx : x \u2208 s \u222a t\nhxc : x \u2264 c\n\u22a2 x \u2208 s"}, {"tactic": "cases hx", "annotated_tactic": ["cases hx", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : Preorder \u03b2\na : \u03b1\nf : \u03b1 \u2192 \u03b2\ns t : Set \u03b1\nc : \u03b1\nh\u2081 : MonotoneOn f s\nh\u2082 : MonotoneOn f t\nhs : IsGreatest s c\nht : IsLeast t c\nx : \u03b1\nhx : x \u2208 s \u222a t\nhxc : x \u2264 c\n\u22a2 x \u2208 s", "state_after": "case inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : Preorder \u03b2\na : \u03b1\nf : \u03b1 \u2192 \u03b2\ns t : Set \u03b1\nc : \u03b1\nh\u2081 : MonotoneOn f s\nh\u2082 : MonotoneOn f t\nhs : IsGreatest s c\nht : IsLeast t c\nx : \u03b1\nhxc : x \u2264 c\nh\u271d : x \u2208 s\n\u22a2 x \u2208 s\n\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : Preorder \u03b2\na : \u03b1\nf : \u03b1 \u2192 \u03b2\ns t : Set \u03b1\nc : \u03b1\nh\u2081 : MonotoneOn f s\nh\u2082 : MonotoneOn f t\nhs : IsGreatest s c\nht : IsLeast t c\nx : \u03b1\nhxc : x \u2264 c\nh\u271d : x \u2208 t\n\u22a2 x \u2208 s"}, {"tactic": "rcases eq_or_lt_of_le hxc with (rfl | h'x)", "annotated_tactic": ["rcases <a>eq_or_lt_of_le</a> hxc with (rfl | h'x)", [{"full_name": "eq_or_lt_of_le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [414, 9], "def_end_pos": [414, 23]}]], "state_before": "case inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : Preorder \u03b2\na : \u03b1\nf : \u03b1 \u2192 \u03b2\ns t : Set \u03b1\nc : \u03b1\nh\u2081 : MonotoneOn f s\nh\u2082 : MonotoneOn f t\nhs : IsGreatest s c\nht : IsLeast t c\nx : \u03b1\nhxc : x \u2264 c\nh\u271d : x \u2208 t\n\u22a2 x \u2208 s", "state_after": "case inr.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : Preorder \u03b2\na : \u03b1\nf : \u03b1 \u2192 \u03b2\ns t : Set \u03b1\nh\u2081 : MonotoneOn f s\nh\u2082 : MonotoneOn f t\nx : \u03b1\nh\u271d : x \u2208 t\nhs : IsGreatest s x\nht : IsLeast t x\nhxc : x \u2264 x\n\u22a2 x \u2208 s\n\ncase inr.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : Preorder \u03b2\na : \u03b1\nf : \u03b1 \u2192 \u03b2\ns t : Set \u03b1\nc : \u03b1\nh\u2081 : MonotoneOn f s\nh\u2082 : MonotoneOn f t\nhs : IsGreatest s c\nht : IsLeast t c\nx : \u03b1\nhxc : x \u2264 c\nh\u271d : x \u2208 t\nh'x : x < c\n\u22a2 x \u2208 s"}, {"tactic": "exact (lt_irrefl _ (h'x.trans_le (ht.2 (by assumption)))).elim", "annotated_tactic": ["exact (<a>lt_irrefl</a> _ (h'x.trans_le (ht.2 (by assumption)))).<a>elim</a>", [{"full_name": "lt_irrefl", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [79, 9], "def_end_pos": [79, 18]}, {"full_name": "False.elim", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [223, 21], "def_end_pos": [223, 31]}]], "state_before": "case inr.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : Preorder \u03b2\na : \u03b1\nf : \u03b1 \u2192 \u03b2\ns t : Set \u03b1\nc : \u03b1\nh\u2081 : MonotoneOn f s\nh\u2082 : MonotoneOn f t\nhs : IsGreatest s c\nht : IsLeast t c\nx : \u03b1\nhxc : x \u2264 c\nh\u271d : x \u2208 t\nh'x : x < c\n\u22a2 x \u2208 s", "state_after": "no goals"}, {"tactic": "assumption", "annotated_tactic": ["assumption", []], "state_before": "case inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : Preorder \u03b2\na : \u03b1\nf : \u03b1 \u2192 \u03b2\ns t : Set \u03b1\nc : \u03b1\nh\u2081 : MonotoneOn f s\nh\u2082 : MonotoneOn f t\nhs : IsGreatest s c\nht : IsLeast t c\nx : \u03b1\nhxc : x \u2264 c\nh\u271d : x \u2208 s\n\u22a2 x \u2208 s", "state_after": "no goals"}, {"tactic": "exact hs.1", "annotated_tactic": ["exact hs.1", []], "state_before": "case inr.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : Preorder \u03b2\na : \u03b1\nf : \u03b1 \u2192 \u03b2\ns t : Set \u03b1\nh\u2081 : MonotoneOn f s\nh\u2082 : MonotoneOn f t\nx : \u03b1\nh\u271d : x \u2208 t\nhs : IsGreatest s x\nht : IsLeast t x\nhxc : x \u2264 x\n\u22a2 x \u2208 s", "state_after": "no goals"}, {"tactic": "assumption", "annotated_tactic": ["assumption", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : Preorder \u03b2\na : \u03b1\nf : \u03b1 \u2192 \u03b2\ns t : Set \u03b1\nc : \u03b1\nh\u2081 : MonotoneOn f s\nh\u2082 : MonotoneOn f t\nhs : IsGreatest s c\nht : IsLeast t c\nx : \u03b1\nhxc : x \u2264 c\nh\u271d : x \u2208 t\nh'x : x < c\n\u22a2 x \u2208 t", "state_after": "no goals"}, {"tactic": "intro x hx hxc", "annotated_tactic": ["intro x hx hxc", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : Preorder \u03b2\na : \u03b1\nf : \u03b1 \u2192 \u03b2\ns t : Set \u03b1\nc : \u03b1\nh\u2081 : MonotoneOn f s\nh\u2082 : MonotoneOn f t\nhs : IsGreatest s c\nht : IsLeast t c\nA : \u2200 (x : \u03b1), x \u2208 s \u222a t \u2192 x \u2264 c \u2192 x \u2208 s\n\u22a2 \u2200 (x : \u03b1), x \u2208 s \u222a t \u2192 c \u2264 x \u2192 x \u2208 t", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : Preorder \u03b2\na : \u03b1\nf : \u03b1 \u2192 \u03b2\ns t : Set \u03b1\nc : \u03b1\nh\u2081 : MonotoneOn f s\nh\u2082 : MonotoneOn f t\nhs : IsGreatest s c\nht : IsLeast t c\nA : \u2200 (x : \u03b1), x \u2208 s \u222a t \u2192 x \u2264 c \u2192 x \u2208 s\nx : \u03b1\nhx : x \u2208 s \u222a t\nhxc : c \u2264 x\n\u22a2 x \u2208 t"}, {"tactic": "exact hx", "annotated_tactic": ["exact hx", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : Preorder \u03b2\na : \u03b1\nf : \u03b1 \u2192 \u03b2\ns t : Set \u03b1\nc : \u03b1\nh\u2081 : MonotoneOn f s\nh\u2082 : MonotoneOn f t\nhs : IsGreatest s c\nht : IsLeast t c\nA : \u2200 (x : \u03b1), x \u2208 s \u222a t \u2192 x \u2264 c \u2192 x \u2208 s\nx : \u03b1\nhx\u271d : x \u2208 s \u222a t\nhxc : c \u2264 x\nhx : x \u2208 t\n\u22a2 x \u2208 t", "state_after": "no goals"}, {"tactic": "rcases eq_or_lt_of_le hxc with (rfl | h'x)", "annotated_tactic": ["rcases <a>eq_or_lt_of_le</a> hxc with (rfl | h'x)", [{"full_name": "eq_or_lt_of_le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [414, 9], "def_end_pos": [414, 23]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : Preorder \u03b2\na : \u03b1\nf : \u03b1 \u2192 \u03b2\ns t : Set \u03b1\nc : \u03b1\nh\u2081 : MonotoneOn f s\nh\u2082 : MonotoneOn f t\nhs : IsGreatest s c\nht : IsLeast t c\nA : \u2200 (x : \u03b1), x \u2208 s \u222a t \u2192 x \u2264 c \u2192 x \u2208 s\nx : \u03b1\nhx\u271d : x \u2208 s \u222a t\nhxc : c \u2264 x\nhx : x \u2208 s\n\u22a2 x \u2208 t", "state_after": "case inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : Preorder \u03b2\na : \u03b1\nf : \u03b1 \u2192 \u03b2\ns t : Set \u03b1\nc : \u03b1\nh\u2081 : MonotoneOn f s\nh\u2082 : MonotoneOn f t\nhs : IsGreatest s c\nht : IsLeast t c\nA : \u2200 (x : \u03b1), x \u2208 s \u222a t \u2192 x \u2264 c \u2192 x \u2208 s\nhx\u271d : c \u2208 s \u222a t\nhxc : c \u2264 c\nhx : c \u2208 s\n\u22a2 c \u2208 t\n\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : Preorder \u03b2\na : \u03b1\nf : \u03b1 \u2192 \u03b2\ns t : Set \u03b1\nc : \u03b1\nh\u2081 : MonotoneOn f s\nh\u2082 : MonotoneOn f t\nhs : IsGreatest s c\nht : IsLeast t c\nA : \u2200 (x : \u03b1), x \u2208 s \u222a t \u2192 x \u2264 c \u2192 x \u2208 s\nx : \u03b1\nhx\u271d : x \u2208 s \u222a t\nhxc : c \u2264 x\nhx : x \u2208 s\nh'x : c < x\n\u22a2 x \u2208 t"}, {"tactic": "exact (lt_irrefl _ (h'x.trans_le (hs.2 hx))).elim", "annotated_tactic": ["exact (<a>lt_irrefl</a> _ (h'x.trans_le (hs.2 hx))).<a>elim</a>", [{"full_name": "lt_irrefl", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [79, 9], "def_end_pos": [79, 18]}, {"full_name": "False.elim", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [223, 21], "def_end_pos": [223, 31]}]], "state_before": "case inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : Preorder \u03b2\na : \u03b1\nf : \u03b1 \u2192 \u03b2\ns t : Set \u03b1\nc : \u03b1\nh\u2081 : MonotoneOn f s\nh\u2082 : MonotoneOn f t\nhs : IsGreatest s c\nht : IsLeast t c\nA : \u2200 (x : \u03b1), x \u2208 s \u222a t \u2192 x \u2264 c \u2192 x \u2208 s\nx : \u03b1\nhx\u271d : x \u2208 s \u222a t\nhxc : c \u2264 x\nhx : x \u2208 s\nh'x : c < x\n\u22a2 x \u2208 t", "state_after": "no goals"}, {"tactic": "exact ht.1", "annotated_tactic": ["exact ht.1", []], "state_before": "case inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : Preorder \u03b2\na : \u03b1\nf : \u03b1 \u2192 \u03b2\ns t : Set \u03b1\nc : \u03b1\nh\u2081 : MonotoneOn f s\nh\u2082 : MonotoneOn f t\nhs : IsGreatest s c\nht : IsLeast t c\nA : \u2200 (x : \u03b1), x \u2208 s \u222a t \u2192 x \u2264 c \u2192 x \u2208 s\nhx\u271d : c \u2208 s \u222a t\nhxc : c \u2264 c\nhx : c \u2208 s\n\u22a2 c \u2208 t", "state_after": "no goals"}, {"tactic": "have xs : x \u2208 s := A _ hx hxc.le", "annotated_tactic": ["have xs : x \u2208 s := A _ hx hxc.le", []], "state_before": "case inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : Preorder \u03b2\na : \u03b1\nf : \u03b1 \u2192 \u03b2\ns t : Set \u03b1\nc : \u03b1\nh\u2081 : MonotoneOn f s\nh\u2082 : MonotoneOn f t\nhs : IsGreatest s c\nht : IsLeast t c\nA : \u2200 (x : \u03b1), x \u2208 s \u222a t \u2192 x \u2264 c \u2192 x \u2208 s\nB : \u2200 (x : \u03b1), x \u2208 s \u222a t \u2192 c \u2264 x \u2192 x \u2208 t\nx : \u03b1\nhx : x \u2208 s \u222a t\ny : \u03b1\nhy : y \u2208 s \u222a t\nhxy : x \u2264 y\nhxc : x < c\n\u22a2 f x \u2264 f y", "state_after": "case inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : Preorder \u03b2\na : \u03b1\nf : \u03b1 \u2192 \u03b2\ns t : Set \u03b1\nc : \u03b1\nh\u2081 : MonotoneOn f s\nh\u2082 : MonotoneOn f t\nhs : IsGreatest s c\nht : IsLeast t c\nA : \u2200 (x : \u03b1), x \u2208 s \u222a t \u2192 x \u2264 c \u2192 x \u2208 s\nB : \u2200 (x : \u03b1), x \u2208 s \u222a t \u2192 c \u2264 x \u2192 x \u2208 t\nx : \u03b1\nhx : x \u2208 s \u222a t\ny : \u03b1\nhy : y \u2208 s \u222a t\nhxy : x \u2264 y\nhxc : x < c\nxs : x \u2208 s\n\u22a2 f x \u2264 f y"}, {"tactic": "rcases lt_or_le y c with (hyc | hcy)", "annotated_tactic": ["rcases <a>lt_or_le</a> y c with (hyc | hcy)", [{"full_name": "lt_or_le", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [336, 9], "def_end_pos": [336, 17]}]], "state_before": "case inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : Preorder \u03b2\na : \u03b1\nf : \u03b1 \u2192 \u03b2\ns t : Set \u03b1\nc : \u03b1\nh\u2081 : MonotoneOn f s\nh\u2082 : MonotoneOn f t\nhs : IsGreatest s c\nht : IsLeast t c\nA : \u2200 (x : \u03b1), x \u2208 s \u222a t \u2192 x \u2264 c \u2192 x \u2208 s\nB : \u2200 (x : \u03b1), x \u2208 s \u222a t \u2192 c \u2264 x \u2192 x \u2208 t\nx : \u03b1\nhx : x \u2208 s \u222a t\ny : \u03b1\nhy : y \u2208 s \u222a t\nhxy : x \u2264 y\nhxc : x < c\nxs : x \u2208 s\n\u22a2 f x \u2264 f y", "state_after": "case inl.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : Preorder \u03b2\na : \u03b1\nf : \u03b1 \u2192 \u03b2\ns t : Set \u03b1\nc : \u03b1\nh\u2081 : MonotoneOn f s\nh\u2082 : MonotoneOn f t\nhs : IsGreatest s c\nht : IsLeast t c\nA : \u2200 (x : \u03b1), x \u2208 s \u222a t \u2192 x \u2264 c \u2192 x \u2208 s\nB : \u2200 (x : \u03b1), x \u2208 s \u222a t \u2192 c \u2264 x \u2192 x \u2208 t\nx : \u03b1\nhx : x \u2208 s \u222a t\ny : \u03b1\nhy : y \u2208 s \u222a t\nhxy : x \u2264 y\nhxc : x < c\nxs : x \u2208 s\nhyc : y < c\n\u22a2 f x \u2264 f y\n\ncase inl.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : Preorder \u03b2\na : \u03b1\nf : \u03b1 \u2192 \u03b2\ns t : Set \u03b1\nc : \u03b1\nh\u2081 : MonotoneOn f s\nh\u2082 : MonotoneOn f t\nhs : IsGreatest s c\nht : IsLeast t c\nA : \u2200 (x : \u03b1), x \u2208 s \u222a t \u2192 x \u2264 c \u2192 x \u2208 s\nB : \u2200 (x : \u03b1), x \u2208 s \u222a t \u2192 c \u2264 x \u2192 x \u2208 t\nx : \u03b1\nhx : x \u2208 s \u222a t\ny : \u03b1\nhy : y \u2208 s \u222a t\nhxy : x \u2264 y\nhxc : x < c\nxs : x \u2208 s\nhcy : c \u2264 y\n\u22a2 f x \u2264 f y"}, {"tactic": "exact h\u2081 xs (A _ hy hyc.le) hxy", "annotated_tactic": ["exact h\u2081 xs (A _ hy hyc.le) hxy", []], "state_before": "case inl.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : Preorder \u03b2\na : \u03b1\nf : \u03b1 \u2192 \u03b2\ns t : Set \u03b1\nc : \u03b1\nh\u2081 : MonotoneOn f s\nh\u2082 : MonotoneOn f t\nhs : IsGreatest s c\nht : IsLeast t c\nA : \u2200 (x : \u03b1), x \u2208 s \u222a t \u2192 x \u2264 c \u2192 x \u2208 s\nB : \u2200 (x : \u03b1), x \u2208 s \u222a t \u2192 c \u2264 x \u2192 x \u2208 t\nx : \u03b1\nhx : x \u2208 s \u222a t\ny : \u03b1\nhy : y \u2208 s \u222a t\nhxy : x \u2264 y\nhxc : x < c\nxs : x \u2208 s\nhyc : y < c\n\u22a2 f x \u2264 f y", "state_after": "no goals"}, {"tactic": "exact (h\u2081 xs hs.1 hxc.le).trans (h\u2082 ht.1 (B _ hy hcy) hcy)", "annotated_tactic": ["exact (h\u2081 xs hs.1 hxc.le).<a>trans</a> (h\u2082 ht.1 (B _ hy hcy) hcy)", [{"full_name": "LE.le.trans", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [120, 7], "def_end_pos": [120, 18]}]], "state_before": "case inl.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : Preorder \u03b2\na : \u03b1\nf : \u03b1 \u2192 \u03b2\ns t : Set \u03b1\nc : \u03b1\nh\u2081 : MonotoneOn f s\nh\u2082 : MonotoneOn f t\nhs : IsGreatest s c\nht : IsLeast t c\nA : \u2200 (x : \u03b1), x \u2208 s \u222a t \u2192 x \u2264 c \u2192 x \u2208 s\nB : \u2200 (x : \u03b1), x \u2208 s \u222a t \u2192 c \u2264 x \u2192 x \u2208 t\nx : \u03b1\nhx : x \u2208 s \u222a t\ny : \u03b1\nhy : y \u2208 s \u222a t\nhxy : x \u2264 y\nhxc : x < c\nxs : x \u2208 s\nhcy : c \u2264 y\n\u22a2 f x \u2264 f y", "state_after": "no goals"}, {"tactic": "have xt : x \u2208 t := B _ hx hcx", "annotated_tactic": ["have xt : x \u2208 t := B _ hx hcx", []], "state_before": "case inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : Preorder \u03b2\na : \u03b1\nf : \u03b1 \u2192 \u03b2\ns t : Set \u03b1\nc : \u03b1\nh\u2081 : MonotoneOn f s\nh\u2082 : MonotoneOn f t\nhs : IsGreatest s c\nht : IsLeast t c\nA : \u2200 (x : \u03b1), x \u2208 s \u222a t \u2192 x \u2264 c \u2192 x \u2208 s\nB : \u2200 (x : \u03b1), x \u2208 s \u222a t \u2192 c \u2264 x \u2192 x \u2208 t\nx : \u03b1\nhx : x \u2208 s \u222a t\ny : \u03b1\nhy : y \u2208 s \u222a t\nhxy : x \u2264 y\nhcx : c \u2264 x\n\u22a2 f x \u2264 f y", "state_after": "case inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : Preorder \u03b2\na : \u03b1\nf : \u03b1 \u2192 \u03b2\ns t : Set \u03b1\nc : \u03b1\nh\u2081 : MonotoneOn f s\nh\u2082 : MonotoneOn f t\nhs : IsGreatest s c\nht : IsLeast t c\nA : \u2200 (x : \u03b1), x \u2208 s \u222a t \u2192 x \u2264 c \u2192 x \u2208 s\nB : \u2200 (x : \u03b1), x \u2208 s \u222a t \u2192 c \u2264 x \u2192 x \u2208 t\nx : \u03b1\nhx : x \u2208 s \u222a t\ny : \u03b1\nhy : y \u2208 s \u222a t\nhxy : x \u2264 y\nhcx : c \u2264 x\nxt : x \u2208 t\n\u22a2 f x \u2264 f y"}, {"tactic": "have yt : y \u2208 t := B _ hy (hcx.trans hxy)", "annotated_tactic": ["have yt : y \u2208 t := B _ hy (hcx.trans hxy)", []], "state_before": "case inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : Preorder \u03b2\na : \u03b1\nf : \u03b1 \u2192 \u03b2\ns t : Set \u03b1\nc : \u03b1\nh\u2081 : MonotoneOn f s\nh\u2082 : MonotoneOn f t\nhs : IsGreatest s c\nht : IsLeast t c\nA : \u2200 (x : \u03b1), x \u2208 s \u222a t \u2192 x \u2264 c \u2192 x \u2208 s\nB : \u2200 (x : \u03b1), x \u2208 s \u222a t \u2192 c \u2264 x \u2192 x \u2208 t\nx : \u03b1\nhx : x \u2208 s \u222a t\ny : \u03b1\nhy : y \u2208 s \u222a t\nhxy : x \u2264 y\nhcx : c \u2264 x\nxt : x \u2208 t\n\u22a2 f x \u2264 f y", "state_after": "case inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : Preorder \u03b2\na : \u03b1\nf : \u03b1 \u2192 \u03b2\ns t : Set \u03b1\nc : \u03b1\nh\u2081 : MonotoneOn f s\nh\u2082 : MonotoneOn f t\nhs : IsGreatest s c\nht : IsLeast t c\nA : \u2200 (x : \u03b1), x \u2208 s \u222a t \u2192 x \u2264 c \u2192 x \u2208 s\nB : \u2200 (x : \u03b1), x \u2208 s \u222a t \u2192 c \u2264 x \u2192 x \u2208 t\nx : \u03b1\nhx : x \u2208 s \u222a t\ny : \u03b1\nhy : y \u2208 s \u222a t\nhxy : x \u2264 y\nhcx : c \u2264 x\nxt : x \u2208 t\nyt : y \u2208 t\n\u22a2 f x \u2264 f y"}, {"tactic": "exact h\u2082 xt yt hxy", "annotated_tactic": ["exact h\u2082 xt yt hxy", []], "state_before": "case inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : Preorder \u03b2\na : \u03b1\nf : \u03b1 \u2192 \u03b2\ns t : Set \u03b1\nc : \u03b1\nh\u2081 : MonotoneOn f s\nh\u2082 : MonotoneOn f t\nhs : IsGreatest s c\nht : IsLeast t c\nA : \u2200 (x : \u03b1), x \u2208 s \u222a t \u2192 x \u2264 c \u2192 x \u2208 s\nB : \u2200 (x : \u03b1), x \u2208 s \u222a t \u2192 c \u2264 x \u2192 x \u2208 t\nx : \u03b1\nhx : x \u2208 s \u222a t\ny : \u03b1\nhy : y \u2208 s \u222a t\nhxy : x \u2264 y\nhcx : c \u2264 x\nxt : x \u2208 t\nyt : y \u2208 t\n\u22a2 f x \u2264 f y", "state_after": "no goals"}]

https://github.com/leanprover-community/mathlib4

3ce43c18f614b76e161f911b75a3e1ef641620ff

Mathlib/LinearAlgebra/AffineSpace/AffineMap.lean

AffineMap.vsub_apply

[]

https://github.com/leanprover-community/mathlib4

3ce43c18f614b76e161f911b75a3e1ef641620ff

Mathlib/AlgebraicGeometry/Spec.lean

AlgebraicGeometry.StructureSheaf.isLocalizedModule_toPushforwardStalkAlgHom_aux

[{"tactic": "obtain \u27e8U, hp, s, e\u27e9 := TopCat.Presheaf.germ_exist\n (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf S).val) _ y", "annotated_tactic": ["obtain \u27e8U, hp, s, e\u27e9 := <a>TopCat.Presheaf.germ_exist</a>\n -- Porting note : originally the first variable does not need to be explicit\n (<a>Spec.topMap</a> (<a>algebraMap</a> \u2191R \u2191S) _* (<a>structureSheaf</a> S).<a>val</a>) _ y", [{"full_name": "TopCat.Presheaf.germ_exist", "def_path": "Mathlib/Topology/Sheaves/Stalks.lean", "def_pos": [415, 9], "def_end_pos": [415, 19]}, {"full_name": "AlgebraicGeometry.Spec.topMap", "def_path": "Mathlib/AlgebraicGeometry/Spec.lean", "def_pos": [61, 5], "def_end_pos": [61, 16]}, {"full_name": "algebraMap", "def_path": "Mathlib/Algebra/Algebra/Basic.lean", "def_pos": [125, 5], "def_end_pos": [125, 15]}, {"full_name": "AlgebraicGeometry.Spec.structureSheaf", "def_path": "Mathlib/AlgebraicGeometry/StructureSheaf.lean", "def_pos": [268, 5], "def_end_pos": [268, 24]}, {"full_name": "CategoryTheory.Sheaf.val", "def_path": "Mathlib/CategoryTheory/Sites/Sheaf.lean", "def_pos": [282, 3], "def_end_pos": [282, 6]}]], "state_before": "R S : CommRingCat\nf : R \u27f6 S\np : PrimeSpectrum \u2191R\ninst\u271d : Algebra \u2191R \u2191S\ny : \u2191(TopCat.Presheaf.stalk (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) p)\n\u22a2 \u2203 x, x.2 \u2022 y = \u2191(toPushforwardStalkAlgHom R S p) x.1", "state_after": "case intro.intro.intro\nR S : CommRingCat\nf : R \u27f6 S\np : PrimeSpectrum \u2191R\ninst\u271d : Algebra \u2191R \u2191S\ny : \u2191(TopCat.Presheaf.stalk (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) p)\nU : TopologicalSpace.Opens \u2191(Spec.topObj R)\nhp : p \u2208 U\ns : (forget CommRingCat).obj ((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).obj (op U))\ne :\n \u2191(TopCat.Presheaf.germ (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) { val := p, property := hp }) s = y\n\u22a2 \u2203 x, x.2 \u2022 y = \u2191(toPushforwardStalkAlgHom R S p) x.1"}, {"tactic": "obtain \u27e8_, \u27e8r, rfl\u27e9, hpr : p \u2208 PrimeSpectrum.basicOpen r, hrU : PrimeSpectrum.basicOpen r \u2264 U\u27e9 :=\n PrimeSpectrum.isTopologicalBasis_basic_opens.exists_subset_of_mem_open (show p \u2208 U from hp) U.2", "annotated_tactic": ["obtain \u27e8_, \u27e8r, rfl\u27e9, hpr : p \u2208 <a>PrimeSpectrum.basicOpen</a> r, hrU : <a>PrimeSpectrum.basicOpen</a> r \u2264 U\u27e9 :=\n PrimeSpectrum.isTopologicalBasis_basic_opens.exists_subset_of_mem_open (show p \u2208 U from hp) U.2", [{"full_name": "PrimeSpectrum.basicOpen", "def_path": "Mathlib/AlgebraicGeometry/PrimeSpectrum/Basic.lean", "def_pos": [752, 5], "def_end_pos": [752, 14]}, {"full_name": "PrimeSpectrum.basicOpen", "def_path": "Mathlib/AlgebraicGeometry/PrimeSpectrum/Basic.lean", "def_pos": [752, 5], "def_end_pos": [752, 14]}]], "state_before": "case intro.intro.intro\nR S : CommRingCat\nf : R \u27f6 S\np : PrimeSpectrum \u2191R\ninst\u271d : Algebra \u2191R \u2191S\ny : \u2191(TopCat.Presheaf.stalk (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) p)\nU : TopologicalSpace.Opens \u2191(Spec.topObj R)\nhp : p \u2208 U\ns : (forget CommRingCat).obj ((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).obj (op U))\ne :\n \u2191(TopCat.Presheaf.germ (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) { val := p, property := hp }) s = y\n\u22a2 \u2203 x, x.2 \u2022 y = \u2191(toPushforwardStalkAlgHom R S p) x.1", "state_after": "case intro.intro.intro.intro.intro.intro.intro\nR S : CommRingCat\nf : R \u27f6 S\np : PrimeSpectrum \u2191R\ninst\u271d : Algebra \u2191R \u2191S\ny : \u2191(TopCat.Presheaf.stalk (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) p)\nU : TopologicalSpace.Opens \u2191(Spec.topObj R)\nhp : p \u2208 U\ns : (forget CommRingCat).obj ((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).obj (op U))\ne :\n \u2191(TopCat.Presheaf.germ (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) { val := p, property := hp }) s = y\nr : \u2191R\nhpr : p \u2208 PrimeSpectrum.basicOpen r\nhrU : PrimeSpectrum.basicOpen r \u2264 U\n\u22a2 \u2203 x, x.2 \u2022 y = \u2191(toPushforwardStalkAlgHom R S p) x.1"}, {"tactic": "replace e :=\n ((Spec.topMap (algebraMap R S) _* (structureSheaf S).1).germ_res_apply (homOfLE hrU)\n \u27e8p, hpr\u27e9 _).trans e", "annotated_tactic": ["replace e :=\n ((<a>Spec.topMap</a> (<a>algebraMap</a> R S) _* (<a>structureSheaf</a> S).1).<a>germ_res_apply</a> (<a>homOfLE</a> hrU)\n \u27e8p, hpr\u27e9 _).<a>trans</a> e", [{"full_name": "AlgebraicGeometry.Spec.topMap", "def_path": "Mathlib/AlgebraicGeometry/Spec.lean", "def_pos": [61, 5], "def_end_pos": [61, 16]}, {"full_name": "algebraMap", "def_path": "Mathlib/Algebra/Algebra/Basic.lean", "def_pos": [125, 5], "def_end_pos": [125, 15]}, {"full_name": "AlgebraicGeometry.Spec.structureSheaf", "def_path": "Mathlib/AlgebraicGeometry/StructureSheaf.lean", "def_pos": [268, 5], "def_end_pos": [268, 24]}, {"full_name": "TopCat.Presheaf.germ_res_apply", "def_path": "Mathlib/Topology/Sheaves/Stalks.lean", "def_pos": [114, 9], "def_end_pos": [114, 23]}, {"full_name": "CategoryTheory.homOfLE", "def_path": "Mathlib/CategoryTheory/Category/Preorder.lean", "def_pos": [67, 5], "def_end_pos": [67, 12]}, {"full_name": "Eq.trans", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [322, 9], "def_end_pos": [322, 17]}]], "state_before": "case intro.intro.intro.intro.intro.intro.intro\nR S : CommRingCat\nf : R \u27f6 S\np : PrimeSpectrum \u2191R\ninst\u271d : Algebra \u2191R \u2191S\ny : \u2191(TopCat.Presheaf.stalk (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) p)\nU : TopologicalSpace.Opens \u2191(Spec.topObj R)\nhp : p \u2208 U\ns : (forget CommRingCat).obj ((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).obj (op U))\ne :\n \u2191(TopCat.Presheaf.germ (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) { val := p, property := hp }) s = y\nr : \u2191R\nhpr : p \u2208 PrimeSpectrum.basicOpen r\nhrU : PrimeSpectrum.basicOpen r \u2264 U\n\u22a2 \u2203 x, x.2 \u2022 y = \u2191(toPushforwardStalkAlgHom R S p) x.1", "state_after": "case intro.intro.intro.intro.intro.intro.intro\nR S : CommRingCat\nf : R \u27f6 S\np : PrimeSpectrum \u2191R\ninst\u271d : Algebra \u2191R \u2191S\ny : \u2191(TopCat.Presheaf.stalk (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) p)\nU : TopologicalSpace.Opens \u2191(Spec.topObj R)\nhp : p \u2208 U\ns : (forget CommRingCat).obj ((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).obj (op U))\nr : \u2191R\nhpr : p \u2208 PrimeSpectrum.basicOpen r\nhrU : PrimeSpectrum.basicOpen r \u2264 U\ne :\n \u2191(TopCat.Presheaf.germ (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) { val := p, property := hpr })\n (\u2191((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).map (homOfLE hrU).op) s) =\n y\n\u22a2 \u2203 x, x.2 \u2022 y = \u2191(toPushforwardStalkAlgHom R S p) x.1"}, {"tactic": "set s' := (Spec.topMap (algebraMap R S) _* (structureSheaf S).1).map (homOfLE hrU).op s with h", "annotated_tactic": ["set s' := (<a>Spec.topMap</a> (<a>algebraMap</a> R S) _* (<a>structureSheaf</a> S).1).<a>map</a> (<a>homOfLE</a> hrU).<a>op</a> s with h", [{"full_name": "AlgebraicGeometry.Spec.topMap", "def_path": "Mathlib/AlgebraicGeometry/Spec.lean", "def_pos": [61, 5], "def_end_pos": [61, 16]}, {"full_name": "algebraMap", "def_path": "Mathlib/Algebra/Algebra/Basic.lean", "def_pos": [125, 5], "def_end_pos": [125, 15]}, {"full_name": "AlgebraicGeometry.Spec.structureSheaf", "def_path": "Mathlib/AlgebraicGeometry/StructureSheaf.lean", "def_pos": [268, 5], "def_end_pos": [268, 24]}, {"full_name": "Prefunctor.map", "def_path": "Mathlib/Combinatorics/Quiver/Basic.lean", "def_pos": [64, 3], "def_end_pos": [64, 6]}, {"full_name": "CategoryTheory.homOfLE", "def_path": "Mathlib/CategoryTheory/Category/Preorder.lean", "def_pos": [67, 5], "def_end_pos": [67, 12]}, {"full_name": "Quiver.Hom.op", "def_path": "Mathlib/Combinatorics/Quiver/Basic.lean", "def_pos": [152, 5], "def_end_pos": [152, 11]}]], "state_before": "case intro.intro.intro.intro.intro.intro.intro\nR S : CommRingCat\nf : R \u27f6 S\np : PrimeSpectrum \u2191R\ninst\u271d : Algebra \u2191R \u2191S\ny : \u2191(TopCat.Presheaf.stalk (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) p)\nU : TopologicalSpace.Opens \u2191(Spec.topObj R)\nhp : p \u2208 U\ns : (forget CommRingCat).obj ((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).obj (op U))\nr : \u2191R\nhpr : p \u2208 PrimeSpectrum.basicOpen r\nhrU : PrimeSpectrum.basicOpen r \u2264 U\ne :\n \u2191(TopCat.Presheaf.germ (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) { val := p, property := hpr })\n (\u2191((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).map (homOfLE hrU).op) s) =\n y\n\u22a2 \u2203 x, x.2 \u2022 y = \u2191(toPushforwardStalkAlgHom R S p) x.1", "state_after": "case intro.intro.intro.intro.intro.intro.intro\nR S : CommRingCat\nf : R \u27f6 S\np : PrimeSpectrum \u2191R\ninst\u271d : Algebra \u2191R \u2191S\ny : \u2191(TopCat.Presheaf.stalk (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) p)\nU : TopologicalSpace.Opens \u2191(Spec.topObj R)\nhp : p \u2208 U\ns : (forget CommRingCat).obj ((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).obj (op U))\nr : \u2191R\nhpr : p \u2208 PrimeSpectrum.basicOpen r\nhrU : PrimeSpectrum.basicOpen r \u2264 U\ns' : (forget CommRingCat).obj\n ((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).obj (op (PrimeSpectrum.basicOpen r))) :=\n \u2191((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).map (homOfLE hrU).op) s\ne :\n \u2191(TopCat.Presheaf.germ (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) { val := p, property := hpr }) s' =\n y\nh : s' = \u2191((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).map (homOfLE hrU).op) s\n\u22a2 \u2203 x, x.2 \u2022 y = \u2191(toPushforwardStalkAlgHom R S p) x.1"}, {"tactic": "replace e : ((Spec.topMap (algebraMap R S) _* (structureSheaf S).val).germ \u27e8p, hpr\u27e9) s' = y", "annotated_tactic": ["replace e : ((<a>Spec.topMap</a> (<a>algebraMap</a> R S) _* (<a>structureSheaf</a> S).<a>val</a>).<a>germ</a> \u27e8p, hpr\u27e9) s' = y", [{"full_name": "AlgebraicGeometry.Spec.topMap", "def_path": "Mathlib/AlgebraicGeometry/Spec.lean", "def_pos": [61, 5], "def_end_pos": [61, 16]}, {"full_name": "algebraMap", "def_path": "Mathlib/Algebra/Algebra/Basic.lean", "def_pos": [125, 5], "def_end_pos": [125, 15]}, {"full_name": "AlgebraicGeometry.Spec.structureSheaf", "def_path": "Mathlib/AlgebraicGeometry/StructureSheaf.lean", "def_pos": [268, 5], "def_end_pos": [268, 24]}, {"full_name": "CategoryTheory.Sheaf.val", "def_path": "Mathlib/CategoryTheory/Sites/Sheaf.lean", "def_pos": [282, 3], "def_end_pos": [282, 6]}, {"full_name": "TopCat.Presheaf.germ", "def_path": "Mathlib/Topology/Sheaves/Stalks.lean", "def_pos": [101, 5], "def_end_pos": [101, 9]}]], "state_before": "case intro.intro.intro.intro.intro.intro.intro\nR S : CommRingCat\nf : R \u27f6 S\np : PrimeSpectrum \u2191R\ninst\u271d : Algebra \u2191R \u2191S\ny : \u2191(TopCat.Presheaf.stalk (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) p)\nU : TopologicalSpace.Opens \u2191(Spec.topObj R)\nhp : p \u2208 U\ns : (forget CommRingCat).obj ((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).obj (op U))\nr : \u2191R\nhpr : p \u2208 PrimeSpectrum.basicOpen r\nhrU : PrimeSpectrum.basicOpen r \u2264 U\ns' : (forget CommRingCat).obj\n ((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).obj (op (PrimeSpectrum.basicOpen r))) :=\n \u2191((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).map (homOfLE hrU).op) s\ne :\n \u2191(TopCat.Presheaf.germ (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) { val := p, property := hpr }) s' =\n y\nh : s' = \u2191((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).map (homOfLE hrU).op) s\n\u22a2 \u2203 x, x.2 \u2022 y = \u2191(toPushforwardStalkAlgHom R S p) x.1", "state_after": "case e\nR S : CommRingCat\nf : R \u27f6 S\np : PrimeSpectrum \u2191R\ninst\u271d : Algebra \u2191R \u2191S\ny : \u2191(TopCat.Presheaf.stalk (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) p)\nU : TopologicalSpace.Opens \u2191(Spec.topObj R)\nhp : p \u2208 U\ns : (forget CommRingCat).obj ((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).obj (op U))\nr : \u2191R\nhpr : p \u2208 PrimeSpectrum.basicOpen r\nhrU : PrimeSpectrum.basicOpen r \u2264 U\ns' : (forget CommRingCat).obj\n ((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).obj (op (PrimeSpectrum.basicOpen r))) :=\n \u2191((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).map (homOfLE hrU).op) s\ne :\n \u2191(TopCat.Presheaf.germ (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) { val := p, property := hpr }) s' =\n y\nh : s' = \u2191((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).map (homOfLE hrU).op) s\n\u22a2 \u2191(TopCat.Presheaf.germ (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) { val := p, property := hpr }) s' =\n y\n\ncase intro.intro.intro.intro.intro.intro.intro\nR S : CommRingCat\nf : R \u27f6 S\np : PrimeSpectrum \u2191R\ninst\u271d : Algebra \u2191R \u2191S\ny : \u2191(TopCat.Presheaf.stalk (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) p)\nU : TopologicalSpace.Opens \u2191(Spec.topObj R)\nhp : p \u2208 U\ns : (forget CommRingCat).obj ((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).obj (op U))\nr : \u2191R\nhpr : p \u2208 PrimeSpectrum.basicOpen r\nhrU : PrimeSpectrum.basicOpen r \u2264 U\ns' : (forget CommRingCat).obj\n ((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).obj (op (PrimeSpectrum.basicOpen r))) :=\n \u2191((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).map (homOfLE hrU).op) s\nh : s' = \u2191((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).map (homOfLE hrU).op) s\ne :\n \u2191(TopCat.Presheaf.germ (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) { val := p, property := hpr }) s' =\n y\n\u22a2 \u2203 x, x.2 \u2022 y = \u2191(toPushforwardStalkAlgHom R S p) x.1"}, {"tactic": "clear_value s'", "annotated_tactic": ["clear_value s'", []], "state_before": "case intro.intro.intro.intro.intro.intro.intro\nR S : CommRingCat\nf : R \u27f6 S\np : PrimeSpectrum \u2191R\ninst\u271d : Algebra \u2191R \u2191S\ny : \u2191(TopCat.Presheaf.stalk (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) p)\nU : TopologicalSpace.Opens \u2191(Spec.topObj R)\nhp : p \u2208 U\ns : (forget CommRingCat).obj ((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).obj (op U))\nr : \u2191R\nhpr : p \u2208 PrimeSpectrum.basicOpen r\nhrU : PrimeSpectrum.basicOpen r \u2264 U\ns' : (forget CommRingCat).obj\n ((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).obj (op (PrimeSpectrum.basicOpen r))) :=\n \u2191((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).map (homOfLE hrU).op) s\nh : s' = \u2191((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).map (homOfLE hrU).op) s\ne :\n \u2191(TopCat.Presheaf.germ (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) { val := p, property := hpr }) s' =\n y\n\u22a2 \u2203 x, x.2 \u2022 y = \u2191(toPushforwardStalkAlgHom R S p) x.1", "state_after": "case intro.intro.intro.intro.intro.intro.intro\nR S : CommRingCat\nf : R \u27f6 S\np : PrimeSpectrum \u2191R\ninst\u271d : Algebra \u2191R \u2191S\ny : \u2191(TopCat.Presheaf.stalk (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) p)\nU : TopologicalSpace.Opens \u2191(Spec.topObj R)\nhp : p \u2208 U\ns : (forget CommRingCat).obj ((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).obj (op U))\nr : \u2191R\nhpr : p \u2208 PrimeSpectrum.basicOpen r\nhrU : PrimeSpectrum.basicOpen r \u2264 U\ns' :\n (forget CommRingCat).obj\n ((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).obj (op (PrimeSpectrum.basicOpen r)))\nh : s' = \u2191((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).map (homOfLE hrU).op) s\ne :\n \u2191(TopCat.Presheaf.germ (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) { val := p, property := hpr }) s' =\n y\n\u22a2 \u2203 x, x.2 \u2022 y = \u2191(toPushforwardStalkAlgHom R S p) x.1"}, {"tactic": "clear! U", "annotated_tactic": ["clear! U", []], "state_before": "case intro.intro.intro.intro.intro.intro.intro\nR S : CommRingCat\nf : R \u27f6 S\np : PrimeSpectrum \u2191R\ninst\u271d : Algebra \u2191R \u2191S\ny : \u2191(TopCat.Presheaf.stalk (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) p)\nU : TopologicalSpace.Opens \u2191(Spec.topObj R)\nhp : p \u2208 U\ns : (forget CommRingCat).obj ((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).obj (op U))\nr : \u2191R\nhpr : p \u2208 PrimeSpectrum.basicOpen r\nhrU : PrimeSpectrum.basicOpen r \u2264 U\ns' :\n (forget CommRingCat).obj\n ((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).obj (op (PrimeSpectrum.basicOpen r)))\nh : s' = \u2191((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).map (homOfLE hrU).op) s\ne :\n \u2191(TopCat.Presheaf.germ (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) { val := p, property := hpr }) s' =\n y\n\u22a2 \u2203 x, x.2 \u2022 y = \u2191(toPushforwardStalkAlgHom R S p) x.1", "state_after": "case intro.intro.intro.intro.intro.intro.intro\nR S : CommRingCat\nf : R \u27f6 S\np : PrimeSpectrum \u2191R\ninst\u271d : Algebra \u2191R \u2191S\ny : \u2191(TopCat.Presheaf.stalk (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) p)\nr : \u2191R\nhpr : p \u2208 PrimeSpectrum.basicOpen r\ns' :\n (forget CommRingCat).obj\n ((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).obj (op (PrimeSpectrum.basicOpen r)))\ne :\n \u2191(TopCat.Presheaf.germ (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) { val := p, property := hpr }) s' =\n y\n\u22a2 \u2203 x, x.2 \u2022 y = \u2191(toPushforwardStalkAlgHom R S p) x.1"}, {"tactic": "obtain \u27e8\u27e8s, \u27e8_, n, rfl\u27e9\u27e9, hsn\u27e9 :=\n @IsLocalization.surj _ _ _ _ _ _\n (StructureSheaf.IsLocalization.to_basicOpen S <| algebraMap R S r) s'", "annotated_tactic": ["obtain \u27e8\u27e8s, \u27e8_, n, rfl\u27e9\u27e9, hsn\u27e9 :=\n @<a>IsLocalization.surj</a> _ _ _ _ _ _\n (<a>StructureSheaf.IsLocalization.to_basicOpen</a> S <| <a>algebraMap</a> R S r) s'", [{"full_name": "IsLocalization.surj", "def_path": "Mathlib/RingTheory/Localization/Basic.lean", "def_pos": [125, 9], "def_end_pos": [125, 13]}, {"full_name": "AlgebraicGeometry.StructureSheaf.IsLocalization.to_basicOpen", "def_path": "Mathlib/AlgebraicGeometry/StructureSheaf.lean", "def_pos": [969, 10], "def_end_pos": [969, 37]}, {"full_name": "algebraMap", "def_path": "Mathlib/Algebra/Algebra/Basic.lean", "def_pos": [125, 5], "def_end_pos": [125, 15]}]], "state_before": "case intro.intro.intro.intro.intro.intro.intro\nR S : CommRingCat\nf : R \u27f6 S\np : PrimeSpectrum \u2191R\ninst\u271d : Algebra \u2191R \u2191S\ny : \u2191(TopCat.Presheaf.stalk (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) p)\nr : \u2191R\nhpr : p \u2208 PrimeSpectrum.basicOpen r\ns' :\n (forget CommRingCat).obj\n ((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).obj (op (PrimeSpectrum.basicOpen r)))\ne :\n \u2191(TopCat.Presheaf.germ (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) { val := p, property := hpr }) s' =\n y\n\u22a2 \u2203 x, x.2 \u2022 y = \u2191(toPushforwardStalkAlgHom R S p) x.1", "state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.mk.mk.intro\nR S : CommRingCat\nf : R \u27f6 S\np : PrimeSpectrum \u2191R\ninst\u271d : Algebra \u2191R \u2191S\ny : \u2191(TopCat.Presheaf.stalk (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) p)\nr : \u2191R\nhpr : p \u2208 PrimeSpectrum.basicOpen r\ns' :\n (forget CommRingCat).obj\n ((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).obj (op (PrimeSpectrum.basicOpen r)))\ne :\n \u2191(TopCat.Presheaf.germ (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) { val := p, property := hpr }) s' =\n y\ns : \u2191S\nn : \u2115\nhsn :\n s' *\n \u2191(algebraMap \u2191S \u2191((structureSheaf \u2191S).val.obj (op (PrimeSpectrum.basicOpen (\u2191(algebraMap \u2191R \u2191S) r)))))\n \u2191(s,\n { val := (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) n,\n property :=\n (_ :\n \u2203 y,\n (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) y =\n (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) n) }).2 =\n \u2191(algebraMap \u2191S \u2191((structureSheaf \u2191S).val.obj (op (PrimeSpectrum.basicOpen (\u2191(algebraMap \u2191R \u2191S) r)))))\n (s,\n { val := (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) n,\n property :=\n (_ :\n \u2203 y,\n (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) y =\n (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) n) }).1\n\u22a2 \u2203 x, x.2 \u2022 y = \u2191(toPushforwardStalkAlgHom R S p) x.1"}, {"tactic": "refine' \u27e8\u27e8s, \u27e8r, hpr\u27e9 ^ n\u27e9, _\u27e9", "annotated_tactic": ["refine' \u27e8\u27e8s, \u27e8r, hpr\u27e9 ^ n\u27e9, _\u27e9", []], "state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.mk.mk.intro\nR S : CommRingCat\nf : R \u27f6 S\np : PrimeSpectrum \u2191R\ninst\u271d : Algebra \u2191R \u2191S\ny : \u2191(TopCat.Presheaf.stalk (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) p)\nr : \u2191R\nhpr : p \u2208 PrimeSpectrum.basicOpen r\ns' :\n (forget CommRingCat).obj\n ((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).obj (op (PrimeSpectrum.basicOpen r)))\ne :\n \u2191(TopCat.Presheaf.germ (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) { val := p, property := hpr }) s' =\n y\ns : \u2191S\nn : \u2115\nhsn :\n s' *\n \u2191(algebraMap \u2191S \u2191((structureSheaf \u2191S).val.obj (op (PrimeSpectrum.basicOpen (\u2191(algebraMap \u2191R \u2191S) r)))))\n \u2191(s,\n { val := (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) n,\n property :=\n (_ :\n \u2203 y,\n (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) y =\n (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) n) }).2 =\n \u2191(algebraMap \u2191S \u2191((structureSheaf \u2191S).val.obj (op (PrimeSpectrum.basicOpen (\u2191(algebraMap \u2191R \u2191S) r)))))\n (s,\n { val := (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) n,\n property :=\n (_ :\n \u2203 y,\n (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) y =\n (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) n) }).1\n\u22a2 \u2203 x, x.2 \u2022 y = \u2191(toPushforwardStalkAlgHom R S p) x.1", "state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.mk.mk.intro\nR S : CommRingCat\nf : R \u27f6 S\np : PrimeSpectrum \u2191R\ninst\u271d : Algebra \u2191R \u2191S\ny : \u2191(TopCat.Presheaf.stalk (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) p)\nr : \u2191R\nhpr : p \u2208 PrimeSpectrum.basicOpen r\ns' :\n (forget CommRingCat).obj\n ((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).obj (op (PrimeSpectrum.basicOpen r)))\ne :\n \u2191(TopCat.Presheaf.germ (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) { val := p, property := hpr }) s' =\n y\ns : \u2191S\nn : \u2115\nhsn :\n s' *\n \u2191(algebraMap \u2191S \u2191((structureSheaf \u2191S).val.obj (op (PrimeSpectrum.basicOpen (\u2191(algebraMap \u2191R \u2191S) r)))))\n \u2191(s,\n { val := (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) n,\n property :=\n (_ :\n \u2203 y,\n (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) y =\n (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) n) }).2 =\n \u2191(algebraMap \u2191S \u2191((structureSheaf \u2191S).val.obj (op (PrimeSpectrum.basicOpen (\u2191(algebraMap \u2191R \u2191S) r)))))\n (s,\n { val := (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) n,\n property :=\n (_ :\n \u2203 y,\n (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) y =\n (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) n) }).1\n\u22a2 (s, { val := r, property := hpr } ^ n).2 \u2022 y =\n \u2191(toPushforwardStalkAlgHom R S p) (s, { val := r, property := hpr } ^ n).1"}, {"tactic": "rw [Submonoid.smul_def, Algebra.smul_def, algebraMap_pushforward_stalk, toPushforwardStalk,\n comp_apply, comp_apply]", "annotated_tactic": ["rw [<a>Submonoid.smul_def</a>, <a>Algebra.smul_def</a>, <a>algebraMap_pushforward_stalk</a>, <a>toPushforwardStalk</a>,\n <a>comp_apply</a>, <a>comp_apply</a>]", [{"full_name": "Submonoid.smul_def", "def_path": "Mathlib/GroupTheory/Submonoid/Operations.lean", "def_pos": [1529, 9], "def_end_pos": [1529, 17]}, {"full_name": "Algebra.smul_def", "def_path": "Mathlib/Algebra/Algebra/Basic.lean", "def_pos": [347, 9], "def_end_pos": [347, 17]}, {"full_name": "AlgebraicGeometry.StructureSheaf.algebraMap_pushforward_stalk", "def_path": "Mathlib/AlgebraicGeometry/Spec.lean", "def_pos": [388, 9], "def_end_pos": [388, 37]}, {"full_name": "AlgebraicGeometry.StructureSheaf.toPushforwardStalk", "def_path": "Mathlib/AlgebraicGeometry/Spec.lean", "def_pos": [367, 5], "def_end_pos": [367, 23]}, {"full_name": "CategoryTheory.comp_apply", "def_path": "Mathlib/CategoryTheory/ConcreteCategory/Basic.lean", "def_pos": [137, 17], "def_end_pos": [137, 27]}, {"full_name": "CategoryTheory.comp_apply", "def_path": "Mathlib/CategoryTheory/ConcreteCategory/Basic.lean", "def_pos": [137, 17], "def_end_pos": [137, 27]}]], "state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.mk.mk.intro\nR S : CommRingCat\nf : R \u27f6 S\np : PrimeSpectrum \u2191R\ninst\u271d : Algebra \u2191R \u2191S\ny : \u2191(TopCat.Presheaf.stalk (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) p)\nr : \u2191R\nhpr : p \u2208 PrimeSpectrum.basicOpen r\ns' :\n (forget CommRingCat).obj\n ((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).obj (op (PrimeSpectrum.basicOpen r)))\ne :\n \u2191(TopCat.Presheaf.germ (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) { val := p, property := hpr }) s' =\n y\ns : \u2191S\nn : \u2115\nhsn :\n s' *\n \u2191(algebraMap \u2191S \u2191((structureSheaf \u2191S).val.obj (op (PrimeSpectrum.basicOpen (\u2191(algebraMap \u2191R \u2191S) r)))))\n \u2191(s,\n { val := (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) n,\n property :=\n (_ :\n \u2203 y,\n (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) y =\n (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) n) }).2 =\n \u2191(algebraMap \u2191S \u2191((structureSheaf \u2191S).val.obj (op (PrimeSpectrum.basicOpen (\u2191(algebraMap \u2191R \u2191S) r)))))\n (s,\n { val := (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) n,\n property :=\n (_ :\n \u2203 y,\n (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) y =\n (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) n) }).1\n\u22a2 (s, { val := r, property := hpr } ^ n).2 \u2022 y =\n \u2191(toPushforwardStalkAlgHom R S p) (s, { val := r, property := hpr } ^ n).1", "state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.mk.mk.intro\nR S : CommRingCat\nf : R \u27f6 S\np : PrimeSpectrum \u2191R\ninst\u271d : Algebra \u2191R \u2191S\ny : \u2191(TopCat.Presheaf.stalk (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) p)\nr : \u2191R\nhpr : p \u2208 PrimeSpectrum.basicOpen r\ns' :\n (forget CommRingCat).obj\n ((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).obj (op (PrimeSpectrum.basicOpen r)))\ne :\n \u2191(TopCat.Presheaf.germ (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) { val := p, property := hpr }) s' =\n y\ns : \u2191S\nn : \u2115\nhsn :\n s' *\n \u2191(algebraMap \u2191S \u2191((structureSheaf \u2191S).val.obj (op (PrimeSpectrum.basicOpen (\u2191(algebraMap \u2191R \u2191S) r)))))\n \u2191(s,\n { val := (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) n,\n property :=\n (_ :\n \u2203 y,\n (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) y =\n (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) n) }).2 =\n \u2191(algebraMap \u2191S \u2191((structureSheaf \u2191S).val.obj (op (PrimeSpectrum.basicOpen (\u2191(algebraMap \u2191R \u2191S) r)))))\n (s,\n { val := (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) n,\n property :=\n (_ :\n \u2203 y,\n (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) y =\n (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) n) }).1\n\u22a2 \u2191(TopCat.Presheaf.germ (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) { val := p, property := trivial })\n (\u2191(toOpen \u2191S \u22a4) (\u2191(algebraMap \u2191R \u2191S) \u2191(s, { val := r, property := hpr } ^ n).2)) *\n y =\n \u2191(toPushforwardStalkAlgHom R S p) (s, { val := r, property := hpr } ^ n).1"}, {"tactic": "iterate 2\n erw [\u2190 (Spec.topMap (algebraMap R S) _* (structureSheaf S).1).germ_res_apply (homOfLE le_top)\n \u27e8p, hpr\u27e9]", "annotated_tactic": ["iterate 2\n erw [\u2190 (<a>Spec.topMap</a> (<a>algebraMap</a> R S) _* (<a>structureSheaf</a> S).1).<a>germ_res_apply</a> (<a>homOfLE</a> <a>le_top</a>)\n \u27e8p, hpr\u27e9]", [{"full_name": "AlgebraicGeometry.Spec.topMap", "def_path": "Mathlib/AlgebraicGeometry/Spec.lean", "def_pos": [61, 5], "def_end_pos": [61, 16]}, {"full_name": "algebraMap", "def_path": "Mathlib/Algebra/Algebra/Basic.lean", "def_pos": [125, 5], "def_end_pos": [125, 15]}, {"full_name": "AlgebraicGeometry.Spec.structureSheaf", "def_path": "Mathlib/AlgebraicGeometry/StructureSheaf.lean", "def_pos": [268, 5], "def_end_pos": [268, 24]}, {"full_name": "TopCat.Presheaf.germ_res_apply", "def_path": "Mathlib/Topology/Sheaves/Stalks.lean", "def_pos": [114, 9], "def_end_pos": [114, 23]}, {"full_name": "CategoryTheory.homOfLE", "def_path": "Mathlib/CategoryTheory/Category/Preorder.lean", "def_pos": [67, 5], "def_end_pos": [67, 12]}, {"full_name": "le_top", "def_path": "Mathlib/Order/BoundedOrder.lean", "def_pos": [98, 9], "def_end_pos": [98, 15]}]], "state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.mk.mk.intro\nR S : CommRingCat\nf : R \u27f6 S\np : PrimeSpectrum \u2191R\ninst\u271d : Algebra \u2191R \u2191S\ny : \u2191(TopCat.Presheaf.stalk (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) p)\nr : \u2191R\nhpr : p \u2208 PrimeSpectrum.basicOpen r\ns' :\n (forget CommRingCat).obj\n ((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).obj (op (PrimeSpectrum.basicOpen r)))\ne :\n \u2191(TopCat.Presheaf.germ (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) { val := p, property := hpr }) s' =\n y\ns : \u2191S\nn : \u2115\nhsn :\n s' *\n \u2191(algebraMap \u2191S \u2191((structureSheaf \u2191S).val.obj (op (PrimeSpectrum.basicOpen (\u2191(algebraMap \u2191R \u2191S) r)))))\n \u2191(s,\n { val := (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) n,\n property :=\n (_ :\n \u2203 y,\n (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) y =\n (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) n) }).2 =\n \u2191(algebraMap \u2191S \u2191((structureSheaf \u2191S).val.obj (op (PrimeSpectrum.basicOpen (\u2191(algebraMap \u2191R \u2191S) r)))))\n (s,\n { val := (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) n,\n property :=\n (_ :\n \u2203 y,\n (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) y =\n (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) n) }).1\n\u22a2 \u2191(TopCat.Presheaf.germ (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) { val := p, property := trivial })\n (\u2191(toOpen \u2191S \u22a4) (\u2191(algebraMap \u2191R \u2191S) \u2191(s, { val := r, property := hpr } ^ n).2)) *\n y =\n \u2191(toPushforwardStalkAlgHom R S p) (s, { val := r, property := hpr } ^ n).1", "state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.mk.mk.intro\nR S : CommRingCat\nf : R \u27f6 S\np : PrimeSpectrum \u2191R\ninst\u271d : Algebra \u2191R \u2191S\ny : \u2191(TopCat.Presheaf.stalk (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) p)\nr : \u2191R\nhpr : p \u2208 PrimeSpectrum.basicOpen r\ns' :\n (forget CommRingCat).obj\n ((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).obj (op (PrimeSpectrum.basicOpen r)))\ne :\n \u2191(TopCat.Presheaf.germ (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) { val := p, property := hpr }) s' =\n y\ns : \u2191S\nn : \u2115\nhsn :\n s' *\n \u2191(algebraMap \u2191S \u2191((structureSheaf \u2191S).val.obj (op (PrimeSpectrum.basicOpen (\u2191(algebraMap \u2191R \u2191S) r)))))\n \u2191(s,\n { val := (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) n,\n property :=\n (_ :\n \u2203 y,\n (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) y =\n (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) n) }).2 =\n \u2191(algebraMap \u2191S \u2191((structureSheaf \u2191S).val.obj (op (PrimeSpectrum.basicOpen (\u2191(algebraMap \u2191R \u2191S) r)))))\n (s,\n { val := (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) n,\n property :=\n (_ :\n \u2203 y,\n (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) y =\n (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) n) }).1\n\u22a2 \u2191(TopCat.Presheaf.germ (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) { val := p, property := hpr })\n (\u2191((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).map\n (homOfLE (_ : PrimeSpectrum.basicOpen r \u2264 \u22a4)).op)\n (\u2191(toOpen \u2191S \u22a4) (\u2191(algebraMap \u2191R \u2191S) \u2191(s, { val := r, property := hpr } ^ n).2))) *\n y =\n \u2191(TopCat.Presheaf.germ (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) { val := p, property := hpr })\n (\u2191((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).map\n (homOfLE (_ : PrimeSpectrum.basicOpen r \u2264 \u22a4)).op)\n (\u2191(toOpen \u2191S \u22a4) (s, { val := r, property := hpr } ^ n).1))"}, {"tactic": "rw [\u2190 e]", "annotated_tactic": ["rw [\u2190 e]", []], "state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.mk.mk.intro\nR S : CommRingCat\nf : R \u27f6 S\np : PrimeSpectrum \u2191R\ninst\u271d : Algebra \u2191R \u2191S\ny : \u2191(TopCat.Presheaf.stalk (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) p)\nr : \u2191R\nhpr : p \u2208 PrimeSpectrum.basicOpen r\ns' :\n (forget CommRingCat).obj\n ((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).obj (op (PrimeSpectrum.basicOpen r)))\ne :\n \u2191(TopCat.Presheaf.germ (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) { val := p, property := hpr }) s' =\n y\ns : \u2191S\nn : \u2115\nhsn :\n s' *\n \u2191(algebraMap \u2191S \u2191((structureSheaf \u2191S).val.obj (op (PrimeSpectrum.basicOpen (\u2191(algebraMap \u2191R \u2191S) r)))))\n \u2191(s,\n { val := (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) n,\n property :=\n (_ :\n \u2203 y,\n (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) y =\n (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) n) }).2 =\n \u2191(algebraMap \u2191S \u2191((structureSheaf \u2191S).val.obj (op (PrimeSpectrum.basicOpen (\u2191(algebraMap \u2191R \u2191S) r)))))\n (s,\n { val := (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) n,\n property :=\n (_ :\n \u2203 y,\n (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) y =\n (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) n) }).1\n\u22a2 \u2191(TopCat.Presheaf.germ (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) { val := p, property := hpr })\n (\u2191((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).map\n (homOfLE (_ : PrimeSpectrum.basicOpen r \u2264 \u22a4)).op)\n (\u2191(toOpen \u2191S \u22a4) (\u2191(algebraMap \u2191R \u2191S) \u2191(s, { val := r, property := hpr } ^ n).2))) *\n y =\n \u2191(TopCat.Presheaf.germ (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) { val := p, property := hpr })\n (\u2191((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).map\n (homOfLE (_ : PrimeSpectrum.basicOpen r \u2264 \u22a4)).op)\n (\u2191(toOpen \u2191S \u22a4) (s, { val := r, property := hpr } ^ n).1))", "state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.mk.mk.intro\nR S : CommRingCat\nf : R \u27f6 S\np : PrimeSpectrum \u2191R\ninst\u271d : Algebra \u2191R \u2191S\ny : \u2191(TopCat.Presheaf.stalk (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) p)\nr : \u2191R\nhpr : p \u2208 PrimeSpectrum.basicOpen r\ns' :\n (forget CommRingCat).obj\n ((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).obj (op (PrimeSpectrum.basicOpen r)))\ne :\n \u2191(TopCat.Presheaf.germ (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) { val := p, property := hpr }) s' =\n y\ns : \u2191S\nn : \u2115\nhsn :\n s' *\n \u2191(algebraMap \u2191S \u2191((structureSheaf \u2191S).val.obj (op (PrimeSpectrum.basicOpen (\u2191(algebraMap \u2191R \u2191S) r)))))\n \u2191(s,\n { val := (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) n,\n property :=\n (_ :\n \u2203 y,\n (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) y =\n (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) n) }).2 =\n \u2191(algebraMap \u2191S \u2191((structureSheaf \u2191S).val.obj (op (PrimeSpectrum.basicOpen (\u2191(algebraMap \u2191R \u2191S) r)))))\n (s,\n { val := (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) n,\n property :=\n (_ :\n \u2203 y,\n (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) y =\n (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) n) }).1\n\u22a2 \u2191(TopCat.Presheaf.germ (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) { val := p, property := hpr })\n (\u2191((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).map\n (homOfLE (_ : PrimeSpectrum.basicOpen r \u2264 \u22a4)).op)\n (\u2191(toOpen \u2191S \u22a4) (\u2191(algebraMap \u2191R \u2191S) \u2191(s, { val := r, property := hpr } ^ n).2))) *\n \u2191(TopCat.Presheaf.germ (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) { val := p, property := hpr })\n s' =\n \u2191(TopCat.Presheaf.germ (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) { val := p, property := hpr })\n (\u2191((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).map\n (homOfLE (_ : PrimeSpectrum.basicOpen r \u2264 \u22a4)).op)\n (\u2191(toOpen \u2191S \u22a4) (s, { val := r, property := hpr } ^ n).1))"}, {"tactic": "let f := TopCat.Presheaf.germ (Spec.topMap (algebraMap R S) _* (structureSheaf S).val) \u27e8p, hpr\u27e9", "annotated_tactic": ["let f := <a>TopCat.Presheaf.germ</a> (<a>Spec.topMap</a> (<a>algebraMap</a> R S) _* (<a>structureSheaf</a> S).<a>val</a>) \u27e8p, hpr\u27e9", [{"full_name": "TopCat.Presheaf.germ", "def_path": "Mathlib/Topology/Sheaves/Stalks.lean", "def_pos": [101, 5], "def_end_pos": [101, 9]}, {"full_name": "AlgebraicGeometry.Spec.topMap", "def_path": "Mathlib/AlgebraicGeometry/Spec.lean", "def_pos": [61, 5], "def_end_pos": [61, 16]}, {"full_name": "algebraMap", "def_path": "Mathlib/Algebra/Algebra/Basic.lean", "def_pos": [125, 5], "def_end_pos": [125, 15]}, {"full_name": "AlgebraicGeometry.Spec.structureSheaf", "def_path": "Mathlib/AlgebraicGeometry/StructureSheaf.lean", "def_pos": [268, 5], "def_end_pos": [268, 24]}, {"full_name": "CategoryTheory.Sheaf.val", "def_path": "Mathlib/CategoryTheory/Sites/Sheaf.lean", "def_pos": [282, 3], "def_end_pos": [282, 6]}]], "state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.mk.mk.intro\nR S : CommRingCat\nf : R \u27f6 S\np : PrimeSpectrum \u2191R\ninst\u271d : Algebra \u2191R \u2191S\ny : \u2191(TopCat.Presheaf.stalk (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) p)\nr : \u2191R\nhpr : p \u2208 PrimeSpectrum.basicOpen r\ns' :\n (forget CommRingCat).obj\n ((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).obj (op (PrimeSpectrum.basicOpen r)))\ne :\n \u2191(TopCat.Presheaf.germ (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) { val := p, property := hpr }) s' =\n y\ns : \u2191S\nn : \u2115\nhsn :\n s' *\n \u2191(algebraMap \u2191S \u2191((structureSheaf \u2191S).val.obj (op (PrimeSpectrum.basicOpen (\u2191(algebraMap \u2191R \u2191S) r)))))\n \u2191(s,\n { val := (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) n,\n property :=\n (_ :\n \u2203 y,\n (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) y =\n (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) n) }).2 =\n \u2191(algebraMap \u2191S \u2191((structureSheaf \u2191S).val.obj (op (PrimeSpectrum.basicOpen (\u2191(algebraMap \u2191R \u2191S) r)))))\n (s,\n { val := (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) n,\n property :=\n (_ :\n \u2203 y,\n (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) y =\n (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) n) }).1\n\u22a2 \u2191(TopCat.Presheaf.germ (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) { val := p, property := hpr })\n (\u2191((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).map\n (homOfLE (_ : PrimeSpectrum.basicOpen r \u2264 \u22a4)).op)\n (\u2191(toOpen \u2191S \u22a4) (\u2191(algebraMap \u2191R \u2191S) \u2191(s, { val := r, property := hpr } ^ n).2))) *\n \u2191(TopCat.Presheaf.germ (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) { val := p, property := hpr })\n s' =\n \u2191(TopCat.Presheaf.germ (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) { val := p, property := hpr })\n (\u2191((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).map\n (homOfLE (_ : PrimeSpectrum.basicOpen r \u2264 \u22a4)).op)\n (\u2191(toOpen \u2191S \u22a4) (s, { val := r, property := hpr } ^ n).1))", "state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.mk.mk.intro\nR S : CommRingCat\nf\u271d : R \u27f6 S\np : PrimeSpectrum \u2191R\ninst\u271d : Algebra \u2191R \u2191S\ny : \u2191(TopCat.Presheaf.stalk (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) p)\nr : \u2191R\nhpr : p \u2208 PrimeSpectrum.basicOpen r\ns' :\n (forget CommRingCat).obj\n ((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).obj (op (PrimeSpectrum.basicOpen r)))\ne :\n \u2191(TopCat.Presheaf.germ (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) { val := p, property := hpr }) s' =\n y\ns : \u2191S\nn : \u2115\nhsn :\n s' *\n \u2191(algebraMap \u2191S \u2191((structureSheaf \u2191S).val.obj (op (PrimeSpectrum.basicOpen (\u2191(algebraMap \u2191R \u2191S) r)))))\n \u2191(s,\n { val := (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) n,\n property :=\n (_ :\n \u2203 y,\n (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) y =\n (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) n) }).2 =\n \u2191(algebraMap \u2191S \u2191((structureSheaf \u2191S).val.obj (op (PrimeSpectrum.basicOpen (\u2191(algebraMap \u2191R \u2191S) r)))))\n (s,\n { val := (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) n,\n property :=\n (_ :\n \u2203 y,\n (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) y =\n (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) n) }).1\nf : (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).obj (op (PrimeSpectrum.basicOpen r)) \u27f6\n TopCat.Presheaf.stalk (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) \u2191{ val := p, property := hpr } :=\n TopCat.Presheaf.germ (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) { val := p, property := hpr }\n\u22a2 \u2191(TopCat.Presheaf.germ (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) { val := p, property := hpr })\n (\u2191((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).map\n (homOfLE (_ : PrimeSpectrum.basicOpen r \u2264 \u22a4)).op)\n (\u2191(toOpen \u2191S \u22a4) (\u2191(algebraMap \u2191R \u2191S) \u2191(s, { val := r, property := hpr } ^ n).2))) *\n \u2191(TopCat.Presheaf.germ (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) { val := p, property := hpr })\n s' =\n \u2191(TopCat.Presheaf.germ (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) { val := p, property := hpr })\n (\u2191((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).map\n (homOfLE (_ : PrimeSpectrum.basicOpen r \u2264 \u22a4)).op)\n (\u2191(toOpen \u2191S \u22a4) (s, { val := r, property := hpr } ^ n).1))"}, {"tactic": "change f _ * f _ = f _", "annotated_tactic": ["change f _ * f _ = f _", []], "state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.mk.mk.intro\nR S : CommRingCat\nf\u271d : R \u27f6 S\np : PrimeSpectrum \u2191R\ninst\u271d : Algebra \u2191R \u2191S\ny : \u2191(TopCat.Presheaf.stalk (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) p)\nr : \u2191R\nhpr : p \u2208 PrimeSpectrum.basicOpen r\ns' :\n (forget CommRingCat).obj\n ((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).obj (op (PrimeSpectrum.basicOpen r)))\ne :\n \u2191(TopCat.Presheaf.germ (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) { val := p, property := hpr }) s' =\n y\ns : \u2191S\nn : \u2115\nhsn :\n s' *\n \u2191(algebraMap \u2191S \u2191((structureSheaf \u2191S).val.obj (op (PrimeSpectrum.basicOpen (\u2191(algebraMap \u2191R \u2191S) r)))))\n \u2191(s,\n { val := (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) n,\n property :=\n (_ :\n \u2203 y,\n (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) y =\n (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) n) }).2 =\n \u2191(algebraMap \u2191S \u2191((structureSheaf \u2191S).val.obj (op (PrimeSpectrum.basicOpen (\u2191(algebraMap \u2191R \u2191S) r)))))\n (s,\n { val := (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) n,\n property :=\n (_ :\n \u2203 y,\n (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) y =\n (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) n) }).1\nf : (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).obj (op (PrimeSpectrum.basicOpen r)) \u27f6\n TopCat.Presheaf.stalk (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) \u2191{ val := p, property := hpr } :=\n TopCat.Presheaf.germ (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) { val := p, property := hpr }\n\u22a2 \u2191(TopCat.Presheaf.germ (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) { val := p, property := hpr })\n (\u2191((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).map\n (homOfLE (_ : PrimeSpectrum.basicOpen r \u2264 \u22a4)).op)\n (\u2191(toOpen \u2191S \u22a4) (\u2191(algebraMap \u2191R \u2191S) \u2191(s, { val := r, property := hpr } ^ n).2))) *\n \u2191(TopCat.Presheaf.germ (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) { val := p, property := hpr })\n s' =\n \u2191(TopCat.Presheaf.germ (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) { val := p, property := hpr })\n (\u2191((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).map\n (homOfLE (_ : PrimeSpectrum.basicOpen r \u2264 \u22a4)).op)\n (\u2191(toOpen \u2191S \u22a4) (s, { val := r, property := hpr } ^ n).1))", "state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.mk.mk.intro\nR S : CommRingCat\nf\u271d : R \u27f6 S\np : PrimeSpectrum \u2191R\ninst\u271d : Algebra \u2191R \u2191S\ny : \u2191(TopCat.Presheaf.stalk (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) p)\nr : \u2191R\nhpr : p \u2208 PrimeSpectrum.basicOpen r\ns' :\n (forget CommRingCat).obj\n ((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).obj (op (PrimeSpectrum.basicOpen r)))\ne :\n \u2191(TopCat.Presheaf.germ (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) { val := p, property := hpr }) s' =\n y\ns : \u2191S\nn : \u2115\nhsn :\n s' *\n \u2191(algebraMap \u2191S \u2191((structureSheaf \u2191S).val.obj (op (PrimeSpectrum.basicOpen (\u2191(algebraMap \u2191R \u2191S) r)))))\n \u2191(s,\n { val := (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) n,\n property :=\n (_ :\n \u2203 y,\n (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) y =\n (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) n) }).2 =\n \u2191(algebraMap \u2191S \u2191((structureSheaf \u2191S).val.obj (op (PrimeSpectrum.basicOpen (\u2191(algebraMap \u2191R \u2191S) r)))))\n (s,\n { val := (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) n,\n property :=\n (_ :\n \u2203 y,\n (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) y =\n (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) n) }).1\nf : (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).obj (op (PrimeSpectrum.basicOpen r)) \u27f6\n TopCat.Presheaf.stalk (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) \u2191{ val := p, property := hpr } :=\n TopCat.Presheaf.germ (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) { val := p, property := hpr }\n\u22a2 \u2191f\n (\u2191((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).map\n (homOfLE (_ : PrimeSpectrum.basicOpen r \u2264 \u22a4)).op)\n (\u2191(toOpen \u2191S \u22a4) (\u2191(algebraMap \u2191R \u2191S) \u2191(s, { val := r, property := hpr } ^ n).2))) *\n \u2191f s' =\n \u2191f\n (\u2191((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).map\n (homOfLE (_ : PrimeSpectrum.basicOpen r \u2264 \u22a4)).op)\n (\u2191(toOpen \u2191S \u22a4) (s, { val := r, property := hpr } ^ n).1))"}, {"tactic": "rw [\u2190 map_mul, mul_comm]", "annotated_tactic": ["rw [\u2190 <a>map_mul</a>, <a>mul_comm</a>]", [{"full_name": "map_mul", "def_path": "Mathlib/Algebra/Hom/Group/Defs.lean", "def_pos": [299, 9], "def_end_pos": [299, 16]}, {"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [302, 9], "def_end_pos": [302, 17]}]], "state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.mk.mk.intro\nR S : CommRingCat\nf\u271d : R \u27f6 S\np : PrimeSpectrum \u2191R\ninst\u271d : Algebra \u2191R \u2191S\ny : \u2191(TopCat.Presheaf.stalk (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) p)\nr : \u2191R\nhpr : p \u2208 PrimeSpectrum.basicOpen r\ns' :\n (forget CommRingCat).obj\n ((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).obj (op (PrimeSpectrum.basicOpen r)))\ne :\n \u2191(TopCat.Presheaf.germ (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) { val := p, property := hpr }) s' =\n y\ns : \u2191S\nn : \u2115\nhsn :\n s' *\n \u2191(algebraMap \u2191S \u2191((structureSheaf \u2191S).val.obj (op (PrimeSpectrum.basicOpen (\u2191(algebraMap \u2191R \u2191S) r)))))\n \u2191(s,\n { val := (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) n,\n property :=\n (_ :\n \u2203 y,\n (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) y =\n (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) n) }).2 =\n \u2191(algebraMap \u2191S \u2191((structureSheaf \u2191S).val.obj (op (PrimeSpectrum.basicOpen (\u2191(algebraMap \u2191R \u2191S) r)))))\n (s,\n { val := (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) n,\n property :=\n (_ :\n \u2203 y,\n (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) y =\n (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) n) }).1\nf : (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).obj (op (PrimeSpectrum.basicOpen r)) \u27f6\n TopCat.Presheaf.stalk (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) \u2191{ val := p, property := hpr } :=\n TopCat.Presheaf.germ (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) { val := p, property := hpr }\n\u22a2 \u2191f\n (\u2191((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).map\n (homOfLE (_ : PrimeSpectrum.basicOpen r \u2264 \u22a4)).op)\n (\u2191(toOpen \u2191S \u22a4) (\u2191(algebraMap \u2191R \u2191S) \u2191(s, { val := r, property := hpr } ^ n).2))) *\n \u2191f s' =\n \u2191f\n (\u2191((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).map\n (homOfLE (_ : PrimeSpectrum.basicOpen r \u2264 \u22a4)).op)\n (\u2191(toOpen \u2191S \u22a4) (s, { val := r, property := hpr } ^ n).1))", "state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.mk.mk.intro\nR S : CommRingCat\nf\u271d : R \u27f6 S\np : PrimeSpectrum \u2191R\ninst\u271d : Algebra \u2191R \u2191S\ny : \u2191(TopCat.Presheaf.stalk (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) p)\nr : \u2191R\nhpr : p \u2208 PrimeSpectrum.basicOpen r\ns' :\n (forget CommRingCat).obj\n ((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).obj (op (PrimeSpectrum.basicOpen r)))\ne :\n \u2191(TopCat.Presheaf.germ (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) { val := p, property := hpr }) s' =\n y\ns : \u2191S\nn : \u2115\nhsn :\n s' *\n \u2191(algebraMap \u2191S \u2191((structureSheaf \u2191S).val.obj (op (PrimeSpectrum.basicOpen (\u2191(algebraMap \u2191R \u2191S) r)))))\n \u2191(s,\n { val := (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) n,\n property :=\n (_ :\n \u2203 y,\n (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) y =\n (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) n) }).2 =\n \u2191(algebraMap \u2191S \u2191((structureSheaf \u2191S).val.obj (op (PrimeSpectrum.basicOpen (\u2191(algebraMap \u2191R \u2191S) r)))))\n (s,\n { val := (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) n,\n property :=\n (_ :\n \u2203 y,\n (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) y =\n (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) n) }).1\nf : (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).obj (op (PrimeSpectrum.basicOpen r)) \u27f6\n TopCat.Presheaf.stalk (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) \u2191{ val := p, property := hpr } :=\n TopCat.Presheaf.germ (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) { val := p, property := hpr }\n\u22a2 \u2191f\n (s' *\n \u2191((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).map\n (homOfLE (_ : PrimeSpectrum.basicOpen r \u2264 \u22a4)).op)\n (\u2191(toOpen \u2191S \u22a4) (\u2191(algebraMap \u2191R \u2191S) \u2191(s, { val := r, property := hpr } ^ n).2))) =\n \u2191f\n (\u2191((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).map\n (homOfLE (_ : PrimeSpectrum.basicOpen r \u2264 \u22a4)).op)\n (\u2191(toOpen \u2191S \u22a4) (s, { val := r, property := hpr } ^ n).1))"}, {"tactic": "dsimp only [Subtype.coe_mk] at hsn", "annotated_tactic": ["dsimp only [<a>Subtype.coe_mk</a>] at hsn", [{"full_name": "Subtype.coe_mk", "def_path": "Mathlib/Data/Subtype.lean", "def_pos": [99, 9], "def_end_pos": [99, 15]}]], "state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.mk.mk.intro\nR S : CommRingCat\nf\u271d : R \u27f6 S\np : PrimeSpectrum \u2191R\ninst\u271d : Algebra \u2191R \u2191S\ny : \u2191(TopCat.Presheaf.stalk (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) p)\nr : \u2191R\nhpr : p \u2208 PrimeSpectrum.basicOpen r\ns' :\n (forget CommRingCat).obj\n ((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).obj (op (PrimeSpectrum.basicOpen r)))\ne :\n \u2191(TopCat.Presheaf.germ (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) { val := p, property := hpr }) s' =\n y\ns : \u2191S\nn : \u2115\nhsn :\n s' *\n \u2191(algebraMap \u2191S \u2191((structureSheaf \u2191S).val.obj (op (PrimeSpectrum.basicOpen (\u2191(algebraMap \u2191R \u2191S) r)))))\n \u2191(s,\n { val := (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) n,\n property :=\n (_ :\n \u2203 y,\n (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) y =\n (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) n) }).2 =\n \u2191(algebraMap \u2191S \u2191((structureSheaf \u2191S).val.obj (op (PrimeSpectrum.basicOpen (\u2191(algebraMap \u2191R \u2191S) r)))))\n (s,\n { val := (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) n,\n property :=\n (_ :\n \u2203 y,\n (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) y =\n (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) n) }).1\nf : (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).obj (op (PrimeSpectrum.basicOpen r)) \u27f6\n TopCat.Presheaf.stalk (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) \u2191{ val := p, property := hpr } :=\n TopCat.Presheaf.germ (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) { val := p, property := hpr }\n\u22a2 \u2191f\n (s' *\n \u2191((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).map\n (homOfLE (_ : PrimeSpectrum.basicOpen r \u2264 \u22a4)).op)\n (\u2191(toOpen \u2191S \u22a4) (\u2191(algebraMap \u2191R \u2191S) \u2191(s, { val := r, property := hpr } ^ n).2))) =\n \u2191f\n (\u2191((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).map\n (homOfLE (_ : PrimeSpectrum.basicOpen r \u2264 \u22a4)).op)\n (\u2191(toOpen \u2191S \u22a4) (s, { val := r, property := hpr } ^ n).1))", "state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.mk.mk.intro\nR S : CommRingCat\nf\u271d : R \u27f6 S\np : PrimeSpectrum \u2191R\ninst\u271d : Algebra \u2191R \u2191S\ny : \u2191(TopCat.Presheaf.stalk (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) p)\nr : \u2191R\nhpr : p \u2208 PrimeSpectrum.basicOpen r\ns' :\n (forget CommRingCat).obj\n ((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).obj (op (PrimeSpectrum.basicOpen r)))\ne :\n \u2191(TopCat.Presheaf.germ (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) { val := p, property := hpr }) s' =\n y\ns : \u2191S\nn : \u2115\nhsn :\n s' *\n \u2191(algebraMap \u2191S \u2191((structureSheaf \u2191S).val.obj (op (PrimeSpectrum.basicOpen (\u2191(algebraMap \u2191R \u2191S) r)))))\n (\u2191(algebraMap \u2191R \u2191S) r ^ n) =\n \u2191(algebraMap \u2191S \u2191((structureSheaf \u2191S).val.obj (op (PrimeSpectrum.basicOpen (\u2191(algebraMap \u2191R \u2191S) r))))) s\nf : (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).obj (op (PrimeSpectrum.basicOpen r)) \u27f6\n TopCat.Presheaf.stalk (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) \u2191{ val := p, property := hpr } :=\n TopCat.Presheaf.germ (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) { val := p, property := hpr }\n\u22a2 \u2191f\n (s' *\n \u2191((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).map\n (homOfLE (_ : PrimeSpectrum.basicOpen r \u2264 \u22a4)).op)\n (\u2191(toOpen \u2191S \u22a4) (\u2191(algebraMap \u2191R \u2191S) \u2191(s, { val := r, property := hpr } ^ n).2))) =\n \u2191f\n (\u2191((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).map\n (homOfLE (_ : PrimeSpectrum.basicOpen r \u2264 \u22a4)).op)\n (\u2191(toOpen \u2191S \u22a4) (s, { val := r, property := hpr } ^ n).1))"}, {"tactic": "rw [\u2190 map_pow (algebraMap R S)] at hsn", "annotated_tactic": ["rw [\u2190 <a>map_pow</a> (<a>algebraMap</a> R S)] at hsn", [{"full_name": "map_pow", "def_path": "Mathlib/Algebra/Hom/Group/Defs.lean", "def_pos": [435, 9], "def_end_pos": [435, 16]}, {"full_name": "algebraMap", "def_path": "Mathlib/Algebra/Algebra/Basic.lean", "def_pos": [125, 5], "def_end_pos": [125, 15]}]], "state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.mk.mk.intro\nR S : CommRingCat\nf\u271d : R \u27f6 S\np : PrimeSpectrum \u2191R\ninst\u271d : Algebra \u2191R \u2191S\ny : \u2191(TopCat.Presheaf.stalk (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) p)\nr : \u2191R\nhpr : p \u2208 PrimeSpectrum.basicOpen r\ns' :\n (forget CommRingCat).obj\n ((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).obj (op (PrimeSpectrum.basicOpen r)))\ne :\n \u2191(TopCat.Presheaf.germ (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) { val := p, property := hpr }) s' =\n y\ns : \u2191S\nn : \u2115\nhsn :\n s' *\n \u2191(algebraMap \u2191S \u2191((structureSheaf \u2191S).val.obj (op (PrimeSpectrum.basicOpen (\u2191(algebraMap \u2191R \u2191S) r)))))\n (\u2191(algebraMap \u2191R \u2191S) r ^ n) =\n \u2191(algebraMap \u2191S \u2191((structureSheaf \u2191S).val.obj (op (PrimeSpectrum.basicOpen (\u2191(algebraMap \u2191R \u2191S) r))))) s\nf : (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).obj (op (PrimeSpectrum.basicOpen r)) \u27f6\n TopCat.Presheaf.stalk (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) \u2191{ val := p, property := hpr } :=\n TopCat.Presheaf.germ (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) { val := p, property := hpr }\n\u22a2 \u2191f\n (s' *\n \u2191((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).map\n (homOfLE (_ : PrimeSpectrum.basicOpen r \u2264 \u22a4)).op)\n (\u2191(toOpen \u2191S \u22a4) (\u2191(algebraMap \u2191R \u2191S) \u2191(s, { val := r, property := hpr } ^ n).2))) =\n \u2191f\n (\u2191((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).map\n (homOfLE (_ : PrimeSpectrum.basicOpen r \u2264 \u22a4)).op)\n (\u2191(toOpen \u2191S \u22a4) (s, { val := r, property := hpr } ^ n).1))", "state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.mk.mk.intro\nR S : CommRingCat\nf\u271d : R \u27f6 S\np : PrimeSpectrum \u2191R\ninst\u271d : Algebra \u2191R \u2191S\ny : \u2191(TopCat.Presheaf.stalk (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) p)\nr : \u2191R\nhpr : p \u2208 PrimeSpectrum.basicOpen r\ns' :\n (forget CommRingCat).obj\n ((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).obj (op (PrimeSpectrum.basicOpen r)))\ne :\n \u2191(TopCat.Presheaf.germ (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) { val := p, property := hpr }) s' =\n y\ns : \u2191S\nn : \u2115\nhsn :\n s' *\n \u2191(algebraMap \u2191S \u2191((structureSheaf \u2191S).val.obj (op (PrimeSpectrum.basicOpen (\u2191(algebraMap \u2191R \u2191S) r)))))\n (\u2191(algebraMap \u2191R \u2191S) (r ^ n)) =\n \u2191(algebraMap \u2191S \u2191((structureSheaf \u2191S).val.obj (op (PrimeSpectrum.basicOpen (\u2191(algebraMap \u2191R \u2191S) r))))) s\nf : (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).obj (op (PrimeSpectrum.basicOpen r)) \u27f6\n TopCat.Presheaf.stalk (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) \u2191{ val := p, property := hpr } :=\n TopCat.Presheaf.germ (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) { val := p, property := hpr }\n\u22a2 \u2191f\n (s' *\n \u2191((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).map\n (homOfLE (_ : PrimeSpectrum.basicOpen r \u2264 \u22a4)).op)\n (\u2191(toOpen \u2191S \u22a4) (\u2191(algebraMap \u2191R \u2191S) \u2191(s, { val := r, property := hpr } ^ n).2))) =\n \u2191f\n (\u2191((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).map\n (homOfLE (_ : PrimeSpectrum.basicOpen r \u2264 \u22a4)).op)\n (\u2191(toOpen \u2191S \u22a4) (s, { val := r, property := hpr } ^ n).1))"}, {"tactic": "congr 1", "annotated_tactic": ["congr 1", []], "state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.mk.mk.intro\nR S : CommRingCat\nf\u271d : R \u27f6 S\np : PrimeSpectrum \u2191R\ninst\u271d : Algebra \u2191R \u2191S\ny : \u2191(TopCat.Presheaf.stalk (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) p)\nr : \u2191R\nhpr : p \u2208 PrimeSpectrum.basicOpen r\ns' :\n (forget CommRingCat).obj\n ((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).obj (op (PrimeSpectrum.basicOpen r)))\ne :\n \u2191(TopCat.Presheaf.germ (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) { val := p, property := hpr }) s' =\n y\ns : \u2191S\nn : \u2115\nhsn :\n s' *\n \u2191(algebraMap \u2191S \u2191((structureSheaf \u2191S).val.obj (op (PrimeSpectrum.basicOpen (\u2191(algebraMap \u2191R \u2191S) r)))))\n (\u2191(algebraMap \u2191R \u2191S) (r ^ n)) =\n \u2191(algebraMap \u2191S \u2191((structureSheaf \u2191S).val.obj (op (PrimeSpectrum.basicOpen (\u2191(algebraMap \u2191R \u2191S) r))))) s\nf : (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).obj (op (PrimeSpectrum.basicOpen r)) \u27f6\n TopCat.Presheaf.stalk (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) \u2191{ val := p, property := hpr } :=\n TopCat.Presheaf.germ (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) { val := p, property := hpr }\n\u22a2 \u2191f\n (s' *\n \u2191((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).map\n (homOfLE (_ : PrimeSpectrum.basicOpen r \u2264 \u22a4)).op)\n (\u2191(toOpen \u2191S \u22a4) (\u2191(algebraMap \u2191R \u2191S) \u2191(s, { val := r, property := hpr } ^ n).2))) =\n \u2191f\n (\u2191((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).map\n (homOfLE (_ : PrimeSpectrum.basicOpen r \u2264 \u22a4)).op)\n (\u2191(toOpen \u2191S \u22a4) (s, { val := r, property := hpr } ^ n).1))", "state_after": "no goals"}, {"tactic": "rw [h]", "annotated_tactic": ["rw [h]", []], "state_before": "case e\nR S : CommRingCat\nf : R \u27f6 S\np : PrimeSpectrum \u2191R\ninst\u271d : Algebra \u2191R \u2191S\ny : \u2191(TopCat.Presheaf.stalk (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) p)\nU : TopologicalSpace.Opens \u2191(Spec.topObj R)\nhp : p \u2208 U\ns : (forget CommRingCat).obj ((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).obj (op U))\nr : \u2191R\nhpr : p \u2208 PrimeSpectrum.basicOpen r\nhrU : PrimeSpectrum.basicOpen r \u2264 U\ns' : (forget CommRingCat).obj\n ((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).obj (op (PrimeSpectrum.basicOpen r))) :=\n \u2191((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).map (homOfLE hrU).op) s\ne :\n \u2191(TopCat.Presheaf.germ (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) { val := p, property := hpr }) s' =\n y\nh : s' = \u2191((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).map (homOfLE hrU).op) s\n\u22a2 \u2191(TopCat.Presheaf.germ (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) { val := p, property := hpr }) s' =\n y", "state_after": "case e\nR S : CommRingCat\nf : R \u27f6 S\np : PrimeSpectrum \u2191R\ninst\u271d : Algebra \u2191R \u2191S\ny : \u2191(TopCat.Presheaf.stalk (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) p)\nU : TopologicalSpace.Opens \u2191(Spec.topObj R)\nhp : p \u2208 U\ns : (forget CommRingCat).obj ((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).obj (op U))\nr : \u2191R\nhpr : p \u2208 PrimeSpectrum.basicOpen r\nhrU : PrimeSpectrum.basicOpen r \u2264 U\ns' : (forget CommRingCat).obj\n ((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).obj (op (PrimeSpectrum.basicOpen r))) :=\n \u2191((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).map (homOfLE hrU).op) s\ne :\n \u2191(TopCat.Presheaf.germ (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) { val := p, property := hpr }) s' =\n y\nh : s' = \u2191((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).map (homOfLE hrU).op) s\n\u22a2 \u2191(TopCat.Presheaf.germ (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) { val := p, property := hpr })\n (\u2191((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).map (homOfLE hrU).op) s) =\n y"}, {"tactic": "exact e", "annotated_tactic": ["exact e", []], "state_before": "case e\nR S : CommRingCat\nf : R \u27f6 S\np : PrimeSpectrum \u2191R\ninst\u271d : Algebra \u2191R \u2191S\ny : \u2191(TopCat.Presheaf.stalk (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) p)\nU : TopologicalSpace.Opens \u2191(Spec.topObj R)\nhp : p \u2208 U\ns : (forget CommRingCat).obj ((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).obj (op U))\nr : \u2191R\nhpr : p \u2208 PrimeSpectrum.basicOpen r\nhrU : PrimeSpectrum.basicOpen r \u2264 U\ns' : (forget CommRingCat).obj\n ((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).obj (op (PrimeSpectrum.basicOpen r))) :=\n \u2191((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).map (homOfLE hrU).op) s\ne :\n \u2191(TopCat.Presheaf.germ (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) { val := p, property := hpr }) s' =\n y\nh : s' = \u2191((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).map (homOfLE hrU).op) s\n\u22a2 \u2191(TopCat.Presheaf.germ (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) { val := p, property := hpr })\n (\u2191((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).map (homOfLE hrU).op) s) =\n y", "state_after": "no goals"}, {"tactic": "erw [\u2190 (Spec.topMap (algebraMap R S) _* (structureSheaf S).1).germ_res_apply (homOfLE le_top)\n \u27e8p, hpr\u27e9]", "annotated_tactic": ["erw [\u2190 (<a>Spec.topMap</a> (<a>algebraMap</a> R S) _* (<a>structureSheaf</a> S).1).<a>germ_res_apply</a> (<a>homOfLE</a> <a>le_top</a>)\n \u27e8p, hpr\u27e9]", [{"full_name": "AlgebraicGeometry.Spec.topMap", "def_path": "Mathlib/AlgebraicGeometry/Spec.lean", "def_pos": [61, 5], "def_end_pos": [61, 16]}, {"full_name": "algebraMap", "def_path": "Mathlib/Algebra/Algebra/Basic.lean", "def_pos": [125, 5], "def_end_pos": [125, 15]}, {"full_name": "AlgebraicGeometry.Spec.structureSheaf", "def_path": "Mathlib/AlgebraicGeometry/StructureSheaf.lean", "def_pos": [268, 5], "def_end_pos": [268, 24]}, {"full_name": "TopCat.Presheaf.germ_res_apply", "def_path": "Mathlib/Topology/Sheaves/Stalks.lean", "def_pos": [114, 9], "def_end_pos": [114, 23]}, {"full_name": "CategoryTheory.homOfLE", "def_path": "Mathlib/CategoryTheory/Category/Preorder.lean", "def_pos": [67, 5], "def_end_pos": [67, 12]}, {"full_name": "le_top", "def_path": "Mathlib/Order/BoundedOrder.lean", "def_pos": [98, 9], "def_end_pos": [98, 15]}]], "state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.mk.mk.intro\nR S : CommRingCat\nf : R \u27f6 S\np : PrimeSpectrum \u2191R\ninst\u271d : Algebra \u2191R \u2191S\ny : \u2191(TopCat.Presheaf.stalk (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) p)\nr : \u2191R\nhpr : p \u2208 PrimeSpectrum.basicOpen r\ns' :\n (forget CommRingCat).obj\n ((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).obj (op (PrimeSpectrum.basicOpen r)))\ne :\n \u2191(TopCat.Presheaf.germ (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) { val := p, property := hpr }) s' =\n y\ns : \u2191S\nn : \u2115\nhsn :\n s' *\n \u2191(algebraMap \u2191S \u2191((structureSheaf \u2191S).val.obj (op (PrimeSpectrum.basicOpen (\u2191(algebraMap \u2191R \u2191S) r)))))\n \u2191(s,\n { val := (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) n,\n property :=\n (_ :\n \u2203 y,\n (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) y =\n (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) n) }).2 =\n \u2191(algebraMap \u2191S \u2191((structureSheaf \u2191S).val.obj (op (PrimeSpectrum.basicOpen (\u2191(algebraMap \u2191R \u2191S) r)))))\n (s,\n { val := (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) n,\n property :=\n (_ :\n \u2203 y,\n (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) y =\n (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) n) }).1\n\u22a2 \u2191(TopCat.Presheaf.germ (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) { val := p, property := hpr })\n (\u2191((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).map\n (homOfLE (_ : PrimeSpectrum.basicOpen r \u2264 \u22a4)).op)\n (\u2191(toOpen \u2191S \u22a4) (\u2191(algebraMap \u2191R \u2191S) \u2191(s, { val := r, property := hpr } ^ n).2))) *\n y =\n \u2191(toPushforwardStalkAlgHom R S p) (s, { val := r, property := hpr } ^ n).1", "state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.mk.mk.intro\nR S : CommRingCat\nf : R \u27f6 S\np : PrimeSpectrum \u2191R\ninst\u271d : Algebra \u2191R \u2191S\ny : \u2191(TopCat.Presheaf.stalk (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) p)\nr : \u2191R\nhpr : p \u2208 PrimeSpectrum.basicOpen r\ns' :\n (forget CommRingCat).obj\n ((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).obj (op (PrimeSpectrum.basicOpen r)))\ne :\n \u2191(TopCat.Presheaf.germ (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) { val := p, property := hpr }) s' =\n y\ns : \u2191S\nn : \u2115\nhsn :\n s' *\n \u2191(algebraMap \u2191S \u2191((structureSheaf \u2191S).val.obj (op (PrimeSpectrum.basicOpen (\u2191(algebraMap \u2191R \u2191S) r)))))\n \u2191(s,\n { val := (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) n,\n property :=\n (_ :\n \u2203 y,\n (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) y =\n (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) n) }).2 =\n \u2191(algebraMap \u2191S \u2191((structureSheaf \u2191S).val.obj (op (PrimeSpectrum.basicOpen (\u2191(algebraMap \u2191R \u2191S) r)))))\n (s,\n { val := (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) n,\n property :=\n (_ :\n \u2203 y,\n (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) y =\n (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) n) }).1\n\u22a2 \u2191(TopCat.Presheaf.germ (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) { val := p, property := hpr })\n (\u2191((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).map\n (homOfLE (_ : PrimeSpectrum.basicOpen r \u2264 \u22a4)).op)\n (\u2191(toOpen \u2191S \u22a4) (\u2191(algebraMap \u2191R \u2191S) \u2191(s, { val := r, property := hpr } ^ n).2))) *\n y =\n \u2191(TopCat.Presheaf.germ (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) { val := p, property := hpr })\n (\u2191((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).map\n (homOfLE (_ : PrimeSpectrum.basicOpen r \u2264 \u22a4)).op)\n (\u2191(toOpen \u2191S \u22a4) (s, { val := r, property := hpr } ^ n).1))"}]

https://github.com/leanprover-community/mathlib4

3ce43c18f614b76e161f911b75a3e1ef641620ff

Mathlib/Data/Matrix/Basic.lean

Matrix.algebraMap_eq_smul

[]

https://github.com/leanprover-community/mathlib4

3ce43c18f614b76e161f911b75a3e1ef641620ff

Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean

Real.cos_sub_two_pi

[]

https://github.com/leanprover-community/mathlib4

3ce43c18f614b76e161f911b75a3e1ef641620ff

Mathlib/LinearAlgebra/Prod.lean

LinearMap.coprod_map_prod

[{"tactic": "simp only [LinearMap.coprod_apply, Submodule.coe_sup, Submodule.map_coe]", "annotated_tactic": ["simp only [<a>LinearMap.coprod_apply</a>, <a>Submodule.coe_sup</a>, <a>Submodule.map_coe</a>]", [{"full_name": "LinearMap.coprod_apply", "def_path": "Mathlib/LinearAlgebra/Prod.lean", "def_pos": [215, 9], "def_end_pos": [215, 21]}, {"full_name": "Submodule.coe_sup", "def_path": "Mathlib/LinearAlgebra/Span.lean", "def_pos": [384, 9], "def_end_pos": [384, 16]}, {"full_name": "Submodule.map_coe", "def_path": "Mathlib/LinearAlgebra/Basic.lean", "def_pos": [567, 9], "def_end_pos": [567, 16]}]], "state_before": "R : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\nS\u271d : Type u_3\ninst\u271d\u00b9\u00b3 : Semiring R\ninst\u271d\u00b9\u00b2 : Semiring S\u271d\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2083\ninst\u271d\u2078 : AddCommMonoid M\u2084\ninst\u271d\u2077 : AddCommMonoid M\u2085\ninst\u271d\u2076 : AddCommMonoid M\u2086\ninst\u271d\u2075 : Module R M\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M\u2084\ninst\u271d\u00b9 : Module R M\u2085\ninst\u271d : Module R M\u2086\nf\u271d : M \u2192\u2097[R] M\u2082\nf : M \u2192\u2097[R] M\u2083\ng : M\u2082 \u2192\u2097[R] M\u2083\nS : Submodule R M\nS' : Submodule R M\u2082\n\u22a2 \u2191(Submodule.map (coprod f g) (Submodule.prod S S')) = \u2191(Submodule.map f S \u2294 Submodule.map g S')", "state_after": "R : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\nS\u271d : Type u_3\ninst\u271d\u00b9\u00b3 : Semiring R\ninst\u271d\u00b9\u00b2 : Semiring S\u271d\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2083\ninst\u271d\u2078 : AddCommMonoid M\u2084\ninst\u271d\u2077 : AddCommMonoid M\u2085\ninst\u271d\u2076 : AddCommMonoid M\u2086\ninst\u271d\u2075 : Module R M\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M\u2084\ninst\u271d\u00b9 : Module R M\u2085\ninst\u271d : Module R M\u2086\nf\u271d : M \u2192\u2097[R] M\u2082\nf : M \u2192\u2097[R] M\u2083\ng : M\u2082 \u2192\u2097[R] M\u2083\nS : Submodule R M\nS' : Submodule R M\u2082\n\u22a2 (fun a => \u2191f a.1 + \u2191g a.2) '' \u2191(Submodule.prod S S') = (fun a => \u2191f a) '' \u2191S + (fun a => \u2191g a) '' \u2191S'"}, {"tactic": "rw [\u2190 Set.image2_add, Set.image2_image_left, Set.image2_image_right]", "annotated_tactic": ["rw [\u2190 <a>Set.image2_add</a>, <a>Set.image2_image_left</a>, <a>Set.image2_image_right</a>]", [{"full_name": "Set.image2_add", "def_path": "Mathlib/Data/Set/Pointwise/Basic.lean", "def_pos": [329, 3], "def_end_pos": [329, 14]}, {"full_name": "Set.image2_image_left", "def_path": "Mathlib/Data/Set/NAry.lean", "def_pos": [286, 9], "def_end_pos": [286, 26]}, {"full_name": "Set.image2_image_right", "def_path": "Mathlib/Data/Set/NAry.lean", "def_pos": [295, 9], "def_end_pos": [295, 27]}]], "state_before": "R : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\nS\u271d : Type u_3\ninst\u271d\u00b9\u00b3 : Semiring R\ninst\u271d\u00b9\u00b2 : Semiring S\u271d\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2083\ninst\u271d\u2078 : AddCommMonoid M\u2084\ninst\u271d\u2077 : AddCommMonoid M\u2085\ninst\u271d\u2076 : AddCommMonoid M\u2086\ninst\u271d\u2075 : Module R M\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M\u2084\ninst\u271d\u00b9 : Module R M\u2085\ninst\u271d : Module R M\u2086\nf\u271d : M \u2192\u2097[R] M\u2082\nf : M \u2192\u2097[R] M\u2083\ng : M\u2082 \u2192\u2097[R] M\u2083\nS : Submodule R M\nS' : Submodule R M\u2082\n\u22a2 (fun a => \u2191f a.1 + \u2191g a.2) '' \u2191(Submodule.prod S S') = (fun a => \u2191f a) '' \u2191S + (fun a => \u2191g a) '' \u2191S'", "state_after": "R : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\nS\u271d : Type u_3\ninst\u271d\u00b9\u00b3 : Semiring R\ninst\u271d\u00b9\u00b2 : Semiring S\u271d\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2083\ninst\u271d\u2078 : AddCommMonoid M\u2084\ninst\u271d\u2077 : AddCommMonoid M\u2085\ninst\u271d\u2076 : AddCommMonoid M\u2086\ninst\u271d\u2075 : Module R M\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M\u2084\ninst\u271d\u00b9 : Module R M\u2085\ninst\u271d : Module R M\u2086\nf\u271d : M \u2192\u2097[R] M\u2082\nf : M \u2192\u2097[R] M\u2083\ng : M\u2082 \u2192\u2097[R] M\u2083\nS : Submodule R M\nS' : Submodule R M\u2082\n\u22a2 (fun a => \u2191f a.1 + \u2191g a.2) '' \u2191(Submodule.prod S S') = Set.image2 (fun a b => \u2191f a + \u2191g b) \u2191S \u2191S'"}, {"tactic": "exact Set.image_prod fun m m\u2082 => f m + g m\u2082", "annotated_tactic": ["exact <a>Set.image_prod</a> fun m m\u2082 => f m + g m\u2082", [{"full_name": "Set.image_prod", "def_path": "Mathlib/Data/Set/NAry.lean", "def_pos": [98, 7], "def_end_pos": [98, 17]}]], "state_before": "R : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\nS\u271d : Type u_3\ninst\u271d\u00b9\u00b3 : Semiring R\ninst\u271d\u00b9\u00b2 : Semiring S\u271d\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2083\ninst\u271d\u2078 : AddCommMonoid M\u2084\ninst\u271d\u2077 : AddCommMonoid M\u2085\ninst\u271d\u2076 : AddCommMonoid M\u2086\ninst\u271d\u2075 : Module R M\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M\u2084\ninst\u271d\u00b9 : Module R M\u2085\ninst\u271d : Module R M\u2086\nf\u271d : M \u2192\u2097[R] M\u2082\nf : M \u2192\u2097[R] M\u2083\ng : M\u2082 \u2192\u2097[R] M\u2083\nS : Submodule R M\nS' : Submodule R M\u2082\n\u22a2 (fun a => \u2191f a.1 + \u2191g a.2) '' \u2191(Submodule.prod S S') = Set.image2 (fun a b => \u2191f a + \u2191g b) \u2191S \u2191S'", "state_after": "no goals"}]

https://github.com/leanprover-community/mathlib4

3ce43c18f614b76e161f911b75a3e1ef641620ff

Mathlib/Data/Sign.lean

SignType.range_eq

[{"tactic": "classical rw [\u2190 Fintype.coe_image_univ, univ_eq]", "annotated_tactic": ["classical rw [\u2190 <a>Fintype.coe_image_univ</a>, <a>univ_eq</a>]", [{"full_name": "Fintype.coe_image_univ", "def_path": "Mathlib/Data/Fintype/Basic.lean", "def_pos": [978, 9], "def_end_pos": [978, 31]}, {"full_name": "SignType.univ_eq", "def_path": "Mathlib/Data/Sign.lean", "def_pos": [281, 9], "def_end_pos": [281, 16]}]], "state_before": "\u03b1 : Type u_1\nf : SignType \u2192 \u03b1\n\u22a2 Set.range f = {f zero, f neg, f pos}", "state_after": "\u03b1 : Type u_1\nf : SignType \u2192 \u03b1\n\u22a2 \u2191(Finset.image f {0, -1, 1}) = {f zero, f neg, f pos}"}, {"tactic": "classical simp [Finset.coe_insert]", "annotated_tactic": ["classical simp [<a>Finset.coe_insert</a>]", [{"full_name": "Finset.coe_insert", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1113, 9], "def_end_pos": [1113, 19]}]], "state_before": "\u03b1 : Type u_1\nf : SignType \u2192 \u03b1\n\u22a2 \u2191(Finset.image f {0, -1, 1}) = {f zero, f neg, f pos}", "state_after": "no goals"}, {"tactic": "rw [\u2190 Fintype.coe_image_univ, univ_eq]", "annotated_tactic": ["rw [\u2190 <a>Fintype.coe_image_univ</a>, <a>univ_eq</a>]", [{"full_name": "Fintype.coe_image_univ", "def_path": "Mathlib/Data/Fintype/Basic.lean", "def_pos": [978, 9], "def_end_pos": [978, 31]}, {"full_name": "SignType.univ_eq", "def_path": "Mathlib/Data/Sign.lean", "def_pos": [281, 9], "def_end_pos": [281, 16]}]], "state_before": "\u03b1 : Type u_1\nf : SignType \u2192 \u03b1\n\u22a2 Set.range f = {f zero, f neg, f pos}", "state_after": "\u03b1 : Type u_1\nf : SignType \u2192 \u03b1\n\u22a2 \u2191(Finset.image f {0, -1, 1}) = {f zero, f neg, f pos}"}, {"tactic": "simp [Finset.coe_insert]", "annotated_tactic": ["simp [<a>Finset.coe_insert</a>]", [{"full_name": "Finset.coe_insert", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1113, 9], "def_end_pos": [1113, 19]}]], "state_before": "\u03b1 : Type u_1\nf : SignType \u2192 \u03b1\n\u22a2 \u2191(Finset.image f {0, -1, 1}) = {f zero, f neg, f pos}", "state_after": "no goals"}]

https://github.com/leanprover-community/mathlib4

3ce43c18f614b76e161f911b75a3e1ef641620ff

Mathlib/Algebra/DirectSum/Module.lean

DirectSum.isInternal_submodule_iff_independent_and_iSup_eq_top

[]

https://github.com/leanprover-community/mathlib4

3ce43c18f614b76e161f911b75a3e1ef641620ff

Mathlib/MeasureTheory/Function/LpSeminorm.lean

MeasurableEmbedding.snorm_map_measure

[{"tactic": "by_cases hp_zero : p = 0", "annotated_tactic": ["by_cases hp_zero : p = 0", []], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\n\u03b2 : Type u_5\nm\u03b2 : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ng\u271d : \u03b2 \u2192 E\ng : \u03b2 \u2192 F\nhf : MeasurableEmbedding f\n\u22a2 snorm g p (Measure.map f \u03bc) = snorm (g \u2218 f) p \u03bc", "state_after": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\n\u03b2 : Type u_5\nm\u03b2 : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ng\u271d : \u03b2 \u2192 E\ng : \u03b2 \u2192 F\nhf : MeasurableEmbedding f\nhp_zero : p = 0\n\u22a2 snorm g p (Measure.map f \u03bc) = snorm (g \u2218 f) p \u03bc\n\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\n\u03b2 : Type u_5\nm\u03b2 : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ng\u271d : \u03b2 \u2192 E\ng : \u03b2 \u2192 F\nhf : MeasurableEmbedding f\nhp_zero : \u00acp = 0\n\u22a2 snorm g p (Measure.map f \u03bc) = snorm (g \u2218 f) p \u03bc"}, {"tactic": "by_cases hp : p = \u221e", "annotated_tactic": ["by_cases hp : p = \u221e", []], "state_before": "case neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\n\u03b2 : Type u_5\nm\u03b2 : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ng\u271d : \u03b2 \u2192 E\ng : \u03b2 \u2192 F\nhf : MeasurableEmbedding f\nhp_zero : \u00acp = 0\n\u22a2 snorm g p (Measure.map f \u03bc) = snorm (g \u2218 f) p \u03bc", "state_after": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\n\u03b2 : Type u_5\nm\u03b2 : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ng\u271d : \u03b2 \u2192 E\ng : \u03b2 \u2192 F\nhf : MeasurableEmbedding f\nhp_zero : \u00acp = 0\nhp : p = \u22a4\n\u22a2 snorm g p (Measure.map f \u03bc) = snorm (g \u2218 f) p \u03bc\n\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\n\u03b2 : Type u_5\nm\u03b2 : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ng\u271d : \u03b2 \u2192 E\ng : \u03b2 \u2192 F\nhf : MeasurableEmbedding f\nhp_zero : \u00acp = 0\nhp : \u00acp = \u22a4\n\u22a2 snorm g p (Measure.map f \u03bc) = snorm (g \u2218 f) p \u03bc"}, {"tactic": "simp only [hp_zero, snorm_exponent_zero]", "annotated_tactic": ["simp only [hp_zero, <a>snorm_exponent_zero</a>]", [{"full_name": "MeasureTheory.snorm_exponent_zero", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [176, 9], "def_end_pos": [176, 28]}]], "state_before": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\n\u03b2 : Type u_5\nm\u03b2 : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ng\u271d : \u03b2 \u2192 E\ng : \u03b2 \u2192 F\nhf : MeasurableEmbedding f\nhp_zero : p = 0\n\u22a2 snorm g p (Measure.map f \u03bc) = snorm (g \u2218 f) p \u03bc", "state_after": "no goals"}, {"tactic": "simp_rw [hp, snorm_exponent_top]", "annotated_tactic": ["simp_rw [hp, <a>snorm_exponent_top</a>]", [{"full_name": "MeasureTheory.snorm_exponent_top", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [103, 9], "def_end_pos": [103, 27]}]], "state_before": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\n\u03b2 : Type u_5\nm\u03b2 : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ng\u271d : \u03b2 \u2192 E\ng : \u03b2 \u2192 F\nhf : MeasurableEmbedding f\nhp_zero : \u00acp = 0\nhp : p = \u22a4\n\u22a2 snorm g p (Measure.map f \u03bc) = snorm (g \u2218 f) p \u03bc", "state_after": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\n\u03b2 : Type u_5\nm\u03b2 : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ng\u271d : \u03b2 \u2192 E\ng : \u03b2 \u2192 F\nhf : MeasurableEmbedding f\nhp_zero : \u00acp = 0\nhp : p = \u22a4\n\u22a2 snormEssSup g (Measure.map f \u03bc) = snormEssSup (g \u2218 f) \u03bc"}, {"tactic": "exact hf.essSup_map_measure", "annotated_tactic": ["exact hf.essSup_map_measure", []], "state_before": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\n\u03b2 : Type u_5\nm\u03b2 : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ng\u271d : \u03b2 \u2192 E\ng : \u03b2 \u2192 F\nhf : MeasurableEmbedding f\nhp_zero : \u00acp = 0\nhp : p = \u22a4\n\u22a2 snormEssSup g (Measure.map f \u03bc) = snormEssSup (g \u2218 f) \u03bc", "state_after": "no goals"}, {"tactic": "simp_rw [snorm_eq_lintegral_rpow_nnnorm hp_zero hp]", "annotated_tactic": ["simp_rw [<a>snorm_eq_lintegral_rpow_nnnorm</a> hp_zero hp]", [{"full_name": "MeasureTheory.snorm_eq_lintegral_rpow_nnnorm", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [92, 9], "def_end_pos": [92, 39]}]], "state_before": "case neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\n\u03b2 : Type u_5\nm\u03b2 : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ng\u271d : \u03b2 \u2192 E\ng : \u03b2 \u2192 F\nhf : MeasurableEmbedding f\nhp_zero : \u00acp = 0\nhp : \u00acp = \u22a4\n\u22a2 snorm g p (Measure.map f \u03bc) = snorm (g \u2218 f) p \u03bc", "state_after": "case neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\n\u03b2 : Type u_5\nm\u03b2 : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ng\u271d : \u03b2 \u2192 E\ng : \u03b2 \u2192 F\nhf : MeasurableEmbedding f\nhp_zero : \u00acp = 0\nhp : \u00acp = \u22a4\n\u22a2 (\u222b\u207b (x : \u03b2), \u2191\u2016g x\u2016\u208a ^ ENNReal.toReal p \u2202Measure.map f \u03bc) ^ (1 / ENNReal.toReal p) =\n (\u222b\u207b (x : \u03b1), \u2191\u2016(g \u2218 f) x\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc) ^ (1 / ENNReal.toReal p)"}, {"tactic": "rw [hf.lintegral_map]", "annotated_tactic": ["rw [hf.lintegral_map]", []], "state_before": "case neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\n\u03b2 : Type u_5\nm\u03b2 : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ng\u271d : \u03b2 \u2192 E\ng : \u03b2 \u2192 F\nhf : MeasurableEmbedding f\nhp_zero : \u00acp = 0\nhp : \u00acp = \u22a4\n\u22a2 (\u222b\u207b (x : \u03b2), \u2191\u2016g x\u2016\u208a ^ ENNReal.toReal p \u2202Measure.map f \u03bc) ^ (1 / ENNReal.toReal p) =\n (\u222b\u207b (x : \u03b1), \u2191\u2016(g \u2218 f) x\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc) ^ (1 / ENNReal.toReal p)", "state_after": "case neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\n\u03b2 : Type u_5\nm\u03b2 : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ng\u271d : \u03b2 \u2192 E\ng : \u03b2 \u2192 F\nhf : MeasurableEmbedding f\nhp_zero : \u00acp = 0\nhp : \u00acp = \u22a4\n\u22a2 (\u222b\u207b (a : \u03b1), \u2191\u2016g (f a)\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc) ^ (1 / ENNReal.toReal p) =\n (\u222b\u207b (x : \u03b1), \u2191\u2016(g \u2218 f) x\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc) ^ (1 / ENNReal.toReal p)"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\n\u03b2 : Type u_5\nm\u03b2 : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ng\u271d : \u03b2 \u2192 E\ng : \u03b2 \u2192 F\nhf : MeasurableEmbedding f\nhp_zero : \u00acp = 0\nhp : \u00acp = \u22a4\n\u22a2 (\u222b\u207b (a : \u03b1), \u2191\u2016g (f a)\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc) ^ (1 / ENNReal.toReal p) =\n (\u222b\u207b (x : \u03b1), \u2191\u2016(g \u2218 f) x\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc) ^ (1 / ENNReal.toReal p)", "state_after": "no goals"}]

https://github.com/leanprover-community/mathlib4

3ce43c18f614b76e161f911b75a3e1ef641620ff

Mathlib/RingTheory/DedekindDomain/AdicValuation.lean

IsDedekindDomain.HeightOneSpectrum.adicValued_apply

[]

https://github.com/leanprover-community/mathlib4

3ce43c18f614b76e161f911b75a3e1ef641620ff

Mathlib/Data/List/Rotate.lean

List.rotate'_length

[{"tactic": "rw [rotate'_eq_drop_append_take le_rfl]", "annotated_tactic": ["rw [<a>rotate'_eq_drop_append_take</a> <a>le_rfl</a>]", [{"full_name": "List.rotate'_eq_drop_append_take", "def_path": "Mathlib/Data/List/Rotate.lean", "def_pos": [65, 9], "def_end_pos": [65, 36]}, {"full_name": "le_rfl", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [170, 9], "def_end_pos": [170, 15]}]], "state_before": "\u03b1 : Type u\nl : List \u03b1\n\u22a2 rotate' l (length l) = l", "state_after": "\u03b1 : Type u\nl : List \u03b1\n\u22a2 drop (length l) l ++ take (length l) l = l"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "\u03b1 : Type u\nl : List \u03b1\n\u22a2 drop (length l) l ++ take (length l) l = l", "state_after": "no goals"}]

https://github.com/leanprover-community/mathlib4

3ce43c18f614b76e161f911b75a3e1ef641620ff

Mathlib/CategoryTheory/Monad/Limits.lean

CategoryTheory.Monad.hasLimit_of_comp_forget_hasLimit

[]

https://github.com/leanprover/std4

869c615eb10130c0637a7bc038e2b80253559913

lake-packages/std/Std/Data/Int/DivMod.lean

Int.fdiv_eq_ediv

[{"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "a\u271d : Nat\nx\u271d : 0 \u2264 0\n\u22a2 fdiv -[a\u271d+1] 0 = -[a\u271d+1] / 0", "state_after": "no goals"}]

https://github.com/leanprover/std4

869c615eb10130c0637a7bc038e2b80253559913

lake-packages/std/Std/Data/Nat/Lemmas.lean

Nat.lt_succ

[]

https://github.com/leanprover-community/mathlib4

3ce43c18f614b76e161f911b75a3e1ef641620ff

Mathlib/Data/Set/Pointwise/Basic.lean

Set.Nonempty.zero_div

[{"tactic": "simpa [mem_div] using hs", "annotated_tactic": ["simpa [<a>mem_div</a>] using hs", [{"full_name": "Set.mem_div", "def_path": "Mathlib/Data/Set/Pointwise/Basic.lean", "def_pos": [603, 9], "def_end_pos": [603, 16]}]], "state_before": "F : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d : GroupWithZero \u03b1\ns t : Set \u03b1\nhs : Set.Nonempty s\n\u22a2 0 \u2286 0 / s", "state_after": "no goals"}]

https://github.com/leanprover-community/mathlib4

3ce43c18f614b76e161f911b75a3e1ef641620ff

Mathlib/Data/Set/Lattice.lean

Set.iUnion_inter

[]

https://github.com/leanprover-community/mathlib4

3ce43c18f614b76e161f911b75a3e1ef641620ff

Mathlib/Analysis/Convex/Cone/Proper.lean

ProperCone.mem_comap

[]

https://github.com/leanprover-community/mathlib4

3ce43c18f614b76e161f911b75a3e1ef641620ff

Mathlib/Order/Filter/Basic.lean

Filter.empty_mem_iff_bot

[]

https://github.com/leanprover-community/mathlib4

3ce43c18f614b76e161f911b75a3e1ef641620ff

Mathlib/MeasureTheory/Group/FundamentalDomain.lean

MeasureTheory.IsFundamentalDomain.nullMeasurableSet_smul

[]

https://github.com/leanprover-community/mathlib4

3ce43c18f614b76e161f911b75a3e1ef641620ff

Mathlib/Init/Data/Bool/Lemmas.lean

Bool.coe_sort_true

[{"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "\u22a2 (true = true) = True", "state_after": "no goals"}]

https://github.com/leanprover-community/mathlib4

3ce43c18f614b76e161f911b75a3e1ef641620ff

Mathlib/Logic/Basic.lean

not_forall_not

[]

https://github.com/leanprover-community/mathlib4

3ce43c18f614b76e161f911b75a3e1ef641620ff

Mathlib/Computability/Halting.lean

ComputablePred.rice

[{"tactic": "cases' h with _ h", "annotated_tactic": ["cases' h with _ h", []], "state_before": "\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nC : Set (\u2115 \u2192. \u2115)\nh : ComputablePred fun c => eval c \u2208 C\nf g : \u2115 \u2192. \u2115\nhf : Nat.Partrec f\nhg : Nat.Partrec g\nfC : f \u2208 C\n\u22a2 g \u2208 C", "state_after": "case intro\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nC : Set (\u2115 \u2192. \u2115)\nf g : \u2115 \u2192. \u2115\nhf : Nat.Partrec f\nhg : Nat.Partrec g\nfC : f \u2208 C\nw\u271d : DecidablePred fun c => eval c \u2208 C\nh : Computable fun a => decide ((fun c => eval c \u2208 C) a)\n\u22a2 g \u2208 C"}, {"tactic": "obtain \u27e8c, e\u27e9 :=\n fixed_point\u2082\n (Partrec.cond (h.comp fst) ((Partrec.nat_iff.2 hg).comp snd).to\u2082\n ((Partrec.nat_iff.2 hf).comp snd).to\u2082).to\u2082", "annotated_tactic": ["obtain \u27e8c, e\u27e9 :=\n <a>fixed_point\u2082</a>\n (<a>Partrec.cond</a> (h.comp <a>fst</a>) ((<a>Partrec.nat_iff</a>.2 hg).<a>comp</a> <a>snd</a>).<a>to\u2082</a>\n ((<a>Partrec.nat_iff</a>.2 hf).<a>comp</a> <a>snd</a>).<a>to\u2082</a>).<a>to\u2082</a>", [{"full_name": "Nat.Partrec.Code.fixed_point\u2082", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [1188, 9], "def_end_pos": [1188, 21]}, {"full_name": "Partrec.cond", "def_path": "Mathlib/Computability/Halting.lean", "def_pos": [116, 9], "def_end_pos": [116, 13]}, {"full_name": "Computable.fst", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [302, 9], "def_end_pos": [302, 12]}, {"full_name": "Partrec.nat_iff", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [485, 9], "def_end_pos": [485, 16]}, {"full_name": "Partrec.comp", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [480, 16], "def_end_pos": [480, 20]}, {"full_name": "Computable.snd", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [306, 9], "def_end_pos": [306, 12]}, {"full_name": "Partrec.to\u2082", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [466, 9], "def_end_pos": [466, 12]}, {"full_name": "Partrec.nat_iff", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [485, 9], "def_end_pos": [485, 16]}, {"full_name": "Partrec.comp", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [480, 16], "def_end_pos": [480, 20]}, {"full_name": "Computable.snd", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [306, 9], "def_end_pos": [306, 12]}, {"full_name": "Partrec.to\u2082", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [466, 9], "def_end_pos": [466, 12]}, {"full_name": "Partrec.to\u2082", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [466, 9], "def_end_pos": [466, 12]}]], "state_before": "case intro\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nC : Set (\u2115 \u2192. \u2115)\nf g : \u2115 \u2192. \u2115\nhf : Nat.Partrec f\nhg : Nat.Partrec g\nfC : f \u2208 C\nw\u271d : DecidablePred fun c => eval c \u2208 C\nh : Computable fun a => decide ((fun c => eval c \u2208 C) a)\n\u22a2 g \u2208 C", "state_after": "case intro.intro\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nC : Set (\u2115 \u2192. \u2115)\nf g : \u2115 \u2192. \u2115\nhf : Nat.Partrec f\nhg : Nat.Partrec g\nfC : f \u2208 C\nw\u271d : DecidablePred fun c => eval c \u2208 C\nh : Computable fun a => decide ((fun c => eval c \u2208 C) a)\nc : Code\ne :\n eval c = fun b =>\n bif decide ((fun c => eval c \u2208 C) (c, b).1) then (fun a b => g (a, b).2) (c, b).1 (c, b).2\n else (fun a b => f (a, b).2) (c, b).1 (c, b).2\n\u22a2 g \u2208 C"}, {"tactic": "simp at e", "annotated_tactic": ["simp at e", []], "state_before": "case intro.intro\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nC : Set (\u2115 \u2192. \u2115)\nf g : \u2115 \u2192. \u2115\nhf : Nat.Partrec f\nhg : Nat.Partrec g\nfC : f \u2208 C\nw\u271d : DecidablePred fun c => eval c \u2208 C\nh : Computable fun a => decide ((fun c => eval c \u2208 C) a)\nc : Code\ne :\n eval c = fun b =>\n bif decide ((fun c => eval c \u2208 C) (c, b).1) then (fun a b => g (a, b).2) (c, b).1 (c, b).2\n else (fun a b => f (a, b).2) (c, b).1 (c, b).2\n\u22a2 g \u2208 C", "state_after": "case intro.intro\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nC : Set (\u2115 \u2192. \u2115)\nf g : \u2115 \u2192. \u2115\nhf : Nat.Partrec f\nhg : Nat.Partrec g\nfC : f \u2208 C\nw\u271d : DecidablePred fun c => eval c \u2208 C\nh : Computable fun a => decide ((fun c => eval c \u2208 C) a)\nc : Code\ne : eval c = fun b => if eval c \u2208 C then g b else f b\n\u22a2 g \u2208 C"}, {"tactic": "by_cases H : eval c \u2208 C", "annotated_tactic": ["by_cases H : <a>eval</a> c \u2208 C", [{"full_name": "Nat.Partrec.Code.eval", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [620, 5], "def_end_pos": [620, 9]}]], "state_before": "case intro.intro\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nC : Set (\u2115 \u2192. \u2115)\nf g : \u2115 \u2192. \u2115\nhf : Nat.Partrec f\nhg : Nat.Partrec g\nfC : f \u2208 C\nw\u271d : DecidablePred fun c => eval c \u2208 C\nh : Computable fun a => decide ((fun c => eval c \u2208 C) a)\nc : Code\ne : eval c = fun b => if eval c \u2208 C then g b else f b\n\u22a2 g \u2208 C", "state_after": "case pos\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nC : Set (\u2115 \u2192. \u2115)\nf g : \u2115 \u2192. \u2115\nhf : Nat.Partrec f\nhg : Nat.Partrec g\nfC : f \u2208 C\nw\u271d : DecidablePred fun c => eval c \u2208 C\nh : Computable fun a => decide ((fun c => eval c \u2208 C) a)\nc : Code\ne : eval c = fun b => if eval c \u2208 C then g b else f b\nH : eval c \u2208 C\n\u22a2 g \u2208 C\n\ncase neg\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nC : Set (\u2115 \u2192. \u2115)\nf g : \u2115 \u2192. \u2115\nhf : Nat.Partrec f\nhg : Nat.Partrec g\nfC : f \u2208 C\nw\u271d : DecidablePred fun c => eval c \u2208 C\nh : Computable fun a => decide ((fun c => eval c \u2208 C) a)\nc : Code\ne : eval c = fun b => if eval c \u2208 C then g b else f b\nH : \u00aceval c \u2208 C\n\u22a2 g \u2208 C"}, {"tactic": "simp only [H, if_true] at e", "annotated_tactic": ["simp only [H, <a>if_true</a>] at e", [{"full_name": "if_true", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [727, 17], "def_end_pos": [727, 24]}]], "state_before": "case pos\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nC : Set (\u2115 \u2192. \u2115)\nf g : \u2115 \u2192. \u2115\nhf : Nat.Partrec f\nhg : Nat.Partrec g\nfC : f \u2208 C\nw\u271d : DecidablePred fun c => eval c \u2208 C\nh : Computable fun a => decide ((fun c => eval c \u2208 C) a)\nc : Code\ne : eval c = fun b => if eval c \u2208 C then g b else f b\nH : eval c \u2208 C\n\u22a2 g \u2208 C", "state_after": "case pos\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nC : Set (\u2115 \u2192. \u2115)\nf g : \u2115 \u2192. \u2115\nhf : Nat.Partrec f\nhg : Nat.Partrec g\nfC : f \u2208 C\nw\u271d : DecidablePred fun c => eval c \u2208 C\nh : Computable fun a => decide ((fun c => eval c \u2208 C) a)\nc : Code\nH : eval c \u2208 C\ne : eval c = fun b => g b\n\u22a2 g \u2208 C"}, {"tactic": "change (fun b => g b) \u2208 C", "annotated_tactic": ["change (fun b => g b) \u2208 C", []], "state_before": "case pos\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nC : Set (\u2115 \u2192. \u2115)\nf g : \u2115 \u2192. \u2115\nhf : Nat.Partrec f\nhg : Nat.Partrec g\nfC : f \u2208 C\nw\u271d : DecidablePred fun c => eval c \u2208 C\nh : Computable fun a => decide ((fun c => eval c \u2208 C) a)\nc : Code\nH : eval c \u2208 C\ne : eval c = fun b => g b\n\u22a2 g \u2208 C", "state_after": "case pos\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nC : Set (\u2115 \u2192. \u2115)\nf g : \u2115 \u2192. \u2115\nhf : Nat.Partrec f\nhg : Nat.Partrec g\nfC : f \u2208 C\nw\u271d : DecidablePred fun c => eval c \u2208 C\nh : Computable fun a => decide ((fun c => eval c \u2208 C) a)\nc : Code\nH : eval c \u2208 C\ne : eval c = fun b => g b\n\u22a2 (fun b => g b) \u2208 C"}, {"tactic": "rwa [\u2190 e]", "annotated_tactic": ["rwa [\u2190 e]", []], "state_before": "case pos\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nC : Set (\u2115 \u2192. \u2115)\nf g : \u2115 \u2192. \u2115\nhf : Nat.Partrec f\nhg : Nat.Partrec g\nfC : f \u2208 C\nw\u271d : DecidablePred fun c => eval c \u2208 C\nh : Computable fun a => decide ((fun c => eval c \u2208 C) a)\nc : Code\nH : eval c \u2208 C\ne : eval c = fun b => g b\n\u22a2 (fun b => g b) \u2208 C", "state_after": "no goals"}, {"tactic": "simp only [H, if_false] at e", "annotated_tactic": ["simp only [H, <a>if_false</a>] at e", [{"full_name": "if_false", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [729, 17], "def_end_pos": [729, 25]}]], "state_before": "case neg\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nC : Set (\u2115 \u2192. \u2115)\nf g : \u2115 \u2192. \u2115\nhf : Nat.Partrec f\nhg : Nat.Partrec g\nfC : f \u2208 C\nw\u271d : DecidablePred fun c => eval c \u2208 C\nh : Computable fun a => decide ((fun c => eval c \u2208 C) a)\nc : Code\ne : eval c = fun b => if eval c \u2208 C then g b else f b\nH : \u00aceval c \u2208 C\n\u22a2 g \u2208 C", "state_after": "case neg\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nC : Set (\u2115 \u2192. \u2115)\nf g : \u2115 \u2192. \u2115\nhf : Nat.Partrec f\nhg : Nat.Partrec g\nfC : f \u2208 C\nw\u271d : DecidablePred fun c => eval c \u2208 C\nh : Computable fun a => decide ((fun c => eval c \u2208 C) a)\nc : Code\nH : \u00aceval c \u2208 C\ne : eval c = fun b => f b\n\u22a2 g \u2208 C"}, {"tactic": "rw [e] at H", "annotated_tactic": ["rw [e] at H", []], "state_before": "case neg\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nC : Set (\u2115 \u2192. \u2115)\nf g : \u2115 \u2192. \u2115\nhf : Nat.Partrec f\nhg : Nat.Partrec g\nfC : f \u2208 C\nw\u271d : DecidablePred fun c => eval c \u2208 C\nh : Computable fun a => decide ((fun c => eval c \u2208 C) a)\nc : Code\nH : \u00aceval c \u2208 C\ne : eval c = fun b => f b\n\u22a2 g \u2208 C", "state_after": "case neg\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nC : Set (\u2115 \u2192. \u2115)\nf g : \u2115 \u2192. \u2115\nhf : Nat.Partrec f\nhg : Nat.Partrec g\nfC : f \u2208 C\nw\u271d : DecidablePred fun c => eval c \u2208 C\nh : Computable fun a => decide ((fun c => eval c \u2208 C) a)\nc : Code\nH : \u00ac(fun b => f b) \u2208 C\ne : eval c = fun b => f b\n\u22a2 g \u2208 C"}, {"tactic": "contradiction", "annotated_tactic": ["contradiction", []], "state_before": "case neg\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nC : Set (\u2115 \u2192. \u2115)\nf g : \u2115 \u2192. \u2115\nhf : Nat.Partrec f\nhg : Nat.Partrec g\nfC : f \u2208 C\nw\u271d : DecidablePred fun c => eval c \u2208 C\nh : Computable fun a => decide ((fun c => eval c \u2208 C) a)\nc : Code\nH : \u00ac(fun b => f b) \u2208 C\ne : eval c = fun b => f b\n\u22a2 g \u2208 C", "state_after": "no goals"}]

https://github.com/leanprover-community/mathlib4

3ce43c18f614b76e161f911b75a3e1ef641620ff

Mathlib/LinearAlgebra/Matrix/GeneralLinearGroup.lean

Matrix.GeneralLinearGroup.coe_inv

[]

https://github.com/leanprover-community/mathlib4

3ce43c18f614b76e161f911b75a3e1ef641620ff

Mathlib/Order/CompleteLattice.lean

iSup_congr_Prop

[{"tactic": "obtain rfl := propext pq", "annotated_tactic": ["obtain rfl := <a>propext</a> pq", [{"full_name": "propext", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [1142, 7], "def_end_pos": [1142, 14]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b2\u2082 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Sort u_5\n\u03b9' : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba' : \u03b9' \u2192 Sort u_8\ninst\u271d : SupSet \u03b1\nf\u271d g : \u03b9 \u2192 \u03b1\np q : Prop\nf\u2081 : p \u2192 \u03b1\nf\u2082 : q \u2192 \u03b1\npq : p \u2194 q\nf : \u2200 (x : q), f\u2081 (_ : p) = f\u2082 x\n\u22a2 iSup f\u2081 = iSup f\u2082", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b2\u2082 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Sort u_5\n\u03b9' : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba' : \u03b9' \u2192 Sort u_8\ninst\u271d : SupSet \u03b1\nf\u271d g : \u03b9 \u2192 \u03b1\np : Prop\nf\u2081 f\u2082 : p \u2192 \u03b1\npq : p \u2194 p\nf : \u2200 (x : p), f\u2081 (_ : p) = f\u2082 x\n\u22a2 iSup f\u2081 = iSup f\u2082"}, {"tactic": "congr with x", "annotated_tactic": ["congr with x", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b2\u2082 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Sort u_5\n\u03b9' : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba' : \u03b9' \u2192 Sort u_8\ninst\u271d : SupSet \u03b1\nf\u271d g : \u03b9 \u2192 \u03b1\np : Prop\nf\u2081 f\u2082 : p \u2192 \u03b1\npq : p \u2194 p\nf : \u2200 (x : p), f\u2081 (_ : p) = f\u2082 x\n\u22a2 iSup f\u2081 = iSup f\u2082", "state_after": "case e_s.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b2\u2082 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Sort u_5\n\u03b9' : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba' : \u03b9' \u2192 Sort u_8\ninst\u271d : SupSet \u03b1\nf\u271d g : \u03b9 \u2192 \u03b1\np : Prop\nf\u2081 f\u2082 : p \u2192 \u03b1\npq : p \u2194 p\nf : \u2200 (x : p), f\u2081 (_ : p) = f\u2082 x\nx : p\n\u22a2 f\u2081 x = f\u2082 x"}, {"tactic": "apply f", "annotated_tactic": ["apply f", []], "state_before": "case e_s.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b2\u2082 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Sort u_5\n\u03b9' : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba' : \u03b9' \u2192 Sort u_8\ninst\u271d : SupSet \u03b1\nf\u271d g : \u03b9 \u2192 \u03b1\np : Prop\nf\u2081 f\u2082 : p \u2192 \u03b1\npq : p \u2194 p\nf : \u2200 (x : p), f\u2081 (_ : p) = f\u2082 x\nx : p\n\u22a2 f\u2081 x = f\u2082 x", "state_after": "no goals"}]

https://github.com/leanprover-community/mathlib4

3ce43c18f614b76e161f911b75a3e1ef641620ff

Mathlib/Topology/UniformSpace/UniformEmbedding.lean

closedEmbedding_of_spaced_out

[{"tactic": "rcases @DiscreteTopology.eq_bot \u03b1 _ _ with rfl", "annotated_tactic": ["rcases @<a>DiscreteTopology.eq_bot</a> \u03b1 _ _ with rfl", [{"full_name": "DiscreteTopology.eq_bot", "def_path": "Mathlib/Topology/Order.lean", "def_pos": [274, 3], "def_end_pos": [274, 9]}]], "state_before": "\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u2075 : UniformSpace \u03b1\u271d\ninst\u271d\u2074 : UniformSpace \u03b2\ninst\u271d\u00b3 : UniformSpace \u03b3\n\u03b1 : Type u_1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : DiscreteTopology \u03b1\ninst\u271d : SeparatedSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set (\u03b2 \u00d7 \u03b2)\nhs : s \u2208 \ud835\udce4 \u03b2\nhf : Pairwise fun x y => \u00ac(f x, f y) \u2208 s\n\u22a2 ClosedEmbedding f", "state_after": "\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u2074 : UniformSpace \u03b1\u271d\ninst\u271d\u00b3 : UniformSpace \u03b2\ninst\u271d\u00b2 : UniformSpace \u03b3\n\u03b1 : Type u_1\ninst\u271d\u00b9 : SeparatedSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set (\u03b2 \u00d7 \u03b2)\nhs : s \u2208 \ud835\udce4 \u03b2\nhf : Pairwise fun x y => \u00ac(f x, f y) \u2208 s\ninst\u271d : DiscreteTopology \u03b1\n\u22a2 ClosedEmbedding f"}, {"tactic": "let _ : UniformSpace \u03b1 := \u22a5", "annotated_tactic": ["let _ : <a>UniformSpace</a> \u03b1 := \u22a5", [{"full_name": "UniformSpace", "def_path": "Mathlib/Topology/UniformSpace/Basic.lean", "def_pos": [305, 7], "def_end_pos": [305, 19]}]], "state_before": "\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u2074 : UniformSpace \u03b1\u271d\ninst\u271d\u00b3 : UniformSpace \u03b2\ninst\u271d\u00b2 : UniformSpace \u03b3\n\u03b1 : Type u_1\ninst\u271d\u00b9 : SeparatedSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set (\u03b2 \u00d7 \u03b2)\nhs : s \u2208 \ud835\udce4 \u03b2\nhf : Pairwise fun x y => \u00ac(f x, f y) \u2208 s\ninst\u271d : DiscreteTopology \u03b1\n\u22a2 ClosedEmbedding f", "state_after": "\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u2074 : UniformSpace \u03b1\u271d\ninst\u271d\u00b3 : UniformSpace \u03b2\ninst\u271d\u00b2 : UniformSpace \u03b3\n\u03b1 : Type u_1\ninst\u271d\u00b9 : SeparatedSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set (\u03b2 \u00d7 \u03b2)\nhs : s \u2208 \ud835\udce4 \u03b2\nhf : Pairwise fun x y => \u00ac(f x, f y) \u2208 s\ninst\u271d : DiscreteTopology \u03b1\nx\u271d : UniformSpace \u03b1 := \u22a5\n\u22a2 ClosedEmbedding f"}, {"tactic": "exact\n { (uniformEmbedding_of_spaced_out hs hf).embedding with\n closed_range := isClosed_range_of_spaced_out hs hf }", "annotated_tactic": ["exact\n { (<a>uniformEmbedding_of_spaced_out</a> hs hf).<a>embedding</a> with\n closed_range := <a>isClosed_range_of_spaced_out</a> hs hf }", [{"full_name": "uniformEmbedding_of_spaced_out", "def_path": "Mathlib/Topology/UniformSpace/UniformEmbedding.lean", "def_pos": [218, 9], "def_end_pos": [218, 39]}, {"full_name": "UniformEmbedding.embedding", "def_path": "Mathlib/Topology/UniformSpace/UniformEmbedding.lean", "def_pos": [224, 19], "def_end_pos": [224, 45]}, {"full_name": "isClosed_range_of_spaced_out", "def_path": "Mathlib/Topology/UniformSpace/Separation.lean", "def_pos": [237, 9], "def_end_pos": [237, 37]}]], "state_before": "\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u2074 : UniformSpace \u03b1\u271d\ninst\u271d\u00b3 : UniformSpace \u03b2\ninst\u271d\u00b2 : UniformSpace \u03b3\n\u03b1 : Type u_1\ninst\u271d\u00b9 : SeparatedSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set (\u03b2 \u00d7 \u03b2)\nhs : s \u2208 \ud835\udce4 \u03b2\nhf : Pairwise fun x y => \u00ac(f x, f y) \u2208 s\ninst\u271d : DiscreteTopology \u03b1\nx\u271d : UniformSpace \u03b1 := \u22a5\n\u22a2 ClosedEmbedding f", "state_after": "no goals"}]

https://github.com/leanprover/std4

869c615eb10130c0637a7bc038e2b80253559913

lake-packages/std/Std/Data/List/Lemmas.lean

List.get?_append

[{"tactic": "have hn' : n < (l\u2081 ++ l\u2082).length := Nat.lt_of_lt_of_le hn <|\n length_append .. \u25b8 Nat.le_add_right ..", "annotated_tactic": ["have hn' : n < (l\u2081 ++ l\u2082).<a>length</a> := <a>Nat.lt_of_lt_of_le</a> hn <|\n <a>length_append</a> .. \u25b8 <a>Nat.le_add_right</a> ..", [{"full_name": "List.length", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2232, 5], "def_end_pos": [2232, 16]}, {"full_name": "Nat.lt_of_lt_of_le", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [259, 19], "def_end_pos": [259, 33]}, {"full_name": "List.length_append", "def_path": "lake-packages/lean4/src/lean/Init/Data/List/Basic.lean", "def_pos": [790, 17], "def_end_pos": [790, 30]}, {"full_name": "Nat.le_add_right", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [340, 9], "def_end_pos": [340, 21]}]], "state_before": "\u03b1 : Type u_1\nl\u2081 l\u2082 : List \u03b1\nn : Nat\nhn : n < length l\u2081\n\u22a2 get? (l\u2081 ++ l\u2082) n = get? l\u2081 n", "state_after": "\u03b1 : Type u_1\nl\u2081 l\u2082 : List \u03b1\nn : Nat\nhn : n < length l\u2081\nhn' : n < length (l\u2081 ++ l\u2082)\n\u22a2 get? (l\u2081 ++ l\u2082) n = get? l\u2081 n"}, {"tactic": "rw [get?_eq_get hn, get?_eq_get hn', get_append]", "annotated_tactic": ["rw [<a>get?_eq_get</a> hn, <a>get?_eq_get</a> hn', <a>get_append</a>]", [{"full_name": "List.get?_eq_get", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [581, 9], "def_end_pos": [581, 20]}, {"full_name": "List.get?_eq_get", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [581, 9], "def_end_pos": [581, 20]}, {"full_name": "List.get_append", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [677, 9], "def_end_pos": [677, 19]}]], "state_before": "\u03b1 : Type u_1\nl\u2081 l\u2082 : List \u03b1\nn : Nat\nhn : n < length l\u2081\nhn' : n < length (l\u2081 ++ l\u2082)\n\u22a2 get? (l\u2081 ++ l\u2082) n = get? l\u2081 n", "state_after": "no goals"}]

https://github.com/leanprover-community/mathlib4

3ce43c18f614b76e161f911b75a3e1ef641620ff

Mathlib/Algebra/Order/Nonneg/Ring.lean

Nonneg.mk_eq_zero

[]

https://github.com/leanprover-community/mathlib4

3ce43c18f614b76e161f911b75a3e1ef641620ff

Mathlib/Topology/Sets/Opens.lean

TopologicalSpace.Opens.eq_bot_or_top

[{"tactic": "subst h", "annotated_tactic": ["subst h", []], "state_before": "\u03b9 : Type u_1\n\u03b1\u271d : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\n\u03b1 : Type u_5\nt : TopologicalSpace \u03b1\nh : t = \u22a4\nU : Opens \u03b1\n\u22a2 U = \u22a5 \u2228 U = \u22a4", "state_after": "\u03b9 : Type u_1\n\u03b1\u271d : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\n\u03b1 : Type u_5\nU : Opens \u03b1\n\u22a2 U = \u22a5 \u2228 U = \u22a4"}, {"tactic": "letI : TopologicalSpace \u03b1 := \u22a4", "annotated_tactic": ["letI : <a>TopologicalSpace</a> \u03b1 := \u22a4", [{"full_name": "TopologicalSpace", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [70, 7], "def_end_pos": [70, 23]}]], "state_before": "\u03b9 : Type u_1\n\u03b1\u271d : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\n\u03b1 : Type u_5\nU : Opens \u03b1\n\u22a2 U = \u22a5 \u2228 U = \u22a4", "state_after": "\u03b9 : Type u_1\n\u03b1\u271d : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\n\u03b1 : Type u_5\nU : Opens \u03b1\nthis : TopologicalSpace \u03b1 := \u22a4\n\u22a2 U = \u22a5 \u2228 U = \u22a4"}, {"tactic": "rw [\u2190 coe_eq_empty, \u2190 coe_eq_univ, \u2190 isOpen_top_iff]", "annotated_tactic": ["rw [\u2190 <a>coe_eq_empty</a>, \u2190 <a>coe_eq_univ</a>, \u2190 <a>isOpen_top_iff</a>]", [{"full_name": "TopologicalSpace.Opens.coe_eq_empty", "def_path": "Mathlib/Topology/Sets/Opens.lean", "def_pos": [191, 9], "def_end_pos": [191, 21]}, {"full_name": "TopologicalSpace.Opens.coe_eq_univ", "def_path": "Mathlib/Topology/Sets/Opens.lean", "def_pos": [203, 9], "def_end_pos": [203, 20]}, {"full_name": "TopologicalSpace.isOpen_top_iff", "def_path": "Mathlib/Topology/Order.lean", "def_pos": [258, 9], "def_end_pos": [258, 40]}]], "state_before": "\u03b9 : Type u_1\n\u03b1\u271d : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\n\u03b1 : Type u_5\nU : Opens \u03b1\nthis : TopologicalSpace \u03b1 := \u22a4\n\u22a2 U = \u22a5 \u2228 U = \u22a4", "state_after": "\u03b9 : Type u_1\n\u03b1\u271d : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\n\u03b1 : Type u_5\nU : Opens \u03b1\nthis : TopologicalSpace \u03b1 := \u22a4\n\u22a2 IsOpen \u2191U"}, {"tactic": "exact U.2", "annotated_tactic": ["exact U.2", []], "state_before": "\u03b9 : Type u_1\n\u03b1\u271d : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\n\u03b1 : Type u_5\nU : Opens \u03b1\nthis : TopologicalSpace \u03b1 := \u22a4\n\u22a2 IsOpen \u2191U", "state_after": "no goals"}]

https://github.com/leanprover-community/mathlib4

3ce43c18f614b76e161f911b75a3e1ef641620ff

Mathlib/Data/MvPolynomial/Basic.lean

MvPolynomial.X_pow_eq_monomial

[{"tactic": "simp [X, monomial_pow]", "annotated_tactic": ["simp [<a>X</a>, <a>monomial_pow</a>]", [{"full_name": "MvPolynomial.X", "def_path": "Mathlib/Data/MvPolynomial/Basic.lean", "def_pos": [193, 5], "def_end_pos": [193, 6]}, {"full_name": "MvPolynomial.monomial_pow", "def_path": "Mathlib/Data/MvPolynomial/Basic.lean", "def_pos": [302, 9], "def_end_pos": [302, 21]}]], "state_before": "R : Type u\nS\u2081 : Type v\nS\u2082 : Type w\nS\u2083 : Type x\n\u03c3 : Type u_1\na a' a\u2081 a\u2082 : R\ne : \u2115\nn m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : CommSemiring S\u2081\np q : MvPolynomial \u03c3 R\n\u22a2 X n ^ e = \u2191(monomial fun\u2080 | n => e) 1", "state_after": "no goals"}]

https://github.com/leanprover-community/mathlib4

3ce43c18f614b76e161f911b75a3e1ef641620ff

Mathlib/ModelTheory/Substructures.lean

FirstOrder.Language.Substructure.range_subtype

[{"tactic": "ext x", "annotated_tactic": ["ext x", []], "state_before": "L : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\nS\u271d S : Substructure L M\n\u22a2 Hom.range (Embedding.toHom (subtype S)) = S", "state_after": "case h\nL : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\nS\u271d S : Substructure L M\nx : M\n\u22a2 x \u2208 Hom.range (Embedding.toHom (subtype S)) \u2194 x \u2208 S"}, {"tactic": "simp only [Hom.mem_range, Embedding.coe_toHom, coeSubtype]", "annotated_tactic": ["simp only [<a>Hom.mem_range</a>, <a>Embedding.coe_toHom</a>, <a>coeSubtype</a>]", [{"full_name": "FirstOrder.Language.Hom.mem_range", "def_path": "Mathlib/ModelTheory/Substructures.lean", "def_pos": [830, 9], "def_end_pos": [830, 18]}, {"full_name": "FirstOrder.Language.Embedding.coe_toHom", "def_path": "Mathlib/ModelTheory/Basic.lean", "def_pos": [641, 9], "def_end_pos": [641, 18]}, {"full_name": "FirstOrder.Language.Substructure.coeSubtype", "def_path": "Mathlib/ModelTheory/Substructures.lean", "def_pos": [670, 9], "def_end_pos": [670, 19]}]], "state_before": "case h\nL : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\nS\u271d S : Substructure L M\nx : M\n\u22a2 x \u2208 Hom.range (Embedding.toHom (subtype S)) \u2194 x \u2208 S", "state_after": "case h\nL : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\nS\u271d S : Substructure L M\nx : M\n\u22a2 (\u2203 y, \u2191y = x) \u2194 x \u2208 S"}, {"tactic": "refine' \u27e8_, fun h => \u27e8\u27e8x, h\u27e9, rfl\u27e9\u27e9", "annotated_tactic": ["refine' \u27e8_, fun h => \u27e8\u27e8x, h\u27e9, <a>rfl</a>\u27e9\u27e9", [{"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "case h\nL : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\nS\u271d S : Substructure L M\nx : M\n\u22a2 (\u2203 y, \u2191y = x) \u2194 x \u2208 S", "state_after": "case h\nL : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\nS\u271d S : Substructure L M\nx : M\n\u22a2 (\u2203 y, \u2191y = x) \u2192 x \u2208 S"}, {"tactic": "rintro \u27e8\u27e8y, hy\u27e9, rfl\u27e9", "annotated_tactic": ["rintro \u27e8\u27e8y, hy\u27e9, rfl\u27e9", []], "state_before": "case h\nL : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\nS\u271d S : Substructure L M\nx : M\n\u22a2 (\u2203 y, \u2191y = x) \u2192 x \u2208 S", "state_after": "case h.intro.mk\nL : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\nS\u271d S : Substructure L M\ny : M\nhy : y \u2208 S\n\u22a2 \u2191{ val := y, property := hy } \u2208 S"}, {"tactic": "exact hy", "annotated_tactic": ["exact hy", []], "state_before": "case h.intro.mk\nL : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\nS\u271d S : Substructure L M\ny : M\nhy : y \u2208 S\n\u22a2 \u2191{ val := y, property := hy } \u2208 S", "state_after": "no goals"}]

https://github.com/leanprover-community/mathlib4

3ce43c18f614b76e161f911b75a3e1ef641620ff

Mathlib/Order/CompleteLattice.lean

sSup_univ

[]

https://github.com/leanprover-community/mathlib4

3ce43c18f614b76e161f911b75a3e1ef641620ff

Mathlib/Topology/UnitInterval.lean

unitInterval.le_one'

[]

https://github.com/leanprover-community/mathlib4

3ce43c18f614b76e161f911b75a3e1ef641620ff

Mathlib/FieldTheory/Subfield.lean

Subfield.closure_union

[]

https://github.com/leanprover-community/mathlib4

3ce43c18f614b76e161f911b75a3e1ef641620ff

Mathlib/LinearAlgebra/FreeModule/Finite/Rank.lean

FiniteDimensional.finrank_finsupp

[{"tactic": "rw [finrank, rank_finsupp_self, \u2190 mk_toNat_eq_card, toNat_lift]", "annotated_tactic": ["rw [<a>finrank</a>, <a>rank_finsupp_self</a>, \u2190 <a>mk_toNat_eq_card</a>, <a>toNat_lift</a>]", [{"full_name": "FiniteDimensional.finrank", "def_path": "Mathlib/LinearAlgebra/Finrank.lean", "def_pos": [58, 19], "def_end_pos": [58, 26]}, {"full_name": "rank_finsupp_self", "def_path": "Mathlib/LinearAlgebra/FreeModule/Rank.lean", "def_pos": [52, 9], "def_end_pos": [52, 26]}, {"full_name": "Cardinal.mk_toNat_eq_card", "def_path": "Mathlib/SetTheory/Cardinal/Basic.lean", "def_pos": [1782, 9], "def_end_pos": [1782, 25]}, {"full_name": "Cardinal.toNat_lift", "def_path": "Mathlib/SetTheory/Cardinal/Basic.lean", "def_pos": [1817, 9], "def_end_pos": [1817, 19]}]], "state_before": "R : Type u\nM : Type v\nN : Type w\ninst\u271d\u00b9\u2070 : Ring R\ninst\u271d\u2079 : StrongRankCondition R\ninst\u271d\u2078 : AddCommGroup M\ninst\u271d\u2077 : Module R M\ninst\u271d\u2076 : Module.Free R M\ninst\u271d\u2075 : Module.Finite R M\ninst\u271d\u2074 : AddCommGroup N\ninst\u271d\u00b3 : Module R N\ninst\u271d\u00b2 : Module.Free R N\ninst\u271d\u00b9 : Module.Finite R N\n\u03b9 : Type v\ninst\u271d : Fintype \u03b9\n\u22a2 finrank R (\u03b9 \u2192\u2080 R) = card \u03b9", "state_after": "no goals"}]

https://github.com/leanprover-community/mathlib4

3ce43c18f614b76e161f911b75a3e1ef641620ff

Mathlib/Order/WellFoundedSet.lean

Set.IsPwo.isWf

[{"tactic": "simpa only [\u2190 lt_iff_le_not_le] using h.wellFoundedOn", "annotated_tactic": ["simpa only [\u2190 <a>lt_iff_le_not_le</a>] using h.wellFoundedOn", [{"full_name": "lt_iff_le_not_le", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [60, 9], "def_end_pos": [60, 25]}]], "state_before": "\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03c0 : \u03b9 \u2192 Type u_5\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\ns t : Set \u03b1\nh : IsPwo s\n\u22a2 IsWf s", "state_after": "no goals"}]

https://github.com/leanprover-community/mathlib4

3ce43c18f614b76e161f911b75a3e1ef641620ff

Mathlib/Topology/Category/TopCat/Limits/Pullbacks.lean

TopCat.pullback_snd_range

[{"tactic": "ext y", "annotated_tactic": ["ext y", []], "state_before": "J : Type v\ninst\u271d : SmallCategory J\nX\u271d Y\u271d Z : TopCat\nX Y S : TopCat\nf : X \u27f6 S\ng : Y \u27f6 S\n\u22a2 Set.range \u2191pullback.snd = {y | \u2203 x, \u2191f x = \u2191g y}", "state_after": "case h\nJ : Type v\ninst\u271d : SmallCategory J\nX\u271d Y\u271d Z : TopCat\nX Y S : TopCat\nf : X \u27f6 S\ng : Y \u27f6 S\ny : (forget TopCat).obj Y\n\u22a2 y \u2208 Set.range \u2191pullback.snd \u2194 y \u2208 {y | \u2203 x, \u2191f x = \u2191g y}"}, {"tactic": "constructor", "annotated_tactic": ["constructor", []], "state_before": "case h\nJ : Type v\ninst\u271d : SmallCategory J\nX\u271d Y\u271d Z : TopCat\nX Y S : TopCat\nf : X \u27f6 S\ng : Y \u27f6 S\ny : (forget TopCat).obj Y\n\u22a2 y \u2208 Set.range \u2191pullback.snd \u2194 y \u2208 {y | \u2203 x, \u2191f x = \u2191g y}", "state_after": "case h.mp\nJ : Type v\ninst\u271d : SmallCategory J\nX\u271d Y\u271d Z : TopCat\nX Y S : TopCat\nf : X \u27f6 S\ng : Y \u27f6 S\ny : (forget TopCat).obj Y\n\u22a2 y \u2208 Set.range \u2191pullback.snd \u2192 y \u2208 {y | \u2203 x, \u2191f x = \u2191g y}\n\ncase h.mpr\nJ : Type v\ninst\u271d : SmallCategory J\nX\u271d Y\u271d Z : TopCat\nX Y S : TopCat\nf : X \u27f6 S\ng : Y \u27f6 S\ny : (forget TopCat).obj Y\n\u22a2 y \u2208 {y | \u2203 x, \u2191f x = \u2191g y} \u2192 y \u2208 Set.range \u2191pullback.snd"}, {"tactic": "rintro \u27e8x, rfl\u27e9", "annotated_tactic": ["rintro \u27e8x, rfl\u27e9", []], "state_before": "case h.mp\nJ : Type v\ninst\u271d : SmallCategory J\nX\u271d Y\u271d Z : TopCat\nX Y S : TopCat\nf : X \u27f6 S\ng : Y \u27f6 S\ny : (forget TopCat).obj Y\n\u22a2 y \u2208 Set.range \u2191pullback.snd \u2192 y \u2208 {y | \u2203 x, \u2191f x = \u2191g y}", "state_after": "case h.mp.intro\nJ : Type v\ninst\u271d : SmallCategory J\nX\u271d Y\u271d Z : TopCat\nX Y S : TopCat\nf : X \u27f6 S\ng : Y \u27f6 S\nx : (forget TopCat).obj (pullback f g)\n\u22a2 \u2191pullback.snd x \u2208 {y | \u2203 x, \u2191f x = \u2191g y}"}, {"tactic": "use (pullback.fst : pullback f g \u27f6 _) x", "annotated_tactic": ["use (<a>pullback.fst</a> : <a>pullback</a> f g \u27f6 _) x", [{"full_name": "CategoryTheory.Limits.pullback.fst", "def_path": "Mathlib/CategoryTheory/Limits/Shapes/Pullbacks.lean", "def_pos": [1113, 8], "def_end_pos": [1113, 20]}, {"full_name": "CategoryTheory.Limits.pullback", "def_path": "Mathlib/CategoryTheory/Limits/Shapes/Pullbacks.lean", "def_pos": [1103, 8], "def_end_pos": [1103, 16]}]], "state_before": "case h.mp.intro\nJ : Type v\ninst\u271d : SmallCategory J\nX\u271d Y\u271d Z : TopCat\nX Y S : TopCat\nf : X \u27f6 S\ng : Y \u27f6 S\nx : (forget TopCat).obj (pullback f g)\n\u22a2 \u2191pullback.snd x \u2208 {y | \u2203 x, \u2191f x = \u2191g y}", "state_after": "case h\nJ : Type v\ninst\u271d : SmallCategory J\nX\u271d Y\u271d Z : TopCat\nX Y S : TopCat\nf : X \u27f6 S\ng : Y \u27f6 S\nx : (forget TopCat).obj (pullback f g)\n\u22a2 \u2191f (\u2191pullback.fst x) = \u2191g (\u2191pullback.snd x)"}, {"tactic": "exact ConcreteCategory.congr_hom pullback.condition x", "annotated_tactic": ["exact <a>ConcreteCategory.congr_hom</a> <a>pullback.condition</a> x", [{"full_name": "CategoryTheory.ConcreteCategory.congr_hom", "def_path": "Mathlib/CategoryTheory/ConcreteCategory/Basic.lean", "def_pos": [144, 9], "def_end_pos": [144, 35]}, {"full_name": "CategoryTheory.Limits.pullback.condition", "def_path": "Mathlib/CategoryTheory/Limits/Shapes/Pullbacks.lean", "def_pos": [1209, 9], "def_end_pos": [1209, 27]}]], "state_before": "case h\nJ : Type v\ninst\u271d : SmallCategory J\nX\u271d Y\u271d Z : TopCat\nX Y S : TopCat\nf : X \u27f6 S\ng : Y \u27f6 S\nx : (forget TopCat).obj (pullback f g)\n\u22a2 \u2191f (\u2191pullback.fst x) = \u2191g (\u2191pullback.snd x)", "state_after": "no goals"}, {"tactic": "rintro \u27e8x, eq\u27e9", "annotated_tactic": ["rintro \u27e8x, eq\u27e9", []], "state_before": "case h.mpr\nJ : Type v\ninst\u271d : SmallCategory J\nX\u271d Y\u271d Z : TopCat\nX Y S : TopCat\nf : X \u27f6 S\ng : Y \u27f6 S\ny : (forget TopCat).obj Y\n\u22a2 y \u2208 {y | \u2203 x, \u2191f x = \u2191g y} \u2192 y \u2208 Set.range \u2191pullback.snd", "state_after": "case h.mpr.intro\nJ : Type v\ninst\u271d : SmallCategory J\nX\u271d Y\u271d Z : TopCat\nX Y S : TopCat\nf : X \u27f6 S\ng : Y \u27f6 S\ny : (forget TopCat).obj Y\nx : \u2191X\neq : \u2191f x = \u2191g y\n\u22a2 y \u2208 Set.range \u2191pullback.snd"}, {"tactic": "use (TopCat.pullbackIsoProdSubtype f g).inv \u27e8\u27e8x, y\u27e9, eq\u27e9", "annotated_tactic": ["use (<a>TopCat.pullbackIsoProdSubtype</a> f g).<a>inv</a> \u27e8\u27e8x, y\u27e9, eq\u27e9", [{"full_name": "TopCat.pullbackIsoProdSubtype", "def_path": "Mathlib/Topology/Category/TopCat/Limits/Pullbacks.lean", "def_pos": [96, 5], "def_end_pos": [96, 27]}, {"full_name": "CategoryTheory.Iso.inv", "def_path": "Mathlib/CategoryTheory/Iso.lean", "def_pos": [55, 3], "def_end_pos": [55, 6]}]], "state_before": "case h.mpr.intro\nJ : Type v\ninst\u271d : SmallCategory J\nX\u271d Y\u271d Z : TopCat\nX Y S : TopCat\nf : X \u27f6 S\ng : Y \u27f6 S\ny : (forget TopCat).obj Y\nx : \u2191X\neq : \u2191f x = \u2191g y\n\u22a2 y \u2208 Set.range \u2191pullback.snd", "state_after": "case h\nJ : Type v\ninst\u271d : SmallCategory J\nX\u271d Y\u271d Z : TopCat\nX Y S : TopCat\nf : X \u27f6 S\ng : Y \u27f6 S\ny : (forget TopCat).obj Y\nx : \u2191X\neq : \u2191f x = \u2191g y\n\u22a2 \u2191pullback.snd (\u2191(pullbackIsoProdSubtype f g).inv { val := (x, y), property := eq }) = y"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case h\nJ : Type v\ninst\u271d : SmallCategory J\nX\u271d Y\u271d Z : TopCat\nX Y S : TopCat\nf : X \u27f6 S\ng : Y \u27f6 S\ny : (forget TopCat).obj Y\nx : \u2191X\neq : \u2191f x = \u2191g y\n\u22a2 \u2191pullback.snd (\u2191(pullbackIsoProdSubtype f g).inv { val := (x, y), property := eq }) = y", "state_after": "no goals"}]

https://github.com/leanprover-community/mathlib4

3ce43c18f614b76e161f911b75a3e1ef641620ff

Mathlib/Analysis/Calculus/FDeriv/Mul.lean

HasFDerivAt.mul_const'

[]

https://github.com/leanprover-community/mathlib4

3ce43c18f614b76e161f911b75a3e1ef641620ff

Mathlib/RingTheory/PowerSeries/Basic.lean

MvPowerSeries.coeff_index_single_self_X

[]

https://github.com/leanprover-community/mathlib4

3ce43c18f614b76e161f911b75a3e1ef641620ff

Mathlib/MeasureTheory/Measure/MeasureSpace.lean

MeasureTheory.Measure.restrict_toMeasurable

[{"tactic": "rw [restrict_apply ht, restrict_apply ht, inter_comm, measure_toMeasurable_inter ht h,\n inter_comm]", "annotated_tactic": ["rw [<a>restrict_apply</a> ht, <a>restrict_apply</a> ht, <a>inter_comm</a>, <a>measure_toMeasurable_inter</a> ht h,\n <a>inter_comm</a>]", [{"full_name": "MeasureTheory.Measure.restrict_apply", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1533, 9], "def_end_pos": [1533, 23]}, {"full_name": "MeasureTheory.Measure.restrict_apply", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1533, 9], "def_end_pos": [1533, 23]}, {"full_name": "Set.inter_comm", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [940, 9], "def_end_pos": [940, 19]}, {"full_name": "MeasureTheory.Measure.measure_toMeasurable_inter", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [740, 9], "def_end_pos": [740, 35]}, {"full_name": "Set.inter_comm", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [940, 9], "def_end_pos": [940, 19]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t\u271d : Set \u03b1\nh : \u2191\u2191\u03bc s \u2260 \u22a4\nt : Set \u03b1\nht : MeasurableSet t\n\u22a2 \u2191\u2191(restrict \u03bc (toMeasurable \u03bc s)) t = \u2191\u2191(restrict \u03bc s) t", "state_after": "no goals"}]

https://github.com/leanprover-community/mathlib4

3ce43c18f614b76e161f911b75a3e1ef641620ff

Mathlib/RingTheory/UniqueFactorizationDomain.lean

WfDvdMonoid.exists_factors

[{"tactic": "rw [s.prod_cons i]", "annotated_tactic": ["rw [s.prod_cons i]", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : CommMonoidWithZero \u03b1\ninst\u271d : WfDvdMonoid \u03b1\na\u271d a i : \u03b1\nha0 : a \u2260 0\nhi : Irreducible i\nih : a \u2260 0 \u2192 \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Irreducible b) \u2227 Multiset.prod f ~\u1d64 a\nx\u271d : i * a \u2260 0\ns : Multiset \u03b1\nhs : (\u2200 (b : \u03b1), b \u2208 s \u2192 Irreducible b) \u2227 Multiset.prod s ~\u1d64 a\n\u22a2 Multiset.prod (i ::\u2098 s) ~\u1d64 i * a", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b9 : CommMonoidWithZero \u03b1\ninst\u271d : WfDvdMonoid \u03b1\na\u271d a i : \u03b1\nha0 : a \u2260 0\nhi : Irreducible i\nih : a \u2260 0 \u2192 \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Irreducible b) \u2227 Multiset.prod f ~\u1d64 a\nx\u271d : i * a \u2260 0\ns : Multiset \u03b1\nhs : (\u2200 (b : \u03b1), b \u2208 s \u2192 Irreducible b) \u2227 Multiset.prod s ~\u1d64 a\n\u22a2 i * Multiset.prod s ~\u1d64 i * a"}, {"tactic": "exact hs.2.mul_left i", "annotated_tactic": ["exact hs.2.<a>mul_left</a> i", [{"full_name": "Associated.mul_left", "def_path": "Mathlib/Algebra/Associated.lean", "def_pos": [528, 9], "def_end_pos": [528, 28]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : CommMonoidWithZero \u03b1\ninst\u271d : WfDvdMonoid \u03b1\na\u271d a i : \u03b1\nha0 : a \u2260 0\nhi : Irreducible i\nih : a \u2260 0 \u2192 \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Irreducible b) \u2227 Multiset.prod f ~\u1d64 a\nx\u271d : i * a \u2260 0\ns : Multiset \u03b1\nhs : (\u2200 (b : \u03b1), b \u2208 s \u2192 Irreducible b) \u2227 Multiset.prod s ~\u1d64 a\n\u22a2 i * Multiset.prod s ~\u1d64 i * a", "state_after": "no goals"}]

https://github.com/leanprover-community/mathlib4

3ce43c18f614b76e161f911b75a3e1ef641620ff

Mathlib/Geometry/Manifold/SmoothManifoldWithCorners.lean

ModelWithCorners.mk_coe

[]

https://github.com/leanprover-community/mathlib4

3ce43c18f614b76e161f911b75a3e1ef641620ff

Mathlib/Logic/Equiv/Basic.lean

Equiv.isEmpty_congr

[]

https://github.com/leanprover-community/mathlib4

3ce43c18f614b76e161f911b75a3e1ef641620ff

Mathlib/Data/Polynomial/Degree/Definitions.lean

Polynomial.natDegree_mem_support_of_nonzero

[{"tactic": "rw [mem_support_iff]", "annotated_tactic": ["rw [<a>mem_support_iff</a>]", [{"full_name": "Polynomial.mem_support_iff", "def_path": "Mathlib/Data/Polynomial/Basic.lean", "def_pos": [732, 9], "def_end_pos": [732, 24]}]], "state_before": "R : Type u\nS : Type v\na b c d : R\nn m : \u2115\ninst\u271d : Semiring R\np q : R[X]\n\u03b9 : Type u_1\nH : p \u2260 0\n\u22a2 natDegree p \u2208 support p", "state_after": "R : Type u\nS : Type v\na b c d : R\nn m : \u2115\ninst\u271d : Semiring R\np q : R[X]\n\u03b9 : Type u_1\nH : p \u2260 0\n\u22a2 coeff p (natDegree p) \u2260 0"}, {"tactic": "exact (not_congr leadingCoeff_eq_zero).mpr H", "annotated_tactic": ["exact (<a>not_congr</a> <a>leadingCoeff_eq_zero</a>).<a>mpr</a> H", [{"full_name": "not_congr", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [22, 9], "def_end_pos": [22, 18]}, {"full_name": "Polynomial.leadingCoeff_eq_zero", "def_path": "Mathlib/Data/Polynomial/Degree/Definitions.lean", "def_pos": [670, 9], "def_end_pos": [670, 29]}, {"full_name": "Iff.mpr", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [92, 3], "def_end_pos": [92, 6]}]], "state_before": "R : Type u\nS : Type v\na b c d : R\nn m : \u2115\ninst\u271d : Semiring R\np q : R[X]\n\u03b9 : Type u_1\nH : p \u2260 0\n\u22a2 coeff p (natDegree p) \u2260 0", "state_after": "no goals"}]

https://github.com/leanprover-community/mathlib4

3ce43c18f614b76e161f911b75a3e1ef641620ff

Mathlib/MeasureTheory/Integral/Lebesgue.lean

MeasureTheory.lintegral_eq_zero_iff

[]

https://github.com/leanprover-community/mathlib4

3ce43c18f614b76e161f911b75a3e1ef641620ff

Mathlib/LinearAlgebra/Matrix/PosDef.lean

Matrix.PosDef.det_pos

[{"tactic": "rw [hM.isHermitian.det_eq_prod_eigenvalues]", "annotated_tactic": ["rw [hM.isHermitian.det_eq_prod_eigenvalues]", []], "state_before": "m : Type u_1\nn : Type u_2\nR : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2076 : Fintype m\ninst\u271d\u2075 : Fintype n\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : PartialOrder R\ninst\u271d\u00b2 : StarOrderedRing R\ninst\u271d\u00b9 : IsROrC \ud835\udd5c\nM : Matrix n n \u211d\nhM : PosDef M\ninst\u271d : DecidableEq n\n\u22a2 0 < det M", "state_after": "m : Type u_1\nn : Type u_2\nR : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2076 : Fintype m\ninst\u271d\u2075 : Fintype n\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : PartialOrder R\ninst\u271d\u00b2 : StarOrderedRing R\ninst\u271d\u00b9 : IsROrC \ud835\udd5c\nM : Matrix n n \u211d\nhM : PosDef M\ninst\u271d : DecidableEq n\n\u22a2 0 < Finset.prod Finset.univ fun i => \u2191(IsHermitian.eigenvalues (_ : IsHermitian M) i)"}, {"tactic": "apply Finset.prod_pos", "annotated_tactic": ["apply <a>Finset.prod_pos</a>", [{"full_name": "Finset.prod_pos", "def_path": "Mathlib/Algebra/BigOperators/Order.lean", "def_pos": [648, 9], "def_end_pos": [648, 17]}]], "state_before": "m : Type u_1\nn : Type u_2\nR : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2076 : Fintype m\ninst\u271d\u2075 : Fintype n\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : PartialOrder R\ninst\u271d\u00b2 : StarOrderedRing R\ninst\u271d\u00b9 : IsROrC \ud835\udd5c\nM : Matrix n n \u211d\nhM : PosDef M\ninst\u271d : DecidableEq n\n\u22a2 0 < Finset.prod Finset.univ fun i => \u2191(IsHermitian.eigenvalues (_ : IsHermitian M) i)", "state_after": "case h0\nm : Type u_1\nn : Type u_2\nR : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2076 : Fintype m\ninst\u271d\u2075 : Fintype n\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : PartialOrder R\ninst\u271d\u00b2 : StarOrderedRing R\ninst\u271d\u00b9 : IsROrC \ud835\udd5c\nM : Matrix n n \u211d\nhM : PosDef M\ninst\u271d : DecidableEq n\n\u22a2 \u2200 (i : n), i \u2208 Finset.univ \u2192 0 < \u2191(IsHermitian.eigenvalues (_ : IsHermitian M) i)"}, {"tactic": "intro i _", "annotated_tactic": ["intro i _", []], "state_before": "case h0\nm : Type u_1\nn : Type u_2\nR : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2076 : Fintype m\ninst\u271d\u2075 : Fintype n\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : PartialOrder R\ninst\u271d\u00b2 : StarOrderedRing R\ninst\u271d\u00b9 : IsROrC \ud835\udd5c\nM : Matrix n n \u211d\nhM : PosDef M\ninst\u271d : DecidableEq n\n\u22a2 \u2200 (i : n), i \u2208 Finset.univ \u2192 0 < \u2191(IsHermitian.eigenvalues (_ : IsHermitian M) i)", "state_after": "case h0\nm : Type u_1\nn : Type u_2\nR : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2076 : Fintype m\ninst\u271d\u2075 : Fintype n\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : PartialOrder R\ninst\u271d\u00b2 : StarOrderedRing R\ninst\u271d\u00b9 : IsROrC \ud835\udd5c\nM : Matrix n n \u211d\nhM : PosDef M\ninst\u271d : DecidableEq n\ni : n\na\u271d : i \u2208 Finset.univ\n\u22a2 0 < \u2191(IsHermitian.eigenvalues (_ : IsHermitian M) i)"}, {"tactic": "rw [hM.isHermitian.eigenvalues_eq]", "annotated_tactic": ["rw [hM.isHermitian.eigenvalues_eq]", []], "state_before": "case h0\nm : Type u_1\nn : Type u_2\nR : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2076 : Fintype m\ninst\u271d\u2075 : Fintype n\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : PartialOrder R\ninst\u271d\u00b2 : StarOrderedRing R\ninst\u271d\u00b9 : IsROrC \ud835\udd5c\nM : Matrix n n \u211d\nhM : PosDef M\ninst\u271d : DecidableEq n\ni : n\na\u271d : i \u2208 Finset.univ\n\u22a2 0 < \u2191(IsHermitian.eigenvalues (_ : IsHermitian M) i)", "state_after": "case h0\nm : Type u_1\nn : Type u_2\nR : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2076 : Fintype m\ninst\u271d\u2075 : Fintype n\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : PartialOrder R\ninst\u271d\u00b2 : StarOrderedRing R\ninst\u271d\u00b9 : IsROrC \ud835\udd5c\nM : Matrix n n \u211d\nhM : PosDef M\ninst\u271d : DecidableEq n\ni : n\na\u271d : i \u2208 Finset.univ\n\u22a2 0 <\n \u2191(\u2191IsROrC.re\n (star ((IsHermitian.eigenvectorMatrix (_ : IsHermitian M))\u1d40 i) \u2b1d\u1d65\n mulVec M ((IsHermitian.eigenvectorMatrix (_ : IsHermitian M))\u1d40 i)))"}, {"tactic": "refine hM.2 _ fun h => ?_", "annotated_tactic": ["refine hM.2 _ fun h => ?_", []], "state_before": "case h0\nm : Type u_1\nn : Type u_2\nR : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2076 : Fintype m\ninst\u271d\u2075 : Fintype n\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : PartialOrder R\ninst\u271d\u00b2 : StarOrderedRing R\ninst\u271d\u00b9 : IsROrC \ud835\udd5c\nM : Matrix n n \u211d\nhM : PosDef M\ninst\u271d : DecidableEq n\ni : n\na\u271d : i \u2208 Finset.univ\n\u22a2 0 <\n \u2191(\u2191IsROrC.re\n (star ((IsHermitian.eigenvectorMatrix (_ : IsHermitian M))\u1d40 i) \u2b1d\u1d65\n mulVec M ((IsHermitian.eigenvectorMatrix (_ : IsHermitian M))\u1d40 i)))", "state_after": "case h0\nm : Type u_1\nn : Type u_2\nR : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2076 : Fintype m\ninst\u271d\u2075 : Fintype n\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : PartialOrder R\ninst\u271d\u00b2 : StarOrderedRing R\ninst\u271d\u00b9 : IsROrC \ud835\udd5c\nM : Matrix n n \u211d\nhM : PosDef M\ninst\u271d : DecidableEq n\ni : n\na\u271d : i \u2208 Finset.univ\nh : (IsHermitian.eigenvectorMatrix (_ : IsHermitian M))\u1d40 i = 0\n\u22a2 False"}, {"tactic": "have h_det : hM.isHermitian.eigenvectorMatrix\u1d40.det = 0 :=\n Matrix.det_eq_zero_of_row_eq_zero i fun j => congr_fun h j", "annotated_tactic": ["have h_det : hM.isHermitian.eigenvectorMatrix\u1d40.<a>det</a> = 0 :=\n <a>Matrix.det_eq_zero_of_row_eq_zero</a> i fun j => <a>congr_fun</a> h j", [{"full_name": "Matrix.det", "def_path": "Mathlib/LinearAlgebra/Matrix/Determinant.lean", "def_pos": [65, 8], "def_end_pos": [65, 11]}, {"full_name": "Matrix.det_eq_zero_of_row_eq_zero", "def_path": "Mathlib/LinearAlgebra/Matrix/Determinant.lean", "def_pos": [363, 9], "def_end_pos": [363, 35]}, {"full_name": "congr_fun", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [42, 7], "def_end_pos": [42, 16]}]], "state_before": "case h0\nm : Type u_1\nn : Type u_2\nR : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2076 : Fintype m\ninst\u271d\u2075 : Fintype n\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : PartialOrder R\ninst\u271d\u00b2 : StarOrderedRing R\ninst\u271d\u00b9 : IsROrC \ud835\udd5c\nM : Matrix n n \u211d\nhM : PosDef M\ninst\u271d : DecidableEq n\ni : n\na\u271d : i \u2208 Finset.univ\nh : (IsHermitian.eigenvectorMatrix (_ : IsHermitian M))\u1d40 i = 0\n\u22a2 False", "state_after": "case h0\nm : Type u_1\nn : Type u_2\nR : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2076 : Fintype m\ninst\u271d\u2075 : Fintype n\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : PartialOrder R\ninst\u271d\u00b2 : StarOrderedRing R\ninst\u271d\u00b9 : IsROrC \ud835\udd5c\nM : Matrix n n \u211d\nhM : PosDef M\ninst\u271d : DecidableEq n\ni : n\na\u271d : i \u2208 Finset.univ\nh : (IsHermitian.eigenvectorMatrix (_ : IsHermitian M))\u1d40 i = 0\nh_det : det (IsHermitian.eigenvectorMatrix (_ : IsHermitian M))\u1d40 = 0\n\u22a2 False"}, {"tactic": "simpa only [h_det, not_isUnit_zero] using\n isUnit_det_of_invertible hM.isHermitian.eigenvectorMatrix\u1d40", "annotated_tactic": ["simpa only [h_det, <a>not_isUnit_zero</a>] using\n <a>isUnit_det_of_invertible</a> hM.isHermitian.eigenvectorMatrix\u1d40", [{"full_name": "not_isUnit_zero", "def_path": "Mathlib/Algebra/GroupWithZero/Units/Basic.lean", "def_pos": [76, 9], "def_end_pos": [76, 24]}, {"full_name": "Matrix.isUnit_det_of_invertible", "def_path": "Mathlib/LinearAlgebra/Matrix/NonsingularInverse.lean", "def_pos": [158, 9], "def_end_pos": [158, 33]}]], "state_before": "case h0\nm : Type u_1\nn : Type u_2\nR : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2076 : Fintype m\ninst\u271d\u2075 : Fintype n\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : PartialOrder R\ninst\u271d\u00b2 : StarOrderedRing R\ninst\u271d\u00b9 : IsROrC \ud835\udd5c\nM : Matrix n n \u211d\nhM : PosDef M\ninst\u271d : DecidableEq n\ni : n\na\u271d : i \u2208 Finset.univ\nh : (IsHermitian.eigenvectorMatrix (_ : IsHermitian M))\u1d40 i = 0\nh_det : det (IsHermitian.eigenvectorMatrix (_ : IsHermitian M))\u1d40 = 0\n\u22a2 False", "state_after": "no goals"}]

https://github.com/leanprover-community/mathlib4

3ce43c18f614b76e161f911b75a3e1ef641620ff

Mathlib/NumberTheory/PellMatiyasevic.lean

Pell.xy_coprime

[{"tactic": "let p := pell_eq a1 n", "annotated_tactic": ["let p := <a>pell_eq</a> a1 n", [{"full_name": "Pell.pell_eq", "def_path": "Mathlib/NumberTheory/PellMatiyasevic.lean", "def_pos": [240, 9], "def_end_pos": [240, 16]}]], "state_before": "a : \u2115\na1 : 1 < a\nn k : \u2115\nx\u271d : Nat.Prime k\nkx : k \u2223 xn a1 n\nky : k \u2223 yn a1 n\n\u22a2 k \u2223 1", "state_after": "a : \u2115\na1 : 1 < a\nn k : \u2115\nx\u271d : Nat.Prime k\nkx : k \u2223 xn a1 n\nky : k \u2223 yn a1 n\np : xn a1 n * xn a1 n - Pell.d a1 * yn a1 n * yn a1 n = 1 := pell_eq a1 n\n\u22a2 k \u2223 1"}, {"tactic": "rw [\u2190 p]", "annotated_tactic": ["rw [\u2190 p]", []], "state_before": "a : \u2115\na1 : 1 < a\nn k : \u2115\nx\u271d : Nat.Prime k\nkx : k \u2223 xn a1 n\nky : k \u2223 yn a1 n\np : xn a1 n * xn a1 n - Pell.d a1 * yn a1 n * yn a1 n = 1 := pell_eq a1 n\n\u22a2 k \u2223 1", "state_after": "a : \u2115\na1 : 1 < a\nn k : \u2115\nx\u271d : Nat.Prime k\nkx : k \u2223 xn a1 n\nky : k \u2223 yn a1 n\np : xn a1 n * xn a1 n - Pell.d a1 * yn a1 n * yn a1 n = 1 := pell_eq a1 n\n\u22a2 k \u2223 xn a1 n * xn a1 n - Pell.d a1 * yn a1 n * yn a1 n"}, {"tactic": "exact Nat.dvd_sub (le_of_lt <| Nat.lt_of_sub_eq_succ p) (kx.mul_left _) (ky.mul_left _)", "annotated_tactic": ["exact <a>Nat.dvd_sub</a> (<a>le_of_lt</a> <| <a>Nat.lt_of_sub_eq_succ</a> p) (kx.mul_left _) (ky.mul_left _)", [{"full_name": "Nat.dvd_sub", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [890, 9], "def_end_pos": [890, 16]}, {"full_name": "le_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [110, 9], "def_end_pos": [110, 17]}, {"full_name": "Nat.lt_of_sub_eq_succ", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [391, 19], "def_end_pos": [391, 36]}]], "state_before": "a : \u2115\na1 : 1 < a\nn k : \u2115\nx\u271d : Nat.Prime k\nkx : k \u2223 xn a1 n\nky : k \u2223 yn a1 n\np : xn a1 n * xn a1 n - Pell.d a1 * yn a1 n * yn a1 n = 1 := pell_eq a1 n\n\u22a2 k \u2223 xn a1 n * xn a1 n - Pell.d a1 * yn a1 n * yn a1 n", "state_after": "no goals"}]

https://github.com/leanprover-community/mathlib4

3ce43c18f614b76e161f911b75a3e1ef641620ff

Mathlib/Init/Logic.lean

iff_self_iff

[]

https://github.com/leanprover-community/mathlib4

3ce43c18f614b76e161f911b75a3e1ef641620ff

Mathlib/LinearAlgebra/Matrix/Transvection.lean

Matrix.det_transvection_of_ne

[{"tactic": "rw [\u2190 updateRow_eq_transvection i j, det_updateRow_add_smul_self _ h, det_one]", "annotated_tactic": ["rw [\u2190 <a>updateRow_eq_transvection</a> i j, <a>det_updateRow_add_smul_self</a> _ h, <a>det_one</a>]", [{"full_name": "Matrix.updateRow_eq_transvection", "def_path": "Mathlib/LinearAlgebra/Matrix/Transvection.lean", "def_pos": [96, 9], "def_end_pos": [96, 34]}, {"full_name": "Matrix.det_updateRow_add_smul_self", "def_path": "Mathlib/LinearAlgebra/Matrix/Determinant.lean", "def_pos": [457, 9], "def_end_pos": [457, 36]}, {"full_name": "Matrix.det_one", "def_path": "Mathlib/LinearAlgebra/Matrix/Determinant.lean", "def_pos": [97, 9], "def_end_pos": [97, 16]}]], "state_before": "n : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u2074 : Field \ud835\udd5c\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : DecidableEq p\ninst\u271d\u00b9 : CommRing R\ni j : n\ninst\u271d : Fintype n\nh : i \u2260 j\nc : R\n\u22a2 det (transvection i j c) = 1", "state_after": "no goals"}]

https://github.com/leanprover-community/mathlib4

3ce43c18f614b76e161f911b75a3e1ef641620ff

Mathlib/Topology/FiberBundle/Trivialization.lean

Pretrivialization.symm_trans_source_eq

[{"tactic": "rw [LocalEquiv.trans_source, e'.source_eq, LocalEquiv.symm_source, e.target_eq, inter_comm,\n e.preimage_symm_proj_inter, inter_comm]", "annotated_tactic": ["rw [<a>LocalEquiv.trans_source</a>, e'.source_eq, <a>LocalEquiv.symm_source</a>, e.target_eq, <a>inter_comm</a>,\n e.preimage_symm_proj_inter, <a>inter_comm</a>]", [{"full_name": "LocalEquiv.trans_source", "def_path": "Mathlib/Logic/Equiv/LocalEquiv.lean", "def_pos": [707, 9], "def_end_pos": [707, 21]}, {"full_name": "LocalEquiv.symm_source", "def_path": "Mathlib/Logic/Equiv/LocalEquiv.lean", "def_pos": [317, 9], "def_end_pos": [317, 20]}, {"full_name": "Set.inter_comm", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [940, 9], "def_end_pos": [940, 19]}, {"full_name": "Set.inter_comm", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [940, 9], "def_end_pos": [940, 19]}]], "state_before": "\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\nZ : Type u_5\ninst\u271d\u00b9 : TopologicalSpace B\ninst\u271d : TopologicalSpace F\nproj : Z \u2192 B\ne\u271d : Pretrivialization F proj\nx : Z\ne e' : Pretrivialization F proj\n\u22a2 (LocalEquiv.trans (LocalEquiv.symm e.toLocalEquiv) e'.toLocalEquiv).source = (e.baseSet \u2229 e'.baseSet) \u00d7\u02e2 univ", "state_after": "no goals"}]

https://github.com/leanprover-community/mathlib4

3ce43c18f614b76e161f911b75a3e1ef641620ff

Mathlib/Order/Filter/Pi.lean

Filter.hasBasis_pi

[{"tactic": "simpa [Set.pi_def] using hasBasis_iInf' fun i => (h i).comap (eval i : (\u2200 j, \u03b1 j) \u2192 \u03b1 i)", "annotated_tactic": ["simpa [<a>Set.pi_def</a>] using <a>hasBasis_iInf'</a> fun i => (h i).<a>comap</a> (<a>eval</a> i : (\u2200 j, \u03b1 j) \u2192 \u03b1 i)", [{"full_name": "Set.pi_def", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [2163, 9], "def_end_pos": [2163, 15]}, {"full_name": "Filter.hasBasis_iInf'", "def_path": "Mathlib/Order/Filter/Bases.lean", "def_pos": [497, 9], "def_end_pos": [497, 23]}, {"full_name": "Filter.HasBasis.comap", "def_path": "Mathlib/Order/Filter/Bases.lean", "def_pos": [793, 9], "def_end_pos": [793, 23]}, {"full_name": "Function.eval", "def_path": "Mathlib/Logic/Function/Basic.lean", "def_pos": [29, 24], "def_end_pos": [29, 28]}]], "state_before": "\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\nf f\u2081 f\u2082 : (i : \u03b9) \u2192 Filter (\u03b1 i)\ns\u271d : (i : \u03b9) \u2192 Set (\u03b1 i)\n\u03b9' : \u03b9 \u2192 Type\ns : (i : \u03b9) \u2192 \u03b9' i \u2192 Set (\u03b1 i)\np : (i : \u03b9) \u2192 \u03b9' i \u2192 Prop\nh : \u2200 (i : \u03b9), HasBasis (f i) (p i) (s i)\n\u22a2 HasBasis (pi f) (fun If => Set.Finite If.1 \u2227 \u2200 (i : \u03b9), i \u2208 If.1 \u2192 p i (If.2 i)) fun If =>\n Set.pi If.1 fun i => s i (If.2 i)", "state_after": "no goals"}]

https://github.com/leanprover-community/mathlib4

3ce43c18f614b76e161f911b75a3e1ef641620ff

Mathlib/Data/Int/Cast/Lemmas.lean

Int.cast_one_le_of_pos

[{"tactic": "exact_mod_cast Int.add_one_le_of_lt h", "annotated_tactic": ["exact_mod_cast <a>Int.add_one_le_of_lt</a> h", [{"full_name": "Int.add_one_le_of_lt", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [1255, 9], "def_end_pos": [1255, 25]}]], "state_before": "F : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d : LinearOrderedRing \u03b1\na b n : \u2124\nh : 0 < a\n\u22a2 1 \u2264 \u2191a", "state_after": "no goals"}]

https://github.com/leanprover-community/mathlib4

3ce43c18f614b76e161f911b75a3e1ef641620ff

Mathlib/MeasureTheory/Measure/MeasureSpace.lean

MeasureTheory.iUnion_spanningSets

[{"tactic": "simp_rw [spanningSets, iUnion_accumulate, \u03bc.toFiniteSpanningSetsIn.spanning]", "annotated_tactic": ["simp_rw [<a>spanningSets</a>, <a>iUnion_accumulate</a>, \u03bc.toFiniteSpanningSetsIn.spanning]", [{"full_name": "MeasureTheory.spanningSets", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [3316, 5], "def_end_pos": [3316, 17]}, {"full_name": "Set.iUnion_accumulate", "def_path": "Mathlib/Data/Set/Accumulate.lean", "def_pos": [51, 9], "def_end_pos": [51, 26]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\n\u22a2 \u22c3 i, spanningSets \u03bc i = univ", "state_after": "no goals"}]