url
stringclasses 3
values | commit
stringclasses 3
values | file_path
stringlengths 20
79
| full_name
stringlengths 3
115
| start
sequence | end
sequence | traced_tactics
stringlengths 2
997k
|
|---|---|---|---|---|---|---|
https://github.com/leanprover-community/mathlib4 | 3ce43c18f614b76e161f911b75a3e1ef641620ff | Mathlib/Analysis/Normed/Group/Basic.lean | nndist_nnnorm_nnnorm_le' | [
970,
1
] | [
971,
48
] | [] |
https://github.com/leanprover-community/mathlib4 | 3ce43c18f614b76e161f911b75a3e1ef641620ff | Mathlib/Topology/Homeomorph.lean | Homeomorph.quotientMap | [
237,
11
] | [
239,
47
] | [{"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 | [
54,
1
] | [
112,
35
] | [{"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' | [
537,
1
] | [
552,
87
] | [{"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 | [
663,
1
] | [
665,
57
] | [] |
https://github.com/leanprover-community/mathlib4 | 3ce43c18f614b76e161f911b75a3e1ef641620ff | Mathlib/CategoryTheory/Preadditive/AdditiveFunctor.lean | CategoryTheory.Functor.map_nsmul | [
89,
1
] | [
90,
63
] | [] |
https://github.com/leanprover-community/mathlib4 | 3ce43c18f614b76e161f911b75a3e1ef641620ff | Mathlib/Analysis/SumIntegralComparisons.lean | MonotoneOn.integral_le_sum | [
165,
1
] | [
168,
31
] | [{"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 | [
94,
1
] | [
95,
23
] | [] |
https://github.com/leanprover-community/mathlib4 | 3ce43c18f614b76e161f911b75a3e1ef641620ff | Mathlib/Data/Polynomial/Degree/Definitions.lean | Polynomial.coeff_mul_X_sub_C | [
532,
1
] | [
533,
89
] | [{"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 | [
135,
1
] | [
137,
22
] | [] |
https://github.com/leanprover-community/mathlib4 | 3ce43c18f614b76e161f911b75a3e1ef641620ff | Mathlib/Data/Polynomial/Expand.lean | Polynomial.expand_eq_zero | [
138,
1
] | [
139,
45
] | [] |
https://github.com/leanprover-community/mathlib4 | 3ce43c18f614b76e161f911b75a3e1ef641620ff | Mathlib/NumberTheory/LegendreSymbol/JacobiSymbol.lean | jacobiSym.value_at | [
325,
1
] | [
331,
79
] | [{"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 | [
1524,
1
] | [
1525,
84
] | [] |
https://github.com/leanprover-community/mathlib4 | 3ce43c18f614b76e161f911b75a3e1ef641620ff | Mathlib/RingTheory/RingHomProperties.lean | RingHom.RespectsIso.cancel_left_isIso | [
50,
1
] | [
54,
89
] | [{"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 | [
2522,
1
] | [
2524,
69
] | [{"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 | [
2307,
1
] | [
2307,
77
] | [{"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 | [
1141,
1
] | [
1148,
54
] | [{"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 | [
900,
1
] | [
901,
47
] | [{"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 | [
232,
1
] | [
243,
13
] | [{"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 | [
1121,
1
] | [
1123,
26
] | [] |
https://github.com/leanprover-community/mathlib4 | 3ce43c18f614b76e161f911b75a3e1ef641620ff | Mathlib/Data/Matrix/PEquiv.lean | PEquiv.toMatrix_injective | [
124,
1
] | [
140,
33
] | [{"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 | [
204,
11
] | [
204,
94
] | [] |
https://github.com/leanprover-community/mathlib4 | 3ce43c18f614b76e161f911b75a3e1ef641620ff | Mathlib/ModelTheory/Substructures.lean | FirstOrder.Language.Substructure.closure_mono | [
289,
1
] | [
290,
25
] | [] |
https://github.com/leanprover-community/mathlib4 | 3ce43c18f614b76e161f911b75a3e1ef641620ff | Mathlib/Algebra/Order/Ring/Defs.lean | nonneg_of_mul_nonneg_left | [
812,
1
] | [
813,
64
] | [] |
https://github.com/leanprover-community/mathlib4 | 3ce43c18f614b76e161f911b75a3e1ef641620ff | Mathlib/Data/Sigma/Basic.lean | Sigma.mk.inj_iff | [
56,
1
] | [
59,
49
] | [{"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 | [
557,
1
] | [
558,
13
] | [] |
https://github.com/leanprover-community/mathlib4 | 3ce43c18f614b76e161f911b75a3e1ef641620ff | Mathlib/Order/Filter/AtTopBot.lean | Filter.Ici_mem_atTop | [
57,
1
] | [
58,
14
] | [] |
https://github.com/leanprover-community/mathlib4 | 3ce43c18f614b76e161f911b75a3e1ef641620ff | Mathlib/CategoryTheory/Limits/Shapes/ZeroMorphisms.lean | CategoryTheory.Limits.zero_comp | [
73,
1
] | [
75,
33
] | [] |
https://github.com/leanprover-community/mathlib4 | 3ce43c18f614b76e161f911b75a3e1ef641620ff | Mathlib/Data/Set/Pointwise/Interval.lean | Set.preimage_neg_uIcc | [
437,
1
] | [
438,
72
] | [{"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 | [
59,
1
] | [
61,
45
] | [{"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 | [
215,
1
] | [
219,
75
] | [{"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 | [
520,
1
] | [
522,
36
] | [] |
https://github.com/leanprover-community/mathlib4 | 3ce43c18f614b76e161f911b75a3e1ef641620ff | Mathlib/Algebra/Group/Units.lean | Units.mk_val | [
164,
1
] | [
165,
10
] | [] |
https://github.com/leanprover-community/mathlib4 | 3ce43c18f614b76e161f911b75a3e1ef641620ff | Mathlib/Data/Set/Sups.lean | Set.infs_inter_subset_right | [
335,
1
] | [
336,
28
] | [] |
https://github.com/leanprover-community/mathlib4 | 3ce43c18f614b76e161f911b75a3e1ef641620ff | Mathlib/RingTheory/DiscreteValuationRing/Basic.lean | DiscreteValuationRing.addVal_uniformizer | [
428,
1
] | [
430,
30
] | [{"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 | [
255,
1
] | [
257,
85
] | [{"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 | [
85,
1
] | [
86,
6
] | [] |
https://github.com/leanprover-community/mathlib4 | 3ce43c18f614b76e161f911b75a3e1ef641620ff | Mathlib/MeasureTheory/MeasurableSpace/Defs.lean | MeasurableSet.empty | [
80,
1
] | [
81,
40
] | [] |
https://github.com/leanprover-community/mathlib4 | 3ce43c18f614b76e161f911b75a3e1ef641620ff | Mathlib/Analysis/Convex/Between.lean | wbtw_or_wbtw_smul_vadd_of_nonneg | [
770,
1
] | [
774,
70
] | [{"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 | [
396,
1
] | [
403,
43
] | [{"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 | [
1115,
1
] | [
1116,
30
] | [] |
https://github.com/leanprover-community/mathlib4 | 3ce43c18f614b76e161f911b75a3e1ef641620ff | Mathlib/Algebra/Order/Group/Defs.lean | mul_inv_lt_iff_lt_mul | [
305,
1
] | [
306,
54
] | [{"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"}] |
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