file_path
stringlengths 11
79
| full_name
stringlengths 2
100
| traced_tactics
list | end
list | commit
stringclasses 4
values | url
stringclasses 4
values | start
list |
|---|---|---|---|---|---|---|
Mathlib/Data/Set/Pointwise/Basic.lean
|
Set.Nonempty.one_mem_div
|
[] |
[
1182,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1180,
1
] |
Mathlib/LinearAlgebra/Matrix/Determinant.lean
|
Matrix.det_permutation
|
[
{
"state_after": "no goals",
"state_before": "m : Type ?u.1185437\nn : Type u_1\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\ninst✝² : DecidableEq m\ninst✝¹ : Fintype m\nR : Type v\ninst✝ : CommRing R\nσ : Perm n\n⊢ det (PEquiv.toMatrix (toPEquiv σ)) = ↑↑(↑sign σ)",
"tactic": "rw [← Matrix.mul_one (σ.toPEquiv.toMatrix : Matrix n n R), PEquiv.toPEquiv_mul_matrix,\n det_permute, det_one, mul_one]"
}
] |
[
276,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
273,
1
] |
Std/Data/List/Lemmas.lean
|
List.iota_eq_reverse_range'
|
[
{
"state_after": "no goals",
"state_before": "n : Nat\n⊢ iota (n + 1) = reverse (range' 1 (n + 1))",
"tactic": "simp [iota, range'_concat, iota_eq_reverse_range' n, reverse_append, Nat.add_comm]"
}
] |
[
1934,
99
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
1932,
1
] |
Mathlib/Data/Finset/Basic.lean
|
Finset.mem_coe
|
[] |
[
212,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
211,
1
] |
Mathlib/Computability/Reduce.lean
|
ManyOneDegree.ind_on
|
[] |
[
380,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
378,
11
] |
Mathlib/Order/Partition/Equipartition.lean
|
Finpartition.bot_isEquipartition
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : DecidableEq α\ns t : Finset α\nP : Finpartition s\n⊢ ∀ (a : Finset α), a ∈ ↑⊥.parts → Finset.card a = 1 ∨ Finset.card a = 1 + 1",
"tactic": "simp"
}
] |
[
74,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
73,
1
] |
Mathlib/LinearAlgebra/AffineSpace/Independent.lean
|
affineIndependent_iff_indicator_eq_of_affineCombination_eq
|
[
{
"state_after": "case mp\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\n⊢ AffineIndependent k p →\n ∀ (s1 s2 : Finset ι) (w1 w2 : ι → k),\n ∑ i in s1, w1 i = 1 →\n ∑ i in s2, w2 i = 1 →\n ↑(Finset.affineCombination k s1 p) w1 = ↑(Finset.affineCombination k s2 p) w2 →\n Set.indicator (↑s1) w1 = Set.indicator (↑s2) w2\n\ncase mpr\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\n⊢ (∀ (s1 s2 : Finset ι) (w1 w2 : ι → k),\n ∑ i in s1, w1 i = 1 →\n ∑ i in s2, w2 i = 1 →\n ↑(Finset.affineCombination k s1 p) w1 = ↑(Finset.affineCombination k s2 p) w2 →\n Set.indicator (↑s1) w1 = Set.indicator (↑s2) w2) →\n AffineIndependent k p",
"state_before": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\n⊢ AffineIndependent k p ↔\n ∀ (s1 s2 : Finset ι) (w1 w2 : ι → k),\n ∑ i in s1, w1 i = 1 →\n ∑ i in s2, w2 i = 1 →\n ↑(Finset.affineCombination k s1 p) w1 = ↑(Finset.affineCombination k s2 p) w2 →\n Set.indicator (↑s1) w1 = Set.indicator (↑s2) w2",
"tactic": "constructor"
},
{
"state_after": "case mp\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\nha : AffineIndependent k p\ns1 s2 : Finset ι\nw1 w2 : ι → k\nhw1 : ∑ i in s1, w1 i = 1\nhw2 : ∑ i in s2, w2 i = 1\nheq : ↑(Finset.affineCombination k s1 p) w1 = ↑(Finset.affineCombination k s2 p) w2\n⊢ Set.indicator (↑s1) w1 = Set.indicator (↑s2) w2",
"state_before": "case mp\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\n⊢ AffineIndependent k p →\n ∀ (s1 s2 : Finset ι) (w1 w2 : ι → k),\n ∑ i in s1, w1 i = 1 →\n ∑ i in s2, w2 i = 1 →\n ↑(Finset.affineCombination k s1 p) w1 = ↑(Finset.affineCombination k s2 p) w2 →\n Set.indicator (↑s1) w1 = Set.indicator (↑s2) w2",
"tactic": "intro ha s1 s2 w1 w2 hw1 hw2 heq"
},
{
"state_after": "case mp.h\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\nha : AffineIndependent k p\ns1 s2 : Finset ι\nw1 w2 : ι → k\nhw1 : ∑ i in s1, w1 i = 1\nhw2 : ∑ i in s2, w2 i = 1\nheq : ↑(Finset.affineCombination k s1 p) w1 = ↑(Finset.affineCombination k s2 p) w2\ni : ι\n⊢ Set.indicator (↑s1) w1 i = Set.indicator (↑s2) w2 i",
"state_before": "case mp\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\nha : AffineIndependent k p\ns1 s2 : Finset ι\nw1 w2 : ι → k\nhw1 : ∑ i in s1, w1 i = 1\nhw2 : ∑ i in s2, w2 i = 1\nheq : ↑(Finset.affineCombination k s1 p) w1 = ↑(Finset.affineCombination k s2 p) w2\n⊢ Set.indicator (↑s1) w1 = Set.indicator (↑s2) w2",
"tactic": "ext i"
},
{
"state_after": "case pos\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\nha : AffineIndependent k p\ns1 s2 : Finset ι\nw1 w2 : ι → k\nhw1 : ∑ i in s1, w1 i = 1\nhw2 : ∑ i in s2, w2 i = 1\nheq : ↑(Finset.affineCombination k s1 p) w1 = ↑(Finset.affineCombination k s2 p) w2\ni : ι\nhi : i ∈ s1 ∪ s2\n⊢ Set.indicator (↑s1) w1 i = Set.indicator (↑s2) w2 i\n\ncase neg\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\nha : AffineIndependent k p\ns1 s2 : Finset ι\nw1 w2 : ι → k\nhw1 : ∑ i in s1, w1 i = 1\nhw2 : ∑ i in s2, w2 i = 1\nheq : ↑(Finset.affineCombination k s1 p) w1 = ↑(Finset.affineCombination k s2 p) w2\ni : ι\nhi : ¬i ∈ s1 ∪ s2\n⊢ Set.indicator (↑s1) w1 i = Set.indicator (↑s2) w2 i",
"state_before": "case mp.h\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\nha : AffineIndependent k p\ns1 s2 : Finset ι\nw1 w2 : ι → k\nhw1 : ∑ i in s1, w1 i = 1\nhw2 : ∑ i in s2, w2 i = 1\nheq : ↑(Finset.affineCombination k s1 p) w1 = ↑(Finset.affineCombination k s2 p) w2\ni : ι\n⊢ Set.indicator (↑s1) w1 i = Set.indicator (↑s2) w2 i",
"tactic": "by_cases hi : i ∈ s1 ∪ s2"
},
{
"state_after": "case pos\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\nha : AffineIndependent k p\ns1 s2 : Finset ι\nw1 w2 : ι → k\nhw1 : ∑ i in s1, w1 i = 1\nhw2 : ∑ i in s2, w2 i = 1\nheq : ↑(Finset.affineCombination k s1 p) w1 = ↑(Finset.affineCombination k s2 p) w2\ni : ι\nhi : i ∈ s1 ∪ s2\n⊢ Set.indicator (↑s1) w1 i - Set.indicator (↑s2) w2 i = 0",
"state_before": "case pos\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\nha : AffineIndependent k p\ns1 s2 : Finset ι\nw1 w2 : ι → k\nhw1 : ∑ i in s1, w1 i = 1\nhw2 : ∑ i in s2, w2 i = 1\nheq : ↑(Finset.affineCombination k s1 p) w1 = ↑(Finset.affineCombination k s2 p) w2\ni : ι\nhi : i ∈ s1 ∪ s2\n⊢ Set.indicator (↑s1) w1 i = Set.indicator (↑s2) w2 i",
"tactic": "rw [← sub_eq_zero]"
},
{
"state_after": "case pos\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\nha : AffineIndependent k p\ns1 s2 : Finset ι\nw1 w2 : ι → k\nhw1 : ∑ i in s1 ∪ s2, Set.indicator (↑s1) (fun i => w1 i) i = 1\nhw2 : ∑ i in s2, w2 i = 1\nheq : ↑(Finset.affineCombination k s1 p) w1 = ↑(Finset.affineCombination k s2 p) w2\ni : ι\nhi : i ∈ s1 ∪ s2\n⊢ Set.indicator (↑s1) w1 i - Set.indicator (↑s2) w2 i = 0",
"state_before": "case pos\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\nha : AffineIndependent k p\ns1 s2 : Finset ι\nw1 w2 : ι → k\nhw1 : ∑ i in s1, w1 i = 1\nhw2 : ∑ i in s2, w2 i = 1\nheq : ↑(Finset.affineCombination k s1 p) w1 = ↑(Finset.affineCombination k s2 p) w2\ni : ι\nhi : i ∈ s1 ∪ s2\n⊢ Set.indicator (↑s1) w1 i - Set.indicator (↑s2) w2 i = 0",
"tactic": "rw [Set.sum_indicator_subset _ (Finset.subset_union_left s1 s2)] at hw1"
},
{
"state_after": "case pos\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\nha : AffineIndependent k p\ns1 s2 : Finset ι\nw1 w2 : ι → k\nhw1 : ∑ i in s1 ∪ s2, Set.indicator (↑s1) (fun i => w1 i) i = 1\nhw2 : ∑ i in s1 ∪ s2, Set.indicator (↑s2) (fun i => w2 i) i = 1\nheq : ↑(Finset.affineCombination k s1 p) w1 = ↑(Finset.affineCombination k s2 p) w2\ni : ι\nhi : i ∈ s1 ∪ s2\n⊢ Set.indicator (↑s1) w1 i - Set.indicator (↑s2) w2 i = 0",
"state_before": "case pos\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\nha : AffineIndependent k p\ns1 s2 : Finset ι\nw1 w2 : ι → k\nhw1 : ∑ i in s1 ∪ s2, Set.indicator (↑s1) (fun i => w1 i) i = 1\nhw2 : ∑ i in s2, w2 i = 1\nheq : ↑(Finset.affineCombination k s1 p) w1 = ↑(Finset.affineCombination k s2 p) w2\ni : ι\nhi : i ∈ s1 ∪ s2\n⊢ Set.indicator (↑s1) w1 i - Set.indicator (↑s2) w2 i = 0",
"tactic": "rw [Set.sum_indicator_subset _ (Finset.subset_union_right s1 s2)] at hw2"
},
{
"state_after": "case pos\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\nha : AffineIndependent k p\ns1 s2 : Finset ι\nw1 w2 : ι → k\nhw1 : ∑ i in s1 ∪ s2, Set.indicator (↑s1) (fun i => w1 i) i = 1\nhw2 : ∑ i in s1 ∪ s2, Set.indicator (↑s2) (fun i => w2 i) i = 1\nheq : ↑(Finset.affineCombination k s1 p) w1 = ↑(Finset.affineCombination k s2 p) w2\ni : ι\nhi : i ∈ s1 ∪ s2\nhws : ∑ i in s1 ∪ s2, (Set.indicator (↑s1) w1 - Set.indicator (↑s2) w2) i = 0\n⊢ Set.indicator (↑s1) w1 i - Set.indicator (↑s2) w2 i = 0",
"state_before": "case pos\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\nha : AffineIndependent k p\ns1 s2 : Finset ι\nw1 w2 : ι → k\nhw1 : ∑ i in s1 ∪ s2, Set.indicator (↑s1) (fun i => w1 i) i = 1\nhw2 : ∑ i in s1 ∪ s2, Set.indicator (↑s2) (fun i => w2 i) i = 1\nheq : ↑(Finset.affineCombination k s1 p) w1 = ↑(Finset.affineCombination k s2 p) w2\ni : ι\nhi : i ∈ s1 ∪ s2\n⊢ Set.indicator (↑s1) w1 i - Set.indicator (↑s2) w2 i = 0",
"tactic": "have hws : (∑ i in s1 ∪ s2, (Set.indicator (↑s1) w1 - Set.indicator (↑s2) w2) i) = 0 := by\n simp [hw1, hw2]"
},
{
"state_after": "case pos\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\nha : AffineIndependent k p\ns1 s2 : Finset ι\nw1 w2 : ι → k\nhw1 : ∑ i in s1 ∪ s2, Set.indicator (↑s1) (fun i => w1 i) i = 1\nhw2 : ∑ i in s1 ∪ s2, Set.indicator (↑s2) (fun i => w2 i) i = 1\nheq : ↑(Finset.weightedVSub (s1 ∪ s2) p) (Set.indicator (↑s1) w1 - Set.indicator (↑s2) w2) = 0\ni : ι\nhi : i ∈ s1 ∪ s2\nhws : ∑ i in s1 ∪ s2, (Set.indicator (↑s1) w1 - Set.indicator (↑s2) w2) i = 0\n⊢ Set.indicator (↑s1) w1 i - Set.indicator (↑s2) w2 i = 0",
"state_before": "case pos\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\nha : AffineIndependent k p\ns1 s2 : Finset ι\nw1 w2 : ι → k\nhw1 : ∑ i in s1 ∪ s2, Set.indicator (↑s1) (fun i => w1 i) i = 1\nhw2 : ∑ i in s1 ∪ s2, Set.indicator (↑s2) (fun i => w2 i) i = 1\nheq : ↑(Finset.affineCombination k s1 p) w1 = ↑(Finset.affineCombination k s2 p) w2\ni : ι\nhi : i ∈ s1 ∪ s2\nhws : ∑ i in s1 ∪ s2, (Set.indicator (↑s1) w1 - Set.indicator (↑s2) w2) i = 0\n⊢ Set.indicator (↑s1) w1 i - Set.indicator (↑s2) w2 i = 0",
"tactic": "rw [Finset.affineCombination_indicator_subset _ _ (Finset.subset_union_left s1 s2),\n Finset.affineCombination_indicator_subset _ _ (Finset.subset_union_right s1 s2),\n ← @vsub_eq_zero_iff_eq V, Finset.affineCombination_vsub] at heq"
},
{
"state_after": "no goals",
"state_before": "case pos\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\nha : AffineIndependent k p\ns1 s2 : Finset ι\nw1 w2 : ι → k\nhw1 : ∑ i in s1 ∪ s2, Set.indicator (↑s1) (fun i => w1 i) i = 1\nhw2 : ∑ i in s1 ∪ s2, Set.indicator (↑s2) (fun i => w2 i) i = 1\nheq : ↑(Finset.weightedVSub (s1 ∪ s2) p) (Set.indicator (↑s1) w1 - Set.indicator (↑s2) w2) = 0\ni : ι\nhi : i ∈ s1 ∪ s2\nhws : ∑ i in s1 ∪ s2, (Set.indicator (↑s1) w1 - Set.indicator (↑s2) w2) i = 0\n⊢ Set.indicator (↑s1) w1 i - Set.indicator (↑s2) w2 i = 0",
"tactic": "exact ha (s1 ∪ s2) (Set.indicator (↑s1) w1 - Set.indicator (↑s2) w2) hws heq i hi"
},
{
"state_after": "no goals",
"state_before": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\nha : AffineIndependent k p\ns1 s2 : Finset ι\nw1 w2 : ι → k\nhw1 : ∑ i in s1 ∪ s2, Set.indicator (↑s1) (fun i => w1 i) i = 1\nhw2 : ∑ i in s1 ∪ s2, Set.indicator (↑s2) (fun i => w2 i) i = 1\nheq : ↑(Finset.affineCombination k s1 p) w1 = ↑(Finset.affineCombination k s2 p) w2\ni : ι\nhi : i ∈ s1 ∪ s2\n⊢ ∑ i in s1 ∪ s2, (Set.indicator (↑s1) w1 - Set.indicator (↑s2) w2) i = 0",
"tactic": "simp [hw1, hw2]"
},
{
"state_after": "case neg\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\nha : AffineIndependent k p\ns1 s2 : Finset ι\nw1 w2 : ι → k\nhw1 : ∑ i in s1, w1 i = 1\nhw2 : ∑ i in s2, w2 i = 1\nheq : ↑(Finset.affineCombination k s1 p) w1 = ↑(Finset.affineCombination k s2 p) w2\ni : ι\nhi : ¬i ∈ ↑s1 ∪ ↑s2\n⊢ Set.indicator (↑s1) w1 i = Set.indicator (↑s2) w2 i",
"state_before": "case neg\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\nha : AffineIndependent k p\ns1 s2 : Finset ι\nw1 w2 : ι → k\nhw1 : ∑ i in s1, w1 i = 1\nhw2 : ∑ i in s2, w2 i = 1\nheq : ↑(Finset.affineCombination k s1 p) w1 = ↑(Finset.affineCombination k s2 p) w2\ni : ι\nhi : ¬i ∈ s1 ∪ s2\n⊢ Set.indicator (↑s1) w1 i = Set.indicator (↑s2) w2 i",
"tactic": "rw [← Finset.mem_coe, Finset.coe_union] at hi"
},
{
"state_after": "case neg\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\nha : AffineIndependent k p\ns1 s2 : Finset ι\nw1 w2 : ι → k\nhw1 : ∑ i in s1, w1 i = 1\nhw2 : ∑ i in s2, w2 i = 1\nheq : ↑(Finset.affineCombination k s1 p) w1 = ↑(Finset.affineCombination k s2 p) w2\ni : ι\nhi : ¬i ∈ ↑s1 ∪ ↑s2\nh₁ : Set.indicator (↑s1) w1 i = 0\n⊢ Set.indicator (↑s1) w1 i = Set.indicator (↑s2) w2 i",
"state_before": "case neg\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\nha : AffineIndependent k p\ns1 s2 : Finset ι\nw1 w2 : ι → k\nhw1 : ∑ i in s1, w1 i = 1\nhw2 : ∑ i in s2, w2 i = 1\nheq : ↑(Finset.affineCombination k s1 p) w1 = ↑(Finset.affineCombination k s2 p) w2\ni : ι\nhi : ¬i ∈ ↑s1 ∪ ↑s2\n⊢ Set.indicator (↑s1) w1 i = Set.indicator (↑s2) w2 i",
"tactic": "have h₁ : Set.indicator (↑s1) w1 i = 0 := by\n simp only [Set.indicator, Finset.mem_coe, ite_eq_right_iff]\n intro h\n by_contra\n exact (mt (@Set.mem_union_left _ i ↑s1 ↑s2) hi) h"
},
{
"state_after": "case neg\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\nha : AffineIndependent k p\ns1 s2 : Finset ι\nw1 w2 : ι → k\nhw1 : ∑ i in s1, w1 i = 1\nhw2 : ∑ i in s2, w2 i = 1\nheq : ↑(Finset.affineCombination k s1 p) w1 = ↑(Finset.affineCombination k s2 p) w2\ni : ι\nhi : ¬i ∈ ↑s1 ∪ ↑s2\nh₁ : Set.indicator (↑s1) w1 i = 0\nh₂ : Set.indicator (↑s2) w2 i = 0\n⊢ Set.indicator (↑s1) w1 i = Set.indicator (↑s2) w2 i",
"state_before": "case neg\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\nha : AffineIndependent k p\ns1 s2 : Finset ι\nw1 w2 : ι → k\nhw1 : ∑ i in s1, w1 i = 1\nhw2 : ∑ i in s2, w2 i = 1\nheq : ↑(Finset.affineCombination k s1 p) w1 = ↑(Finset.affineCombination k s2 p) w2\ni : ι\nhi : ¬i ∈ ↑s1 ∪ ↑s2\nh₁ : Set.indicator (↑s1) w1 i = 0\n⊢ Set.indicator (↑s1) w1 i = Set.indicator (↑s2) w2 i",
"tactic": "have h₂ : Set.indicator (↑s2) w2 i = 0 := by\n simp only [Set.indicator, Finset.mem_coe, ite_eq_right_iff]\n intro h\n by_contra\n exact ( mt (@Set.mem_union_right _ i ↑s2 ↑s1) hi) h"
},
{
"state_after": "no goals",
"state_before": "case neg\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\nha : AffineIndependent k p\ns1 s2 : Finset ι\nw1 w2 : ι → k\nhw1 : ∑ i in s1, w1 i = 1\nhw2 : ∑ i in s2, w2 i = 1\nheq : ↑(Finset.affineCombination k s1 p) w1 = ↑(Finset.affineCombination k s2 p) w2\ni : ι\nhi : ¬i ∈ ↑s1 ∪ ↑s2\nh₁ : Set.indicator (↑s1) w1 i = 0\nh₂ : Set.indicator (↑s2) w2 i = 0\n⊢ Set.indicator (↑s1) w1 i = Set.indicator (↑s2) w2 i",
"tactic": "simp [h₁, h₂]"
},
{
"state_after": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\nha : AffineIndependent k p\ns1 s2 : Finset ι\nw1 w2 : ι → k\nhw1 : ∑ i in s1, w1 i = 1\nhw2 : ∑ i in s2, w2 i = 1\nheq : ↑(Finset.affineCombination k s1 p) w1 = ↑(Finset.affineCombination k s2 p) w2\ni : ι\nhi : ¬i ∈ ↑s1 ∪ ↑s2\n⊢ i ∈ s1 → w1 i = 0",
"state_before": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\nha : AffineIndependent k p\ns1 s2 : Finset ι\nw1 w2 : ι → k\nhw1 : ∑ i in s1, w1 i = 1\nhw2 : ∑ i in s2, w2 i = 1\nheq : ↑(Finset.affineCombination k s1 p) w1 = ↑(Finset.affineCombination k s2 p) w2\ni : ι\nhi : ¬i ∈ ↑s1 ∪ ↑s2\n⊢ Set.indicator (↑s1) w1 i = 0",
"tactic": "simp only [Set.indicator, Finset.mem_coe, ite_eq_right_iff]"
},
{
"state_after": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\nha : AffineIndependent k p\ns1 s2 : Finset ι\nw1 w2 : ι → k\nhw1 : ∑ i in s1, w1 i = 1\nhw2 : ∑ i in s2, w2 i = 1\nheq : ↑(Finset.affineCombination k s1 p) w1 = ↑(Finset.affineCombination k s2 p) w2\ni : ι\nhi : ¬i ∈ ↑s1 ∪ ↑s2\nh : i ∈ s1\n⊢ w1 i = 0",
"state_before": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\nha : AffineIndependent k p\ns1 s2 : Finset ι\nw1 w2 : ι → k\nhw1 : ∑ i in s1, w1 i = 1\nhw2 : ∑ i in s2, w2 i = 1\nheq : ↑(Finset.affineCombination k s1 p) w1 = ↑(Finset.affineCombination k s2 p) w2\ni : ι\nhi : ¬i ∈ ↑s1 ∪ ↑s2\n⊢ i ∈ s1 → w1 i = 0",
"tactic": "intro h"
},
{
"state_after": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\nha : AffineIndependent k p\ns1 s2 : Finset ι\nw1 w2 : ι → k\nhw1 : ∑ i in s1, w1 i = 1\nhw2 : ∑ i in s2, w2 i = 1\nheq : ↑(Finset.affineCombination k s1 p) w1 = ↑(Finset.affineCombination k s2 p) w2\ni : ι\nhi : ¬i ∈ ↑s1 ∪ ↑s2\nh : i ∈ s1\nx✝ : ¬w1 i = 0\n⊢ False",
"state_before": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\nha : AffineIndependent k p\ns1 s2 : Finset ι\nw1 w2 : ι → k\nhw1 : ∑ i in s1, w1 i = 1\nhw2 : ∑ i in s2, w2 i = 1\nheq : ↑(Finset.affineCombination k s1 p) w1 = ↑(Finset.affineCombination k s2 p) w2\ni : ι\nhi : ¬i ∈ ↑s1 ∪ ↑s2\nh : i ∈ s1\n⊢ w1 i = 0",
"tactic": "by_contra"
},
{
"state_after": "no goals",
"state_before": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\nha : AffineIndependent k p\ns1 s2 : Finset ι\nw1 w2 : ι → k\nhw1 : ∑ i in s1, w1 i = 1\nhw2 : ∑ i in s2, w2 i = 1\nheq : ↑(Finset.affineCombination k s1 p) w1 = ↑(Finset.affineCombination k s2 p) w2\ni : ι\nhi : ¬i ∈ ↑s1 ∪ ↑s2\nh : i ∈ s1\nx✝ : ¬w1 i = 0\n⊢ False",
"tactic": "exact (mt (@Set.mem_union_left _ i ↑s1 ↑s2) hi) h"
},
{
"state_after": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\nha : AffineIndependent k p\ns1 s2 : Finset ι\nw1 w2 : ι → k\nhw1 : ∑ i in s1, w1 i = 1\nhw2 : ∑ i in s2, w2 i = 1\nheq : ↑(Finset.affineCombination k s1 p) w1 = ↑(Finset.affineCombination k s2 p) w2\ni : ι\nhi : ¬i ∈ ↑s1 ∪ ↑s2\nh₁ : Set.indicator (↑s1) w1 i = 0\n⊢ i ∈ s2 → w2 i = 0",
"state_before": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\nha : AffineIndependent k p\ns1 s2 : Finset ι\nw1 w2 : ι → k\nhw1 : ∑ i in s1, w1 i = 1\nhw2 : ∑ i in s2, w2 i = 1\nheq : ↑(Finset.affineCombination k s1 p) w1 = ↑(Finset.affineCombination k s2 p) w2\ni : ι\nhi : ¬i ∈ ↑s1 ∪ ↑s2\nh₁ : Set.indicator (↑s1) w1 i = 0\n⊢ Set.indicator (↑s2) w2 i = 0",
"tactic": "simp only [Set.indicator, Finset.mem_coe, ite_eq_right_iff]"
},
{
"state_after": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\nha : AffineIndependent k p\ns1 s2 : Finset ι\nw1 w2 : ι → k\nhw1 : ∑ i in s1, w1 i = 1\nhw2 : ∑ i in s2, w2 i = 1\nheq : ↑(Finset.affineCombination k s1 p) w1 = ↑(Finset.affineCombination k s2 p) w2\ni : ι\nhi : ¬i ∈ ↑s1 ∪ ↑s2\nh₁ : Set.indicator (↑s1) w1 i = 0\nh : i ∈ s2\n⊢ w2 i = 0",
"state_before": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\nha : AffineIndependent k p\ns1 s2 : Finset ι\nw1 w2 : ι → k\nhw1 : ∑ i in s1, w1 i = 1\nhw2 : ∑ i in s2, w2 i = 1\nheq : ↑(Finset.affineCombination k s1 p) w1 = ↑(Finset.affineCombination k s2 p) w2\ni : ι\nhi : ¬i ∈ ↑s1 ∪ ↑s2\nh₁ : Set.indicator (↑s1) w1 i = 0\n⊢ i ∈ s2 → w2 i = 0",
"tactic": "intro h"
},
{
"state_after": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\nha : AffineIndependent k p\ns1 s2 : Finset ι\nw1 w2 : ι → k\nhw1 : ∑ i in s1, w1 i = 1\nhw2 : ∑ i in s2, w2 i = 1\nheq : ↑(Finset.affineCombination k s1 p) w1 = ↑(Finset.affineCombination k s2 p) w2\ni : ι\nhi : ¬i ∈ ↑s1 ∪ ↑s2\nh₁ : Set.indicator (↑s1) w1 i = 0\nh : i ∈ s2\nx✝ : ¬w2 i = 0\n⊢ False",
"state_before": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\nha : AffineIndependent k p\ns1 s2 : Finset ι\nw1 w2 : ι → k\nhw1 : ∑ i in s1, w1 i = 1\nhw2 : ∑ i in s2, w2 i = 1\nheq : ↑(Finset.affineCombination k s1 p) w1 = ↑(Finset.affineCombination k s2 p) w2\ni : ι\nhi : ¬i ∈ ↑s1 ∪ ↑s2\nh₁ : Set.indicator (↑s1) w1 i = 0\nh : i ∈ s2\n⊢ w2 i = 0",
"tactic": "by_contra"
},
{
"state_after": "no goals",
"state_before": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\nha : AffineIndependent k p\ns1 s2 : Finset ι\nw1 w2 : ι → k\nhw1 : ∑ i in s1, w1 i = 1\nhw2 : ∑ i in s2, w2 i = 1\nheq : ↑(Finset.affineCombination k s1 p) w1 = ↑(Finset.affineCombination k s2 p) w2\ni : ι\nhi : ¬i ∈ ↑s1 ∪ ↑s2\nh₁ : Set.indicator (↑s1) w1 i = 0\nh : i ∈ s2\nx✝ : ¬w2 i = 0\n⊢ False",
"tactic": "exact ( mt (@Set.mem_union_right _ i ↑s2 ↑s1) hi) h"
},
{
"state_after": "case mpr\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\nha :\n ∀ (s1 s2 : Finset ι) (w1 w2 : ι → k),\n ∑ i in s1, w1 i = 1 →\n ∑ i in s2, w2 i = 1 →\n ↑(Finset.affineCombination k s1 p) w1 = ↑(Finset.affineCombination k s2 p) w2 →\n Set.indicator (↑s1) w1 = Set.indicator (↑s2) w2\ns : Finset ι\nw : ι → k\nhw : ∑ i in s, w i = 0\nhs : ↑(Finset.weightedVSub s p) w = 0\ni0 : ι\nhi0 : i0 ∈ s\n⊢ w i0 = 0",
"state_before": "case mpr\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\n⊢ (∀ (s1 s2 : Finset ι) (w1 w2 : ι → k),\n ∑ i in s1, w1 i = 1 →\n ∑ i in s2, w2 i = 1 →\n ↑(Finset.affineCombination k s1 p) w1 = ↑(Finset.affineCombination k s2 p) w2 →\n Set.indicator (↑s1) w1 = Set.indicator (↑s2) w2) →\n AffineIndependent k p",
"tactic": "intro ha s w hw hs i0 hi0"
},
{
"state_after": "case mpr\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\nha :\n ∀ (s1 s2 : Finset ι) (w1 w2 : ι → k),\n ∑ i in s1, w1 i = 1 →\n ∑ i in s2, w2 i = 1 →\n ↑(Finset.affineCombination k s1 p) w1 = ↑(Finset.affineCombination k s2 p) w2 →\n Set.indicator (↑s1) w1 = Set.indicator (↑s2) w2\ns : Finset ι\nw : ι → k\nhw : ∑ i in s, w i = 0\nhs : ↑(Finset.weightedVSub s p) w = 0\ni0 : ι\nhi0 : i0 ∈ s\nw1 : ι → k := update (const ι 0) i0 1\n⊢ w i0 = 0",
"state_before": "case mpr\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\nha :\n ∀ (s1 s2 : Finset ι) (w1 w2 : ι → k),\n ∑ i in s1, w1 i = 1 →\n ∑ i in s2, w2 i = 1 →\n ↑(Finset.affineCombination k s1 p) w1 = ↑(Finset.affineCombination k s2 p) w2 →\n Set.indicator (↑s1) w1 = Set.indicator (↑s2) w2\ns : Finset ι\nw : ι → k\nhw : ∑ i in s, w i = 0\nhs : ↑(Finset.weightedVSub s p) w = 0\ni0 : ι\nhi0 : i0 ∈ s\n⊢ w i0 = 0",
"tactic": "let w1 : ι → k := Function.update (Function.const ι 0) i0 1"
},
{
"state_after": "case mpr\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\nha :\n ∀ (s1 s2 : Finset ι) (w1 w2 : ι → k),\n ∑ i in s1, w1 i = 1 →\n ∑ i in s2, w2 i = 1 →\n ↑(Finset.affineCombination k s1 p) w1 = ↑(Finset.affineCombination k s2 p) w2 →\n Set.indicator (↑s1) w1 = Set.indicator (↑s2) w2\ns : Finset ι\nw : ι → k\nhw : ∑ i in s, w i = 0\nhs : ↑(Finset.weightedVSub s p) w = 0\ni0 : ι\nhi0 : i0 ∈ s\nw1 : ι → k := update (const ι 0) i0 1\nhw1 : ∑ i in s, w1 i = 1\n⊢ w i0 = 0",
"state_before": "case mpr\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\nha :\n ∀ (s1 s2 : Finset ι) (w1 w2 : ι → k),\n ∑ i in s1, w1 i = 1 →\n ∑ i in s2, w2 i = 1 →\n ↑(Finset.affineCombination k s1 p) w1 = ↑(Finset.affineCombination k s2 p) w2 →\n Set.indicator (↑s1) w1 = Set.indicator (↑s2) w2\ns : Finset ι\nw : ι → k\nhw : ∑ i in s, w i = 0\nhs : ↑(Finset.weightedVSub s p) w = 0\ni0 : ι\nhi0 : i0 ∈ s\nw1 : ι → k := update (const ι 0) i0 1\n⊢ w i0 = 0",
"tactic": "have hw1 : (∑ i in s, w1 i) = 1 := by\n rw [Finset.sum_update_of_mem hi0]\n simp only [Finset.sum_const_zero, add_zero, const_apply]"
},
{
"state_after": "case mpr\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\nha :\n ∀ (s1 s2 : Finset ι) (w1 w2 : ι → k),\n ∑ i in s1, w1 i = 1 →\n ∑ i in s2, w2 i = 1 →\n ↑(Finset.affineCombination k s1 p) w1 = ↑(Finset.affineCombination k s2 p) w2 →\n Set.indicator (↑s1) w1 = Set.indicator (↑s2) w2\ns : Finset ι\nw : ι → k\nhw : ∑ i in s, w i = 0\nhs : ↑(Finset.weightedVSub s p) w = 0\ni0 : ι\nhi0 : i0 ∈ s\nw1 : ι → k := update (const ι 0) i0 1\nhw1 : ∑ i in s, w1 i = 1\nhw1s : ↑(Finset.affineCombination k s p) w1 = p i0\n⊢ w i0 = 0",
"state_before": "case mpr\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\nha :\n ∀ (s1 s2 : Finset ι) (w1 w2 : ι → k),\n ∑ i in s1, w1 i = 1 →\n ∑ i in s2, w2 i = 1 →\n ↑(Finset.affineCombination k s1 p) w1 = ↑(Finset.affineCombination k s2 p) w2 →\n Set.indicator (↑s1) w1 = Set.indicator (↑s2) w2\ns : Finset ι\nw : ι → k\nhw : ∑ i in s, w i = 0\nhs : ↑(Finset.weightedVSub s p) w = 0\ni0 : ι\nhi0 : i0 ∈ s\nw1 : ι → k := update (const ι 0) i0 1\nhw1 : ∑ i in s, w1 i = 1\n⊢ w i0 = 0",
"tactic": "have hw1s : s.affineCombination k p w1 = p i0 :=\n s.affineCombination_of_eq_one_of_eq_zero w1 p hi0 (Function.update_same _ _ _)\n fun _ _ hne => Function.update_noteq hne _ _"
},
{
"state_after": "case mpr\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\nha :\n ∀ (s1 s2 : Finset ι) (w1 w2 : ι → k),\n ∑ i in s1, w1 i = 1 →\n ∑ i in s2, w2 i = 1 →\n ↑(Finset.affineCombination k s1 p) w1 = ↑(Finset.affineCombination k s2 p) w2 →\n Set.indicator (↑s1) w1 = Set.indicator (↑s2) w2\ns : Finset ι\nw : ι → k\nhw : ∑ i in s, w i = 0\nhs : ↑(Finset.weightedVSub s p) w = 0\ni0 : ι\nhi0 : i0 ∈ s\nw1 : ι → k := update (const ι 0) i0 1\nhw1 : ∑ i in s, w1 i = 1\nhw1s : ↑(Finset.affineCombination k s p) w1 = p i0\nw2 : ι → k := w + w1\n⊢ w i0 = 0",
"state_before": "case mpr\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\nha :\n ∀ (s1 s2 : Finset ι) (w1 w2 : ι → k),\n ∑ i in s1, w1 i = 1 →\n ∑ i in s2, w2 i = 1 →\n ↑(Finset.affineCombination k s1 p) w1 = ↑(Finset.affineCombination k s2 p) w2 →\n Set.indicator (↑s1) w1 = Set.indicator (↑s2) w2\ns : Finset ι\nw : ι → k\nhw : ∑ i in s, w i = 0\nhs : ↑(Finset.weightedVSub s p) w = 0\ni0 : ι\nhi0 : i0 ∈ s\nw1 : ι → k := update (const ι 0) i0 1\nhw1 : ∑ i in s, w1 i = 1\nhw1s : ↑(Finset.affineCombination k s p) w1 = p i0\n⊢ w i0 = 0",
"tactic": "let w2 := w + w1"
},
{
"state_after": "case mpr\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\nha :\n ∀ (s1 s2 : Finset ι) (w1 w2 : ι → k),\n ∑ i in s1, w1 i = 1 →\n ∑ i in s2, w2 i = 1 →\n ↑(Finset.affineCombination k s1 p) w1 = ↑(Finset.affineCombination k s2 p) w2 →\n Set.indicator (↑s1) w1 = Set.indicator (↑s2) w2\ns : Finset ι\nw : ι → k\nhw : ∑ i in s, w i = 0\nhs : ↑(Finset.weightedVSub s p) w = 0\ni0 : ι\nhi0 : i0 ∈ s\nw1 : ι → k := update (const ι 0) i0 1\nhw1 : ∑ i in s, w1 i = 1\nhw1s : ↑(Finset.affineCombination k s p) w1 = p i0\nw2 : ι → k := w + w1\nhw2 : ∑ i in s, w2 i = 1\n⊢ w i0 = 0",
"state_before": "case mpr\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\nha :\n ∀ (s1 s2 : Finset ι) (w1 w2 : ι → k),\n ∑ i in s1, w1 i = 1 →\n ∑ i in s2, w2 i = 1 →\n ↑(Finset.affineCombination k s1 p) w1 = ↑(Finset.affineCombination k s2 p) w2 →\n Set.indicator (↑s1) w1 = Set.indicator (↑s2) w2\ns : Finset ι\nw : ι → k\nhw : ∑ i in s, w i = 0\nhs : ↑(Finset.weightedVSub s p) w = 0\ni0 : ι\nhi0 : i0 ∈ s\nw1 : ι → k := update (const ι 0) i0 1\nhw1 : ∑ i in s, w1 i = 1\nhw1s : ↑(Finset.affineCombination k s p) w1 = p i0\nw2 : ι → k := w + w1\n⊢ w i0 = 0",
"tactic": "have hw2 : (∑ i in s, w2 i) = 1 := by\n simp_all only [Pi.add_apply, Finset.sum_add_distrib, zero_add]"
},
{
"state_after": "case mpr\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\nha :\n ∀ (s1 s2 : Finset ι) (w1 w2 : ι → k),\n ∑ i in s1, w1 i = 1 →\n ∑ i in s2, w2 i = 1 →\n ↑(Finset.affineCombination k s1 p) w1 = ↑(Finset.affineCombination k s2 p) w2 →\n Set.indicator (↑s1) w1 = Set.indicator (↑s2) w2\ns : Finset ι\nw : ι → k\nhw : ∑ i in s, w i = 0\nhs : ↑(Finset.weightedVSub s p) w = 0\ni0 : ι\nhi0 : i0 ∈ s\nw1 : ι → k := update (const ι 0) i0 1\nhw1 : ∑ i in s, w1 i = 1\nhw1s : ↑(Finset.affineCombination k s p) w1 = p i0\nw2 : ι → k := w + w1\nhw2 : ∑ i in s, w2 i = 1\nhw2s : ↑(Finset.affineCombination k s p) w2 = p i0\n⊢ w i0 = 0",
"state_before": "case mpr\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\nha :\n ∀ (s1 s2 : Finset ι) (w1 w2 : ι → k),\n ∑ i in s1, w1 i = 1 →\n ∑ i in s2, w2 i = 1 →\n ↑(Finset.affineCombination k s1 p) w1 = ↑(Finset.affineCombination k s2 p) w2 →\n Set.indicator (↑s1) w1 = Set.indicator (↑s2) w2\ns : Finset ι\nw : ι → k\nhw : ∑ i in s, w i = 0\nhs : ↑(Finset.weightedVSub s p) w = 0\ni0 : ι\nhi0 : i0 ∈ s\nw1 : ι → k := update (const ι 0) i0 1\nhw1 : ∑ i in s, w1 i = 1\nhw1s : ↑(Finset.affineCombination k s p) w1 = p i0\nw2 : ι → k := w + w1\nhw2 : ∑ i in s, w2 i = 1\n⊢ w i0 = 0",
"tactic": "have hw2s : s.affineCombination k p w2 = p i0 := by\n simp_all only [← Finset.weightedVSub_vadd_affineCombination, zero_vadd]"
},
{
"state_after": "case mpr\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\ns : Finset ι\nw : ι → k\nhw : ∑ i in s, w i = 0\nhs : ↑(Finset.weightedVSub s p) w = 0\ni0 : ι\nhi0 : i0 ∈ s\nw1 : ι → k := update (const ι 0) i0 1\nhw1 : ∑ i in s, w1 i = 1\nhw1s : ↑(Finset.affineCombination k s p) w1 = p i0\nw2 : ι → k := w + w1\nhw2 : ∑ i in s, w2 i = 1\nhw2s : ↑(Finset.affineCombination k s p) w2 = p i0\nha : Set.indicator (↑s) w2 = Set.indicator (↑s) w1\n⊢ w i0 = 0",
"state_before": "case mpr\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\nha :\n ∀ (s1 s2 : Finset ι) (w1 w2 : ι → k),\n ∑ i in s1, w1 i = 1 →\n ∑ i in s2, w2 i = 1 →\n ↑(Finset.affineCombination k s1 p) w1 = ↑(Finset.affineCombination k s2 p) w2 →\n Set.indicator (↑s1) w1 = Set.indicator (↑s2) w2\ns : Finset ι\nw : ι → k\nhw : ∑ i in s, w i = 0\nhs : ↑(Finset.weightedVSub s p) w = 0\ni0 : ι\nhi0 : i0 ∈ s\nw1 : ι → k := update (const ι 0) i0 1\nhw1 : ∑ i in s, w1 i = 1\nhw1s : ↑(Finset.affineCombination k s p) w1 = p i0\nw2 : ι → k := w + w1\nhw2 : ∑ i in s, w2 i = 1\nhw2s : ↑(Finset.affineCombination k s p) w2 = p i0\n⊢ w i0 = 0",
"tactic": "replace ha := ha s s w2 w1 hw2 hw1 (hw1s.symm ▸ hw2s)"
},
{
"state_after": "case mpr\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\ns : Finset ι\nw : ι → k\nhw : ∑ i in s, w i = 0\nhs : ↑(Finset.weightedVSub s p) w = 0\ni0 : ι\nhi0 : i0 ∈ s\nw1 : ι → k := update (const ι 0) i0 1\nhw1 : ∑ i in s, w1 i = 1\nhw1s : ↑(Finset.affineCombination k s p) w1 = p i0\nw2 : ι → k := w + w1\nhw2 : ∑ i in s, w2 i = 1\nhw2s : ↑(Finset.affineCombination k s p) w2 = p i0\nha : Set.indicator (↑s) w2 = Set.indicator (↑s) w1\nhws : w2 i0 - w1 i0 = 0\n⊢ w i0 = 0",
"state_before": "case mpr\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\ns : Finset ι\nw : ι → k\nhw : ∑ i in s, w i = 0\nhs : ↑(Finset.weightedVSub s p) w = 0\ni0 : ι\nhi0 : i0 ∈ s\nw1 : ι → k := update (const ι 0) i0 1\nhw1 : ∑ i in s, w1 i = 1\nhw1s : ↑(Finset.affineCombination k s p) w1 = p i0\nw2 : ι → k := w + w1\nhw2 : ∑ i in s, w2 i = 1\nhw2s : ↑(Finset.affineCombination k s p) w2 = p i0\nha : Set.indicator (↑s) w2 = Set.indicator (↑s) w1\n⊢ w i0 = 0",
"tactic": "have hws : w2 i0 - w1 i0 = 0 := by\n rw [← Finset.mem_coe] at hi0\n rw [← Set.indicator_of_mem hi0 w2, ← Set.indicator_of_mem hi0 w1, ha, sub_self]"
},
{
"state_after": "no goals",
"state_before": "case mpr\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\ns : Finset ι\nw : ι → k\nhw : ∑ i in s, w i = 0\nhs : ↑(Finset.weightedVSub s p) w = 0\ni0 : ι\nhi0 : i0 ∈ s\nw1 : ι → k := update (const ι 0) i0 1\nhw1 : ∑ i in s, w1 i = 1\nhw1s : ↑(Finset.affineCombination k s p) w1 = p i0\nw2 : ι → k := w + w1\nhw2 : ∑ i in s, w2 i = 1\nhw2s : ↑(Finset.affineCombination k s p) w2 = p i0\nha : Set.indicator (↑s) w2 = Set.indicator (↑s) w1\nhws : w2 i0 - w1 i0 = 0\n⊢ w i0 = 0",
"tactic": "simpa using hws"
},
{
"state_after": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\nha :\n ∀ (s1 s2 : Finset ι) (w1 w2 : ι → k),\n ∑ i in s1, w1 i = 1 →\n ∑ i in s2, w2 i = 1 →\n ↑(Finset.affineCombination k s1 p) w1 = ↑(Finset.affineCombination k s2 p) w2 →\n Set.indicator (↑s1) w1 = Set.indicator (↑s2) w2\ns : Finset ι\nw : ι → k\nhw : ∑ i in s, w i = 0\nhs : ↑(Finset.weightedVSub s p) w = 0\ni0 : ι\nhi0 : i0 ∈ s\nw1 : ι → k := update (const ι 0) i0 1\n⊢ 1 + ∑ x in s \\ {i0}, const ι 0 x = 1",
"state_before": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\nha :\n ∀ (s1 s2 : Finset ι) (w1 w2 : ι → k),\n ∑ i in s1, w1 i = 1 →\n ∑ i in s2, w2 i = 1 →\n ↑(Finset.affineCombination k s1 p) w1 = ↑(Finset.affineCombination k s2 p) w2 →\n Set.indicator (↑s1) w1 = Set.indicator (↑s2) w2\ns : Finset ι\nw : ι → k\nhw : ∑ i in s, w i = 0\nhs : ↑(Finset.weightedVSub s p) w = 0\ni0 : ι\nhi0 : i0 ∈ s\nw1 : ι → k := update (const ι 0) i0 1\n⊢ ∑ i in s, w1 i = 1",
"tactic": "rw [Finset.sum_update_of_mem hi0]"
},
{
"state_after": "no goals",
"state_before": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\nha :\n ∀ (s1 s2 : Finset ι) (w1 w2 : ι → k),\n ∑ i in s1, w1 i = 1 →\n ∑ i in s2, w2 i = 1 →\n ↑(Finset.affineCombination k s1 p) w1 = ↑(Finset.affineCombination k s2 p) w2 →\n Set.indicator (↑s1) w1 = Set.indicator (↑s2) w2\ns : Finset ι\nw : ι → k\nhw : ∑ i in s, w i = 0\nhs : ↑(Finset.weightedVSub s p) w = 0\ni0 : ι\nhi0 : i0 ∈ s\nw1 : ι → k := update (const ι 0) i0 1\n⊢ 1 + ∑ x in s \\ {i0}, const ι 0 x = 1",
"tactic": "simp only [Finset.sum_const_zero, add_zero, const_apply]"
},
{
"state_after": "no goals",
"state_before": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\nha :\n ∀ (s1 s2 : Finset ι) (w1 w2 : ι → k),\n ∑ i in s1, w1 i = 1 →\n ∑ i in s2, w2 i = 1 →\n ↑(Finset.affineCombination k s1 p) w1 = ↑(Finset.affineCombination k s2 p) w2 →\n Set.indicator (↑s1) w1 = Set.indicator (↑s2) w2\ns : Finset ι\nw : ι → k\nhw : ∑ i in s, w i = 0\nhs : ↑(Finset.weightedVSub s p) w = 0\ni0 : ι\nhi0 : i0 ∈ s\nw1 : ι → k := update (const ι 0) i0 1\nhw1 : ∑ i in s, w1 i = 1\nhw1s : ↑(Finset.affineCombination k s p) w1 = p i0\nw2 : ι → k := w + w1\n⊢ ∑ i in s, w2 i = 1",
"tactic": "simp_all only [Pi.add_apply, Finset.sum_add_distrib, zero_add]"
},
{
"state_after": "no goals",
"state_before": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\nha :\n ∀ (s1 s2 : Finset ι) (w1 w2 : ι → k),\n ∑ i in s1, w1 i = 1 →\n ∑ i in s2, w2 i = 1 →\n ↑(Finset.affineCombination k s1 p) w1 = ↑(Finset.affineCombination k s2 p) w2 →\n Set.indicator (↑s1) w1 = Set.indicator (↑s2) w2\ns : Finset ι\nw : ι → k\nhw : ∑ i in s, w i = 0\nhs : ↑(Finset.weightedVSub s p) w = 0\ni0 : ι\nhi0 : i0 ∈ s\nw1 : ι → k := update (const ι 0) i0 1\nhw1 : ∑ i in s, w1 i = 1\nhw1s : ↑(Finset.affineCombination k s p) w1 = p i0\nw2 : ι → k := w + w1\nhw2 : ∑ i in s, w2 i = 1\n⊢ ↑(Finset.affineCombination k s p) w2 = p i0",
"tactic": "simp_all only [← Finset.weightedVSub_vadd_affineCombination, zero_vadd]"
},
{
"state_after": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\ns : Finset ι\nw : ι → k\nhw : ∑ i in s, w i = 0\nhs : ↑(Finset.weightedVSub s p) w = 0\ni0 : ι\nhi0 : i0 ∈ ↑s\nw1 : ι → k := update (const ι 0) i0 1\nhw1 : ∑ i in s, w1 i = 1\nhw1s : ↑(Finset.affineCombination k s p) w1 = p i0\nw2 : ι → k := w + w1\nhw2 : ∑ i in s, w2 i = 1\nhw2s : ↑(Finset.affineCombination k s p) w2 = p i0\nha : Set.indicator (↑s) w2 = Set.indicator (↑s) w1\n⊢ w2 i0 - w1 i0 = 0",
"state_before": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\ns : Finset ι\nw : ι → k\nhw : ∑ i in s, w i = 0\nhs : ↑(Finset.weightedVSub s p) w = 0\ni0 : ι\nhi0 : i0 ∈ s\nw1 : ι → k := update (const ι 0) i0 1\nhw1 : ∑ i in s, w1 i = 1\nhw1s : ↑(Finset.affineCombination k s p) w1 = p i0\nw2 : ι → k := w + w1\nhw2 : ∑ i in s, w2 i = 1\nhw2s : ↑(Finset.affineCombination k s p) w2 = p i0\nha : Set.indicator (↑s) w2 = Set.indicator (↑s) w1\n⊢ w2 i0 - w1 i0 = 0",
"tactic": "rw [← Finset.mem_coe] at hi0"
},
{
"state_after": "no goals",
"state_before": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\ns : Finset ι\nw : ι → k\nhw : ∑ i in s, w i = 0\nhs : ↑(Finset.weightedVSub s p) w = 0\ni0 : ι\nhi0 : i0 ∈ ↑s\nw1 : ι → k := update (const ι 0) i0 1\nhw1 : ∑ i in s, w1 i = 1\nhw1s : ↑(Finset.affineCombination k s p) w1 = p i0\nw2 : ι → k := w + w1\nhw2 : ∑ i in s, w2 i = 1\nhw2s : ↑(Finset.affineCombination k s p) w2 = p i0\nha : Set.indicator (↑s) w2 = Set.indicator (↑s) w1\n⊢ w2 i0 - w1 i0 = 0",
"tactic": "rw [← Set.indicator_of_mem hi0 w2, ← Set.indicator_of_mem hi0 w1, ha, sub_self]"
}
] |
[
234,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
184,
1
] |
Mathlib/Analysis/Convex/Between.lean
|
affineSegment_eq_segment
|
[
{
"state_after": "no goals",
"state_before": "R : Type u_2\nV : Type u_1\nV' : Type ?u.5184\nP : Type ?u.5187\nP' : Type ?u.5190\ninst✝⁶ : OrderedRing R\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddTorsor V P\ninst✝² : AddCommGroup V'\ninst✝¹ : Module R V'\ninst✝ : AddTorsor V' P'\nx y : V\n⊢ affineSegment R x y = segment R x y",
"tactic": "rw [segment_eq_image_lineMap, affineSegment]"
}
] |
[
52,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
51,
1
] |
Mathlib/MeasureTheory/Measure/OuterMeasure.lean
|
MeasureTheory.OuterMeasure.map_sup
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ✝ : Type ?u.78827\nR : Type ?u.78830\nR' : Type ?u.78833\nms : Set (OuterMeasure α)\nm✝ : OuterMeasure α\nβ : Type u_1\nf : α → β\nm m' : OuterMeasure α\ns : Set β\n⊢ ↑(↑(map f) (m ⊔ m')) s = ↑(↑(map f) m ⊔ ↑(map f) m') s",
"tactic": "simp only [map_apply, sup_apply]"
}
] |
[
478,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
477,
1
] |
Mathlib/NumberTheory/Padics/Hensel.lean
|
T_lt_one
|
[
{
"state_after": "p : ℕ\ninst✝ : Fact (Nat.Prime p)\nF : Polynomial ℤ_[p]\na : ℤ_[p]\nhnorm : ‖Polynomial.eval a F‖ < ‖Polynomial.eval a (↑Polynomial.derivative F)‖ ^ 2\nhnsol : Polynomial.eval a F ≠ 0\nh : ‖Polynomial.eval a F‖ / ‖Polynomial.eval a (↑Polynomial.derivative F)‖ ^ 2 < 1\n⊢ T_gen p F a < 1",
"state_before": "p : ℕ\ninst✝ : Fact (Nat.Prime p)\nF : Polynomial ℤ_[p]\na : ℤ_[p]\nhnorm : ‖Polynomial.eval a F‖ < ‖Polynomial.eval a (↑Polynomial.derivative F)‖ ^ 2\nhnsol : Polynomial.eval a F ≠ 0\n⊢ T_gen p F a < 1",
"tactic": "have h := (div_lt_one (deriv_sq_norm_pos hnorm)).2 hnorm"
},
{
"state_after": "p : ℕ\ninst✝ : Fact (Nat.Prime p)\nF : Polynomial ℤ_[p]\na : ℤ_[p]\nhnorm : ‖Polynomial.eval a F‖ < ‖Polynomial.eval a (↑Polynomial.derivative F)‖ ^ 2\nhnsol : Polynomial.eval a F ≠ 0\nh : ‖Polynomial.eval a F‖ / ‖Polynomial.eval a (↑Polynomial.derivative F)‖ ^ 2 < 1\n⊢ ‖Polynomial.eval a F‖ / ‖Polynomial.eval a (↑Polynomial.derivative F)‖ ^ 2 < 1",
"state_before": "p : ℕ\ninst✝ : Fact (Nat.Prime p)\nF : Polynomial ℤ_[p]\na : ℤ_[p]\nhnorm : ‖Polynomial.eval a F‖ < ‖Polynomial.eval a (↑Polynomial.derivative F)‖ ^ 2\nhnsol : Polynomial.eval a F ≠ 0\nh : ‖Polynomial.eval a F‖ / ‖Polynomial.eval a (↑Polynomial.derivative F)‖ ^ 2 < 1\n⊢ T_gen p F a < 1",
"tactic": "rw [T_def]"
},
{
"state_after": "no goals",
"state_before": "p : ℕ\ninst✝ : Fact (Nat.Prime p)\nF : Polynomial ℤ_[p]\na : ℤ_[p]\nhnorm : ‖Polynomial.eval a F‖ < ‖Polynomial.eval a (↑Polynomial.derivative F)‖ ^ 2\nhnsol : Polynomial.eval a F ≠ 0\nh : ‖Polynomial.eval a F‖ / ‖Polynomial.eval a (↑Polynomial.derivative F)‖ ^ 2 < 1\n⊢ ‖Polynomial.eval a F‖ / ‖Polynomial.eval a (↑Polynomial.derivative F)‖ ^ 2 < 1",
"tactic": "exact h"
}
] |
[
131,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
129,
9
] |
Mathlib/Data/Set/Prod.lean
|
Set.mk_preimage_prod
|
[] |
[
221,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
219,
1
] |
Mathlib/Analysis/Normed/Group/Pointwise.lean
|
singleton_div_ball_one
|
[
{
"state_after": "no goals",
"state_before": "E : Type u_1\ninst✝ : SeminormedCommGroup E\nε δ : ℝ\ns t : Set E\nx y : E\n⊢ {x} / ball 1 δ = ball x δ",
"tactic": "rw [singleton_div_ball, div_one]"
}
] |
[
135,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
134,
1
] |
Mathlib/Data/Nat/Factorization/Basic.lean
|
Nat.ord_compl_dvd
|
[] |
[
379,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
378,
1
] |
Mathlib/Data/List/Destutter.lean
|
List.mem_destutter'
|
[
{
"state_after": "case nil\nα : Type u_1\nl : List α\nR : α → α → Prop\ninst✝ : DecidableRel R\na✝ b a : α\n⊢ a ∈ destutter' R a []\n\ncase cons\nα : Type u_1\nl✝ : List α\nR : α → α → Prop\ninst✝ : DecidableRel R\na✝ b✝ a b : α\nl : List α\nhl : a ∈ destutter' R a l\n⊢ a ∈ destutter' R a (b :: l)",
"state_before": "α : Type u_1\nl : List α\nR : α → α → Prop\ninst✝ : DecidableRel R\na✝ b a : α\n⊢ a ∈ destutter' R a l",
"tactic": "induction' l with b l hl"
},
{
"state_after": "case cons\nα : Type u_1\nl✝ : List α\nR : α → α → Prop\ninst✝ : DecidableRel R\na✝ b✝ a b : α\nl : List α\nhl : a ∈ destutter' R a l\n⊢ a ∈ if R a b then a :: destutter' R b l else destutter' R a l",
"state_before": "case cons\nα : Type u_1\nl✝ : List α\nR : α → α → Prop\ninst✝ : DecidableRel R\na✝ b✝ a b : α\nl : List α\nhl : a ∈ destutter' R a l\n⊢ a ∈ destutter' R a (b :: l)",
"tactic": "rw [destutter']"
},
{
"state_after": "case cons.inl\nα : Type u_1\nl✝ : List α\nR : α → α → Prop\ninst✝ : DecidableRel R\na✝ b✝ a b : α\nl : List α\nhl : a ∈ destutter' R a l\nh✝ : R a b\n⊢ a ∈ a :: destutter' R b l\n\ncase cons.inr\nα : Type u_1\nl✝ : List α\nR : α → α → Prop\ninst✝ : DecidableRel R\na✝ b✝ a b : α\nl : List α\nhl : a ∈ destutter' R a l\nh✝ : ¬R a b\n⊢ a ∈ destutter' R a l",
"state_before": "case cons\nα : Type u_1\nl✝ : List α\nR : α → α → Prop\ninst✝ : DecidableRel R\na✝ b✝ a b : α\nl : List α\nhl : a ∈ destutter' R a l\n⊢ a ∈ if R a b then a :: destutter' R b l else destutter' R a l",
"tactic": "split_ifs"
},
{
"state_after": "no goals",
"state_before": "case nil\nα : Type u_1\nl : List α\nR : α → α → Prop\ninst✝ : DecidableRel R\na✝ b a : α\n⊢ a ∈ destutter' R a []",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "case cons.inl\nα : Type u_1\nl✝ : List α\nR : α → α → Prop\ninst✝ : DecidableRel R\na✝ b✝ a b : α\nl : List α\nhl : a ∈ destutter' R a l\nh✝ : R a b\n⊢ a ∈ a :: destutter' R b l",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "case cons.inr\nα : Type u_1\nl✝ : List α\nR : α → α → Prop\ninst✝ : DecidableRel R\na✝ b✝ a b : α\nl : List α\nhl : a ∈ destutter' R a l\nh✝ : ¬R a b\n⊢ a ∈ destutter' R a l",
"tactic": "assumption"
}
] |
[
82,
15
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
76,
1
] |
Mathlib/Algebra/GroupPower/Order.lean
|
MonoidHom.map_neg_one
|
[
{
"state_after": "no goals",
"state_before": "β : Type ?u.293119\nA : Type ?u.293122\nG : Type ?u.293125\nM : Type u_1\nR : Type u_2\ninst✝³ : Ring R\ninst✝² : Monoid M\ninst✝¹ : LinearOrder M\ninst✝ : CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1\nf : R →* M\n⊢ ↑f (-1) ^ Nat.succ 1 = 1",
"tactic": "rw [← map_pow, neg_one_sq, map_one]"
}
] |
[
801,
84
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
800,
1
] |
Mathlib/RingTheory/Ideal/QuotientOperations.lean
|
Ideal.kerLiftAlg_toRingHom
|
[] |
[
300,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
298,
1
] |
Mathlib/CategoryTheory/Sites/Grothendieck.lean
|
CategoryTheory.GrothendieckTopology.pullback_stable
|
[] |
[
131,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
130,
1
] |
Mathlib/LinearAlgebra/Prod.lean
|
LinearMap.tailings_zero
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM₂ : Type w\nV₂ : Type w'\nM₃ : Type y\nV₃ : Type y'\nM₄ : Type z\nι : Type x\nM₅ : Type ?u.564691\nM₆ : Type ?u.564694\ninst✝⁴ : Ring R\nN : Type u_1\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\nf : M × N →ₗ[R] M\ni : Injective ↑f\n⊢ tailings f i 0 = tailing f i 0",
"tactic": "simp [tailings]"
}
] |
[
965,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
964,
1
] |
Mathlib/Combinatorics/SimpleGraph/Connectivity.lean
|
SimpleGraph.Walk.end_mem_support
|
[
{
"state_after": "no goals",
"state_before": "V : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nu v : V\np : Walk G u v\n⊢ v ∈ support p",
"tactic": "induction p <;> simp [*]"
}
] |
[
579,
98
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
579,
1
] |
Mathlib/Data/Finsupp/Basic.lean
|
IsSMulRegular.finsupp
|
[] |
[
1508,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1506,
1
] |
Mathlib/Algebra/GroupPower/Lemmas.lean
|
Commute.self_cast_int_mul_cast_int_mul
|
[] |
[
1197,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1196,
1
] |
Mathlib/Data/Complex/Module.lean
|
AlgHom.map_coe_real_complex
|
[] |
[
123,
15
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
122,
1
] |
Mathlib/CategoryTheory/Adjunction/Opposites.lean
|
CategoryTheory.Adjunction.leftAdjointUniq_trans
|
[
{
"state_after": "case w.h\nC : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nF F' F'' : C ⥤ D\nG : D ⥤ C\nadj1 : F ⊣ G\nadj2 : F' ⊣ G\nadj3 : F'' ⊣ G\nx✝ : C\n⊢ ((leftAdjointUniq adj1 adj2).hom ≫ (leftAdjointUniq adj2 adj3).hom).app x✝ = (leftAdjointUniq adj1 adj3).hom.app x✝",
"state_before": "C : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nF F' F'' : C ⥤ D\nG : D ⥤ C\nadj1 : F ⊣ G\nadj2 : F' ⊣ G\nadj3 : F'' ⊣ G\n⊢ (leftAdjointUniq adj1 adj2).hom ≫ (leftAdjointUniq adj2 adj3).hom = (leftAdjointUniq adj1 adj3).hom",
"tactic": "ext"
},
{
"state_after": "case w.h.a\nC : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nF F' F'' : C ⥤ D\nG : D ⥤ C\nadj1 : F ⊣ G\nadj2 : F' ⊣ G\nadj3 : F'' ⊣ G\nx✝ : C\n⊢ (((leftAdjointUniq adj1 adj2).hom ≫ (leftAdjointUniq adj2 adj3).hom).app x✝).op =\n ((leftAdjointUniq adj1 adj3).hom.app x✝).op",
"state_before": "case w.h\nC : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nF F' F'' : C ⥤ D\nG : D ⥤ C\nadj1 : F ⊣ G\nadj2 : F' ⊣ G\nadj3 : F'' ⊣ G\nx✝ : C\n⊢ ((leftAdjointUniq adj1 adj2).hom ≫ (leftAdjointUniq adj2 adj3).hom).app x✝ = (leftAdjointUniq adj1 adj3).hom.app x✝",
"tactic": "apply Quiver.Hom.op_inj"
},
{
"state_after": "case w.h.a.a\nC : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nF F' F'' : C ⥤ D\nG : D ⥤ C\nadj1 : F ⊣ G\nadj2 : F' ⊣ G\nadj3 : F'' ⊣ G\nx✝ : C\n⊢ coyoneda.map (((leftAdjointUniq adj1 adj2).hom ≫ (leftAdjointUniq adj2 adj3).hom).app x✝).op =\n coyoneda.map ((leftAdjointUniq adj1 adj3).hom.app x✝).op",
"state_before": "case w.h.a\nC : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nF F' F'' : C ⥤ D\nG : D ⥤ C\nadj1 : F ⊣ G\nadj2 : F' ⊣ G\nadj3 : F'' ⊣ G\nx✝ : C\n⊢ (((leftAdjointUniq adj1 adj2).hom ≫ (leftAdjointUniq adj2 adj3).hom).app x✝).op =\n ((leftAdjointUniq adj1 adj3).hom.app x✝).op",
"tactic": "apply coyoneda.map_injective"
},
{
"state_after": "case w.h.a.a.w.h.h\nC : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nF F' F'' : C ⥤ D\nG : D ⥤ C\nadj1 : F ⊣ G\nadj2 : F' ⊣ G\nadj3 : F'' ⊣ G\nx✝¹ : C\nx✝ : D\na✝ : (coyoneda.obj (F''.obj x✝¹).op).obj x✝\n⊢ (coyoneda.map (((leftAdjointUniq adj1 adj2).hom ≫ (leftAdjointUniq adj2 adj3).hom).app x✝¹).op).app x✝ a✝ =\n (coyoneda.map ((leftAdjointUniq adj1 adj3).hom.app x✝¹).op).app x✝ a✝",
"state_before": "case w.h.a.a\nC : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nF F' F'' : C ⥤ D\nG : D ⥤ C\nadj1 : F ⊣ G\nadj2 : F' ⊣ G\nadj3 : F'' ⊣ G\nx✝ : C\n⊢ coyoneda.map (((leftAdjointUniq adj1 adj2).hom ≫ (leftAdjointUniq adj2 adj3).hom).app x✝).op =\n coyoneda.map ((leftAdjointUniq adj1 adj3).hom.app x✝).op",
"tactic": "ext"
},
{
"state_after": "case w.h.a.a.w.h.h\nC : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nF F' F'' : C ⥤ D\nG : D ⥤ C\nadj1 : F ⊣ G\nadj2 : F' ⊣ G\nadj3 : F'' ⊣ G\nx✝¹ : C\nx✝ : D\na✝ : (coyoneda.obj (F''.obj x✝¹).op).obj x✝\n⊢ (coyoneda.map (((leftAdjointUniq adj1 adj2).hom ≫ (leftAdjointUniq adj2 adj3).hom).app x✝¹).op).app x✝ a✝ =\n (coyoneda.map ((leftAdjointUniq adj1 adj3).hom.app x✝¹).op).app x✝ a✝",
"state_before": "case w.h.a.a.w.h.h\nC : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nF F' F'' : C ⥤ D\nG : D ⥤ C\nadj1 : F ⊣ G\nadj2 : F' ⊣ G\nadj3 : F'' ⊣ G\nx✝¹ : C\nx✝ : D\na✝ : (coyoneda.obj (F''.obj x✝¹).op).obj x✝\n⊢ (coyoneda.map (((leftAdjointUniq adj1 adj2).hom ≫ (leftAdjointUniq adj2 adj3).hom).app x✝¹).op).app x✝ a✝ =\n (coyoneda.map ((leftAdjointUniq adj1 adj3).hom.app x✝¹).op).app x✝ a✝",
"tactic": "funext"
},
{
"state_after": "no goals",
"state_before": "case w.h.a.a.w.h.h\nC : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nF F' F'' : C ⥤ D\nG : D ⥤ C\nadj1 : F ⊣ G\nadj2 : F' ⊣ G\nadj3 : F'' ⊣ G\nx✝¹ : C\nx✝ : D\na✝ : (coyoneda.obj (F''.obj x✝¹).op).obj x✝\n⊢ (coyoneda.map (((leftAdjointUniq adj1 adj2).hom ≫ (leftAdjointUniq adj2 adj3).hom).app x✝¹).op).app x✝ a✝ =\n (coyoneda.map ((leftAdjointUniq adj1 adj3).hom.app x✝¹).op).app x✝ a✝",
"tactic": "simp [leftAdjointsCoyonedaEquiv, leftAdjointUniq]"
}
] |
[
193,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
184,
1
] |
Mathlib/Combinatorics/SimpleGraph/Connectivity.lean
|
SimpleGraph.Walk.darts_cons
|
[] |
[
690,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
689,
1
] |
Mathlib/Analysis/Complex/Basic.lean
|
Complex.restrictScalars_one_smulRight
|
[
{
"state_after": "case h\nE : Type ?u.361704\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nx z : ℂ\n⊢ ↑(restrictScalars ℝ (smulRight 1 x)) z = ↑(x • 1) z",
"state_before": "E : Type ?u.361704\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nx : ℂ\n⊢ restrictScalars ℝ (smulRight 1 x) = x • 1",
"tactic": "ext1 z"
},
{
"state_after": "case h\nE : Type ?u.361704\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nx z : ℂ\n⊢ z * x = x * z",
"state_before": "case h\nE : Type ?u.361704\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nx z : ℂ\n⊢ ↑(restrictScalars ℝ (smulRight 1 x)) z = ↑(x • 1) z",
"tactic": "dsimp"
},
{
"state_after": "no goals",
"state_before": "case h\nE : Type ?u.361704\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nx z : ℂ\n⊢ z * x = x * z",
"tactic": "apply mul_comm"
}
] |
[
307,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
302,
1
] |
Mathlib/Topology/IsLocallyHomeomorph.lean
|
IsLocallyHomeomorphOn.continuousOn
|
[] |
[
72,
54
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
71,
11
] |
Mathlib/RingTheory/AdjoinRoot.lean
|
AdjoinRoot.coe_injective'
|
[] |
[
421,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
420,
1
] |
Mathlib/Analysis/Calculus/Deriv/Add.lean
|
HasDerivWithinAt.sum
|
[] |
[
177,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
175,
1
] |
Mathlib/Topology/Algebra/StarSubalgebra.lean
|
StarAlgHomClass.ext_topologicalClosure
|
[
{
"state_after": "R : Type u_3\nA : Type u_4\nB : Type u_1\ninst✝¹⁴ : CommSemiring R\ninst✝¹³ : StarRing R\ninst✝¹² : TopologicalSpace A\ninst✝¹¹ : Semiring A\ninst✝¹⁰ : Algebra R A\ninst✝⁹ : StarRing A\ninst✝⁸ : StarModule R A\ninst✝⁷ : TopologicalSemiring A\ninst✝⁶ : ContinuousStar A\ninst✝⁵ : TopologicalSpace B\ninst✝⁴ : Semiring B\ninst✝³ : Algebra R B\ninst✝² : StarRing B\ninst✝¹ : T2Space B\nF : Type u_2\nS : StarSubalgebra R A\ninst✝ : StarAlgHomClass F R { x // x ∈ topologicalClosure S } B\nφ ψ : F\nhφ : Continuous ↑φ\nhψ : Continuous ↑ψ\nh :\n ∀ (x : { x // x ∈ S }),\n ↑φ (↑(inclusion (_ : S ≤ topologicalClosure S)) x) = ↑ψ (↑(inclusion (_ : S ≤ topologicalClosure S)) x)\nthis :\n (let src := ↑φ;\n {\n toAlgHom :=\n {\n toRingHom :=\n {\n toMonoidHom :=\n { toOneHom := { toFun := ↑φ, map_one' := (_ : OneHom.toFun (↑↑↑↑φ) 1 = 1) },\n map_mul' :=\n (_ :\n ∀ (x y : { x // x ∈ topologicalClosure S }),\n OneHom.toFun (↑↑↑↑φ) (x * y) = OneHom.toFun (↑↑↑↑φ) x * OneHom.toFun (↑↑↑↑φ) y) },\n map_zero' := (_ : OneHom.toFun (↑↑↑↑φ) 0 = 0),\n map_add' :=\n (_ :\n ∀ (x y : { x // x ∈ topologicalClosure S }),\n OneHom.toFun (↑↑↑↑φ) (x + y) = OneHom.toFun (↑↑↑↑φ) x + OneHom.toFun (↑↑↑↑φ) y) },\n commutes' :=\n (_ :\n ∀ (r : R),\n OneHom.toFun (↑↑↑src) (↑(algebraMap R { x // x ∈ topologicalClosure S }) r) = ↑(algebraMap R B) r) },\n map_star' := (_ : ∀ (r : { x // x ∈ topologicalClosure S }), ↑φ (star r) = star (↑φ r)) }) =\n let src := ↑ψ;\n {\n toAlgHom :=\n {\n toRingHom :=\n {\n toMonoidHom :=\n { toOneHom := { toFun := ↑ψ, map_one' := (_ : OneHom.toFun (↑↑↑↑ψ) 1 = 1) },\n map_mul' :=\n (_ :\n ∀ (x y : { x // x ∈ topologicalClosure S }),\n OneHom.toFun (↑↑↑↑ψ) (x * y) = OneHom.toFun (↑↑↑↑ψ) x * OneHom.toFun (↑↑↑↑ψ) y) },\n map_zero' := (_ : OneHom.toFun (↑↑↑↑ψ) 0 = 0),\n map_add' :=\n (_ :\n ∀ (x y : { x // x ∈ topologicalClosure S }),\n OneHom.toFun (↑↑↑↑ψ) (x + y) = OneHom.toFun (↑↑↑↑ψ) x + OneHom.toFun (↑↑↑↑ψ) y) },\n commutes' :=\n (_ :\n ∀ (r : R),\n OneHom.toFun (↑↑↑src) (↑(algebraMap R { x // x ∈ topologicalClosure S }) r) = ↑(algebraMap R B) r) },\n map_star' := (_ : ∀ (r : { x // x ∈ topologicalClosure S }), ↑ψ (star r) = star (↑ψ r)) }\n⊢ φ = ψ",
"state_before": "R : Type u_3\nA : Type u_4\nB : Type u_1\ninst✝¹⁴ : CommSemiring R\ninst✝¹³ : StarRing R\ninst✝¹² : TopologicalSpace A\ninst✝¹¹ : Semiring A\ninst✝¹⁰ : Algebra R A\ninst✝⁹ : StarRing A\ninst✝⁸ : StarModule R A\ninst✝⁷ : TopologicalSemiring A\ninst✝⁶ : ContinuousStar A\ninst✝⁵ : TopologicalSpace B\ninst✝⁴ : Semiring B\ninst✝³ : Algebra R B\ninst✝² : StarRing B\ninst✝¹ : T2Space B\nF : Type u_2\nS : StarSubalgebra R A\ninst✝ : StarAlgHomClass F R { x // x ∈ topologicalClosure S } B\nφ ψ : F\nhφ : Continuous ↑φ\nhψ : Continuous ↑ψ\nh :\n ∀ (x : { x // x ∈ S }),\n ↑φ (↑(inclusion (_ : S ≤ topologicalClosure S)) x) = ↑ψ (↑(inclusion (_ : S ≤ topologicalClosure S)) x)\n⊢ φ = ψ",
"tactic": "have : (φ : S.topologicalClosure →⋆ₐ[R] B) = (ψ : S.topologicalClosure →⋆ₐ[R] B) := by\n refine StarAlgHom.ext_topologicalClosure (R := R) (A := A) (B := B) hφ hψ (StarAlgHom.ext ?_)\n simpa only [StarAlgHom.coe_comp, StarAlgHom.coe_coe] using h"
},
{
"state_after": "R : Type u_3\nA : Type u_4\nB : Type u_1\ninst✝¹⁴ : CommSemiring R\ninst✝¹³ : StarRing R\ninst✝¹² : TopologicalSpace A\ninst✝¹¹ : Semiring A\ninst✝¹⁰ : Algebra R A\ninst✝⁹ : StarRing A\ninst✝⁸ : StarModule R A\ninst✝⁷ : TopologicalSemiring A\ninst✝⁶ : ContinuousStar A\ninst✝⁵ : TopologicalSpace B\ninst✝⁴ : Semiring B\ninst✝³ : Algebra R B\ninst✝² : StarRing B\ninst✝¹ : T2Space B\nF : Type u_2\nS : StarSubalgebra R A\ninst✝ : StarAlgHomClass F R { x // x ∈ topologicalClosure S } B\nφ ψ : F\nhφ : Continuous ↑φ\nhψ : Continuous ↑ψ\nh :\n ∀ (x : { x // x ∈ S }),\n ↑φ (↑(inclusion (_ : S ≤ topologicalClosure S)) x) = ↑ψ (↑(inclusion (_ : S ≤ topologicalClosure S)) x)\nthis :\n (let src := ↑φ;\n {\n toAlgHom :=\n {\n toRingHom :=\n {\n toMonoidHom :=\n { toOneHom := { toFun := ↑φ, map_one' := (_ : OneHom.toFun (↑↑↑↑φ) 1 = 1) },\n map_mul' :=\n (_ :\n ∀ (x y : { x // x ∈ topologicalClosure S }),\n OneHom.toFun (↑↑↑↑φ) (x * y) = OneHom.toFun (↑↑↑↑φ) x * OneHom.toFun (↑↑↑↑φ) y) },\n map_zero' := (_ : OneHom.toFun (↑↑↑↑φ) 0 = 0),\n map_add' :=\n (_ :\n ∀ (x y : { x // x ∈ topologicalClosure S }),\n OneHom.toFun (↑↑↑↑φ) (x + y) = OneHom.toFun (↑↑↑↑φ) x + OneHom.toFun (↑↑↑↑φ) y) },\n commutes' :=\n (_ :\n ∀ (r : R),\n OneHom.toFun (↑↑↑src) (↑(algebraMap R { x // x ∈ topologicalClosure S }) r) = ↑(algebraMap R B) r) },\n map_star' := (_ : ∀ (r : { x // x ∈ topologicalClosure S }), ↑φ (star r) = star (↑φ r)) }) =\n let src := ↑ψ;\n {\n toAlgHom :=\n {\n toRingHom :=\n {\n toMonoidHom :=\n { toOneHom := { toFun := ↑ψ, map_one' := (_ : OneHom.toFun (↑↑↑↑ψ) 1 = 1) },\n map_mul' :=\n (_ :\n ∀ (x y : { x // x ∈ topologicalClosure S }),\n OneHom.toFun (↑↑↑↑ψ) (x * y) = OneHom.toFun (↑↑↑↑ψ) x * OneHom.toFun (↑↑↑↑ψ) y) },\n map_zero' := (_ : OneHom.toFun (↑↑↑↑ψ) 0 = 0),\n map_add' :=\n (_ :\n ∀ (x y : { x // x ∈ topologicalClosure S }),\n OneHom.toFun (↑↑↑↑ψ) (x + y) = OneHom.toFun (↑↑↑↑ψ) x + OneHom.toFun (↑↑↑↑ψ) y) },\n commutes' :=\n (_ :\n ∀ (r : R),\n OneHom.toFun (↑↑↑src) (↑(algebraMap R { x // x ∈ topologicalClosure S }) r) = ↑(algebraMap R B) r) },\n map_star' := (_ : ∀ (r : { x // x ∈ topologicalClosure S }), ↑ψ (star r) = star (↑ψ r)) }\n⊢ ↑(let src := ↑φ;\n {\n toAlgHom :=\n {\n toRingHom :=\n {\n toMonoidHom :=\n { toOneHom := { toFun := ↑φ, map_one' := (_ : OneHom.toFun (↑↑↑↑φ) 1 = 1) },\n map_mul' :=\n (_ :\n ∀ (x y : { x // x ∈ topologicalClosure S }),\n OneHom.toFun (↑↑↑↑φ) (x * y) = OneHom.toFun (↑↑↑↑φ) x * OneHom.toFun (↑↑↑↑φ) y) },\n map_zero' := (_ : OneHom.toFun (↑↑↑↑φ) 0 = 0),\n map_add' :=\n (_ :\n ∀ (x y : { x // x ∈ topologicalClosure S }),\n OneHom.toFun (↑↑↑↑φ) (x + y) = OneHom.toFun (↑↑↑↑φ) x + OneHom.toFun (↑↑↑↑φ) y) },\n commutes' :=\n (_ :\n ∀ (r : R),\n OneHom.toFun (↑↑↑src) (↑(algebraMap R { x // x ∈ topologicalClosure S }) r) = ↑(algebraMap R B) r) },\n map_star' := (_ : ∀ (r : { x // x ∈ topologicalClosure S }), ↑φ (star r) = star (↑φ r)) }) =\n ↑ψ",
"state_before": "R : Type u_3\nA : Type u_4\nB : Type u_1\ninst✝¹⁴ : CommSemiring R\ninst✝¹³ : StarRing R\ninst✝¹² : TopologicalSpace A\ninst✝¹¹ : Semiring A\ninst✝¹⁰ : Algebra R A\ninst✝⁹ : StarRing A\ninst✝⁸ : StarModule R A\ninst✝⁷ : TopologicalSemiring A\ninst✝⁶ : ContinuousStar A\ninst✝⁵ : TopologicalSpace B\ninst✝⁴ : Semiring B\ninst✝³ : Algebra R B\ninst✝² : StarRing B\ninst✝¹ : T2Space B\nF : Type u_2\nS : StarSubalgebra R A\ninst✝ : StarAlgHomClass F R { x // x ∈ topologicalClosure S } B\nφ ψ : F\nhφ : Continuous ↑φ\nhψ : Continuous ↑ψ\nh :\n ∀ (x : { x // x ∈ S }),\n ↑φ (↑(inclusion (_ : S ≤ topologicalClosure S)) x) = ↑ψ (↑(inclusion (_ : S ≤ topologicalClosure S)) x)\nthis :\n (let src := ↑φ;\n {\n toAlgHom :=\n {\n toRingHom :=\n {\n toMonoidHom :=\n { toOneHom := { toFun := ↑φ, map_one' := (_ : OneHom.toFun (↑↑↑↑φ) 1 = 1) },\n map_mul' :=\n (_ :\n ∀ (x y : { x // x ∈ topologicalClosure S }),\n OneHom.toFun (↑↑↑↑φ) (x * y) = OneHom.toFun (↑↑↑↑φ) x * OneHom.toFun (↑↑↑↑φ) y) },\n map_zero' := (_ : OneHom.toFun (↑↑↑↑φ) 0 = 0),\n map_add' :=\n (_ :\n ∀ (x y : { x // x ∈ topologicalClosure S }),\n OneHom.toFun (↑↑↑↑φ) (x + y) = OneHom.toFun (↑↑↑↑φ) x + OneHom.toFun (↑↑↑↑φ) y) },\n commutes' :=\n (_ :\n ∀ (r : R),\n OneHom.toFun (↑↑↑src) (↑(algebraMap R { x // x ∈ topologicalClosure S }) r) = ↑(algebraMap R B) r) },\n map_star' := (_ : ∀ (r : { x // x ∈ topologicalClosure S }), ↑φ (star r) = star (↑φ r)) }) =\n let src := ↑ψ;\n {\n toAlgHom :=\n {\n toRingHom :=\n {\n toMonoidHom :=\n { toOneHom := { toFun := ↑ψ, map_one' := (_ : OneHom.toFun (↑↑↑↑ψ) 1 = 1) },\n map_mul' :=\n (_ :\n ∀ (x y : { x // x ∈ topologicalClosure S }),\n OneHom.toFun (↑↑↑↑ψ) (x * y) = OneHom.toFun (↑↑↑↑ψ) x * OneHom.toFun (↑↑↑↑ψ) y) },\n map_zero' := (_ : OneHom.toFun (↑↑↑↑ψ) 0 = 0),\n map_add' :=\n (_ :\n ∀ (x y : { x // x ∈ topologicalClosure S }),\n OneHom.toFun (↑↑↑↑ψ) (x + y) = OneHom.toFun (↑↑↑↑ψ) x + OneHom.toFun (↑↑↑↑ψ) y) },\n commutes' :=\n (_ :\n ∀ (r : R),\n OneHom.toFun (↑↑↑src) (↑(algebraMap R { x // x ∈ topologicalClosure S }) r) = ↑(algebraMap R B) r) },\n map_star' := (_ : ∀ (r : { x // x ∈ topologicalClosure S }), ↑ψ (star r) = star (↑ψ r)) }\n⊢ φ = ψ",
"tactic": "rw [FunLike.ext'_iff, ← StarAlgHom.coe_coe]"
},
{
"state_after": "no goals",
"state_before": "R : Type u_3\nA : Type u_4\nB : Type u_1\ninst✝¹⁴ : CommSemiring R\ninst✝¹³ : StarRing R\ninst✝¹² : TopologicalSpace A\ninst✝¹¹ : Semiring A\ninst✝¹⁰ : Algebra R A\ninst✝⁹ : StarRing A\ninst✝⁸ : StarModule R A\ninst✝⁷ : TopologicalSemiring A\ninst✝⁶ : ContinuousStar A\ninst✝⁵ : TopologicalSpace B\ninst✝⁴ : Semiring B\ninst✝³ : Algebra R B\ninst✝² : StarRing B\ninst✝¹ : T2Space B\nF : Type u_2\nS : StarSubalgebra R A\ninst✝ : StarAlgHomClass F R { x // x ∈ topologicalClosure S } B\nφ ψ : F\nhφ : Continuous ↑φ\nhψ : Continuous ↑ψ\nh :\n ∀ (x : { x // x ∈ S }),\n ↑φ (↑(inclusion (_ : S ≤ topologicalClosure S)) x) = ↑ψ (↑(inclusion (_ : S ≤ topologicalClosure S)) x)\nthis :\n (let src := ↑φ;\n {\n toAlgHom :=\n {\n toRingHom :=\n {\n toMonoidHom :=\n { toOneHom := { toFun := ↑φ, map_one' := (_ : OneHom.toFun (↑↑↑↑φ) 1 = 1) },\n map_mul' :=\n (_ :\n ∀ (x y : { x // x ∈ topologicalClosure S }),\n OneHom.toFun (↑↑↑↑φ) (x * y) = OneHom.toFun (↑↑↑↑φ) x * OneHom.toFun (↑↑↑↑φ) y) },\n map_zero' := (_ : OneHom.toFun (↑↑↑↑φ) 0 = 0),\n map_add' :=\n (_ :\n ∀ (x y : { x // x ∈ topologicalClosure S }),\n OneHom.toFun (↑↑↑↑φ) (x + y) = OneHom.toFun (↑↑↑↑φ) x + OneHom.toFun (↑↑↑↑φ) y) },\n commutes' :=\n (_ :\n ∀ (r : R),\n OneHom.toFun (↑↑↑src) (↑(algebraMap R { x // x ∈ topologicalClosure S }) r) = ↑(algebraMap R B) r) },\n map_star' := (_ : ∀ (r : { x // x ∈ topologicalClosure S }), ↑φ (star r) = star (↑φ r)) }) =\n let src := ↑ψ;\n {\n toAlgHom :=\n {\n toRingHom :=\n {\n toMonoidHom :=\n { toOneHom := { toFun := ↑ψ, map_one' := (_ : OneHom.toFun (↑↑↑↑ψ) 1 = 1) },\n map_mul' :=\n (_ :\n ∀ (x y : { x // x ∈ topologicalClosure S }),\n OneHom.toFun (↑↑↑↑ψ) (x * y) = OneHom.toFun (↑↑↑↑ψ) x * OneHom.toFun (↑↑↑↑ψ) y) },\n map_zero' := (_ : OneHom.toFun (↑↑↑↑ψ) 0 = 0),\n map_add' :=\n (_ :\n ∀ (x y : { x // x ∈ topologicalClosure S }),\n OneHom.toFun (↑↑↑↑ψ) (x + y) = OneHom.toFun (↑↑↑↑ψ) x + OneHom.toFun (↑↑↑↑ψ) y) },\n commutes' :=\n (_ :\n ∀ (r : R),\n OneHom.toFun (↑↑↑src) (↑(algebraMap R { x // x ∈ topologicalClosure S }) r) = ↑(algebraMap R B) r) },\n map_star' := (_ : ∀ (r : { x // x ∈ topologicalClosure S }), ↑ψ (star r) = star (↑ψ r)) }\n⊢ ↑(let src := ↑φ;\n {\n toAlgHom :=\n {\n toRingHom :=\n {\n toMonoidHom :=\n { toOneHom := { toFun := ↑φ, map_one' := (_ : OneHom.toFun (↑↑↑↑φ) 1 = 1) },\n map_mul' :=\n (_ :\n ∀ (x y : { x // x ∈ topologicalClosure S }),\n OneHom.toFun (↑↑↑↑φ) (x * y) = OneHom.toFun (↑↑↑↑φ) x * OneHom.toFun (↑↑↑↑φ) y) },\n map_zero' := (_ : OneHom.toFun (↑↑↑↑φ) 0 = 0),\n map_add' :=\n (_ :\n ∀ (x y : { x // x ∈ topologicalClosure S }),\n OneHom.toFun (↑↑↑↑φ) (x + y) = OneHom.toFun (↑↑↑↑φ) x + OneHom.toFun (↑↑↑↑φ) y) },\n commutes' :=\n (_ :\n ∀ (r : R),\n OneHom.toFun (↑↑↑src) (↑(algebraMap R { x // x ∈ topologicalClosure S }) r) = ↑(algebraMap R B) r) },\n map_star' := (_ : ∀ (r : { x // x ∈ topologicalClosure S }), ↑φ (star r) = star (↑φ r)) }) =\n ↑ψ",
"tactic": "apply congrArg _ this"
},
{
"state_after": "R : Type u_3\nA : Type u_4\nB : Type u_1\ninst✝¹⁴ : CommSemiring R\ninst✝¹³ : StarRing R\ninst✝¹² : TopologicalSpace A\ninst✝¹¹ : Semiring A\ninst✝¹⁰ : Algebra R A\ninst✝⁹ : StarRing A\ninst✝⁸ : StarModule R A\ninst✝⁷ : TopologicalSemiring A\ninst✝⁶ : ContinuousStar A\ninst✝⁵ : TopologicalSpace B\ninst✝⁴ : Semiring B\ninst✝³ : Algebra R B\ninst✝² : StarRing B\ninst✝¹ : T2Space B\nF : Type u_2\nS : StarSubalgebra R A\ninst✝ : StarAlgHomClass F R { x // x ∈ topologicalClosure S } B\nφ ψ : F\nhφ : Continuous ↑φ\nhψ : Continuous ↑ψ\nh :\n ∀ (x : { x // x ∈ S }),\n ↑φ (↑(inclusion (_ : S ≤ topologicalClosure S)) x) = ↑ψ (↑(inclusion (_ : S ≤ topologicalClosure S)) x)\n⊢ ∀ (x : { x // x ∈ S }),\n ↑(StarAlgHom.comp\n (let src := ↑φ;\n {\n toAlgHom :=\n {\n toRingHom :=\n {\n toMonoidHom :=\n { toOneHom := { toFun := ↑φ, map_one' := (_ : OneHom.toFun (↑↑↑↑φ) 1 = 1) },\n map_mul' :=\n (_ :\n ∀ (x y : { x // x ∈ topologicalClosure S }),\n OneHom.toFun (↑↑↑↑φ) (x * y) = OneHom.toFun (↑↑↑↑φ) x * OneHom.toFun (↑↑↑↑φ) y) },\n map_zero' := (_ : OneHom.toFun (↑↑↑↑φ) 0 = 0),\n map_add' :=\n (_ :\n ∀ (x y : { x // x ∈ topologicalClosure S }),\n OneHom.toFun (↑↑↑↑φ) (x + y) = OneHom.toFun (↑↑↑↑φ) x + OneHom.toFun (↑↑↑↑φ) y) },\n commutes' :=\n (_ :\n ∀ (r : R),\n OneHom.toFun (↑↑↑src) (↑(algebraMap R { x // x ∈ topologicalClosure S }) r) =\n ↑(algebraMap R B) r) },\n map_star' := (_ : ∀ (r : { x // x ∈ topologicalClosure S }), ↑φ (star r) = star (↑φ r)) })\n (inclusion (_ : S ≤ topologicalClosure S)))\n x =\n ↑(StarAlgHom.comp\n (let src := ↑ψ;\n {\n toAlgHom :=\n {\n toRingHom :=\n {\n toMonoidHom :=\n { toOneHom := { toFun := ↑ψ, map_one' := (_ : OneHom.toFun (↑↑↑↑ψ) 1 = 1) },\n map_mul' :=\n (_ :\n ∀ (x y : { x // x ∈ topologicalClosure S }),\n OneHom.toFun (↑↑↑↑ψ) (x * y) = OneHom.toFun (↑↑↑↑ψ) x * OneHom.toFun (↑↑↑↑ψ) y) },\n map_zero' := (_ : OneHom.toFun (↑↑↑↑ψ) 0 = 0),\n map_add' :=\n (_ :\n ∀ (x y : { x // x ∈ topologicalClosure S }),\n OneHom.toFun (↑↑↑↑ψ) (x + y) = OneHom.toFun (↑↑↑↑ψ) x + OneHom.toFun (↑↑↑↑ψ) y) },\n commutes' :=\n (_ :\n ∀ (r : R),\n OneHom.toFun (↑↑↑src) (↑(algebraMap R { x // x ∈ topologicalClosure S }) r) =\n ↑(algebraMap R B) r) },\n map_star' := (_ : ∀ (r : { x // x ∈ topologicalClosure S }), ↑ψ (star r) = star (↑ψ r)) })\n (inclusion (_ : S ≤ topologicalClosure S)))\n x",
"state_before": "R : Type u_3\nA : Type u_4\nB : Type u_1\ninst✝¹⁴ : CommSemiring R\ninst✝¹³ : StarRing R\ninst✝¹² : TopologicalSpace A\ninst✝¹¹ : Semiring A\ninst✝¹⁰ : Algebra R A\ninst✝⁹ : StarRing A\ninst✝⁸ : StarModule R A\ninst✝⁷ : TopologicalSemiring A\ninst✝⁶ : ContinuousStar A\ninst✝⁵ : TopologicalSpace B\ninst✝⁴ : Semiring B\ninst✝³ : Algebra R B\ninst✝² : StarRing B\ninst✝¹ : T2Space B\nF : Type u_2\nS : StarSubalgebra R A\ninst✝ : StarAlgHomClass F R { x // x ∈ topologicalClosure S } B\nφ ψ : F\nhφ : Continuous ↑φ\nhψ : Continuous ↑ψ\nh :\n ∀ (x : { x // x ∈ S }),\n ↑φ (↑(inclusion (_ : S ≤ topologicalClosure S)) x) = ↑ψ (↑(inclusion (_ : S ≤ topologicalClosure S)) x)\n⊢ (let src := ↑φ;\n {\n toAlgHom :=\n {\n toRingHom :=\n {\n toMonoidHom :=\n { toOneHom := { toFun := ↑φ, map_one' := (_ : OneHom.toFun (↑↑↑↑φ) 1 = 1) },\n map_mul' :=\n (_ :\n ∀ (x y : { x // x ∈ topologicalClosure S }),\n OneHom.toFun (↑↑↑↑φ) (x * y) = OneHom.toFun (↑↑↑↑φ) x * OneHom.toFun (↑↑↑↑φ) y) },\n map_zero' := (_ : OneHom.toFun (↑↑↑↑φ) 0 = 0),\n map_add' :=\n (_ :\n ∀ (x y : { x // x ∈ topologicalClosure S }),\n OneHom.toFun (↑↑↑↑φ) (x + y) = OneHom.toFun (↑↑↑↑φ) x + OneHom.toFun (↑↑↑↑φ) y) },\n commutes' :=\n (_ :\n ∀ (r : R),\n OneHom.toFun (↑↑↑src) (↑(algebraMap R { x // x ∈ topologicalClosure S }) r) = ↑(algebraMap R B) r) },\n map_star' := (_ : ∀ (r : { x // x ∈ topologicalClosure S }), ↑φ (star r) = star (↑φ r)) }) =\n let src := ↑ψ;\n {\n toAlgHom :=\n {\n toRingHom :=\n {\n toMonoidHom :=\n { toOneHom := { toFun := ↑ψ, map_one' := (_ : OneHom.toFun (↑↑↑↑ψ) 1 = 1) },\n map_mul' :=\n (_ :\n ∀ (x y : { x // x ∈ topologicalClosure S }),\n OneHom.toFun (↑↑↑↑ψ) (x * y) = OneHom.toFun (↑↑↑↑ψ) x * OneHom.toFun (↑↑↑↑ψ) y) },\n map_zero' := (_ : OneHom.toFun (↑↑↑↑ψ) 0 = 0),\n map_add' :=\n (_ :\n ∀ (x y : { x // x ∈ topologicalClosure S }),\n OneHom.toFun (↑↑↑↑ψ) (x + y) = OneHom.toFun (↑↑↑↑ψ) x + OneHom.toFun (↑↑↑↑ψ) y) },\n commutes' :=\n (_ :\n ∀ (r : R),\n OneHom.toFun (↑↑↑src) (↑(algebraMap R { x // x ∈ topologicalClosure S }) r) = ↑(algebraMap R B) r) },\n map_star' := (_ : ∀ (r : { x // x ∈ topologicalClosure S }), ↑ψ (star r) = star (↑ψ r)) }",
"tactic": "refine StarAlgHom.ext_topologicalClosure (R := R) (A := A) (B := B) hφ hψ (StarAlgHom.ext ?_)"
},
{
"state_after": "no goals",
"state_before": "R : Type u_3\nA : Type u_4\nB : Type u_1\ninst✝¹⁴ : CommSemiring R\ninst✝¹³ : StarRing R\ninst✝¹² : TopologicalSpace A\ninst✝¹¹ : Semiring A\ninst✝¹⁰ : Algebra R A\ninst✝⁹ : StarRing A\ninst✝⁸ : StarModule R A\ninst✝⁷ : TopologicalSemiring A\ninst✝⁶ : ContinuousStar A\ninst✝⁵ : TopologicalSpace B\ninst✝⁴ : Semiring B\ninst✝³ : Algebra R B\ninst✝² : StarRing B\ninst✝¹ : T2Space B\nF : Type u_2\nS : StarSubalgebra R A\ninst✝ : StarAlgHomClass F R { x // x ∈ topologicalClosure S } B\nφ ψ : F\nhφ : Continuous ↑φ\nhψ : Continuous ↑ψ\nh :\n ∀ (x : { x // x ∈ S }),\n ↑φ (↑(inclusion (_ : S ≤ topologicalClosure S)) x) = ↑ψ (↑(inclusion (_ : S ≤ topologicalClosure S)) x)\n⊢ ∀ (x : { x // x ∈ S }),\n ↑(StarAlgHom.comp\n (let src := ↑φ;\n {\n toAlgHom :=\n {\n toRingHom :=\n {\n toMonoidHom :=\n { toOneHom := { toFun := ↑φ, map_one' := (_ : OneHom.toFun (↑↑↑↑φ) 1 = 1) },\n map_mul' :=\n (_ :\n ∀ (x y : { x // x ∈ topologicalClosure S }),\n OneHom.toFun (↑↑↑↑φ) (x * y) = OneHom.toFun (↑↑↑↑φ) x * OneHom.toFun (↑↑↑↑φ) y) },\n map_zero' := (_ : OneHom.toFun (↑↑↑↑φ) 0 = 0),\n map_add' :=\n (_ :\n ∀ (x y : { x // x ∈ topologicalClosure S }),\n OneHom.toFun (↑↑↑↑φ) (x + y) = OneHom.toFun (↑↑↑↑φ) x + OneHom.toFun (↑↑↑↑φ) y) },\n commutes' :=\n (_ :\n ∀ (r : R),\n OneHom.toFun (↑↑↑src) (↑(algebraMap R { x // x ∈ topologicalClosure S }) r) =\n ↑(algebraMap R B) r) },\n map_star' := (_ : ∀ (r : { x // x ∈ topologicalClosure S }), ↑φ (star r) = star (↑φ r)) })\n (inclusion (_ : S ≤ topologicalClosure S)))\n x =\n ↑(StarAlgHom.comp\n (let src := ↑ψ;\n {\n toAlgHom :=\n {\n toRingHom :=\n {\n toMonoidHom :=\n { toOneHom := { toFun := ↑ψ, map_one' := (_ : OneHom.toFun (↑↑↑↑ψ) 1 = 1) },\n map_mul' :=\n (_ :\n ∀ (x y : { x // x ∈ topologicalClosure S }),\n OneHom.toFun (↑↑↑↑ψ) (x * y) = OneHom.toFun (↑↑↑↑ψ) x * OneHom.toFun (↑↑↑↑ψ) y) },\n map_zero' := (_ : OneHom.toFun (↑↑↑↑ψ) 0 = 0),\n map_add' :=\n (_ :\n ∀ (x y : { x // x ∈ topologicalClosure S }),\n OneHom.toFun (↑↑↑↑ψ) (x + y) = OneHom.toFun (↑↑↑↑ψ) x + OneHom.toFun (↑↑↑↑ψ) y) },\n commutes' :=\n (_ :\n ∀ (r : R),\n OneHom.toFun (↑↑↑src) (↑(algebraMap R { x // x ∈ topologicalClosure S }) r) =\n ↑(algebraMap R B) r) },\n map_star' := (_ : ∀ (r : { x // x ∈ topologicalClosure S }), ↑ψ (star r) = star (↑ψ r)) })\n (inclusion (_ : S ≤ topologicalClosure S)))\n x",
"tactic": "simpa only [StarAlgHom.coe_comp, StarAlgHom.coe_coe] using h"
}
] |
[
159,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
149,
1
] |
Mathlib/Data/Finset/Basic.lean
|
Finset.le_eq_subset
|
[] |
[
397,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
396,
1
] |
Mathlib/Geometry/Euclidean/Basic.lean
|
EuclideanGeometry.inter_eq_singleton_orthogonalProjection
|
[
{
"state_after": "V : Type u_1\nP : Type u_2\ninst✝⁵ : NormedAddCommGroup V\ninst✝⁴ : InnerProductSpace ℝ V\ninst✝³ : MetricSpace P\ninst✝² : NormedAddTorsor V P\ns : AffineSubspace ℝ P\ninst✝¹ : Nonempty { x // x ∈ s }\ninst✝ : CompleteSpace { x // x ∈ direction s }\np : P\n⊢ ↑s ∩ ↑(mk' p (direction s)ᗮ) = {orthogonalProjectionFn s p}",
"state_before": "V : Type u_1\nP : Type u_2\ninst✝⁵ : NormedAddCommGroup V\ninst✝⁴ : InnerProductSpace ℝ V\ninst✝³ : MetricSpace P\ninst✝² : NormedAddTorsor V P\ns : AffineSubspace ℝ P\ninst✝¹ : Nonempty { x // x ∈ s }\ninst✝ : CompleteSpace { x // x ∈ direction s }\np : P\n⊢ ↑s ∩ ↑(mk' p (direction s)ᗮ) = {↑(↑(orthogonalProjection s) p)}",
"tactic": "rw [← orthogonalProjectionFn_eq]"
},
{
"state_after": "no goals",
"state_before": "V : Type u_1\nP : Type u_2\ninst✝⁵ : NormedAddCommGroup V\ninst✝⁴ : InnerProductSpace ℝ V\ninst✝³ : MetricSpace P\ninst✝² : NormedAddTorsor V P\ns : AffineSubspace ℝ P\ninst✝¹ : Nonempty { x // x ∈ s }\ninst✝ : CompleteSpace { x // x ∈ direction s }\np : P\n⊢ ↑s ∩ ↑(mk' p (direction s)ᗮ) = {orthogonalProjectionFn s p}",
"tactic": "exact inter_eq_singleton_orthogonalProjectionFn p"
}
] |
[
345,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
341,
1
] |
Mathlib/GroupTheory/Submonoid/Inverses.lean
|
Submonoid.leftInvEquiv_symm_mul
|
[
{
"state_after": "case h.e'_2.h.e'_6.h.e'_3\nM : Type u_1\ninst✝ : CommMonoid M\nS : Submonoid M\nhS : S ≤ IsUnit.submonoid M\nx : { x // x ∈ S }\n⊢ x = ↑(leftInvEquiv S hS) (↑(MulEquiv.symm (leftInvEquiv S hS)) x)",
"state_before": "M : Type u_1\ninst✝ : CommMonoid M\nS : Submonoid M\nhS : S ≤ IsUnit.submonoid M\nx : { x // x ∈ S }\n⊢ ↑(↑(MulEquiv.symm (leftInvEquiv S hS)) x) * ↑x = 1",
"tactic": "convert S.mul_leftInvEquiv hS ((S.leftInvEquiv hS).symm x)"
},
{
"state_after": "no goals",
"state_before": "case h.e'_2.h.e'_6.h.e'_3\nM : Type u_1\ninst✝ : CommMonoid M\nS : Submonoid M\nhS : S ≤ IsUnit.submonoid M\nx : { x // x ∈ S }\n⊢ x = ↑(leftInvEquiv S hS) (↑(MulEquiv.symm (leftInvEquiv S hS)) x)",
"tactic": "simp"
}
] |
[
211,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
209,
1
] |
Mathlib/Data/Finset/LocallyFinite.lean
|
Finset.Ioc_eq_empty_of_le
|
[] |
[
114,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
113,
1
] |
Mathlib/NumberTheory/ArithmeticFunction.lean
|
Nat.ArithmeticFunction.zero_apply
|
[] |
[
107,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
106,
1
] |
Mathlib/Order/SuccPred/Basic.lean
|
Order.Iio_succ
|
[] |
[
367,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
366,
1
] |
Mathlib/GroupTheory/Commensurable.lean
|
Commensurable.commensurator_mem_iff
|
[] |
[
112,
79
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
111,
1
] |
Mathlib/Algebra/Algebra/Tower.lean
|
AlgHom.restrictScalars_apply
|
[] |
[
194,
91
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
194,
1
] |
Mathlib/Combinatorics/SimpleGraph/Coloring.lean
|
SimpleGraph.colorable_set_nonempty_of_colorable
|
[] |
[
246,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
244,
1
] |
Mathlib/Order/Iterate.lean
|
Monotone.iterate_comp_le_of_le
|
[] |
[
105,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
103,
1
] |
Mathlib/Analysis/Calculus/Series.lean
|
tendstoUniformly_tsum_nat
|
[] |
[
78,
82
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
74,
1
] |
Mathlib/Data/Dfinsupp/Basic.lean
|
Dfinsupp.sumAddHom_singleAddHom
|
[] |
[
2068,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2066,
1
] |
Mathlib/Order/Lattice.lean
|
SemilatticeInf.ext
|
[
{
"state_after": "α✝ : Type u\nβ : Type v\ninst✝ : SemilatticeInf α✝\na b c d : α✝\nα : Type u_1\nA B : SemilatticeInf α\nH : ∀ (x y : α), x ≤ y ↔ x ≤ y\nss : toInf = toInf\n⊢ A = B",
"state_before": "α✝ : Type u\nβ : Type v\ninst✝ : SemilatticeInf α✝\na b c d : α✝\nα : Type u_1\nA B : SemilatticeInf α\nH : ∀ (x y : α), x ≤ y ↔ x ≤ y\n⊢ A = B",
"tactic": "have ss : A.toInf = B.toInf := by ext; apply SemilatticeInf.ext_inf H"
},
{
"state_after": "case mk\nα✝ : Type u\nβ : Type v\ninst✝ : SemilatticeInf α✝\na b c d : α✝\nα : Type u_1\nB : SemilatticeInf α\ntoInf✝ : Inf α\ntoPartialOrder✝ : PartialOrder α\ninf_le_left✝ : ∀ (a b : α), a ⊓ b ≤ a\ninf_le_right✝ : ∀ (a b : α), a ⊓ b ≤ b\nle_inf✝ : ∀ (a b c : α), a ≤ b → a ≤ c → a ≤ b ⊓ c\nH : ∀ (x y : α), x ≤ y ↔ x ≤ y\nss : toInf = toInf\n⊢ mk inf_le_left✝ inf_le_right✝ le_inf✝ = B",
"state_before": "α✝ : Type u\nβ : Type v\ninst✝ : SemilatticeInf α✝\na b c d : α✝\nα : Type u_1\nA B : SemilatticeInf α\nH : ∀ (x y : α), x ≤ y ↔ x ≤ y\nss : toInf = toInf\n⊢ A = B",
"tactic": "cases A"
},
{
"state_after": "case mk.mk\nα✝ : Type u\nβ : Type v\ninst✝ : SemilatticeInf α✝\na b c d : α✝\nα : Type u_1\ntoInf✝¹ : Inf α\ntoPartialOrder✝¹ : PartialOrder α\ninf_le_left✝¹ : ∀ (a b : α), a ⊓ b ≤ a\ninf_le_right✝¹ : ∀ (a b : α), a ⊓ b ≤ b\nle_inf✝¹ : ∀ (a b c : α), a ≤ b → a ≤ c → a ≤ b ⊓ c\ntoInf✝ : Inf α\ntoPartialOrder✝ : PartialOrder α\ninf_le_left✝ : ∀ (a b : α), a ⊓ b ≤ a\ninf_le_right✝ : ∀ (a b : α), a ⊓ b ≤ b\nle_inf✝ : ∀ (a b c : α), a ≤ b → a ≤ c → a ≤ b ⊓ c\nH : ∀ (x y : α), x ≤ y ↔ x ≤ y\nss : toInf = toInf\n⊢ mk inf_le_left✝¹ inf_le_right✝¹ le_inf✝¹ = mk inf_le_left✝ inf_le_right✝ le_inf✝",
"state_before": "case mk\nα✝ : Type u\nβ : Type v\ninst✝ : SemilatticeInf α✝\na b c d : α✝\nα : Type u_1\nB : SemilatticeInf α\ntoInf✝ : Inf α\ntoPartialOrder✝ : PartialOrder α\ninf_le_left✝ : ∀ (a b : α), a ⊓ b ≤ a\ninf_le_right✝ : ∀ (a b : α), a ⊓ b ≤ b\nle_inf✝ : ∀ (a b c : α), a ≤ b → a ≤ c → a ≤ b ⊓ c\nH : ∀ (x y : α), x ≤ y ↔ x ≤ y\nss : toInf = toInf\n⊢ mk inf_le_left✝ inf_le_right✝ le_inf✝ = B",
"tactic": "cases B"
},
{
"state_after": "case mk.mk.refl\nα✝ : Type u\nβ : Type v\ninst✝ : SemilatticeInf α✝\na b c d : α✝\nα : Type u_1\ntoInf✝¹ : Inf α\ntoPartialOrder✝ : PartialOrder α\ninf_le_left✝¹ : ∀ (a b : α), a ⊓ b ≤ a\ninf_le_right✝¹ : ∀ (a b : α), a ⊓ b ≤ b\nle_inf✝¹ : ∀ (a b c : α), a ≤ b → a ≤ c → a ≤ b ⊓ c\ntoInf✝ : Inf α\ninf_le_left✝ : ∀ (a b : α), a ⊓ b ≤ a\ninf_le_right✝ : ∀ (a b : α), a ⊓ b ≤ b\nle_inf✝ : ∀ (a b c : α), a ≤ b → a ≤ c → a ≤ b ⊓ c\nH : ∀ (x y : α), x ≤ y ↔ x ≤ y\nss : toInf = toInf\n⊢ mk inf_le_left✝¹ inf_le_right✝¹ le_inf✝¹ = mk inf_le_left✝ inf_le_right✝ le_inf✝",
"state_before": "case mk.mk\nα✝ : Type u\nβ : Type v\ninst✝ : SemilatticeInf α✝\na b c d : α✝\nα : Type u_1\ntoInf✝¹ : Inf α\ntoPartialOrder✝¹ : PartialOrder α\ninf_le_left✝¹ : ∀ (a b : α), a ⊓ b ≤ a\ninf_le_right✝¹ : ∀ (a b : α), a ⊓ b ≤ b\nle_inf✝¹ : ∀ (a b c : α), a ≤ b → a ≤ c → a ≤ b ⊓ c\ntoInf✝ : Inf α\ntoPartialOrder✝ : PartialOrder α\ninf_le_left✝ : ∀ (a b : α), a ⊓ b ≤ a\ninf_le_right✝ : ∀ (a b : α), a ⊓ b ≤ b\nle_inf✝ : ∀ (a b c : α), a ≤ b → a ≤ c → a ≤ b ⊓ c\nH : ∀ (x y : α), x ≤ y ↔ x ≤ y\nss : toInf = toInf\n⊢ mk inf_le_left✝¹ inf_le_right✝¹ le_inf✝¹ = mk inf_le_left✝ inf_le_right✝ le_inf✝",
"tactic": "cases PartialOrder.ext H"
},
{
"state_after": "no goals",
"state_before": "case mk.mk.refl\nα✝ : Type u\nβ : Type v\ninst✝ : SemilatticeInf α✝\na b c d : α✝\nα : Type u_1\ntoInf✝¹ : Inf α\ntoPartialOrder✝ : PartialOrder α\ninf_le_left✝¹ : ∀ (a b : α), a ⊓ b ≤ a\ninf_le_right✝¹ : ∀ (a b : α), a ⊓ b ≤ b\nle_inf✝¹ : ∀ (a b c : α), a ≤ b → a ≤ c → a ≤ b ⊓ c\ntoInf✝ : Inf α\ninf_le_left✝ : ∀ (a b : α), a ⊓ b ≤ a\ninf_le_right✝ : ∀ (a b : α), a ⊓ b ≤ b\nle_inf✝ : ∀ (a b c : α), a ≤ b → a ≤ c → a ≤ b ⊓ c\nH : ∀ (x y : α), x ≤ y ↔ x ≤ y\nss : toInf = toInf\n⊢ mk inf_le_left✝¹ inf_le_right✝¹ le_inf✝¹ = mk inf_le_left✝ inf_le_right✝ le_inf✝",
"tactic": "congr"
},
{
"state_after": "case inf.h.h\nα✝ : Type u\nβ : Type v\ninst✝ : SemilatticeInf α✝\na b c d : α✝\nα : Type u_1\nA B : SemilatticeInf α\nH : ∀ (x y : α), x ≤ y ↔ x ≤ y\nx✝¹ x✝ : α\n⊢ x✝¹ ⊓ x✝ = x✝¹ ⊓ x✝",
"state_before": "α✝ : Type u\nβ : Type v\ninst✝ : SemilatticeInf α✝\na b c d : α✝\nα : Type u_1\nA B : SemilatticeInf α\nH : ∀ (x y : α), x ≤ y ↔ x ≤ y\n⊢ toInf = toInf",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case inf.h.h\nα✝ : Type u\nβ : Type v\ninst✝ : SemilatticeInf α✝\na b c d : α✝\nα : Type u_1\nA B : SemilatticeInf α\nH : ∀ (x y : α), x ≤ y ↔ x ≤ y\nx✝¹ x✝ : α\n⊢ x✝¹ ⊓ x✝ = x✝¹ ⊓ x✝",
"tactic": "apply SemilatticeInf.ext_inf H"
}
] |
[
577,
8
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
570,
1
] |
Mathlib/Analysis/Normed/Group/Basic.lean
|
preimage_mul_ball
|
[
{
"state_after": "case h\n𝓕 : Type ?u.743408\n𝕜 : Type ?u.743411\nα : Type ?u.743414\nι : Type ?u.743417\nκ : Type ?u.743420\nE : Type u_1\nF : Type ?u.743426\nG : Type ?u.743429\ninst✝¹ : SeminormedCommGroup E\ninst✝ : SeminormedCommGroup F\na✝ a₁ a₂ b✝ b₁ b₂ : E\nr✝ r₁ r₂ : ℝ\na b : E\nr : ℝ\nc : E\n⊢ c ∈ (fun x x_1 => x * x_1) b ⁻¹' ball a r ↔ c ∈ ball (a / b) r",
"state_before": "𝓕 : Type ?u.743408\n𝕜 : Type ?u.743411\nα : Type ?u.743414\nι : Type ?u.743417\nκ : Type ?u.743420\nE : Type u_1\nF : Type ?u.743426\nG : Type ?u.743429\ninst✝¹ : SeminormedCommGroup E\ninst✝ : SeminormedCommGroup F\na✝ a₁ a₂ b✝ b₁ b₂ : E\nr✝ r₁ r₂ : ℝ\na b : E\nr : ℝ\n⊢ (fun x x_1 => x * x_1) b ⁻¹' ball a r = ball (a / b) r",
"tactic": "ext c"
},
{
"state_after": "no goals",
"state_before": "case h\n𝓕 : Type ?u.743408\n𝕜 : Type ?u.743411\nα : Type ?u.743414\nι : Type ?u.743417\nκ : Type ?u.743420\nE : Type u_1\nF : Type ?u.743426\nG : Type ?u.743429\ninst✝¹ : SeminormedCommGroup E\ninst✝ : SeminormedCommGroup F\na✝ a₁ a₂ b✝ b₁ b₂ : E\nr✝ r₁ r₂ : ℝ\na b : E\nr : ℝ\nc : E\n⊢ c ∈ (fun x x_1 => x * x_1) b ⁻¹' ball a r ↔ c ∈ ball (a / b) r",
"tactic": "simp only [dist_eq_norm_div, Set.mem_preimage, mem_ball, div_div_eq_mul_div, mul_comm]"
}
] |
[
1528,
89
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1526,
1
] |
Std/Data/Nat/Lemmas.lean
|
Nat.sub_mul_div
|
[
{
"state_after": "no goals",
"state_before": "x n p : Nat\nh₁ : n * p ≤ x\n⊢ (x - n * p) / n = x / n - p",
"tactic": "match eq_zero_or_pos n with\n| .inl h₀ => rw [h₀, Nat.div_zero, Nat.div_zero, Nat.zero_sub]\n| .inr h₀ => induction p with\n | zero => rw [Nat.mul_zero, Nat.sub_zero, Nat.sub_zero]\n | succ p IH =>\n have h₂ : n * p ≤ x := Nat.le_trans (Nat.mul_le_mul_left _ (le_succ _)) h₁\n have h₃ : x - n * p ≥ n := by\n apply Nat.le_of_add_le_add_right\n rw [Nat.sub_add_cancel h₂, Nat.add_comm]\n rw [mul_succ] at h₁\n exact h₁\n rw [sub_succ, ← IH h₂, div_eq_sub_div h₀ h₃]\n simp [add_one, Nat.pred_succ, mul_succ, Nat.sub_sub]"
},
{
"state_after": "no goals",
"state_before": "x n p : Nat\nh₁ : n * p ≤ x\nh₀ : n = 0\n⊢ (x - n * p) / n = x / n - p",
"tactic": "rw [h₀, Nat.div_zero, Nat.div_zero, Nat.zero_sub]"
},
{
"state_after": "no goals",
"state_before": "x n p : Nat\nh₁ : n * p ≤ x\nh₀ : n > 0\n⊢ (x - n * p) / n = x / n - p",
"tactic": "induction p with\n| zero => rw [Nat.mul_zero, Nat.sub_zero, Nat.sub_zero]\n| succ p IH =>\nhave h₂ : n * p ≤ x := Nat.le_trans (Nat.mul_le_mul_left _ (le_succ _)) h₁\nhave h₃ : x - n * p ≥ n := by\napply Nat.le_of_add_le_add_right\nrw [Nat.sub_add_cancel h₂, Nat.add_comm]\nrw [mul_succ] at h₁\nexact h₁\nrw [sub_succ, ← IH h₂, div_eq_sub_div h₀ h₃]\nsimp [add_one, Nat.pred_succ, mul_succ, Nat.sub_sub]"
},
{
"state_after": "no goals",
"state_before": "case zero\nx n : Nat\nh₀ : n > 0\nh₁ : n * zero ≤ x\n⊢ (x - n * zero) / n = x / n - zero",
"tactic": "rw [Nat.mul_zero, Nat.sub_zero, Nat.sub_zero]"
},
{
"state_after": "case succ\nx n : Nat\nh₀ : n > 0\np : Nat\nIH : n * p ≤ x → (x - n * p) / n = x / n - p\nh₁ : n * succ p ≤ x\nh₂ : n * p ≤ x\n⊢ (x - n * succ p) / n = x / n - succ p",
"state_before": "case succ\nx n : Nat\nh₀ : n > 0\np : Nat\nIH : n * p ≤ x → (x - n * p) / n = x / n - p\nh₁ : n * succ p ≤ x\n⊢ (x - n * succ p) / n = x / n - succ p",
"tactic": "have h₂ : n * p ≤ x := Nat.le_trans (Nat.mul_le_mul_left _ (le_succ _)) h₁"
},
{
"state_after": "case succ\nx n : Nat\nh₀ : n > 0\np : Nat\nIH : n * p ≤ x → (x - n * p) / n = x / n - p\nh₁ : n * succ p ≤ x\nh₂ : n * p ≤ x\nh₃ : x - n * p ≥ n\n⊢ (x - n * succ p) / n = x / n - succ p",
"state_before": "case succ\nx n : Nat\nh₀ : n > 0\np : Nat\nIH : n * p ≤ x → (x - n * p) / n = x / n - p\nh₁ : n * succ p ≤ x\nh₂ : n * p ≤ x\n⊢ (x - n * succ p) / n = x / n - succ p",
"tactic": "have h₃ : x - n * p ≥ n := by\n apply Nat.le_of_add_le_add_right\n rw [Nat.sub_add_cancel h₂, Nat.add_comm]\n rw [mul_succ] at h₁\n exact h₁"
},
{
"state_after": "case succ\nx n : Nat\nh₀ : n > 0\np : Nat\nIH : n * p ≤ x → (x - n * p) / n = x / n - p\nh₁ : n * succ p ≤ x\nh₂ : n * p ≤ x\nh₃ : x - n * p ≥ n\n⊢ (x - n * succ p) / n = pred ((x - n * p - n) / n + 1)",
"state_before": "case succ\nx n : Nat\nh₀ : n > 0\np : Nat\nIH : n * p ≤ x → (x - n * p) / n = x / n - p\nh₁ : n * succ p ≤ x\nh₂ : n * p ≤ x\nh₃ : x - n * p ≥ n\n⊢ (x - n * succ p) / n = x / n - succ p",
"tactic": "rw [sub_succ, ← IH h₂, div_eq_sub_div h₀ h₃]"
},
{
"state_after": "no goals",
"state_before": "case succ\nx n : Nat\nh₀ : n > 0\np : Nat\nIH : n * p ≤ x → (x - n * p) / n = x / n - p\nh₁ : n * succ p ≤ x\nh₂ : n * p ≤ x\nh₃ : x - n * p ≥ n\n⊢ (x - n * succ p) / n = pred ((x - n * p - n) / n + 1)",
"tactic": "simp [add_one, Nat.pred_succ, mul_succ, Nat.sub_sub]"
},
{
"state_after": "case a\nx n : Nat\nh₀ : n > 0\np : Nat\nIH : n * p ≤ x → (x - n * p) / n = x / n - p\nh₁ : n * succ p ≤ x\nh₂ : n * p ≤ x\n⊢ n + ?b ≤ x - n * p + ?b\n\ncase b\nx n : Nat\nh₀ : n > 0\np : Nat\nIH : n * p ≤ x → (x - n * p) / n = x / n - p\nh₁ : n * succ p ≤ x\nh₂ : n * p ≤ x\n⊢ Nat",
"state_before": "x n : Nat\nh₀ : n > 0\np : Nat\nIH : n * p ≤ x → (x - n * p) / n = x / n - p\nh₁ : n * succ p ≤ x\nh₂ : n * p ≤ x\n⊢ x - n * p ≥ n",
"tactic": "apply Nat.le_of_add_le_add_right"
},
{
"state_after": "case a\nx n : Nat\nh₀ : n > 0\np : Nat\nIH : n * p ≤ x → (x - n * p) / n = x / n - p\nh₁ : n * succ p ≤ x\nh₂ : n * p ≤ x\n⊢ n * p + n ≤ x",
"state_before": "case a\nx n : Nat\nh₀ : n > 0\np : Nat\nIH : n * p ≤ x → (x - n * p) / n = x / n - p\nh₁ : n * succ p ≤ x\nh₂ : n * p ≤ x\n⊢ n + ?b ≤ x - n * p + ?b\n\ncase b\nx n : Nat\nh₀ : n > 0\np : Nat\nIH : n * p ≤ x → (x - n * p) / n = x / n - p\nh₁ : n * succ p ≤ x\nh₂ : n * p ≤ x\n⊢ Nat",
"tactic": "rw [Nat.sub_add_cancel h₂, Nat.add_comm]"
},
{
"state_after": "case a\nx n : Nat\nh₀ : n > 0\np : Nat\nIH : n * p ≤ x → (x - n * p) / n = x / n - p\nh₁ : n * p + n ≤ x\nh₂ : n * p ≤ x\n⊢ n * p + n ≤ x",
"state_before": "case a\nx n : Nat\nh₀ : n > 0\np : Nat\nIH : n * p ≤ x → (x - n * p) / n = x / n - p\nh₁ : n * succ p ≤ x\nh₂ : n * p ≤ x\n⊢ n * p + n ≤ x",
"tactic": "rw [mul_succ] at h₁"
},
{
"state_after": "no goals",
"state_before": "case a\nx n : Nat\nh₀ : n > 0\np : Nat\nIH : n * p ≤ x → (x - n * p) / n = x / n - p\nh₁ : n * p + n ≤ x\nh₂ : n * p ≤ x\n⊢ n * p + n ≤ x",
"tactic": "exact h₁"
}
] |
[
558,
59
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
545,
1
] |
Mathlib/Topology/Spectral/Hom.lean
|
SpectralMap.coe_comp_continuousMap
|
[
{
"state_after": "no goals",
"state_before": "F : Type ?u.8928\nα : Type u_3\nβ : Type u_1\nγ : Type u_2\nδ : Type ?u.8940\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\ninst✝¹ : TopologicalSpace γ\ninst✝ : TopologicalSpace δ\nf : SpectralMap β γ\ng : SpectralMap α β\n⊢ ↑f ∘ ↑g = ↑(ContinuousMap.mk ↑f) ∘ ↑(ContinuousMap.mk ↑g)",
"tactic": "rfl"
}
] |
[
192,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
190,
1
] |
Mathlib/Data/Polynomial/RingDivision.lean
|
Polynomial.mem_roots_sub_C
|
[] |
[
643,
88
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
641,
1
] |
Mathlib/Probability/ProbabilityMassFunction/Constructions.lean
|
Pmf.monad_map_eq_map
|
[] |
[
49,
91
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
49,
1
] |
Mathlib/Probability/Kernel/Basic.lean
|
ProbabilityTheory.kernel.set_lintegral_piecewise
|
[
{
"state_after": "α : Type u_2\nβ : Type u_1\nι : Type ?u.1469486\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ η : { x // x ∈ kernel α β }\ns : Set α\nhs : MeasurableSet s\ninst✝ : DecidablePred fun x => x ∈ s\na : α\ng : β → ℝ≥0∞\nt : Set β\n⊢ (∫⁻ (b : β) in t, g b ∂if a ∈ s then ↑κ a else ↑η a) =\n if a ∈ s then ∫⁻ (b : β) in t, g b ∂↑κ a else ∫⁻ (b : β) in t, g b ∂↑η a",
"state_before": "α : Type u_2\nβ : Type u_1\nι : Type ?u.1469486\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ η : { x // x ∈ kernel α β }\ns : Set α\nhs : MeasurableSet s\ninst✝ : DecidablePred fun x => x ∈ s\na : α\ng : β → ℝ≥0∞\nt : Set β\n⊢ (∫⁻ (b : β) in t, g b ∂↑(piecewise hs κ η) a) =\n if a ∈ s then ∫⁻ (b : β) in t, g b ∂↑κ a else ∫⁻ (b : β) in t, g b ∂↑η a",
"tactic": "simp_rw [piecewise_apply]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type u_1\nι : Type ?u.1469486\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ η : { x // x ∈ kernel α β }\ns : Set α\nhs : MeasurableSet s\ninst✝ : DecidablePred fun x => x ∈ s\na : α\ng : β → ℝ≥0∞\nt : Set β\n⊢ (∫⁻ (b : β) in t, g b ∂if a ∈ s then ↑κ a else ↑η a) =\n if a ∈ s then ∫⁻ (b : β) in t, g b ∂↑κ a else ∫⁻ (b : β) in t, g b ∂↑η a",
"tactic": "split_ifs <;> rfl"
}
] |
[
656,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
653,
1
] |
Mathlib/Data/Finset/PImage.lean
|
Finset.pimage_mono
|
[] |
[
120,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
119,
1
] |
Mathlib/Analysis/NormedSpace/FiniteDimension.lean
|
summable_of_isBigO'
|
[] |
[
686,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
683,
1
] |
Mathlib/Data/Finsupp/Basic.lean
|
Finsupp.mapDomain_notin_range
|
[
{
"state_after": "α : Type u_1\nβ : Type u_3\nγ : Type ?u.158167\nι : Type ?u.158170\nM : Type u_2\nM' : Type ?u.158176\nN : Type ?u.158179\nP : Type ?u.158182\nG : Type ?u.158185\nH : Type ?u.158188\nR : Type ?u.158191\nS : Type ?u.158194\ninst✝ : AddCommMonoid M\nv v₁ v₂ : α →₀ M\nf : α → β\nx : α →₀ M\na : β\nh : ¬a ∈ Set.range f\n⊢ ∑ a_1 in x.support, ↑(single (f a_1) (↑x a_1)) a = 0",
"state_before": "α : Type u_1\nβ : Type u_3\nγ : Type ?u.158167\nι : Type ?u.158170\nM : Type u_2\nM' : Type ?u.158176\nN : Type ?u.158179\nP : Type ?u.158182\nG : Type ?u.158185\nH : Type ?u.158188\nR : Type ?u.158191\nS : Type ?u.158194\ninst✝ : AddCommMonoid M\nv v₁ v₂ : α →₀ M\nf : α → β\nx : α →₀ M\na : β\nh : ¬a ∈ Set.range f\n⊢ ↑(mapDomain f x) a = 0",
"tactic": "rw [mapDomain, sum_apply, sum]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_3\nγ : Type ?u.158167\nι : Type ?u.158170\nM : Type u_2\nM' : Type ?u.158176\nN : Type ?u.158179\nP : Type ?u.158182\nG : Type ?u.158185\nH : Type ?u.158188\nR : Type ?u.158191\nS : Type ?u.158194\ninst✝ : AddCommMonoid M\nv v₁ v₂ : α →₀ M\nf : α → β\nx : α →₀ M\na : β\nh : ¬a ∈ Set.range f\n⊢ ∑ a_1 in x.support, ↑(single (f a_1) (↑x a_1)) a = 0",
"tactic": "exact Finset.sum_eq_zero fun a' _ => single_eq_of_ne fun eq => h <| eq ▸ Set.mem_range_self _"
}
] |
[
464,
96
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
461,
1
] |
Mathlib/GroupTheory/Submonoid/Operations.lean
|
Submonoid.closure_closure_coe_preimage
|
[
{
"state_after": "case refine'_1\nM : Type u_1\nN : Type ?u.95294\nP : Type ?u.95297\ninst✝³ : MulOneClass M\ninst✝² : MulOneClass N\ninst✝¹ : MulOneClass P\nS : Submonoid M\nA : Type ?u.95318\ninst✝ : SetLike A M\nhA : SubmonoidClass A M\nS' : A\ns : Set M\nx✝¹ : { x // x ∈ ↑(closure s) }\nx : M\nhx : x ∈ ↑(closure s)\nx✝ : { val := x, property := hx } ∈ ⊤\n⊢ { val := 1, property := (_ : 1 ∈ closure s) } ∈ closure (Subtype.val ⁻¹' s)\n\ncase refine'_2\nM : Type u_1\nN : Type ?u.95294\nP : Type ?u.95297\ninst✝³ : MulOneClass M\ninst✝² : MulOneClass N\ninst✝¹ : MulOneClass P\nS : Submonoid M\nA : Type ?u.95318\ninst✝ : SetLike A M\nhA : SubmonoidClass A M\nS' : A\ns : Set M\nx✝¹ : { x // x ∈ ↑(closure s) }\nx : M\nhx : x ∈ ↑(closure s)\nx✝ : { val := x, property := hx } ∈ ⊤\ng₁ : M\ng₂ : g₁ ∈ closure s\nhg₁ : M\nhg₂ : hg₁ ∈ closure s\n⊢ { val := g₁, property := g₂ } ∈ closure (Subtype.val ⁻¹' s) →\n { val := hg₁, property := hg₂ } ∈ closure (Subtype.val ⁻¹' s) →\n { val := g₁ * hg₁, property := (_ : g₁ * hg₁ ∈ closure s) } ∈ closure (Subtype.val ⁻¹' s)",
"state_before": "M : Type u_1\nN : Type ?u.95294\nP : Type ?u.95297\ninst✝³ : MulOneClass M\ninst✝² : MulOneClass N\ninst✝¹ : MulOneClass P\nS : Submonoid M\nA : Type ?u.95318\ninst✝ : SetLike A M\nhA : SubmonoidClass A M\nS' : A\ns : Set M\nx✝¹ : { x // x ∈ ↑(closure s) }\nx : M\nhx : x ∈ ↑(closure s)\nx✝ : { val := x, property := hx } ∈ ⊤\n⊢ { val := x, property := hx } ∈ closure (Subtype.val ⁻¹' s)",
"tactic": "refine' closure_induction' _ (fun g hg => subset_closure hg) _ (fun g₁ g₂ hg₁ hg₂ => _) hx"
},
{
"state_after": "no goals",
"state_before": "case refine'_1\nM : Type u_1\nN : Type ?u.95294\nP : Type ?u.95297\ninst✝³ : MulOneClass M\ninst✝² : MulOneClass N\ninst✝¹ : MulOneClass P\nS : Submonoid M\nA : Type ?u.95318\ninst✝ : SetLike A M\nhA : SubmonoidClass A M\nS' : A\ns : Set M\nx✝¹ : { x // x ∈ ↑(closure s) }\nx : M\nhx : x ∈ ↑(closure s)\nx✝ : { val := x, property := hx } ∈ ⊤\n⊢ { val := 1, property := (_ : 1 ∈ closure s) } ∈ closure (Subtype.val ⁻¹' s)",
"tactic": "exact Submonoid.one_mem _"
},
{
"state_after": "no goals",
"state_before": "case refine'_2\nM : Type u_1\nN : Type ?u.95294\nP : Type ?u.95297\ninst✝³ : MulOneClass M\ninst✝² : MulOneClass N\ninst✝¹ : MulOneClass P\nS : Submonoid M\nA : Type ?u.95318\ninst✝ : SetLike A M\nhA : SubmonoidClass A M\nS' : A\ns : Set M\nx✝¹ : { x // x ∈ ↑(closure s) }\nx : M\nhx : x ∈ ↑(closure s)\nx✝ : { val := x, property := hx } ∈ ⊤\ng₁ : M\ng₂ : g₁ ∈ closure s\nhg₁ : M\nhg₂ : hg₁ ∈ closure s\n⊢ { val := g₁, property := g₂ } ∈ closure (Subtype.val ⁻¹' s) →\n { val := hg₁, property := hg₂ } ∈ closure (Subtype.val ⁻¹' s) →\n { val := g₁ * hg₁, property := (_ : g₁ * hg₁ ∈ closure s) } ∈ closure (Subtype.val ⁻¹' s)",
"tactic": "exact Submonoid.mul_mem _"
}
] |
[
833,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
828,
1
] |
Mathlib/RingTheory/AdjoinRoot.lean
|
AdjoinRoot.Minpoly.toAdjoin_apply'
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\nK : Type w\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nx : S\na : AdjoinRoot (minpoly R x)\n⊢ ↑(aeval { val := x, property := (_ : x ∈ adjoin R {x}) }) (minpoly R x) = 0",
"tactic": "simp [← Subalgebra.coe_eq_zero, aeval_subalgebra_coe]"
}
] |
[
632,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
628,
1
] |
Mathlib/RingTheory/Subring/Basic.lean
|
Subring.map_id
|
[] |
[
609,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
608,
1
] |
Mathlib/Algebra/Order/Monoid/WithTop.lean
|
WithBot.add_lt_add_left
|
[] |
[
686,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
684,
11
] |
Mathlib/Algebra/Hom/Equiv/Basic.lean
|
MulEquiv.equivLike_inv_eq_symm
|
[] |
[
281,
77
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
281,
1
] |
Mathlib/Algebra/GCDMonoid/Basic.lean
|
lcm_dvd_lcm_mul_right_right
|
[] |
[
861,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
860,
1
] |
Mathlib/Topology/MetricSpace/Basic.lean
|
Metric.ediam_univ_eq_top_iff_noncompact
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nX : Type ?u.534313\nι : Type ?u.534316\ninst✝¹ : PseudoMetricSpace α\ns : Set α\nx y z : α\ninst✝ : ProperSpace α\n⊢ EMetric.diam univ = ⊤ ↔ NoncompactSpace α",
"tactic": "rw [← not_compactSpace_iff, compactSpace_iff_bounded_univ, bounded_iff_ediam_ne_top,\n Classical.not_not]"
}
] |
[
2662,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2659,
1
] |
Mathlib/Algebra/DirectSum/Internal.lean
|
DirectSum.coe_mul_apply
|
[
{
"state_after": "ι : Type u_1\nσ : Type u_3\nS : Type ?u.100155\nR : Type u_2\ninst✝⁶ : DecidableEq ι\ninst✝⁵ : Semiring R\ninst✝⁴ : SetLike σ R\ninst✝³ : AddSubmonoidClass σ R\nA : ι → σ\ninst✝² : AddMonoid ι\ninst✝¹ : SetLike.GradedMonoid A\ninst✝ : (i : ι) → (x : { x // x ∈ A i }) → Decidable (x ≠ 0)\nr r' : ⨁ (i : ι), { x // x ∈ A i }\nn : ι\n⊢ ∑ i in Dfinsupp.support r ×ˢ Dfinsupp.support r',\n ↑(↑(↑(of (fun i => (fun i => { x // x ∈ A i }) i) (i.fst + i.snd)) (GradedMonoid.GMul.mul (↑r i.fst) (↑r' i.snd)))\n n) =\n ∑ ij in Finset.filter (fun ij => ij.fst + ij.snd = n) (Dfinsupp.support r ×ˢ Dfinsupp.support r'),\n ↑(↑r ij.fst) * ↑(↑r' ij.snd)",
"state_before": "ι : Type u_1\nσ : Type u_3\nS : Type ?u.100155\nR : Type u_2\ninst✝⁶ : DecidableEq ι\ninst✝⁵ : Semiring R\ninst✝⁴ : SetLike σ R\ninst✝³ : AddSubmonoidClass σ R\nA : ι → σ\ninst✝² : AddMonoid ι\ninst✝¹ : SetLike.GradedMonoid A\ninst✝ : (i : ι) → (x : { x // x ∈ A i }) → Decidable (x ≠ 0)\nr r' : ⨁ (i : ι), { x // x ∈ A i }\nn : ι\n⊢ ↑(↑(r * r') n) =\n ∑ ij in Finset.filter (fun ij => ij.fst + ij.snd = n) (Dfinsupp.support r ×ˢ Dfinsupp.support r'),\n ↑(↑r ij.fst) * ↑(↑r' ij.snd)",
"tactic": "rw [mul_eq_sum_support_ghas_mul, Dfinsupp.finset_sum_apply, AddSubmonoidClass.coe_finset_sum]"
},
{
"state_after": "no goals",
"state_before": "ι : Type u_1\nσ : Type u_3\nS : Type ?u.100155\nR : Type u_2\ninst✝⁶ : DecidableEq ι\ninst✝⁵ : Semiring R\ninst✝⁴ : SetLike σ R\ninst✝³ : AddSubmonoidClass σ R\nA : ι → σ\ninst✝² : AddMonoid ι\ninst✝¹ : SetLike.GradedMonoid A\ninst✝ : (i : ι) → (x : { x // x ∈ A i }) → Decidable (x ≠ 0)\nr r' : ⨁ (i : ι), { x // x ∈ A i }\nn : ι\n⊢ ∑ i in Dfinsupp.support r ×ˢ Dfinsupp.support r',\n ↑(↑(↑(of (fun i => (fun i => { x // x ∈ A i }) i) (i.fst + i.snd)) (GradedMonoid.GMul.mul (↑r i.fst) (↑r' i.snd)))\n n) =\n ∑ ij in Finset.filter (fun ij => ij.fst + ij.snd = n) (Dfinsupp.support r ×ˢ Dfinsupp.support r'),\n ↑(↑r ij.fst) * ↑(↑r' ij.snd)",
"tactic": "simp_rw [coe_of_apply, apply_ite, ZeroMemClass.coe_zero, ← Finset.sum_filter, SetLike.coe_gMul]"
}
] |
[
164,
98
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
158,
1
] |
Mathlib/Analysis/NormedSpace/AddTorsor.lean
|
dist_midpoint_left
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.60686\nV : Type u_3\nP : Type u_1\nW : Type ?u.60695\nQ : Type ?u.60698\ninst✝⁹ : SeminormedAddCommGroup V\ninst✝⁸ : PseudoMetricSpace P\ninst✝⁷ : NormedAddTorsor V P\ninst✝⁶ : NormedAddCommGroup W\ninst✝⁵ : MetricSpace Q\ninst✝⁴ : NormedAddTorsor W Q\n𝕜 : Type u_2\ninst✝³ : NormedField 𝕜\ninst✝² : NormedSpace 𝕜 V\ninst✝¹ : NormedSpace 𝕜 W\ninst✝ : Invertible 2\np₁ p₂ : P\n⊢ dist (midpoint 𝕜 p₁ p₂) p₁ = ‖2‖⁻¹ * dist p₁ p₂",
"tactic": "rw [dist_comm, dist_left_midpoint]"
}
] |
[
182,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
181,
1
] |
Mathlib/Data/Sym/Sym2.lean
|
Sym2.exists
|
[] |
[
133,
54
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
131,
11
] |
Mathlib/SetTheory/Game/PGame.lean
|
PGame.relabel_moveRight
|
[
{
"state_after": "no goals",
"state_before": "x : PGame\nxl' xr' : Type u_1\nel : xl' ≃ LeftMoves x\ner : xr' ≃ RightMoves x\nj : RightMoves x\n⊢ moveRight (relabel el er) (↑er.symm j) = moveRight x j",
"tactic": "simp"
}
] |
[
1166,
90
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1165,
1
] |
Mathlib/Combinatorics/SimpleGraph/Connectivity.lean
|
SimpleGraph.Walk.toSubgraph_append
|
[
{
"state_after": "no goals",
"state_before": "V : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nu v w : V\np : Walk G u v\nq : Walk G v w\n⊢ Walk.toSubgraph (append p q) = Walk.toSubgraph p ⊔ Walk.toSubgraph q",
"tactic": "induction p <;> simp [*, sup_assoc]"
}
] |
[
2254,
100
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2253,
1
] |
Mathlib/MeasureTheory/Measure/OuterMeasure.lean
|
MeasureTheory.OuterMeasure.sInf_eq_boundedBy_sInfGen
|
[
{
"state_after": "case refine'_1\nα : Type u_1\nm : Set (OuterMeasure α)\n⊢ sInf m ≤ boundedBy (sInfGen m)\n\ncase refine'_2\nα : Type u_1\nm : Set (OuterMeasure α)\n⊢ boundedBy (sInfGen m) ≤ sInf m",
"state_before": "α : Type u_1\nm : Set (OuterMeasure α)\n⊢ sInf m = boundedBy (sInfGen m)",
"tactic": "refine' le_antisymm _ _"
},
{
"state_after": "case refine'_1\nα : Type u_1\nm : Set (OuterMeasure α)\ns : Set α\nμ : OuterMeasure α\nhμ : μ ∈ m\n⊢ ↑(sInf m) s ≤ ↑μ s",
"state_before": "case refine'_1\nα : Type u_1\nm : Set (OuterMeasure α)\n⊢ sInf m ≤ boundedBy (sInfGen m)",
"tactic": "refine' le_boundedBy.2 fun s => le_iInf₂ fun μ hμ => _"
},
{
"state_after": "no goals",
"state_before": "case refine'_1\nα : Type u_1\nm : Set (OuterMeasure α)\ns : Set α\nμ : OuterMeasure α\nhμ : μ ∈ m\n⊢ ↑(sInf m) s ≤ ↑μ s",
"tactic": "apply sInf_le hμ"
},
{
"state_after": "case refine'_2\nα : Type u_1\nm : Set (OuterMeasure α)\n⊢ ∀ (b : OuterMeasure α), b ∈ m → boundedBy (sInfGen m) ≤ b",
"state_before": "case refine'_2\nα : Type u_1\nm : Set (OuterMeasure α)\n⊢ boundedBy (sInfGen m) ≤ sInf m",
"tactic": "refine' le_sInf _"
},
{
"state_after": "case refine'_2\nα : Type u_1\nm : Set (OuterMeasure α)\nμ : OuterMeasure α\nhμ : μ ∈ m\nt : Set α\n⊢ ↑(boundedBy (sInfGen m)) t ≤ ↑μ t",
"state_before": "case refine'_2\nα : Type u_1\nm : Set (OuterMeasure α)\n⊢ ∀ (b : OuterMeasure α), b ∈ m → boundedBy (sInfGen m) ≤ b",
"tactic": "intro μ hμ t"
},
{
"state_after": "no goals",
"state_before": "case refine'_2\nα : Type u_1\nm : Set (OuterMeasure α)\nμ : OuterMeasure α\nhμ : μ ∈ m\nt : Set α\n⊢ ↑(boundedBy (sInfGen m)) t ≤ ↑μ t",
"tactic": "refine' le_trans (boundedBy_le t) (iInf₂_le μ hμ)"
}
] |
[
1155,
54
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1148,
1
] |
Mathlib/Data/Bool/Basic.lean
|
Bool.decide_coe
|
[
{
"state_after": "case false\nh : Decidable (false = true)\n⊢ decide (false = true) = false\n\ncase true\nh : Decidable (true = true)\n⊢ decide (true = true) = true",
"state_before": "b : Bool\nh : Decidable (b = true)\n⊢ decide (b = true) = b",
"tactic": "cases b"
},
{
"state_after": "no goals",
"state_before": "case false\nh : Decidable (false = true)\n⊢ decide (false = true) = false",
"tactic": "exact decide_eq_false $ λ j => by cases j"
},
{
"state_after": "no goals",
"state_before": "h : Decidable (false = true)\nj : false = true\n⊢ False",
"tactic": "cases j"
},
{
"state_after": "no goals",
"state_before": "case true\nh : Decidable (true = true)\n⊢ decide (true = true) = true",
"tactic": "exact decide_eq_true $ rfl"
}
] |
[
40,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
37,
1
] |
Mathlib/Topology/Algebra/ConstMulAction.lean
|
ContinuousOn.const_smul
|
[] |
[
106,
76
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
105,
1
] |
Std/WF.lean
|
WellFounded.fix_eq_fixC
|
[
{
"state_after": "case h.h.h.h.h.h\nα : Sort u_1\nC : α → Sort u_2\nr : α → α → Prop\nhwf : WellFounded r\nF : (x : α) → ((y : α) → r y x → C y) → C x\nx : α\n⊢ fix hwf F x = WellFounded.fixC hwf F x",
"state_before": "⊢ @fix = @WellFounded.fixC",
"tactic": "funext α C r hwf F x"
},
{
"state_after": "no goals",
"state_before": "case h.h.h.h.h.h\nα : Sort u_1\nC : α → Sort u_2\nr : α → α → Prop\nhwf : WellFounded r\nF : (x : α) → ((y : α) → r y x → C y) → C x\nx : α\n⊢ fix hwf F x = WellFounded.fixC hwf F x",
"tactic": "rw [fix, fixF_eq_fixFC, fixC]"
}
] |
[
111,
32
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
109,
18
] |
Mathlib/Data/List/Sublists.lean
|
List.mem_sublists'
|
[
{
"state_after": "case nil\nα : Type u\nβ : Type v\nγ : Type w\ns✝ s : List α\n⊢ s ∈ sublists' [] ↔ s <+ []\n\ncase cons\nα : Type u\nβ : Type v\nγ : Type w\ns✝ : List α\na : α\nt : List α\nIH : ∀ {s : List α}, s ∈ sublists' t ↔ s <+ t\ns : List α\n⊢ s ∈ sublists' (a :: t) ↔ s <+ a :: t",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ns t : List α\n⊢ s ∈ sublists' t ↔ s <+ t",
"tactic": "induction' t with a t IH generalizing s"
},
{
"state_after": "case cons\nα : Type u\nβ : Type v\nγ : Type w\ns✝ : List α\na : α\nt : List α\nIH : ∀ {s : List α}, s ∈ sublists' t ↔ s <+ t\ns : List α\n⊢ (s <+ t ∨ ∃ a_1, a_1 <+ t ∧ a :: a_1 = s) ↔ s <+ a :: t",
"state_before": "case cons\nα : Type u\nβ : Type v\nγ : Type w\ns✝ : List α\na : α\nt : List α\nIH : ∀ {s : List α}, s ∈ sublists' t ↔ s <+ t\ns : List α\n⊢ s ∈ sublists' (a :: t) ↔ s <+ a :: t",
"tactic": "simp only [sublists'_cons, mem_append, IH, mem_map]"
},
{
"state_after": "case cons.mp\nα : Type u\nβ : Type v\nγ : Type w\ns✝ : List α\na : α\nt : List α\nIH : ∀ {s : List α}, s ∈ sublists' t ↔ s <+ t\ns : List α\nh : s <+ t ∨ ∃ a_1, a_1 <+ t ∧ a :: a_1 = s\n⊢ s <+ a :: t\n\ncase cons.mpr\nα : Type u\nβ : Type v\nγ : Type w\ns✝ : List α\na : α\nt : List α\nIH : ∀ {s : List α}, s ∈ sublists' t ↔ s <+ t\ns : List α\nh : s <+ a :: t\n⊢ s <+ t ∨ ∃ a_1, a_1 <+ t ∧ a :: a_1 = s",
"state_before": "case cons\nα : Type u\nβ : Type v\nγ : Type w\ns✝ : List α\na : α\nt : List α\nIH : ∀ {s : List α}, s ∈ sublists' t ↔ s <+ t\ns : List α\n⊢ (s <+ t ∨ ∃ a_1, a_1 <+ t ∧ a :: a_1 = s) ↔ s <+ a :: t",
"tactic": "constructor <;> intro h"
},
{
"state_after": "case cons.mp.inl\nα : Type u\nβ : Type v\nγ : Type w\ns✝ : List α\na : α\nt : List α\nIH : ∀ {s : List α}, s ∈ sublists' t ↔ s <+ t\ns : List α\nh : s <+ t\n⊢ s <+ a :: t\n\ncase cons.mp.inr.intro.intro\nα : Type u\nβ : Type v\nγ : Type w\ns✝ : List α\na : α\nt : List α\nIH : ∀ {s : List α}, s ∈ sublists' t ↔ s <+ t\ns : List α\nh : s <+ t\n⊢ a :: s <+ a :: t\n\ncase cons.mpr\nα : Type u\nβ : Type v\nγ : Type w\ns✝ : List α\na : α\nt : List α\nIH : ∀ {s : List α}, s ∈ sublists' t ↔ s <+ t\ns : List α\nh : s <+ a :: t\n⊢ s <+ t ∨ ∃ a_1, a_1 <+ t ∧ a :: a_1 = s",
"state_before": "case cons.mp\nα : Type u\nβ : Type v\nγ : Type w\ns✝ : List α\na : α\nt : List α\nIH : ∀ {s : List α}, s ∈ sublists' t ↔ s <+ t\ns : List α\nh : s <+ t ∨ ∃ a_1, a_1 <+ t ∧ a :: a_1 = s\n⊢ s <+ a :: t\n\ncase cons.mpr\nα : Type u\nβ : Type v\nγ : Type w\ns✝ : List α\na : α\nt : List α\nIH : ∀ {s : List α}, s ∈ sublists' t ↔ s <+ t\ns : List α\nh : s <+ a :: t\n⊢ s <+ t ∨ ∃ a_1, a_1 <+ t ∧ a :: a_1 = s",
"tactic": "rcases h with (h | ⟨s, h, rfl⟩)"
},
{
"state_after": "case nil\nα : Type u\nβ : Type v\nγ : Type w\ns✝ s : List α\n⊢ s = [] ↔ s <+ []",
"state_before": "case nil\nα : Type u\nβ : Type v\nγ : Type w\ns✝ s : List α\n⊢ s ∈ sublists' [] ↔ s <+ []",
"tactic": "simp only [sublists'_nil, mem_singleton]"
},
{
"state_after": "no goals",
"state_before": "case nil\nα : Type u\nβ : Type v\nγ : Type w\ns✝ s : List α\n⊢ s = [] ↔ s <+ []",
"tactic": "exact ⟨fun h => by rw [h], eq_nil_of_sublist_nil⟩"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ns✝ s : List α\nh : s = []\n⊢ s <+ []",
"tactic": "rw [h]"
},
{
"state_after": "no goals",
"state_before": "case cons.mp.inl\nα : Type u\nβ : Type v\nγ : Type w\ns✝ : List α\na : α\nt : List α\nIH : ∀ {s : List α}, s ∈ sublists' t ↔ s <+ t\ns : List α\nh : s <+ t\n⊢ s <+ a :: t",
"tactic": "exact sublist_cons_of_sublist _ h"
},
{
"state_after": "no goals",
"state_before": "case cons.mp.inr.intro.intro\nα : Type u\nβ : Type v\nγ : Type w\ns✝ : List α\na : α\nt : List α\nIH : ∀ {s : List α}, s ∈ sublists' t ↔ s <+ t\ns : List α\nh : s <+ t\n⊢ a :: s <+ a :: t",
"tactic": "exact h.cons_cons _"
},
{
"state_after": "case cons.mpr.cons\nα : Type u\nβ : Type v\nγ : Type w\ns✝ : List α\na : α\nt : List α\nIH : ∀ {s : List α}, s ∈ sublists' t ↔ s <+ t\ns : List α\nh : s <+ t\n⊢ s <+ t ∨ ∃ a_1, a_1 <+ t ∧ a :: a_1 = s\n\ncase cons.mpr.cons₂\nα : Type u\nβ : Type v\nγ : Type w\ns✝ : List α\na : α\nt : List α\nIH : ∀ {s : List α}, s ∈ sublists' t ↔ s <+ t\ns : List α\nh : s <+ t\n⊢ a :: s <+ t ∨ ∃ a_1, a_1 <+ t ∧ a :: a_1 = a :: s",
"state_before": "case cons.mpr\nα : Type u\nβ : Type v\nγ : Type w\ns✝ : List α\na : α\nt : List α\nIH : ∀ {s : List α}, s ∈ sublists' t ↔ s <+ t\ns : List α\nh : s <+ a :: t\n⊢ s <+ t ∨ ∃ a_1, a_1 <+ t ∧ a :: a_1 = s",
"tactic": "cases' h with _ _ _ h s _ _ h"
},
{
"state_after": "no goals",
"state_before": "case cons.mpr.cons\nα : Type u\nβ : Type v\nγ : Type w\ns✝ : List α\na : α\nt : List α\nIH : ∀ {s : List α}, s ∈ sublists' t ↔ s <+ t\ns : List α\nh : s <+ t\n⊢ s <+ t ∨ ∃ a_1, a_1 <+ t ∧ a :: a_1 = s",
"tactic": "exact Or.inl h"
},
{
"state_after": "no goals",
"state_before": "case cons.mpr.cons₂\nα : Type u\nβ : Type v\nγ : Type w\ns✝ : List α\na : α\nt : List α\nIH : ∀ {s : List α}, s ∈ sublists' t ↔ s <+ t\ns : List α\nh : s <+ t\n⊢ a :: s <+ t ∨ ∃ a_1, a_1 <+ t ∧ a :: a_1 = a :: s",
"tactic": "exact Or.inr ⟨s, h, rfl⟩"
}
] |
[
95,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
85,
1
] |
Mathlib/Order/Filter/NAry.lean
|
Filter.NeBot.of_map₂_right
|
[] |
[
143,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
142,
1
] |
Mathlib/Algebra/Homology/HomologicalComplex.lean
|
HomologicalComplex.Hom.sqFrom_left
|
[] |
[
563,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
562,
1
] |
Mathlib/Data/Bitvec/Lemmas.lean
|
Bitvec.bits_toNat_decide
|
[
{
"state_after": "n : ℕ\n⊢ bitsToNat [decide (n % 2 = 1)] = n % 2",
"state_before": "n : ℕ\n⊢ Bitvec.toNat (decide (n % 2 = 1) ::ᵥ Vector.nil) = n % 2",
"tactic": "simp [bitsToNat_toList]"
},
{
"state_after": "n : ℕ\n⊢ List.foldl (fun r b => r + r + bif b then 1 else 0) (0 + 0 + bif decide (n % 2 = 1) then 1 else 0) [] = n % 2",
"state_before": "n : ℕ\n⊢ bitsToNat [decide (n % 2 = 1)] = n % 2",
"tactic": "unfold bitsToNat addLsb List.foldl"
},
{
"state_after": "no goals",
"state_before": "n : ℕ\n⊢ List.foldl (fun r b => r + r + bif b then 1 else 0) (0 + 0 + bif decide (n % 2 = 1) then 1 else 0) [] = n % 2",
"tactic": "simp [Nat.cond_decide_mod_two, -Bool.cond_decide]"
}
] |
[
65,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
62,
1
] |
Mathlib/LinearAlgebra/Prod.lean
|
LinearEquiv.coe_prod
|
[] |
[
793,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
791,
1
] |
Mathlib/LinearAlgebra/SesquilinearForm.lean
|
LinearMap.IsAlt.neg
|
[
{
"state_after": "R : Type u_1\nR₁ : Type u_3\nR₂ : Type ?u.184837\nR₃ : Type ?u.184840\nM : Type ?u.184843\nM₁ : Type u_2\nM₂ : Type ?u.184849\nMₗ₁ : Type ?u.184852\nMₗ₁' : Type ?u.184855\nMₗ₂ : Type ?u.184858\nMₗ₂' : Type ?u.184861\nK : Type ?u.184864\nK₁ : Type ?u.184867\nK₂ : Type ?u.184870\nV : Type ?u.184873\nV₁ : Type ?u.184876\nV₂ : Type ?u.184879\nn : Type ?u.184882\ninst✝³ : CommRing R\ninst✝² : CommSemiring R₁\ninst✝¹ : AddCommMonoid M₁\ninst✝ : Module R₁ M₁\nI₁ I₂ I : R₁ →+* R\nB : M₁ →ₛₗ[I₁] M₁ →ₛₗ[I₂] R\nH : IsAlt B\nx y : M₁\nH1 : ↑(↑B (y + x)) (y + x) = 0\n⊢ -↑(↑B x) y = ↑(↑B y) x",
"state_before": "R : Type u_1\nR₁ : Type u_3\nR₂ : Type ?u.184837\nR₃ : Type ?u.184840\nM : Type ?u.184843\nM₁ : Type u_2\nM₂ : Type ?u.184849\nMₗ₁ : Type ?u.184852\nMₗ₁' : Type ?u.184855\nMₗ₂ : Type ?u.184858\nMₗ₂' : Type ?u.184861\nK : Type ?u.184864\nK₁ : Type ?u.184867\nK₂ : Type ?u.184870\nV : Type ?u.184873\nV₁ : Type ?u.184876\nV₂ : Type ?u.184879\nn : Type ?u.184882\ninst✝³ : CommRing R\ninst✝² : CommSemiring R₁\ninst✝¹ : AddCommMonoid M₁\ninst✝ : Module R₁ M₁\nI₁ I₂ I : R₁ →+* R\nB : M₁ →ₛₗ[I₁] M₁ →ₛₗ[I₂] R\nH : IsAlt B\nx y : M₁\n⊢ -↑(↑B x) y = ↑(↑B y) x",
"tactic": "have H1 : B (y + x) (y + x) = 0 := self_eq_zero H (y + x)"
},
{
"state_after": "R : Type u_1\nR₁ : Type u_3\nR₂ : Type ?u.184837\nR₃ : Type ?u.184840\nM : Type ?u.184843\nM₁ : Type u_2\nM₂ : Type ?u.184849\nMₗ₁ : Type ?u.184852\nMₗ₁' : Type ?u.184855\nMₗ₂ : Type ?u.184858\nMₗ₂' : Type ?u.184861\nK : Type ?u.184864\nK₁ : Type ?u.184867\nK₂ : Type ?u.184870\nV : Type ?u.184873\nV₁ : Type ?u.184876\nV₂ : Type ?u.184879\nn : Type ?u.184882\ninst✝³ : CommRing R\ninst✝² : CommSemiring R₁\ninst✝¹ : AddCommMonoid M₁\ninst✝ : Module R₁ M₁\nI₁ I₂ I : R₁ →+* R\nB : M₁ →ₛₗ[I₁] M₁ →ₛₗ[I₂] R\nH : IsAlt B\nx y : M₁\nH1 : ↑(↑B x) y + ↑(↑B y) x = 0\n⊢ -↑(↑B x) y = ↑(↑B y) x",
"state_before": "R : Type u_1\nR₁ : Type u_3\nR₂ : Type ?u.184837\nR₃ : Type ?u.184840\nM : Type ?u.184843\nM₁ : Type u_2\nM₂ : Type ?u.184849\nMₗ₁ : Type ?u.184852\nMₗ₁' : Type ?u.184855\nMₗ₂ : Type ?u.184858\nMₗ₂' : Type ?u.184861\nK : Type ?u.184864\nK₁ : Type ?u.184867\nK₂ : Type ?u.184870\nV : Type ?u.184873\nV₁ : Type ?u.184876\nV₂ : Type ?u.184879\nn : Type ?u.184882\ninst✝³ : CommRing R\ninst✝² : CommSemiring R₁\ninst✝¹ : AddCommMonoid M₁\ninst✝ : Module R₁ M₁\nI₁ I₂ I : R₁ →+* R\nB : M₁ →ₛₗ[I₁] M₁ →ₛₗ[I₂] R\nH : IsAlt B\nx y : M₁\nH1 : ↑(↑B (y + x)) (y + x) = 0\n⊢ -↑(↑B x) y = ↑(↑B y) x",
"tactic": "simp [map_add, self_eq_zero H] at H1"
},
{
"state_after": "R : Type u_1\nR₁ : Type u_3\nR₂ : Type ?u.184837\nR₃ : Type ?u.184840\nM : Type ?u.184843\nM₁ : Type u_2\nM₂ : Type ?u.184849\nMₗ₁ : Type ?u.184852\nMₗ₁' : Type ?u.184855\nMₗ₂ : Type ?u.184858\nMₗ₂' : Type ?u.184861\nK : Type ?u.184864\nK₁ : Type ?u.184867\nK₂ : Type ?u.184870\nV : Type ?u.184873\nV₁ : Type ?u.184876\nV₂ : Type ?u.184879\nn : Type ?u.184882\ninst✝³ : CommRing R\ninst✝² : CommSemiring R₁\ninst✝¹ : AddCommMonoid M₁\ninst✝ : Module R₁ M₁\nI₁ I₂ I : R₁ →+* R\nB : M₁ →ₛₗ[I₁] M₁ →ₛₗ[I₂] R\nH : IsAlt B\nx y : M₁\nH1 : -↑(↑B x) y = ↑(↑B y) x\n⊢ -↑(↑B x) y = ↑(↑B y) x",
"state_before": "R : Type u_1\nR₁ : Type u_3\nR₂ : Type ?u.184837\nR₃ : Type ?u.184840\nM : Type ?u.184843\nM₁ : Type u_2\nM₂ : Type ?u.184849\nMₗ₁ : Type ?u.184852\nMₗ₁' : Type ?u.184855\nMₗ₂ : Type ?u.184858\nMₗ₂' : Type ?u.184861\nK : Type ?u.184864\nK₁ : Type ?u.184867\nK₂ : Type ?u.184870\nV : Type ?u.184873\nV₁ : Type ?u.184876\nV₂ : Type ?u.184879\nn : Type ?u.184882\ninst✝³ : CommRing R\ninst✝² : CommSemiring R₁\ninst✝¹ : AddCommMonoid M₁\ninst✝ : Module R₁ M₁\nI₁ I₂ I : R₁ →+* R\nB : M₁ →ₛₗ[I₁] M₁ →ₛₗ[I₂] R\nH : IsAlt B\nx y : M₁\nH1 : ↑(↑B x) y + ↑(↑B y) x = 0\n⊢ -↑(↑B x) y = ↑(↑B y) x",
"tactic": "rw [add_eq_zero_iff_neg_eq] at H1"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\nR₁ : Type u_3\nR₂ : Type ?u.184837\nR₃ : Type ?u.184840\nM : Type ?u.184843\nM₁ : Type u_2\nM₂ : Type ?u.184849\nMₗ₁ : Type ?u.184852\nMₗ₁' : Type ?u.184855\nMₗ₂ : Type ?u.184858\nMₗ₂' : Type ?u.184861\nK : Type ?u.184864\nK₁ : Type ?u.184867\nK₂ : Type ?u.184870\nV : Type ?u.184873\nV₁ : Type ?u.184876\nV₂ : Type ?u.184879\nn : Type ?u.184882\ninst✝³ : CommRing R\ninst✝² : CommSemiring R₁\ninst✝¹ : AddCommMonoid M₁\ninst✝ : Module R₁ M₁\nI₁ I₂ I : R₁ →+* R\nB : M₁ →ₛₗ[I₁] M₁ →ₛₗ[I₂] R\nH : IsAlt B\nx y : M₁\nH1 : -↑(↑B x) y = ↑(↑B y) x\n⊢ -↑(↑B x) y = ↑(↑B y) x",
"tactic": "exact H1"
}
] |
[
282,
11
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
278,
1
] |
Mathlib/Order/RelIso/Basic.lean
|
RelIso.cast_trans
|
[
{
"state_after": "α✝ : Type ?u.81083\nβ : Type ?u.81086\nγ✝ : Type ?u.81089\nδ : Type ?u.81092\nr✝ : α✝ → α✝ → Prop\ns✝ : β → β → Prop\nt✝ : γ✝ → γ✝ → Prop\nu : δ → δ → Prop\nα γ : Type u\nr : α → α → Prop\nt : γ → γ → Prop\nx : α\ns : α → α → Prop\nh₁' : α = γ\nh₂ : HEq r s\nh₂' : HEq s t\n⊢ ↑(RelIso.trans (RelIso.cast (_ : α = α) h₂) (RelIso.cast h₁' h₂')) x = ↑(RelIso.cast (_ : α = γ) (_ : HEq r t)) x",
"state_before": "α✝ : Type ?u.81083\nβ✝ : Type ?u.81086\nγ✝ : Type ?u.81089\nδ : Type ?u.81092\nr✝ : α✝ → α✝ → Prop\ns✝ : β✝ → β✝ → Prop\nt✝ : γ✝ → γ✝ → Prop\nu : δ → δ → Prop\nα β γ : Type u\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\nh₁ : α = β\nh₁' : β = γ\nh₂ : HEq r s\nh₂' : HEq s t\nx : α\n⊢ ↑(RelIso.trans (RelIso.cast h₁ h₂) (RelIso.cast h₁' h₂')) x = ↑(RelIso.cast (_ : α = γ) (_ : HEq r t)) x",
"tactic": "subst h₁"
},
{
"state_after": "no goals",
"state_before": "α✝ : Type ?u.81083\nβ : Type ?u.81086\nγ✝ : Type ?u.81089\nδ : Type ?u.81092\nr✝ : α✝ → α✝ → Prop\ns✝ : β → β → Prop\nt✝ : γ✝ → γ✝ → Prop\nu : δ → δ → Prop\nα γ : Type u\nr : α → α → Prop\nt : γ → γ → Prop\nx : α\ns : α → α → Prop\nh₁' : α = γ\nh₂ : HEq r s\nh₂' : HEq s t\n⊢ ↑(RelIso.trans (RelIso.cast (_ : α = α) h₂) (RelIso.cast h₁' h₂')) x = ↑(RelIso.cast (_ : α = γ) (_ : HEq r t)) x",
"tactic": "rfl"
}
] |
[
771,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
768,
11
] |
Mathlib/MeasureTheory/Measure/MeasureSpace.lean
|
MeasureTheory.Measure.exists_subset_measure_lt_top
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.731841\nγ : Type ?u.731844\nδ : Type ?u.731847\nι : Type ?u.731850\nR : Type ?u.731853\nR' : Type ?u.731856\nm0 : MeasurableSpace α\ninst✝² : MeasurableSpace β\ninst✝¹ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t : Set α\ninst✝ : SigmaFinite μ\nr : ℝ≥0∞\nhs : MeasurableSet s\nh's✝ : r < ↑↑μ s\nh's : ∃ i, r < ↑↑(restrict μ (spanningSets μ i)) s\n⊢ ∃ t, MeasurableSet t ∧ t ⊆ s ∧ r < ↑↑μ t ∧ ↑↑μ t < ⊤",
"state_before": "α : Type u_1\nβ : Type ?u.731841\nγ : Type ?u.731844\nδ : Type ?u.731847\nι : Type ?u.731850\nR : Type ?u.731853\nR' : Type ?u.731856\nm0 : MeasurableSpace α\ninst✝² : MeasurableSpace β\ninst✝¹ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t : Set α\ninst✝ : SigmaFinite μ\nr : ℝ≥0∞\nhs : MeasurableSet s\nh's : r < ↑↑μ s\n⊢ ∃ t, MeasurableSet t ∧ t ⊆ s ∧ r < ↑↑μ t ∧ ↑↑μ t < ⊤",
"tactic": "rw [← iSup_restrict_spanningSets hs,\n @lt_iSup_iff _ _ _ r fun i : ℕ => μ.restrict (spanningSets μ i) s] at h's"
},
{
"state_after": "case intro\nα : Type u_1\nβ : Type ?u.731841\nγ : Type ?u.731844\nδ : Type ?u.731847\nι : Type ?u.731850\nR : Type ?u.731853\nR' : Type ?u.731856\nm0 : MeasurableSpace α\ninst✝² : MeasurableSpace β\ninst✝¹ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t : Set α\ninst✝ : SigmaFinite μ\nr : ℝ≥0∞\nhs : MeasurableSet s\nh's : r < ↑↑μ s\nn : ℕ\nhn : r < ↑↑(restrict μ (spanningSets μ n)) s\n⊢ ∃ t, MeasurableSet t ∧ t ⊆ s ∧ r < ↑↑μ t ∧ ↑↑μ t < ⊤",
"state_before": "α : Type u_1\nβ : Type ?u.731841\nγ : Type ?u.731844\nδ : Type ?u.731847\nι : Type ?u.731850\nR : Type ?u.731853\nR' : Type ?u.731856\nm0 : MeasurableSpace α\ninst✝² : MeasurableSpace β\ninst✝¹ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t : Set α\ninst✝ : SigmaFinite μ\nr : ℝ≥0∞\nhs : MeasurableSet s\nh's✝ : r < ↑↑μ s\nh's : ∃ i, r < ↑↑(restrict μ (spanningSets μ i)) s\n⊢ ∃ t, MeasurableSet t ∧ t ⊆ s ∧ r < ↑↑μ t ∧ ↑↑μ t < ⊤",
"tactic": "rcases h's with ⟨n, hn⟩"
},
{
"state_after": "case intro\nα : Type u_1\nβ : Type ?u.731841\nγ : Type ?u.731844\nδ : Type ?u.731847\nι : Type ?u.731850\nR : Type ?u.731853\nR' : Type ?u.731856\nm0 : MeasurableSpace α\ninst✝² : MeasurableSpace β\ninst✝¹ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t : Set α\ninst✝ : SigmaFinite μ\nr : ℝ≥0∞\nhs : MeasurableSet s\nh's : r < ↑↑μ s\nn : ℕ\nhn : r < ↑↑μ (s ∩ spanningSets μ n)\n⊢ ∃ t, MeasurableSet t ∧ t ⊆ s ∧ r < ↑↑μ t ∧ ↑↑μ t < ⊤",
"state_before": "case intro\nα : Type u_1\nβ : Type ?u.731841\nγ : Type ?u.731844\nδ : Type ?u.731847\nι : Type ?u.731850\nR : Type ?u.731853\nR' : Type ?u.731856\nm0 : MeasurableSpace α\ninst✝² : MeasurableSpace β\ninst✝¹ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t : Set α\ninst✝ : SigmaFinite μ\nr : ℝ≥0∞\nhs : MeasurableSet s\nh's : r < ↑↑μ s\nn : ℕ\nhn : r < ↑↑(restrict μ (spanningSets μ n)) s\n⊢ ∃ t, MeasurableSet t ∧ t ⊆ s ∧ r < ↑↑μ t ∧ ↑↑μ t < ⊤",
"tactic": "simp only [restrict_apply hs] at hn"
},
{
"state_after": "case intro\nα : Type u_1\nβ : Type ?u.731841\nγ : Type ?u.731844\nδ : Type ?u.731847\nι : Type ?u.731850\nR : Type ?u.731853\nR' : Type ?u.731856\nm0 : MeasurableSpace α\ninst✝² : MeasurableSpace β\ninst✝¹ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t : Set α\ninst✝ : SigmaFinite μ\nr : ℝ≥0∞\nhs : MeasurableSet s\nh's : r < ↑↑μ s\nn : ℕ\nhn : r < ↑↑μ (s ∩ spanningSets μ n)\n⊢ ↑↑μ (s ∩ spanningSets μ n) < ⊤",
"state_before": "case intro\nα : Type u_1\nβ : Type ?u.731841\nγ : Type ?u.731844\nδ : Type ?u.731847\nι : Type ?u.731850\nR : Type ?u.731853\nR' : Type ?u.731856\nm0 : MeasurableSpace α\ninst✝² : MeasurableSpace β\ninst✝¹ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t : Set α\ninst✝ : SigmaFinite μ\nr : ℝ≥0∞\nhs : MeasurableSet s\nh's : r < ↑↑μ s\nn : ℕ\nhn : r < ↑↑μ (s ∩ spanningSets μ n)\n⊢ ∃ t, MeasurableSet t ∧ t ⊆ s ∧ r < ↑↑μ t ∧ ↑↑μ t < ⊤",
"tactic": "refine'\n ⟨s ∩ spanningSets μ n, hs.inter (measurable_spanningSets _ _), inter_subset_left _ _, hn, _⟩"
},
{
"state_after": "no goals",
"state_before": "case intro\nα : Type u_1\nβ : Type ?u.731841\nγ : Type ?u.731844\nδ : Type ?u.731847\nι : Type ?u.731850\nR : Type ?u.731853\nR' : Type ?u.731856\nm0 : MeasurableSpace α\ninst✝² : MeasurableSpace β\ninst✝¹ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t : Set α\ninst✝ : SigmaFinite μ\nr : ℝ≥0∞\nhs : MeasurableSet s\nh's : r < ↑↑μ s\nn : ℕ\nhn : r < ↑↑μ (s ∩ spanningSets μ n)\n⊢ ↑↑μ (s ∩ spanningSets μ n) < ⊤",
"tactic": "exact (measure_mono (inter_subset_right _ _)).trans_lt (measure_spanningSets_lt_top _ _)"
}
] |
[
3551,
91
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
3543,
1
] |
Mathlib/Analysis/InnerProductSpace/Orthogonal.lean
|
Submodule.orthogonal_orthogonal_monotone
|
[] |
[
165,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
164,
1
] |
Mathlib/GroupTheory/SemidirectProduct.lean
|
SemidirectProduct.map_left
|
[] |
[
284,
74
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
284,
1
] |
Mathlib/Algebra/Group/Units.lean
|
Units.ext
|
[
{
"state_after": "α : Type u\ninst✝ : Monoid α\nv i₁ : α\nvi₁ : v * i₁ = 1\niv₁ : i₁ * v = 1\nv' i₂ : α\nvi₂ : v' * i₂ = 1\niv₂ : i₂ * v' = 1\ne : v = v'\n⊢ { val := v, inv := i₁, val_inv := vi₁, inv_val := iv₁ } = { val := v', inv := i₂, val_inv := vi₂, inv_val := iv₂ }",
"state_before": "α : Type u\ninst✝ : Monoid α\nv i₁ : α\nvi₁ : v * i₁ = 1\niv₁ : i₁ * v = 1\nv' i₂ : α\nvi₂ : v' * i₂ = 1\niv₂ : i₂ * v' = 1\ne : ↑{ val := v, inv := i₁, val_inv := vi₁, inv_val := iv₁ } = ↑{ val := v', inv := i₂, val_inv := vi₂, inv_val := iv₂ }\n⊢ { val := v, inv := i₁, val_inv := vi₁, inv_val := iv₁ } = { val := v', inv := i₂, val_inv := vi₂, inv_val := iv₂ }",
"tactic": "simp only at e"
},
{
"state_after": "α : Type u\ninst✝ : Monoid α\nv i₁ : α\nvi₁ : v * i₁ = 1\niv₁ : i₁ * v = 1\ni₂ : α\nvi₂ : v * i₂ = 1\niv₂ : i₂ * v = 1\n⊢ { val := v, inv := i₁, val_inv := vi₁, inv_val := iv₁ } = { val := v, inv := i₂, val_inv := vi₂, inv_val := iv₂ }",
"state_before": "α : Type u\ninst✝ : Monoid α\nv i₁ : α\nvi₁ : v * i₁ = 1\niv₁ : i₁ * v = 1\nv' i₂ : α\nvi₂ : v' * i₂ = 1\niv₂ : i₂ * v' = 1\ne : v = v'\n⊢ { val := v, inv := i₁, val_inv := vi₁, inv_val := iv₁ } = { val := v', inv := i₂, val_inv := vi₂, inv_val := iv₂ }",
"tactic": "subst v'"
},
{
"state_after": "case e_inv\nα : Type u\ninst✝ : Monoid α\nv i₁ : α\nvi₁ : v * i₁ = 1\niv₁ : i₁ * v = 1\ni₂ : α\nvi₂ : v * i₂ = 1\niv₂ : i₂ * v = 1\n⊢ i₁ = i₂",
"state_before": "α : Type u\ninst✝ : Monoid α\nv i₁ : α\nvi₁ : v * i₁ = 1\niv₁ : i₁ * v = 1\ni₂ : α\nvi₂ : v * i₂ = 1\niv₂ : i₂ * v = 1\n⊢ { val := v, inv := i₁, val_inv := vi₁, inv_val := iv₁ } = { val := v, inv := i₂, val_inv := vi₂, inv_val := iv₂ }",
"tactic": "congr"
},
{
"state_after": "no goals",
"state_before": "case e_inv\nα : Type u\ninst✝ : Monoid α\nv i₁ : α\nvi₁ : v * i₁ = 1\niv₁ : i₁ * v = 1\ni₂ : α\nvi₂ : v * i₂ = 1\niv₂ : i₂ * v = 1\n⊢ i₁ = i₂",
"tactic": "simpa only [iv₂, vi₁, one_mul, mul_one] using mul_assoc i₂ v i₁"
}
] |
[
138,
68
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
135,
1
] |
Mathlib/LinearAlgebra/Matrix/Hermitian.lean
|
Matrix.IsHermitian.coe_re_diag
|
[] |
[
270,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
268,
1
] |
Mathlib/Data/Nat/Fib.lean
|
Nat.fib_add_two
|
[
{
"state_after": "no goals",
"state_before": "n : ℕ\n⊢ fib (n + 2) = fib n + fib (n + 1)",
"tactic": "simp [fib, Function.iterate_succ_apply']"
}
] |
[
94,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
93,
1
] |
Mathlib/LinearAlgebra/Matrix/ZPow.lean
|
Matrix.zpow_neg_one
|
[
{
"state_after": "case h.e'_3.h.e'_3\nn' : Type u_1\ninst✝² : DecidableEq n'\ninst✝¹ : Fintype n'\nR : Type u_2\ninst✝ : CommRing R\nA : M\n⊢ A = DivInvMonoid.zpow (↑(Nat.succ 0)) A",
"state_before": "n' : Type u_1\ninst✝² : DecidableEq n'\ninst✝¹ : Fintype n'\nR : Type u_2\ninst✝ : CommRing R\nA : M\n⊢ A ^ (-1) = A⁻¹",
"tactic": "convert DivInvMonoid.zpow_neg' 0 A"
},
{
"state_after": "no goals",
"state_before": "case h.e'_3.h.e'_3\nn' : Type u_1\ninst✝² : DecidableEq n'\ninst✝¹ : Fintype n'\nR : Type u_2\ninst✝ : CommRing R\nA : M\n⊢ A = DivInvMonoid.zpow (↑(Nat.succ 0)) A",
"tactic": "simp only [zpow_one, Int.ofNat_zero, Int.ofNat_succ, zpow_eq_pow, zero_add]"
}
] |
[
109,
78
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
107,
1
] |
Mathlib/Analysis/MeanInequalities.lean
|
NNReal.isGreatest_Lp
|
[
{
"state_after": "case left\nι : Type u\ns : Finset ι\nf : ι → ℝ≥0\np q : ℝ\nhpq : Real.IsConjugateExponent p q\n⊢ (∑ i in s, f i ^ p) ^ (1 / p) ∈ (fun g => ∑ i in s, f i * g i) '' {g | ∑ i in s, g i ^ q ≤ 1}\n\ncase right\nι : Type u\ns : Finset ι\nf : ι → ℝ≥0\np q : ℝ\nhpq : Real.IsConjugateExponent p q\n⊢ (∑ i in s, f i ^ p) ^ (1 / p) ∈ upperBounds ((fun g => ∑ i in s, f i * g i) '' {g | ∑ i in s, g i ^ q ≤ 1})",
"state_before": "ι : Type u\ns : Finset ι\nf : ι → ℝ≥0\np q : ℝ\nhpq : Real.IsConjugateExponent p q\n⊢ IsGreatest ((fun g => ∑ i in s, f i * g i) '' {g | ∑ i in s, g i ^ q ≤ 1}) ((∑ i in s, f i ^ p) ^ (1 / p))",
"tactic": "constructor"
},
{
"state_after": "case left\nι : Type u\ns : Finset ι\nf : ι → ℝ≥0\np q : ℝ\nhpq : Real.IsConjugateExponent p q\n⊢ (fun i => f i ^ p / f i / (∑ i in s, f i ^ p) ^ (1 / q)) ∈ {g | ∑ i in s, g i ^ q ≤ 1} ∧\n ((fun g => ∑ i in s, f i * g i) fun i => f i ^ p / f i / (∑ i in s, f i ^ p) ^ (1 / q)) =\n (∑ i in s, f i ^ p) ^ (1 / p)",
"state_before": "case left\nι : Type u\ns : Finset ι\nf : ι → ℝ≥0\np q : ℝ\nhpq : Real.IsConjugateExponent p q\n⊢ (∑ i in s, f i ^ p) ^ (1 / p) ∈ (fun g => ∑ i in s, f i * g i) '' {g | ∑ i in s, g i ^ q ≤ 1}",
"tactic": "use fun i => f i ^ p / f i / (∑ i in s, f i ^ p) ^ (1 / q)"
},
{
"state_after": "case pos\nι : Type u\ns : Finset ι\nf : ι → ℝ≥0\np q : ℝ\nhpq : Real.IsConjugateExponent p q\nhf : ∑ i in s, f i ^ p = 0\n⊢ (fun i => f i ^ p / f i / (∑ i in s, f i ^ p) ^ (1 / q)) ∈ {g | ∑ i in s, g i ^ q ≤ 1} ∧\n ((fun g => ∑ i in s, f i * g i) fun i => f i ^ p / f i / (∑ i in s, f i ^ p) ^ (1 / q)) =\n (∑ i in s, f i ^ p) ^ (1 / p)\n\ncase neg\nι : Type u\ns : Finset ι\nf : ι → ℝ≥0\np q : ℝ\nhpq : Real.IsConjugateExponent p q\nhf : ¬∑ i in s, f i ^ p = 0\n⊢ (fun i => f i ^ p / f i / (∑ i in s, f i ^ p) ^ (1 / q)) ∈ {g | ∑ i in s, g i ^ q ≤ 1} ∧\n ((fun g => ∑ i in s, f i * g i) fun i => f i ^ p / f i / (∑ i in s, f i ^ p) ^ (1 / q)) =\n (∑ i in s, f i ^ p) ^ (1 / p)",
"state_before": "case left\nι : Type u\ns : Finset ι\nf : ι → ℝ≥0\np q : ℝ\nhpq : Real.IsConjugateExponent p q\n⊢ (fun i => f i ^ p / f i / (∑ i in s, f i ^ p) ^ (1 / q)) ∈ {g | ∑ i in s, g i ^ q ≤ 1} ∧\n ((fun g => ∑ i in s, f i * g i) fun i => f i ^ p / f i / (∑ i in s, f i ^ p) ^ (1 / q)) =\n (∑ i in s, f i ^ p) ^ (1 / p)",
"tactic": "by_cases hf : (∑ i in s, f i ^ p) = 0"
},
{
"state_after": "no goals",
"state_before": "case pos\nι : Type u\ns : Finset ι\nf : ι → ℝ≥0\np q : ℝ\nhpq : Real.IsConjugateExponent p q\nhf : ∑ i in s, f i ^ p = 0\n⊢ (fun i => f i ^ p / f i / (∑ i in s, f i ^ p) ^ (1 / q)) ∈ {g | ∑ i in s, g i ^ q ≤ 1} ∧\n ((fun g => ∑ i in s, f i * g i) fun i => f i ^ p / f i / (∑ i in s, f i ^ p) ^ (1 / q)) =\n (∑ i in s, f i ^ p) ^ (1 / p)",
"tactic": "simp [hf, hpq.ne_zero, hpq.symm.ne_zero]"
},
{
"state_after": "case neg\nι : Type u\ns : Finset ι\nf : ι → ℝ≥0\np q : ℝ\nhpq : Real.IsConjugateExponent p q\nhf : ¬∑ i in s, f i ^ p = 0\nA : p + q - q ≠ 0\n⊢ (fun i => f i ^ p / f i / (∑ i in s, f i ^ p) ^ (1 / q)) ∈ {g | ∑ i in s, g i ^ q ≤ 1} ∧\n ((fun g => ∑ i in s, f i * g i) fun i => f i ^ p / f i / (∑ i in s, f i ^ p) ^ (1 / q)) =\n (∑ i in s, f i ^ p) ^ (1 / p)",
"state_before": "case neg\nι : Type u\ns : Finset ι\nf : ι → ℝ≥0\np q : ℝ\nhpq : Real.IsConjugateExponent p q\nhf : ¬∑ i in s, f i ^ p = 0\n⊢ (fun i => f i ^ p / f i / (∑ i in s, f i ^ p) ^ (1 / q)) ∈ {g | ∑ i in s, g i ^ q ≤ 1} ∧\n ((fun g => ∑ i in s, f i * g i) fun i => f i ^ p / f i / (∑ i in s, f i ^ p) ^ (1 / q)) =\n (∑ i in s, f i ^ p) ^ (1 / p)",
"tactic": "have A : p + q - q ≠ 0 := by simp [hpq.ne_zero]"
},
{
"state_after": "case neg\nι : Type u\ns : Finset ι\nf : ι → ℝ≥0\np q : ℝ\nhpq : Real.IsConjugateExponent p q\nhf : ¬∑ i in s, f i ^ p = 0\nA : p + q - q ≠ 0\nB : ∀ (y : ℝ≥0), y * y ^ p / y = y ^ p\n⊢ (fun i => f i ^ p / f i / (∑ i in s, f i ^ p) ^ (1 / q)) ∈ {g | ∑ i in s, g i ^ q ≤ 1} ∧\n ((fun g => ∑ i in s, f i * g i) fun i => f i ^ p / f i / (∑ i in s, f i ^ p) ^ (1 / q)) =\n (∑ i in s, f i ^ p) ^ (1 / p)",
"state_before": "case neg\nι : Type u\ns : Finset ι\nf : ι → ℝ≥0\np q : ℝ\nhpq : Real.IsConjugateExponent p q\nhf : ¬∑ i in s, f i ^ p = 0\nA : p + q - q ≠ 0\n⊢ (fun i => f i ^ p / f i / (∑ i in s, f i ^ p) ^ (1 / q)) ∈ {g | ∑ i in s, g i ^ q ≤ 1} ∧\n ((fun g => ∑ i in s, f i * g i) fun i => f i ^ p / f i / (∑ i in s, f i ^ p) ^ (1 / q)) =\n (∑ i in s, f i ^ p) ^ (1 / p)",
"tactic": "have B : ∀ y : ℝ≥0, y * y ^ p / y = y ^ p := by\n refine' fun y => mul_div_cancel_left_of_imp fun h => _\n simp [h, hpq.ne_zero]"
},
{
"state_after": "case neg\nι : Type u\ns : Finset ι\nf : ι → ℝ≥0\np q : ℝ\nhpq : Real.IsConjugateExponent p q\nhf : ¬∑ i in s, f i ^ p = 0\nA : p + q - q ≠ 0\nB : ∀ (y : ℝ≥0), y * y ^ p / y = y ^ p\n⊢ (∑ i in s, f i ^ p) / (∑ i in s, f i ^ p) ^ (1 / q) = (∑ i in s, f i ^ p) ^ (1 / p)",
"state_before": "case neg\nι : Type u\ns : Finset ι\nf : ι → ℝ≥0\np q : ℝ\nhpq : Real.IsConjugateExponent p q\nhf : ¬∑ i in s, f i ^ p = 0\nA : p + q - q ≠ 0\nB : ∀ (y : ℝ≥0), y * y ^ p / y = y ^ p\n⊢ (fun i => f i ^ p / f i / (∑ i in s, f i ^ p) ^ (1 / q)) ∈ {g | ∑ i in s, g i ^ q ≤ 1} ∧\n ((fun g => ∑ i in s, f i * g i) fun i => f i ^ p / f i / (∑ i in s, f i ^ p) ^ (1 / q)) =\n (∑ i in s, f i ^ p) ^ (1 / p)",
"tactic": "simp only [Set.mem_setOf_eq, div_rpow, ← sum_div, ← rpow_mul,\n div_mul_cancel _ hpq.symm.ne_zero, rpow_one, div_le_iff hf, one_mul, hpq.mul_eq_add, ←\n rpow_sub' _ A, _root_.add_sub_cancel, le_refl, true_and_iff, ← mul_div_assoc, B]"
},
{
"state_after": "case neg\nι : Type u\ns : Finset ι\nf : ι → ℝ≥0\np q : ℝ\nhpq : Real.IsConjugateExponent p q\nhf : ¬∑ i in s, f i ^ p = 0\nA : p + q - q ≠ 0\nB : ∀ (y : ℝ≥0), y * y ^ p / y = y ^ p\n⊢ (∑ i in s, f i ^ p) ^ (1 / q) ≠ 0",
"state_before": "case neg\nι : Type u\ns : Finset ι\nf : ι → ℝ≥0\np q : ℝ\nhpq : Real.IsConjugateExponent p q\nhf : ¬∑ i in s, f i ^ p = 0\nA : p + q - q ≠ 0\nB : ∀ (y : ℝ≥0), y * y ^ p / y = y ^ p\n⊢ (∑ i in s, f i ^ p) / (∑ i in s, f i ^ p) ^ (1 / q) = (∑ i in s, f i ^ p) ^ (1 / p)",
"tactic": "rw [div_eq_iff, ← rpow_add hf, hpq.inv_add_inv_conj, rpow_one]"
},
{
"state_after": "no goals",
"state_before": "case neg\nι : Type u\ns : Finset ι\nf : ι → ℝ≥0\np q : ℝ\nhpq : Real.IsConjugateExponent p q\nhf : ¬∑ i in s, f i ^ p = 0\nA : p + q - q ≠ 0\nB : ∀ (y : ℝ≥0), y * y ^ p / y = y ^ p\n⊢ (∑ i in s, f i ^ p) ^ (1 / q) ≠ 0",
"tactic": "simpa [hpq.symm.ne_zero] using hf"
},
{
"state_after": "no goals",
"state_before": "ι : Type u\ns : Finset ι\nf : ι → ℝ≥0\np q : ℝ\nhpq : Real.IsConjugateExponent p q\nhf : ¬∑ i in s, f i ^ p = 0\n⊢ p + q - q ≠ 0",
"tactic": "simp [hpq.ne_zero]"
},
{
"state_after": "ι : Type u\ns : Finset ι\nf : ι → ℝ≥0\np q : ℝ\nhpq : Real.IsConjugateExponent p q\nhf : ¬∑ i in s, f i ^ p = 0\nA : p + q - q ≠ 0\ny : ℝ≥0\nh : y = 0\n⊢ y ^ p = 0",
"state_before": "ι : Type u\ns : Finset ι\nf : ι → ℝ≥0\np q : ℝ\nhpq : Real.IsConjugateExponent p q\nhf : ¬∑ i in s, f i ^ p = 0\nA : p + q - q ≠ 0\n⊢ ∀ (y : ℝ≥0), y * y ^ p / y = y ^ p",
"tactic": "refine' fun y => mul_div_cancel_left_of_imp fun h => _"
},
{
"state_after": "no goals",
"state_before": "ι : Type u\ns : Finset ι\nf : ι → ℝ≥0\np q : ℝ\nhpq : Real.IsConjugateExponent p q\nhf : ¬∑ i in s, f i ^ p = 0\nA : p + q - q ≠ 0\ny : ℝ≥0\nh : y = 0\n⊢ y ^ p = 0",
"tactic": "simp [h, hpq.ne_zero]"
},
{
"state_after": "case right.intro.intro\nι : Type u\ns : Finset ι\nf : ι → ℝ≥0\np q : ℝ\nhpq : Real.IsConjugateExponent p q\ng : ι → ℝ≥0\nhg : g ∈ {g | ∑ i in s, g i ^ q ≤ 1}\n⊢ (fun g => ∑ i in s, f i * g i) g ≤ (∑ i in s, f i ^ p) ^ (1 / p)",
"state_before": "case right\nι : Type u\ns : Finset ι\nf : ι → ℝ≥0\np q : ℝ\nhpq : Real.IsConjugateExponent p q\n⊢ (∑ i in s, f i ^ p) ^ (1 / p) ∈ upperBounds ((fun g => ∑ i in s, f i * g i) '' {g | ∑ i in s, g i ^ q ≤ 1})",
"tactic": "rintro _ ⟨g, hg, rfl⟩"
},
{
"state_after": "case right.intro.intro\nι : Type u\ns : Finset ι\nf : ι → ℝ≥0\np q : ℝ\nhpq : Real.IsConjugateExponent p q\ng : ι → ℝ≥0\nhg : g ∈ {g | ∑ i in s, g i ^ q ≤ 1}\n⊢ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) ≤ (∑ i in s, f i ^ p) ^ (1 / p)",
"state_before": "case right.intro.intro\nι : Type u\ns : Finset ι\nf : ι → ℝ≥0\np q : ℝ\nhpq : Real.IsConjugateExponent p q\ng : ι → ℝ≥0\nhg : g ∈ {g | ∑ i in s, g i ^ q ≤ 1}\n⊢ (fun g => ∑ i in s, f i * g i) g ≤ (∑ i in s, f i ^ p) ^ (1 / p)",
"tactic": "apply le_trans (inner_le_Lp_mul_Lq s f g hpq)"
},
{
"state_after": "no goals",
"state_before": "case right.intro.intro\nι : Type u\ns : Finset ι\nf : ι → ℝ≥0\np q : ℝ\nhpq : Real.IsConjugateExponent p q\ng : ι → ℝ≥0\nhg : g ∈ {g | ∑ i in s, g i ^ q ≤ 1}\n⊢ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) ≤ (∑ i in s, f i ^ p) ^ (1 / p)",
"tactic": "simpa only [mul_one] using\n mul_le_mul_left' (NNReal.rpow_le_one hg (le_of_lt hpq.symm.one_div_pos)) _"
}
] |
[
471,
81
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
452,
1
] |
Mathlib/LinearAlgebra/Span.lean
|
Submodule.coe_scott_continuous
|
[] |
[
333,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
331,
1
] |
Mathlib/Topology/MetricSpace/Basic.lean
|
Metric.diam_closedBall
|
[] |
[
2725,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2724,
1
] |
Mathlib/Analysis/Normed/Group/Quotient.lean
|
Submodule.Quotient.norm_mk_le
|
[] |
[
464,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
463,
1
] |
Mathlib/Analysis/NormedSpace/LinearIsometry.lean
|
SemilinearIsometryClass.ediam_range
|
[] |
[
116,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
114,
1
] |
Mathlib/Data/Complex/Basic.lean
|
Complex.normSq_pos
|
[] |
[
644,
80
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
643,
1
] |
Mathlib/Data/Fin/Tuple/Sort.lean
|
Tuple.graph.card
|
[
{
"state_after": "n : ℕ\nα : Type u_1\ninst✝ : LinearOrder α\nf : Fin n → α\n⊢ Finset.card Finset.univ = n\n\ncase H\nn : ℕ\nα : Type u_1\ninst✝ : LinearOrder α\nf : Fin n → α\n⊢ Function.Injective fun i => (f i, i)",
"state_before": "n : ℕ\nα : Type u_1\ninst✝ : LinearOrder α\nf : Fin n → α\n⊢ Finset.card (graph f) = n",
"tactic": "rw [graph, Finset.card_image_of_injective]"
},
{
"state_after": "no goals",
"state_before": "n : ℕ\nα : Type u_1\ninst✝ : LinearOrder α\nf : Fin n → α\n⊢ Finset.card Finset.univ = n",
"tactic": "exact Finset.card_fin _"
},
{
"state_after": "case H\nn : ℕ\nα : Type u_1\ninst✝ : LinearOrder α\nf : Fin n → α\na₁✝ a₂✝ : Fin n\n⊢ (fun i => (f i, i)) a₁✝ = (fun i => (f i, i)) a₂✝ → a₁✝ = a₂✝",
"state_before": "case H\nn : ℕ\nα : Type u_1\ninst✝ : LinearOrder α\nf : Fin n → α\n⊢ Function.Injective fun i => (f i, i)",
"tactic": "intro _ _"
},
{
"state_after": "case H\nn : ℕ\nα : Type u_1\ninst✝ : LinearOrder α\nf : Fin n → α\na₁✝ a₂✝ : Fin n\n⊢ (f a₁✝, a₁✝) = (f a₂✝, a₂✝) → a₁✝ = a₂✝",
"state_before": "case H\nn : ℕ\nα : Type u_1\ninst✝ : LinearOrder α\nf : Fin n → α\na₁✝ a₂✝ : Fin n\n⊢ (fun i => (f i, i)) a₁✝ = (fun i => (f i, i)) a₂✝ → a₁✝ = a₂✝",
"tactic": "dsimp only"
},
{
"state_after": "case H\nn : ℕ\nα : Type u_1\ninst✝ : LinearOrder α\nf : Fin n → α\na₁✝ a₂✝ : Fin n\n⊢ (f a₁✝, a₁✝).fst = (f a₂✝, a₂✝).fst ∧ (f a₁✝, a₁✝).snd = (f a₂✝, a₂✝).snd → a₁✝ = a₂✝",
"state_before": "case H\nn : ℕ\nα : Type u_1\ninst✝ : LinearOrder α\nf : Fin n → α\na₁✝ a₂✝ : Fin n\n⊢ (f a₁✝, a₁✝) = (f a₂✝, a₂✝) → a₁✝ = a₂✝",
"tactic": "rw [Prod.ext_iff]"
},
{
"state_after": "no goals",
"state_before": "case H\nn : ℕ\nα : Type u_1\ninst✝ : LinearOrder α\nf : Fin n → α\na₁✝ a₂✝ : Fin n\n⊢ (f a₁✝, a₁✝).fst = (f a₂✝, a₂✝).fst ∧ (f a₁✝, a₁✝).snd = (f a₂✝, a₂✝).snd → a₁✝ = a₂✝",
"tactic": "simp"
}
] |
[
60,
9
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
53,
1
] |
Mathlib/Algebra/MonoidAlgebra/Basic.lean
|
AddMonoidAlgebra.mapDomain_single
|
[] |
[
1223,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1221,
1
] |
Mathlib/AlgebraicGeometry/PrimeSpectrum/Basic.lean
|
PrimeSpectrum.gc
|
[] |
[
195,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
192,
1
] |
Mathlib/Data/Finset/Option.lean
|
Finset.mem_eraseNone
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.7248\ns : Finset (Option α)\nx : α\n⊢ x ∈ ↑eraseNone s ↔ some x ∈ s",
"tactic": "simp [eraseNone]"
}
] |
[
94,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
93,
1
] |
Mathlib/Data/Seq/Computation.lean
|
Computation.destruct_empty
|
[] |
[
156,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
155,
1
] |
Mathlib/SetTheory/Game/PGame.lean
|
PGame.Fuzzy.swap
|
[] |
[
906,
11
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
905,
1
] |
Mathlib/Data/Real/ENNReal.lean
|
ENNReal.iInf_ne_top
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.38037\na b c d : ℝ≥0∞\nr p q : ℝ≥0\ninst✝ : CompleteLattice α\nf : ℝ≥0∞ → α\n⊢ (⨅ (x : ℝ≥0∞) (_ : x ≠ ⊤), f x) = ⨅ (x : ℝ≥0), f ↑x",
"tactic": "rw [iInf_subtype', cinfi_ne_top]"
}
] |
[
434,
85
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
433,
1
] |
Mathlib/GroupTheory/Perm/Support.lean
|
Equiv.Perm.coe_support_eq_set_support
|
[
{
"state_after": "case h\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf✝ g f : Perm α\nx✝ : α\n⊢ x✝ ∈ ↑(support f) ↔ x✝ ∈ {x | ↑f x ≠ x}",
"state_before": "α : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf✝ g f : Perm α\n⊢ ↑(support f) = {x | ↑f x ≠ x}",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case h\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf✝ g f : Perm α\nx✝ : α\n⊢ x✝ ∈ ↑(support f) ↔ x✝ ∈ {x | ↑f x ≠ x}",
"tactic": "simp"
}
] |
[
302,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
300,
1
] |
Mathlib/Order/BooleanAlgebra.lean
|
compl_compl
|
[] |
[
637,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
636,
1
] |
Mathlib/Analysis/Calculus/Inverse.lean
|
HasStrictFDerivAt.approximates_deriv_on_nhds
|
[
{
"state_after": "case inl\n𝕜 : Type u_1\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type ?u.502454\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type ?u.502557\ninst✝¹ : NormedAddCommGroup G'\ninst✝ : NormedSpace 𝕜 G'\nε : ℝ\nf : E → F\nf' : E →L[𝕜] F\na : E\nhf : HasStrictFDerivAt f f' a\nc : ℝ≥0\nhE : Subsingleton E\n⊢ ∃ s, s ∈ 𝓝 a ∧ ApproximatesLinearOn f f' s c\n\ncase inr\n𝕜 : Type u_1\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type ?u.502454\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type ?u.502557\ninst✝¹ : NormedAddCommGroup G'\ninst✝ : NormedSpace 𝕜 G'\nε : ℝ\nf : E → F\nf' : E →L[𝕜] F\na : E\nhf : HasStrictFDerivAt f f' a\nc : ℝ≥0\nhc : 0 < c\n⊢ ∃ s, s ∈ 𝓝 a ∧ ApproximatesLinearOn f f' s c",
"state_before": "𝕜 : Type u_1\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type ?u.502454\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type ?u.502557\ninst✝¹ : NormedAddCommGroup G'\ninst✝ : NormedSpace 𝕜 G'\nε : ℝ\nf : E → F\nf' : E →L[𝕜] F\na : E\nhf : HasStrictFDerivAt f f' a\nc : ℝ≥0\nhc : Subsingleton E ∨ 0 < c\n⊢ ∃ s, s ∈ 𝓝 a ∧ ApproximatesLinearOn f f' s c",
"tactic": "cases' hc with hE hc"
},
{
"state_after": "case inr\n𝕜 : Type u_1\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type ?u.502454\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type ?u.502557\ninst✝¹ : NormedAddCommGroup G'\ninst✝ : NormedSpace 𝕜 G'\nε : ℝ\nf : E → F\nf' : E →L[𝕜] F\na : E\nhf : HasStrictFDerivAt f f' a\nc : ℝ≥0\nhc : 0 < c\nthis : ∀ᶠ (x : E × E) in 𝓝 (a, a), ‖f x.fst - f x.snd - ↑f' (x.fst - x.snd)‖ ≤ (fun a => ↑a) c * ‖x.fst - x.snd‖\n⊢ ∃ s, s ∈ 𝓝 a ∧ ApproximatesLinearOn f f' s c",
"state_before": "case inr\n𝕜 : Type u_1\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type ?u.502454\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type ?u.502557\ninst✝¹ : NormedAddCommGroup G'\ninst✝ : NormedSpace 𝕜 G'\nε : ℝ\nf : E → F\nf' : E →L[𝕜] F\na : E\nhf : HasStrictFDerivAt f f' a\nc : ℝ≥0\nhc : 0 < c\n⊢ ∃ s, s ∈ 𝓝 a ∧ ApproximatesLinearOn f f' s c",
"tactic": "have := hf.def hc"
},
{
"state_after": "case inr\n𝕜 : Type u_1\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type ?u.502454\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type ?u.502557\ninst✝¹ : NormedAddCommGroup G'\ninst✝ : NormedSpace 𝕜 G'\nε : ℝ\nf : E → F\nf' : E →L[𝕜] F\na : E\nhf : HasStrictFDerivAt f f' a\nc : ℝ≥0\nhc : 0 < c\nthis : ∃ t, t ∈ 𝓝 a ∧ t ×ˢ t ⊆ {x | ‖f x.fst - f x.snd - ↑f' (x.fst - x.snd)‖ ≤ (fun a => ↑a) c * ‖x.fst - x.snd‖}\n⊢ ∃ s, s ∈ 𝓝 a ∧ ApproximatesLinearOn f f' s c",
"state_before": "case inr\n𝕜 : Type u_1\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type ?u.502454\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type ?u.502557\ninst✝¹ : NormedAddCommGroup G'\ninst✝ : NormedSpace 𝕜 G'\nε : ℝ\nf : E → F\nf' : E →L[𝕜] F\na : E\nhf : HasStrictFDerivAt f f' a\nc : ℝ≥0\nhc : 0 < c\nthis : ∀ᶠ (x : E × E) in 𝓝 (a, a), ‖f x.fst - f x.snd - ↑f' (x.fst - x.snd)‖ ≤ (fun a => ↑a) c * ‖x.fst - x.snd‖\n⊢ ∃ s, s ∈ 𝓝 a ∧ ApproximatesLinearOn f f' s c",
"tactic": "rw [nhds_prod_eq, Filter.Eventually, mem_prod_same_iff] at this"
},
{
"state_after": "case inr.intro.intro\n𝕜 : Type u_1\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type ?u.502454\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type ?u.502557\ninst✝¹ : NormedAddCommGroup G'\ninst✝ : NormedSpace 𝕜 G'\nε : ℝ\nf : E → F\nf' : E →L[𝕜] F\na : E\nhf : HasStrictFDerivAt f f' a\nc : ℝ≥0\nhc : 0 < c\ns : Set E\nhas : s ∈ 𝓝 a\nhs : s ×ˢ s ⊆ {x | ‖f x.fst - f x.snd - ↑f' (x.fst - x.snd)‖ ≤ (fun a => ↑a) c * ‖x.fst - x.snd‖}\n⊢ ∃ s, s ∈ 𝓝 a ∧ ApproximatesLinearOn f f' s c",
"state_before": "case inr\n𝕜 : Type u_1\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type ?u.502454\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type ?u.502557\ninst✝¹ : NormedAddCommGroup G'\ninst✝ : NormedSpace 𝕜 G'\nε : ℝ\nf : E → F\nf' : E →L[𝕜] F\na : E\nhf : HasStrictFDerivAt f f' a\nc : ℝ≥0\nhc : 0 < c\nthis : ∃ t, t ∈ 𝓝 a ∧ t ×ˢ t ⊆ {x | ‖f x.fst - f x.snd - ↑f' (x.fst - x.snd)‖ ≤ (fun a => ↑a) c * ‖x.fst - x.snd‖}\n⊢ ∃ s, s ∈ 𝓝 a ∧ ApproximatesLinearOn f f' s c",
"tactic": "rcases this with ⟨s, has, hs⟩"
},
{
"state_after": "no goals",
"state_before": "case inr.intro.intro\n𝕜 : Type u_1\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type ?u.502454\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type ?u.502557\ninst✝¹ : NormedAddCommGroup G'\ninst✝ : NormedSpace 𝕜 G'\nε : ℝ\nf : E → F\nf' : E →L[𝕜] F\na : E\nhf : HasStrictFDerivAt f f' a\nc : ℝ≥0\nhc : 0 < c\ns : Set E\nhas : s ∈ 𝓝 a\nhs : s ×ˢ s ⊆ {x | ‖f x.fst - f x.snd - ↑f' (x.fst - x.snd)‖ ≤ (fun a => ↑a) c * ‖x.fst - x.snd‖}\n⊢ ∃ s, s ∈ 𝓝 a ∧ ApproximatesLinearOn f f' s c",
"tactic": "exact ⟨s, has, fun x hx y hy => hs (mk_mem_prod hx hy)⟩"
},
{
"state_after": "case inl\n𝕜 : Type u_1\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type ?u.502454\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type ?u.502557\ninst✝¹ : NormedAddCommGroup G'\ninst✝ : NormedSpace 𝕜 G'\nε : ℝ\nf : E → F\nf' : E →L[𝕜] F\na : E\nhf : HasStrictFDerivAt f f' a\nc : ℝ≥0\nhE : Subsingleton E\nx : E\nx✝¹ : x ∈ univ\ny : E\nx✝ : y ∈ univ\n⊢ ‖f x - f y - ↑f' (x - y)‖ ≤ ↑c * ‖x - y‖",
"state_before": "case inl\n𝕜 : Type u_1\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type ?u.502454\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type ?u.502557\ninst✝¹ : NormedAddCommGroup G'\ninst✝ : NormedSpace 𝕜 G'\nε : ℝ\nf : E → F\nf' : E →L[𝕜] F\na : E\nhf : HasStrictFDerivAt f f' a\nc : ℝ≥0\nhE : Subsingleton E\n⊢ ∃ s, s ∈ 𝓝 a ∧ ApproximatesLinearOn f f' s c",
"tactic": "refine' ⟨univ, IsOpen.mem_nhds isOpen_univ trivial, fun x _ y _ => _⟩"
},
{
"state_after": "no goals",
"state_before": "case inl\n𝕜 : Type u_1\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type ?u.502454\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type ?u.502557\ninst✝¹ : NormedAddCommGroup G'\ninst✝ : NormedSpace 𝕜 G'\nε : ℝ\nf : E → F\nf' : E →L[𝕜] F\na : E\nhf : HasStrictFDerivAt f f' a\nc : ℝ≥0\nhE : Subsingleton E\nx : E\nx✝¹ : x ∈ univ\ny : E\nx✝ : y ∈ univ\n⊢ ‖f x - f y - ↑f' (x - y)‖ ≤ ↑c * ‖x - y‖",
"tactic": "simp [@Subsingleton.elim E hE x y]"
}
] |
[
549,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
540,
1
] |
Mathlib/MeasureTheory/Integral/IntervalIntegral.lean
|
intervalIntegral.integral_eq_zero_iff_of_nonneg_ae
|
[
{
"state_after": "case inl\nι : Type ?u.19452276\n𝕜 : Type ?u.19452279\nE : Type ?u.19452282\nF : Type ?u.19452285\nA : Type ?u.19452288\ninst✝² : NormedAddCommGroup E\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\nf g : ℝ → ℝ\na b : ℝ\nμ : MeasureTheory.Measure ℝ\nhfi : IntervalIntegrable f μ a b\nhab : a ≤ b\nhf : 0 ≤ᵐ[Measure.restrict μ (Ioc a b)] f\n⊢ (∫ (x : ℝ) in a..b, f x ∂μ) = 0 ↔ f =ᵐ[Measure.restrict μ (Ioc a b)] 0\n\ncase inr\nι : Type ?u.19452276\n𝕜 : Type ?u.19452279\nE : Type ?u.19452282\nF : Type ?u.19452285\nA : Type ?u.19452288\ninst✝² : NormedAddCommGroup E\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\nf g : ℝ → ℝ\na b : ℝ\nμ : MeasureTheory.Measure ℝ\nhfi : IntervalIntegrable f μ a b\nhab : b ≤ a\nhf : 0 ≤ᵐ[Measure.restrict μ (Ioc b a)] f\n⊢ (∫ (x : ℝ) in a..b, f x ∂μ) = 0 ↔ f =ᵐ[Measure.restrict μ (Ioc b a)] 0",
"state_before": "ι : Type ?u.19452276\n𝕜 : Type ?u.19452279\nE : Type ?u.19452282\nF : Type ?u.19452285\nA : Type ?u.19452288\ninst✝² : NormedAddCommGroup E\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\nf g : ℝ → ℝ\na b : ℝ\nμ : MeasureTheory.Measure ℝ\nhf : 0 ≤ᵐ[Measure.restrict μ (Ioc a b ∪ Ioc b a)] f\nhfi : IntervalIntegrable f μ a b\n⊢ (∫ (x : ℝ) in a..b, f x ∂μ) = 0 ↔ f =ᵐ[Measure.restrict μ (Ioc a b ∪ Ioc b a)] 0",
"tactic": "cases' le_total a b with hab hab <;>\n simp only [Ioc_eq_empty hab.not_lt, empty_union, union_empty] at hf ⊢"
},
{
"state_after": "no goals",
"state_before": "case inl\nι : Type ?u.19452276\n𝕜 : Type ?u.19452279\nE : Type ?u.19452282\nF : Type ?u.19452285\nA : Type ?u.19452288\ninst✝² : NormedAddCommGroup E\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\nf g : ℝ → ℝ\na b : ℝ\nμ : MeasureTheory.Measure ℝ\nhfi : IntervalIntegrable f μ a b\nhab : a ≤ b\nhf : 0 ≤ᵐ[Measure.restrict μ (Ioc a b)] f\n⊢ (∫ (x : ℝ) in a..b, f x ∂μ) = 0 ↔ f =ᵐ[Measure.restrict μ (Ioc a b)] 0",
"tactic": "exact integral_eq_zero_iff_of_le_of_nonneg_ae hab hf hfi"
},
{
"state_after": "no goals",
"state_before": "case inr\nι : Type ?u.19452276\n𝕜 : Type ?u.19452279\nE : Type ?u.19452282\nF : Type ?u.19452285\nA : Type ?u.19452288\ninst✝² : NormedAddCommGroup E\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\nf g : ℝ → ℝ\na b : ℝ\nμ : MeasureTheory.Measure ℝ\nhfi : IntervalIntegrable f μ a b\nhab : b ≤ a\nhf : 0 ≤ᵐ[Measure.restrict μ (Ioc b a)] f\n⊢ (∫ (x : ℝ) in a..b, f x ∂μ) = 0 ↔ f =ᵐ[Measure.restrict μ (Ioc b a)] 0",
"tactic": "rw [integral_symm, neg_eq_zero, integral_eq_zero_iff_of_le_of_nonneg_ae hab hf hfi.symm]"
}
] |
[
1273,
93
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1267,
1
] |
Mathlib/Algebra/Order/Kleene.lean
|
kstar_mul_le_self
|
[] |
[
219,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
218,
1
] |
Mathlib/GroupTheory/Subgroup/ZPowers.lean
|
AddSubgroup.range_zmultiplesHom
|
[] |
[
94,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
93,
1
] |
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