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/NumberTheory/Padics/PadicVal.lean
|
one_le_padicValNat_of_dvd
|
[
294,
1
] |
[
297,
13
] |
[{"tactic": "rwa [\u2190 PartENat.coe_le_coe, padicValNat_def' hp.out.ne_one hn, \u2190 pow_dvd_iff_le_multiplicity,\n pow_one]", "annotated_tactic": ["rwa [\u2190 <a>PartENat.coe_le_coe</a>, <a>padicValNat_def'</a> hp.out.ne_one hn, \u2190 <a>pow_dvd_iff_le_multiplicity</a>,\n <a>pow_one</a>]", [{"full_name": "PartENat.coe_le_coe", "def_path": "Mathlib/Data/Nat/PartENat.lean", "def_pos": [266, 9], "def_end_pos": [266, 19]}, {"full_name": "padicValNat_def'", "def_path": "Mathlib/NumberTheory/Padics/PadicVal.lean", "def_pos": [284, 9], "def_end_pos": [284, 25]}, {"full_name": "multiplicity.pow_dvd_iff_le_multiplicity", "def_path": "Mathlib/RingTheory/Multiplicity.lean", "def_pos": [146, 9], "def_end_pos": [146, 36]}, {"full_name": "pow_one", "def_path": "Mathlib/Algebra/GroupPower/Basic.lean", "def_pos": [97, 9], "def_end_pos": [97, 16]}]], "state_before": "p n : \u2115\nhp : Fact (Nat.Prime p)\nhn : 0 < n\ndiv : p \u2223 n\n\u22a2 1 \u2264 padicValNat p n", "state_after": "no goals"}]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/LinearAlgebra/QuadraticForm/Prod.lean
|
QuadraticForm.nonneg_prod_iff
|
[
116,
1
] |
[
126,
37
] |
[{"tactic": "simp_rw [Prod.forall, prod_apply]", "annotated_tactic": ["simp_rw [<a>Prod.forall</a>, <a>prod_apply</a>]", [{"full_name": "Prod.forall", "def_path": "Mathlib/Data/Prod/Basic.lean", "def_pos": [36, 9], "def_end_pos": [36, 17]}, {"full_name": "QuadraticForm.prod_apply", "def_path": "Mathlib/LinearAlgebra/QuadraticForm/Prod.lean", "def_pos": [51, 3], "def_end_pos": [51, 9]}]], "state_before": "\u03b9 : Type u_1\nR\u271d : Type u_2\nM\u2081 : Type u_3\nM\u2082 : Type u_4\nN\u2081 : Type u_5\nN\u2082 : Type u_6\nM\u1d62 : \u03b9 \u2192 Type u_7\nN\u1d62 : \u03b9 \u2192 Type u_8\ninst\u271d\u00b9\u00b9 : CommSemiring R\u271d\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2081\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid N\u2081\ninst\u271d\u2077 : AddCommMonoid N\u2082\ninst\u271d\u2076 : Module R\u271d M\u2081\ninst\u271d\u2075 : Module R\u271d M\u2082\ninst\u271d\u2074 : Module R\u271d N\u2081\ninst\u271d\u00b3 : Module R\u271d N\u2082\nR : Type u_9\ninst\u271d\u00b2 : OrderedCommRing R\ninst\u271d\u00b9 : Module R M\u2081\ninst\u271d : Module R M\u2082\nQ\u2081 : QuadraticForm R M\u2081\nQ\u2082 : QuadraticForm R M\u2082\n\u22a2 (\u2200 (x : M\u2081 \u00d7 M\u2082), 0 \u2264 \u2191(prod Q\u2081 Q\u2082) x) \u2194 (\u2200 (x : M\u2081), 0 \u2264 \u2191Q\u2081 x) \u2227 \u2200 (x : M\u2082), 0 \u2264 \u2191Q\u2082 x", "state_after": "\u03b9 : Type u_1\nR\u271d : Type u_2\nM\u2081 : Type u_3\nM\u2082 : Type u_4\nN\u2081 : Type u_5\nN\u2082 : Type u_6\nM\u1d62 : \u03b9 \u2192 Type u_7\nN\u1d62 : \u03b9 \u2192 Type u_8\ninst\u271d\u00b9\u00b9 : CommSemiring R\u271d\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2081\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid N\u2081\ninst\u271d\u2077 : AddCommMonoid N\u2082\ninst\u271d\u2076 : Module R\u271d M\u2081\ninst\u271d\u2075 : Module R\u271d M\u2082\ninst\u271d\u2074 : Module R\u271d N\u2081\ninst\u271d\u00b3 : Module R\u271d N\u2082\nR : Type u_9\ninst\u271d\u00b2 : OrderedCommRing R\ninst\u271d\u00b9 : Module R M\u2081\ninst\u271d : Module R M\u2082\nQ\u2081 : QuadraticForm R M\u2081\nQ\u2082 : QuadraticForm R M\u2082\n\u22a2 (\u2200 (a : M\u2081) (b : M\u2082), 0 \u2264 \u2191Q\u2081 a + \u2191Q\u2082 b) \u2194 (\u2200 (x : M\u2081), 0 \u2264 \u2191Q\u2081 x) \u2227 \u2200 (x : M\u2082), 0 \u2264 \u2191Q\u2082 x"}, {"tactic": "constructor", "annotated_tactic": ["constructor", []], "state_before": "\u03b9 : Type u_1\nR\u271d : Type u_2\nM\u2081 : Type u_3\nM\u2082 : Type u_4\nN\u2081 : Type u_5\nN\u2082 : Type u_6\nM\u1d62 : \u03b9 \u2192 Type u_7\nN\u1d62 : \u03b9 \u2192 Type u_8\ninst\u271d\u00b9\u00b9 : CommSemiring R\u271d\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2081\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid N\u2081\ninst\u271d\u2077 : AddCommMonoid N\u2082\ninst\u271d\u2076 : Module R\u271d M\u2081\ninst\u271d\u2075 : Module R\u271d M\u2082\ninst\u271d\u2074 : Module R\u271d N\u2081\ninst\u271d\u00b3 : Module R\u271d N\u2082\nR : Type u_9\ninst\u271d\u00b2 : OrderedCommRing R\ninst\u271d\u00b9 : Module R M\u2081\ninst\u271d : Module R M\u2082\nQ\u2081 : QuadraticForm R M\u2081\nQ\u2082 : QuadraticForm R M\u2082\n\u22a2 (\u2200 (a : M\u2081) (b : M\u2082), 0 \u2264 \u2191Q\u2081 a + \u2191Q\u2082 b) \u2194 (\u2200 (x : M\u2081), 0 \u2264 \u2191Q\u2081 x) \u2227 \u2200 (x : M\u2082), 0 \u2264 \u2191Q\u2082 x", "state_after": "case mp\n\u03b9 : Type u_1\nR\u271d : Type u_2\nM\u2081 : Type u_3\nM\u2082 : Type u_4\nN\u2081 : Type u_5\nN\u2082 : Type u_6\nM\u1d62 : \u03b9 \u2192 Type u_7\nN\u1d62 : \u03b9 \u2192 Type u_8\ninst\u271d\u00b9\u00b9 : CommSemiring R\u271d\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2081\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid N\u2081\ninst\u271d\u2077 : AddCommMonoid N\u2082\ninst\u271d\u2076 : Module R\u271d M\u2081\ninst\u271d\u2075 : Module R\u271d M\u2082\ninst\u271d\u2074 : Module R\u271d N\u2081\ninst\u271d\u00b3 : Module R\u271d N\u2082\nR : Type u_9\ninst\u271d\u00b2 : OrderedCommRing R\ninst\u271d\u00b9 : Module R M\u2081\ninst\u271d : Module R M\u2082\nQ\u2081 : QuadraticForm R M\u2081\nQ\u2082 : QuadraticForm R M\u2082\n\u22a2 (\u2200 (a : M\u2081) (b : M\u2082), 0 \u2264 \u2191Q\u2081 a + \u2191Q\u2082 b) \u2192 (\u2200 (x : M\u2081), 0 \u2264 \u2191Q\u2081 x) \u2227 \u2200 (x : M\u2082), 0 \u2264 \u2191Q\u2082 x\n\ncase mpr\n\u03b9 : Type u_1\nR\u271d : Type u_2\nM\u2081 : Type u_3\nM\u2082 : Type u_4\nN\u2081 : Type u_5\nN\u2082 : Type u_6\nM\u1d62 : \u03b9 \u2192 Type u_7\nN\u1d62 : \u03b9 \u2192 Type u_8\ninst\u271d\u00b9\u00b9 : CommSemiring R\u271d\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2081\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid N\u2081\ninst\u271d\u2077 : AddCommMonoid N\u2082\ninst\u271d\u2076 : Module R\u271d M\u2081\ninst\u271d\u2075 : Module R\u271d M\u2082\ninst\u271d\u2074 : Module R\u271d N\u2081\ninst\u271d\u00b3 : Module R\u271d N\u2082\nR : Type u_9\ninst\u271d\u00b2 : OrderedCommRing R\ninst\u271d\u00b9 : Module R M\u2081\ninst\u271d : Module R M\u2082\nQ\u2081 : QuadraticForm R M\u2081\nQ\u2082 : QuadraticForm R M\u2082\n\u22a2 ((\u2200 (x : M\u2081), 0 \u2264 \u2191Q\u2081 x) \u2227 \u2200 (x : M\u2082), 0 \u2264 \u2191Q\u2082 x) \u2192 \u2200 (a : M\u2081) (b : M\u2082), 0 \u2264 \u2191Q\u2081 a + \u2191Q\u2082 b"}, {"tactic": "intro h", "annotated_tactic": ["intro h", []], "state_before": "case mp\n\u03b9 : Type u_1\nR\u271d : Type u_2\nM\u2081 : Type u_3\nM\u2082 : Type u_4\nN\u2081 : Type u_5\nN\u2082 : Type u_6\nM\u1d62 : \u03b9 \u2192 Type u_7\nN\u1d62 : \u03b9 \u2192 Type u_8\ninst\u271d\u00b9\u00b9 : CommSemiring R\u271d\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2081\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid N\u2081\ninst\u271d\u2077 : AddCommMonoid N\u2082\ninst\u271d\u2076 : Module R\u271d M\u2081\ninst\u271d\u2075 : Module R\u271d M\u2082\ninst\u271d\u2074 : Module R\u271d N\u2081\ninst\u271d\u00b3 : Module R\u271d N\u2082\nR : Type u_9\ninst\u271d\u00b2 : OrderedCommRing R\ninst\u271d\u00b9 : Module R M\u2081\ninst\u271d : Module R M\u2082\nQ\u2081 : QuadraticForm R M\u2081\nQ\u2082 : QuadraticForm R M\u2082\n\u22a2 (\u2200 (a : M\u2081) (b : M\u2082), 0 \u2264 \u2191Q\u2081 a + \u2191Q\u2082 b) \u2192 (\u2200 (x : M\u2081), 0 \u2264 \u2191Q\u2081 x) \u2227 \u2200 (x : M\u2082), 0 \u2264 \u2191Q\u2082 x", "state_after": "case mp\n\u03b9 : Type u_1\nR\u271d : Type u_2\nM\u2081 : Type u_3\nM\u2082 : Type u_4\nN\u2081 : Type u_5\nN\u2082 : Type u_6\nM\u1d62 : \u03b9 \u2192 Type u_7\nN\u1d62 : \u03b9 \u2192 Type u_8\ninst\u271d\u00b9\u00b9 : CommSemiring R\u271d\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2081\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid N\u2081\ninst\u271d\u2077 : AddCommMonoid N\u2082\ninst\u271d\u2076 : Module R\u271d M\u2081\ninst\u271d\u2075 : Module R\u271d M\u2082\ninst\u271d\u2074 : Module R\u271d N\u2081\ninst\u271d\u00b3 : Module R\u271d N\u2082\nR : Type u_9\ninst\u271d\u00b2 : OrderedCommRing R\ninst\u271d\u00b9 : Module R M\u2081\ninst\u271d : Module R M\u2082\nQ\u2081 : QuadraticForm R M\u2081\nQ\u2082 : QuadraticForm R M\u2082\nh : \u2200 (a : M\u2081) (b : M\u2082), 0 \u2264 \u2191Q\u2081 a + \u2191Q\u2082 b\n\u22a2 (\u2200 (x : M\u2081), 0 \u2264 \u2191Q\u2081 x) \u2227 \u2200 (x : M\u2082), 0 \u2264 \u2191Q\u2082 x"}, {"tactic": "constructor", "annotated_tactic": ["constructor", []], "state_before": "case mp\n\u03b9 : Type u_1\nR\u271d : Type u_2\nM\u2081 : Type u_3\nM\u2082 : Type u_4\nN\u2081 : Type u_5\nN\u2082 : Type u_6\nM\u1d62 : \u03b9 \u2192 Type u_7\nN\u1d62 : \u03b9 \u2192 Type u_8\ninst\u271d\u00b9\u00b9 : CommSemiring R\u271d\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2081\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid N\u2081\ninst\u271d\u2077 : AddCommMonoid N\u2082\ninst\u271d\u2076 : Module R\u271d M\u2081\ninst\u271d\u2075 : Module R\u271d M\u2082\ninst\u271d\u2074 : Module R\u271d N\u2081\ninst\u271d\u00b3 : Module R\u271d N\u2082\nR : Type u_9\ninst\u271d\u00b2 : OrderedCommRing R\ninst\u271d\u00b9 : Module R M\u2081\ninst\u271d : Module R M\u2082\nQ\u2081 : QuadraticForm R M\u2081\nQ\u2082 : QuadraticForm R M\u2082\nh : \u2200 (a : M\u2081) (b : M\u2082), 0 \u2264 \u2191Q\u2081 a + \u2191Q\u2082 b\n\u22a2 (\u2200 (x : M\u2081), 0 \u2264 \u2191Q\u2081 x) \u2227 \u2200 (x : M\u2082), 0 \u2264 \u2191Q\u2082 x", "state_after": "case mp.left\n\u03b9 : Type u_1\nR\u271d : Type u_2\nM\u2081 : Type u_3\nM\u2082 : Type u_4\nN\u2081 : Type u_5\nN\u2082 : Type u_6\nM\u1d62 : \u03b9 \u2192 Type u_7\nN\u1d62 : \u03b9 \u2192 Type u_8\ninst\u271d\u00b9\u00b9 : CommSemiring R\u271d\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2081\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid N\u2081\ninst\u271d\u2077 : AddCommMonoid N\u2082\ninst\u271d\u2076 : Module R\u271d M\u2081\ninst\u271d\u2075 : Module R\u271d M\u2082\ninst\u271d\u2074 : Module R\u271d N\u2081\ninst\u271d\u00b3 : Module R\u271d N\u2082\nR : Type u_9\ninst\u271d\u00b2 : OrderedCommRing R\ninst\u271d\u00b9 : Module R M\u2081\ninst\u271d : Module R M\u2082\nQ\u2081 : QuadraticForm R M\u2081\nQ\u2082 : QuadraticForm R M\u2082\nh : \u2200 (a : M\u2081) (b : M\u2082), 0 \u2264 \u2191Q\u2081 a + \u2191Q\u2082 b\n\u22a2 \u2200 (x : M\u2081), 0 \u2264 \u2191Q\u2081 x\n\ncase mp.right\n\u03b9 : Type u_1\nR\u271d : Type u_2\nM\u2081 : Type u_3\nM\u2082 : Type u_4\nN\u2081 : Type u_5\nN\u2082 : Type u_6\nM\u1d62 : \u03b9 \u2192 Type u_7\nN\u1d62 : \u03b9 \u2192 Type u_8\ninst\u271d\u00b9\u00b9 : CommSemiring R\u271d\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2081\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid N\u2081\ninst\u271d\u2077 : AddCommMonoid N\u2082\ninst\u271d\u2076 : Module R\u271d M\u2081\ninst\u271d\u2075 : Module R\u271d M\u2082\ninst\u271d\u2074 : Module R\u271d N\u2081\ninst\u271d\u00b3 : Module R\u271d N\u2082\nR : Type u_9\ninst\u271d\u00b2 : OrderedCommRing R\ninst\u271d\u00b9 : Module R M\u2081\ninst\u271d : Module R M\u2082\nQ\u2081 : QuadraticForm R M\u2081\nQ\u2082 : QuadraticForm R M\u2082\nh : \u2200 (a : M\u2081) (b : M\u2082), 0 \u2264 \u2191Q\u2081 a + \u2191Q\u2082 b\n\u22a2 \u2200 (x : M\u2082), 0 \u2264 \u2191Q\u2082 x"}, {"tactic": "intro x", "annotated_tactic": ["intro x", []], "state_before": "case mp.left\n\u03b9 : Type u_1\nR\u271d : Type u_2\nM\u2081 : Type u_3\nM\u2082 : Type u_4\nN\u2081 : Type u_5\nN\u2082 : Type u_6\nM\u1d62 : \u03b9 \u2192 Type u_7\nN\u1d62 : \u03b9 \u2192 Type u_8\ninst\u271d\u00b9\u00b9 : CommSemiring R\u271d\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2081\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid N\u2081\ninst\u271d\u2077 : AddCommMonoid N\u2082\ninst\u271d\u2076 : Module R\u271d M\u2081\ninst\u271d\u2075 : Module R\u271d M\u2082\ninst\u271d\u2074 : Module R\u271d N\u2081\ninst\u271d\u00b3 : Module R\u271d N\u2082\nR : Type u_9\ninst\u271d\u00b2 : OrderedCommRing R\ninst\u271d\u00b9 : Module R M\u2081\ninst\u271d : Module R M\u2082\nQ\u2081 : QuadraticForm R M\u2081\nQ\u2082 : QuadraticForm R M\u2082\nh : \u2200 (a : M\u2081) (b : M\u2082), 0 \u2264 \u2191Q\u2081 a + \u2191Q\u2082 b\n\u22a2 \u2200 (x : M\u2081), 0 \u2264 \u2191Q\u2081 x", "state_after": "case mp.left\n\u03b9 : Type u_1\nR\u271d : Type u_2\nM\u2081 : Type u_3\nM\u2082 : Type u_4\nN\u2081 : Type u_5\nN\u2082 : Type u_6\nM\u1d62 : \u03b9 \u2192 Type u_7\nN\u1d62 : \u03b9 \u2192 Type u_8\ninst\u271d\u00b9\u00b9 : CommSemiring R\u271d\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2081\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid N\u2081\ninst\u271d\u2077 : AddCommMonoid N\u2082\ninst\u271d\u2076 : Module R\u271d M\u2081\ninst\u271d\u2075 : Module R\u271d M\u2082\ninst\u271d\u2074 : Module R\u271d N\u2081\ninst\u271d\u00b3 : Module R\u271d N\u2082\nR : Type u_9\ninst\u271d\u00b2 : OrderedCommRing R\ninst\u271d\u00b9 : Module R M\u2081\ninst\u271d : Module R M\u2082\nQ\u2081 : QuadraticForm R M\u2081\nQ\u2082 : QuadraticForm R M\u2082\nh : \u2200 (a : M\u2081) (b : M\u2082), 0 \u2264 \u2191Q\u2081 a + \u2191Q\u2082 b\nx : M\u2081\n\u22a2 0 \u2264 \u2191Q\u2081 x"}, {"tactic": "simpa only [add_zero, map_zero] using h x 0", "annotated_tactic": ["simpa only [<a>add_zero</a>, <a>map_zero</a>] using h x 0", [{"full_name": "add_zero", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [469, 3], "def_end_pos": [469, 14]}, {"full_name": "QuadraticForm.map_zero", "def_path": "Mathlib/LinearAlgebra/QuadraticForm/Basic.lean", "def_pos": [237, 9], "def_end_pos": [237, 17]}]], "state_before": "case mp.left\n\u03b9 : Type u_1\nR\u271d : Type u_2\nM\u2081 : Type u_3\nM\u2082 : Type u_4\nN\u2081 : Type u_5\nN\u2082 : Type u_6\nM\u1d62 : \u03b9 \u2192 Type u_7\nN\u1d62 : \u03b9 \u2192 Type u_8\ninst\u271d\u00b9\u00b9 : CommSemiring R\u271d\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2081\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid N\u2081\ninst\u271d\u2077 : AddCommMonoid N\u2082\ninst\u271d\u2076 : Module R\u271d M\u2081\ninst\u271d\u2075 : Module R\u271d M\u2082\ninst\u271d\u2074 : Module R\u271d N\u2081\ninst\u271d\u00b3 : Module R\u271d N\u2082\nR : Type u_9\ninst\u271d\u00b2 : OrderedCommRing R\ninst\u271d\u00b9 : Module R M\u2081\ninst\u271d : Module R M\u2082\nQ\u2081 : QuadraticForm R M\u2081\nQ\u2082 : QuadraticForm R M\u2082\nh : \u2200 (a : M\u2081) (b : M\u2082), 0 \u2264 \u2191Q\u2081 a + \u2191Q\u2082 b\nx : M\u2081\n\u22a2 0 \u2264 \u2191Q\u2081 x", "state_after": "no goals"}, {"tactic": "intro x", "annotated_tactic": ["intro x", []], "state_before": "case mp.right\n\u03b9 : Type u_1\nR\u271d : Type u_2\nM\u2081 : Type u_3\nM\u2082 : Type u_4\nN\u2081 : Type u_5\nN\u2082 : Type u_6\nM\u1d62 : \u03b9 \u2192 Type u_7\nN\u1d62 : \u03b9 \u2192 Type u_8\ninst\u271d\u00b9\u00b9 : CommSemiring R\u271d\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2081\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid N\u2081\ninst\u271d\u2077 : AddCommMonoid N\u2082\ninst\u271d\u2076 : Module R\u271d M\u2081\ninst\u271d\u2075 : Module R\u271d M\u2082\ninst\u271d\u2074 : Module R\u271d N\u2081\ninst\u271d\u00b3 : Module R\u271d N\u2082\nR : Type u_9\ninst\u271d\u00b2 : OrderedCommRing R\ninst\u271d\u00b9 : Module R M\u2081\ninst\u271d : Module R M\u2082\nQ\u2081 : QuadraticForm R M\u2081\nQ\u2082 : QuadraticForm R M\u2082\nh : \u2200 (a : M\u2081) (b : M\u2082), 0 \u2264 \u2191Q\u2081 a + \u2191Q\u2082 b\n\u22a2 \u2200 (x : M\u2082), 0 \u2264 \u2191Q\u2082 x", "state_after": "case mp.right\n\u03b9 : Type u_1\nR\u271d : Type u_2\nM\u2081 : Type u_3\nM\u2082 : Type u_4\nN\u2081 : Type u_5\nN\u2082 : Type u_6\nM\u1d62 : \u03b9 \u2192 Type u_7\nN\u1d62 : \u03b9 \u2192 Type u_8\ninst\u271d\u00b9\u00b9 : CommSemiring R\u271d\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2081\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid N\u2081\ninst\u271d\u2077 : AddCommMonoid N\u2082\ninst\u271d\u2076 : Module R\u271d M\u2081\ninst\u271d\u2075 : Module R\u271d M\u2082\ninst\u271d\u2074 : Module R\u271d N\u2081\ninst\u271d\u00b3 : Module R\u271d N\u2082\nR : Type u_9\ninst\u271d\u00b2 : OrderedCommRing R\ninst\u271d\u00b9 : Module R M\u2081\ninst\u271d : Module R M\u2082\nQ\u2081 : QuadraticForm R M\u2081\nQ\u2082 : QuadraticForm R M\u2082\nh : \u2200 (a : M\u2081) (b : M\u2082), 0 \u2264 \u2191Q\u2081 a + \u2191Q\u2082 b\nx : M\u2082\n\u22a2 0 \u2264 \u2191Q\u2082 x"}, {"tactic": "simpa only [zero_add, map_zero] using h 0 x", "annotated_tactic": ["simpa only [<a>zero_add</a>, <a>map_zero</a>] using h 0 x", [{"full_name": "zero_add", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [463, 3], "def_end_pos": [463, 14]}, {"full_name": "QuadraticForm.map_zero", "def_path": "Mathlib/LinearAlgebra/QuadraticForm/Basic.lean", "def_pos": [237, 9], "def_end_pos": [237, 17]}]], "state_before": "case mp.right\n\u03b9 : Type u_1\nR\u271d : Type u_2\nM\u2081 : Type u_3\nM\u2082 : Type u_4\nN\u2081 : Type u_5\nN\u2082 : Type u_6\nM\u1d62 : \u03b9 \u2192 Type u_7\nN\u1d62 : \u03b9 \u2192 Type u_8\ninst\u271d\u00b9\u00b9 : CommSemiring R\u271d\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2081\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid N\u2081\ninst\u271d\u2077 : AddCommMonoid N\u2082\ninst\u271d\u2076 : Module R\u271d M\u2081\ninst\u271d\u2075 : Module R\u271d M\u2082\ninst\u271d\u2074 : Module R\u271d N\u2081\ninst\u271d\u00b3 : Module R\u271d N\u2082\nR : Type u_9\ninst\u271d\u00b2 : OrderedCommRing R\ninst\u271d\u00b9 : Module R M\u2081\ninst\u271d : Module R M\u2082\nQ\u2081 : QuadraticForm R M\u2081\nQ\u2082 : QuadraticForm R M\u2082\nh : \u2200 (a : M\u2081) (b : M\u2082), 0 \u2264 \u2191Q\u2081 a + \u2191Q\u2082 b\nx : M\u2082\n\u22a2 0 \u2264 \u2191Q\u2082 x", "state_after": "no goals"}, {"tactic": "rintro \u27e8h\u2081, h\u2082\u27e9 x\u2081 x\u2082", "annotated_tactic": ["rintro \u27e8h\u2081, h\u2082\u27e9 x\u2081 x\u2082", []], "state_before": "case mpr\n\u03b9 : Type u_1\nR\u271d : Type u_2\nM\u2081 : Type u_3\nM\u2082 : Type u_4\nN\u2081 : Type u_5\nN\u2082 : Type u_6\nM\u1d62 : \u03b9 \u2192 Type u_7\nN\u1d62 : \u03b9 \u2192 Type u_8\ninst\u271d\u00b9\u00b9 : CommSemiring R\u271d\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2081\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid N\u2081\ninst\u271d\u2077 : AddCommMonoid N\u2082\ninst\u271d\u2076 : Module R\u271d M\u2081\ninst\u271d\u2075 : Module R\u271d M\u2082\ninst\u271d\u2074 : Module R\u271d N\u2081\ninst\u271d\u00b3 : Module R\u271d N\u2082\nR : Type u_9\ninst\u271d\u00b2 : OrderedCommRing R\ninst\u271d\u00b9 : Module R M\u2081\ninst\u271d : Module R M\u2082\nQ\u2081 : QuadraticForm R M\u2081\nQ\u2082 : QuadraticForm R M\u2082\n\u22a2 ((\u2200 (x : M\u2081), 0 \u2264 \u2191Q\u2081 x) \u2227 \u2200 (x : M\u2082), 0 \u2264 \u2191Q\u2082 x) \u2192 \u2200 (a : M\u2081) (b : M\u2082), 0 \u2264 \u2191Q\u2081 a + \u2191Q\u2082 b", "state_after": "case mpr.intro\n\u03b9 : Type u_1\nR\u271d : Type u_2\nM\u2081 : Type u_3\nM\u2082 : Type u_4\nN\u2081 : Type u_5\nN\u2082 : Type u_6\nM\u1d62 : \u03b9 \u2192 Type u_7\nN\u1d62 : \u03b9 \u2192 Type u_8\ninst\u271d\u00b9\u00b9 : CommSemiring R\u271d\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2081\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid N\u2081\ninst\u271d\u2077 : AddCommMonoid N\u2082\ninst\u271d\u2076 : Module R\u271d M\u2081\ninst\u271d\u2075 : Module R\u271d M\u2082\ninst\u271d\u2074 : Module R\u271d N\u2081\ninst\u271d\u00b3 : Module R\u271d N\u2082\nR : Type u_9\ninst\u271d\u00b2 : OrderedCommRing R\ninst\u271d\u00b9 : Module R M\u2081\ninst\u271d : Module R M\u2082\nQ\u2081 : QuadraticForm R M\u2081\nQ\u2082 : QuadraticForm R M\u2082\nh\u2081 : \u2200 (x : M\u2081), 0 \u2264 \u2191Q\u2081 x\nh\u2082 : \u2200 (x : M\u2082), 0 \u2264 \u2191Q\u2082 x\nx\u2081 : M\u2081\nx\u2082 : M\u2082\n\u22a2 0 \u2264 \u2191Q\u2081 x\u2081 + \u2191Q\u2082 x\u2082"}, {"tactic": "exact add_nonneg (h\u2081 x\u2081) (h\u2082 x\u2082)", "annotated_tactic": ["exact <a>add_nonneg</a> (h\u2081 x\u2081) (h\u2082 x\u2082)", [{"full_name": "add_nonneg", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [1092, 24], "def_end_pos": [1092, 34]}]], "state_before": "case mpr.intro\n\u03b9 : Type u_1\nR\u271d : Type u_2\nM\u2081 : Type u_3\nM\u2082 : Type u_4\nN\u2081 : Type u_5\nN\u2082 : Type u_6\nM\u1d62 : \u03b9 \u2192 Type u_7\nN\u1d62 : \u03b9 \u2192 Type u_8\ninst\u271d\u00b9\u00b9 : CommSemiring R\u271d\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2081\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid N\u2081\ninst\u271d\u2077 : AddCommMonoid N\u2082\ninst\u271d\u2076 : Module R\u271d M\u2081\ninst\u271d\u2075 : Module R\u271d M\u2082\ninst\u271d\u2074 : Module R\u271d N\u2081\ninst\u271d\u00b3 : Module R\u271d N\u2082\nR : Type u_9\ninst\u271d\u00b2 : OrderedCommRing R\ninst\u271d\u00b9 : Module R M\u2081\ninst\u271d : Module R M\u2082\nQ\u2081 : QuadraticForm R M\u2081\nQ\u2082 : QuadraticForm R M\u2082\nh\u2081 : \u2200 (x : M\u2081), 0 \u2264 \u2191Q\u2081 x\nh\u2082 : \u2200 (x : M\u2082), 0 \u2264 \u2191Q\u2082 x\nx\u2081 : M\u2081\nx\u2082 : M\u2082\n\u22a2 0 \u2264 \u2191Q\u2081 x\u2081 + \u2191Q\u2082 x\u2082", "state_after": "no goals"}]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Order/Filter/Pointwise.lean
|
Filter.Tendsto.mul_mul
|
[
630,
1
] |
[
632,
51
] |
[]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Analysis/MeanInequalitiesPow.lean
|
NNReal.arith_mean_le_rpow_mean
|
[
166,
1
] |
[
170,
39
] |
[{"tactic": "exact_mod_cast\n Real.arith_mean_le_rpow_mean s _ _ (fun i _ => (w i).coe_nonneg) (by exact_mod_cast hw')\n (fun i _ => (z i).coe_nonneg) hp", "annotated_tactic": ["exact_mod_cast\n <a>Real.arith_mean_le_rpow_mean</a> s _ _ (fun i _ => (w i).<a>coe_nonneg</a>) (by exact_mod_cast hw')\n (fun i _ => (z i).<a>coe_nonneg</a>) hp", [{"full_name": "Real.arith_mean_le_rpow_mean", "def_path": "Mathlib/Analysis/MeanInequalitiesPow.lean", "def_pos": [102, 9], "def_end_pos": [102, 32]}, {"full_name": "NNReal.coe_nonneg", "def_path": "Mathlib/Data/Real/NNReal.lean", "def_pos": [134, 9], "def_end_pos": [134, 19]}, {"full_name": "NNReal.coe_nonneg", "def_path": "Mathlib/Data/Real/NNReal.lean", "def_pos": [134, 9], "def_end_pos": [134, 19]}]], "state_before": "\u03b9 : Type u\ns : Finset \u03b9\nw z : \u03b9 \u2192 \u211d\u22650\nhw' : \u2211 i in s, w i = 1\np : \u211d\nhp : 1 \u2264 p\n\u22a2 \u2211 i in s, w i * z i \u2264 (\u2211 i in s, w i * z i ^ p) ^ (1 / p)", "state_after": "no goals"}, {"tactic": "exact_mod_cast hw'", "annotated_tactic": ["exact_mod_cast hw'", []], "state_before": "\u03b9 : Type u\ns : Finset \u03b9\nw z : \u03b9 \u2192 \u211d\u22650\nhw' : \u2211 i in s, w i = 1\np : \u211d\nhp : 1 \u2264 p\n\u22a2 \u2211 i in s, \u2191(w i) = 1", "state_after": "no goals"}]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Topology/Order.lean
|
denseRange_discrete
|
[
297,
1
] |
[
298,
56
] |
[{"tactic": "rw [DenseRange, dense_discrete, range_iff_surjective]", "annotated_tactic": ["rw [<a>DenseRange</a>, <a>dense_discrete</a>, <a>range_iff_surjective</a>]", [{"full_name": "DenseRange", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [1830, 5], "def_end_pos": [1830, 15]}, {"full_name": "dense_discrete", "def_path": "Mathlib/Topology/Order.lean", "def_pos": [294, 17], "def_end_pos": [294, 31]}, {"full_name": "Set.range_iff_surjective", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [709, 9], "def_end_pos": [709, 29]}]], "state_before": "\u03b1 : Type u_2\nt t\u2081 t\u2082 : TopologicalSpace \u03b1\ns : Set \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : DiscreteTopology \u03b1\n\u03b9 : Type u_1\nf : \u03b9 \u2192 \u03b1\n\u22a2 DenseRange f \u2194 Surjective f", "state_after": "no goals"}]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean
|
Real.volume_Ioi
|
[
145,
1
] |
[
150,
63
] |
[{"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\na : \u211d\nn : \u2115\n\u22a2 \u2191n = \u2191\u2191volume (Ioo a (a + \u2191n))", "state_after": "no goals"}]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Topology/ContinuousFunction/Weierstrass.lean
|
continuousMap_mem_polynomialFunctions_closure
|
[
86,
1
] |
[
89,
7
] |
[{"tactic": "rw [polynomialFunctions_closure_eq_top _ _]", "annotated_tactic": ["rw [<a>polynomialFunctions_closure_eq_top</a> _ _]", [{"full_name": "polynomialFunctions_closure_eq_top", "def_path": "Mathlib/Topology/ContinuousFunction/Weierstrass.lean", "def_pos": [54, 9], "def_end_pos": [54, 43]}]], "state_before": "a b : \u211d\nf : C(\u2191(Set.Icc a b), \u211d)\n\u22a2 f \u2208 Subalgebra.topologicalClosure (polynomialFunctions (Set.Icc a b))", "state_after": "a b : \u211d\nf : C(\u2191(Set.Icc a b), \u211d)\n\u22a2 f \u2208 \u22a4"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "a b : \u211d\nf : C(\u2191(Set.Icc a b), \u211d)\n\u22a2 f \u2208 \u22a4", "state_after": "no goals"}]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Data/Multiset/Basic.lean
|
Multiset.subset_zero
|
[
431,
1
] |
[
432,
64
] |
[]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/MeasureTheory/Measure/MeasureSpace.lean
|
MeasureTheory.one_le_prob_iff
|
[
3071,
1
] |
[
3072,
63
] |
[]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Algebra/GroupPower/Lemmas.lean
|
pow_bit1_nonpos_iff
|
[
724,
1
] |
[
736,
22
] |
[{"tactic": "simp only [le_iff_lt_or_eq, pow_bit1_neg_iff]", "annotated_tactic": ["simp only [<a>le_iff_lt_or_eq</a>, <a>pow_bit1_neg_iff</a>]", [{"full_name": "le_iff_lt_or_eq", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [232, 9], "def_end_pos": [232, 24]}, {"full_name": "pow_bit1_neg_iff", "def_path": "Mathlib/Algebra/GroupPower/Lemmas.lean", "def_pos": [714, 9], "def_end_pos": [714, 25]}]], "state_before": "\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : LinearOrderedRing R\na : R\nn : \u2115\n\u22a2 a ^ bit1 n \u2264 0 \u2194 a \u2264 0", "state_after": "\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : LinearOrderedRing R\na : R\nn : \u2115\n\u22a2 a < 0 \u2228 a ^ bit1 n = 0 \u2194 a < 0 \u2228 a = 0"}, {"tactic": "refine' \u27e8_, _\u27e9", "annotated_tactic": ["refine' \u27e8_, _\u27e9", []], "state_before": "\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : LinearOrderedRing R\na : R\nn : \u2115\n\u22a2 a < 0 \u2228 a ^ bit1 n = 0 \u2194 a < 0 \u2228 a = 0", "state_after": "case refine'_1\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : LinearOrderedRing R\na : R\nn : \u2115\n\u22a2 a < 0 \u2228 a ^ bit1 n = 0 \u2192 a < 0 \u2228 a = 0\n\ncase refine'_2\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : LinearOrderedRing R\na : R\nn : \u2115\n\u22a2 a < 0 \u2228 a = 0 \u2192 a < 0 \u2228 a ^ bit1 n = 0"}, {"tactic": "rintro (hpos | hz)", "annotated_tactic": ["rintro (hpos | hz)", []], "state_before": "case refine'_1\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : LinearOrderedRing R\na : R\nn : \u2115\n\u22a2 a < 0 \u2228 a ^ bit1 n = 0 \u2192 a < 0 \u2228 a = 0", "state_after": "case refine'_1.inl\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : LinearOrderedRing R\na : R\nn : \u2115\nhpos : a < 0\n\u22a2 a < 0 \u2228 a = 0\n\ncase refine'_1.inr\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : LinearOrderedRing R\na : R\nn : \u2115\nhz : a ^ bit1 n = 0\n\u22a2 a < 0 \u2228 a = 0"}, {"tactic": "left", "annotated_tactic": ["left", []], "state_before": "case refine'_1.inl\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : LinearOrderedRing R\na : R\nn : \u2115\nhpos : a < 0\n\u22a2 a < 0 \u2228 a = 0", "state_after": "case refine'_1.inl.h\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : LinearOrderedRing R\na : R\nn : \u2115\nhpos : a < 0\n\u22a2 a < 0"}, {"tactic": "exact hpos", "annotated_tactic": ["exact hpos", []], "state_before": "case refine'_1.inl.h\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : LinearOrderedRing R\na : R\nn : \u2115\nhpos : a < 0\n\u22a2 a < 0", "state_after": "no goals"}, {"tactic": "right", "annotated_tactic": ["right", []], "state_before": "case refine'_1.inr\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : LinearOrderedRing R\na : R\nn : \u2115\nhz : a ^ bit1 n = 0\n\u22a2 a < 0 \u2228 a = 0", "state_after": "case refine'_1.inr.h\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : LinearOrderedRing R\na : R\nn : \u2115\nhz : a ^ bit1 n = 0\n\u22a2 a = 0"}, {"tactic": "exact (pow_eq_zero_iff'.1 hz).1", "annotated_tactic": ["exact (<a>pow_eq_zero_iff'</a>.1 hz).1", [{"full_name": "pow_eq_zero_iff'", "def_path": "Mathlib/Algebra/GroupPower/Ring.lean", "def_pos": [69, 9], "def_end_pos": [69, 25]}]], "state_before": "case refine'_1.inr.h\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : LinearOrderedRing R\na : R\nn : \u2115\nhz : a ^ bit1 n = 0\n\u22a2 a = 0", "state_after": "no goals"}, {"tactic": "rintro (hneg | hz)", "annotated_tactic": ["rintro (hneg | hz)", []], "state_before": "case refine'_2\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : LinearOrderedRing R\na : R\nn : \u2115\n\u22a2 a < 0 \u2228 a = 0 \u2192 a < 0 \u2228 a ^ bit1 n = 0", "state_after": "case refine'_2.inl\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : LinearOrderedRing R\na : R\nn : \u2115\nhneg : a < 0\n\u22a2 a < 0 \u2228 a ^ bit1 n = 0\n\ncase refine'_2.inr\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : LinearOrderedRing R\na : R\nn : \u2115\nhz : a = 0\n\u22a2 a < 0 \u2228 a ^ bit1 n = 0"}, {"tactic": "left", "annotated_tactic": ["left", []], "state_before": "case refine'_2.inl\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : LinearOrderedRing R\na : R\nn : \u2115\nhneg : a < 0\n\u22a2 a < 0 \u2228 a ^ bit1 n = 0", "state_after": "case refine'_2.inl.h\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : LinearOrderedRing R\na : R\nn : \u2115\nhneg : a < 0\n\u22a2 a < 0"}, {"tactic": "exact hneg", "annotated_tactic": ["exact hneg", []], "state_before": "case refine'_2.inl.h\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : LinearOrderedRing R\na : R\nn : \u2115\nhneg : a < 0\n\u22a2 a < 0", "state_after": "no goals"}, {"tactic": "right", "annotated_tactic": ["right", []], "state_before": "case refine'_2.inr\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : LinearOrderedRing R\na : R\nn : \u2115\nhz : a = 0\n\u22a2 a < 0 \u2228 a ^ bit1 n = 0", "state_after": "case refine'_2.inr.h\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : LinearOrderedRing R\na : R\nn : \u2115\nhz : a = 0\n\u22a2 a ^ bit1 n = 0"}, {"tactic": "simp [hz, bit1]", "annotated_tactic": ["simp [hz, <a>bit1</a>]", [{"full_name": "bit1", "def_path": "Mathlib/Init/ZeroOne.lean", "def_pos": [39, 34], "def_end_pos": [39, 38]}]], "state_before": "case refine'_2.inr.h\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : LinearOrderedRing R\na : R\nn : \u2115\nhz : a = 0\n\u22a2 a ^ bit1 n = 0", "state_after": "no goals"}]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Geometry/Euclidean/Angle/Oriented/Affine.lean
|
EuclideanGeometry.oangle_self_left_right
|
[
69,
1
] |
[
70,
18
] |
[]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/CategoryTheory/Functor/Basic.lean
|
CategoryTheory.Functor.comp_id
|
[
129,
11
] |
[
129,
71
] |
[{"tactic": "cases F", "annotated_tactic": ["cases F", []], "state_before": "C : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nE : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} E\nF : C \u2964 D\n\u22a2 F \u22d9 \ud835\udfed D = F", "state_after": "case mk\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nE : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} E\ntoPrefunctor\u271d : C \u2964q D\nmap_id\u271d : \u2200 (X : C), toPrefunctor\u271d.map (\ud835\udfd9 X) = \ud835\udfd9 (toPrefunctor\u271d.obj X)\nmap_comp\u271d : \u2200 {X Y Z : C} (f : X \u27f6 Y) (g : Y \u27f6 Z), toPrefunctor\u271d.map (f \u226b g) = toPrefunctor\u271d.map f \u226b toPrefunctor\u271d.map g\n\u22a2 mk toPrefunctor\u271d \u22d9 \ud835\udfed D = mk toPrefunctor\u271d"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case mk\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nE : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} E\ntoPrefunctor\u271d : C \u2964q D\nmap_id\u271d : \u2200 (X : C), toPrefunctor\u271d.map (\ud835\udfd9 X) = \ud835\udfd9 (toPrefunctor\u271d.obj X)\nmap_comp\u271d : \u2200 {X Y Z : C} (f : X \u27f6 Y) (g : Y \u27f6 Z), toPrefunctor\u271d.map (f \u226b g) = toPrefunctor\u271d.map f \u226b toPrefunctor\u271d.map g\n\u22a2 mk toPrefunctor\u271d \u22d9 \ud835\udfed D = mk toPrefunctor\u271d", "state_after": "no goals"}]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/CategoryTheory/Monoidal/Mon_.lean
|
Mon_.mul_leftUnitor
|
[
460,
1
] |
[
466,
47
] |
[{"tactic": "rw [\u2190 Category.comp_id (\u03bb_ (\ud835\udfd9_ C)).hom, \u2190 Category.id_comp M.mul, tensor_comp]", "annotated_tactic": ["rw [\u2190 <a>Category.comp_id</a> (\u03bb_ (\ud835\udfd9_ C)).<a>hom</a>, \u2190 <a>Category.id_comp</a> M.mul, <a>tensor_comp</a>]", [{"full_name": "CategoryTheory.Category.comp_id", "def_path": "Mathlib/CategoryTheory/Category/Basic.lean", "def_pos": [157, 3], "def_end_pos": [157, 10]}, {"full_name": "CategoryTheory.Iso.hom", "def_path": "Mathlib/CategoryTheory/Iso.lean", "def_pos": [53, 3], "def_end_pos": [53, 6]}, {"full_name": "CategoryTheory.Category.id_comp", "def_path": "Mathlib/CategoryTheory/Category/Basic.lean", "def_pos": [155, 3], "def_end_pos": [155, 10]}, {"full_name": "CategoryTheory.MonoidalCategory.tensor_comp", "def_path": "Mathlib/CategoryTheory/Monoidal/Category.lean", "def_pos": [93, 3], "def_end_pos": [93, 14]}]], "state_before": "C : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM : Mon_ C\n\u22a2 (tensor_\u03bc C (\ud835\udfd9_ C, M.X) (\ud835\udfd9_ C, M.X) \u226b ((\u03bb_ (\ud835\udfd9_ C)).hom \u2297 M.mul)) \u226b (\u03bb_ M.X).hom =\n ((\u03bb_ M.X).hom \u2297 (\u03bb_ M.X).hom) \u226b M.mul", "state_after": "C : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM : Mon_ C\n\u22a2 (tensor_\u03bc C (\ud835\udfd9_ C, M.X) (\ud835\udfd9_ C, M.X) \u226b ((\u03bb_ (\ud835\udfd9_ C)).hom \u2297 \ud835\udfd9 (M.X \u2297 M.X)) \u226b (\ud835\udfd9 (\ud835\udfd9_ C) \u2297 M.mul)) \u226b (\u03bb_ M.X).hom =\n ((\u03bb_ M.X).hom \u2297 (\u03bb_ M.X).hom) \u226b \ud835\udfd9 (M.X \u2297 M.X) \u226b M.mul"}, {"tactic": "slice_lhs 3 4 => rw [leftUnitor_naturality]", "annotated_tactic": ["slice_lhs 3 4 => rw [<a>leftUnitor_naturality</a>]", [{"full_name": "CategoryTheory.MonoidalCategory.leftUnitor_naturality", "def_path": "Mathlib/CategoryTheory/Monoidal/Category.lean", "def_pos": [118, 3], "def_end_pos": [118, 24]}]], "state_before": "C : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM : Mon_ C\n\u22a2 (tensor_\u03bc C (\ud835\udfd9_ C, M.X) (\ud835\udfd9_ C, M.X) \u226b ((\u03bb_ (\ud835\udfd9_ C)).hom \u2297 \ud835\udfd9 (M.X \u2297 M.X)) \u226b (\ud835\udfd9 (\ud835\udfd9_ C) \u2297 M.mul)) \u226b (\u03bb_ M.X).hom =\n ((\u03bb_ M.X).hom \u2297 (\u03bb_ M.X).hom) \u226b \ud835\udfd9 (M.X \u2297 M.X) \u226b M.mul", "state_after": "C : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM : Mon_ C\n\u22a2 tensor_\u03bc C (\ud835\udfd9_ C, M.X) (\ud835\udfd9_ C, M.X) \u226b ((\u03bb_ (\ud835\udfd9_ C)).hom \u2297 \ud835\udfd9 (M.X \u2297 M.X)) \u226b (\u03bb_ (M.X \u2297 M.X)).hom \u226b M.mul =\n ((\u03bb_ M.X).hom \u2297 (\u03bb_ M.X).hom) \u226b \ud835\udfd9 (M.X \u2297 M.X) \u226b M.mul"}, {"tactic": "slice_lhs 1 3 => rw [\u2190 leftUnitor_monoidal]", "annotated_tactic": ["slice_lhs 1 3 => rw [\u2190 <a>leftUnitor_monoidal</a>]", [{"full_name": "CategoryTheory.leftUnitor_monoidal", "def_path": "Mathlib/CategoryTheory/Monoidal/Braided.lean", "def_pos": [577, 9], "def_end_pos": [577, 28]}]], "state_before": "C : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM : Mon_ C\n\u22a2 tensor_\u03bc C (\ud835\udfd9_ C, M.X) (\ud835\udfd9_ C, M.X) \u226b ((\u03bb_ (\ud835\udfd9_ C)).hom \u2297 \ud835\udfd9 (M.X \u2297 M.X)) \u226b (\u03bb_ (M.X \u2297 M.X)).hom \u226b M.mul =\n ((\u03bb_ M.X).hom \u2297 (\u03bb_ M.X).hom) \u226b \ud835\udfd9 (M.X \u2297 M.X) \u226b M.mul", "state_after": "C : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM : Mon_ C\n\u22a2 ((\u03bb_ M.X).hom \u2297 (\u03bb_ M.X).hom) \u226b M.mul = ((\u03bb_ M.X).hom \u2297 (\u03bb_ M.X).hom) \u226b \ud835\udfd9 (M.X \u2297 M.X) \u226b M.mul"}, {"tactic": "simp only [Category.assoc, Category.id_comp]", "annotated_tactic": ["simp only [<a>Category.assoc</a>, <a>Category.id_comp</a>]", [{"full_name": "CategoryTheory.Category.assoc", "def_path": "Mathlib/CategoryTheory/Category/Basic.lean", "def_pos": [159, 3], "def_end_pos": [159, 8]}, {"full_name": "CategoryTheory.Category.id_comp", "def_path": "Mathlib/CategoryTheory/Category/Basic.lean", "def_pos": [155, 3], "def_end_pos": [155, 10]}]], "state_before": "C : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM : Mon_ C\n\u22a2 ((\u03bb_ M.X).hom \u2297 (\u03bb_ M.X).hom) \u226b M.mul = ((\u03bb_ M.X).hom \u2297 (\u03bb_ M.X).hom) \u226b \ud835\udfd9 (M.X \u2297 M.X) \u226b M.mul", "state_after": "no goals"}]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/LinearAlgebra/Multilinear/Basic.lean
|
MultilinearMap.coe_inj
|
[
142,
1
] |
[
143,
20
] |
[]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Algebra/Order/Ring/Lemmas.lean
|
le_of_le_mul_of_le_one_of_nonneg_right
|
[
951,
1
] |
[
953,
41
] |
[]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Algebra/Order/ToIntervalMod.lean
|
toIcoMod_add_left'
|
[
495,
1
] |
[
496,
47
] |
[{"tactic": "rw [add_comm, toIcoMod_add_right', add_comm]", "annotated_tactic": ["rw [<a>add_comm</a>, <a>toIcoMod_add_right'</a>, <a>add_comm</a>]", [{"full_name": "add_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [301, 3], "def_end_pos": [301, 14]}, {"full_name": "toIcoMod_add_right'", "def_path": "Mathlib/Algebra/Order/ToIntervalMod.lean", "def_pos": [475, 9], "def_end_pos": [475, 28]}, {"full_name": "add_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [301, 3], "def_end_pos": [301, 14]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : LinearOrderedAddCommGroup \u03b1\nh\u03b1 : Archimedean \u03b1\np : \u03b1\nhp : 0 < p\na\u271d b\u271d c : \u03b1\nn : \u2124\na b : \u03b1\n\u22a2 toIcoMod hp (p + a) b = p + toIcoMod hp a b", "state_after": "no goals"}]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Order/Filter/Extr.lean
|
IsExtrOn.inter
|
[
301,
1
] |
[
302,
39
] |
[]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/MeasureTheory/Measure/NullMeasurable.lean
|
MeasureTheory.measure_union₀'
|
[
335,
1
] |
[
336,
98
] |
[{"tactic": "rw [union_comm, measure_union\u2080 hs (AEDisjoint.symm hd), add_comm]", "annotated_tactic": ["rw [<a>union_comm</a>, <a>measure_union\u2080</a> hs (<a>AEDisjoint.symm</a> hd), <a>add_comm</a>]", [{"full_name": "Set.union_comm", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [786, 9], "def_end_pos": [786, 19]}, {"full_name": "MeasureTheory.measure_union\u2080", "def_path": "Mathlib/MeasureTheory/Measure/NullMeasurable.lean", "def_pos": [331, 9], "def_end_pos": [331, 23]}, {"full_name": "MeasureTheory.AEDisjoint.symm", "def_path": "Mathlib/MeasureTheory/Measure/AEDisjoint.lean", "def_pos": [55, 19], "def_end_pos": [55, 23]}, {"full_name": "add_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [301, 3], "def_end_pos": [301, 14]}]], "state_before": "\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nhs : NullMeasurableSet s\nhd : AEDisjoint \u03bc s t\n\u22a2 \u2191\u2191\u03bc (s \u222a t) = \u2191\u2191\u03bc s + \u2191\u2191\u03bc t", "state_after": "no goals"}]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/RingTheory/GradedAlgebra/HomogeneousIdeal.lean
|
Ideal.homogeneousCore'_le
|
[
123,
1
] |
[
124,
47
] |
[]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Order/Hom/Set.lean
|
OrderIso.symm_preimage_preimage
|
[
51,
1
] |
[
52,
37
] |
[]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Algebra/QuaternionBasis.lean
|
QuaternionAlgebra.Basis.lift_one
|
[
118,
1
] |
[
118,
65
] |
[{"tactic": "simp [lift]", "annotated_tactic": ["simp [<a>lift</a>]", [{"full_name": "QuaternionAlgebra.Basis.lift", "def_path": "Mathlib/Algebra/QuaternionBasis.lean", "def_pos": [111, 5], "def_end_pos": [111, 9]}]], "state_before": "R : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : Ring A\ninst\u271d\u00b2 : Ring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nc\u2081 c\u2082 : R\nq : Basis A c\u2081 c\u2082\n\u22a2 lift q 1 = 1", "state_after": "no goals"}]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Algebra/Group/Commute/Units.lean
|
Commute.units_inv_left_iff
|
[
44,
1
] |
[
45,
37
] |
[]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Topology/NhdsSet.lean
|
monotone_nhdsSet
|
[
132,
1
] |
[
132,
87
] |
[]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Tactic/NormNum/Result.lean
|
Mathlib.Meta.NormNum.IsInt.neg_to_eq
|
[
175,
1
] |
[
177,
64
] |
[{"tactic": "simp [Int.negOfNat_eq, Int.cast_neg]", "annotated_tactic": ["simp [<a>Int.negOfNat_eq</a>, <a>Int.cast_neg</a>]", [{"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.cast_neg", "def_path": "Mathlib/Data/Int/Cast/Basic.lean", "def_pos": [83, 9], "def_end_pos": [83, 17]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : Ring \u03b1\nn : \u2115\n\u22a2 \u2191(Int.negOfNat n) = -\u2191n", "state_after": "no goals"}]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/CategoryTheory/Subobject/Limits.lean
|
CategoryTheory.Limits.factorThruKernelSubobject_comp_kernelSubobjectIso
|
[
141,
1
] |
[
143,
42
] |
[{"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "C : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nX Y Z : C\ninst\u271d\u00b9 : HasZeroMorphisms C\nf : X \u27f6 Y\ninst\u271d : HasKernel f\nW : C\nh : W \u27f6 X\nw : h \u226b f = 0\n\u22a2 (factorThruKernelSubobject f h w \u226b (kernelSubobjectIso f).hom) \u226b kernel.\u03b9 f = kernel.lift f h w \u226b kernel.\u03b9 f", "state_after": "no goals"}]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Order/LocallyFinite.lean
|
WithBot.Ioo_coe_coe
|
[
1206,
1
] |
[
1207,
6
] |
[]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/RingTheory/Subring/Basic.lean
|
Subring.coe_eq_zero_iff
|
[
450,
1
] |
[
451,
87
] |
[]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/CategoryTheory/Limits/Shapes/Pullbacks.lean
|
CategoryTheory.Limits.pullbackConeOfLeftIso_snd
|
[
1623,
1
] |
[
1623,
81
] |
[]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Data/Finset/Basic.lean
|
Finset.insert_inj
|
[
1204,
1
] |
[
1205,
101
] |
[]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Order/Filter/AtTopBot.lean
|
Filter.tendsto_atTop_of_add_const_right
|
[
748,
1
] |
[
750,
93
] |
[]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Data/Polynomial/Eval.lean
|
Polynomial.nat_cast_mul_comp
|
[
636,
1
] |
[
637,
50
] |
[{"tactic": "rw [\u2190 C_eq_nat_cast, C_mul_comp, C_eq_nat_cast]", "annotated_tactic": ["rw [\u2190 <a>C_eq_nat_cast</a>, <a>C_mul_comp</a>, <a>C_eq_nat_cast</a>]", [{"full_name": "Polynomial.C_eq_nat_cast", "def_path": "Mathlib/Data/Polynomial/Basic.lean", "def_pos": [551, 9], "def_end_pos": [551, 22]}, {"full_name": "Polynomial.C_mul_comp", "def_path": "Mathlib/Data/Polynomial/Eval.lean", "def_pos": [626, 9], "def_end_pos": [626, 19]}, {"full_name": "Polynomial.C_eq_nat_cast", "def_path": "Mathlib/Data/Polynomial/Basic.lean", "def_pos": [551, 9], "def_end_pos": [551, 22]}]], "state_before": "R : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n\u271d : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nn : \u2115\n\u22a2 comp (\u2191n * p) r = \u2191n * comp p r", "state_after": "no goals"}]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Algebra/Order/Group/Defs.lean
|
inv_lt'
|
[
389,
1
] |
[
389,
73
] |
[{"tactic": "rw [\u2190 inv_lt_inv_iff, inv_inv]", "annotated_tactic": ["rw [\u2190 <a>inv_lt_inv_iff</a>, <a>inv_inv</a>]", [{"full_name": "inv_lt_inv_iff", "def_path": "Mathlib/Algebra/Order/Group/Defs.lean", "def_pos": [382, 9], "def_end_pos": [382, 23]}, {"full_name": "inv_inv", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [800, 9], "def_end_pos": [800, 16]}]], "state_before": "\u03b1 : Type u\ninst\u271d\u00b3 : Group \u03b1\ninst\u271d\u00b2 : LT \u03b1\ninst\u271d\u00b9 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x < x_1\ninst\u271d : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x < x_1\na b c d : \u03b1\n\u22a2 a\u207b\u00b9 < b \u2194 b\u207b\u00b9 < a", "state_after": "no goals"}]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Data/Rat/Defs.lean
|
Rat.mkRat_one
|
[
259,
1
] |
[
260,
38
] |
[{"tactic": "simp [Rat.mkRat_eq, Rat.divInt_one]", "annotated_tactic": ["simp [<a>Rat.mkRat_eq</a>, <a>Rat.divInt_one</a>]", [{"full_name": "Rat.mkRat_eq", "def_path": "Mathlib/Data/Rat/Defs.lean", "def_pos": [64, 9], "def_end_pos": [64, 17]}, {"full_name": "Rat.divInt_one", "def_path": "Mathlib/Data/Rat/Defs.lean", "def_pos": [253, 9], "def_end_pos": [253, 19]}]], "state_before": "a b c : \u211a\nn : \u2124\n\u22a2 mkRat n 1 = \u2191n", "state_after": "no goals"}]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Analysis/Convolution.lean
|
MeasureTheory.AEStronglyMeasurable.convolution_integrand'
|
[
194,
1
] |
[
198,
75
] |
[]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Data/Analysis/Topology.lean
|
Ctop.Realizer.isOpen
|
[
157,
11
] |
[
158,
80
] |
[{"tactic": "simpa using F.mem_nhds.2 \u27e8s, m, Subset.refl _\u27e9", "annotated_tactic": ["simpa using F.mem_nhds.2 \u27e8s, m, <a>Subset.refl</a> _\u27e9", [{"full_name": "Set.Subset.refl", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [354, 9], "def_end_pos": [354, 20]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03c3 : Type u_3\n\u03c4 : Type u_4\ninst\u271d : TopologicalSpace \u03b1\nF : Realizer \u03b1\ns : F.\u03c3\na : \u03b1\nm : a \u2208 f F.F s\n\u22a2 \ud835\udcdd a \u2264 \ud835\udcdf (f F.F s)", "state_after": "no goals"}]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Combinatorics/Young/SemistandardTableau.lean
|
Ssyt.copy_eq
|
[
106,
1
] |
[
108,
17
] |
[]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/MeasureTheory/Measure/AEMeasurable.lean
|
AEMeasurable.ae_inf_principal_eq_mk
|
[
70,
1
] |
[
71,
28
] |
[]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/NumberTheory/PellMatiyasevic.lean
|
Pell.eq_pellZd
|
[
334,
1
] |
[
341,
32
] |
[{"tactic": "rw [Zsqrtd.coe_nat_val]", "annotated_tactic": ["rw [<a>Zsqrtd.coe_nat_val</a>]", [{"full_name": "Zsqrtd.coe_nat_val", "def_path": "Mathlib/NumberTheory/Zsqrtd/Basic.lean", "def_pos": [288, 9], "def_end_pos": [288, 20]}]], "state_before": "a : \u2115\na1 : 1 < a\nb : \u2124\u221a\u2191(Pell.d a1)\nb1 : 1 \u2264 b\nhp : IsPell b\nn : \u2115\nh : b \u2264 \u2191n\n\u22a2 \u2191n \u2264 pellZd a1 n", "state_after": "a : \u2115\na1 : 1 < a\nb : \u2124\u221a\u2191(Pell.d a1)\nb1 : 1 \u2264 b\nhp : IsPell b\nn : \u2115\nh : b \u2264 \u2191n\n\u22a2 { re := \u2191n, im := 0 } \u2264 pellZd a1 n"}, {"tactic": "exact\n Zsqrtd.le_of_le_le (Int.ofNat_le_ofNat_of_le <| le_of_lt <| n_lt_xn _ _)\n (Int.ofNat_zero_le _)", "annotated_tactic": ["exact\n <a>Zsqrtd.le_of_le_le</a> (<a>Int.ofNat_le_ofNat_of_le</a> <| <a>le_of_lt</a> <| <a>n_lt_xn</a> _ _)\n (<a>Int.ofNat_zero_le</a> _)", [{"full_name": "Zsqrtd.le_of_le_le", "def_path": "Mathlib/NumberTheory/Zsqrtd/Basic.lean", "def_pos": [734, 9], "def_end_pos": [734, 20]}, {"full_name": "Int.ofNat_le_ofNat_of_le", "def_path": "Mathlib/Init/Data/Int/Order.lean", "def_pos": [29, 30], "def_end_pos": [29, 50]}, {"full_name": "le_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [110, 9], "def_end_pos": [110, 17]}, {"full_name": "Pell.n_lt_xn", "def_path": "Mathlib/NumberTheory/PellMatiyasevic.lean", "def_pos": [279, 9], "def_end_pos": [279, 16]}, {"full_name": "Int.ofNat_zero_le", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [582, 9], "def_end_pos": [582, 22]}]], "state_before": "a : \u2115\na1 : 1 < a\nb : \u2124\u221a\u2191(Pell.d a1)\nb1 : 1 \u2264 b\nhp : IsPell b\nn : \u2115\nh : b \u2264 \u2191n\n\u22a2 { re := \u2191n, im := 0 } \u2264 pellZd a1 n", "state_after": "no goals"}]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Probability/Kernel/Composition.lean
|
ProbabilityTheory.kernel.lintegral_prod
|
[
928,
1
] |
[
931,
99
] |
[{"tactic": "simp_rw [prod, lintegral_compProd _ _ _ hg, swapLeft_apply, prodMkLeft_apply, Prod.swap_prod_mk]", "annotated_tactic": ["simp_rw [<a>prod</a>, <a>lintegral_compProd</a> _ _ _ hg, <a>swapLeft_apply</a>, <a>prodMkLeft_apply</a>, <a>Prod.swap_prod_mk</a>]", [{"full_name": "ProbabilityTheory.kernel.prod", "def_path": "Mathlib/Probability/Kernel/Composition.lean", "def_pos": [916, 19], "def_end_pos": [916, 23]}, {"full_name": "ProbabilityTheory.kernel.lintegral_compProd", "def_path": "Mathlib/Probability/Kernel/Composition.lean", "def_pos": [432, 9], "def_end_pos": [432, 27]}, {"full_name": "ProbabilityTheory.kernel.swapLeft_apply", "def_path": "Mathlib/Probability/Kernel/Composition.lean", "def_pos": [716, 9], "def_end_pos": [716, 23]}, {"full_name": "ProbabilityTheory.kernel.prodMkLeft_apply", "def_path": "Mathlib/Probability/Kernel/Composition.lean", "def_pos": [685, 9], "def_end_pos": [685, 25]}, {"full_name": "Prod.swap_prod_mk", "def_path": "Mathlib/Data/Prod/Basic.lean", "def_pos": [182, 9], "def_end_pos": [182, 21]}]], "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\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\n\u03b7 : { x // x \u2208 kernel \u03b1 \u03b3 }\ninst\u271d : IsSFiniteKernel \u03b7\na : \u03b1\ng : \u03b2 \u00d7 \u03b3 \u2192 \u211d\u22650\u221e\nhg : Measurable g\n\u22a2 \u222b\u207b (c : \u03b2 \u00d7 \u03b3), g c \u2202\u2191(\u03ba \u00d7\u2096 \u03b7) a = \u222b\u207b (b : \u03b2), \u222b\u207b (c : \u03b3), g (b, c) \u2202\u2191\u03b7 a \u2202\u2191\u03ba a", "state_after": "no goals"}]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Algebra/Quaternion.lean
|
QuaternionAlgebra.coe_basisOneIJK_repr
|
[
569,
1
] |
[
571,
6
] |
[]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/RepresentationTheory/Action.lean
|
Action.tensor_v
|
[
495,
1
] |
[
496,
6
] |
[]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/GroupTheory/OrderOfElement.lean
|
Subgroup.pow_index_mem
|
[
986,
1
] |
[
987,
93
] |
[{"tactic": "rw [\u2190 eq_one_iff, QuotientGroup.mk_pow H, index, pow_card_eq_one']", "annotated_tactic": ["rw [\u2190 <a>eq_one_iff</a>, <a>QuotientGroup.mk_pow</a> H, <a>index</a>, <a>pow_card_eq_one'</a>]", [{"full_name": "QuotientGroup.eq_one_iff", "def_path": "Mathlib/GroupTheory/QuotientGroup.lean", "def_pos": [120, 9], "def_end_pos": [120, 19]}, {"full_name": "QuotientGroup.mk_pow", "def_path": "Mathlib/GroupTheory/QuotientGroup.lean", "def_pos": [183, 9], "def_end_pos": [183, 15]}, {"full_name": "Subgroup.index", "def_path": "Mathlib/GroupTheory/Index.lean", "def_pos": [46, 19], "def_end_pos": [46, 24]}, {"full_name": "pow_card_eq_one'", "def_path": "Mathlib/GroupTheory/OrderOfElement.lean", "def_pos": [974, 9], "def_end_pos": [974, 25]}]], "state_before": "G\u271d : Type u_1\nH\u271d : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b3 : Group G\u271d\nx y : G\u271d\nn : \u2115\ninst\u271d\u00b2 : Fintype G\u271d\nG : Type u_6\ninst\u271d\u00b9 : Group G\nH : Subgroup G\ninst\u271d : Normal H\ng : G\n\u22a2 g ^ index H \u2208 H", "state_after": "no goals"}]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Topology/Algebra/Order/UpperLower.lean
|
IsUpperSet.interior
|
[
101,
11
] |
[
103,
24
] |
[{"tactic": "rw [\u2190 isLowerSet_compl, \u2190 closure_compl]", "annotated_tactic": ["rw [\u2190 <a>isLowerSet_compl</a>, \u2190 <a>closure_compl</a>]", [{"full_name": "isLowerSet_compl", "def_path": "Mathlib/Order/UpperLower/Basic.lean", "def_pos": [97, 9], "def_end_pos": [97, 25]}, {"full_name": "closure_compl", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [572, 9], "def_end_pos": [572, 22]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : HasUpperLowerClosure \u03b1\ns : Set \u03b1\nh : IsUpperSet s\n\u22a2 IsUpperSet (interior s)", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : HasUpperLowerClosure \u03b1\ns : Set \u03b1\nh : IsUpperSet s\n\u22a2 IsLowerSet (closure s\u1d9c)"}, {"tactic": "exact h.compl.closure", "annotated_tactic": ["exact h.compl.closure", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : HasUpperLowerClosure \u03b1\ns : Set \u03b1\nh : IsUpperSet s\n\u22a2 IsLowerSet (closure s\u1d9c)", "state_after": "no goals"}]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/CategoryTheory/Filtered/Basic.lean
|
CategoryTheory.IsFiltered.sup_objs_exists
|
[
237,
1
] |
[
246,
79
] |
[{"tactic": "induction' O using Finset.induction with X O' nm h", "annotated_tactic": ["induction' O using <a>Finset.induction</a> with X O' nm h", [{"full_name": "Finset.induction", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1240, 19], "def_end_pos": [1240, 28]}]], "state_before": "C : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : IsFiltered C\nO : Finset C\n\u22a2 \u2203 S, \u2200 {X : C}, X \u2208 O \u2192 _root_.Nonempty (X \u27f6 S)", "state_after": "case empty\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : IsFiltered C\n\u22a2 \u2203 S, \u2200 {X : C}, X \u2208 \u2205 \u2192 _root_.Nonempty (X \u27f6 S)\n\ncase insert\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : IsFiltered C\nX : C\nO' : Finset C\nnm : \u00acX \u2208 O'\nh : \u2203 S, \u2200 {X : C}, X \u2208 O' \u2192 _root_.Nonempty (X \u27f6 S)\n\u22a2 \u2203 S, \u2200 {X_1 : C}, X_1 \u2208 insert X O' \u2192 _root_.Nonempty (X_1 \u27f6 S)"}, {"tactic": "exact \u27e8Classical.choice IsFiltered.Nonempty, by intro; simp\u27e9", "annotated_tactic": ["exact \u27e8<a>Classical.choice</a> <a>IsFiltered.Nonempty</a>, by intro; simp\u27e9", [{"full_name": "Classical.choice", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [712, 7], "def_end_pos": [712, 23]}, {"full_name": "CategoryTheory.IsFiltered.Nonempty", "def_path": "Mathlib/CategoryTheory/Filtered/Basic.lean", "def_pos": [86, 4], "def_end_pos": [86, 12]}]], "state_before": "case empty\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : IsFiltered C\n\u22a2 \u2203 S, \u2200 {X : C}, X \u2208 \u2205 \u2192 _root_.Nonempty (X \u27f6 S)", "state_after": "no goals"}, {"tactic": "intro", "annotated_tactic": ["intro", []], "state_before": "C : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : IsFiltered C\n\u22a2 \u2200 {X : C}, X \u2208 \u2205 \u2192 _root_.Nonempty (X \u27f6 Classical.choice (_ : _root_.Nonempty C))", "state_after": "C : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : IsFiltered C\nX\u271d : C\n\u22a2 X\u271d \u2208 \u2205 \u2192 _root_.Nonempty (X\u271d \u27f6 Classical.choice (_ : _root_.Nonempty C))"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "C : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : IsFiltered C\nX\u271d : C\n\u22a2 X\u271d \u2208 \u2205 \u2192 _root_.Nonempty (X\u271d \u27f6 Classical.choice (_ : _root_.Nonempty C))", "state_after": "no goals"}, {"tactic": "obtain \u27e8S', w'\u27e9 := h", "annotated_tactic": ["obtain \u27e8S', w'\u27e9 := h", []], "state_before": "case insert\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : IsFiltered C\nX : C\nO' : Finset C\nnm : \u00acX \u2208 O'\nh : \u2203 S, \u2200 {X : C}, X \u2208 O' \u2192 _root_.Nonempty (X \u27f6 S)\n\u22a2 \u2203 S, \u2200 {X_1 : C}, X_1 \u2208 insert X O' \u2192 _root_.Nonempty (X_1 \u27f6 S)", "state_after": "case insert.intro\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : IsFiltered C\nX : C\nO' : Finset C\nnm : \u00acX \u2208 O'\nS' : C\nw' : \u2200 {X : C}, X \u2208 O' \u2192 _root_.Nonempty (X \u27f6 S')\n\u22a2 \u2203 S, \u2200 {X_1 : C}, X_1 \u2208 insert X O' \u2192 _root_.Nonempty (X_1 \u27f6 S)"}, {"tactic": "use max X S'", "annotated_tactic": ["use <a>max</a> X S'", [{"full_name": "CategoryTheory.IsFiltered.max", "def_path": "Mathlib/CategoryTheory/Filtered/Basic.lean", "def_pos": [149, 19], "def_end_pos": [149, 22]}]], "state_before": "case insert.intro\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : IsFiltered C\nX : C\nO' : Finset C\nnm : \u00acX \u2208 O'\nS' : C\nw' : \u2200 {X : C}, X \u2208 O' \u2192 _root_.Nonempty (X \u27f6 S')\n\u22a2 \u2203 S, \u2200 {X_1 : C}, X_1 \u2208 insert X O' \u2192 _root_.Nonempty (X_1 \u27f6 S)", "state_after": "case h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : IsFiltered C\nX : C\nO' : Finset C\nnm : \u00acX \u2208 O'\nS' : C\nw' : \u2200 {X : C}, X \u2208 O' \u2192 _root_.Nonempty (X \u27f6 S')\n\u22a2 \u2200 {X_1 : C}, X_1 \u2208 insert X O' \u2192 _root_.Nonempty (X_1 \u27f6 max X S')"}, {"tactic": "rintro Y mY", "annotated_tactic": ["rintro Y mY", []], "state_before": "case h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : IsFiltered C\nX : C\nO' : Finset C\nnm : \u00acX \u2208 O'\nS' : C\nw' : \u2200 {X : C}, X \u2208 O' \u2192 _root_.Nonempty (X \u27f6 S')\n\u22a2 \u2200 {X_1 : C}, X_1 \u2208 insert X O' \u2192 _root_.Nonempty (X_1 \u27f6 max X S')", "state_after": "case h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : IsFiltered C\nX : C\nO' : Finset C\nnm : \u00acX \u2208 O'\nS' : C\nw' : \u2200 {X : C}, X \u2208 O' \u2192 _root_.Nonempty (X \u27f6 S')\nY : C\nmY : Y \u2208 insert X O'\n\u22a2 _root_.Nonempty (Y \u27f6 max X S')"}, {"tactic": "obtain rfl | h := eq_or_ne Y X", "annotated_tactic": ["obtain rfl | h := <a>eq_or_ne</a> Y X", [{"full_name": "eq_or_ne", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [209, 9], "def_end_pos": [209, 17]}]], "state_before": "case h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : IsFiltered C\nX : C\nO' : Finset C\nnm : \u00acX \u2208 O'\nS' : C\nw' : \u2200 {X : C}, X \u2208 O' \u2192 _root_.Nonempty (X \u27f6 S')\nY : C\nmY : Y \u2208 insert X O'\n\u22a2 _root_.Nonempty (Y \u27f6 max X S')", "state_after": "case h.inl\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : IsFiltered C\nO' : Finset C\nS' : C\nw' : \u2200 {X : C}, X \u2208 O' \u2192 _root_.Nonempty (X \u27f6 S')\nY : C\nnm : \u00acY \u2208 O'\nmY : Y \u2208 insert Y O'\n\u22a2 _root_.Nonempty (Y \u27f6 max Y S')\n\ncase h.inr\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : IsFiltered C\nX : C\nO' : Finset C\nnm : \u00acX \u2208 O'\nS' : C\nw' : \u2200 {X : C}, X \u2208 O' \u2192 _root_.Nonempty (X \u27f6 S')\nY : C\nmY : Y \u2208 insert X O'\nh : Y \u2260 X\n\u22a2 _root_.Nonempty (Y \u27f6 max X S')"}, {"tactic": "exact \u27e8leftToMax _ _\u27e9", "annotated_tactic": ["exact \u27e8<a>leftToMax</a> _ _\u27e9", [{"full_name": "CategoryTheory.IsFiltered.leftToMax", "def_path": "Mathlib/CategoryTheory/Filtered/Basic.lean", "def_pos": [156, 19], "def_end_pos": [156, 28]}]], "state_before": "case h.inl\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : IsFiltered C\nO' : Finset C\nS' : C\nw' : \u2200 {X : C}, X \u2208 O' \u2192 _root_.Nonempty (X \u27f6 S')\nY : C\nnm : \u00acY \u2208 O'\nmY : Y \u2208 insert Y O'\n\u22a2 _root_.Nonempty (Y \u27f6 max Y S')", "state_after": "no goals"}, {"tactic": "exact \u27e8(w' (Finset.mem_of_mem_insert_of_ne mY h)).some \u226b rightToMax _ _\u27e9", "annotated_tactic": ["exact \u27e8(w' (<a>Finset.mem_of_mem_insert_of_ne</a> mY h)).<a>some</a> \u226b <a>rightToMax</a> _ _\u27e9", [{"full_name": "Finset.mem_of_mem_insert_of_ne", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1099, 9], "def_end_pos": [1099, 32]}, {"full_name": "Nonempty.some", "def_path": "Mathlib/Logic/Nonempty.lean", "def_pos": [128, 29], "def_end_pos": [128, 42]}, {"full_name": "CategoryTheory.IsFiltered.rightToMax", "def_path": "Mathlib/CategoryTheory/Filtered/Basic.lean", "def_pos": [163, 19], "def_end_pos": [163, 29]}]], "state_before": "case h.inr\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : IsFiltered C\nX : C\nO' : Finset C\nnm : \u00acX \u2208 O'\nS' : C\nw' : \u2200 {X : C}, X \u2208 O' \u2192 _root_.Nonempty (X \u27f6 S')\nY : C\nmY : Y \u2208 insert X O'\nh : Y \u2260 X\n\u22a2 _root_.Nonempty (Y \u27f6 max X S')", "state_after": "no goals"}]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Topology/ContinuousFunction/Bounded.lean
|
BoundedContinuousFunction.restrict_apply
|
[
404,
1
] |
[
404,
86
] |
[]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/GroupTheory/Abelianization.lean
|
commutator_eq_normalClosure
|
[
53,
1
] |
[
54,
61
] |
[{"tactic": "simp [commutator, Subgroup.commutator_def', commutatorSet]", "annotated_tactic": ["simp [<a>commutator</a>, <a>Subgroup.commutator_def'</a>, <a>commutatorSet</a>]", [{"full_name": "commutator", "def_path": "Mathlib/GroupTheory/Abelianization.lean", "def_pos": [39, 5], "def_end_pos": [39, 15]}, {"full_name": "Subgroup.commutator_def'", "def_path": "Mathlib/GroupTheory/Commutator.lean", "def_pos": [142, 9], "def_end_pos": [142, 24]}, {"full_name": "commutatorSet", "def_path": "Mathlib/GroupTheory/Commutator.lean", "def_pos": [246, 5], "def_end_pos": [246, 18]}]], "state_before": "G : Type u\ninst\u271d : Group G\n\u22a2 commutator G = Subgroup.normalClosure (commutatorSet G)", "state_after": "no goals"}]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Analysis/Convex/Side.lean
|
AffineSubspace.wOppSide_lineMap_left
|
[
361,
1
] |
[
363,
40
] |
[]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Data/Real/Hyperreal.lean
|
Hyperreal.not_infiniteNeg_add_infinitePos
|
[
543,
1
] |
[
545,
55
] |
[]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Algebra/Homology/ImageToKernel.lean
|
homology.map_comp
|
[
337,
1
] |
[
341,
92
] |
[{"tactic": "ext", "annotated_tactic": ["ext", []], "state_before": "\u03b9 : Type u_1\nV : Type u\ninst\u271d\u00b9\u2079 : Category.{v, u} V\ninst\u271d\u00b9\u2078 : HasZeroMorphisms V\nA B C : V\nf : A \u27f6 B\ninst\u271d\u00b9\u2077 : HasImage f\ng : B \u27f6 C\ninst\u271d\u00b9\u2076 : HasKernel g\nw : f \u226b g = 0\nA' B' C' : V\nf' : A' \u27f6 B'\ninst\u271d\u00b9\u2075 : HasImage f'\ng' : B' \u27f6 C'\ninst\u271d\u00b9\u2074 : HasKernel g'\nw' : f' \u226b g' = 0\n\u03b1 : Arrow.mk f \u27f6 Arrow.mk f'\ninst\u271d\u00b9\u00b3 : HasImageMap \u03b1\n\u03b2 : Arrow.mk g \u27f6 Arrow.mk g'\nA\u2081 B\u2081 C\u2081 : V\nf\u2081 : A\u2081 \u27f6 B\u2081\ninst\u271d\u00b9\u00b2 : HasImage f\u2081\ng\u2081 : B\u2081 \u27f6 C\u2081\ninst\u271d\u00b9\u00b9 : HasKernel g\u2081\nw\u2081 : f\u2081 \u226b g\u2081 = 0\nA\u2082 B\u2082 C\u2082 : V\nf\u2082 : A\u2082 \u27f6 B\u2082\ninst\u271d\u00b9\u2070 : HasImage f\u2082\ng\u2082 : B\u2082 \u27f6 C\u2082\ninst\u271d\u2079 : HasKernel g\u2082\nw\u2082 : f\u2082 \u226b g\u2082 = 0\nA\u2083 B\u2083 C\u2083 : V\nf\u2083 : A\u2083 \u27f6 B\u2083\ninst\u271d\u2078 : HasImage f\u2083\ng\u2083 : B\u2083 \u27f6 C\u2083\ninst\u271d\u2077 : HasKernel g\u2083\nw\u2083 : f\u2083 \u226b g\u2083 = 0\n\u03b1\u2081 : Arrow.mk f\u2081 \u27f6 Arrow.mk f\u2082\ninst\u271d\u2076 : HasImageMap \u03b1\u2081\n\u03b2\u2081 : Arrow.mk g\u2081 \u27f6 Arrow.mk g\u2082\n\u03b1\u2082 : Arrow.mk f\u2082 \u27f6 Arrow.mk f\u2083\ninst\u271d\u2075 : HasImageMap \u03b1\u2082\n\u03b2\u2082 : Arrow.mk g\u2082 \u27f6 Arrow.mk g\u2083\ninst\u271d\u2074 : HasCokernel (imageToKernel f g w)\ninst\u271d\u00b3 : HasCokernel (imageToKernel f' g' w')\ninst\u271d\u00b2 : HasCokernel (imageToKernel f\u2081 g\u2081 w\u2081)\ninst\u271d\u00b9 : HasCokernel (imageToKernel f\u2082 g\u2082 w\u2082)\ninst\u271d : HasCokernel (imageToKernel f\u2083 g\u2083 w\u2083)\np\u2081 : \u03b1\u2081.right = \u03b2\u2081.left\np\u2082 : \u03b1\u2082.right = \u03b2\u2082.left\n\u22a2 map w\u2081 w\u2082 \u03b1\u2081 \u03b2\u2081 p\u2081 \u226b map w\u2082 w\u2083 \u03b1\u2082 \u03b2\u2082 p\u2082 = map w\u2081 w\u2083 (\u03b1\u2081 \u226b \u03b1\u2082) (\u03b2\u2081 \u226b \u03b2\u2082) (_ : (\u03b1\u2081 \u226b \u03b1\u2082).right = (\u03b2\u2081 \u226b \u03b2\u2082).left)", "state_after": "case p\n\u03b9 : Type u_1\nV : Type u\ninst\u271d\u00b9\u2079 : Category.{v, u} V\ninst\u271d\u00b9\u2078 : HasZeroMorphisms V\nA B C : V\nf : A \u27f6 B\ninst\u271d\u00b9\u2077 : HasImage f\ng : B \u27f6 C\ninst\u271d\u00b9\u2076 : HasKernel g\nw : f \u226b g = 0\nA' B' C' : V\nf' : A' \u27f6 B'\ninst\u271d\u00b9\u2075 : HasImage f'\ng' : B' \u27f6 C'\ninst\u271d\u00b9\u2074 : HasKernel g'\nw' : f' \u226b g' = 0\n\u03b1 : Arrow.mk f \u27f6 Arrow.mk f'\ninst\u271d\u00b9\u00b3 : HasImageMap \u03b1\n\u03b2 : Arrow.mk g \u27f6 Arrow.mk g'\nA\u2081 B\u2081 C\u2081 : V\nf\u2081 : A\u2081 \u27f6 B\u2081\ninst\u271d\u00b9\u00b2 : HasImage f\u2081\ng\u2081 : B\u2081 \u27f6 C\u2081\ninst\u271d\u00b9\u00b9 : HasKernel g\u2081\nw\u2081 : f\u2081 \u226b g\u2081 = 0\nA\u2082 B\u2082 C\u2082 : V\nf\u2082 : A\u2082 \u27f6 B\u2082\ninst\u271d\u00b9\u2070 : HasImage f\u2082\ng\u2082 : B\u2082 \u27f6 C\u2082\ninst\u271d\u2079 : HasKernel g\u2082\nw\u2082 : f\u2082 \u226b g\u2082 = 0\nA\u2083 B\u2083 C\u2083 : V\nf\u2083 : A\u2083 \u27f6 B\u2083\ninst\u271d\u2078 : HasImage f\u2083\ng\u2083 : B\u2083 \u27f6 C\u2083\ninst\u271d\u2077 : HasKernel g\u2083\nw\u2083 : f\u2083 \u226b g\u2083 = 0\n\u03b1\u2081 : Arrow.mk f\u2081 \u27f6 Arrow.mk f\u2082\ninst\u271d\u2076 : HasImageMap \u03b1\u2081\n\u03b2\u2081 : Arrow.mk g\u2081 \u27f6 Arrow.mk g\u2082\n\u03b1\u2082 : Arrow.mk f\u2082 \u27f6 Arrow.mk f\u2083\ninst\u271d\u2075 : HasImageMap \u03b1\u2082\n\u03b2\u2082 : Arrow.mk g\u2082 \u27f6 Arrow.mk g\u2083\ninst\u271d\u2074 : HasCokernel (imageToKernel f g w)\ninst\u271d\u00b3 : HasCokernel (imageToKernel f' g' w')\ninst\u271d\u00b2 : HasCokernel (imageToKernel f\u2081 g\u2081 w\u2081)\ninst\u271d\u00b9 : HasCokernel (imageToKernel f\u2082 g\u2082 w\u2082)\ninst\u271d : HasCokernel (imageToKernel f\u2083 g\u2083 w\u2083)\np\u2081 : \u03b1\u2081.right = \u03b2\u2081.left\np\u2082 : \u03b1\u2082.right = \u03b2\u2082.left\n\u22a2 \u03c0 f\u2081 g\u2081 w\u2081 \u226b map w\u2081 w\u2082 \u03b1\u2081 \u03b2\u2081 p\u2081 \u226b map w\u2082 w\u2083 \u03b1\u2082 \u03b2\u2082 p\u2082 =\n \u03c0 f\u2081 g\u2081 w\u2081 \u226b map w\u2081 w\u2083 (\u03b1\u2081 \u226b \u03b1\u2082) (\u03b2\u2081 \u226b \u03b2\u2082) (_ : (\u03b1\u2081 \u226b \u03b1\u2082).right = (\u03b2\u2081 \u226b \u03b2\u2082).left)"}, {"tactic": "simp only [kernelSubobjectMap_comp, homology.\u03c0_map_assoc, homology.\u03c0_map, Category.assoc]", "annotated_tactic": ["simp only [<a>kernelSubobjectMap_comp</a>, <a>homology.\u03c0_map_assoc</a>, <a>homology.\u03c0_map</a>, <a>Category.assoc</a>]", [{"full_name": "CategoryTheory.Limits.kernelSubobjectMap_comp", "def_path": "Mathlib/CategoryTheory/Subobject/Limits.lean", "def_pos": [168, 9], "def_end_pos": [168, 32]}, {"full_name": "homology.\u03c0_map_assoc", "def_path": "Mathlib/Algebra/Homology/ImageToKernel.lean", "def_pos": [292, 3], "def_end_pos": [292, 25]}, {"full_name": "homology.\u03c0_map", "def_path": "Mathlib/Algebra/Homology/ImageToKernel.lean", "def_pos": [293, 9], "def_end_pos": [293, 23]}, {"full_name": "CategoryTheory.Category.assoc", "def_path": "Mathlib/CategoryTheory/Category/Basic.lean", "def_pos": [159, 3], "def_end_pos": [159, 8]}]], "state_before": "case p\n\u03b9 : Type u_1\nV : Type u\ninst\u271d\u00b9\u2079 : Category.{v, u} V\ninst\u271d\u00b9\u2078 : HasZeroMorphisms V\nA B C : V\nf : A \u27f6 B\ninst\u271d\u00b9\u2077 : HasImage f\ng : B \u27f6 C\ninst\u271d\u00b9\u2076 : HasKernel g\nw : f \u226b g = 0\nA' B' C' : V\nf' : A' \u27f6 B'\ninst\u271d\u00b9\u2075 : HasImage f'\ng' : B' \u27f6 C'\ninst\u271d\u00b9\u2074 : HasKernel g'\nw' : f' \u226b g' = 0\n\u03b1 : Arrow.mk f \u27f6 Arrow.mk f'\ninst\u271d\u00b9\u00b3 : HasImageMap \u03b1\n\u03b2 : Arrow.mk g \u27f6 Arrow.mk g'\nA\u2081 B\u2081 C\u2081 : V\nf\u2081 : A\u2081 \u27f6 B\u2081\ninst\u271d\u00b9\u00b2 : HasImage f\u2081\ng\u2081 : B\u2081 \u27f6 C\u2081\ninst\u271d\u00b9\u00b9 : HasKernel g\u2081\nw\u2081 : f\u2081 \u226b g\u2081 = 0\nA\u2082 B\u2082 C\u2082 : V\nf\u2082 : A\u2082 \u27f6 B\u2082\ninst\u271d\u00b9\u2070 : HasImage f\u2082\ng\u2082 : B\u2082 \u27f6 C\u2082\ninst\u271d\u2079 : HasKernel g\u2082\nw\u2082 : f\u2082 \u226b g\u2082 = 0\nA\u2083 B\u2083 C\u2083 : V\nf\u2083 : A\u2083 \u27f6 B\u2083\ninst\u271d\u2078 : HasImage f\u2083\ng\u2083 : B\u2083 \u27f6 C\u2083\ninst\u271d\u2077 : HasKernel g\u2083\nw\u2083 : f\u2083 \u226b g\u2083 = 0\n\u03b1\u2081 : Arrow.mk f\u2081 \u27f6 Arrow.mk f\u2082\ninst\u271d\u2076 : HasImageMap \u03b1\u2081\n\u03b2\u2081 : Arrow.mk g\u2081 \u27f6 Arrow.mk g\u2082\n\u03b1\u2082 : Arrow.mk f\u2082 \u27f6 Arrow.mk f\u2083\ninst\u271d\u2075 : HasImageMap \u03b1\u2082\n\u03b2\u2082 : Arrow.mk g\u2082 \u27f6 Arrow.mk g\u2083\ninst\u271d\u2074 : HasCokernel (imageToKernel f g w)\ninst\u271d\u00b3 : HasCokernel (imageToKernel f' g' w')\ninst\u271d\u00b2 : HasCokernel (imageToKernel f\u2081 g\u2081 w\u2081)\ninst\u271d\u00b9 : HasCokernel (imageToKernel f\u2082 g\u2082 w\u2082)\ninst\u271d : HasCokernel (imageToKernel f\u2083 g\u2083 w\u2083)\np\u2081 : \u03b1\u2081.right = \u03b2\u2081.left\np\u2082 : \u03b1\u2082.right = \u03b2\u2082.left\n\u22a2 \u03c0 f\u2081 g\u2081 w\u2081 \u226b map w\u2081 w\u2082 \u03b1\u2081 \u03b2\u2081 p\u2081 \u226b map w\u2082 w\u2083 \u03b1\u2082 \u03b2\u2082 p\u2082 =\n \u03c0 f\u2081 g\u2081 w\u2081 \u226b map w\u2081 w\u2083 (\u03b1\u2081 \u226b \u03b1\u2082) (\u03b2\u2081 \u226b \u03b2\u2082) (_ : (\u03b1\u2081 \u226b \u03b1\u2082).right = (\u03b2\u2081 \u226b \u03b2\u2082).left)", "state_after": "no goals"}]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Topology/Instances/ENNReal.lean
|
Summable.toNNReal
|
[
1305,
1
] |
[
1308,
84
] |
[{"tactic": "apply NNReal.summable_coe.1", "annotated_tactic": ["apply <a>NNReal.summable_coe</a>.1", [{"full_name": "NNReal.summable_coe", "def_path": "Mathlib/Topology/Instances/NNReal.lean", "def_pos": [170, 9], "def_end_pos": [170, 21]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u211d\nhf : Summable f\n\u22a2 Summable fun n => Real.toNNReal (f n)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u211d\nhf : Summable f\n\u22a2 Summable fun a => \u2191(Real.toNNReal (f a))"}, {"tactic": "refine' summable_of_nonneg_of_le (fun n => NNReal.coe_nonneg _) (fun n => _) hf.abs", "annotated_tactic": ["refine' <a>summable_of_nonneg_of_le</a> (fun n => <a>NNReal.coe_nonneg</a> _) (fun n => _) hf.abs", [{"full_name": "summable_of_nonneg_of_le", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [1297, 9], "def_end_pos": [1297, 33]}, {"full_name": "NNReal.coe_nonneg", "def_path": "Mathlib/Data/Real/NNReal.lean", "def_pos": [134, 9], "def_end_pos": [134, 19]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u211d\nhf : Summable f\n\u22a2 Summable fun a => \u2191(Real.toNNReal (f a))", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u211d\nhf : Summable f\nn : \u03b1\n\u22a2 \u2191(Real.toNNReal (f n)) \u2264 |f n|"}, {"tactic": "simp only [le_abs_self, Real.coe_toNNReal', max_le_iff, abs_nonneg, and_self_iff]", "annotated_tactic": ["simp only [<a>le_abs_self</a>, <a>Real.coe_toNNReal'</a>, <a>max_le_iff</a>, <a>abs_nonneg</a>, <a>and_self_iff</a>]", [{"full_name": "le_abs_self", "def_path": "Mathlib/Algebra/Order/Group/Abs.lean", "def_pos": [57, 9], "def_end_pos": [57, 20]}, {"full_name": "Real.coe_toNNReal'", "def_path": "Mathlib/Data/Real/NNReal.lean", "def_pos": [615, 9], "def_end_pos": [615, 22]}, {"full_name": "max_le_iff", "def_path": "Mathlib/Order/MinMax.lean", "def_pos": [48, 9], "def_end_pos": [48, 19]}, {"full_name": "abs_nonneg", "def_path": "Mathlib/Algebra/Order/Group/Abs.lean", "def_pos": [169, 9], "def_end_pos": [169, 19]}, {"full_name": "and_self_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [155, 9], "def_end_pos": [155, 21]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u211d\nhf : Summable f\nn : \u03b1\n\u22a2 \u2191(Real.toNNReal (f n)) \u2264 |f n|", "state_after": "no goals"}]
|
https://github.com/leanprover/std4
|
869c615eb10130c0637a7bc038e2b80253559913
|
lake-packages/std/Std/Data/Nat/Lemmas.lean
|
Nat.mul_self_sub_mul_self_eq
|
[
540,
11
] |
[
542,
72
] |
[{"tactic": "rw [Nat.mul_sub_left_distrib, Nat.right_distrib, Nat.right_distrib,\n Nat.mul_comm b a, Nat.add_comm (a*a) (a*b), Nat.add_sub_add_left]", "annotated_tactic": ["rw [<a>Nat.mul_sub_left_distrib</a>, <a>Nat.right_distrib</a>, <a>Nat.right_distrib</a>,\n <a>Nat.mul_comm</a> b a, <a>Nat.add_comm</a> (a*a) (a*b), <a>Nat.add_sub_add_left</a>]", [{"full_name": "Nat.mul_sub_left_distrib", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [716, 19], "def_end_pos": [716, 39]}, {"full_name": "Nat.right_distrib", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [189, 19], "def_end_pos": [189, 32]}, {"full_name": "Nat.right_distrib", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [189, 19], "def_end_pos": [189, 32]}, {"full_name": "Nat.mul_comm", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [174, 19], "def_end_pos": [174, 27]}, {"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_sub_add_left", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [591, 19], "def_end_pos": [591, 35]}]], "state_before": "a b : Nat\n\u22a2 a * a - b * b = (a + b) * (a - b)", "state_after": "no goals"}]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Algebra/Lie/Submodule.lean
|
LieIdeal.mem_comap
|
[
965,
1
] |
[
966,
10
] |
[]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Order/Filter/Extr.lean
|
IsMaxOn.sup
|
[
544,
1
] |
[
545,
24
] |
[]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/MeasureTheory/MeasurableSpace/Basic.lean
|
measurable_iff_comap_le
|
[
221,
1
] |
[
223,
27
] |
[]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean
|
MeasureTheory.StronglyMeasurable.finStronglyMeasurable
|
[
333,
11
] |
[
336,
37
] |
[{"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b3 : Countable \u03b9\nf g : \u03b1 \u2192 \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : Zero \u03b2\nm0 : MeasurableSpace \u03b1\nhf : StronglyMeasurable f\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\n\u22a2 \u2200 (x : \u03b1), x \u2208 univ\u1d9c \u2192 f x = 0", "state_after": "no goals"}, {"tactic": "rwa [Measure.restrict_univ]", "annotated_tactic": ["rwa [<a>Measure.restrict_univ</a>]", [{"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\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b3 : Countable \u03b9\nf g : \u03b1 \u2192 \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : Zero \u03b2\nm0 : MeasurableSpace \u03b1\nhf : StronglyMeasurable f\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\n\u22a2 SigmaFinite (Measure.restrict \u03bc univ)", "state_after": "no goals"}]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Analysis/Convex/Segment.lean
|
Convex.mem_Ico
|
[
597,
1
] |
[
610,
81
] |
[{"tactic": "refine' \u27e8fun hz => _, _\u27e9", "annotated_tactic": ["refine' \u27e8fun hz => _, _\u27e9", []], "state_before": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d : LinearOrderedField \ud835\udd5c\nx y z : \ud835\udd5c\nh : x < y\n\u22a2 z \u2208 Ico x y \u2194 \u2203 a b, 0 < a \u2227 0 \u2264 b \u2227 a + b = 1 \u2227 a * x + b * y = z", "state_after": "case refine'_1\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d : LinearOrderedField \ud835\udd5c\nx y z : \ud835\udd5c\nh : x < y\nhz : z \u2208 Ico x y\n\u22a2 \u2203 a b, 0 < a \u2227 0 \u2264 b \u2227 a + b = 1 \u2227 a * x + b * y = z\n\ncase refine'_2\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d : LinearOrderedField \ud835\udd5c\nx y z : \ud835\udd5c\nh : x < y\n\u22a2 (\u2203 a b, 0 < a \u2227 0 \u2264 b \u2227 a + b = 1 \u2227 a * x + b * y = z) \u2192 z \u2208 Ico x y"}, {"tactic": "obtain \u27e8a, b, ha, hb, hab, rfl\u27e9 := (Convex.mem_Icc h.le).1 (Ico_subset_Icc_self hz)", "annotated_tactic": ["obtain \u27e8a, b, ha, hb, hab, rfl\u27e9 := (<a>Convex.mem_Icc</a> h.le).1 (<a>Ico_subset_Icc_self</a> hz)", [{"full_name": "Convex.mem_Icc", "def_path": "Mathlib/Analysis/Convex/Segment.lean", "def_pos": [561, 9], "def_end_pos": [561, 23]}, {"full_name": "Set.Ico_subset_Icc_self", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [510, 9], "def_end_pos": [510, 28]}]], "state_before": "case refine'_1\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d : LinearOrderedField \ud835\udd5c\nx y z : \ud835\udd5c\nh : x < y\nhz : z \u2208 Ico x y\n\u22a2 \u2203 a b, 0 < a \u2227 0 \u2264 b \u2227 a + b = 1 \u2227 a * x + b * y = z", "state_after": "case refine'_1.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d : LinearOrderedField \ud835\udd5c\nx y : \ud835\udd5c\nh : x < y\na b : \ud835\udd5c\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\nhz : a * x + b * y \u2208 Ico x y\n\u22a2 \u2203 a_1 b_1, 0 < a_1 \u2227 0 \u2264 b_1 \u2227 a_1 + b_1 = 1 \u2227 a_1 * x + b_1 * y = a * x + b * y"}, {"tactic": "obtain rfl | ha' := ha.eq_or_lt", "annotated_tactic": ["obtain rfl | ha' := ha.eq_or_lt", []], "state_before": "case refine'_1.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d : LinearOrderedField \ud835\udd5c\nx y : \ud835\udd5c\nh : x < y\na b : \ud835\udd5c\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\nhz : a * x + b * y \u2208 Ico x y\n\u22a2 \u2203 a_1 b_1, 0 < a_1 \u2227 0 \u2264 b_1 \u2227 a_1 + b_1 = 1 \u2227 a_1 * x + b_1 * y = a * x + b * y", "state_after": "case refine'_1.intro.intro.intro.intro.intro.inl\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d : LinearOrderedField \ud835\udd5c\nx y : \ud835\udd5c\nh : x < y\nb : \ud835\udd5c\nhb : 0 \u2264 b\nha : 0 \u2264 0\nhab : 0 + b = 1\nhz : 0 * x + b * y \u2208 Ico x y\n\u22a2 \u2203 a b_1, 0 < a \u2227 0 \u2264 b_1 \u2227 a + b_1 = 1 \u2227 a * x + b_1 * y = 0 * x + b * y\n\ncase refine'_1.intro.intro.intro.intro.intro.inr\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d : LinearOrderedField \ud835\udd5c\nx y : \ud835\udd5c\nh : x < y\na b : \ud835\udd5c\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\nhz : a * x + b * y \u2208 Ico x y\nha' : 0 < a\n\u22a2 \u2203 a_1 b_1, 0 < a_1 \u2227 0 \u2264 b_1 \u2227 a_1 + b_1 = 1 \u2227 a_1 * x + b_1 * y = a * x + b * y"}, {"tactic": "rw [zero_add] at hab", "annotated_tactic": ["rw [<a>zero_add</a>] at hab", [{"full_name": "zero_add", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [463, 3], "def_end_pos": [463, 14]}]], "state_before": "case refine'_1.intro.intro.intro.intro.intro.inl\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d : LinearOrderedField \ud835\udd5c\nx y : \ud835\udd5c\nh : x < y\nb : \ud835\udd5c\nhb : 0 \u2264 b\nha : 0 \u2264 0\nhab : 0 + b = 1\nhz : 0 * x + b * y \u2208 Ico x y\n\u22a2 \u2203 a b_1, 0 < a \u2227 0 \u2264 b_1 \u2227 a + b_1 = 1 \u2227 a * x + b_1 * y = 0 * x + b * y", "state_after": "case refine'_1.intro.intro.intro.intro.intro.inl\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d : LinearOrderedField \ud835\udd5c\nx y : \ud835\udd5c\nh : x < y\nb : \ud835\udd5c\nhb : 0 \u2264 b\nha : 0 \u2264 0\nhab : b = 1\nhz : 0 * x + b * y \u2208 Ico x y\n\u22a2 \u2203 a b_1, 0 < a \u2227 0 \u2264 b_1 \u2227 a + b_1 = 1 \u2227 a * x + b_1 * y = 0 * x + b * y"}, {"tactic": "rw [hab, one_mul, zero_mul, zero_add] at hz", "annotated_tactic": ["rw [hab, <a>one_mul</a>, <a>zero_mul</a>, <a>zero_add</a>] at hz", [{"full_name": "one_mul", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [464, 9], "def_end_pos": [464, 16]}, {"full_name": "MulZeroClass.zero_mul", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [36, 3], "def_end_pos": [36, 11]}, {"full_name": "zero_add", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [463, 3], "def_end_pos": [463, 14]}]], "state_before": "case refine'_1.intro.intro.intro.intro.intro.inl\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d : LinearOrderedField \ud835\udd5c\nx y : \ud835\udd5c\nh : x < y\nb : \ud835\udd5c\nhb : 0 \u2264 b\nha : 0 \u2264 0\nhab : b = 1\nhz : 0 * x + b * y \u2208 Ico x y\n\u22a2 \u2203 a b_1, 0 < a \u2227 0 \u2264 b_1 \u2227 a + b_1 = 1 \u2227 a * x + b_1 * y = 0 * x + b * y", "state_after": "case refine'_1.intro.intro.intro.intro.intro.inl\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d : LinearOrderedField \ud835\udd5c\nx y : \ud835\udd5c\nh : x < y\nb : \ud835\udd5c\nhb : 0 \u2264 b\nha : 0 \u2264 0\nhab : b = 1\nhz : y \u2208 Ico x y\n\u22a2 \u2203 a b_1, 0 < a \u2227 0 \u2264 b_1 \u2227 a + b_1 = 1 \u2227 a * x + b_1 * y = 0 * x + b * y"}, {"tactic": "exact (hz.2.ne rfl).elim", "annotated_tactic": ["exact (hz.2.<a>ne</a> <a>rfl</a>).<a>elim</a>", [{"full_name": "LT.lt.ne", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [152, 7], "def_end_pos": [152, 15]}, {"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}, {"full_name": "False.elim", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [223, 21], "def_end_pos": [223, 31]}]], "state_before": "case refine'_1.intro.intro.intro.intro.intro.inl\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d : LinearOrderedField \ud835\udd5c\nx y : \ud835\udd5c\nh : x < y\nb : \ud835\udd5c\nhb : 0 \u2264 b\nha : 0 \u2264 0\nhab : b = 1\nhz : y \u2208 Ico x y\n\u22a2 \u2203 a b_1, 0 < a \u2227 0 \u2264 b_1 \u2227 a + b_1 = 1 \u2227 a * x + b_1 * y = 0 * x + b * y", "state_after": "no goals"}, {"tactic": "exact \u27e8a, b, ha', hb, hab, rfl\u27e9", "annotated_tactic": ["exact \u27e8a, b, ha', hb, hab, <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]}]], "state_before": "case refine'_1.intro.intro.intro.intro.intro.inr\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d : LinearOrderedField \ud835\udd5c\nx y : \ud835\udd5c\nh : x < y\na b : \ud835\udd5c\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\nhz : a * x + b * y \u2208 Ico x y\nha' : 0 < a\n\u22a2 \u2203 a_1 b_1, 0 < a_1 \u2227 0 \u2264 b_1 \u2227 a_1 + b_1 = 1 \u2227 a_1 * x + b_1 * y = a * x + b * y", "state_after": "no goals"}, {"tactic": "rintro \u27e8a, b, ha, hb, hab, rfl\u27e9", "annotated_tactic": ["rintro \u27e8a, b, ha, hb, hab, rfl\u27e9", []], "state_before": "case refine'_2\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d : LinearOrderedField \ud835\udd5c\nx y z : \ud835\udd5c\nh : x < y\n\u22a2 (\u2203 a b, 0 < a \u2227 0 \u2264 b \u2227 a + b = 1 \u2227 a * x + b * y = z) \u2192 z \u2208 Ico x y", "state_after": "case refine'_2.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d : LinearOrderedField \ud835\udd5c\nx y : \ud835\udd5c\nh : x < y\na b : \ud835\udd5c\nha : 0 < a\nhb : 0 \u2264 b\nhab : a + b = 1\n\u22a2 a * x + b * y \u2208 Ico x y"}, {"tactic": "obtain rfl | hb' := hb.eq_or_lt", "annotated_tactic": ["obtain rfl | hb' := hb.eq_or_lt", []], "state_before": "case refine'_2.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d : LinearOrderedField \ud835\udd5c\nx y : \ud835\udd5c\nh : x < y\na b : \ud835\udd5c\nha : 0 < a\nhb : 0 \u2264 b\nhab : a + b = 1\n\u22a2 a * x + b * y \u2208 Ico x y", "state_after": "case refine'_2.intro.intro.intro.intro.intro.inl\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d : LinearOrderedField \ud835\udd5c\nx y : \ud835\udd5c\nh : x < y\na : \ud835\udd5c\nha : 0 < a\nhb : 0 \u2264 0\nhab : a + 0 = 1\n\u22a2 a * x + 0 * y \u2208 Ico x y\n\ncase refine'_2.intro.intro.intro.intro.intro.inr\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d : LinearOrderedField \ud835\udd5c\nx y : \ud835\udd5c\nh : x < y\na b : \ud835\udd5c\nha : 0 < a\nhb : 0 \u2264 b\nhab : a + b = 1\nhb' : 0 < b\n\u22a2 a * x + b * y \u2208 Ico x y"}, {"tactic": "rw [add_zero] at hab", "annotated_tactic": ["rw [<a>add_zero</a>] at hab", [{"full_name": "add_zero", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [469, 3], "def_end_pos": [469, 14]}]], "state_before": "case refine'_2.intro.intro.intro.intro.intro.inl\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d : LinearOrderedField \ud835\udd5c\nx y : \ud835\udd5c\nh : x < y\na : \ud835\udd5c\nha : 0 < a\nhb : 0 \u2264 0\nhab : a + 0 = 1\n\u22a2 a * x + 0 * y \u2208 Ico x y", "state_after": "case refine'_2.intro.intro.intro.intro.intro.inl\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d : LinearOrderedField \ud835\udd5c\nx y : \ud835\udd5c\nh : x < y\na : \ud835\udd5c\nha : 0 < a\nhb : 0 \u2264 0\nhab : a = 1\n\u22a2 a * x + 0 * y \u2208 Ico x y"}, {"tactic": "rwa [hab, one_mul, zero_mul, add_zero, left_mem_Ico]", "annotated_tactic": ["rwa [hab, <a>one_mul</a>, <a>zero_mul</a>, <a>add_zero</a>, <a>left_mem_Ico</a>]", [{"full_name": "one_mul", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [464, 9], "def_end_pos": [464, 16]}, {"full_name": "MulZeroClass.zero_mul", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [36, 3], "def_end_pos": [36, 11]}, {"full_name": "add_zero", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [469, 3], "def_end_pos": [469, 14]}, {"full_name": "Set.left_mem_Ico", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [186, 9], "def_end_pos": [186, 21]}]], "state_before": "case refine'_2.intro.intro.intro.intro.intro.inl\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d : LinearOrderedField \ud835\udd5c\nx y : \ud835\udd5c\nh : x < y\na : \ud835\udd5c\nha : 0 < a\nhb : 0 \u2264 0\nhab : a = 1\n\u22a2 a * x + 0 * y \u2208 Ico x y", "state_after": "no goals"}, {"tactic": "exact Ioo_subset_Ico_self ((Convex.mem_Ioo h).2 \u27e8a, b, ha, hb', hab, rfl\u27e9)", "annotated_tactic": ["exact <a>Ioo_subset_Ico_self</a> ((<a>Convex.mem_Ioo</a> h).2 \u27e8a, b, ha, hb', hab, <a>rfl</a>\u27e9)", [{"full_name": "Set.Ioo_subset_Ico_self", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [504, 9], "def_end_pos": [504, 28]}, {"full_name": "Convex.mem_Ioo", "def_path": "Mathlib/Analysis/Convex/Segment.lean", "def_pos": [570, 9], "def_end_pos": [570, 23]}, {"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 refine'_2.intro.intro.intro.intro.intro.inr\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d : LinearOrderedField \ud835\udd5c\nx y : \ud835\udd5c\nh : x < y\na b : \ud835\udd5c\nha : 0 < a\nhb : 0 \u2264 b\nhab : a + b = 1\nhb' : 0 < b\n\u22a2 a * x + b * y \u2208 Ico x y", "state_after": "no goals"}]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/GroupTheory/Perm/Support.lean
|
Equiv.Perm.Disjoint.conj
|
[
121,
1
] |
[
122,
24
] |
[]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Geometry/Manifold/MFDeriv.lean
|
Smooth.mdifferentiable
|
[
693,
1
] |
[
694,
38
] |
[]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Analysis/NormedSpace/ENorm.lean
|
ENorm.eq_zero_iff
|
[
102,
1
] |
[
103,
47
] |
[]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Algebra/Order/Hom/Monoid.lean
|
OrderMonoidHom.id_comp
|
[
451,
1
] |
[
452,
6
] |
[]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/MeasureTheory/Function/AEEqFun.lean
|
MeasureTheory.AEEqFun.le_sup_right
|
[
541,
11
] |
[
545,
21
] |
[{"tactic": "rw [\u2190 coeFn_le]", "annotated_tactic": ["rw [\u2190 <a>coeFn_le</a>]", [{"full_name": "MeasureTheory.AEEqFun.coeFn_le", "def_path": "Mathlib/MeasureTheory/Function/AEEqFun.lean", "def_pos": [513, 9], "def_end_pos": [513, 17]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2075 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : TopologicalSpace \u03b3\ninst\u271d\u00b2 : TopologicalSpace \u03b4\ninst\u271d\u00b9 : SemilatticeSup \u03b2\ninst\u271d : ContinuousSup \u03b2\nf g : \u03b1 \u2192\u2098[\u03bc] \u03b2\n\u22a2 g \u2264 f \u2294 g", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2075 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : TopologicalSpace \u03b3\ninst\u271d\u00b2 : TopologicalSpace \u03b4\ninst\u271d\u00b9 : SemilatticeSup \u03b2\ninst\u271d : ContinuousSup \u03b2\nf g : \u03b1 \u2192\u2098[\u03bc] \u03b2\n\u22a2 \u2191g \u2264\u1d50[\u03bc] \u2191(f \u2294 g)"}, {"tactic": "filter_upwards [coeFn_sup f g] with _ ha", "annotated_tactic": ["filter_upwards [<a>coeFn_sup</a> f g] with _ ha", [{"full_name": "MeasureTheory.AEEqFun.coeFn_sup", "def_path": "Mathlib/MeasureTheory/Function/AEEqFun.lean", "def_pos": [530, 9], "def_end_pos": [530, 18]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2075 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : TopologicalSpace \u03b3\ninst\u271d\u00b2 : TopologicalSpace \u03b4\ninst\u271d\u00b9 : SemilatticeSup \u03b2\ninst\u271d : ContinuousSup \u03b2\nf g : \u03b1 \u2192\u2098[\u03bc] \u03b2\n\u22a2 \u2191g \u2264\u1d50[\u03bc] \u2191(f \u2294 g)", "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2075 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : TopologicalSpace \u03b3\ninst\u271d\u00b2 : TopologicalSpace \u03b4\ninst\u271d\u00b9 : SemilatticeSup \u03b2\ninst\u271d : ContinuousSup \u03b2\nf g : \u03b1 \u2192\u2098[\u03bc] \u03b2\na\u271d : \u03b1\nha : \u2191(f \u2294 g) a\u271d = \u2191f a\u271d \u2294 \u2191g a\u271d\n\u22a2 \u2191g a\u271d \u2264 \u2191(f \u2294 g) a\u271d"}, {"tactic": "rw [ha]", "annotated_tactic": ["rw [ha]", []], "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2075 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : TopologicalSpace \u03b3\ninst\u271d\u00b2 : TopologicalSpace \u03b4\ninst\u271d\u00b9 : SemilatticeSup \u03b2\ninst\u271d : ContinuousSup \u03b2\nf g : \u03b1 \u2192\u2098[\u03bc] \u03b2\na\u271d : \u03b1\nha : \u2191(f \u2294 g) a\u271d = \u2191f a\u271d \u2294 \u2191g a\u271d\n\u22a2 \u2191g a\u271d \u2264 \u2191(f \u2294 g) a\u271d", "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2075 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : TopologicalSpace \u03b3\ninst\u271d\u00b2 : TopologicalSpace \u03b4\ninst\u271d\u00b9 : SemilatticeSup \u03b2\ninst\u271d : ContinuousSup \u03b2\nf g : \u03b1 \u2192\u2098[\u03bc] \u03b2\na\u271d : \u03b1\nha : \u2191(f \u2294 g) a\u271d = \u2191f a\u271d \u2294 \u2191g a\u271d\n\u22a2 \u2191g a\u271d \u2264 \u2191f a\u271d \u2294 \u2191g a\u271d"}, {"tactic": "exact le_sup_right", "annotated_tactic": ["exact <a>le_sup_right</a>", [{"full_name": "le_sup_right", "def_path": "Mathlib/Order/Lattice.lean", "def_pos": [141, 9], "def_end_pos": [141, 21]}]], "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2075 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : TopologicalSpace \u03b3\ninst\u271d\u00b2 : TopologicalSpace \u03b4\ninst\u271d\u00b9 : SemilatticeSup \u03b2\ninst\u271d : ContinuousSup \u03b2\nf g : \u03b1 \u2192\u2098[\u03bc] \u03b2\na\u271d : \u03b1\nha : \u2191(f \u2294 g) a\u271d = \u2191f a\u271d \u2294 \u2191g a\u271d\n\u22a2 \u2191g a\u271d \u2264 \u2191f a\u271d \u2294 \u2191g a\u271d", "state_after": "no goals"}]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Data/Matrix/Block.lean
|
Matrix.blockDiagonal_apply'
|
[
368,
1
] |
[
370,
6
] |
[]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Topology/MetricSpace/Infsep.lean
|
Set.einfsep_zero
|
[
55,
1
] |
[
57,
71
] |
[{"tactic": "simp_rw [einfsep, \u2190 _root_.bot_eq_zero, iInf_eq_bot, iInf_lt_iff]", "annotated_tactic": ["simp_rw [<a>einfsep</a>, \u2190 <a>_root_.bot_eq_zero</a>, <a>iInf_eq_bot</a>, <a>iInf_lt_iff</a>]", [{"full_name": "Set.einfsep", "def_path": "Mathlib/Topology/MetricSpace/Infsep.lean", "def_pos": [42, 19], "def_end_pos": [42, 26]}, {"full_name": "bot_eq_zero", "def_path": "Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean", "def_pos": [223, 3], "def_end_pos": [223, 14]}, {"full_name": "iInf_eq_bot", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [1770, 9], "def_end_pos": [1770, 20]}, {"full_name": "iInf_lt_iff", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [672, 9], "def_end_pos": [672, 20]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : EDist \u03b1\nx y : \u03b1\ns t : Set \u03b1\n\u22a2 einfsep s = 0 \u2194 \u2200 (C : \u211d\u22650\u221e), 0 < C \u2192 \u2203 x x_1 y x_2 _hxy, edist x y < C", "state_after": "no goals"}]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Data/Set/Intervals/Basic.lean
|
Set.Ico_inter_Iio
|
[
1853,
1
] |
[
1854,
62
] |
[{"tactic": "simp (config := { contextual := true }) [iff_def]", "annotated_tactic": ["simp (config := { contextual := <a>true</a> }) [<a>iff_def</a>]", [{"full_name": "Bool.true", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [549, 5], "def_end_pos": [549, 9]}, {"full_name": "iff_def", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [38, 9], "def_end_pos": [38, 16]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\n\u22a2 \u2200 (x : \u03b1), x \u2208 Ico a b \u2229 Iio c \u2194 x \u2208 Ico a (min b c)", "state_after": "no goals"}]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Data/Finset/LocallyFinite.lean
|
Finset.Ioc_insert_left
|
[
585,
1
] |
[
586,
100
] |
[{"tactic": "rw [\u2190 coe_inj, coe_insert, coe_Ioc, coe_Icc, Set.insert_eq, Set.union_comm, Set.Ioc_union_left h]", "annotated_tactic": ["rw [\u2190 <a>coe_inj</a>, <a>coe_insert</a>, <a>coe_Ioc</a>, <a>coe_Icc</a>, <a>Set.insert_eq</a>, <a>Set.union_comm</a>, <a>Set.Ioc_union_left</a> h]", [{"full_name": "Finset.coe_inj", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [244, 9], "def_end_pos": [244, 16]}, {"full_name": "Finset.coe_insert", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1113, 9], "def_end_pos": [1113, 19]}, {"full_name": "Finset.coe_Ioc", "def_path": "Mathlib/Order/LocallyFinite.lean", "def_pos": [356, 9], "def_end_pos": [356, 16]}, {"full_name": "Finset.coe_Icc", "def_path": "Mathlib/Order/LocallyFinite.lean", "def_pos": [346, 9], "def_end_pos": [346, 16]}, {"full_name": "Set.insert_eq", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1310, 9], "def_end_pos": [1310, 18]}, {"full_name": "Set.union_comm", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [786, 9], "def_end_pos": [786, 19]}, {"full_name": "Set.Ioc_union_left", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [880, 9], "def_end_pos": [880, 23]}]], "state_before": "\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : LocallyFiniteOrder \u03b1\na b c : \u03b1\ninst\u271d : DecidableEq \u03b1\nh : a \u2264 b\n\u22a2 insert a (Ioc a b) = Icc a b", "state_after": "no goals"}]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/RingTheory/PowerSeries/Basic.lean
|
Polynomial.coe_X
|
[
2693,
1
] |
[
2694,
19
] |
[]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Data/ZMod/Basic.lean
|
ZMod.valMinAbs_mem_Ioc
|
[
1008,
1
] |
[
1016,
24
] |
[{"tactic": "simp_rw [valMinAbs_def_pos, Nat.le_div_two_iff_mul_two_le]", "annotated_tactic": ["simp_rw [<a>valMinAbs_def_pos</a>, <a>Nat.le_div_two_iff_mul_two_le</a>]", [{"full_name": "ZMod.valMinAbs_def_pos", "def_path": "Mathlib/Data/ZMod/Basic.lean", "def_pos": [963, 9], "def_end_pos": [963, 26]}, {"full_name": "Nat.le_div_two_iff_mul_two_le", "def_path": "Mathlib/Data/ZMod/Basic.lean", "def_pos": [984, 9], "def_end_pos": [984, 45]}]], "state_before": "n : \u2115\ninst\u271d : NeZero n\nx : ZMod n\n\u22a2 valMinAbs x * 2 \u2208 Set.Ioc (-\u2191n) \u2191n", "state_after": "n : \u2115\ninst\u271d : NeZero n\nx : ZMod n\n\u22a2 (if \u2191(val x) * 2 \u2264 \u2191n then \u2191(val x) else \u2191(val x) - \u2191n) * 2 \u2208 Set.Ioc (-\u2191n) \u2191n"}, {"tactic": "split_ifs with h", "annotated_tactic": ["split_ifs with h", []], "state_before": "n : \u2115\ninst\u271d : NeZero n\nx : ZMod n\n\u22a2 (if \u2191(val x) * 2 \u2264 \u2191n then \u2191(val x) else \u2191(val x) - \u2191n) * 2 \u2208 Set.Ioc (-\u2191n) \u2191n", "state_after": "case pos\nn : \u2115\ninst\u271d : NeZero n\nx : ZMod n\nh : \u2191(val x) * 2 \u2264 \u2191n\n\u22a2 \u2191(val x) * 2 \u2208 Set.Ioc (-\u2191n) \u2191n\n\ncase neg\nn : \u2115\ninst\u271d : NeZero n\nx : ZMod n\nh : \u00ac\u2191(val x) * 2 \u2264 \u2191n\n\u22a2 (\u2191(val x) - \u2191n) * 2 \u2208 Set.Ioc (-\u2191n) \u2191n"}, {"tactic": "refine' \u27e8(neg_lt_zero.2 <| by exact_mod_cast NeZero.pos n).trans_le (mul_nonneg _ _), h\u27e9", "annotated_tactic": ["refine' \u27e8(<a>neg_lt_zero</a>.2 <| by exact_mod_cast <a>NeZero.pos</a> n).<a>trans_le</a> (<a>mul_nonneg</a> _ _), h\u27e9", [{"full_name": "neg_lt_zero", "def_path": "Mathlib/Algebra/Order/Group/Defs.lean", "def_pos": [601, 24], "def_end_pos": [601, 35]}, {"full_name": "NeZero.pos", "def_path": "Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean", "def_pos": [311, 9], "def_end_pos": [311, 12]}, {"full_name": "LT.lt.trans_le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [148, 7], "def_end_pos": [148, 21]}, {"full_name": "mul_nonneg", "def_path": "Mathlib/Algebra/Order/Ring/Lemmas.lean", "def_pos": [380, 7], "def_end_pos": [380, 17]}]], "state_before": "case pos\nn : \u2115\ninst\u271d : NeZero n\nx : ZMod n\nh : \u2191(val x) * 2 \u2264 \u2191n\n\u22a2 \u2191(val x) * 2 \u2208 Set.Ioc (-\u2191n) \u2191n", "state_after": "case pos.refine'_1\nn : \u2115\ninst\u271d : NeZero n\nx : ZMod n\nh : \u2191(val x) * 2 \u2264 \u2191n\n\u22a2 0 \u2264 \u2191(val x)\n\ncase pos.refine'_2\nn : \u2115\ninst\u271d : NeZero n\nx : ZMod n\nh : \u2191(val x) * 2 \u2264 \u2191n\n\u22a2 0 \u2264 2"}, {"tactic": "exacts [Nat.cast_nonneg _, zero_le_two]", "annotated_tactic": ["exacts [<a>Nat.cast_nonneg</a> _, <a>zero_le_two</a>]", [{"full_name": "Nat.cast_nonneg", "def_path": "Mathlib/Data/Nat/Cast/Order.lean", "def_pos": [44, 9], "def_end_pos": [44, 20]}, {"full_name": "zero_le_two", "def_path": "Mathlib/Algebra/Order/Monoid/NatCast.lean", "def_pos": [32, 7], "def_end_pos": [32, 18]}]], "state_before": "case pos.refine'_1\nn : \u2115\ninst\u271d : NeZero n\nx : ZMod n\nh : \u2191(val x) * 2 \u2264 \u2191n\n\u22a2 0 \u2264 \u2191(val x)\n\ncase pos.refine'_2\nn : \u2115\ninst\u271d : NeZero n\nx : ZMod n\nh : \u2191(val x) * 2 \u2264 \u2191n\n\u22a2 0 \u2264 2", "state_after": "no goals"}, {"tactic": "exact_mod_cast NeZero.pos n", "annotated_tactic": ["exact_mod_cast <a>NeZero.pos</a> n", [{"full_name": "NeZero.pos", "def_path": "Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean", "def_pos": [311, 9], "def_end_pos": [311, 12]}]], "state_before": "n : \u2115\ninst\u271d : NeZero n\nx : ZMod n\nh : \u2191(val x) * 2 \u2264 \u2191n\n\u22a2 0 < \u2191n", "state_after": "no goals"}, {"tactic": "refine' \u27e8_, le_trans (mul_nonpos_of_nonpos_of_nonneg _ zero_le_two) <| Nat.cast_nonneg _\u27e9", "annotated_tactic": ["refine' \u27e8_, <a>le_trans</a> (<a>mul_nonpos_of_nonpos_of_nonneg</a> _ <a>zero_le_two</a>) <| <a>Nat.cast_nonneg</a> _\u27e9", [{"full_name": "le_trans", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [56, 9], "def_end_pos": [56, 17]}, {"full_name": "mul_nonpos_of_nonpos_of_nonneg", "def_path": "Mathlib/Algebra/Order/Ring/Lemmas.lean", "def_pos": [392, 9], "def_end_pos": [392, 39]}, {"full_name": "zero_le_two", "def_path": "Mathlib/Algebra/Order/Monoid/NatCast.lean", "def_pos": [32, 7], "def_end_pos": [32, 18]}, {"full_name": "Nat.cast_nonneg", "def_path": "Mathlib/Data/Nat/Cast/Order.lean", "def_pos": [44, 9], "def_end_pos": [44, 20]}]], "state_before": "case neg\nn : \u2115\ninst\u271d : NeZero n\nx : ZMod n\nh : \u00ac\u2191(val x) * 2 \u2264 \u2191n\n\u22a2 (\u2191(val x) - \u2191n) * 2 \u2208 Set.Ioc (-\u2191n) \u2191n", "state_after": "case neg.refine'_1\nn : \u2115\ninst\u271d : NeZero n\nx : ZMod n\nh : \u00ac\u2191(val x) * 2 \u2264 \u2191n\n\u22a2 -\u2191n < (\u2191(val x) - \u2191n) * 2\n\ncase neg.refine'_2\nn : \u2115\ninst\u271d : NeZero n\nx : ZMod n\nh : \u00ac\u2191(val x) * 2 \u2264 \u2191n\n\u22a2 \u2191(val x) - \u2191n \u2264 0"}, {"tactic": "linarith only [h]", "annotated_tactic": ["linarith only [h]", []], "state_before": "case neg.refine'_1\nn : \u2115\ninst\u271d : NeZero n\nx : ZMod n\nh : \u00ac\u2191(val x) * 2 \u2264 \u2191n\n\u22a2 -\u2191n < (\u2191(val x) - \u2191n) * 2", "state_after": "no goals"}, {"tactic": "rw [sub_nonpos, Int.ofNat_le]", "annotated_tactic": ["rw [<a>sub_nonpos</a>, <a>Int.ofNat_le</a>]", [{"full_name": "sub_nonpos", "def_path": "Mathlib/Algebra/Order/Group/Defs.lean", "def_pos": [730, 30], "def_end_pos": [730, 40]}, {"full_name": "Int.ofNat_le", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [574, 28], "def_end_pos": [574, 36]}]], "state_before": "case neg.refine'_2\nn : \u2115\ninst\u271d : NeZero n\nx : ZMod n\nh : \u00ac\u2191(val x) * 2 \u2264 \u2191n\n\u22a2 \u2191(val x) - \u2191n \u2264 0", "state_after": "case neg.refine'_2\nn : \u2115\ninst\u271d : NeZero n\nx : ZMod n\nh : \u00ac\u2191(val x) * 2 \u2264 \u2191n\n\u22a2 val x \u2264 n"}, {"tactic": "exact x.val_lt.le", "annotated_tactic": ["exact x.val_lt.le", []], "state_before": "case neg.refine'_2\nn : \u2115\ninst\u271d : NeZero n\nx : ZMod n\nh : \u00ac\u2191(val x) * 2 \u2264 \u2191n\n\u22a2 val x \u2264 n", "state_after": "no goals"}]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/CategoryTheory/Limits/Shapes/Biproducts.lean
|
CategoryTheory.Limits.biprod.associator_inv_natural
|
[
2045,
1
] |
[
2048,
12
] |
[{"tactic": "aesop_cat", "annotated_tactic": ["aesop_cat", []], "state_before": "J : Type w\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\nP Q : C\ninst\u271d : HasBinaryBiproducts C\nU V W X Y Z : C\nf : U \u27f6 X\ng : V \u27f6 Y\nh : W \u27f6 Z\n\u22a2 map f (map g h) \u226b (associator X Y Z).inv = (associator U V W).inv \u226b map (map f g) h", "state_after": "no goals"}]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/CategoryTheory/Monoidal/CommMon_.lean
|
CommMon_.forget₂_Mon_obj_one
|
[
93,
1
] |
[
94,
6
] |
[]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Analysis/Analytic/Basic.lean
|
HasFPowerSeriesAt.eq_zero_of_eventually
|
[
1158,
1
] |
[
1160,
24
] |
[]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/GroupTheory/HNNExtension.lean
|
HNNExtension.NormalWord.t_pow_smul_eq_unitsSMul
|
[
504,
1
] |
[
507,
72
] |
[{"tactic": "rcases Int.units_eq_one_or u with (rfl | rfl) <;>\n simp [instHSMul, SMul.smul, MulAction.toEndHom, Equiv.Perm.inv_def]", "annotated_tactic": ["rcases <a>Int.units_eq_one_or</a> u with (rfl | rfl) <;>\n simp [<a>instHSMul</a>, <a>SMul.smul</a>, <a>MulAction.toEndHom</a>, <a>Equiv.Perm.inv_def</a>]", [{"full_name": "Int.units_eq_one_or", "def_path": "Mathlib/Data/Int/Units.lean", "def_pos": [30, 9], "def_end_pos": [30, 24]}, {"full_name": "instHSMul", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [96, 10], "def_end_pos": [96, 19]}, {"full_name": "SMul.smul", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [82, 3], "def_end_pos": [82, 7]}, {"full_name": "MulAction.toEndHom", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [1183, 5], "def_end_pos": [1183, 23]}, {"full_name": "Equiv.Perm.inv_def", "def_path": "Mathlib/GroupTheory/Perm/Basic.lean", "def_pos": [93, 9], "def_end_pos": [93, 16]}]], "state_before": "G : Type u_1\ninst\u271d\u00b3 : Group G\nA B : Subgroup G\n\u03c6 : { x // x \u2208 A } \u2243* { x // x \u2208 B }\nH : Type u_2\ninst\u271d\u00b2 : Group H\nM : Type u_3\ninst\u271d\u00b9 : Monoid M\nd : TransversalPair G A B\ninst\u271d : DecidableEq G\nu : \u2124\u02e3\nw : NormalWord d\n\u22a2 t ^ \u2191u \u2022 w = unitsSMul \u03c6 u w", "state_after": "no goals"}]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Algebra/Order/Ring/Lemmas.lean
|
neg_of_mul_pos_left
|
[
572,
1
] |
[
573,
84
] |
[]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/RingTheory/Ideal/Operations.lean
|
Ideal.add_eq_sup
|
[
415,
1
] |
[
416,
6
] |
[]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Data/Finset/Basic.lean
|
Finset.filter_empty
|
[
2780,
1
] |
[
2781,
38
] |
[]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Algebra/GroupWithZero/Basic.lean
|
mul_eq_mul_left_iff
|
[
172,
1
] |
[
173,
65
] |
[{"tactic": "by_cases ha : a = 0 <;> [simp [ha]; simp [mul_right_inj', ha]]", "annotated_tactic": ["by_cases ha : a = 0 <;> [simp [ha]; simp [<a>mul_right_inj'</a>, ha]]", [{"full_name": "mul_right_inj'", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [131, 9], "def_end_pos": [131, 23]}]], "state_before": "\u03b1 : Type u_1\nM\u2080 : Type u_2\nG\u2080 : Type u_3\nM\u2080' : Type u_4\nG\u2080' : Type u_5\nF : Type u_6\nF' : Type u_7\ninst\u271d : CancelMonoidWithZero M\u2080\na b c : M\u2080\n\u22a2 a * b = a * c \u2194 b = c \u2228 a = 0", "state_after": "no goals"}]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Topology/Algebra/Group/Basic.lean
|
isOpenMap_inv
|
[
340,
1
] |
[
341,
31
] |
[]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Analysis/NormedSpace/Exponential.lean
|
expSeries_summable'
|
[
426,
1
] |
[
427,
57
] |
[]
|
https://github.com/leanprover/std4
|
869c615eb10130c0637a7bc038e2b80253559913
|
lake-packages/std/Std/Data/Int/Lemmas.lean
|
Int.neg_add_le_right_of_le_add
|
[
1013,
11
] |
[
1015,
40
] |
[{"tactic": "rw [Int.add_comm] at h", "annotated_tactic": ["rw [<a>Int.add_comm</a>] at h", [{"full_name": "Int.add_comm", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [218, 19], "def_end_pos": [218, 27]}]], "state_before": "a b c : Int\nh : a \u2264 b + c\n\u22a2 -c + a \u2264 b", "state_after": "a b c : Int\nh : a \u2264 c + b\n\u22a2 -c + a \u2264 b"}, {"tactic": "exact Int.neg_add_le_left_of_le_add h", "annotated_tactic": ["exact <a>Int.neg_add_le_left_of_le_add</a> h", [{"full_name": "Int.neg_add_le_left_of_le_add", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [1005, 19], "def_end_pos": [1005, 44]}]], "state_before": "a b c : Int\nh : a \u2264 c + b\n\u22a2 -c + a \u2264 b", "state_after": "no goals"}]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Combinatorics/SimpleGraph/Density.lean
|
SimpleGraph.card_interedges_add_card_interedges_compl
|
[
375,
1
] |
[
383,
56
] |
[{"tactic": "rw [\u2190 card_product, interedges_def, interedges_def]", "annotated_tactic": ["rw [\u2190 <a>card_product</a>, <a>interedges_def</a>, <a>interedges_def</a>]", [{"full_name": "Finset.card_product", "def_path": "Mathlib/Data/Finset/Prod.lean", "def_pos": [139, 9], "def_end_pos": [139, 21]}, {"full_name": "SimpleGraph.interedges_def", "def_path": "Mathlib/Combinatorics/SimpleGraph/Density.lean", "def_pos": [312, 9], "def_end_pos": [312, 23]}, {"full_name": "SimpleGraph.interedges_def", "def_path": "Mathlib/Combinatorics/SimpleGraph/Density.lean", "def_pos": [312, 9], "def_end_pos": [312, 23]}]], "state_before": "\ud835\udd5c : Type u_1\n\u03b9 : Type u_2\n\u03ba : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\nG : SimpleGraph \u03b1\ninst\u271d\u00b9 : DecidableRel G.Adj\ns s\u2081 s\u2082 t t\u2081 t\u2082 : Finset \u03b1\na b : \u03b1\ninst\u271d : DecidableEq \u03b1\nh : Disjoint s t\n\u22a2 card (interedges G s t) + card (interedges G\u1d9c s t) = card s * card t", "state_after": "\ud835\udd5c : Type u_1\n\u03b9 : Type u_2\n\u03ba : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\nG : SimpleGraph \u03b1\ninst\u271d\u00b9 : DecidableRel G.Adj\ns s\u2081 s\u2082 t t\u2081 t\u2082 : Finset \u03b1\na b : \u03b1\ninst\u271d : DecidableEq \u03b1\nh : Disjoint s t\n\u22a2 card (filter (fun e => Adj G e.1 e.2) (s \u00d7\u02e2 t)) + card (filter (fun e => Adj G\u1d9c e.1 e.2) (s \u00d7\u02e2 t)) = card (s \u00d7\u02e2 t)"}, {"tactic": "have : ((s \u00d7\u02e2 t).filter fun e \u21a6 G\u1d9c.Adj e.1 e.2) = (s \u00d7\u02e2 t).filter fun e \u21a6 \u00acG.Adj e.1 e.2 := by\n refine' filter_congr fun x hx \u21a6 _\n rw [mem_product] at hx\n rw [compl_adj, and_iff_right (h.forall_ne_finset hx.1 hx.2)]", "annotated_tactic": ["have : ((s \u00d7\u02e2 t).<a>filter</a> fun e \u21a6 G\u1d9c.<a>Adj</a> e.1 e.2) = (s \u00d7\u02e2 t).<a>filter</a> fun e \u21a6 \u00acG.Adj e.1 e.2 := by\n refine' <a>filter_congr</a> fun x hx \u21a6 _\n rw [<a>mem_product</a>] at hx\n rw [<a>compl_adj</a>, <a>and_iff_right</a> (h.forall_ne_finset hx.1 hx.2)]", [{"full_name": "Finset.filter", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [2691, 5], "def_end_pos": [2691, 11]}, {"full_name": "SimpleGraph.Adj", "def_path": "Mathlib/Combinatorics/SimpleGraph/Basic.lean", "def_pos": [132, 3], "def_end_pos": [132, 6]}, {"full_name": "Finset.filter", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [2691, 5], "def_end_pos": [2691, 11]}, {"full_name": "Finset.filter_congr", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [2774, 9], "def_end_pos": [2774, 21]}, {"full_name": "Finset.mem_product", "def_path": "Mathlib/Data/Finset/Prod.lean", "def_pos": [53, 9], "def_end_pos": [53, 20]}, {"full_name": "SimpleGraph.compl_adj", "def_path": "Mathlib/Combinatorics/SimpleGraph/Basic.lean", "def_pos": [299, 9], "def_end_pos": [299, 18]}, {"full_name": "and_iff_right", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [206, 9], "def_end_pos": [206, 22]}]], "state_before": "\ud835\udd5c : Type u_1\n\u03b9 : Type u_2\n\u03ba : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\nG : SimpleGraph \u03b1\ninst\u271d\u00b9 : DecidableRel G.Adj\ns s\u2081 s\u2082 t t\u2081 t\u2082 : Finset \u03b1\na b : \u03b1\ninst\u271d : DecidableEq \u03b1\nh : Disjoint s t\n\u22a2 card (filter (fun e => Adj G e.1 e.2) (s \u00d7\u02e2 t)) + card (filter (fun e => Adj G\u1d9c e.1 e.2) (s \u00d7\u02e2 t)) = card (s \u00d7\u02e2 t)", "state_after": "\ud835\udd5c : Type u_1\n\u03b9 : Type u_2\n\u03ba : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\nG : SimpleGraph \u03b1\ninst\u271d\u00b9 : DecidableRel G.Adj\ns s\u2081 s\u2082 t t\u2081 t\u2082 : Finset \u03b1\na b : \u03b1\ninst\u271d : DecidableEq \u03b1\nh : Disjoint s t\nthis : filter (fun e => Adj G\u1d9c e.1 e.2) (s \u00d7\u02e2 t) = filter (fun e => \u00acAdj G e.1 e.2) (s \u00d7\u02e2 t)\n\u22a2 card (filter (fun e => Adj G e.1 e.2) (s \u00d7\u02e2 t)) + card (filter (fun e => Adj G\u1d9c e.1 e.2) (s \u00d7\u02e2 t)) = card (s \u00d7\u02e2 t)"}, {"tactic": "rw [this, \u2190 card_union_eq, filter_union_filter_neg_eq]", "annotated_tactic": ["rw [this, \u2190 <a>card_union_eq</a>, <a>filter_union_filter_neg_eq</a>]", [{"full_name": "Finset.card_union_eq", "def_path": "Mathlib/Data/Finset/Card.lean", "def_pos": [431, 9], "def_end_pos": [431, 22]}, {"full_name": "Finset.filter_union_filter_neg_eq", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [3012, 9], "def_end_pos": [3012, 35]}]], "state_before": "\ud835\udd5c : Type u_1\n\u03b9 : Type u_2\n\u03ba : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\nG : SimpleGraph \u03b1\ninst\u271d\u00b9 : DecidableRel G.Adj\ns s\u2081 s\u2082 t t\u2081 t\u2082 : Finset \u03b1\na b : \u03b1\ninst\u271d : DecidableEq \u03b1\nh : Disjoint s t\nthis : filter (fun e => Adj G\u1d9c e.1 e.2) (s \u00d7\u02e2 t) = filter (fun e => \u00acAdj G e.1 e.2) (s \u00d7\u02e2 t)\n\u22a2 card (filter (fun e => Adj G e.1 e.2) (s \u00d7\u02e2 t)) + card (filter (fun e => Adj G\u1d9c e.1 e.2) (s \u00d7\u02e2 t)) = card (s \u00d7\u02e2 t)", "state_after": "\ud835\udd5c : Type u_1\n\u03b9 : Type u_2\n\u03ba : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\nG : SimpleGraph \u03b1\ninst\u271d\u00b9 : DecidableRel G.Adj\ns s\u2081 s\u2082 t t\u2081 t\u2082 : Finset \u03b1\na b : \u03b1\ninst\u271d : DecidableEq \u03b1\nh : Disjoint s t\nthis : filter (fun e => Adj G\u1d9c e.1 e.2) (s \u00d7\u02e2 t) = filter (fun e => \u00acAdj G e.1 e.2) (s \u00d7\u02e2 t)\n\u22a2 Disjoint (filter (fun e => Adj G e.1 e.2) (s \u00d7\u02e2 t)) (filter (fun e => \u00acAdj G e.1 e.2) (s \u00d7\u02e2 t))"}, {"tactic": "exact disjoint_filter.2 fun _ _ \u21a6 Classical.not_not.2", "annotated_tactic": ["exact <a>disjoint_filter</a>.2 fun _ _ \u21a6 <a>Classical.not_not</a>.2", [{"full_name": "Finset.disjoint_filter", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [2822, 9], "def_end_pos": [2822, 24]}, {"full_name": "Classical.not_not", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [683, 24], "def_end_pos": [683, 31]}]], "state_before": "\ud835\udd5c : Type u_1\n\u03b9 : Type u_2\n\u03ba : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\nG : SimpleGraph \u03b1\ninst\u271d\u00b9 : DecidableRel G.Adj\ns s\u2081 s\u2082 t t\u2081 t\u2082 : Finset \u03b1\na b : \u03b1\ninst\u271d : DecidableEq \u03b1\nh : Disjoint s t\nthis : filter (fun e => Adj G\u1d9c e.1 e.2) (s \u00d7\u02e2 t) = filter (fun e => \u00acAdj G e.1 e.2) (s \u00d7\u02e2 t)\n\u22a2 Disjoint (filter (fun e => Adj G e.1 e.2) (s \u00d7\u02e2 t)) (filter (fun e => \u00acAdj G e.1 e.2) (s \u00d7\u02e2 t))", "state_after": "no goals"}, {"tactic": "refine' filter_congr fun x hx \u21a6 _", "annotated_tactic": ["refine' <a>filter_congr</a> fun x hx \u21a6 _", [{"full_name": "Finset.filter_congr", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [2774, 9], "def_end_pos": [2774, 21]}]], "state_before": "\ud835\udd5c : Type u_1\n\u03b9 : Type u_2\n\u03ba : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\nG : SimpleGraph \u03b1\ninst\u271d\u00b9 : DecidableRel G.Adj\ns s\u2081 s\u2082 t t\u2081 t\u2082 : Finset \u03b1\na b : \u03b1\ninst\u271d : DecidableEq \u03b1\nh : Disjoint s t\n\u22a2 filter (fun e => Adj G\u1d9c e.1 e.2) (s \u00d7\u02e2 t) = filter (fun e => \u00acAdj G e.1 e.2) (s \u00d7\u02e2 t)", "state_after": "\ud835\udd5c : Type u_1\n\u03b9 : Type u_2\n\u03ba : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\nG : SimpleGraph \u03b1\ninst\u271d\u00b9 : DecidableRel G.Adj\ns s\u2081 s\u2082 t t\u2081 t\u2082 : Finset \u03b1\na b : \u03b1\ninst\u271d : DecidableEq \u03b1\nh : Disjoint s t\nx : \u03b1 \u00d7 \u03b1\nhx : x \u2208 s \u00d7\u02e2 t\n\u22a2 Adj G\u1d9c x.1 x.2 \u2194 \u00acAdj G x.1 x.2"}, {"tactic": "rw [mem_product] at hx", "annotated_tactic": ["rw [<a>mem_product</a>] at hx", [{"full_name": "Finset.mem_product", "def_path": "Mathlib/Data/Finset/Prod.lean", "def_pos": [53, 9], "def_end_pos": [53, 20]}]], "state_before": "\ud835\udd5c : Type u_1\n\u03b9 : Type u_2\n\u03ba : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\nG : SimpleGraph \u03b1\ninst\u271d\u00b9 : DecidableRel G.Adj\ns s\u2081 s\u2082 t t\u2081 t\u2082 : Finset \u03b1\na b : \u03b1\ninst\u271d : DecidableEq \u03b1\nh : Disjoint s t\nx : \u03b1 \u00d7 \u03b1\nhx : x \u2208 s \u00d7\u02e2 t\n\u22a2 Adj G\u1d9c x.1 x.2 \u2194 \u00acAdj G x.1 x.2", "state_after": "\ud835\udd5c : Type u_1\n\u03b9 : Type u_2\n\u03ba : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\nG : SimpleGraph \u03b1\ninst\u271d\u00b9 : DecidableRel G.Adj\ns s\u2081 s\u2082 t t\u2081 t\u2082 : Finset \u03b1\na b : \u03b1\ninst\u271d : DecidableEq \u03b1\nh : Disjoint s t\nx : \u03b1 \u00d7 \u03b1\nhx : x.1 \u2208 s \u2227 x.2 \u2208 t\n\u22a2 Adj G\u1d9c x.1 x.2 \u2194 \u00acAdj G x.1 x.2"}, {"tactic": "rw [compl_adj, and_iff_right (h.forall_ne_finset hx.1 hx.2)]", "annotated_tactic": ["rw [<a>compl_adj</a>, <a>and_iff_right</a> (h.forall_ne_finset hx.1 hx.2)]", [{"full_name": "SimpleGraph.compl_adj", "def_path": "Mathlib/Combinatorics/SimpleGraph/Basic.lean", "def_pos": [299, 9], "def_end_pos": [299, 18]}, {"full_name": "and_iff_right", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [206, 9], "def_end_pos": [206, 22]}]], "state_before": "\ud835\udd5c : Type u_1\n\u03b9 : Type u_2\n\u03ba : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\nG : SimpleGraph \u03b1\ninst\u271d\u00b9 : DecidableRel G.Adj\ns s\u2081 s\u2082 t t\u2081 t\u2082 : Finset \u03b1\na b : \u03b1\ninst\u271d : DecidableEq \u03b1\nh : Disjoint s t\nx : \u03b1 \u00d7 \u03b1\nhx : x.1 \u2208 s \u2227 x.2 \u2208 t\n\u22a2 Adj G\u1d9c x.1 x.2 \u2194 \u00acAdj G x.1 x.2", "state_after": "no goals"}]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/RingTheory/PowerSeries/Basic.lean
|
MvPowerSeries.mul_one
|
[
279,
11
] |
[
280,
61
] |
[{"tactic": "simpa using coeff_add_mul_monomial n 0 \u03c6 1", "annotated_tactic": ["simpa using <a>coeff_add_mul_monomial</a> n 0 \u03c6 1", [{"full_name": "MvPowerSeries.coeff_add_mul_monomial", "def_path": "Mathlib/RingTheory/PowerSeries/Basic.lean", "def_pos": [259, 9], "def_end_pos": [259, 31]}]], "state_before": "\u03c3 : Type u_1\nR : Type u_2\ninst\u271d : Semiring R\nm n\u271d : \u03c3 \u2192\u2080 \u2115\n\u03c6 \u03c8 : MvPowerSeries \u03c3 R\nn : \u03c3 \u2192\u2080 \u2115\n\u22a2 \u2191(coeff R n) (\u03c6 * 1) = \u2191(coeff R n) \u03c6", "state_after": "no goals"}]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/RingTheory/FiniteType.lean
|
Algebra.FiniteType.trans
|
[
127,
1
] |
[
129,
26
] |
[]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean
|
eq_one_or_one_lt
|
[
250,
1
] |
[
251,
39
] |
[]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Order/Filter/EventuallyConst.lean
|
Filter.eventuallyConst_set
|
[
70,
1
] |
[
72,
23
] |
[]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Algebra/Order/ToIntervalMod.lean
|
toIcoDiv_add_left
|
[
316,
1
] |
[
317,
36
] |
[{"tactic": "rw [add_comm, toIcoDiv_add_right]", "annotated_tactic": ["rw [<a>add_comm</a>, <a>toIcoDiv_add_right</a>]", [{"full_name": "add_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [301, 3], "def_end_pos": [301, 14]}, {"full_name": "toIcoDiv_add_right", "def_path": "Mathlib/Algebra/Order/ToIntervalMod.lean", "def_pos": [296, 9], "def_end_pos": [296, 27]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : LinearOrderedAddCommGroup \u03b1\nh\u03b1 : Archimedean \u03b1\np : \u03b1\nhp : 0 < p\na\u271d b\u271d c : \u03b1\nn : \u2124\na b : \u03b1\n\u22a2 toIcoDiv hp a (p + b) = toIcoDiv hp a b + 1", "state_after": "no goals"}]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/LinearAlgebra/AffineSpace/Combination.lean
|
Finset.centroid_eq_of_inj_on_of_image_eq
|
[
975,
1
] |
[
980,
56
] |
[{"tactic": "classical rw [s.centroid_eq_centroid_image_of_inj_on k hi rfl,\n s\u2082.centroid_eq_centroid_image_of_inj_on k hi\u2082 he]", "annotated_tactic": ["classical rw [s.centroid_eq_centroid_image_of_inj_on k hi <a>rfl</a>,\n s\u2082.centroid_eq_centroid_image_of_inj_on k hi\u2082 he]", [{"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\ns : Finset \u03b9\n\u03b9\u2082 : Type u_5\ns\u2082 : Finset \u03b9\u2082\np : \u03b9 \u2192 P\nhi : \u2200 (i : \u03b9), i \u2208 s \u2192 \u2200 (j : \u03b9), j \u2208 s \u2192 p i = p j \u2192 i = j\np\u2082 : \u03b9\u2082 \u2192 P\nhi\u2082 : \u2200 (i : \u03b9\u2082), i \u2208 s\u2082 \u2192 \u2200 (j : \u03b9\u2082), j \u2208 s\u2082 \u2192 p\u2082 i = p\u2082 j \u2192 i = j\nhe : p '' \u2191s = p\u2082 '' \u2191s\u2082\n\u22a2 centroid k s p = centroid k s\u2082 p\u2082", "state_after": "no goals"}, {"tactic": "rw [s.centroid_eq_centroid_image_of_inj_on k hi rfl,\ns\u2082.centroid_eq_centroid_image_of_inj_on k hi\u2082 he]", "annotated_tactic": ["rw [s.centroid_eq_centroid_image_of_inj_on k hi <a>rfl</a>,\n s\u2082.centroid_eq_centroid_image_of_inj_on k hi\u2082 he]", [{"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\ns : Finset \u03b9\n\u03b9\u2082 : Type u_5\ns\u2082 : Finset \u03b9\u2082\np : \u03b9 \u2192 P\nhi : \u2200 (i : \u03b9), i \u2208 s \u2192 \u2200 (j : \u03b9), j \u2208 s \u2192 p i = p j \u2192 i = j\np\u2082 : \u03b9\u2082 \u2192 P\nhi\u2082 : \u2200 (i : \u03b9\u2082), i \u2208 s\u2082 \u2192 \u2200 (j : \u03b9\u2082), j \u2208 s\u2082 \u2192 p\u2082 i = p\u2082 j \u2192 i = j\nhe : p '' \u2191s = p\u2082 '' \u2191s\u2082\n\u22a2 centroid k s p = centroid k s\u2082 p\u2082", "state_after": "no goals"}]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Topology/Order.lean
|
generateFrom_iInter
|
[
1008,
1
] |
[
1010,
33
] |
[]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Topology/MetricSpace/Contracting.lean
|
ContractingWith.dist_fixedPoint_le
|
[
320,
1
] |
[
321,
53
] |
[]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Order/Category/Preord.lean
|
Preord.coe_of
|
[
54,
1
] |
[
55,
6
] |
[]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Order/ConditionallyCompleteLattice/Basic.lean
|
exists_lt_of_lt_ciSup
|
[
1007,
1
] |
[
1009,
9
] |
[]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Analysis/Convex/StrictConvexSpace.lean
|
openSegment_subset_ball_of_ne
|
[
168,
1
] |
[
170,
73
] |
[]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/CategoryTheory/Sites/SheafOfTypes.lean
|
CategoryTheory.Presieve.isSeparatedFor_and_exists_isAmalgamation_iff_isSheafFor
|
[
580,
1
] |
[
593,
58
] |
[{"tactic": "rw [IsSeparatedFor, \u2190 forall_and]", "annotated_tactic": ["rw [<a>IsSeparatedFor</a>, \u2190 <a>forall_and</a>]", [{"full_name": "CategoryTheory.Presieve.IsSeparatedFor", "def_path": "Mathlib/CategoryTheory/Sites/SheafOfTypes.lean", "def_pos": [412, 5], "def_end_pos": [412, 19]}, {"full_name": "forall_and", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [426, 9], "def_end_pos": [426, 19]}]], "state_before": "C : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\n\u22a2 (IsSeparatedFor P R \u2227\n \u2200 (x : FamilyOfElements P R), FamilyOfElements.Compatible x \u2192 \u2203 t, FamilyOfElements.IsAmalgamation x t) \u2194\n IsSheafFor P R", "state_after": "C : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\n\u22a2 (\u2200 (x : FamilyOfElements P R),\n (\u2200 (t\u2081 t\u2082 : P.obj (op X)),\n FamilyOfElements.IsAmalgamation x t\u2081 \u2192 FamilyOfElements.IsAmalgamation x t\u2082 \u2192 t\u2081 = t\u2082) \u2227\n (FamilyOfElements.Compatible x \u2192 \u2203 t, FamilyOfElements.IsAmalgamation x t)) \u2194\n IsSheafFor P R"}, {"tactic": "apply forall_congr'", "annotated_tactic": ["apply <a>forall_congr'</a>", [{"full_name": "forall_congr'", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [376, 9], "def_end_pos": [376, 22]}]], "state_before": "C : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\n\u22a2 (\u2200 (x : FamilyOfElements P R),\n (\u2200 (t\u2081 t\u2082 : P.obj (op X)),\n FamilyOfElements.IsAmalgamation x t\u2081 \u2192 FamilyOfElements.IsAmalgamation x t\u2082 \u2192 t\u2081 = t\u2082) \u2227\n (FamilyOfElements.Compatible x \u2192 \u2203 t, FamilyOfElements.IsAmalgamation x t)) \u2194\n IsSheafFor P R", "state_after": "case h\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\n\u22a2 \u2200 (a : FamilyOfElements P R),\n (\u2200 (t\u2081 t\u2082 : P.obj (op X)), FamilyOfElements.IsAmalgamation a t\u2081 \u2192 FamilyOfElements.IsAmalgamation a t\u2082 \u2192 t\u2081 = t\u2082) \u2227\n (FamilyOfElements.Compatible a \u2192 \u2203 t, FamilyOfElements.IsAmalgamation a t) \u2194\n FamilyOfElements.Compatible a \u2192 \u2203! t, FamilyOfElements.IsAmalgamation a t"}, {"tactic": "intro x", "annotated_tactic": ["intro x", []], "state_before": "case h\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\n\u22a2 \u2200 (a : FamilyOfElements P R),\n (\u2200 (t\u2081 t\u2082 : P.obj (op X)), FamilyOfElements.IsAmalgamation a t\u2081 \u2192 FamilyOfElements.IsAmalgamation a t\u2082 \u2192 t\u2081 = t\u2082) \u2227\n (FamilyOfElements.Compatible a \u2192 \u2203 t, FamilyOfElements.IsAmalgamation a t) \u2194\n FamilyOfElements.Compatible a \u2192 \u2203! t, FamilyOfElements.IsAmalgamation a t", "state_after": "case h\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nx : FamilyOfElements P R\n\u22a2 (\u2200 (t\u2081 t\u2082 : P.obj (op X)), FamilyOfElements.IsAmalgamation x t\u2081 \u2192 FamilyOfElements.IsAmalgamation x t\u2082 \u2192 t\u2081 = t\u2082) \u2227\n (FamilyOfElements.Compatible x \u2192 \u2203 t, FamilyOfElements.IsAmalgamation x t) \u2194\n FamilyOfElements.Compatible x \u2192 \u2203! t, FamilyOfElements.IsAmalgamation x t"}, {"tactic": "constructor", "annotated_tactic": ["constructor", []], "state_before": "case h\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nx : FamilyOfElements P R\n\u22a2 (\u2200 (t\u2081 t\u2082 : P.obj (op X)), FamilyOfElements.IsAmalgamation x t\u2081 \u2192 FamilyOfElements.IsAmalgamation x t\u2082 \u2192 t\u2081 = t\u2082) \u2227\n (FamilyOfElements.Compatible x \u2192 \u2203 t, FamilyOfElements.IsAmalgamation x t) \u2194\n FamilyOfElements.Compatible x \u2192 \u2203! t, FamilyOfElements.IsAmalgamation x t", "state_after": "case h.mp\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nx : FamilyOfElements P R\n\u22a2 (\u2200 (t\u2081 t\u2082 : P.obj (op X)), FamilyOfElements.IsAmalgamation x t\u2081 \u2192 FamilyOfElements.IsAmalgamation x t\u2082 \u2192 t\u2081 = t\u2082) \u2227\n (FamilyOfElements.Compatible x \u2192 \u2203 t, FamilyOfElements.IsAmalgamation x t) \u2192\n FamilyOfElements.Compatible x \u2192 \u2203! t, FamilyOfElements.IsAmalgamation x t\n\ncase h.mpr\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nx : FamilyOfElements P R\n\u22a2 (FamilyOfElements.Compatible x \u2192 \u2203! t, FamilyOfElements.IsAmalgamation x t) \u2192\n (\u2200 (t\u2081 t\u2082 : P.obj (op X)), FamilyOfElements.IsAmalgamation x t\u2081 \u2192 FamilyOfElements.IsAmalgamation x t\u2082 \u2192 t\u2081 = t\u2082) \u2227\n (FamilyOfElements.Compatible x \u2192 \u2203 t, FamilyOfElements.IsAmalgamation x t)"}, {"tactic": "intro z hx", "annotated_tactic": ["intro z hx", []], "state_before": "case h.mp\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nx : FamilyOfElements P R\n\u22a2 (\u2200 (t\u2081 t\u2082 : P.obj (op X)), FamilyOfElements.IsAmalgamation x t\u2081 \u2192 FamilyOfElements.IsAmalgamation x t\u2082 \u2192 t\u2081 = t\u2082) \u2227\n (FamilyOfElements.Compatible x \u2192 \u2203 t, FamilyOfElements.IsAmalgamation x t) \u2192\n FamilyOfElements.Compatible x \u2192 \u2203! t, FamilyOfElements.IsAmalgamation x t", "state_after": "case h.mp\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nx : FamilyOfElements P R\nz :\n (\u2200 (t\u2081 t\u2082 : P.obj (op X)), FamilyOfElements.IsAmalgamation x t\u2081 \u2192 FamilyOfElements.IsAmalgamation x t\u2082 \u2192 t\u2081 = t\u2082) \u2227\n (FamilyOfElements.Compatible x \u2192 \u2203 t, FamilyOfElements.IsAmalgamation x t)\nhx : FamilyOfElements.Compatible x\n\u22a2 \u2203! t, FamilyOfElements.IsAmalgamation x t"}, {"tactic": "exact exists_unique_of_exists_of_unique (z.2 hx) z.1", "annotated_tactic": ["exact <a>exists_unique_of_exists_of_unique</a> (z.2 hx) z.1", [{"full_name": "exists_unique_of_exists_of_unique", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [244, 9], "def_end_pos": [244, 42]}]], "state_before": "case h.mp\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nx : FamilyOfElements P R\nz :\n (\u2200 (t\u2081 t\u2082 : P.obj (op X)), FamilyOfElements.IsAmalgamation x t\u2081 \u2192 FamilyOfElements.IsAmalgamation x t\u2082 \u2192 t\u2081 = t\u2082) \u2227\n (FamilyOfElements.Compatible x \u2192 \u2203 t, FamilyOfElements.IsAmalgamation x t)\nhx : FamilyOfElements.Compatible x\n\u22a2 \u2203! t, FamilyOfElements.IsAmalgamation x t", "state_after": "no goals"}, {"tactic": "intro h", "annotated_tactic": ["intro h", []], "state_before": "case h.mpr\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nx : FamilyOfElements P R\n\u22a2 (FamilyOfElements.Compatible x \u2192 \u2203! t, FamilyOfElements.IsAmalgamation x t) \u2192\n (\u2200 (t\u2081 t\u2082 : P.obj (op X)), FamilyOfElements.IsAmalgamation x t\u2081 \u2192 FamilyOfElements.IsAmalgamation x t\u2082 \u2192 t\u2081 = t\u2082) \u2227\n (FamilyOfElements.Compatible x \u2192 \u2203 t, FamilyOfElements.IsAmalgamation x t)", "state_after": "case h.mpr\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nx : FamilyOfElements P R\nh : FamilyOfElements.Compatible x \u2192 \u2203! t, FamilyOfElements.IsAmalgamation x t\n\u22a2 (\u2200 (t\u2081 t\u2082 : P.obj (op X)), FamilyOfElements.IsAmalgamation x t\u2081 \u2192 FamilyOfElements.IsAmalgamation x t\u2082 \u2192 t\u2081 = t\u2082) \u2227\n (FamilyOfElements.Compatible x \u2192 \u2203 t, FamilyOfElements.IsAmalgamation x t)"}, {"tactic": "refine' \u27e8_, ExistsUnique.exists \u2218 h\u27e9", "annotated_tactic": ["refine' \u27e8_, <a>ExistsUnique.exists</a> \u2218 h\u27e9", [{"full_name": "ExistsUnique.exists", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [248, 9], "def_end_pos": [248, 28]}]], "state_before": "case h.mpr\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nx : FamilyOfElements P R\nh : FamilyOfElements.Compatible x \u2192 \u2203! t, FamilyOfElements.IsAmalgamation x t\n\u22a2 (\u2200 (t\u2081 t\u2082 : P.obj (op X)), FamilyOfElements.IsAmalgamation x t\u2081 \u2192 FamilyOfElements.IsAmalgamation x t\u2082 \u2192 t\u2081 = t\u2082) \u2227\n (FamilyOfElements.Compatible x \u2192 \u2203 t, FamilyOfElements.IsAmalgamation x t)", "state_after": "case h.mpr\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nx : FamilyOfElements P R\nh : FamilyOfElements.Compatible x \u2192 \u2203! t, FamilyOfElements.IsAmalgamation x t\n\u22a2 \u2200 (t\u2081 t\u2082 : P.obj (op X)), FamilyOfElements.IsAmalgamation x t\u2081 \u2192 FamilyOfElements.IsAmalgamation x t\u2082 \u2192 t\u2081 = t\u2082"}, {"tactic": "intro t\u2081 t\u2082 ht\u2081 ht\u2082", "annotated_tactic": ["intro t\u2081 t\u2082 ht\u2081 ht\u2082", []], "state_before": "case h.mpr\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nx : FamilyOfElements P R\nh : FamilyOfElements.Compatible x \u2192 \u2203! t, FamilyOfElements.IsAmalgamation x t\n\u22a2 \u2200 (t\u2081 t\u2082 : P.obj (op X)), FamilyOfElements.IsAmalgamation x t\u2081 \u2192 FamilyOfElements.IsAmalgamation x t\u2082 \u2192 t\u2081 = t\u2082", "state_after": "case h.mpr\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nx : FamilyOfElements P R\nh : FamilyOfElements.Compatible x \u2192 \u2203! t, FamilyOfElements.IsAmalgamation x t\nt\u2081 t\u2082 : P.obj (op X)\nht\u2081 : FamilyOfElements.IsAmalgamation x t\u2081\nht\u2082 : FamilyOfElements.IsAmalgamation x t\u2082\n\u22a2 t\u2081 = t\u2082"}, {"tactic": "apply (h _).unique ht\u2081 ht\u2082", "annotated_tactic": ["apply (h _).<a>unique</a> ht\u2081 ht\u2082", [{"full_name": "ExistsUnique.unique", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [252, 9], "def_end_pos": [252, 28]}]], "state_before": "case h.mpr\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nx : FamilyOfElements P R\nh : FamilyOfElements.Compatible x \u2192 \u2203! t, FamilyOfElements.IsAmalgamation x t\nt\u2081 t\u2082 : P.obj (op X)\nht\u2081 : FamilyOfElements.IsAmalgamation x t\u2081\nht\u2082 : FamilyOfElements.IsAmalgamation x t\u2082\n\u22a2 t\u2081 = t\u2082", "state_after": "C : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nx : FamilyOfElements P R\nh : FamilyOfElements.Compatible x \u2192 \u2203! t, FamilyOfElements.IsAmalgamation x t\nt\u2081 t\u2082 : P.obj (op X)\nht\u2081 : FamilyOfElements.IsAmalgamation x t\u2081\nht\u2082 : FamilyOfElements.IsAmalgamation x t\u2082\n\u22a2 FamilyOfElements.Compatible x"}, {"tactic": "exact is_compatible_of_exists_amalgamation x \u27e8_, ht\u2082\u27e9", "annotated_tactic": ["exact <a>is_compatible_of_exists_amalgamation</a> x \u27e8_, ht\u2082\u27e9", [{"full_name": "CategoryTheory.Presieve.is_compatible_of_exists_amalgamation", "def_path": "Mathlib/CategoryTheory/Sites/SheafOfTypes.lean", "def_pos": [391, 9], "def_end_pos": [391, 45]}]], "state_before": "C : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nx : FamilyOfElements P R\nh : FamilyOfElements.Compatible x \u2192 \u2203! t, FamilyOfElements.IsAmalgamation x t\nt\u2081 t\u2082 : P.obj (op X)\nht\u2081 : FamilyOfElements.IsAmalgamation x t\u2081\nht\u2082 : FamilyOfElements.IsAmalgamation x t\u2082\n\u22a2 FamilyOfElements.Compatible x", "state_after": "no goals"}]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/LinearAlgebra/Lagrange.lean
|
Lagrange.nodal_empty
|
[
520,
1
] |
[
521,
6
] |
[{"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "R : Type u_1\ninst\u271d : CommRing R\n\u03b9 : Type u_2\ns : Finset \u03b9\nv : \u03b9 \u2192 R\n\u22a2 nodal \u2205 v = 1", "state_after": "no goals"}]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Topology/Category/Profinite/Basic.lean
|
Profinite.to_compHausToTopCat
|
[
172,
1
] |
[
174,
6
] |
[]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Tactic/CategoryTheory/BicategoryCoherence.lean
|
Mathlib.Tactic.BicategoryCoherence.bicategoricalComp_refl
|
[
243,
1
] |
[
244,
34
] |
[{"tactic": "dsimp [bicategoricalComp]", "annotated_tactic": ["dsimp [<a>bicategoricalComp</a>]", [{"full_name": "Mathlib.Tactic.BicategoryCoherence.bicategoricalComp", "def_path": "Mathlib/Tactic/CategoryTheory/BicategoryCoherence.lean", "def_pos": [210, 5], "def_end_pos": [210, 22]}]], "state_before": "B : Type u\ninst\u271d : Bicategory B\na b c d e : B\nf g h : a \u27f6 b\n\u03b7 : f \u27f6 g\n\u03b8 : g \u27f6 h\n\u22a2 \u03b7 \u2297\u226b \u03b8 = \u03b7 \u226b \u03b8", "state_after": "B : Type u\ninst\u271d : Bicategory B\na b c d e : B\nf g h : a \u27f6 b\n\u03b7 : f \u27f6 g\n\u03b8 : g \u27f6 h\n\u22a2 \u03b7 \u226b \ud835\udfd9 g \u226b \u03b8 = \u03b7 \u226b \u03b8"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "B : Type u\ninst\u271d : Bicategory B\na b c d e : B\nf g h : a \u27f6 b\n\u03b7 : f \u27f6 g\n\u03b8 : g \u27f6 h\n\u22a2 \u03b7 \u226b \ud835\udfd9 g \u226b \u03b8 = \u03b7 \u226b \u03b8", "state_after": "no goals"}]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Algebra/IndicatorFunction.lean
|
Set.mulIndicator_union_of_not_mem_inter
|
[
371,
1
] |
[
373,
86
] |
[{"tactic": "rw [\u2190 mulIndicator_union_mul_inter_apply f s t, mulIndicator_of_not_mem h, mul_one]", "annotated_tactic": ["rw [\u2190 <a>mulIndicator_union_mul_inter_apply</a> f s t, <a>mulIndicator_of_not_mem</a> h, <a>mul_one</a>]", [{"full_name": "Set.mulIndicator_union_mul_inter_apply", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [357, 9], "def_end_pos": [357, 43]}, {"full_name": "Set.mulIndicator_of_not_mem", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [74, 9], "def_end_pos": [74, 32]}, {"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 u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d : MulOneClass M\ns t : Set \u03b1\nf\u271d g : \u03b1 \u2192 M\na : \u03b1\nh : \u00aca \u2208 s \u2229 t\nf : \u03b1 \u2192 M\n\u22a2 mulIndicator (s \u222a t) f a = mulIndicator s f a * mulIndicator t f a", "state_after": "no goals"}]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Algebra/Order/Group/Defs.lean
|
inv_le_inv'
|
[
1239,
1
] |
[
1241,
29
] |
[]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Data/Complex/Basic.lean
|
Complex.neg_im
|
[
249,
1
] |
[
250,
6
] |
[]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Combinatorics/SimpleGraph/Regularity/Bound.lean
|
SzemerediRegularity.one_le_m_coe
|
[
120,
1
] |
[
121,
33
] |
[]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Algebra/Module/Torsion.lean
|
Submodule.torsionBySet_isTorsionBySet
|
[
356,
1
] |
[
357,
52
] |
[]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Topology/Sets/Opens.lean
|
TopologicalSpace.Opens.coe_finset_sup
|
[
212,
1
] |
[
213,
79
] |
[]
|
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