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/Order/OmegaCompletePartialOrder.lean
|
OmegaCompletePartialOrder.ωSup_le_ωSup_of_le
|
[
{
"state_after": "case intro\nα : Type u\nβ : Type v\nγ : Type ?u.8099\ninst✝ : OmegaCompletePartialOrder α\nc₀ c₁ : Chain α\nh✝ : c₀ ≤ c₁\ni w✝ : ℕ\nh : ↑c₀ i ≤ ↑c₁ w✝\n⊢ ↑c₀ i ≤ ωSup c₁",
"state_before": "α : Type u\nβ : Type v\nγ : Type ?u.8099\ninst✝ : OmegaCompletePartialOrder α\nc₀ c₁ : Chain α\nh : c₀ ≤ c₁\ni : ℕ\n⊢ ↑c₀ i ≤ ωSup c₁",
"tactic": "obtain ⟨_, h⟩ := h i"
},
{
"state_after": "no goals",
"state_before": "case intro\nα : Type u\nβ : Type v\nγ : Type ?u.8099\ninst✝ : OmegaCompletePartialOrder α\nc₀ c₁ : Chain α\nh✝ : c₀ ≤ c₁\ni w✝ : ℕ\nh : ↑c₀ i ≤ ↑c₁ w✝\n⊢ ↑c₀ i ≤ ωSup c₁",
"tactic": "exact le_trans h (le_ωSup _ _)"
}
] |
[
218,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
215,
1
] |
Mathlib/Data/Set/Function.lean
|
Set.InjOn.mono
|
[] |
[
629,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
628,
1
] |
Mathlib/RingTheory/Subring/Basic.lean
|
Subring.mk'_toAddSubgroup
|
[] |
[
324,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
322,
1
] |
Mathlib/Data/Num/Lemmas.lean
|
PosNum.le_to_nat
|
[
{
"state_after": "α : Type ?u.168918\nm n : PosNum\n⊢ ¬↑n < ↑m ↔ m ≤ n",
"state_before": "α : Type ?u.168918\nm n : PosNum\n⊢ ↑m ≤ ↑n ↔ m ≤ n",
"tactic": "rw [← not_lt]"
},
{
"state_after": "no goals",
"state_before": "α : Type ?u.168918\nm n : PosNum\n⊢ ¬↑n < ↑m ↔ m ≤ n",
"tactic": "exact not_congr lt_to_nat"
}
] |
[
199,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
198,
1
] |
Mathlib/Data/Fin/VecNotation.lean
|
Matrix.tail_sub
|
[] |
[
522,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
521,
1
] |
Mathlib/GroupTheory/Submonoid/Pointwise.lean
|
AddSubmonoid.mul_bot
|
[
{
"state_after": "α : Type ?u.249186\nG : Type ?u.249189\nM : Type ?u.249192\nR : Type u_1\nA : Type ?u.249198\ninst✝² : Monoid M\ninst✝¹ : AddMonoid A\ninst✝ : NonUnitalNonAssocSemiring R\nS : AddSubmonoid R\nm : R\nx✝ : m ∈ S\nn : R\nhn : n = 0\n⊢ m * n = 0",
"state_before": "α : Type ?u.249186\nG : Type ?u.249189\nM : Type ?u.249192\nR : Type u_1\nA : Type ?u.249198\ninst✝² : Monoid M\ninst✝¹ : AddMonoid A\ninst✝ : NonUnitalNonAssocSemiring R\nS : AddSubmonoid R\nm : R\nx✝ : m ∈ S\nn : R\nhn : n ∈ ⊥\n⊢ m * n ∈ ⊥",
"tactic": "rw [AddSubmonoid.mem_bot] at hn ⊢"
},
{
"state_after": "no goals",
"state_before": "α : Type ?u.249186\nG : Type ?u.249189\nM : Type ?u.249192\nR : Type u_1\nA : Type ?u.249198\ninst✝² : Monoid M\ninst✝¹ : AddMonoid A\ninst✝ : NonUnitalNonAssocSemiring R\nS : AddSubmonoid R\nm : R\nx✝ : m ∈ S\nn : R\nhn : n = 0\n⊢ m * n = 0",
"tactic": "rw [hn, mul_zero]"
}
] |
[
576,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
574,
1
] |
Mathlib/Data/Fintype/BigOperators.lean
|
Finset.prod_univ_pi
|
[
{
"state_after": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.15155\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\ninst✝ : CommMonoid β\nδ : α → Type u_3\nt : (a : α) → Finset (δ a)\nf : ((a : α) → a ∈ univ → δ a) → β\n⊢ ∀ (a : (a : α) → a ∈ univ → δ a) (ha : a ∈ pi univ t), (fun x x_1 a => x a (_ : a ∈ univ)) a ha ∈ Fintype.piFinset t",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.15155\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\ninst✝ : CommMonoid β\nδ : α → Type u_3\nt : (a : α) → Finset (δ a)\nf : ((a : α) → a ∈ univ → δ a) → β\n⊢ ∏ x in pi univ t, f x = ∏ x in Fintype.piFinset t, f fun a x_1 => x a",
"tactic": "refine prod_bij (fun x _ a => x a (mem_univ _)) ?_ (by simp)\n (by simp (config := { contextual := true }) [Function.funext_iff]) fun x hx =>\n ⟨fun a _ => x a, by simp_all⟩"
},
{
"state_after": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.15155\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\ninst✝ : CommMonoid β\nδ : α → Type u_3\nt : (a : α) → Finset (δ a)\nf : ((a : α) → a ∈ univ → δ a) → β\na : (a : α) → a ∈ univ → δ a\nha : a ∈ pi univ t\n⊢ (fun x x_1 a => x a (_ : a ∈ univ)) a ha ∈ Fintype.piFinset t",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.15155\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\ninst✝ : CommMonoid β\nδ : α → Type u_3\nt : (a : α) → Finset (δ a)\nf : ((a : α) → a ∈ univ → δ a) → β\n⊢ ∀ (a : (a : α) → a ∈ univ → δ a) (ha : a ∈ pi univ t), (fun x x_1 a => x a (_ : a ∈ univ)) a ha ∈ Fintype.piFinset t",
"tactic": "intro a ha"
},
{
"state_after": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.15155\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\ninst✝ : CommMonoid β\nδ : α → Type u_3\nt : (a : α) → Finset (δ a)\nf : ((a : α) → a ∈ univ → δ a) → β\na : (a : α) → a ∈ univ → δ a\nha : a ∈ pi univ t\n⊢ ∃ a_1, (∀ (a : α) (h : a ∈ univ), a_1 a h ∈ t a) ∧ (fun a => a_1 a (_ : a ∈ univ)) = fun a_2 => a a_2 (_ : a_2 ∈ univ)",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.15155\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\ninst✝ : CommMonoid β\nδ : α → Type u_3\nt : (a : α) → Finset (δ a)\nf : ((a : α) → a ∈ univ → δ a) → β\na : (a : α) → a ∈ univ → δ a\nha : a ∈ pi univ t\n⊢ (fun x x_1 a => x a (_ : a ∈ univ)) a ha ∈ Fintype.piFinset t",
"tactic": "simp only [Fintype.piFinset, mem_map, mem_pi, Function.Embedding.coeFn_mk]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.15155\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\ninst✝ : CommMonoid β\nδ : α → Type u_3\nt : (a : α) → Finset (δ a)\nf : ((a : α) → a ∈ univ → δ a) → β\na : (a : α) → a ∈ univ → δ a\nha : a ∈ pi univ t\n⊢ ∃ a_1, (∀ (a : α) (h : a ∈ univ), a_1 a h ∈ t a) ∧ (fun a => a_1 a (_ : a ∈ univ)) = fun a_2 => a a_2 (_ : a_2 ∈ univ)",
"tactic": "exact ⟨a, by simpa using ha, by simp⟩"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.15155\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\ninst✝ : CommMonoid β\nδ : α → Type u_3\nt : (a : α) → Finset (δ a)\nf : ((a : α) → a ∈ univ → δ a) → β\n⊢ ∀ (a : (a : α) → a ∈ univ → δ a) (ha : a ∈ pi univ t),\n f a = f fun a_1 x => (fun x x_1 a => x a (_ : a ∈ univ)) a ha a_1",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.15155\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\ninst✝ : CommMonoid β\nδ : α → Type u_3\nt : (a : α) → Finset (δ a)\nf : ((a : α) → a ∈ univ → δ a) → β\n⊢ ∀ (a₁ a₂ : (a : α) → a ∈ univ → δ a) (ha₁ : a₁ ∈ pi univ t) (ha₂ : a₂ ∈ pi univ t),\n (fun x x_1 a => x a (_ : a ∈ univ)) a₁ ha₁ = (fun x x_1 a => x a (_ : a ∈ univ)) a₂ ha₂ → a₁ = a₂",
"tactic": "simp (config := { contextual := true }) [Function.funext_iff]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.15155\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\ninst✝ : CommMonoid β\nδ : α → Type u_3\nt : (a : α) → Finset (δ a)\nf : ((a : α) → a ∈ univ → δ a) → β\nx : (a : α) → δ a\nhx : x ∈ Fintype.piFinset t\n⊢ ∃ ha, x = (fun x x_1 a => x a (_ : a ∈ univ)) (fun a x_1 => x a) ha",
"tactic": "simp_all"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.15155\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\ninst✝ : CommMonoid β\nδ : α → Type u_3\nt : (a : α) → Finset (δ a)\nf : ((a : α) → a ∈ univ → δ a) → β\na : (a : α) → a ∈ univ → δ a\nha : a ∈ pi univ t\n⊢ ∀ (a_1 : α) (h : a_1 ∈ univ), a a_1 h ∈ t a_1",
"tactic": "simpa using ha"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.15155\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\ninst✝ : CommMonoid β\nδ : α → Type u_3\nt : (a : α) → Finset (δ a)\nf : ((a : α) → a ∈ univ → δ a) → β\na : (a : α) → a ∈ univ → δ a\nha : a ∈ pi univ t\n⊢ (fun a_1 => a a_1 (_ : a_1 ∈ univ)) = fun a_1 => a a_1 (_ : a_1 ∈ univ)",
"tactic": "simp"
}
] |
[
184,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
175,
1
] |
Mathlib/Algebra/Hom/Equiv/Basic.lean
|
MulEquiv.piCongrRight_symm
|
[] |
[
668,
95
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
667,
1
] |
Mathlib/Data/Polynomial/Basic.lean
|
Polynomial.C_mul_monomial
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\na b : R\nm n : ℕ\ninst✝ : Semiring R\np q : R[X]\n⊢ ↑C a * ↑(monomial n) b = ↑(monomial n) (a * b)",
"tactic": "simp only [← monomial_zero_left, monomial_mul_monomial, zero_add]"
}
] |
[
541,
68
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
540,
1
] |
Mathlib/Analysis/NormedSpace/Pointwise.lean
|
smul_closedBall'
|
[
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\nE : Type u_2\ninst✝² : NormedField 𝕜\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nc : 𝕜\nhc : c ≠ 0\nx : E\nr : ℝ\n⊢ c • closedBall x r = closedBall (c • x) (‖c‖ * r)",
"tactic": "simp only [← ball_union_sphere, Set.smul_set_union, _root_.smul_ball hc, smul_sphere' hc]"
}
] |
[
112,
92
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
110,
1
] |
Mathlib/Data/Bool/Basic.lean
|
Bool.false_le
|
[] |
[
336,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
335,
1
] |
Mathlib/Data/Matrix/Basic.lean
|
Matrix.diagonal_injective
|
[
{
"state_after": "no goals",
"state_before": "l : Type ?u.47215\nm : Type ?u.47218\nn : Type u_1\no : Type ?u.47224\nm' : o → Type ?u.47229\nn' : o → Type ?u.47234\nR : Type ?u.47237\nS : Type ?u.47240\nα : Type v\nβ : Type w\nγ : Type ?u.47247\ninst✝¹ : DecidableEq n\ninst✝ : Zero α\nd₁ d₂ : n → α\nh : diagonal d₁ = diagonal d₂\ni : n\n⊢ d₁ i = d₂ i",
"tactic": "simpa using Matrix.ext_iff.mpr h i i"
}
] |
[
455,
73
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
454,
1
] |
Mathlib/NumberTheory/Zsqrtd/Basic.lean
|
Zsqrtd.smul_re
|
[
{
"state_after": "no goals",
"state_before": "d a : ℤ\nb : ℤ√d\n⊢ (↑a * b).re = a * b.re",
"tactic": "simp"
}
] |
[
319,
70
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
319,
1
] |
Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean
|
MeasureTheory.Lp.simpleFunc.memℒp
|
[] |
[
630,
85
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
629,
11
] |
Mathlib/Topology/UniformSpace/Cauchy.lean
|
Filter.HasBasis.totallyBounded_iff
|
[] |
[
488,
80
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
484,
1
] |
Mathlib/Order/Bounds/Basic.lean
|
isGreatest_Iic
|
[] |
[
515,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
514,
1
] |
Mathlib/Data/MvPolynomial/Basic.lean
|
MvPolynomial.eval₂_comp_right
|
[
{
"state_after": "case h_C\nR : Type u\nS₁ : Type v\nS₂✝ : Type w\nS₃ : Type x\nσ : Type u_2\na a' a₁ a₂ : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝² : CommSemiring R\ninst✝¹ : CommSemiring S₁\np✝ q : MvPolynomial σ R\nf✝ : R →+* S₁\nS₂ : Type u_1\ninst✝ : CommSemiring S₂\nk : S₁ →+* S₂\nf : R →+* S₁\ng : σ → S₁\np : MvPolynomial σ R\n⊢ ∀ (a : R), ↑k (eval₂ f g (↑C a)) = eval₂ k (↑k ∘ g) (↑(map f) (↑C a))\n\ncase h_add\nR : Type u\nS₁ : Type v\nS₂✝ : Type w\nS₃ : Type x\nσ : Type u_2\na a' a₁ a₂ : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝² : CommSemiring R\ninst✝¹ : CommSemiring S₁\np✝ q : MvPolynomial σ R\nf✝ : R →+* S₁\nS₂ : Type u_1\ninst✝ : CommSemiring S₂\nk : S₁ →+* S₂\nf : R →+* S₁\ng : σ → S₁\np : MvPolynomial σ R\n⊢ ∀ (p q : MvPolynomial σ R),\n ↑k (eval₂ f g p) = eval₂ k (↑k ∘ g) (↑(map f) p) →\n ↑k (eval₂ f g q) = eval₂ k (↑k ∘ g) (↑(map f) q) → ↑k (eval₂ f g (p + q)) = eval₂ k (↑k ∘ g) (↑(map f) (p + q))\n\ncase h_X\nR : Type u\nS₁ : Type v\nS₂✝ : Type w\nS₃ : Type x\nσ : Type u_2\na a' a₁ a₂ : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝² : CommSemiring R\ninst✝¹ : CommSemiring S₁\np✝ q : MvPolynomial σ R\nf✝ : R →+* S₁\nS₂ : Type u_1\ninst✝ : CommSemiring S₂\nk : S₁ →+* S₂\nf : R →+* S₁\ng : σ → S₁\np : MvPolynomial σ R\n⊢ ∀ (p : MvPolynomial σ R) (n : σ),\n ↑k (eval₂ f g p) = eval₂ k (↑k ∘ g) (↑(map f) p) → ↑k (eval₂ f g (p * X n)) = eval₂ k (↑k ∘ g) (↑(map f) (p * X n))",
"state_before": "R : Type u\nS₁ : Type v\nS₂✝ : Type w\nS₃ : Type x\nσ : Type u_2\na a' a₁ a₂ : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝² : CommSemiring R\ninst✝¹ : CommSemiring S₁\np✝ q : MvPolynomial σ R\nf✝ : R →+* S₁\nS₂ : Type u_1\ninst✝ : CommSemiring S₂\nk : S₁ →+* S₂\nf : R →+* S₁\ng : σ → S₁\np : MvPolynomial σ R\n⊢ ↑k (eval₂ f g p) = eval₂ k (↑k ∘ g) (↑(map f) p)",
"tactic": "apply MvPolynomial.induction_on p"
},
{
"state_after": "case h_C\nR : Type u\nS₁ : Type v\nS₂✝ : Type w\nS₃ : Type x\nσ : Type u_2\na a' a₁ a₂ : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝² : CommSemiring R\ninst✝¹ : CommSemiring S₁\np✝ q : MvPolynomial σ R\nf✝ : R →+* S₁\nS₂ : Type u_1\ninst✝ : CommSemiring S₂\nk : S₁ →+* S₂\nf : R →+* S₁\ng : σ → S₁\np : MvPolynomial σ R\nr : R\n⊢ ↑k (eval₂ f g (↑C r)) = eval₂ k (↑k ∘ g) (↑(map f) (↑C r))",
"state_before": "case h_C\nR : Type u\nS₁ : Type v\nS₂✝ : Type w\nS₃ : Type x\nσ : Type u_2\na a' a₁ a₂ : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝² : CommSemiring R\ninst✝¹ : CommSemiring S₁\np✝ q : MvPolynomial σ R\nf✝ : R →+* S₁\nS₂ : Type u_1\ninst✝ : CommSemiring S₂\nk : S₁ →+* S₂\nf : R →+* S₁\ng : σ → S₁\np : MvPolynomial σ R\n⊢ ∀ (a : R), ↑k (eval₂ f g (↑C a)) = eval₂ k (↑k ∘ g) (↑(map f) (↑C a))",
"tactic": "intro r"
},
{
"state_after": "no goals",
"state_before": "case h_C\nR : Type u\nS₁ : Type v\nS₂✝ : Type w\nS₃ : Type x\nσ : Type u_2\na a' a₁ a₂ : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝² : CommSemiring R\ninst✝¹ : CommSemiring S₁\np✝ q : MvPolynomial σ R\nf✝ : R →+* S₁\nS₂ : Type u_1\ninst✝ : CommSemiring S₂\nk : S₁ →+* S₂\nf : R →+* S₁\ng : σ → S₁\np : MvPolynomial σ R\nr : R\n⊢ ↑k (eval₂ f g (↑C r)) = eval₂ k (↑k ∘ g) (↑(map f) (↑C r))",
"tactic": "rw [eval₂_C, map_C, eval₂_C]"
},
{
"state_after": "case h_add\nR : Type u\nS₁ : Type v\nS₂✝ : Type w\nS₃ : Type x\nσ : Type u_2\na a' a₁ a₂ : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝² : CommSemiring R\ninst✝¹ : CommSemiring S₁\np✝¹ q✝ : MvPolynomial σ R\nf✝ : R →+* S₁\nS₂ : Type u_1\ninst✝ : CommSemiring S₂\nk : S₁ →+* S₂\nf : R →+* S₁\ng : σ → S₁\np✝ p q : MvPolynomial σ R\nhp : ↑k (eval₂ f g p) = eval₂ k (↑k ∘ g) (↑(map f) p)\nhq : ↑k (eval₂ f g q) = eval₂ k (↑k ∘ g) (↑(map f) q)\n⊢ ↑k (eval₂ f g (p + q)) = eval₂ k (↑k ∘ g) (↑(map f) (p + q))",
"state_before": "case h_add\nR : Type u\nS₁ : Type v\nS₂✝ : Type w\nS₃ : Type x\nσ : Type u_2\na a' a₁ a₂ : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝² : CommSemiring R\ninst✝¹ : CommSemiring S₁\np✝ q : MvPolynomial σ R\nf✝ : R →+* S₁\nS₂ : Type u_1\ninst✝ : CommSemiring S₂\nk : S₁ →+* S₂\nf : R →+* S₁\ng : σ → S₁\np : MvPolynomial σ R\n⊢ ∀ (p q : MvPolynomial σ R),\n ↑k (eval₂ f g p) = eval₂ k (↑k ∘ g) (↑(map f) p) →\n ↑k (eval₂ f g q) = eval₂ k (↑k ∘ g) (↑(map f) q) → ↑k (eval₂ f g (p + q)) = eval₂ k (↑k ∘ g) (↑(map f) (p + q))",
"tactic": "intro p q hp hq"
},
{
"state_after": "no goals",
"state_before": "case h_add\nR : Type u\nS₁ : Type v\nS₂✝ : Type w\nS₃ : Type x\nσ : Type u_2\na a' a₁ a₂ : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝² : CommSemiring R\ninst✝¹ : CommSemiring S₁\np✝¹ q✝ : MvPolynomial σ R\nf✝ : R →+* S₁\nS₂ : Type u_1\ninst✝ : CommSemiring S₂\nk : S₁ →+* S₂\nf : R →+* S₁\ng : σ → S₁\np✝ p q : MvPolynomial σ R\nhp : ↑k (eval₂ f g p) = eval₂ k (↑k ∘ g) (↑(map f) p)\nhq : ↑k (eval₂ f g q) = eval₂ k (↑k ∘ g) (↑(map f) q)\n⊢ ↑k (eval₂ f g (p + q)) = eval₂ k (↑k ∘ g) (↑(map f) (p + q))",
"tactic": "rw [eval₂_add, k.map_add, (map f).map_add, eval₂_add, hp, hq]"
},
{
"state_after": "case h_X\nR : Type u\nS₁ : Type v\nS₂✝ : Type w\nS₃ : Type x\nσ : Type u_2\na a' a₁ a₂ : R\ne : ℕ\nn m : σ\ns✝ : σ →₀ ℕ\ninst✝² : CommSemiring R\ninst✝¹ : CommSemiring S₁\np✝¹ q : MvPolynomial σ R\nf✝ : R →+* S₁\nS₂ : Type u_1\ninst✝ : CommSemiring S₂\nk : S₁ →+* S₂\nf : R →+* S₁\ng : σ → S₁\np✝ p : MvPolynomial σ R\ns : σ\nhp : ↑k (eval₂ f g p) = eval₂ k (↑k ∘ g) (↑(map f) p)\n⊢ ↑k (eval₂ f g (p * X s)) = eval₂ k (↑k ∘ g) (↑(map f) (p * X s))",
"state_before": "case h_X\nR : Type u\nS₁ : Type v\nS₂✝ : Type w\nS₃ : Type x\nσ : Type u_2\na a' a₁ a₂ : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝² : CommSemiring R\ninst✝¹ : CommSemiring S₁\np✝ q : MvPolynomial σ R\nf✝ : R →+* S₁\nS₂ : Type u_1\ninst✝ : CommSemiring S₂\nk : S₁ →+* S₂\nf : R →+* S₁\ng : σ → S₁\np : MvPolynomial σ R\n⊢ ∀ (p : MvPolynomial σ R) (n : σ),\n ↑k (eval₂ f g p) = eval₂ k (↑k ∘ g) (↑(map f) p) → ↑k (eval₂ f g (p * X n)) = eval₂ k (↑k ∘ g) (↑(map f) (p * X n))",
"tactic": "intro p s hp"
},
{
"state_after": "case h_X\nR : Type u\nS₁ : Type v\nS₂✝ : Type w\nS₃ : Type x\nσ : Type u_2\na a' a₁ a₂ : R\ne : ℕ\nn m : σ\ns✝ : σ →₀ ℕ\ninst✝² : CommSemiring R\ninst✝¹ : CommSemiring S₁\np✝¹ q : MvPolynomial σ R\nf✝ : R →+* S₁\nS₂ : Type u_1\ninst✝ : CommSemiring S₂\nk : S₁ →+* S₂\nf : R →+* S₁\ng : σ → S₁\np✝ p : MvPolynomial σ R\ns : σ\nhp : ↑k (eval₂ f g p) = eval₂ k (↑k ∘ g) (↑(map f) p)\n⊢ eval₂ k (↑k ∘ g) (↑(map f) p) * ↑k (g s) = eval₂ k (↑k ∘ g) (↑(map f) p) * (↑k ∘ g) s",
"state_before": "case h_X\nR : Type u\nS₁ : Type v\nS₂✝ : Type w\nS₃ : Type x\nσ : Type u_2\na a' a₁ a₂ : R\ne : ℕ\nn m : σ\ns✝ : σ →₀ ℕ\ninst✝² : CommSemiring R\ninst✝¹ : CommSemiring S₁\np✝¹ q : MvPolynomial σ R\nf✝ : R →+* S₁\nS₂ : Type u_1\ninst✝ : CommSemiring S₂\nk : S₁ →+* S₂\nf : R →+* S₁\ng : σ → S₁\np✝ p : MvPolynomial σ R\ns : σ\nhp : ↑k (eval₂ f g p) = eval₂ k (↑k ∘ g) (↑(map f) p)\n⊢ ↑k (eval₂ f g (p * X s)) = eval₂ k (↑k ∘ g) (↑(map f) (p * X s))",
"tactic": "rw [eval₂_mul, k.map_mul, (map f).map_mul, eval₂_mul, map_X, hp, eval₂_X, eval₂_X]"
},
{
"state_after": "no goals",
"state_before": "case h_X\nR : Type u\nS₁ : Type v\nS₂✝ : Type w\nS₃ : Type x\nσ : Type u_2\na a' a₁ a₂ : R\ne : ℕ\nn m : σ\ns✝ : σ →₀ ℕ\ninst✝² : CommSemiring R\ninst✝¹ : CommSemiring S₁\np✝¹ q : MvPolynomial σ R\nf✝ : R →+* S₁\nS₂ : Type u_1\ninst✝ : CommSemiring S₂\nk : S₁ →+* S₂\nf : R →+* S₁\ng : σ → S₁\np✝ p : MvPolynomial σ R\ns : σ\nhp : ↑k (eval₂ f g p) = eval₂ k (↑k ∘ g) (↑(map f) p)\n⊢ eval₂ k (↑k ∘ g) (↑(map f) p) * ↑k (g s) = eval₂ k (↑k ∘ g) (↑(map f) p) * (↑k ∘ g) s",
"tactic": "rfl"
}
] |
[
1254,
8
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1245,
1
] |
Mathlib/Algebra/Symmetrized.lean
|
SymAlg.sym_ne_one_iff
|
[] |
[
258,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
257,
1
] |
Mathlib/Topology/Separation.lean
|
RegularSpace.ofBasis
|
[] |
[
1512,
76
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1509,
1
] |
Mathlib/Algebra/GroupWithZero/Units/Lemmas.lean
|
MonoidWithZero.inverse_apply
|
[] |
[
270,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
268,
1
] |
Mathlib/Analysis/NormedSpace/Exponential.lean
|
exp_add_of_commute_of_mem_ball
|
[
{
"state_after": "𝕂 : Type u_1\n𝔸 : Type u_2\n𝔹 : Type ?u.101113\ninst✝⁶ : NontriviallyNormedField 𝕂\ninst✝⁵ : NormedRing 𝔸\ninst✝⁴ : NormedRing 𝔹\ninst✝³ : NormedAlgebra 𝕂 𝔸\ninst✝² : NormedAlgebra 𝕂 𝔹\ninst✝¹ : CompleteSpace 𝔸\ninst✝ : CharZero 𝕂\nx y : 𝔸\nhxy : Commute x y\nhx : x ∈ EMetric.ball 0 (FormalMultilinearSeries.radius (expSeries 𝕂 𝔸))\nhy : y ∈ EMetric.ball 0 (FormalMultilinearSeries.radius (expSeries 𝕂 𝔸))\n⊢ (fun x => ∑' (n : ℕ), (↑n !)⁻¹ • x ^ n) (x + y) =\n ∑' (n : ℕ), ∑ kl in Finset.Nat.antidiagonal n, (↑kl.fst !)⁻¹ • x ^ kl.fst * (↑kl.snd !)⁻¹ • y ^ kl.snd",
"state_before": "𝕂 : Type u_1\n𝔸 : Type u_2\n𝔹 : Type ?u.101113\ninst✝⁶ : NontriviallyNormedField 𝕂\ninst✝⁵ : NormedRing 𝔸\ninst✝⁴ : NormedRing 𝔹\ninst✝³ : NormedAlgebra 𝕂 𝔸\ninst✝² : NormedAlgebra 𝕂 𝔹\ninst✝¹ : CompleteSpace 𝔸\ninst✝ : CharZero 𝕂\nx y : 𝔸\nhxy : Commute x y\nhx : x ∈ EMetric.ball 0 (FormalMultilinearSeries.radius (expSeries 𝕂 𝔸))\nhy : y ∈ EMetric.ball 0 (FormalMultilinearSeries.radius (expSeries 𝕂 𝔸))\n⊢ exp 𝕂 (x + y) = exp 𝕂 x * exp 𝕂 y",
"tactic": "rw [exp_eq_tsum,\n tsum_mul_tsum_eq_tsum_sum_antidiagonal_of_summable_norm\n (norm_expSeries_summable_of_mem_ball' x hx) (norm_expSeries_summable_of_mem_ball' y hy)]"
},
{
"state_after": "𝕂 : Type u_1\n𝔸 : Type u_2\n𝔹 : Type ?u.101113\ninst✝⁶ : NontriviallyNormedField 𝕂\ninst✝⁵ : NormedRing 𝔸\ninst✝⁴ : NormedRing 𝔹\ninst✝³ : NormedAlgebra 𝕂 𝔸\ninst✝² : NormedAlgebra 𝕂 𝔹\ninst✝¹ : CompleteSpace 𝔸\ninst✝ : CharZero 𝕂\nx y : 𝔸\nhxy : Commute x y\nhx : x ∈ EMetric.ball 0 (FormalMultilinearSeries.radius (expSeries 𝕂 𝔸))\nhy : y ∈ EMetric.ball 0 (FormalMultilinearSeries.radius (expSeries 𝕂 𝔸))\n⊢ (∑' (n : ℕ), (↑n !)⁻¹ • (x + y) ^ n) =\n ∑' (n : ℕ), ∑ kl in Finset.Nat.antidiagonal n, (↑kl.fst !)⁻¹ • x ^ kl.fst * (↑kl.snd !)⁻¹ • y ^ kl.snd",
"state_before": "𝕂 : Type u_1\n𝔸 : Type u_2\n𝔹 : Type ?u.101113\ninst✝⁶ : NontriviallyNormedField 𝕂\ninst✝⁵ : NormedRing 𝔸\ninst✝⁴ : NormedRing 𝔹\ninst✝³ : NormedAlgebra 𝕂 𝔸\ninst✝² : NormedAlgebra 𝕂 𝔹\ninst✝¹ : CompleteSpace 𝔸\ninst✝ : CharZero 𝕂\nx y : 𝔸\nhxy : Commute x y\nhx : x ∈ EMetric.ball 0 (FormalMultilinearSeries.radius (expSeries 𝕂 𝔸))\nhy : y ∈ EMetric.ball 0 (FormalMultilinearSeries.radius (expSeries 𝕂 𝔸))\n⊢ (fun x => ∑' (n : ℕ), (↑n !)⁻¹ • x ^ n) (x + y) =\n ∑' (n : ℕ), ∑ kl in Finset.Nat.antidiagonal n, (↑kl.fst !)⁻¹ • x ^ kl.fst * (↑kl.snd !)⁻¹ • y ^ kl.snd",
"tactic": "dsimp only"
},
{
"state_after": "𝕂 : Type u_1\n𝔸 : Type u_2\n𝔹 : Type ?u.101113\ninst✝⁶ : NontriviallyNormedField 𝕂\ninst✝⁵ : NormedRing 𝔸\ninst✝⁴ : NormedRing 𝔹\ninst✝³ : NormedAlgebra 𝕂 𝔸\ninst✝² : NormedAlgebra 𝕂 𝔹\ninst✝¹ : CompleteSpace 𝔸\ninst✝ : CharZero 𝕂\nx y : 𝔸\nhxy : Commute x y\nhx : x ∈ EMetric.ball 0 (FormalMultilinearSeries.radius (expSeries 𝕂 𝔸))\nhy : y ∈ EMetric.ball 0 (FormalMultilinearSeries.radius (expSeries 𝕂 𝔸))\n⊢ (∑' (x_1 : ℕ),\n ∑ x_2 in Finset.Nat.antidiagonal x_1, (↑x_1 !)⁻¹ • Nat.choose x_1 x_2.fst • (x ^ x_2.fst * y ^ x_2.snd)) =\n ∑' (n : ℕ), ∑ kl in Finset.Nat.antidiagonal n, (↑kl.fst !)⁻¹ • x ^ kl.fst * (↑kl.snd !)⁻¹ • y ^ kl.snd",
"state_before": "𝕂 : Type u_1\n𝔸 : Type u_2\n𝔹 : Type ?u.101113\ninst✝⁶ : NontriviallyNormedField 𝕂\ninst✝⁵ : NormedRing 𝔸\ninst✝⁴ : NormedRing 𝔹\ninst✝³ : NormedAlgebra 𝕂 𝔸\ninst✝² : NormedAlgebra 𝕂 𝔹\ninst✝¹ : CompleteSpace 𝔸\ninst✝ : CharZero 𝕂\nx y : 𝔸\nhxy : Commute x y\nhx : x ∈ EMetric.ball 0 (FormalMultilinearSeries.radius (expSeries 𝕂 𝔸))\nhy : y ∈ EMetric.ball 0 (FormalMultilinearSeries.radius (expSeries 𝕂 𝔸))\n⊢ (∑' (n : ℕ), (↑n !)⁻¹ • (x + y) ^ n) =\n ∑' (n : ℕ), ∑ kl in Finset.Nat.antidiagonal n, (↑kl.fst !)⁻¹ • x ^ kl.fst * (↑kl.snd !)⁻¹ • y ^ kl.snd",
"tactic": "conv_lhs =>\n congr\n ext\n rw [hxy.add_pow' _, Finset.smul_sum]"
},
{
"state_after": "𝕂 : Type u_1\n𝔸 : Type u_2\n𝔹 : Type ?u.101113\ninst✝⁶ : NontriviallyNormedField 𝕂\ninst✝⁵ : NormedRing 𝔸\ninst✝⁴ : NormedRing 𝔹\ninst✝³ : NormedAlgebra 𝕂 𝔸\ninst✝² : NormedAlgebra 𝕂 𝔹\ninst✝¹ : CompleteSpace 𝔸\ninst✝ : CharZero 𝕂\nx y : 𝔸\nhxy : Commute x y\nhx : x ∈ EMetric.ball 0 (FormalMultilinearSeries.radius (expSeries 𝕂 𝔸))\nhy : y ∈ EMetric.ball 0 (FormalMultilinearSeries.radius (expSeries 𝕂 𝔸))\nn : ℕ\nkl : ℕ × ℕ\nhkl : kl ∈ Finset.Nat.antidiagonal n\n⊢ (↑n !)⁻¹ • Nat.choose n kl.fst • (x ^ kl.fst * y ^ kl.snd) = (↑kl.fst !)⁻¹ • x ^ kl.fst * (↑kl.snd !)⁻¹ • y ^ kl.snd",
"state_before": "𝕂 : Type u_1\n𝔸 : Type u_2\n𝔹 : Type ?u.101113\ninst✝⁶ : NontriviallyNormedField 𝕂\ninst✝⁵ : NormedRing 𝔸\ninst✝⁴ : NormedRing 𝔹\ninst✝³ : NormedAlgebra 𝕂 𝔸\ninst✝² : NormedAlgebra 𝕂 𝔹\ninst✝¹ : CompleteSpace 𝔸\ninst✝ : CharZero 𝕂\nx y : 𝔸\nhxy : Commute x y\nhx : x ∈ EMetric.ball 0 (FormalMultilinearSeries.radius (expSeries 𝕂 𝔸))\nhy : y ∈ EMetric.ball 0 (FormalMultilinearSeries.radius (expSeries 𝕂 𝔸))\n⊢ (∑' (x_1 : ℕ),\n ∑ x_2 in Finset.Nat.antidiagonal x_1, (↑x_1 !)⁻¹ • Nat.choose x_1 x_2.fst • (x ^ x_2.fst * y ^ x_2.snd)) =\n ∑' (n : ℕ), ∑ kl in Finset.Nat.antidiagonal n, (↑kl.fst !)⁻¹ • x ^ kl.fst * (↑kl.snd !)⁻¹ • y ^ kl.snd",
"tactic": "refine' tsum_congr fun n => Finset.sum_congr rfl fun kl hkl => _"
},
{
"state_after": "𝕂 : Type u_1\n𝔸 : Type u_2\n𝔹 : Type ?u.101113\ninst✝⁶ : NontriviallyNormedField 𝕂\ninst✝⁵ : NormedRing 𝔸\ninst✝⁴ : NormedRing 𝔹\ninst✝³ : NormedAlgebra 𝕂 𝔸\ninst✝² : NormedAlgebra 𝕂 𝔹\ninst✝¹ : CompleteSpace 𝔸\ninst✝ : CharZero 𝕂\nx y : 𝔸\nhxy : Commute x y\nhx : x ∈ EMetric.ball 0 (FormalMultilinearSeries.radius (expSeries 𝕂 𝔸))\nhy : y ∈ EMetric.ball 0 (FormalMultilinearSeries.radius (expSeries 𝕂 𝔸))\nn : ℕ\nkl : ℕ × ℕ\nhkl : kl ∈ Finset.Nat.antidiagonal n\n⊢ ((↑n !)⁻¹ * (↑n ! / (↑kl.fst ! * ↑kl.snd !))) • (x ^ kl.fst * y ^ kl.snd) =\n ((↑kl.fst !)⁻¹ * (↑kl.snd !)⁻¹) • (x ^ kl.fst * y ^ kl.snd)",
"state_before": "𝕂 : Type u_1\n𝔸 : Type u_2\n𝔹 : Type ?u.101113\ninst✝⁶ : NontriviallyNormedField 𝕂\ninst✝⁵ : NormedRing 𝔸\ninst✝⁴ : NormedRing 𝔹\ninst✝³ : NormedAlgebra 𝕂 𝔸\ninst✝² : NormedAlgebra 𝕂 𝔹\ninst✝¹ : CompleteSpace 𝔸\ninst✝ : CharZero 𝕂\nx y : 𝔸\nhxy : Commute x y\nhx : x ∈ EMetric.ball 0 (FormalMultilinearSeries.radius (expSeries 𝕂 𝔸))\nhy : y ∈ EMetric.ball 0 (FormalMultilinearSeries.radius (expSeries 𝕂 𝔸))\nn : ℕ\nkl : ℕ × ℕ\nhkl : kl ∈ Finset.Nat.antidiagonal n\n⊢ (↑n !)⁻¹ • Nat.choose n kl.fst • (x ^ kl.fst * y ^ kl.snd) = (↑kl.fst !)⁻¹ • x ^ kl.fst * (↑kl.snd !)⁻¹ • y ^ kl.snd",
"tactic": "rw [nsmul_eq_smul_cast 𝕂, smul_smul, smul_mul_smul, ← Finset.Nat.mem_antidiagonal.mp hkl,\n Nat.cast_add_choose, Finset.Nat.mem_antidiagonal.mp hkl]"
},
{
"state_after": "case e_a\n𝕂 : Type u_1\n𝔸 : Type u_2\n𝔹 : Type ?u.101113\ninst✝⁶ : NontriviallyNormedField 𝕂\ninst✝⁵ : NormedRing 𝔸\ninst✝⁴ : NormedRing 𝔹\ninst✝³ : NormedAlgebra 𝕂 𝔸\ninst✝² : NormedAlgebra 𝕂 𝔹\ninst✝¹ : CompleteSpace 𝔸\ninst✝ : CharZero 𝕂\nx y : 𝔸\nhxy : Commute x y\nhx : x ∈ EMetric.ball 0 (FormalMultilinearSeries.radius (expSeries 𝕂 𝔸))\nhy : y ∈ EMetric.ball 0 (FormalMultilinearSeries.radius (expSeries 𝕂 𝔸))\nn : ℕ\nkl : ℕ × ℕ\nhkl : kl ∈ Finset.Nat.antidiagonal n\n⊢ (↑n !)⁻¹ * (↑n ! / (↑kl.fst ! * ↑kl.snd !)) = (↑kl.fst !)⁻¹ * (↑kl.snd !)⁻¹",
"state_before": "𝕂 : Type u_1\n𝔸 : Type u_2\n𝔹 : Type ?u.101113\ninst✝⁶ : NontriviallyNormedField 𝕂\ninst✝⁵ : NormedRing 𝔸\ninst✝⁴ : NormedRing 𝔹\ninst✝³ : NormedAlgebra 𝕂 𝔸\ninst✝² : NormedAlgebra 𝕂 𝔹\ninst✝¹ : CompleteSpace 𝔸\ninst✝ : CharZero 𝕂\nx y : 𝔸\nhxy : Commute x y\nhx : x ∈ EMetric.ball 0 (FormalMultilinearSeries.radius (expSeries 𝕂 𝔸))\nhy : y ∈ EMetric.ball 0 (FormalMultilinearSeries.radius (expSeries 𝕂 𝔸))\nn : ℕ\nkl : ℕ × ℕ\nhkl : kl ∈ Finset.Nat.antidiagonal n\n⊢ ((↑n !)⁻¹ * (↑n ! / (↑kl.fst ! * ↑kl.snd !))) • (x ^ kl.fst * y ^ kl.snd) =\n ((↑kl.fst !)⁻¹ * (↑kl.snd !)⁻¹) • (x ^ kl.fst * y ^ kl.snd)",
"tactic": "congr 1"
},
{
"state_after": "case e_a\n𝕂 : Type u_1\n𝔸 : Type u_2\n𝔹 : Type ?u.101113\ninst✝⁶ : NontriviallyNormedField 𝕂\ninst✝⁵ : NormedRing 𝔸\ninst✝⁴ : NormedRing 𝔹\ninst✝³ : NormedAlgebra 𝕂 𝔸\ninst✝² : NormedAlgebra 𝕂 𝔹\ninst✝¹ : CompleteSpace 𝔸\ninst✝ : CharZero 𝕂\nx y : 𝔸\nhxy : Commute x y\nhx : x ∈ EMetric.ball 0 (FormalMultilinearSeries.radius (expSeries 𝕂 𝔸))\nhy : y ∈ EMetric.ball 0 (FormalMultilinearSeries.radius (expSeries 𝕂 𝔸))\nn : ℕ\nkl : ℕ × ℕ\nhkl : kl ∈ Finset.Nat.antidiagonal n\nthis : ↑n ! ≠ 0\n⊢ (↑n !)⁻¹ * (↑n ! / (↑kl.fst ! * ↑kl.snd !)) = (↑kl.fst !)⁻¹ * (↑kl.snd !)⁻¹",
"state_before": "case e_a\n𝕂 : Type u_1\n𝔸 : Type u_2\n𝔹 : Type ?u.101113\ninst✝⁶ : NontriviallyNormedField 𝕂\ninst✝⁵ : NormedRing 𝔸\ninst✝⁴ : NormedRing 𝔹\ninst✝³ : NormedAlgebra 𝕂 𝔸\ninst✝² : NormedAlgebra 𝕂 𝔹\ninst✝¹ : CompleteSpace 𝔸\ninst✝ : CharZero 𝕂\nx y : 𝔸\nhxy : Commute x y\nhx : x ∈ EMetric.ball 0 (FormalMultilinearSeries.radius (expSeries 𝕂 𝔸))\nhy : y ∈ EMetric.ball 0 (FormalMultilinearSeries.radius (expSeries 𝕂 𝔸))\nn : ℕ\nkl : ℕ × ℕ\nhkl : kl ∈ Finset.Nat.antidiagonal n\n⊢ (↑n !)⁻¹ * (↑n ! / (↑kl.fst ! * ↑kl.snd !)) = (↑kl.fst !)⁻¹ * (↑kl.snd !)⁻¹",
"tactic": "have : (n ! : 𝕂) ≠ 0 := Nat.cast_ne_zero.mpr n.factorial_ne_zero"
},
{
"state_after": "no goals",
"state_before": "case e_a\n𝕂 : Type u_1\n𝔸 : Type u_2\n𝔹 : Type ?u.101113\ninst✝⁶ : NontriviallyNormedField 𝕂\ninst✝⁵ : NormedRing 𝔸\ninst✝⁴ : NormedRing 𝔹\ninst✝³ : NormedAlgebra 𝕂 𝔸\ninst✝² : NormedAlgebra 𝕂 𝔹\ninst✝¹ : CompleteSpace 𝔸\ninst✝ : CharZero 𝕂\nx y : 𝔸\nhxy : Commute x y\nhx : x ∈ EMetric.ball 0 (FormalMultilinearSeries.radius (expSeries 𝕂 𝔸))\nhy : y ∈ EMetric.ball 0 (FormalMultilinearSeries.radius (expSeries 𝕂 𝔸))\nn : ℕ\nkl : ℕ × ℕ\nhkl : kl ∈ Finset.Nat.antidiagonal n\nthis : ↑n ! ≠ 0\n⊢ (↑n !)⁻¹ * (↑n ! / (↑kl.fst ! * ↑kl.snd !)) = (↑kl.fst !)⁻¹ * (↑kl.snd !)⁻¹",
"tactic": "field_simp [this]"
}
] |
[
281,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
265,
1
] |
Mathlib/SetTheory/Ordinal/Arithmetic.lean
|
Ordinal.sup_le_sup
|
[
{
"state_after": "case intro\nα : Type ?u.305800\nβ : Type ?u.305803\nγ : Type ?u.305806\nr✝ : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\nι ι' : Type u\nr : ι → ι → Prop\nr' : ι' → ι' → Prop\ninst✝¹ : IsWellOrder ι r\ninst✝ : IsWellOrder ι' r'\no : Ordinal\nho : type r = o\nho' : type r' = o\nf : (a : Ordinal) → a < o → Ordinal\ni : ι\nj : ι'\nhj : typein r' j = typein r i\n⊢ familyOfBFamily' r ho f i ≤ sup (familyOfBFamily' r' ho' f)",
"state_before": "α : Type ?u.305800\nβ : Type ?u.305803\nγ : Type ?u.305806\nr✝ : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\nι ι' : Type u\nr : ι → ι → Prop\nr' : ι' → ι' → Prop\ninst✝¹ : IsWellOrder ι r\ninst✝ : IsWellOrder ι' r'\no : Ordinal\nho : type r = o\nho' : type r' = o\nf : (a : Ordinal) → a < o → Ordinal\ni : ι\n⊢ familyOfBFamily' r ho f i ≤ sup (familyOfBFamily' r' ho' f)",
"tactic": "cases'\n typein_surj r'\n (by\n rw [ho', ← ho]\n exact typein_lt_type r i) with\n j hj"
},
{
"state_after": "case intro\nα : Type ?u.305800\nβ : Type ?u.305803\nγ : Type ?u.305806\nr✝ : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\nι ι' : Type u\nr : ι → ι → Prop\nr' : ι' → ι' → Prop\ninst✝¹ : IsWellOrder ι r\ninst✝ : IsWellOrder ι' r'\no : Ordinal\nho : type r = o\nho' : type r' = o\nf : (a : Ordinal) → a < o → Ordinal\ni : ι\nj : ι'\nhj : typein r' j = typein r i\n⊢ f (typein r' j) (_ : typein r' j < o) ≤ sup fun i => f (typein r' i) (_ : typein r' i < o)",
"state_before": "case intro\nα : Type ?u.305800\nβ : Type ?u.305803\nγ : Type ?u.305806\nr✝ : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\nι ι' : Type u\nr : ι → ι → Prop\nr' : ι' → ι' → Prop\ninst✝¹ : IsWellOrder ι r\ninst✝ : IsWellOrder ι' r'\no : Ordinal\nho : type r = o\nho' : type r' = o\nf : (a : Ordinal) → a < o → Ordinal\ni : ι\nj : ι'\nhj : typein r' j = typein r i\n⊢ familyOfBFamily' r ho f i ≤ sup (familyOfBFamily' r' ho' f)",
"tactic": "simp_rw [familyOfBFamily', ← hj]"
},
{
"state_after": "no goals",
"state_before": "case intro\nα : Type ?u.305800\nβ : Type ?u.305803\nγ : Type ?u.305806\nr✝ : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\nι ι' : Type u\nr : ι → ι → Prop\nr' : ι' → ι' → Prop\ninst✝¹ : IsWellOrder ι r\ninst✝ : IsWellOrder ι' r'\no : Ordinal\nho : type r = o\nho' : type r' = o\nf : (a : Ordinal) → a < o → Ordinal\ni : ι\nj : ι'\nhj : typein r' j = typein r i\n⊢ f (typein r' j) (_ : typein r' j < o) ≤ sup fun i => f (typein r' i) (_ : typein r' i < o)",
"tactic": "apply le_sup"
},
{
"state_after": "α : Type ?u.305800\nβ : Type ?u.305803\nγ : Type ?u.305806\nr✝ : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\nι ι' : Type u\nr : ι → ι → Prop\nr' : ι' → ι' → Prop\ninst✝¹ : IsWellOrder ι r\ninst✝ : IsWellOrder ι' r'\no : Ordinal\nho : type r = o\nho' : type r' = o\nf : (a : Ordinal) → a < o → Ordinal\ni : ι\n⊢ ?m.306085 < type r\n\nα : Type ?u.305800\nβ : Type ?u.305803\nγ : Type ?u.305806\nr✝ : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\nι ι' : Type u\nr : ι → ι → Prop\nr' : ι' → ι' → Prop\ninst✝¹ : IsWellOrder ι r\ninst✝ : IsWellOrder ι' r'\no : Ordinal\nho : type r = o\nho' : type r' = o\nf : (a : Ordinal) → a < o → Ordinal\ni : ι\n⊢ Ordinal\n\nα : Type ?u.305800\nβ : Type ?u.305803\nγ : Type ?u.305806\nr✝ : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\nι ι' : Type u\nr : ι → ι → Prop\nr' : ι' → ι' → Prop\ninst✝¹ : IsWellOrder ι r\ninst✝ : IsWellOrder ι' r'\no : Ordinal\nho : type r = o\nho' : type r' = o\nf : (a : Ordinal) → a < o → Ordinal\ni : ι\n⊢ Ordinal",
"state_before": "α : Type ?u.305800\nβ : Type ?u.305803\nγ : Type ?u.305806\nr✝ : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\nι ι' : Type u\nr : ι → ι → Prop\nr' : ι' → ι' → Prop\ninst✝¹ : IsWellOrder ι r\ninst✝ : IsWellOrder ι' r'\no : Ordinal\nho : type r = o\nho' : type r' = o\nf : (a : Ordinal) → a < o → Ordinal\ni : ι\n⊢ ?m.306085 < type r'",
"tactic": "rw [ho', ← ho]"
},
{
"state_after": "no goals",
"state_before": "α : Type ?u.305800\nβ : Type ?u.305803\nγ : Type ?u.305806\nr✝ : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\nι ι' : Type u\nr : ι → ι → Prop\nr' : ι' → ι' → Prop\ninst✝¹ : IsWellOrder ι r\ninst✝ : IsWellOrder ι' r'\no : Ordinal\nho : type r = o\nho' : type r' = o\nf : (a : Ordinal) → a < o → Ordinal\ni : ι\n⊢ ?m.306085 < type r\n\nα : Type ?u.305800\nβ : Type ?u.305803\nγ : Type ?u.305806\nr✝ : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\nι ι' : Type u\nr : ι → ι → Prop\nr' : ι' → ι' → Prop\ninst✝¹ : IsWellOrder ι r\ninst✝ : IsWellOrder ι' r'\no : Ordinal\nho : type r = o\nho' : type r' = o\nf : (a : Ordinal) → a < o → Ordinal\ni : ι\n⊢ Ordinal\n\nα : Type ?u.305800\nβ : Type ?u.305803\nγ : Type ?u.305806\nr✝ : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\nι ι' : Type u\nr : ι → ι → Prop\nr' : ι' → ι' → Prop\ninst✝¹ : IsWellOrder ι r\ninst✝ : IsWellOrder ι' r'\no : Ordinal\nho : type r = o\nho' : type r' = o\nf : (a : Ordinal) → a < o → Ordinal\ni : ι\n⊢ Ordinal",
"tactic": "exact typein_lt_type r i"
}
] |
[
1427,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1415,
9
] |
Mathlib/LinearAlgebra/SModEq.lean
|
SModEq.comap
|
[] |
[
107,
83
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
105,
1
] |
Mathlib/Combinatorics/SimpleGraph/Connectivity.lean
|
SimpleGraph.Walk.concat_eq_append
|
[] |
[
193,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
192,
1
] |
Mathlib/Analysis/Complex/Basic.lean
|
Complex.ofRealClm_coe
|
[] |
[
404,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
403,
1
] |
Mathlib/Topology/Algebra/Group/Basic.lean
|
map_mul_right_nhds
|
[] |
[
832,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
831,
1
] |
Mathlib/Analysis/Calculus/Deriv/Add.lean
|
derivWithin_neg
|
[] |
[
277,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
276,
1
] |
Mathlib/Data/PNat/Xgcd.lean
|
PNat.XgcdType.reduce_isSpecial
|
[
{
"state_after": "u✝ x✝ : XgcdType\nu : XgcdType := x✝\nh : r u = 0\nhs : IsSpecial x✝\n⊢ IsSpecial (finish u)",
"state_before": "u✝ x✝ : XgcdType\nu : XgcdType := x✝\nh : r u = 0\nhs : IsSpecial x✝\n⊢ IsSpecial (reduce x✝)",
"tactic": "rw [reduce_a h]"
},
{
"state_after": "no goals",
"state_before": "u✝ x✝ : XgcdType\nu : XgcdType := x✝\nh : r u = 0\nhs : IsSpecial x✝\n⊢ IsSpecial (finish u)",
"tactic": "exact u.finish_isSpecial hs"
},
{
"state_after": "u✝ x✝ : XgcdType\nu : XgcdType := x✝\nh : ¬r u = 0\nhs : IsSpecial x✝\nthis : sizeOf (step u) < sizeOf u\n⊢ IsSpecial (reduce x✝)",
"state_before": "u✝ x✝ : XgcdType\nu : XgcdType := x✝\nh : ¬r u = 0\nhs : IsSpecial x✝\n⊢ IsSpecial (reduce x✝)",
"tactic": "have : SizeOf.sizeOf u.step < SizeOf.sizeOf u := u.step_wf h"
},
{
"state_after": "u✝ x✝ : XgcdType\nu : XgcdType := x✝\nh : ¬r u = 0\nhs : IsSpecial x✝\nthis : sizeOf (step u) < sizeOf u\n⊢ IsSpecial (flip (reduce (step u)))",
"state_before": "u✝ x✝ : XgcdType\nu : XgcdType := x✝\nh : ¬r u = 0\nhs : IsSpecial x✝\nthis : sizeOf (step u) < sizeOf u\n⊢ IsSpecial (reduce x✝)",
"tactic": "rw [reduce_b h]"
},
{
"state_after": "no goals",
"state_before": "u✝ x✝ : XgcdType\nu : XgcdType := x✝\nh : ¬r u = 0\nhs : IsSpecial x✝\nthis : sizeOf (step u) < sizeOf u\n⊢ IsSpecial (flip (reduce (step u)))",
"tactic": "exact (flip_isSpecial _).mpr (reduce_isSpecial _ (u.step_isSpecial hs))"
}
] |
[
386,
78
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
377,
1
] |
Mathlib/Data/Matrix/Basis.lean
|
Matrix.StdBasisMatrix.apply_of_row_ne
|
[
{
"state_after": "no goals",
"state_before": "l : Type ?u.22747\nm : Type u_1\nn : Type u_3\nR : Type ?u.22756\nα : Type u_2\ninst✝³ : DecidableEq l\ninst✝² : DecidableEq m\ninst✝¹ : DecidableEq n\ninst✝ : Semiring α\ni✝ : m\nj✝ : n\nc : α\ni'✝ : m\nj'✝ : n\ni i' : m\nhi : i ≠ i'\nj j' : n\na : α\n⊢ stdBasisMatrix i j a i' j' = 0",
"tactic": "simp [hi]"
}
] |
[
137,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
136,
1
] |
Mathlib/Topology/Algebra/Order/MonotoneContinuity.lean
|
StrictMonoOn.continuousAt_of_exists_between
|
[] |
[
220,
95
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
215,
1
] |
Mathlib/Data/PFun.lean
|
PFun.image_union
|
[] |
[
421,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
420,
1
] |
src/lean/Init/SimpLemmas.lean
|
Bool.beq_to_eq
|
[
{
"state_after": "no goals",
"state_before": "a b : Bool\n⊢ ((a == b) = true) = (a = b)",
"tactic": "cases a <;> cases b <;> decide"
}
] |
[
132,
58
] |
d5348dfac847a56a4595fb6230fd0708dcb4e7e9
|
https://github.com/leanprover/lean4
|
[
131,
9
] |
Mathlib/MeasureTheory/Function/SimpleFuncDense.lean
|
MeasureTheory.SimpleFunc.edist_approxOn_y0_le
|
[] |
[
189,
91
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
182,
1
] |
Mathlib/Algebra/Algebra/Hom.lean
|
AlgHom.congr_arg
|
[] |
[
219,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
218,
11
] |
Std/Data/List/Lemmas.lean
|
List.disjoint_take_drop
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nm n : Nat\nx✝¹ : Nodup []\nx✝ : m ≤ n\n⊢ Disjoint (take m []) (drop n [])",
"tactic": "simp"
},
{
"state_after": "case succ.zero\nα : Type u_1\nx : α\nxs : List α\nhl : Nodup (x :: xs)\nn✝ : Nat\nh : succ n✝ ≤ zero\n⊢ False\n\ncase succ.succ\nα : Type u_1\nx : α\nxs : List α\nhl : Nodup (x :: xs)\nn✝¹ n✝ : Nat\nh : succ n✝¹ ≤ succ n✝\n⊢ ¬x ∈ drop n✝ xs ∧ Disjoint (take n✝¹ xs) (drop n✝ xs)",
"state_before": "α : Type u_1\nm n : Nat\nx : α\nxs : List α\nhl : Nodup (x :: xs)\nh : m ≤ n\n⊢ Disjoint (take m (x :: xs)) (drop n (x :: xs))",
"tactic": "cases m <;> cases n <;> simp only [disjoint_cons_left, mem_cons, disjoint_cons_right,\n drop, true_or, eq_self_iff_true, not_true, false_and, not_mem_nil, disjoint_nil_left, take]"
},
{
"state_after": "no goals",
"state_before": "case succ.zero\nα : Type u_1\nx : α\nxs : List α\nhl : Nodup (x :: xs)\nn✝ : Nat\nh : succ n✝ ≤ zero\n⊢ False",
"tactic": "case succ.zero => cases h"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nx : α\nxs : List α\nhl : Nodup (x :: xs)\nn✝ : Nat\nh : succ n✝ ≤ zero\n⊢ False",
"tactic": "cases h"
},
{
"state_after": "no goals",
"state_before": "case succ.succ\nα : Type u_1\nx : α\nxs : List α\nhl : Nodup (x :: xs)\nn✝¹ n✝ : Nat\nh : succ n✝¹ ≤ succ n✝\n⊢ ¬x ∈ drop n✝ xs ∧ Disjoint (take n✝¹ xs) (drop n✝ xs)",
"tactic": "cases hl with | cons h₀ h₁ =>\nrefine ⟨fun h => h₀ _ (mem_of_mem_drop h) rfl, ?_⟩\nexact disjoint_take_drop h₁ (Nat.le_of_succ_le_succ h)"
},
{
"state_after": "case succ.succ.cons\nα : Type u_1\nx : α\nxs : List α\nn✝¹ n✝ : Nat\nh : succ n✝¹ ≤ succ n✝\nh₁ : Pairwise (fun x x_1 => x ≠ x_1) xs\nh₀ : ∀ (a' : α), a' ∈ xs → x ≠ a'\n⊢ Disjoint (take n✝¹ xs) (drop n✝ xs)",
"state_before": "case succ.succ.cons\nα : Type u_1\nx : α\nxs : List α\nn✝¹ n✝ : Nat\nh : succ n✝¹ ≤ succ n✝\nh₁ : Pairwise (fun x x_1 => x ≠ x_1) xs\nh₀ : ∀ (a' : α), a' ∈ xs → x ≠ a'\n⊢ ¬x ∈ drop n✝ xs ∧ Disjoint (take n✝¹ xs) (drop n✝ xs)",
"tactic": "refine ⟨fun h => h₀ _ (mem_of_mem_drop h) rfl, ?_⟩"
},
{
"state_after": "no goals",
"state_before": "case succ.succ.cons\nα : Type u_1\nx : α\nxs : List α\nn✝¹ n✝ : Nat\nh : succ n✝¹ ≤ succ n✝\nh₁ : Pairwise (fun x x_1 => x ≠ x_1) xs\nh₀ : ∀ (a' : α), a' ∈ xs → x ≠ a'\n⊢ Disjoint (take n✝¹ xs) (drop n✝ xs)",
"tactic": "exact disjoint_take_drop h₁ (Nat.le_of_succ_le_succ h)"
}
] |
[
1761,
61
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
1753,
1
] |
Mathlib/Topology/Inseparable.lean
|
SeparationQuotient.continuous_lift
|
[
{
"state_after": "no goals",
"state_before": "X : Type u_1\nY : Type u_2\nZ : Type ?u.105005\nα : Type ?u.105008\nι : Type ?u.105011\nπ : ι → Type ?u.105016\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace Y\ninst✝¹ : TopologicalSpace Z\ninst✝ : (i : ι) → TopologicalSpace (π i)\nx y z : X\ns : Set X\nf✝ : X → Y\nt : Set (SeparationQuotient X)\nf : X → Y\nhf : ∀ (x y : X), (x ~ᵢ y) → f x = f y\n⊢ Continuous (lift f hf) ↔ Continuous f",
"tactic": "simp only [continuous_iff_continuousOn_univ, continuousOn_lift, preimage_univ]"
}
] |
[
575,
81
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
573,
1
] |
Mathlib/Data/Real/EReal.lean
|
EReal.toReal_add
|
[
{
"state_after": "case intro\ny : EReal\nhy : y ≠ ⊤\nh'y : y ≠ ⊥\nx : ℝ\nhx : ↑x ≠ ⊤\nh'x : ↑x ≠ ⊥\n⊢ toReal (↑x + y) = toReal ↑x + toReal y",
"state_before": "x y : EReal\nhx : x ≠ ⊤\nh'x : x ≠ ⊥\nhy : y ≠ ⊤\nh'y : y ≠ ⊥\n⊢ toReal (x + y) = toReal x + toReal y",
"tactic": "lift x to ℝ using ⟨hx, h'x⟩"
},
{
"state_after": "case intro.intro\nx : ℝ\nhx : ↑x ≠ ⊤\nh'x : ↑x ≠ ⊥\ny : ℝ\nhy : ↑y ≠ ⊤\nh'y : ↑y ≠ ⊥\n⊢ toReal (↑x + ↑y) = toReal ↑x + toReal ↑y",
"state_before": "case intro\ny : EReal\nhy : y ≠ ⊤\nh'y : y ≠ ⊥\nx : ℝ\nhx : ↑x ≠ ⊤\nh'x : ↑x ≠ ⊥\n⊢ toReal (↑x + y) = toReal ↑x + toReal y",
"tactic": "lift y to ℝ using ⟨hy, h'y⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\nx : ℝ\nhx : ↑x ≠ ⊤\nh'x : ↑x ≠ ⊥\ny : ℝ\nhy : ↑y ≠ ⊤\nh'y : ↑y ≠ ⊥\n⊢ toReal (↑x + ↑y) = toReal ↑x + toReal ↑y",
"tactic": "rfl"
}
] |
[
676,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
672,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/StrictInitial.lean
|
CategoryTheory.Limits.IsTerminal.isIso_from
|
[] |
[
192,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
191,
1
] |
Mathlib/Logic/Function/Basic.lean
|
Function.sometimes_spec
|
[
{
"state_after": "no goals",
"state_before": "p : Prop\nα : Sort u_1\ninst✝ : Nonempty α\nP : α → Prop\nf : p → α\na : p\nh : P (f a)\n⊢ P (sometimes f)",
"tactic": "rwa [sometimes_eq]"
}
] |
[
979,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
977,
1
] |
Mathlib/LinearAlgebra/Matrix/SesquilinearForm.lean
|
isAdjointPair_toLinearMap₂
|
[
{
"state_after": "R : Type u_1\nR₁ : Type ?u.2189106\nR₂ : Type ?u.2189109\nM : Type ?u.2189112\nM₁ : Type u_2\nM₂ : Type u_3\nM₁' : Type ?u.2189121\nM₂' : Type ?u.2189124\nn : Type u_4\nm : Type ?u.2189130\nn' : Type u_5\nm' : Type ?u.2189136\nι : Type ?u.2189139\ninst✝⁸ : CommRing R\ninst✝⁷ : AddCommMonoid M₁\ninst✝⁶ : Module R M₁\ninst✝⁵ : AddCommMonoid M₂\ninst✝⁴ : Module R M₂\ninst✝³ : Fintype n\ninst✝² : Fintype n'\nb₁ : Basis n R M₁\nb₂ : Basis n' R M₂\nJ J₂ : Matrix n n R\nJ' : Matrix n' n' R\nA : Matrix n' n R\nA' : Matrix n n' R\nA₁ : Matrix n n R\ninst✝¹ : DecidableEq n\ninst✝ : DecidableEq n'\n⊢ comp (↑(toLinearMap₂ b₂ b₂) J') (↑(toLin b₁ b₂) A) = compl₂ (↑(toLinearMap₂ b₁ b₁) J) (↑(toLin b₂ b₁) A') ↔\n Matrix.IsAdjointPair J J' A A'",
"state_before": "R : Type u_1\nR₁ : Type ?u.2189106\nR₂ : Type ?u.2189109\nM : Type ?u.2189112\nM₁ : Type u_2\nM₂ : Type u_3\nM₁' : Type ?u.2189121\nM₂' : Type ?u.2189124\nn : Type u_4\nm : Type ?u.2189130\nn' : Type u_5\nm' : Type ?u.2189136\nι : Type ?u.2189139\ninst✝⁸ : CommRing R\ninst✝⁷ : AddCommMonoid M₁\ninst✝⁶ : Module R M₁\ninst✝⁵ : AddCommMonoid M₂\ninst✝⁴ : Module R M₂\ninst✝³ : Fintype n\ninst✝² : Fintype n'\nb₁ : Basis n R M₁\nb₂ : Basis n' R M₂\nJ J₂ : Matrix n n R\nJ' : Matrix n' n' R\nA : Matrix n' n R\nA' : Matrix n n' R\nA₁ : Matrix n n R\ninst✝¹ : DecidableEq n\ninst✝ : DecidableEq n'\n⊢ LinearMap.IsAdjointPair (↑(toLinearMap₂ b₁ b₁) J) (↑(toLinearMap₂ b₂ b₂) J') (↑(toLin b₁ b₂) A) (↑(toLin b₂ b₁) A') ↔\n Matrix.IsAdjointPair J J' A A'",
"tactic": "rw [isAdjointPair_iff_comp_eq_compl₂]"
},
{
"state_after": "R : Type u_1\nR₁ : Type ?u.2189106\nR₂ : Type ?u.2189109\nM : Type ?u.2189112\nM₁ : Type u_2\nM₂ : Type u_3\nM₁' : Type ?u.2189121\nM₂' : Type ?u.2189124\nn : Type u_4\nm : Type ?u.2189130\nn' : Type u_5\nm' : Type ?u.2189136\nι : Type ?u.2189139\ninst✝⁸ : CommRing R\ninst✝⁷ : AddCommMonoid M₁\ninst✝⁶ : Module R M₁\ninst✝⁵ : AddCommMonoid M₂\ninst✝⁴ : Module R M₂\ninst✝³ : Fintype n\ninst✝² : Fintype n'\nb₁ : Basis n R M₁\nb₂ : Basis n' R M₂\nJ J₂ : Matrix n n R\nJ' : Matrix n' n' R\nA : Matrix n' n R\nA' : Matrix n n' R\nA₁ : Matrix n n R\ninst✝¹ : DecidableEq n\ninst✝ : DecidableEq n'\nh : ∀ (B B' : M₁ →ₗ[R] M₂ →ₗ[R] R), B = B' ↔ ↑(toMatrix₂ b₁ b₂) B = ↑(toMatrix₂ b₁ b₂) B'\n⊢ Aᵀ ⬝ J' = J ⬝ A' ↔ Matrix.IsAdjointPair J J' A A'",
"state_before": "R : Type u_1\nR₁ : Type ?u.2189106\nR₂ : Type ?u.2189109\nM : Type ?u.2189112\nM₁ : Type u_2\nM₂ : Type u_3\nM₁' : Type ?u.2189121\nM₂' : Type ?u.2189124\nn : Type u_4\nm : Type ?u.2189130\nn' : Type u_5\nm' : Type ?u.2189136\nι : Type ?u.2189139\ninst✝⁸ : CommRing R\ninst✝⁷ : AddCommMonoid M₁\ninst✝⁶ : Module R M₁\ninst✝⁵ : AddCommMonoid M₂\ninst✝⁴ : Module R M₂\ninst✝³ : Fintype n\ninst✝² : Fintype n'\nb₁ : Basis n R M₁\nb₂ : Basis n' R M₂\nJ J₂ : Matrix n n R\nJ' : Matrix n' n' R\nA : Matrix n' n R\nA' : Matrix n n' R\nA₁ : Matrix n n R\ninst✝¹ : DecidableEq n\ninst✝ : DecidableEq n'\nh : ∀ (B B' : M₁ →ₗ[R] M₂ →ₗ[R] R), B = B' ↔ ↑(toMatrix₂ b₁ b₂) B = ↑(toMatrix₂ b₁ b₂) B'\n⊢ comp (↑(toLinearMap₂ b₂ b₂) J') (↑(toLin b₁ b₂) A) = compl₂ (↑(toLinearMap₂ b₁ b₁) J) (↑(toLin b₂ b₁) A') ↔\n Matrix.IsAdjointPair J J' A A'",
"tactic": "simp_rw [h, LinearMap.toMatrix₂_comp b₂ b₂, LinearMap.toMatrix₂_compl₂ b₁ b₁,\n LinearMap.toMatrix_toLin, LinearMap.toMatrix₂_toLinearMap₂]"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\nR₁ : Type ?u.2189106\nR₂ : Type ?u.2189109\nM : Type ?u.2189112\nM₁ : Type u_2\nM₂ : Type u_3\nM₁' : Type ?u.2189121\nM₂' : Type ?u.2189124\nn : Type u_4\nm : Type ?u.2189130\nn' : Type u_5\nm' : Type ?u.2189136\nι : Type ?u.2189139\ninst✝⁸ : CommRing R\ninst✝⁷ : AddCommMonoid M₁\ninst✝⁶ : Module R M₁\ninst✝⁵ : AddCommMonoid M₂\ninst✝⁴ : Module R M₂\ninst✝³ : Fintype n\ninst✝² : Fintype n'\nb₁ : Basis n R M₁\nb₂ : Basis n' R M₂\nJ J₂ : Matrix n n R\nJ' : Matrix n' n' R\nA : Matrix n' n R\nA' : Matrix n n' R\nA₁ : Matrix n n R\ninst✝¹ : DecidableEq n\ninst✝ : DecidableEq n'\nh : ∀ (B B' : M₁ →ₗ[R] M₂ →ₗ[R] R), B = B' ↔ ↑(toMatrix₂ b₁ b₂) B = ↑(toMatrix₂ b₁ b₂) B'\n⊢ Aᵀ ⬝ J' = J ⬝ A' ↔ Matrix.IsAdjointPair J J' A A'",
"tactic": "rfl"
},
{
"state_after": "R : Type u_1\nR₁ : Type ?u.2189106\nR₂ : Type ?u.2189109\nM : Type ?u.2189112\nM₁ : Type u_2\nM₂ : Type u_3\nM₁' : Type ?u.2189121\nM₂' : Type ?u.2189124\nn : Type u_4\nm : Type ?u.2189130\nn' : Type u_5\nm' : Type ?u.2189136\nι : Type ?u.2189139\ninst✝⁸ : CommRing R\ninst✝⁷ : AddCommMonoid M₁\ninst✝⁶ : Module R M₁\ninst✝⁵ : AddCommMonoid M₂\ninst✝⁴ : Module R M₂\ninst✝³ : Fintype n\ninst✝² : Fintype n'\nb₁ : Basis n R M₁\nb₂ : Basis n' R M₂\nJ J₂ : Matrix n n R\nJ' : Matrix n' n' R\nA : Matrix n' n R\nA' : Matrix n n' R\nA₁ : Matrix n n R\ninst✝¹ : DecidableEq n\ninst✝ : DecidableEq n'\nB B' : M₁ →ₗ[R] M₂ →ₗ[R] R\n⊢ B = B' ↔ ↑(toMatrix₂ b₁ b₂) B = ↑(toMatrix₂ b₁ b₂) B'",
"state_before": "R : Type u_1\nR₁ : Type ?u.2189106\nR₂ : Type ?u.2189109\nM : Type ?u.2189112\nM₁ : Type u_2\nM₂ : Type u_3\nM₁' : Type ?u.2189121\nM₂' : Type ?u.2189124\nn : Type u_4\nm : Type ?u.2189130\nn' : Type u_5\nm' : Type ?u.2189136\nι : Type ?u.2189139\ninst✝⁸ : CommRing R\ninst✝⁷ : AddCommMonoid M₁\ninst✝⁶ : Module R M₁\ninst✝⁵ : AddCommMonoid M₂\ninst✝⁴ : Module R M₂\ninst✝³ : Fintype n\ninst✝² : Fintype n'\nb₁ : Basis n R M₁\nb₂ : Basis n' R M₂\nJ J₂ : Matrix n n R\nJ' : Matrix n' n' R\nA : Matrix n' n R\nA' : Matrix n n' R\nA₁ : Matrix n n R\ninst✝¹ : DecidableEq n\ninst✝ : DecidableEq n'\n⊢ ∀ (B B' : M₁ →ₗ[R] M₂ →ₗ[R] R), B = B' ↔ ↑(toMatrix₂ b₁ b₂) B = ↑(toMatrix₂ b₁ b₂) B'",
"tactic": "intro B B'"
},
{
"state_after": "case mp\nR : Type u_1\nR₁ : Type ?u.2189106\nR₂ : Type ?u.2189109\nM : Type ?u.2189112\nM₁ : Type u_2\nM₂ : Type u_3\nM₁' : Type ?u.2189121\nM₂' : Type ?u.2189124\nn : Type u_4\nm : Type ?u.2189130\nn' : Type u_5\nm' : Type ?u.2189136\nι : Type ?u.2189139\ninst✝⁸ : CommRing R\ninst✝⁷ : AddCommMonoid M₁\ninst✝⁶ : Module R M₁\ninst✝⁵ : AddCommMonoid M₂\ninst✝⁴ : Module R M₂\ninst✝³ : Fintype n\ninst✝² : Fintype n'\nb₁ : Basis n R M₁\nb₂ : Basis n' R M₂\nJ J₂ : Matrix n n R\nJ' : Matrix n' n' R\nA : Matrix n' n R\nA' : Matrix n n' R\nA₁ : Matrix n n R\ninst✝¹ : DecidableEq n\ninst✝ : DecidableEq n'\nB B' : M₁ →ₗ[R] M₂ →ₗ[R] R\nh : B = B'\n⊢ ↑(toMatrix₂ b₁ b₂) B = ↑(toMatrix₂ b₁ b₂) B'\n\ncase mpr\nR : Type u_1\nR₁ : Type ?u.2189106\nR₂ : Type ?u.2189109\nM : Type ?u.2189112\nM₁ : Type u_2\nM₂ : Type u_3\nM₁' : Type ?u.2189121\nM₂' : Type ?u.2189124\nn : Type u_4\nm : Type ?u.2189130\nn' : Type u_5\nm' : Type ?u.2189136\nι : Type ?u.2189139\ninst✝⁸ : CommRing R\ninst✝⁷ : AddCommMonoid M₁\ninst✝⁶ : Module R M₁\ninst✝⁵ : AddCommMonoid M₂\ninst✝⁴ : Module R M₂\ninst✝³ : Fintype n\ninst✝² : Fintype n'\nb₁ : Basis n R M₁\nb₂ : Basis n' R M₂\nJ J₂ : Matrix n n R\nJ' : Matrix n' n' R\nA : Matrix n' n R\nA' : Matrix n n' R\nA₁ : Matrix n n R\ninst✝¹ : DecidableEq n\ninst✝ : DecidableEq n'\nB B' : M₁ →ₗ[R] M₂ →ₗ[R] R\nh : ↑(toMatrix₂ b₁ b₂) B = ↑(toMatrix₂ b₁ b₂) B'\n⊢ B = B'",
"state_before": "R : Type u_1\nR₁ : Type ?u.2189106\nR₂ : Type ?u.2189109\nM : Type ?u.2189112\nM₁ : Type u_2\nM₂ : Type u_3\nM₁' : Type ?u.2189121\nM₂' : Type ?u.2189124\nn : Type u_4\nm : Type ?u.2189130\nn' : Type u_5\nm' : Type ?u.2189136\nι : Type ?u.2189139\ninst✝⁸ : CommRing R\ninst✝⁷ : AddCommMonoid M₁\ninst✝⁶ : Module R M₁\ninst✝⁵ : AddCommMonoid M₂\ninst✝⁴ : Module R M₂\ninst✝³ : Fintype n\ninst✝² : Fintype n'\nb₁ : Basis n R M₁\nb₂ : Basis n' R M₂\nJ J₂ : Matrix n n R\nJ' : Matrix n' n' R\nA : Matrix n' n R\nA' : Matrix n n' R\nA₁ : Matrix n n R\ninst✝¹ : DecidableEq n\ninst✝ : DecidableEq n'\nB B' : M₁ →ₗ[R] M₂ →ₗ[R] R\n⊢ B = B' ↔ ↑(toMatrix₂ b₁ b₂) B = ↑(toMatrix₂ b₁ b₂) B'",
"tactic": "constructor <;> intro h"
},
{
"state_after": "no goals",
"state_before": "case mp\nR : Type u_1\nR₁ : Type ?u.2189106\nR₂ : Type ?u.2189109\nM : Type ?u.2189112\nM₁ : Type u_2\nM₂ : Type u_3\nM₁' : Type ?u.2189121\nM₂' : Type ?u.2189124\nn : Type u_4\nm : Type ?u.2189130\nn' : Type u_5\nm' : Type ?u.2189136\nι : Type ?u.2189139\ninst✝⁸ : CommRing R\ninst✝⁷ : AddCommMonoid M₁\ninst✝⁶ : Module R M₁\ninst✝⁵ : AddCommMonoid M₂\ninst✝⁴ : Module R M₂\ninst✝³ : Fintype n\ninst✝² : Fintype n'\nb₁ : Basis n R M₁\nb₂ : Basis n' R M₂\nJ J₂ : Matrix n n R\nJ' : Matrix n' n' R\nA : Matrix n' n R\nA' : Matrix n n' R\nA₁ : Matrix n n R\ninst✝¹ : DecidableEq n\ninst✝ : DecidableEq n'\nB B' : M₁ →ₗ[R] M₂ →ₗ[R] R\nh : B = B'\n⊢ ↑(toMatrix₂ b₁ b₂) B = ↑(toMatrix₂ b₁ b₂) B'",
"tactic": "rw [h]"
},
{
"state_after": "no goals",
"state_before": "case mpr\nR : Type u_1\nR₁ : Type ?u.2189106\nR₂ : Type ?u.2189109\nM : Type ?u.2189112\nM₁ : Type u_2\nM₂ : Type u_3\nM₁' : Type ?u.2189121\nM₂' : Type ?u.2189124\nn : Type u_4\nm : Type ?u.2189130\nn' : Type u_5\nm' : Type ?u.2189136\nι : Type ?u.2189139\ninst✝⁸ : CommRing R\ninst✝⁷ : AddCommMonoid M₁\ninst✝⁶ : Module R M₁\ninst✝⁵ : AddCommMonoid M₂\ninst✝⁴ : Module R M₂\ninst✝³ : Fintype n\ninst✝² : Fintype n'\nb₁ : Basis n R M₁\nb₂ : Basis n' R M₂\nJ J₂ : Matrix n n R\nJ' : Matrix n' n' R\nA : Matrix n' n R\nA' : Matrix n n' R\nA₁ : Matrix n n R\ninst✝¹ : DecidableEq n\ninst✝ : DecidableEq n'\nB B' : M₁ →ₗ[R] M₂ →ₗ[R] R\nh : ↑(toMatrix₂ b₁ b₂) B = ↑(toMatrix₂ b₁ b₂) B'\n⊢ B = B'",
"tactic": "exact (LinearMap.toMatrix₂ b₁ b₂).injective h"
}
] |
[
591,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
577,
1
] |
Mathlib/Order/Iterate.lean
|
Monotone.seq_le_seq
|
[
{
"state_after": "case zero\nα : Type u_1\ninst✝ : Preorder α\nf : α → α\nx y : ℕ → α\nhf : Monotone f\nn : ℕ\nh₀ : x 0 ≤ y 0\nhx✝ : ∀ (k : ℕ), k < n → x (k + 1) ≤ f (x k)\nhy✝ : ∀ (k : ℕ), k < n → f (y k) ≤ y (k + 1)\nhx : ∀ (k : ℕ), k < Nat.zero → x (k + 1) ≤ f (x k)\nhy : ∀ (k : ℕ), k < Nat.zero → f (y k) ≤ y (k + 1)\n⊢ x Nat.zero ≤ y Nat.zero\n\ncase succ\nα : Type u_1\ninst✝ : Preorder α\nf : α → α\nx y : ℕ → α\nhf : Monotone f\nn✝ : ℕ\nh₀ : x 0 ≤ y 0\nhx✝ : ∀ (k : ℕ), k < n✝ → x (k + 1) ≤ f (x k)\nhy✝ : ∀ (k : ℕ), k < n✝ → f (y k) ≤ y (k + 1)\nn : ℕ\nihn : (∀ (k : ℕ), k < n → x (k + 1) ≤ f (x k)) → (∀ (k : ℕ), k < n → f (y k) ≤ y (k + 1)) → x n ≤ y n\nhx : ∀ (k : ℕ), k < Nat.succ n → x (k + 1) ≤ f (x k)\nhy : ∀ (k : ℕ), k < Nat.succ n → f (y k) ≤ y (k + 1)\n⊢ x (Nat.succ n) ≤ y (Nat.succ n)",
"state_before": "α : Type u_1\ninst✝ : Preorder α\nf : α → α\nx y : ℕ → α\nhf : Monotone f\nn : ℕ\nh₀ : x 0 ≤ y 0\nhx : ∀ (k : ℕ), k < n → x (k + 1) ≤ f (x k)\nhy : ∀ (k : ℕ), k < n → f (y k) ≤ y (k + 1)\n⊢ x n ≤ y n",
"tactic": "induction' n with n ihn"
},
{
"state_after": "no goals",
"state_before": "case zero\nα : Type u_1\ninst✝ : Preorder α\nf : α → α\nx y : ℕ → α\nhf : Monotone f\nn : ℕ\nh₀ : x 0 ≤ y 0\nhx✝ : ∀ (k : ℕ), k < n → x (k + 1) ≤ f (x k)\nhy✝ : ∀ (k : ℕ), k < n → f (y k) ≤ y (k + 1)\nhx : ∀ (k : ℕ), k < Nat.zero → x (k + 1) ≤ f (x k)\nhy : ∀ (k : ℕ), k < Nat.zero → f (y k) ≤ y (k + 1)\n⊢ x Nat.zero ≤ y Nat.zero",
"tactic": "exact h₀"
},
{
"state_after": "case succ.refine'_1\nα : Type u_1\ninst✝ : Preorder α\nf : α → α\nx y : ℕ → α\nhf : Monotone f\nn✝ : ℕ\nh₀ : x 0 ≤ y 0\nhx✝ : ∀ (k : ℕ), k < n✝ → x (k + 1) ≤ f (x k)\nhy✝ : ∀ (k : ℕ), k < n✝ → f (y k) ≤ y (k + 1)\nn : ℕ\nihn : (∀ (k : ℕ), k < n → x (k + 1) ≤ f (x k)) → (∀ (k : ℕ), k < n → f (y k) ≤ y (k + 1)) → x n ≤ y n\nhx : ∀ (k : ℕ), k < Nat.succ n → x (k + 1) ≤ f (x k)\nhy : ∀ (k : ℕ), k < Nat.succ n → f (y k) ≤ y (k + 1)\n⊢ ∀ (k : ℕ), k < n → x (k + 1) ≤ f (x k)\n\ncase succ.refine'_2\nα : Type u_1\ninst✝ : Preorder α\nf : α → α\nx y : ℕ → α\nhf : Monotone f\nn✝ : ℕ\nh₀ : x 0 ≤ y 0\nhx✝ : ∀ (k : ℕ), k < n✝ → x (k + 1) ≤ f (x k)\nhy✝ : ∀ (k : ℕ), k < n✝ → f (y k) ≤ y (k + 1)\nn : ℕ\nihn : (∀ (k : ℕ), k < n → x (k + 1) ≤ f (x k)) → (∀ (k : ℕ), k < n → f (y k) ≤ y (k + 1)) → x n ≤ y n\nhx : ∀ (k : ℕ), k < Nat.succ n → x (k + 1) ≤ f (x k)\nhy : ∀ (k : ℕ), k < Nat.succ n → f (y k) ≤ y (k + 1)\n⊢ ∀ (k : ℕ), k < n → f (y k) ≤ y (k + 1)",
"state_before": "case succ\nα : Type u_1\ninst✝ : Preorder α\nf : α → α\nx y : ℕ → α\nhf : Monotone f\nn✝ : ℕ\nh₀ : x 0 ≤ y 0\nhx✝ : ∀ (k : ℕ), k < n✝ → x (k + 1) ≤ f (x k)\nhy✝ : ∀ (k : ℕ), k < n✝ → f (y k) ≤ y (k + 1)\nn : ℕ\nihn : (∀ (k : ℕ), k < n → x (k + 1) ≤ f (x k)) → (∀ (k : ℕ), k < n → f (y k) ≤ y (k + 1)) → x n ≤ y n\nhx : ∀ (k : ℕ), k < Nat.succ n → x (k + 1) ≤ f (x k)\nhy : ∀ (k : ℕ), k < Nat.succ n → f (y k) ≤ y (k + 1)\n⊢ x (Nat.succ n) ≤ y (Nat.succ n)",
"tactic": "refine' (hx _ n.lt_succ_self).trans ((hf $ ihn _ _).trans (hy _ n.lt_succ_self))"
},
{
"state_after": "case succ.refine'_2\nα : Type u_1\ninst✝ : Preorder α\nf : α → α\nx y : ℕ → α\nhf : Monotone f\nn✝ : ℕ\nh₀ : x 0 ≤ y 0\nhx✝ : ∀ (k : ℕ), k < n✝ → x (k + 1) ≤ f (x k)\nhy✝ : ∀ (k : ℕ), k < n✝ → f (y k) ≤ y (k + 1)\nn : ℕ\nihn : (∀ (k : ℕ), k < n → x (k + 1) ≤ f (x k)) → (∀ (k : ℕ), k < n → f (y k) ≤ y (k + 1)) → x n ≤ y n\nhx : ∀ (k : ℕ), k < Nat.succ n → x (k + 1) ≤ f (x k)\nhy : ∀ (k : ℕ), k < Nat.succ n → f (y k) ≤ y (k + 1)\n⊢ ∀ (k : ℕ), k < n → f (y k) ≤ y (k + 1)",
"state_before": "case succ.refine'_1\nα : Type u_1\ninst✝ : Preorder α\nf : α → α\nx y : ℕ → α\nhf : Monotone f\nn✝ : ℕ\nh₀ : x 0 ≤ y 0\nhx✝ : ∀ (k : ℕ), k < n✝ → x (k + 1) ≤ f (x k)\nhy✝ : ∀ (k : ℕ), k < n✝ → f (y k) ≤ y (k + 1)\nn : ℕ\nihn : (∀ (k : ℕ), k < n → x (k + 1) ≤ f (x k)) → (∀ (k : ℕ), k < n → f (y k) ≤ y (k + 1)) → x n ≤ y n\nhx : ∀ (k : ℕ), k < Nat.succ n → x (k + 1) ≤ f (x k)\nhy : ∀ (k : ℕ), k < Nat.succ n → f (y k) ≤ y (k + 1)\n⊢ ∀ (k : ℕ), k < n → x (k + 1) ≤ f (x k)\n\ncase succ.refine'_2\nα : Type u_1\ninst✝ : Preorder α\nf : α → α\nx y : ℕ → α\nhf : Monotone f\nn✝ : ℕ\nh₀ : x 0 ≤ y 0\nhx✝ : ∀ (k : ℕ), k < n✝ → x (k + 1) ≤ f (x k)\nhy✝ : ∀ (k : ℕ), k < n✝ → f (y k) ≤ y (k + 1)\nn : ℕ\nihn : (∀ (k : ℕ), k < n → x (k + 1) ≤ f (x k)) → (∀ (k : ℕ), k < n → f (y k) ≤ y (k + 1)) → x n ≤ y n\nhx : ∀ (k : ℕ), k < Nat.succ n → x (k + 1) ≤ f (x k)\nhy : ∀ (k : ℕ), k < Nat.succ n → f (y k) ≤ y (k + 1)\n⊢ ∀ (k : ℕ), k < n → f (y k) ≤ y (k + 1)",
"tactic": "exact fun k hk => hx _ (hk.trans n.lt_succ_self)"
},
{
"state_after": "no goals",
"state_before": "case succ.refine'_2\nα : Type u_1\ninst✝ : Preorder α\nf : α → α\nx y : ℕ → α\nhf : Monotone f\nn✝ : ℕ\nh₀ : x 0 ≤ y 0\nhx✝ : ∀ (k : ℕ), k < n✝ → x (k + 1) ≤ f (x k)\nhy✝ : ∀ (k : ℕ), k < n✝ → f (y k) ≤ y (k + 1)\nn : ℕ\nihn : (∀ (k : ℕ), k < n → x (k + 1) ≤ f (x k)) → (∀ (k : ℕ), k < n → f (y k) ≤ y (k + 1)) → x n ≤ y n\nhx : ∀ (k : ℕ), k < Nat.succ n → x (k + 1) ≤ f (x k)\nhy : ∀ (k : ℕ), k < Nat.succ n → f (y k) ≤ y (k + 1)\n⊢ ∀ (k : ℕ), k < n → f (y k) ≤ y (k + 1)",
"tactic": "exact fun k hk => hy _ (hk.trans n.lt_succ_self)"
}
] |
[
50,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
44,
1
] |
Mathlib/SetTheory/Cardinal/Basic.lean
|
Cardinal.toNat_le_of_le_of_lt_aleph0
|
[] |
[
1705,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1703,
1
] |
Mathlib/Geometry/Euclidean/Basic.lean
|
EuclideanGeometry.dist_smul_vadd_sq
|
[
{
"state_after": "V : Type u_2\nP : Type u_1\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\nr : ℝ\nv : V\np₁ p₂ : P\n⊢ r * (r * inner v v) + 2 * (r * inner v (p₁ -ᵥ p₂)) + inner (p₁ -ᵥ p₂) (p₁ -ᵥ p₂) =\n inner v v * r * r + 2 * inner v (p₁ -ᵥ p₂) * r + inner (p₁ -ᵥ p₂) (p₁ -ᵥ p₂)",
"state_before": "V : Type u_2\nP : Type u_1\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\nr : ℝ\nv : V\np₁ p₂ : P\n⊢ dist (r • v +ᵥ p₁) p₂ * dist (r • v +ᵥ p₁) p₂ =\n inner v v * r * r + 2 * inner v (p₁ -ᵥ p₂) * r + inner (p₁ -ᵥ p₂) (p₁ -ᵥ p₂)",
"tactic": "rw [dist_eq_norm_vsub V _ p₂, ← real_inner_self_eq_norm_mul_norm, vadd_vsub_assoc,\n real_inner_add_add_self, real_inner_smul_left, real_inner_smul_left, real_inner_smul_right]"
},
{
"state_after": "no goals",
"state_before": "V : Type u_2\nP : Type u_1\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\nr : ℝ\nv : V\np₁ p₂ : P\n⊢ r * (r * inner v v) + 2 * (r * inner v (p₁ -ᵥ p₂)) + inner (p₁ -ᵥ p₂) (p₁ -ᵥ p₂) =\n inner v v * r * r + 2 * inner v (p₁ -ᵥ p₂) * r + inner (p₁ -ᵥ p₂) (p₁ -ᵥ p₂)",
"tactic": "ring"
}
] |
[
139,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
134,
1
] |
Mathlib/LinearAlgebra/FiniteDimensional.lean
|
LinearMap.ker_eq_bot_iff_range_eq_top
|
[
{
"state_after": "no goals",
"state_before": "K : Type u\nV : Type v\ninst✝⁵ : DivisionRing K\ninst✝⁴ : AddCommGroup V\ninst✝³ : Module K V\nV₂ : Type v'\ninst✝² : AddCommGroup V₂\ninst✝¹ : Module K V₂\ninst✝ : FiniteDimensional K V\nf : V →ₗ[K] V\n⊢ ker f = ⊥ ↔ range f = ⊤",
"tactic": "rw [range_eq_top, ker_eq_bot, injective_iff_surjective]"
}
] |
[
926,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
924,
1
] |
Mathlib/Data/Set/Sups.lean
|
Set.iUnion_image_sup_left
|
[] |
[
179,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
178,
1
] |
Mathlib/LinearAlgebra/Determinant.lean
|
Basis.det_map'
|
[] |
[
635,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
633,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/CommSq.lean
|
CategoryTheory.IsPullback.paste_horiz
|
[] |
[
519,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
515,
1
] |
Mathlib/AlgebraicTopology/FundamentalGroupoid/Basic.lean
|
Path.Homotopy.continuous_transAssocReparamAux
|
[
{
"state_after": "case refine'_9\nX : Type u\nY : Type v\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nx₀ x₁ : X\nx : ↑I\nhx : ↑x = 1 / 4\n⊢ 2 * ↑x = if ↑x ≤ 1 / 2 then ↑x + 1 / 4 else 1 / 2 * (↑x + 1)",
"state_before": "case refine'_9\nX : Type u\nY : Type v\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nx₀ x₁ : X\n⊢ ∀ (x : ↑I), ↑x = 1 / 4 → 2 * ↑x = if ↑x ≤ 1 / 2 then ↑x + 1 / 4 else 1 / 2 * (↑x + 1)",
"tactic": "intro x hx"
},
{
"state_after": "case refine'_9\nX : Type u\nY : Type v\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nx₀ x₁ : X\nx : ↑I\nhx : ↑x = 1 / 4\n⊢ 2 * 4⁻¹ = if 4⁻¹ ≤ 2⁻¹ then 4⁻¹ + 4⁻¹ else 2⁻¹ * (4⁻¹ + 1)",
"state_before": "case refine'_9\nX : Type u\nY : Type v\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nx₀ x₁ : X\nx : ↑I\nhx : ↑x = 1 / 4\n⊢ 2 * ↑x = if ↑x ≤ 1 / 2 then ↑x + 1 / 4 else 1 / 2 * (↑x + 1)",
"tactic": "simp [hx]"
},
{
"state_after": "no goals",
"state_before": "case refine'_9\nX : Type u\nY : Type v\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nx₀ x₁ : X\nx : ↑I\nhx : ↑x = 1 / 4\n⊢ 2 * 4⁻¹ = if 4⁻¹ ≤ 2⁻¹ then 4⁻¹ + 4⁻¹ else 2⁻¹ * (4⁻¹ + 1)",
"tactic": "norm_num"
}
] |
[
210,
15
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
201,
1
] |
Mathlib/Geometry/Euclidean/Sphere/Basic.lean
|
EuclideanGeometry.inner_nonneg_of_dist_le_radius
|
[
{
"state_after": "case inl\nV : Type u_2\nP : Type u_1\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\ns : Sphere P\np₁ p₂ : P\nhp₁ : p₁ ∈ s\nhp₂ : dist p₂ s.center ≤ s.radius\nh : 0 < inner (p₁ -ᵥ p₂) (p₁ -ᵥ s.center)\n⊢ 0 ≤ inner (p₁ -ᵥ p₂) (p₁ -ᵥ s.center)\n\ncase inr\nV : Type u_2\nP : Type u_1\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\ns : Sphere P\np₁ : P\nhp₁ : p₁ ∈ s\nhp₂ : dist p₁ s.center ≤ s.radius\n⊢ 0 ≤ inner (p₁ -ᵥ p₁) (p₁ -ᵥ s.center)",
"state_before": "V : Type u_2\nP : Type u_1\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\ns : Sphere P\np₁ p₂ : P\nhp₁ : p₁ ∈ s\nhp₂ : dist p₂ s.center ≤ s.radius\n⊢ 0 ≤ inner (p₁ -ᵥ p₂) (p₁ -ᵥ s.center)",
"tactic": "rcases inner_pos_or_eq_of_dist_le_radius hp₁ hp₂ with (h | rfl)"
},
{
"state_after": "no goals",
"state_before": "case inl\nV : Type u_2\nP : Type u_1\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\ns : Sphere P\np₁ p₂ : P\nhp₁ : p₁ ∈ s\nhp₂ : dist p₂ s.center ≤ s.radius\nh : 0 < inner (p₁ -ᵥ p₂) (p₁ -ᵥ s.center)\n⊢ 0 ≤ inner (p₁ -ᵥ p₂) (p₁ -ᵥ s.center)",
"tactic": "exact h.le"
},
{
"state_after": "no goals",
"state_before": "case inr\nV : Type u_2\nP : Type u_1\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\ns : Sphere P\np₁ : P\nhp₁ : p₁ ∈ s\nhp₂ : dist p₁ s.center ≤ s.radius\n⊢ 0 ≤ inner (p₁ -ᵥ p₁) (p₁ -ᵥ s.center)",
"tactic": "simp"
}
] |
[
366,
9
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
362,
1
] |
Mathlib/MeasureTheory/Measure/Stieltjes.lean
|
StieltjesFunction.length_Ioc
|
[
{
"state_after": "f : StieltjesFunction\na b a' b' : ℝ\nh : Ioc a b ⊆ Ioc a' b'\n⊢ Real.toNNReal (↑f b - ↑f a) ≤ Real.toNNReal (↑f b' - ↑f a')",
"state_before": "f : StieltjesFunction\na b : ℝ\n⊢ length f (Ioc a b) = ofReal (↑f b - ↑f a)",
"tactic": "refine'\n le_antisymm (iInf_le_of_le a <| iInf₂_le b Subset.rfl)\n (le_iInf fun a' => le_iInf fun b' => le_iInf fun h => ENNReal.coe_le_coe.2 _)"
},
{
"state_after": "case inl\nf : StieltjesFunction\na b a' b' : ℝ\nh : Ioc a b ⊆ Ioc a' b'\nab : b ≤ a\n⊢ Real.toNNReal (↑f b - ↑f a) ≤ Real.toNNReal (↑f b' - ↑f a')\n\ncase inr\nf : StieltjesFunction\na b a' b' : ℝ\nh : Ioc a b ⊆ Ioc a' b'\nab : a < b\n⊢ Real.toNNReal (↑f b - ↑f a) ≤ Real.toNNReal (↑f b' - ↑f a')",
"state_before": "f : StieltjesFunction\na b a' b' : ℝ\nh : Ioc a b ⊆ Ioc a' b'\n⊢ Real.toNNReal (↑f b - ↑f a) ≤ Real.toNNReal (↑f b' - ↑f a')",
"tactic": "cases' le_or_lt b a with ab ab"
},
{
"state_after": "case inr.intro\nf : StieltjesFunction\na b a' b' : ℝ\nh : Ioc a b ⊆ Ioc a' b'\nab : a < b\nh₁ : b ≤ b'\nh₂ : a' ≤ a\n⊢ Real.toNNReal (↑f b - ↑f a) ≤ Real.toNNReal (↑f b' - ↑f a')",
"state_before": "case inr\nf : StieltjesFunction\na b a' b' : ℝ\nh : Ioc a b ⊆ Ioc a' b'\nab : a < b\n⊢ Real.toNNReal (↑f b - ↑f a) ≤ Real.toNNReal (↑f b' - ↑f a')",
"tactic": "cases' (Ioc_subset_Ioc_iff ab).1 h with h₁ h₂"
},
{
"state_after": "no goals",
"state_before": "case inr.intro\nf : StieltjesFunction\na b a' b' : ℝ\nh : Ioc a b ⊆ Ioc a' b'\nab : a < b\nh₁ : b ≤ b'\nh₂ : a' ≤ a\n⊢ Real.toNNReal (↑f b - ↑f a) ≤ Real.toNNReal (↑f b' - ↑f a')",
"tactic": "exact Real.toNNReal_le_toNNReal (sub_le_sub (f.mono h₁) (f.mono h₂))"
},
{
"state_after": "case inl\nf : StieltjesFunction\na b a' b' : ℝ\nh : Ioc a b ⊆ Ioc a' b'\nab : b ≤ a\n⊢ 0 ≤ Real.toNNReal (↑f b' - ↑f a')",
"state_before": "case inl\nf : StieltjesFunction\na b a' b' : ℝ\nh : Ioc a b ⊆ Ioc a' b'\nab : b ≤ a\n⊢ Real.toNNReal (↑f b - ↑f a) ≤ Real.toNNReal (↑f b' - ↑f a')",
"tactic": "rw [Real.toNNReal_of_nonpos (sub_nonpos.2 (f.mono ab))]"
},
{
"state_after": "no goals",
"state_before": "case inl\nf : StieltjesFunction\na b a' b' : ℝ\nh : Ioc a b ⊆ Ioc a' b'\nab : b ≤ a\n⊢ 0 ≤ Real.toNNReal (↑f b' - ↑f a')",
"tactic": "apply zero_le"
}
] |
[
330,
71
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
322,
1
] |
Mathlib/Topology/MetricSpace/Basic.lean
|
Metric.diam_le_of_forall_dist_le_of_nonempty
|
[] |
[
2634,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2629,
1
] |
Mathlib/Analysis/Calculus/Deriv/Pow.lean
|
differentiableOn_pow
|
[] |
[
87,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
86,
1
] |
Mathlib/Data/Real/Hyperreal.lean
|
Hyperreal.epsilon_lt_pos
|
[] |
[
218,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
217,
1
] |
Mathlib/Data/Real/ENatENNReal.lean
|
ENat.toENNReal_sub
|
[] |
[
121,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
120,
1
] |
Mathlib/AlgebraicGeometry/StructureSheaf.lean
|
AlgebraicGeometry.StructureSheaf.const_mul_rev
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\ninst✝ : CommRing R\nf g : R\nU : Opens ↑(PrimeSpectrum.Top R)\nhu₁ : ∀ (x : ↑(PrimeSpectrum.Top R)), x ∈ U → g ∈ Ideal.primeCompl x.asIdeal\nhu₂ : ∀ (x : ↑(PrimeSpectrum.Top R)), x ∈ U → f ∈ Ideal.primeCompl x.asIdeal\n⊢ const R f g U hu₁ * const R g f U hu₂ = 1",
"tactic": "rw [const_mul, const_congr R rfl (mul_comm g f), const_self]"
}
] |
[
397,
63
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
396,
1
] |
Mathlib/Data/Ordmap/Ordset.lean
|
Ordset.empty_iff
|
[
{
"state_after": "case refl\nα : Type u_1\ninst✝ : Preorder α\n⊢ empty ↑∅ = true",
"state_before": "α : Type u_1\ninst✝ : Preorder α\ns : Ordset α\nh : s = ∅\n⊢ empty ↑s = true",
"tactic": "cases h"
},
{
"state_after": "no goals",
"state_before": "case refl\nα : Type u_1\ninst✝ : Preorder α\n⊢ empty ↑∅ = true",
"tactic": "exact rfl"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : Preorder α\ns : Ordset α\nh : empty ↑s = true\n⊢ s = ∅",
"tactic": "cases s with | mk s_val _ => cases s_val <;> [rfl; cases h]"
},
{
"state_after": "no goals",
"state_before": "case mk\nα : Type u_1\ninst✝ : Preorder α\ns_val : Ordnode α\nproperty✝ : Valid s_val\nh : empty ↑{ val := s_val, property := property✝ } = true\n⊢ { val := s_val, property := property✝ } = ∅",
"tactic": "cases s_val <;> [rfl; cases h]"
}
] |
[
1728,
77
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1726,
1
] |
Mathlib/Analysis/Asymptotics/Asymptotics.lean
|
Asymptotics.IsLittleO.trans_eventuallyEq
|
[] |
[
403,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
401,
1
] |
Mathlib/Logic/Basic.lean
|
not_iff_comm
|
[] |
[
433,
69
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
433,
1
] |
Mathlib/Data/List/Indexes.lean
|
List.mapIdx_eq_ofFn
|
[
{
"state_after": "case nil\nα✝ : Type u\nβ✝ : Type v\nα : Type u_1\nβ : Type u_2\nf✝ f : ℕ → α → β\n⊢ mapIdx f [] = ofFn fun i => f (↑i) (get [] i)\n\ncase cons\nα✝ : Type u\nβ✝ : Type v\nα : Type u_1\nβ : Type u_2\nf✝ : ℕ → α → β\nhd : α\ntl : List α\nIH : ∀ (f : ℕ → α → β), mapIdx f tl = ofFn fun i => f (↑i) (get tl i)\nf : ℕ → α → β\n⊢ mapIdx f (hd :: tl) = ofFn fun i => f (↑i) (get (hd :: tl) i)",
"state_before": "α✝ : Type u\nβ✝ : Type v\nα : Type u_1\nβ : Type u_2\nl : List α\nf : ℕ → α → β\n⊢ mapIdx f l = ofFn fun i => f (↑i) (get l i)",
"tactic": "induction' l with hd tl IH generalizing f"
},
{
"state_after": "no goals",
"state_before": "case nil\nα✝ : Type u\nβ✝ : Type v\nα : Type u_1\nβ : Type u_2\nf✝ f : ℕ → α → β\n⊢ mapIdx f [] = ofFn fun i => f (↑i) (get [] i)",
"tactic": "rfl"
},
{
"state_after": "no goals",
"state_before": "case cons\nα✝ : Type u\nβ✝ : Type v\nα : Type u_1\nβ : Type u_2\nf✝ : ℕ → α → β\nhd : α\ntl : List α\nIH : ∀ (f : ℕ → α → β), mapIdx f tl = ofFn fun i => f (↑i) (get tl i)\nf : ℕ → α → β\n⊢ mapIdx f (hd :: tl) = ofFn fun i => f (↑i) (get (hd :: tl) i)",
"tactic": "simp [IH]"
}
] |
[
209,
14
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
205,
1
] |
Mathlib/Algebra/Opposites.lean
|
MulOpposite.op_injective
|
[] |
[
153,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
152,
1
] |
Mathlib/Data/Matrix/Kronecker.lean
|
Matrix.det_kroneckerMapBilinear
|
[
{
"state_after": "no goals",
"state_before": "R : Type u_1\nα : Type u_4\nα' : Type ?u.61413\nβ : Type u_5\nβ' : Type ?u.61419\nγ : Type u_6\nγ' : Type ?u.61425\nl : Type ?u.61428\nm : Type u_2\nn : Type u_3\np : Type ?u.61437\nq : Type ?u.61440\nr : Type ?u.61443\nl' : Type ?u.61446\nm' : Type ?u.61449\nn' : Type ?u.61452\np' : Type ?u.61455\ninst✝¹⁰ : CommSemiring R\ninst✝⁹ : Fintype m\ninst✝⁸ : Fintype n\ninst✝⁷ : DecidableEq m\ninst✝⁶ : DecidableEq n\ninst✝⁵ : CommRing α\ninst✝⁴ : CommRing β\ninst✝³ : CommRing γ\ninst✝² : Module R α\ninst✝¹ : Module R β\ninst✝ : Module R γ\nf : α →ₗ[R] β →ₗ[R] γ\nh_comm : ∀ (a b : α) (a' b' : β), ↑(↑f (a * b)) (a' * b') = ↑(↑f a) a' * ↑(↑f b) b'\nA : Matrix m m α\nB : Matrix n n β\n⊢ det (↑(↑(kroneckerMapBilinear f) A) B) = det (↑(↑(kroneckerMapBilinear f) A) 1 ⬝ ↑(↑(kroneckerMapBilinear f) 1) B)",
"tactic": "rw [← kroneckerMapBilinear_mul_mul f h_comm, Matrix.mul_one, Matrix.one_mul]"
},
{
"state_after": "R : Type u_1\nα : Type u_4\nα' : Type ?u.61413\nβ : Type u_5\nβ' : Type ?u.61419\nγ : Type u_6\nγ' : Type ?u.61425\nl : Type ?u.61428\nm : Type u_2\nn : Type u_3\np : Type ?u.61437\nq : Type ?u.61440\nr : Type ?u.61443\nl' : Type ?u.61446\nm' : Type ?u.61449\nn' : Type ?u.61452\np' : Type ?u.61455\ninst✝¹⁰ : CommSemiring R\ninst✝⁹ : Fintype m\ninst✝⁸ : Fintype n\ninst✝⁷ : DecidableEq m\ninst✝⁶ : DecidableEq n\ninst✝⁵ : CommRing α\ninst✝⁴ : CommRing β\ninst✝³ : CommRing γ\ninst✝² : Module R α\ninst✝¹ : Module R β\ninst✝ : Module R γ\nf : α →ₗ[R] β →ₗ[R] γ\nh_comm : ∀ (a b : α) (a' b' : β), ↑(↑f (a * b)) (a' * b') = ↑(↑f a) a' * ↑(↑f b) b'\nA : Matrix m m α\nB : Matrix n n β\n⊢ ∀ (x : β), ↑(↑f 0) x = 0\n\nR : Type u_1\nα : Type u_4\nα' : Type ?u.61413\nβ : Type u_5\nβ' : Type ?u.61419\nγ : Type u_6\nγ' : Type ?u.61425\nl : Type ?u.61428\nm : Type u_2\nn : Type u_3\np : Type ?u.61437\nq : Type ?u.61440\nr : Type ?u.61443\nl' : Type ?u.61446\nm' : Type ?u.61449\nn' : Type ?u.61452\np' : Type ?u.61455\ninst✝¹⁰ : CommSemiring R\ninst✝⁹ : Fintype m\ninst✝⁸ : Fintype n\ninst✝⁷ : DecidableEq m\ninst✝⁶ : DecidableEq n\ninst✝⁵ : CommRing α\ninst✝⁴ : CommRing β\ninst✝³ : CommRing γ\ninst✝² : Module R α\ninst✝¹ : Module R β\ninst✝ : Module R γ\nf : α →ₗ[R] β →ₗ[R] γ\nh_comm : ∀ (a b : α) (a' b' : β), ↑(↑f (a * b)) (a' * b') = ↑(↑f a) a' * ↑(↑f b) b'\nA : Matrix m m α\nB : Matrix n n β\n⊢ ∀ (x : α), ↑(↑f x) 0 = 0",
"state_before": "R : Type u_1\nα : Type u_4\nα' : Type ?u.61413\nβ : Type u_5\nβ' : Type ?u.61419\nγ : Type u_6\nγ' : Type ?u.61425\nl : Type ?u.61428\nm : Type u_2\nn : Type u_3\np : Type ?u.61437\nq : Type ?u.61440\nr : Type ?u.61443\nl' : Type ?u.61446\nm' : Type ?u.61449\nn' : Type ?u.61452\np' : Type ?u.61455\ninst✝¹⁰ : CommSemiring R\ninst✝⁹ : Fintype m\ninst✝⁸ : Fintype n\ninst✝⁷ : DecidableEq m\ninst✝⁶ : DecidableEq n\ninst✝⁵ : CommRing α\ninst✝⁴ : CommRing β\ninst✝³ : CommRing γ\ninst✝² : Module R α\ninst✝¹ : Module R β\ninst✝ : Module R γ\nf : α →ₗ[R] β →ₗ[R] γ\nh_comm : ∀ (a b : α) (a' b' : β), ↑(↑f (a * b)) (a' * b') = ↑(↑f a) a' * ↑(↑f b) b'\nA : Matrix m m α\nB : Matrix n n β\n⊢ det (↑(↑(kroneckerMapBilinear f) A) 1 ⬝ ↑(↑(kroneckerMapBilinear f) 1) B) =\n det (blockDiagonal fun x => map A fun a => ↑(↑f a) 1) * det (blockDiagonal fun x => map B fun b => ↑(↑f 1) b)",
"tactic": "rw [det_mul, ← diagonal_one, ← diagonal_one, kroneckerMapBilinear_apply_apply,\n kroneckerMap_diagonal_right _ fun _ => _, kroneckerMapBilinear_apply_apply,\n kroneckerMap_diagonal_left _ fun _ => _, det_reindex_self]"
},
{
"state_after": "R : Type u_1\nα : Type u_4\nα' : Type ?u.61413\nβ : Type u_5\nβ' : Type ?u.61419\nγ : Type u_6\nγ' : Type ?u.61425\nl : Type ?u.61428\nm : Type u_2\nn : Type u_3\np : Type ?u.61437\nq : Type ?u.61440\nr : Type ?u.61443\nl' : Type ?u.61446\nm' : Type ?u.61449\nn' : Type ?u.61452\np' : Type ?u.61455\ninst✝¹⁰ : CommSemiring R\ninst✝⁹ : Fintype m\ninst✝⁸ : Fintype n\ninst✝⁷ : DecidableEq m\ninst✝⁶ : DecidableEq n\ninst✝⁵ : CommRing α\ninst✝⁴ : CommRing β\ninst✝³ : CommRing γ\ninst✝² : Module R α\ninst✝¹ : Module R β\ninst✝ : Module R γ\nf : α →ₗ[R] β →ₗ[R] γ\nh_comm : ∀ (a b : α) (a' b' : β), ↑(↑f (a * b)) (a' * b') = ↑(↑f a) a' * ↑(↑f b) b'\nA : Matrix m m α\nB : Matrix n n β\nx✝ : β\n⊢ ↑(↑f 0) x✝ = 0",
"state_before": "R : Type u_1\nα : Type u_4\nα' : Type ?u.61413\nβ : Type u_5\nβ' : Type ?u.61419\nγ : Type u_6\nγ' : Type ?u.61425\nl : Type ?u.61428\nm : Type u_2\nn : Type u_3\np : Type ?u.61437\nq : Type ?u.61440\nr : Type ?u.61443\nl' : Type ?u.61446\nm' : Type ?u.61449\nn' : Type ?u.61452\np' : Type ?u.61455\ninst✝¹⁰ : CommSemiring R\ninst✝⁹ : Fintype m\ninst✝⁸ : Fintype n\ninst✝⁷ : DecidableEq m\ninst✝⁶ : DecidableEq n\ninst✝⁵ : CommRing α\ninst✝⁴ : CommRing β\ninst✝³ : CommRing γ\ninst✝² : Module R α\ninst✝¹ : Module R β\ninst✝ : Module R γ\nf : α →ₗ[R] β →ₗ[R] γ\nh_comm : ∀ (a b : α) (a' b' : β), ↑(↑f (a * b)) (a' * b') = ↑(↑f a) a' * ↑(↑f b) b'\nA : Matrix m m α\nB : Matrix n n β\n⊢ ∀ (x : β), ↑(↑f 0) x = 0",
"tactic": "intro"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\nα : Type u_4\nα' : Type ?u.61413\nβ : Type u_5\nβ' : Type ?u.61419\nγ : Type u_6\nγ' : Type ?u.61425\nl : Type ?u.61428\nm : Type u_2\nn : Type u_3\np : Type ?u.61437\nq : Type ?u.61440\nr : Type ?u.61443\nl' : Type ?u.61446\nm' : Type ?u.61449\nn' : Type ?u.61452\np' : Type ?u.61455\ninst✝¹⁰ : CommSemiring R\ninst✝⁹ : Fintype m\ninst✝⁸ : Fintype n\ninst✝⁷ : DecidableEq m\ninst✝⁶ : DecidableEq n\ninst✝⁵ : CommRing α\ninst✝⁴ : CommRing β\ninst✝³ : CommRing γ\ninst✝² : Module R α\ninst✝¹ : Module R β\ninst✝ : Module R γ\nf : α →ₗ[R] β →ₗ[R] γ\nh_comm : ∀ (a b : α) (a' b' : β), ↑(↑f (a * b)) (a' * b') = ↑(↑f a) a' * ↑(↑f b) b'\nA : Matrix m m α\nB : Matrix n n β\nx✝ : β\n⊢ ↑(↑f 0) x✝ = 0",
"tactic": "exact LinearMap.map_zero₂ _ _"
},
{
"state_after": "R : Type u_1\nα : Type u_4\nα' : Type ?u.61413\nβ : Type u_5\nβ' : Type ?u.61419\nγ : Type u_6\nγ' : Type ?u.61425\nl : Type ?u.61428\nm : Type u_2\nn : Type u_3\np : Type ?u.61437\nq : Type ?u.61440\nr : Type ?u.61443\nl' : Type ?u.61446\nm' : Type ?u.61449\nn' : Type ?u.61452\np' : Type ?u.61455\ninst✝¹⁰ : CommSemiring R\ninst✝⁹ : Fintype m\ninst✝⁸ : Fintype n\ninst✝⁷ : DecidableEq m\ninst✝⁶ : DecidableEq n\ninst✝⁵ : CommRing α\ninst✝⁴ : CommRing β\ninst✝³ : CommRing γ\ninst✝² : Module R α\ninst✝¹ : Module R β\ninst✝ : Module R γ\nf : α →ₗ[R] β →ₗ[R] γ\nh_comm : ∀ (a b : α) (a' b' : β), ↑(↑f (a * b)) (a' * b') = ↑(↑f a) a' * ↑(↑f b) b'\nA : Matrix m m α\nB : Matrix n n β\nx✝ : α\n⊢ ↑(↑f x✝) 0 = 0",
"state_before": "R : Type u_1\nα : Type u_4\nα' : Type ?u.61413\nβ : Type u_5\nβ' : Type ?u.61419\nγ : Type u_6\nγ' : Type ?u.61425\nl : Type ?u.61428\nm : Type u_2\nn : Type u_3\np : Type ?u.61437\nq : Type ?u.61440\nr : Type ?u.61443\nl' : Type ?u.61446\nm' : Type ?u.61449\nn' : Type ?u.61452\np' : Type ?u.61455\ninst✝¹⁰ : CommSemiring R\ninst✝⁹ : Fintype m\ninst✝⁸ : Fintype n\ninst✝⁷ : DecidableEq m\ninst✝⁶ : DecidableEq n\ninst✝⁵ : CommRing α\ninst✝⁴ : CommRing β\ninst✝³ : CommRing γ\ninst✝² : Module R α\ninst✝¹ : Module R β\ninst✝ : Module R γ\nf : α →ₗ[R] β →ₗ[R] γ\nh_comm : ∀ (a b : α) (a' b' : β), ↑(↑f (a * b)) (a' * b') = ↑(↑f a) a' * ↑(↑f b) b'\nA : Matrix m m α\nB : Matrix n n β\n⊢ ∀ (x : α), ↑(↑f x) 0 = 0",
"tactic": "intro"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\nα : Type u_4\nα' : Type ?u.61413\nβ : Type u_5\nβ' : Type ?u.61419\nγ : Type u_6\nγ' : Type ?u.61425\nl : Type ?u.61428\nm : Type u_2\nn : Type u_3\np : Type ?u.61437\nq : Type ?u.61440\nr : Type ?u.61443\nl' : Type ?u.61446\nm' : Type ?u.61449\nn' : Type ?u.61452\np' : Type ?u.61455\ninst✝¹⁰ : CommSemiring R\ninst✝⁹ : Fintype m\ninst✝⁸ : Fintype n\ninst✝⁷ : DecidableEq m\ninst✝⁶ : DecidableEq n\ninst✝⁵ : CommRing α\ninst✝⁴ : CommRing β\ninst✝³ : CommRing γ\ninst✝² : Module R α\ninst✝¹ : Module R β\ninst✝ : Module R γ\nf : α →ₗ[R] β →ₗ[R] γ\nh_comm : ∀ (a b : α) (a' b' : β), ↑(↑f (a * b)) (a' * b') = ↑(↑f a) a' * ↑(↑f b) b'\nA : Matrix m m α\nB : Matrix n n β\nx✝ : α\n⊢ ↑(↑f x✝) 0 = 0",
"tactic": "exact map_zero _"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\nα : Type u_4\nα' : Type ?u.61413\nβ : Type u_5\nβ' : Type ?u.61419\nγ : Type u_6\nγ' : Type ?u.61425\nl : Type ?u.61428\nm : Type u_2\nn : Type u_3\np : Type ?u.61437\nq : Type ?u.61440\nr : Type ?u.61443\nl' : Type ?u.61446\nm' : Type ?u.61449\nn' : Type ?u.61452\np' : Type ?u.61455\ninst✝¹⁰ : CommSemiring R\ninst✝⁹ : Fintype m\ninst✝⁸ : Fintype n\ninst✝⁷ : DecidableEq m\ninst✝⁶ : DecidableEq n\ninst✝⁵ : CommRing α\ninst✝⁴ : CommRing β\ninst✝³ : CommRing γ\ninst✝² : Module R α\ninst✝¹ : Module R β\ninst✝ : Module R γ\nf : α →ₗ[R] β →ₗ[R] γ\nh_comm : ∀ (a b : α) (a' b' : β), ↑(↑f (a * b)) (a' * b') = ↑(↑f a) a' * ↑(↑f b) b'\nA : Matrix m m α\nB : Matrix n n β\n⊢ det (blockDiagonal fun x => map A fun a => ↑(↑f a) 1) * det (blockDiagonal fun x => map B fun b => ↑(↑f 1) b) =\n det (map A fun a => ↑(↑f a) 1) ^ Fintype.card n * det (map B fun b => ↑(↑f 1) b) ^ Fintype.card m",
"tactic": "simp_rw [det_blockDiagonal, Finset.prod_const, Finset.card_univ]"
}
] |
[
256,
81
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
239,
1
] |
Mathlib/Order/Filter/Bases.lean
|
FilterBasis.hasBasis
|
[] |
[
343,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
341,
11
] |
Mathlib/RepresentationTheory/Basic.lean
|
Representation.ofMulAction_def
|
[] |
[
276,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
275,
1
] |
Mathlib/Data/List/Perm.lean
|
List.Perm.pmap
|
[
{
"state_after": "no goals",
"state_before": "α : Type uu\nβ : Type vv\nl₁✝ l₂✝ : List α\np✝ : α → Prop\nf : (a : α) → p✝ a → β\nl₁ l₂ : List α\np : l₁ ~ l₂\nH₁ : ∀ (a : α), a ∈ l₁ → p✝ a\nH₂ : ∀ (a : α), a ∈ l₂ → p✝ a\n⊢ List.pmap f l₁ H₁ ~ List.pmap f l₂ H₂",
"tactic": "induction p with\n| nil => simp\n| cons x _p IH => simp [IH, Perm.cons]\n| swap x y => simp [swap]\n| trans _p₁ p₂ IH₁ IH₂ =>\n refine' IH₁.trans IH₂\n exact fun a m => H₂ a (p₂.subset m)"
},
{
"state_after": "no goals",
"state_before": "case nil\nα : Type uu\nβ : Type vv\nl₁✝ l₂✝ : List α\np : α → Prop\nf : (a : α) → p a → β\nl₁ l₂ : List α\nH₁ H₂ : ∀ (a : α), a ∈ [] → p a\n⊢ List.pmap f [] H₁ ~ List.pmap f [] H₂",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "case cons\nα : Type uu\nβ : Type vv\nl₁✝¹ l₂✝¹ : List α\np : α → Prop\nf : (a : α) → p a → β\nl₁ l₂ : List α\nx : α\nl₁✝ l₂✝ : List α\n_p : l₁✝ ~ l₂✝\nIH : ∀ {H₁ : ∀ (a : α), a ∈ l₁✝ → p a} {H₂ : ∀ (a : α), a ∈ l₂✝ → p a}, List.pmap f l₁✝ H₁ ~ List.pmap f l₂✝ H₂\nH₁ : ∀ (a : α), a ∈ x :: l₁✝ → p a\nH₂ : ∀ (a : α), a ∈ x :: l₂✝ → p a\n⊢ List.pmap f (x :: l₁✝) H₁ ~ List.pmap f (x :: l₂✝) H₂",
"tactic": "simp [IH, Perm.cons]"
},
{
"state_after": "no goals",
"state_before": "case swap\nα : Type uu\nβ : Type vv\nl₁✝ l₂✝ : List α\np : α → Prop\nf : (a : α) → p a → β\nl₁ l₂ : List α\nx y : α\nl✝ : List α\nH₁ : ∀ (a : α), a ∈ y :: x :: l✝ → p a\nH₂ : ∀ (a : α), a ∈ x :: y :: l✝ → p a\n⊢ List.pmap f (y :: x :: l✝) H₁ ~ List.pmap f (x :: y :: l✝) H₂",
"tactic": "simp [swap]"
},
{
"state_after": "case trans\nα : Type uu\nβ : Type vv\nl₁✝¹ l₂✝¹ : List α\np : α → Prop\nf : (a : α) → p a → β\nl₁ l₂ l₁✝ l₂✝ l₃✝ : List α\n_p₁ : l₁✝ ~ l₂✝\np₂ : l₂✝ ~ l₃✝\nIH₁ : ∀ {H₁ : ∀ (a : α), a ∈ l₁✝ → p a} {H₂ : ∀ (a : α), a ∈ l₂✝ → p a}, List.pmap f l₁✝ H₁ ~ List.pmap f l₂✝ H₂\nIH₂ : ∀ {H₁ : ∀ (a : α), a ∈ l₂✝ → p a} {H₂ : ∀ (a : α), a ∈ l₃✝ → p a}, List.pmap f l₂✝ H₁ ~ List.pmap f l₃✝ H₂\nH₁ : ∀ (a : α), a ∈ l₁✝ → p a\nH₂ : ∀ (a : α), a ∈ l₃✝ → p a\n⊢ ∀ (a : α), a ∈ l₂✝ → p a",
"state_before": "case trans\nα : Type uu\nβ : Type vv\nl₁✝¹ l₂✝¹ : List α\np : α → Prop\nf : (a : α) → p a → β\nl₁ l₂ l₁✝ l₂✝ l₃✝ : List α\n_p₁ : l₁✝ ~ l₂✝\np₂ : l₂✝ ~ l₃✝\nIH₁ : ∀ {H₁ : ∀ (a : α), a ∈ l₁✝ → p a} {H₂ : ∀ (a : α), a ∈ l₂✝ → p a}, List.pmap f l₁✝ H₁ ~ List.pmap f l₂✝ H₂\nIH₂ : ∀ {H₁ : ∀ (a : α), a ∈ l₂✝ → p a} {H₂ : ∀ (a : α), a ∈ l₃✝ → p a}, List.pmap f l₂✝ H₁ ~ List.pmap f l₃✝ H₂\nH₁ : ∀ (a : α), a ∈ l₁✝ → p a\nH₂ : ∀ (a : α), a ∈ l₃✝ → p a\n⊢ List.pmap f l₁✝ H₁ ~ List.pmap f l₃✝ H₂",
"tactic": "refine' IH₁.trans IH₂"
},
{
"state_after": "no goals",
"state_before": "case trans\nα : Type uu\nβ : Type vv\nl₁✝¹ l₂✝¹ : List α\np : α → Prop\nf : (a : α) → p a → β\nl₁ l₂ l₁✝ l₂✝ l₃✝ : List α\n_p₁ : l₁✝ ~ l₂✝\np₂ : l₂✝ ~ l₃✝\nIH₁ : ∀ {H₁ : ∀ (a : α), a ∈ l₁✝ → p a} {H₂ : ∀ (a : α), a ∈ l₂✝ → p a}, List.pmap f l₁✝ H₁ ~ List.pmap f l₂✝ H₂\nIH₂ : ∀ {H₁ : ∀ (a : α), a ∈ l₂✝ → p a} {H₂ : ∀ (a : α), a ∈ l₃✝ → p a}, List.pmap f l₂✝ H₁ ~ List.pmap f l₃✝ H₂\nH₁ : ∀ (a : α), a ∈ l₁✝ → p a\nH₂ : ∀ (a : α), a ∈ l₃✝ → p a\n⊢ ∀ (a : α), a ∈ l₂✝ → p a",
"tactic": "exact fun a m => H₂ a (p₂.subset m)"
}
] |
[
271,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
263,
1
] |
Mathlib/Algebra/BigOperators/Ring.lean
|
Finset.prod_pow_eq_pow_sum
|
[
{
"state_after": "case empty\nα : Type u\nβ : Type v\nγ : Type w\ns s₁ s₂ : Finset α\na : α\nb : β\nf✝ g : α → β\ninst✝ : CommMonoid β\nx : β\nf : α → ℕ\n⊢ ∏ i in ∅, x ^ f i = x ^ ∑ x in ∅, f x\n\ncase insert\nα : Type u\nβ : Type v\nγ : Type w\ns s₁ s₂ : Finset α\na : α\nb : β\nf✝ g : α → β\ninst✝ : CommMonoid β\nx : β\nf : α → ℕ\n⊢ ∀ ⦃a : α⦄ {s : Finset α},\n ¬a ∈ s → ∏ i in s, x ^ f i = x ^ ∑ x in s, f x → ∏ i in insert a s, x ^ f i = x ^ ∑ x in insert a s, f x",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ns s₁ s₂ : Finset α\na : α\nb : β\nf✝ g : α → β\ninst✝ : CommMonoid β\nx : β\nf : α → ℕ\n⊢ ∀ {s : Finset α}, ∏ i in s, x ^ f i = x ^ ∑ x in s, f x",
"tactic": "apply Finset.induction"
},
{
"state_after": "no goals",
"state_before": "case empty\nα : Type u\nβ : Type v\nγ : Type w\ns s₁ s₂ : Finset α\na : α\nb : β\nf✝ g : α → β\ninst✝ : CommMonoid β\nx : β\nf : α → ℕ\n⊢ ∏ i in ∅, x ^ f i = x ^ ∑ x in ∅, f x",
"tactic": "simp"
},
{
"state_after": "case insert\nα : Type u\nβ : Type v\nγ : Type w\ns✝ s₁ s₂ : Finset α\na✝ : α\nb : β\nf✝ g : α → β\ninst✝ : CommMonoid β\nx : β\nf : α → ℕ\na : α\ns : Finset α\nhas : ¬a ∈ s\nH : ∏ i in s, x ^ f i = x ^ ∑ x in s, f x\n⊢ ∏ i in insert a s, x ^ f i = x ^ ∑ x in insert a s, f x",
"state_before": "case insert\nα : Type u\nβ : Type v\nγ : Type w\ns s₁ s₂ : Finset α\na : α\nb : β\nf✝ g : α → β\ninst✝ : CommMonoid β\nx : β\nf : α → ℕ\n⊢ ∀ ⦃a : α⦄ {s : Finset α},\n ¬a ∈ s → ∏ i in s, x ^ f i = x ^ ∑ x in s, f x → ∏ i in insert a s, x ^ f i = x ^ ∑ x in insert a s, f x",
"tactic": "intro a s has H"
},
{
"state_after": "no goals",
"state_before": "case insert\nα : Type u\nβ : Type v\nγ : Type w\ns✝ s₁ s₂ : Finset α\na✝ : α\nb : β\nf✝ g : α → β\ninst✝ : CommMonoid β\nx : β\nf : α → ℕ\na : α\ns : Finset α\nhas : ¬a ∈ s\nH : ∏ i in s, x ^ f i = x ^ ∑ x in s, f x\n⊢ ∏ i in insert a s, x ^ f i = x ^ ∑ x in insert a s, f x",
"tactic": "rw [Finset.prod_insert has, Finset.sum_insert has, pow_add, H]"
}
] |
[
45,
67
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
40,
1
] |
Mathlib/Data/Polynomial/EraseLead.lean
|
Polynomial.map_natDegree_eq_natDegree
|
[
{
"state_after": "no goals",
"state_before": "R : Type u_3\ninst✝² : Semiring R\nf : R[X]\nS : Type u_1\nF : Type u_2\ninst✝¹ : Semiring S\ninst✝ : AddMonoidHomClass F R[X] S[X]\nφ : F\np : R[X]\nφ_mon_nat : ∀ (n : ℕ) (c : R), c ≠ 0 → natDegree (↑φ (↑(monomial n) c)) = n\n⊢ ∀ (n : ℕ) (c : R), c ≠ 0 → natDegree (↑φ (↑(monomial n) c)) = n - 0",
"tactic": "simpa"
}
] |
[
289,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
285,
1
] |
Mathlib/Data/Set/Finite.lean
|
Set.Finite.toFinset_insert
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nι : Sort w\nγ : Type x\ninst✝ : DecidableEq α\ns : Set α\na : α\nhs : Set.Finite (insert a s)\n⊢ ∀ (a_1 : α), a_1 ∈ Finite.toFinset hs ↔ a_1 ∈ insert a (Finite.toFinset (_ : Set.Finite s))",
"tactic": "simp"
}
] |
[
1061,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1059,
1
] |
Mathlib/Data/Real/ENNReal.lean
|
ENNReal.mul_lt_mul_right
|
[] |
[
1044,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1043,
1
] |
Mathlib/Order/Filter/Pointwise.lean
|
Filter.vsub_mem_vsub
|
[] |
[
1093,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1092,
1
] |
Mathlib/Data/Polynomial/Laurent.lean
|
LaurentPolynomial.degree_zero
|
[] |
[
485,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
484,
1
] |
Mathlib/Algebra/Ring/Prod.lean
|
RingHom.fst_comp_prod
|
[] |
[
232,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
231,
1
] |
Mathlib/Algebra/Group/TypeTags.lean
|
toAdd_ofAdd
|
[] |
[
103,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
102,
1
] |
Mathlib/Order/Filter/Germ.lean
|
Filter.Germ.liftRel_const
|
[] |
[
325,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
323,
1
] |
Mathlib/Logic/Equiv/Defs.lean
|
Equiv.Perm.congr_fun
|
[] |
[
142,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
141,
11
] |
Mathlib/Order/CompleteLattice.lean
|
le_iSup'
|
[] |
[
761,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
760,
1
] |
Mathlib/FieldTheory/Separable.lean
|
Polynomial.exists_finset_of_splits
|
[
{
"state_after": "case intro\nF : Type u\ninst✝¹ : Field F\nK : Type v\ninst✝ : Field K\ni✝ i : F →+* K\nf : F[X]\nsep : Separable f\nsp : Splits i f\ns : Multiset K\nh : map i f = ↑C (↑i (leadingCoeff f)) * Multiset.prod (Multiset.map (fun a => X - ↑C a) s)\n⊢ ∃ s, map i f = ↑C (↑i (leadingCoeff f)) * ∏ a in s, (X - ↑C a)",
"state_before": "F : Type u\ninst✝¹ : Field F\nK : Type v\ninst✝ : Field K\ni✝ i : F →+* K\nf : F[X]\nsep : Separable f\nsp : Splits i f\n⊢ ∃ s, map i f = ↑C (↑i (leadingCoeff f)) * ∏ a in s, (X - ↑C a)",
"tactic": "obtain ⟨s, h⟩ := (splits_iff_exists_multiset _).1 sp"
},
{
"state_after": "case intro\nF : Type u\ninst✝¹ : Field F\nK : Type v\ninst✝ : Field K\ni✝ i : F →+* K\nf : F[X]\nsep : Separable f\nsp : Splits i f\ns : Multiset K\nh : map i f = ↑C (↑i (leadingCoeff f)) * Multiset.prod (Multiset.map (fun a => X - ↑C a) s)\n⊢ map i f = ↑C (↑i (leadingCoeff f)) * ∏ a in Multiset.toFinset s, (X - ↑C a)",
"state_before": "case intro\nF : Type u\ninst✝¹ : Field F\nK : Type v\ninst✝ : Field K\ni✝ i : F →+* K\nf : F[X]\nsep : Separable f\nsp : Splits i f\ns : Multiset K\nh : map i f = ↑C (↑i (leadingCoeff f)) * Multiset.prod (Multiset.map (fun a => X - ↑C a) s)\n⊢ ∃ s, map i f = ↑C (↑i (leadingCoeff f)) * ∏ a in s, (X - ↑C a)",
"tactic": "use s.toFinset"
},
{
"state_after": "case intro\nF : Type u\ninst✝¹ : Field F\nK : Type v\ninst✝ : Field K\ni✝ i : F →+* K\nf : F[X]\nsep : Separable f\nsp : Splits i f\ns : Multiset K\nh : map i f = ↑C (↑i (leadingCoeff f)) * Multiset.prod (Multiset.map (fun a => X - ↑C a) s)\n⊢ Multiset.Nodup s",
"state_before": "case intro\nF : Type u\ninst✝¹ : Field F\nK : Type v\ninst✝ : Field K\ni✝ i : F →+* K\nf : F[X]\nsep : Separable f\nsp : Splits i f\ns : Multiset K\nh : map i f = ↑C (↑i (leadingCoeff f)) * Multiset.prod (Multiset.map (fun a => X - ↑C a) s)\n⊢ map i f = ↑C (↑i (leadingCoeff f)) * ∏ a in Multiset.toFinset s, (X - ↑C a)",
"tactic": "rw [h, Finset.prod_eq_multiset_prod, ← Multiset.toFinset_eq]"
},
{
"state_after": "case intro.hs\nF : Type u\ninst✝¹ : Field F\nK : Type v\ninst✝ : Field K\ni✝ i : F →+* K\nf : F[X]\nsep : Separable f\nsp : Splits i f\ns : Multiset K\nh : map i f = ↑C (↑i (leadingCoeff f)) * Multiset.prod (Multiset.map (fun a => X - ↑C a) s)\n⊢ Separable (Multiset.prod (Multiset.map (fun a => X - ↑C a) s))",
"state_before": "case intro\nF : Type u\ninst✝¹ : Field F\nK : Type v\ninst✝ : Field K\ni✝ i : F →+* K\nf : F[X]\nsep : Separable f\nsp : Splits i f\ns : Multiset K\nh : map i f = ↑C (↑i (leadingCoeff f)) * Multiset.prod (Multiset.map (fun a => X - ↑C a) s)\n⊢ Multiset.Nodup s",
"tactic": "apply nodup_of_separable_prod"
},
{
"state_after": "case intro.hs.h\nF : Type u\ninst✝¹ : Field F\nK : Type v\ninst✝ : Field K\ni✝ i : F →+* K\nf : F[X]\nsep : Separable f\nsp : Splits i f\ns : Multiset K\nh : map i f = ↑C (↑i (leadingCoeff f)) * Multiset.prod (Multiset.map (fun a => X - ↑C a) s)\n⊢ Separable (?intro.hs.f * Multiset.prod (Multiset.map (fun a => X - ↑C a) s))\n\ncase intro.hs.f\nF : Type u\ninst✝¹ : Field F\nK : Type v\ninst✝ : Field K\ni✝ i : F →+* K\nf : F[X]\nsep : Separable f\nsp : Splits i f\ns : Multiset K\nh : map i f = ↑C (↑i (leadingCoeff f)) * Multiset.prod (Multiset.map (fun a => X - ↑C a) s)\n⊢ K[X]",
"state_before": "case intro.hs\nF : Type u\ninst✝¹ : Field F\nK : Type v\ninst✝ : Field K\ni✝ i : F →+* K\nf : F[X]\nsep : Separable f\nsp : Splits i f\ns : Multiset K\nh : map i f = ↑C (↑i (leadingCoeff f)) * Multiset.prod (Multiset.map (fun a => X - ↑C a) s)\n⊢ Separable (Multiset.prod (Multiset.map (fun a => X - ↑C a) s))",
"tactic": "apply Separable.of_mul_right"
},
{
"state_after": "case intro.hs.h\nF : Type u\ninst✝¹ : Field F\nK : Type v\ninst✝ : Field K\ni✝ i : F →+* K\nf : F[X]\nsep : Separable f\nsp : Splits i f\ns : Multiset K\nh : map i f = ↑C (↑i (leadingCoeff f)) * Multiset.prod (Multiset.map (fun a => X - ↑C a) s)\n⊢ Separable (map i f)",
"state_before": "case intro.hs.h\nF : Type u\ninst✝¹ : Field F\nK : Type v\ninst✝ : Field K\ni✝ i : F →+* K\nf : F[X]\nsep : Separable f\nsp : Splits i f\ns : Multiset K\nh : map i f = ↑C (↑i (leadingCoeff f)) * Multiset.prod (Multiset.map (fun a => X - ↑C a) s)\n⊢ Separable (?intro.hs.f * Multiset.prod (Multiset.map (fun a => X - ↑C a) s))\n\ncase intro.hs.f\nF : Type u\ninst✝¹ : Field F\nK : Type v\ninst✝ : Field K\ni✝ i : F →+* K\nf : F[X]\nsep : Separable f\nsp : Splits i f\ns : Multiset K\nh : map i f = ↑C (↑i (leadingCoeff f)) * Multiset.prod (Multiset.map (fun a => X - ↑C a) s)\n⊢ K[X]",
"tactic": "rw [← h]"
},
{
"state_after": "no goals",
"state_before": "case intro.hs.h\nF : Type u\ninst✝¹ : Field F\nK : Type v\ninst✝ : Field K\ni✝ i : F →+* K\nf : F[X]\nsep : Separable f\nsp : Splits i f\ns : Multiset K\nh : map i f = ↑C (↑i (leadingCoeff f)) * Multiset.prod (Multiset.map (fun a => X - ↑C a) s)\n⊢ Separable (map i f)",
"tactic": "exact sep.map"
}
] |
[
460,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
452,
1
] |
Mathlib/MeasureTheory/Group/Arithmetic.lean
|
Measurable.pow_const
|
[] |
[
232,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
231,
1
] |
Mathlib/GroupTheory/QuotientGroup.lean
|
QuotientGroup.rangeKerLift_injective
|
[
{
"state_after": "no goals",
"state_before": "G : Type u\ninst✝¹ : Group G\nN : Subgroup G\nnN : Subgroup.Normal N\nH : Type v\ninst✝ : Group H\nφ : G →* H\na✝ b✝ : G ⧸ ker φ\na b : G\nh : ↑(rangeRestrict φ) a = ↑(rangeRestrict φ) b\n⊢ Setoid.r a b",
"tactic": "rw [leftRel_apply, ← ker_rangeRestrict, mem_ker, φ.rangeRestrict.map_mul, ← h,\n φ.rangeRestrict.map_inv, inv_mul_self]"
}
] |
[
385,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
381,
1
] |
Mathlib/Topology/Algebra/Module/Basic.lean
|
ContinuousLinearMap.map_smul_of_tower
|
[] |
[
512,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
509,
1
] |
Mathlib/NumberTheory/Bernoulli.lean
|
bernoulli'_one
|
[
{
"state_after": "A : Type ?u.182698\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\n⊢ 1 - ∑ k in range 1, ↑(Nat.choose 1 k) / (↑1 - ↑k + 1) * bernoulli' k = 1 / 2",
"state_before": "A : Type ?u.182698\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\n⊢ bernoulli' 1 = 1 / 2",
"tactic": "rw [bernoulli'_def]"
},
{
"state_after": "no goals",
"state_before": "A : Type ?u.182698\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\n⊢ 1 - ∑ k in range 1, ↑(Nat.choose 1 k) / (↑1 - ↑k + 1) * bernoulli' k = 1 / 2",
"tactic": "norm_num"
}
] |
[
114,
11
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
112,
1
] |
Mathlib/Analysis/Asymptotics/SpecificAsymptotics.lean
|
Asymptotics.isLittleO_sum_range_of_tendsto_zero
|
[
{
"state_after": "α : Type u_1\ninst✝ : NormedAddCommGroup α\nf : ℕ → α\nh : Tendsto f atTop (𝓝 0)\nthis :\n Tendsto (fun n => ∑ i in range n, 1) atTop atTop → (fun n => ∑ i in range n, f i) =o[atTop] fun n => ∑ i in range n, 1\n⊢ (fun n => ∑ i in range n, f i) =o[atTop] fun n => ↑n",
"state_before": "α : Type u_1\ninst✝ : NormedAddCommGroup α\nf : ℕ → α\nh : Tendsto f atTop (𝓝 0)\n⊢ (fun n => ∑ i in range n, f i) =o[atTop] fun n => ↑n",
"tactic": "have := ((isLittleO_one_iff ℝ).2 h).sum_range fun i => zero_le_one"
},
{
"state_after": "α : Type u_1\ninst✝ : NormedAddCommGroup α\nf : ℕ → α\nh : Tendsto f atTop (𝓝 0)\nthis : Tendsto (fun n => ↑n) atTop atTop → (fun n => ∑ i in range n, f i) =o[atTop] fun n => ↑n\n⊢ (fun n => ∑ i in range n, f i) =o[atTop] fun n => ↑n",
"state_before": "α : Type u_1\ninst✝ : NormedAddCommGroup α\nf : ℕ → α\nh : Tendsto f atTop (𝓝 0)\nthis :\n Tendsto (fun n => ∑ i in range n, 1) atTop atTop → (fun n => ∑ i in range n, f i) =o[atTop] fun n => ∑ i in range n, 1\n⊢ (fun n => ∑ i in range n, f i) =o[atTop] fun n => ↑n",
"tactic": "simp only [sum_const, card_range, Nat.smul_one_eq_coe] at this"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : NormedAddCommGroup α\nf : ℕ → α\nh : Tendsto f atTop (𝓝 0)\nthis : Tendsto (fun n => ↑n) atTop atTop → (fun n => ∑ i in range n, f i) =o[atTop] fun n => ↑n\n⊢ (fun n => ∑ i in range n, f i) =o[atTop] fun n => ↑n",
"tactic": "exact this tendsto_nat_cast_atTop_atTop"
}
] |
[
147,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
142,
1
] |
Mathlib/Control/Applicative.lean
|
Functor.Comp.map_pure
|
[
{
"state_after": "no goals",
"state_before": "F : Type u → Type w\nG : Type v → Type u\ninst✝³ : Applicative F\ninst✝² : Applicative G\ninst✝¹ : LawfulApplicative F\ninst✝ : LawfulApplicative G\nα β γ : Type v\nf : α → β\nx : α\n⊢ run (f <$> pure x) = run (pure (f x))",
"tactic": "simp"
}
] |
[
97,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
96,
1
] |
Mathlib/Algebra/Order/Monoid/Lemmas.lean
|
AntitoneOn.const_mul'
|
[] |
[
1307,
90
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1306,
1
] |
Mathlib/SetTheory/Cardinal/Basic.lean
|
Cardinal.two_le_iff
|
[
{
"state_after": "no goals",
"state_before": "α β : Type u\n⊢ 2 ≤ (#α) ↔ ∃ x y, x ≠ y",
"tactic": "rw [← Nat.cast_two, nat_succ, succ_le_iff, Nat.cast_one, one_lt_iff_nontrivial, nontrivial_iff]"
}
] |
[
2230,
98
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2229,
1
] |
Mathlib/MeasureTheory/Integral/IntervalIntegral.lean
|
intervalIntegral.inv_smul_integral_comp_div
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.14846073\n𝕜 : Type ?u.14846076\nE : Type u_1\nF : Type ?u.14846082\nA : Type ?u.14846085\ninst✝² : NormedAddCommGroup E\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\na b c✝ d : ℝ\nf : ℝ → E\nc : ℝ\n⊢ (c⁻¹ • ∫ (x : ℝ) in a..b, f (x / c)) = ∫ (x : ℝ) in a / c..b / c, f x",
"tactic": "by_cases hc : c = 0 <;> simp [hc, integral_comp_div]"
}
] |
[
733,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
731,
1
] |
Mathlib/Algebra/GroupPower/Basic.lean
|
pow_bit0'
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.37531\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u₁\nS : Type u₂\ninst✝¹ : Monoid M\ninst✝ : AddMonoid A\na : M\nn : ℕ\n⊢ a ^ bit0 n = (a * a) ^ n",
"tactic": "rw [pow_bit0, (Commute.refl a).mul_pow]"
}
] |
[
211,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
210,
1
] |
Mathlib/Data/Finsupp/Basic.lean
|
Finsupp.subtypeDomain_sum
|
[] |
[
1109,
64
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1107,
1
] |
Mathlib/Order/Hom/Set.lean
|
StrictMono.orderIsoOfSurjective_symm_apply_self
|
[] |
[
129,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
127,
1
] |
Mathlib/MeasureTheory/Function/L1Space.lean
|
MeasureTheory.integrable_of_integrable_trim
|
[
{
"state_after": "case intro\nα : Type u_1\nβ : Type ?u.1259200\nγ : Type ?u.1259203\nδ : Type ?u.1259206\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝³ : MeasurableSpace δ\ninst✝² : NormedAddCommGroup β\ninst✝¹ : NormedAddCommGroup γ\nH : Type u_2\ninst✝ : NormedAddCommGroup H\nm0 : MeasurableSpace α\nμ' : Measure α\nf : α → H\nhm : m ≤ m0\nhf_meas_ae : AEStronglyMeasurable f (Measure.trim μ' hm)\nhf : HasFiniteIntegral f\n⊢ Integrable f",
"state_before": "α : Type u_1\nβ : Type ?u.1259200\nγ : Type ?u.1259203\nδ : Type ?u.1259206\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝³ : MeasurableSpace δ\ninst✝² : NormedAddCommGroup β\ninst✝¹ : NormedAddCommGroup γ\nH : Type u_2\ninst✝ : NormedAddCommGroup H\nm0 : MeasurableSpace α\nμ' : Measure α\nf : α → H\nhm : m ≤ m0\nhf_int : Integrable f\n⊢ Integrable f",
"tactic": "obtain ⟨hf_meas_ae, hf⟩ := hf_int"
},
{
"state_after": "case intro\nα : Type u_1\nβ : Type ?u.1259200\nγ : Type ?u.1259203\nδ : Type ?u.1259206\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝³ : MeasurableSpace δ\ninst✝² : NormedAddCommGroup β\ninst✝¹ : NormedAddCommGroup γ\nH : Type u_2\ninst✝ : NormedAddCommGroup H\nm0 : MeasurableSpace α\nμ' : Measure α\nf : α → H\nhm : m ≤ m0\nhf_meas_ae : AEStronglyMeasurable f (Measure.trim μ' hm)\nhf : HasFiniteIntegral f\n⊢ HasFiniteIntegral f",
"state_before": "case intro\nα : Type u_1\nβ : Type ?u.1259200\nγ : Type ?u.1259203\nδ : Type ?u.1259206\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝³ : MeasurableSpace δ\ninst✝² : NormedAddCommGroup β\ninst✝¹ : NormedAddCommGroup γ\nH : Type u_2\ninst✝ : NormedAddCommGroup H\nm0 : MeasurableSpace α\nμ' : Measure α\nf : α → H\nhm : m ≤ m0\nhf_meas_ae : AEStronglyMeasurable f (Measure.trim μ' hm)\nhf : HasFiniteIntegral f\n⊢ Integrable f",
"tactic": "refine' ⟨aestronglyMeasurable_of_aestronglyMeasurable_trim hm hf_meas_ae, _⟩"
},
{
"state_after": "case intro\nα : Type u_1\nβ : Type ?u.1259200\nγ : Type ?u.1259203\nδ : Type ?u.1259206\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝³ : MeasurableSpace δ\ninst✝² : NormedAddCommGroup β\ninst✝¹ : NormedAddCommGroup γ\nH : Type u_2\ninst✝ : NormedAddCommGroup H\nm0 : MeasurableSpace α\nμ' : Measure α\nf : α → H\nhm : m ≤ m0\nhf_meas_ae : AEStronglyMeasurable f (Measure.trim μ' hm)\nhf : (∫⁻ (a : α), ↑‖f a‖₊ ∂Measure.trim μ' hm) < ⊤\n⊢ (∫⁻ (a : α), ↑‖f a‖₊ ∂μ') < ⊤",
"state_before": "case intro\nα : Type u_1\nβ : Type ?u.1259200\nγ : Type ?u.1259203\nδ : Type ?u.1259206\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝³ : MeasurableSpace δ\ninst✝² : NormedAddCommGroup β\ninst✝¹ : NormedAddCommGroup γ\nH : Type u_2\ninst✝ : NormedAddCommGroup H\nm0 : MeasurableSpace α\nμ' : Measure α\nf : α → H\nhm : m ≤ m0\nhf_meas_ae : AEStronglyMeasurable f (Measure.trim μ' hm)\nhf : HasFiniteIntegral f\n⊢ HasFiniteIntegral f",
"tactic": "rw [HasFiniteIntegral] at hf ⊢"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.1259200\nγ : Type ?u.1259203\nδ : Type ?u.1259206\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝³ : MeasurableSpace δ\ninst✝² : NormedAddCommGroup β\ninst✝¹ : NormedAddCommGroup γ\nH : Type u_2\ninst✝ : NormedAddCommGroup H\nm0 : MeasurableSpace α\nμ' : Measure α\nf : α → H\nhm : m ≤ m0\nhf_meas_ae : AEStronglyMeasurable f (Measure.trim μ' hm)\nhf : (∫⁻ (a : α), ↑‖f a‖₊ ∂Measure.trim μ' hm) < ⊤\n⊢ AEMeasurable fun a => ↑‖f a‖₊",
"state_before": "case intro\nα : Type u_1\nβ : Type ?u.1259200\nγ : Type ?u.1259203\nδ : Type ?u.1259206\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝³ : MeasurableSpace δ\ninst✝² : NormedAddCommGroup β\ninst✝¹ : NormedAddCommGroup γ\nH : Type u_2\ninst✝ : NormedAddCommGroup H\nm0 : MeasurableSpace α\nμ' : Measure α\nf : α → H\nhm : m ≤ m0\nhf_meas_ae : AEStronglyMeasurable f (Measure.trim μ' hm)\nhf : (∫⁻ (a : α), ↑‖f a‖₊ ∂Measure.trim μ' hm) < ⊤\n⊢ (∫⁻ (a : α), ↑‖f a‖₊ ∂μ') < ⊤",
"tactic": "rwa [lintegral_trim_ae hm _] at hf"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.1259200\nγ : Type ?u.1259203\nδ : Type ?u.1259206\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝³ : MeasurableSpace δ\ninst✝² : NormedAddCommGroup β\ninst✝¹ : NormedAddCommGroup γ\nH : Type u_2\ninst✝ : NormedAddCommGroup H\nm0 : MeasurableSpace α\nμ' : Measure α\nf : α → H\nhm : m ≤ m0\nhf_meas_ae : AEStronglyMeasurable f (Measure.trim μ' hm)\nhf : (∫⁻ (a : α), ↑‖f a‖₊ ∂Measure.trim μ' hm) < ⊤\n⊢ AEMeasurable fun a => ↑‖f a‖₊",
"tactic": "exact AEStronglyMeasurable.ennnorm hf_meas_ae"
}
] |
[
1175,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1169,
1
] |
Mathlib/Order/Lattice.lean
|
sup_lt_iff
|
[] |
[
865,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
863,
1
] |
Mathlib/RingTheory/AdjoinRoot.lean
|
AdjoinRoot.lift_of
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\nK : Type w\ninst✝¹ : CommRing R\nf : R[X]\ninst✝ : CommRing S\ni : R →+* S\na : S\nh : eval₂ i a f = 0\nx : R\n⊢ ↑(lift i a h) (↑(of f) x) = ↑i x",
"tactic": "rw [← mk_C x, lift_mk, eval₂_C]"
}
] |
[
294,
83
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
294,
1
] |
Mathlib/Algebra/Order/Floor.lean
|
Int.ceil_intCast
|
[
{
"state_after": "no goals",
"state_before": "F : Type ?u.210900\nα : Type u_1\nβ : Type ?u.210906\ninst✝¹ : LinearOrderedRing α\ninst✝ : FloorRing α\nz✝ : ℤ\na✝ : α\nz a : ℤ\n⊢ ⌈↑z⌉ ≤ a ↔ z ≤ a",
"tactic": "rw [ceil_le, Int.cast_le]"
}
] |
[
1131,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1130,
1
] |
Mathlib/Topology/UniformSpace/AbstractCompletion.lean
|
AbstractCompletion.map_comp
|
[] |
[
238,
63
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
236,
1
] |
Mathlib/NumberTheory/ArithmeticFunction.lean
|
Nat.ArithmeticFunction.intCoe_mul
|
[
{
"state_after": "case h\nR : Type u_1\ninst✝ : Ring R\nf g : ArithmeticFunction ℤ\nn : ℕ\n⊢ ↑↑(f * g) n = ↑(↑f * ↑g) n",
"state_before": "R : Type u_1\ninst✝ : Ring R\nf g : ArithmeticFunction ℤ\n⊢ ↑(f * g) = ↑f * ↑g",
"tactic": "ext n"
},
{
"state_after": "no goals",
"state_before": "case h\nR : Type u_1\ninst✝ : Ring R\nf g : ArithmeticFunction ℤ\nn : ℕ\n⊢ ↑↑(f * g) n = ↑(↑f * ↑g) n",
"tactic": "simp"
}
] |
[
295,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
292,
1
] |
Mathlib/Algebra/Hom/Ring.lean
|
NonUnitalRingHom.coe_mulHom_injective
|
[] |
[
199,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
198,
1
] |
Mathlib/Data/List/Card.lean
|
List.card_cons_of_mem
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Sort ?u.20954\ninst✝¹ : DecidableEq α\ninst✝ : DecidableEq β\na : α\nas : List α\nh : a ∈ as\n⊢ card (a :: as) = card as",
"tactic": "simp [card, h]"
}
] |
[
82,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
81,
9
] |
Mathlib/Algebra/Associated.lean
|
Associates.mk_eq_mk_iff_associated
|
[] |
[
750,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
749,
1
] |
Std/Data/Int/Lemmas.lean
|
Int.mul_lt_mul
|
[] |
[
1190,
95
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
1188,
11
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean
|
Real.Angle.two_nsmul_toReal_eq_two_mul_sub_two_pi
|
[
{
"state_after": "θ : Angle\n⊢ toReal (2 • ↑(toReal θ)) = 2 * toReal θ - 2 * π ↔ π / 2 < toReal θ",
"state_before": "θ : Angle\n⊢ toReal (2 • θ) = 2 * toReal θ - 2 * π ↔ π / 2 < toReal θ",
"tactic": "nth_rw 1 [← coe_toReal θ]"
},
{
"state_after": "θ : Angle\n⊢ π < 2 * toReal θ ∧ 2 * toReal θ ≤ 3 * π ↔ π / 2 < toReal θ",
"state_before": "θ : Angle\n⊢ toReal (2 • ↑(toReal θ)) = 2 * toReal θ - 2 * π ↔ π / 2 < toReal θ",
"tactic": "rw [← coe_nsmul, two_nsmul, ← two_mul, toReal_coe_eq_self_sub_two_pi_iff, Set.mem_Ioc]"
},
{
"state_after": "no goals",
"state_before": "θ : Angle\n⊢ π < 2 * toReal θ ∧ 2 * toReal θ ≤ 3 * π ↔ π / 2 < toReal θ",
"tactic": "exact\n ⟨fun h => by linarith, fun h =>\n ⟨(div_lt_iff' (zero_lt_two' ℝ)).1 h, by linarith [pi_pos, toReal_le_pi θ]⟩⟩"
},
{
"state_after": "no goals",
"state_before": "θ : Angle\nh : π < 2 * toReal θ ∧ 2 * toReal θ ≤ 3 * π\n⊢ π / 2 < toReal θ",
"tactic": "linarith"
},
{
"state_after": "no goals",
"state_before": "θ : Angle\nh : π / 2 < toReal θ\n⊢ 2 * toReal θ ≤ 3 * π",
"tactic": "linarith [pi_pos, toReal_le_pi θ]"
}
] |
[
704,
82
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
698,
1
] |
Mathlib/SetTheory/Ordinal/Arithmetic.lean
|
Ordinal.bsup_eq_blsub_iff_lt_bsup
|
[
{
"state_after": "α : Type ?u.356399\nβ : Type ?u.356402\nγ : Type ?u.356405\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\no : Ordinal\nf : (a : Ordinal) → a < o → Ordinal\nh : bsup o f = blsub o f\ni : Ordinal\n⊢ ∀ (hi : i < o), f i hi < blsub o f",
"state_before": "α : Type ?u.356399\nβ : Type ?u.356402\nγ : Type ?u.356405\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\no : Ordinal\nf : (a : Ordinal) → a < o → Ordinal\nh : bsup o f = blsub o f\ni : Ordinal\n⊢ ∀ (hi : i < o), f i hi < bsup o f",
"tactic": "rw [h]"
},
{
"state_after": "no goals",
"state_before": "α : Type ?u.356399\nβ : Type ?u.356402\nγ : Type ?u.356405\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\no : Ordinal\nf : (a : Ordinal) → a < o → Ordinal\nh : bsup o f = blsub o f\ni : Ordinal\n⊢ ∀ (hi : i < o), f i hi < blsub o f",
"tactic": "apply lt_blsub"
}
] |
[
1867,
73
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1863,
1
] |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.