url
stringclasses 3
values | commit
stringclasses 3
values | file_path
stringlengths 20
79
| full_name
stringlengths 3
115
| start
list | end
list | traced_tactics
stringlengths 2
997k
|
|---|---|---|---|---|---|---|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/FieldTheory/Finite/Basic.lean
|
FiniteField.sum_subgroup_pow_eq_zero
|
[
178,
1
] |
[
213,
12
] |
[{"tactic": "nontriviality K", "annotated_tactic": ["nontriviality K", []], "state_before": "K : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : CommRing K\ninst\u271d\u00b9 : NoZeroDivisors K\nG : Subgroup K\u02e3\ninst\u271d : Fintype { x // x \u2208 G }\nk : \u2115\nk_pos : k \u2260 0\nk_lt_card_G : k < Fintype.card { x // x \u2208 G }\n\u22a2 \u2211 x : { x // x \u2208 G }, \u2191\u2191x ^ k = 0", "state_after": "K : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : CommRing K\ninst\u271d\u00b9 : NoZeroDivisors K\nG : Subgroup K\u02e3\ninst\u271d : Fintype { x // x \u2208 G }\nk : \u2115\nk_pos : k \u2260 0\nk_lt_card_G : k < Fintype.card { x // x \u2208 G }\n\u271d : Nontrivial K\n\u22a2 \u2211 x : { x // x \u2208 G }, \u2191\u2191x ^ k = 0"}, {"tactic": "have := NoZeroDivisors.to_isDomain K", "annotated_tactic": ["have := <a>NoZeroDivisors.to_isDomain</a> K", [{"full_name": "NoZeroDivisors.to_isDomain", "def_path": "Mathlib/Algebra/Ring/Basic.lean", "def_pos": [191, 7], "def_end_pos": [191, 33]}]], "state_before": "K : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : CommRing K\ninst\u271d\u00b9 : NoZeroDivisors K\nG : Subgroup K\u02e3\ninst\u271d : Fintype { x // x \u2208 G }\nk : \u2115\nk_pos : k \u2260 0\nk_lt_card_G : k < Fintype.card { x // x \u2208 G }\n\u271d : Nontrivial K\n\u22a2 \u2211 x : { x // x \u2208 G }, \u2191\u2191x ^ k = 0", "state_after": "K : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : CommRing K\ninst\u271d\u00b9 : NoZeroDivisors K\nG : Subgroup K\u02e3\ninst\u271d : Fintype { x // x \u2208 G }\nk : \u2115\nk_pos : k \u2260 0\nk_lt_card_G : k < Fintype.card { x // x \u2208 G }\n\u271d : Nontrivial K\nthis : IsDomain K\n\u22a2 \u2211 x : { x // x \u2208 G }, \u2191\u2191x ^ k = 0"}, {"tactic": "rcases (exists_pow_ne_one_of_isCyclic k_pos k_lt_card_G) with \u27e8a, ha\u27e9", "annotated_tactic": ["rcases (<a>exists_pow_ne_one_of_isCyclic</a> k_pos k_lt_card_G) with \u27e8a, ha\u27e9", [{"full_name": "exists_pow_ne_one_of_isCyclic", "def_path": "Mathlib/GroupTheory/SpecificGroups/Cyclic.lean", "def_pos": [156, 9], "def_end_pos": [156, 38]}]], "state_before": "K : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : CommRing K\ninst\u271d\u00b9 : NoZeroDivisors K\nG : Subgroup K\u02e3\ninst\u271d : Fintype { x // x \u2208 G }\nk : \u2115\nk_pos : k \u2260 0\nk_lt_card_G : k < Fintype.card { x // x \u2208 G }\n\u271d : Nontrivial K\nthis : IsDomain K\n\u22a2 \u2211 x : { x // x \u2208 G }, \u2191\u2191x ^ k = 0", "state_after": "case intro\nK : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : CommRing K\ninst\u271d\u00b9 : NoZeroDivisors K\nG : Subgroup K\u02e3\ninst\u271d : Fintype { x // x \u2208 G }\nk : \u2115\nk_pos : k \u2260 0\nk_lt_card_G : k < Fintype.card { x // x \u2208 G }\n\u271d : Nontrivial K\nthis : IsDomain K\na : { x // x \u2208 G }\nha : a ^ k \u2260 1\n\u22a2 \u2211 x : { x // x \u2208 G }, \u2191\u2191x ^ k = 0"}, {"tactic": "rw [Finset.sum_eq_multiset_sum]", "annotated_tactic": ["rw [<a>Finset.sum_eq_multiset_sum</a>]", [{"full_name": "Finset.sum_eq_multiset_sum", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [183, 3], "def_end_pos": [183, 14]}]], "state_before": "case intro\nK : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : CommRing K\ninst\u271d\u00b9 : NoZeroDivisors K\nG : Subgroup K\u02e3\ninst\u271d : Fintype { x // x \u2208 G }\nk : \u2115\nk_pos : k \u2260 0\nk_lt_card_G : k < Fintype.card { x // x \u2208 G }\n\u271d : Nontrivial K\nthis : IsDomain K\na : { x // x \u2208 G }\nha : a ^ k \u2260 1\n\u22a2 \u2211 x : { x // x \u2208 G }, \u2191\u2191x ^ k = 0", "state_after": "case intro\nK : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : CommRing K\ninst\u271d\u00b9 : NoZeroDivisors K\nG : Subgroup K\u02e3\ninst\u271d : Fintype { x // x \u2208 G }\nk : \u2115\nk_pos : k \u2260 0\nk_lt_card_G : k < Fintype.card { x // x \u2208 G }\n\u271d : Nontrivial K\nthis : IsDomain K\na : { x // x \u2208 G }\nha : a ^ k \u2260 1\n\u22a2 Multiset.sum (Multiset.map (fun x => \u2191\u2191x ^ k) univ.val) = 0"}, {"tactic": "have h_multiset_map :\n Finset.univ.val.map (fun x : G => ((x : K\u02e3) : K) ^ k) =\n Finset.univ.val.map (fun x : G => ((x : K\u02e3) : K) ^ k * ((a : K\u02e3) : K) ^ k) := by\n simp_rw [\u2190 mul_pow]\n have as_comp :\n (fun x : \u21a5G => (((x : K\u02e3) : K) * ((a : K\u02e3) : K)) ^ k)\n = (fun x : \u21a5G => ((x : K\u02e3) : K) ^ k) \u2218 fun x : \u21a5G => x * a := by\n funext x\n simp only [Function.comp_apply, Submonoid.coe_mul, Subgroup.coe_toSubmonoid, Units.val_mul]\n rw [as_comp, \u2190 Multiset.map_map]\n congr\n rw [eq_comm]\n exact Multiset.map_univ_val_equiv (Equiv.mulRight a)", "annotated_tactic": ["have h_multiset_map :\n Finset.univ.val.map (fun x : G => ((x : K\u02e3) : K) ^ k) =\n Finset.univ.val.map (fun x : G => ((x : K\u02e3) : K) ^ k * ((a : K\u02e3) : K) ^ k) := by\n simp_rw [\u2190 <a>mul_pow</a>]\n have as_comp :\n (fun x : \u21a5G => (((x : K\u02e3) : K) * ((a : K\u02e3) : K)) ^ k)\n = (fun x : \u21a5G => ((x : K\u02e3) : K) ^ k) \u2218 fun x : \u21a5G => x * a := by\n funext x\n simp only [<a>Function.comp_apply</a>, <a>Submonoid.coe_mul</a>, <a>Subgroup.coe_toSubmonoid</a>, <a>Units.val_mul</a>]\n rw [as_comp, \u2190 <a>Multiset.map_map</a>]\n congr\n rw [<a>eq_comm</a>]\n exact <a>Multiset.map_univ_val_equiv</a> (<a>Equiv.mulRight</a> a)", [{"full_name": "mul_pow", "def_path": "Mathlib/Algebra/GroupPower/Basic.lean", "def_pos": [257, 9], "def_end_pos": [257, 16]}, {"full_name": "Function.comp_apply", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [33, 17], "def_end_pos": [33, 36]}, {"full_name": "Submonoid.coe_mul", "def_path": "Mathlib/GroupTheory/Submonoid/Operations.lean", "def_pos": [678, 9], "def_end_pos": [678, 16]}, {"full_name": "Subgroup.coe_toSubmonoid", "def_path": "Mathlib/GroupTheory/Subgroup/Basic.lean", "def_pos": [436, 9], "def_end_pos": [436, 24]}, {"full_name": "Units.val_mul", "def_path": "Mathlib/Algebra/Group/Units.lean", "def_pos": [229, 9], "def_end_pos": [229, 16]}, {"full_name": "Multiset.map_map", "def_path": "Mathlib/Data/Multiset/Basic.lean", "def_pos": [1283, 9], "def_end_pos": [1283, 16]}, {"full_name": "eq_comm", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [104, 9], "def_end_pos": [104, 16]}, {"full_name": "Multiset.map_univ_val_equiv", "def_path": "Mathlib/Data/Fintype/Basic.lean", "def_pos": [1228, 9], "def_end_pos": [1228, 27]}, {"full_name": "Equiv.mulRight", "def_path": "Mathlib/Algebra/Hom/Equiv/Units/Basic.lean", "def_pos": [152, 15], "def_end_pos": [152, 23]}]], "state_before": "case intro\nK : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : CommRing K\ninst\u271d\u00b9 : NoZeroDivisors K\nG : Subgroup K\u02e3\ninst\u271d : Fintype { x // x \u2208 G }\nk : \u2115\nk_pos : k \u2260 0\nk_lt_card_G : k < Fintype.card { x // x \u2208 G }\n\u271d : Nontrivial K\nthis : IsDomain K\na : { x // x \u2208 G }\nha : a ^ k \u2260 1\n\u22a2 Multiset.sum (Multiset.map (fun x => \u2191\u2191x ^ k) univ.val) = 0", "state_after": "case intro\nK : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : CommRing K\ninst\u271d\u00b9 : NoZeroDivisors K\nG : Subgroup K\u02e3\ninst\u271d : Fintype { x // x \u2208 G }\nk : \u2115\nk_pos : k \u2260 0\nk_lt_card_G : k < Fintype.card { x // x \u2208 G }\n\u271d : Nontrivial K\nthis : IsDomain K\na : { x // x \u2208 G }\nha : a ^ k \u2260 1\nh_multiset_map : Multiset.map (fun x => \u2191\u2191x ^ k) univ.val = Multiset.map (fun x => \u2191\u2191x ^ k * \u2191\u2191a ^ k) univ.val\n\u22a2 Multiset.sum (Multiset.map (fun x => \u2191\u2191x ^ k) univ.val) = 0"}, {"tactic": "have h_multiset_map_sum :\n (Multiset.map (fun x : G => ((x : K\u02e3) : K) ^ k) Finset.univ.val).sum =\n (Multiset.map (fun x : G => ((x : K\u02e3) : K) ^ k * ((a : K\u02e3) : K) ^ k) Finset.univ.val).sum", "annotated_tactic": ["have h_multiset_map_sum :\n (<a>Multiset.map</a> (fun x : G => ((x : K\u02e3) : K) ^ k) Finset.univ.val).<a>sum</a> =\n (<a>Multiset.map</a> (fun x : G => ((x : K\u02e3) : K) ^ k * ((a : K\u02e3) : K) ^ k) Finset.univ.val).<a>sum</a>", [{"full_name": "Multiset.map", "def_path": "Mathlib/Data/Multiset/Basic.lean", "def_pos": [1142, 5], "def_end_pos": [1142, 8]}, {"full_name": "Multiset.sum", "def_path": "Mathlib/Algebra/BigOperators/Multiset/Basic.lean", "def_pos": [41, 3], "def_end_pos": [41, 14]}, {"full_name": "Multiset.map", "def_path": "Mathlib/Data/Multiset/Basic.lean", "def_pos": [1142, 5], "def_end_pos": [1142, 8]}, {"full_name": "Multiset.sum", "def_path": "Mathlib/Algebra/BigOperators/Multiset/Basic.lean", "def_pos": [41, 3], "def_end_pos": [41, 14]}]], "state_before": "case intro\nK : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : CommRing K\ninst\u271d\u00b9 : NoZeroDivisors K\nG : Subgroup K\u02e3\ninst\u271d : Fintype { x // x \u2208 G }\nk : \u2115\nk_pos : k \u2260 0\nk_lt_card_G : k < Fintype.card { x // x \u2208 G }\n\u271d : Nontrivial K\nthis : IsDomain K\na : { x // x \u2208 G }\nha : a ^ k \u2260 1\nh_multiset_map : Multiset.map (fun x => \u2191\u2191x ^ k) univ.val = Multiset.map (fun x => \u2191\u2191x ^ k * \u2191\u2191a ^ k) univ.val\n\u22a2 Multiset.sum (Multiset.map (fun x => \u2191\u2191x ^ k) univ.val) = 0", "state_after": "case h_multiset_map_sum\nK : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : CommRing K\ninst\u271d\u00b9 : NoZeroDivisors K\nG : Subgroup K\u02e3\ninst\u271d : Fintype { x // x \u2208 G }\nk : \u2115\nk_pos : k \u2260 0\nk_lt_card_G : k < Fintype.card { x // x \u2208 G }\n\u271d : Nontrivial K\nthis : IsDomain K\na : { x // x \u2208 G }\nha : a ^ k \u2260 1\nh_multiset_map : Multiset.map (fun x => \u2191\u2191x ^ k) univ.val = Multiset.map (fun x => \u2191\u2191x ^ k * \u2191\u2191a ^ k) univ.val\n\u22a2 Multiset.sum (Multiset.map (fun x => \u2191\u2191x ^ k) univ.val) =\n Multiset.sum (Multiset.map (fun x => \u2191\u2191x ^ k * \u2191\u2191a ^ k) univ.val)\n\ncase intro\nK : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : CommRing K\ninst\u271d\u00b9 : NoZeroDivisors K\nG : Subgroup K\u02e3\ninst\u271d : Fintype { x // x \u2208 G }\nk : \u2115\nk_pos : k \u2260 0\nk_lt_card_G : k < Fintype.card { x // x \u2208 G }\n\u271d : Nontrivial K\nthis : IsDomain K\na : { x // x \u2208 G }\nha : a ^ k \u2260 1\nh_multiset_map : Multiset.map (fun x => \u2191\u2191x ^ k) univ.val = Multiset.map (fun x => \u2191\u2191x ^ k * \u2191\u2191a ^ k) univ.val\nh_multiset_map_sum :\n Multiset.sum (Multiset.map (fun x => \u2191\u2191x ^ k) univ.val) =\n Multiset.sum (Multiset.map (fun x => \u2191\u2191x ^ k * \u2191\u2191a ^ k) univ.val)\n\u22a2 Multiset.sum (Multiset.map (fun x => \u2191\u2191x ^ k) univ.val) = 0"}, {"tactic": "rw [h_multiset_map]", "annotated_tactic": ["rw [h_multiset_map]", []], "state_before": "case h_multiset_map_sum\nK : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : CommRing K\ninst\u271d\u00b9 : NoZeroDivisors K\nG : Subgroup K\u02e3\ninst\u271d : Fintype { x // x \u2208 G }\nk : \u2115\nk_pos : k \u2260 0\nk_lt_card_G : k < Fintype.card { x // x \u2208 G }\n\u271d : Nontrivial K\nthis : IsDomain K\na : { x // x \u2208 G }\nha : a ^ k \u2260 1\nh_multiset_map : Multiset.map (fun x => \u2191\u2191x ^ k) univ.val = Multiset.map (fun x => \u2191\u2191x ^ k * \u2191\u2191a ^ k) univ.val\n\u22a2 Multiset.sum (Multiset.map (fun x => \u2191\u2191x ^ k) univ.val) =\n Multiset.sum (Multiset.map (fun x => \u2191\u2191x ^ k * \u2191\u2191a ^ k) univ.val)\n\ncase intro\nK : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : CommRing K\ninst\u271d\u00b9 : NoZeroDivisors K\nG : Subgroup K\u02e3\ninst\u271d : Fintype { x // x \u2208 G }\nk : \u2115\nk_pos : k \u2260 0\nk_lt_card_G : k < Fintype.card { x // x \u2208 G }\n\u271d : Nontrivial K\nthis : IsDomain K\na : { x // x \u2208 G }\nha : a ^ k \u2260 1\nh_multiset_map : Multiset.map (fun x => \u2191\u2191x ^ k) univ.val = Multiset.map (fun x => \u2191\u2191x ^ k * \u2191\u2191a ^ k) univ.val\nh_multiset_map_sum :\n Multiset.sum (Multiset.map (fun x => \u2191\u2191x ^ k) univ.val) =\n Multiset.sum (Multiset.map (fun x => \u2191\u2191x ^ k * \u2191\u2191a ^ k) univ.val)\n\u22a2 Multiset.sum (Multiset.map (fun x => \u2191\u2191x ^ k) univ.val) = 0", "state_after": "case intro\nK : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : CommRing K\ninst\u271d\u00b9 : NoZeroDivisors K\nG : Subgroup K\u02e3\ninst\u271d : Fintype { x // x \u2208 G }\nk : \u2115\nk_pos : k \u2260 0\nk_lt_card_G : k < Fintype.card { x // x \u2208 G }\n\u271d : Nontrivial K\nthis : IsDomain K\na : { x // x \u2208 G }\nha : a ^ k \u2260 1\nh_multiset_map : Multiset.map (fun x => \u2191\u2191x ^ k) univ.val = Multiset.map (fun x => \u2191\u2191x ^ k * \u2191\u2191a ^ k) univ.val\nh_multiset_map_sum :\n Multiset.sum (Multiset.map (fun x => \u2191\u2191x ^ k) univ.val) =\n Multiset.sum (Multiset.map (fun x => \u2191\u2191x ^ k * \u2191\u2191a ^ k) univ.val)\n\u22a2 Multiset.sum (Multiset.map (fun x => \u2191\u2191x ^ k) univ.val) = 0"}, {"tactic": "rw [Multiset.sum_map_mul_right] at h_multiset_map_sum", "annotated_tactic": ["rw [<a>Multiset.sum_map_mul_right</a>] at h_multiset_map_sum", [{"full_name": "Multiset.sum_map_mul_right", "def_path": "Mathlib/Algebra/BigOperators/Multiset/Basic.lean", "def_pos": [340, 9], "def_end_pos": [340, 26]}]], "state_before": "case intro\nK : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : CommRing K\ninst\u271d\u00b9 : NoZeroDivisors K\nG : Subgroup K\u02e3\ninst\u271d : Fintype { x // x \u2208 G }\nk : \u2115\nk_pos : k \u2260 0\nk_lt_card_G : k < Fintype.card { x // x \u2208 G }\n\u271d : Nontrivial K\nthis : IsDomain K\na : { x // x \u2208 G }\nha : a ^ k \u2260 1\nh_multiset_map : Multiset.map (fun x => \u2191\u2191x ^ k) univ.val = Multiset.map (fun x => \u2191\u2191x ^ k * \u2191\u2191a ^ k) univ.val\nh_multiset_map_sum :\n Multiset.sum (Multiset.map (fun x => \u2191\u2191x ^ k) univ.val) =\n Multiset.sum (Multiset.map (fun x => \u2191\u2191x ^ k * \u2191\u2191a ^ k) univ.val)\n\u22a2 Multiset.sum (Multiset.map (fun x => \u2191\u2191x ^ k) univ.val) = 0", "state_after": "case intro\nK : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : CommRing K\ninst\u271d\u00b9 : NoZeroDivisors K\nG : Subgroup K\u02e3\ninst\u271d : Fintype { x // x \u2208 G }\nk : \u2115\nk_pos : k \u2260 0\nk_lt_card_G : k < Fintype.card { x // x \u2208 G }\n\u271d : Nontrivial K\nthis : IsDomain K\na : { x // x \u2208 G }\nha : a ^ k \u2260 1\nh_multiset_map : Multiset.map (fun x => \u2191\u2191x ^ k) univ.val = Multiset.map (fun x => \u2191\u2191x ^ k * \u2191\u2191a ^ k) univ.val\nh_multiset_map_sum :\n Multiset.sum (Multiset.map (fun x => \u2191\u2191x ^ k) univ.val) =\n Multiset.sum (Multiset.map (fun x => \u2191\u2191x ^ k) univ.val) * \u2191\u2191a ^ k\n\u22a2 Multiset.sum (Multiset.map (fun x => \u2191\u2191x ^ k) univ.val) = 0"}, {"tactic": "have hzero : (((a : K\u02e3) : K) ^ k - 1 : K)\n * (Multiset.map (fun i : G => (i.val : K) ^ k) Finset.univ.val).sum = 0 := by\n rw [sub_mul, mul_comm, \u2190 h_multiset_map_sum, one_mul, sub_self]", "annotated_tactic": ["have hzero : (((a : K\u02e3) : K) ^ k - 1 : K)\n * (<a>Multiset.map</a> (fun i : G => (i.val : K) ^ k) Finset.univ.val).<a>sum</a> = 0 := by\n rw [<a>sub_mul</a>, <a>mul_comm</a>, \u2190 h_multiset_map_sum, <a>one_mul</a>, <a>sub_self</a>]", [{"full_name": "Multiset.map", "def_path": "Mathlib/Data/Multiset/Basic.lean", "def_pos": [1142, 5], "def_end_pos": [1142, 8]}, {"full_name": "Multiset.sum", "def_path": "Mathlib/Algebra/BigOperators/Multiset/Basic.lean", "def_pos": [41, 3], "def_end_pos": [41, 14]}, {"full_name": "sub_mul", "def_path": "Mathlib/Algebra/Ring/Defs.lean", "def_pos": [372, 7], "def_end_pos": [372, 14]}, {"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [302, 9], "def_end_pos": [302, 17]}, {"full_name": "one_mul", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [464, 9], "def_end_pos": [464, 16]}, {"full_name": "sub_self", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [734, 30], "def_end_pos": [734, 38]}]], "state_before": "case intro\nK : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : CommRing K\ninst\u271d\u00b9 : NoZeroDivisors K\nG : Subgroup K\u02e3\ninst\u271d : Fintype { x // x \u2208 G }\nk : \u2115\nk_pos : k \u2260 0\nk_lt_card_G : k < Fintype.card { x // x \u2208 G }\n\u271d : Nontrivial K\nthis : IsDomain K\na : { x // x \u2208 G }\nha : a ^ k \u2260 1\nh_multiset_map : Multiset.map (fun x => \u2191\u2191x ^ k) univ.val = Multiset.map (fun x => \u2191\u2191x ^ k * \u2191\u2191a ^ k) univ.val\nh_multiset_map_sum :\n Multiset.sum (Multiset.map (fun x => \u2191\u2191x ^ k) univ.val) =\n Multiset.sum (Multiset.map (fun x => \u2191\u2191x ^ k) univ.val) * \u2191\u2191a ^ k\n\u22a2 Multiset.sum (Multiset.map (fun x => \u2191\u2191x ^ k) univ.val) = 0", "state_after": "case intro\nK : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : CommRing K\ninst\u271d\u00b9 : NoZeroDivisors K\nG : Subgroup K\u02e3\ninst\u271d : Fintype { x // x \u2208 G }\nk : \u2115\nk_pos : k \u2260 0\nk_lt_card_G : k < Fintype.card { x // x \u2208 G }\n\u271d : Nontrivial K\nthis : IsDomain K\na : { x // x \u2208 G }\nha : a ^ k \u2260 1\nh_multiset_map : Multiset.map (fun x => \u2191\u2191x ^ k) univ.val = Multiset.map (fun x => \u2191\u2191x ^ k * \u2191\u2191a ^ k) univ.val\nh_multiset_map_sum :\n Multiset.sum (Multiset.map (fun x => \u2191\u2191x ^ k) univ.val) =\n Multiset.sum (Multiset.map (fun x => \u2191\u2191x ^ k) univ.val) * \u2191\u2191a ^ k\nhzero : (\u2191\u2191a ^ k - 1) * Multiset.sum (Multiset.map (fun i => \u2191\u2191i ^ k) univ.val) = 0\n\u22a2 Multiset.sum (Multiset.map (fun x => \u2191\u2191x ^ k) univ.val) = 0"}, {"tactic": "rw [mul_eq_zero] at hzero", "annotated_tactic": ["rw [<a>mul_eq_zero</a>] at hzero", [{"full_name": "mul_eq_zero", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [240, 9], "def_end_pos": [240, 20]}]], "state_before": "case intro\nK : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : CommRing K\ninst\u271d\u00b9 : NoZeroDivisors K\nG : Subgroup K\u02e3\ninst\u271d : Fintype { x // x \u2208 G }\nk : \u2115\nk_pos : k \u2260 0\nk_lt_card_G : k < Fintype.card { x // x \u2208 G }\n\u271d : Nontrivial K\nthis : IsDomain K\na : { x // x \u2208 G }\nha : a ^ k \u2260 1\nh_multiset_map : Multiset.map (fun x => \u2191\u2191x ^ k) univ.val = Multiset.map (fun x => \u2191\u2191x ^ k * \u2191\u2191a ^ k) univ.val\nh_multiset_map_sum :\n Multiset.sum (Multiset.map (fun x => \u2191\u2191x ^ k) univ.val) =\n Multiset.sum (Multiset.map (fun x => \u2191\u2191x ^ k) univ.val) * \u2191\u2191a ^ k\nhzero : (\u2191\u2191a ^ k - 1) * Multiset.sum (Multiset.map (fun i => \u2191\u2191i ^ k) univ.val) = 0\n\u22a2 Multiset.sum (Multiset.map (fun x => \u2191\u2191x ^ k) univ.val) = 0", "state_after": "case intro\nK : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : CommRing K\ninst\u271d\u00b9 : NoZeroDivisors K\nG : Subgroup K\u02e3\ninst\u271d : Fintype { x // x \u2208 G }\nk : \u2115\nk_pos : k \u2260 0\nk_lt_card_G : k < Fintype.card { x // x \u2208 G }\n\u271d : Nontrivial K\nthis : IsDomain K\na : { x // x \u2208 G }\nha : a ^ k \u2260 1\nh_multiset_map : Multiset.map (fun x => \u2191\u2191x ^ k) univ.val = Multiset.map (fun x => \u2191\u2191x ^ k * \u2191\u2191a ^ k) univ.val\nh_multiset_map_sum :\n Multiset.sum (Multiset.map (fun x => \u2191\u2191x ^ k) univ.val) =\n Multiset.sum (Multiset.map (fun x => \u2191\u2191x ^ k) univ.val) * \u2191\u2191a ^ k\nhzero : \u2191\u2191a ^ k - 1 = 0 \u2228 Multiset.sum (Multiset.map (fun i => \u2191\u2191i ^ k) univ.val) = 0\n\u22a2 Multiset.sum (Multiset.map (fun x => \u2191\u2191x ^ k) univ.val) = 0"}, {"tactic": "rcases hzero with h | h", "annotated_tactic": ["rcases hzero with h | h", []], "state_before": "case intro\nK : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : CommRing K\ninst\u271d\u00b9 : NoZeroDivisors K\nG : Subgroup K\u02e3\ninst\u271d : Fintype { x // x \u2208 G }\nk : \u2115\nk_pos : k \u2260 0\nk_lt_card_G : k < Fintype.card { x // x \u2208 G }\n\u271d : Nontrivial K\nthis : IsDomain K\na : { x // x \u2208 G }\nha : a ^ k \u2260 1\nh_multiset_map : Multiset.map (fun x => \u2191\u2191x ^ k) univ.val = Multiset.map (fun x => \u2191\u2191x ^ k * \u2191\u2191a ^ k) univ.val\nh_multiset_map_sum :\n Multiset.sum (Multiset.map (fun x => \u2191\u2191x ^ k) univ.val) =\n Multiset.sum (Multiset.map (fun x => \u2191\u2191x ^ k) univ.val) * \u2191\u2191a ^ k\nhzero : \u2191\u2191a ^ k - 1 = 0 \u2228 Multiset.sum (Multiset.map (fun i => \u2191\u2191i ^ k) univ.val) = 0\n\u22a2 Multiset.sum (Multiset.map (fun x => \u2191\u2191x ^ k) univ.val) = 0", "state_after": "case intro.inl\nK : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : CommRing K\ninst\u271d\u00b9 : NoZeroDivisors K\nG : Subgroup K\u02e3\ninst\u271d : Fintype { x // x \u2208 G }\nk : \u2115\nk_pos : k \u2260 0\nk_lt_card_G : k < Fintype.card { x // x \u2208 G }\n\u271d : Nontrivial K\nthis : IsDomain K\na : { x // x \u2208 G }\nha : a ^ k \u2260 1\nh_multiset_map : Multiset.map (fun x => \u2191\u2191x ^ k) univ.val = Multiset.map (fun x => \u2191\u2191x ^ k * \u2191\u2191a ^ k) univ.val\nh_multiset_map_sum :\n Multiset.sum (Multiset.map (fun x => \u2191\u2191x ^ k) univ.val) =\n Multiset.sum (Multiset.map (fun x => \u2191\u2191x ^ k) univ.val) * \u2191\u2191a ^ k\nh : \u2191\u2191a ^ k - 1 = 0\n\u22a2 Multiset.sum (Multiset.map (fun x => \u2191\u2191x ^ k) univ.val) = 0\n\ncase intro.inr\nK : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : CommRing K\ninst\u271d\u00b9 : NoZeroDivisors K\nG : Subgroup K\u02e3\ninst\u271d : Fintype { x // x \u2208 G }\nk : \u2115\nk_pos : k \u2260 0\nk_lt_card_G : k < Fintype.card { x // x \u2208 G }\n\u271d : Nontrivial K\nthis : IsDomain K\na : { x // x \u2208 G }\nha : a ^ k \u2260 1\nh_multiset_map : Multiset.map (fun x => \u2191\u2191x ^ k) univ.val = Multiset.map (fun x => \u2191\u2191x ^ k * \u2191\u2191a ^ k) univ.val\nh_multiset_map_sum :\n Multiset.sum (Multiset.map (fun x => \u2191\u2191x ^ k) univ.val) =\n Multiset.sum (Multiset.map (fun x => \u2191\u2191x ^ k) univ.val) * \u2191\u2191a ^ k\nh : Multiset.sum (Multiset.map (fun i => \u2191\u2191i ^ k) univ.val) = 0\n\u22a2 Multiset.sum (Multiset.map (fun x => \u2191\u2191x ^ k) univ.val) = 0"}, {"tactic": "simp_rw [\u2190 mul_pow]", "annotated_tactic": ["simp_rw [\u2190 <a>mul_pow</a>]", [{"full_name": "mul_pow", "def_path": "Mathlib/Algebra/GroupPower/Basic.lean", "def_pos": [257, 9], "def_end_pos": [257, 16]}]], "state_before": "K : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : CommRing K\ninst\u271d\u00b9 : NoZeroDivisors K\nG : Subgroup K\u02e3\ninst\u271d : Fintype { x // x \u2208 G }\nk : \u2115\nk_pos : k \u2260 0\nk_lt_card_G : k < Fintype.card { x // x \u2208 G }\n\u271d : Nontrivial K\nthis : IsDomain K\na : { x // x \u2208 G }\nha : a ^ k \u2260 1\n\u22a2 Multiset.map (fun x => \u2191\u2191x ^ k) univ.val = Multiset.map (fun x => \u2191\u2191x ^ k * \u2191\u2191a ^ k) univ.val", "state_after": "K : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : CommRing K\ninst\u271d\u00b9 : NoZeroDivisors K\nG : Subgroup K\u02e3\ninst\u271d : Fintype { x // x \u2208 G }\nk : \u2115\nk_pos : k \u2260 0\nk_lt_card_G : k < Fintype.card { x // x \u2208 G }\n\u271d : Nontrivial K\nthis : IsDomain K\na : { x // x \u2208 G }\nha : a ^ k \u2260 1\n\u22a2 Multiset.map (fun x => \u2191\u2191x ^ k) univ.val = Multiset.map (fun x => (\u2191\u2191x * \u2191\u2191a) ^ k) univ.val"}, {"tactic": "have as_comp :\n (fun x : \u21a5G => (((x : K\u02e3) : K) * ((a : K\u02e3) : K)) ^ k)\n = (fun x : \u21a5G => ((x : K\u02e3) : K) ^ k) \u2218 fun x : \u21a5G => x * a := by\n funext x\n simp only [Function.comp_apply, Submonoid.coe_mul, Subgroup.coe_toSubmonoid, Units.val_mul]", "annotated_tactic": ["have as_comp :\n (fun x : \u21a5G => (((x : K\u02e3) : K) * ((a : K\u02e3) : K)) ^ k)\n = (fun x : \u21a5G => ((x : K\u02e3) : K) ^ k) \u2218 fun x : \u21a5G => x * a := by\n funext x\n simp only [<a>Function.comp_apply</a>, <a>Submonoid.coe_mul</a>, <a>Subgroup.coe_toSubmonoid</a>, <a>Units.val_mul</a>]", [{"full_name": "Function.comp_apply", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [33, 17], "def_end_pos": [33, 36]}, {"full_name": "Submonoid.coe_mul", "def_path": "Mathlib/GroupTheory/Submonoid/Operations.lean", "def_pos": [678, 9], "def_end_pos": [678, 16]}, {"full_name": "Subgroup.coe_toSubmonoid", "def_path": "Mathlib/GroupTheory/Subgroup/Basic.lean", "def_pos": [436, 9], "def_end_pos": [436, 24]}, {"full_name": "Units.val_mul", "def_path": "Mathlib/Algebra/Group/Units.lean", "def_pos": [229, 9], "def_end_pos": [229, 16]}]], "state_before": "K : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : CommRing K\ninst\u271d\u00b9 : NoZeroDivisors K\nG : Subgroup K\u02e3\ninst\u271d : Fintype { x // x \u2208 G }\nk : \u2115\nk_pos : k \u2260 0\nk_lt_card_G : k < Fintype.card { x // x \u2208 G }\n\u271d : Nontrivial K\nthis : IsDomain K\na : { x // x \u2208 G }\nha : a ^ k \u2260 1\n\u22a2 Multiset.map (fun x => \u2191\u2191x ^ k) univ.val = Multiset.map (fun x => (\u2191\u2191x * \u2191\u2191a) ^ k) univ.val", "state_after": "K : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : CommRing K\ninst\u271d\u00b9 : NoZeroDivisors K\nG : Subgroup K\u02e3\ninst\u271d : Fintype { x // x \u2208 G }\nk : \u2115\nk_pos : k \u2260 0\nk_lt_card_G : k < Fintype.card { x // x \u2208 G }\n\u271d : Nontrivial K\nthis : IsDomain K\na : { x // x \u2208 G }\nha : a ^ k \u2260 1\nas_comp : (fun x => (\u2191\u2191x * \u2191\u2191a) ^ k) = (fun x => \u2191\u2191x ^ k) \u2218 fun x => x * a\n\u22a2 Multiset.map (fun x => \u2191\u2191x ^ k) univ.val = Multiset.map (fun x => (\u2191\u2191x * \u2191\u2191a) ^ k) univ.val"}, {"tactic": "rw [as_comp, \u2190 Multiset.map_map]", "annotated_tactic": ["rw [as_comp, \u2190 <a>Multiset.map_map</a>]", [{"full_name": "Multiset.map_map", "def_path": "Mathlib/Data/Multiset/Basic.lean", "def_pos": [1283, 9], "def_end_pos": [1283, 16]}]], "state_before": "K : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : CommRing K\ninst\u271d\u00b9 : NoZeroDivisors K\nG : Subgroup K\u02e3\ninst\u271d : Fintype { x // x \u2208 G }\nk : \u2115\nk_pos : k \u2260 0\nk_lt_card_G : k < Fintype.card { x // x \u2208 G }\n\u271d : Nontrivial K\nthis : IsDomain K\na : { x // x \u2208 G }\nha : a ^ k \u2260 1\nas_comp : (fun x => (\u2191\u2191x * \u2191\u2191a) ^ k) = (fun x => \u2191\u2191x ^ k) \u2218 fun x => x * a\n\u22a2 Multiset.map (fun x => \u2191\u2191x ^ k) univ.val = Multiset.map (fun x => (\u2191\u2191x * \u2191\u2191a) ^ k) univ.val", "state_after": "K : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : CommRing K\ninst\u271d\u00b9 : NoZeroDivisors K\nG : Subgroup K\u02e3\ninst\u271d : Fintype { x // x \u2208 G }\nk : \u2115\nk_pos : k \u2260 0\nk_lt_card_G : k < Fintype.card { x // x \u2208 G }\n\u271d : Nontrivial K\nthis : IsDomain K\na : { x // x \u2208 G }\nha : a ^ k \u2260 1\nas_comp : (fun x => (\u2191\u2191x * \u2191\u2191a) ^ k) = (fun x => \u2191\u2191x ^ k) \u2218 fun x => x * a\n\u22a2 Multiset.map (fun x => \u2191\u2191x ^ k) univ.val = Multiset.map (fun x => \u2191\u2191x ^ k) (Multiset.map (fun x => x * a) univ.val)"}, {"tactic": "congr", "annotated_tactic": ["congr", []], "state_before": "K : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : CommRing K\ninst\u271d\u00b9 : NoZeroDivisors K\nG : Subgroup K\u02e3\ninst\u271d : Fintype { x // x \u2208 G }\nk : \u2115\nk_pos : k \u2260 0\nk_lt_card_G : k < Fintype.card { x // x \u2208 G }\n\u271d : Nontrivial K\nthis : IsDomain K\na : { x // x \u2208 G }\nha : a ^ k \u2260 1\nas_comp : (fun x => (\u2191\u2191x * \u2191\u2191a) ^ k) = (fun x => \u2191\u2191x ^ k) \u2218 fun x => x * a\n\u22a2 Multiset.map (fun x => \u2191\u2191x ^ k) univ.val = Multiset.map (fun x => \u2191\u2191x ^ k) (Multiset.map (fun x => x * a) univ.val)", "state_after": "case e_s\nK : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : CommRing K\ninst\u271d\u00b9 : NoZeroDivisors K\nG : Subgroup K\u02e3\ninst\u271d : Fintype { x // x \u2208 G }\nk : \u2115\nk_pos : k \u2260 0\nk_lt_card_G : k < Fintype.card { x // x \u2208 G }\n\u271d : Nontrivial K\nthis : IsDomain K\na : { x // x \u2208 G }\nha : a ^ k \u2260 1\nas_comp : (fun x => (\u2191\u2191x * \u2191\u2191a) ^ k) = (fun x => \u2191\u2191x ^ k) \u2218 fun x => x * a\n\u22a2 univ.val = Multiset.map (fun x => x * a) univ.val"}, {"tactic": "rw [eq_comm]", "annotated_tactic": ["rw [<a>eq_comm</a>]", [{"full_name": "eq_comm", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [104, 9], "def_end_pos": [104, 16]}]], "state_before": "case e_s\nK : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : CommRing K\ninst\u271d\u00b9 : NoZeroDivisors K\nG : Subgroup K\u02e3\ninst\u271d : Fintype { x // x \u2208 G }\nk : \u2115\nk_pos : k \u2260 0\nk_lt_card_G : k < Fintype.card { x // x \u2208 G }\n\u271d : Nontrivial K\nthis : IsDomain K\na : { x // x \u2208 G }\nha : a ^ k \u2260 1\nas_comp : (fun x => (\u2191\u2191x * \u2191\u2191a) ^ k) = (fun x => \u2191\u2191x ^ k) \u2218 fun x => x * a\n\u22a2 univ.val = Multiset.map (fun x => x * a) univ.val", "state_after": "case e_s\nK : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : CommRing K\ninst\u271d\u00b9 : NoZeroDivisors K\nG : Subgroup K\u02e3\ninst\u271d : Fintype { x // x \u2208 G }\nk : \u2115\nk_pos : k \u2260 0\nk_lt_card_G : k < Fintype.card { x // x \u2208 G }\n\u271d : Nontrivial K\nthis : IsDomain K\na : { x // x \u2208 G }\nha : a ^ k \u2260 1\nas_comp : (fun x => (\u2191\u2191x * \u2191\u2191a) ^ k) = (fun x => \u2191\u2191x ^ k) \u2218 fun x => x * a\n\u22a2 Multiset.map (fun x => x * a) univ.val = univ.val"}, {"tactic": "exact Multiset.map_univ_val_equiv (Equiv.mulRight a)", "annotated_tactic": ["exact <a>Multiset.map_univ_val_equiv</a> (<a>Equiv.mulRight</a> a)", [{"full_name": "Multiset.map_univ_val_equiv", "def_path": "Mathlib/Data/Fintype/Basic.lean", "def_pos": [1228, 9], "def_end_pos": [1228, 27]}, {"full_name": "Equiv.mulRight", "def_path": "Mathlib/Algebra/Hom/Equiv/Units/Basic.lean", "def_pos": [152, 15], "def_end_pos": [152, 23]}]], "state_before": "case e_s\nK : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : CommRing K\ninst\u271d\u00b9 : NoZeroDivisors K\nG : Subgroup K\u02e3\ninst\u271d : Fintype { x // x \u2208 G }\nk : \u2115\nk_pos : k \u2260 0\nk_lt_card_G : k < Fintype.card { x // x \u2208 G }\n\u271d : Nontrivial K\nthis : IsDomain K\na : { x // x \u2208 G }\nha : a ^ k \u2260 1\nas_comp : (fun x => (\u2191\u2191x * \u2191\u2191a) ^ k) = (fun x => \u2191\u2191x ^ k) \u2218 fun x => x * a\n\u22a2 Multiset.map (fun x => x * a) univ.val = univ.val", "state_after": "no goals"}, {"tactic": "funext x", "annotated_tactic": ["funext x", []], "state_before": "K : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : CommRing K\ninst\u271d\u00b9 : NoZeroDivisors K\nG : Subgroup K\u02e3\ninst\u271d : Fintype { x // x \u2208 G }\nk : \u2115\nk_pos : k \u2260 0\nk_lt_card_G : k < Fintype.card { x // x \u2208 G }\n\u271d : Nontrivial K\nthis : IsDomain K\na : { x // x \u2208 G }\nha : a ^ k \u2260 1\n\u22a2 (fun x => (\u2191\u2191x * \u2191\u2191a) ^ k) = (fun x => \u2191\u2191x ^ k) \u2218 fun x => x * a", "state_after": "case h\nK : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : CommRing K\ninst\u271d\u00b9 : NoZeroDivisors K\nG : Subgroup K\u02e3\ninst\u271d : Fintype { x // x \u2208 G }\nk : \u2115\nk_pos : k \u2260 0\nk_lt_card_G : k < Fintype.card { x // x \u2208 G }\n\u271d : Nontrivial K\nthis : IsDomain K\na : { x // x \u2208 G }\nha : a ^ k \u2260 1\nx : { x // x \u2208 G }\n\u22a2 (\u2191\u2191x * \u2191\u2191a) ^ k = ((fun x => \u2191\u2191x ^ k) \u2218 fun x => x * a) x"}, {"tactic": "simp only [Function.comp_apply, Submonoid.coe_mul, Subgroup.coe_toSubmonoid, Units.val_mul]", "annotated_tactic": ["simp only [<a>Function.comp_apply</a>, <a>Submonoid.coe_mul</a>, <a>Subgroup.coe_toSubmonoid</a>, <a>Units.val_mul</a>]", [{"full_name": "Function.comp_apply", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [33, 17], "def_end_pos": [33, 36]}, {"full_name": "Submonoid.coe_mul", "def_path": "Mathlib/GroupTheory/Submonoid/Operations.lean", "def_pos": [678, 9], "def_end_pos": [678, 16]}, {"full_name": "Subgroup.coe_toSubmonoid", "def_path": "Mathlib/GroupTheory/Subgroup/Basic.lean", "def_pos": [436, 9], "def_end_pos": [436, 24]}, {"full_name": "Units.val_mul", "def_path": "Mathlib/Algebra/Group/Units.lean", "def_pos": [229, 9], "def_end_pos": [229, 16]}]], "state_before": "case h\nK : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : CommRing K\ninst\u271d\u00b9 : NoZeroDivisors K\nG : Subgroup K\u02e3\ninst\u271d : Fintype { x // x \u2208 G }\nk : \u2115\nk_pos : k \u2260 0\nk_lt_card_G : k < Fintype.card { x // x \u2208 G }\n\u271d : Nontrivial K\nthis : IsDomain K\na : { x // x \u2208 G }\nha : a ^ k \u2260 1\nx : { x // x \u2208 G }\n\u22a2 (\u2191\u2191x * \u2191\u2191a) ^ k = ((fun x => \u2191\u2191x ^ k) \u2218 fun x => x * a) x", "state_after": "no goals"}, {"tactic": "rw [sub_mul, mul_comm, \u2190 h_multiset_map_sum, one_mul, sub_self]", "annotated_tactic": ["rw [<a>sub_mul</a>, <a>mul_comm</a>, \u2190 h_multiset_map_sum, <a>one_mul</a>, <a>sub_self</a>]", [{"full_name": "sub_mul", "def_path": "Mathlib/Algebra/Ring/Defs.lean", "def_pos": [372, 7], "def_end_pos": [372, 14]}, {"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [302, 9], "def_end_pos": [302, 17]}, {"full_name": "one_mul", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [464, 9], "def_end_pos": [464, 16]}, {"full_name": "sub_self", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [734, 30], "def_end_pos": [734, 38]}]], "state_before": "K : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : CommRing K\ninst\u271d\u00b9 : NoZeroDivisors K\nG : Subgroup K\u02e3\ninst\u271d : Fintype { x // x \u2208 G }\nk : \u2115\nk_pos : k \u2260 0\nk_lt_card_G : k < Fintype.card { x // x \u2208 G }\n\u271d : Nontrivial K\nthis : IsDomain K\na : { x // x \u2208 G }\nha : a ^ k \u2260 1\nh_multiset_map : Multiset.map (fun x => \u2191\u2191x ^ k) univ.val = Multiset.map (fun x => \u2191\u2191x ^ k * \u2191\u2191a ^ k) univ.val\nh_multiset_map_sum :\n Multiset.sum (Multiset.map (fun x => \u2191\u2191x ^ k) univ.val) =\n Multiset.sum (Multiset.map (fun x => \u2191\u2191x ^ k) univ.val) * \u2191\u2191a ^ k\n\u22a2 (\u2191\u2191a ^ k - 1) * Multiset.sum (Multiset.map (fun i => \u2191\u2191i ^ k) univ.val) = 0", "state_after": "no goals"}, {"tactic": "contrapose! ha", "annotated_tactic": ["contrapose! ha", []], "state_before": "case intro.inl\nK : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : CommRing K\ninst\u271d\u00b9 : NoZeroDivisors K\nG : Subgroup K\u02e3\ninst\u271d : Fintype { x // x \u2208 G }\nk : \u2115\nk_pos : k \u2260 0\nk_lt_card_G : k < Fintype.card { x // x \u2208 G }\n\u271d : Nontrivial K\nthis : IsDomain K\na : { x // x \u2208 G }\nha : a ^ k \u2260 1\nh_multiset_map : Multiset.map (fun x => \u2191\u2191x ^ k) univ.val = Multiset.map (fun x => \u2191\u2191x ^ k * \u2191\u2191a ^ k) univ.val\nh_multiset_map_sum :\n Multiset.sum (Multiset.map (fun x => \u2191\u2191x ^ k) univ.val) =\n Multiset.sum (Multiset.map (fun x => \u2191\u2191x ^ k) univ.val) * \u2191\u2191a ^ k\nh : \u2191\u2191a ^ k - 1 = 0\n\u22a2 Multiset.sum (Multiset.map (fun x => \u2191\u2191x ^ k) univ.val) = 0", "state_after": "case intro.inl\nK : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : CommRing K\ninst\u271d\u00b9 : NoZeroDivisors K\nG : Subgroup K\u02e3\ninst\u271d : Fintype { x // x \u2208 G }\nk : \u2115\nk_pos : k \u2260 0\nk_lt_card_G : k < Fintype.card { x // x \u2208 G }\n\u271d : Nontrivial K\nthis : IsDomain K\na : { x // x \u2208 G }\nh_multiset_map : Multiset.map (fun x => \u2191\u2191x ^ k) univ.val = Multiset.map (fun x => \u2191\u2191x ^ k * \u2191\u2191a ^ k) univ.val\nh_multiset_map_sum :\n Multiset.sum (Multiset.map (fun x => \u2191\u2191x ^ k) univ.val) =\n Multiset.sum (Multiset.map (fun x => \u2191\u2191x ^ k) univ.val) * \u2191\u2191a ^ k\nh : \u2191\u2191a ^ k - 1 = 0\nha : Multiset.sum (Multiset.map (fun x => \u2191\u2191x ^ k) univ.val) \u2260 0\n\u22a2 a ^ k = 1"}, {"tactic": "ext", "annotated_tactic": ["ext", []], "state_before": "case intro.inl\nK : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : CommRing K\ninst\u271d\u00b9 : NoZeroDivisors K\nG : Subgroup K\u02e3\ninst\u271d : Fintype { x // x \u2208 G }\nk : \u2115\nk_pos : k \u2260 0\nk_lt_card_G : k < Fintype.card { x // x \u2208 G }\n\u271d : Nontrivial K\nthis : IsDomain K\na : { x // x \u2208 G }\nh_multiset_map : Multiset.map (fun x => \u2191\u2191x ^ k) univ.val = Multiset.map (fun x => \u2191\u2191x ^ k * \u2191\u2191a ^ k) univ.val\nh_multiset_map_sum :\n Multiset.sum (Multiset.map (fun x => \u2191\u2191x ^ k) univ.val) =\n Multiset.sum (Multiset.map (fun x => \u2191\u2191x ^ k) univ.val) * \u2191\u2191a ^ k\nh : \u2191\u2191a ^ k - 1 = 0\nha : Multiset.sum (Multiset.map (fun x => \u2191\u2191x ^ k) univ.val) \u2260 0\n\u22a2 a ^ k = 1", "state_after": "case intro.inl.a.a\nK : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : CommRing K\ninst\u271d\u00b9 : NoZeroDivisors K\nG : Subgroup K\u02e3\ninst\u271d : Fintype { x // x \u2208 G }\nk : \u2115\nk_pos : k \u2260 0\nk_lt_card_G : k < Fintype.card { x // x \u2208 G }\n\u271d : Nontrivial K\nthis : IsDomain K\na : { x // x \u2208 G }\nh_multiset_map : Multiset.map (fun x => \u2191\u2191x ^ k) univ.val = Multiset.map (fun x => \u2191\u2191x ^ k * \u2191\u2191a ^ k) univ.val\nh_multiset_map_sum :\n Multiset.sum (Multiset.map (fun x => \u2191\u2191x ^ k) univ.val) =\n Multiset.sum (Multiset.map (fun x => \u2191\u2191x ^ k) univ.val) * \u2191\u2191a ^ k\nh : \u2191\u2191a ^ k - 1 = 0\nha : Multiset.sum (Multiset.map (fun x => \u2191\u2191x ^ k) univ.val) \u2260 0\n\u22a2 \u2191\u2191(a ^ k) = \u2191\u21911"}, {"tactic": "rw [\u2190sub_eq_zero]", "annotated_tactic": ["rw [\u2190<a>sub_eq_zero</a>]", [{"full_name": "sub_eq_zero", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [801, 3], "def_end_pos": [801, 14]}]], "state_before": "case intro.inl.a.a\nK : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : CommRing K\ninst\u271d\u00b9 : NoZeroDivisors K\nG : Subgroup K\u02e3\ninst\u271d : Fintype { x // x \u2208 G }\nk : \u2115\nk_pos : k \u2260 0\nk_lt_card_G : k < Fintype.card { x // x \u2208 G }\n\u271d : Nontrivial K\nthis : IsDomain K\na : { x // x \u2208 G }\nh_multiset_map : Multiset.map (fun x => \u2191\u2191x ^ k) univ.val = Multiset.map (fun x => \u2191\u2191x ^ k * \u2191\u2191a ^ k) univ.val\nh_multiset_map_sum :\n Multiset.sum (Multiset.map (fun x => \u2191\u2191x ^ k) univ.val) =\n Multiset.sum (Multiset.map (fun x => \u2191\u2191x ^ k) univ.val) * \u2191\u2191a ^ k\nh : \u2191\u2191a ^ k - 1 = 0\nha : Multiset.sum (Multiset.map (fun x => \u2191\u2191x ^ k) univ.val) \u2260 0\n\u22a2 \u2191\u2191(a ^ k) = \u2191\u21911", "state_after": "case intro.inl.a.a\nK : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : CommRing K\ninst\u271d\u00b9 : NoZeroDivisors K\nG : Subgroup K\u02e3\ninst\u271d : Fintype { x // x \u2208 G }\nk : \u2115\nk_pos : k \u2260 0\nk_lt_card_G : k < Fintype.card { x // x \u2208 G }\n\u271d : Nontrivial K\nthis : IsDomain K\na : { x // x \u2208 G }\nh_multiset_map : Multiset.map (fun x => \u2191\u2191x ^ k) univ.val = Multiset.map (fun x => \u2191\u2191x ^ k * \u2191\u2191a ^ k) univ.val\nh_multiset_map_sum :\n Multiset.sum (Multiset.map (fun x => \u2191\u2191x ^ k) univ.val) =\n Multiset.sum (Multiset.map (fun x => \u2191\u2191x ^ k) univ.val) * \u2191\u2191a ^ k\nh : \u2191\u2191a ^ k - 1 = 0\nha : Multiset.sum (Multiset.map (fun x => \u2191\u2191x ^ k) univ.val) \u2260 0\n\u22a2 \u2191\u2191(a ^ k) - \u2191\u21911 = 0"}, {"tactic": "simp_rw [SubmonoidClass.coe_pow, Units.val_pow_eq_pow_val, OneMemClass.coe_one,\n Units.val_one, h]", "annotated_tactic": ["simp_rw [<a>SubmonoidClass.coe_pow</a>, <a>Units.val_pow_eq_pow_val</a>, <a>OneMemClass.coe_one</a>,\n <a>Units.val_one</a>, h]", [{"full_name": "SubmonoidClass.coe_pow", "def_path": "Mathlib/GroupTheory/Submonoid/Operations.lean", "def_pos": [562, 9], "def_end_pos": [562, 16]}, {"full_name": "Units.val_pow_eq_pow_val", "def_path": "Mathlib/Algebra/Hom/Units.lean", "def_pos": [118, 9], "def_end_pos": [118, 27]}, {"full_name": "OneMemClass.coe_one", "def_path": "Mathlib/GroupTheory/Submonoid/Operations.lean", "def_pos": [521, 9], "def_end_pos": [521, 16]}, {"full_name": "Units.val_one", "def_path": "Mathlib/Algebra/Group/Units.lean", "def_pos": [235, 9], "def_end_pos": [235, 16]}]], "state_before": "case intro.inl.a.a\nK : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : CommRing K\ninst\u271d\u00b9 : NoZeroDivisors K\nG : Subgroup K\u02e3\ninst\u271d : Fintype { x // x \u2208 G }\nk : \u2115\nk_pos : k \u2260 0\nk_lt_card_G : k < Fintype.card { x // x \u2208 G }\n\u271d : Nontrivial K\nthis : IsDomain K\na : { x // x \u2208 G }\nh_multiset_map : Multiset.map (fun x => \u2191\u2191x ^ k) univ.val = Multiset.map (fun x => \u2191\u2191x ^ k * \u2191\u2191a ^ k) univ.val\nh_multiset_map_sum :\n Multiset.sum (Multiset.map (fun x => \u2191\u2191x ^ k) univ.val) =\n Multiset.sum (Multiset.map (fun x => \u2191\u2191x ^ k) univ.val) * \u2191\u2191a ^ k\nh : \u2191\u2191a ^ k - 1 = 0\nha : Multiset.sum (Multiset.map (fun x => \u2191\u2191x ^ k) univ.val) \u2260 0\n\u22a2 \u2191\u2191(a ^ k) - \u2191\u21911 = 0", "state_after": "no goals"}, {"tactic": "exact h", "annotated_tactic": ["exact h", []], "state_before": "case intro.inr\nK : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : CommRing K\ninst\u271d\u00b9 : NoZeroDivisors K\nG : Subgroup K\u02e3\ninst\u271d : Fintype { x // x \u2208 G }\nk : \u2115\nk_pos : k \u2260 0\nk_lt_card_G : k < Fintype.card { x // x \u2208 G }\n\u271d : Nontrivial K\nthis : IsDomain K\na : { x // x \u2208 G }\nha : a ^ k \u2260 1\nh_multiset_map : Multiset.map (fun x => \u2191\u2191x ^ k) univ.val = Multiset.map (fun x => \u2191\u2191x ^ k * \u2191\u2191a ^ k) univ.val\nh_multiset_map_sum :\n Multiset.sum (Multiset.map (fun x => \u2191\u2191x ^ k) univ.val) =\n Multiset.sum (Multiset.map (fun x => \u2191\u2191x ^ k) univ.val) * \u2191\u2191a ^ k\nh : Multiset.sum (Multiset.map (fun i => \u2191\u2191i ^ k) univ.val) = 0\n\u22a2 Multiset.sum (Multiset.map (fun x => \u2191\u2191x ^ k) univ.val) = 0", "state_after": "no goals"}]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Data/List/Perm.lean
|
List.Perm.append_right
|
[
107,
1
] |
[
112,
35
] |
[]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Logic/Basic.lean
|
Decidable.eq_or_ne
|
[
203,
1
] |
[
203,
92
] |
[]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/LinearAlgebra/Dual.lean
|
Module.Dual.transpose_comp
|
[
152,
1
] |
[
154,
6
] |
[]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/NumberTheory/Dioph.lean
|
Poly.induction
|
[
244,
1
] |
[
248,
65
] |
[{"tactic": "cases' f with f pf", "annotated_tactic": ["cases' f with f pf", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nC : Poly \u03b1 \u2192 Prop\nH1 : \u2200 (i : \u03b1), C (proj i)\nH2 : \u2200 (n : \u2124), C (const n)\nH3 : \u2200 (f g : Poly \u03b1), C f \u2192 C g \u2192 C (f - g)\nH4 : \u2200 (f g : Poly \u03b1), C f \u2192 C g \u2192 C (f * g)\nf : Poly \u03b1\n\u22a2 C f", "state_after": "case mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nC : Poly \u03b1 \u2192 Prop\nH1 : \u2200 (i : \u03b1), C (proj i)\nH2 : \u2200 (n : \u2124), C (const n)\nH3 : \u2200 (f g : Poly \u03b1), C f \u2192 C g \u2192 C (f - g)\nH4 : \u2200 (f g : Poly \u03b1), C f \u2192 C g \u2192 C (f * g)\nf : (\u03b1 \u2192 \u2115) \u2192 \u2124\npf : IsPoly f\n\u22a2 C { val := f, property := pf }"}, {"tactic": "induction' pf with i n f g pf pg ihf ihg f g pf pg ihf ihg", "annotated_tactic": ["induction' pf with i n f g pf pg ihf ihg f g pf pg ihf ihg", []], "state_before": "case mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nC : Poly \u03b1 \u2192 Prop\nH1 : \u2200 (i : \u03b1), C (proj i)\nH2 : \u2200 (n : \u2124), C (const n)\nH3 : \u2200 (f g : Poly \u03b1), C f \u2192 C g \u2192 C (f - g)\nH4 : \u2200 (f g : Poly \u03b1), C f \u2192 C g \u2192 C (f * g)\nf : (\u03b1 \u2192 \u2115) \u2192 \u2124\npf : IsPoly f\n\u22a2 C { val := f, property := pf }", "state_after": "case mk.proj\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nC : Poly \u03b1 \u2192 Prop\nH1 : \u2200 (i : \u03b1), C (proj i)\nH2 : \u2200 (n : \u2124), C (const n)\nH3 : \u2200 (f g : Poly \u03b1), C f \u2192 C g \u2192 C (f - g)\nH4 : \u2200 (f g : Poly \u03b1), C f \u2192 C g \u2192 C (f * g)\nf : (\u03b1 \u2192 \u2115) \u2192 \u2124\ni : \u03b1\n\u22a2 C { val := fun x => \u2191(x i), property := (_ : IsPoly fun x => \u2191(x i)) }\n\ncase mk.const\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nC : Poly \u03b1 \u2192 Prop\nH1 : \u2200 (i : \u03b1), C (proj i)\nH2 : \u2200 (n : \u2124), C (const n)\nH3 : \u2200 (f g : Poly \u03b1), C f \u2192 C g \u2192 C (f - g)\nH4 : \u2200 (f g : Poly \u03b1), C f \u2192 C g \u2192 C (f * g)\nf : (\u03b1 \u2192 \u2115) \u2192 \u2124\nn : \u2124\n\u22a2 C { val := fun x => n, property := (_ : IsPoly fun x => n) }\n\ncase mk.sub\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nC : Poly \u03b1 \u2192 Prop\nH1 : \u2200 (i : \u03b1), C (proj i)\nH2 : \u2200 (n : \u2124), C (const n)\nH3 : \u2200 (f g : Poly \u03b1), C f \u2192 C g \u2192 C (f - g)\nH4 : \u2200 (f g : Poly \u03b1), C f \u2192 C g \u2192 C (f * g)\nf\u271d f g : (\u03b1 \u2192 \u2115) \u2192 \u2124\npf : IsPoly f\npg : IsPoly g\nihf : C { val := f, property := pf }\nihg : C { val := g, property := pg }\n\u22a2 C { val := fun x => f x - g x, property := (_ : IsPoly fun x => f x - g x) }\n\ncase mk.mul\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nC : Poly \u03b1 \u2192 Prop\nH1 : \u2200 (i : \u03b1), C (proj i)\nH2 : \u2200 (n : \u2124), C (const n)\nH3 : \u2200 (f g : Poly \u03b1), C f \u2192 C g \u2192 C (f - g)\nH4 : \u2200 (f g : Poly \u03b1), C f \u2192 C g \u2192 C (f * g)\nf\u271d f g : (\u03b1 \u2192 \u2115) \u2192 \u2124\npf : IsPoly f\npg : IsPoly g\nihf : C { val := f, property := pf }\nihg : C { val := g, property := pg }\n\u22a2 C { val := fun x => f x * g x, property := (_ : IsPoly fun x => f x * g x) }"}, {"tactic": "apply H1", "annotated_tactic": ["apply H1", []], "state_before": "case mk.proj\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nC : Poly \u03b1 \u2192 Prop\nH1 : \u2200 (i : \u03b1), C (proj i)\nH2 : \u2200 (n : \u2124), C (const n)\nH3 : \u2200 (f g : Poly \u03b1), C f \u2192 C g \u2192 C (f - g)\nH4 : \u2200 (f g : Poly \u03b1), C f \u2192 C g \u2192 C (f * g)\nf : (\u03b1 \u2192 \u2115) \u2192 \u2124\ni : \u03b1\n\u22a2 C { val := fun x => \u2191(x i), property := (_ : IsPoly fun x => \u2191(x i)) }\n\ncase mk.const\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nC : Poly \u03b1 \u2192 Prop\nH1 : \u2200 (i : \u03b1), C (proj i)\nH2 : \u2200 (n : \u2124), C (const n)\nH3 : \u2200 (f g : Poly \u03b1), C f \u2192 C g \u2192 C (f - g)\nH4 : \u2200 (f g : Poly \u03b1), C f \u2192 C g \u2192 C (f * g)\nf : (\u03b1 \u2192 \u2115) \u2192 \u2124\nn : \u2124\n\u22a2 C { val := fun x => n, property := (_ : IsPoly fun x => n) }\n\ncase mk.sub\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nC : Poly \u03b1 \u2192 Prop\nH1 : \u2200 (i : \u03b1), C (proj i)\nH2 : \u2200 (n : \u2124), C (const n)\nH3 : \u2200 (f g : Poly \u03b1), C f \u2192 C g \u2192 C (f - g)\nH4 : \u2200 (f g : Poly \u03b1), C f \u2192 C g \u2192 C (f * g)\nf\u271d f g : (\u03b1 \u2192 \u2115) \u2192 \u2124\npf : IsPoly f\npg : IsPoly g\nihf : C { val := f, property := pf }\nihg : C { val := g, property := pg }\n\u22a2 C { val := fun x => f x - g x, property := (_ : IsPoly fun x => f x - g x) }\n\ncase mk.mul\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nC : Poly \u03b1 \u2192 Prop\nH1 : \u2200 (i : \u03b1), C (proj i)\nH2 : \u2200 (n : \u2124), C (const n)\nH3 : \u2200 (f g : Poly \u03b1), C f \u2192 C g \u2192 C (f - g)\nH4 : \u2200 (f g : Poly \u03b1), C f \u2192 C g \u2192 C (f * g)\nf\u271d f g : (\u03b1 \u2192 \u2115) \u2192 \u2124\npf : IsPoly f\npg : IsPoly g\nihf : C { val := f, property := pf }\nihg : C { val := g, property := pg }\n\u22a2 C { val := fun x => f x * g x, property := (_ : IsPoly fun x => f x * g x) }", "state_after": "case mk.const\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nC : Poly \u03b1 \u2192 Prop\nH1 : \u2200 (i : \u03b1), C (proj i)\nH2 : \u2200 (n : \u2124), C (const n)\nH3 : \u2200 (f g : Poly \u03b1), C f \u2192 C g \u2192 C (f - g)\nH4 : \u2200 (f g : Poly \u03b1), C f \u2192 C g \u2192 C (f * g)\nf : (\u03b1 \u2192 \u2115) \u2192 \u2124\nn : \u2124\n\u22a2 C { val := fun x => n, property := (_ : IsPoly fun x => n) }\n\ncase mk.sub\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nC : Poly \u03b1 \u2192 Prop\nH1 : \u2200 (i : \u03b1), C (proj i)\nH2 : \u2200 (n : \u2124), C (const n)\nH3 : \u2200 (f g : Poly \u03b1), C f \u2192 C g \u2192 C (f - g)\nH4 : \u2200 (f g : Poly \u03b1), C f \u2192 C g \u2192 C (f * g)\nf\u271d f g : (\u03b1 \u2192 \u2115) \u2192 \u2124\npf : IsPoly f\npg : IsPoly g\nihf : C { val := f, property := pf }\nihg : C { val := g, property := pg }\n\u22a2 C { val := fun x => f x - g x, property := (_ : IsPoly fun x => f x - g x) }\n\ncase mk.mul\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nC : Poly \u03b1 \u2192 Prop\nH1 : \u2200 (i : \u03b1), C (proj i)\nH2 : \u2200 (n : \u2124), C (const n)\nH3 : \u2200 (f g : Poly \u03b1), C f \u2192 C g \u2192 C (f - g)\nH4 : \u2200 (f g : Poly \u03b1), C f \u2192 C g \u2192 C (f * g)\nf\u271d f g : (\u03b1 \u2192 \u2115) \u2192 \u2124\npf : IsPoly f\npg : IsPoly g\nihf : C { val := f, property := pf }\nihg : C { val := g, property := pg }\n\u22a2 C { val := fun x => f x * g x, property := (_ : IsPoly fun x => f x * g x) }"}, {"tactic": "apply H2", "annotated_tactic": ["apply H2", []], "state_before": "case mk.const\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nC : Poly \u03b1 \u2192 Prop\nH1 : \u2200 (i : \u03b1), C (proj i)\nH2 : \u2200 (n : \u2124), C (const n)\nH3 : \u2200 (f g : Poly \u03b1), C f \u2192 C g \u2192 C (f - g)\nH4 : \u2200 (f g : Poly \u03b1), C f \u2192 C g \u2192 C (f * g)\nf : (\u03b1 \u2192 \u2115) \u2192 \u2124\nn : \u2124\n\u22a2 C { val := fun x => n, property := (_ : IsPoly fun x => n) }\n\ncase mk.sub\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nC : Poly \u03b1 \u2192 Prop\nH1 : \u2200 (i : \u03b1), C (proj i)\nH2 : \u2200 (n : \u2124), C (const n)\nH3 : \u2200 (f g : Poly \u03b1), C f \u2192 C g \u2192 C (f - g)\nH4 : \u2200 (f g : Poly \u03b1), C f \u2192 C g \u2192 C (f * g)\nf\u271d f g : (\u03b1 \u2192 \u2115) \u2192 \u2124\npf : IsPoly f\npg : IsPoly g\nihf : C { val := f, property := pf }\nihg : C { val := g, property := pg }\n\u22a2 C { val := fun x => f x - g x, property := (_ : IsPoly fun x => f x - g x) }\n\ncase mk.mul\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nC : Poly \u03b1 \u2192 Prop\nH1 : \u2200 (i : \u03b1), C (proj i)\nH2 : \u2200 (n : \u2124), C (const n)\nH3 : \u2200 (f g : Poly \u03b1), C f \u2192 C g \u2192 C (f - g)\nH4 : \u2200 (f g : Poly \u03b1), C f \u2192 C g \u2192 C (f * g)\nf\u271d f g : (\u03b1 \u2192 \u2115) \u2192 \u2124\npf : IsPoly f\npg : IsPoly g\nihf : C { val := f, property := pf }\nihg : C { val := g, property := pg }\n\u22a2 C { val := fun x => f x * g x, property := (_ : IsPoly fun x => f x * g x) }", "state_after": "case mk.sub\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nC : Poly \u03b1 \u2192 Prop\nH1 : \u2200 (i : \u03b1), C (proj i)\nH2 : \u2200 (n : \u2124), C (const n)\nH3 : \u2200 (f g : Poly \u03b1), C f \u2192 C g \u2192 C (f - g)\nH4 : \u2200 (f g : Poly \u03b1), C f \u2192 C g \u2192 C (f * g)\nf\u271d f g : (\u03b1 \u2192 \u2115) \u2192 \u2124\npf : IsPoly f\npg : IsPoly g\nihf : C { val := f, property := pf }\nihg : C { val := g, property := pg }\n\u22a2 C { val := fun x => f x - g x, property := (_ : IsPoly fun x => f x - g x) }\n\ncase mk.mul\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nC : Poly \u03b1 \u2192 Prop\nH1 : \u2200 (i : \u03b1), C (proj i)\nH2 : \u2200 (n : \u2124), C (const n)\nH3 : \u2200 (f g : Poly \u03b1), C f \u2192 C g \u2192 C (f - g)\nH4 : \u2200 (f g : Poly \u03b1), C f \u2192 C g \u2192 C (f * g)\nf\u271d f g : (\u03b1 \u2192 \u2115) \u2192 \u2124\npf : IsPoly f\npg : IsPoly g\nihf : C { val := f, property := pf }\nihg : C { val := g, property := pg }\n\u22a2 C { val := fun x => f x * g x, property := (_ : IsPoly fun x => f x * g x) }"}, {"tactic": "apply H3 _ _ ihf ihg", "annotated_tactic": ["apply H3 _ _ ihf ihg", []], "state_before": "case mk.sub\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nC : Poly \u03b1 \u2192 Prop\nH1 : \u2200 (i : \u03b1), C (proj i)\nH2 : \u2200 (n : \u2124), C (const n)\nH3 : \u2200 (f g : Poly \u03b1), C f \u2192 C g \u2192 C (f - g)\nH4 : \u2200 (f g : Poly \u03b1), C f \u2192 C g \u2192 C (f * g)\nf\u271d f g : (\u03b1 \u2192 \u2115) \u2192 \u2124\npf : IsPoly f\npg : IsPoly g\nihf : C { val := f, property := pf }\nihg : C { val := g, property := pg }\n\u22a2 C { val := fun x => f x - g x, property := (_ : IsPoly fun x => f x - g x) }\n\ncase mk.mul\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nC : Poly \u03b1 \u2192 Prop\nH1 : \u2200 (i : \u03b1), C (proj i)\nH2 : \u2200 (n : \u2124), C (const n)\nH3 : \u2200 (f g : Poly \u03b1), C f \u2192 C g \u2192 C (f - g)\nH4 : \u2200 (f g : Poly \u03b1), C f \u2192 C g \u2192 C (f * g)\nf\u271d f g : (\u03b1 \u2192 \u2115) \u2192 \u2124\npf : IsPoly f\npg : IsPoly g\nihf : C { val := f, property := pf }\nihg : C { val := g, property := pg }\n\u22a2 C { val := fun x => f x * g x, property := (_ : IsPoly fun x => f x * g x) }", "state_after": "case mk.mul\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nC : Poly \u03b1 \u2192 Prop\nH1 : \u2200 (i : \u03b1), C (proj i)\nH2 : \u2200 (n : \u2124), C (const n)\nH3 : \u2200 (f g : Poly \u03b1), C f \u2192 C g \u2192 C (f - g)\nH4 : \u2200 (f g : Poly \u03b1), C f \u2192 C g \u2192 C (f * g)\nf\u271d f g : (\u03b1 \u2192 \u2115) \u2192 \u2124\npf : IsPoly f\npg : IsPoly g\nihf : C { val := f, property := pf }\nihg : C { val := g, property := pg }\n\u22a2 C { val := fun x => f x * g x, property := (_ : IsPoly fun x => f x * g x) }"}, {"tactic": "apply H4 _ _ ihf ihg", "annotated_tactic": ["apply H4 _ _ ihf ihg", []], "state_before": "case mk.mul\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nC : Poly \u03b1 \u2192 Prop\nH1 : \u2200 (i : \u03b1), C (proj i)\nH2 : \u2200 (n : \u2124), C (const n)\nH3 : \u2200 (f g : Poly \u03b1), C f \u2192 C g \u2192 C (f - g)\nH4 : \u2200 (f g : Poly \u03b1), C f \u2192 C g \u2192 C (f * g)\nf\u271d f g : (\u03b1 \u2192 \u2115) \u2192 \u2124\npf : IsPoly f\npg : IsPoly g\nihf : C { val := f, property := pf }\nihg : C { val := g, property := pg }\n\u22a2 C { val := fun x => f x * g x, property := (_ : IsPoly fun x => f x * g x) }", "state_after": "no goals"}]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Analysis/Calculus/ContDiff.lean
|
ContinuousLinearMap.norm_iteratedFDerivWithin_le_of_bilinear_of_le_one
|
[
2499,
1
] |
[
2507,
13
] |
[{"tactic": "apply (B.norm_iteratedFDerivWithin_le_of_bilinear hf hg hs hx hn).trans", "annotated_tactic": ["apply (B.norm_iteratedFDerivWithin_le_of_bilinear hf hg hs hx hn).<a>trans</a>", [{"full_name": "LE.le.trans", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [120, 7], "def_end_pos": [120, 18]}]], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nD : Type uD\ninst\u271d\u2079 : NormedAddCommGroup D\ninst\u271d\u2078 : NormedSpace \ud835\udd5c D\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns\u271d s\u2081 t u : Set E\nf\u271d f\u2081 : E \u2192 F\ng\u271d : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nb : E \u00d7 F \u2192 G\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nB : E \u2192L[\ud835\udd5c] F \u2192L[\ud835\udd5c] G\nf : D \u2192 E\ng : D \u2192 F\nN : \u2115\u221e\ns : Set D\nx : D\nhf : ContDiffOn \ud835\udd5c N f s\nhg : ContDiffOn \ud835\udd5c N g s\nhs : UniqueDiffOn \ud835\udd5c s\nhx : x \u2208 s\nn : \u2115\nhn : \u2191n \u2264 N\nhB : \u2016B\u2016 \u2264 1\n\u22a2 \u2016iteratedFDerivWithin \ud835\udd5c n (fun y => \u2191(\u2191B (f y)) (g y)) s x\u2016 \u2264\n \u2211 i in Finset.range (n + 1),\n \u2191(Nat.choose n i) * \u2016iteratedFDerivWithin \ud835\udd5c i f s x\u2016 * \u2016iteratedFDerivWithin \ud835\udd5c (n - i) g s x\u2016", "state_after": "\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nD : Type uD\ninst\u271d\u2079 : NormedAddCommGroup D\ninst\u271d\u2078 : NormedSpace \ud835\udd5c D\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns\u271d s\u2081 t u : Set E\nf\u271d f\u2081 : E \u2192 F\ng\u271d : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nb : E \u00d7 F \u2192 G\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nB : E \u2192L[\ud835\udd5c] F \u2192L[\ud835\udd5c] G\nf : D \u2192 E\ng : D \u2192 F\nN : \u2115\u221e\ns : Set D\nx : D\nhf : ContDiffOn \ud835\udd5c N f s\nhg : ContDiffOn \ud835\udd5c N g s\nhs : UniqueDiffOn \ud835\udd5c s\nhx : x \u2208 s\nn : \u2115\nhn : \u2191n \u2264 N\nhB : \u2016B\u2016 \u2264 1\n\u22a2 \u2016B\u2016 *\n \u2211 i in Finset.range (n + 1),\n \u2191(Nat.choose n i) * \u2016iteratedFDerivWithin \ud835\udd5c i f s x\u2016 * \u2016iteratedFDerivWithin \ud835\udd5c (n - i) g s x\u2016 \u2264\n \u2211 i in Finset.range (n + 1),\n \u2191(Nat.choose n i) * \u2016iteratedFDerivWithin \ud835\udd5c i f s x\u2016 * \u2016iteratedFDerivWithin \ud835\udd5c (n - i) g s x\u2016"}, {"tactic": "apply mul_le_of_le_one_left (Finset.sum_nonneg' fun i => ?_) hB", "annotated_tactic": ["apply <a>mul_le_of_le_one_left</a> (<a>Finset.sum_nonneg'</a> fun i => ?_) hB", [{"full_name": "mul_le_of_le_one_left", "def_path": "Mathlib/Algebra/Order/Ring/Lemmas.lean", "def_pos": [666, 9], "def_end_pos": [666, 30]}, {"full_name": "Finset.sum_nonneg'", "def_path": "Mathlib/Algebra/BigOperators/Order.lean", "def_pos": [144, 15], "def_end_pos": [144, 33]}]], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nD : Type uD\ninst\u271d\u2079 : NormedAddCommGroup D\ninst\u271d\u2078 : NormedSpace \ud835\udd5c D\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns\u271d s\u2081 t u : Set E\nf\u271d f\u2081 : E \u2192 F\ng\u271d : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nb : E \u00d7 F \u2192 G\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nB : E \u2192L[\ud835\udd5c] F \u2192L[\ud835\udd5c] G\nf : D \u2192 E\ng : D \u2192 F\nN : \u2115\u221e\ns : Set D\nx : D\nhf : ContDiffOn \ud835\udd5c N f s\nhg : ContDiffOn \ud835\udd5c N g s\nhs : UniqueDiffOn \ud835\udd5c s\nhx : x \u2208 s\nn : \u2115\nhn : \u2191n \u2264 N\nhB : \u2016B\u2016 \u2264 1\n\u22a2 \u2016B\u2016 *\n \u2211 i in Finset.range (n + 1),\n \u2191(Nat.choose n i) * \u2016iteratedFDerivWithin \ud835\udd5c i f s x\u2016 * \u2016iteratedFDerivWithin \ud835\udd5c (n - i) g s x\u2016 \u2264\n \u2211 i in Finset.range (n + 1),\n \u2191(Nat.choose n i) * \u2016iteratedFDerivWithin \ud835\udd5c i f s x\u2016 * \u2016iteratedFDerivWithin \ud835\udd5c (n - i) g s x\u2016", "state_after": "\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nD : Type uD\ninst\u271d\u2079 : NormedAddCommGroup D\ninst\u271d\u2078 : NormedSpace \ud835\udd5c D\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns\u271d s\u2081 t u : Set E\nf\u271d f\u2081 : E \u2192 F\ng\u271d : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nb : E \u00d7 F \u2192 G\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nB : E \u2192L[\ud835\udd5c] F \u2192L[\ud835\udd5c] G\nf : D \u2192 E\ng : D \u2192 F\nN : \u2115\u221e\ns : Set D\nx : D\nhf : ContDiffOn \ud835\udd5c N f s\nhg : ContDiffOn \ud835\udd5c N g s\nhs : UniqueDiffOn \ud835\udd5c s\nhx : x \u2208 s\nn : \u2115\nhn : \u2191n \u2264 N\nhB : \u2016B\u2016 \u2264 1\ni : \u2115\n\u22a2 0 \u2264 \u2191(Nat.choose n i) * \u2016iteratedFDerivWithin \ud835\udd5c i f s x\u2016 * \u2016iteratedFDerivWithin \ud835\udd5c (n - i) g s x\u2016"}, {"tactic": "positivity", "annotated_tactic": ["positivity", []], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nD : Type uD\ninst\u271d\u2079 : NormedAddCommGroup D\ninst\u271d\u2078 : NormedSpace \ud835\udd5c D\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns\u271d s\u2081 t u : Set E\nf\u271d f\u2081 : E \u2192 F\ng\u271d : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nb : E \u00d7 F \u2192 G\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nB : E \u2192L[\ud835\udd5c] F \u2192L[\ud835\udd5c] G\nf : D \u2192 E\ng : D \u2192 F\nN : \u2115\u221e\ns : Set D\nx : D\nhf : ContDiffOn \ud835\udd5c N f s\nhg : ContDiffOn \ud835\udd5c N g s\nhs : UniqueDiffOn \ud835\udd5c s\nhx : x \u2208 s\nn : \u2115\nhn : \u2191n \u2264 N\nhB : \u2016B\u2016 \u2264 1\ni : \u2115\n\u22a2 0 \u2264 \u2191(Nat.choose n i) * \u2016iteratedFDerivWithin \ud835\udd5c i f s x\u2016 * \u2016iteratedFDerivWithin \ud835\udd5c (n - i) g s x\u2016", "state_after": "no goals"}]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Combinatorics/Additive/Energy.lean
|
Finset.multiplicativeEnergy_univ_left
|
[
144,
1
] |
[
159,
25
] |
[{"tactic": "simp only [multiplicativeEnergy, univ_product_univ, Fintype.card, sq, \u2190 card_product]", "annotated_tactic": ["simp only [<a>multiplicativeEnergy</a>, <a>univ_product_univ</a>, <a>Fintype.card</a>, <a>sq</a>, \u2190 <a>card_product</a>]", [{"full_name": "Finset.multiplicativeEnergy", "def_path": "Mathlib/Combinatorics/Additive/Energy.lean", "def_pos": [43, 5], "def_end_pos": [43, 25]}, {"full_name": "Finset.univ_product_univ", "def_path": "Mathlib/Data/Fintype/Prod.lean", "def_pos": [48, 9], "def_end_pos": [48, 33]}, {"full_name": "Fintype.card", "def_path": "Mathlib/Data/Fintype/Card.lean", "def_pos": [61, 5], "def_end_pos": [61, 9]}, {"full_name": "sq", "def_path": "Mathlib/Algebra/GroupPower/Basic.lean", "def_pos": [106, 7], "def_end_pos": [106, 9]}, {"full_name": "Finset.card_product", "def_path": "Mathlib/Data/Finset/Prod.lean", "def_pos": [139, 9], "def_end_pos": [139, 21]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : Fintype \u03b1\ns t : Finset \u03b1\n\u22a2 multiplicativeEnergy univ t = Fintype.card \u03b1 * card t ^ 2", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : Fintype \u03b1\ns t : Finset \u03b1\n\u22a2 card (filter (fun x => x.1.1 * x.2.1 = x.1.2 * x.2.2) (univ \u00d7\u02e2 t \u00d7\u02e2 t)) = card (univ \u00d7\u02e2 t \u00d7\u02e2 t)"}, {"tactic": "let f : \u03b1 \u00d7 \u03b1 \u00d7 \u03b1 \u2192 (\u03b1 \u00d7 \u03b1) \u00d7 \u03b1 \u00d7 \u03b1 := fun x => ((x.1 * x.2.2, x.1 * x.2.1), x.2)", "annotated_tactic": ["let f : \u03b1 \u00d7 \u03b1 \u00d7 \u03b1 \u2192 (\u03b1 \u00d7 \u03b1) \u00d7 \u03b1 \u00d7 \u03b1 := fun x => ((x.1 * x.2.2, x.1 * x.2.1), x.2)", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : Fintype \u03b1\ns t : Finset \u03b1\n\u22a2 card (filter (fun x => x.1.1 * x.2.1 = x.1.2 * x.2.2) (univ \u00d7\u02e2 t \u00d7\u02e2 t)) = card (univ \u00d7\u02e2 t \u00d7\u02e2 t)", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : Fintype \u03b1\ns t : Finset \u03b1\nf : \u03b1 \u00d7 \u03b1 \u00d7 \u03b1 \u2192 (\u03b1 \u00d7 \u03b1) \u00d7 \u03b1 \u00d7 \u03b1 := fun x => ((x.1 * x.2.2, x.1 * x.2.1), x.2)\n\u22a2 card (filter (fun x => x.1.1 * x.2.1 = x.1.2 * x.2.2) (univ \u00d7\u02e2 t \u00d7\u02e2 t)) = card (univ \u00d7\u02e2 t \u00d7\u02e2 t)"}, {"tactic": "have : (\u2191((univ : Finset \u03b1) \u00d7\u02e2 t \u00d7\u02e2 t) : Set (\u03b1 \u00d7 \u03b1 \u00d7 \u03b1)).InjOn f := by\n rintro \u27e8a\u2081, b\u2081, c\u2081\u27e9 _ \u27e8a\u2082, b\u2082, c\u2082\u27e9 h\u2082 h\n simp_rw [Prod.ext_iff] at h\n obtain \u27e8h, rfl, rfl\u27e9 := h\n rw [mul_right_cancel h.1]", "annotated_tactic": ["have : (\u2191((<a>univ</a> : <a>Finset</a> \u03b1) \u00d7\u02e2 t \u00d7\u02e2 t) : <a>Set</a> (\u03b1 \u00d7 \u03b1 \u00d7 \u03b1)).<a>InjOn</a> f := by\n rintro \u27e8a\u2081, b\u2081, c\u2081\u27e9 _ \u27e8a\u2082, b\u2082, c\u2082\u27e9 h\u2082 h\n simp_rw [<a>Prod.ext_iff</a>] at h\n obtain \u27e8h, rfl, rfl\u27e9 := h\n rw [<a>mul_right_cancel</a> h.1]", [{"full_name": "Finset.univ", "def_path": "Mathlib/Data/Fintype/Basic.lean", "def_pos": [67, 5], "def_end_pos": [67, 9]}, {"full_name": "Finset", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [138, 11], "def_end_pos": [138, 17]}, {"full_name": "Set", "def_path": "Mathlib/Init/Set.lean", "def_pos": [38, 5], "def_end_pos": [38, 8]}, {"full_name": "Set.InjOn", "def_path": "Mathlib/Data/Set/Function.lean", "def_pos": [603, 5], "def_end_pos": [603, 10]}, {"full_name": "Prod.ext_iff", "def_path": "Mathlib/Data/Prod/Basic.lean", "def_pos": [124, 9], "def_end_pos": [124, 16]}, {"full_name": "mul_right_cancel", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [207, 9], "def_end_pos": [207, 25]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : Fintype \u03b1\ns t : Finset \u03b1\nf : \u03b1 \u00d7 \u03b1 \u00d7 \u03b1 \u2192 (\u03b1 \u00d7 \u03b1) \u00d7 \u03b1 \u00d7 \u03b1 := fun x => ((x.1 * x.2.2, x.1 * x.2.1), x.2)\n\u22a2 card (filter (fun x => x.1.1 * x.2.1 = x.1.2 * x.2.2) (univ \u00d7\u02e2 t \u00d7\u02e2 t)) = card (univ \u00d7\u02e2 t \u00d7\u02e2 t)", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : Fintype \u03b1\ns t : Finset \u03b1\nf : \u03b1 \u00d7 \u03b1 \u00d7 \u03b1 \u2192 (\u03b1 \u00d7 \u03b1) \u00d7 \u03b1 \u00d7 \u03b1 := fun x => ((x.1 * x.2.2, x.1 * x.2.1), x.2)\nthis : Set.InjOn f \u2191(univ \u00d7\u02e2 t \u00d7\u02e2 t)\n\u22a2 card (filter (fun x => x.1.1 * x.2.1 = x.1.2 * x.2.2) (univ \u00d7\u02e2 t \u00d7\u02e2 t)) = card (univ \u00d7\u02e2 t \u00d7\u02e2 t)"}, {"tactic": "rw [\u2190 card_image_of_injOn this]", "annotated_tactic": ["rw [\u2190 <a>card_image_of_injOn</a> this]", [{"full_name": "Finset.card_image_of_injOn", "def_path": "Mathlib/Data/Finset/Card.lean", "def_pos": [232, 9], "def_end_pos": [232, 28]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : Fintype \u03b1\ns t : Finset \u03b1\nf : \u03b1 \u00d7 \u03b1 \u00d7 \u03b1 \u2192 (\u03b1 \u00d7 \u03b1) \u00d7 \u03b1 \u00d7 \u03b1 := fun x => ((x.1 * x.2.2, x.1 * x.2.1), x.2)\nthis : Set.InjOn f \u2191(univ \u00d7\u02e2 t \u00d7\u02e2 t)\n\u22a2 card (filter (fun x => x.1.1 * x.2.1 = x.1.2 * x.2.2) (univ \u00d7\u02e2 t \u00d7\u02e2 t)) = card (univ \u00d7\u02e2 t \u00d7\u02e2 t)", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : Fintype \u03b1\ns t : Finset \u03b1\nf : \u03b1 \u00d7 \u03b1 \u00d7 \u03b1 \u2192 (\u03b1 \u00d7 \u03b1) \u00d7 \u03b1 \u00d7 \u03b1 := fun x => ((x.1 * x.2.2, x.1 * x.2.1), x.2)\nthis : Set.InjOn f \u2191(univ \u00d7\u02e2 t \u00d7\u02e2 t)\n\u22a2 card (filter (fun x => x.1.1 * x.2.1 = x.1.2 * x.2.2) (univ \u00d7\u02e2 t \u00d7\u02e2 t)) = card (image f (univ \u00d7\u02e2 t \u00d7\u02e2 t))"}, {"tactic": "congr with a", "annotated_tactic": ["congr with a", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : Fintype \u03b1\ns t : Finset \u03b1\nf : \u03b1 \u00d7 \u03b1 \u00d7 \u03b1 \u2192 (\u03b1 \u00d7 \u03b1) \u00d7 \u03b1 \u00d7 \u03b1 := fun x => ((x.1 * x.2.2, x.1 * x.2.1), x.2)\nthis : Set.InjOn f \u2191(univ \u00d7\u02e2 t \u00d7\u02e2 t)\n\u22a2 card (filter (fun x => x.1.1 * x.2.1 = x.1.2 * x.2.2) (univ \u00d7\u02e2 t \u00d7\u02e2 t)) = card (image f (univ \u00d7\u02e2 t \u00d7\u02e2 t))", "state_after": "case e_s.a\n\u03b1 : Type u_1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : Fintype \u03b1\ns t : Finset \u03b1\nf : \u03b1 \u00d7 \u03b1 \u00d7 \u03b1 \u2192 (\u03b1 \u00d7 \u03b1) \u00d7 \u03b1 \u00d7 \u03b1 := fun x => ((x.1 * x.2.2, x.1 * x.2.1), x.2)\nthis : Set.InjOn f \u2191(univ \u00d7\u02e2 t \u00d7\u02e2 t)\na : (\u03b1 \u00d7 \u03b1) \u00d7 \u03b1 \u00d7 \u03b1\n\u22a2 a \u2208 filter (fun x => x.1.1 * x.2.1 = x.1.2 * x.2.2) (univ \u00d7\u02e2 t \u00d7\u02e2 t) \u2194 a \u2208 image f (univ \u00d7\u02e2 t \u00d7\u02e2 t)"}, {"tactic": "simp only [mem_filter, mem_product, mem_univ, true_and_iff, mem_image, exists_prop,\n Prod.exists]", "annotated_tactic": ["simp only [<a>mem_filter</a>, <a>mem_product</a>, <a>mem_univ</a>, <a>true_and_iff</a>, <a>mem_image</a>, <a>exists_prop</a>,\n <a>Prod.exists</a>]", [{"full_name": "Finset.mem_filter", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [2708, 9], "def_end_pos": [2708, 19]}, {"full_name": "Finset.mem_product", "def_path": "Mathlib/Data/Finset/Prod.lean", "def_pos": [53, 9], "def_end_pos": [53, 20]}, {"full_name": "Finset.mem_univ", "def_path": "Mathlib/Data/Fintype/Basic.lean", "def_pos": [72, 9], "def_end_pos": [72, 17]}, {"full_name": "true_and_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [147, 9], "def_end_pos": [147, 21]}, {"full_name": "Finset.mem_image", "def_path": "Mathlib/Data/Finset/Image.lean", "def_pos": [330, 9], "def_end_pos": [330, 18]}, {"full_name": "exists_prop", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [485, 17], "def_end_pos": [485, 28]}, {"full_name": "Prod.exists", "def_path": "Mathlib/Data/Prod/Basic.lean", "def_pos": [41, 9], "def_end_pos": [41, 17]}]], "state_before": "case e_s.a\n\u03b1 : Type u_1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : Fintype \u03b1\ns t : Finset \u03b1\nf : \u03b1 \u00d7 \u03b1 \u00d7 \u03b1 \u2192 (\u03b1 \u00d7 \u03b1) \u00d7 \u03b1 \u00d7 \u03b1 := fun x => ((x.1 * x.2.2, x.1 * x.2.1), x.2)\nthis : Set.InjOn f \u2191(univ \u00d7\u02e2 t \u00d7\u02e2 t)\na : (\u03b1 \u00d7 \u03b1) \u00d7 \u03b1 \u00d7 \u03b1\n\u22a2 a \u2208 filter (fun x => x.1.1 * x.2.1 = x.1.2 * x.2.2) (univ \u00d7\u02e2 t \u00d7\u02e2 t) \u2194 a \u2208 image f (univ \u00d7\u02e2 t \u00d7\u02e2 t)", "state_after": "case e_s.a\n\u03b1 : Type u_1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : Fintype \u03b1\ns t : Finset \u03b1\nf : \u03b1 \u00d7 \u03b1 \u00d7 \u03b1 \u2192 (\u03b1 \u00d7 \u03b1) \u00d7 \u03b1 \u00d7 \u03b1 := fun x => ((x.1 * x.2.2, x.1 * x.2.1), x.2)\nthis : Set.InjOn f \u2191(univ \u00d7\u02e2 t \u00d7\u02e2 t)\na : (\u03b1 \u00d7 \u03b1) \u00d7 \u03b1 \u00d7 \u03b1\n\u22a2 (a.2.1 \u2208 t \u2227 a.2.2 \u2208 t) \u2227 a.1.1 * a.2.1 = a.1.2 * a.2.2 \u2194\n \u2203 a_1 a_2 b, (a_2 \u2208 t \u2227 b \u2208 t) \u2227 ((a_1 * b, a_1 * a_2), a_2, b) = a"}, {"tactic": "refine' \u27e8fun h => \u27e8a.1.1 * a.2.2\u207b\u00b9, _, _, h.1, by simp [mul_right_comm, h.2]\u27e9, _\u27e9", "annotated_tactic": ["refine' \u27e8fun h => \u27e8a.1.1 * a.2.2\u207b\u00b9, _, _, h.1, by simp [<a>mul_right_comm</a>, h.2]\u27e9, _\u27e9", [{"full_name": "mul_right_comm", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [102, 9], "def_end_pos": [102, 23]}]], "state_before": "case e_s.a\n\u03b1 : Type u_1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : Fintype \u03b1\ns t : Finset \u03b1\nf : \u03b1 \u00d7 \u03b1 \u00d7 \u03b1 \u2192 (\u03b1 \u00d7 \u03b1) \u00d7 \u03b1 \u00d7 \u03b1 := fun x => ((x.1 * x.2.2, x.1 * x.2.1), x.2)\nthis : Set.InjOn f \u2191(univ \u00d7\u02e2 t \u00d7\u02e2 t)\na : (\u03b1 \u00d7 \u03b1) \u00d7 \u03b1 \u00d7 \u03b1\n\u22a2 (a.2.1 \u2208 t \u2227 a.2.2 \u2208 t) \u2227 a.1.1 * a.2.1 = a.1.2 * a.2.2 \u2194\n \u2203 a_1 a_2 b, (a_2 \u2208 t \u2227 b \u2208 t) \u2227 ((a_1 * b, a_1 * a_2), a_2, b) = a", "state_after": "case e_s.a\n\u03b1 : Type u_1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : Fintype \u03b1\ns t : Finset \u03b1\nf : \u03b1 \u00d7 \u03b1 \u00d7 \u03b1 \u2192 (\u03b1 \u00d7 \u03b1) \u00d7 \u03b1 \u00d7 \u03b1 := fun x => ((x.1 * x.2.2, x.1 * x.2.1), x.2)\nthis : Set.InjOn f \u2191(univ \u00d7\u02e2 t \u00d7\u02e2 t)\na : (\u03b1 \u00d7 \u03b1) \u00d7 \u03b1 \u00d7 \u03b1\n\u22a2 (\u2203 a_1 a_2 b, (a_2 \u2208 t \u2227 b \u2208 t) \u2227 ((a_1 * b, a_1 * a_2), a_2, b) = a) \u2192\n (a.2.1 \u2208 t \u2227 a.2.2 \u2208 t) \u2227 a.1.1 * a.2.1 = a.1.2 * a.2.2"}, {"tactic": "rintro \u27e8b, c, d, hcd, rfl\u27e9", "annotated_tactic": ["rintro \u27e8b, c, d, hcd, rfl\u27e9", []], "state_before": "case e_s.a\n\u03b1 : Type u_1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : Fintype \u03b1\ns t : Finset \u03b1\nf : \u03b1 \u00d7 \u03b1 \u00d7 \u03b1 \u2192 (\u03b1 \u00d7 \u03b1) \u00d7 \u03b1 \u00d7 \u03b1 := fun x => ((x.1 * x.2.2, x.1 * x.2.1), x.2)\nthis : Set.InjOn f \u2191(univ \u00d7\u02e2 t \u00d7\u02e2 t)\na : (\u03b1 \u00d7 \u03b1) \u00d7 \u03b1 \u00d7 \u03b1\n\u22a2 (\u2203 a_1 a_2 b, (a_2 \u2208 t \u2227 b \u2208 t) \u2227 ((a_1 * b, a_1 * a_2), a_2, b) = a) \u2192\n (a.2.1 \u2208 t \u2227 a.2.2 \u2208 t) \u2227 a.1.1 * a.2.1 = a.1.2 * a.2.2", "state_after": "case e_s.a.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : Fintype \u03b1\ns t : Finset \u03b1\nf : \u03b1 \u00d7 \u03b1 \u00d7 \u03b1 \u2192 (\u03b1 \u00d7 \u03b1) \u00d7 \u03b1 \u00d7 \u03b1 := fun x => ((x.1 * x.2.2, x.1 * x.2.1), x.2)\nthis : Set.InjOn f \u2191(univ \u00d7\u02e2 t \u00d7\u02e2 t)\nb c d : \u03b1\nhcd : c \u2208 t \u2227 d \u2208 t\n\u22a2 (((b * d, b * c), c, d).2.1 \u2208 t \u2227 ((b * d, b * c), c, d).2.2 \u2208 t) \u2227\n ((b * d, b * c), c, d).1.1 * ((b * d, b * c), c, d).2.1 = ((b * d, b * c), c, d).1.2 * ((b * d, b * c), c, d).2.2"}, {"tactic": "simpa [mul_right_comm]", "annotated_tactic": ["simpa [<a>mul_right_comm</a>]", [{"full_name": "mul_right_comm", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [102, 9], "def_end_pos": [102, 23]}]], "state_before": "case e_s.a.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : Fintype \u03b1\ns t : Finset \u03b1\nf : \u03b1 \u00d7 \u03b1 \u00d7 \u03b1 \u2192 (\u03b1 \u00d7 \u03b1) \u00d7 \u03b1 \u00d7 \u03b1 := fun x => ((x.1 * x.2.2, x.1 * x.2.1), x.2)\nthis : Set.InjOn f \u2191(univ \u00d7\u02e2 t \u00d7\u02e2 t)\nb c d : \u03b1\nhcd : c \u2208 t \u2227 d \u2208 t\n\u22a2 (((b * d, b * c), c, d).2.1 \u2208 t \u2227 ((b * d, b * c), c, d).2.2 \u2208 t) \u2227\n ((b * d, b * c), c, d).1.1 * ((b * d, b * c), c, d).2.1 = ((b * d, b * c), c, d).1.2 * ((b * d, b * c), c, d).2.2", "state_after": "no goals"}, {"tactic": "rintro \u27e8a\u2081, b\u2081, c\u2081\u27e9 _ \u27e8a\u2082, b\u2082, c\u2082\u27e9 h\u2082 h", "annotated_tactic": ["rintro \u27e8a\u2081, b\u2081, c\u2081\u27e9 _ \u27e8a\u2082, b\u2082, c\u2082\u27e9 h\u2082 h", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : Fintype \u03b1\ns t : Finset \u03b1\nf : \u03b1 \u00d7 \u03b1 \u00d7 \u03b1 \u2192 (\u03b1 \u00d7 \u03b1) \u00d7 \u03b1 \u00d7 \u03b1 := fun x => ((x.1 * x.2.2, x.1 * x.2.1), x.2)\n\u22a2 Set.InjOn f \u2191(univ \u00d7\u02e2 t \u00d7\u02e2 t)", "state_after": "case mk.mk.mk.mk\n\u03b1 : Type u_1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : Fintype \u03b1\ns t : Finset \u03b1\nf : \u03b1 \u00d7 \u03b1 \u00d7 \u03b1 \u2192 (\u03b1 \u00d7 \u03b1) \u00d7 \u03b1 \u00d7 \u03b1 := fun x => ((x.1 * x.2.2, x.1 * x.2.1), x.2)\na\u2081 b\u2081 c\u2081 : \u03b1\na\u271d : (a\u2081, b\u2081, c\u2081) \u2208 \u2191(univ \u00d7\u02e2 t \u00d7\u02e2 t)\na\u2082 b\u2082 c\u2082 : \u03b1\nh\u2082 : (a\u2082, b\u2082, c\u2082) \u2208 \u2191(univ \u00d7\u02e2 t \u00d7\u02e2 t)\nh : f (a\u2081, b\u2081, c\u2081) = f (a\u2082, b\u2082, c\u2082)\n\u22a2 (a\u2081, b\u2081, c\u2081) = (a\u2082, b\u2082, c\u2082)"}, {"tactic": "simp_rw [Prod.ext_iff] at h", "annotated_tactic": ["simp_rw [<a>Prod.ext_iff</a>] at h", [{"full_name": "Prod.ext_iff", "def_path": "Mathlib/Data/Prod/Basic.lean", "def_pos": [124, 9], "def_end_pos": [124, 16]}]], "state_before": "case mk.mk.mk.mk\n\u03b1 : Type u_1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : Fintype \u03b1\ns t : Finset \u03b1\nf : \u03b1 \u00d7 \u03b1 \u00d7 \u03b1 \u2192 (\u03b1 \u00d7 \u03b1) \u00d7 \u03b1 \u00d7 \u03b1 := fun x => ((x.1 * x.2.2, x.1 * x.2.1), x.2)\na\u2081 b\u2081 c\u2081 : \u03b1\na\u271d : (a\u2081, b\u2081, c\u2081) \u2208 \u2191(univ \u00d7\u02e2 t \u00d7\u02e2 t)\na\u2082 b\u2082 c\u2082 : \u03b1\nh\u2082 : (a\u2082, b\u2082, c\u2082) \u2208 \u2191(univ \u00d7\u02e2 t \u00d7\u02e2 t)\nh : f (a\u2081, b\u2081, c\u2081) = f (a\u2082, b\u2082, c\u2082)\n\u22a2 (a\u2081, b\u2081, c\u2081) = (a\u2082, b\u2082, c\u2082)", "state_after": "case mk.mk.mk.mk\n\u03b1 : Type u_1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : Fintype \u03b1\ns t : Finset \u03b1\nf : \u03b1 \u00d7 \u03b1 \u00d7 \u03b1 \u2192 (\u03b1 \u00d7 \u03b1) \u00d7 \u03b1 \u00d7 \u03b1 := fun x => ((x.1 * x.2.2, x.1 * x.2.1), x.2)\na\u2081 b\u2081 c\u2081 : \u03b1\na\u271d : (a\u2081, b\u2081, c\u2081) \u2208 \u2191(univ \u00d7\u02e2 t \u00d7\u02e2 t)\na\u2082 b\u2082 c\u2082 : \u03b1\nh\u2082 : (a\u2082, b\u2082, c\u2082) \u2208 \u2191(univ \u00d7\u02e2 t \u00d7\u02e2 t)\nh : (a\u2081 * c\u2081 = a\u2082 * c\u2082 \u2227 a\u2081 * b\u2081 = a\u2082 * b\u2082) \u2227 b\u2081 = b\u2082 \u2227 c\u2081 = c\u2082\n\u22a2 (a\u2081, b\u2081, c\u2081) = (a\u2082, b\u2082, c\u2082)"}, {"tactic": "obtain \u27e8h, rfl, rfl\u27e9 := h", "annotated_tactic": ["obtain \u27e8h, rfl, rfl\u27e9 := h", []], "state_before": "case mk.mk.mk.mk\n\u03b1 : Type u_1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : Fintype \u03b1\ns t : Finset \u03b1\nf : \u03b1 \u00d7 \u03b1 \u00d7 \u03b1 \u2192 (\u03b1 \u00d7 \u03b1) \u00d7 \u03b1 \u00d7 \u03b1 := fun x => ((x.1 * x.2.2, x.1 * x.2.1), x.2)\na\u2081 b\u2081 c\u2081 : \u03b1\na\u271d : (a\u2081, b\u2081, c\u2081) \u2208 \u2191(univ \u00d7\u02e2 t \u00d7\u02e2 t)\na\u2082 b\u2082 c\u2082 : \u03b1\nh\u2082 : (a\u2082, b\u2082, c\u2082) \u2208 \u2191(univ \u00d7\u02e2 t \u00d7\u02e2 t)\nh : (a\u2081 * c\u2081 = a\u2082 * c\u2082 \u2227 a\u2081 * b\u2081 = a\u2082 * b\u2082) \u2227 b\u2081 = b\u2082 \u2227 c\u2081 = c\u2082\n\u22a2 (a\u2081, b\u2081, c\u2081) = (a\u2082, b\u2082, c\u2082)", "state_after": "case mk.mk.mk.mk.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : Fintype \u03b1\ns t : Finset \u03b1\nf : \u03b1 \u00d7 \u03b1 \u00d7 \u03b1 \u2192 (\u03b1 \u00d7 \u03b1) \u00d7 \u03b1 \u00d7 \u03b1 := fun x => ((x.1 * x.2.2, x.1 * x.2.1), x.2)\na\u2081 b\u2081 c\u2081 : \u03b1\na\u271d : (a\u2081, b\u2081, c\u2081) \u2208 \u2191(univ \u00d7\u02e2 t \u00d7\u02e2 t)\na\u2082 : \u03b1\nh\u2082 : (a\u2082, b\u2081, c\u2081) \u2208 \u2191(univ \u00d7\u02e2 t \u00d7\u02e2 t)\nh : a\u2081 * c\u2081 = a\u2082 * c\u2081 \u2227 a\u2081 * b\u2081 = a\u2082 * b\u2081\n\u22a2 (a\u2081, b\u2081, c\u2081) = (a\u2082, b\u2081, c\u2081)"}, {"tactic": "rw [mul_right_cancel h.1]", "annotated_tactic": ["rw [<a>mul_right_cancel</a> h.1]", [{"full_name": "mul_right_cancel", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [207, 9], "def_end_pos": [207, 25]}]], "state_before": "case mk.mk.mk.mk.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : Fintype \u03b1\ns t : Finset \u03b1\nf : \u03b1 \u00d7 \u03b1 \u00d7 \u03b1 \u2192 (\u03b1 \u00d7 \u03b1) \u00d7 \u03b1 \u00d7 \u03b1 := fun x => ((x.1 * x.2.2, x.1 * x.2.1), x.2)\na\u2081 b\u2081 c\u2081 : \u03b1\na\u271d : (a\u2081, b\u2081, c\u2081) \u2208 \u2191(univ \u00d7\u02e2 t \u00d7\u02e2 t)\na\u2082 : \u03b1\nh\u2082 : (a\u2082, b\u2081, c\u2081) \u2208 \u2191(univ \u00d7\u02e2 t \u00d7\u02e2 t)\nh : a\u2081 * c\u2081 = a\u2082 * c\u2081 \u2227 a\u2081 * b\u2081 = a\u2082 * b\u2081\n\u22a2 (a\u2081, b\u2081, c\u2081) = (a\u2082, b\u2081, c\u2081)", "state_after": "no goals"}, {"tactic": "simp [mul_right_comm, h.2]", "annotated_tactic": ["simp [<a>mul_right_comm</a>, h.2]", [{"full_name": "mul_right_comm", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [102, 9], "def_end_pos": [102, 23]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : Fintype \u03b1\ns t : Finset \u03b1\nf : \u03b1 \u00d7 \u03b1 \u00d7 \u03b1 \u2192 (\u03b1 \u00d7 \u03b1) \u00d7 \u03b1 \u00d7 \u03b1 := fun x => ((x.1 * x.2.2, x.1 * x.2.1), x.2)\nthis : Set.InjOn f \u2191(univ \u00d7\u02e2 t \u00d7\u02e2 t)\na : (\u03b1 \u00d7 \u03b1) \u00d7 \u03b1 \u00d7 \u03b1\nh : (a.2.1 \u2208 t \u2227 a.2.2 \u2208 t) \u2227 a.1.1 * a.2.1 = a.1.2 * a.2.2\n\u22a2 ((a.1.1 * a.2.2\u207b\u00b9 * a.2.2, a.1.1 * a.2.2\u207b\u00b9 * a.2.1), a.2.1, a.2.2) = a", "state_after": "no goals"}]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/RingTheory/Polynomial/Content.lean
|
Polynomial.isPrimitive_of_dvd
|
[
66,
1
] |
[
67,
72
] |
[]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Geometry/Euclidean/Angle/Unoriented/Affine.lean
|
EuclideanGeometry.angle_le_pi
|
[
136,
8
] |
[
137,
18
] |
[]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Combinatorics/SimpleGraph/Connectivity.lean
|
SimpleGraph.Walk.length_cons
|
[
385,
1
] |
[
386,
44
] |
[]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Topology/MetricSpace/Lipschitz.lean
|
LipschitzWith.uncurry
|
[
280,
11
] |
[
287,
73
] |
[{"tactic": "rintro \u27e8a\u2081, b\u2081\u27e9 \u27e8a\u2082, b\u2082\u27e9", "annotated_tactic": ["rintro \u27e8a\u2081, b\u2081\u27e9 \u27e8a\u2082, b\u2082\u27e9", []], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d\u00b2 : PseudoEMetricSpace \u03b1\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b2\ninst\u271d : PseudoEMetricSpace \u03b3\nK : \u211d\u22650\nf\u271d : \u03b1 \u2192 \u03b2\nx y : \u03b1\nr : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nK\u03b1 K\u03b2 : \u211d\u22650\nh\u03b1 : \u2200 (b : \u03b2), LipschitzWith K\u03b1 fun a => f a b\nh\u03b2 : \u2200 (a : \u03b1), LipschitzWith K\u03b2 (f a)\n\u22a2 LipschitzWith (K\u03b1 + K\u03b2) (uncurry f)", "state_after": "case mk.mk\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d\u00b2 : PseudoEMetricSpace \u03b1\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b2\ninst\u271d : PseudoEMetricSpace \u03b3\nK : \u211d\u22650\nf\u271d : \u03b1 \u2192 \u03b2\nx y : \u03b1\nr : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nK\u03b1 K\u03b2 : \u211d\u22650\nh\u03b1 : \u2200 (b : \u03b2), LipschitzWith K\u03b1 fun a => f a b\nh\u03b2 : \u2200 (a : \u03b1), LipschitzWith K\u03b2 (f a)\na\u2081 : \u03b1\nb\u2081 : \u03b2\na\u2082 : \u03b1\nb\u2082 : \u03b2\n\u22a2 edist (uncurry f (a\u2081, b\u2081)) (uncurry f (a\u2082, b\u2082)) \u2264 \u2191(K\u03b1 + K\u03b2) * edist (a\u2081, b\u2081) (a\u2082, b\u2082)"}, {"tactic": "simp only [Function.uncurry, ENNReal.coe_add, add_mul]", "annotated_tactic": ["simp only [<a>Function.uncurry</a>, <a>ENNReal.coe_add</a>, <a>add_mul</a>]", [{"full_name": "Function.uncurry", "def_path": "Mathlib/Init/Function.lean", "def_pos": [217, 5], "def_end_pos": [217, 12]}, {"full_name": "ENNReal.coe_add", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [386, 28], "def_end_pos": [386, 35]}, {"full_name": "add_mul", "def_path": "Mathlib/Algebra/Ring/Defs.lean", "def_pos": [91, 7], "def_end_pos": [91, 14]}]], "state_before": "case mk.mk\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d\u00b2 : PseudoEMetricSpace \u03b1\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b2\ninst\u271d : PseudoEMetricSpace \u03b3\nK : \u211d\u22650\nf\u271d : \u03b1 \u2192 \u03b2\nx y : \u03b1\nr : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nK\u03b1 K\u03b2 : \u211d\u22650\nh\u03b1 : \u2200 (b : \u03b2), LipschitzWith K\u03b1 fun a => f a b\nh\u03b2 : \u2200 (a : \u03b1), LipschitzWith K\u03b2 (f a)\na\u2081 : \u03b1\nb\u2081 : \u03b2\na\u2082 : \u03b1\nb\u2082 : \u03b2\n\u22a2 edist (uncurry f (a\u2081, b\u2081)) (uncurry f (a\u2082, b\u2082)) \u2264 \u2191(K\u03b1 + K\u03b2) * edist (a\u2081, b\u2081) (a\u2082, b\u2082)", "state_after": "case mk.mk\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d\u00b2 : PseudoEMetricSpace \u03b1\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b2\ninst\u271d : PseudoEMetricSpace \u03b3\nK : \u211d\u22650\nf\u271d : \u03b1 \u2192 \u03b2\nx y : \u03b1\nr : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nK\u03b1 K\u03b2 : \u211d\u22650\nh\u03b1 : \u2200 (b : \u03b2), LipschitzWith K\u03b1 fun a => f a b\nh\u03b2 : \u2200 (a : \u03b1), LipschitzWith K\u03b2 (f a)\na\u2081 : \u03b1\nb\u2081 : \u03b2\na\u2082 : \u03b1\nb\u2082 : \u03b2\n\u22a2 edist (f a\u2081 b\u2081) (f a\u2082 b\u2082) \u2264 \u2191K\u03b1 * edist (a\u2081, b\u2081) (a\u2082, b\u2082) + \u2191K\u03b2 * edist (a\u2081, b\u2081) (a\u2082, b\u2082)"}, {"tactic": "apply le_trans (edist_triangle _ (f a\u2082 b\u2081) _)", "annotated_tactic": ["apply <a>le_trans</a> (<a>edist_triangle</a> _ (f a\u2082 b\u2081) _)", [{"full_name": "le_trans", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [56, 9], "def_end_pos": [56, 17]}, {"full_name": "PseudoEMetricSpace.edist_triangle", "def_path": "Mathlib/Topology/EMetricSpace/Basic.lean", "def_pos": [80, 3], "def_end_pos": [80, 17]}]], "state_before": "case mk.mk\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d\u00b2 : PseudoEMetricSpace \u03b1\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b2\ninst\u271d : PseudoEMetricSpace \u03b3\nK : \u211d\u22650\nf\u271d : \u03b1 \u2192 \u03b2\nx y : \u03b1\nr : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nK\u03b1 K\u03b2 : \u211d\u22650\nh\u03b1 : \u2200 (b : \u03b2), LipschitzWith K\u03b1 fun a => f a b\nh\u03b2 : \u2200 (a : \u03b1), LipschitzWith K\u03b2 (f a)\na\u2081 : \u03b1\nb\u2081 : \u03b2\na\u2082 : \u03b1\nb\u2082 : \u03b2\n\u22a2 edist (f a\u2081 b\u2081) (f a\u2082 b\u2082) \u2264 \u2191K\u03b1 * edist (a\u2081, b\u2081) (a\u2082, b\u2082) + \u2191K\u03b2 * edist (a\u2081, b\u2081) (a\u2082, b\u2082)", "state_after": "case mk.mk\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d\u00b2 : PseudoEMetricSpace \u03b1\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b2\ninst\u271d : PseudoEMetricSpace \u03b3\nK : \u211d\u22650\nf\u271d : \u03b1 \u2192 \u03b2\nx y : \u03b1\nr : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nK\u03b1 K\u03b2 : \u211d\u22650\nh\u03b1 : \u2200 (b : \u03b2), LipschitzWith K\u03b1 fun a => f a b\nh\u03b2 : \u2200 (a : \u03b1), LipschitzWith K\u03b2 (f a)\na\u2081 : \u03b1\nb\u2081 : \u03b2\na\u2082 : \u03b1\nb\u2082 : \u03b2\n\u22a2 edist (f a\u2081 b\u2081) (f a\u2082 b\u2081) + edist (f a\u2082 b\u2081) (f a\u2082 b\u2082) \u2264 \u2191K\u03b1 * edist (a\u2081, b\u2081) (a\u2082, b\u2082) + \u2191K\u03b2 * edist (a\u2081, b\u2081) (a\u2082, b\u2082)"}, {"tactic": "exact\n add_le_add (le_trans (h\u03b1 _ _ _) <| ENNReal.mul_left_mono <| le_max_left _ _)\n (le_trans (h\u03b2 _ _ _) <| ENNReal.mul_left_mono <| le_max_right _ _)", "annotated_tactic": ["exact\n <a>add_le_add</a> (<a>le_trans</a> (h\u03b1 _ _ _) <| <a>ENNReal.mul_left_mono</a> <| <a>le_max_left</a> _ _)\n (<a>le_trans</a> (h\u03b2 _ _ _) <| <a>ENNReal.mul_left_mono</a> <| <a>le_max_right</a> _ _)", [{"full_name": "add_le_add", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [205, 15], "def_end_pos": [205, 25]}, {"full_name": "le_trans", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [56, 9], "def_end_pos": [56, 17]}, {"full_name": "ENNReal.mul_left_mono", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1012, 9], "def_end_pos": [1012, 22]}, {"full_name": "le_max_left", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [54, 9], "def_end_pos": [54, 20]}, {"full_name": "le_trans", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [56, 9], "def_end_pos": [56, 17]}, {"full_name": "ENNReal.mul_left_mono", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1012, 9], "def_end_pos": [1012, 22]}, {"full_name": "le_max_right", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [61, 9], "def_end_pos": [61, 21]}]], "state_before": "case mk.mk\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d\u00b2 : PseudoEMetricSpace \u03b1\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b2\ninst\u271d : PseudoEMetricSpace \u03b3\nK : \u211d\u22650\nf\u271d : \u03b1 \u2192 \u03b2\nx y : \u03b1\nr : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nK\u03b1 K\u03b2 : \u211d\u22650\nh\u03b1 : \u2200 (b : \u03b2), LipschitzWith K\u03b1 fun a => f a b\nh\u03b2 : \u2200 (a : \u03b1), LipschitzWith K\u03b2 (f a)\na\u2081 : \u03b1\nb\u2081 : \u03b2\na\u2082 : \u03b1\nb\u2082 : \u03b2\n\u22a2 edist (f a\u2081 b\u2081) (f a\u2082 b\u2081) + edist (f a\u2082 b\u2081) (f a\u2082 b\u2082) \u2264 \u2191K\u03b1 * edist (a\u2081, b\u2081) (a\u2082, b\u2082) + \u2191K\u03b2 * edist (a\u2081, b\u2081) (a\u2082, b\u2082)", "state_after": "no goals"}]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Algebra/Order/LatticeGroup.lean
|
LatticeOrderedCommGroup.abs_abs_div_abs_le
|
[
637,
1
] |
[
646,
27
] |
[{"tactic": "rw [abs_eq_sup_inv, sup_le_iff]", "annotated_tactic": ["rw [<a>abs_eq_sup_inv</a>, <a>sup_le_iff</a>]", [{"full_name": "abs_eq_sup_inv", "def_path": "Mathlib/Algebra/Order/Group/Abs.lean", "def_pos": [34, 9], "def_end_pos": [34, 23]}, {"full_name": "sup_le_iff", "def_path": "Mathlib/Order/Lattice.lean", "def_pos": [172, 9], "def_end_pos": [172, 19]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : Lattice \u03b1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na b : \u03b1\n\u22a2 ||a| / |b|| \u2264 |a / b|", "state_after": "\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : Lattice \u03b1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na b : \u03b1\n\u22a2 |a| / |b| \u2264 |a / b| \u2227 (|a| / |b|)\u207b\u00b9 \u2264 |a / b|"}, {"tactic": "constructor", "annotated_tactic": ["constructor", []], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : Lattice \u03b1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na b : \u03b1\n\u22a2 |a| / |b| \u2264 |a / b| \u2227 (|a| / |b|)\u207b\u00b9 \u2264 |a / b|", "state_after": "case left\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : Lattice \u03b1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na b : \u03b1\n\u22a2 |a| / |b| \u2264 |a / b|\n\ncase right\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : Lattice \u03b1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na b : \u03b1\n\u22a2 (|a| / |b|)\u207b\u00b9 \u2264 |a / b|"}, {"tactic": "apply div_le_iff_le_mul.2", "annotated_tactic": ["apply <a>div_le_iff_le_mul</a>.2", [{"full_name": "div_le_iff_le_mul", "def_path": "Mathlib/Algebra/Order/Group/Defs.lean", "def_pos": [751, 9], "def_end_pos": [751, 26]}]], "state_before": "case left\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : Lattice \u03b1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na b : \u03b1\n\u22a2 |a| / |b| \u2264 |a / b|", "state_after": "case left\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : Lattice \u03b1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na b : \u03b1\n\u22a2 |a| \u2264 |a / b| * |b|"}, {"tactic": "convert mabs_mul_le (a / b) b", "annotated_tactic": ["convert <a>mabs_mul_le</a> (a / b) b", [{"full_name": "LatticeOrderedCommGroup.mabs_mul_le", "def_path": "Mathlib/Algebra/Order/LatticeGroup.lean", "def_pos": [627, 9], "def_end_pos": [627, 20]}]], "state_before": "case left\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : Lattice \u03b1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na b : \u03b1\n\u22a2 |a| \u2264 |a / b| * |b|", "state_after": "case h.e'_3.h.e'_3\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : Lattice \u03b1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na b : \u03b1\n\u22a2 a = a / b * b"}, {"tactic": "rw [div_mul_cancel']", "annotated_tactic": ["rw [<a>div_mul_cancel'</a>]", [{"full_name": "div_mul_cancel'", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [729, 9], "def_end_pos": [729, 24]}]], "state_before": "case h.e'_3.h.e'_3\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : Lattice \u03b1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na b : \u03b1\n\u22a2 a = a / b * b", "state_after": "no goals"}, {"tactic": "rw [div_eq_mul_inv, mul_inv_rev, inv_inv, mul_inv_le_iff_le_mul, abs_div_comm]", "annotated_tactic": ["rw [<a>div_eq_mul_inv</a>, <a>mul_inv_rev</a>, <a>inv_inv</a>, <a>mul_inv_le_iff_le_mul</a>, <a>abs_div_comm</a>]", [{"full_name": "div_eq_mul_inv", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [977, 9], "def_end_pos": [977, 23]}, {"full_name": "mul_inv_rev", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [1050, 9], "def_end_pos": [1050, 20]}, {"full_name": "inv_inv", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [800, 9], "def_end_pos": [800, 16]}, {"full_name": "mul_inv_le_iff_le_mul", "def_path": "Mathlib/Algebra/Order/Group/Defs.lean", "def_pos": [242, 9], "def_end_pos": [242, 30]}, {"full_name": "LatticeOrderedGroup.abs_div_comm", "def_path": "Mathlib/Algebra/Order/LatticeGroup.lean", "def_pos": [361, 9], "def_end_pos": [361, 21]}]], "state_before": "case right\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : Lattice \u03b1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na b : \u03b1\n\u22a2 (|a| / |b|)\u207b\u00b9 \u2264 |a / b|", "state_after": "case right\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : Lattice \u03b1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na b : \u03b1\n\u22a2 |b| \u2264 |b / a| * |a|"}, {"tactic": "convert mabs_mul_le (b / a) a", "annotated_tactic": ["convert <a>mabs_mul_le</a> (b / a) a", [{"full_name": "LatticeOrderedCommGroup.mabs_mul_le", "def_path": "Mathlib/Algebra/Order/LatticeGroup.lean", "def_pos": [627, 9], "def_end_pos": [627, 20]}]], "state_before": "case right\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : Lattice \u03b1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na b : \u03b1\n\u22a2 |b| \u2264 |b / a| * |a|", "state_after": "case h.e'_3.h.e'_3\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : Lattice \u03b1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na b : \u03b1\n\u22a2 b = b / a * a"}, {"tactic": "rw [div_mul_cancel']", "annotated_tactic": ["rw [<a>div_mul_cancel'</a>]", [{"full_name": "div_mul_cancel'", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [729, 9], "def_end_pos": [729, 24]}]], "state_before": "case h.e'_3.h.e'_3\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : Lattice \u03b1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na b : \u03b1\n\u22a2 b = b / a * a", "state_after": "no goals"}]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Combinatorics/SimpleGraph/Connectivity.lean
|
SimpleGraph.Walk.transfer_eq_map_of_le
|
[
1772,
1
] |
[
1774,
27
] |
[{"tactic": "induction p <;> simp [*]", "annotated_tactic": ["induction p <;> simp [*]", []], "state_before": "V : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nu v : V\np : Walk G u v\nH : SimpleGraph V\nhp : \u2200 (e : Sym2 V), e \u2208 edges p \u2192 e \u2208 edgeSet H\nGH : G \u2264 H\n\u22a2 Walk.transfer p H hp = Walk.map (Hom.mapSpanningSubgraphs GH) p", "state_after": "no goals"}]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Topology/Separation.lean
|
Ne.nhdsWithin_diff_singleton
|
[
407,
1
] |
[
410,
58
] |
[{"tactic": "rw [diff_eq, inter_comm, nhdsWithin_inter_of_mem]", "annotated_tactic": ["rw [<a>diff_eq</a>, <a>inter_comm</a>, <a>nhdsWithin_inter_of_mem</a>]", [{"full_name": "Set.diff_eq", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1814, 9], "def_end_pos": [1814, 16]}, {"full_name": "Set.inter_comm", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [940, 9], "def_end_pos": [940, 19]}, {"full_name": "nhdsWithin_inter_of_mem", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [270, 9], "def_end_pos": [270, 32]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : T1Space \u03b1\nx y : \u03b1\nh : x \u2260 y\ns : Set \u03b1\n\u22a2 \ud835\udcdd[s \\ {y}] x = \ud835\udcdd[s] x", "state_after": "\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : T1Space \u03b1\nx y : \u03b1\nh : x \u2260 y\ns : Set \u03b1\n\u22a2 {y}\u1d9c \u2208 \ud835\udcdd[s] x"}, {"tactic": "exact mem_nhdsWithin_of_mem_nhds (isOpen_ne.mem_nhds h)", "annotated_tactic": ["exact <a>mem_nhdsWithin_of_mem_nhds</a> (isOpen_ne.mem_nhds h)", [{"full_name": "mem_nhdsWithin_of_mem_nhds", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [147, 9], "def_end_pos": [147, 35]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : T1Space \u03b1\nx y : \u03b1\nh : x \u2260 y\ns : Set \u03b1\n\u22a2 {y}\u1d9c \u2208 \ud835\udcdd[s] x", "state_after": "no goals"}]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/RingTheory/NonUnitalSubsemiring/Basic.lean
|
NonUnitalRingHom.sclosure_preimage_le
|
[
928,
1
] |
[
930,
81
] |
[]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Order/WellFounded.lean
|
WellFounded.min_mem
|
[
70,
1
] |
[
73,
4
] |
[]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Analysis/BoxIntegral/Basic.lean
|
BoxIntegral.HasIntegral.of_mul
|
[
202,
1
] |
[
209,
62
] |
[{"tactic": "refine' hasIntegral_iff.2 fun \u03b5 h\u03b5 => _", "annotated_tactic": ["refine' <a>hasIntegral_iff</a>.2 fun \u03b5 h\u03b5 => _", [{"full_name": "BoxIntegral.hasIntegral_iff", "def_path": "Mathlib/Analysis/BoxIntegral/Basic.lean", "def_pos": [193, 9], "def_end_pos": [193, 24]}]], "state_before": "\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\nI J : Box \u03b9\n\u03c0 : TaggedPrepartition I\ninst\u271d : Fintype \u03b9\nl : IntegrationParams\nf g : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\na : \u211d\nh :\n \u2200 (\u03b5 : \u211d),\n 0 < \u03b5 \u2192\n \u2203 r,\n (\u2200 (c : \u211d\u22650), RCond l (r c)) \u2227\n \u2200 (c : \u211d\u22650) (\u03c0 : TaggedPrepartition I),\n MemBaseSet l I c (r c) \u03c0 \u2192 IsPartition \u03c0 \u2192 dist (integralSum f vol \u03c0) y \u2264 a * \u03b5\n\u22a2 HasIntegral I l f vol y", "state_after": "\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\nI J : Box \u03b9\n\u03c0 : TaggedPrepartition I\ninst\u271d : Fintype \u03b9\nl : IntegrationParams\nf g : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\na : \u211d\nh :\n \u2200 (\u03b5 : \u211d),\n 0 < \u03b5 \u2192\n \u2203 r,\n (\u2200 (c : \u211d\u22650), RCond l (r c)) \u2227\n \u2200 (c : \u211d\u22650) (\u03c0 : TaggedPrepartition I),\n MemBaseSet l I c (r c) \u03c0 \u2192 IsPartition \u03c0 \u2192 dist (integralSum f vol \u03c0) y \u2264 a * \u03b5\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\n\u22a2 \u2203 r,\n (\u2200 (c : \u211d\u22650), RCond l (r c)) \u2227\n \u2200 (c : \u211d\u22650) (\u03c0 : TaggedPrepartition I),\n MemBaseSet l I c (r c) \u03c0 \u2192 IsPartition \u03c0 \u2192 dist (integralSum f vol \u03c0) y \u2264 \u03b5"}, {"tactic": "rcases exists_pos_mul_lt h\u03b5 a with \u27e8\u03b5', h\u03b5', ha\u27e9", "annotated_tactic": ["rcases <a>exists_pos_mul_lt</a> h\u03b5 a with \u27e8\u03b5', h\u03b5', ha\u27e9", [{"full_name": "exists_pos_mul_lt", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [565, 9], "def_end_pos": [565, 26]}]], "state_before": "\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\nI J : Box \u03b9\n\u03c0 : TaggedPrepartition I\ninst\u271d : Fintype \u03b9\nl : IntegrationParams\nf g : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\na : \u211d\nh :\n \u2200 (\u03b5 : \u211d),\n 0 < \u03b5 \u2192\n \u2203 r,\n (\u2200 (c : \u211d\u22650), RCond l (r c)) \u2227\n \u2200 (c : \u211d\u22650) (\u03c0 : TaggedPrepartition I),\n MemBaseSet l I c (r c) \u03c0 \u2192 IsPartition \u03c0 \u2192 dist (integralSum f vol \u03c0) y \u2264 a * \u03b5\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\n\u22a2 \u2203 r,\n (\u2200 (c : \u211d\u22650), RCond l (r c)) \u2227\n \u2200 (c : \u211d\u22650) (\u03c0 : TaggedPrepartition I),\n MemBaseSet l I c (r c) \u03c0 \u2192 IsPartition \u03c0 \u2192 dist (integralSum f vol \u03c0) y \u2264 \u03b5", "state_after": "case intro.intro\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\nI J : Box \u03b9\n\u03c0 : TaggedPrepartition I\ninst\u271d : Fintype \u03b9\nl : IntegrationParams\nf g : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\na : \u211d\nh :\n \u2200 (\u03b5 : \u211d),\n 0 < \u03b5 \u2192\n \u2203 r,\n (\u2200 (c : \u211d\u22650), RCond l (r c)) \u2227\n \u2200 (c : \u211d\u22650) (\u03c0 : TaggedPrepartition I),\n MemBaseSet l I c (r c) \u03c0 \u2192 IsPartition \u03c0 \u2192 dist (integralSum f vol \u03c0) y \u2264 a * \u03b5\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\n\u03b5' : \u211d\nh\u03b5' : 0 < \u03b5'\nha : a * \u03b5' < \u03b5\n\u22a2 \u2203 r,\n (\u2200 (c : \u211d\u22650), RCond l (r c)) \u2227\n \u2200 (c : \u211d\u22650) (\u03c0 : TaggedPrepartition I),\n MemBaseSet l I c (r c) \u03c0 \u2192 IsPartition \u03c0 \u2192 dist (integralSum f vol \u03c0) y \u2264 \u03b5"}, {"tactic": "rcases h \u03b5' h\u03b5' with \u27e8r, hr, H\u27e9", "annotated_tactic": ["rcases h \u03b5' h\u03b5' with \u27e8r, hr, H\u27e9", []], "state_before": "case intro.intro\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\nI J : Box \u03b9\n\u03c0 : TaggedPrepartition I\ninst\u271d : Fintype \u03b9\nl : IntegrationParams\nf g : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\na : \u211d\nh :\n \u2200 (\u03b5 : \u211d),\n 0 < \u03b5 \u2192\n \u2203 r,\n (\u2200 (c : \u211d\u22650), RCond l (r c)) \u2227\n \u2200 (c : \u211d\u22650) (\u03c0 : TaggedPrepartition I),\n MemBaseSet l I c (r c) \u03c0 \u2192 IsPartition \u03c0 \u2192 dist (integralSum f vol \u03c0) y \u2264 a * \u03b5\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\n\u03b5' : \u211d\nh\u03b5' : 0 < \u03b5'\nha : a * \u03b5' < \u03b5\n\u22a2 \u2203 r,\n (\u2200 (c : \u211d\u22650), RCond l (r c)) \u2227\n \u2200 (c : \u211d\u22650) (\u03c0 : TaggedPrepartition I),\n MemBaseSet l I c (r c) \u03c0 \u2192 IsPartition \u03c0 \u2192 dist (integralSum f vol \u03c0) y \u2264 \u03b5", "state_after": "case intro.intro.intro.intro\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\nI J : Box \u03b9\n\u03c0 : TaggedPrepartition I\ninst\u271d : Fintype \u03b9\nl : IntegrationParams\nf g : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\na : \u211d\nh :\n \u2200 (\u03b5 : \u211d),\n 0 < \u03b5 \u2192\n \u2203 r,\n (\u2200 (c : \u211d\u22650), RCond l (r c)) \u2227\n \u2200 (c : \u211d\u22650) (\u03c0 : TaggedPrepartition I),\n MemBaseSet l I c (r c) \u03c0 \u2192 IsPartition \u03c0 \u2192 dist (integralSum f vol \u03c0) y \u2264 a * \u03b5\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\n\u03b5' : \u211d\nh\u03b5' : 0 < \u03b5'\nha : a * \u03b5' < \u03b5\nr : \u211d\u22650 \u2192 (\u03b9 \u2192 \u211d) \u2192 \u2191(Set.Ioi 0)\nhr : \u2200 (c : \u211d\u22650), RCond l (r c)\nH :\n \u2200 (c : \u211d\u22650) (\u03c0 : TaggedPrepartition I),\n MemBaseSet l I c (r c) \u03c0 \u2192 IsPartition \u03c0 \u2192 dist (integralSum f vol \u03c0) y \u2264 a * \u03b5'\n\u22a2 \u2203 r,\n (\u2200 (c : \u211d\u22650), RCond l (r c)) \u2227\n \u2200 (c : \u211d\u22650) (\u03c0 : TaggedPrepartition I),\n MemBaseSet l I c (r c) \u03c0 \u2192 IsPartition \u03c0 \u2192 dist (integralSum f vol \u03c0) y \u2264 \u03b5"}, {"tactic": "exact \u27e8r, hr, fun c \u03c0 h\u03c0 h\u03c0p => (H c \u03c0 h\u03c0 h\u03c0p).trans ha.le\u27e9", "annotated_tactic": ["exact \u27e8r, hr, fun c \u03c0 h\u03c0 h\u03c0p => (H c \u03c0 h\u03c0 h\u03c0p).<a>trans</a> ha.le\u27e9", [{"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\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\nI J : Box \u03b9\n\u03c0 : TaggedPrepartition I\ninst\u271d : Fintype \u03b9\nl : IntegrationParams\nf g : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\na : \u211d\nh :\n \u2200 (\u03b5 : \u211d),\n 0 < \u03b5 \u2192\n \u2203 r,\n (\u2200 (c : \u211d\u22650), RCond l (r c)) \u2227\n \u2200 (c : \u211d\u22650) (\u03c0 : TaggedPrepartition I),\n MemBaseSet l I c (r c) \u03c0 \u2192 IsPartition \u03c0 \u2192 dist (integralSum f vol \u03c0) y \u2264 a * \u03b5\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\n\u03b5' : \u211d\nh\u03b5' : 0 < \u03b5'\nha : a * \u03b5' < \u03b5\nr : \u211d\u22650 \u2192 (\u03b9 \u2192 \u211d) \u2192 \u2191(Set.Ioi 0)\nhr : \u2200 (c : \u211d\u22650), RCond l (r c)\nH :\n \u2200 (c : \u211d\u22650) (\u03c0 : TaggedPrepartition I),\n MemBaseSet l I c (r c) \u03c0 \u2192 IsPartition \u03c0 \u2192 dist (integralSum f vol \u03c0) y \u2264 a * \u03b5'\n\u22a2 \u2203 r,\n (\u2200 (c : \u211d\u22650), RCond l (r c)) \u2227\n \u2200 (c : \u211d\u22650) (\u03c0 : TaggedPrepartition I),\n MemBaseSet l I c (r c) \u03c0 \u2192 IsPartition \u03c0 \u2192 dist (integralSum f vol \u03c0) y \u2264 \u03b5", "state_after": "no goals"}]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/RingTheory/PrincipalIdealDomain.lean
|
gcd_isUnit_iff
|
[
384,
1
] |
[
385,
93
] |
[{"tactic": "rw [IsCoprime, \u2190 Ideal.mem_span_pair, \u2190 span_gcd, \u2190 span_singleton_eq_top, eq_top_iff_one]", "annotated_tactic": ["rw [<a>IsCoprime</a>, \u2190 <a>Ideal.mem_span_pair</a>, \u2190 <a>span_gcd</a>, \u2190 <a>span_singleton_eq_top</a>, <a>eq_top_iff_one</a>]", [{"full_name": "IsCoprime", "def_path": "Mathlib/RingTheory/Coprime/Basic.lean", "def_pos": [38, 5], "def_end_pos": [38, 14]}, {"full_name": "Ideal.mem_span_pair", "def_path": "Mathlib/RingTheory/Ideal/Basic.lean", "def_pos": [364, 9], "def_end_pos": [364, 22]}, {"full_name": "span_gcd", "def_path": "Mathlib/RingTheory/PrincipalIdealDomain.lean", "def_pos": [353, 9], "def_end_pos": [353, 17]}, {"full_name": "Ideal.span_singleton_eq_top", "def_path": "Mathlib/RingTheory/Ideal/Basic.lean", "def_pos": [520, 9], "def_end_pos": [520, 30]}, {"full_name": "Ideal.eq_top_iff_one", "def_path": "Mathlib/RingTheory/Ideal/Basic.lean", "def_pos": [92, 9], "def_end_pos": [92, 23]}]], "state_before": "R : Type u\nM : Type v\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : IsDomain R\ninst\u271d\u00b9 : IsPrincipalIdealRing R\ninst\u271d : GCDMonoid R\nx y : R\n\u22a2 IsUnit (gcd x y) \u2194 IsCoprime x y", "state_after": "no goals"}]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/MeasureTheory/Constructions/Prod/Integral.lean
|
MeasureTheory.integral_integral_symm
|
[
493,
1
] |
[
495,
38
] |
[]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Data/Seq/Computation.lean
|
Computation.exists_of_mem_bind
|
[
839,
1
] |
[
843,
22
] |
[]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/GroupTheory/FiniteAbelian.lean
|
CommGroup.finite_of_fg_torsion
|
[
95,
1
] |
[
96,
96
] |
[]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/SetTheory/Ordinal/NaturalOps.lean
|
Ordinal.zero_nmul
|
[
573,
1
] |
[
573,
66
] |
[{"tactic": "rw [nmul_comm, nmul_zero]", "annotated_tactic": ["rw [<a>nmul_comm</a>, <a>nmul_zero</a>]", [{"full_name": "Ordinal.nmul_comm", "def_path": "Mathlib/SetTheory/Ordinal/NaturalOps.lean", "def_pos": [553, 9], "def_end_pos": [553, 18]}, {"full_name": "Ordinal.nmul_zero", "def_path": "Mathlib/SetTheory/Ordinal/NaturalOps.lean", "def_pos": [567, 9], "def_end_pos": [567, 18]}]], "state_before": "a\u271d b c d : Ordinal.{u}\na : Ordinal.{u_1}\n\u22a2 0 \u2a33 a = 0", "state_after": "no goals"}]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Data/Polynomial/Degree/Definitions.lean
|
Polynomial.natDegree_X_pow_le
|
[
1322,
1
] |
[
1324,
34
] |
[{"tactic": "nontriviality R", "annotated_tactic": ["nontriviality R", []], "state_before": "R\u271d : Type u\nS : Type v\na b c d : R\u271d\nn\u271d\u00b9 m : \u2115\ninst\u271d\u00b2 : Semiring R\u271d\ninst\u271d\u00b9 : Nontrivial R\u271d\np q : R\u271d[X]\nn\u271d : \u2115\nR : Type u_1\ninst\u271d : Semiring R\nn : \u2115\n\u22a2 natDegree (X ^ n) \u2264 n", "state_after": "R\u271d : Type u\nS : Type v\na b c d : R\u271d\nn\u271d\u00b9 m : \u2115\ninst\u271d\u00b2 : Semiring R\u271d\ninst\u271d\u00b9 : Nontrivial R\u271d\np q : R\u271d[X]\nn\u271d : \u2115\nR : Type u_1\ninst\u271d : Semiring R\nn : \u2115\n\u271d : Nontrivial R\n\u22a2 natDegree (X ^ n) \u2264 n"}, {"tactic": "rw [Polynomial.natDegree_X_pow]", "annotated_tactic": ["rw [<a>Polynomial.natDegree_X_pow</a>]", [{"full_name": "Polynomial.natDegree_X_pow", "def_path": "Mathlib/Data/Polynomial/Degree/Definitions.lean", "def_pos": [1305, 9], "def_end_pos": [1305, 24]}]], "state_before": "R\u271d : Type u\nS : Type v\na b c d : R\u271d\nn\u271d\u00b9 m : \u2115\ninst\u271d\u00b2 : Semiring R\u271d\ninst\u271d\u00b9 : Nontrivial R\u271d\np q : R\u271d[X]\nn\u271d : \u2115\nR : Type u_1\ninst\u271d : Semiring R\nn : \u2115\n\u271d : Nontrivial R\n\u22a2 natDegree (X ^ n) \u2264 n", "state_after": "no goals"}]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/NumberTheory/LegendreSymbol/JacobiSymbol.lean
|
jacobiSym.list_prod_left
|
[
253,
1
] |
[
256,
64
] |
[{"tactic": "induction' l with n l' ih", "annotated_tactic": ["induction' l with n l' ih", []], "state_before": "l : List \u2124\nn : \u2115\n\u22a2 J(List.prod l | n) = List.prod (List.map (fun a => J(a | n)) l)", "state_after": "case nil\nn : \u2115\n\u22a2 J(List.prod [] | n) = List.prod (List.map (fun a => J(a | n)) [])\n\ncase cons\nn\u271d : \u2115\nn : \u2124\nl' : List \u2124\nih : J(List.prod l' | n\u271d) = List.prod (List.map (fun a => J(a | n\u271d)) l')\n\u22a2 J(List.prod (n :: l') | n\u271d) = List.prod (List.map (fun a => J(a | n\u271d)) (n :: l'))"}, {"tactic": "simp only [List.prod_nil, List.map_nil, one_left]", "annotated_tactic": ["simp only [<a>List.prod_nil</a>, <a>List.map_nil</a>, <a>one_left</a>]", [{"full_name": "List.prod_nil", "def_path": "Mathlib/Data/List/BigOperators/Basic.lean", "def_pos": [29, 9], "def_end_pos": [29, 17]}, {"full_name": "List.map_nil", "def_path": "lake-packages/std/Std/Data/List/Init/Lemmas.lean", "def_pos": [87, 17], "def_end_pos": [87, 24]}, {"full_name": "jacobiSym.one_left", "def_path": "Mathlib/NumberTheory/LegendreSymbol/JacobiSymbol.lean", "def_pos": [143, 9], "def_end_pos": [143, 17]}]], "state_before": "case nil\nn : \u2115\n\u22a2 J(List.prod [] | n) = List.prod (List.map (fun a => J(a | n)) [])", "state_after": "no goals"}, {"tactic": "rw [List.map, List.prod_cons, List.prod_cons, mul_left, ih]", "annotated_tactic": ["rw [<a>List.map</a>, <a>List.prod_cons</a>, <a>List.prod_cons</a>, <a>mul_left</a>, ih]", [{"full_name": "List.map", "def_path": "lake-packages/lean4/src/lean/Init/Data/List/Basic.lean", "def_pos": [151, 19], "def_end_pos": [151, 22]}, {"full_name": "List.prod_cons", "def_path": "Mathlib/Data/List/BigOperators/Basic.lean", "def_pos": [41, 9], "def_end_pos": [41, 18]}, {"full_name": "List.prod_cons", "def_path": "Mathlib/Data/List/BigOperators/Basic.lean", "def_pos": [41, 9], "def_end_pos": [41, 18]}, {"full_name": "jacobiSym.mul_left", "def_path": "Mathlib/NumberTheory/LegendreSymbol/JacobiSymbol.lean", "def_pos": [153, 9], "def_end_pos": [153, 17]}]], "state_before": "case cons\nn\u271d : \u2115\nn : \u2124\nl' : List \u2124\nih : J(List.prod l' | n\u271d) = List.prod (List.map (fun a => J(a | n\u271d)) l')\n\u22a2 J(List.prod (n :: l') | n\u271d) = List.prod (List.map (fun a => J(a | n\u271d)) (n :: l'))", "state_after": "no goals"}]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Analysis/Calculus/ContDiff.lean
|
ContDiffAt.hasStrictDerivAt'
|
[
1990,
1
] |
[
1992,
31
] |
[]
|
https://github.com/leanprover/std4
|
869c615eb10130c0637a7bc038e2b80253559913
|
lake-packages/std/Std/Data/List/Lemmas.lean
|
List.Sublist.filter
|
[
1353,
1
] |
[
1354,
48
] |
[{"tactic": "rw [\u2190 filterMap_eq_filter]", "annotated_tactic": ["rw [\u2190 <a>filterMap_eq_filter</a>]", [{"full_name": "List.filterMap_eq_filter", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [1293, 9], "def_end_pos": [1293, 28]}]], "state_before": "\u03b1 : Type u_1\np : \u03b1 \u2192 Bool\nl\u2081 l\u2082 : List \u03b1\ns : l\u2081 <+ l\u2082\n\u22a2 List.filter p l\u2081 <+ List.filter p l\u2082", "state_after": "\u03b1 : Type u_1\np : \u03b1 \u2192 Bool\nl\u2081 l\u2082 : List \u03b1\ns : l\u2081 <+ l\u2082\n\u22a2 List.filterMap (Option.guard fun x => p x = true) l\u2081 <+ List.filterMap (Option.guard fun x => p x = true) l\u2082"}, {"tactic": "apply s.filterMap", "annotated_tactic": ["apply s.filterMap", []], "state_before": "\u03b1 : Type u_1\np : \u03b1 \u2192 Bool\nl\u2081 l\u2082 : List \u03b1\ns : l\u2081 <+ l\u2082\n\u22a2 List.filterMap (Option.guard fun x => p x = true) l\u2081 <+ List.filterMap (Option.guard fun x => p x = true) l\u2082", "state_after": "no goals"}]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Topology/ContinuousOn.lean
|
tendsto_nhdsWithin_mono_right
|
[
370,
1
] |
[
372,
39
] |
[]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Topology/Algebra/InfiniteSum/Basic.lean
|
HasSum.update'
|
[
439,
1
] |
[
449,
58
] |
[{"tactic": "have h := hf.add (hasSum_ite_eq b x)", "annotated_tactic": ["have h := hf.add (<a>hasSum_ite_eq</a> b x)", [{"full_name": "hasSum_ite_eq", "def_path": "Mathlib/Topology/Algebra/InfiniteSum/Basic.lean", "def_pos": [218, 9], "def_end_pos": [218, 22]}]], "state_before": "\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2076 : AddCommMonoid \u03b1\u271d\ninst\u271d\u2075 : TopologicalSpace \u03b1\u271d\nf\u271d g : \u03b2\u271d \u2192 \u03b1\u271d\na\u271d b\u271d : \u03b1\u271d\ns : Finset \u03b2\u271d\ninst\u271d\u2074 : ContinuousAdd \u03b1\u271d\n\u03b1 : Type u_5\n\u03b2 : Type u_6\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : AddCommMonoid \u03b1\ninst\u271d\u00b9 : T2Space \u03b1\ninst\u271d : ContinuousAdd \u03b1\nf : \u03b2 \u2192 \u03b1\na a' : \u03b1\nhf : HasSum f a\nb : \u03b2\nx : \u03b1\nhf' : HasSum (update f b x) a'\nthis : \u2200 (b' : \u03b2), (f b' + if b' = b then x else 0) = update f b x b' + if b' = b then f b else 0\n\u22a2 a + x = a' + f b", "state_after": "\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2076 : AddCommMonoid \u03b1\u271d\ninst\u271d\u2075 : TopologicalSpace \u03b1\u271d\nf\u271d g : \u03b2\u271d \u2192 \u03b1\u271d\na\u271d b\u271d : \u03b1\u271d\ns : Finset \u03b2\u271d\ninst\u271d\u2074 : ContinuousAdd \u03b1\u271d\n\u03b1 : Type u_5\n\u03b2 : Type u_6\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : AddCommMonoid \u03b1\ninst\u271d\u00b9 : T2Space \u03b1\ninst\u271d : ContinuousAdd \u03b1\nf : \u03b2 \u2192 \u03b1\na a' : \u03b1\nhf : HasSum f a\nb : \u03b2\nx : \u03b1\nhf' : HasSum (update f b x) a'\nthis : \u2200 (b' : \u03b2), (f b' + if b' = b then x else 0) = update f b x b' + if b' = b then f b else 0\nh : HasSum (fun b_1 => f b_1 + if b_1 = b then x else 0) (a + x)\n\u22a2 a + x = a' + f b"}, {"tactic": "simp_rw [this] at h", "annotated_tactic": ["simp_rw [this] at h", []], "state_before": "\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2076 : AddCommMonoid \u03b1\u271d\ninst\u271d\u2075 : TopologicalSpace \u03b1\u271d\nf\u271d g : \u03b2\u271d \u2192 \u03b1\u271d\na\u271d b\u271d : \u03b1\u271d\ns : Finset \u03b2\u271d\ninst\u271d\u2074 : ContinuousAdd \u03b1\u271d\n\u03b1 : Type u_5\n\u03b2 : Type u_6\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : AddCommMonoid \u03b1\ninst\u271d\u00b9 : T2Space \u03b1\ninst\u271d : ContinuousAdd \u03b1\nf : \u03b2 \u2192 \u03b1\na a' : \u03b1\nhf : HasSum f a\nb : \u03b2\nx : \u03b1\nhf' : HasSum (update f b x) a'\nthis : \u2200 (b' : \u03b2), (f b' + if b' = b then x else 0) = update f b x b' + if b' = b then f b else 0\nh : HasSum (fun b_1 => f b_1 + if b_1 = b then x else 0) (a + x)\n\u22a2 a + x = a' + f b", "state_after": "\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2076 : AddCommMonoid \u03b1\u271d\ninst\u271d\u2075 : TopologicalSpace \u03b1\u271d\nf\u271d g : \u03b2\u271d \u2192 \u03b1\u271d\na\u271d b\u271d : \u03b1\u271d\ns : Finset \u03b2\u271d\ninst\u271d\u2074 : ContinuousAdd \u03b1\u271d\n\u03b1 : Type u_5\n\u03b2 : Type u_6\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : AddCommMonoid \u03b1\ninst\u271d\u00b9 : T2Space \u03b1\ninst\u271d : ContinuousAdd \u03b1\nf : \u03b2 \u2192 \u03b1\na a' : \u03b1\nhf : HasSum f a\nb : \u03b2\nx : \u03b1\nhf' : HasSum (update f b x) a'\nthis : \u2200 (b' : \u03b2), (f b' + if b' = b then x else 0) = update f b x b' + if b' = b then f b else 0\nh : HasSum (fun b_1 => update f b x b_1 + if b_1 = b then f b else 0) (a + x)\n\u22a2 a + x = a' + f b"}, {"tactic": "exact HasSum.unique h (hf'.add (hasSum_ite_eq b (f b)))", "annotated_tactic": ["exact <a>HasSum.unique</a> h (hf'.add (<a>hasSum_ite_eq</a> b (f b)))", [{"full_name": "HasSum.unique", "def_path": "Mathlib/Topology/Algebra/InfiniteSum/Basic.lean", "def_pos": [302, 9], "def_end_pos": [302, 22]}, {"full_name": "hasSum_ite_eq", "def_path": "Mathlib/Topology/Algebra/InfiniteSum/Basic.lean", "def_pos": [218, 9], "def_end_pos": [218, 22]}]], "state_before": "\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2076 : AddCommMonoid \u03b1\u271d\ninst\u271d\u2075 : TopologicalSpace \u03b1\u271d\nf\u271d g : \u03b2\u271d \u2192 \u03b1\u271d\na\u271d b\u271d : \u03b1\u271d\ns : Finset \u03b2\u271d\ninst\u271d\u2074 : ContinuousAdd \u03b1\u271d\n\u03b1 : Type u_5\n\u03b2 : Type u_6\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : AddCommMonoid \u03b1\ninst\u271d\u00b9 : T2Space \u03b1\ninst\u271d : ContinuousAdd \u03b1\nf : \u03b2 \u2192 \u03b1\na a' : \u03b1\nhf : HasSum f a\nb : \u03b2\nx : \u03b1\nhf' : HasSum (update f b x) a'\nthis : \u2200 (b' : \u03b2), (f b' + if b' = b then x else 0) = update f b x b' + if b' = b then f b else 0\nh : HasSum (fun b_1 => update f b x b_1 + if b_1 = b then f b else 0) (a + x)\n\u22a2 a + x = a' + f b", "state_after": "no goals"}, {"tactic": "intro b'", "annotated_tactic": ["intro b'", []], "state_before": "\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2076 : AddCommMonoid \u03b1\u271d\ninst\u271d\u2075 : TopologicalSpace \u03b1\u271d\nf\u271d g : \u03b2\u271d \u2192 \u03b1\u271d\na\u271d b\u271d : \u03b1\u271d\ns : Finset \u03b2\u271d\ninst\u271d\u2074 : ContinuousAdd \u03b1\u271d\n\u03b1 : Type u_5\n\u03b2 : Type u_6\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : AddCommMonoid \u03b1\ninst\u271d\u00b9 : T2Space \u03b1\ninst\u271d : ContinuousAdd \u03b1\nf : \u03b2 \u2192 \u03b1\na a' : \u03b1\nhf : HasSum f a\nb : \u03b2\nx : \u03b1\nhf' : HasSum (update f b x) a'\n\u22a2 \u2200 (b' : \u03b2), (f b' + if b' = b then x else 0) = update f b x b' + if b' = b then f b else 0", "state_after": "\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2076 : AddCommMonoid \u03b1\u271d\ninst\u271d\u2075 : TopologicalSpace \u03b1\u271d\nf\u271d g : \u03b2\u271d \u2192 \u03b1\u271d\na\u271d b\u271d : \u03b1\u271d\ns : Finset \u03b2\u271d\ninst\u271d\u2074 : ContinuousAdd \u03b1\u271d\n\u03b1 : Type u_5\n\u03b2 : Type u_6\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : AddCommMonoid \u03b1\ninst\u271d\u00b9 : T2Space \u03b1\ninst\u271d : ContinuousAdd \u03b1\nf : \u03b2 \u2192 \u03b1\na a' : \u03b1\nhf : HasSum f a\nb : \u03b2\nx : \u03b1\nhf' : HasSum (update f b x) a'\nb' : \u03b2\n\u22a2 (f b' + if b' = b then x else 0) = update f b x b' + if b' = b then f b else 0"}, {"tactic": "split_ifs with hb'", "annotated_tactic": ["split_ifs with hb'", []], "state_before": "\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2076 : AddCommMonoid \u03b1\u271d\ninst\u271d\u2075 : TopologicalSpace \u03b1\u271d\nf\u271d g : \u03b2\u271d \u2192 \u03b1\u271d\na\u271d b\u271d : \u03b1\u271d\ns : Finset \u03b2\u271d\ninst\u271d\u2074 : ContinuousAdd \u03b1\u271d\n\u03b1 : Type u_5\n\u03b2 : Type u_6\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : AddCommMonoid \u03b1\ninst\u271d\u00b9 : T2Space \u03b1\ninst\u271d : ContinuousAdd \u03b1\nf : \u03b2 \u2192 \u03b1\na a' : \u03b1\nhf : HasSum f a\nb : \u03b2\nx : \u03b1\nhf' : HasSum (update f b x) a'\nb' : \u03b2\n\u22a2 (f b' + if b' = b then x else 0) = update f b x b' + if b' = b then f b else 0", "state_after": "case pos\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2076 : AddCommMonoid \u03b1\u271d\ninst\u271d\u2075 : TopologicalSpace \u03b1\u271d\nf\u271d g : \u03b2\u271d \u2192 \u03b1\u271d\na\u271d b\u271d : \u03b1\u271d\ns : Finset \u03b2\u271d\ninst\u271d\u2074 : ContinuousAdd \u03b1\u271d\n\u03b1 : Type u_5\n\u03b2 : Type u_6\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : AddCommMonoid \u03b1\ninst\u271d\u00b9 : T2Space \u03b1\ninst\u271d : ContinuousAdd \u03b1\nf : \u03b2 \u2192 \u03b1\na a' : \u03b1\nhf : HasSum f a\nb : \u03b2\nx : \u03b1\nhf' : HasSum (update f b x) a'\nb' : \u03b2\nhb' : b' = b\n\u22a2 f b' + x = update f b x b' + f b\n\ncase neg\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2076 : AddCommMonoid \u03b1\u271d\ninst\u271d\u2075 : TopologicalSpace \u03b1\u271d\nf\u271d g : \u03b2\u271d \u2192 \u03b1\u271d\na\u271d b\u271d : \u03b1\u271d\ns : Finset \u03b2\u271d\ninst\u271d\u2074 : ContinuousAdd \u03b1\u271d\n\u03b1 : Type u_5\n\u03b2 : Type u_6\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : AddCommMonoid \u03b1\ninst\u271d\u00b9 : T2Space \u03b1\ninst\u271d : ContinuousAdd \u03b1\nf : \u03b2 \u2192 \u03b1\na a' : \u03b1\nhf : HasSum f a\nb : \u03b2\nx : \u03b1\nhf' : HasSum (update f b x) a'\nb' : \u03b2\nhb' : \u00acb' = b\n\u22a2 f b' + 0 = update f b x b' + 0"}, {"tactic": "simpa only [Function.update_apply, hb', eq_self_iff_true] using add_comm (f b) x", "annotated_tactic": ["simpa only [<a>Function.update_apply</a>, hb', <a>eq_self_iff_true</a>] using <a>add_comm</a> (f b) x", [{"full_name": "Function.update_apply", "def_path": "Mathlib/Logic/Function/Basic.lean", "def_pos": [565, 9], "def_end_pos": [565, 21]}, {"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": "add_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [301, 3], "def_end_pos": [301, 14]}]], "state_before": "case pos\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2076 : AddCommMonoid \u03b1\u271d\ninst\u271d\u2075 : TopologicalSpace \u03b1\u271d\nf\u271d g : \u03b2\u271d \u2192 \u03b1\u271d\na\u271d b\u271d : \u03b1\u271d\ns : Finset \u03b2\u271d\ninst\u271d\u2074 : ContinuousAdd \u03b1\u271d\n\u03b1 : Type u_5\n\u03b2 : Type u_6\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : AddCommMonoid \u03b1\ninst\u271d\u00b9 : T2Space \u03b1\ninst\u271d : ContinuousAdd \u03b1\nf : \u03b2 \u2192 \u03b1\na a' : \u03b1\nhf : HasSum f a\nb : \u03b2\nx : \u03b1\nhf' : HasSum (update f b x) a'\nb' : \u03b2\nhb' : b' = b\n\u22a2 f b' + x = update f b x b' + f b", "state_after": "no goals"}, {"tactic": "simp only [Function.update_apply, hb', if_false]", "annotated_tactic": ["simp only [<a>Function.update_apply</a>, hb', <a>if_false</a>]", [{"full_name": "Function.update_apply", "def_path": "Mathlib/Logic/Function/Basic.lean", "def_pos": [565, 9], "def_end_pos": [565, 21]}, {"full_name": "if_false", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [729, 17], "def_end_pos": [729, 25]}]], "state_before": "case neg\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2076 : AddCommMonoid \u03b1\u271d\ninst\u271d\u2075 : TopologicalSpace \u03b1\u271d\nf\u271d g : \u03b2\u271d \u2192 \u03b1\u271d\na\u271d b\u271d : \u03b1\u271d\ns : Finset \u03b2\u271d\ninst\u271d\u2074 : ContinuousAdd \u03b1\u271d\n\u03b1 : Type u_5\n\u03b2 : Type u_6\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : AddCommMonoid \u03b1\ninst\u271d\u00b9 : T2Space \u03b1\ninst\u271d : ContinuousAdd \u03b1\nf : \u03b2 \u2192 \u03b1\na a' : \u03b1\nhf : HasSum f a\nb : \u03b2\nx : \u03b1\nhf' : HasSum (update f b x) a'\nb' : \u03b2\nhb' : \u00acb' = b\n\u22a2 f b' + 0 = update f b x b' + 0", "state_after": "no goals"}]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/NumberTheory/Zsqrtd/Basic.lean
|
Zsqrtd.eq_of_smul_eq_smul_left
|
[
375,
11
] |
[
378,
82
] |
[{"tactic": "rw [ext] at h \u22a2", "annotated_tactic": ["rw [<a>ext</a>] at h \u22a2", [{"full_name": "Zsqrtd.ext", "def_path": "Mathlib/NumberTheory/Zsqrtd/Basic.lean", "def_pos": [43, 9], "def_end_pos": [43, 12]}]], "state_before": "d a : \u2124\nb c : \u2124\u221ad\nha : a \u2260 0\nh : \u2191a * b = \u2191a * c\n\u22a2 b = c", "state_after": "d a : \u2124\nb c : \u2124\u221ad\nha : a \u2260 0\nh : (\u2191a * b).re = (\u2191a * c).re \u2227 (\u2191a * b).im = (\u2191a * c).im\n\u22a2 b.re = c.re \u2227 b.im = c.im"}, {"tactic": "apply And.imp _ _ h <;> simpa only [smul_re, smul_im] using mul_left_cancel\u2080 ha", "annotated_tactic": ["apply <a>And.imp</a> _ _ h <;> simpa only [<a>smul_re</a>, <a>smul_im</a>] using <a>mul_left_cancel\u2080</a> ha", [{"full_name": "And.imp", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [150, 9], "def_end_pos": [150, 16]}, {"full_name": "Zsqrtd.smul_re", "def_path": "Mathlib/NumberTheory/Zsqrtd/Basic.lean", "def_pos": [313, 9], "def_end_pos": [313, 16]}, {"full_name": "Zsqrtd.smul_im", "def_path": "Mathlib/NumberTheory/Zsqrtd/Basic.lean", "def_pos": [316, 9], "def_end_pos": [316, 16]}, {"full_name": "mul_left_cancel\u2080", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [51, 9], "def_end_pos": [51, 25]}]], "state_before": "d a : \u2124\nb c : \u2124\u221ad\nha : a \u2260 0\nh : (\u2191a * b).re = (\u2191a * c).re \u2227 (\u2191a * b).im = (\u2191a * c).im\n\u22a2 b.re = c.re \u2227 b.im = c.im", "state_after": "no goals"}]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Probability/Kernel/Basic.lean
|
ProbabilityTheory.kernel.piecewise_apply
|
[
628,
1
] |
[
629,
6
] |
[]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/LinearAlgebra/LinearPMap.lean
|
LinearMap.toPMap_domain
|
[
685,
1
] |
[
686,
6
] |
[]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Topology/Algebra/InfiniteSum/Order.lean
|
tsum_le_of_sum_le
|
[
114,
1
] |
[
115,
34
] |
[]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Algebra/Order/Chebyshev.lean
|
Monovary.sum_smul_sum_le_card_smul_sum
|
[
80,
1
] |
[
82,
51
] |
[]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Topology/Connected/Basic.lean
|
isConnected_connectedComponentIn_iff
|
[
628,
1
] |
[
631,
18
] |
[{"tactic": "simp_rw [\u2190 connectedComponentIn_nonempty_iff, IsConnected, isPreconnected_connectedComponentIn,\n and_true_iff]", "annotated_tactic": ["simp_rw [\u2190 <a>connectedComponentIn_nonempty_iff</a>, <a>IsConnected</a>, <a>isPreconnected_connectedComponentIn</a>,\n <a>and_true_iff</a>]", [{"full_name": "connectedComponentIn_nonempty_iff", "def_path": "Mathlib/Topology/Connected/Basic.lean", "def_pos": [602, 9], "def_end_pos": [602, 42]}, {"full_name": "IsConnected", "def_path": "Mathlib/Topology/Connected/Basic.lean", "def_pos": [57, 5], "def_end_pos": [57, 16]}, {"full_name": "isPreconnected_connectedComponentIn", "def_path": "Mathlib/Topology/Connected/Basic.lean", "def_pos": [617, 9], "def_end_pos": [617, 44]}, {"full_name": "and_true_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [145, 9], "def_end_pos": [145, 21]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\nx : \u03b1\nF : Set \u03b1\n\u22a2 IsConnected (connectedComponentIn F x) \u2194 x \u2208 F", "state_after": "no goals"}]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Algebra/MonoidAlgebra/Basic.lean
|
AddMonoidAlgebra.of'_apply
|
[
1625,
1
] |
[
1626,
6
] |
[]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/MeasureTheory/Function/LpSeminorm.lean
|
MeasureTheory.limsup_trim
|
[
996,
1
] |
[
1006,
36
] |
[{"tactic": "simp_rw [limsup_eq]", "annotated_tactic": ["simp_rw [<a>limsup_eq</a>]", [{"full_name": "Filter.limsup_eq", "def_path": "Mathlib/Order/LiminfLimsup.lean", "def_pos": [446, 9], "def_end_pos": [446, 18]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nhm : m \u2264 m0\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\n\u22a2 limsup f (Measure.ae (Measure.trim \u03bd hm)) = limsup f (Measure.ae \u03bd)", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nhm : m \u2264 m0\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\n\u22a2 sInf {a | \u2200\u1d50 (n : \u03b1) \u2202Measure.trim \u03bd hm, f n \u2264 a} = sInf {a | \u2200\u1d50 (n : \u03b1) \u2202\u03bd, f n \u2264 a}"}, {"tactic": "suffices h_set_eq : { a : \u211d\u22650\u221e | \u2200\u1d50 n \u2202\u03bd.trim hm, f n \u2264 a } = { a : \u211d\u22650\u221e | \u2200\u1d50 n \u2202\u03bd, f n \u2264 a }", "annotated_tactic": ["suffices h_set_eq : { a : \u211d\u22650\u221e | \u2200\u1d50 n \u2202\u03bd.trim hm, f n \u2264 a } = { a : \u211d\u22650\u221e | \u2200\u1d50 n \u2202\u03bd, f n \u2264 a }", []], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nhm : m \u2264 m0\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\n\u22a2 sInf {a | \u2200\u1d50 (n : \u03b1) \u2202Measure.trim \u03bd hm, f n \u2264 a} = sInf {a | \u2200\u1d50 (n : \u03b1) \u2202\u03bd, f n \u2264 a}", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nhm : m \u2264 m0\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nh_set_eq : {a | \u2200\u1d50 (n : \u03b1) \u2202Measure.trim \u03bd hm, f n \u2264 a} = {a | \u2200\u1d50 (n : \u03b1) \u2202\u03bd, f n \u2264 a}\n\u22a2 sInf {a | \u2200\u1d50 (n : \u03b1) \u2202Measure.trim \u03bd hm, f n \u2264 a} = sInf {a | \u2200\u1d50 (n : \u03b1) \u2202\u03bd, f n \u2264 a}\n\ncase h_set_eq\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nhm : m \u2264 m0\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\n\u22a2 {a | \u2200\u1d50 (n : \u03b1) \u2202Measure.trim \u03bd hm, f n \u2264 a} = {a | \u2200\u1d50 (n : \u03b1) \u2202\u03bd, f n \u2264 a}"}, {"tactic": "ext1 a", "annotated_tactic": ["ext1 a", []], "state_before": "case h_set_eq\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nhm : m \u2264 m0\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\n\u22a2 {a | \u2200\u1d50 (n : \u03b1) \u2202Measure.trim \u03bd hm, f n \u2264 a} = {a | \u2200\u1d50 (n : \u03b1) \u2202\u03bd, f n \u2264 a}", "state_after": "case h_set_eq.h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nhm : m \u2264 m0\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\na : \u211d\u22650\u221e\n\u22a2 a \u2208 {a | \u2200\u1d50 (n : \u03b1) \u2202Measure.trim \u03bd hm, f n \u2264 a} \u2194 a \u2208 {a | \u2200\u1d50 (n : \u03b1) \u2202\u03bd, f n \u2264 a}"}, {"tactic": "suffices h_meas_eq : \u03bd { x | \u00acf x \u2264 a } = \u03bd.trim hm { x | \u00acf x \u2264 a }", "annotated_tactic": ["suffices h_meas_eq : \u03bd { x | \u00acf x \u2264 a } = \u03bd.trim hm { x | \u00acf x \u2264 a }", []], "state_before": "case h_set_eq.h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nhm : m \u2264 m0\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\na : \u211d\u22650\u221e\n\u22a2 a \u2208 {a | \u2200\u1d50 (n : \u03b1) \u2202Measure.trim \u03bd hm, f n \u2264 a} \u2194 a \u2208 {a | \u2200\u1d50 (n : \u03b1) \u2202\u03bd, f n \u2264 a}", "state_after": "case h_set_eq.h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nhm : m \u2264 m0\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\na : \u211d\u22650\u221e\nh_meas_eq : \u2191\u2191\u03bd {x | \u00acf x \u2264 a} = \u2191\u2191(Measure.trim \u03bd hm) {x | \u00acf x \u2264 a}\n\u22a2 a \u2208 {a | \u2200\u1d50 (n : \u03b1) \u2202Measure.trim \u03bd hm, f n \u2264 a} \u2194 a \u2208 {a | \u2200\u1d50 (n : \u03b1) \u2202\u03bd, f n \u2264 a}\n\ncase h_meas_eq\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nhm : m \u2264 m0\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\na : \u211d\u22650\u221e\n\u22a2 \u2191\u2191\u03bd {x | \u00acf x \u2264 a} = \u2191\u2191(Measure.trim \u03bd hm) {x | \u00acf x \u2264 a}"}, {"tactic": "refine' (trim_measurableSet_eq hm _).symm", "annotated_tactic": ["refine' (<a>trim_measurableSet_eq</a> hm _).<a>symm</a>", [{"full_name": "MeasureTheory.trim_measurableSet_eq", "def_path": "Mathlib/MeasureTheory/Measure/Trim.lean", "def_pos": [53, 9], "def_end_pos": [53, 30]}, {"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_meas_eq\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nhm : m \u2264 m0\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\na : \u211d\u22650\u221e\n\u22a2 \u2191\u2191\u03bd {x | \u00acf x \u2264 a} = \u2191\u2191(Measure.trim \u03bd hm) {x | \u00acf x \u2264 a}", "state_after": "case h_meas_eq\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nhm : m \u2264 m0\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\na : \u211d\u22650\u221e\n\u22a2 MeasurableSet {x | \u00acf x \u2264 a}"}, {"tactic": "refine' @MeasurableSet.compl _ _ m (@measurableSet_le \u211d\u22650\u221e _ _ _ _ m _ _ _ _ _ hf _)", "annotated_tactic": ["refine' @<a>MeasurableSet.compl</a> _ _ m (@<a>measurableSet_le</a> \u211d\u22650\u221e _ _ _ _ m _ _ _ _ _ hf _)", [{"full_name": "MeasurableSet.compl", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [87, 19], "def_end_pos": [87, 38]}, {"full_name": "measurableSet_le", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [559, 9], "def_end_pos": [559, 25]}]], "state_before": "case h_meas_eq\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nhm : m \u2264 m0\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\na : \u211d\u22650\u221e\n\u22a2 MeasurableSet {x | \u00acf x \u2264 a}", "state_after": "case h_meas_eq\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nhm : m \u2264 m0\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\na : \u211d\u22650\u221e\n\u22a2 Measurable fun x => a"}, {"tactic": "exact @measurable_const _ _ _ m _", "annotated_tactic": ["exact @<a>measurable_const</a> _ _ _ m _", [{"full_name": "measurable_const", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [570, 9], "def_end_pos": [570, 25]}]], "state_before": "case h_meas_eq\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nhm : m \u2264 m0\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\na : \u211d\u22650\u221e\n\u22a2 Measurable fun x => a", "state_after": "no goals"}, {"tactic": "rw [h_set_eq]", "annotated_tactic": ["rw [h_set_eq]", []], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nhm : m \u2264 m0\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nh_set_eq : {a | \u2200\u1d50 (n : \u03b1) \u2202Measure.trim \u03bd hm, f n \u2264 a} = {a | \u2200\u1d50 (n : \u03b1) \u2202\u03bd, f n \u2264 a}\n\u22a2 sInf {a | \u2200\u1d50 (n : \u03b1) \u2202Measure.trim \u03bd hm, f n \u2264 a} = sInf {a | \u2200\u1d50 (n : \u03b1) \u2202\u03bd, f n \u2264 a}", "state_after": "no goals"}, {"tactic": "simp_rw [Set.mem_setOf_eq, ae_iff, h_meas_eq]", "annotated_tactic": ["simp_rw [<a>Set.mem_setOf_eq</a>, <a>ae_iff</a>, h_meas_eq]", [{"full_name": "Set.mem_setOf_eq", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [256, 29], "def_end_pos": [256, 41]}, {"full_name": "MeasureTheory.ae_iff", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [388, 9], "def_end_pos": [388, 15]}]], "state_before": "case h_set_eq.h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nhm : m \u2264 m0\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\na : \u211d\u22650\u221e\nh_meas_eq : \u2191\u2191\u03bd {x | \u00acf x \u2264 a} = \u2191\u2191(Measure.trim \u03bd hm) {x | \u00acf x \u2264 a}\n\u22a2 a \u2208 {a | \u2200\u1d50 (n : \u03b1) \u2202Measure.trim \u03bd hm, f n \u2264 a} \u2194 a \u2208 {a | \u2200\u1d50 (n : \u03b1) \u2202\u03bd, f n \u2264 a}", "state_after": "case h_set_eq.h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nhm : m \u2264 m0\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\na : \u211d\u22650\u221e\nh_meas_eq : \u2191\u2191\u03bd {x | \u00acf x \u2264 a} = \u2191\u2191(Measure.trim \u03bd hm) {x | \u00acf x \u2264 a}\n\u22a2 (\u2200\u1d50 (n : \u03b1) \u2202Measure.trim \u03bd hm, f n \u2264 a) \u2194 \u2191\u2191(Measure.trim \u03bd hm) {a_1 | \u00acf a_1 \u2264 a} = 0"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case h_set_eq.h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nhm : m \u2264 m0\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\na : \u211d\u22650\u221e\nh_meas_eq : \u2191\u2191\u03bd {x | \u00acf x \u2264 a} = \u2191\u2191(Measure.trim \u03bd hm) {x | \u00acf x \u2264 a}\n\u22a2 (\u2200\u1d50 (n : \u03b1) \u2202Measure.trim \u03bd hm, f n \u2264 a) \u2194 \u2191\u2191(Measure.trim \u03bd hm) {a_1 | \u00acf a_1 \u2264 a} = 0", "state_after": "no goals"}]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Data/MvPolynomial/Comap.lean
|
MvPolynomial.comap_eq_id_of_eq_id
|
[
83,
1
] |
[
87,
29
] |
[{"tactic": "convert comap_id_apply x", "annotated_tactic": ["convert <a>comap_id_apply</a> x", [{"full_name": "MvPolynomial.comap_id_apply", "def_path": "Mathlib/Data/MvPolynomial/Comap.lean", "def_pos": [48, 9], "def_end_pos": [48, 23]}]], "state_before": "\u03c3 : Type u_1\n\u03c4 : Type u_2\n\u03c5 : Type u_3\nR : Type u_4\ninst\u271d : CommSemiring R\nf : MvPolynomial \u03c3 R \u2192\u2090[R] MvPolynomial \u03c3 R\nhf : \u2200 (\u03c6 : MvPolynomial \u03c3 R), \u2191f \u03c6 = \u03c6\nx : \u03c3 \u2192 R\n\u22a2 comap f x = x", "state_after": "case h.e'_2.h.e'_5\n\u03c3 : Type u_1\n\u03c4 : Type u_2\n\u03c5 : Type u_3\nR : Type u_4\ninst\u271d : CommSemiring R\nf : MvPolynomial \u03c3 R \u2192\u2090[R] MvPolynomial \u03c3 R\nhf : \u2200 (\u03c6 : MvPolynomial \u03c3 R), \u2191f \u03c6 = \u03c6\nx : \u03c3 \u2192 R\n\u22a2 f = AlgHom.id R (MvPolynomial \u03c3 R)"}, {"tactic": "ext1 \u03c6", "annotated_tactic": ["ext1 \u03c6", []], "state_before": "case h.e'_2.h.e'_5\n\u03c3 : Type u_1\n\u03c4 : Type u_2\n\u03c5 : Type u_3\nR : Type u_4\ninst\u271d : CommSemiring R\nf : MvPolynomial \u03c3 R \u2192\u2090[R] MvPolynomial \u03c3 R\nhf : \u2200 (\u03c6 : MvPolynomial \u03c3 R), \u2191f \u03c6 = \u03c6\nx : \u03c3 \u2192 R\n\u22a2 f = AlgHom.id R (MvPolynomial \u03c3 R)", "state_after": "case h.e'_2.h.e'_5.hf\n\u03c3 : Type u_1\n\u03c4 : Type u_2\n\u03c5 : Type u_3\nR : Type u_4\ninst\u271d : CommSemiring R\nf : MvPolynomial \u03c3 R \u2192\u2090[R] MvPolynomial \u03c3 R\nhf : \u2200 (\u03c6 : MvPolynomial \u03c3 R), \u2191f \u03c6 = \u03c6\nx : \u03c3 \u2192 R\n\u03c6 : \u03c3\n\u22a2 \u2191f (X \u03c6) = \u2191(AlgHom.id R (MvPolynomial \u03c3 R)) (X \u03c6)"}, {"tactic": "simp [hf, AlgHom.id_apply]", "annotated_tactic": ["simp [hf, <a>AlgHom.id_apply</a>]", [{"full_name": "AlgHom.id_apply", "def_path": "Mathlib/Algebra/Algebra/Hom.lean", "def_pos": [322, 9], "def_end_pos": [322, 17]}]], "state_before": "case h.e'_2.h.e'_5.hf\n\u03c3 : Type u_1\n\u03c4 : Type u_2\n\u03c5 : Type u_3\nR : Type u_4\ninst\u271d : CommSemiring R\nf : MvPolynomial \u03c3 R \u2192\u2090[R] MvPolynomial \u03c3 R\nhf : \u2200 (\u03c6 : MvPolynomial \u03c3 R), \u2191f \u03c6 = \u03c6\nx : \u03c3 \u2192 R\n\u03c6 : \u03c3\n\u22a2 \u2191f (X \u03c6) = \u2191(AlgHom.id R (MvPolynomial \u03c3 R)) (X \u03c6)", "state_after": "no goals"}]
|
https://github.com/leanprover/std4
|
869c615eb10130c0637a7bc038e2b80253559913
|
lake-packages/std/Std/Data/Int/Lemmas.lean
|
Int.subNatNat_add_add
|
[
121,
1
] |
[
128,
30
] |
[{"tactic": "apply subNatNat_elim m n (fun m n i => subNatNat (m + k) (n + k) = i)", "annotated_tactic": ["apply <a>subNatNat_elim</a> m n (fun m n i => <a>subNatNat</a> (m + k) (n + k) = i)", [{"full_name": "Int.subNatNat_elim", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [101, 9], "def_end_pos": [101, 23]}, {"full_name": "Int.subNatNat", "def_path": "lake-packages/lean4/src/lean/Init/Data/Int/Basic.lean", "def_pos": [86, 5], "def_end_pos": [86, 14]}]], "state_before": "m n k : Nat\n\u22a2 subNatNat (m + k) (n + k) = subNatNat m n", "state_after": "case hp\nm n k : Nat\n\u22a2 \u2200 (i n : Nat), subNatNat (n + i + k) (n + k) = \u2191i\n\ncase hn\nm n k : Nat\n\u22a2 \u2200 (i m : Nat), subNatNat (m + k) (m + i + 1 + k) = -[i+1]"}, {"tactic": "intro i j", "annotated_tactic": ["intro i j", []], "state_before": "case hp\nm n k : Nat\n\u22a2 \u2200 (i n : Nat), subNatNat (n + i + k) (n + k) = \u2191i", "state_after": "case hp\nm n k i j : Nat\n\u22a2 subNatNat (j + i + k) (j + k) = \u2191i"}, {"tactic": "rw [Nat.add_assoc, Nat.add_comm i k, \u2190 Nat.add_assoc]", "annotated_tactic": ["rw [<a>Nat.add_assoc</a>, <a>Nat.add_comm</a> i k, \u2190 <a>Nat.add_assoc</a>]", [{"full_name": "Nat.add_assoc", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [138, 19], "def_end_pos": [138, 28]}, {"full_name": "Nat.add_comm", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [131, 19], "def_end_pos": [131, 27]}, {"full_name": "Nat.add_assoc", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [138, 19], "def_end_pos": [138, 28]}]], "state_before": "case hp\nm n k i j : Nat\n\u22a2 subNatNat (j + i + k) (j + k) = \u2191i", "state_after": "case hp\nm n k i j : Nat\n\u22a2 subNatNat (j + k + i) (j + k) = \u2191i"}, {"tactic": "exact subNatNat_add_left", "annotated_tactic": ["exact <a>subNatNat_add_left</a>", [{"full_name": "Int.subNatNat_add_left", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [114, 9], "def_end_pos": [114, 27]}]], "state_before": "case hp\nm n k i j : Nat\n\u22a2 subNatNat (j + k + i) (j + k) = \u2191i", "state_after": "no goals"}, {"tactic": "intro i j", "annotated_tactic": ["intro i j", []], "state_before": "case hn\nm n k : Nat\n\u22a2 \u2200 (i m : Nat), subNatNat (m + k) (m + i + 1 + k) = -[i+1]", "state_after": "case hn\nm n k i j : Nat\n\u22a2 subNatNat (j + k) (j + i + 1 + k) = -[i+1]"}, {"tactic": "rw [Nat.add_assoc j i 1, Nat.add_comm j (i+1), Nat.add_assoc, Nat.add_comm (i+1) (j+k)]", "annotated_tactic": ["rw [<a>Nat.add_assoc</a> j i 1, <a>Nat.add_comm</a> j (i+1), <a>Nat.add_assoc</a>, <a>Nat.add_comm</a> (i+1) (j+k)]", [{"full_name": "Nat.add_assoc", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [138, 19], "def_end_pos": [138, 28]}, {"full_name": "Nat.add_comm", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [131, 19], "def_end_pos": [131, 27]}, {"full_name": "Nat.add_assoc", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [138, 19], "def_end_pos": [138, 28]}, {"full_name": "Nat.add_comm", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [131, 19], "def_end_pos": [131, 27]}]], "state_before": "case hn\nm n k i j : Nat\n\u22a2 subNatNat (j + k) (j + i + 1 + k) = -[i+1]", "state_after": "case hn\nm n k i j : Nat\n\u22a2 subNatNat (j + k) (j + k + (i + 1)) = -[i+1]"}, {"tactic": "exact subNatNat_add_right", "annotated_tactic": ["exact <a>subNatNat_add_right</a>", [{"full_name": "Int.subNatNat_add_right", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [118, 9], "def_end_pos": [118, 28]}]], "state_before": "case hn\nm n k i j : Nat\n\u22a2 subNatNat (j + k) (j + k + (i + 1)) = -[i+1]", "state_after": "no goals"}]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/MeasureTheory/Integral/Bochner.lean
|
MeasureTheory.snorm_one_le_of_le
|
[
1940,
1
] |
[
1984,
48
] |
[{"tactic": "by_cases hr : r = 0", "annotated_tactic": ["by_cases hr : r = 0", []], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nr : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhfint : Integrable f\nhfint' : 0 \u2264 \u222b (x : \u03b1), f x \u2202\u03bc\nhf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 \u2191r\n\u22a2 snorm f 1 \u03bc \u2264 2 * \u2191\u2191\u03bc univ * \u2191r", "state_after": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nr : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhfint : Integrable f\nhfint' : 0 \u2264 \u222b (x : \u03b1), f x \u2202\u03bc\nhf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 \u2191r\nhr : r = 0\n\u22a2 snorm f 1 \u03bc \u2264 2 * \u2191\u2191\u03bc univ * \u2191r\n\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nr : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhfint : Integrable f\nhfint' : 0 \u2264 \u222b (x : \u03b1), f x \u2202\u03bc\nhf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 \u2191r\nhr : \u00acr = 0\n\u22a2 snorm f 1 \u03bc \u2264 2 * \u2191\u2191\u03bc univ * \u2191r"}, {"tactic": "by_cases h\u03bc : IsFiniteMeasure \u03bc", "annotated_tactic": ["by_cases h\u03bc : <a>IsFiniteMeasure</a> \u03bc", [{"full_name": "MeasureTheory.IsFiniteMeasure", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2850, 7], "def_end_pos": [2850, 22]}]], "state_before": "case neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nr : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhfint : Integrable f\nhfint' : 0 \u2264 \u222b (x : \u03b1), f x \u2202\u03bc\nhf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 \u2191r\nhr : \u00acr = 0\n\u22a2 snorm f 1 \u03bc \u2264 2 * \u2191\u2191\u03bc univ * \u2191r", "state_after": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nr : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhfint : Integrable f\nhfint' : 0 \u2264 \u222b (x : \u03b1), f x \u2202\u03bc\nhf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 \u2191r\nhr : \u00acr = 0\nh\u03bc : IsFiniteMeasure \u03bc\n\u22a2 snorm f 1 \u03bc \u2264 2 * \u2191\u2191\u03bc univ * \u2191r\n\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nr : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhfint : Integrable f\nhfint' : 0 \u2264 \u222b (x : \u03b1), f x \u2202\u03bc\nhf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 \u2191r\nhr : \u00acr = 0\nh\u03bc : \u00acIsFiniteMeasure \u03bc\n\u22a2 snorm f 1 \u03bc \u2264 2 * \u2191\u2191\u03bc univ * \u2191r"}, {"tactic": "swap", "annotated_tactic": ["swap", []], "state_before": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nr : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhfint : Integrable f\nhfint' : 0 \u2264 \u222b (x : \u03b1), f x \u2202\u03bc\nhf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 \u2191r\nhr : \u00acr = 0\nh\u03bc : IsFiniteMeasure \u03bc\n\u22a2 snorm f 1 \u03bc \u2264 2 * \u2191\u2191\u03bc univ * \u2191r\n\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nr : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhfint : Integrable f\nhfint' : 0 \u2264 \u222b (x : \u03b1), f x \u2202\u03bc\nhf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 \u2191r\nhr : \u00acr = 0\nh\u03bc : \u00acIsFiniteMeasure \u03bc\n\u22a2 snorm f 1 \u03bc \u2264 2 * \u2191\u2191\u03bc univ * \u2191r", "state_after": "case neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nr : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhfint : Integrable f\nhfint' : 0 \u2264 \u222b (x : \u03b1), f x \u2202\u03bc\nhf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 \u2191r\nhr : \u00acr = 0\nh\u03bc : \u00acIsFiniteMeasure \u03bc\n\u22a2 snorm f 1 \u03bc \u2264 2 * \u2191\u2191\u03bc univ * \u2191r\n\ncase pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nr : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhfint : Integrable f\nhfint' : 0 \u2264 \u222b (x : \u03b1), f x \u2202\u03bc\nhf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 \u2191r\nhr : \u00acr = 0\nh\u03bc : IsFiniteMeasure \u03bc\n\u22a2 snorm f 1 \u03bc \u2264 2 * \u2191\u2191\u03bc univ * \u2191r"}, {"tactic": "haveI := h\u03bc", "annotated_tactic": ["haveI := h\u03bc", []], "state_before": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nr : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhfint : Integrable f\nhfint' : 0 \u2264 \u222b (x : \u03b1), f x \u2202\u03bc\nhf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 \u2191r\nhr : \u00acr = 0\nh\u03bc : IsFiniteMeasure \u03bc\n\u22a2 snorm f 1 \u03bc \u2264 2 * \u2191\u2191\u03bc univ * \u2191r", "state_after": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nr : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhfint : Integrable f\nhfint' : 0 \u2264 \u222b (x : \u03b1), f x \u2202\u03bc\nhf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 \u2191r\nhr : \u00acr = 0\nh\u03bc this : IsFiniteMeasure \u03bc\n\u22a2 snorm f 1 \u03bc \u2264 2 * \u2191\u2191\u03bc univ * \u2191r"}, {"tactic": "rw [integral_eq_integral_pos_part_sub_integral_neg_part hfint, sub_nonneg] at hfint'", "annotated_tactic": ["rw [<a>integral_eq_integral_pos_part_sub_integral_neg_part</a> hfint, <a>sub_nonneg</a>] at hfint'", [{"full_name": "MeasureTheory.integral_eq_integral_pos_part_sub_integral_neg_part", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [1194, 9], "def_end_pos": [1194, 60]}, {"full_name": "sub_nonneg", "def_path": "Mathlib/Algebra/Order/Group/Defs.lean", "def_pos": [720, 30], "def_end_pos": [720, 40]}]], "state_before": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nr : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhfint : Integrable f\nhfint' : 0 \u2264 \u222b (x : \u03b1), f x \u2202\u03bc\nhf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 \u2191r\nhr : \u00acr = 0\nh\u03bc this : IsFiniteMeasure \u03bc\n\u22a2 snorm f 1 \u03bc \u2264 2 * \u2191\u2191\u03bc univ * \u2191r", "state_after": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nr : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhfint : Integrable f\nhfint' : \u222b (a : \u03b1), \u2191(Real.toNNReal (-f a)) \u2202\u03bc \u2264 \u222b (a : \u03b1), \u2191(Real.toNNReal (f a)) \u2202\u03bc\nhf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 \u2191r\nhr : \u00acr = 0\nh\u03bc this : IsFiniteMeasure \u03bc\n\u22a2 snorm f 1 \u03bc \u2264 2 * \u2191\u2191\u03bc univ * \u2191r"}, {"tactic": "have hposbdd : \u222b \u03c9, max (f \u03c9) 0 \u2202\u03bc \u2264 (\u03bc Set.univ).toReal \u2022 (r : \u211d) := by\n rw [\u2190 integral_const]\n refine' integral_mono_ae hfint.real_toNNReal (integrable_const (r : \u211d)) _\n filter_upwards [hf] with \u03c9 h\u03c9 using Real.toNNReal_le_iff_le_coe.2 h\u03c9", "annotated_tactic": ["have hposbdd : \u222b \u03c9, <a>max</a> (f \u03c9) 0 \u2202\u03bc \u2264 (\u03bc <a>Set.univ</a>).<a>toReal</a> \u2022 (r : \u211d) := by\n rw [\u2190 <a>integral_const</a>]\n refine' <a>integral_mono_ae</a> hfint.real_toNNReal (<a>integrable_const</a> (r : \u211d)) _\n filter_upwards [hf] with \u03c9 h\u03c9 using <a>Real.toNNReal_le_iff_le_coe</a>.2 h\u03c9", [{"full_name": "Max.max", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1090, 3], "def_end_pos": [1090, 6]}, {"full_name": "Set.univ", "def_path": "Mathlib/Init/Set.lean", "def_pos": [90, 5], "def_end_pos": [90, 9]}, {"full_name": "ENNReal.toReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [168, 15], "def_end_pos": [168, 21]}, {"full_name": "MeasureTheory.integral_const", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [1409, 9], "def_end_pos": [1409, 23]}, {"full_name": "MeasureTheory.integral_mono_ae", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [1342, 9], "def_end_pos": [1342, 25]}, {"full_name": "MeasureTheory.integrable_const", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [506, 9], "def_end_pos": [506, 25]}, {"full_name": "Real.toNNReal_le_iff_le_coe", "def_path": "Mathlib/Data/Real/NNReal.lean", "def_pos": [685, 9], "def_end_pos": [685, 31]}]], "state_before": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nr : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhfint : Integrable f\nhfint' : \u222b (a : \u03b1), \u2191(Real.toNNReal (-f a)) \u2202\u03bc \u2264 \u222b (a : \u03b1), \u2191(Real.toNNReal (f a)) \u2202\u03bc\nhf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 \u2191r\nhr : \u00acr = 0\nh\u03bc this : IsFiniteMeasure \u03bc\n\u22a2 snorm f 1 \u03bc \u2264 2 * \u2191\u2191\u03bc univ * \u2191r", "state_after": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nr : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhfint : Integrable f\nhfint' : \u222b (a : \u03b1), \u2191(Real.toNNReal (-f a)) \u2202\u03bc \u2264 \u222b (a : \u03b1), \u2191(Real.toNNReal (f a)) \u2202\u03bc\nhf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 \u2191r\nhr : \u00acr = 0\nh\u03bc this : IsFiniteMeasure \u03bc\nhposbdd : \u222b (\u03c9 : \u03b1), max (f \u03c9) 0 \u2202\u03bc \u2264 ENNReal.toReal (\u2191\u2191\u03bc univ) \u2022 \u2191r\n\u22a2 snorm f 1 \u03bc \u2264 2 * \u2191\u2191\u03bc univ * \u2191r"}, {"tactic": "rw [Mem\u2112p.snorm_eq_integral_rpow_norm one_ne_zero ENNReal.one_ne_top\n (mem\u2112p_one_iff_integrable.2 hfint),\n ENNReal.ofReal_le_iff_le_toReal\n (ENNReal.mul_ne_top (ENNReal.mul_ne_top ENNReal.two_ne_top <| @measure_ne_top _ _ _ h\u03bc _)\n ENNReal.coe_ne_top)]", "annotated_tactic": ["rw [<a>Mem\u2112p.snorm_eq_integral_rpow_norm</a> <a>one_ne_zero</a> <a>ENNReal.one_ne_top</a>\n (<a>mem\u2112p_one_iff_integrable</a>.2 hfint),\n <a>ENNReal.ofReal_le_iff_le_toReal</a>\n (<a>ENNReal.mul_ne_top</a> (<a>ENNReal.mul_ne_top</a> <a>ENNReal.two_ne_top</a> <| @<a>measure_ne_top</a> _ _ _ h\u03bc _)\n <a>ENNReal.coe_ne_top</a>)]", [{"full_name": "MeasureTheory.Mem\u2112p.snorm_eq_integral_rpow_norm", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [1321, 9], "def_end_pos": [1321, 42]}, {"full_name": "one_ne_zero", "def_path": "Mathlib/Algebra/NeZero.lean", "def_pos": [55, 15], "def_end_pos": [55, 26]}, {"full_name": "ENNReal.one_ne_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [340, 17], "def_end_pos": [340, 27]}, {"full_name": "MeasureTheory.mem\u2112p_one_iff_integrable", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [453, 9], "def_end_pos": [453, 33]}, {"full_name": "ENNReal.ofReal_le_iff_le_toReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [2189, 9], "def_end_pos": [2189, 32]}, {"full_name": "ENNReal.mul_ne_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [615, 9], "def_end_pos": [615, 19]}, {"full_name": "ENNReal.mul_ne_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [615, 9], "def_end_pos": [615, 19]}, {"full_name": "ENNReal.two_ne_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [431, 9], "def_end_pos": [431, 19]}, {"full_name": "MeasureTheory.measure_ne_top", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2875, 9], "def_end_pos": [2875, 23]}, {"full_name": "ENNReal.coe_ne_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [302, 17], "def_end_pos": [302, 27]}]], "state_before": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nr : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhfint : Integrable f\nhfint' : \u222b (a : \u03b1), \u2191(Real.toNNReal (-f a)) \u2202\u03bc \u2264 \u222b (a : \u03b1), \u2191(Real.toNNReal (f a)) \u2202\u03bc\nhf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 \u2191r\nhr : \u00acr = 0\nh\u03bc this : IsFiniteMeasure \u03bc\nhposbdd : \u222b (\u03c9 : \u03b1), max (f \u03c9) 0 \u2202\u03bc \u2264 ENNReal.toReal (\u2191\u2191\u03bc univ) \u2022 \u2191r\n\u22a2 snorm f 1 \u03bc \u2264 2 * \u2191\u2191\u03bc univ * \u2191r", "state_after": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nr : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhfint : Integrable f\nhfint' : \u222b (a : \u03b1), \u2191(Real.toNNReal (-f a)) \u2202\u03bc \u2264 \u222b (a : \u03b1), \u2191(Real.toNNReal (f a)) \u2202\u03bc\nhf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 \u2191r\nhr : \u00acr = 0\nh\u03bc this : IsFiniteMeasure \u03bc\nhposbdd : \u222b (\u03c9 : \u03b1), max (f \u03c9) 0 \u2202\u03bc \u2264 ENNReal.toReal (\u2191\u2191\u03bc univ) \u2022 \u2191r\n\u22a2 (\u222b (a : \u03b1), \u2016f a\u2016 ^ ENNReal.toReal 1 \u2202\u03bc) ^ (ENNReal.toReal 1)\u207b\u00b9 \u2264 ENNReal.toReal (2 * \u2191\u2191\u03bc univ * \u2191r)"}, {"tactic": "simp_rw [ENNReal.one_toReal, _root_.inv_one, Real.rpow_one, Real.norm_eq_abs, \u2190\n max_zero_add_max_neg_zero_eq_abs_self, \u2190 Real.coe_toNNReal']", "annotated_tactic": ["simp_rw [<a>ENNReal.one_toReal</a>, <a>_root_.inv_one</a>, <a>Real.rpow_one</a>, <a>Real.norm_eq_abs</a>, \u2190\n <a>max_zero_add_max_neg_zero_eq_abs_self</a>, \u2190 <a>Real.coe_toNNReal'</a>]", [{"full_name": "ENNReal.one_toReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [230, 17], "def_end_pos": [230, 27]}, {"full_name": "inv_one", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [1015, 9], "def_end_pos": [1015, 16]}, {"full_name": "Real.rpow_one", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/Real.lean", "def_pos": [126, 9], "def_end_pos": [126, 17]}, {"full_name": "Real.norm_eq_abs", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [1761, 9], "def_end_pos": [1761, 20]}, {"full_name": "max_zero_add_max_neg_zero_eq_abs_self", "def_path": "Mathlib/Algebra/Order/Ring/Abs.lean", "def_pos": [78, 9], "def_end_pos": [78, 46]}, {"full_name": "Real.coe_toNNReal'", "def_path": "Mathlib/Data/Real/NNReal.lean", "def_pos": [615, 9], "def_end_pos": [615, 22]}]], "state_before": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nr : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhfint : Integrable f\nhfint' : \u222b (a : \u03b1), \u2191(Real.toNNReal (-f a)) \u2202\u03bc \u2264 \u222b (a : \u03b1), \u2191(Real.toNNReal (f a)) \u2202\u03bc\nhf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 \u2191r\nhr : \u00acr = 0\nh\u03bc this : IsFiniteMeasure \u03bc\nhposbdd : \u222b (\u03c9 : \u03b1), max (f \u03c9) 0 \u2202\u03bc \u2264 ENNReal.toReal (\u2191\u2191\u03bc univ) \u2022 \u2191r\n\u22a2 (\u222b (a : \u03b1), \u2016f a\u2016 ^ ENNReal.toReal 1 \u2202\u03bc) ^ (ENNReal.toReal 1)\u207b\u00b9 \u2264 ENNReal.toReal (2 * \u2191\u2191\u03bc univ * \u2191r)", "state_after": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nr : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhfint : Integrable f\nhfint' : \u222b (a : \u03b1), \u2191(Real.toNNReal (-f a)) \u2202\u03bc \u2264 \u222b (a : \u03b1), \u2191(Real.toNNReal (f a)) \u2202\u03bc\nhf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 \u2191r\nhr : \u00acr = 0\nh\u03bc this : IsFiniteMeasure \u03bc\nhposbdd : \u222b (\u03c9 : \u03b1), max (f \u03c9) 0 \u2202\u03bc \u2264 ENNReal.toReal (\u2191\u2191\u03bc univ) \u2022 \u2191r\n\u22a2 \u222b (a : \u03b1), \u2191(Real.toNNReal (f a)) + \u2191(Real.toNNReal (-f a)) \u2202\u03bc \u2264 ENNReal.toReal (2 * \u2191\u2191\u03bc univ * \u2191r)"}, {"tactic": "rw [integral_add hfint.real_toNNReal]", "annotated_tactic": ["rw [<a>integral_add</a> hfint.real_toNNReal]", [{"full_name": "MeasureTheory.integral_add", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [868, 9], "def_end_pos": [868, 21]}]], "state_before": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nr : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhfint : Integrable f\nhfint' : \u222b (a : \u03b1), \u2191(Real.toNNReal (-f a)) \u2202\u03bc \u2264 \u222b (a : \u03b1), \u2191(Real.toNNReal (f a)) \u2202\u03bc\nhf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 \u2191r\nhr : \u00acr = 0\nh\u03bc this : IsFiniteMeasure \u03bc\nhposbdd : \u222b (\u03c9 : \u03b1), max (f \u03c9) 0 \u2202\u03bc \u2264 ENNReal.toReal (\u2191\u2191\u03bc univ) \u2022 \u2191r\n\u22a2 \u222b (a : \u03b1), \u2191(Real.toNNReal (f a)) + \u2191(Real.toNNReal (-f a)) \u2202\u03bc \u2264 ENNReal.toReal (2 * \u2191\u2191\u03bc univ * \u2191r)", "state_after": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nr : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhfint : Integrable f\nhfint' : \u222b (a : \u03b1), \u2191(Real.toNNReal (-f a)) \u2202\u03bc \u2264 \u222b (a : \u03b1), \u2191(Real.toNNReal (f a)) \u2202\u03bc\nhf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 \u2191r\nhr : \u00acr = 0\nh\u03bc this : IsFiniteMeasure \u03bc\nhposbdd : \u222b (\u03c9 : \u03b1), max (f \u03c9) 0 \u2202\u03bc \u2264 ENNReal.toReal (\u2191\u2191\u03bc univ) \u2022 \u2191r\n\u22a2 \u222b (a : \u03b1), \u2191(Real.toNNReal (f a)) \u2202\u03bc + \u222b (a : \u03b1), \u2191(Real.toNNReal (-f a)) \u2202\u03bc \u2264 ENNReal.toReal (2 * \u2191\u2191\u03bc univ * \u2191r)\n\ncase pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nr : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhfint : Integrable f\nhfint' : \u222b (a : \u03b1), \u2191(Real.toNNReal (-f a)) \u2202\u03bc \u2264 \u222b (a : \u03b1), \u2191(Real.toNNReal (f a)) \u2202\u03bc\nhf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 \u2191r\nhr : \u00acr = 0\nh\u03bc this : IsFiniteMeasure \u03bc\nhposbdd : \u222b (\u03c9 : \u03b1), max (f \u03c9) 0 \u2202\u03bc \u2264 ENNReal.toReal (\u2191\u2191\u03bc univ) \u2022 \u2191r\n\u22a2 Integrable fun a => \u2191(Real.toNNReal (-f a))"}, {"tactic": "suffices f =\u1d50[\u03bc] 0 by\n rw [snorm_congr_ae this, snorm_zero, hr, ENNReal.coe_zero, mul_zero]", "annotated_tactic": ["suffices f =\u1d50[\u03bc] 0 by\n rw [<a>snorm_congr_ae</a> this, <a>snorm_zero</a>, hr, <a>ENNReal.coe_zero</a>, <a>mul_zero</a>]", [{"full_name": "MeasureTheory.snorm_congr_ae", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [549, 9], "def_end_pos": [549, 23]}, {"full_name": "MeasureTheory.snorm_zero", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [201, 9], "def_end_pos": [201, 19]}, {"full_name": "ENNReal.coe_zero", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [215, 28], "def_end_pos": [215, 36]}, {"full_name": "MulZeroClass.mul_zero", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [38, 3], "def_end_pos": [38, 11]}]], "state_before": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nr : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhfint : Integrable f\nhfint' : 0 \u2264 \u222b (x : \u03b1), f x \u2202\u03bc\nhf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 \u2191r\nhr : r = 0\n\u22a2 snorm f 1 \u03bc \u2264 2 * \u2191\u2191\u03bc univ * \u2191r", "state_after": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nr : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhfint : Integrable f\nhfint' : 0 \u2264 \u222b (x : \u03b1), f x \u2202\u03bc\nhf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 \u2191r\nhr : r = 0\n\u22a2 f =\u1d50[\u03bc] 0"}, {"tactic": "rw [hr] at hf", "annotated_tactic": ["rw [hr] at hf", []], "state_before": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nr : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhfint : Integrable f\nhfint' : 0 \u2264 \u222b (x : \u03b1), f x \u2202\u03bc\nhf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 \u2191r\nhr : r = 0\n\u22a2 f =\u1d50[\u03bc] 0", "state_after": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nr : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhfint : Integrable f\nhfint' : 0 \u2264 \u222b (x : \u03b1), f x \u2202\u03bc\nhf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 \u21910\nhr : r = 0\n\u22a2 f =\u1d50[\u03bc] 0"}, {"tactic": "norm_cast at hf", "annotated_tactic": ["norm_cast at hf", []], "state_before": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nr : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhfint : Integrable f\nhfint' : 0 \u2264 \u222b (x : \u03b1), f x \u2202\u03bc\nhf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 \u21910\nhr : r = 0\n\u22a2 f =\u1d50[\u03bc] 0", "state_after": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nr : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhfint : Integrable f\nhfint' : 0 \u2264 \u222b (x : \u03b1), f x \u2202\u03bc\nhr : r = 0\nhf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 0\n\u22a2 f =\u1d50[\u03bc] 0"}, {"tactic": "have hnegf : \u222b x, -f x \u2202\u03bc = 0 := by\n rw [integral_neg, neg_eq_zero]\n exact le_antisymm (integral_nonpos_of_ae hf) hfint'", "annotated_tactic": ["have hnegf : \u222b x, -f x \u2202\u03bc = 0 := by\n rw [<a>integral_neg</a>, <a>neg_eq_zero</a>]\n exact <a>le_antisymm</a> (<a>integral_nonpos_of_ae</a> hf) hfint'", [{"full_name": "MeasureTheory.integral_neg", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [890, 9], "def_end_pos": [890, 21]}, {"full_name": "neg_eq_zero", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [422, 3], "def_end_pos": [422, 14]}, {"full_name": "le_antisymm", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [188, 9], "def_end_pos": [188, 20]}, {"full_name": "MeasureTheory.integral_nonpos_of_ae", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [1255, 9], "def_end_pos": [1255, 30]}]], "state_before": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nr : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhfint : Integrable f\nhfint' : 0 \u2264 \u222b (x : \u03b1), f x \u2202\u03bc\nhr : r = 0\nhf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 0\n\u22a2 f =\u1d50[\u03bc] 0", "state_after": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nr : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhfint : Integrable f\nhfint' : 0 \u2264 \u222b (x : \u03b1), f x \u2202\u03bc\nhr : r = 0\nhf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 0\nhnegf : \u222b (x : \u03b1), -f x \u2202\u03bc = 0\n\u22a2 f =\u1d50[\u03bc] 0"}, {"tactic": "have := (integral_eq_zero_iff_of_nonneg_ae ?_ hfint.neg).1 hnegf", "annotated_tactic": ["have := (<a>integral_eq_zero_iff_of_nonneg_ae</a> ?_ hfint.neg).1 hnegf", [{"full_name": "MeasureTheory.integral_eq_zero_iff_of_nonneg_ae", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [1265, 9], "def_end_pos": [1265, 42]}]], "state_before": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nr : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhfint : Integrable f\nhfint' : 0 \u2264 \u222b (x : \u03b1), f x \u2202\u03bc\nhr : r = 0\nhf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 0\nhnegf : \u222b (x : \u03b1), -f x \u2202\u03bc = 0\n\u22a2 f =\u1d50[\u03bc] 0", "state_after": "case pos.refine_2\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nr : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhfint : Integrable f\nhfint' : 0 \u2264 \u222b (x : \u03b1), f x \u2202\u03bc\nhr : r = 0\nhf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 0\nhnegf : \u222b (x : \u03b1), -f x \u2202\u03bc = 0\nthis : -f =\u1d50[\u03bc] 0\n\u22a2 f =\u1d50[\u03bc] 0\n\ncase pos.refine_1\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nr : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhfint : Integrable f\nhfint' : 0 \u2264 \u222b (x : \u03b1), f x \u2202\u03bc\nhr : r = 0\nhf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 0\nhnegf : \u222b (x : \u03b1), -f x \u2202\u03bc = 0\n\u22a2 0 \u2264\u1d50[\u03bc] -f"}, {"tactic": "rw [snorm_congr_ae this, snorm_zero, hr, ENNReal.coe_zero, mul_zero]", "annotated_tactic": ["rw [<a>snorm_congr_ae</a> this, <a>snorm_zero</a>, hr, <a>ENNReal.coe_zero</a>, <a>mul_zero</a>]", [{"full_name": "MeasureTheory.snorm_congr_ae", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [549, 9], "def_end_pos": [549, 23]}, {"full_name": "MeasureTheory.snorm_zero", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [201, 9], "def_end_pos": [201, 19]}, {"full_name": "ENNReal.coe_zero", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [215, 28], "def_end_pos": [215, 36]}, {"full_name": "MulZeroClass.mul_zero", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [38, 3], "def_end_pos": [38, 11]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nr : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhfint : Integrable f\nhfint' : 0 \u2264 \u222b (x : \u03b1), f x \u2202\u03bc\nhf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 \u2191r\nhr : r = 0\nthis : f =\u1d50[\u03bc] 0\n\u22a2 snorm f 1 \u03bc \u2264 2 * \u2191\u2191\u03bc univ * \u2191r", "state_after": "no goals"}, {"tactic": "rw [integral_neg, neg_eq_zero]", "annotated_tactic": ["rw [<a>integral_neg</a>, <a>neg_eq_zero</a>]", [{"full_name": "MeasureTheory.integral_neg", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [890, 9], "def_end_pos": [890, 21]}, {"full_name": "neg_eq_zero", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [422, 3], "def_end_pos": [422, 14]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nr : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhfint : Integrable f\nhfint' : 0 \u2264 \u222b (x : \u03b1), f x \u2202\u03bc\nhr : r = 0\nhf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 0\n\u22a2 \u222b (x : \u03b1), -f x \u2202\u03bc = 0", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nr : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhfint : Integrable f\nhfint' : 0 \u2264 \u222b (x : \u03b1), f x \u2202\u03bc\nhr : r = 0\nhf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 0\n\u22a2 \u222b (a : \u03b1), f a \u2202\u03bc = 0"}, {"tactic": "exact le_antisymm (integral_nonpos_of_ae hf) hfint'", "annotated_tactic": ["exact <a>le_antisymm</a> (<a>integral_nonpos_of_ae</a> hf) hfint'", [{"full_name": "le_antisymm", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [188, 9], "def_end_pos": [188, 20]}, {"full_name": "MeasureTheory.integral_nonpos_of_ae", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [1255, 9], "def_end_pos": [1255, 30]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nr : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhfint : Integrable f\nhfint' : 0 \u2264 \u222b (x : \u03b1), f x \u2202\u03bc\nhr : r = 0\nhf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 0\n\u22a2 \u222b (a : \u03b1), f a \u2202\u03bc = 0", "state_after": "no goals"}, {"tactic": "filter_upwards [this] with \u03c9 h\u03c9", "annotated_tactic": ["filter_upwards [this] with \u03c9 h\u03c9", []], "state_before": "case pos.refine_2\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nr : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhfint : Integrable f\nhfint' : 0 \u2264 \u222b (x : \u03b1), f x \u2202\u03bc\nhr : r = 0\nhf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 0\nhnegf : \u222b (x : \u03b1), -f x \u2202\u03bc = 0\nthis : -f =\u1d50[\u03bc] 0\n\u22a2 f =\u1d50[\u03bc] 0", "state_after": "case h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nr : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhfint : Integrable f\nhfint' : 0 \u2264 \u222b (x : \u03b1), f x \u2202\u03bc\nhr : r = 0\nhf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 0\nhnegf : \u222b (x : \u03b1), -f x \u2202\u03bc = 0\nthis : -f =\u1d50[\u03bc] 0\n\u03c9 : \u03b1\nh\u03c9 : (-f) \u03c9 = OfNat.ofNat 0 \u03c9\n\u22a2 f \u03c9 = OfNat.ofNat 0 \u03c9"}, {"tactic": "rwa [Pi.neg_apply, Pi.zero_apply, neg_eq_zero] at h\u03c9", "annotated_tactic": ["rwa [<a>Pi.neg_apply</a>, <a>Pi.zero_apply</a>, <a>neg_eq_zero</a>] at h\u03c9", [{"full_name": "Pi.neg_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [170, 3], "def_end_pos": [170, 14]}, {"full_name": "Pi.zero_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [46, 3], "def_end_pos": [46, 14]}, {"full_name": "neg_eq_zero", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [422, 3], "def_end_pos": [422, 14]}]], "state_before": "case h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nr : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhfint : Integrable f\nhfint' : 0 \u2264 \u222b (x : \u03b1), f x \u2202\u03bc\nhr : r = 0\nhf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 0\nhnegf : \u222b (x : \u03b1), -f x \u2202\u03bc = 0\nthis : -f =\u1d50[\u03bc] 0\n\u03c9 : \u03b1\nh\u03c9 : (-f) \u03c9 = OfNat.ofNat 0 \u03c9\n\u22a2 f \u03c9 = OfNat.ofNat 0 \u03c9", "state_after": "no goals"}, {"tactic": "filter_upwards [hf] with \u03c9 h\u03c9", "annotated_tactic": ["filter_upwards [hf] with \u03c9 h\u03c9", []], "state_before": "case pos.refine_1\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nr : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhfint : Integrable f\nhfint' : 0 \u2264 \u222b (x : \u03b1), f x \u2202\u03bc\nhr : r = 0\nhf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 0\nhnegf : \u222b (x : \u03b1), -f x \u2202\u03bc = 0\n\u22a2 0 \u2264\u1d50[\u03bc] -f", "state_after": "case h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nr : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhfint : Integrable f\nhfint' : 0 \u2264 \u222b (x : \u03b1), f x \u2202\u03bc\nhr : r = 0\nhf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 0\nhnegf : \u222b (x : \u03b1), -f x \u2202\u03bc = 0\n\u03c9 : \u03b1\nh\u03c9 : f \u03c9 \u2264 0\n\u22a2 OfNat.ofNat 0 \u03c9 \u2264 (-f) \u03c9"}, {"tactic": "rwa [Pi.zero_apply, Pi.neg_apply, Right.nonneg_neg_iff]", "annotated_tactic": ["rwa [<a>Pi.zero_apply</a>, <a>Pi.neg_apply</a>, <a>Right.nonneg_neg_iff</a>]", [{"full_name": "Pi.zero_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [46, 3], "def_end_pos": [46, 14]}, {"full_name": "Pi.neg_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [170, 3], "def_end_pos": [170, 14]}, {"full_name": "Right.nonneg_neg_iff", "def_path": "Mathlib/Algebra/Order/Group/Defs.lean", "def_pos": [222, 3], "def_end_pos": [222, 14]}]], "state_before": "case h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nr : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhfint : Integrable f\nhfint' : 0 \u2264 \u222b (x : \u03b1), f x \u2202\u03bc\nhr : r = 0\nhf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 0\nhnegf : \u222b (x : \u03b1), -f x \u2202\u03bc = 0\n\u03c9 : \u03b1\nh\u03c9 : f \u03c9 \u2264 0\n\u22a2 OfNat.ofNat 0 \u03c9 \u2264 (-f) \u03c9", "state_after": "no goals"}, {"tactic": "have : \u03bc Set.univ = \u221e := by\n by_contra h\u03bc'\n exact h\u03bc (IsFiniteMeasure.mk <| lt_top_iff_ne_top.2 h\u03bc')", "annotated_tactic": ["have : \u03bc <a>Set.univ</a> = \u221e := by\n by_contra h\u03bc'\n exact h\u03bc (<a>IsFiniteMeasure.mk</a> <| <a>lt_top_iff_ne_top</a>.2 h\u03bc')", [{"full_name": "Set.univ", "def_path": "Mathlib/Init/Set.lean", "def_pos": [90, 5], "def_end_pos": [90, 9]}, {"full_name": "MeasureTheory.IsFiniteMeasure.mk", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2850, 23], "def_end_pos": [2850, 38]}, {"full_name": "lt_top_iff_ne_top", "def_path": "Mathlib/Order/BoundedOrder.lean", "def_pos": [173, 9], "def_end_pos": [173, 26]}]], "state_before": "case neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nr : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhfint : Integrable f\nhfint' : 0 \u2264 \u222b (x : \u03b1), f x \u2202\u03bc\nhf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 \u2191r\nhr : \u00acr = 0\nh\u03bc : \u00acIsFiniteMeasure \u03bc\n\u22a2 snorm f 1 \u03bc \u2264 2 * \u2191\u2191\u03bc univ * \u2191r", "state_after": "case neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nr : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhfint : Integrable f\nhfint' : 0 \u2264 \u222b (x : \u03b1), f x \u2202\u03bc\nhf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 \u2191r\nhr : \u00acr = 0\nh\u03bc : \u00acIsFiniteMeasure \u03bc\nthis : \u2191\u2191\u03bc univ = \u22a4\n\u22a2 snorm f 1 \u03bc \u2264 2 * \u2191\u2191\u03bc univ * \u2191r"}, {"tactic": "rw [this, ENNReal.mul_top', if_neg, ENNReal.top_mul', if_neg]", "annotated_tactic": ["rw [this, <a>ENNReal.mul_top'</a>, <a>if_neg</a>, <a>ENNReal.top_mul'</a>, <a>if_neg</a>]", [{"full_name": "ENNReal.mul_top'", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [577, 9], "def_end_pos": [577, 17]}, {"full_name": "if_neg", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [795, 9], "def_end_pos": [795, 15]}, {"full_name": "ENNReal.top_mul'", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [583, 9], "def_end_pos": [583, 17]}, {"full_name": "if_neg", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [795, 9], "def_end_pos": [795, 15]}]], "state_before": "case neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nr : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhfint : Integrable f\nhfint' : 0 \u2264 \u222b (x : \u03b1), f x \u2202\u03bc\nhf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 \u2191r\nhr : \u00acr = 0\nh\u03bc : \u00acIsFiniteMeasure \u03bc\nthis : \u2191\u2191\u03bc univ = \u22a4\n\u22a2 snorm f 1 \u03bc \u2264 2 * \u2191\u2191\u03bc univ * \u2191r", "state_after": "case neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nr : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhfint : Integrable f\nhfint' : 0 \u2264 \u222b (x : \u03b1), f x \u2202\u03bc\nhf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 \u2191r\nhr : \u00acr = 0\nh\u03bc : \u00acIsFiniteMeasure \u03bc\nthis : \u2191\u2191\u03bc univ = \u22a4\n\u22a2 snorm f 1 \u03bc \u2264 \u22a4\n\ncase neg.hnc\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nr : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhfint : Integrable f\nhfint' : 0 \u2264 \u222b (x : \u03b1), f x \u2202\u03bc\nhf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 \u2191r\nhr : \u00acr = 0\nh\u03bc : \u00acIsFiniteMeasure \u03bc\nthis : \u2191\u2191\u03bc univ = \u22a4\n\u22a2 \u00ac\u2191r = 0\n\ncase neg.hnc\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nr : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhfint : Integrable f\nhfint' : 0 \u2264 \u222b (x : \u03b1), f x \u2202\u03bc\nhf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 \u2191r\nhr : \u00acr = 0\nh\u03bc : \u00acIsFiniteMeasure \u03bc\nthis : \u2191\u2191\u03bc univ = \u22a4\n\u22a2 \u00ac2 = 0"}, {"tactic": "by_contra h\u03bc'", "annotated_tactic": ["by_contra h\u03bc'", []], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nr : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhfint : Integrable f\nhfint' : 0 \u2264 \u222b (x : \u03b1), f x \u2202\u03bc\nhf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 \u2191r\nhr : \u00acr = 0\nh\u03bc : \u00acIsFiniteMeasure \u03bc\n\u22a2 \u2191\u2191\u03bc univ = \u22a4", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nr : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhfint : Integrable f\nhfint' : 0 \u2264 \u222b (x : \u03b1), f x \u2202\u03bc\nhf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 \u2191r\nhr : \u00acr = 0\nh\u03bc : \u00acIsFiniteMeasure \u03bc\nh\u03bc' : \u00ac\u2191\u2191\u03bc univ = \u22a4\n\u22a2 False"}, {"tactic": "exact h\u03bc (IsFiniteMeasure.mk <| lt_top_iff_ne_top.2 h\u03bc')", "annotated_tactic": ["exact h\u03bc (<a>IsFiniteMeasure.mk</a> <| <a>lt_top_iff_ne_top</a>.2 h\u03bc')", [{"full_name": "MeasureTheory.IsFiniteMeasure.mk", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2850, 23], "def_end_pos": [2850, 38]}, {"full_name": "lt_top_iff_ne_top", "def_path": "Mathlib/Order/BoundedOrder.lean", "def_pos": [173, 9], "def_end_pos": [173, 26]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nr : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhfint : Integrable f\nhfint' : 0 \u2264 \u222b (x : \u03b1), f x \u2202\u03bc\nhf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 \u2191r\nhr : \u00acr = 0\nh\u03bc : \u00acIsFiniteMeasure \u03bc\nh\u03bc' : \u00ac\u2191\u2191\u03bc univ = \u22a4\n\u22a2 False", "state_after": "no goals"}, {"tactic": "exact le_top", "annotated_tactic": ["exact <a>le_top</a>", [{"full_name": "le_top", "def_path": "Mathlib/Order/BoundedOrder.lean", "def_pos": [98, 9], "def_end_pos": [98, 15]}]], "state_before": "case neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nr : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhfint : Integrable f\nhfint' : 0 \u2264 \u222b (x : \u03b1), f x \u2202\u03bc\nhf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 \u2191r\nhr : \u00acr = 0\nh\u03bc : \u00acIsFiniteMeasure \u03bc\nthis : \u2191\u2191\u03bc univ = \u22a4\n\u22a2 snorm f 1 \u03bc \u2264 \u22a4", "state_after": "no goals"}, {"tactic": "simp [hr]", "annotated_tactic": ["simp [hr]", []], "state_before": "case neg.hnc\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nr : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhfint : Integrable f\nhfint' : 0 \u2264 \u222b (x : \u03b1), f x \u2202\u03bc\nhf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 \u2191r\nhr : \u00acr = 0\nh\u03bc : \u00acIsFiniteMeasure \u03bc\nthis : \u2191\u2191\u03bc univ = \u22a4\n\u22a2 \u00ac\u2191r = 0", "state_after": "no goals"}, {"tactic": "norm_num", "annotated_tactic": ["norm_num", []], "state_before": "case neg.hnc\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nr : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhfint : Integrable f\nhfint' : 0 \u2264 \u222b (x : \u03b1), f x \u2202\u03bc\nhf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 \u2191r\nhr : \u00acr = 0\nh\u03bc : \u00acIsFiniteMeasure \u03bc\nthis : \u2191\u2191\u03bc univ = \u22a4\n\u22a2 \u00ac2 = 0", "state_after": "no goals"}, {"tactic": "rw [\u2190 integral_const]", "annotated_tactic": ["rw [\u2190 <a>integral_const</a>]", [{"full_name": "MeasureTheory.integral_const", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [1409, 9], "def_end_pos": [1409, 23]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nr : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhfint : Integrable f\nhfint' : \u222b (a : \u03b1), \u2191(Real.toNNReal (-f a)) \u2202\u03bc \u2264 \u222b (a : \u03b1), \u2191(Real.toNNReal (f a)) \u2202\u03bc\nhf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 \u2191r\nhr : \u00acr = 0\nh\u03bc this : IsFiniteMeasure \u03bc\n\u22a2 \u222b (\u03c9 : \u03b1), max (f \u03c9) 0 \u2202\u03bc \u2264 ENNReal.toReal (\u2191\u2191\u03bc univ) \u2022 \u2191r", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nr : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhfint : Integrable f\nhfint' : \u222b (a : \u03b1), \u2191(Real.toNNReal (-f a)) \u2202\u03bc \u2264 \u222b (a : \u03b1), \u2191(Real.toNNReal (f a)) \u2202\u03bc\nhf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 \u2191r\nhr : \u00acr = 0\nh\u03bc this : IsFiniteMeasure \u03bc\n\u22a2 \u222b (\u03c9 : \u03b1), max (f \u03c9) 0 \u2202\u03bc \u2264 \u222b (x : \u03b1), \u2191r \u2202\u03bc"}, {"tactic": "refine' integral_mono_ae hfint.real_toNNReal (integrable_const (r : \u211d)) _", "annotated_tactic": ["refine' <a>integral_mono_ae</a> hfint.real_toNNReal (<a>integrable_const</a> (r : \u211d)) _", [{"full_name": "MeasureTheory.integral_mono_ae", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [1342, 9], "def_end_pos": [1342, 25]}, {"full_name": "MeasureTheory.integrable_const", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [506, 9], "def_end_pos": [506, 25]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nr : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhfint : Integrable f\nhfint' : \u222b (a : \u03b1), \u2191(Real.toNNReal (-f a)) \u2202\u03bc \u2264 \u222b (a : \u03b1), \u2191(Real.toNNReal (f a)) \u2202\u03bc\nhf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 \u2191r\nhr : \u00acr = 0\nh\u03bc this : IsFiniteMeasure \u03bc\n\u22a2 \u222b (\u03c9 : \u03b1), max (f \u03c9) 0 \u2202\u03bc \u2264 \u222b (x : \u03b1), \u2191r \u2202\u03bc", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nr : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhfint : Integrable f\nhfint' : \u222b (a : \u03b1), \u2191(Real.toNNReal (-f a)) \u2202\u03bc \u2264 \u222b (a : \u03b1), \u2191(Real.toNNReal (f a)) \u2202\u03bc\nhf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 \u2191r\nhr : \u00acr = 0\nh\u03bc this : IsFiniteMeasure \u03bc\n\u22a2 (fun \u03c9 => max (f \u03c9) 0) \u2264\u1d50[\u03bc] fun x => \u2191r"}, {"tactic": "filter_upwards [hf] with \u03c9 h\u03c9 using Real.toNNReal_le_iff_le_coe.2 h\u03c9", "annotated_tactic": ["filter_upwards [hf] with \u03c9 h\u03c9 using <a>Real.toNNReal_le_iff_le_coe</a>.2 h\u03c9", [{"full_name": "Real.toNNReal_le_iff_le_coe", "def_path": "Mathlib/Data/Real/NNReal.lean", "def_pos": [685, 9], "def_end_pos": [685, 31]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nr : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhfint : Integrable f\nhfint' : \u222b (a : \u03b1), \u2191(Real.toNNReal (-f a)) \u2202\u03bc \u2264 \u222b (a : \u03b1), \u2191(Real.toNNReal (f a)) \u2202\u03bc\nhf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 \u2191r\nhr : \u00acr = 0\nh\u03bc this : IsFiniteMeasure \u03bc\n\u22a2 (fun \u03c9 => max (f \u03c9) 0) \u2264\u1d50[\u03bc] fun x => \u2191r", "state_after": "no goals"}, {"tactic": "simp only [Real.coe_toNNReal', ENNReal.toReal_mul, ENNReal.one_toReal, ENNReal.coe_toReal,\n ge_iff_le, Left.nonneg_neg_iff, Left.neg_nonpos_iff, toReal_ofNat] at hfint' \u22a2", "annotated_tactic": ["simp only [<a>Real.coe_toNNReal'</a>, <a>ENNReal.toReal_mul</a>, <a>ENNReal.one_toReal</a>, <a>ENNReal.coe_toReal</a>,\n <a>ge_iff_le</a>, <a>Left.nonneg_neg_iff</a>, <a>Left.neg_nonpos_iff</a>, <a>toReal_ofNat</a>] at hfint' \u22a2", [{"full_name": "Real.coe_toNNReal'", "def_path": "Mathlib/Data/Real/NNReal.lean", "def_pos": [615, 9], "def_end_pos": [615, 22]}, {"full_name": "ENNReal.toReal_mul", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [2296, 9], "def_end_pos": [2296, 19]}, {"full_name": "ENNReal.one_toReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [230, 17], "def_end_pos": [230, 27]}, {"full_name": "ENNReal.coe_toReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [236, 17], "def_end_pos": [236, 27]}, {"full_name": "ge_iff_le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [359, 9], "def_end_pos": [359, 18]}, {"full_name": "Left.nonneg_neg_iff", "def_path": "Mathlib/Algebra/Order/Group/Defs.lean", "def_pos": [105, 3], "def_end_pos": [105, 14]}, {"full_name": "Left.neg_nonpos_iff", "def_path": "Mathlib/Algebra/Order/Group/Defs.lean", "def_pos": [97, 3], "def_end_pos": [97, 14]}, {"full_name": "ENNReal.toReal_ofNat", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [736, 17], "def_end_pos": [736, 29]}]], "state_before": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nr : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhfint : Integrable f\nhfint' : \u222b (a : \u03b1), \u2191(Real.toNNReal (-f a)) \u2202\u03bc \u2264 \u222b (a : \u03b1), \u2191(Real.toNNReal (f a)) \u2202\u03bc\nhf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 \u2191r\nhr : \u00acr = 0\nh\u03bc this : IsFiniteMeasure \u03bc\nhposbdd : \u222b (\u03c9 : \u03b1), max (f \u03c9) 0 \u2202\u03bc \u2264 ENNReal.toReal (\u2191\u2191\u03bc univ) \u2022 \u2191r\n\u22a2 \u222b (a : \u03b1), \u2191(Real.toNNReal (f a)) \u2202\u03bc + \u222b (a : \u03b1), \u2191(Real.toNNReal (-f a)) \u2202\u03bc \u2264 ENNReal.toReal (2 * \u2191\u2191\u03bc univ * \u2191r)", "state_after": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nr : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhfint : Integrable f\nhfint' : \u222b (a : \u03b1), max (-f a) 0 \u2202\u03bc \u2264 \u222b (a : \u03b1), max (f a) 0 \u2202\u03bc\nhf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 \u2191r\nhr : \u00acr = 0\nh\u03bc this : IsFiniteMeasure \u03bc\nhposbdd : \u222b (\u03c9 : \u03b1), max (f \u03c9) 0 \u2202\u03bc \u2264 ENNReal.toReal (\u2191\u2191\u03bc univ) \u2022 \u2191r\n\u22a2 \u222b (a : \u03b1), max (f a) 0 \u2202\u03bc + \u222b (a : \u03b1), max (-f a) 0 \u2202\u03bc \u2264 2 * ENNReal.toReal (\u2191\u2191\u03bc univ) * \u2191r"}, {"tactic": "refine' (add_le_add_left hfint' _).trans _", "annotated_tactic": ["refine' (<a>add_le_add_left</a> hfint' _).<a>trans</a> _", [{"full_name": "add_le_add_left", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [49, 15], "def_end_pos": [49, 30]}, {"full_name": "LE.le.trans", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [120, 7], "def_end_pos": [120, 18]}]], "state_before": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nr : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhfint : Integrable f\nhfint' : \u222b (a : \u03b1), max (-f a) 0 \u2202\u03bc \u2264 \u222b (a : \u03b1), max (f a) 0 \u2202\u03bc\nhf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 \u2191r\nhr : \u00acr = 0\nh\u03bc this : IsFiniteMeasure \u03bc\nhposbdd : \u222b (\u03c9 : \u03b1), max (f \u03c9) 0 \u2202\u03bc \u2264 ENNReal.toReal (\u2191\u2191\u03bc univ) \u2022 \u2191r\n\u22a2 \u222b (a : \u03b1), max (f a) 0 \u2202\u03bc + \u222b (a : \u03b1), max (-f a) 0 \u2202\u03bc \u2264 2 * ENNReal.toReal (\u2191\u2191\u03bc univ) * \u2191r", "state_after": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nr : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhfint : Integrable f\nhfint' : \u222b (a : \u03b1), max (-f a) 0 \u2202\u03bc \u2264 \u222b (a : \u03b1), max (f a) 0 \u2202\u03bc\nhf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 \u2191r\nhr : \u00acr = 0\nh\u03bc this : IsFiniteMeasure \u03bc\nhposbdd : \u222b (\u03c9 : \u03b1), max (f \u03c9) 0 \u2202\u03bc \u2264 ENNReal.toReal (\u2191\u2191\u03bc univ) \u2022 \u2191r\n\u22a2 \u222b (a : \u03b1), max (f a) 0 \u2202\u03bc + \u222b (a : \u03b1), max (f a) 0 \u2202\u03bc \u2264 2 * ENNReal.toReal (\u2191\u2191\u03bc univ) * \u2191r"}, {"tactic": "rwa [\u2190 two_mul, mul_assoc, mul_le_mul_left (two_pos : (0 : \u211d) < 2)]", "annotated_tactic": ["rwa [\u2190 <a>two_mul</a>, <a>mul_assoc</a>, <a>mul_le_mul_left</a> (<a>two_pos</a> : (0 : \u211d) < 2)]", [{"full_name": "two_mul", "def_path": "Mathlib/Algebra/Ring/Defs.lean", "def_pos": [177, 9], "def_end_pos": [177, 16]}, {"full_name": "mul_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [264, 9], "def_end_pos": [264, 18]}, {"full_name": "mul_le_mul_left", "def_path": "Mathlib/Algebra/Order/Ring/Lemmas.lean", "def_pos": [209, 9], "def_end_pos": [209, 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 pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nr : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhfint : Integrable f\nhfint' : \u222b (a : \u03b1), max (-f a) 0 \u2202\u03bc \u2264 \u222b (a : \u03b1), max (f a) 0 \u2202\u03bc\nhf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 \u2191r\nhr : \u00acr = 0\nh\u03bc this : IsFiniteMeasure \u03bc\nhposbdd : \u222b (\u03c9 : \u03b1), max (f \u03c9) 0 \u2202\u03bc \u2264 ENNReal.toReal (\u2191\u2191\u03bc univ) \u2022 \u2191r\n\u22a2 \u222b (a : \u03b1), max (f a) 0 \u2202\u03bc + \u222b (a : \u03b1), max (f a) 0 \u2202\u03bc \u2264 2 * ENNReal.toReal (\u2191\u2191\u03bc univ) * \u2191r", "state_after": "no goals"}, {"tactic": "exact hfint.neg.sup (integrable_zero _ _ \u03bc)", "annotated_tactic": ["exact hfint.neg.sup (<a>integrable_zero</a> _ _ \u03bc)", [{"full_name": "MeasureTheory.integrable_zero", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [662, 9], "def_end_pos": [662, 24]}]], "state_before": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nr : \u211d\u22650\nf : \u03b1 \u2192 \u211d\nhfint : Integrable f\nhfint' : \u222b (a : \u03b1), \u2191(Real.toNNReal (-f a)) \u2202\u03bc \u2264 \u222b (a : \u03b1), \u2191(Real.toNNReal (f a)) \u2202\u03bc\nhf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 \u2191r\nhr : \u00acr = 0\nh\u03bc this : IsFiniteMeasure \u03bc\nhposbdd : \u222b (\u03c9 : \u03b1), max (f \u03c9) 0 \u2202\u03bc \u2264 ENNReal.toReal (\u2191\u2191\u03bc univ) \u2022 \u2191r\n\u22a2 Integrable fun a => \u2191(Real.toNNReal (-f a))", "state_after": "no goals"}]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Data/Set/Intervals/Group.lean
|
Set.pairwise_disjoint_Ioo_zpow
|
[
221,
1
] |
[
223,
64
] |
[{"tactic": "simpa only [one_mul] using pairwise_disjoint_Ioo_mul_zpow 1 b", "annotated_tactic": ["simpa only [<a>one_mul</a>] using <a>pairwise_disjoint_Ioo_mul_zpow</a> 1 b", [{"full_name": "one_mul", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [464, 9], "def_end_pos": [464, 16]}, {"full_name": "Set.pairwise_disjoint_Ioo_mul_zpow", "def_path": "Mathlib/Data/Set/Intervals/Group.lean", "def_pos": [200, 9], "def_end_pos": [200, 39]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : OrderedCommGroup \u03b1\na b : \u03b1\n\u22a2 Pairwise (Disjoint on fun n => Ioo (b ^ n) (b ^ (n + 1)))", "state_after": "no goals"}]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Algebra/Hom/Freiman.lean
|
FreimanHom.cancel_right_on
|
[
262,
1
] |
[
264,
32
] |
[{"tactic": "simp [hf.cancel_right hf']", "annotated_tactic": ["simp [hf.cancel_right hf']", []], "state_before": "F : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\nG : Type u_6\ninst\u271d\u2075 : FunLike F \u03b1 fun x => \u03b2\ninst\u271d\u2074 : CommMonoid \u03b1\ninst\u271d\u00b3 : CommMonoid \u03b2\ninst\u271d\u00b2 : CommMonoid \u03b3\ninst\u271d\u00b9 : CommMonoid \u03b4\ninst\u271d : CommGroup G\nA : Set \u03b1\nB : Set \u03b2\nC : Set \u03b3\nn : \u2115\na b c d : \u03b1\ng\u2081 g\u2082 : B \u2192*[n] \u03b3\nf : A \u2192*[n] \u03b2\nhf : Set.SurjOn (\u2191f) A B\nhf' : Set.MapsTo (\u2191f) A B\n\u22a2 Set.EqOn (\u2191(FreimanHom.comp g\u2081 f hf')) (\u2191(FreimanHom.comp g\u2082 f hf')) A \u2194 Set.EqOn (\u2191g\u2081) (\u2191g\u2082) B", "state_after": "no goals"}]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Data/Real/ENNReal.lean
|
ENNReal.inv_three_add_inv_three
|
[
1775,
1
] |
[
1777,
79
] |
[{"tactic": "ring", "annotated_tactic": ["ring", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\n\u22a2 3\u207b\u00b9 + 3\u207b\u00b9 + 3\u207b\u00b9 = 3 * 3\u207b\u00b9", "state_after": "no goals"}, {"tactic": "decide", "annotated_tactic": ["decide", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\n\u22a2 3 \u2260 0", "state_after": "no goals"}]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Analysis/SpecialFunctions/Gaussian.lean
|
Real.Gamma_one_half_eq
|
[
326,
1
] |
[
339,
79
] |
[{"tactic": "rw [Gamma_eq_integral one_half_pos, \u2190 integral_comp_rpow_Ioi_of_pos zero_lt_two]", "annotated_tactic": ["rw [<a>Gamma_eq_integral</a> <a>one_half_pos</a>, \u2190 <a>integral_comp_rpow_Ioi_of_pos</a> <a>zero_lt_two</a>]", [{"full_name": "Real.Gamma_eq_integral", "def_path": "Mathlib/Analysis/SpecialFunctions/Gamma/Basic.lean", "def_pos": [474, 9], "def_end_pos": [474, 26]}, {"full_name": "one_half_pos", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [508, 9], "def_end_pos": [508, 21]}, {"full_name": "MeasureTheory.integral_comp_rpow_Ioi_of_pos", "def_path": "Mathlib/MeasureTheory/Integral/IntegralEqImproper.lean", "def_pos": [869, 9], "def_end_pos": [869, 38]}, {"full_name": "zero_lt_two", "def_path": "Mathlib/Algebra/Order/Monoid/NatCast.lean", "def_pos": [71, 15], "def_end_pos": [71, 26]}]], "state_before": "\u22a2 Gamma (1 / 2) = sqrt \u03c0", "state_after": "\u22a2 \u222b (x : \u211d) in Ioi 0, (2 * x ^ (2 - 1)) \u2022 (rexp (-x ^ 2) * (x ^ 2) ^ (1 / 2 - 1)) = sqrt \u03c0"}, {"tactic": "convert congr_arg (fun x : \u211d => 2 * x) (integral_gaussian_Ioi 1) using 1", "annotated_tactic": ["convert <a>congr_arg</a> (fun x : \u211d => 2 * x) (<a>integral_gaussian_Ioi</a> 1) using 1", [{"full_name": "congr_arg", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [43, 7], "def_end_pos": [43, 16]}, {"full_name": "integral_gaussian_Ioi", "def_path": "Mathlib/Analysis/SpecialFunctions/Gaussian.lean", "def_pos": [310, 9], "def_end_pos": [310, 30]}]], "state_before": "\u22a2 \u222b (x : \u211d) in Ioi 0, (2 * x ^ (2 - 1)) \u2022 (rexp (-x ^ 2) * (x ^ 2) ^ (1 / 2 - 1)) = sqrt \u03c0", "state_after": "case h.e'_2\n\n\u22a2 \u222b (x : \u211d) in Ioi 0, (2 * x ^ (2 - 1)) \u2022 (rexp (-x ^ 2) * (x ^ 2) ^ (1 / 2 - 1)) =\n 2 * \u222b (x : \u211d) in Ioi 0, rexp (-1 * x ^ 2)\n\ncase h.e'_3\n\n\u22a2 sqrt \u03c0 = 2 * (sqrt (\u03c0 / 1) / 2)"}, {"tactic": "rw [\u2190 integral_mul_left]", "annotated_tactic": ["rw [\u2190 <a>integral_mul_left</a>]", [{"full_name": "MeasureTheory.integral_mul_left", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [923, 9], "def_end_pos": [923, 26]}]], "state_before": "case h.e'_2\n\n\u22a2 \u222b (x : \u211d) in Ioi 0, (2 * x ^ (2 - 1)) \u2022 (rexp (-x ^ 2) * (x ^ 2) ^ (1 / 2 - 1)) =\n 2 * \u222b (x : \u211d) in Ioi 0, rexp (-1 * x ^ 2)", "state_after": "case h.e'_2\n\n\u22a2 \u222b (x : \u211d) in Ioi 0, (2 * x ^ (2 - 1)) \u2022 (rexp (-x ^ 2) * (x ^ 2) ^ (1 / 2 - 1)) =\n \u222b (a : \u211d) in Ioi 0, 2 * rexp (-1 * a ^ 2)"}, {"tactic": "refine' set_integral_congr measurableSet_Ioi fun x hx => _", "annotated_tactic": ["refine' <a>set_integral_congr</a> <a>measurableSet_Ioi</a> fun x hx => _", [{"full_name": "MeasureTheory.set_integral_congr", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [87, 9], "def_end_pos": [87, 27]}, {"full_name": "measurableSet_Ioi", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [579, 9], "def_end_pos": [579, 26]}]], "state_before": "case h.e'_2\n\n\u22a2 \u222b (x : \u211d) in Ioi 0, (2 * x ^ (2 - 1)) \u2022 (rexp (-x ^ 2) * (x ^ 2) ^ (1 / 2 - 1)) =\n \u222b (a : \u211d) in Ioi 0, 2 * rexp (-1 * a ^ 2)", "state_after": "case h.e'_2\nx : \u211d\nhx : x \u2208 Ioi 0\n\u22a2 (2 * x ^ (2 - 1)) \u2022 (rexp (-x ^ 2) * (x ^ 2) ^ (1 / 2 - 1)) = 2 * rexp (-1 * x ^ 2)"}, {"tactic": "have : (x ^ (2 : \u211d)) ^ (1 / (2 : \u211d) - 1) = x\u207b\u00b9 := by\n rw [\u2190 rpow_mul (le_of_lt hx)]\n norm_num\n rw [rpow_neg (le_of_lt hx), rpow_one]", "annotated_tactic": ["have : (x ^ (2 : \u211d)) ^ (1 / (2 : \u211d) - 1) = x\u207b\u00b9 := by\n rw [\u2190 <a>rpow_mul</a> (<a>le_of_lt</a> hx)]\n norm_num\n rw [<a>rpow_neg</a> (<a>le_of_lt</a> hx), <a>rpow_one</a>]", [{"full_name": "Real.rpow_mul", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/Real.lean", "def_pos": [317, 9], "def_end_pos": [317, 17]}, {"full_name": "le_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [110, 9], "def_end_pos": [110, 17]}, {"full_name": "Real.rpow_neg", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/Real.lean", "def_pos": [221, 9], "def_end_pos": [221, 17]}, {"full_name": "le_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [110, 9], "def_end_pos": [110, 17]}, {"full_name": "Real.rpow_one", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/Real.lean", "def_pos": [126, 9], "def_end_pos": [126, 17]}]], "state_before": "case h.e'_2\nx : \u211d\nhx : x \u2208 Ioi 0\n\u22a2 (2 * x ^ (2 - 1)) \u2022 (rexp (-x ^ 2) * (x ^ 2) ^ (1 / 2 - 1)) = 2 * rexp (-1 * x ^ 2)", "state_after": "case h.e'_2\nx : \u211d\nhx : x \u2208 Ioi 0\nthis : (x ^ 2) ^ (1 / 2 - 1) = x\u207b\u00b9\n\u22a2 (2 * x ^ (2 - 1)) \u2022 (rexp (-x ^ 2) * (x ^ 2) ^ (1 / 2 - 1)) = 2 * rexp (-1 * x ^ 2)"}, {"tactic": "rw [smul_eq_mul, this]", "annotated_tactic": ["rw [<a>smul_eq_mul</a>, this]", [{"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'_2\nx : \u211d\nhx : x \u2208 Ioi 0\nthis : (x ^ 2) ^ (1 / 2 - 1) = x\u207b\u00b9\n\u22a2 (2 * x ^ (2 - 1)) \u2022 (rexp (-x ^ 2) * (x ^ 2) ^ (1 / 2 - 1)) = 2 * rexp (-1 * x ^ 2)", "state_after": "case h.e'_2\nx : \u211d\nhx : x \u2208 Ioi 0\nthis : (x ^ 2) ^ (1 / 2 - 1) = x\u207b\u00b9\n\u22a2 2 * x ^ (2 - 1) * (rexp (-x ^ 2) * x\u207b\u00b9) = 2 * rexp (-1 * x ^ 2)"}, {"tactic": "field_simp [(ne_of_lt (show 0 < x from hx)).symm]", "annotated_tactic": ["field_simp [(<a>ne_of_lt</a> (show 0 < x from hx)).<a>symm</a>]", [{"full_name": "ne_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [101, 9], "def_end_pos": [101, 17]}, {"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.e'_2\nx : \u211d\nhx : x \u2208 Ioi 0\nthis : (x ^ 2) ^ (1 / 2 - 1) = x\u207b\u00b9\n\u22a2 2 * x ^ (2 - 1) * (rexp (-x ^ 2) * x\u207b\u00b9) = 2 * rexp (-1 * x ^ 2)", "state_after": "case h.e'_2\nx : \u211d\nhx : x \u2208 Ioi 0\nthis : (x ^ 2) ^ (1 / 2 - 1) = x\u207b\u00b9\n\u22a2 2 * x ^ (2 - 1) * rexp (-x ^ 2) = 2 * rexp (-x ^ 2) * x"}, {"tactic": "norm_num", "annotated_tactic": ["norm_num", []], "state_before": "case h.e'_2\nx : \u211d\nhx : x \u2208 Ioi 0\nthis : (x ^ 2) ^ (1 / 2 - 1) = x\u207b\u00b9\n\u22a2 2 * x ^ (2 - 1) * rexp (-x ^ 2) = 2 * rexp (-x ^ 2) * x", "state_after": "case h.e'_2\nx : \u211d\nhx : x \u2208 Ioi 0\nthis : (x ^ 2) ^ (1 / 2 - 1) = x\u207b\u00b9\n\u22a2 2 * x * rexp (-x ^ 2) = 2 * rexp (-x ^ 2) * x"}, {"tactic": "ring", "annotated_tactic": ["ring", []], "state_before": "case h.e'_2\nx : \u211d\nhx : x \u2208 Ioi 0\nthis : (x ^ 2) ^ (1 / 2 - 1) = x\u207b\u00b9\n\u22a2 2 * x * rexp (-x ^ 2) = 2 * rexp (-x ^ 2) * x", "state_after": "no goals"}, {"tactic": "rw [\u2190 rpow_mul (le_of_lt hx)]", "annotated_tactic": ["rw [\u2190 <a>rpow_mul</a> (<a>le_of_lt</a> hx)]", [{"full_name": "Real.rpow_mul", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/Real.lean", "def_pos": [317, 9], "def_end_pos": [317, 17]}, {"full_name": "le_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [110, 9], "def_end_pos": [110, 17]}]], "state_before": "x : \u211d\nhx : x \u2208 Ioi 0\n\u22a2 (x ^ 2) ^ (1 / 2 - 1) = x\u207b\u00b9", "state_after": "x : \u211d\nhx : x \u2208 Ioi 0\n\u22a2 x ^ (2 * (1 / 2 - 1)) = x\u207b\u00b9"}, {"tactic": "norm_num", "annotated_tactic": ["norm_num", []], "state_before": "x : \u211d\nhx : x \u2208 Ioi 0\n\u22a2 x ^ (2 * (1 / 2 - 1)) = x\u207b\u00b9", "state_after": "x : \u211d\nhx : x \u2208 Ioi 0\n\u22a2 x ^ (-1) = x\u207b\u00b9"}, {"tactic": "rw [rpow_neg (le_of_lt hx), rpow_one]", "annotated_tactic": ["rw [<a>rpow_neg</a> (<a>le_of_lt</a> hx), <a>rpow_one</a>]", [{"full_name": "Real.rpow_neg", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/Real.lean", "def_pos": [221, 9], "def_end_pos": [221, 17]}, {"full_name": "le_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [110, 9], "def_end_pos": [110, 17]}, {"full_name": "Real.rpow_one", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/Real.lean", "def_pos": [126, 9], "def_end_pos": [126, 17]}]], "state_before": "x : \u211d\nhx : x \u2208 Ioi 0\n\u22a2 x ^ (-1) = x\u207b\u00b9", "state_after": "no goals"}, {"tactic": "rw [div_one, \u2190 mul_div_assoc, mul_comm, mul_div_cancel _ (two_ne_zero' \u211d)]", "annotated_tactic": ["rw [<a>div_one</a>, \u2190 <a>mul_div_assoc</a>, <a>mul_comm</a>, <a>mul_div_cancel</a> _ (<a>two_ne_zero'</a> \u211d)]", [{"full_name": "div_one", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [340, 9], "def_end_pos": [340, 16]}, {"full_name": "mul_div_assoc", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [306, 9], "def_end_pos": [306, 22]}, {"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [302, 9], "def_end_pos": [302, 17]}, {"full_name": "mul_div_cancel", "def_path": "Mathlib/Algebra/GroupWithZero/Units/Lemmas.lean", "def_pos": [71, 9], "def_end_pos": [71, 23]}, {"full_name": "two_ne_zero'", "def_path": "Mathlib/Algebra/NeZero.lean", "def_pos": [81, 7], "def_end_pos": [81, 19]}]], "state_before": "case h.e'_3\n\n\u22a2 sqrt \u03c0 = 2 * (sqrt (\u03c0 / 1) / 2)", "state_after": "no goals"}]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/ModelTheory/Syntax.lean
|
FirstOrder.Language.BoundedFormula.not_all_isAtomic
|
[
692,
1
] |
[
693,
12
] |
[{"tactic": "cases con", "annotated_tactic": ["cases con", []], "state_before": "L : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nn l : \u2115\n\u03c6\u271d \u03c8 : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv : \u03b1 \u2192 M\nxs : Fin l \u2192 M\n\u03c6 : BoundedFormula L \u03b1 (n + 1)\ncon : IsAtomic (all \u03c6)\n\u22a2 False", "state_after": "no goals"}]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Topology/MetricSpace/Isometry.lean
|
Isometry.preimage_setOf_dist
|
[
231,
1
] |
[
234,
20
] |
[{"tactic": "ext y", "annotated_tactic": ["ext y", []], "state_before": "\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : Isometry f\nx : \u03b1\np : \u211d \u2192 Prop\n\u22a2 f \u207b\u00b9' {y | p (dist y (f x))} = {y | p (dist y x)}", "state_after": "case h\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : Isometry f\nx : \u03b1\np : \u211d \u2192 Prop\ny : \u03b1\n\u22a2 y \u2208 f \u207b\u00b9' {y | p (dist y (f x))} \u2194 y \u2208 {y | p (dist y x)}"}, {"tactic": "simp [hf.dist_eq]", "annotated_tactic": ["simp [hf.dist_eq]", []], "state_before": "case h\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : Isometry f\nx : \u03b1\np : \u211d \u2192 Prop\ny : \u03b1\n\u22a2 y \u2208 f \u207b\u00b9' {y | p (dist y (f x))} \u2194 y \u2208 {y | p (dist y x)}", "state_after": "no goals"}]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/MeasureTheory/Group/Prod.lean
|
MeasureTheory.measure_eq_div_smul
|
[
352,
1
] |
[
356,
94
] |
[{"tactic": "ext1 t ht", "annotated_tactic": ["ext1 t ht", []], "state_before": "G : Type u_1\ninst\u271d\u2077 : MeasurableSpace G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : SigmaFinite \u03bc\ns : Set G\ninst\u271d\u00b2 : MeasurableInv G\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulLeftInvariant \u03bd\nhs : MeasurableSet s\nh2s : \u2191\u2191\u03bd s \u2260 0\nh3s : \u2191\u2191\u03bd s \u2260 \u22a4\n\u22a2 \u03bc = (\u2191\u2191\u03bc s / \u2191\u2191\u03bd s) \u2022 \u03bd", "state_after": "case h\nG : Type u_1\ninst\u271d\u2077 : MeasurableSpace G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : SigmaFinite \u03bc\ns : Set G\ninst\u271d\u00b2 : MeasurableInv G\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulLeftInvariant \u03bd\nhs : MeasurableSet s\nh2s : \u2191\u2191\u03bd s \u2260 0\nh3s : \u2191\u2191\u03bd s \u2260 \u22a4\nt : Set G\nht : MeasurableSet t\n\u22a2 \u2191\u2191\u03bc t = \u2191\u2191((\u2191\u2191\u03bc s / \u2191\u2191\u03bd s) \u2022 \u03bd) t"}, {"tactic": "rw [smul_apply, smul_eq_mul, mul_comm, \u2190 mul_div_assoc, mul_comm,\n measure_mul_measure_eq \u03bc \u03bd hs ht h2s h3s, mul_div_assoc, ENNReal.mul_div_cancel' h2s h3s]", "annotated_tactic": ["rw [<a>smul_apply</a>, <a>smul_eq_mul</a>, <a>mul_comm</a>, \u2190 <a>mul_div_assoc</a>, <a>mul_comm</a>,\n <a>measure_mul_measure_eq</a> \u03bc \u03bd hs ht h2s h3s, <a>mul_div_assoc</a>, <a>ENNReal.mul_div_cancel'</a> h2s h3s]", [{"full_name": "MeasureTheory.Measure.smul_apply", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [841, 9], "def_end_pos": [841, 19]}, {"full_name": "smul_eq_mul", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [93, 9], "def_end_pos": [93, 20]}, {"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [302, 9], "def_end_pos": [302, 17]}, {"full_name": "mul_div_assoc", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [306, 9], "def_end_pos": [306, 22]}, {"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [302, 9], "def_end_pos": [302, 17]}, {"full_name": "MeasureTheory.measure_mul_measure_eq", "def_path": "Mathlib/MeasureTheory/Group/Prod.lean", "def_pos": [336, 9], "def_end_pos": [336, 31]}, {"full_name": "mul_div_assoc", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [306, 9], "def_end_pos": [306, 22]}, {"full_name": "ENNReal.mul_div_cancel'", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1432, 19], "def_end_pos": [1432, 34]}]], "state_before": "case h\nG : Type u_1\ninst\u271d\u2077 : MeasurableSpace G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : SigmaFinite \u03bc\ns : Set G\ninst\u271d\u00b2 : MeasurableInv G\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulLeftInvariant \u03bd\nhs : MeasurableSet s\nh2s : \u2191\u2191\u03bd s \u2260 0\nh3s : \u2191\u2191\u03bd s \u2260 \u22a4\nt : Set G\nht : MeasurableSet t\n\u22a2 \u2191\u2191\u03bc t = \u2191\u2191((\u2191\u2191\u03bc s / \u2191\u2191\u03bd s) \u2022 \u03bd) t", "state_after": "no goals"}]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Topology/Category/TopCat/Limits/Products.lean
|
TopCat.range_prod_map
|
[
247,
1
] |
[
268,
16
] |
[{"tactic": "ext x", "annotated_tactic": ["ext x", []], "state_before": "J : Type v\ninst\u271d : SmallCategory J\nW X Y Z : TopCat\nf : W \u27f6 Y\ng : X \u27f6 Z\n\u22a2 Set.range \u2191(prod.map f g) = \u2191prod.fst \u207b\u00b9' Set.range \u2191f \u2229 \u2191prod.snd \u207b\u00b9' Set.range \u2191g", "state_after": "case h\nJ : Type v\ninst\u271d : SmallCategory J\nW X Y Z : TopCat\nf : W \u27f6 Y\ng : X \u27f6 Z\nx : (forget TopCat).obj (Y \u2a2f Z)\n\u22a2 x \u2208 Set.range \u2191(prod.map f g) \u2194 x \u2208 \u2191prod.fst \u207b\u00b9' Set.range \u2191f \u2229 \u2191prod.snd \u207b\u00b9' Set.range \u2191g"}, {"tactic": "constructor", "annotated_tactic": ["constructor", []], "state_before": "case h\nJ : Type v\ninst\u271d : SmallCategory J\nW X Y Z : TopCat\nf : W \u27f6 Y\ng : X \u27f6 Z\nx : (forget TopCat).obj (Y \u2a2f Z)\n\u22a2 x \u2208 Set.range \u2191(prod.map f g) \u2194 x \u2208 \u2191prod.fst \u207b\u00b9' Set.range \u2191f \u2229 \u2191prod.snd \u207b\u00b9' Set.range \u2191g", "state_after": "case h.mp\nJ : Type v\ninst\u271d : SmallCategory J\nW X Y Z : TopCat\nf : W \u27f6 Y\ng : X \u27f6 Z\nx : (forget TopCat).obj (Y \u2a2f Z)\n\u22a2 x \u2208 Set.range \u2191(prod.map f g) \u2192 x \u2208 \u2191prod.fst \u207b\u00b9' Set.range \u2191f \u2229 \u2191prod.snd \u207b\u00b9' Set.range \u2191g\n\ncase h.mpr\nJ : Type v\ninst\u271d : SmallCategory J\nW X Y Z : TopCat\nf : W \u27f6 Y\ng : X \u27f6 Z\nx : (forget TopCat).obj (Y \u2a2f Z)\n\u22a2 x \u2208 \u2191prod.fst \u207b\u00b9' Set.range \u2191f \u2229 \u2191prod.snd \u207b\u00b9' Set.range \u2191g \u2192 x \u2208 Set.range \u2191(prod.map f g)"}, {"tactic": "rintro \u27e8y, rfl\u27e9", "annotated_tactic": ["rintro \u27e8y, rfl\u27e9", []], "state_before": "case h.mp\nJ : Type v\ninst\u271d : SmallCategory J\nW X Y Z : TopCat\nf : W \u27f6 Y\ng : X \u27f6 Z\nx : (forget TopCat).obj (Y \u2a2f Z)\n\u22a2 x \u2208 Set.range \u2191(prod.map f g) \u2192 x \u2208 \u2191prod.fst \u207b\u00b9' Set.range \u2191f \u2229 \u2191prod.snd \u207b\u00b9' Set.range \u2191g", "state_after": "case h.mp.intro\nJ : Type v\ninst\u271d : SmallCategory J\nW X Y Z : TopCat\nf : W \u27f6 Y\ng : X \u27f6 Z\ny : (forget TopCat).obj (W \u2a2f X)\n\u22a2 \u2191(prod.map f g) y \u2208 \u2191prod.fst \u207b\u00b9' Set.range \u2191f \u2229 \u2191prod.snd \u207b\u00b9' Set.range \u2191g"}, {"tactic": "simp only [Set.mem_preimage, Set.mem_range, Set.mem_inter_iff, \u2190 comp_apply]", "annotated_tactic": ["simp only [<a>Set.mem_preimage</a>, <a>Set.mem_range</a>, <a>Set.mem_inter_iff</a>, \u2190 <a>comp_apply</a>]", [{"full_name": "Set.mem_preimage", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [64, 9], "def_end_pos": [64, 21]}, {"full_name": "Set.mem_range", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [673, 9], "def_end_pos": [673, 18]}, {"full_name": "Set.mem_inter_iff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [909, 9], "def_end_pos": [909, 22]}, {"full_name": "CategoryTheory.comp_apply", "def_path": "Mathlib/CategoryTheory/ConcreteCategory/Basic.lean", "def_pos": [137, 17], "def_end_pos": [137, 27]}]], "state_before": "case h.mp.intro\nJ : Type v\ninst\u271d : SmallCategory J\nW X Y Z : TopCat\nf : W \u27f6 Y\ng : X \u27f6 Z\ny : (forget TopCat).obj (W \u2a2f X)\n\u22a2 \u2191(prod.map f g) y \u2208 \u2191prod.fst \u207b\u00b9' Set.range \u2191f \u2229 \u2191prod.snd \u207b\u00b9' Set.range \u2191g", "state_after": "case h.mp.intro\nJ : Type v\ninst\u271d : SmallCategory J\nW X Y Z : TopCat\nf : W \u27f6 Y\ng : X \u27f6 Z\ny : (forget TopCat).obj (W \u2a2f X)\n\u22a2 (\u2203 y_1, \u2191f y_1 = \u2191(prod.map f g \u226b prod.fst) y) \u2227 \u2203 y_1, \u2191g y_1 = \u2191(prod.map f g \u226b prod.snd) y"}, {"tactic": "simp only [Limits.prod.map_fst, Limits.prod.map_snd, exists_apply_eq_apply, comp_apply,\n and_self_iff]", "annotated_tactic": ["simp only [<a>Limits.prod.map_fst</a>, <a>Limits.prod.map_snd</a>, <a>exists_apply_eq_apply</a>, <a>comp_apply</a>,\n <a>and_self_iff</a>]", [{"full_name": "CategoryTheory.Limits.prod.map_fst", "def_path": "Mathlib/CategoryTheory/Limits/Shapes/BinaryProducts.lean", "def_pos": [718, 9], "def_end_pos": [718, 21]}, {"full_name": "CategoryTheory.Limits.prod.map_snd", "def_path": "Mathlib/CategoryTheory/Limits/Shapes/BinaryProducts.lean", "def_pos": [725, 9], "def_end_pos": [725, 21]}, {"full_name": "exists_apply_eq_apply", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [488, 17], "def_end_pos": [488, 38]}, {"full_name": "CategoryTheory.comp_apply", "def_path": "Mathlib/CategoryTheory/ConcreteCategory/Basic.lean", "def_pos": [137, 17], "def_end_pos": [137, 27]}, {"full_name": "and_self_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [155, 9], "def_end_pos": [155, 21]}]], "state_before": "case h.mp.intro\nJ : Type v\ninst\u271d : SmallCategory J\nW X Y Z : TopCat\nf : W \u27f6 Y\ng : X \u27f6 Z\ny : (forget TopCat).obj (W \u2a2f X)\n\u22a2 (\u2203 y_1, \u2191f y_1 = \u2191(prod.map f g \u226b prod.fst) y) \u2227 \u2203 y_1, \u2191g y_1 = \u2191(prod.map f g \u226b prod.snd) y", "state_after": "no goals"}, {"tactic": "rintro \u27e8\u27e8x\u2081, hx\u2081\u27e9, \u27e8x\u2082, hx\u2082\u27e9\u27e9", "annotated_tactic": ["rintro \u27e8\u27e8x\u2081, hx\u2081\u27e9, \u27e8x\u2082, hx\u2082\u27e9\u27e9", []], "state_before": "case h.mpr\nJ : Type v\ninst\u271d : SmallCategory J\nW X Y Z : TopCat\nf : W \u27f6 Y\ng : X \u27f6 Z\nx : (forget TopCat).obj (Y \u2a2f Z)\n\u22a2 x \u2208 \u2191prod.fst \u207b\u00b9' Set.range \u2191f \u2229 \u2191prod.snd \u207b\u00b9' Set.range \u2191g \u2192 x \u2208 Set.range \u2191(prod.map f g)", "state_after": "case h.mpr.intro.intro.intro\nJ : Type v\ninst\u271d : SmallCategory J\nW X Y Z : TopCat\nf : W \u27f6 Y\ng : X \u27f6 Z\nx : (forget TopCat).obj (Y \u2a2f Z)\nx\u2081 : (forget TopCat).obj W\nhx\u2081 : \u2191f x\u2081 = \u2191prod.fst x\nx\u2082 : (forget TopCat).obj X\nhx\u2082 : \u2191g x\u2082 = \u2191prod.snd x\n\u22a2 x \u2208 Set.range \u2191(prod.map f g)"}, {"tactic": "use (prodIsoProd W X).inv (x\u2081, x\u2082)", "annotated_tactic": ["use (<a>prodIsoProd</a> W X).<a>inv</a> (x\u2081, x\u2082)", [{"full_name": "TopCat.prodIsoProd", "def_path": "Mathlib/Topology/Category/TopCat/Limits/Products.lean", "def_pos": [197, 5], "def_end_pos": [197, 16]}, {"full_name": "CategoryTheory.Iso.inv", "def_path": "Mathlib/CategoryTheory/Iso.lean", "def_pos": [55, 3], "def_end_pos": [55, 6]}]], "state_before": "case h.mpr.intro.intro.intro\nJ : Type v\ninst\u271d : SmallCategory J\nW X Y Z : TopCat\nf : W \u27f6 Y\ng : X \u27f6 Z\nx : (forget TopCat).obj (Y \u2a2f Z)\nx\u2081 : (forget TopCat).obj W\nhx\u2081 : \u2191f x\u2081 = \u2191prod.fst x\nx\u2082 : (forget TopCat).obj X\nhx\u2082 : \u2191g x\u2082 = \u2191prod.snd x\n\u22a2 x \u2208 Set.range \u2191(prod.map f g)", "state_after": "case h\nJ : Type v\ninst\u271d : SmallCategory J\nW X Y Z : TopCat\nf : W \u27f6 Y\ng : X \u27f6 Z\nx : (forget TopCat).obj (Y \u2a2f Z)\nx\u2081 : (forget TopCat).obj W\nhx\u2081 : \u2191f x\u2081 = \u2191prod.fst x\nx\u2082 : (forget TopCat).obj X\nhx\u2082 : \u2191g x\u2082 = \u2191prod.snd x\n\u22a2 \u2191(prod.map f g) (\u2191(prodIsoProd W X).inv (x\u2081, x\u2082)) = x"}, {"tactic": "apply Concrete.limit_ext", "annotated_tactic": ["apply <a>Concrete.limit_ext</a>", [{"full_name": "CategoryTheory.Limits.Concrete.limit_ext", "def_path": "Mathlib/CategoryTheory/Limits/ConcreteCategory.lean", "def_pos": [55, 9], "def_end_pos": [55, 27]}]], "state_before": "case h\nJ : Type v\ninst\u271d : SmallCategory J\nW X Y Z : TopCat\nf : W \u27f6 Y\ng : X \u27f6 Z\nx : (forget TopCat).obj (Y \u2a2f Z)\nx\u2081 : (forget TopCat).obj W\nhx\u2081 : \u2191f x\u2081 = \u2191prod.fst x\nx\u2082 : (forget TopCat).obj X\nhx\u2082 : \u2191g x\u2082 = \u2191prod.snd x\n\u22a2 \u2191(prod.map f g) (\u2191(prodIsoProd W X).inv (x\u2081, x\u2082)) = x", "state_after": "case h.a\nJ : Type v\ninst\u271d : SmallCategory J\nW X Y Z : TopCat\nf : W \u27f6 Y\ng : X \u27f6 Z\nx : (forget TopCat).obj (Y \u2a2f Z)\nx\u2081 : (forget TopCat).obj W\nhx\u2081 : \u2191f x\u2081 = \u2191prod.fst x\nx\u2082 : (forget TopCat).obj X\nhx\u2082 : \u2191g x\u2082 = \u2191prod.snd x\n\u22a2 \u2200 (j : Discrete WalkingPair),\n \u2191(limit.\u03c0 (pair Y Z) j) (\u2191(prod.map f g) (\u2191(prodIsoProd W X).inv (x\u2081, x\u2082))) = \u2191(limit.\u03c0 (pair Y Z) j) x"}, {"tactic": "rintro \u27e8\u27e8\u27e9\u27e9", "annotated_tactic": ["rintro \u27e8\u27e8\u27e9\u27e9", []], "state_before": "case h.a\nJ : Type v\ninst\u271d : SmallCategory J\nW X Y Z : TopCat\nf : W \u27f6 Y\ng : X \u27f6 Z\nx : (forget TopCat).obj (Y \u2a2f Z)\nx\u2081 : (forget TopCat).obj W\nhx\u2081 : \u2191f x\u2081 = \u2191prod.fst x\nx\u2082 : (forget TopCat).obj X\nhx\u2082 : \u2191g x\u2082 = \u2191prod.snd x\n\u22a2 \u2200 (j : Discrete WalkingPair),\n \u2191(limit.\u03c0 (pair Y Z) j) (\u2191(prod.map f g) (\u2191(prodIsoProd W X).inv (x\u2081, x\u2082))) = \u2191(limit.\u03c0 (pair Y Z) j) x", "state_after": "case h.a.mk.left\nJ : Type v\ninst\u271d : SmallCategory J\nW X Y Z : TopCat\nf : W \u27f6 Y\ng : X \u27f6 Z\nx : (forget TopCat).obj (Y \u2a2f Z)\nx\u2081 : (forget TopCat).obj W\nhx\u2081 : \u2191f x\u2081 = \u2191prod.fst x\nx\u2082 : (forget TopCat).obj X\nhx\u2082 : \u2191g x\u2082 = \u2191prod.snd x\n\u22a2 \u2191(limit.\u03c0 (pair Y Z) { as := WalkingPair.left }) (\u2191(prod.map f g) (\u2191(prodIsoProd W X).inv (x\u2081, x\u2082))) =\n \u2191(limit.\u03c0 (pair Y Z) { as := WalkingPair.left }) x\n\ncase h.a.mk.right\nJ : Type v\ninst\u271d : SmallCategory J\nW X Y Z : TopCat\nf : W \u27f6 Y\ng : X \u27f6 Z\nx : (forget TopCat).obj (Y \u2a2f Z)\nx\u2081 : (forget TopCat).obj W\nhx\u2081 : \u2191f x\u2081 = \u2191prod.fst x\nx\u2082 : (forget TopCat).obj X\nhx\u2082 : \u2191g x\u2082 = \u2191prod.snd x\n\u22a2 \u2191(limit.\u03c0 (pair Y Z) { as := WalkingPair.right }) (\u2191(prod.map f g) (\u2191(prodIsoProd W X).inv (x\u2081, x\u2082))) =\n \u2191(limit.\u03c0 (pair Y Z) { as := WalkingPair.right }) x"}, {"tactic": "simp only [\u2190 comp_apply, Category.assoc]", "annotated_tactic": ["simp only [\u2190 <a>comp_apply</a>, <a>Category.assoc</a>]", [{"full_name": "CategoryTheory.comp_apply", "def_path": "Mathlib/CategoryTheory/ConcreteCategory/Basic.lean", "def_pos": [137, 17], "def_end_pos": [137, 27]}, {"full_name": "CategoryTheory.Category.assoc", "def_path": "Mathlib/CategoryTheory/Category/Basic.lean", "def_pos": [159, 3], "def_end_pos": [159, 8]}]], "state_before": "case h.a.mk.left\nJ : Type v\ninst\u271d : SmallCategory J\nW X Y Z : TopCat\nf : W \u27f6 Y\ng : X \u27f6 Z\nx : (forget TopCat).obj (Y \u2a2f Z)\nx\u2081 : (forget TopCat).obj W\nhx\u2081 : \u2191f x\u2081 = \u2191prod.fst x\nx\u2082 : (forget TopCat).obj X\nhx\u2082 : \u2191g x\u2082 = \u2191prod.snd x\n\u22a2 \u2191(limit.\u03c0 (pair Y Z) { as := WalkingPair.left }) (\u2191(prod.map f g) (\u2191(prodIsoProd W X).inv (x\u2081, x\u2082))) =\n \u2191(limit.\u03c0 (pair Y Z) { as := WalkingPair.left }) x", "state_after": "case h.a.mk.left\nJ : Type v\ninst\u271d : SmallCategory J\nW X Y Z : TopCat\nf : W \u27f6 Y\ng : X \u27f6 Z\nx : (forget TopCat).obj (Y \u2a2f Z)\nx\u2081 : (forget TopCat).obj W\nhx\u2081 : \u2191f x\u2081 = \u2191prod.fst x\nx\u2082 : (forget TopCat).obj X\nhx\u2082 : \u2191g x\u2082 = \u2191prod.snd x\n\u22a2 \u2191((prodIsoProd W X).inv \u226b prod.map f g \u226b limit.\u03c0 (pair Y Z) { as := WalkingPair.left }) (x\u2081, x\u2082) =\n \u2191(limit.\u03c0 (pair Y Z) { as := WalkingPair.left }) x"}, {"tactic": "erw [Limits.prod.map_fst]", "annotated_tactic": ["erw [<a>Limits.prod.map_fst</a>]", [{"full_name": "CategoryTheory.Limits.prod.map_fst", "def_path": "Mathlib/CategoryTheory/Limits/Shapes/BinaryProducts.lean", "def_pos": [718, 9], "def_end_pos": [718, 21]}]], "state_before": "case h.a.mk.left\nJ : Type v\ninst\u271d : SmallCategory J\nW X Y Z : TopCat\nf : W \u27f6 Y\ng : X \u27f6 Z\nx : (forget TopCat).obj (Y \u2a2f Z)\nx\u2081 : (forget TopCat).obj W\nhx\u2081 : \u2191f x\u2081 = \u2191prod.fst x\nx\u2082 : (forget TopCat).obj X\nhx\u2082 : \u2191g x\u2082 = \u2191prod.snd x\n\u22a2 \u2191((prodIsoProd W X).inv \u226b prod.map f g \u226b limit.\u03c0 (pair Y Z) { as := WalkingPair.left }) (x\u2081, x\u2082) =\n \u2191(limit.\u03c0 (pair Y Z) { as := WalkingPair.left }) x", "state_after": "case h.a.mk.left\nJ : Type v\ninst\u271d : SmallCategory J\nW X Y Z : TopCat\nf : W \u27f6 Y\ng : X \u27f6 Z\nx : (forget TopCat).obj (Y \u2a2f Z)\nx\u2081 : (forget TopCat).obj W\nhx\u2081 : \u2191f x\u2081 = \u2191prod.fst x\nx\u2082 : (forget TopCat).obj X\nhx\u2082 : \u2191g x\u2082 = \u2191prod.snd x\n\u22a2 \u2191((prodIsoProd W X).inv \u226b prod.fst \u226b f) (x\u2081, x\u2082) = \u2191(limit.\u03c0 (pair Y Z) { as := WalkingPair.left }) x"}, {"tactic": "rw [TopCat.prodIsoProd_inv_fst_assoc,TopCat.comp_app]", "annotated_tactic": ["rw [<a>TopCat.prodIsoProd_inv_fst_assoc</a>,<a>TopCat.comp_app</a>]", [{"full_name": "TopCat.prodIsoProd_inv_fst_assoc", "def_path": "Mathlib/Topology/Category/TopCat/Limits/Products.lean", "def_pos": [226, 3], "def_end_pos": [226, 25]}, {"full_name": "TopCat.comp_app", "def_path": "Mathlib/Topology/Category/TopCat/Basic.lean", "def_pos": [72, 9], "def_end_pos": [72, 17]}]], "state_before": "case h.a.mk.left\nJ : Type v\ninst\u271d : SmallCategory J\nW X Y Z : TopCat\nf : W \u27f6 Y\ng : X \u27f6 Z\nx : (forget TopCat).obj (Y \u2a2f Z)\nx\u2081 : (forget TopCat).obj W\nhx\u2081 : \u2191f x\u2081 = \u2191prod.fst x\nx\u2082 : (forget TopCat).obj X\nhx\u2082 : \u2191g x\u2082 = \u2191prod.snd x\n\u22a2 \u2191((prodIsoProd W X).inv \u226b prod.fst \u226b f) (x\u2081, x\u2082) = \u2191(limit.\u03c0 (pair Y Z) { as := WalkingPair.left }) x", "state_after": "case h.a.mk.left\nJ : Type v\ninst\u271d : SmallCategory J\nW X Y Z : TopCat\nf : W \u27f6 Y\ng : X \u27f6 Z\nx : (forget TopCat).obj (Y \u2a2f Z)\nx\u2081 : (forget TopCat).obj W\nhx\u2081 : \u2191f x\u2081 = \u2191prod.fst x\nx\u2082 : (forget TopCat).obj X\nhx\u2082 : \u2191g x\u2082 = \u2191prod.snd x\n\u22a2 \u2191f (\u2191prodFst (x\u2081, x\u2082)) = \u2191(limit.\u03c0 (pair Y Z) { as := WalkingPair.left }) x"}, {"tactic": "exact hx\u2081", "annotated_tactic": ["exact hx\u2081", []], "state_before": "case h.a.mk.left\nJ : Type v\ninst\u271d : SmallCategory J\nW X Y Z : TopCat\nf : W \u27f6 Y\ng : X \u27f6 Z\nx : (forget TopCat).obj (Y \u2a2f Z)\nx\u2081 : (forget TopCat).obj W\nhx\u2081 : \u2191f x\u2081 = \u2191prod.fst x\nx\u2082 : (forget TopCat).obj X\nhx\u2082 : \u2191g x\u2082 = \u2191prod.snd x\n\u22a2 \u2191f (\u2191prodFst (x\u2081, x\u2082)) = \u2191(limit.\u03c0 (pair Y Z) { as := WalkingPair.left }) x", "state_after": "no goals"}, {"tactic": "simp only [\u2190 comp_apply, Category.assoc]", "annotated_tactic": ["simp only [\u2190 <a>comp_apply</a>, <a>Category.assoc</a>]", [{"full_name": "CategoryTheory.comp_apply", "def_path": "Mathlib/CategoryTheory/ConcreteCategory/Basic.lean", "def_pos": [137, 17], "def_end_pos": [137, 27]}, {"full_name": "CategoryTheory.Category.assoc", "def_path": "Mathlib/CategoryTheory/Category/Basic.lean", "def_pos": [159, 3], "def_end_pos": [159, 8]}]], "state_before": "case h.a.mk.right\nJ : Type v\ninst\u271d : SmallCategory J\nW X Y Z : TopCat\nf : W \u27f6 Y\ng : X \u27f6 Z\nx : (forget TopCat).obj (Y \u2a2f Z)\nx\u2081 : (forget TopCat).obj W\nhx\u2081 : \u2191f x\u2081 = \u2191prod.fst x\nx\u2082 : (forget TopCat).obj X\nhx\u2082 : \u2191g x\u2082 = \u2191prod.snd x\n\u22a2 \u2191(limit.\u03c0 (pair Y Z) { as := WalkingPair.right }) (\u2191(prod.map f g) (\u2191(prodIsoProd W X).inv (x\u2081, x\u2082))) =\n \u2191(limit.\u03c0 (pair Y Z) { as := WalkingPair.right }) x", "state_after": "case h.a.mk.right\nJ : Type v\ninst\u271d : SmallCategory J\nW X Y Z : TopCat\nf : W \u27f6 Y\ng : X \u27f6 Z\nx : (forget TopCat).obj (Y \u2a2f Z)\nx\u2081 : (forget TopCat).obj W\nhx\u2081 : \u2191f x\u2081 = \u2191prod.fst x\nx\u2082 : (forget TopCat).obj X\nhx\u2082 : \u2191g x\u2082 = \u2191prod.snd x\n\u22a2 \u2191((prodIsoProd W X).inv \u226b prod.map f g \u226b limit.\u03c0 (pair Y Z) { as := WalkingPair.right }) (x\u2081, x\u2082) =\n \u2191(limit.\u03c0 (pair Y Z) { as := WalkingPair.right }) x"}, {"tactic": "erw [Limits.prod.map_snd]", "annotated_tactic": ["erw [<a>Limits.prod.map_snd</a>]", [{"full_name": "CategoryTheory.Limits.prod.map_snd", "def_path": "Mathlib/CategoryTheory/Limits/Shapes/BinaryProducts.lean", "def_pos": [725, 9], "def_end_pos": [725, 21]}]], "state_before": "case h.a.mk.right\nJ : Type v\ninst\u271d : SmallCategory J\nW X Y Z : TopCat\nf : W \u27f6 Y\ng : X \u27f6 Z\nx : (forget TopCat).obj (Y \u2a2f Z)\nx\u2081 : (forget TopCat).obj W\nhx\u2081 : \u2191f x\u2081 = \u2191prod.fst x\nx\u2082 : (forget TopCat).obj X\nhx\u2082 : \u2191g x\u2082 = \u2191prod.snd x\n\u22a2 \u2191((prodIsoProd W X).inv \u226b prod.map f g \u226b limit.\u03c0 (pair Y Z) { as := WalkingPair.right }) (x\u2081, x\u2082) =\n \u2191(limit.\u03c0 (pair Y Z) { as := WalkingPair.right }) x", "state_after": "case h.a.mk.right\nJ : Type v\ninst\u271d : SmallCategory J\nW X Y Z : TopCat\nf : W \u27f6 Y\ng : X \u27f6 Z\nx : (forget TopCat).obj (Y \u2a2f Z)\nx\u2081 : (forget TopCat).obj W\nhx\u2081 : \u2191f x\u2081 = \u2191prod.fst x\nx\u2082 : (forget TopCat).obj X\nhx\u2082 : \u2191g x\u2082 = \u2191prod.snd x\n\u22a2 \u2191((prodIsoProd W X).inv \u226b prod.snd \u226b g) (x\u2081, x\u2082) = \u2191(limit.\u03c0 (pair Y Z) { as := WalkingPair.right }) x"}, {"tactic": "rw [TopCat.prodIsoProd_inv_snd_assoc,TopCat.comp_app]", "annotated_tactic": ["rw [<a>TopCat.prodIsoProd_inv_snd_assoc</a>,<a>TopCat.comp_app</a>]", [{"full_name": "TopCat.prodIsoProd_inv_snd_assoc", "def_path": "Mathlib/Topology/Category/TopCat/Limits/Products.lean", "def_pos": [231, 3], "def_end_pos": [231, 25]}, {"full_name": "TopCat.comp_app", "def_path": "Mathlib/Topology/Category/TopCat/Basic.lean", "def_pos": [72, 9], "def_end_pos": [72, 17]}]], "state_before": "case h.a.mk.right\nJ : Type v\ninst\u271d : SmallCategory J\nW X Y Z : TopCat\nf : W \u27f6 Y\ng : X \u27f6 Z\nx : (forget TopCat).obj (Y \u2a2f Z)\nx\u2081 : (forget TopCat).obj W\nhx\u2081 : \u2191f x\u2081 = \u2191prod.fst x\nx\u2082 : (forget TopCat).obj X\nhx\u2082 : \u2191g x\u2082 = \u2191prod.snd x\n\u22a2 \u2191((prodIsoProd W X).inv \u226b prod.snd \u226b g) (x\u2081, x\u2082) = \u2191(limit.\u03c0 (pair Y Z) { as := WalkingPair.right }) x", "state_after": "case h.a.mk.right\nJ : Type v\ninst\u271d : SmallCategory J\nW X Y Z : TopCat\nf : W \u27f6 Y\ng : X \u27f6 Z\nx : (forget TopCat).obj (Y \u2a2f Z)\nx\u2081 : (forget TopCat).obj W\nhx\u2081 : \u2191f x\u2081 = \u2191prod.fst x\nx\u2082 : (forget TopCat).obj X\nhx\u2082 : \u2191g x\u2082 = \u2191prod.snd x\n\u22a2 \u2191g (\u2191prodSnd (x\u2081, x\u2082)) = \u2191(limit.\u03c0 (pair Y Z) { as := WalkingPair.right }) x"}, {"tactic": "exact hx\u2082", "annotated_tactic": ["exact hx\u2082", []], "state_before": "case h.a.mk.right\nJ : Type v\ninst\u271d : SmallCategory J\nW X Y Z : TopCat\nf : W \u27f6 Y\ng : X \u27f6 Z\nx : (forget TopCat).obj (Y \u2a2f Z)\nx\u2081 : (forget TopCat).obj W\nhx\u2081 : \u2191f x\u2081 = \u2191prod.fst x\nx\u2082 : (forget TopCat).obj X\nhx\u2082 : \u2191g x\u2082 = \u2191prod.snd x\n\u22a2 \u2191g (\u2191prodSnd (x\u2081, x\u2082)) = \u2191(limit.\u03c0 (pair Y Z) { as := WalkingPair.right }) x", "state_after": "no goals"}]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Data/QPF/Multivariate/Basic.lean
|
MvQPF.liftP_iff
|
[
126,
1
] |
[
138,
26
] |
[{"tactic": "constructor", "annotated_tactic": ["constructor", []], "state_before": "n : \u2115\nF : TypeVec.{u} n \u2192 Type u_1\ninst\u271d : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\np : \u2983i : Fin2 n\u2984 \u2192 \u03b1 i \u2192 Prop\nx : F \u03b1\n\u22a2 LiftP p x \u2194 \u2203 a f, x = abs { fst := a, snd := f } \u2227 \u2200 (i : Fin2 n) (j : MvPFunctor.B (P F) a i), p (f i j)", "state_after": "case mp\nn : \u2115\nF : TypeVec.{u} n \u2192 Type u_1\ninst\u271d : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\np : \u2983i : Fin2 n\u2984 \u2192 \u03b1 i \u2192 Prop\nx : F \u03b1\n\u22a2 LiftP p x \u2192 \u2203 a f, x = abs { fst := a, snd := f } \u2227 \u2200 (i : Fin2 n) (j : MvPFunctor.B (P F) a i), p (f i j)\n\ncase mpr\nn : \u2115\nF : TypeVec.{u} n \u2192 Type u_1\ninst\u271d : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\np : \u2983i : Fin2 n\u2984 \u2192 \u03b1 i \u2192 Prop\nx : F \u03b1\n\u22a2 (\u2203 a f, x = abs { fst := a, snd := f } \u2227 \u2200 (i : Fin2 n) (j : MvPFunctor.B (P F) a i), p (f i j)) \u2192 LiftP p x"}, {"tactic": "rintro \u27e8a, f, h\u2080, h\u2081\u27e9", "annotated_tactic": ["rintro \u27e8a, f, h\u2080, h\u2081\u27e9", []], "state_before": "case mpr\nn : \u2115\nF : TypeVec.{u} n \u2192 Type u_1\ninst\u271d : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\np : \u2983i : Fin2 n\u2984 \u2192 \u03b1 i \u2192 Prop\nx : F \u03b1\n\u22a2 (\u2203 a f, x = abs { fst := a, snd := f } \u2227 \u2200 (i : Fin2 n) (j : MvPFunctor.B (P F) a i), p (f i j)) \u2192 LiftP p x", "state_after": "case mpr.intro.intro.intro\nn : \u2115\nF : TypeVec.{u} n \u2192 Type u_1\ninst\u271d : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\np : \u2983i : Fin2 n\u2984 \u2192 \u03b1 i \u2192 Prop\nx : F \u03b1\na : (P F).A\nf : MvPFunctor.B (P F) a \u27f9 \u03b1\nh\u2080 : x = abs { fst := a, snd := f }\nh\u2081 : \u2200 (i : Fin2 n) (j : MvPFunctor.B (P F) a i), p (f i j)\n\u22a2 LiftP p x"}, {"tactic": "use abs \u27e8a, fun i j => \u27e8f i j, h\u2081 i j\u27e9\u27e9", "annotated_tactic": ["use <a>abs</a> \u27e8a, fun i j => \u27e8f i j, h\u2081 i j\u27e9\u27e9", [{"full_name": "MvQPF.abs", "def_path": "Mathlib/Data/QPF/Multivariate/Basic.lean", "def_pos": [87, 3], "def_end_pos": [87, 6]}]], "state_before": "case mpr.intro.intro.intro\nn : \u2115\nF : TypeVec.{u} n \u2192 Type u_1\ninst\u271d : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\np : \u2983i : Fin2 n\u2984 \u2192 \u03b1 i \u2192 Prop\nx : F \u03b1\na : (P F).A\nf : MvPFunctor.B (P F) a \u27f9 \u03b1\nh\u2080 : x = abs { fst := a, snd := f }\nh\u2081 : \u2200 (i : Fin2 n) (j : MvPFunctor.B (P F) a i), p (f i j)\n\u22a2 LiftP p x", "state_after": "case h\nn : \u2115\nF : TypeVec.{u} n \u2192 Type u_1\ninst\u271d : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\np : \u2983i : Fin2 n\u2984 \u2192 \u03b1 i \u2192 Prop\nx : F \u03b1\na : (P F).A\nf : MvPFunctor.B (P F) a \u27f9 \u03b1\nh\u2080 : x = abs { fst := a, snd := f }\nh\u2081 : \u2200 (i : Fin2 n) (j : MvPFunctor.B (P F) a i), p (f i j)\n\u22a2 (fun i => Subtype.val) <$$> abs { fst := a, snd := fun i j => { val := f i j, property := (_ : p (f i j)) } } = x"}, {"tactic": "rw [\u2190 abs_map, h\u2080]", "annotated_tactic": ["rw [\u2190 <a>abs_map</a>, h\u2080]", [{"full_name": "MvQPF.abs_map", "def_path": "Mathlib/Data/QPF/Multivariate/Basic.lean", "def_pos": [90, 3], "def_end_pos": [90, 10]}]], "state_before": "case h\nn : \u2115\nF : TypeVec.{u} n \u2192 Type u_1\ninst\u271d : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\np : \u2983i : Fin2 n\u2984 \u2192 \u03b1 i \u2192 Prop\nx : F \u03b1\na : (P F).A\nf : MvPFunctor.B (P F) a \u27f9 \u03b1\nh\u2080 : x = abs { fst := a, snd := f }\nh\u2081 : \u2200 (i : Fin2 n) (j : MvPFunctor.B (P F) a i), p (f i j)\n\u22a2 (fun i => Subtype.val) <$$> abs { fst := a, snd := fun i j => { val := f i j, property := (_ : p (f i j)) } } = x", "state_after": "case h\nn : \u2115\nF : TypeVec.{u} n \u2192 Type u_1\ninst\u271d : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\np : \u2983i : Fin2 n\u2984 \u2192 \u03b1 i \u2192 Prop\nx : F \u03b1\na : (P F).A\nf : MvPFunctor.B (P F) a \u27f9 \u03b1\nh\u2080 : x = abs { fst := a, snd := f }\nh\u2081 : \u2200 (i : Fin2 n) (j : MvPFunctor.B (P F) a i), p (f i j)\n\u22a2 abs ((fun i => Subtype.val) <$$> { fst := a, snd := fun i j => { val := f i j, property := (_ : p (f i j)) } }) =\n abs { fst := a, snd := f }"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case h\nn : \u2115\nF : TypeVec.{u} n \u2192 Type u_1\ninst\u271d : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\np : \u2983i : Fin2 n\u2984 \u2192 \u03b1 i \u2192 Prop\nx : F \u03b1\na : (P F).A\nf : MvPFunctor.B (P F) a \u27f9 \u03b1\nh\u2080 : x = abs { fst := a, snd := f }\nh\u2081 : \u2200 (i : Fin2 n) (j : MvPFunctor.B (P F) a i), p (f i j)\n\u22a2 abs ((fun i => Subtype.val) <$$> { fst := a, snd := fun i j => { val := f i j, property := (_ : p (f i j)) } }) =\n abs { fst := a, snd := f }", "state_after": "no goals"}, {"tactic": "rintro \u27e8y, hy\u27e9", "annotated_tactic": ["rintro \u27e8y, hy\u27e9", []], "state_before": "case mp\nn : \u2115\nF : TypeVec.{u} n \u2192 Type u_1\ninst\u271d : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\np : \u2983i : Fin2 n\u2984 \u2192 \u03b1 i \u2192 Prop\nx : F \u03b1\n\u22a2 LiftP p x \u2192 \u2203 a f, x = abs { fst := a, snd := f } \u2227 \u2200 (i : Fin2 n) (j : MvPFunctor.B (P F) a i), p (f i j)", "state_after": "case mp.intro\nn : \u2115\nF : TypeVec.{u} n \u2192 Type u_1\ninst\u271d : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\np : \u2983i : Fin2 n\u2984 \u2192 \u03b1 i \u2192 Prop\nx : F \u03b1\ny : F fun i => Subtype p\nhy : (fun i => Subtype.val) <$$> y = x\n\u22a2 \u2203 a f, x = abs { fst := a, snd := f } \u2227 \u2200 (i : Fin2 n) (j : MvPFunctor.B (P F) a i), p (f i j)"}, {"tactic": "cases' h : repr y with a f", "annotated_tactic": ["cases' h : <a>repr</a> y with a f", [{"full_name": "MvQPF.repr", "def_path": "Mathlib/Data/QPF/Multivariate/Basic.lean", "def_pos": [88, 3], "def_end_pos": [88, 7]}]], "state_before": "case mp.intro\nn : \u2115\nF : TypeVec.{u} n \u2192 Type u_1\ninst\u271d : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\np : \u2983i : Fin2 n\u2984 \u2192 \u03b1 i \u2192 Prop\nx : F \u03b1\ny : F fun i => Subtype p\nhy : (fun i => Subtype.val) <$$> y = x\n\u22a2 \u2203 a f, x = abs { fst := a, snd := f } \u2227 \u2200 (i : Fin2 n) (j : MvPFunctor.B (P F) a i), p (f i j)", "state_after": "case mp.intro.mk\nn : \u2115\nF : TypeVec.{u} n \u2192 Type u_1\ninst\u271d : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\np : \u2983i : Fin2 n\u2984 \u2192 \u03b1 i \u2192 Prop\nx : F \u03b1\ny : F fun i => Subtype p\nhy : (fun i => Subtype.val) <$$> y = x\na : (P F).A\nf : MvPFunctor.B (P F) a \u27f9 fun i => Subtype p\nh : repr y = { fst := a, snd := f }\n\u22a2 \u2203 a f, x = abs { fst := a, snd := f } \u2227 \u2200 (i : Fin2 n) (j : MvPFunctor.B (P F) a i), p (f i j)"}, {"tactic": "use a, fun i j => (f i j).val", "annotated_tactic": ["use a, fun i j => (f i j).<a>val</a>", [{"full_name": "Subtype.val", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [564, 3], "def_end_pos": [564, 6]}]], "state_before": "case mp.intro.mk\nn : \u2115\nF : TypeVec.{u} n \u2192 Type u_1\ninst\u271d : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\np : \u2983i : Fin2 n\u2984 \u2192 \u03b1 i \u2192 Prop\nx : F \u03b1\ny : F fun i => Subtype p\nhy : (fun i => Subtype.val) <$$> y = x\na : (P F).A\nf : MvPFunctor.B (P F) a \u27f9 fun i => Subtype p\nh : repr y = { fst := a, snd := f }\n\u22a2 \u2203 a f, x = abs { fst := a, snd := f } \u2227 \u2200 (i : Fin2 n) (j : MvPFunctor.B (P F) a i), p (f i j)", "state_after": "case h\nn : \u2115\nF : TypeVec.{u} n \u2192 Type u_1\ninst\u271d : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\np : \u2983i : Fin2 n\u2984 \u2192 \u03b1 i \u2192 Prop\nx : F \u03b1\ny : F fun i => Subtype p\nhy : (fun i => Subtype.val) <$$> y = x\na : (P F).A\nf : MvPFunctor.B (P F) a \u27f9 fun i => Subtype p\nh : repr y = { fst := a, snd := f }\n\u22a2 x = abs { fst := a, snd := fun i j => \u2191(f i j) } \u2227 \u2200 (i : Fin2 n) (j : MvPFunctor.B (P F) a i), p \u2191(f i j)"}, {"tactic": "constructor", "annotated_tactic": ["constructor", []], "state_before": "case h\nn : \u2115\nF : TypeVec.{u} n \u2192 Type u_1\ninst\u271d : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\np : \u2983i : Fin2 n\u2984 \u2192 \u03b1 i \u2192 Prop\nx : F \u03b1\ny : F fun i => Subtype p\nhy : (fun i => Subtype.val) <$$> y = x\na : (P F).A\nf : MvPFunctor.B (P F) a \u27f9 fun i => Subtype p\nh : repr y = { fst := a, snd := f }\n\u22a2 x = abs { fst := a, snd := fun i j => \u2191(f i j) } \u2227 \u2200 (i : Fin2 n) (j : MvPFunctor.B (P F) a i), p \u2191(f i j)", "state_after": "case h.left\nn : \u2115\nF : TypeVec.{u} n \u2192 Type u_1\ninst\u271d : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\np : \u2983i : Fin2 n\u2984 \u2192 \u03b1 i \u2192 Prop\nx : F \u03b1\ny : F fun i => Subtype p\nhy : (fun i => Subtype.val) <$$> y = x\na : (P F).A\nf : MvPFunctor.B (P F) a \u27f9 fun i => Subtype p\nh : repr y = { fst := a, snd := f }\n\u22a2 x = abs { fst := a, snd := fun i j => \u2191(f i j) }\n\ncase h.right\nn : \u2115\nF : TypeVec.{u} n \u2192 Type u_1\ninst\u271d : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\np : \u2983i : Fin2 n\u2984 \u2192 \u03b1 i \u2192 Prop\nx : F \u03b1\ny : F fun i => Subtype p\nhy : (fun i => Subtype.val) <$$> y = x\na : (P F).A\nf : MvPFunctor.B (P F) a \u27f9 fun i => Subtype p\nh : repr y = { fst := a, snd := f }\n\u22a2 \u2200 (i : Fin2 n) (j : MvPFunctor.B (P F) a i), p \u2191(f i j)"}, {"tactic": "intro i j", "annotated_tactic": ["intro i j", []], "state_before": "case h.right\nn : \u2115\nF : TypeVec.{u} n \u2192 Type u_1\ninst\u271d : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\np : \u2983i : Fin2 n\u2984 \u2192 \u03b1 i \u2192 Prop\nx : F \u03b1\ny : F fun i => Subtype p\nhy : (fun i => Subtype.val) <$$> y = x\na : (P F).A\nf : MvPFunctor.B (P F) a \u27f9 fun i => Subtype p\nh : repr y = { fst := a, snd := f }\n\u22a2 \u2200 (i : Fin2 n) (j : MvPFunctor.B (P F) a i), p \u2191(f i j)", "state_after": "case h.right\nn : \u2115\nF : TypeVec.{u} n \u2192 Type u_1\ninst\u271d : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\np : \u2983i : Fin2 n\u2984 \u2192 \u03b1 i \u2192 Prop\nx : F \u03b1\ny : F fun i => Subtype p\nhy : (fun i => Subtype.val) <$$> y = x\na : (P F).A\nf : MvPFunctor.B (P F) a \u27f9 fun i => Subtype p\nh : repr y = { fst := a, snd := f }\ni : Fin2 n\nj : MvPFunctor.B (P F) a i\n\u22a2 p \u2191(f i j)"}, {"tactic": "apply (f i j).property", "annotated_tactic": ["apply (f i j).<a>property</a>", [{"full_name": "Subtype.property", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [567, 3], "def_end_pos": [567, 11]}]], "state_before": "case h.right\nn : \u2115\nF : TypeVec.{u} n \u2192 Type u_1\ninst\u271d : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\np : \u2983i : Fin2 n\u2984 \u2192 \u03b1 i \u2192 Prop\nx : F \u03b1\ny : F fun i => Subtype p\nhy : (fun i => Subtype.val) <$$> y = x\na : (P F).A\nf : MvPFunctor.B (P F) a \u27f9 fun i => Subtype p\nh : repr y = { fst := a, snd := f }\ni : Fin2 n\nj : MvPFunctor.B (P F) a i\n\u22a2 p \u2191(f i j)", "state_after": "no goals"}, {"tactic": "rw [\u2190 hy, \u2190 abs_repr y, h, \u2190 abs_map]", "annotated_tactic": ["rw [\u2190 hy, \u2190 <a>abs_repr</a> y, h, \u2190 <a>abs_map</a>]", [{"full_name": "MvQPF.abs_repr", "def_path": "Mathlib/Data/QPF/Multivariate/Basic.lean", "def_pos": [89, 3], "def_end_pos": [89, 11]}, {"full_name": "MvQPF.abs_map", "def_path": "Mathlib/Data/QPF/Multivariate/Basic.lean", "def_pos": [90, 3], "def_end_pos": [90, 10]}]], "state_before": "case h.left\nn : \u2115\nF : TypeVec.{u} n \u2192 Type u_1\ninst\u271d : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\np : \u2983i : Fin2 n\u2984 \u2192 \u03b1 i \u2192 Prop\nx : F \u03b1\ny : F fun i => Subtype p\nhy : (fun i => Subtype.val) <$$> y = x\na : (P F).A\nf : MvPFunctor.B (P F) a \u27f9 fun i => Subtype p\nh : repr y = { fst := a, snd := f }\n\u22a2 x = abs { fst := a, snd := fun i j => \u2191(f i j) }", "state_after": "case h.left\nn : \u2115\nF : TypeVec.{u} n \u2192 Type u_1\ninst\u271d : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\np : \u2983i : Fin2 n\u2984 \u2192 \u03b1 i \u2192 Prop\nx : F \u03b1\ny : F fun i => Subtype p\nhy : (fun i => Subtype.val) <$$> y = x\na : (P F).A\nf : MvPFunctor.B (P F) a \u27f9 fun i => Subtype p\nh : repr y = { fst := a, snd := f }\n\u22a2 abs ((fun i => Subtype.val) <$$> { fst := a, snd := f }) = abs { fst := a, snd := fun i j => \u2191(f i j) }"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case h.left\nn : \u2115\nF : TypeVec.{u} n \u2192 Type u_1\ninst\u271d : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\np : \u2983i : Fin2 n\u2984 \u2192 \u03b1 i \u2192 Prop\nx : F \u03b1\ny : F fun i => Subtype p\nhy : (fun i => Subtype.val) <$$> y = x\na : (P F).A\nf : MvPFunctor.B (P F) a \u27f9 fun i => Subtype p\nh : repr y = { fst := a, snd := f }\n\u22a2 abs ((fun i => Subtype.val) <$$> { fst := a, snd := f }) = abs { fst := a, snd := fun i j => \u2191(f i j) }", "state_after": "no goals"}]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Algebra/Associated.lean
|
Irreducible.dvd_comm
|
[
245,
1
] |
[
247,
35
] |
[]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/RingTheory/Subring/Basic.lean
|
Subring.prod_mono_right
|
[
1126,
1
] |
[
1127,
24
] |
[]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Data/Finset/Powerset.lean
|
Finset.powersetCard_sup
|
[
303,
1
] |
[
318,
56
] |
[{"tactic": "apply le_antisymm", "annotated_tactic": ["apply <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\ns t : Finset \u03b1\ninst\u271d : DecidableEq \u03b1\nu : Finset \u03b1\nn : \u2115\nhn : n < card u\n\u22a2 sup (powersetCard (Nat.succ n) u) id = u", "state_after": "case a\n\u03b1 : Type u_1\ns t : Finset \u03b1\ninst\u271d : DecidableEq \u03b1\nu : Finset \u03b1\nn : \u2115\nhn : n < card u\n\u22a2 sup (powersetCard (Nat.succ n) u) id \u2264 u\n\ncase a\n\u03b1 : Type u_1\ns t : Finset \u03b1\ninst\u271d : DecidableEq \u03b1\nu : Finset \u03b1\nn : \u2115\nhn : n < card u\n\u22a2 u \u2264 sup (powersetCard (Nat.succ n) u) id"}, {"tactic": "simp_rw [Finset.sup_le_iff, mem_powersetCard]", "annotated_tactic": ["simp_rw [<a>Finset.sup_le_iff</a>, <a>mem_powersetCard</a>]", [{"full_name": "Finset.sup_le_iff", "def_path": "Mathlib/Data/Finset/Lattice.lean", "def_pos": [99, 19], "def_end_pos": [99, 29]}, {"full_name": "Finset.mem_powersetCard", "def_path": "Mathlib/Data/Finset/Powerset.lean", "def_pos": [203, 9], "def_end_pos": [203, 25]}]], "state_before": "case a\n\u03b1 : Type u_1\ns t : Finset \u03b1\ninst\u271d : DecidableEq \u03b1\nu : Finset \u03b1\nn : \u2115\nhn : n < card u\n\u22a2 sup (powersetCard (Nat.succ n) u) id \u2264 u", "state_after": "case a\n\u03b1 : Type u_1\ns t : Finset \u03b1\ninst\u271d : DecidableEq \u03b1\nu : Finset \u03b1\nn : \u2115\nhn : n < card u\n\u22a2 \u2200 (b : Finset \u03b1), b \u2286 u \u2227 card b = Nat.succ n \u2192 id b \u2264 u"}, {"tactic": "rintro x \u27e8h, -\u27e9", "annotated_tactic": ["rintro x \u27e8h, -\u27e9", []], "state_before": "case a\n\u03b1 : Type u_1\ns t : Finset \u03b1\ninst\u271d : DecidableEq \u03b1\nu : Finset \u03b1\nn : \u2115\nhn : n < card u\n\u22a2 \u2200 (b : Finset \u03b1), b \u2286 u \u2227 card b = Nat.succ n \u2192 id b \u2264 u", "state_after": "case a.intro\n\u03b1 : Type u_1\ns t : Finset \u03b1\ninst\u271d : DecidableEq \u03b1\nu : Finset \u03b1\nn : \u2115\nhn : n < card u\nx : Finset \u03b1\nh : x \u2286 u\n\u22a2 id x \u2264 u"}, {"tactic": "exact h", "annotated_tactic": ["exact h", []], "state_before": "case a.intro\n\u03b1 : Type u_1\ns t : Finset \u03b1\ninst\u271d : DecidableEq \u03b1\nu : Finset \u03b1\nn : \u2115\nhn : n < card u\nx : Finset \u03b1\nh : x \u2286 u\n\u22a2 id x \u2264 u", "state_after": "no goals"}, {"tactic": "rw [sup_eq_biUnion, le_iff_subset, subset_iff]", "annotated_tactic": ["rw [<a>sup_eq_biUnion</a>, <a>le_iff_subset</a>, <a>subset_iff</a>]", [{"full_name": "Finset.sup_eq_biUnion", "def_path": "Mathlib/Data/Finset/Lattice.lean", "def_pos": [1876, 9], "def_end_pos": [1876, 23]}, {"full_name": "Finset.le_iff_subset", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [402, 9], "def_end_pos": [402, 22]}, {"full_name": "Finset.subset_iff", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [371, 9], "def_end_pos": [371, 19]}]], "state_before": "case a\n\u03b1 : Type u_1\ns t : Finset \u03b1\ninst\u271d : DecidableEq \u03b1\nu : Finset \u03b1\nn : \u2115\nhn : n < card u\n\u22a2 u \u2264 sup (powersetCard (Nat.succ n) u) id", "state_after": "case a\n\u03b1 : Type u_1\ns t : Finset \u03b1\ninst\u271d : DecidableEq \u03b1\nu : Finset \u03b1\nn : \u2115\nhn : n < card u\n\u22a2 \u2200 \u2983x : \u03b1\u2984, x \u2208 u \u2192 x \u2208 Finset.biUnion (powersetCard (Nat.succ n) u) id"}, {"tactic": "cases' (Nat.succ_le_of_lt hn).eq_or_lt with h' h'", "annotated_tactic": ["cases' (<a>Nat.succ_le_of_lt</a> hn).<a>eq_or_lt</a> with h' h'", [{"full_name": "Nat.succ_le_of_lt", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [313, 9], "def_end_pos": [313, 22]}, {"full_name": "LE.le.eq_or_lt", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [428, 7], "def_end_pos": [428, 21]}]], "state_before": "case a\n\u03b1 : Type u_1\ns t : Finset \u03b1\ninst\u271d : DecidableEq \u03b1\nu : Finset \u03b1\nn : \u2115\nhn : n < card u\n\u22a2 \u2200 \u2983x : \u03b1\u2984, x \u2208 u \u2192 x \u2208 Finset.biUnion (powersetCard (Nat.succ n) u) id", "state_after": "case a.inl\n\u03b1 : Type u_1\ns t : Finset \u03b1\ninst\u271d : DecidableEq \u03b1\nu : Finset \u03b1\nn : \u2115\nhn : n < card u\nh' : Nat.succ n = card u\n\u22a2 \u2200 \u2983x : \u03b1\u2984, x \u2208 u \u2192 x \u2208 Finset.biUnion (powersetCard (Nat.succ n) u) id\n\ncase a.inr\n\u03b1 : Type u_1\ns t : Finset \u03b1\ninst\u271d : DecidableEq \u03b1\nu : Finset \u03b1\nn : \u2115\nhn : n < card u\nh' : Nat.succ n < card u\n\u22a2 \u2200 \u2983x : \u03b1\u2984, x \u2208 u \u2192 x \u2208 Finset.biUnion (powersetCard (Nat.succ n) u) id"}, {"tactic": "simp [h']", "annotated_tactic": ["simp [h']", []], "state_before": "case a.inl\n\u03b1 : Type u_1\ns t : Finset \u03b1\ninst\u271d : DecidableEq \u03b1\nu : Finset \u03b1\nn : \u2115\nhn : n < card u\nh' : Nat.succ n = card u\n\u22a2 \u2200 \u2983x : \u03b1\u2984, x \u2208 u \u2192 x \u2208 Finset.biUnion (powersetCard (Nat.succ n) u) id", "state_after": "no goals"}, {"tactic": "intro x hx", "annotated_tactic": ["intro x hx", []], "state_before": "case a.inr\n\u03b1 : Type u_1\ns t : Finset \u03b1\ninst\u271d : DecidableEq \u03b1\nu : Finset \u03b1\nn : \u2115\nhn : n < card u\nh' : Nat.succ n < card u\n\u22a2 \u2200 \u2983x : \u03b1\u2984, x \u2208 u \u2192 x \u2208 Finset.biUnion (powersetCard (Nat.succ n) u) id", "state_after": "case a.inr\n\u03b1 : Type u_1\ns t : Finset \u03b1\ninst\u271d : DecidableEq \u03b1\nu : Finset \u03b1\nn : \u2115\nhn : n < card u\nh' : Nat.succ n < card u\nx : \u03b1\nhx : x \u2208 u\n\u22a2 x \u2208 Finset.biUnion (powersetCard (Nat.succ n) u) id"}, {"tactic": "simp only [mem_biUnion, exists_prop, id.def]", "annotated_tactic": ["simp only [<a>mem_biUnion</a>, <a>exists_prop</a>, <a>id.def</a>]", [{"full_name": "Finset.mem_biUnion", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [3613, 9], "def_end_pos": [3613, 20]}, {"full_name": "exists_prop", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [485, 17], "def_end_pos": [485, 28]}, {"full_name": "id.def", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [527, 9], "def_end_pos": [527, 15]}]], "state_before": "case a.inr\n\u03b1 : Type u_1\ns t : Finset \u03b1\ninst\u271d : DecidableEq \u03b1\nu : Finset \u03b1\nn : \u2115\nhn : n < card u\nh' : Nat.succ n < card u\nx : \u03b1\nhx : x \u2208 u\n\u22a2 x \u2208 Finset.biUnion (powersetCard (Nat.succ n) u) id", "state_after": "case a.inr\n\u03b1 : Type u_1\ns t : Finset \u03b1\ninst\u271d : DecidableEq \u03b1\nu : Finset \u03b1\nn : \u2115\nhn : n < card u\nh' : Nat.succ n < card u\nx : \u03b1\nhx : x \u2208 u\n\u22a2 \u2203 a, a \u2208 powersetCard (Nat.succ n) u \u2227 x \u2208 a"}, {"tactic": "obtain \u27e8t, ht\u27e9 : \u2203 t, t \u2208 powersetCard n (u.erase x) := powersetCard_nonempty\n (le_trans (Nat.le_pred_of_lt hn) pred_card_le_card_erase)", "annotated_tactic": ["obtain \u27e8t, ht\u27e9 : \u2203 t, t \u2208 <a>powersetCard</a> n (u.erase x) := <a>powersetCard_nonempty</a>\n (<a>le_trans</a> (<a>Nat.le_pred_of_lt</a> hn) <a>pred_card_le_card_erase</a>)", [{"full_name": "Finset.powersetCard", "def_path": "Mathlib/Data/Finset/Powerset.lean", "def_pos": [197, 5], "def_end_pos": [197, 17]}, {"full_name": "Finset.powersetCard_nonempty", "def_path": "Mathlib/Data/Finset/Powerset.lean", "def_pos": [254, 9], "def_end_pos": [254, 30]}, {"full_name": "le_trans", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [56, 9], "def_end_pos": [56, 17]}, {"full_name": "Nat.le_pred_of_lt", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [268, 9], "def_end_pos": [268, 22]}, {"full_name": "Finset.pred_card_le_card_erase", "def_path": "Mathlib/Data/Finset/Card.lean", "def_pos": [162, 9], "def_end_pos": [162, 32]}]], "state_before": "case a.inr\n\u03b1 : Type u_1\ns t : Finset \u03b1\ninst\u271d : DecidableEq \u03b1\nu : Finset \u03b1\nn : \u2115\nhn : n < card u\nh' : Nat.succ n < card u\nx : \u03b1\nhx : x \u2208 u\n\u22a2 \u2203 a, a \u2208 powersetCard (Nat.succ n) u \u2227 x \u2208 a", "state_after": "case a.inr.intro\n\u03b1 : Type u_1\ns t\u271d : Finset \u03b1\ninst\u271d : DecidableEq \u03b1\nu : Finset \u03b1\nn : \u2115\nhn : n < card u\nh' : Nat.succ n < card u\nx : \u03b1\nhx : x \u2208 u\nt : Finset \u03b1\nht : t \u2208 powersetCard n (erase u x)\n\u22a2 \u2203 a, a \u2208 powersetCard (Nat.succ n) u \u2227 x \u2208 a"}, {"tactic": "refine' \u27e8insert x t, _, mem_insert_self _ _\u27e9", "annotated_tactic": ["refine' \u27e8<a>insert</a> x t, _, <a>mem_insert_self</a> _ _\u27e9", [{"full_name": "Insert.insert", "def_path": "lake-packages/std/Std/Classes/SetNotation.lean", "def_pos": [69, 3], "def_end_pos": [69, 9]}, {"full_name": "Finset.mem_insert_self", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1091, 9], "def_end_pos": [1091, 24]}]], "state_before": "case a.inr.intro\n\u03b1 : Type u_1\ns t\u271d : Finset \u03b1\ninst\u271d : DecidableEq \u03b1\nu : Finset \u03b1\nn : \u2115\nhn : n < card u\nh' : Nat.succ n < card u\nx : \u03b1\nhx : x \u2208 u\nt : Finset \u03b1\nht : t \u2208 powersetCard n (erase u x)\n\u22a2 \u2203 a, a \u2208 powersetCard (Nat.succ n) u \u2227 x \u2208 a", "state_after": "case a.inr.intro\n\u03b1 : Type u_1\ns t\u271d : Finset \u03b1\ninst\u271d : DecidableEq \u03b1\nu : Finset \u03b1\nn : \u2115\nhn : n < card u\nh' : Nat.succ n < card u\nx : \u03b1\nhx : x \u2208 u\nt : Finset \u03b1\nht : t \u2208 powersetCard n (erase u x)\n\u22a2 insert x t \u2208 powersetCard (Nat.succ n) u"}, {"tactic": "rw [\u2190 insert_erase hx, powersetCard_succ_insert (not_mem_erase _ _)]", "annotated_tactic": ["rw [\u2190 <a>insert_erase</a> hx, <a>powersetCard_succ_insert</a> (<a>not_mem_erase</a> _ _)]", [{"full_name": "Finset.insert_erase", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1957, 9], "def_end_pos": [1957, 21]}, {"full_name": "Finset.powersetCard_succ_insert", "def_path": "Mathlib/Data/Finset/Powerset.lean", "def_pos": [240, 9], "def_end_pos": [240, 33]}, {"full_name": "Finset.not_mem_erase", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1891, 9], "def_end_pos": [1891, 22]}]], "state_before": "case a.inr.intro\n\u03b1 : Type u_1\ns t\u271d : Finset \u03b1\ninst\u271d : DecidableEq \u03b1\nu : Finset \u03b1\nn : \u2115\nhn : n < card u\nh' : Nat.succ n < card u\nx : \u03b1\nhx : x \u2208 u\nt : Finset \u03b1\nht : t \u2208 powersetCard n (erase u x)\n\u22a2 insert x t \u2208 powersetCard (Nat.succ n) u", "state_after": "case a.inr.intro\n\u03b1 : Type u_1\ns t\u271d : Finset \u03b1\ninst\u271d : DecidableEq \u03b1\nu : Finset \u03b1\nn : \u2115\nhn : n < card u\nh' : Nat.succ n < card u\nx : \u03b1\nhx : x \u2208 u\nt : Finset \u03b1\nht : t \u2208 powersetCard n (erase u x)\n\u22a2 insert x t \u2208 powersetCard (Nat.succ n) (erase u x) \u222a image (insert x) (powersetCard n (erase u x))"}, {"tactic": "exact mem_union_right _ (mem_image_of_mem _ ht)", "annotated_tactic": ["exact <a>mem_union_right</a> _ (<a>mem_image_of_mem</a> _ ht)", [{"full_name": "Finset.mem_union_right", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1386, 9], "def_end_pos": [1386, 24]}, {"full_name": "Finset.mem_image_of_mem", "def_path": "Mathlib/Data/Finset/Image.lean", "def_pos": [334, 9], "def_end_pos": [334, 25]}]], "state_before": "case a.inr.intro\n\u03b1 : Type u_1\ns t\u271d : Finset \u03b1\ninst\u271d : DecidableEq \u03b1\nu : Finset \u03b1\nn : \u2115\nhn : n < card u\nh' : Nat.succ n < card u\nx : \u03b1\nhx : x \u2208 u\nt : Finset \u03b1\nht : t \u2208 powersetCard n (erase u x)\n\u22a2 insert x t \u2208 powersetCard (Nat.succ n) (erase u x) \u222a image (insert x) (powersetCard n (erase u x))", "state_after": "no goals"}]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/RingTheory/WittVector/Compare.lean
|
WittVector.toPadicInt_comp_fromPadicInt
|
[
183,
1
] |
[
189,
71
] |
[{"tactic": "rw [\u2190 PadicInt.toZModPow_eq_iff_ext]", "annotated_tactic": ["rw [\u2190 <a>PadicInt.toZModPow_eq_iff_ext</a>]", [{"full_name": "PadicInt.toZModPow_eq_iff_ext", "def_path": "Mathlib/NumberTheory/Padics/RingHoms.lean", "def_pos": [676, 9], "def_end_pos": [676, 29]}]], "state_before": "p : \u2115\nhp : Fact (Nat.Prime p)\n\u22a2 RingHom.comp (toPadicInt p) (fromPadicInt p) = RingHom.id \u2124_[p]", "state_after": "p : \u2115\nhp : Fact (Nat.Prime p)\n\u22a2 \u2200 (n : \u2115),\n RingHom.comp (PadicInt.toZModPow n) (RingHom.comp (toPadicInt p) (fromPadicInt p)) =\n RingHom.comp (PadicInt.toZModPow n) (RingHom.id \u2124_[p])"}, {"tactic": "intro n", "annotated_tactic": ["intro n", []], "state_before": "p : \u2115\nhp : Fact (Nat.Prime p)\n\u22a2 \u2200 (n : \u2115),\n RingHom.comp (PadicInt.toZModPow n) (RingHom.comp (toPadicInt p) (fromPadicInt p)) =\n RingHom.comp (PadicInt.toZModPow n) (RingHom.id \u2124_[p])", "state_after": "p : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\n\u22a2 RingHom.comp (PadicInt.toZModPow n) (RingHom.comp (toPadicInt p) (fromPadicInt p)) =\n RingHom.comp (PadicInt.toZModPow n) (RingHom.id \u2124_[p])"}, {"tactic": "rw [\u2190 RingHom.comp_assoc, toPadicInt, PadicInt.lift_spec]", "annotated_tactic": ["rw [\u2190 <a>RingHom.comp_assoc</a>, <a>toPadicInt</a>, <a>PadicInt.lift_spec</a>]", [{"full_name": "RingHom.comp_assoc", "def_path": "Mathlib/Algebra/Hom/Ring/Defs.lean", "def_pos": [662, 9], "def_end_pos": [662, 19]}, {"full_name": "WittVector.toPadicInt", "def_path": "Mathlib/RingTheory/WittVector/Compare.lean", "def_pos": [163, 5], "def_end_pos": [163, 15]}, {"full_name": "PadicInt.lift_spec", "def_path": "Mathlib/NumberTheory/Padics/RingHoms.lean", "def_pos": [637, 9], "def_end_pos": [637, 18]}]], "state_before": "p : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\n\u22a2 RingHom.comp (PadicInt.toZModPow n) (RingHom.comp (toPadicInt p) (fromPadicInt p)) =\n RingHom.comp (PadicInt.toZModPow n) (RingHom.id \u2124_[p])", "state_after": "p : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\n\u22a2 RingHom.comp (toZModPow p n) (fromPadicInt p) = RingHom.comp (PadicInt.toZModPow n) (RingHom.id \u2124_[p])"}, {"tactic": "simp only [fromPadicInt, toZModPow, RingHom.comp_id]", "annotated_tactic": ["simp only [<a>fromPadicInt</a>, <a>toZModPow</a>, <a>RingHom.comp_id</a>]", [{"full_name": "WittVector.fromPadicInt", "def_path": "Mathlib/RingTheory/WittVector/Compare.lean", "def_pos": [178, 5], "def_end_pos": [178, 17]}, {"full_name": "WittVector.toZModPow", "def_path": "Mathlib/RingTheory/WittVector/Compare.lean", "def_pos": [145, 5], "def_end_pos": [145, 14]}, {"full_name": "RingHom.comp_id", "def_path": "Mathlib/Algebra/Hom/Ring/Defs.lean", "def_pos": [678, 9], "def_end_pos": [678, 16]}]], "state_before": "p : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\n\u22a2 RingHom.comp (toZModPow p n) (fromPadicInt p) = RingHom.comp (PadicInt.toZModPow n) (RingHom.id \u2124_[p])", "state_after": "p : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\n\u22a2 RingHom.comp (RingHom.comp (RingEquiv.toRingHom (RingEquiv.symm (zmodEquivTrunc p n))) (truncate n))\n (lift (fun k => RingHom.comp (RingEquiv.toRingHom (zmodEquivTrunc p k)) (PadicInt.toZModPow k))\n (_ :\n \u2200 (k\u2081 k\u2082 : \u2115) (hk : k\u2081 \u2264 k\u2082),\n RingHom.comp (TruncatedWittVector.truncate hk)\n (RingHom.comp (RingEquiv.toRingHom (zmodEquivTrunc p k\u2082)) (PadicInt.toZModPow k\u2082)) =\n RingHom.comp (RingEquiv.toRingHom (zmodEquivTrunc p k\u2081)) (PadicInt.toZModPow k\u2081))) =\n PadicInt.toZModPow n"}, {"tactic": "rw [RingHom.comp_assoc, truncate_comp_lift, \u2190 RingHom.comp_assoc]", "annotated_tactic": ["rw [<a>RingHom.comp_assoc</a>, <a>truncate_comp_lift</a>, \u2190 <a>RingHom.comp_assoc</a>]", [{"full_name": "RingHom.comp_assoc", "def_path": "Mathlib/Algebra/Hom/Ring/Defs.lean", "def_pos": [662, 9], "def_end_pos": [662, 19]}, {"full_name": "WittVector.truncate_comp_lift", "def_path": "Mathlib/RingTheory/WittVector/Truncated.lean", "def_pos": [502, 9], "def_end_pos": [502, 27]}, {"full_name": "RingHom.comp_assoc", "def_path": "Mathlib/Algebra/Hom/Ring/Defs.lean", "def_pos": [662, 9], "def_end_pos": [662, 19]}]], "state_before": "p : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\n\u22a2 RingHom.comp (RingHom.comp (RingEquiv.toRingHom (RingEquiv.symm (zmodEquivTrunc p n))) (truncate n))\n (lift (fun k => RingHom.comp (RingEquiv.toRingHom (zmodEquivTrunc p k)) (PadicInt.toZModPow k))\n (_ :\n \u2200 (k\u2081 k\u2082 : \u2115) (hk : k\u2081 \u2264 k\u2082),\n RingHom.comp (TruncatedWittVector.truncate hk)\n (RingHom.comp (RingEquiv.toRingHom (zmodEquivTrunc p k\u2082)) (PadicInt.toZModPow k\u2082)) =\n RingHom.comp (RingEquiv.toRingHom (zmodEquivTrunc p k\u2081)) (PadicInt.toZModPow k\u2081))) =\n PadicInt.toZModPow n", "state_after": "p : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\n\u22a2 RingHom.comp\n (RingHom.comp (RingEquiv.toRingHom (RingEquiv.symm (zmodEquivTrunc p n)))\n (RingEquiv.toRingHom (zmodEquivTrunc p n)))\n (PadicInt.toZModPow n) =\n PadicInt.toZModPow n"}, {"tactic": "simp only [RingEquiv.symm_toRingHom_comp_toRingHom, RingHom.id_comp]", "annotated_tactic": ["simp only [<a>RingEquiv.symm_toRingHom_comp_toRingHom</a>, <a>RingHom.id_comp</a>]", [{"full_name": "RingEquiv.symm_toRingHom_comp_toRingHom", "def_path": "Mathlib/Algebra/Ring/Equiv.lean", "def_pos": [785, 9], "def_end_pos": [785, 38]}, {"full_name": "RingHom.id_comp", "def_path": "Mathlib/Algebra/Hom/Ring/Defs.lean", "def_pos": [683, 9], "def_end_pos": [683, 16]}]], "state_before": "p : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\n\u22a2 RingHom.comp\n (RingHom.comp (RingEquiv.toRingHom (RingEquiv.symm (zmodEquivTrunc p n)))\n (RingEquiv.toRingHom (zmodEquivTrunc p n)))\n (PadicInt.toZModPow n) =\n PadicInt.toZModPow n", "state_after": "no goals"}]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Algebra/CharP/Basic.lean
|
add_pow_char_pow
|
[
277,
1
] |
[
279,
54
] |
[]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Data/Set/Basic.lean
|
Set.insert_nonempty
|
[
1217,
1
] |
[
1218,
22
] |
[]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Data/Nat/Factorization/PrimePow.lean
|
IsPrimePow.exists_ord_compl_eq_one
|
[
64,
1
] |
[
74,
7
] |
[{"tactic": "rcases eq_or_ne n 0 with (rfl | hn0)", "annotated_tactic": ["rcases <a>eq_or_ne</a> n 0 with (rfl | hn0)", [{"full_name": "eq_or_ne", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [209, 9], "def_end_pos": [209, 17]}]], "state_before": "R : Type u_1\ninst\u271d : CommMonoidWithZero R\nn\u271d p : R\nk n : \u2115\nh : IsPrimePow n\n\u22a2 \u2203 p, Nat.Prime p \u2227 n / p ^ \u2191(Nat.factorization n) p = 1", "state_after": "case inl\nR : Type u_1\ninst\u271d : CommMonoidWithZero R\nn p : R\nk : \u2115\nh : IsPrimePow 0\n\u22a2 \u2203 p, Nat.Prime p \u2227 0 / p ^ \u2191(Nat.factorization 0) p = 1\n\ncase inr\nR : Type u_1\ninst\u271d : CommMonoidWithZero R\nn\u271d p : R\nk n : \u2115\nh : IsPrimePow n\nhn0 : n \u2260 0\n\u22a2 \u2203 p, Nat.Prime p \u2227 n / p ^ \u2191(Nat.factorization n) p = 1"}, {"tactic": "rcases isPrimePow_iff_factorization_eq_single.mp h with \u27e8p, k, hk0, h1\u27e9", "annotated_tactic": ["rcases isPrimePow_iff_factorization_eq_single.mp h with \u27e8p, k, hk0, h1\u27e9", []], "state_before": "case inr\nR : Type u_1\ninst\u271d : CommMonoidWithZero R\nn\u271d p : R\nk n : \u2115\nh : IsPrimePow n\nhn0 : n \u2260 0\n\u22a2 \u2203 p, Nat.Prime p \u2227 n / p ^ \u2191(Nat.factorization n) p = 1", "state_after": "case inr.intro.intro.intro\nR : Type u_1\ninst\u271d : CommMonoidWithZero R\nn\u271d p\u271d : R\nk\u271d n : \u2115\nh : IsPrimePow n\nhn0 : n \u2260 0\np k : \u2115\nhk0 : 0 < k\nh1 : Nat.factorization n = fun\u2080 | p => k\n\u22a2 \u2203 p, Nat.Prime p \u2227 n / p ^ \u2191(Nat.factorization n) p = 1"}, {"tactic": "rcases em' p.Prime with (pp | pp)", "annotated_tactic": ["rcases <a>em'</a> p.Prime with (pp | pp)", [{"full_name": "em'", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [197, 9], "def_end_pos": [197, 12]}]], "state_before": "case inr.intro.intro.intro\nR : Type u_1\ninst\u271d : CommMonoidWithZero R\nn\u271d p\u271d : R\nk\u271d n : \u2115\nh : IsPrimePow n\nhn0 : n \u2260 0\np k : \u2115\nhk0 : 0 < k\nh1 : Nat.factorization n = fun\u2080 | p => k\n\u22a2 \u2203 p, Nat.Prime p \u2227 n / p ^ \u2191(Nat.factorization n) p = 1", "state_after": "case inr.intro.intro.intro.inl\nR : Type u_1\ninst\u271d : CommMonoidWithZero R\nn\u271d p\u271d : R\nk\u271d n : \u2115\nh : IsPrimePow n\nhn0 : n \u2260 0\np k : \u2115\nhk0 : 0 < k\nh1 : Nat.factorization n = fun\u2080 | p => k\npp : \u00acNat.Prime p\n\u22a2 \u2203 p, Nat.Prime p \u2227 n / p ^ \u2191(Nat.factorization n) p = 1\n\ncase inr.intro.intro.intro.inr\nR : Type u_1\ninst\u271d : CommMonoidWithZero R\nn\u271d p\u271d : R\nk\u271d n : \u2115\nh : IsPrimePow n\nhn0 : n \u2260 0\np k : \u2115\nhk0 : 0 < k\nh1 : Nat.factorization n = fun\u2080 | p => k\npp : Nat.Prime p\n\u22a2 \u2203 p, Nat.Prime p \u2227 n / p ^ \u2191(Nat.factorization n) p = 1"}, {"tactic": "refine' \u27e8p, pp, _\u27e9", "annotated_tactic": ["refine' \u27e8p, pp, _\u27e9", []], "state_before": "case inr.intro.intro.intro.inr\nR : Type u_1\ninst\u271d : CommMonoidWithZero R\nn\u271d p\u271d : R\nk\u271d n : \u2115\nh : IsPrimePow n\nhn0 : n \u2260 0\np k : \u2115\nhk0 : 0 < k\nh1 : Nat.factorization n = fun\u2080 | p => k\npp : Nat.Prime p\n\u22a2 \u2203 p, Nat.Prime p \u2227 n / p ^ \u2191(Nat.factorization n) p = 1", "state_after": "case inr.intro.intro.intro.inr\nR : Type u_1\ninst\u271d : CommMonoidWithZero R\nn\u271d p\u271d : R\nk\u271d n : \u2115\nh : IsPrimePow n\nhn0 : n \u2260 0\np k : \u2115\nhk0 : 0 < k\nh1 : Nat.factorization n = fun\u2080 | p => k\npp : Nat.Prime p\n\u22a2 n / p ^ \u2191(Nat.factorization n) p = 1"}, {"tactic": "refine' Nat.eq_of_factorization_eq (Nat.ord_compl_pos p hn0).ne' (by simp) fun q => _", "annotated_tactic": ["refine' <a>Nat.eq_of_factorization_eq</a> (<a>Nat.ord_compl_pos</a> p hn0).<a>ne'</a> (by simp) fun q => _", [{"full_name": "Nat.eq_of_factorization_eq", "def_path": "Mathlib/Data/Nat/Factorization/Basic.lean", "def_pos": [107, 9], "def_end_pos": [107, 31]}, {"full_name": "Nat.ord_compl_pos", "def_path": "Mathlib/Data/Nat/Factorization/Basic.lean", "def_pos": [388, 9], "def_end_pos": [388, 22]}, {"full_name": "LT.lt.ne'", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [328, 9], "def_end_pos": [328, 12]}]], "state_before": "case inr.intro.intro.intro.inr\nR : Type u_1\ninst\u271d : CommMonoidWithZero R\nn\u271d p\u271d : R\nk\u271d n : \u2115\nh : IsPrimePow n\nhn0 : n \u2260 0\np k : \u2115\nhk0 : 0 < k\nh1 : Nat.factorization n = fun\u2080 | p => k\npp : Nat.Prime p\n\u22a2 n / p ^ \u2191(Nat.factorization n) p = 1", "state_after": "case inr.intro.intro.intro.inr\nR : Type u_1\ninst\u271d : CommMonoidWithZero R\nn\u271d p\u271d : R\nk\u271d n : \u2115\nh : IsPrimePow n\nhn0 : n \u2260 0\np k : \u2115\nhk0 : 0 < k\nh1 : Nat.factorization n = fun\u2080 | p => k\npp : Nat.Prime p\nq : \u2115\n\u22a2 \u2191(Nat.factorization (n / p ^ \u2191(Nat.factorization n) p)) q = \u2191(Nat.factorization 1) q"}, {"tactic": "rw [Nat.factorization_ord_compl n p, h1]", "annotated_tactic": ["rw [<a>Nat.factorization_ord_compl</a> n p, h1]", [{"full_name": "Nat.factorization_ord_compl", "def_path": "Mathlib/Data/Nat/Factorization/Basic.lean", "def_pos": [535, 9], "def_end_pos": [535, 32]}]], "state_before": "case inr.intro.intro.intro.inr\nR : Type u_1\ninst\u271d : CommMonoidWithZero R\nn\u271d p\u271d : R\nk\u271d n : \u2115\nh : IsPrimePow n\nhn0 : n \u2260 0\np k : \u2115\nhk0 : 0 < k\nh1 : Nat.factorization n = fun\u2080 | p => k\npp : Nat.Prime p\nq : \u2115\n\u22a2 \u2191(Nat.factorization (n / p ^ \u2191(Nat.factorization n) p)) q = \u2191(Nat.factorization 1) q", "state_after": "case inr.intro.intro.intro.inr\nR : Type u_1\ninst\u271d : CommMonoidWithZero R\nn\u271d p\u271d : R\nk\u271d n : \u2115\nh : IsPrimePow n\nhn0 : n \u2260 0\np k : \u2115\nhk0 : 0 < k\nh1 : Nat.factorization n = fun\u2080 | p => k\npp : Nat.Prime p\nq : \u2115\n\u22a2 \u2191(Finsupp.erase p fun\u2080 | p => k) q = \u2191(Nat.factorization 1) q"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case inr.intro.intro.intro.inr\nR : Type u_1\ninst\u271d : CommMonoidWithZero R\nn\u271d p\u271d : R\nk\u271d n : \u2115\nh : IsPrimePow n\nhn0 : n \u2260 0\np k : \u2115\nhk0 : 0 < k\nh1 : Nat.factorization n = fun\u2080 | p => k\npp : Nat.Prime p\nq : \u2115\n\u22a2 \u2191(Finsupp.erase p fun\u2080 | p => k) q = \u2191(Nat.factorization 1) q", "state_after": "no goals"}, {"tactic": "cases not_isPrimePow_zero h", "annotated_tactic": ["cases <a>not_isPrimePow_zero</a> h", [{"full_name": "not_isPrimePow_zero", "def_path": "Mathlib/Algebra/IsPrimePow.lean", "def_pos": [38, 9], "def_end_pos": [38, 28]}]], "state_before": "case inl\nR : Type u_1\ninst\u271d : CommMonoidWithZero R\nn p : R\nk : \u2115\nh : IsPrimePow 0\n\u22a2 \u2203 p, Nat.Prime p \u2227 0 / p ^ \u2191(Nat.factorization 0) p = 1", "state_after": "no goals"}, {"tactic": "refine' absurd _ hk0.ne'", "annotated_tactic": ["refine' <a>absurd</a> _ hk0.ne'", [{"full_name": "absurd", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [233, 21], "def_end_pos": [233, 27]}]], "state_before": "case inr.intro.intro.intro.inl\nR : Type u_1\ninst\u271d : CommMonoidWithZero R\nn\u271d p\u271d : R\nk\u271d n : \u2115\nh : IsPrimePow n\nhn0 : n \u2260 0\np k : \u2115\nhk0 : 0 < k\nh1 : Nat.factorization n = fun\u2080 | p => k\npp : \u00acNat.Prime p\n\u22a2 \u2203 p, Nat.Prime p \u2227 n / p ^ \u2191(Nat.factorization n) p = 1", "state_after": "case inr.intro.intro.intro.inl\nR : Type u_1\ninst\u271d : CommMonoidWithZero R\nn\u271d p\u271d : R\nk\u271d n : \u2115\nh : IsPrimePow n\nhn0 : n \u2260 0\np k : \u2115\nhk0 : 0 < k\nh1 : Nat.factorization n = fun\u2080 | p => k\npp : \u00acNat.Prime p\n\u22a2 k = 0"}, {"tactic": "simp [\u2190 Nat.factorization_eq_zero_of_non_prime n pp, h1]", "annotated_tactic": ["simp [\u2190 <a>Nat.factorization_eq_zero_of_non_prime</a> n pp, h1]", [{"full_name": "Nat.factorization_eq_zero_of_non_prime", "def_path": "Mathlib/Data/Nat/Factorization/Basic.lean", "def_pos": [159, 9], "def_end_pos": [159, 43]}]], "state_before": "case inr.intro.intro.intro.inl\nR : Type u_1\ninst\u271d : CommMonoidWithZero R\nn\u271d p\u271d : R\nk\u271d n : \u2115\nh : IsPrimePow n\nhn0 : n \u2260 0\np k : \u2115\nhk0 : 0 < k\nh1 : Nat.factorization n = fun\u2080 | p => k\npp : \u00acNat.Prime p\n\u22a2 k = 0", "state_after": "no goals"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "R : Type u_1\ninst\u271d : CommMonoidWithZero R\nn\u271d p\u271d : R\nk\u271d n : \u2115\nh : IsPrimePow n\nhn0 : n \u2260 0\np k : \u2115\nhk0 : 0 < k\nh1 : Nat.factorization n = fun\u2080 | p => k\npp : Nat.Prime p\n\u22a2 1 \u2260 0", "state_after": "no goals"}]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Topology/Algebra/Module/FiniteDimension.lean
|
LinearMap.coe_toContinuousLinearMap
|
[
300,
1
] |
[
302,
6
] |
[]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Analysis/Convex/Between.lean
|
affineSegment_eq_segment
|
[
48,
1
] |
[
49,
47
] |
[{"tactic": "rw [segment_eq_image_lineMap, affineSegment]", "annotated_tactic": ["rw [<a>segment_eq_image_lineMap</a>, <a>affineSegment</a>]", [{"full_name": "segment_eq_image_lineMap", "def_path": "Mathlib/Analysis/Convex/Segment.lean", "def_pos": [221, 9], "def_end_pos": [221, 33]}, {"full_name": "affineSegment", "def_path": "Mathlib/Analysis/Convex/Between.lean", "def_pos": [44, 5], "def_end_pos": [44, 18]}]], "state_before": "R : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : OrderedRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\nx y : V\n\u22a2 affineSegment R x y = segment R x y", "state_after": "no goals"}]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/NumberTheory/PellMatiyasevic.lean
|
Pell.xn_succ
|
[
139,
1
] |
[
140,
6
] |
[]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Algebra/Order/Monoid/Lemmas.lean
|
Right.mul_lt_one_of_le_of_lt
|
[
903,
1
] |
[
906,
31
] |
[]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Data/Nat/Pow.lean
|
Nat.one_lt_pow
|
[
69,
1
] |
[
71,
36
] |
[{"tactic": "rw [\u2190 one_pow n]", "annotated_tactic": ["rw [\u2190 <a>one_pow</a> n]", [{"full_name": "one_pow", "def_path": "Mathlib/Algebra/GroupPower/Basic.lean", "def_pos": [90, 9], "def_end_pos": [90, 16]}]], "state_before": "n m : \u2115\nh\u2080 : 0 < n\nh\u2081 : 1 < m\n\u22a2 1 < m ^ n", "state_after": "n m : \u2115\nh\u2080 : 0 < n\nh\u2081 : 1 < m\n\u22a2 1 ^ n < m ^ n"}, {"tactic": "exact pow_lt_pow_of_lt_left h\u2081 h\u2080", "annotated_tactic": ["exact <a>pow_lt_pow_of_lt_left</a> h\u2081 h\u2080", [{"full_name": "Nat.pow_lt_pow_of_lt_left", "def_path": "Mathlib/Data/Nat/Pow.lean", "def_pos": [27, 9], "def_end_pos": [27, 30]}]], "state_before": "n m : \u2115\nh\u2080 : 0 < n\nh\u2081 : 1 < m\n\u22a2 1 ^ n < m ^ n", "state_after": "no goals"}]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Data/Polynomial/AlgebraMap.lean
|
Polynomial.aeval_X_left_apply
|
[
270,
1
] |
[
271,
41
] |
[]
|
https://github.com/leanprover/std4
|
869c615eb10130c0637a7bc038e2b80253559913
|
lake-packages/std/Std/Data/List/Lemmas.lean
|
List.findIdx_get?_eq_get_of_exists
|
[
1440,
1
] |
[
1442,
46
] |
[]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Topology/Algebra/Nonarchimedean/Bases.lean
|
RingSubgroupsBasis.hasBasis_nhds_zero
|
[
146,
1
] |
[
154,
33
] |
[{"tactic": "intro s", "annotated_tactic": ["intro s", []], "state_before": "A : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : Ring A\ninst\u271d : Nonempty \u03b9\nB : \u03b9 \u2192 AddSubgroup A\nhB : RingSubgroupsBasis B\n\u22a2 \u2200 (t : Set A), t \u2208 \ud835\udcdd 0 \u2194 \u2203 i, True \u2227 \u2191(B i) \u2286 t", "state_after": "A : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : Ring A\ninst\u271d : Nonempty \u03b9\nB : \u03b9 \u2192 AddSubgroup A\nhB : RingSubgroupsBasis B\ns : Set A\n\u22a2 s \u2208 \ud835\udcdd 0 \u2194 \u2203 i, True \u2227 \u2191(B i) \u2286 s"}, {"tactic": "rw [hB.toRingFilterBasis.toAddGroupFilterBasis.nhds_zero_hasBasis.mem_iff]", "annotated_tactic": ["rw [hB.toRingFilterBasis.toAddGroupFilterBasis.nhds_zero_hasBasis.mem_iff]", []], "state_before": "A : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : Ring A\ninst\u271d : Nonempty \u03b9\nB : \u03b9 \u2192 AddSubgroup A\nhB : RingSubgroupsBasis B\ns : Set A\n\u22a2 s \u2208 \ud835\udcdd 0 \u2194 \u2203 i, True \u2227 \u2191(B i) \u2286 s", "state_after": "A : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : Ring A\ninst\u271d : Nonempty \u03b9\nB : \u03b9 \u2192 AddSubgroup A\nhB : RingSubgroupsBasis B\ns : Set A\n\u22a2 (\u2203 i, i \u2208 RingFilterBasis.toAddGroupFilterBasis \u2227 id i \u2286 s) \u2194 \u2203 i, True \u2227 \u2191(B i) \u2286 s"}, {"tactic": "constructor", "annotated_tactic": ["constructor", []], "state_before": "A : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : Ring A\ninst\u271d : Nonempty \u03b9\nB : \u03b9 \u2192 AddSubgroup A\nhB : RingSubgroupsBasis B\ns : Set A\n\u22a2 (\u2203 i, i \u2208 RingFilterBasis.toAddGroupFilterBasis \u2227 id i \u2286 s) \u2194 \u2203 i, True \u2227 \u2191(B i) \u2286 s", "state_after": "case mp\nA : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : Ring A\ninst\u271d : Nonempty \u03b9\nB : \u03b9 \u2192 AddSubgroup A\nhB : RingSubgroupsBasis B\ns : Set A\n\u22a2 (\u2203 i, i \u2208 RingFilterBasis.toAddGroupFilterBasis \u2227 id i \u2286 s) \u2192 \u2203 i, True \u2227 \u2191(B i) \u2286 s\n\ncase mpr\nA : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : Ring A\ninst\u271d : Nonempty \u03b9\nB : \u03b9 \u2192 AddSubgroup A\nhB : RingSubgroupsBasis B\ns : Set A\n\u22a2 (\u2203 i, True \u2227 \u2191(B i) \u2286 s) \u2192 \u2203 i, i \u2208 RingFilterBasis.toAddGroupFilterBasis \u2227 id i \u2286 s"}, {"tactic": "rintro \u27e8-, \u27e8i, rfl\u27e9, hi\u27e9", "annotated_tactic": ["rintro \u27e8-, \u27e8i, rfl\u27e9, hi\u27e9", []], "state_before": "case mp\nA : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : Ring A\ninst\u271d : Nonempty \u03b9\nB : \u03b9 \u2192 AddSubgroup A\nhB : RingSubgroupsBasis B\ns : Set A\n\u22a2 (\u2203 i, i \u2208 RingFilterBasis.toAddGroupFilterBasis \u2227 id i \u2286 s) \u2192 \u2203 i, True \u2227 \u2191(B i) \u2286 s", "state_after": "case mp.intro.intro.intro\nA : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : Ring A\ninst\u271d : Nonempty \u03b9\nB : \u03b9 \u2192 AddSubgroup A\nhB : RingSubgroupsBasis B\ns : Set A\ni : \u03b9\nhi : id \u2191(B i) \u2286 s\n\u22a2 \u2203 i, True \u2227 \u2191(B i) \u2286 s"}, {"tactic": "exact \u27e8i, trivial, hi\u27e9", "annotated_tactic": ["exact \u27e8i, <a>trivial</a>, hi\u27e9", [{"full_name": "trivial", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [514, 31], "def_end_pos": [514, 38]}]], "state_before": "case mp.intro.intro.intro\nA : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : Ring A\ninst\u271d : Nonempty \u03b9\nB : \u03b9 \u2192 AddSubgroup A\nhB : RingSubgroupsBasis B\ns : Set A\ni : \u03b9\nhi : id \u2191(B i) \u2286 s\n\u22a2 \u2203 i, True \u2227 \u2191(B i) \u2286 s", "state_after": "no goals"}, {"tactic": "rintro \u27e8i, -, hi\u27e9", "annotated_tactic": ["rintro \u27e8i, -, hi\u27e9", []], "state_before": "case mpr\nA : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : Ring A\ninst\u271d : Nonempty \u03b9\nB : \u03b9 \u2192 AddSubgroup A\nhB : RingSubgroupsBasis B\ns : Set A\n\u22a2 (\u2203 i, True \u2227 \u2191(B i) \u2286 s) \u2192 \u2203 i, i \u2208 RingFilterBasis.toAddGroupFilterBasis \u2227 id i \u2286 s", "state_after": "case mpr.intro.intro\nA : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : Ring A\ninst\u271d : Nonempty \u03b9\nB : \u03b9 \u2192 AddSubgroup A\nhB : RingSubgroupsBasis B\ns : Set A\ni : \u03b9\nhi : \u2191(B i) \u2286 s\n\u22a2 \u2203 i, i \u2208 RingFilterBasis.toAddGroupFilterBasis \u2227 id i \u2286 s"}, {"tactic": "exact \u27e8B i, \u27e8i, rfl\u27e9, hi\u27e9", "annotated_tactic": ["exact \u27e8B i, \u27e8i, <a>rfl</a>\u27e9, hi\u27e9", [{"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "case mpr.intro.intro\nA : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : Ring A\ninst\u271d : Nonempty \u03b9\nB : \u03b9 \u2192 AddSubgroup A\nhB : RingSubgroupsBasis B\ns : Set A\ni : \u03b9\nhi : \u2191(B i) \u2286 s\n\u22a2 \u2203 i, i \u2208 RingFilterBasis.toAddGroupFilterBasis \u2227 id i \u2286 s", "state_after": "no goals"}]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/MeasureTheory/Measure/VectorMeasure.lean
|
MeasureTheory.Measure.toSignedMeasure_add
|
[
465,
1
] |
[
471,
41
] |
[{"tactic": "ext i hi", "annotated_tactic": ["ext i hi", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\n\u22a2 toSignedMeasure (\u03bc + \u03bd) = toSignedMeasure \u03bc + toSignedMeasure \u03bd", "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\ni : Set \u03b1\nhi : MeasurableSet i\n\u22a2 \u2191(toSignedMeasure (\u03bc + \u03bd)) i = \u2191(toSignedMeasure \u03bc + toSignedMeasure \u03bd) i"}, {"tactic": "rw [toSignedMeasure_apply_measurable hi, add_apply,\n ENNReal.toReal_add (ne_of_lt (measure_lt_top _ _)) (ne_of_lt (measure_lt_top _ _)),\n VectorMeasure.add_apply, toSignedMeasure_apply_measurable hi,\n toSignedMeasure_apply_measurable hi]", "annotated_tactic": ["rw [<a>toSignedMeasure_apply_measurable</a> hi, <a>add_apply</a>,\n <a>ENNReal.toReal_add</a> (<a>ne_of_lt</a> (<a>measure_lt_top</a> _ _)) (<a>ne_of_lt</a> (<a>measure_lt_top</a> _ _)),\n <a>VectorMeasure.add_apply</a>, <a>toSignedMeasure_apply_measurable</a> hi,\n <a>toSignedMeasure_apply_measurable</a> hi]", [{"full_name": "MeasureTheory.Measure.toSignedMeasure_apply_measurable", "def_path": "Mathlib/MeasureTheory/Measure/VectorMeasure.lean", "def_pos": [435, 9], "def_end_pos": [435, 41]}, {"full_name": "MeasureTheory.Measure.add_apply", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [798, 9], "def_end_pos": [798, 18]}, {"full_name": "ENNReal.toReal_add", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1997, 9], "def_end_pos": [1997, 19]}, {"full_name": "ne_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [101, 9], "def_end_pos": [101, 17]}, {"full_name": "MeasureTheory.measure_lt_top", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2866, 9], "def_end_pos": [2866, 23]}, {"full_name": "ne_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [101, 9], "def_end_pos": [101, 17]}, {"full_name": "MeasureTheory.measure_lt_top", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2866, 9], "def_end_pos": [2866, 23]}, {"full_name": "MeasureTheory.VectorMeasure.add_apply", "def_path": "Mathlib/MeasureTheory/Measure/VectorMeasure.lean", "def_pos": [314, 9], "def_end_pos": [314, 18]}, {"full_name": "MeasureTheory.Measure.toSignedMeasure_apply_measurable", "def_path": "Mathlib/MeasureTheory/Measure/VectorMeasure.lean", "def_pos": [435, 9], "def_end_pos": [435, 41]}, {"full_name": "MeasureTheory.Measure.toSignedMeasure_apply_measurable", "def_path": "Mathlib/MeasureTheory/Measure/VectorMeasure.lean", "def_pos": [435, 9], "def_end_pos": [435, 41]}]], "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\ni : Set \u03b1\nhi : MeasurableSet i\n\u22a2 \u2191(toSignedMeasure (\u03bc + \u03bd)) i = \u2191(toSignedMeasure \u03bc + toSignedMeasure \u03bd) i", "state_after": "no goals"}]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/MeasureTheory/Measure/MeasureSpace.lean
|
MeasureTheory.union_ae_eq_left_iff_ae_subset
|
[
309,
1
] |
[
315,
67
] |
[{"tactic": "rw [ae_le_set]", "annotated_tactic": ["rw [<a>ae_le_set</a>]", [{"full_name": "MeasureTheory.ae_le_set", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [466, 9], "def_end_pos": [466, 18]}]], "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\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns s\u2081 s\u2082 t : Set \u03b1\n\u22a2 s \u222a t =\u1d50[\u03bc] s \u2194 t \u2264\u1d50[\u03bc] s", "state_after": "\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\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns s\u2081 s\u2082 t : Set \u03b1\n\u22a2 s \u222a t =\u1d50[\u03bc] s \u2194 \u2191\u2191\u03bc (t \\ s) = 0"}, {"tactic": "refine'\n \u27e8fun h => by simpa only [union_diff_left] using (ae_eq_set.mp h).1, fun h =>\n eventuallyLE_antisymm_iff.mpr\n \u27e8by rwa [ae_le_set, union_diff_left],\n HasSubset.Subset.eventuallyLE <| subset_union_left s t\u27e9\u27e9", "annotated_tactic": ["refine'\n \u27e8fun h => by simpa only [<a>union_diff_left</a>] using (ae_eq_set.mp h).1, fun h =>\n eventuallyLE_antisymm_iff.mpr\n \u27e8by rwa [<a>ae_le_set</a>, <a>union_diff_left</a>],\n <a>HasSubset.Subset.eventuallyLE</a> <| <a>subset_union_left</a> s t\u27e9\u27e9", [{"full_name": "Set.union_diff_left", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1866, 9], "def_end_pos": [1866, 24]}, {"full_name": "MeasureTheory.ae_le_set", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [466, 9], "def_end_pos": [466, 18]}, {"full_name": "Set.union_diff_left", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1866, 9], "def_end_pos": [1866, 24]}, {"full_name": "HasSubset.Subset.eventuallyLE", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [3239, 9], "def_end_pos": [3239, 38]}, {"full_name": "Set.subset_union_left", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [829, 9], "def_end_pos": [829, 26]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns s\u2081 s\u2082 t : Set \u03b1\n\u22a2 s \u222a t =\u1d50[\u03bc] s \u2194 \u2191\u2191\u03bc (t \\ s) = 0", "state_after": "no goals"}, {"tactic": "simpa only [union_diff_left] using (ae_eq_set.mp h).1", "annotated_tactic": ["simpa only [<a>union_diff_left</a>] using (ae_eq_set.mp h).1", [{"full_name": "Set.union_diff_left", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1866, 9], "def_end_pos": [1866, 24]}]], "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\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns s\u2081 s\u2082 t : Set \u03b1\nh : s \u222a t =\u1d50[\u03bc] s\n\u22a2 \u2191\u2191\u03bc (t \\ s) = 0", "state_after": "no goals"}, {"tactic": "rwa [ae_le_set, union_diff_left]", "annotated_tactic": ["rwa [<a>ae_le_set</a>, <a>union_diff_left</a>]", [{"full_name": "MeasureTheory.ae_le_set", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [466, 9], "def_end_pos": [466, 18]}, {"full_name": "Set.union_diff_left", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1866, 9], "def_end_pos": [1866, 24]}]], "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\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns s\u2081 s\u2082 t : Set \u03b1\nh : \u2191\u2191\u03bc (t \\ s) = 0\n\u22a2 s \u222a t \u2264\u1d50[\u03bc] s", "state_after": "no goals"}]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/CategoryTheory/Types.lean
|
CategoryTheory.FunctorToTypes.naturality
|
[
151,
1
] |
[
152,
31
] |
[]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Data/Nat/Interval.lean
|
Nat.card_Ico
|
[
103,
1
] |
[
104,
27
] |
[]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Analysis/SpecificLimits/Normed.lean
|
SeminormedAddCommGroup.cauchySeq_of_le_geometric
|
[
375,
8
] |
[
377,
69
] |
[{"tactic": "simpa [dist_eq_norm] using h", "annotated_tactic": ["simpa [<a>dist_eq_norm</a>] using h", [{"full_name": "dist_eq_norm", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [383, 7], "def_end_pos": [383, 19]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\ninst\u271d : SeminormedAddCommGroup \u03b1\nr\u271d C\u271d : \u211d\nf : \u2115 \u2192 \u03b1\nC r : \u211d\nhr : r < 1\nu : \u2115 \u2192 \u03b1\nh : \u2200 (n : \u2115), \u2016u n - u (n + 1)\u2016 \u2264 C * r ^ n\n\u22a2 \u2200 (n : \u2115), dist (u n) (u (n + 1)) \u2264 C * r ^ n", "state_after": "no goals"}]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Data/Real/Basic.lean
|
Real.ind_mk
|
[
360,
11
] |
[
363,
12
] |
[{"tactic": "cases' x with x", "annotated_tactic": ["cases' x with x", []], "state_before": "x\u271d y : \u211d\nC : \u211d \u2192 Prop\nx : \u211d\nh : \u2200 (y : CauSeq \u211a abs), C (mk y)\n\u22a2 C x", "state_after": "case ofCauchy\nx\u271d y : \u211d\nC : \u211d \u2192 Prop\nh : \u2200 (y : CauSeq \u211a abs), C (mk y)\nx : Cauchy abs\n\u22a2 C { cauchy := x }"}, {"tactic": "induction' x using Quot.induction_on with x", "annotated_tactic": ["induction' x using <a>Quot.induction_on</a> with x", [{"full_name": "Quot.induction_on", "def_path": "Mathlib/Data/Quot.lean", "def_pos": [46, 19], "def_end_pos": [46, 31]}]], "state_before": "case ofCauchy\nx\u271d y : \u211d\nC : \u211d \u2192 Prop\nh : \u2200 (y : CauSeq \u211a abs), C (mk y)\nx : Cauchy abs\n\u22a2 C { cauchy := x }", "state_after": "case ofCauchy.h\nx\u271d y : \u211d\nC : \u211d \u2192 Prop\nh : \u2200 (y : CauSeq \u211a abs), C (mk y)\nx : CauSeq \u211a abs\n\u22a2 C { cauchy := Quot.mk Setoid.r x }"}, {"tactic": "exact h x", "annotated_tactic": ["exact h x", []], "state_before": "case ofCauchy.h\nx\u271d y : \u211d\nC : \u211d \u2192 Prop\nh : \u2200 (y : CauSeq \u211a abs), C (mk y)\nx : CauSeq \u211a abs\n\u22a2 C { cauchy := Quot.mk Setoid.r x }", "state_after": "no goals"}]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/NumberTheory/LegendreSymbol/JacobiSymbol.lean
|
qrSign.sq_eq_one
|
[
380,
1
] |
[
381,
76
] |
[{"tactic": "rw [neg_one_pow hm hn, \u2190 pow_mul, mul_comm, pow_mul, neg_one_sq, one_pow]", "annotated_tactic": ["rw [<a>neg_one_pow</a> hm hn, \u2190 <a>pow_mul</a>, <a>mul_comm</a>, <a>pow_mul</a>, <a>neg_one_sq</a>, <a>one_pow</a>]", [{"full_name": "qrSign.neg_one_pow", "def_path": "Mathlib/NumberTheory/LegendreSymbol/JacobiSymbol.lean", "def_pos": [371, 9], "def_end_pos": [371, 20]}, {"full_name": "pow_mul", "def_path": "Mathlib/Algebra/GroupPower/Basic.lean", "def_pos": [125, 9], "def_end_pos": [125, 16]}, {"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [302, 9], "def_end_pos": [302, 17]}, {"full_name": "pow_mul", "def_path": "Mathlib/Algebra/GroupPower/Basic.lean", "def_pos": [125, 9], "def_end_pos": [125, 16]}, {"full_name": "neg_one_sq", "def_path": "Mathlib/Algebra/GroupPower/Ring.lean", "def_pos": [223, 9], "def_end_pos": [223, 19]}, {"full_name": "one_pow", "def_path": "Mathlib/Algebra/GroupPower/Basic.lean", "def_pos": [90, 9], "def_end_pos": [90, 16]}]], "state_before": "m n : \u2115\nhm : Odd m\nhn : Odd n\n\u22a2 qrSign m n ^ 2 = 1", "state_after": "no goals"}]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Analysis/Convex/Gauge.lean
|
continuous_gauge
|
[
411,
1
] |
[
428,
75
] |
[{"tactic": "have ha : Absorbent \u211d s := absorbent_nhds_zero hs\u2080", "annotated_tactic": ["have ha : <a>Absorbent</a> \u211d s := <a>absorbent_nhds_zero</a> hs\u2080", [{"full_name": "Absorbent", "def_path": "Mathlib/Analysis/LocallyConvex/Basic.lean", "def_pos": [121, 5], "def_end_pos": [121, 14]}, {"full_name": "absorbent_nhds_zero", "def_path": "Mathlib/Analysis/LocallyConvex/Basic.lean", "def_pos": [338, 9], "def_end_pos": [338, 28]}]], "state_before": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : Module \u211d E\ns t : Set E\na : \u211d\ninst\u271d\u00b2 : TopologicalSpace E\ninst\u271d\u00b9 : TopologicalAddGroup E\ninst\u271d : ContinuousSMul \u211d E\nhc : Convex \u211d s\nhs\u2080 : s \u2208 \ud835\udcdd 0\n\u22a2 Continuous (gauge s)", "state_after": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : Module \u211d E\ns t : Set E\na : \u211d\ninst\u271d\u00b2 : TopologicalSpace E\ninst\u271d\u00b9 : TopologicalAddGroup E\ninst\u271d : ContinuousSMul \u211d E\nhc : Convex \u211d s\nhs\u2080 : s \u2208 \ud835\udcdd 0\nha : Absorbent \u211d s\n\u22a2 Continuous (gauge s)"}, {"tactic": "simp only [continuous_iff_continuousAt, ContinuousAt, (nhds_basis_Icc_pos _).tendsto_right_iff]", "annotated_tactic": ["simp only [<a>continuous_iff_continuousAt</a>, <a>ContinuousAt</a>, (<a>nhds_basis_Icc_pos</a> _).<a>tendsto_right_iff</a>]", [{"full_name": "continuous_iff_continuousAt", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [1713, 9], "def_end_pos": [1713, 36]}, {"full_name": "ContinuousAt", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [1620, 5], "def_end_pos": [1620, 17]}, {"full_name": "nhds_basis_Icc_pos", "def_path": "Mathlib/Topology/Order/Basic.lean", "def_pos": [1998, 9], "def_end_pos": [1998, 27]}, {"full_name": "Filter.HasBasis.tendsto_right_iff", "def_path": "Mathlib/Order/Filter/Bases.lean", "def_pos": [864, 9], "def_end_pos": [864, 35]}]], "state_before": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : Module \u211d E\ns t : Set E\na : \u211d\ninst\u271d\u00b2 : TopologicalSpace E\ninst\u271d\u00b9 : TopologicalAddGroup E\ninst\u271d : ContinuousSMul \u211d E\nhc : Convex \u211d s\nhs\u2080 : s \u2208 \ud835\udcdd 0\nha : Absorbent \u211d s\n\u22a2 Continuous (gauge s)", "state_after": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : Module \u211d E\ns t : Set E\na : \u211d\ninst\u271d\u00b2 : TopologicalSpace E\ninst\u271d\u00b9 : TopologicalAddGroup E\ninst\u271d : ContinuousSMul \u211d E\nhc : Convex \u211d s\nhs\u2080 : s \u2208 \ud835\udcdd 0\nha : Absorbent \u211d s\n\u22a2 \u2200 (x : E) (i : \u211d), 0 < i \u2192 \u2200\u1da0 (x_1 : E) in \ud835\udcdd x, gauge s x_1 \u2208 Icc (gauge s x - i) (gauge s x + i)"}, {"tactic": "intro x \u03b5 h\u03b5\u2080", "annotated_tactic": ["intro x \u03b5 h\u03b5\u2080", []], "state_before": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : Module \u211d E\ns t : Set E\na : \u211d\ninst\u271d\u00b2 : TopologicalSpace E\ninst\u271d\u00b9 : TopologicalAddGroup E\ninst\u271d : ContinuousSMul \u211d E\nhc : Convex \u211d s\nhs\u2080 : s \u2208 \ud835\udcdd 0\nha : Absorbent \u211d s\n\u22a2 \u2200 (x : E) (i : \u211d), 0 < i \u2192 \u2200\u1da0 (x_1 : E) in \ud835\udcdd x, gauge s x_1 \u2208 Icc (gauge s x - i) (gauge s x + i)", "state_after": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : Module \u211d E\ns t : Set E\na : \u211d\ninst\u271d\u00b2 : TopologicalSpace E\ninst\u271d\u00b9 : TopologicalAddGroup E\ninst\u271d : ContinuousSMul \u211d E\nhc : Convex \u211d s\nhs\u2080 : s \u2208 \ud835\udcdd 0\nha : Absorbent \u211d s\nx : E\n\u03b5 : \u211d\nh\u03b5\u2080 : 0 < \u03b5\n\u22a2 \u2200\u1da0 (x_1 : E) in \ud835\udcdd x, gauge s x_1 \u2208 Icc (gauge s x - \u03b5) (gauge s x + \u03b5)"}, {"tactic": "rw [\u2190 map_add_left_nhds_zero, eventually_map]", "annotated_tactic": ["rw [\u2190 <a>map_add_left_nhds_zero</a>, <a>eventually_map</a>]", [{"full_name": "map_add_left_nhds_zero", "def_path": "Mathlib/Topology/Algebra/Group/Basic.lean", "def_pos": [822, 3], "def_end_pos": [822, 14]}, {"full_name": "Filter.eventually_map", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1845, 9], "def_end_pos": [1845, 23]}]], "state_before": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : Module \u211d E\ns t : Set E\na : \u211d\ninst\u271d\u00b2 : TopologicalSpace E\ninst\u271d\u00b9 : TopologicalAddGroup E\ninst\u271d : ContinuousSMul \u211d E\nhc : Convex \u211d s\nhs\u2080 : s \u2208 \ud835\udcdd 0\nha : Absorbent \u211d s\nx : E\n\u03b5 : \u211d\nh\u03b5\u2080 : 0 < \u03b5\n\u22a2 \u2200\u1da0 (x_1 : E) in \ud835\udcdd x, gauge s x_1 \u2208 Icc (gauge s x - \u03b5) (gauge s x + \u03b5)", "state_after": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : Module \u211d E\ns t : Set E\na : \u211d\ninst\u271d\u00b2 : TopologicalSpace E\ninst\u271d\u00b9 : TopologicalAddGroup E\ninst\u271d : ContinuousSMul \u211d E\nhc : Convex \u211d s\nhs\u2080 : s \u2208 \ud835\udcdd 0\nha : Absorbent \u211d s\nx : E\n\u03b5 : \u211d\nh\u03b5\u2080 : 0 < \u03b5\n\u22a2 \u2200\u1da0 (a : E) in \ud835\udcdd 0, gauge s ((fun x x_1 => x + x_1) x a) \u2208 Icc (gauge s x - \u03b5) (gauge s x + \u03b5)"}, {"tactic": "have : \u03b5 \u2022 s \u2229 -(\u03b5 \u2022 s) \u2208 \ud835\udcdd 0", "annotated_tactic": ["have : \u03b5 \u2022 s \u2229 -(\u03b5 \u2022 s) \u2208 \ud835\udcdd 0", []], "state_before": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : Module \u211d E\ns t : Set E\na : \u211d\ninst\u271d\u00b2 : TopologicalSpace E\ninst\u271d\u00b9 : TopologicalAddGroup E\ninst\u271d : ContinuousSMul \u211d E\nhc : Convex \u211d s\nhs\u2080 : s \u2208 \ud835\udcdd 0\nha : Absorbent \u211d s\nx : E\n\u03b5 : \u211d\nh\u03b5\u2080 : 0 < \u03b5\n\u22a2 \u2200\u1da0 (a : E) in \ud835\udcdd 0, gauge s ((fun x x_1 => x + x_1) x a) \u2208 Icc (gauge s x - \u03b5) (gauge s x + \u03b5)", "state_after": "case this\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : Module \u211d E\ns t : Set E\na : \u211d\ninst\u271d\u00b2 : TopologicalSpace E\ninst\u271d\u00b9 : TopologicalAddGroup E\ninst\u271d : ContinuousSMul \u211d E\nhc : Convex \u211d s\nhs\u2080 : s \u2208 \ud835\udcdd 0\nha : Absorbent \u211d s\nx : E\n\u03b5 : \u211d\nh\u03b5\u2080 : 0 < \u03b5\n\u22a2 \u03b5 \u2022 s \u2229 -(\u03b5 \u2022 s) \u2208 \ud835\udcdd 0\n\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : Module \u211d E\ns t : Set E\na : \u211d\ninst\u271d\u00b2 : TopologicalSpace E\ninst\u271d\u00b9 : TopologicalAddGroup E\ninst\u271d : ContinuousSMul \u211d E\nhc : Convex \u211d s\nhs\u2080 : s \u2208 \ud835\udcdd 0\nha : Absorbent \u211d s\nx : E\n\u03b5 : \u211d\nh\u03b5\u2080 : 0 < \u03b5\nthis : \u03b5 \u2022 s \u2229 -(\u03b5 \u2022 s) \u2208 \ud835\udcdd 0\n\u22a2 \u2200\u1da0 (a : E) in \ud835\udcdd 0, gauge s ((fun x x_1 => x + x_1) x a) \u2208 Icc (gauge s x - \u03b5) (gauge s x + \u03b5)"}, {"tactic": "filter_upwards [this] with y hy", "annotated_tactic": ["filter_upwards [this] with y hy", []], "state_before": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : Module \u211d E\ns t : Set E\na : \u211d\ninst\u271d\u00b2 : TopologicalSpace E\ninst\u271d\u00b9 : TopologicalAddGroup E\ninst\u271d : ContinuousSMul \u211d E\nhc : Convex \u211d s\nhs\u2080 : s \u2208 \ud835\udcdd 0\nha : Absorbent \u211d s\nx : E\n\u03b5 : \u211d\nh\u03b5\u2080 : 0 < \u03b5\nthis : \u03b5 \u2022 s \u2229 -(\u03b5 \u2022 s) \u2208 \ud835\udcdd 0\n\u22a2 \u2200\u1da0 (a : E) in \ud835\udcdd 0, gauge s ((fun x x_1 => x + x_1) x a) \u2208 Icc (gauge s x - \u03b5) (gauge s x + \u03b5)", "state_after": "case h\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : Module \u211d E\ns t : Set E\na : \u211d\ninst\u271d\u00b2 : TopologicalSpace E\ninst\u271d\u00b9 : TopologicalAddGroup E\ninst\u271d : ContinuousSMul \u211d E\nhc : Convex \u211d s\nhs\u2080 : s \u2208 \ud835\udcdd 0\nha : Absorbent \u211d s\nx : E\n\u03b5 : \u211d\nh\u03b5\u2080 : 0 < \u03b5\nthis : \u03b5 \u2022 s \u2229 -(\u03b5 \u2022 s) \u2208 \ud835\udcdd 0\ny : E\nhy : y \u2208 \u03b5 \u2022 s \u2229 -(\u03b5 \u2022 s)\n\u22a2 gauge s (x + y) \u2208 Icc (gauge s x - \u03b5) (gauge s x + \u03b5)"}, {"tactic": "constructor", "annotated_tactic": ["constructor", []], "state_before": "case h\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : Module \u211d E\ns t : Set E\na : \u211d\ninst\u271d\u00b2 : TopologicalSpace E\ninst\u271d\u00b9 : TopologicalAddGroup E\ninst\u271d : ContinuousSMul \u211d E\nhc : Convex \u211d s\nhs\u2080 : s \u2208 \ud835\udcdd 0\nha : Absorbent \u211d s\nx : E\n\u03b5 : \u211d\nh\u03b5\u2080 : 0 < \u03b5\nthis : \u03b5 \u2022 s \u2229 -(\u03b5 \u2022 s) \u2208 \ud835\udcdd 0\ny : E\nhy : y \u2208 \u03b5 \u2022 s \u2229 -(\u03b5 \u2022 s)\n\u22a2 gauge s (x + y) \u2208 Icc (gauge s x - \u03b5) (gauge s x + \u03b5)", "state_after": "case h.left\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : Module \u211d E\ns t : Set E\na : \u211d\ninst\u271d\u00b2 : TopologicalSpace E\ninst\u271d\u00b9 : TopologicalAddGroup E\ninst\u271d : ContinuousSMul \u211d E\nhc : Convex \u211d s\nhs\u2080 : s \u2208 \ud835\udcdd 0\nha : Absorbent \u211d s\nx : E\n\u03b5 : \u211d\nh\u03b5\u2080 : 0 < \u03b5\nthis : \u03b5 \u2022 s \u2229 -(\u03b5 \u2022 s) \u2208 \ud835\udcdd 0\ny : E\nhy : y \u2208 \u03b5 \u2022 s \u2229 -(\u03b5 \u2022 s)\n\u22a2 gauge s x - \u03b5 \u2264 gauge s (x + y)\n\ncase h.right\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : Module \u211d E\ns t : Set E\na : \u211d\ninst\u271d\u00b2 : TopologicalSpace E\ninst\u271d\u00b9 : TopologicalAddGroup E\ninst\u271d : ContinuousSMul \u211d E\nhc : Convex \u211d s\nhs\u2080 : s \u2208 \ud835\udcdd 0\nha : Absorbent \u211d s\nx : E\n\u03b5 : \u211d\nh\u03b5\u2080 : 0 < \u03b5\nthis : \u03b5 \u2022 s \u2229 -(\u03b5 \u2022 s) \u2208 \ud835\udcdd 0\ny : E\nhy : y \u2208 \u03b5 \u2022 s \u2229 -(\u03b5 \u2022 s)\n\u22a2 gauge s (x + y) \u2264 gauge s x + \u03b5"}, {"tactic": "exact inter_mem ((set_smul_mem_nhds_zero_iff h\u03b5\u2080.ne').2 hs\u2080)\n (neg_mem_nhds_zero _ ((set_smul_mem_nhds_zero_iff h\u03b5\u2080.ne').2 hs\u2080))", "annotated_tactic": ["exact <a>inter_mem</a> ((<a>set_smul_mem_nhds_zero_iff</a> h\u03b5\u2080.ne').2 hs\u2080)\n (<a>neg_mem_nhds_zero</a> _ ((<a>set_smul_mem_nhds_zero_iff</a> h\u03b5\u2080.ne').2 hs\u2080))", [{"full_name": "Filter.inter_mem", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [156, 9], "def_end_pos": [156, 18]}, {"full_name": "set_smul_mem_nhds_zero_iff", "def_path": "Mathlib/Topology/Algebra/ConstMulAction.lean", "def_pos": [576, 9], "def_end_pos": [576, 35]}, {"full_name": "neg_mem_nhds_zero", "def_path": "Mathlib/Topology/Algebra/Group/Basic.lean", "def_pos": [641, 3], "def_end_pos": [641, 14]}, {"full_name": "set_smul_mem_nhds_zero_iff", "def_path": "Mathlib/Topology/Algebra/ConstMulAction.lean", "def_pos": [576, 9], "def_end_pos": [576, 35]}]], "state_before": "case this\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : Module \u211d E\ns t : Set E\na : \u211d\ninst\u271d\u00b2 : TopologicalSpace E\ninst\u271d\u00b9 : TopologicalAddGroup E\ninst\u271d : ContinuousSMul \u211d E\nhc : Convex \u211d s\nhs\u2080 : s \u2208 \ud835\udcdd 0\nha : Absorbent \u211d s\nx : E\n\u03b5 : \u211d\nh\u03b5\u2080 : 0 < \u03b5\n\u22a2 \u03b5 \u2022 s \u2229 -(\u03b5 \u2022 s) \u2208 \ud835\udcdd 0", "state_after": "no goals"}, {"tactic": "rw [sub_le_iff_le_add]", "annotated_tactic": ["rw [<a>sub_le_iff_le_add</a>]", [{"full_name": "sub_le_iff_le_add", "def_path": "Mathlib/Algebra/Order/Group/Defs.lean", "def_pos": [750, 3], "def_end_pos": [750, 14]}]], "state_before": "case h.left\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : Module \u211d E\ns t : Set E\na : \u211d\ninst\u271d\u00b2 : TopologicalSpace E\ninst\u271d\u00b9 : TopologicalAddGroup E\ninst\u271d : ContinuousSMul \u211d E\nhc : Convex \u211d s\nhs\u2080 : s \u2208 \ud835\udcdd 0\nha : Absorbent \u211d s\nx : E\n\u03b5 : \u211d\nh\u03b5\u2080 : 0 < \u03b5\nthis : \u03b5 \u2022 s \u2229 -(\u03b5 \u2022 s) \u2208 \ud835\udcdd 0\ny : E\nhy : y \u2208 \u03b5 \u2022 s \u2229 -(\u03b5 \u2022 s)\n\u22a2 gauge s x - \u03b5 \u2264 gauge s (x + y)", "state_after": "case h.left\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : Module \u211d E\ns t : Set E\na : \u211d\ninst\u271d\u00b2 : TopologicalSpace E\ninst\u271d\u00b9 : TopologicalAddGroup E\ninst\u271d : ContinuousSMul \u211d E\nhc : Convex \u211d s\nhs\u2080 : s \u2208 \ud835\udcdd 0\nha : Absorbent \u211d s\nx : E\n\u03b5 : \u211d\nh\u03b5\u2080 : 0 < \u03b5\nthis : \u03b5 \u2022 s \u2229 -(\u03b5 \u2022 s) \u2208 \ud835\udcdd 0\ny : E\nhy : y \u2208 \u03b5 \u2022 s \u2229 -(\u03b5 \u2022 s)\n\u22a2 gauge s x \u2264 gauge s (x + y) + \u03b5"}, {"tactic": "calc\n gauge s x = gauge s (x + y + (-y)) := by simp\n _ \u2264 gauge s (x + y) + gauge s (-y) := gauge_add_le hc ha _ _\n _ \u2264 gauge s (x + y) + \u03b5 := add_le_add_left (gauge_le_of_mem h\u03b5\u2080.le (mem_neg.1 hy.2)) _", "annotated_tactic": ["calc\n <a>gauge</a> s x = <a>gauge</a> s (x + y + (-y)) := by simp\n _ \u2264 <a>gauge</a> s (x + y) + <a>gauge</a> s (-y) := <a>gauge_add_le</a> hc ha _ _\n _ \u2264 <a>gauge</a> s (x + y) + \u03b5 := <a>add_le_add_left</a> (<a>gauge_le_of_mem</a> h\u03b5\u2080.le (<a>mem_neg</a>.1 hy.2)) _", [{"full_name": "gauge", "def_path": "Mathlib/Analysis/Convex/Gauge.lean", "def_pos": [57, 5], "def_end_pos": [57, 10]}, {"full_name": "gauge", "def_path": "Mathlib/Analysis/Convex/Gauge.lean", "def_pos": [57, 5], "def_end_pos": [57, 10]}, {"full_name": "gauge", "def_path": "Mathlib/Analysis/Convex/Gauge.lean", "def_pos": [57, 5], "def_end_pos": [57, 10]}, {"full_name": "gauge", "def_path": "Mathlib/Analysis/Convex/Gauge.lean", "def_pos": [57, 5], "def_end_pos": [57, 10]}, {"full_name": "gauge_add_le", "def_path": "Mathlib/Analysis/Convex/Gauge.lean", "def_pos": [204, 9], "def_end_pos": [204, 21]}, {"full_name": "gauge", "def_path": "Mathlib/Analysis/Convex/Gauge.lean", "def_pos": [57, 5], "def_end_pos": [57, 10]}, {"full_name": "add_le_add_left", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [49, 15], "def_end_pos": [49, 30]}, {"full_name": "gauge_le_of_mem", "def_path": "Mathlib/Analysis/Convex/Gauge.lean", "def_pos": [145, 9], "def_end_pos": [145, 24]}, {"full_name": "Set.mem_neg", "def_path": "Mathlib/Data/Set/Pointwise/Basic.lean", "def_pos": [179, 3], "def_end_pos": [179, 14]}]], "state_before": "case h.left\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : Module \u211d E\ns t : Set E\na : \u211d\ninst\u271d\u00b2 : TopologicalSpace E\ninst\u271d\u00b9 : TopologicalAddGroup E\ninst\u271d : ContinuousSMul \u211d E\nhc : Convex \u211d s\nhs\u2080 : s \u2208 \ud835\udcdd 0\nha : Absorbent \u211d s\nx : E\n\u03b5 : \u211d\nh\u03b5\u2080 : 0 < \u03b5\nthis : \u03b5 \u2022 s \u2229 -(\u03b5 \u2022 s) \u2208 \ud835\udcdd 0\ny : E\nhy : y \u2208 \u03b5 \u2022 s \u2229 -(\u03b5 \u2022 s)\n\u22a2 gauge s x \u2264 gauge s (x + y) + \u03b5", "state_after": "no goals"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : Module \u211d E\ns t : Set E\na : \u211d\ninst\u271d\u00b2 : TopologicalSpace E\ninst\u271d\u00b9 : TopologicalAddGroup E\ninst\u271d : ContinuousSMul \u211d E\nhc : Convex \u211d s\nhs\u2080 : s \u2208 \ud835\udcdd 0\nha : Absorbent \u211d s\nx : E\n\u03b5 : \u211d\nh\u03b5\u2080 : 0 < \u03b5\nthis : \u03b5 \u2022 s \u2229 -(\u03b5 \u2022 s) \u2208 \ud835\udcdd 0\ny : E\nhy : y \u2208 \u03b5 \u2022 s \u2229 -(\u03b5 \u2022 s)\n\u22a2 gauge s x = gauge s (x + y + -y)", "state_after": "no goals"}, {"tactic": "calc\n gauge s (x + y) \u2264 gauge s x + gauge s y := gauge_add_le hc ha _ _\n _ \u2264 gauge s x + \u03b5 := add_le_add_left (gauge_le_of_mem h\u03b5\u2080.le hy.1) _", "annotated_tactic": ["calc\n <a>gauge</a> s (x + y) \u2264 <a>gauge</a> s x + <a>gauge</a> s y := <a>gauge_add_le</a> hc ha _ _\n _ \u2264 <a>gauge</a> s x + \u03b5 := <a>add_le_add_left</a> (<a>gauge_le_of_mem</a> h\u03b5\u2080.le hy.1) _", [{"full_name": "gauge", "def_path": "Mathlib/Analysis/Convex/Gauge.lean", "def_pos": [57, 5], "def_end_pos": [57, 10]}, {"full_name": "gauge", "def_path": "Mathlib/Analysis/Convex/Gauge.lean", "def_pos": [57, 5], "def_end_pos": [57, 10]}, {"full_name": "gauge", "def_path": "Mathlib/Analysis/Convex/Gauge.lean", "def_pos": [57, 5], "def_end_pos": [57, 10]}, {"full_name": "gauge_add_le", "def_path": "Mathlib/Analysis/Convex/Gauge.lean", "def_pos": [204, 9], "def_end_pos": [204, 21]}, {"full_name": "gauge", "def_path": "Mathlib/Analysis/Convex/Gauge.lean", "def_pos": [57, 5], "def_end_pos": [57, 10]}, {"full_name": "add_le_add_left", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [49, 15], "def_end_pos": [49, 30]}, {"full_name": "gauge_le_of_mem", "def_path": "Mathlib/Analysis/Convex/Gauge.lean", "def_pos": [145, 9], "def_end_pos": [145, 24]}]], "state_before": "case h.right\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : Module \u211d E\ns t : Set E\na : \u211d\ninst\u271d\u00b2 : TopologicalSpace E\ninst\u271d\u00b9 : TopologicalAddGroup E\ninst\u271d : ContinuousSMul \u211d E\nhc : Convex \u211d s\nhs\u2080 : s \u2208 \ud835\udcdd 0\nha : Absorbent \u211d s\nx : E\n\u03b5 : \u211d\nh\u03b5\u2080 : 0 < \u03b5\nthis : \u03b5 \u2022 s \u2229 -(\u03b5 \u2022 s) \u2208 \ud835\udcdd 0\ny : E\nhy : y \u2208 \u03b5 \u2022 s \u2229 -(\u03b5 \u2022 s)\n\u22a2 gauge s (x + y) \u2264 gauge s x + \u03b5", "state_after": "no goals"}]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Analysis/Calculus/FDeriv/Basic.lean
|
fderiv_id
|
[
1110,
1
] |
[
1111,
40
] |
[]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/RepresentationTheory/GroupCohomology/Resolution.lean
|
GroupCohomology.Resolution.diagonalSucc_inv_single_left
|
[
221,
1
] |
[
240,
33
] |
[{"tactic": "refine' f.induction _ _", "annotated_tactic": ["refine' f.induction _ _", []], "state_before": "k G : Type u\ninst\u271d\u00b9 : CommRing k\nn : \u2115\ninst\u271d : Group G\ng : G\nf : (Fin n \u2192 G) \u2192\u2080 k\nr : k\n\u22a2 \u2191(diagonalSucc k G n).inv.hom ((fun\u2080 | g => r) \u2297\u209c[k] f) =\n \u2191(\u2191(Finsupp.lift ((Fin (n + 1) \u2192 G) \u2192\u2080 k) k (Fin n \u2192 G)) fun f => fun\u2080 | g \u2022 partialProd f => r) f", "state_after": "case refine'_1\nk G : Type u\ninst\u271d\u00b9 : CommRing k\nn : \u2115\ninst\u271d : Group G\ng : G\nf : (Fin n \u2192 G) \u2192\u2080 k\nr : k\n\u22a2 \u2191(diagonalSucc k G n).inv.hom ((fun\u2080 | g => r) \u2297\u209c[k] 0) =\n \u2191(\u2191(Finsupp.lift ((Fin (n + 1) \u2192 G) \u2192\u2080 k) k (Fin n \u2192 G)) fun f => fun\u2080 | g \u2022 partialProd f => r) 0\n\ncase refine'_2\nk G : Type u\ninst\u271d\u00b9 : CommRing k\nn : \u2115\ninst\u271d : Group G\ng : G\nf : (Fin n \u2192 G) \u2192\u2080 k\nr : k\n\u22a2 \u2200 (a : Fin n \u2192 G) (b : k) (f : (Fin n \u2192 G) \u2192\u2080 k),\n \u00aca \u2208 f.support \u2192\n b \u2260 0 \u2192\n \u2191(diagonalSucc k G n).inv.hom ((fun\u2080 | g => r) \u2297\u209c[k] f) =\n \u2191(\u2191(Finsupp.lift ((Fin (n + 1) \u2192 G) \u2192\u2080 k) k (Fin n \u2192 G)) fun f => fun\u2080 | g \u2022 partialProd f => r) f \u2192\n \u2191(diagonalSucc k G n).inv.hom ((fun\u2080 | g => r) \u2297\u209c[k] ((fun\u2080 | a => b) + f)) =\n \u2191(\u2191(Finsupp.lift ((Fin (n + 1) \u2192 G) \u2192\u2080 k) k (Fin n \u2192 G)) fun f => fun\u2080 | g \u2022 partialProd f => r)\n ((fun\u2080 | a => b) + f)"}, {"tactic": "rw [TensorProduct.tmul_zero, map_zero]", "annotated_tactic": ["rw [<a>TensorProduct.tmul_zero</a>, <a>map_zero</a>]", [{"full_name": "TensorProduct.tmul_zero", "def_path": "Mathlib/LinearAlgebra/TensorProduct.lean", "def_pos": [162, 9], "def_end_pos": [162, 18]}, {"full_name": "map_zero", "def_path": "Mathlib/Algebra/Hom/Group/Defs.lean", "def_pos": [201, 3], "def_end_pos": [201, 14]}]], "state_before": "case refine'_1\nk G : Type u\ninst\u271d\u00b9 : CommRing k\nn : \u2115\ninst\u271d : Group G\ng : G\nf : (Fin n \u2192 G) \u2192\u2080 k\nr : k\n\u22a2 \u2191(diagonalSucc k G n).inv.hom ((fun\u2080 | g => r) \u2297\u209c[k] 0) =\n \u2191(\u2191(Finsupp.lift ((Fin (n + 1) \u2192 G) \u2192\u2080 k) k (Fin n \u2192 G)) fun f => fun\u2080 | g \u2022 partialProd f => r) 0", "state_after": "case refine'_1\nk G : Type u\ninst\u271d\u00b9 : CommRing k\nn : \u2115\ninst\u271d : Group G\ng : G\nf : (Fin n \u2192 G) \u2192\u2080 k\nr : k\n\u22a2 \u2191(diagonalSucc k G n).inv.hom 0 = 0"}, {"tactic": "erw [map_zero]", "annotated_tactic": ["erw [<a>map_zero</a>]", [{"full_name": "map_zero", "def_path": "Mathlib/Algebra/Hom/Group/Defs.lean", "def_pos": [201, 3], "def_end_pos": [201, 14]}]], "state_before": "case refine'_1\nk G : Type u\ninst\u271d\u00b9 : CommRing k\nn : \u2115\ninst\u271d : Group G\ng : G\nf : (Fin n \u2192 G) \u2192\u2080 k\nr : k\n\u22a2 \u2191(diagonalSucc k G n).inv.hom 0 = 0", "state_after": "no goals"}, {"tactic": "intro _ _ _ _ _ hx", "annotated_tactic": ["intro _ _ _ _ _ hx", []], "state_before": "case refine'_2\nk G : Type u\ninst\u271d\u00b9 : CommRing k\nn : \u2115\ninst\u271d : Group G\ng : G\nf : (Fin n \u2192 G) \u2192\u2080 k\nr : k\n\u22a2 \u2200 (a : Fin n \u2192 G) (b : k) (f : (Fin n \u2192 G) \u2192\u2080 k),\n \u00aca \u2208 f.support \u2192\n b \u2260 0 \u2192\n \u2191(diagonalSucc k G n).inv.hom ((fun\u2080 | g => r) \u2297\u209c[k] f) =\n \u2191(\u2191(Finsupp.lift ((Fin (n + 1) \u2192 G) \u2192\u2080 k) k (Fin n \u2192 G)) fun f => fun\u2080 | g \u2022 partialProd f => r) f \u2192\n \u2191(diagonalSucc k G n).inv.hom ((fun\u2080 | g => r) \u2297\u209c[k] ((fun\u2080 | a => b) + f)) =\n \u2191(\u2191(Finsupp.lift ((Fin (n + 1) \u2192 G) \u2192\u2080 k) k (Fin n \u2192 G)) fun f => fun\u2080 | g \u2022 partialProd f => r)\n ((fun\u2080 | a => b) + f)", "state_after": "case refine'_2\nk G : Type u\ninst\u271d\u00b9 : CommRing k\nn : \u2115\ninst\u271d : Group G\ng : G\nf : (Fin n \u2192 G) \u2192\u2080 k\nr : k\na\u271d\u00b2 : Fin n \u2192 G\nb\u271d : k\nf\u271d : (Fin n \u2192 G) \u2192\u2080 k\na\u271d\u00b9 : \u00aca\u271d\u00b2 \u2208 f\u271d.support\na\u271d : b\u271d \u2260 0\nhx :\n \u2191(diagonalSucc k G n).inv.hom ((fun\u2080 | g => r) \u2297\u209c[k] f\u271d) =\n \u2191(\u2191(Finsupp.lift ((Fin (n + 1) \u2192 G) \u2192\u2080 k) k (Fin n \u2192 G)) fun f => fun\u2080 | g \u2022 partialProd f => r) f\u271d\n\u22a2 \u2191(diagonalSucc k G n).inv.hom ((fun\u2080 | g => r) \u2297\u209c[k] ((fun\u2080 | a\u271d\u00b2 => b\u271d) + f\u271d)) =\n \u2191(\u2191(Finsupp.lift ((Fin (n + 1) \u2192 G) \u2192\u2080 k) k (Fin n \u2192 G)) fun f => fun\u2080 | g \u2022 partialProd f => r)\n ((fun\u2080 | a\u271d\u00b2 => b\u271d) + f\u271d)"}, {"tactic": "rw [TensorProduct.tmul_add, map_add]", "annotated_tactic": ["rw [<a>TensorProduct.tmul_add</a>, <a>map_add</a>]", [{"full_name": "TensorProduct.tmul_add", "def_path": "Mathlib/LinearAlgebra/TensorProduct.lean", "def_pos": [168, 9], "def_end_pos": [168, 17]}, {"full_name": "map_add", "def_path": "Mathlib/Algebra/Hom/Group/Defs.lean", "def_pos": [298, 3], "def_end_pos": [298, 14]}]], "state_before": "case refine'_2\nk G : Type u\ninst\u271d\u00b9 : CommRing k\nn : \u2115\ninst\u271d : Group G\ng : G\nf : (Fin n \u2192 G) \u2192\u2080 k\nr : k\na\u271d\u00b2 : Fin n \u2192 G\nb\u271d : k\nf\u271d : (Fin n \u2192 G) \u2192\u2080 k\na\u271d\u00b9 : \u00aca\u271d\u00b2 \u2208 f\u271d.support\na\u271d : b\u271d \u2260 0\nhx :\n \u2191(diagonalSucc k G n).inv.hom ((fun\u2080 | g => r) \u2297\u209c[k] f\u271d) =\n \u2191(\u2191(Finsupp.lift ((Fin (n + 1) \u2192 G) \u2192\u2080 k) k (Fin n \u2192 G)) fun f => fun\u2080 | g \u2022 partialProd f => r) f\u271d\n\u22a2 \u2191(diagonalSucc k G n).inv.hom ((fun\u2080 | g => r) \u2297\u209c[k] ((fun\u2080 | a\u271d\u00b2 => b\u271d) + f\u271d)) =\n \u2191(\u2191(Finsupp.lift ((Fin (n + 1) \u2192 G) \u2192\u2080 k) k (Fin n \u2192 G)) fun f => fun\u2080 | g \u2022 partialProd f => r)\n ((fun\u2080 | a\u271d\u00b2 => b\u271d) + f\u271d)", "state_after": "case refine'_2\nk G : Type u\ninst\u271d\u00b9 : CommRing k\nn : \u2115\ninst\u271d : Group G\ng : G\nf : (Fin n \u2192 G) \u2192\u2080 k\nr : k\na\u271d\u00b2 : Fin n \u2192 G\nb\u271d : k\nf\u271d : (Fin n \u2192 G) \u2192\u2080 k\na\u271d\u00b9 : \u00aca\u271d\u00b2 \u2208 f\u271d.support\na\u271d : b\u271d \u2260 0\nhx :\n \u2191(diagonalSucc k G n).inv.hom ((fun\u2080 | g => r) \u2297\u209c[k] f\u271d) =\n \u2191(\u2191(Finsupp.lift ((Fin (n + 1) \u2192 G) \u2192\u2080 k) k (Fin n \u2192 G)) fun f => fun\u2080 | g \u2022 partialProd f => r) f\u271d\n\u22a2 \u2191(diagonalSucc k G n).inv.hom (((fun\u2080 | g => r) \u2297\u209c[k] fun\u2080 | a\u271d\u00b2 => b\u271d) + (fun\u2080 | g => r) \u2297\u209c[k] f\u271d) =\n (\u2191(\u2191(Finsupp.lift ((Fin (n + 1) \u2192 G) \u2192\u2080 k) k (Fin n \u2192 G)) fun f => fun\u2080 | g \u2022 partialProd f => r)\n fun\u2080 | a\u271d\u00b2 => b\u271d) +\n \u2191(\u2191(Finsupp.lift ((Fin (n + 1) \u2192 G) \u2192\u2080 k) k (Fin n \u2192 G)) fun f => fun\u2080 | g \u2022 partialProd f => r) f\u271d"}, {"tactic": "erw [map_add, hx]", "annotated_tactic": ["erw [<a>map_add</a>, hx]", [{"full_name": "map_add", "def_path": "Mathlib/Algebra/Hom/Group/Defs.lean", "def_pos": [298, 3], "def_end_pos": [298, 14]}]], "state_before": "case refine'_2\nk G : Type u\ninst\u271d\u00b9 : CommRing k\nn : \u2115\ninst\u271d : Group G\ng : G\nf : (Fin n \u2192 G) \u2192\u2080 k\nr : k\na\u271d\u00b2 : Fin n \u2192 G\nb\u271d : k\nf\u271d : (Fin n \u2192 G) \u2192\u2080 k\na\u271d\u00b9 : \u00aca\u271d\u00b2 \u2208 f\u271d.support\na\u271d : b\u271d \u2260 0\nhx :\n \u2191(diagonalSucc k G n).inv.hom ((fun\u2080 | g => r) \u2297\u209c[k] f\u271d) =\n \u2191(\u2191(Finsupp.lift ((Fin (n + 1) \u2192 G) \u2192\u2080 k) k (Fin n \u2192 G)) fun f => fun\u2080 | g \u2022 partialProd f => r) f\u271d\n\u22a2 \u2191(diagonalSucc k G n).inv.hom (((fun\u2080 | g => r) \u2297\u209c[k] fun\u2080 | a\u271d\u00b2 => b\u271d) + (fun\u2080 | g => r) \u2297\u209c[k] f\u271d) =\n (\u2191(\u2191(Finsupp.lift ((Fin (n + 1) \u2192 G) \u2192\u2080 k) k (Fin n \u2192 G)) fun f => fun\u2080 | g \u2022 partialProd f => r)\n fun\u2080 | a\u271d\u00b2 => b\u271d) +\n \u2191(\u2191(Finsupp.lift ((Fin (n + 1) \u2192 G) \u2192\u2080 k) k (Fin n \u2192 G)) fun f => fun\u2080 | g \u2022 partialProd f => r) f\u271d", "state_after": "case refine'_2\nk G : Type u\ninst\u271d\u00b9 : CommRing k\nn : \u2115\ninst\u271d : Group G\ng : G\nf : (Fin n \u2192 G) \u2192\u2080 k\nr : k\na\u271d\u00b2 : Fin n \u2192 G\nb\u271d : k\nf\u271d : (Fin n \u2192 G) \u2192\u2080 k\na\u271d\u00b9 : \u00aca\u271d\u00b2 \u2208 f\u271d.support\na\u271d : b\u271d \u2260 0\nhx :\n \u2191(diagonalSucc k G n).inv.hom ((fun\u2080 | g => r) \u2297\u209c[k] f\u271d) =\n \u2191(\u2191(Finsupp.lift ((Fin (n + 1) \u2192 G) \u2192\u2080 k) k (Fin n \u2192 G)) fun f => fun\u2080 | g \u2022 partialProd f => r) f\u271d\n\u22a2 \u2191(diagonalSucc k G n).inv.hom ((fun\u2080 | g => r) \u2297\u209c[k] fun\u2080 | a\u271d\u00b2 => b\u271d) +\n \u2191(\u2191(Finsupp.lift ((Fin (n + 1) \u2192 G) \u2192\u2080 k) k (Fin n \u2192 G)) fun f => fun\u2080 | g \u2022 partialProd f => r) f\u271d =\n (\u2191(\u2191(Finsupp.lift ((Fin (n + 1) \u2192 G) \u2192\u2080 k) k (Fin n \u2192 G)) fun f => fun\u2080 | g \u2022 partialProd f => r)\n fun\u2080 | a\u271d\u00b2 => b\u271d) +\n \u2191(\u2191(Finsupp.lift ((Fin (n + 1) \u2192 G) \u2192\u2080 k) k (Fin n \u2192 G)) fun f => fun\u2080 | g \u2022 partialProd f => r) f\u271d"}, {"tactic": "simp_rw [lift_apply, smul_single, smul_eq_mul]", "annotated_tactic": ["simp_rw [<a>lift_apply</a>, <a>smul_single</a>, <a>smul_eq_mul</a>]", [{"full_name": "Finsupp.lift_apply", "def_path": "Mathlib/LinearAlgebra/Finsupp.lean", "def_pos": [407, 9], "def_end_pos": [407, 19]}, {"full_name": "Finsupp.smul_single", "def_path": "Mathlib/Data/Finsupp/Basic.lean", "def_pos": [1577, 9], "def_end_pos": [1577, 20]}, {"full_name": "smul_eq_mul", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [93, 9], "def_end_pos": [93, 20]}]], "state_before": "case refine'_2\nk G : Type u\ninst\u271d\u00b9 : CommRing k\nn : \u2115\ninst\u271d : Group G\ng : G\nf : (Fin n \u2192 G) \u2192\u2080 k\nr : k\na\u271d\u00b2 : Fin n \u2192 G\nb\u271d : k\nf\u271d : (Fin n \u2192 G) \u2192\u2080 k\na\u271d\u00b9 : \u00aca\u271d\u00b2 \u2208 f\u271d.support\na\u271d : b\u271d \u2260 0\nhx :\n \u2191(diagonalSucc k G n).inv.hom ((fun\u2080 | g => r) \u2297\u209c[k] f\u271d) =\n \u2191(\u2191(Finsupp.lift ((Fin (n + 1) \u2192 G) \u2192\u2080 k) k (Fin n \u2192 G)) fun f => fun\u2080 | g \u2022 partialProd f => r) f\u271d\n\u22a2 \u2191(diagonalSucc k G n).inv.hom ((fun\u2080 | g => r) \u2297\u209c[k] fun\u2080 | a\u271d\u00b2 => b\u271d) +\n \u2191(\u2191(Finsupp.lift ((Fin (n + 1) \u2192 G) \u2192\u2080 k) k (Fin n \u2192 G)) fun f => fun\u2080 | g \u2022 partialProd f => r) f\u271d =\n (\u2191(\u2191(Finsupp.lift ((Fin (n + 1) \u2192 G) \u2192\u2080 k) k (Fin n \u2192 G)) fun f => fun\u2080 | g \u2022 partialProd f => r)\n fun\u2080 | a\u271d\u00b2 => b\u271d) +\n \u2191(\u2191(Finsupp.lift ((Fin (n + 1) \u2192 G) \u2192\u2080 k) k (Fin n \u2192 G)) fun f => fun\u2080 | g \u2022 partialProd f => r) f\u271d", "state_after": "case refine'_2\nk G : Type u\ninst\u271d\u00b9 : CommRing k\nn : \u2115\ninst\u271d : Group G\ng : G\nf : (Fin n \u2192 G) \u2192\u2080 k\nr : k\na\u271d\u00b2 : Fin n \u2192 G\nb\u271d : k\nf\u271d : (Fin n \u2192 G) \u2192\u2080 k\na\u271d\u00b9 : \u00aca\u271d\u00b2 \u2208 f\u271d.support\na\u271d : b\u271d \u2260 0\nhx :\n \u2191(diagonalSucc k G n).inv.hom ((fun\u2080 | g => r) \u2297\u209c[k] f\u271d) =\n \u2191(\u2191(Finsupp.lift ((Fin (n + 1) \u2192 G) \u2192\u2080 k) k (Fin n \u2192 G)) fun f => fun\u2080 | g \u2022 partialProd f => r) f\u271d\n\u22a2 (\u2191(diagonalSucc k G n).inv.hom ((fun\u2080 | g => r) \u2297\u209c[k] fun\u2080 | a\u271d\u00b2 => b\u271d) +\n sum f\u271d fun x r_1 => fun\u2080 | g \u2022 partialProd x => r_1 * r) =\n (sum (fun\u2080 | a\u271d\u00b2 => b\u271d) fun x r_1 => fun\u2080 | g \u2022 partialProd x => r_1 * r) +\n sum f\u271d fun x r_1 => fun\u2080 | g \u2022 partialProd x => r_1 * r"}, {"tactic": "erw [diagonalSucc_inv_single_single]", "annotated_tactic": ["erw [<a>diagonalSucc_inv_single_single</a>]", [{"full_name": "GroupCohomology.Resolution.diagonalSucc_inv_single_single", "def_path": "Mathlib/RepresentationTheory/GroupCohomology/Resolution.lean", "def_pos": [200, 9], "def_end_pos": [200, 39]}]], "state_before": "case refine'_2\nk G : Type u\ninst\u271d\u00b9 : CommRing k\nn : \u2115\ninst\u271d : Group G\ng : G\nf : (Fin n \u2192 G) \u2192\u2080 k\nr : k\na\u271d\u00b2 : Fin n \u2192 G\nb\u271d : k\nf\u271d : (Fin n \u2192 G) \u2192\u2080 k\na\u271d\u00b9 : \u00aca\u271d\u00b2 \u2208 f\u271d.support\na\u271d : b\u271d \u2260 0\nhx :\n \u2191(diagonalSucc k G n).inv.hom ((fun\u2080 | g => r) \u2297\u209c[k] f\u271d) =\n \u2191(\u2191(Finsupp.lift ((Fin (n + 1) \u2192 G) \u2192\u2080 k) k (Fin n \u2192 G)) fun f => fun\u2080 | g \u2022 partialProd f => r) f\u271d\n\u22a2 (\u2191(diagonalSucc k G n).inv.hom ((fun\u2080 | g => r) \u2297\u209c[k] fun\u2080 | a\u271d\u00b2 => b\u271d) +\n sum f\u271d fun x r_1 => fun\u2080 | g \u2022 partialProd x => r_1 * r) =\n (sum (fun\u2080 | a\u271d\u00b2 => b\u271d) fun x r_1 => fun\u2080 | g \u2022 partialProd x => r_1 * r) +\n sum f\u271d fun x r_1 => fun\u2080 | g \u2022 partialProd x => r_1 * r", "state_after": "case refine'_2\nk G : Type u\ninst\u271d\u00b9 : CommRing k\nn : \u2115\ninst\u271d : Group G\ng : G\nf : (Fin n \u2192 G) \u2192\u2080 k\nr : k\na\u271d\u00b2 : Fin n \u2192 G\nb\u271d : k\nf\u271d : (Fin n \u2192 G) \u2192\u2080 k\na\u271d\u00b9 : \u00aca\u271d\u00b2 \u2208 f\u271d.support\na\u271d : b\u271d \u2260 0\nhx :\n \u2191(diagonalSucc k G n).inv.hom ((fun\u2080 | g => r) \u2297\u209c[k] f\u271d) =\n \u2191(\u2191(Finsupp.lift ((Fin (n + 1) \u2192 G) \u2192\u2080 k) k (Fin n \u2192 G)) fun f => fun\u2080 | g \u2022 partialProd f => r) f\u271d\n\u22a2 ((fun\u2080 | g \u2022 partialProd a\u271d\u00b2 => r * b\u271d) + sum f\u271d fun x r_1 => fun\u2080 | g \u2022 partialProd x => r_1 * r) =\n (sum (fun\u2080 | a\u271d\u00b2 => b\u271d) fun x r_1 => fun\u2080 | g \u2022 partialProd x => r_1 * r) +\n sum f\u271d fun x r_1 => fun\u2080 | g \u2022 partialProd x => r_1 * r"}, {"tactic": "rw [sum_single_index, mul_comm]", "annotated_tactic": ["rw [<a>sum_single_index</a>, <a>mul_comm</a>]", [{"full_name": "Finsupp.sum_single_index", "def_path": "Mathlib/Algebra/BigOperators/Finsupp.lean", "def_pos": [74, 3], "def_end_pos": [74, 14]}, {"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [302, 9], "def_end_pos": [302, 17]}]], "state_before": "case refine'_2\nk G : Type u\ninst\u271d\u00b9 : CommRing k\nn : \u2115\ninst\u271d : Group G\ng : G\nf : (Fin n \u2192 G) \u2192\u2080 k\nr : k\na\u271d\u00b2 : Fin n \u2192 G\nb\u271d : k\nf\u271d : (Fin n \u2192 G) \u2192\u2080 k\na\u271d\u00b9 : \u00aca\u271d\u00b2 \u2208 f\u271d.support\na\u271d : b\u271d \u2260 0\nhx :\n \u2191(diagonalSucc k G n).inv.hom ((fun\u2080 | g => r) \u2297\u209c[k] f\u271d) =\n \u2191(\u2191(Finsupp.lift ((Fin (n + 1) \u2192 G) \u2192\u2080 k) k (Fin n \u2192 G)) fun f => fun\u2080 | g \u2022 partialProd f => r) f\u271d\n\u22a2 ((fun\u2080 | g \u2022 partialProd a\u271d\u00b2 => r * b\u271d) + sum f\u271d fun x r_1 => fun\u2080 | g \u2022 partialProd x => r_1 * r) =\n (sum (fun\u2080 | a\u271d\u00b2 => b\u271d) fun x r_1 => fun\u2080 | g \u2022 partialProd x => r_1 * r) +\n sum f\u271d fun x r_1 => fun\u2080 | g \u2022 partialProd x => r_1 * r", "state_after": "case refine'_2\nk G : Type u\ninst\u271d\u00b9 : CommRing k\nn : \u2115\ninst\u271d : Group G\ng : G\nf : (Fin n \u2192 G) \u2192\u2080 k\nr : k\na\u271d\u00b2 : Fin n \u2192 G\nb\u271d : k\nf\u271d : (Fin n \u2192 G) \u2192\u2080 k\na\u271d\u00b9 : \u00aca\u271d\u00b2 \u2208 f\u271d.support\na\u271d : b\u271d \u2260 0\nhx :\n \u2191(diagonalSucc k G n).inv.hom ((fun\u2080 | g => r) \u2297\u209c[k] f\u271d) =\n \u2191(\u2191(Finsupp.lift ((Fin (n + 1) \u2192 G) \u2192\u2080 k) k (Fin n \u2192 G)) fun f => fun\u2080 | g \u2022 partialProd f => r) f\u271d\n\u22a2 (fun\u2080 | g \u2022 partialProd a\u271d\u00b2 => 0 * r) = 0"}, {"tactic": "rw [zero_mul, single_zero]", "annotated_tactic": ["rw [<a>zero_mul</a>, <a>single_zero</a>]", [{"full_name": "MulZeroClass.zero_mul", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [36, 3], "def_end_pos": [36, 11]}, {"full_name": "Finsupp.single_zero", "def_path": "Mathlib/Data/Finsupp/Defs.lean", "def_pos": [343, 9], "def_end_pos": [343, 20]}]], "state_before": "case refine'_2\nk G : Type u\ninst\u271d\u00b9 : CommRing k\nn : \u2115\ninst\u271d : Group G\ng : G\nf : (Fin n \u2192 G) \u2192\u2080 k\nr : k\na\u271d\u00b2 : Fin n \u2192 G\nb\u271d : k\nf\u271d : (Fin n \u2192 G) \u2192\u2080 k\na\u271d\u00b9 : \u00aca\u271d\u00b2 \u2208 f\u271d.support\na\u271d : b\u271d \u2260 0\nhx :\n \u2191(diagonalSucc k G n).inv.hom ((fun\u2080 | g => r) \u2297\u209c[k] f\u271d) =\n \u2191(\u2191(Finsupp.lift ((Fin (n + 1) \u2192 G) \u2192\u2080 k) k (Fin n \u2192 G)) fun f => fun\u2080 | g \u2022 partialProd f => r) f\u271d\n\u22a2 (fun\u2080 | g \u2022 partialProd a\u271d\u00b2 => 0 * r) = 0", "state_after": "no goals"}]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/RingTheory/UniqueFactorizationDomain.lean
|
UniqueFactorizationMonoid.count_normalizedFactors_eq
|
[
988,
1
] |
[
997,
54
] |
[{"tactic": "by_cases hx0 : x = 0", "annotated_tactic": ["by_cases hx0 : x = 0", []], "state_before": "\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : Nontrivial R\ninst\u271d\u00b9 : NormalizationMonoid R\ndec_dvd : DecidableRel Dvd.dvd\ninst\u271d : DecidableEq R\np x : R\nhp : Irreducible p\nhnorm : \u2191normalize p = p\nn : \u2115\nhle : p ^ n \u2223 x\nhlt : \u00acp ^ (n + 1) \u2223 x\nthis : DecidableRel fun x x_1 => x \u2223 x_1 := fun x x_1 => Classical.propDecidable ((fun x x_2 => x \u2223 x_2) x x_1)\n\u22a2 count p (normalizedFactors x) = n", "state_after": "case pos\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : Nontrivial R\ninst\u271d\u00b9 : NormalizationMonoid R\ndec_dvd : DecidableRel Dvd.dvd\ninst\u271d : DecidableEq R\np x : R\nhp : Irreducible p\nhnorm : \u2191normalize p = p\nn : \u2115\nhle : p ^ n \u2223 x\nhlt : \u00acp ^ (n + 1) \u2223 x\nthis : DecidableRel fun x x_1 => x \u2223 x_1 := fun x x_1 => Classical.propDecidable ((fun x x_2 => x \u2223 x_2) x x_1)\nhx0 : x = 0\n\u22a2 count p (normalizedFactors x) = n\n\ncase neg\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : Nontrivial R\ninst\u271d\u00b9 : NormalizationMonoid R\ndec_dvd : DecidableRel Dvd.dvd\ninst\u271d : DecidableEq R\np x : R\nhp : Irreducible p\nhnorm : \u2191normalize p = p\nn : \u2115\nhle : p ^ n \u2223 x\nhlt : \u00acp ^ (n + 1) \u2223 x\nthis : DecidableRel fun x x_1 => x \u2223 x_1 := fun x x_1 => Classical.propDecidable ((fun x x_2 => x \u2223 x_2) x x_1)\nhx0 : \u00acx = 0\n\u22a2 count p (normalizedFactors x) = n"}, {"tactic": "rw [\u2190 PartENat.natCast_inj]", "annotated_tactic": ["rw [\u2190 <a>PartENat.natCast_inj</a>]", [{"full_name": "PartENat.natCast_inj", "def_path": "Mathlib/Data/Nat/PartENat.lean", "def_pos": [112, 9], "def_end_pos": [112, 20]}]], "state_before": "case neg\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : Nontrivial R\ninst\u271d\u00b9 : NormalizationMonoid R\ndec_dvd : DecidableRel Dvd.dvd\ninst\u271d : DecidableEq R\np x : R\nhp : Irreducible p\nhnorm : \u2191normalize p = p\nn : \u2115\nhle : p ^ n \u2223 x\nhlt : \u00acp ^ (n + 1) \u2223 x\nthis : DecidableRel fun x x_1 => x \u2223 x_1 := fun x x_1 => Classical.propDecidable ((fun x x_2 => x \u2223 x_2) x x_1)\nhx0 : \u00acx = 0\n\u22a2 count p (normalizedFactors x) = n", "state_after": "case neg\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : Nontrivial R\ninst\u271d\u00b9 : NormalizationMonoid R\ndec_dvd : DecidableRel Dvd.dvd\ninst\u271d : DecidableEq R\np x : R\nhp : Irreducible p\nhnorm : \u2191normalize p = p\nn : \u2115\nhle : p ^ n \u2223 x\nhlt : \u00acp ^ (n + 1) \u2223 x\nthis : DecidableRel fun x x_1 => x \u2223 x_1 := fun x x_1 => Classical.propDecidable ((fun x x_2 => x \u2223 x_2) x x_1)\nhx0 : \u00acx = 0\n\u22a2 \u2191(count p (normalizedFactors x)) = \u2191n"}, {"tactic": "convert (multiplicity_eq_count_normalizedFactors hp hx0).symm", "annotated_tactic": ["convert (<a>multiplicity_eq_count_normalizedFactors</a> hp hx0).<a>symm</a>", [{"full_name": "UniqueFactorizationMonoid.multiplicity_eq_count_normalizedFactors", "def_path": "Mathlib/RingTheory/UniqueFactorizationDomain.lean", "def_pos": [972, 9], "def_end_pos": [972, 48]}, {"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 neg\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : Nontrivial R\ninst\u271d\u00b9 : NormalizationMonoid R\ndec_dvd : DecidableRel Dvd.dvd\ninst\u271d : DecidableEq R\np x : R\nhp : Irreducible p\nhnorm : \u2191normalize p = p\nn : \u2115\nhle : p ^ n \u2223 x\nhlt : \u00acp ^ (n + 1) \u2223 x\nthis : DecidableRel fun x x_1 => x \u2223 x_1 := fun x x_1 => Classical.propDecidable ((fun x x_2 => x \u2223 x_2) x x_1)\nhx0 : \u00acx = 0\n\u22a2 \u2191(count p (normalizedFactors x)) = \u2191n", "state_after": "case h.e'_2.h.e'_3.h.e'_3\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : Nontrivial R\ninst\u271d\u00b9 : NormalizationMonoid R\ndec_dvd : DecidableRel Dvd.dvd\ninst\u271d : DecidableEq R\np x : R\nhp : Irreducible p\nhnorm : \u2191normalize p = p\nn : \u2115\nhle : p ^ n \u2223 x\nhlt : \u00acp ^ (n + 1) \u2223 x\nthis : DecidableRel fun x x_1 => x \u2223 x_1 := fun x x_1 => Classical.propDecidable ((fun x x_2 => x \u2223 x_2) x x_1)\nhx0 : \u00acx = 0\n\u22a2 p = \u2191normalize p\n\ncase h.e'_3\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : Nontrivial R\ninst\u271d\u00b9 : NormalizationMonoid R\ndec_dvd : DecidableRel Dvd.dvd\ninst\u271d : DecidableEq R\np x : R\nhp : Irreducible p\nhnorm : \u2191normalize p = p\nn : \u2115\nhle : p ^ n \u2223 x\nhlt : \u00acp ^ (n + 1) \u2223 x\nthis : DecidableRel fun x x_1 => x \u2223 x_1 := fun x x_1 => Classical.propDecidable ((fun x x_2 => x \u2223 x_2) x x_1)\nhx0 : \u00acx = 0\n\u22a2 \u2191n = multiplicity p x"}, {"tactic": "exact (multiplicity.eq_coe_iff.mpr \u27e8hle, hlt\u27e9).symm", "annotated_tactic": ["exact (multiplicity.eq_coe_iff.mpr \u27e8hle, hlt\u27e9).<a>symm</a>", [{"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'_3\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : Nontrivial R\ninst\u271d\u00b9 : NormalizationMonoid R\ndec_dvd : DecidableRel Dvd.dvd\ninst\u271d : DecidableEq R\np x : R\nhp : Irreducible p\nhnorm : \u2191normalize p = p\nn : \u2115\nhle : p ^ n \u2223 x\nhlt : \u00acp ^ (n + 1) \u2223 x\nthis : DecidableRel fun x x_1 => x \u2223 x_1 := fun x x_1 => Classical.propDecidable ((fun x x_2 => x \u2223 x_2) x x_1)\nhx0 : \u00acx = 0\n\u22a2 \u2191n = multiplicity p x", "state_after": "no goals"}, {"tactic": "simp [hx0] at hlt", "annotated_tactic": ["simp [hx0] at hlt", []], "state_before": "case pos\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : Nontrivial R\ninst\u271d\u00b9 : NormalizationMonoid R\ndec_dvd : DecidableRel Dvd.dvd\ninst\u271d : DecidableEq R\np x : R\nhp : Irreducible p\nhnorm : \u2191normalize p = p\nn : \u2115\nhle : p ^ n \u2223 x\nhlt : \u00acp ^ (n + 1) \u2223 x\nthis : DecidableRel fun x x_1 => x \u2223 x_1 := fun x x_1 => Classical.propDecidable ((fun x x_2 => x \u2223 x_2) x x_1)\nhx0 : x = 0\n\u22a2 count p (normalizedFactors x) = n", "state_after": "no goals"}, {"tactic": "exact hnorm.symm", "annotated_tactic": ["exact hnorm.symm", []], "state_before": "case h.e'_2.h.e'_3.h.e'_3\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : Nontrivial R\ninst\u271d\u00b9 : NormalizationMonoid R\ndec_dvd : DecidableRel Dvd.dvd\ninst\u271d : DecidableEq R\np x : R\nhp : Irreducible p\nhnorm : \u2191normalize p = p\nn : \u2115\nhle : p ^ n \u2223 x\nhlt : \u00acp ^ (n + 1) \u2223 x\nthis : DecidableRel fun x x_1 => x \u2223 x_1 := fun x x_1 => Classical.propDecidable ((fun x x_2 => x \u2223 x_2) x x_1)\nhx0 : \u00acx = 0\n\u22a2 p = \u2191normalize p", "state_after": "no goals"}]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Analysis/SpecialFunctions/Log/Basic.lean
|
Real.eq_one_of_pos_of_log_eq_zero
|
[
271,
1
] |
[
272,
92
] |
[]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Probability/Kernel/Composition.lean
|
ProbabilityTheory.kernel.set_lintegral_compProd_univ_right
|
[
462,
1
] |
[
466,
89
] |
[{"tactic": "simp_rw [set_lintegral_compProd \u03ba \u03b7 a hf hs MeasurableSet.univ, Measure.restrict_univ]", "annotated_tactic": ["simp_rw [<a>set_lintegral_compProd</a> \u03ba \u03b7 a hf hs <a>MeasurableSet.univ</a>, <a>Measure.restrict_univ</a>]", [{"full_name": "ProbabilityTheory.kernel.set_lintegral_compProd", "def_path": "Mathlib/Probability/Kernel/Composition.lean", "def_pos": [454, 9], "def_end_pos": [454, 31]}, {"full_name": "MeasurableSet.univ", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [101, 19], "def_end_pos": [101, 37]}, {"full_name": "MeasureTheory.Measure.restrict_univ", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1701, 9], "def_end_pos": [1701, 22]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03b3 : Type u_4\nm\u03b3 : MeasurableSpace \u03b3\ns\u271d : Set (\u03b2 \u00d7 \u03b3)\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\ninst\u271d : IsSFiniteKernel \u03b7\na : \u03b1\nf : \u03b2 \u00d7 \u03b3 \u2192 \u211d\u22650\u221e\nhf : Measurable f\ns : Set \u03b2\nhs : MeasurableSet s\n\u22a2 \u222b\u207b (z : \u03b2 \u00d7 \u03b3) in s \u00d7\u02e2 Set.univ, f z \u2202\u2191(\u03ba \u2297\u2096 \u03b7) a = \u222b\u207b (x : \u03b2) in s, \u222b\u207b (y : \u03b3), f (x, y) \u2202\u2191\u03b7 (a, x) \u2202\u2191\u03ba a", "state_after": "no goals"}]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Topology/Separation.lean
|
compl_singleton_mem_nhdsSet_iff
|
[
691,
1
] |
[
692,
70
] |
[{"tactic": "rw [isOpen_compl_singleton.mem_nhdsSet, subset_compl_singleton_iff]", "annotated_tactic": ["rw [isOpen_compl_singleton.mem_nhdsSet, <a>subset_compl_singleton_iff</a>]", [{"full_name": "Set.subset_compl_singleton_iff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1799, 9], "def_end_pos": [1799, 35]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : T1Space \u03b1\nx : \u03b1\ns : Set \u03b1\n\u22a2 {x}\u1d9c \u2208 \ud835\udcdd\u02e2 s \u2194 \u00acx \u2208 s", "state_after": "no goals"}]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Algebra/BigOperators/Multiset/Basic.lean
|
Multiset.prod_induction_nonempty
|
[
230,
1
] |
[
239,
70
] |
[{"tactic": "induction' s using Multiset.induction_on with a s hsa", "annotated_tactic": ["induction' s using <a>Multiset.induction_on</a> with a s hsa", [{"full_name": "Multiset.induction_on", "def_path": "Mathlib/Data/Multiset/Basic.lean", "def_pos": [160, 19], "def_end_pos": [160, 31]}]], "state_before": "\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d : CommMonoid \u03b1\ns t : Multiset \u03b1\na : \u03b1\nm : Multiset \u03b9\nf g : \u03b9 \u2192 \u03b1\np : \u03b1 \u2192 Prop\np_mul : \u2200 (a b : \u03b1), p a \u2192 p b \u2192 p (a * b)\nhs : s \u2260 \u2205\np_s : \u2200 (a : \u03b1), a \u2208 s \u2192 p a\n\u22a2 p (prod s)", "state_after": "case empty\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d : CommMonoid \u03b1\ns t : Multiset \u03b1\na : \u03b1\nm : Multiset \u03b9\nf g : \u03b9 \u2192 \u03b1\np : \u03b1 \u2192 Prop\np_mul : \u2200 (a b : \u03b1), p a \u2192 p b \u2192 p (a * b)\nhs\u271d : s \u2260 \u2205\np_s\u271d : \u2200 (a : \u03b1), a \u2208 s \u2192 p a\nhs : 0 \u2260 \u2205\np_s : \u2200 (a : \u03b1), a \u2208 0 \u2192 p a\n\u22a2 p (prod 0)\n\ncase cons\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d : CommMonoid \u03b1\ns\u271d t : Multiset \u03b1\na\u271d : \u03b1\nm : Multiset \u03b9\nf g : \u03b9 \u2192 \u03b1\np : \u03b1 \u2192 Prop\np_mul : \u2200 (a b : \u03b1), p a \u2192 p b \u2192 p (a * b)\nhs\u271d : s\u271d \u2260 \u2205\np_s\u271d : \u2200 (a : \u03b1), a \u2208 s\u271d \u2192 p a\na : \u03b1\ns : Multiset \u03b1\nhsa : s \u2260 \u2205 \u2192 (\u2200 (a : \u03b1), a \u2208 s \u2192 p a) \u2192 p (prod s)\nhs : a ::\u2098 s \u2260 \u2205\np_s : \u2200 (a_1 : \u03b1), a_1 \u2208 a ::\u2098 s \u2192 p a_1\n\u22a2 p (prod (a ::\u2098 s))"}, {"tactic": "rw [prod_cons]", "annotated_tactic": ["rw [<a>prod_cons</a>]", [{"full_name": "Multiset.prod_cons", "def_path": "Mathlib/Algebra/BigOperators/Multiset/Basic.lean", "def_pos": [83, 9], "def_end_pos": [83, 18]}]], "state_before": "case cons\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d : CommMonoid \u03b1\ns\u271d t : Multiset \u03b1\na\u271d : \u03b1\nm : Multiset \u03b9\nf g : \u03b9 \u2192 \u03b1\np : \u03b1 \u2192 Prop\np_mul : \u2200 (a b : \u03b1), p a \u2192 p b \u2192 p (a * b)\nhs\u271d : s\u271d \u2260 \u2205\np_s\u271d : \u2200 (a : \u03b1), a \u2208 s\u271d \u2192 p a\na : \u03b1\ns : Multiset \u03b1\nhsa : s \u2260 \u2205 \u2192 (\u2200 (a : \u03b1), a \u2208 s \u2192 p a) \u2192 p (prod s)\nhs : a ::\u2098 s \u2260 \u2205\np_s : \u2200 (a_1 : \u03b1), a_1 \u2208 a ::\u2098 s \u2192 p a_1\n\u22a2 p (prod (a ::\u2098 s))", "state_after": "case cons\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d : CommMonoid \u03b1\ns\u271d t : Multiset \u03b1\na\u271d : \u03b1\nm : Multiset \u03b9\nf g : \u03b9 \u2192 \u03b1\np : \u03b1 \u2192 Prop\np_mul : \u2200 (a b : \u03b1), p a \u2192 p b \u2192 p (a * b)\nhs\u271d : s\u271d \u2260 \u2205\np_s\u271d : \u2200 (a : \u03b1), a \u2208 s\u271d \u2192 p a\na : \u03b1\ns : Multiset \u03b1\nhsa : s \u2260 \u2205 \u2192 (\u2200 (a : \u03b1), a \u2208 s \u2192 p a) \u2192 p (prod s)\nhs : a ::\u2098 s \u2260 \u2205\np_s : \u2200 (a_1 : \u03b1), a_1 \u2208 a ::\u2098 s \u2192 p a_1\n\u22a2 p (a * prod s)"}, {"tactic": "by_cases hs_empty : s = \u2205", "annotated_tactic": ["by_cases hs_empty : s = \u2205", []], "state_before": "case cons\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d : CommMonoid \u03b1\ns\u271d t : Multiset \u03b1\na\u271d : \u03b1\nm : Multiset \u03b9\nf g : \u03b9 \u2192 \u03b1\np : \u03b1 \u2192 Prop\np_mul : \u2200 (a b : \u03b1), p a \u2192 p b \u2192 p (a * b)\nhs\u271d : s\u271d \u2260 \u2205\np_s\u271d : \u2200 (a : \u03b1), a \u2208 s\u271d \u2192 p a\na : \u03b1\ns : Multiset \u03b1\nhsa : s \u2260 \u2205 \u2192 (\u2200 (a : \u03b1), a \u2208 s \u2192 p a) \u2192 p (prod s)\nhs : a ::\u2098 s \u2260 \u2205\np_s : \u2200 (a_1 : \u03b1), a_1 \u2208 a ::\u2098 s \u2192 p a_1\n\u22a2 p (a * prod s)", "state_after": "case pos\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d : CommMonoid \u03b1\ns\u271d t : Multiset \u03b1\na\u271d : \u03b1\nm : Multiset \u03b9\nf g : \u03b9 \u2192 \u03b1\np : \u03b1 \u2192 Prop\np_mul : \u2200 (a b : \u03b1), p a \u2192 p b \u2192 p (a * b)\nhs\u271d : s\u271d \u2260 \u2205\np_s\u271d : \u2200 (a : \u03b1), a \u2208 s\u271d \u2192 p a\na : \u03b1\ns : Multiset \u03b1\nhsa : s \u2260 \u2205 \u2192 (\u2200 (a : \u03b1), a \u2208 s \u2192 p a) \u2192 p (prod s)\nhs : a ::\u2098 s \u2260 \u2205\np_s : \u2200 (a_1 : \u03b1), a_1 \u2208 a ::\u2098 s \u2192 p a_1\nhs_empty : s = \u2205\n\u22a2 p (a * prod s)\n\ncase neg\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d : CommMonoid \u03b1\ns\u271d t : Multiset \u03b1\na\u271d : \u03b1\nm : Multiset \u03b9\nf g : \u03b9 \u2192 \u03b1\np : \u03b1 \u2192 Prop\np_mul : \u2200 (a b : \u03b1), p a \u2192 p b \u2192 p (a * b)\nhs\u271d : s\u271d \u2260 \u2205\np_s\u271d : \u2200 (a : \u03b1), a \u2208 s\u271d \u2192 p a\na : \u03b1\ns : Multiset \u03b1\nhsa : s \u2260 \u2205 \u2192 (\u2200 (a : \u03b1), a \u2208 s \u2192 p a) \u2192 p (prod s)\nhs : a ::\u2098 s \u2260 \u2205\np_s : \u2200 (a_1 : \u03b1), a_1 \u2208 a ::\u2098 s \u2192 p a_1\nhs_empty : \u00acs = \u2205\n\u22a2 p (a * prod s)"}, {"tactic": "have hps : \u2200 x, x \u2208 s \u2192 p x := fun x hxs => p_s x (mem_cons_of_mem hxs)", "annotated_tactic": ["have hps : \u2200 x, x \u2208 s \u2192 p x := fun x hxs => p_s x (<a>mem_cons_of_mem</a> hxs)", [{"full_name": "Multiset.mem_cons_of_mem", "def_path": "Mathlib/Data/Multiset/Basic.lean", "def_pos": [239, 9], "def_end_pos": [239, 24]}]], "state_before": "case neg\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d : CommMonoid \u03b1\ns\u271d t : Multiset \u03b1\na\u271d : \u03b1\nm : Multiset \u03b9\nf g : \u03b9 \u2192 \u03b1\np : \u03b1 \u2192 Prop\np_mul : \u2200 (a b : \u03b1), p a \u2192 p b \u2192 p (a * b)\nhs\u271d : s\u271d \u2260 \u2205\np_s\u271d : \u2200 (a : \u03b1), a \u2208 s\u271d \u2192 p a\na : \u03b1\ns : Multiset \u03b1\nhsa : s \u2260 \u2205 \u2192 (\u2200 (a : \u03b1), a \u2208 s \u2192 p a) \u2192 p (prod s)\nhs : a ::\u2098 s \u2260 \u2205\np_s : \u2200 (a_1 : \u03b1), a_1 \u2208 a ::\u2098 s \u2192 p a_1\nhs_empty : \u00acs = \u2205\n\u22a2 p (a * prod s)", "state_after": "case neg\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d : CommMonoid \u03b1\ns\u271d t : Multiset \u03b1\na\u271d : \u03b1\nm : Multiset \u03b9\nf g : \u03b9 \u2192 \u03b1\np : \u03b1 \u2192 Prop\np_mul : \u2200 (a b : \u03b1), p a \u2192 p b \u2192 p (a * b)\nhs\u271d : s\u271d \u2260 \u2205\np_s\u271d : \u2200 (a : \u03b1), a \u2208 s\u271d \u2192 p a\na : \u03b1\ns : Multiset \u03b1\nhsa : s \u2260 \u2205 \u2192 (\u2200 (a : \u03b1), a \u2208 s \u2192 p a) \u2192 p (prod s)\nhs : a ::\u2098 s \u2260 \u2205\np_s : \u2200 (a_1 : \u03b1), a_1 \u2208 a ::\u2098 s \u2192 p a_1\nhs_empty : \u00acs = \u2205\nhps : \u2200 (x : \u03b1), x \u2208 s \u2192 p x\n\u22a2 p (a * prod s)"}, {"tactic": "exact p_mul a s.prod (p_s a (mem_cons_self a s)) (hsa hs_empty hps)", "annotated_tactic": ["exact p_mul a s.prod (p_s a (<a>mem_cons_self</a> a s)) (hsa hs_empty hps)", [{"full_name": "Multiset.mem_cons_self", "def_path": "Mathlib/Data/Multiset/Basic.lean", "def_pos": [244, 9], "def_end_pos": [244, 22]}]], "state_before": "case neg\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d : CommMonoid \u03b1\ns\u271d t : Multiset \u03b1\na\u271d : \u03b1\nm : Multiset \u03b9\nf g : \u03b9 \u2192 \u03b1\np : \u03b1 \u2192 Prop\np_mul : \u2200 (a b : \u03b1), p a \u2192 p b \u2192 p (a * b)\nhs\u271d : s\u271d \u2260 \u2205\np_s\u271d : \u2200 (a : \u03b1), a \u2208 s\u271d \u2192 p a\na : \u03b1\ns : Multiset \u03b1\nhsa : s \u2260 \u2205 \u2192 (\u2200 (a : \u03b1), a \u2208 s \u2192 p a) \u2192 p (prod s)\nhs : a ::\u2098 s \u2260 \u2205\np_s : \u2200 (a_1 : \u03b1), a_1 \u2208 a ::\u2098 s \u2192 p a_1\nhs_empty : \u00acs = \u2205\nhps : \u2200 (x : \u03b1), x \u2208 s \u2192 p x\n\u22a2 p (a * prod s)", "state_after": "no goals"}, {"tactic": "simp at hs", "annotated_tactic": ["simp at hs", []], "state_before": "case empty\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d : CommMonoid \u03b1\ns t : Multiset \u03b1\na : \u03b1\nm : Multiset \u03b9\nf g : \u03b9 \u2192 \u03b1\np : \u03b1 \u2192 Prop\np_mul : \u2200 (a b : \u03b1), p a \u2192 p b \u2192 p (a * b)\nhs\u271d : s \u2260 \u2205\np_s\u271d : \u2200 (a : \u03b1), a \u2208 s \u2192 p a\nhs : 0 \u2260 \u2205\np_s : \u2200 (a : \u03b1), a \u2208 0 \u2192 p a\n\u22a2 p (prod 0)", "state_after": "no goals"}, {"tactic": "simp [hs_empty, p_s a]", "annotated_tactic": ["simp [hs_empty, p_s a]", []], "state_before": "case pos\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d : CommMonoid \u03b1\ns\u271d t : Multiset \u03b1\na\u271d : \u03b1\nm : Multiset \u03b9\nf g : \u03b9 \u2192 \u03b1\np : \u03b1 \u2192 Prop\np_mul : \u2200 (a b : \u03b1), p a \u2192 p b \u2192 p (a * b)\nhs\u271d : s\u271d \u2260 \u2205\np_s\u271d : \u2200 (a : \u03b1), a \u2208 s\u271d \u2192 p a\na : \u03b1\ns : Multiset \u03b1\nhsa : s \u2260 \u2205 \u2192 (\u2200 (a : \u03b1), a \u2208 s \u2192 p a) \u2192 p (prod s)\nhs : a ::\u2098 s \u2260 \u2205\np_s : \u2200 (a_1 : \u03b1), a_1 \u2208 a ::\u2098 s \u2192 p a_1\nhs_empty : s = \u2205\n\u22a2 p (a * prod s)", "state_after": "no goals"}]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Data/Fintype/Basic.lean
|
Finset.univ_eq_empty
|
[
114,
1
] |
[
115,
26
] |
[]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Topology/Algebra/Order/LeftRightLim.lean
|
Monotone.rightLim_le
|
[
151,
1
] |
[
152,
23
] |
[]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Computability/TuringMachine.lean
|
Turing.Tape.map_mk'
|
[
718,
1
] |
[
721,
24
] |
[{"tactic": "simp only [Tape.mk', Tape.map, ListBlank.head_map, eq_self_iff_true, and_self_iff,\n ListBlank.tail_map]", "annotated_tactic": ["simp only [<a>Tape.mk'</a>, <a>Tape.map</a>, <a>ListBlank.head_map</a>, <a>eq_self_iff_true</a>, <a>and_self_iff</a>,\n <a>ListBlank.tail_map</a>]", [{"full_name": "Turing.Tape.mk'", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [545, 5], "def_end_pos": [545, 13]}, {"full_name": "Turing.Tape.map", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [682, 5], "def_end_pos": [682, 13]}, {"full_name": "Turing.ListBlank.head_map", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [395, 9], "def_end_pos": [395, 27]}, {"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": "and_self_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [155, 9], "def_end_pos": [155, 21]}, {"full_name": "Turing.ListBlank.tail_map", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [402, 9], "def_end_pos": [402, 27]}]], "state_before": "\u0393 : Type u_1\n\u0393' : Type u_2\ninst\u271d\u00b9 : Inhabited \u0393\ninst\u271d : Inhabited \u0393'\nf : PointedMap \u0393 \u0393'\nL R : ListBlank \u0393\n\u22a2 map f (mk' L R) = mk' (ListBlank.map f L) (ListBlank.map f R)", "state_after": "no goals"}]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Data/Multiset/Basic.lean
|
Multiset.nsmul_singleton
|
[
964,
1
] |
[
965,
49
] |
[{"tactic": "rw [\u2190 replicate_one, nsmul_replicate, mul_one]", "annotated_tactic": ["rw [\u2190 <a>replicate_one</a>, <a>nsmul_replicate</a>, <a>mul_one</a>]", [{"full_name": "Multiset.replicate_one", "def_path": "Mathlib/Data/Multiset/Basic.lean", "def_pos": [910, 9], "def_end_pos": [910, 22]}, {"full_name": "Multiset.nsmul_replicate", "def_path": "Mathlib/Data/Multiset/Basic.lean", "def_pos": [960, 9], "def_end_pos": [960, 24]}, {"full_name": "mul_one", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [470, 9], "def_end_pos": [470, 16]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type v\n\u03b3 : Type u_2\na : \u03b1\nn : \u2115\n\u22a2 n \u2022 {a} = replicate n a", "state_after": "no goals"}]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/MeasureTheory/Measure/MeasureSpace.lean
|
MeasurableEquiv.map_apply_eq_iff_map_symm_apply_eq
|
[
4246,
1
] |
[
4247,
73
] |
[{"tactic": "rw [\u2190 (map_measurableEquiv_injective e).eq_iff, map_map_symm, eq_comm]", "annotated_tactic": ["rw [\u2190 (<a>map_measurableEquiv_injective</a> e).<a>eq_iff</a>, <a>map_map_symm</a>, <a>eq_comm</a>]", [{"full_name": "MeasurableEquiv.map_measurableEquiv_injective", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [4240, 9], "def_end_pos": [4240, 38]}, {"full_name": "Function.Injective.eq_iff", "def_path": "Mathlib/Logic/Function/Basic.lean", "def_pos": [95, 9], "def_end_pos": [95, 25]}, {"full_name": "MeasurableEquiv.map_map_symm", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [4236, 9], "def_end_pos": [4236, 21]}, {"full_name": "eq_comm", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [104, 9], "def_end_pos": [104, 16]}]], "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\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSpace \u03b2\n\u03bc : Measure \u03b1\n\u03bd : Measure \u03b2\ne : \u03b1 \u2243\u1d50 \u03b2\n\u22a2 Measure.map (\u2191e) \u03bc = \u03bd \u2194 Measure.map (\u2191(symm e)) \u03bd = \u03bc", "state_after": "no goals"}]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Algebra/IndicatorFunction.lean
|
Set.mulIndicator_of_mem
|
[
68,
1
] |
[
69,
11
] |
[]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Algebra/Group/OrderSynonym.lean
|
toLex_pow
|
[
281,
1
] |
[
281,
81
] |
[]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Analysis/NormedSpace/Spectrum.lean
|
spectrum.mem_resolventSet_of_norm_lt
|
[
112,
1
] |
[
113,
63
] |
[{"tactic": "rwa [norm_one, mul_one]", "annotated_tactic": ["rwa [<a>norm_one</a>, <a>mul_one</a>]", [{"full_name": "NormOneClass.norm_one", "def_path": "Mathlib/Analysis/Normed/Field/Basic.lean", "def_pos": [134, 3], "def_end_pos": [134, 11]}, {"full_name": "mul_one", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [470, 9], "def_end_pos": [470, 16]}]], "state_before": "\ud835\udd5c : Type u_1\nA : Type u_2\ninst\u271d\u2074 : NormedField \ud835\udd5c\ninst\u271d\u00b3 : NormedRing A\ninst\u271d\u00b2 : NormedAlgebra \ud835\udd5c A\ninst\u271d\u00b9 : CompleteSpace A\ninst\u271d : NormOneClass A\na : A\nk : \ud835\udd5c\nh : \u2016a\u2016 < \u2016k\u2016\n\u22a2 \u2016a\u2016 * \u20161\u2016 < \u2016k\u2016", "state_after": "no goals"}]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Data/DFinsupp/Basic.lean
|
DFinsupp.support_subset_iff
|
[
1190,
1
] |
[
1192,
67
] |
[{"tactic": "simp [Set.subset_def]", "annotated_tactic": ["simp [<a>Set.subset_def</a>]", [{"full_name": "Set.subset_def", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [345, 9], "def_end_pos": [345, 19]}]], "state_before": "\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0)\ns : Set \u03b9\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\n\u22a2 \u2191(support f) \u2286 s \u2194 \u2200 (i : \u03b9), \u00aci \u2208 s \u2192 \u2191f i = 0", "state_after": "\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0)\ns : Set \u03b9\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\n\u22a2 (\u2200 (x : \u03b9), \u00ac\u2191f x = 0 \u2192 x \u2208 s) \u2194 \u2200 (i : \u03b9), \u00aci \u2208 s \u2192 \u2191f i = 0"}, {"tactic": "exact forall_congr' fun i => not_imp_comm", "annotated_tactic": ["exact <a>forall_congr'</a> fun i => <a>not_imp_comm</a>", [{"full_name": "forall_congr'", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [376, 9], "def_end_pos": [376, 22]}, {"full_name": "not_imp_comm", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [265, 9], "def_end_pos": [265, 21]}]], "state_before": "\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0)\ns : Set \u03b9\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\n\u22a2 (\u2200 (x : \u03b9), \u00ac\u2191f x = 0 \u2192 x \u2208 s) \u2194 \u2200 (i : \u03b9), \u00aci \u2208 s \u2192 \u2191f i = 0", "state_after": "no goals"}]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/LinearAlgebra/Finsupp.lean
|
LinearMap.splittingOfFinsuppSurjective_splits
|
[
1270,
1
] |
[
1278,
19
] |
[{"tactic": "refine lhom_ext' fun x => ext_ring <| Finsupp.ext fun y => ?_", "annotated_tactic": ["refine <a>lhom_ext'</a> fun x => <a>ext_ring</a> <| <a>Finsupp.ext</a> fun y => ?_", [{"full_name": "Finsupp.lhom_ext'", "def_path": "Mathlib/LinearAlgebra/Finsupp.lean", "def_pos": [75, 9], "def_end_pos": [75, 18]}, {"full_name": "LinearMap.ext_ring", "def_path": "Mathlib/Algebra/Module/LinearMap.lean", "def_pos": [496, 9], "def_end_pos": [496, 17]}, {"full_name": "Finsupp.ext", "def_path": "Mathlib/Data/Finsupp/Defs.lean", "def_pos": [134, 9], "def_end_pos": [134, 12]}]], "state_before": "R : Type u_1\nM : Type u_2\nN : Type u_3\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : AddCommMonoid N\ninst\u271d : Module R N\n\u03b1 : Type u_4\nf : M \u2192\u2097[R] \u03b1 \u2192\u2080 R\ns : Surjective \u2191f\n\u22a2 comp f (splittingOfFinsuppSurjective f s) = id", "state_after": "R : Type u_1\nM : Type u_2\nN : Type u_3\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : AddCommMonoid N\ninst\u271d : Module R N\n\u03b1 : Type u_4\nf : M \u2192\u2097[R] \u03b1 \u2192\u2080 R\ns : Surjective \u2191f\nx y : \u03b1\n\u22a2 \u2191(\u2191(comp (comp f (splittingOfFinsuppSurjective f s)) (lsingle x)) 1) y = \u2191(\u2191(comp id (lsingle x)) 1) y"}, {"tactic": "dsimp [splittingOfFinsuppSurjective]", "annotated_tactic": ["dsimp [<a>splittingOfFinsuppSurjective</a>]", [{"full_name": "LinearMap.splittingOfFinsuppSurjective", "def_path": "Mathlib/LinearAlgebra/Finsupp.lean", "def_pos": [1266, 5], "def_end_pos": [1266, 33]}]], "state_before": "R : Type u_1\nM : Type u_2\nN : Type u_3\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : AddCommMonoid N\ninst\u271d : Module R N\n\u03b1 : Type u_4\nf : M \u2192\u2097[R] \u03b1 \u2192\u2080 R\ns : Surjective \u2191f\nx y : \u03b1\n\u22a2 \u2191(\u2191(comp (comp f (splittingOfFinsuppSurjective f s)) (lsingle x)) 1) y = \u2191(\u2191(comp id (lsingle x)) 1) y", "state_after": "R : Type u_1\nM : Type u_2\nN : Type u_3\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : AddCommMonoid N\ninst\u271d : Module R N\n\u03b1 : Type u_4\nf : M \u2192\u2097[R] \u03b1 \u2192\u2080 R\ns : Surjective \u2191f\nx y : \u03b1\n\u22a2 \u2191(\u2191f (sum (fun\u2080 | x => 1) fun x r => r \u2022 Exists.choose (_ : \u2203 a, \u2191f a = fun\u2080 | x => 1))) y = (\u2191fun\u2080 | x => 1) y"}, {"tactic": "congr", "annotated_tactic": ["congr", []], "state_before": "R : Type u_1\nM : Type u_2\nN : Type u_3\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : AddCommMonoid N\ninst\u271d : Module R N\n\u03b1 : Type u_4\nf : M \u2192\u2097[R] \u03b1 \u2192\u2080 R\ns : Surjective \u2191f\nx y : \u03b1\n\u22a2 \u2191(\u2191f (sum (fun\u2080 | x => 1) fun x r => r \u2022 Exists.choose (_ : \u2203 a, \u2191f a = fun\u2080 | x => 1))) y = (\u2191fun\u2080 | x => 1) y", "state_after": "case e_a\nR : Type u_1\nM : Type u_2\nN : Type u_3\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : AddCommMonoid N\ninst\u271d : Module R N\n\u03b1 : Type u_4\nf : M \u2192\u2097[R] \u03b1 \u2192\u2080 R\ns : Surjective \u2191f\nx y : \u03b1\n\u22a2 \u2191f (sum (fun\u2080 | x => 1) fun x r => r \u2022 Exists.choose (_ : \u2203 a, \u2191f a = fun\u2080 | x => 1)) = fun\u2080 | x => 1"}, {"tactic": "rw [sum_single_index, one_smul]", "annotated_tactic": ["rw [<a>sum_single_index</a>, <a>one_smul</a>]", [{"full_name": "Finsupp.sum_single_index", "def_path": "Mathlib/Algebra/BigOperators/Finsupp.lean", "def_pos": [74, 3], "def_end_pos": [74, 14]}, {"full_name": "one_smul", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [492, 9], "def_end_pos": [492, 17]}]], "state_before": "case e_a\nR : Type u_1\nM : Type u_2\nN : Type u_3\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : AddCommMonoid N\ninst\u271d : Module R N\n\u03b1 : Type u_4\nf : M \u2192\u2097[R] \u03b1 \u2192\u2080 R\ns : Surjective \u2191f\nx y : \u03b1\n\u22a2 \u2191f (sum (fun\u2080 | x => 1) fun x r => r \u2022 Exists.choose (_ : \u2203 a, \u2191f a = fun\u2080 | x => 1)) = fun\u2080 | x => 1", "state_after": "case e_a\nR : Type u_1\nM : Type u_2\nN : Type u_3\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : AddCommMonoid N\ninst\u271d : Module R N\n\u03b1 : Type u_4\nf : M \u2192\u2097[R] \u03b1 \u2192\u2080 R\ns : Surjective \u2191f\nx y : \u03b1\n\u22a2 \u2191f (Exists.choose (_ : \u2203 a, \u2191f a = fun\u2080 | x => 1)) = fun\u2080 | x => 1\n\ncase e_a\nR : Type u_1\nM : Type u_2\nN : Type u_3\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : AddCommMonoid N\ninst\u271d : Module R N\n\u03b1 : Type u_4\nf : M \u2192\u2097[R] \u03b1 \u2192\u2080 R\ns : Surjective \u2191f\nx y : \u03b1\n\u22a2 0 \u2022 Exists.choose (_ : \u2203 a, \u2191f a = fun\u2080 | x => 1) = 0"}, {"tactic": "exact (s (Finsupp.single x 1)).choose_spec", "annotated_tactic": ["exact (s (<a>Finsupp.single</a> x 1)).<a>choose_spec</a>", [{"full_name": "Finsupp.single", "def_path": "Mathlib/Data/Finsupp/Defs.lean", "def_pos": [289, 5], "def_end_pos": [289, 11]}, {"full_name": "Exists.choose_spec", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [445, 9], "def_end_pos": [445, 27]}]], "state_before": "case e_a\nR : Type u_1\nM : Type u_2\nN : Type u_3\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : AddCommMonoid N\ninst\u271d : Module R N\n\u03b1 : Type u_4\nf : M \u2192\u2097[R] \u03b1 \u2192\u2080 R\ns : Surjective \u2191f\nx y : \u03b1\n\u22a2 \u2191f (Exists.choose (_ : \u2203 a, \u2191f a = fun\u2080 | x => 1)) = fun\u2080 | x => 1", "state_after": "no goals"}, {"tactic": "rw [zero_smul]", "annotated_tactic": ["rw [<a>zero_smul</a>]", [{"full_name": "zero_smul", "def_path": "Mathlib/Algebra/SMulWithZero.lean", "def_pos": [70, 9], "def_end_pos": [70, 18]}]], "state_before": "case e_a\nR : Type u_1\nM : Type u_2\nN : Type u_3\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : AddCommMonoid N\ninst\u271d : Module R N\n\u03b1 : Type u_4\nf : M \u2192\u2097[R] \u03b1 \u2192\u2080 R\ns : Surjective \u2191f\nx y : \u03b1\n\u22a2 0 \u2022 Exists.choose (_ : \u2203 a, \u2191f a = fun\u2080 | x => 1) = 0", "state_after": "no goals"}]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Tactic/NormNum/Pow.lean
|
Mathlib.Meta.NormNum.intPow_negOfNat_bit1
|
[
111,
1
] |
[
116,
44
] |
[{"tactic": "rw [\u2190 hb, Int.negOfNat_eq, Int.negOfNat_eq, pow_eq, pow_succ, pow_mul, neg_pow_two, \u2190 pow_mul,\n two_mul, pow_add, \u2190 hc, \u2190 h1]", "annotated_tactic": ["rw [\u2190 hb, <a>Int.negOfNat_eq</a>, <a>Int.negOfNat_eq</a>, <a>pow_eq</a>, <a>pow_succ</a>, <a>pow_mul</a>, <a>neg_pow_two</a>, \u2190 <a>pow_mul</a>,\n <a>two_mul</a>, <a>pow_add</a>, \u2190 hc, \u2190 h1]", [{"full_name": "Int.negOfNat_eq", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [45, 9], "def_end_pos": [45, 20]}, {"full_name": "Int.negOfNat_eq", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [45, 9], "def_end_pos": [45, 20]}, {"full_name": "pow_eq", "def_path": "Mathlib/Algebra/GroupPower/Lemmas.lean", "def_pos": [1252, 17], "def_end_pos": [1252, 23]}, {"full_name": "pow_succ", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [645, 9], "def_end_pos": [645, 17]}, {"full_name": "pow_mul", "def_path": "Mathlib/Algebra/GroupPower/Basic.lean", "def_pos": [125, 9], "def_end_pos": [125, 16]}, {"full_name": "neg_pow_two", "def_path": "Mathlib/Algebra/GroupPower/Ring.lean", "def_pos": [226, 7], "def_end_pos": [226, 18]}, {"full_name": "pow_mul", "def_path": "Mathlib/Algebra/GroupPower/Basic.lean", "def_pos": [125, 9], "def_end_pos": [125, 16]}, {"full_name": "two_mul", "def_path": "Mathlib/Algebra/Ring/Defs.lean", "def_pos": [177, 9], "def_end_pos": [177, 16]}, {"full_name": "pow_add", "def_path": "Mathlib/Algebra/GroupPower/Basic.lean", "def_pos": [118, 9], "def_end_pos": [118, 16]}]], "state_before": "a b' : \u2115\nc' b c : \u2115\nh1 : Nat.pow a b' = c'\nhb : 2 * b' + 1 = b\nhc : c' * (c' * a) = c\n\u22a2 Int.pow (Int.negOfNat a) b = Int.negOfNat c", "state_after": "a b' : \u2115\nc' b c : \u2115\nh1 : Nat.pow a b' = c'\nhb : 2 * b' + 1 = b\nhc : c' * (c' * a) = c\n\u22a2 -Int.ofNat a * (Int.ofNat a ^ b' * Int.ofNat a ^ b') = -Int.ofNat (Nat.pow a b' * (Nat.pow a b' * a))"}, {"tactic": "simp [mul_assoc, mul_comm, mul_left_comm]", "annotated_tactic": ["simp [<a>mul_assoc</a>, <a>mul_comm</a>, <a>mul_left_comm</a>]", [{"full_name": "mul_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [264, 9], "def_end_pos": [264, 18]}, {"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [302, 9], "def_end_pos": [302, 17]}, {"full_name": "mul_left_comm", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [96, 9], "def_end_pos": [96, 22]}]], "state_before": "a b' : \u2115\nc' b c : \u2115\nh1 : Nat.pow a b' = c'\nhb : 2 * b' + 1 = b\nhc : c' * (c' * a) = c\n\u22a2 -Int.ofNat a * (Int.ofNat a ^ b' * Int.ofNat a ^ b') = -Int.ofNat (Nat.pow a b' * (Nat.pow a b' * a))", "state_after": "no goals"}]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Data/Finset/Basic.lean
|
Finset.eq_of_mem_singleton
|
[
682,
1
] |
[
683,
20
] |
[]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/GroupTheory/FreeAbelianGroup.lean
|
FreeAbelianGroup.add_seq
|
[
273,
1
] |
[
275,
17
] |
[]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean
|
Real.Angle.toReal_eq_neg_pi_div_two_iff
|
[
625,
1
] |
[
626,
43
] |
[{"tactic": "rw [\u2190 toReal_inj, toReal_neg_pi_div_two]", "annotated_tactic": ["rw [\u2190 <a>toReal_inj</a>, <a>toReal_neg_pi_div_two</a>]", [{"full_name": "Real.Angle.toReal_inj", "def_path": "Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean", "def_pos": [546, 9], "def_end_pos": [546, 19]}, {"full_name": "Real.Angle.toReal_neg_pi_div_two", "def_path": "Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean", "def_pos": [620, 9], "def_end_pos": [620, 30]}]], "state_before": "\u03b8 : Angle\n\u22a2 toReal \u03b8 = -\u03c0 / 2 \u2194 \u03b8 = \u2191(-\u03c0 / 2)", "state_after": "no goals"}]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Data/Finset/Basic.lean
|
Finset.eq_singleton_iff_unique_mem
|
[
735,
1
] |
[
741,
48
] |
[{"tactic": "constructor <;> intro t", "annotated_tactic": ["constructor <;> intro t", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns\u271d : Finset \u03b1\na\u271d b : \u03b1\ns : Finset \u03b1\na : \u03b1\n\u22a2 s = {a} \u2194 a \u2208 s \u2227 \u2200 (x : \u03b1), x \u2208 s \u2192 x = a", "state_after": "case mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns\u271d : Finset \u03b1\na\u271d b : \u03b1\ns : Finset \u03b1\na : \u03b1\nt : s = {a}\n\u22a2 a \u2208 s \u2227 \u2200 (x : \u03b1), x \u2208 s \u2192 x = a\n\ncase mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns\u271d : Finset \u03b1\na\u271d b : \u03b1\ns : Finset \u03b1\na : \u03b1\nt : a \u2208 s \u2227 \u2200 (x : \u03b1), x \u2208 s \u2192 x = a\n\u22a2 s = {a}"}, {"tactic": "rw [t]", "annotated_tactic": ["rw [t]", []], "state_before": "case mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns\u271d : Finset \u03b1\na\u271d b : \u03b1\ns : Finset \u03b1\na : \u03b1\nt : s = {a}\n\u22a2 a \u2208 s \u2227 \u2200 (x : \u03b1), x \u2208 s \u2192 x = a", "state_after": "case mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns\u271d : Finset \u03b1\na\u271d b : \u03b1\ns : Finset \u03b1\na : \u03b1\nt : s = {a}\n\u22a2 a \u2208 {a} \u2227 \u2200 (x : \u03b1), x \u2208 {a} \u2192 x = a"}, {"tactic": "exact \u27e8Finset.mem_singleton_self _, fun _ => Finset.mem_singleton.1\u27e9", "annotated_tactic": ["exact \u27e8<a>Finset.mem_singleton_self</a> _, fun _ => <a>Finset.mem_singleton</a>.1\u27e9", [{"full_name": "Finset.mem_singleton_self", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [690, 9], "def_end_pos": [690, 27]}, {"full_name": "Finset.mem_singleton", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [678, 9], "def_end_pos": [678, 22]}]], "state_before": "case mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns\u271d : Finset \u03b1\na\u271d b : \u03b1\ns : Finset \u03b1\na : \u03b1\nt : s = {a}\n\u22a2 a \u2208 {a} \u2227 \u2200 (x : \u03b1), x \u2208 {a} \u2192 x = a", "state_after": "no goals"}, {"tactic": "ext", "annotated_tactic": ["ext", []], "state_before": "case mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns\u271d : Finset \u03b1\na\u271d b : \u03b1\ns : Finset \u03b1\na : \u03b1\nt : a \u2208 s \u2227 \u2200 (x : \u03b1), x \u2208 s \u2192 x = a\n\u22a2 s = {a}", "state_after": "case mpr.a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns\u271d : Finset \u03b1\na\u271d\u00b9 b : \u03b1\ns : Finset \u03b1\na : \u03b1\nt : a \u2208 s \u2227 \u2200 (x : \u03b1), x \u2208 s \u2192 x = a\na\u271d : \u03b1\n\u22a2 a\u271d \u2208 s \u2194 a\u271d \u2208 {a}"}, {"tactic": "rw [Finset.mem_singleton]", "annotated_tactic": ["rw [<a>Finset.mem_singleton</a>]", [{"full_name": "Finset.mem_singleton", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [678, 9], "def_end_pos": [678, 22]}]], "state_before": "case mpr.a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns\u271d : Finset \u03b1\na\u271d\u00b9 b : \u03b1\ns : Finset \u03b1\na : \u03b1\nt : a \u2208 s \u2227 \u2200 (x : \u03b1), x \u2208 s \u2192 x = a\na\u271d : \u03b1\n\u22a2 a\u271d \u2208 s \u2194 a\u271d \u2208 {a}", "state_after": "case mpr.a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns\u271d : Finset \u03b1\na\u271d\u00b9 b : \u03b1\ns : Finset \u03b1\na : \u03b1\nt : a \u2208 s \u2227 \u2200 (x : \u03b1), x \u2208 s \u2192 x = a\na\u271d : \u03b1\n\u22a2 a\u271d \u2208 s \u2194 a\u271d = a"}, {"tactic": "exact \u27e8t.right _, fun r => r.symm \u25b8 t.left\u27e9", "annotated_tactic": ["exact \u27e8t.right _, fun r => r.symm \u25b8 t.left\u27e9", []], "state_before": "case mpr.a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns\u271d : Finset \u03b1\na\u271d\u00b9 b : \u03b1\ns : Finset \u03b1\na : \u03b1\nt : a \u2208 s \u2227 \u2200 (x : \u03b1), x \u2208 s \u2192 x = a\na\u271d : \u03b1\n\u22a2 a\u271d \u2208 s \u2194 a\u271d = a", "state_after": "no goals"}]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/AlgebraicTopology/SimplicialObject.lean
|
CategoryTheory.SimplicialObject.δ_comp_δ
|
[
110,
1
] |
[
113,
66
] |
[{"tactic": "dsimp [\u03b4]", "annotated_tactic": ["dsimp [<a>\u03b4</a>]", [{"full_name": "CategoryTheory.SimplicialObject.\u03b4", "def_path": "Mathlib/AlgebraicTopology/SimplicialObject.lean", "def_pos": [88, 5], "def_end_pos": [88, 6]}]], "state_before": "C : Type u\ninst\u271d : Category.{v, u} C\nX : SimplicialObject C\nn : \u2115\ni j : Fin (n + 2)\nH : i \u2264 j\n\u22a2 \u03b4 X (Fin.succ j) \u226b \u03b4 X i = \u03b4 X (Fin.castSucc i) \u226b \u03b4 X j", "state_after": "C : Type u\ninst\u271d : Category.{v, u} C\nX : SimplicialObject C\nn : \u2115\ni j : Fin (n + 2)\nH : i \u2264 j\n\u22a2 X.map (SimplexCategory.\u03b4 (Fin.succ j)).op \u226b X.map (SimplexCategory.\u03b4 i).op =\n X.map (SimplexCategory.\u03b4 (Fin.castSucc i)).op \u226b X.map (SimplexCategory.\u03b4 j).op"}, {"tactic": "simp only [\u2190 X.map_comp, \u2190 op_comp, SimplexCategory.\u03b4_comp_\u03b4 H]", "annotated_tactic": ["simp only [\u2190 X.map_comp, \u2190 <a>op_comp</a>, <a>SimplexCategory.\u03b4_comp_\u03b4</a> H]", [{"full_name": "CategoryTheory.op_comp", "def_path": "Mathlib/CategoryTheory/Opposites.lean", "def_pos": [73, 9], "def_end_pos": [73, 16]}, {"full_name": "SimplexCategory.\u03b4_comp_\u03b4", "def_path": "Mathlib/AlgebraicTopology/SimplexCategory.lean", "def_pos": [214, 9], "def_end_pos": [214, 17]}]], "state_before": "C : Type u\ninst\u271d : Category.{v, u} C\nX : SimplicialObject C\nn : \u2115\ni j : Fin (n + 2)\nH : i \u2264 j\n\u22a2 X.map (SimplexCategory.\u03b4 (Fin.succ j)).op \u226b X.map (SimplexCategory.\u03b4 i).op =\n X.map (SimplexCategory.\u03b4 (Fin.castSucc i)).op \u226b X.map (SimplexCategory.\u03b4 j).op", "state_after": "no goals"}]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Logic/Relation.lean
|
Relation.transitive_reflTransGen
|
[
570,
1
] |
[
570,
83
] |
[]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Logic/Equiv/List.lean
|
Denumerable.denumerable_list_aux
|
[
249,
1
] |
[
260,
61
] |
[{"tactic": "rw [decodeList]", "annotated_tactic": ["rw [<a>decodeList</a>]", [{"full_name": "Encodable.decodeList", "def_path": "Mathlib/Logic/Equiv/List.lean", "def_pos": [37, 5], "def_end_pos": [37, 15]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Denumerable \u03b1\ninst\u271d : Denumerable \u03b2\n\u22a2 \u2203 a, a \u2208 decodeList 0 \u2227 encodeList a = 0", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Denumerable \u03b1\ninst\u271d : Denumerable \u03b2\n\u22a2 \u2203 a, a \u2208 some [] \u2227 encodeList a = 0"}, {"tactic": "exact \u27e8_, rfl, rfl\u27e9", "annotated_tactic": ["exact \u27e8_, <a>rfl</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": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Denumerable \u03b1\ninst\u271d : Denumerable \u03b2\n\u22a2 \u2203 a, a \u2208 some [] \u2227 encodeList a = 0", "state_after": "no goals"}, {"tactic": "cases' e : unpair v with v\u2081 v\u2082", "annotated_tactic": ["cases' e : <a>unpair</a> v with v\u2081 v\u2082", [{"full_name": "Nat.unpair", "def_path": "Mathlib/Data/Nat/Pairing.lean", "def_pos": [44, 5], "def_end_pos": [44, 11]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Denumerable \u03b1\ninst\u271d : Denumerable \u03b2\nv : \u2115\n\u22a2 \u2203 a, a \u2208 decodeList (succ v) \u2227 encodeList a = succ v", "state_after": "case mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Denumerable \u03b1\ninst\u271d : Denumerable \u03b2\nv v\u2081 v\u2082 : \u2115\ne : unpair v = (v\u2081, v\u2082)\n\u22a2 \u2203 a, a \u2208 decodeList (succ v) \u2227 encodeList a = succ v"}, {"tactic": "have h := unpair_right_le v", "annotated_tactic": ["have h := <a>unpair_right_le</a> v", [{"full_name": "Nat.unpair_right_le", "def_path": "Mathlib/Data/Nat/Pairing.lean", "def_pos": [123, 9], "def_end_pos": [123, 24]}]], "state_before": "case mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Denumerable \u03b1\ninst\u271d : Denumerable \u03b2\nv v\u2081 v\u2082 : \u2115\ne : unpair v = (v\u2081, v\u2082)\n\u22a2 \u2203 a, a \u2208 decodeList (succ v) \u2227 encodeList a = succ v", "state_after": "case mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Denumerable \u03b1\ninst\u271d : Denumerable \u03b2\nv v\u2081 v\u2082 : \u2115\ne : unpair v = (v\u2081, v\u2082)\nh : (unpair v).2 \u2264 v\n\u22a2 \u2203 a, a \u2208 decodeList (succ v) \u2227 encodeList a = succ v"}, {"tactic": "rw [e] at h", "annotated_tactic": ["rw [e] at h", []], "state_before": "case mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Denumerable \u03b1\ninst\u271d : Denumerable \u03b2\nv v\u2081 v\u2082 : \u2115\ne : unpair v = (v\u2081, v\u2082)\nh : (unpair v).2 \u2264 v\n\u22a2 \u2203 a, a \u2208 decodeList (succ v) \u2227 encodeList a = succ v", "state_after": "case mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Denumerable \u03b1\ninst\u271d : Denumerable \u03b2\nv v\u2081 v\u2082 : \u2115\ne : unpair v = (v\u2081, v\u2082)\nh : (v\u2081, v\u2082).2 \u2264 v\n\u22a2 \u2203 a, a \u2208 decodeList (succ v) \u2227 encodeList a = succ v"}, {"tactic": "rcases have : v\u2082 < succ v := lt_succ_of_le h\n denumerable_list_aux v\u2082 with\n \u27e8a, h\u2081, h\u2082\u27e9", "annotated_tactic": ["rcases have : v\u2082 < <a>succ</a> v := <a>lt_succ_of_le</a> h\n denumerable_list_aux v\u2082 with\n \u27e8a, h\u2081, h\u2082\u27e9", [{"full_name": "Nat.succ", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1044, 5], "def_end_pos": [1044, 9]}, {"full_name": "Nat.lt_succ_of_le", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [212, 9], "def_end_pos": [212, 22]}]], "state_before": "case mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Denumerable \u03b1\ninst\u271d : Denumerable \u03b2\nv v\u2081 v\u2082 : \u2115\ne : unpair v = (v\u2081, v\u2082)\nh : (v\u2081, v\u2082).2 \u2264 v\n\u22a2 \u2203 a, a \u2208 decodeList (succ v) \u2227 encodeList a = succ v", "state_after": "case mk.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Denumerable \u03b1\ninst\u271d : Denumerable \u03b2\nv v\u2081 v\u2082 : \u2115\ne : unpair v = (v\u2081, v\u2082)\nh : (v\u2081, v\u2082).2 \u2264 v\na : List \u03b1\nh\u2081 : a \u2208 decodeList v\u2082\nh\u2082 : encodeList a = v\u2082\n\u22a2 \u2203 a, a \u2208 decodeList (succ v) \u2227 encodeList a = succ v"}, {"tactic": "rw [Option.mem_def] at h\u2081", "annotated_tactic": ["rw [<a>Option.mem_def</a>] at h\u2081", [{"full_name": "Option.mem_def", "def_path": "lake-packages/std/Std/Data/Option/Basic.lean", "def_pos": [19, 17], "def_end_pos": [19, 24]}]], "state_before": "case mk.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Denumerable \u03b1\ninst\u271d : Denumerable \u03b2\nv v\u2081 v\u2082 : \u2115\ne : unpair v = (v\u2081, v\u2082)\nh : (v\u2081, v\u2082).2 \u2264 v\na : List \u03b1\nh\u2081 : a \u2208 decodeList v\u2082\nh\u2082 : encodeList a = v\u2082\n\u22a2 \u2203 a, a \u2208 decodeList (succ v) \u2227 encodeList a = succ v", "state_after": "case mk.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Denumerable \u03b1\ninst\u271d : Denumerable \u03b2\nv v\u2081 v\u2082 : \u2115\ne : unpair v = (v\u2081, v\u2082)\nh : (v\u2081, v\u2082).2 \u2264 v\na : List \u03b1\nh\u2081 : decodeList v\u2082 = some a\nh\u2082 : encodeList a = v\u2082\n\u22a2 \u2203 a, a \u2208 decodeList (succ v) \u2227 encodeList a = succ v"}, {"tactic": "use ofNat \u03b1 v\u2081 :: a", "annotated_tactic": ["use <a>ofNat</a> \u03b1 v\u2081 :: a", [{"full_name": "Denumerable.ofNat", "def_path": "Mathlib/Logic/Denumerable.lean", "def_pos": [50, 5], "def_end_pos": [50, 10]}]], "state_before": "case mk.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Denumerable \u03b1\ninst\u271d : Denumerable \u03b2\nv v\u2081 v\u2082 : \u2115\ne : unpair v = (v\u2081, v\u2082)\nh : (v\u2081, v\u2082).2 \u2264 v\na : List \u03b1\nh\u2081 : decodeList v\u2082 = some a\nh\u2082 : encodeList a = v\u2082\n\u22a2 \u2203 a, a \u2208 decodeList (succ v) \u2227 encodeList a = succ v", "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Denumerable \u03b1\ninst\u271d : Denumerable \u03b2\nv v\u2081 v\u2082 : \u2115\ne : unpair v = (v\u2081, v\u2082)\nh : (v\u2081, v\u2082).2 \u2264 v\na : List \u03b1\nh\u2081 : decodeList v\u2082 = some a\nh\u2082 : encodeList a = v\u2082\n\u22a2 ofNat \u03b1 v\u2081 :: a \u2208 decodeList (succ v) \u2227 encodeList (ofNat \u03b1 v\u2081 :: a) = succ v"}, {"tactic": "simp [decodeList, e, h\u2082, h\u2081, encodeList, pair_unpair' e]", "annotated_tactic": ["simp [<a>decodeList</a>, e, h\u2082, h\u2081, <a>encodeList</a>, <a>pair_unpair'</a> e]", [{"full_name": "Encodable.decodeList", "def_path": "Mathlib/Logic/Equiv/List.lean", "def_pos": [37, 5], "def_end_pos": [37, 15]}, {"full_name": "Encodable.encodeList", "def_path": "Mathlib/Logic/Equiv/List.lean", "def_pos": [31, 5], "def_end_pos": [31, 15]}, {"full_name": "Nat.pair_unpair'", "def_path": "Mathlib/Data/Nat/Pairing.lean", "def_pos": [60, 9], "def_end_pos": [60, 21]}]], "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Denumerable \u03b1\ninst\u271d : Denumerable \u03b2\nv v\u2081 v\u2082 : \u2115\ne : unpair v = (v\u2081, v\u2082)\nh : (v\u2081, v\u2082).2 \u2264 v\na : List \u03b1\nh\u2081 : decodeList v\u2082 = some a\nh\u2082 : encodeList a = v\u2082\n\u22a2 ofNat \u03b1 v\u2081 :: a \u2208 decodeList (succ v) \u2227 encodeList (ofNat \u03b1 v\u2081 :: a) = succ v", "state_after": "no goals"}]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Data/Finset/LocallyFinite.lean
|
Finset.Ioc_subset_Ioc
|
[
179,
1
] |
[
180,
54
] |
[{"tactic": "simpa [\u2190 coe_subset] using Set.Ioc_subset_Ioc ha hb", "annotated_tactic": ["simpa [\u2190 <a>coe_subset</a>] using <a>Set.Ioc_subset_Ioc</a> ha hb", [{"full_name": "Finset.coe_subset", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [376, 9], "def_end_pos": [376, 19]}, {"full_name": "Set.Ioc_subset_Ioc", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [480, 9], "def_end_pos": [480, 23]}]], "state_before": "\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c x : \u03b1\nha : a\u2082 \u2264 a\u2081\nhb : b\u2081 \u2264 b\u2082\n\u22a2 Ioc a\u2081 b\u2081 \u2286 Ioc a\u2082 b\u2082", "state_after": "no goals"}]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Data/Int/Units.lean
|
Int.isUnit_eq_or_eq_neg
|
[
60,
1
] |
[
61,
58
] |
[]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Combinatorics/Young/YoungDiagram.lean
|
YoungDiagram.get_rowLens
|
[
417,
1
] |
[
418,
89
] |
[{"tactic": "simp only [rowLens, List.get_range, List.get_map]", "annotated_tactic": ["simp only [<a>rowLens</a>, <a>List.get_range</a>, <a>List.get_map</a>]", [{"full_name": "YoungDiagram.rowLens", "def_path": "Mathlib/Combinatorics/Young/YoungDiagram.lean", "def_pos": [411, 5], "def_end_pos": [411, 12]}, {"full_name": "List.get_range", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [2122, 17], "def_end_pos": [2122, 26]}, {"full_name": "List.get_map", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [660, 17], "def_end_pos": [660, 24]}]], "state_before": "\u03bc : YoungDiagram\ni : Fin (List.length (rowLens \u03bc))\n\u22a2 List.get (rowLens \u03bc) i = rowLen \u03bc \u2191i", "state_after": "no goals"}]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Topology/ContinuousOn.lean
|
continuousOn_pi
|
[
618,
1
] |
[
620,
100
] |
[]
|
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