file_path
stringlengths 11
79
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stringlengths 2
100
| traced_tactics
list | end
list | commit
stringclasses 4
values | url
stringclasses 4
values | start
list |
|---|---|---|---|---|---|---|
Mathlib/Data/MvPolynomial/Basic.lean
|
MvPolynomial.map_surjective
|
[
{
"state_after": "case h1\nR : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_1\na a' a₁ a₂ : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\np q : MvPolynomial σ R\nf : R →+* S₁\nhf : Surjective ↑f\ni : σ →₀ ℕ\nfr : S₁\n⊢ ∃ a, ↑(map f) a = ↑(monomial i) fr\n\ncase h2\nR : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_1\na✝ a' a₁ a₂ : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\np q : MvPolynomial σ R\nf : R →+* S₁\nhf : Surjective ↑f\na b : MvPolynomial σ S₁\nha : ∃ a_1, ↑(map f) a_1 = a\nhb : ∃ a, ↑(map f) a = b\n⊢ ∃ a_1, ↑(map f) a_1 = a + b",
"state_before": "R : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_1\na a' a₁ a₂ : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\np✝ q : MvPolynomial σ R\nf : R →+* S₁\nhf : Surjective ↑f\np : MvPolynomial σ S₁\n⊢ ∃ a, ↑(map f) a = p",
"tactic": "induction' p using MvPolynomial.induction_on' with i fr a b ha hb"
},
{
"state_after": "case h1.intro\nR : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_1\na a' a₁ a₂ : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\np q : MvPolynomial σ R\nf : R →+* S₁\nhf : Surjective ↑f\ni : σ →₀ ℕ\nr : R\n⊢ ∃ a, ↑(map f) a = ↑(monomial i) (↑f r)",
"state_before": "case h1\nR : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_1\na a' a₁ a₂ : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\np q : MvPolynomial σ R\nf : R →+* S₁\nhf : Surjective ↑f\ni : σ →₀ ℕ\nfr : S₁\n⊢ ∃ a, ↑(map f) a = ↑(monomial i) fr",
"tactic": "obtain ⟨r, rfl⟩ := hf fr"
},
{
"state_after": "no goals",
"state_before": "case h1.intro\nR : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_1\na a' a₁ a₂ : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\np q : MvPolynomial σ R\nf : R →+* S₁\nhf : Surjective ↑f\ni : σ →₀ ℕ\nr : R\n⊢ ∃ a, ↑(map f) a = ↑(monomial i) (↑f r)",
"tactic": "exact ⟨monomial i r, map_monomial _ _ _⟩"
},
{
"state_after": "case h2.intro\nR : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_1\na✝ a' a₁ a₂ : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\np q : MvPolynomial σ R\nf : R →+* S₁\nhf : Surjective ↑f\nb : MvPolynomial σ S₁\nhb : ∃ a, ↑(map f) a = b\na : MvPolynomial σ R\n⊢ ∃ a_1, ↑(map f) a_1 = ↑(map f) a + b",
"state_before": "case h2\nR : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_1\na✝ a' a₁ a₂ : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\np q : MvPolynomial σ R\nf : R →+* S₁\nhf : Surjective ↑f\na b : MvPolynomial σ S₁\nha : ∃ a_1, ↑(map f) a_1 = a\nhb : ∃ a, ↑(map f) a = b\n⊢ ∃ a_1, ↑(map f) a_1 = a + b",
"tactic": "obtain ⟨a, rfl⟩ := ha"
},
{
"state_after": "case h2.intro.intro\nR : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_1\na✝ a' a₁ a₂ : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\np q : MvPolynomial σ R\nf : R →+* S₁\nhf : Surjective ↑f\na b : MvPolynomial σ R\n⊢ ∃ a_1, ↑(map f) a_1 = ↑(map f) a + ↑(map f) b",
"state_before": "case h2.intro\nR : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_1\na✝ a' a₁ a₂ : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\np q : MvPolynomial σ R\nf : R →+* S₁\nhf : Surjective ↑f\nb : MvPolynomial σ S₁\nhb : ∃ a, ↑(map f) a = b\na : MvPolynomial σ R\n⊢ ∃ a_1, ↑(map f) a_1 = ↑(map f) a + b",
"tactic": "obtain ⟨b, rfl⟩ := hb"
},
{
"state_after": "no goals",
"state_before": "case h2.intro.intro\nR : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_1\na✝ a' a₁ a₂ : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\np q : MvPolynomial σ R\nf : R →+* S₁\nhf : Surjective ↑f\na b : MvPolynomial σ R\n⊢ ∃ a_1, ↑(map f) a_1 = ↑(map f) a + ↑(map f) b",
"tactic": "exact ⟨a + b, RingHom.map_add _ _ _⟩"
}
] |
[
1304,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1297,
1
] |
Mathlib/Algebra/Order/AbsoluteValue.lean
|
AbsoluteValue.pos_iff
|
[] |
[
132,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
131,
11
] |
Mathlib/Order/Filter/Basic.lean
|
Filter.eventually_and
|
[] |
[
1143,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1141,
1
] |
Mathlib/Topology/VectorBundle/Basic.lean
|
Trivialization.comp_continuousLinearEquivAt_eq_coord_change
|
[
{
"state_after": "case h.h\nR : Type u_4\nB : Type u_1\nF : Type u_2\nE : B → Type u_3\ninst✝¹⁰ : NontriviallyNormedField R\ninst✝⁹ : (x : B) → AddCommMonoid (E x)\ninst✝⁸ : (x : B) → Module R (E x)\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedSpace R F\ninst✝⁵ : TopologicalSpace B\ninst✝⁴ : TopologicalSpace (TotalSpace E)\ninst✝³ : (x : B) → TopologicalSpace (E x)\ninst✝² : FiberBundle F E\ne e' : Trivialization F TotalSpace.proj\ninst✝¹ : Trivialization.IsLinear R e\ninst✝ : Trivialization.IsLinear R e'\nb : B\nhb : b ∈ e.baseSet ∩ e'.baseSet\nv : F\n⊢ ↑(ContinuousLinearEquiv.trans (ContinuousLinearEquiv.symm (continuousLinearEquivAt R e b (_ : b ∈ e.baseSet)))\n (continuousLinearEquivAt R e' b (_ : b ∈ e'.baseSet)))\n v =\n ↑(coordChangeL R e e' b) v",
"state_before": "R : Type u_4\nB : Type u_1\nF : Type u_2\nE : B → Type u_3\ninst✝¹⁰ : NontriviallyNormedField R\ninst✝⁹ : (x : B) → AddCommMonoid (E x)\ninst✝⁸ : (x : B) → Module R (E x)\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedSpace R F\ninst✝⁵ : TopologicalSpace B\ninst✝⁴ : TopologicalSpace (TotalSpace E)\ninst✝³ : (x : B) → TopologicalSpace (E x)\ninst✝² : FiberBundle F E\ne e' : Trivialization F TotalSpace.proj\ninst✝¹ : Trivialization.IsLinear R e\ninst✝ : Trivialization.IsLinear R e'\nb : B\nhb : b ∈ e.baseSet ∩ e'.baseSet\n⊢ ContinuousLinearEquiv.trans (ContinuousLinearEquiv.symm (continuousLinearEquivAt R e b (_ : b ∈ e.baseSet)))\n (continuousLinearEquivAt R e' b (_ : b ∈ e'.baseSet)) =\n coordChangeL R e e' b",
"tactic": "ext v"
},
{
"state_after": "case h.h\nR : Type u_4\nB : Type u_1\nF : Type u_2\nE : B → Type u_3\ninst✝¹⁰ : NontriviallyNormedField R\ninst✝⁹ : (x : B) → AddCommMonoid (E x)\ninst✝⁸ : (x : B) → Module R (E x)\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedSpace R F\ninst✝⁵ : TopologicalSpace B\ninst✝⁴ : TopologicalSpace (TotalSpace E)\ninst✝³ : (x : B) → TopologicalSpace (E x)\ninst✝² : FiberBundle F E\ne e' : Trivialization F TotalSpace.proj\ninst✝¹ : Trivialization.IsLinear R e\ninst✝ : Trivialization.IsLinear R e'\nb : B\nhb : b ∈ e.baseSet ∩ e'.baseSet\nv : F\n⊢ ↑(ContinuousLinearEquiv.trans (ContinuousLinearEquiv.symm (continuousLinearEquivAt R e b (_ : b ∈ e.baseSet)))\n (continuousLinearEquivAt R e' b (_ : b ∈ e'.baseSet)))\n v =\n (↑e' (totalSpaceMk b (Trivialization.symm e b v))).snd",
"state_before": "case h.h\nR : Type u_4\nB : Type u_1\nF : Type u_2\nE : B → Type u_3\ninst✝¹⁰ : NontriviallyNormedField R\ninst✝⁹ : (x : B) → AddCommMonoid (E x)\ninst✝⁸ : (x : B) → Module R (E x)\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedSpace R F\ninst✝⁵ : TopologicalSpace B\ninst✝⁴ : TopologicalSpace (TotalSpace E)\ninst✝³ : (x : B) → TopologicalSpace (E x)\ninst✝² : FiberBundle F E\ne e' : Trivialization F TotalSpace.proj\ninst✝¹ : Trivialization.IsLinear R e\ninst✝ : Trivialization.IsLinear R e'\nb : B\nhb : b ∈ e.baseSet ∩ e'.baseSet\nv : F\n⊢ ↑(ContinuousLinearEquiv.trans (ContinuousLinearEquiv.symm (continuousLinearEquivAt R e b (_ : b ∈ e.baseSet)))\n (continuousLinearEquivAt R e' b (_ : b ∈ e'.baseSet)))\n v =\n ↑(coordChangeL R e e' b) v",
"tactic": "rw [coordChangeL_apply e e' hb]"
},
{
"state_after": "no goals",
"state_before": "case h.h\nR : Type u_4\nB : Type u_1\nF : Type u_2\nE : B → Type u_3\ninst✝¹⁰ : NontriviallyNormedField R\ninst✝⁹ : (x : B) → AddCommMonoid (E x)\ninst✝⁸ : (x : B) → Module R (E x)\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedSpace R F\ninst✝⁵ : TopologicalSpace B\ninst✝⁴ : TopologicalSpace (TotalSpace E)\ninst✝³ : (x : B) → TopologicalSpace (E x)\ninst✝² : FiberBundle F E\ne e' : Trivialization F TotalSpace.proj\ninst✝¹ : Trivialization.IsLinear R e\ninst✝ : Trivialization.IsLinear R e'\nb : B\nhb : b ∈ e.baseSet ∩ e'.baseSet\nv : F\n⊢ ↑(ContinuousLinearEquiv.trans (ContinuousLinearEquiv.symm (continuousLinearEquivAt R e b (_ : b ∈ e.baseSet)))\n (continuousLinearEquivAt R e' b (_ : b ∈ e'.baseSet)))\n v =\n (↑e' (totalSpaceMk b (Trivialization.symm e b v))).snd",
"tactic": "rfl"
}
] |
[
542,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
536,
1
] |
Mathlib/Order/OrderIsoNat.lean
|
RelEmbedding.not_wellFounded_of_decreasing_seq
|
[
{
"state_after": "α : Type u_1\nr : α → α → Prop\ninst✝ : IsStrictOrder α r\nf : (fun x x_1 => x > x_1) ↪r r\n⊢ Nonempty ((fun x x_1 => x > x_1) ↪r r)",
"state_before": "α : Type u_1\nr : α → α → Prop\ninst✝ : IsStrictOrder α r\nf : (fun x x_1 => x > x_1) ↪r r\n⊢ ¬WellFounded r",
"tactic": "rw [wellFounded_iff_no_descending_seq, not_isEmpty_iff]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nr : α → α → Prop\ninst✝ : IsStrictOrder α r\nf : (fun x x_1 => x > x_1) ↪r r\n⊢ Nonempty ((fun x x_1 => x > x_1) ↪r r)",
"tactic": "exact ⟨f⟩"
}
] |
[
103,
12
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
101,
1
] |
Mathlib/Order/ConditionallyCompleteLattice/Basic.lean
|
csInf_Ico
|
[] |
[
729,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
728,
1
] |
Mathlib/Topology/Separation.lean
|
QuotientMap.of_surjective_continuous
|
[] |
[
1344,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1342,
1
] |
Mathlib/Data/Complex/Exponential.lean
|
Complex.sin_mul_I
|
[
{
"state_after": "x y : ℂ\nh : I * sin (x * I) = -sinh x\n⊢ sin (x * I) = sinh x * I",
"state_before": "x y : ℂ\n⊢ sin (x * I) = sinh x * I",
"tactic": "have h : I * sin (x * I) = -sinh x := by\n rw [mul_comm, ← sinh_mul_I]\n ring_nf\n simp"
},
{
"state_after": "x y : ℂ\nh : I * sin (x * I) = -sinh x\n⊢ sin (x * I) = -(I * sin (x * I)) * I",
"state_before": "x y : ℂ\nh : I * sin (x * I) = -sinh x\n⊢ sin (x * I) = sinh x * I",
"tactic": "rw [← neg_neg (sinh x), ← h]"
},
{
"state_after": "no goals",
"state_before": "x y : ℂ\nh : I * sin (x * I) = -sinh x\n⊢ sin (x * I) = -(I * sin (x * I)) * I",
"tactic": "ext <;> simp"
},
{
"state_after": "x y : ℂ\n⊢ sinh (x * I * I) = -sinh x",
"state_before": "x y : ℂ\n⊢ I * sin (x * I) = -sinh x",
"tactic": "rw [mul_comm, ← sinh_mul_I]"
},
{
"state_after": "x y : ℂ\n⊢ sinh (x * I ^ 2) = -sinh x",
"state_before": "x y : ℂ\n⊢ sinh (x * I * I) = -sinh x",
"tactic": "ring_nf"
},
{
"state_after": "no goals",
"state_before": "x y : ℂ\n⊢ sinh (x * I ^ 2) = -sinh x",
"tactic": "simp"
}
] |
[
838,
15
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
832,
1
] |
Mathlib/Algebra/CubicDiscriminant.lean
|
Cubic.natDegree_of_a_ne_zero'
|
[] |
[
391,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
390,
1
] |
Mathlib/Analysis/NormedSpace/AffineIsometry.lean
|
AffineIsometryEquiv.pointReflection_involutive
|
[] |
[
795,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
794,
1
] |
Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean
|
Polynomial.eq_cyclotomic_iff
|
[
{
"state_after": "R : Type u_1\ninst✝ : CommRing R\nn : ℕ\nhpos : 0 < n\nP : R[X]\n✝ : Nontrivial R\n⊢ P = cyclotomic n R ↔ P * ∏ i in Nat.properDivisors n, cyclotomic i R = X ^ n - 1",
"state_before": "R : Type u_1\ninst✝ : CommRing R\nn : ℕ\nhpos : 0 < n\nP : R[X]\n⊢ P = cyclotomic n R ↔ P * ∏ i in Nat.properDivisors n, cyclotomic i R = X ^ n - 1",
"tactic": "nontriviality R"
},
{
"state_after": "case refine'_1\nR : Type u_1\ninst✝ : CommRing R\nn : ℕ\nhpos : 0 < n\nP : R[X]\n✝ : Nontrivial R\nhcycl : P = cyclotomic n R\n⊢ P * ∏ i in Nat.properDivisors n, cyclotomic i R = X ^ n - 1\n\ncase refine'_2\nR : Type u_1\ninst✝ : CommRing R\nn : ℕ\nhpos : 0 < n\nP : R[X]\n✝ : Nontrivial R\nhP : P * ∏ i in Nat.properDivisors n, cyclotomic i R = X ^ n - 1\n⊢ P = cyclotomic n R",
"state_before": "R : Type u_1\ninst✝ : CommRing R\nn : ℕ\nhpos : 0 < n\nP : R[X]\n✝ : Nontrivial R\n⊢ P = cyclotomic n R ↔ P * ∏ i in Nat.properDivisors n, cyclotomic i R = X ^ n - 1",
"tactic": "refine' ⟨fun hcycl => _, fun hP => _⟩"
},
{
"state_after": "no goals",
"state_before": "case refine'_1\nR : Type u_1\ninst✝ : CommRing R\nn : ℕ\nhpos : 0 < n\nP : R[X]\n✝ : Nontrivial R\nhcycl : P = cyclotomic n R\n⊢ P * ∏ i in Nat.properDivisors n, cyclotomic i R = X ^ n - 1",
"tactic": "rw [hcycl, ← prod_cyclotomic_eq_X_pow_sub_one hpos R, ← Nat.cons_self_properDivisors hpos.ne',\n Finset.prod_cons]"
},
{
"state_after": "case refine'_2\nR : Type u_1\ninst✝ : CommRing R\nn : ℕ\nhpos : 0 < n\nP : R[X]\n✝ : Nontrivial R\nhP : P * ∏ i in Nat.properDivisors n, cyclotomic i R = X ^ n - 1\nprod_monic : Monic (∏ i in Nat.properDivisors n, cyclotomic i R)\n⊢ P = cyclotomic n R",
"state_before": "case refine'_2\nR : Type u_1\ninst✝ : CommRing R\nn : ℕ\nhpos : 0 < n\nP : R[X]\n✝ : Nontrivial R\nhP : P * ∏ i in Nat.properDivisors n, cyclotomic i R = X ^ n - 1\n⊢ P = cyclotomic n R",
"tactic": "have prod_monic : (∏ i in Nat.properDivisors n, cyclotomic i R).Monic := by\n apply monic_prod_of_monic\n intro i _\n exact cyclotomic.monic i R"
},
{
"state_after": "R : Type u_1\ninst✝ : CommRing R\nn : ℕ\nhpos : 0 < n\nP : R[X]\n✝ : Nontrivial R\nhP : P * ∏ i in Nat.properDivisors n, cyclotomic i R = X ^ n - 1\nprod_monic : Monic (∏ i in Nat.properDivisors n, cyclotomic i R)\n⊢ 0 + (∏ i in Nat.properDivisors n, cyclotomic i R) * P = X ^ n - 1 ∧\n degree 0 < degree (∏ i in Nat.properDivisors n, cyclotomic i R)",
"state_before": "case refine'_2\nR : Type u_1\ninst✝ : CommRing R\nn : ℕ\nhpos : 0 < n\nP : R[X]\n✝ : Nontrivial R\nhP : P * ∏ i in Nat.properDivisors n, cyclotomic i R = X ^ n - 1\nprod_monic : Monic (∏ i in Nat.properDivisors n, cyclotomic i R)\n⊢ P = cyclotomic n R",
"tactic": "rw [@cyclotomic_eq_X_pow_sub_one_div R _ _ hpos, (div_modByMonic_unique P 0 prod_monic _).1]"
},
{
"state_after": "R : Type u_1\ninst✝ : CommRing R\nn : ℕ\nhpos : 0 < n\nP : R[X]\n✝ : Nontrivial R\nhP : P * ∏ i in Nat.properDivisors n, cyclotomic i R = X ^ n - 1\nprod_monic : Monic (∏ i in Nat.properDivisors n, cyclotomic i R)\n⊢ degree 0 < degree (∏ i in Nat.properDivisors n, cyclotomic i R)",
"state_before": "R : Type u_1\ninst✝ : CommRing R\nn : ℕ\nhpos : 0 < n\nP : R[X]\n✝ : Nontrivial R\nhP : P * ∏ i in Nat.properDivisors n, cyclotomic i R = X ^ n - 1\nprod_monic : Monic (∏ i in Nat.properDivisors n, cyclotomic i R)\n⊢ 0 + (∏ i in Nat.properDivisors n, cyclotomic i R) * P = X ^ n - 1 ∧\n degree 0 < degree (∏ i in Nat.properDivisors n, cyclotomic i R)",
"tactic": "refine' ⟨by rwa [zero_add, mul_comm], _⟩"
},
{
"state_after": "R : Type u_1\ninst✝ : CommRing R\nn : ℕ\nhpos : 0 < n\nP : R[X]\n✝ : Nontrivial R\nhP : P * ∏ i in Nat.properDivisors n, cyclotomic i R = X ^ n - 1\nprod_monic : Monic (∏ i in Nat.properDivisors n, cyclotomic i R)\n⊢ degree (∏ i in Nat.properDivisors n, cyclotomic i R) ≠ ⊥",
"state_before": "R : Type u_1\ninst✝ : CommRing R\nn : ℕ\nhpos : 0 < n\nP : R[X]\n✝ : Nontrivial R\nhP : P * ∏ i in Nat.properDivisors n, cyclotomic i R = X ^ n - 1\nprod_monic : Monic (∏ i in Nat.properDivisors n, cyclotomic i R)\n⊢ degree 0 < degree (∏ i in Nat.properDivisors n, cyclotomic i R)",
"tactic": "rw [degree_zero, bot_lt_iff_ne_bot]"
},
{
"state_after": "R : Type u_1\ninst✝ : CommRing R\nn : ℕ\nhpos : 0 < n\nP : R[X]\n✝ : Nontrivial R\nhP : P * ∏ i in Nat.properDivisors n, cyclotomic i R = X ^ n - 1\nprod_monic : Monic (∏ i in Nat.properDivisors n, cyclotomic i R)\nh : degree (∏ i in Nat.properDivisors n, cyclotomic i R) = ⊥\n⊢ False",
"state_before": "R : Type u_1\ninst✝ : CommRing R\nn : ℕ\nhpos : 0 < n\nP : R[X]\n✝ : Nontrivial R\nhP : P * ∏ i in Nat.properDivisors n, cyclotomic i R = X ^ n - 1\nprod_monic : Monic (∏ i in Nat.properDivisors n, cyclotomic i R)\n⊢ degree (∏ i in Nat.properDivisors n, cyclotomic i R) ≠ ⊥",
"tactic": "intro h"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝ : CommRing R\nn : ℕ\nhpos : 0 < n\nP : R[X]\n✝ : Nontrivial R\nhP : P * ∏ i in Nat.properDivisors n, cyclotomic i R = X ^ n - 1\nprod_monic : Monic (∏ i in Nat.properDivisors n, cyclotomic i R)\nh : degree (∏ i in Nat.properDivisors n, cyclotomic i R) = ⊥\n⊢ False",
"tactic": "exact Monic.ne_zero prod_monic (degree_eq_bot.1 h)"
},
{
"state_after": "case hs\nR : Type u_1\ninst✝ : CommRing R\nn : ℕ\nhpos : 0 < n\nP : R[X]\n✝ : Nontrivial R\nhP : P * ∏ i in Nat.properDivisors n, cyclotomic i R = X ^ n - 1\n⊢ ∀ (i : ℕ), i ∈ Nat.properDivisors n → Monic (cyclotomic i R)",
"state_before": "R : Type u_1\ninst✝ : CommRing R\nn : ℕ\nhpos : 0 < n\nP : R[X]\n✝ : Nontrivial R\nhP : P * ∏ i in Nat.properDivisors n, cyclotomic i R = X ^ n - 1\n⊢ Monic (∏ i in Nat.properDivisors n, cyclotomic i R)",
"tactic": "apply monic_prod_of_monic"
},
{
"state_after": "case hs\nR : Type u_1\ninst✝ : CommRing R\nn : ℕ\nhpos : 0 < n\nP : R[X]\n✝ : Nontrivial R\nhP : P * ∏ i in Nat.properDivisors n, cyclotomic i R = X ^ n - 1\ni : ℕ\na✝ : i ∈ Nat.properDivisors n\n⊢ Monic (cyclotomic i R)",
"state_before": "case hs\nR : Type u_1\ninst✝ : CommRing R\nn : ℕ\nhpos : 0 < n\nP : R[X]\n✝ : Nontrivial R\nhP : P * ∏ i in Nat.properDivisors n, cyclotomic i R = X ^ n - 1\n⊢ ∀ (i : ℕ), i ∈ Nat.properDivisors n → Monic (cyclotomic i R)",
"tactic": "intro i _"
},
{
"state_after": "no goals",
"state_before": "case hs\nR : Type u_1\ninst✝ : CommRing R\nn : ℕ\nhpos : 0 < n\nP : R[X]\n✝ : Nontrivial R\nhP : P * ∏ i in Nat.properDivisors n, cyclotomic i R = X ^ n - 1\ni : ℕ\na✝ : i ∈ Nat.properDivisors n\n⊢ Monic (cyclotomic i R)",
"tactic": "exact cyclotomic.monic i R"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝ : CommRing R\nn : ℕ\nhpos : 0 < n\nP : R[X]\n✝ : Nontrivial R\nhP : P * ∏ i in Nat.properDivisors n, cyclotomic i R = X ^ n - 1\nprod_monic : Monic (∏ i in Nat.properDivisors n, cyclotomic i R)\n⊢ 0 + (∏ i in Nat.properDivisors n, cyclotomic i R) * P = X ^ n - 1",
"tactic": "rwa [zero_add, mul_comm]"
}
] |
[
556,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
541,
1
] |
Mathlib/Data/PNat/Prime.lean
|
PNat.Coprime.gcd_mul_right_cancel
|
[
{
"state_after": "m n k : ℕ+\n⊢ Coprime k n → gcd (k * m) n = gcd m n",
"state_before": "m n k : ℕ+\n⊢ Coprime k n → gcd (m * k) n = gcd m n",
"tactic": "rw [mul_comm]"
},
{
"state_after": "no goals",
"state_before": "m n k : ℕ+\n⊢ Coprime k n → gcd (k * m) n = gcd m n",
"tactic": "apply Coprime.gcd_mul_left_cancel"
}
] |
[
214,
54
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
213,
1
] |
Mathlib/Data/List/Cycle.lean
|
List.prev_cons_cons_eq'
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : DecidableEq α\nl : List α\nx y z : α\nh : x ∈ y :: z :: l\nhx : x = y\n⊢ prev (y :: z :: l) x h = getLast (z :: l) (_ : z :: l ≠ [])",
"tactic": "rw [prev, dif_pos hx]"
}
] |
[
220,
92
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
219,
1
] |
Mathlib/Algebra/Order/ToIntervalMod.lean
|
toIocMod_toIcoMod
|
[] |
[
746,
75
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
745,
1
] |
Mathlib/Topology/Separation.lean
|
compl_singleton_mem_nhdsSet_iff
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\ninst✝¹ : TopologicalSpace α\ninst✝ : T1Space α\nx : α\ns : Set α\n⊢ {x}ᶜ ∈ 𝓝ˢ s ↔ ¬x ∈ s",
"tactic": "rw [isOpen_compl_singleton.mem_nhdsSet, subset_compl_singleton_iff]"
}
] |
[
677,
70
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
676,
1
] |
Mathlib/Data/Set/Ncard.lean
|
Set.ncard_le_one_iff_eq
|
[
{
"state_after": "case inl\nα : Type u_1\nβ : Type ?u.153618\nt : Set α\na b x y : α\nf : α → β\nhs : autoParam (Set.Finite ∅) _auto✝\n⊢ ncard ∅ ≤ 1 ↔ ∅ = ∅ ∨ ∃ a, ∅ = {a}\n\ncase inr.intro\nα : Type u_1\nβ : Type ?u.153618\ns t : Set α\na b x✝ y : α\nf : α → β\nhs : autoParam (Set.Finite s) _auto✝\nx : α\nhx : x ∈ s\n⊢ ncard s ≤ 1 ↔ s = ∅ ∨ ∃ a, s = {a}",
"state_before": "α : Type u_1\nβ : Type ?u.153618\ns t : Set α\na b x y : α\nf : α → β\nhs : autoParam (Set.Finite s) _auto✝\n⊢ ncard s ≤ 1 ↔ s = ∅ ∨ ∃ a, s = {a}",
"tactic": "obtain rfl | ⟨x, hx⟩ := s.eq_empty_or_nonempty"
},
{
"state_after": "case inr.intro\nα : Type u_1\nβ : Type ?u.153618\ns t : Set α\na b x✝ y : α\nf : α → β\nhs : autoParam (Set.Finite s) _auto✝\nx : α\nhx : x ∈ s\n⊢ (∀ {a b : α}, a ∈ s → b ∈ s → a = b) ↔ s = ∅ ∨ ∃ a, s = {a}",
"state_before": "case inr.intro\nα : Type u_1\nβ : Type ?u.153618\ns t : Set α\na b x✝ y : α\nf : α → β\nhs : autoParam (Set.Finite s) _auto✝\nx : α\nhx : x ∈ s\n⊢ ncard s ≤ 1 ↔ s = ∅ ∨ ∃ a, s = {a}",
"tactic": "rw [ncard_le_one_iff hs]"
},
{
"state_after": "case inr.intro\nα : Type u_1\nβ : Type ?u.153618\ns t : Set α\na b x✝ y : α\nf : α → β\nhs : autoParam (Set.Finite s) _auto✝\nx : α\nhx : x ∈ s\n⊢ (s = ∅ ∨ ∃ a, s = {a}) → ∀ {a b : α}, a ∈ s → b ∈ s → a = b",
"state_before": "case inr.intro\nα : Type u_1\nβ : Type ?u.153618\ns t : Set α\na b x✝ y : α\nf : α → β\nhs : autoParam (Set.Finite s) _auto✝\nx : α\nhx : x ∈ s\n⊢ (∀ {a b : α}, a ∈ s → b ∈ s → a = b) ↔ s = ∅ ∨ ∃ a, s = {a}",
"tactic": "refine' ⟨fun h ↦ Or.inr ⟨x, (singleton_subset_iff.mpr hx).antisymm' fun y hy ↦ h hy hx⟩, _⟩"
},
{
"state_after": "case inr.intro.inl\nα : Type u_1\nβ : Type ?u.153618\nt : Set α\na b x✝ y : α\nf : α → β\nx : α\nhs : autoParam (Set.Finite ∅) _auto✝\nhx : x ∈ ∅\n⊢ ∀ {a b : α}, a ∈ ∅ → b ∈ ∅ → a = b\n\ncase inr.intro.inr.intro\nα : Type u_1\nβ : Type ?u.153618\nt : Set α\na✝ b x✝ y : α\nf : α → β\nx a : α\nhs : autoParam (Set.Finite {a}) _auto✝\nhx : x ∈ {a}\n⊢ ∀ {a_1 b : α}, a_1 ∈ {a} → b ∈ {a} → a_1 = b",
"state_before": "case inr.intro\nα : Type u_1\nβ : Type ?u.153618\ns t : Set α\na b x✝ y : α\nf : α → β\nhs : autoParam (Set.Finite s) _auto✝\nx : α\nhx : x ∈ s\n⊢ (s = ∅ ∨ ∃ a, s = {a}) → ∀ {a b : α}, a ∈ s → b ∈ s → a = b",
"tactic": "rintro (rfl | ⟨a, rfl⟩)"
},
{
"state_after": "case inr.intro.inr.intro\nα : Type u_1\nβ : Type ?u.153618\nt : Set α\na✝ b x✝ y : α\nf : α → β\nx a : α\nhs : autoParam (Set.Finite {a}) _auto✝\nhx : x = a\n⊢ ∀ {a_1 b : α}, a_1 = a → b = a → a_1 = b",
"state_before": "case inr.intro.inr.intro\nα : Type u_1\nβ : Type ?u.153618\nt : Set α\na✝ b x✝ y : α\nf : α → β\nx a : α\nhs : autoParam (Set.Finite {a}) _auto✝\nhx : x ∈ {a}\n⊢ ∀ {a_1 b : α}, a_1 ∈ {a} → b ∈ {a} → a_1 = b",
"tactic": "simp_rw [mem_singleton_iff] at hx⊢"
},
{
"state_after": "case inr.intro.inr.intro\nα : Type u_1\nβ : Type ?u.153618\nt : Set α\na b x✝ y : α\nf : α → β\nx : α\nhs : autoParam (Set.Finite {x}) _auto✝\n⊢ ∀ {a b : α}, a = x → b = x → a = b",
"state_before": "case inr.intro.inr.intro\nα : Type u_1\nβ : Type ?u.153618\nt : Set α\na✝ b x✝ y : α\nf : α → β\nx a : α\nhs : autoParam (Set.Finite {a}) _auto✝\nhx : x = a\n⊢ ∀ {a_1 b : α}, a_1 = a → b = a → a_1 = b",
"tactic": "subst hx"
},
{
"state_after": "no goals",
"state_before": "case inr.intro.inr.intro\nα : Type u_1\nβ : Type ?u.153618\nt : Set α\na b x✝ y : α\nf : α → β\nx : α\nhs : autoParam (Set.Finite {x}) _auto✝\n⊢ ∀ {a b : α}, a = x → b = x → a = b",
"tactic": "simp only [forall_eq_apply_imp_iff', imp_self, implies_true]"
},
{
"state_after": "no goals",
"state_before": "case inl\nα : Type u_1\nβ : Type ?u.153618\nt : Set α\na b x y : α\nf : α → β\nhs : autoParam (Set.Finite ∅) _auto✝\n⊢ ncard ∅ ≤ 1 ↔ ∅ = ∅ ∨ ∃ a, ∅ = {a}",
"tactic": "exact iff_of_true (by simp) (Or.inl rfl)"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.153618\nt : Set α\na b x y : α\nf : α → β\nhs : autoParam (Set.Finite ∅) _auto✝\n⊢ ncard ∅ ≤ 1",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "case inr.intro.inl\nα : Type u_1\nβ : Type ?u.153618\nt : Set α\na b x✝ y : α\nf : α → β\nx : α\nhs : autoParam (Set.Finite ∅) _auto✝\nhx : x ∈ ∅\n⊢ ∀ {a b : α}, a ∈ ∅ → b ∈ ∅ → a = b",
"tactic": "exact (not_mem_empty _ hx).elim"
}
] |
[
681,
63
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
672,
1
] |
Mathlib/RingTheory/UniqueFactorizationDomain.lean
|
Associates.count_ne_zero_iff_dvd
|
[
{
"state_after": "α : Type u_1\ninst✝² : CancelCommMonoidWithZero α\ndec_irr : (p : Associates α) → Decidable (Irreducible p)\ninst✝¹ : UniqueFactorizationMonoid α\ndec : DecidableEq α\ndec' : DecidableEq (Associates α)\na p : α\nha0 : a ≠ 0\nhp : Irreducible p\ninst✝ : Nontrivial α\n⊢ count (Associates.mk p) (factors (Associates.mk a)) ≠ 0 ↔ p ∣ a",
"state_before": "α : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ndec_irr : (p : Associates α) → Decidable (Irreducible p)\ninst✝ : UniqueFactorizationMonoid α\ndec : DecidableEq α\ndec' : DecidableEq (Associates α)\na p : α\nha0 : a ≠ 0\nhp : Irreducible p\n⊢ count (Associates.mk p) (factors (Associates.mk a)) ≠ 0 ↔ p ∣ a",
"tactic": "nontriviality α"
},
{
"state_after": "α : Type u_1\ninst✝² : CancelCommMonoidWithZero α\ndec_irr : (p : Associates α) → Decidable (Irreducible p)\ninst✝¹ : UniqueFactorizationMonoid α\ndec : DecidableEq α\ndec' : DecidableEq (Associates α)\na p : α\nha0 : a ≠ 0\nhp : Irreducible p\ninst✝ : Nontrivial α\n⊢ count (Associates.mk p) (factors (Associates.mk a)) ≠ 0 ↔ Associates.mk p ≤ Associates.mk a",
"state_before": "α : Type u_1\ninst✝² : CancelCommMonoidWithZero α\ndec_irr : (p : Associates α) → Decidable (Irreducible p)\ninst✝¹ : UniqueFactorizationMonoid α\ndec : DecidableEq α\ndec' : DecidableEq (Associates α)\na p : α\nha0 : a ≠ 0\nhp : Irreducible p\ninst✝ : Nontrivial α\n⊢ count (Associates.mk p) (factors (Associates.mk a)) ≠ 0 ↔ p ∣ a",
"tactic": "rw [← Associates.mk_le_mk_iff_dvd_iff]"
},
{
"state_after": "α : Type u_1\ninst✝² : CancelCommMonoidWithZero α\ndec_irr : (p : Associates α) → Decidable (Irreducible p)\ninst✝¹ : UniqueFactorizationMonoid α\ndec : DecidableEq α\ndec' : DecidableEq (Associates α)\na p : α\nha0 : a ≠ 0\nhp : Irreducible p\ninst✝ : Nontrivial α\nh : Associates.mk p ≤ Associates.mk a\n⊢ count (Associates.mk p) (factors (Associates.mk a)) ≠ 0",
"state_before": "α : Type u_1\ninst✝² : CancelCommMonoidWithZero α\ndec_irr : (p : Associates α) → Decidable (Irreducible p)\ninst✝¹ : UniqueFactorizationMonoid α\ndec : DecidableEq α\ndec' : DecidableEq (Associates α)\na p : α\nha0 : a ≠ 0\nhp : Irreducible p\ninst✝ : Nontrivial α\n⊢ count (Associates.mk p) (factors (Associates.mk a)) ≠ 0 ↔ Associates.mk p ≤ Associates.mk a",
"tactic": "refine'\n ⟨fun h =>\n Associates.le_of_count_ne_zero (Associates.mk_ne_zero.mpr ha0)\n ((Associates.irreducible_mk p).mpr hp) h,\n fun h => _⟩"
},
{
"state_after": "α : Type u_1\ninst✝² : CancelCommMonoidWithZero α\ndec_irr : (p : Associates α) → Decidable (Irreducible p)\ninst✝¹ : UniqueFactorizationMonoid α\ndec : DecidableEq α\ndec' : DecidableEq (Associates α)\na p : α\nha0 : a ≠ 0\nhp : Irreducible p\ninst✝ : Nontrivial α\nh : 1 ≤ count (Associates.mk p) (factors (Associates.mk a))\n⊢ count (Associates.mk p) (factors (Associates.mk a)) ≠ 0",
"state_before": "α : Type u_1\ninst✝² : CancelCommMonoidWithZero α\ndec_irr : (p : Associates α) → Decidable (Irreducible p)\ninst✝¹ : UniqueFactorizationMonoid α\ndec : DecidableEq α\ndec' : DecidableEq (Associates α)\na p : α\nha0 : a ≠ 0\nhp : Irreducible p\ninst✝ : Nontrivial α\nh : Associates.mk p ≤ Associates.mk a\n⊢ count (Associates.mk p) (factors (Associates.mk a)) ≠ 0",
"tactic": "rw [← pow_one (Associates.mk p),\n Associates.prime_pow_dvd_iff_le (Associates.mk_ne_zero.mpr ha0)\n ((Associates.irreducible_mk p).mpr hp)] at h"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝² : CancelCommMonoidWithZero α\ndec_irr : (p : Associates α) → Decidable (Irreducible p)\ninst✝¹ : UniqueFactorizationMonoid α\ndec : DecidableEq α\ndec' : DecidableEq (Associates α)\na p : α\nha0 : a ≠ 0\nhp : Irreducible p\ninst✝ : Nontrivial α\nh : 1 ≤ count (Associates.mk p) (factors (Associates.mk a))\n⊢ count (Associates.mk p) (factors (Associates.mk a)) ≠ 0",
"tactic": "exact (zero_lt_one.trans_le h).ne'"
}
] |
[
1715,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1703,
1
] |
Mathlib/Data/Polynomial/Degree/Definitions.lean
|
Polynomial.coeff_mul_degree_add_degree
|
[
{
"state_after": "case refine'_1\nR : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝ : Semiring R\np✝ q✝ : R[X]\nι : Type ?u.653636\np q : R[X]\n⊢ ∀ (b : ℕ × ℕ),\n b ∈ Nat.antidiagonal (natDegree p + natDegree q) →\n b ≠ (natDegree p, natDegree q) → coeff p b.fst * coeff q b.snd = 0\n\ncase refine'_2\nR : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝ : Semiring R\np✝ q✝ : R[X]\nι : Type ?u.653636\np q : R[X]\n⊢ ¬(natDegree p, natDegree q) ∈ Nat.antidiagonal (natDegree p + natDegree q) →\n coeff p (natDegree p, natDegree q).fst * coeff q (natDegree p, natDegree q).snd = 0",
"state_before": "R : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝ : Semiring R\np✝ q✝ : R[X]\nι : Type ?u.653636\np q : R[X]\n⊢ ∑ x in Nat.antidiagonal (natDegree p + natDegree q), coeff p x.fst * coeff q x.snd =\n coeff p (natDegree p) * coeff q (natDegree q)",
"tactic": "refine' Finset.sum_eq_single (natDegree p, natDegree q) _ _"
},
{
"state_after": "case refine'_1.mk\nR : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝ : Semiring R\np✝ q✝ : R[X]\nι : Type ?u.653636\np q : R[X]\ni j : ℕ\nh₁ : (i, j) ∈ Nat.antidiagonal (natDegree p + natDegree q)\nh₂ : (i, j) ≠ (natDegree p, natDegree q)\n⊢ coeff p (i, j).fst * coeff q (i, j).snd = 0",
"state_before": "case refine'_1\nR : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝ : Semiring R\np✝ q✝ : R[X]\nι : Type ?u.653636\np q : R[X]\n⊢ ∀ (b : ℕ × ℕ),\n b ∈ Nat.antidiagonal (natDegree p + natDegree q) →\n b ≠ (natDegree p, natDegree q) → coeff p b.fst * coeff q b.snd = 0",
"tactic": "rintro ⟨i, j⟩ h₁ h₂"
},
{
"state_after": "case refine'_1.mk\nR : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝ : Semiring R\np✝ q✝ : R[X]\nι : Type ?u.653636\np q : R[X]\ni j : ℕ\nh₁ : (i, j).fst + (i, j).snd = natDegree p + natDegree q\nh₂ : (i, j) ≠ (natDegree p, natDegree q)\n⊢ coeff p (i, j).fst * coeff q (i, j).snd = 0",
"state_before": "case refine'_1.mk\nR : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝ : Semiring R\np✝ q✝ : R[X]\nι : Type ?u.653636\np q : R[X]\ni j : ℕ\nh₁ : (i, j) ∈ Nat.antidiagonal (natDegree p + natDegree q)\nh₂ : (i, j) ≠ (natDegree p, natDegree q)\n⊢ coeff p (i, j).fst * coeff q (i, j).snd = 0",
"tactic": "rw [Nat.mem_antidiagonal] at h₁"
},
{
"state_after": "case pos\nR : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝ : Semiring R\np✝ q✝ : R[X]\nι : Type ?u.653636\np q : R[X]\ni j : ℕ\nh₁ : (i, j).fst + (i, j).snd = natDegree p + natDegree q\nh₂ : (i, j) ≠ (natDegree p, natDegree q)\nH : natDegree p < i\n⊢ coeff p (i, j).fst * coeff q (i, j).snd = 0\n\ncase neg\nR : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝ : Semiring R\np✝ q✝ : R[X]\nι : Type ?u.653636\np q : R[X]\ni j : ℕ\nh₁ : (i, j).fst + (i, j).snd = natDegree p + natDegree q\nh₂ : (i, j) ≠ (natDegree p, natDegree q)\nH : ¬natDegree p < i\n⊢ coeff p (i, j).fst * coeff q (i, j).snd = 0",
"state_before": "case refine'_1.mk\nR : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝ : Semiring R\np✝ q✝ : R[X]\nι : Type ?u.653636\np q : R[X]\ni j : ℕ\nh₁ : (i, j).fst + (i, j).snd = natDegree p + natDegree q\nh₂ : (i, j) ≠ (natDegree p, natDegree q)\n⊢ coeff p (i, j).fst * coeff q (i, j).snd = 0",
"tactic": "by_cases H : natDegree p < i"
},
{
"state_after": "no goals",
"state_before": "case pos\nR : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝ : Semiring R\np✝ q✝ : R[X]\nι : Type ?u.653636\np q : R[X]\ni j : ℕ\nh₁ : (i, j).fst + (i, j).snd = natDegree p + natDegree q\nh₂ : (i, j) ≠ (natDegree p, natDegree q)\nH : natDegree p < i\n⊢ coeff p (i, j).fst * coeff q (i, j).snd = 0",
"tactic": "rw [coeff_eq_zero_of_degree_lt\n (lt_of_le_of_lt degree_le_natDegree (WithBot.coe_lt_coe.2 H)),\n zero_mul]"
},
{
"state_after": "case neg\nR : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝ : Semiring R\np✝ q✝ : R[X]\nι : Type ?u.653636\np q : R[X]\ni j : ℕ\nh₁ : (i, j).fst + (i, j).snd = natDegree p + natDegree q\nh₂ : (i, j) ≠ (natDegree p, natDegree q)\nH : natDegree p = i ∨ i < natDegree p\n⊢ coeff p (i, j).fst * coeff q (i, j).snd = 0",
"state_before": "case neg\nR : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝ : Semiring R\np✝ q✝ : R[X]\nι : Type ?u.653636\np q : R[X]\ni j : ℕ\nh₁ : (i, j).fst + (i, j).snd = natDegree p + natDegree q\nh₂ : (i, j) ≠ (natDegree p, natDegree q)\nH : ¬natDegree p < i\n⊢ coeff p (i, j).fst * coeff q (i, j).snd = 0",
"tactic": "rw [not_lt_iff_eq_or_lt] at H"
},
{
"state_after": "case neg.inl\nR : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝ : Semiring R\np✝ q✝ : R[X]\nι : Type ?u.653636\np q : R[X]\ni j : ℕ\nh₁ : (i, j).fst + (i, j).snd = natDegree p + natDegree q\nh₂ : (i, j) ≠ (natDegree p, natDegree q)\nH : natDegree p = i\n⊢ coeff p (i, j).fst * coeff q (i, j).snd = 0\n\ncase neg.inr\nR : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝ : Semiring R\np✝ q✝ : R[X]\nι : Type ?u.653636\np q : R[X]\ni j : ℕ\nh₁ : (i, j).fst + (i, j).snd = natDegree p + natDegree q\nh₂ : (i, j) ≠ (natDegree p, natDegree q)\nH : i < natDegree p\n⊢ coeff p (i, j).fst * coeff q (i, j).snd = 0",
"state_before": "case neg\nR : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝ : Semiring R\np✝ q✝ : R[X]\nι : Type ?u.653636\np q : R[X]\ni j : ℕ\nh₁ : (i, j).fst + (i, j).snd = natDegree p + natDegree q\nh₂ : (i, j) ≠ (natDegree p, natDegree q)\nH : natDegree p = i ∨ i < natDegree p\n⊢ coeff p (i, j).fst * coeff q (i, j).snd = 0",
"tactic": "cases' H with H H"
},
{
"state_after": "case neg.inl\nR : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝ : Semiring R\np✝ q✝ : R[X]\nι : Type ?u.653636\np q : R[X]\nj : ℕ\nh₁ : (natDegree p, j).fst + (natDegree p, j).snd = natDegree p + natDegree q\nh₂ : (natDegree p, j) ≠ (natDegree p, natDegree q)\n⊢ coeff p (natDegree p, j).fst * coeff q (natDegree p, j).snd = 0",
"state_before": "case neg.inl\nR : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝ : Semiring R\np✝ q✝ : R[X]\nι : Type ?u.653636\np q : R[X]\ni j : ℕ\nh₁ : (i, j).fst + (i, j).snd = natDegree p + natDegree q\nh₂ : (i, j) ≠ (natDegree p, natDegree q)\nH : natDegree p = i\n⊢ coeff p (i, j).fst * coeff q (i, j).snd = 0",
"tactic": "subst H"
},
{
"state_after": "case neg.inl\nR : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝ : Semiring R\np✝ q✝ : R[X]\nι : Type ?u.653636\np q : R[X]\nj : ℕ\nh₁ : (natDegree p, j).snd = natDegree q\nh₂ : (natDegree p, j) ≠ (natDegree p, natDegree q)\n⊢ coeff p (natDegree p, j).fst * coeff q (natDegree p, j).snd = 0",
"state_before": "case neg.inl\nR : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝ : Semiring R\np✝ q✝ : R[X]\nι : Type ?u.653636\np q : R[X]\nj : ℕ\nh₁ : (natDegree p, j).fst + (natDegree p, j).snd = natDegree p + natDegree q\nh₂ : (natDegree p, j) ≠ (natDegree p, natDegree q)\n⊢ coeff p (natDegree p, j).fst * coeff q (natDegree p, j).snd = 0",
"tactic": "rw [add_left_cancel_iff] at h₁"
},
{
"state_after": "case neg.inl\nR : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝ : Semiring R\np✝ q✝ : R[X]\nι : Type ?u.653636\np q : R[X]\nj : ℕ\nh₁ : j = natDegree q\nh₂ : (natDegree p, j) ≠ (natDegree p, natDegree q)\n⊢ coeff p (natDegree p, j).fst * coeff q (natDegree p, j).snd = 0",
"state_before": "case neg.inl\nR : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝ : Semiring R\np✝ q✝ : R[X]\nι : Type ?u.653636\np q : R[X]\nj : ℕ\nh₁ : (natDegree p, j).snd = natDegree q\nh₂ : (natDegree p, j) ≠ (natDegree p, natDegree q)\n⊢ coeff p (natDegree p, j).fst * coeff q (natDegree p, j).snd = 0",
"tactic": "dsimp at h₁"
},
{
"state_after": "case neg.inl\nR : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝ : Semiring R\np✝ q✝ : R[X]\nι : Type ?u.653636\np q : R[X]\nh₂ : (natDegree p, natDegree q) ≠ (natDegree p, natDegree q)\n⊢ coeff p (natDegree p, natDegree q).fst * coeff q (natDegree p, natDegree q).snd = 0",
"state_before": "case neg.inl\nR : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝ : Semiring R\np✝ q✝ : R[X]\nι : Type ?u.653636\np q : R[X]\nj : ℕ\nh₁ : j = natDegree q\nh₂ : (natDegree p, j) ≠ (natDegree p, natDegree q)\n⊢ coeff p (natDegree p, j).fst * coeff q (natDegree p, j).snd = 0",
"tactic": "subst h₁"
},
{
"state_after": "case neg.inl.h\nR : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝ : Semiring R\np✝ q✝ : R[X]\nι : Type ?u.653636\np q : R[X]\nh₂ : (natDegree p, natDegree q) ≠ (natDegree p, natDegree q)\n⊢ False",
"state_before": "case neg.inl\nR : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝ : Semiring R\np✝ q✝ : R[X]\nι : Type ?u.653636\np q : R[X]\nh₂ : (natDegree p, natDegree q) ≠ (natDegree p, natDegree q)\n⊢ coeff p (natDegree p, natDegree q).fst * coeff q (natDegree p, natDegree q).snd = 0",
"tactic": "exfalso"
},
{
"state_after": "no goals",
"state_before": "case neg.inl.h\nR : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝ : Semiring R\np✝ q✝ : R[X]\nι : Type ?u.653636\np q : R[X]\nh₂ : (natDegree p, natDegree q) ≠ (natDegree p, natDegree q)\n⊢ False",
"tactic": "exact h₂ rfl"
},
{
"state_after": "case neg.inr\nR : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝ : Semiring R\np✝ q✝ : R[X]\nι : Type ?u.653636\np q : R[X]\ni j : ℕ\nh₁ : (i, j).fst + (i, j).snd = natDegree p + natDegree q\nh₂ : (i, j) ≠ (natDegree p, natDegree q)\nH : i < natDegree p\n⊢ natDegree q < j",
"state_before": "case neg.inr\nR : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝ : Semiring R\np✝ q✝ : R[X]\nι : Type ?u.653636\np q : R[X]\ni j : ℕ\nh₁ : (i, j).fst + (i, j).snd = natDegree p + natDegree q\nh₂ : (i, j) ≠ (natDegree p, natDegree q)\nH : i < natDegree p\n⊢ coeff p (i, j).fst * coeff q (i, j).snd = 0",
"tactic": "suffices natDegree q < j by\n rw [coeff_eq_zero_of_degree_lt\n (lt_of_le_of_lt degree_le_natDegree (WithBot.coe_lt_coe.2 this)),\n mul_zero]"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝ : Semiring R\np✝ q✝ : R[X]\nι : Type ?u.653636\np q : R[X]\ni j : ℕ\nh₁ : (i, j).fst + (i, j).snd = natDegree p + natDegree q\nh₂ : (i, j) ≠ (natDegree p, natDegree q)\nH : i < natDegree p\nthis : natDegree q < j\n⊢ coeff p (i, j).fst * coeff q (i, j).snd = 0",
"tactic": "rw [coeff_eq_zero_of_degree_lt\n (lt_of_le_of_lt degree_le_natDegree (WithBot.coe_lt_coe.2 this)),\n mul_zero]"
},
{
"state_after": "case neg.inr\nR : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝ : Semiring R\np✝ q✝ : R[X]\nι : Type ?u.653636\np q : R[X]\ni j : ℕ\nh₁ : (i, j).fst + (i, j).snd = natDegree p + natDegree q\nh₂ : (i, j) ≠ (natDegree p, natDegree q)\nH : i < natDegree p\nH' : ¬natDegree q < j\n⊢ False",
"state_before": "case neg.inr\nR : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝ : Semiring R\np✝ q✝ : R[X]\nι : Type ?u.653636\np q : R[X]\ni j : ℕ\nh₁ : (i, j).fst + (i, j).snd = natDegree p + natDegree q\nh₂ : (i, j) ≠ (natDegree p, natDegree q)\nH : i < natDegree p\n⊢ natDegree q < j",
"tactic": "by_contra H'"
},
{
"state_after": "case neg.inr\nR : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝ : Semiring R\np✝ q✝ : R[X]\nι : Type ?u.653636\np q : R[X]\ni j : ℕ\nh₁ : (i, j).fst + (i, j).snd = natDegree p + natDegree q\nh₂ : (i, j) ≠ (natDegree p, natDegree q)\nH : i < natDegree p\nH' : j ≤ natDegree q\n⊢ False",
"state_before": "case neg.inr\nR : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝ : Semiring R\np✝ q✝ : R[X]\nι : Type ?u.653636\np q : R[X]\ni j : ℕ\nh₁ : (i, j).fst + (i, j).snd = natDegree p + natDegree q\nh₂ : (i, j) ≠ (natDegree p, natDegree q)\nH : i < natDegree p\nH' : ¬natDegree q < j\n⊢ False",
"tactic": "rw [not_lt] at H'"
},
{
"state_after": "no goals",
"state_before": "case neg.inr\nR : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝ : Semiring R\np✝ q✝ : R[X]\nι : Type ?u.653636\np q : R[X]\ni j : ℕ\nh₁ : (i, j).fst + (i, j).snd = natDegree p + natDegree q\nh₂ : (i, j) ≠ (natDegree p, natDegree q)\nH : i < natDegree p\nH' : j ≤ natDegree q\n⊢ False",
"tactic": "exact\n ne_of_lt (Nat.lt_of_lt_of_le (Nat.add_lt_add_right H j) (Nat.add_le_add_left H' _))\n h₁"
},
{
"state_after": "case refine'_2\nR : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝ : Semiring R\np✝ q✝ : R[X]\nι : Type ?u.653636\np q : R[X]\nH : ¬(natDegree p, natDegree q) ∈ Nat.antidiagonal (natDegree p + natDegree q)\n⊢ coeff p (natDegree p, natDegree q).fst * coeff q (natDegree p, natDegree q).snd = 0",
"state_before": "case refine'_2\nR : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝ : Semiring R\np✝ q✝ : R[X]\nι : Type ?u.653636\np q : R[X]\n⊢ ¬(natDegree p, natDegree q) ∈ Nat.antidiagonal (natDegree p + natDegree q) →\n coeff p (natDegree p, natDegree q).fst * coeff q (natDegree p, natDegree q).snd = 0",
"tactic": "intro H"
},
{
"state_after": "case refine'_2.h\nR : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝ : Semiring R\np✝ q✝ : R[X]\nι : Type ?u.653636\np q : R[X]\nH : ¬(natDegree p, natDegree q) ∈ Nat.antidiagonal (natDegree p + natDegree q)\n⊢ False",
"state_before": "case refine'_2\nR : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝ : Semiring R\np✝ q✝ : R[X]\nι : Type ?u.653636\np q : R[X]\nH : ¬(natDegree p, natDegree q) ∈ Nat.antidiagonal (natDegree p + natDegree q)\n⊢ coeff p (natDegree p, natDegree q).fst * coeff q (natDegree p, natDegree q).snd = 0",
"tactic": "exfalso"
},
{
"state_after": "case refine'_2.h\nR : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝ : Semiring R\np✝ q✝ : R[X]\nι : Type ?u.653636\np q : R[X]\nH : ¬(natDegree p, natDegree q) ∈ Nat.antidiagonal (natDegree p + natDegree q)\n⊢ (natDegree p, natDegree q) ∈ Nat.antidiagonal (natDegree p + natDegree q)",
"state_before": "case refine'_2.h\nR : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝ : Semiring R\np✝ q✝ : R[X]\nι : Type ?u.653636\np q : R[X]\nH : ¬(natDegree p, natDegree q) ∈ Nat.antidiagonal (natDegree p + natDegree q)\n⊢ False",
"tactic": "apply H"
},
{
"state_after": "no goals",
"state_before": "case refine'_2.h\nR : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝ : Semiring R\np✝ q✝ : R[X]\nι : Type ?u.653636\np q : R[X]\nH : ¬(natDegree p, natDegree q) ∈ Nat.antidiagonal (natDegree p + natDegree q)\n⊢ (natDegree p, natDegree q) ∈ Nat.antidiagonal (natDegree p + natDegree q)",
"tactic": "rw [Nat.mem_antidiagonal]"
}
] |
[
925,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
891,
1
] |
Mathlib/Data/Set/Basic.lean
|
Set.nonempty_of_not_subset
|
[] |
[
483,
14
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
481,
1
] |
Mathlib/Data/Polynomial/IntegralNormalization.lean
|
Polynomial.integralNormalization_eval₂_eq_zero
|
[
{
"state_after": "R : Type u\nS : Type v\na b : R\nm n : ℕ\nι : Type y\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : CommSemiring S\np : R[X]\nf : R →+* S\nz : S\nhz : eval₂ f z p = 0\ninj : ∀ (x : R), ↑f x = 0 → x = 0\n⊢ ∑ n in support p, ↑f (coeff (integralNormalization p) n) * (z * ↑f (leadingCoeff p)) ^ n =\n ∑ i in Finset.attach (support p), ↑f (coeff (integralNormalization p) ↑i * leadingCoeff p ^ ↑i) * z ^ ↑i",
"state_before": "R : Type u\nS : Type v\na b : R\nm n : ℕ\nι : Type y\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : CommSemiring S\np : R[X]\nf : R →+* S\nz : S\nhz : eval₂ f z p = 0\ninj : ∀ (x : R), ↑f x = 0 → x = 0\n⊢ eval₂ f (z * ↑f (leadingCoeff p)) (integralNormalization p) =\n ∑ i in Finset.attach (support p), ↑f (coeff (integralNormalization p) ↑i * leadingCoeff p ^ ↑i) * z ^ ↑i",
"tactic": "rw [eval₂_eq_sum, sum_def, support_integralNormalization]"
},
{
"state_after": "R : Type u\nS : Type v\na b : R\nm n : ℕ\nι : Type y\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : CommSemiring S\np : R[X]\nf : R →+* S\nz : S\nhz : eval₂ f z p = 0\ninj : ∀ (x : R), ↑f x = 0 → x = 0\n⊢ ∑ x in support p, ↑f (coeff (integralNormalization p) x) * (↑f (leadingCoeff p) ^ x * z ^ x) =\n ∑ x in Finset.attach (support p), ↑f (coeff (integralNormalization p) ↑x) * (↑f (leadingCoeff p) ^ ↑x * z ^ ↑x)",
"state_before": "R : Type u\nS : Type v\na b : R\nm n : ℕ\nι : Type y\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : CommSemiring S\np : R[X]\nf : R →+* S\nz : S\nhz : eval₂ f z p = 0\ninj : ∀ (x : R), ↑f x = 0 → x = 0\n⊢ ∑ n in support p, ↑f (coeff (integralNormalization p) n) * (z * ↑f (leadingCoeff p)) ^ n =\n ∑ i in Finset.attach (support p), ↑f (coeff (integralNormalization p) ↑i * leadingCoeff p ^ ↑i) * z ^ ↑i",
"tactic": "simp only [mul_comm z, mul_pow, mul_assoc, RingHom.map_pow, RingHom.map_mul]"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\na b : R\nm n : ℕ\nι : Type y\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : CommSemiring S\np : R[X]\nf : R →+* S\nz : S\nhz : eval₂ f z p = 0\ninj : ∀ (x : R), ↑f x = 0 → x = 0\n⊢ ∑ x in support p, ↑f (coeff (integralNormalization p) x) * (↑f (leadingCoeff p) ^ x * z ^ x) =\n ∑ x in Finset.attach (support p), ↑f (coeff (integralNormalization p) ↑x) * (↑f (leadingCoeff p) ^ ↑x * z ^ ↑x)",
"tactic": "exact Finset.sum_attach.symm"
},
{
"state_after": "case pos\nR : Type u\nS : Type v\na b : R\nm n : ℕ\nι : Type y\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : CommSemiring S\np : R[X]\nf : R →+* S\nz : S\nhz : eval₂ f z p = 0\ninj : ∀ (x : R), ↑f x = 0 → x = 0\nhp : p = 0\n⊢ ∑ i in Finset.attach (support p), ↑f (coeff (integralNormalization p) ↑i * leadingCoeff p ^ ↑i) * z ^ ↑i =\n ∑ i in Finset.attach (support p), ↑f (coeff p ↑i * leadingCoeff p ^ (natDegree p - 1)) * z ^ ↑i\n\ncase neg\nR : Type u\nS : Type v\na b : R\nm n : ℕ\nι : Type y\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : CommSemiring S\np : R[X]\nf : R →+* S\nz : S\nhz : eval₂ f z p = 0\ninj : ∀ (x : R), ↑f x = 0 → x = 0\nhp : ¬p = 0\n⊢ ∑ i in Finset.attach (support p), ↑f (coeff (integralNormalization p) ↑i * leadingCoeff p ^ ↑i) * z ^ ↑i =\n ∑ i in Finset.attach (support p), ↑f (coeff p ↑i * leadingCoeff p ^ (natDegree p - 1)) * z ^ ↑i",
"state_before": "R : Type u\nS : Type v\na b : R\nm n : ℕ\nι : Type y\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : CommSemiring S\np : R[X]\nf : R →+* S\nz : S\nhz : eval₂ f z p = 0\ninj : ∀ (x : R), ↑f x = 0 → x = 0\n⊢ ∑ i in Finset.attach (support p), ↑f (coeff (integralNormalization p) ↑i * leadingCoeff p ^ ↑i) * z ^ ↑i =\n ∑ i in Finset.attach (support p), ↑f (coeff p ↑i * leadingCoeff p ^ (natDegree p - 1)) * z ^ ↑i",
"tactic": "by_cases hp : p = 0"
},
{
"state_after": "case neg\nR : Type u\nS : Type v\na b : R\nm n : ℕ\nι : Type y\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : CommSemiring S\np : R[X]\nf : R →+* S\nz : S\nhz : eval₂ f z p = 0\ninj : ∀ (x : R), ↑f x = 0 → x = 0\nhp : ¬p = 0\none_le_deg : 1 ≤ natDegree p\n⊢ ∑ i in Finset.attach (support p), ↑f (coeff (integralNormalization p) ↑i * leadingCoeff p ^ ↑i) * z ^ ↑i =\n ∑ i in Finset.attach (support p), ↑f (coeff p ↑i * leadingCoeff p ^ (natDegree p - 1)) * z ^ ↑i",
"state_before": "case neg\nR : Type u\nS : Type v\na b : R\nm n : ℕ\nι : Type y\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : CommSemiring S\np : R[X]\nf : R →+* S\nz : S\nhz : eval₂ f z p = 0\ninj : ∀ (x : R), ↑f x = 0 → x = 0\nhp : ¬p = 0\n⊢ ∑ i in Finset.attach (support p), ↑f (coeff (integralNormalization p) ↑i * leadingCoeff p ^ ↑i) * z ^ ↑i =\n ∑ i in Finset.attach (support p), ↑f (coeff p ↑i * leadingCoeff p ^ (natDegree p - 1)) * z ^ ↑i",
"tactic": "have one_le_deg : 1 ≤ natDegree p :=\n Nat.succ_le_of_lt (natDegree_pos_of_eval₂_root hp f hz inj)"
},
{
"state_after": "case neg.e_f.h\nR : Type u\nS : Type v\na b : R\nm n : ℕ\nι : Type y\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : CommSemiring S\np : R[X]\nf : R →+* S\nz : S\nhz : eval₂ f z p = 0\ninj : ∀ (x : R), ↑f x = 0 → x = 0\nhp : ¬p = 0\none_le_deg : 1 ≤ natDegree p\ni : { x // x ∈ support p }\n⊢ ↑f (coeff (integralNormalization p) ↑i * leadingCoeff p ^ ↑i) * z ^ ↑i =\n ↑f (coeff p ↑i * leadingCoeff p ^ (natDegree p - 1)) * z ^ ↑i",
"state_before": "case neg\nR : Type u\nS : Type v\na b : R\nm n : ℕ\nι : Type y\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : CommSemiring S\np : R[X]\nf : R →+* S\nz : S\nhz : eval₂ f z p = 0\ninj : ∀ (x : R), ↑f x = 0 → x = 0\nhp : ¬p = 0\none_le_deg : 1 ≤ natDegree p\n⊢ ∑ i in Finset.attach (support p), ↑f (coeff (integralNormalization p) ↑i * leadingCoeff p ^ ↑i) * z ^ ↑i =\n ∑ i in Finset.attach (support p), ↑f (coeff p ↑i * leadingCoeff p ^ (natDegree p - 1)) * z ^ ↑i",
"tactic": "congr with i"
},
{
"state_after": "case neg.e_f.h.e_a.h.e_6.h\nR : Type u\nS : Type v\na b : R\nm n : ℕ\nι : Type y\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : CommSemiring S\np : R[X]\nf : R →+* S\nz : S\nhz : eval₂ f z p = 0\ninj : ∀ (x : R), ↑f x = 0 → x = 0\nhp : ¬p = 0\none_le_deg : 1 ≤ natDegree p\ni : { x // x ∈ support p }\n⊢ coeff (integralNormalization p) ↑i * leadingCoeff p ^ ↑i = coeff p ↑i * leadingCoeff p ^ (natDegree p - 1)",
"state_before": "case neg.e_f.h\nR : Type u\nS : Type v\na b : R\nm n : ℕ\nι : Type y\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : CommSemiring S\np : R[X]\nf : R →+* S\nz : S\nhz : eval₂ f z p = 0\ninj : ∀ (x : R), ↑f x = 0 → x = 0\nhp : ¬p = 0\none_le_deg : 1 ≤ natDegree p\ni : { x // x ∈ support p }\n⊢ ↑f (coeff (integralNormalization p) ↑i * leadingCoeff p ^ ↑i) * z ^ ↑i =\n ↑f (coeff p ↑i * leadingCoeff p ^ (natDegree p - 1)) * z ^ ↑i",
"tactic": "congr 2"
},
{
"state_after": "case pos\nR : Type u\nS : Type v\na b : R\nm n : ℕ\nι : Type y\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : CommSemiring S\np : R[X]\nf : R →+* S\nz : S\nhz : eval₂ f z p = 0\ninj : ∀ (x : R), ↑f x = 0 → x = 0\nhp : ¬p = 0\none_le_deg : 1 ≤ natDegree p\ni : { x // x ∈ support p }\nhi : ↑i = natDegree p\n⊢ coeff (integralNormalization p) ↑i * leadingCoeff p ^ ↑i = coeff p ↑i * leadingCoeff p ^ (natDegree p - 1)\n\ncase neg\nR : Type u\nS : Type v\na b : R\nm n : ℕ\nι : Type y\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : CommSemiring S\np : R[X]\nf : R →+* S\nz : S\nhz : eval₂ f z p = 0\ninj : ∀ (x : R), ↑f x = 0 → x = 0\nhp : ¬p = 0\none_le_deg : 1 ≤ natDegree p\ni : { x // x ∈ support p }\nhi : ¬↑i = natDegree p\n⊢ coeff (integralNormalization p) ↑i * leadingCoeff p ^ ↑i = coeff p ↑i * leadingCoeff p ^ (natDegree p - 1)",
"state_before": "case neg.e_f.h.e_a.h.e_6.h\nR : Type u\nS : Type v\na b : R\nm n : ℕ\nι : Type y\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : CommSemiring S\np : R[X]\nf : R →+* S\nz : S\nhz : eval₂ f z p = 0\ninj : ∀ (x : R), ↑f x = 0 → x = 0\nhp : ¬p = 0\none_le_deg : 1 ≤ natDegree p\ni : { x // x ∈ support p }\n⊢ coeff (integralNormalization p) ↑i * leadingCoeff p ^ ↑i = coeff p ↑i * leadingCoeff p ^ (natDegree p - 1)",
"tactic": "by_cases hi : i.1 = natDegree p"
},
{
"state_after": "no goals",
"state_before": "case pos\nR : Type u\nS : Type v\na b : R\nm n : ℕ\nι : Type y\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : CommSemiring S\np : R[X]\nf : R →+* S\nz : S\nhz : eval₂ f z p = 0\ninj : ∀ (x : R), ↑f x = 0 → x = 0\nhp : p = 0\n⊢ ∑ i in Finset.attach (support p), ↑f (coeff (integralNormalization p) ↑i * leadingCoeff p ^ ↑i) * z ^ ↑i =\n ∑ i in Finset.attach (support p), ↑f (coeff p ↑i * leadingCoeff p ^ (natDegree p - 1)) * z ^ ↑i",
"tactic": "simp [hp]"
},
{
"state_after": "case pos\nR : Type u\nS : Type v\na b : R\nm n : ℕ\nι : Type y\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : CommSemiring S\np : R[X]\nf : R →+* S\nz : S\nhz : eval₂ f z p = 0\ninj : ∀ (x : R), ↑f x = 0 → x = 0\nhp : ¬p = 0\none_le_deg : 1 ≤ natDegree p\ni : { x // x ∈ support p }\nhi : ↑i = natDegree p\n⊢ degree p = ↑(natDegree p)",
"state_before": "case pos\nR : Type u\nS : Type v\na b : R\nm n : ℕ\nι : Type y\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : CommSemiring S\np : R[X]\nf : R →+* S\nz : S\nhz : eval₂ f z p = 0\ninj : ∀ (x : R), ↑f x = 0 → x = 0\nhp : ¬p = 0\none_le_deg : 1 ≤ natDegree p\ni : { x // x ∈ support p }\nhi : ↑i = natDegree p\n⊢ coeff (integralNormalization p) ↑i * leadingCoeff p ^ ↑i = coeff p ↑i * leadingCoeff p ^ (natDegree p - 1)",
"tactic": "rw [hi, integralNormalization_coeff_degree, one_mul, leadingCoeff, ← pow_succ,\n tsub_add_cancel_of_le one_le_deg]"
},
{
"state_after": "no goals",
"state_before": "case pos\nR : Type u\nS : Type v\na b : R\nm n : ℕ\nι : Type y\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : CommSemiring S\np : R[X]\nf : R →+* S\nz : S\nhz : eval₂ f z p = 0\ninj : ∀ (x : R), ↑f x = 0 → x = 0\nhp : ¬p = 0\none_le_deg : 1 ≤ natDegree p\ni : { x // x ∈ support p }\nhi : ↑i = natDegree p\n⊢ degree p = ↑(natDegree p)",
"tactic": "exact degree_eq_natDegree hp"
},
{
"state_after": "case neg\nR : Type u\nS : Type v\na b : R\nm n : ℕ\nι : Type y\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : CommSemiring S\np : R[X]\nf : R →+* S\nz : S\nhz : eval₂ f z p = 0\ninj : ∀ (x : R), ↑f x = 0 → x = 0\nhp : ¬p = 0\none_le_deg : 1 ≤ natDegree p\ni : { x // x ∈ support p }\nhi : ¬↑i = natDegree p\nthis : ↑i ≤ natDegree p - 1\n⊢ coeff (integralNormalization p) ↑i * leadingCoeff p ^ ↑i = coeff p ↑i * leadingCoeff p ^ (natDegree p - 1)",
"state_before": "case neg\nR : Type u\nS : Type v\na b : R\nm n : ℕ\nι : Type y\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : CommSemiring S\np : R[X]\nf : R →+* S\nz : S\nhz : eval₂ f z p = 0\ninj : ∀ (x : R), ↑f x = 0 → x = 0\nhp : ¬p = 0\none_le_deg : 1 ≤ natDegree p\ni : { x // x ∈ support p }\nhi : ¬↑i = natDegree p\n⊢ coeff (integralNormalization p) ↑i * leadingCoeff p ^ ↑i = coeff p ↑i * leadingCoeff p ^ (natDegree p - 1)",
"tactic": "have : i.1 ≤ p.natDegree - 1 :=\n Nat.le_pred_of_lt (lt_of_le_of_ne (le_natDegree_of_ne_zero (mem_support_iff.mp i.2)) hi)"
},
{
"state_after": "no goals",
"state_before": "case neg\nR : Type u\nS : Type v\na b : R\nm n : ℕ\nι : Type y\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : CommSemiring S\np : R[X]\nf : R →+* S\nz : S\nhz : eval₂ f z p = 0\ninj : ∀ (x : R), ↑f x = 0 → x = 0\nhp : ¬p = 0\none_le_deg : 1 ≤ natDegree p\ni : { x // x ∈ support p }\nhi : ¬↑i = natDegree p\nthis : ↑i ≤ natDegree p - 1\n⊢ coeff (integralNormalization p) ↑i * leadingCoeff p ^ ↑i = coeff p ↑i * leadingCoeff p ^ (natDegree p - 1)",
"tactic": "rw [integralNormalization_coeff_ne_natDegree hi, mul_assoc, ← pow_add,\n tsub_add_cancel_of_le this]"
},
{
"state_after": "R : Type u\nS : Type v\na b : R\nm n : ℕ\nι : Type y\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : CommSemiring S\np : R[X]\nf : R →+* S\nz : S\nhz : eval₂ f z p = 0\ninj : ∀ (x : R), ↑f x = 0 → x = 0\n⊢ ↑f (leadingCoeff p) ^ (natDegree p - 1) * ∑ x in Finset.attach (support p), ↑f (coeff p ↑x) * z ^ ↑x =\n ↑f (leadingCoeff p) ^ (natDegree p - 1) * ∑ n in support p, ↑f (coeff p n) * z ^ n",
"state_before": "R : Type u\nS : Type v\na b : R\nm n : ℕ\nι : Type y\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : CommSemiring S\np : R[X]\nf : R →+* S\nz : S\nhz : eval₂ f z p = 0\ninj : ∀ (x : R), ↑f x = 0 → x = 0\n⊢ ∑ i in Finset.attach (support p), ↑f (coeff p ↑i * leadingCoeff p ^ (natDegree p - 1)) * z ^ ↑i =\n ↑f (leadingCoeff p) ^ (natDegree p - 1) * eval₂ f z p",
"tactic": "simp_rw [eval₂_eq_sum, sum_def, fun i => mul_comm (coeff p i), RingHom.map_mul,\n RingHom.map_pow, mul_assoc, ← Finset.mul_sum]"
},
{
"state_after": "case e_a\nR : Type u\nS : Type v\na b : R\nm n : ℕ\nι : Type y\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : CommSemiring S\np : R[X]\nf : R →+* S\nz : S\nhz : eval₂ f z p = 0\ninj : ∀ (x : R), ↑f x = 0 → x = 0\n⊢ ∑ x in Finset.attach (support p), ↑f (coeff p ↑x) * z ^ ↑x = ∑ n in support p, ↑f (coeff p n) * z ^ n",
"state_before": "R : Type u\nS : Type v\na b : R\nm n : ℕ\nι : Type y\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : CommSemiring S\np : R[X]\nf : R →+* S\nz : S\nhz : eval₂ f z p = 0\ninj : ∀ (x : R), ↑f x = 0 → x = 0\n⊢ ↑f (leadingCoeff p) ^ (natDegree p - 1) * ∑ x in Finset.attach (support p), ↑f (coeff p ↑x) * z ^ ↑x =\n ↑f (leadingCoeff p) ^ (natDegree p - 1) * ∑ n in support p, ↑f (coeff p n) * z ^ n",
"tactic": "congr 1"
},
{
"state_after": "no goals",
"state_before": "case e_a\nR : Type u\nS : Type v\na b : R\nm n : ℕ\nι : Type y\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : CommSemiring S\np : R[X]\nf : R →+* S\nz : S\nhz : eval₂ f z p = 0\ninj : ∀ (x : R), ↑f x = 0 → x = 0\n⊢ ∑ x in Finset.attach (support p), ↑f (coeff p ↑x) * z ^ ↑x = ∑ n in support p, ↑f (coeff p n) * z ^ n",
"tactic": "exact @Finset.sum_attach _ _ p.support _ fun i => f (p.coeff i) * z ^ i"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\na b : R\nm n : ℕ\nι : Type y\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : CommSemiring S\np : R[X]\nf : R →+* S\nz : S\nhz : eval₂ f z p = 0\ninj : ∀ (x : R), ↑f x = 0 → x = 0\n⊢ ↑f (leadingCoeff p) ^ (natDegree p - 1) * eval₂ f z p = 0",
"tactic": "rw [hz, mul_zero]"
}
] |
[
147,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
116,
1
] |
Mathlib/Data/Finset/NoncommProd.lean
|
Finset.noncommProd_mul_distrib_aux
|
[
{
"state_after": "F : Type ?u.217198\nι : Type ?u.217201\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.217210\nf✝ : α → β → β\nop : α → α → α\ninst✝¹ : Monoid β\ninst✝ : Monoid γ\ns : Finset α\nf g : α → β\ncomm_ff : Set.Pairwise ↑s fun x y => Commute (f x) (f y)\ncomm_gg : Set.Pairwise ↑s fun x y => Commute (g x) (g y)\ncomm_gf : Set.Pairwise ↑s fun x y => Commute (g x) (f y)\nx : α\nhx : x ∈ ↑s\ny : α\nhy : y ∈ ↑s\nh : x ≠ y\n⊢ Commute ((f * g) x) ((f * g) y)",
"state_before": "F : Type ?u.217198\nι : Type ?u.217201\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.217210\nf✝ : α → β → β\nop : α → α → α\ninst✝¹ : Monoid β\ninst✝ : Monoid γ\ns : Finset α\nf g : α → β\ncomm_ff : Set.Pairwise ↑s fun x y => Commute (f x) (f y)\ncomm_gg : Set.Pairwise ↑s fun x y => Commute (g x) (g y)\ncomm_gf : Set.Pairwise ↑s fun x y => Commute (g x) (f y)\n⊢ Set.Pairwise ↑s fun x y => Commute ((f * g) x) ((f * g) y)",
"tactic": "intro x hx y hy h"
},
{
"state_after": "case hac.hab\nF : Type ?u.217198\nι : Type ?u.217201\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.217210\nf✝ : α → β → β\nop : α → α → α\ninst✝¹ : Monoid β\ninst✝ : Monoid γ\ns : Finset α\nf g : α → β\ncomm_ff : Set.Pairwise ↑s fun x y => Commute (f x) (f y)\ncomm_gg : Set.Pairwise ↑s fun x y => Commute (g x) (g y)\ncomm_gf : Set.Pairwise ↑s fun x y => Commute (g x) (f y)\nx : α\nhx : x ∈ ↑s\ny : α\nhy : y ∈ ↑s\nh : x ≠ y\n⊢ Commute (f x) (f y)\n\ncase hac.hac\nF : Type ?u.217198\nι : Type ?u.217201\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.217210\nf✝ : α → β → β\nop : α → α → α\ninst✝¹ : Monoid β\ninst✝ : Monoid γ\ns : Finset α\nf g : α → β\ncomm_ff : Set.Pairwise ↑s fun x y => Commute (f x) (f y)\ncomm_gg : Set.Pairwise ↑s fun x y => Commute (g x) (g y)\ncomm_gf : Set.Pairwise ↑s fun x y => Commute (g x) (f y)\nx : α\nhx : x ∈ ↑s\ny : α\nhy : y ∈ ↑s\nh : x ≠ y\n⊢ Commute (f x) (g y)\n\ncase hbc.hab\nF : Type ?u.217198\nι : Type ?u.217201\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.217210\nf✝ : α → β → β\nop : α → α → α\ninst✝¹ : Monoid β\ninst✝ : Monoid γ\ns : Finset α\nf g : α → β\ncomm_ff : Set.Pairwise ↑s fun x y => Commute (f x) (f y)\ncomm_gg : Set.Pairwise ↑s fun x y => Commute (g x) (g y)\ncomm_gf : Set.Pairwise ↑s fun x y => Commute (g x) (f y)\nx : α\nhx : x ∈ ↑s\ny : α\nhy : y ∈ ↑s\nh : x ≠ y\n⊢ Commute (g x) (f y)\n\ncase hbc.hac\nF : Type ?u.217198\nι : Type ?u.217201\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.217210\nf✝ : α → β → β\nop : α → α → α\ninst✝¹ : Monoid β\ninst✝ : Monoid γ\ns : Finset α\nf g : α → β\ncomm_ff : Set.Pairwise ↑s fun x y => Commute (f x) (f y)\ncomm_gg : Set.Pairwise ↑s fun x y => Commute (g x) (g y)\ncomm_gf : Set.Pairwise ↑s fun x y => Commute (g x) (f y)\nx : α\nhx : x ∈ ↑s\ny : α\nhy : y ∈ ↑s\nh : x ≠ y\n⊢ Commute (g x) (g y)",
"state_before": "F : Type ?u.217198\nι : Type ?u.217201\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.217210\nf✝ : α → β → β\nop : α → α → α\ninst✝¹ : Monoid β\ninst✝ : Monoid γ\ns : Finset α\nf g : α → β\ncomm_ff : Set.Pairwise ↑s fun x y => Commute (f x) (f y)\ncomm_gg : Set.Pairwise ↑s fun x y => Commute (g x) (g y)\ncomm_gf : Set.Pairwise ↑s fun x y => Commute (g x) (f y)\nx : α\nhx : x ∈ ↑s\ny : α\nhy : y ∈ ↑s\nh : x ≠ y\n⊢ Commute ((f * g) x) ((f * g) y)",
"tactic": "apply Commute.mul_left <;> apply Commute.mul_right"
},
{
"state_after": "no goals",
"state_before": "case hac.hab\nF : Type ?u.217198\nι : Type ?u.217201\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.217210\nf✝ : α → β → β\nop : α → α → α\ninst✝¹ : Monoid β\ninst✝ : Monoid γ\ns : Finset α\nf g : α → β\ncomm_ff : Set.Pairwise ↑s fun x y => Commute (f x) (f y)\ncomm_gg : Set.Pairwise ↑s fun x y => Commute (g x) (g y)\ncomm_gf : Set.Pairwise ↑s fun x y => Commute (g x) (f y)\nx : α\nhx : x ∈ ↑s\ny : α\nhy : y ∈ ↑s\nh : x ≠ y\n⊢ Commute (f x) (f y)",
"tactic": "exact comm_ff.of_refl hx hy"
},
{
"state_after": "no goals",
"state_before": "case hac.hac\nF : Type ?u.217198\nι : Type ?u.217201\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.217210\nf✝ : α → β → β\nop : α → α → α\ninst✝¹ : Monoid β\ninst✝ : Monoid γ\ns : Finset α\nf g : α → β\ncomm_ff : Set.Pairwise ↑s fun x y => Commute (f x) (f y)\ncomm_gg : Set.Pairwise ↑s fun x y => Commute (g x) (g y)\ncomm_gf : Set.Pairwise ↑s fun x y => Commute (g x) (f y)\nx : α\nhx : x ∈ ↑s\ny : α\nhy : y ∈ ↑s\nh : x ≠ y\n⊢ Commute (f x) (g y)",
"tactic": "exact (comm_gf hy hx h.symm).symm"
},
{
"state_after": "no goals",
"state_before": "case hbc.hab\nF : Type ?u.217198\nι : Type ?u.217201\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.217210\nf✝ : α → β → β\nop : α → α → α\ninst✝¹ : Monoid β\ninst✝ : Monoid γ\ns : Finset α\nf g : α → β\ncomm_ff : Set.Pairwise ↑s fun x y => Commute (f x) (f y)\ncomm_gg : Set.Pairwise ↑s fun x y => Commute (g x) (g y)\ncomm_gf : Set.Pairwise ↑s fun x y => Commute (g x) (f y)\nx : α\nhx : x ∈ ↑s\ny : α\nhy : y ∈ ↑s\nh : x ≠ y\n⊢ Commute (g x) (f y)",
"tactic": "exact comm_gf hx hy h"
},
{
"state_after": "no goals",
"state_before": "case hbc.hac\nF : Type ?u.217198\nι : Type ?u.217201\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.217210\nf✝ : α → β → β\nop : α → α → α\ninst✝¹ : Monoid β\ninst✝ : Monoid γ\ns : Finset α\nf g : α → β\ncomm_ff : Set.Pairwise ↑s fun x y => Commute (f x) (f y)\ncomm_gg : Set.Pairwise ↑s fun x y => Commute (g x) (g y)\ncomm_gf : Set.Pairwise ↑s fun x y => Commute (g x) (f y)\nx : α\nhx : x ∈ ↑s\ny : α\nhy : y ∈ ↑s\nh : x ≠ y\n⊢ Commute (g x) (g y)",
"tactic": "exact comm_gg.of_refl hx hy"
}
] |
[
388,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
378,
1
] |
Mathlib/Topology/Basic.lean
|
IsClosed.preimage
|
[] |
[
1717,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1715,
1
] |
Mathlib/Data/MvPolynomial/Basic.lean
|
MvPolynomial.eval₂Hom_congr
|
[
{
"state_after": "R : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_1\na a' a₁ a₂ : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\np q : MvPolynomial σ R\nf : R →+* S₁\ng : σ → S₁\nf₁ : R →+* S₁\ng₁ : σ → S₁\np₁ : MvPolynomial σ R\n⊢ ↑(eval₂Hom f₁ g₁) p₁ = ↑(eval₂Hom f₁ g₁) p₁",
"state_before": "R : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_1\na a' a₁ a₂ : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\np q : MvPolynomial σ R\nf : R →+* S₁\ng : σ → S₁\nf₁ f₂ : R →+* S₁\ng₁ g₂ : σ → S₁\np₁ p₂ : MvPolynomial σ R\n⊢ f₁ = f₂ → g₁ = g₂ → p₁ = p₂ → ↑(eval₂Hom f₁ g₁) p₁ = ↑(eval₂Hom f₂ g₂) p₂",
"tactic": "rintro rfl rfl rfl"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_1\na a' a₁ a₂ : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\np q : MvPolynomial σ R\nf : R →+* S₁\ng : σ → S₁\nf₁ : R →+* S₁\ng₁ : σ → S₁\np₁ : MvPolynomial σ R\n⊢ ↑(eval₂Hom f₁ g₁) p₁ = ↑(eval₂Hom f₁ g₁) p₁",
"tactic": "rfl"
}
] |
[
1035,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1033,
1
] |
Std/Data/Int/DivMod.lean
|
Int.div_eq_of_eq_mul_right
|
[
{
"state_after": "no goals",
"state_before": "a b c : Int\nH1 : b ≠ 0\nH2 : a = b * c\n⊢ div a b = c",
"tactic": "rw [H2, Int.mul_div_cancel_left _ H1]"
}
] |
[
742,
92
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
741,
11
] |
Mathlib/Analysis/Seminorm.lean
|
Seminorm.comp_mono
|
[] |
[
362,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
361,
1
] |
Mathlib/Algebra/Order/Field/Basic.lean
|
lt_one_div_of_neg
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.159623\nα : Type u_1\nβ : Type ?u.159629\ninst✝ : LinearOrderedField α\na b c d : α\nn : ℤ\nha : a < 0\nhb : b < 0\n⊢ a < 1 / b ↔ b < 1 / a",
"tactic": "simpa using lt_inv_of_neg ha hb"
}
] |
[
832,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
831,
1
] |
Mathlib/Init/Propext.lean
|
iff_eq_eq
|
[] |
[
35,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
34,
1
] |
Mathlib/Topology/Basic.lean
|
IsClosed.closure_subset
|
[] |
[
446,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
445,
1
] |
Mathlib/RingTheory/JacobsonIdeal.lean
|
Ideal.map_jacobson_of_bijective
|
[] |
[
204,
72
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
201,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Deriv.lean
|
ContDiffOn.csin
|
[] |
[
429,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
427,
1
] |
Mathlib/Control/Bitraversable/Lemmas.lean
|
Bitraversable.comp_tsnd
|
[
{
"state_after": "t : Type u → Type u → Type u\ninst✝⁵ : Bitraversable t\nβ : Type u\nF G : Type u → Type u\ninst✝⁴ : Applicative F\ninst✝³ : Applicative G\ninst✝² : IsLawfulBitraversable t\ninst✝¹ : LawfulApplicative F\ninst✝ : LawfulApplicative G\nα β₀ β₁ β₂ : Type u\ng : β₀ → F β₁\ng' : β₁ → G β₂\nx : t α β₀\n⊢ bitraverse (Comp.mk ∘ map pure ∘ pure) (Comp.mk ∘ map g' ∘ g) x = tsnd (Comp.mk ∘ map g' ∘ g) x",
"state_before": "t : Type u → Type u → Type u\ninst✝⁵ : Bitraversable t\nβ : Type u\nF G : Type u → Type u\ninst✝⁴ : Applicative F\ninst✝³ : Applicative G\ninst✝² : IsLawfulBitraversable t\ninst✝¹ : LawfulApplicative F\ninst✝ : LawfulApplicative G\nα β₀ β₁ β₂ : Type u\ng : β₀ → F β₁\ng' : β₁ → G β₂\nx : t α β₀\n⊢ Comp.mk (tsnd g' <$> tsnd g x) = tsnd (Comp.mk ∘ map g' ∘ g) x",
"tactic": "rw [← comp_bitraverse]"
},
{
"state_after": "t : Type u → Type u → Type u\ninst✝⁵ : Bitraversable t\nβ : Type u\nF G : Type u → Type u\ninst✝⁴ : Applicative F\ninst✝³ : Applicative G\ninst✝² : IsLawfulBitraversable t\ninst✝¹ : LawfulApplicative F\ninst✝ : LawfulApplicative G\nα β₀ β₁ β₂ : Type u\ng : β₀ → F β₁\ng' : β₁ → G β₂\nx : t α β₀\n⊢ bitraverse (fun x => Comp.mk (pure (pure x))) (fun x => Comp.mk (g' <$> g x)) x =\n tsnd (fun x => Comp.mk (g' <$> g x)) x",
"state_before": "t : Type u → Type u → Type u\ninst✝⁵ : Bitraversable t\nβ : Type u\nF G : Type u → Type u\ninst✝⁴ : Applicative F\ninst✝³ : Applicative G\ninst✝² : IsLawfulBitraversable t\ninst✝¹ : LawfulApplicative F\ninst✝ : LawfulApplicative G\nα β₀ β₁ β₂ : Type u\ng : β₀ → F β₁\ng' : β₁ → G β₂\nx : t α β₀\n⊢ bitraverse (Comp.mk ∘ map pure ∘ pure) (Comp.mk ∘ map g' ∘ g) x = tsnd (Comp.mk ∘ map g' ∘ g) x",
"tactic": "simp only [Function.comp, map_pure]"
},
{
"state_after": "no goals",
"state_before": "t : Type u → Type u → Type u\ninst✝⁵ : Bitraversable t\nβ : Type u\nF G : Type u → Type u\ninst✝⁴ : Applicative F\ninst✝³ : Applicative G\ninst✝² : IsLawfulBitraversable t\ninst✝¹ : LawfulApplicative F\ninst✝ : LawfulApplicative G\nα β₀ β₁ β₂ : Type u\ng : β₀ → F β₁\ng' : β₁ → G β₂\nx : t α β₀\n⊢ bitraverse (fun x => Comp.mk (pure (pure x))) (fun x => Comp.mk (g' <$> g x)) x =\n tsnd (fun x => Comp.mk (g' <$> g x)) x",
"tactic": "rfl"
}
] |
[
105,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
101,
1
] |
Mathlib/Algebra/Order/ToIntervalMod.lean
|
left_le_toIcoMod
|
[] |
[
99,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
98,
1
] |
Mathlib/Data/Set/Ncard.lean
|
Set.ncard_union_eq
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.122422\ns t : Set α\na b x y : α\nf : α → β\nh : Disjoint s t\nhs : autoParam (Set.Finite s) _auto✝\nht : autoParam (Set.Finite t) _auto✝\n⊢ ncard (s ∪ t) = ncard s + ncard t",
"tactic": "classical\nrw [ncard_eq_toFinset_card _ hs, ncard_eq_toFinset_card _ ht,\n ncard_eq_toFinset_card _ (hs.union ht), Finite.toFinset_union]\nrefine' Finset.card_union_eq _\nrwa [Finite.disjoint_toFinset]"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.122422\ns t : Set α\na b x y : α\nf : α → β\nh : Disjoint s t\nhs : autoParam (Set.Finite s) _auto✝\nht : autoParam (Set.Finite t) _auto✝\n⊢ Finset.card (Finite.toFinset ?hs ∪ Finite.toFinset ?ht) =\n Finset.card (Finite.toFinset hs) + Finset.card (Finite.toFinset ht)\n\ncase hs\nα : Type u_1\nβ : Type ?u.122422\ns t : Set α\na b x y : α\nf : α → β\nh : Disjoint s t\nhs : autoParam (Set.Finite s) _auto✝\nht : autoParam (Set.Finite t) _auto✝\n⊢ Set.Finite s\n\ncase ht\nα : Type u_1\nβ : Type ?u.122422\ns t : Set α\na b x y : α\nf : α → β\nh : Disjoint s t\nhs : autoParam (Set.Finite s) _auto✝\nht : autoParam (Set.Finite t) _auto✝\n⊢ Set.Finite t",
"state_before": "α : Type u_1\nβ : Type ?u.122422\ns t : Set α\na b x y : α\nf : α → β\nh : Disjoint s t\nhs : autoParam (Set.Finite s) _auto✝\nht : autoParam (Set.Finite t) _auto✝\n⊢ ncard (s ∪ t) = ncard s + ncard t",
"tactic": "rw [ncard_eq_toFinset_card _ hs, ncard_eq_toFinset_card _ ht,\n ncard_eq_toFinset_card _ (hs.union ht), Finite.toFinset_union]"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.122422\ns t : Set α\na b x y : α\nf : α → β\nh : Disjoint s t\nhs : autoParam (Set.Finite s) _auto✝\nht : autoParam (Set.Finite t) _auto✝\n⊢ Disjoint (Finite.toFinset hs) (Finite.toFinset ht)",
"state_before": "α : Type u_1\nβ : Type ?u.122422\ns t : Set α\na b x y : α\nf : α → β\nh : Disjoint s t\nhs : autoParam (Set.Finite s) _auto✝\nht : autoParam (Set.Finite t) _auto✝\n⊢ Finset.card (Finite.toFinset ?hs ∪ Finite.toFinset ?ht) =\n Finset.card (Finite.toFinset hs) + Finset.card (Finite.toFinset ht)\n\ncase hs\nα : Type u_1\nβ : Type ?u.122422\ns t : Set α\na b x y : α\nf : α → β\nh : Disjoint s t\nhs : autoParam (Set.Finite s) _auto✝\nht : autoParam (Set.Finite t) _auto✝\n⊢ Set.Finite s\n\ncase ht\nα : Type u_1\nβ : Type ?u.122422\ns t : Set α\na b x y : α\nf : α → β\nh : Disjoint s t\nhs : autoParam (Set.Finite s) _auto✝\nht : autoParam (Set.Finite t) _auto✝\n⊢ Set.Finite t",
"tactic": "refine' Finset.card_union_eq _"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.122422\ns t : Set α\na b x y : α\nf : α → β\nh : Disjoint s t\nhs : autoParam (Set.Finite s) _auto✝\nht : autoParam (Set.Finite t) _auto✝\n⊢ Disjoint (Finite.toFinset hs) (Finite.toFinset ht)",
"tactic": "rwa [Finite.disjoint_toFinset]"
}
] |
[
491,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
485,
1
] |
Mathlib/Data/Matrix/Block.lean
|
Matrix.blockDiagonal'_neg
|
[] |
[
745,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
743,
1
] |
Mathlib/Analysis/Complex/Basic.lean
|
Complex.comap_abs_nhds_zero
|
[] |
[
148,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
147,
1
] |
Mathlib/Computability/Ackermann.lean
|
ack_two
|
[
{
"state_after": "case zero\n\n⊢ ack 2 zero = 2 * zero + 3\n\ncase succ\nn : ℕ\nIH : ack 2 n = 2 * n + 3\n⊢ ack 2 (succ n) = 2 * succ n + 3",
"state_before": "n : ℕ\n⊢ ack 2 n = 2 * n + 3",
"tactic": "induction' n with n IH"
},
{
"state_after": "no goals",
"state_before": "case zero\n\n⊢ ack 2 zero = 2 * zero + 3",
"tactic": "rfl"
},
{
"state_after": "no goals",
"state_before": "case succ\nn : ℕ\nIH : ack 2 n = 2 * n + 3\n⊢ ack 2 (succ n) = 2 * succ n + 3",
"tactic": "simpa [mul_succ]"
}
] |
[
94,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
91,
1
] |
src/lean/Init/Prelude.lean
|
PLift.down_up
|
[] |
[
760,
71
] |
d5348dfac847a56a4595fb6230fd0708dcb4e7e9
|
https://github.com/leanprover/lean4
|
[
760,
1
] |
Mathlib/Algebra/GroupPower/Basic.lean
|
zpow_neg_one
|
[] |
[
301,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
300,
1
] |
Mathlib/Order/WellFounded.lean
|
Function.argmin_le
|
[] |
[
223,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
222,
1
] |
Mathlib/Algebra/Lie/Submodule.lean
|
LieIdeal.incl_idealRange
|
[
{
"state_after": "R : Type u\nL : Type v\nL' : Type w₂\nM : Type w\nM' : Type w₁\ninst✝¹² : CommRing R\ninst✝¹¹ : LieRing L\ninst✝¹⁰ : LieAlgebra R L\ninst✝⁹ : LieRing L'\ninst✝⁸ : LieAlgebra R L'\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\ninst✝³ : AddCommGroup M'\ninst✝² : Module R M'\ninst✝¹ : LieRingModule L M'\ninst✝ : LieModule R L M'\nf : L →ₗ⁅R⁆ L'\nI : LieIdeal R L\nJ : LieIdeal R L'\n⊢ ∃ N, ↑N = ↑I",
"state_before": "R : Type u\nL : Type v\nL' : Type w₂\nM : Type w\nM' : Type w₁\ninst✝¹² : CommRing R\ninst✝¹¹ : LieRing L\ninst✝¹⁰ : LieAlgebra R L\ninst✝⁹ : LieRing L'\ninst✝⁸ : LieAlgebra R L'\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\ninst✝³ : AddCommGroup M'\ninst✝² : Module R M'\ninst✝¹ : LieRingModule L M'\ninst✝ : LieModule R L M'\nf : L →ₗ⁅R⁆ L'\nI : LieIdeal R L\nJ : LieIdeal R L'\n⊢ LieHom.idealRange (incl I) = I",
"tactic": "rw [LieHom.idealRange_eq_lieSpan_range, ← LieSubalgebra.coe_to_submodule, ←\n LieSubmodule.coe_toSubmodule_eq_iff, incl_range, coe_to_lieSubalgebra_to_submodule,\n LieSubmodule.coe_lieSpan_submodule_eq_iff]"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nL : Type v\nL' : Type w₂\nM : Type w\nM' : Type w₁\ninst✝¹² : CommRing R\ninst✝¹¹ : LieRing L\ninst✝¹⁰ : LieAlgebra R L\ninst✝⁹ : LieRing L'\ninst✝⁸ : LieAlgebra R L'\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\ninst✝³ : AddCommGroup M'\ninst✝² : Module R M'\ninst✝¹ : LieRingModule L M'\ninst✝ : LieModule R L M'\nf : L →ₗ⁅R⁆ L'\nI : LieIdeal R L\nJ : LieIdeal R L'\n⊢ ∃ N, ↑N = ↑I",
"tactic": "use I"
}
] |
[
1143,
8
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1139,
1
] |
Mathlib/RingTheory/Valuation/Basic.lean
|
AddValuation.IsEquiv.comap
|
[] |
[
857,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
855,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/Biproducts.lean
|
CategoryTheory.Limits.biprod.hom_ext'
|
[] |
[
1431,
70
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1429,
1
] |
Mathlib/MeasureTheory/Group/Arithmetic.lean
|
measurable_mul_op
|
[] |
[
789,
5
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
788,
1
] |
Mathlib/Data/Real/Irrational.lean
|
Irrational.of_nat_add
|
[] |
[
247,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
246,
1
] |
Mathlib/Algebra/Order/Hom/Basic.lean
|
le_map_div_add_map_div
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.16670\nF : Type u_3\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.16682\nδ : Type ?u.16685\ninst✝³ : Group α\ninst✝² : AddCommSemigroup β\ninst✝¹ : LE β\ninst✝ : MulLEAddHomClass F α β\nf : F\na b c : α\n⊢ ↑f (a / c) ≤ ↑f (a / b) + ↑f (b / c)",
"tactic": "simpa only [div_mul_div_cancel'] using map_mul_le_add f (a / b) (b / c)"
}
] |
[
154,
76
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
152,
1
] |
Mathlib/Combinatorics/SimpleGraph/Connectivity.lean
|
SimpleGraph.Walk.exists_boundary_dart
|
[
{
"state_after": "case nil\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nu v : V\nS : Set V\nuS✝ : u ∈ S\nvS✝ : ¬v ∈ S\nu✝ : V\nuS : u✝ ∈ S\nvS : ¬u✝ ∈ S\n⊢ ∃ d, d ∈ darts nil ∧ d.fst ∈ S ∧ ¬d.snd ∈ S\n\ncase cons\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nu v : V\nS : Set V\nuS✝ : u ∈ S\nvS✝ : ¬v ∈ S\nx y w : V\na : Adj G x y\np' : Walk G y w\nih : y ∈ S → ¬w ∈ S → ∃ d, d ∈ darts p' ∧ d.fst ∈ S ∧ ¬d.snd ∈ S\nuS : x ∈ S\nvS : ¬w ∈ S\n⊢ ∃ d, d ∈ darts (cons a p') ∧ d.fst ∈ S ∧ ¬d.snd ∈ S",
"state_before": "V : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nu v : V\np : Walk G u v\nS : Set V\nuS : u ∈ S\nvS : ¬v ∈ S\n⊢ ∃ d, d ∈ darts p ∧ d.fst ∈ S ∧ ¬d.snd ∈ S",
"tactic": "induction' p with _ x y w a p' ih"
},
{
"state_after": "no goals",
"state_before": "case nil\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nu v : V\nS : Set V\nuS✝ : u ∈ S\nvS✝ : ¬v ∈ S\nu✝ : V\nuS : u✝ ∈ S\nvS : ¬u✝ ∈ S\n⊢ ∃ d, d ∈ darts nil ∧ d.fst ∈ S ∧ ¬d.snd ∈ S",
"tactic": "cases vS uS"
},
{
"state_after": "case pos\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nu v : V\nS : Set V\nuS✝ : u ∈ S\nvS✝ : ¬v ∈ S\nx y w : V\na : Adj G x y\np' : Walk G y w\nih : y ∈ S → ¬w ∈ S → ∃ d, d ∈ darts p' ∧ d.fst ∈ S ∧ ¬d.snd ∈ S\nuS : x ∈ S\nvS : ¬w ∈ S\nh : y ∈ S\n⊢ ∃ d, d ∈ darts (cons a p') ∧ d.fst ∈ S ∧ ¬d.snd ∈ S\n\ncase neg\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nu v : V\nS : Set V\nuS✝ : u ∈ S\nvS✝ : ¬v ∈ S\nx y w : V\na : Adj G x y\np' : Walk G y w\nih : y ∈ S → ¬w ∈ S → ∃ d, d ∈ darts p' ∧ d.fst ∈ S ∧ ¬d.snd ∈ S\nuS : x ∈ S\nvS : ¬w ∈ S\nh : ¬y ∈ S\n⊢ ∃ d, d ∈ darts (cons a p') ∧ d.fst ∈ S ∧ ¬d.snd ∈ S",
"state_before": "case cons\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nu v : V\nS : Set V\nuS✝ : u ∈ S\nvS✝ : ¬v ∈ S\nx y w : V\na : Adj G x y\np' : Walk G y w\nih : y ∈ S → ¬w ∈ S → ∃ d, d ∈ darts p' ∧ d.fst ∈ S ∧ ¬d.snd ∈ S\nuS : x ∈ S\nvS : ¬w ∈ S\n⊢ ∃ d, d ∈ darts (cons a p') ∧ d.fst ∈ S ∧ ¬d.snd ∈ S",
"tactic": "by_cases h : y ∈ S"
},
{
"state_after": "case pos.intro.intro\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nu v : V\nS : Set V\nuS✝ : u ∈ S\nvS✝ : ¬v ∈ S\nx y w : V\na : Adj G x y\np' : Walk G y w\nih : y ∈ S → ¬w ∈ S → ∃ d, d ∈ darts p' ∧ d.fst ∈ S ∧ ¬d.snd ∈ S\nuS : x ∈ S\nvS : ¬w ∈ S\nh : y ∈ S\nd : Dart G\nhd : d ∈ darts p'\nhcd : d.fst ∈ S ∧ ¬d.snd ∈ S\n⊢ ∃ d, d ∈ darts (cons a p') ∧ d.fst ∈ S ∧ ¬d.snd ∈ S",
"state_before": "case pos\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nu v : V\nS : Set V\nuS✝ : u ∈ S\nvS✝ : ¬v ∈ S\nx y w : V\na : Adj G x y\np' : Walk G y w\nih : y ∈ S → ¬w ∈ S → ∃ d, d ∈ darts p' ∧ d.fst ∈ S ∧ ¬d.snd ∈ S\nuS : x ∈ S\nvS : ¬w ∈ S\nh : y ∈ S\n⊢ ∃ d, d ∈ darts (cons a p') ∧ d.fst ∈ S ∧ ¬d.snd ∈ S",
"tactic": "obtain ⟨d, hd, hcd⟩ := ih h vS"
},
{
"state_after": "no goals",
"state_before": "case pos.intro.intro\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nu v : V\nS : Set V\nuS✝ : u ∈ S\nvS✝ : ¬v ∈ S\nx y w : V\na : Adj G x y\np' : Walk G y w\nih : y ∈ S → ¬w ∈ S → ∃ d, d ∈ darts p' ∧ d.fst ∈ S ∧ ¬d.snd ∈ S\nuS : x ∈ S\nvS : ¬w ∈ S\nh : y ∈ S\nd : Dart G\nhd : d ∈ darts p'\nhcd : d.fst ∈ S ∧ ¬d.snd ∈ S\n⊢ ∃ d, d ∈ darts (cons a p') ∧ d.fst ∈ S ∧ ¬d.snd ∈ S",
"tactic": "exact ⟨d, List.Mem.tail _ hd, hcd⟩"
},
{
"state_after": "no goals",
"state_before": "case neg\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nu v : V\nS : Set V\nuS✝ : u ∈ S\nvS✝ : ¬v ∈ S\nx y w : V\na : Adj G x y\np' : Walk G y w\nih : y ∈ S → ¬w ∈ S → ∃ d, d ∈ darts p' ∧ d.fst ∈ S ∧ ¬d.snd ∈ S\nuS : x ∈ S\nvS : ¬w ∈ S\nh : ¬y ∈ S\n⊢ ∃ d, d ∈ darts (cons a p') ∧ d.fst ∈ S ∧ ¬d.snd ∈ S",
"tactic": "exact ⟨⟨(x, y), a⟩, List.Mem.head _, uS, h⟩"
}
] |
[
1262,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1255,
1
] |
Mathlib/CategoryTheory/Equivalence.lean
|
CategoryTheory.IsEquivalence.ofIso_trans
|
[
{
"state_after": "C : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nF G H : C ⥤ D\ne : F ≅ G\ne' : G ≅ H\nhF : IsEquivalence F\n⊢ mk' (inverse F) ((unitIso ≪≫ hcomp e (Iso.refl (inverse F))) ≪≫ hcomp e' (Iso.refl (inverse F)))\n (hcomp (Iso.refl (inverse F)) e'.symm ≪≫ hcomp (Iso.refl (inverse F)) e.symm ≪≫ counitIso) =\n mk' (inverse F) (unitIso ≪≫ hcomp (e ≪≫ e') (Iso.refl (inverse F)))\n (hcomp (Iso.refl (inverse F)) (e'.symm ≪≫ e.symm) ≪≫ counitIso)",
"state_before": "C : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nF G H : C ⥤ D\ne : F ≅ G\ne' : G ≅ H\nhF : IsEquivalence F\n⊢ ofIso e' (ofIso e hF) = ofIso (e ≪≫ e') hF",
"tactic": "dsimp [ofIso]"
},
{
"state_after": "case e_unitIso.w.w.h\nC : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nF G H : C ⥤ D\ne : F ≅ G\ne' : G ≅ H\nhF : IsEquivalence F\nX : C\n⊢ (unitIso.hom.app X ≫ 𝟙 ((inverse F).obj (F.obj X)) ≫ (inverse F).map (e.hom.app X)) ≫\n 𝟙 ((inverse F).obj (G.obj X)) ≫ (inverse F).map (e'.hom.app X) =\n unitIso.hom.app X ≫ 𝟙 ((inverse F).obj (F.obj X)) ≫ (inverse F).map (e.hom.app X ≫ e'.hom.app X)\n\ncase e_counitIso.w.w.h\nC : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nF G H : C ⥤ D\ne : F ≅ G\ne' : G ≅ H\nhF : IsEquivalence F\nX : D\n⊢ (e'.inv.app ((inverse F).obj X) ≫ G.map (𝟙 ((inverse F).obj X))) ≫\n (e.inv.app ((inverse F).obj X) ≫ F.map (𝟙 ((inverse F).obj X))) ≫ counitIso.hom.app X =\n ((e'.inv.app ((inverse F).obj X) ≫ e.inv.app ((inverse F).obj X)) ≫ F.map (𝟙 ((inverse F).obj X))) ≫\n counitIso.hom.app X",
"state_before": "C : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nF G H : C ⥤ D\ne : F ≅ G\ne' : G ≅ H\nhF : IsEquivalence F\n⊢ mk' (inverse F) ((unitIso ≪≫ hcomp e (Iso.refl (inverse F))) ≪≫ hcomp e' (Iso.refl (inverse F)))\n (hcomp (Iso.refl (inverse F)) e'.symm ≪≫ hcomp (Iso.refl (inverse F)) e.symm ≪≫ counitIso) =\n mk' (inverse F) (unitIso ≪≫ hcomp (e ≪≫ e') (Iso.refl (inverse F)))\n (hcomp (Iso.refl (inverse F)) (e'.symm ≪≫ e.symm) ≪≫ counitIso)",
"tactic": "congr 1 <;> ext X <;> dsimp [NatIso.hcomp]"
},
{
"state_after": "no goals",
"state_before": "case e_unitIso.w.w.h\nC : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nF G H : C ⥤ D\ne : F ≅ G\ne' : G ≅ H\nhF : IsEquivalence F\nX : C\n⊢ (unitIso.hom.app X ≫ 𝟙 ((inverse F).obj (F.obj X)) ≫ (inverse F).map (e.hom.app X)) ≫\n 𝟙 ((inverse F).obj (G.obj X)) ≫ (inverse F).map (e'.hom.app X) =\n unitIso.hom.app X ≫ 𝟙 ((inverse F).obj (F.obj X)) ≫ (inverse F).map (e.hom.app X ≫ e'.hom.app X)",
"tactic": "simp only [id_comp, assoc, Functor.map_comp]"
},
{
"state_after": "no goals",
"state_before": "case e_counitIso.w.w.h\nC : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nF G H : C ⥤ D\ne : F ≅ G\ne' : G ≅ H\nhF : IsEquivalence F\nX : D\n⊢ (e'.inv.app ((inverse F).obj X) ≫ G.map (𝟙 ((inverse F).obj X))) ≫\n (e.inv.app ((inverse F).obj X) ≫ F.map (𝟙 ((inverse F).obj X))) ≫ counitIso.hom.app X =\n ((e'.inv.app ((inverse F).obj X) ≫ e.inv.app ((inverse F).obj X)) ≫ F.map (𝟙 ((inverse F).obj X))) ≫\n counitIso.hom.app X",
"tactic": "simp only [Functor.map_id, comp_id, id_comp, assoc]"
}
] |
[
641,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
636,
1
] |
Mathlib/RingTheory/PowerSeries/Basic.lean
|
Polynomial.coe_zero
|
[] |
[
2557,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2556,
1
] |
Mathlib/Tactic/NormNum/Basic.lean
|
Mathlib.Meta.NormNum.ble_eq_false
|
[
{
"state_after": "no goals",
"state_before": "x y : ℕ\n⊢ Nat.ble x y = false ↔ y < x",
"tactic": "rw [← Nat.not_le, ← Bool.not_eq_true, Nat.ble_eq]"
}
] |
[
672,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
671,
1
] |
Mathlib/Control/Bitraversable/Lemmas.lean
|
Bitraversable.comp_tfst
|
[
{
"state_after": "t : Type u → Type u → Type u\ninst✝⁵ : Bitraversable t\nβ✝ : Type u\nF G : Type u → Type u\ninst✝⁴ : Applicative F\ninst✝³ : Applicative G\ninst✝² : IsLawfulBitraversable t\ninst✝¹ : LawfulApplicative F\ninst✝ : LawfulApplicative G\nα₀ α₁ α₂ β : Type u\nf : α₀ → F α₁\nf' : α₁ → G α₂\nx : t α₀ β\n⊢ bitraverse (Comp.mk ∘ map f' ∘ f) (Comp.mk ∘ map pure ∘ pure) x = tfst (Comp.mk ∘ map f' ∘ f) x",
"state_before": "t : Type u → Type u → Type u\ninst✝⁵ : Bitraversable t\nβ✝ : Type u\nF G : Type u → Type u\ninst✝⁴ : Applicative F\ninst✝³ : Applicative G\ninst✝² : IsLawfulBitraversable t\ninst✝¹ : LawfulApplicative F\ninst✝ : LawfulApplicative G\nα₀ α₁ α₂ β : Type u\nf : α₀ → F α₁\nf' : α₁ → G α₂\nx : t α₀ β\n⊢ Comp.mk (tfst f' <$> tfst f x) = tfst (Comp.mk ∘ map f' ∘ f) x",
"tactic": "rw [← comp_bitraverse]"
},
{
"state_after": "no goals",
"state_before": "t : Type u → Type u → Type u\ninst✝⁵ : Bitraversable t\nβ✝ : Type u\nF G : Type u → Type u\ninst✝⁴ : Applicative F\ninst✝³ : Applicative G\ninst✝² : IsLawfulBitraversable t\ninst✝¹ : LawfulApplicative F\ninst✝ : LawfulApplicative G\nα₀ α₁ α₂ β : Type u\nf : α₀ → F α₁\nf' : α₁ → G α₂\nx : t α₀ β\n⊢ bitraverse (Comp.mk ∘ map f' ∘ f) (Comp.mk ∘ map pure ∘ pure) x = tfst (Comp.mk ∘ map f' ∘ f) x",
"tactic": "simp only [Function.comp, tfst, map_pure, Pure.pure]"
}
] |
[
81,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
78,
1
] |
Mathlib/Algebra/Algebra/Equiv.lean
|
AlgEquiv.symm_bijective
|
[] |
[
354,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
353,
1
] |
Mathlib/RingTheory/NonZeroDivisors.lean
|
eq_zero_of_ne_zero_of_mul_left_eq_zero
|
[] |
[
119,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
117,
1
] |
Std/Data/String/Lemmas.lean
|
String.Pos.ne_of_lt
|
[] |
[
121,
88
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
121,
1
] |
Mathlib/Analysis/NormedSpace/Basic.lean
|
interior_sphere
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.89830\nβ : Type ?u.89833\nγ : Type ?u.89836\nι : Type ?u.89839\ninst✝⁶ : NormedField α\ninst✝⁵ : SeminormedAddCommGroup β\nE : Type u_1\ninst✝⁴ : SeminormedAddCommGroup E\ninst✝³ : NormedSpace α E\nF : Type ?u.89870\ninst✝² : SeminormedAddCommGroup F\ninst✝¹ : NormedSpace α F\ninst✝ : NormedSpace ℝ E\nx : E\nr : ℝ\nhr : r ≠ 0\n⊢ interior (sphere x r) = ∅",
"tactic": "rw [← frontier_closedBall x hr, interior_frontier isClosed_ball]"
}
] |
[
160,
67
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
158,
1
] |
Mathlib/Data/Real/CauSeqCompletion.lean
|
CauSeq.Completion.ofRat_mul
|
[] |
[
158,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
156,
1
] |
Std/Logic.lean
|
Decidable.not_and
|
[] |
[
600,
81
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
599,
1
] |
Mathlib/Data/MvPolynomial/Supported.lean
|
MvPolynomial.supported_strictMono
|
[] |
[
135,
68
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
133,
1
] |
Mathlib/Data/Nat/Factors.lean
|
Nat.factors_add_two
|
[
{
"state_after": "no goals",
"state_before": "n : ℕ\n⊢ factors (n + 2) = minFac (n + 2) :: factors ((n + 2) / minFac (n + 2))",
"tactic": "rw [factors]"
}
] |
[
121,
94
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
120,
1
] |
Mathlib/Algebra/Lie/Normalizer.lean
|
LieSubmodule.comap_normalizer
|
[
{
"state_after": "case h\nR : Type u_1\nL : Type u_2\nM : Type u_4\nM' : Type u_3\ninst✝¹⁰ : CommRing R\ninst✝⁹ : LieRing L\ninst✝⁸ : LieAlgebra R L\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\ninst✝³ : AddCommGroup M'\ninst✝² : Module R M'\ninst✝¹ : LieRingModule L M'\ninst✝ : LieModule R L M'\nN N₁ N₂ : LieSubmodule R L M\nf : M' →ₗ⁅R,L⁆ M\nm✝ : M'\n⊢ m✝ ∈ comap f (normalizer N) ↔ m✝ ∈ normalizer (comap f N)",
"state_before": "R : Type u_1\nL : Type u_2\nM : Type u_4\nM' : Type u_3\ninst✝¹⁰ : CommRing R\ninst✝⁹ : LieRing L\ninst✝⁸ : LieAlgebra R L\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\ninst✝³ : AddCommGroup M'\ninst✝² : Module R M'\ninst✝¹ : LieRingModule L M'\ninst✝ : LieModule R L M'\nN N₁ N₂ : LieSubmodule R L M\nf : M' →ₗ⁅R,L⁆ M\n⊢ comap f (normalizer N) = normalizer (comap f N)",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case h\nR : Type u_1\nL : Type u_2\nM : Type u_4\nM' : Type u_3\ninst✝¹⁰ : CommRing R\ninst✝⁹ : LieRing L\ninst✝⁸ : LieAlgebra R L\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\ninst✝³ : AddCommGroup M'\ninst✝² : Module R M'\ninst✝¹ : LieRingModule L M'\ninst✝ : LieModule R L M'\nN N₁ N₂ : LieSubmodule R L M\nf : M' →ₗ⁅R,L⁆ M\nm✝ : M'\n⊢ m✝ ∈ comap f (normalizer N) ↔ m✝ ∈ normalizer (comap f N)",
"tactic": "simp"
}
] |
[
86,
12
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
85,
1
] |
Mathlib/Topology/Algebra/Module/Basic.lean
|
ContinuousLinearEquiv.map_nhds_eq
|
[] |
[
1851,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1850,
1
] |
Mathlib/Data/List/Rotate.lean
|
List.rotate_eq_rotate'
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nl : List α\nn : ℕ\nh : length l = 0\n⊢ rotate l n = rotate' l n",
"tactic": "simp_all [length_eq_zero]"
},
{
"state_after": "α : Type u\nl : List α\nn : ℕ\nh : ¬length l = 0\n⊢ rotate l n = drop (n % length l) l ++ take (n % length l) l",
"state_before": "α : Type u\nl : List α\nn : ℕ\nh : ¬length l = 0\n⊢ rotate l n = rotate' l n",
"tactic": "rw [← rotate'_mod,\n rotate'_eq_drop_append_take (le_of_lt (Nat.mod_lt _ (Nat.pos_of_ne_zero h)))]"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nl : List α\nn : ℕ\nh : ¬length l = 0\n⊢ rotate l n = drop (n % length l) l ++ take (n % length l) l",
"tactic": "simp [rotate]"
}
] |
[
116,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
111,
1
] |
Mathlib/Data/Fin/Tuple/NatAntidiagonal.lean
|
Finset.Nat.antidiagonalTuple_zero_zero
|
[] |
[
238,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
237,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/StrongEpi.lean
|
CategoryTheory.strongEpi_comp
|
[
{
"state_after": "C : Type u\ninst✝³ : Category C\nP Q R : C\nf : P ⟶ Q\ng : Q ⟶ R\ninst✝² : StrongEpi f\ninst✝¹ : StrongEpi g\nX✝ Y✝ : C\nz✝ : X✝ ⟶ Y✝\ninst✝ : Mono z✝\n⊢ HasLiftingProperty (f ≫ g) z✝",
"state_before": "C : Type u\ninst✝² : Category C\nP Q R : C\nf : P ⟶ Q\ng : Q ⟶ R\ninst✝¹ : StrongEpi f\ninst✝ : StrongEpi g\n⊢ ∀ ⦃X Y : C⦄ (z : X ⟶ Y) [inst : Mono z], HasLiftingProperty (f ≫ g) z",
"tactic": "intros"
},
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝³ : Category C\nP Q R : C\nf : P ⟶ Q\ng : Q ⟶ R\ninst✝² : StrongEpi f\ninst✝¹ : StrongEpi g\nX✝ Y✝ : C\nz✝ : X✝ ⟶ Y✝\ninst✝ : Mono z✝\n⊢ HasLiftingProperty (f ≫ g) z✝",
"tactic": "infer_instance"
}
] |
[
106,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
102,
1
] |
Mathlib/Order/Basic.lean
|
GE.ge.le
|
[] |
[
324,
4
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
323,
11
] |
Mathlib/Order/SymmDiff.lean
|
himp_bihimp
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.45050\nα : Type u_1\nβ : Type ?u.45056\nπ : ι → Type ?u.45061\ninst✝ : GeneralizedHeytingAlgebra α\na b c d : α\n⊢ a ⇨ b ⇔ c = (a ⊓ c ⇨ b) ⊓ (a ⊓ b ⇨ c)",
"tactic": "rw [bihimp, himp_inf_distrib, himp_himp, himp_himp]"
}
] |
[
291,
54
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
290,
1
] |
Mathlib/Data/Rat/Cast.lean
|
Rat.cast_inj
|
[
{
"state_after": "F : Type ?u.40042\nι : Type ?u.40045\nα : Type u_1\nβ : Type ?u.40051\ninst✝¹ : DivisionRing α\ninst✝ : CharZero α\nn₁ : ℤ\nd₁ : ℕ\nd₁0 : d₁ ≠ 0\nc₁ : Nat.coprime (Int.natAbs n₁) d₁\nn₂ : ℤ\nd₂ : ℕ\nd₂0 : d₂ ≠ 0\nc₂ : Nat.coprime (Int.natAbs n₂) d₂\nh : ↑(mk' n₁ d₁) = ↑(mk' n₂ d₂)\n⊢ mk' n₁ d₁ = mk' n₂ d₂",
"state_before": "F : Type ?u.40042\nι : Type ?u.40045\nα : Type u_1\nβ : Type ?u.40051\ninst✝¹ : DivisionRing α\ninst✝ : CharZero α\nn₁ : ℤ\nd₁ : ℕ\nd₁0 : d₁ ≠ 0\nc₁ : Nat.coprime (Int.natAbs n₁) d₁\nn₂ : ℤ\nd₂ : ℕ\nd₂0 : d₂ ≠ 0\nc₂ : Nat.coprime (Int.natAbs n₂) d₂\n⊢ ↑(mk' n₁ d₁) = ↑(mk' n₂ d₂) ↔ mk' n₁ d₁ = mk' n₂ d₂",
"tactic": "refine' ⟨fun h => _, congr_arg _⟩"
},
{
"state_after": "F : Type ?u.40042\nι : Type ?u.40045\nα : Type u_1\nβ : Type ?u.40051\ninst✝¹ : DivisionRing α\ninst✝ : CharZero α\nn₁ : ℤ\nd₁ : ℕ\nd₁0 : d₁ ≠ 0\nc₁ : Nat.coprime (Int.natAbs n₁) d₁\nn₂ : ℤ\nd₂ : ℕ\nd₂0 : d₂ ≠ 0\nc₂ : Nat.coprime (Int.natAbs n₂) d₂\nh : ↑(mk' n₁ d₁) = ↑(mk' n₂ d₂)\nd₁a : ↑d₁ ≠ 0\n⊢ mk' n₁ d₁ = mk' n₂ d₂",
"state_before": "F : Type ?u.40042\nι : Type ?u.40045\nα : Type u_1\nβ : Type ?u.40051\ninst✝¹ : DivisionRing α\ninst✝ : CharZero α\nn₁ : ℤ\nd₁ : ℕ\nd₁0 : d₁ ≠ 0\nc₁ : Nat.coprime (Int.natAbs n₁) d₁\nn₂ : ℤ\nd₂ : ℕ\nd₂0 : d₂ ≠ 0\nc₂ : Nat.coprime (Int.natAbs n₂) d₂\nh : ↑(mk' n₁ d₁) = ↑(mk' n₂ d₂)\n⊢ mk' n₁ d₁ = mk' n₂ d₂",
"tactic": "have d₁a : (d₁ : α) ≠ 0 := Nat.cast_ne_zero.2 d₁0"
},
{
"state_after": "F : Type ?u.40042\nι : Type ?u.40045\nα : Type u_1\nβ : Type ?u.40051\ninst✝¹ : DivisionRing α\ninst✝ : CharZero α\nn₁ : ℤ\nd₁ : ℕ\nd₁0 : d₁ ≠ 0\nc₁ : Nat.coprime (Int.natAbs n₁) d₁\nn₂ : ℤ\nd₂ : ℕ\nd₂0 : d₂ ≠ 0\nc₂ : Nat.coprime (Int.natAbs n₂) d₂\nh : ↑(mk' n₁ d₁) = ↑(mk' n₂ d₂)\nd₁a : ↑d₁ ≠ 0\nd₂a : ↑d₂ ≠ 0\n⊢ mk' n₁ d₁ = mk' n₂ d₂",
"state_before": "F : Type ?u.40042\nι : Type ?u.40045\nα : Type u_1\nβ : Type ?u.40051\ninst✝¹ : DivisionRing α\ninst✝ : CharZero α\nn₁ : ℤ\nd₁ : ℕ\nd₁0 : d₁ ≠ 0\nc₁ : Nat.coprime (Int.natAbs n₁) d₁\nn₂ : ℤ\nd₂ : ℕ\nd₂0 : d₂ ≠ 0\nc₂ : Nat.coprime (Int.natAbs n₂) d₂\nh : ↑(mk' n₁ d₁) = ↑(mk' n₂ d₂)\nd₁a : ↑d₁ ≠ 0\n⊢ mk' n₁ d₁ = mk' n₂ d₂",
"tactic": "have d₂a : (d₂ : α) ≠ 0 := Nat.cast_ne_zero.2 d₂0"
},
{
"state_after": "F : Type ?u.40042\nι : Type ?u.40045\nα : Type u_1\nβ : Type ?u.40051\ninst✝¹ : DivisionRing α\ninst✝ : CharZero α\nn₁ : ℤ\nd₁ : ℕ\nd₁0 : d₁ ≠ 0\nc₁ : Nat.coprime (Int.natAbs n₁) d₁\nn₂ : ℤ\nd₂ : ℕ\nd₂0 : d₂ ≠ 0\nc₂ : Nat.coprime (Int.natAbs n₂) d₂\nh : ↑(n₁ /. ↑d₁) = ↑(n₂ /. ↑d₂)\nd₁a : ↑d₁ ≠ 0\nd₂a : ↑d₂ ≠ 0\n⊢ n₁ /. ↑d₁ = n₂ /. ↑d₂",
"state_before": "F : Type ?u.40042\nι : Type ?u.40045\nα : Type u_1\nβ : Type ?u.40051\ninst✝¹ : DivisionRing α\ninst✝ : CharZero α\nn₁ : ℤ\nd₁ : ℕ\nd₁0 : d₁ ≠ 0\nc₁ : Nat.coprime (Int.natAbs n₁) d₁\nn₂ : ℤ\nd₂ : ℕ\nd₂0 : d₂ ≠ 0\nc₂ : Nat.coprime (Int.natAbs n₂) d₂\nh : ↑(mk' n₁ d₁) = ↑(mk' n₂ d₂)\nd₁a : ↑d₁ ≠ 0\nd₂a : ↑d₂ ≠ 0\n⊢ mk' n₁ d₁ = mk' n₂ d₂",
"tactic": "rw [num_den', num_den'] at h⊢"
},
{
"state_after": "F : Type ?u.40042\nι : Type ?u.40045\nα : Type u_1\nβ : Type ?u.40051\ninst✝¹ : DivisionRing α\ninst✝ : CharZero α\nn₁ : ℤ\nd₁ : ℕ\nd₁0 : d₁ ≠ 0\nc₁ : Nat.coprime (Int.natAbs n₁) d₁\nn₂ : ℤ\nd₂ : ℕ\nd₂0 : d₂ ≠ 0\nc₂ : Nat.coprime (Int.natAbs n₂) d₂\nd₁a : ↑d₁ ≠ 0\nd₂a : ↑d₂ ≠ 0\nh : ↑n₁ / ↑d₁ = ↑n₂ / ↑d₂\n⊢ mkRat n₁ d₁ = mkRat n₂ d₂",
"state_before": "F : Type ?u.40042\nι : Type ?u.40045\nα : Type u_1\nβ : Type ?u.40051\ninst✝¹ : DivisionRing α\ninst✝ : CharZero α\nn₁ : ℤ\nd₁ : ℕ\nd₁0 : d₁ ≠ 0\nc₁ : Nat.coprime (Int.natAbs n₁) d₁\nn₂ : ℤ\nd₂ : ℕ\nd₂0 : d₂ ≠ 0\nc₂ : Nat.coprime (Int.natAbs n₂) d₂\nh : ↑(n₁ /. ↑d₁) = ↑(n₂ /. ↑d₂)\nd₁a : ↑d₁ ≠ 0\nd₂a : ↑d₂ ≠ 0\n⊢ n₁ /. ↑d₁ = n₂ /. ↑d₂",
"tactic": "rw [cast_mk_of_ne_zero, cast_mk_of_ne_zero] at h <;> simp [d₁0, d₂0] at h⊢"
},
{
"state_after": "no goals",
"state_before": "F : Type ?u.40042\nι : Type ?u.40045\nα : Type u_1\nβ : Type ?u.40051\ninst✝¹ : DivisionRing α\ninst✝ : CharZero α\nn₁ : ℤ\nd₁ : ℕ\nd₁0 : d₁ ≠ 0\nc₁ : Nat.coprime (Int.natAbs n₁) d₁\nn₂ : ℤ\nd₂ : ℕ\nd₂0 : d₂ ≠ 0\nc₂ : Nat.coprime (Int.natAbs n₂) d₂\nd₁a : ↑d₁ ≠ 0\nd₂a : ↑d₂ ≠ 0\nh : ↑n₁ / ↑d₁ = ↑n₂ / ↑d₂\n⊢ mkRat n₁ d₁ = mkRat n₂ d₂",
"tactic": "rwa [eq_div_iff_mul_eq d₂a, division_def, mul_assoc, (d₁.cast_commute (d₂ : α)).inv_left₀.eq, ←\n mul_assoc, ← division_def, eq_comm, eq_div_iff_mul_eq d₁a, eq_comm, ← Int.cast_ofNat d₁, ←\n Int.cast_mul, ← Int.cast_ofNat d₂, ← Int.cast_mul, Int.cast_inj, ← mkRat_eq_iff d₁0 d₂0] at h"
}
] |
[
203,
100
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
194,
1
] |
Mathlib/Analysis/Convex/Join.lean
|
convexJoin_comm
|
[
{
"state_after": "no goals",
"state_before": "ι : Sort ?u.12394\n𝕜 : Type u_2\nE : Type u_1\ninst✝² : OrderedSemiring 𝕜\ninst✝¹ : AddCommMonoid E\ninst✝ : Module 𝕜 E\ns✝ t✝ s₁ s₂ t₁ t₂ u : Set E\nx y : E\ns t : Set E\n⊢ (⋃ (i₂ : E) (_ : i₂ ∈ t) (i₁ : E) (_ : i₁ ∈ s), segment 𝕜 i₁ i₂) = convexJoin 𝕜 t s",
"tactic": "simp_rw [convexJoin, segment_symm]"
}
] |
[
46,
66
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
45,
1
] |
Mathlib/CategoryTheory/Limits/IsLimit.lean
|
CategoryTheory.Limits.IsLimit.lift_comp_conePointsIsoOfNatIso_inv
|
[
{
"state_after": "no goals",
"state_before": "J : Type u₁\ninst✝² : Category J\nK : Type u₂\ninst✝¹ : Category K\nC : Type u₃\ninst✝ : Category C\nF✝ : J ⥤ C\nt✝ : Cone F✝\nF G : J ⥤ C\nr s : Cone G\nt : Cone F\nP : IsLimit t\nQ : IsLimit s\nw : F ≅ G\n⊢ ∀ (j : J), (lift Q r ≫ (conePointsIsoOfNatIso P Q w).inv) ≫ t.π.app j = map r P w.inv ≫ t.π.app j",
"tactic": "simp"
}
] |
[
337,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
334,
1
] |
Mathlib/Algebra/Group/Basic.lean
|
div_eq_one
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.61400\nβ : Type ?u.61403\nG : Type u_1\ninst✝ : Group G\na b c d : G\nh : a = b\n⊢ a / b = 1",
"tactic": "rw [h, div_self']"
}
] |
[
816,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
815,
1
] |
Mathlib/MeasureTheory/Covering/BesicovitchVectorSpace.lean
|
Besicovitch.multiplicity_le
|
[
{
"state_after": "case h₁\nE : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : FiniteDimensional ℝ E\n⊢ Set.Nonempty\n {N |\n ∃ s, Finset.card s = N ∧ (∀ (c : E), c ∈ s → ‖c‖ ≤ 2) ∧ ∀ (c : E), c ∈ s → ∀ (d : E), d ∈ s → c ≠ d → 1 ≤ ‖c - d‖}\n\ncase h₂\nE : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : FiniteDimensional ℝ E\n⊢ ∀ (b : ℕ),\n b ∈\n {N |\n ∃ s,\n Finset.card s = N ∧\n (∀ (c : E), c ∈ s → ‖c‖ ≤ 2) ∧ ∀ (c : E), c ∈ s → ∀ (d : E), d ∈ s → c ≠ d → 1 ≤ ‖c - d‖} →\n b ≤ 5 ^ finrank ℝ E",
"state_before": "E : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : FiniteDimensional ℝ E\n⊢ multiplicity E ≤ 5 ^ finrank ℝ E",
"tactic": "apply csSup_le"
},
{
"state_after": "no goals",
"state_before": "case h₁\nE : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : FiniteDimensional ℝ E\n⊢ Set.Nonempty\n {N |\n ∃ s, Finset.card s = N ∧ (∀ (c : E), c ∈ s → ‖c‖ ≤ 2) ∧ ∀ (c : E), c ∈ s → ∀ (d : E), d ∈ s → c ≠ d → 1 ≤ ‖c - d‖}",
"tactic": "refine' ⟨0, ⟨∅, by simp⟩⟩"
},
{
"state_after": "no goals",
"state_before": "E : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : FiniteDimensional ℝ E\n⊢ Finset.card ∅ = 0 ∧ (∀ (c : E), c ∈ ∅ → ‖c‖ ≤ 2) ∧ ∀ (c : E), c ∈ ∅ → ∀ (d : E), d ∈ ∅ → c ≠ d → 1 ≤ ‖c - d‖",
"tactic": "simp"
},
{
"state_after": "case h₂.intro.intro\nE : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : FiniteDimensional ℝ E\ns : Finset E\nh : (∀ (c : E), c ∈ s → ‖c‖ ≤ 2) ∧ ∀ (c : E), c ∈ s → ∀ (d : E), d ∈ s → c ≠ d → 1 ≤ ‖c - d‖\n⊢ Finset.card s ≤ 5 ^ finrank ℝ E",
"state_before": "case h₂\nE : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : FiniteDimensional ℝ E\n⊢ ∀ (b : ℕ),\n b ∈\n {N |\n ∃ s,\n Finset.card s = N ∧\n (∀ (c : E), c ∈ s → ‖c‖ ≤ 2) ∧ ∀ (c : E), c ∈ s → ∀ (d : E), d ∈ s → c ≠ d → 1 ≤ ‖c - d‖} →\n b ≤ 5 ^ finrank ℝ E",
"tactic": "rintro _ ⟨s, ⟨rfl, h⟩⟩"
},
{
"state_after": "no goals",
"state_before": "case h₂.intro.intro\nE : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : FiniteDimensional ℝ E\ns : Finset E\nh : (∀ (c : E), c ∈ s → ‖c‖ ≤ 2) ∧ ∀ (c : E), c ∈ s → ∀ (d : E), d ∈ s → c ≠ d → 1 ≤ ‖c - d‖\n⊢ Finset.card s ≤ 5 ^ finrank ℝ E",
"tactic": "exact Besicovitch.card_le_of_separated s h.1 h.2"
}
] |
[
193,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
189,
1
] |
Mathlib/AlgebraicTopology/FundamentalGroupoid/Basic.lean
|
Path.Homotopy.reflTransSymmAux_mem_I
|
[
{
"state_after": "X : Type u\nY : Type v\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nx₀ x₁ : X\nx : ↑I × ↑I\n⊢ (if ↑x.snd ≤ 1 / 2 then ↑x.fst * 2 * ↑x.snd else ↑x.fst * (2 - 2 * ↑x.snd)) ∈ I",
"state_before": "X : Type u\nY : Type v\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nx₀ x₁ : X\nx : ↑I × ↑I\n⊢ reflTransSymmAux x ∈ I",
"tactic": "dsimp only [reflTransSymmAux]"
},
{
"state_after": "case inl\nX : Type u\nY : Type v\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nx₀ x₁ : X\nx : ↑I × ↑I\nh✝ : ↑x.snd ≤ 1 / 2\n⊢ ↑x.fst * 2 * ↑x.snd ∈ I\n\ncase inr\nX : Type u\nY : Type v\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nx₀ x₁ : X\nx : ↑I × ↑I\nh✝ : ¬↑x.snd ≤ 1 / 2\n⊢ ↑x.fst * (2 - 2 * ↑x.snd) ∈ I",
"state_before": "X : Type u\nY : Type v\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nx₀ x₁ : X\nx : ↑I × ↑I\n⊢ (if ↑x.snd ≤ 1 / 2 then ↑x.fst * 2 * ↑x.snd else ↑x.fst * (2 - 2 * ↑x.snd)) ∈ I",
"tactic": "split_ifs"
},
{
"state_after": "case inl.left\nX : Type u\nY : Type v\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nx₀ x₁ : X\nx : ↑I × ↑I\nh✝ : ↑x.snd ≤ 1 / 2\n⊢ 0 ≤ ↑x.fst * 2 * ↑x.snd\n\ncase inl.right\nX : Type u\nY : Type v\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nx₀ x₁ : X\nx : ↑I × ↑I\nh✝ : ↑x.snd ≤ 1 / 2\n⊢ ↑x.fst * 2 * ↑x.snd ≤ 1",
"state_before": "case inl\nX : Type u\nY : Type v\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nx₀ x₁ : X\nx : ↑I × ↑I\nh✝ : ↑x.snd ≤ 1 / 2\n⊢ ↑x.fst * 2 * ↑x.snd ∈ I",
"tactic": "constructor"
},
{
"state_after": "case inl.left.ha\nX : Type u\nY : Type v\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nx₀ x₁ : X\nx : ↑I × ↑I\nh✝ : ↑x.snd ≤ 1 / 2\n⊢ 0 ≤ ↑x.fst * 2\n\ncase inl.left.hb\nX : Type u\nY : Type v\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nx₀ x₁ : X\nx : ↑I × ↑I\nh✝ : ↑x.snd ≤ 1 / 2\n⊢ 0 ≤ ↑x.snd",
"state_before": "case inl.left\nX : Type u\nY : Type v\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nx₀ x₁ : X\nx : ↑I × ↑I\nh✝ : ↑x.snd ≤ 1 / 2\n⊢ 0 ≤ ↑x.fst * 2 * ↑x.snd",
"tactic": "apply mul_nonneg"
},
{
"state_after": "case inl.left.ha.ha\nX : Type u\nY : Type v\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nx₀ x₁ : X\nx : ↑I × ↑I\nh✝ : ↑x.snd ≤ 1 / 2\n⊢ 0 ≤ ↑x.fst\n\ncase inl.left.ha.hb\nX : Type u\nY : Type v\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nx₀ x₁ : X\nx : ↑I × ↑I\nh✝ : ↑x.snd ≤ 1 / 2\n⊢ 0 ≤ 2",
"state_before": "case inl.left.ha\nX : Type u\nY : Type v\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nx₀ x₁ : X\nx : ↑I × ↑I\nh✝ : ↑x.snd ≤ 1 / 2\n⊢ 0 ≤ ↑x.fst * 2",
"tactic": "apply mul_nonneg"
},
{
"state_after": "no goals",
"state_before": "case inl.left.ha.ha\nX : Type u\nY : Type v\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nx₀ x₁ : X\nx : ↑I × ↑I\nh✝ : ↑x.snd ≤ 1 / 2\n⊢ 0 ≤ ↑x.fst",
"tactic": "unit_interval"
},
{
"state_after": "no goals",
"state_before": "case inl.left.ha.hb\nX : Type u\nY : Type v\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nx₀ x₁ : X\nx : ↑I × ↑I\nh✝ : ↑x.snd ≤ 1 / 2\n⊢ 0 ≤ 2",
"tactic": "norm_num"
},
{
"state_after": "no goals",
"state_before": "case inl.left.hb\nX : Type u\nY : Type v\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nx₀ x₁ : X\nx : ↑I × ↑I\nh✝ : ↑x.snd ≤ 1 / 2\n⊢ 0 ≤ ↑x.snd",
"tactic": "unit_interval"
},
{
"state_after": "case inl.right\nX : Type u\nY : Type v\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nx₀ x₁ : X\nx : ↑I × ↑I\nh✝ : ↑x.snd ≤ 1 / 2\n⊢ ↑x.fst * (2 * ↑x.snd) ≤ 1",
"state_before": "case inl.right\nX : Type u\nY : Type v\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nx₀ x₁ : X\nx : ↑I × ↑I\nh✝ : ↑x.snd ≤ 1 / 2\n⊢ ↑x.fst * 2 * ↑x.snd ≤ 1",
"tactic": "rw [mul_assoc]"
},
{
"state_after": "case inl.right.ha\nX : Type u\nY : Type v\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nx₀ x₁ : X\nx : ↑I × ↑I\nh✝ : ↑x.snd ≤ 1 / 2\n⊢ ↑x.fst ≤ 1\n\ncase inl.right.hb'\nX : Type u\nY : Type v\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nx₀ x₁ : X\nx : ↑I × ↑I\nh✝ : ↑x.snd ≤ 1 / 2\n⊢ 0 ≤ 2 * ↑x.snd\n\ncase inl.right.hb\nX : Type u\nY : Type v\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nx₀ x₁ : X\nx : ↑I × ↑I\nh✝ : ↑x.snd ≤ 1 / 2\n⊢ 2 * ↑x.snd ≤ 1",
"state_before": "case inl.right\nX : Type u\nY : Type v\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nx₀ x₁ : X\nx : ↑I × ↑I\nh✝ : ↑x.snd ≤ 1 / 2\n⊢ ↑x.fst * (2 * ↑x.snd) ≤ 1",
"tactic": "apply mul_le_one"
},
{
"state_after": "no goals",
"state_before": "case inl.right.ha\nX : Type u\nY : Type v\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nx₀ x₁ : X\nx : ↑I × ↑I\nh✝ : ↑x.snd ≤ 1 / 2\n⊢ ↑x.fst ≤ 1",
"tactic": "unit_interval"
},
{
"state_after": "case inl.right.hb'.ha\nX : Type u\nY : Type v\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nx₀ x₁ : X\nx : ↑I × ↑I\nh✝ : ↑x.snd ≤ 1 / 2\n⊢ 0 ≤ 2\n\ncase inl.right.hb'.hb\nX : Type u\nY : Type v\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nx₀ x₁ : X\nx : ↑I × ↑I\nh✝ : ↑x.snd ≤ 1 / 2\n⊢ 0 ≤ ↑x.snd",
"state_before": "case inl.right.hb'\nX : Type u\nY : Type v\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nx₀ x₁ : X\nx : ↑I × ↑I\nh✝ : ↑x.snd ≤ 1 / 2\n⊢ 0 ≤ 2 * ↑x.snd",
"tactic": "apply mul_nonneg"
},
{
"state_after": "no goals",
"state_before": "case inl.right.hb'.ha\nX : Type u\nY : Type v\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nx₀ x₁ : X\nx : ↑I × ↑I\nh✝ : ↑x.snd ≤ 1 / 2\n⊢ 0 ≤ 2",
"tactic": "norm_num"
},
{
"state_after": "no goals",
"state_before": "case inl.right.hb'.hb\nX : Type u\nY : Type v\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nx₀ x₁ : X\nx : ↑I × ↑I\nh✝ : ↑x.snd ≤ 1 / 2\n⊢ 0 ≤ ↑x.snd",
"tactic": "unit_interval"
},
{
"state_after": "no goals",
"state_before": "case inl.right.hb\nX : Type u\nY : Type v\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nx₀ x₁ : X\nx : ↑I × ↑I\nh✝ : ↑x.snd ≤ 1 / 2\n⊢ 2 * ↑x.snd ≤ 1",
"tactic": "linarith"
},
{
"state_after": "case inr.left\nX : Type u\nY : Type v\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nx₀ x₁ : X\nx : ↑I × ↑I\nh✝ : ¬↑x.snd ≤ 1 / 2\n⊢ 0 ≤ ↑x.fst * (2 - 2 * ↑x.snd)\n\ncase inr.right\nX : Type u\nY : Type v\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nx₀ x₁ : X\nx : ↑I × ↑I\nh✝ : ¬↑x.snd ≤ 1 / 2\n⊢ ↑x.fst * (2 - 2 * ↑x.snd) ≤ 1",
"state_before": "case inr\nX : Type u\nY : Type v\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nx₀ x₁ : X\nx : ↑I × ↑I\nh✝ : ¬↑x.snd ≤ 1 / 2\n⊢ ↑x.fst * (2 - 2 * ↑x.snd) ∈ I",
"tactic": "constructor"
},
{
"state_after": "case inr.left.ha\nX : Type u\nY : Type v\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nx₀ x₁ : X\nx : ↑I × ↑I\nh✝ : ¬↑x.snd ≤ 1 / 2\n⊢ 0 ≤ ↑x.fst\n\ncase inr.left.hb\nX : Type u\nY : Type v\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nx₀ x₁ : X\nx : ↑I × ↑I\nh✝ : ¬↑x.snd ≤ 1 / 2\n⊢ 0 ≤ 2 - 2 * ↑x.snd",
"state_before": "case inr.left\nX : Type u\nY : Type v\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nx₀ x₁ : X\nx : ↑I × ↑I\nh✝ : ¬↑x.snd ≤ 1 / 2\n⊢ 0 ≤ ↑x.fst * (2 - 2 * ↑x.snd)",
"tactic": "apply mul_nonneg"
},
{
"state_after": "no goals",
"state_before": "case inr.left.hb\nX : Type u\nY : Type v\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nx₀ x₁ : X\nx : ↑I × ↑I\nh✝ : ¬↑x.snd ≤ 1 / 2\n⊢ 0 ≤ 2 - 2 * ↑x.snd",
"tactic": "linarith [unitInterval.nonneg x.2, unitInterval.le_one x.2]"
},
{
"state_after": "no goals",
"state_before": "case inr.left.ha\nX : Type u\nY : Type v\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nx₀ x₁ : X\nx : ↑I × ↑I\nh✝ : ¬↑x.snd ≤ 1 / 2\n⊢ 0 ≤ ↑x.fst",
"tactic": "unit_interval"
},
{
"state_after": "case inr.right.ha\nX : Type u\nY : Type v\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nx₀ x₁ : X\nx : ↑I × ↑I\nh✝ : ¬↑x.snd ≤ 1 / 2\n⊢ ↑x.fst ≤ 1\n\ncase inr.right.hb'\nX : Type u\nY : Type v\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nx₀ x₁ : X\nx : ↑I × ↑I\nh✝ : ¬↑x.snd ≤ 1 / 2\n⊢ 0 ≤ 2 - 2 * ↑x.snd\n\ncase inr.right.hb\nX : Type u\nY : Type v\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nx₀ x₁ : X\nx : ↑I × ↑I\nh✝ : ¬↑x.snd ≤ 1 / 2\n⊢ 2 - 2 * ↑x.snd ≤ 1",
"state_before": "case inr.right\nX : Type u\nY : Type v\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nx₀ x₁ : X\nx : ↑I × ↑I\nh✝ : ¬↑x.snd ≤ 1 / 2\n⊢ ↑x.fst * (2 - 2 * ↑x.snd) ≤ 1",
"tactic": "apply mul_le_one"
},
{
"state_after": "no goals",
"state_before": "case inr.right.ha\nX : Type u\nY : Type v\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nx₀ x₁ : X\nx : ↑I × ↑I\nh✝ : ¬↑x.snd ≤ 1 / 2\n⊢ ↑x.fst ≤ 1",
"tactic": "unit_interval"
},
{
"state_after": "no goals",
"state_before": "case inr.right.hb'\nX : Type u\nY : Type v\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nx₀ x₁ : X\nx : ↑I × ↑I\nh✝ : ¬↑x.snd ≤ 1 / 2\n⊢ 0 ≤ 2 - 2 * ↑x.snd",
"tactic": "linarith [unitInterval.nonneg x.2, unitInterval.le_one x.2]"
},
{
"state_after": "no goals",
"state_before": "case inr.right.hb\nX : Type u\nY : Type v\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nx₀ x₁ : X\nx : ↑I × ↑I\nh✝ : ¬↑x.snd ≤ 1 / 2\n⊢ 2 - 2 * ↑x.snd ≤ 1",
"tactic": "linarith [unitInterval.nonneg x.2, unitInterval.le_one x.2]"
}
] |
[
83,
68
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
60,
1
] |
Mathlib/Data/Rat/Cast.lean
|
Rat.cast_inv_int
|
[
{
"state_after": "case ofNat\nF : Type ?u.33624\nι : Type ?u.33627\nα : Type u_1\nβ : Type ?u.33633\ninst✝ : DivisionRing α\nn : ℕ\n⊢ ↑(↑(Int.ofNat n))⁻¹ = (↑(Int.ofNat n))⁻¹\n\ncase negSucc\nF : Type ?u.33624\nι : Type ?u.33627\nα : Type u_1\nβ : Type ?u.33633\ninst✝ : DivisionRing α\nn : ℕ\n⊢ ↑(↑(Int.negSucc n))⁻¹ = (↑(Int.negSucc n))⁻¹",
"state_before": "F : Type ?u.33624\nι : Type ?u.33627\nα : Type u_1\nβ : Type ?u.33633\ninst✝ : DivisionRing α\nn : ℤ\n⊢ ↑(↑n)⁻¹ = (↑n)⁻¹",
"tactic": "cases' n with n n"
},
{
"state_after": "no goals",
"state_before": "case ofNat\nF : Type ?u.33624\nι : Type ?u.33627\nα : Type u_1\nβ : Type ?u.33633\ninst✝ : DivisionRing α\nn : ℕ\n⊢ ↑(↑(Int.ofNat n))⁻¹ = (↑(Int.ofNat n))⁻¹",
"tactic": "simp [ofInt_eq_cast, cast_inv_nat]"
},
{
"state_after": "no goals",
"state_before": "case negSucc\nF : Type ?u.33624\nι : Type ?u.33627\nα : Type u_1\nβ : Type ?u.33633\ninst✝ : DivisionRing α\nn : ℕ\n⊢ ↑(↑(Int.negSucc n))⁻¹ = (↑(Int.negSucc n))⁻¹",
"tactic": "simp only [ofInt_eq_cast, Int.cast_negSucc, ← Nat.cast_succ, cast_neg, inv_neg, cast_inv_nat]"
}
] |
[
167,
98
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
164,
1
] |
Mathlib/NumberTheory/Padics/PadicIntegers.lean
|
PadicInt.norm_le_one
|
[] |
[
267,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
267,
1
] |
Mathlib/Geometry/Manifold/SmoothManifoldWithCorners.lean
|
modelWithCornersSelf_coe_symm
|
[] |
[
388,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
387,
1
] |
Mathlib/Analysis/SpecialFunctions/Integrals.lean
|
integral_one_div_one_add_sq
|
[
{
"state_after": "no goals",
"state_before": "a b : ℝ\nn : ℕ\n⊢ (∫ (x : ℝ) in a..b, 1 / (1 + x ^ 2)) = arctan b - arctan a",
"tactic": "simp only [one_div, integral_inv_one_add_sq]"
}
] |
[
553,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
551,
1
] |
Mathlib/Data/Seq/Computation.lean
|
Computation.LiftRel.swap
|
[] |
[
1052,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1050,
1
] |
Mathlib/CategoryTheory/Types.lean
|
CategoryTheory.FunctorToTypes.inv_hom_id_app_apply
|
[] |
[
186,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
185,
1
] |
Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean
|
Measurable.iInf_Prop
|
[
{
"state_after": "case h.e'_5.h\nα✝ : Type ?u.1409247\nβ : Type ?u.1409250\nγ : Type ?u.1409253\nγ₂ : Type ?u.1409256\nδ : Type u_2\nι : Sort y\ns t u : Set α✝\ninst✝¹¹ : TopologicalSpace α✝\ninst✝¹⁰ : MeasurableSpace α✝\ninst✝⁹ : BorelSpace α✝\ninst✝⁸ : TopologicalSpace β\ninst✝⁷ : MeasurableSpace β\ninst✝⁶ : BorelSpace β\ninst✝⁵ : TopologicalSpace γ\ninst✝⁴ : MeasurableSpace γ\ninst✝³ : BorelSpace γ\ninst✝² : MeasurableSpace δ\nα : Type u_1\ninst✝¹ : MeasurableSpace α\ninst✝ : CompleteLattice α\np : Prop\nf : δ → α\nhf : Measurable f\nh : p\nx✝ : δ\n⊢ (⨅ (_ : p), f x✝) = f x✝",
"state_before": "α✝ : Type ?u.1409247\nβ : Type ?u.1409250\nγ : Type ?u.1409253\nγ₂ : Type ?u.1409256\nδ : Type u_2\nι : Sort y\ns t u : Set α✝\ninst✝¹¹ : TopologicalSpace α✝\ninst✝¹⁰ : MeasurableSpace α✝\ninst✝⁹ : BorelSpace α✝\ninst✝⁸ : TopologicalSpace β\ninst✝⁷ : MeasurableSpace β\ninst✝⁶ : BorelSpace β\ninst✝⁵ : TopologicalSpace γ\ninst✝⁴ : MeasurableSpace γ\ninst✝³ : BorelSpace γ\ninst✝² : MeasurableSpace δ\nα : Type u_1\ninst✝¹ : MeasurableSpace α\ninst✝ : CompleteLattice α\np : Prop\nf : δ → α\nhf : Measurable f\nh : p\n⊢ Measurable fun b => ⨅ (_ : p), f b",
"tactic": "convert hf"
},
{
"state_after": "case h.e'_5.h\nα✝ : Type ?u.1409247\nβ : Type ?u.1409250\nγ : Type ?u.1409253\nγ₂ : Type ?u.1409256\nδ : Type u_2\nι : Sort y\ns t u : Set α✝\ninst✝¹¹ : TopologicalSpace α✝\ninst✝¹⁰ : MeasurableSpace α✝\ninst✝⁹ : BorelSpace α✝\ninst✝⁸ : TopologicalSpace β\ninst✝⁷ : MeasurableSpace β\ninst✝⁶ : BorelSpace β\ninst✝⁵ : TopologicalSpace γ\ninst✝⁴ : MeasurableSpace γ\ninst✝³ : BorelSpace γ\ninst✝² : MeasurableSpace δ\nα : Type u_1\ninst✝¹ : MeasurableSpace α\ninst✝ : CompleteLattice α\np : Prop\nf : δ → α\nhf : Measurable f\nh : p\nx✝ : δ\n⊢ (⨅ (_ : p), f x✝) = f x✝",
"state_before": "case h.e'_5.h\nα✝ : Type ?u.1409247\nβ : Type ?u.1409250\nγ : Type ?u.1409253\nγ₂ : Type ?u.1409256\nδ : Type u_2\nι : Sort y\ns t u : Set α✝\ninst✝¹¹ : TopologicalSpace α✝\ninst✝¹⁰ : MeasurableSpace α✝\ninst✝⁹ : BorelSpace α✝\ninst✝⁸ : TopologicalSpace β\ninst✝⁷ : MeasurableSpace β\ninst✝⁶ : BorelSpace β\ninst✝⁵ : TopologicalSpace γ\ninst✝⁴ : MeasurableSpace γ\ninst✝³ : BorelSpace γ\ninst✝² : MeasurableSpace δ\nα : Type u_1\ninst✝¹ : MeasurableSpace α\ninst✝ : CompleteLattice α\np : Prop\nf : δ → α\nhf : Measurable f\nh : p\nx✝ : δ\n⊢ (⨅ (_ : p), f x✝) = f x✝",
"tactic": "funext"
},
{
"state_after": "no goals",
"state_before": "case h.e'_5.h\nα✝ : Type ?u.1409247\nβ : Type ?u.1409250\nγ : Type ?u.1409253\nγ₂ : Type ?u.1409256\nδ : Type u_2\nι : Sort y\ns t u : Set α✝\ninst✝¹¹ : TopologicalSpace α✝\ninst✝¹⁰ : MeasurableSpace α✝\ninst✝⁹ : BorelSpace α✝\ninst✝⁸ : TopologicalSpace β\ninst✝⁷ : MeasurableSpace β\ninst✝⁶ : BorelSpace β\ninst✝⁵ : TopologicalSpace γ\ninst✝⁴ : MeasurableSpace γ\ninst✝³ : BorelSpace γ\ninst✝² : MeasurableSpace δ\nα : Type u_1\ninst✝¹ : MeasurableSpace α\ninst✝ : CompleteLattice α\np : Prop\nf : δ → α\nhf : Measurable f\nh : p\nx✝ : δ\n⊢ (⨅ (_ : p), f x✝) = f x✝",
"tactic": "exact iInf_pos h"
},
{
"state_after": "case h.e'_5\nα✝ : Type ?u.1409247\nβ : Type ?u.1409250\nγ : Type ?u.1409253\nγ₂ : Type ?u.1409256\nδ : Type u_2\nι : Sort y\ns t u : Set α✝\ninst✝¹¹ : TopologicalSpace α✝\ninst✝¹⁰ : MeasurableSpace α✝\ninst✝⁹ : BorelSpace α✝\ninst✝⁸ : TopologicalSpace β\ninst✝⁷ : MeasurableSpace β\ninst✝⁶ : BorelSpace β\ninst✝⁵ : TopologicalSpace γ\ninst✝⁴ : MeasurableSpace γ\ninst✝³ : BorelSpace γ\ninst✝² : MeasurableSpace δ\nα : Type u_1\ninst✝¹ : MeasurableSpace α\ninst✝ : CompleteLattice α\np : Prop\nf : δ → α\nhf : Measurable f\nh : ¬p\n⊢ (fun b => ⨅ (_ : p), f b) = fun x => ?convert_5\n\ncase convert_5\nα✝ : Type ?u.1409247\nβ : Type ?u.1409250\nγ : Type ?u.1409253\nγ₂ : Type ?u.1409256\nδ : Type u_2\nι : Sort y\ns t u : Set α✝\ninst✝¹¹ : TopologicalSpace α✝\ninst✝¹⁰ : MeasurableSpace α✝\ninst✝⁹ : BorelSpace α✝\ninst✝⁸ : TopologicalSpace β\ninst✝⁷ : MeasurableSpace β\ninst✝⁶ : BorelSpace β\ninst✝⁵ : TopologicalSpace γ\ninst✝⁴ : MeasurableSpace γ\ninst✝³ : BorelSpace γ\ninst✝² : MeasurableSpace δ\nα : Type u_1\ninst✝¹ : MeasurableSpace α\ninst✝ : CompleteLattice α\np : Prop\nf : δ → α\nhf : Measurable f\nh : ¬p\n⊢ α",
"state_before": "α✝ : Type ?u.1409247\nβ : Type ?u.1409250\nγ : Type ?u.1409253\nγ₂ : Type ?u.1409256\nδ : Type u_2\nι : Sort y\ns t u : Set α✝\ninst✝¹¹ : TopologicalSpace α✝\ninst✝¹⁰ : MeasurableSpace α✝\ninst✝⁹ : BorelSpace α✝\ninst✝⁸ : TopologicalSpace β\ninst✝⁷ : MeasurableSpace β\ninst✝⁶ : BorelSpace β\ninst✝⁵ : TopologicalSpace γ\ninst✝⁴ : MeasurableSpace γ\ninst✝³ : BorelSpace γ\ninst✝² : MeasurableSpace δ\nα : Type u_1\ninst✝¹ : MeasurableSpace α\ninst✝ : CompleteLattice α\np : Prop\nf : δ → α\nhf : Measurable f\nh : ¬p\n⊢ Measurable fun b => ⨅ (_ : p), f b",
"tactic": "convert measurable_const using 1"
},
{
"state_after": "case h.e'_5.h\nα✝ : Type ?u.1409247\nβ : Type ?u.1409250\nγ : Type ?u.1409253\nγ₂ : Type ?u.1409256\nδ : Type u_2\nι : Sort y\ns t u : Set α✝\ninst✝¹¹ : TopologicalSpace α✝\ninst✝¹⁰ : MeasurableSpace α✝\ninst✝⁹ : BorelSpace α✝\ninst✝⁸ : TopologicalSpace β\ninst✝⁷ : MeasurableSpace β\ninst✝⁶ : BorelSpace β\ninst✝⁵ : TopologicalSpace γ\ninst✝⁴ : MeasurableSpace γ\ninst✝³ : BorelSpace γ\ninst✝² : MeasurableSpace δ\nα : Type u_1\ninst✝¹ : MeasurableSpace α\ninst✝ : CompleteLattice α\np : Prop\nf : δ → α\nhf : Measurable f\nh : ¬p\nx✝ : δ\n⊢ (⨅ (_ : p), f x✝) = ?convert_5\n\ncase convert_5\nα✝ : Type ?u.1409247\nβ : Type ?u.1409250\nγ : Type ?u.1409253\nγ₂ : Type ?u.1409256\nδ : Type u_2\nι : Sort y\ns t u : Set α✝\ninst✝¹¹ : TopologicalSpace α✝\ninst✝¹⁰ : MeasurableSpace α✝\ninst✝⁹ : BorelSpace α✝\ninst✝⁸ : TopologicalSpace β\ninst✝⁷ : MeasurableSpace β\ninst✝⁶ : BorelSpace β\ninst✝⁵ : TopologicalSpace γ\ninst✝⁴ : MeasurableSpace γ\ninst✝³ : BorelSpace γ\ninst✝² : MeasurableSpace δ\nα : Type u_1\ninst✝¹ : MeasurableSpace α\ninst✝ : CompleteLattice α\np : Prop\nf : δ → α\nhf : Measurable f\nh : ¬p\n⊢ α",
"state_before": "case h.e'_5\nα✝ : Type ?u.1409247\nβ : Type ?u.1409250\nγ : Type ?u.1409253\nγ₂ : Type ?u.1409256\nδ : Type u_2\nι : Sort y\ns t u : Set α✝\ninst✝¹¹ : TopologicalSpace α✝\ninst✝¹⁰ : MeasurableSpace α✝\ninst✝⁹ : BorelSpace α✝\ninst✝⁸ : TopologicalSpace β\ninst✝⁷ : MeasurableSpace β\ninst✝⁶ : BorelSpace β\ninst✝⁵ : TopologicalSpace γ\ninst✝⁴ : MeasurableSpace γ\ninst✝³ : BorelSpace γ\ninst✝² : MeasurableSpace δ\nα : Type u_1\ninst✝¹ : MeasurableSpace α\ninst✝ : CompleteLattice α\np : Prop\nf : δ → α\nhf : Measurable f\nh : ¬p\n⊢ (fun b => ⨅ (_ : p), f b) = fun x => ?convert_5\n\ncase convert_5\nα✝ : Type ?u.1409247\nβ : Type ?u.1409250\nγ : Type ?u.1409253\nγ₂ : Type ?u.1409256\nδ : Type u_2\nι : Sort y\ns t u : Set α✝\ninst✝¹¹ : TopologicalSpace α✝\ninst✝¹⁰ : MeasurableSpace α✝\ninst✝⁹ : BorelSpace α✝\ninst✝⁸ : TopologicalSpace β\ninst✝⁷ : MeasurableSpace β\ninst✝⁶ : BorelSpace β\ninst✝⁵ : TopologicalSpace γ\ninst✝⁴ : MeasurableSpace γ\ninst✝³ : BorelSpace γ\ninst✝² : MeasurableSpace δ\nα : Type u_1\ninst✝¹ : MeasurableSpace α\ninst✝ : CompleteLattice α\np : Prop\nf : δ → α\nhf : Measurable f\nh : ¬p\n⊢ α",
"tactic": "funext"
},
{
"state_after": "no goals",
"state_before": "case h.e'_5.h\nα✝ : Type ?u.1409247\nβ : Type ?u.1409250\nγ : Type ?u.1409253\nγ₂ : Type ?u.1409256\nδ : Type u_2\nι : Sort y\ns t u : Set α✝\ninst✝¹¹ : TopologicalSpace α✝\ninst✝¹⁰ : MeasurableSpace α✝\ninst✝⁹ : BorelSpace α✝\ninst✝⁸ : TopologicalSpace β\ninst✝⁷ : MeasurableSpace β\ninst✝⁶ : BorelSpace β\ninst✝⁵ : TopologicalSpace γ\ninst✝⁴ : MeasurableSpace γ\ninst✝³ : BorelSpace γ\ninst✝² : MeasurableSpace δ\nα : Type u_1\ninst✝¹ : MeasurableSpace α\ninst✝ : CompleteLattice α\np : Prop\nf : δ → α\nhf : Measurable f\nh : ¬p\nx✝ : δ\n⊢ (⨅ (_ : p), f x✝) = ?convert_5\n\ncase convert_5\nα✝ : Type ?u.1409247\nβ : Type ?u.1409250\nγ : Type ?u.1409253\nγ₂ : Type ?u.1409256\nδ : Type u_2\nι : Sort y\ns t u : Set α✝\ninst✝¹¹ : TopologicalSpace α✝\ninst✝¹⁰ : MeasurableSpace α✝\ninst✝⁹ : BorelSpace α✝\ninst✝⁸ : TopologicalSpace β\ninst✝⁷ : MeasurableSpace β\ninst✝⁶ : BorelSpace β\ninst✝⁵ : TopologicalSpace γ\ninst✝⁴ : MeasurableSpace γ\ninst✝³ : BorelSpace γ\ninst✝² : MeasurableSpace δ\nα : Type u_1\ninst✝¹ : MeasurableSpace α\ninst✝ : CompleteLattice α\np : Prop\nf : δ → α\nhf : Measurable f\nh : ¬p\n⊢ α",
"tactic": "exact iInf_neg h"
}
] |
[
1249,
63
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1246,
1
] |
Mathlib/Topology/MetricSpace/HausdorffDistance.lean
|
Metric.frontier_thickening_disjoint
|
[
{
"state_after": "ι : Sort ?u.92543\nα : Type u\nβ : Type v\ninst✝ : PseudoEMetricSpace α\nδ : ℝ\ns : Set α\nx : α\nA : Set α\nr₁ r₂ : ℝ\nhr : r₁ < r₂\n⊢ Disjoint (frontier (thickening r₁ A)) (frontier (thickening r₂ A))",
"state_before": "ι : Sort ?u.92543\nα : Type u\nβ : Type v\ninst✝ : PseudoEMetricSpace α\nδ : ℝ\ns : Set α\nx : α\nA : Set α\n⊢ Pairwise (Disjoint on fun r => frontier (thickening r A))",
"tactic": "refine' (pairwise_disjoint_on _).2 fun r₁ r₂ hr => _"
},
{
"state_after": "case inl\nι : Sort ?u.92543\nα : Type u\nβ : Type v\ninst✝ : PseudoEMetricSpace α\nδ : ℝ\ns : Set α\nx : α\nA : Set α\nr₁ r₂ : ℝ\nhr : r₁ < r₂\nh₁ : r₁ ≤ 0\n⊢ Disjoint (frontier (thickening r₁ A)) (frontier (thickening r₂ A))\n\ncase inr\nι : Sort ?u.92543\nα : Type u\nβ : Type v\ninst✝ : PseudoEMetricSpace α\nδ : ℝ\ns : Set α\nx : α\nA : Set α\nr₁ r₂ : ℝ\nhr : r₁ < r₂\nh₁ : 0 ≤ r₁\n⊢ Disjoint (frontier (thickening r₁ A)) (frontier (thickening r₂ A))",
"state_before": "ι : Sort ?u.92543\nα : Type u\nβ : Type v\ninst✝ : PseudoEMetricSpace α\nδ : ℝ\ns : Set α\nx : α\nA : Set α\nr₁ r₂ : ℝ\nhr : r₁ < r₂\n⊢ Disjoint (frontier (thickening r₁ A)) (frontier (thickening r₂ A))",
"tactic": "cases' le_total r₁ 0 with h₁ h₁"
},
{
"state_after": "case inr\nι : Sort ?u.92543\nα : Type u\nβ : Type v\ninst✝ : PseudoEMetricSpace α\nδ : ℝ\ns : Set α\nx : α\nA : Set α\nr₁ r₂ : ℝ\nhr : r₁ < r₂\nh₁ : 0 ≤ r₁\nh : ENNReal.ofReal r₁ = ENNReal.ofReal r₂\n⊢ r₁ = r₂",
"state_before": "case inr\nι : Sort ?u.92543\nα : Type u\nβ : Type v\ninst✝ : PseudoEMetricSpace α\nδ : ℝ\ns : Set α\nx : α\nA : Set α\nr₁ r₂ : ℝ\nhr : r₁ < r₂\nh₁ : 0 ≤ r₁\n⊢ Disjoint (frontier (thickening r₁ A)) (frontier (thickening r₂ A))",
"tactic": "refine' ((disjoint_singleton.2 fun h => hr.ne _).preimage _).mono (frontier_thickening_subset _)\n (frontier_thickening_subset _)"
},
{
"state_after": "case inr\nι : Sort ?u.92543\nα : Type u\nβ : Type v\ninst✝ : PseudoEMetricSpace α\nδ : ℝ\ns : Set α\nx : α\nA : Set α\nr₁ r₂ : ℝ\nhr : r₁ < r₂\nh₁ : 0 ≤ r₁\nh : ENNReal.toReal (ENNReal.ofReal r₁) = ENNReal.toReal (ENNReal.ofReal r₂)\n⊢ r₁ = r₂",
"state_before": "case inr\nι : Sort ?u.92543\nα : Type u\nβ : Type v\ninst✝ : PseudoEMetricSpace α\nδ : ℝ\ns : Set α\nx : α\nA : Set α\nr₁ r₂ : ℝ\nhr : r₁ < r₂\nh₁ : 0 ≤ r₁\nh : ENNReal.ofReal r₁ = ENNReal.ofReal r₂\n⊢ r₁ = r₂",
"tactic": "apply_fun ENNReal.toReal at h"
},
{
"state_after": "no goals",
"state_before": "case inr\nι : Sort ?u.92543\nα : Type u\nβ : Type v\ninst✝ : PseudoEMetricSpace α\nδ : ℝ\ns : Set α\nx : α\nA : Set α\nr₁ r₂ : ℝ\nhr : r₁ < r₂\nh₁ : 0 ≤ r₁\nh : ENNReal.toReal (ENNReal.ofReal r₁) = ENNReal.toReal (ENNReal.ofReal r₂)\n⊢ r₁ = r₂",
"tactic": "rwa [ENNReal.toReal_ofReal h₁, ENNReal.toReal_ofReal (h₁.trans hr.le)] at h"
},
{
"state_after": "no goals",
"state_before": "case inl\nι : Sort ?u.92543\nα : Type u\nβ : Type v\ninst✝ : PseudoEMetricSpace α\nδ : ℝ\ns : Set α\nx : α\nA : Set α\nr₁ r₂ : ℝ\nhr : r₁ < r₂\nh₁ : r₁ ≤ 0\n⊢ Disjoint (frontier (thickening r₁ A)) (frontier (thickening r₂ A))",
"tactic": "simp [thickening_of_nonpos h₁]"
}
] |
[
965,
78
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
957,
1
] |
Mathlib/Analysis/SpecialFunctions/Pow/Deriv.lean
|
aux
|
[
{
"state_after": "no goals",
"state_before": "f g : ℂ → ℂ\ns : Set ℂ\nf' g' x c : ℂ\n⊢ ↑((g x * f x ^ (g x - 1)) • ContinuousLinearMap.smulRight 1 f' +\n (f x ^ g x * log (f x)) • ContinuousLinearMap.smulRight 1 g')\n 1 =\n g x * f x ^ (g x - 1) * f' + f x ^ g x * log (f x) * g'",
"tactic": "simp only [Algebra.id.smul_eq_mul, one_mul, ContinuousLinearMap.one_apply,\n ContinuousLinearMap.smulRight_apply, ContinuousLinearMap.add_apply, Pi.smul_apply,\n ContinuousLinearMap.coe_smul']"
}
] |
[
152,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
147,
9
] |
Mathlib/Topology/Homotopy/HomotopyGroup.lean
|
Cube.boundary_one
|
[
{
"state_after": "case h\nX : Type u\ninst✝ : TopologicalSpace X\nn : ℕ\nx✝ : X\nx : I^ 1\n⊢ x ∈ boundary 1 ↔ x ∈ {0, 1}",
"state_before": "X : Type u\ninst✝ : TopologicalSpace X\nn : ℕ\nx : X\n⊢ boundary 1 = {0, 1}",
"tactic": "ext x"
},
{
"state_after": "case h\nX : Type u\ninst✝ : TopologicalSpace X\nn : ℕ\nx✝ : X\nx : I^ 1\n⊢ x 0 = 0 ∨ x 0 = 1 ↔ x 0 = OfNat.ofNat 0 0 ∨ x 0 = OfNat.ofNat 1 0",
"state_before": "case h\nX : Type u\ninst✝ : TopologicalSpace X\nn : ℕ\nx✝ : X\nx : I^ 1\n⊢ x ∈ boundary 1 ↔ x ∈ {0, 1}",
"tactic": "rw [mem_boundary, Fin.exists_fin_one, mem_insert_iff, mem_singleton_iff,\n Function.funext_iff, Function.funext_iff, Fin.forall_fin_one, Fin.forall_fin_one]"
},
{
"state_after": "no goals",
"state_before": "case h\nX : Type u\ninst✝ : TopologicalSpace X\nn : ℕ\nx✝ : X\nx : I^ 1\n⊢ x 0 = 0 ∨ x 0 = 1 ↔ x 0 = OfNat.ofNat 0 0 ∨ x 0 = OfNat.ofNat 1 0",
"tactic": "rfl"
}
] |
[
113,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
109,
1
] |
Std/Data/Int/DivMod.lean
|
Int.ediv_eq_of_eq_mul_right
|
[
{
"state_after": "no goals",
"state_before": "a b c : Int\nH1 : b ≠ 0\nH2 : a = b * c\n⊢ a / b = c",
"tactic": "rw [H2, Int.mul_ediv_cancel_left _ H1]"
}
] |
[
745,
91
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
744,
11
] |
Mathlib/LinearAlgebra/Matrix/NonsingularInverse.lean
|
Matrix.invOf_eq
|
[
{
"state_after": "l : Type ?u.30804\nm : Type u\nn : Type u'\nα : Type v\ninst✝⁴ : Fintype n\ninst✝³ : DecidableEq n\ninst✝² : CommRing α\nA B : Matrix n n α\ninst✝¹ : Invertible (det A)\ninst✝ : Invertible A\nthis : Invertible A := invertibleOfDetInvertible A\n⊢ ⅟A = ⅟(det A) • adjugate A",
"state_before": "l : Type ?u.30804\nm : Type u\nn : Type u'\nα : Type v\ninst✝⁴ : Fintype n\ninst✝³ : DecidableEq n\ninst✝² : CommRing α\nA B : Matrix n n α\ninst✝¹ : Invertible (det A)\ninst✝ : Invertible A\n⊢ ⅟A = ⅟(det A) • adjugate A",
"tactic": "letI := invertibleOfDetInvertible A"
},
{
"state_after": "no goals",
"state_before": "l : Type ?u.30804\nm : Type u\nn : Type u'\nα : Type v\ninst✝⁴ : Fintype n\ninst✝³ : DecidableEq n\ninst✝² : CommRing α\nA B : Matrix n n α\ninst✝¹ : Invertible (det A)\ninst✝ : Invertible A\nthis : Invertible A := invertibleOfDetInvertible A\n⊢ ⅟A = ⅟(det A) • adjugate A",
"tactic": "convert(rfl : ⅟ A = _)"
}
] |
[
113,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
111,
1
] |
Mathlib/Init/Function.lean
|
Function.Injective.comp
|
[] |
[
67,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
65,
1
] |
Mathlib/Data/List/Basic.lean
|
List.enum_singleton
|
[] |
[
3893,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
3892,
1
] |
Mathlib/Combinatorics/Young/YoungDiagram.lean
|
YoungDiagram.rowLen_transpose
|
[
{
"state_after": "no goals",
"state_before": "μ : YoungDiagram\ni : ℕ\n⊢ rowLen (transpose μ) i = colLen μ i",
"tactic": "simp [rowLen, colLen]"
}
] |
[
375,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
374,
1
] |
Mathlib/GroupTheory/SpecificGroups/Cyclic.lean
|
IsCyclic.exponent_eq_card
|
[
{
"state_after": "case intro\nα : Type u\na : α\ninst✝² : Group α\ninst✝¹ : IsCyclic α\ninst✝ : Fintype α\ng : α\nhg : ∀ (x : α), x ∈ Subgroup.zpowers g\n⊢ exponent α = Fintype.card α",
"state_before": "α : Type u\na : α\ninst✝² : Group α\ninst✝¹ : IsCyclic α\ninst✝ : Fintype α\n⊢ exponent α = Fintype.card α",
"tactic": "obtain ⟨g, hg⟩ := IsCyclic.exists_generator (α := α)"
},
{
"state_after": "case intro.a\nα : Type u\na : α\ninst✝² : Group α\ninst✝¹ : IsCyclic α\ninst✝ : Fintype α\ng : α\nhg : ∀ (x : α), x ∈ Subgroup.zpowers g\n⊢ exponent α ∣ Fintype.card α\n\ncase intro.a\nα : Type u\na : α\ninst✝² : Group α\ninst✝¹ : IsCyclic α\ninst✝ : Fintype α\ng : α\nhg : ∀ (x : α), x ∈ Subgroup.zpowers g\n⊢ Fintype.card α ∣ exponent α",
"state_before": "case intro\nα : Type u\na : α\ninst✝² : Group α\ninst✝¹ : IsCyclic α\ninst✝ : Fintype α\ng : α\nhg : ∀ (x : α), x ∈ Subgroup.zpowers g\n⊢ exponent α = Fintype.card α",
"tactic": "apply Nat.dvd_antisymm"
},
{
"state_after": "case intro.a\nα : Type u\na : α\ninst✝² : Group α\ninst✝¹ : IsCyclic α\ninst✝ : Fintype α\ng : α\nhg : ∀ (x : α), x ∈ Subgroup.zpowers g\n⊢ orderOf g ∣ exponent α",
"state_before": "case intro.a\nα : Type u\na : α\ninst✝² : Group α\ninst✝¹ : IsCyclic α\ninst✝ : Fintype α\ng : α\nhg : ∀ (x : α), x ∈ Subgroup.zpowers g\n⊢ Fintype.card α ∣ exponent α",
"tactic": "rw [← orderOf_eq_card_of_forall_mem_zpowers hg]"
},
{
"state_after": "no goals",
"state_before": "case intro.a\nα : Type u\na : α\ninst✝² : Group α\ninst✝¹ : IsCyclic α\ninst✝ : Fintype α\ng : α\nhg : ∀ (x : α), x ∈ Subgroup.zpowers g\n⊢ orderOf g ∣ exponent α",
"tactic": "exact order_dvd_exponent _"
},
{
"state_after": "case intro.a\nα : Type u\na : α\ninst✝² : Group α\ninst✝¹ : IsCyclic α\ninst✝ : Fintype α\ng : α\nhg : ∀ (x : α), x ∈ Subgroup.zpowers g\n⊢ ∀ (b : α), b ∈ Finset.univ → orderOf b ∣ Fintype.card α",
"state_before": "case intro.a\nα : Type u\na : α\ninst✝² : Group α\ninst✝¹ : IsCyclic α\ninst✝ : Fintype α\ng : α\nhg : ∀ (x : α), x ∈ Subgroup.zpowers g\n⊢ exponent α ∣ Fintype.card α",
"tactic": "rw [← lcm_order_eq_exponent, Finset.lcm_dvd_iff]"
},
{
"state_after": "no goals",
"state_before": "case intro.a\nα : Type u\na : α\ninst✝² : Group α\ninst✝¹ : IsCyclic α\ninst✝ : Fintype α\ng : α\nhg : ∀ (x : α), x ∈ Subgroup.zpowers g\n⊢ ∀ (b : α), b ∈ Finset.univ → orderOf b ∣ Fintype.card α",
"tactic": "exact fun b _ => orderOf_dvd_card_univ"
}
] |
[
571,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
564,
1
] |
Mathlib/Algebra/Order/Module.lean
|
bddAbove_smul_iff_of_neg
|
[] |
[
245,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
244,
1
] |
Mathlib/Data/Nat/Digits.lean
|
Nat.digits_zero_succ'
|
[] |
[
105,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
103,
1
] |
Mathlib/Data/Part.lean
|
Part.get_mem
|
[] |
[
108,
11
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
107,
1
] |
Mathlib/Data/List/Forall2.lean
|
List.rel_reverse
|
[
{
"state_after": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.140328\nδ : Type ?u.140331\nR S : α → β → Prop\nP : γ → δ → Prop\nRₐ : α → α → Prop\na✝ : α\nb✝ : β\nl₁✝ : List α\nl₂✝ : List β\nh₁ : R a✝ b✝\nh₂ : Forall₂ R l₁✝ l₂✝\n⊢ Forall₂ R (reverse l₁✝ ++ [a✝]) (reverse l₂✝ ++ [b✝])",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.140328\nδ : Type ?u.140331\nR S : α → β → Prop\nP : γ → δ → Prop\nRₐ : α → α → Prop\na✝ : α\nb✝ : β\nl₁✝ : List α\nl₂✝ : List β\nh₁ : R a✝ b✝\nh₂ : Forall₂ R l₁✝ l₂✝\n⊢ Forall₂ R (reverse (a✝ :: l₁✝)) (reverse (b✝ :: l₂✝))",
"tactic": "simp only [reverse_cons]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.140328\nδ : Type ?u.140331\nR S : α → β → Prop\nP : γ → δ → Prop\nRₐ : α → α → Prop\na✝ : α\nb✝ : β\nl₁✝ : List α\nl₂✝ : List β\nh₁ : R a✝ b✝\nh₂ : Forall₂ R l₁✝ l₂✝\n⊢ Forall₂ R (reverse l₁✝ ++ [a✝]) (reverse l₂✝ ++ [b✝])",
"tactic": "exact rel_append (rel_reverse h₂) (Forall₂.cons h₁ Forall₂.nil)"
}
] |
[
260,
68
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
256,
1
] |
Mathlib/MeasureTheory/Measure/MeasureSpace.lean
|
MeasureTheory.measure_union_congr_of_subset
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.36188\nγ : Type ?u.36191\nδ : Type ?u.36194\nι : Type ?u.36197\nR : Type ?u.36200\nR' : Type ?u.36203\nm : MeasurableSpace α\nμ μ₁ μ₂ : Measure α\ns s₁ s₂ t t₁ t₂ : Set α\nhs : s₁ ⊆ s₂\nhsμ : ↑↑μ s₂ ≤ ↑↑μ s₁\nht : t₁ ⊆ t₂\nhtμ : ↑↑μ t₂ ≤ ↑↑μ t₁\n⊢ ↑↑μ (⋃ (b : Bool), bif b then s₁ else t₁) = ↑↑μ (⋃ (b : Bool), bif b then s₂ else t₂)",
"state_before": "α : Type u_1\nβ : Type ?u.36188\nγ : Type ?u.36191\nδ : Type ?u.36194\nι : Type ?u.36197\nR : Type ?u.36200\nR' : Type ?u.36203\nm : MeasurableSpace α\nμ μ₁ μ₂ : Measure α\ns s₁ s₂ t t₁ t₂ : Set α\nhs : s₁ ⊆ s₂\nhsμ : ↑↑μ s₂ ≤ ↑↑μ s₁\nht : t₁ ⊆ t₂\nhtμ : ↑↑μ t₂ ≤ ↑↑μ t₁\n⊢ ↑↑μ (s₁ ∪ t₁) = ↑↑μ (s₂ ∪ t₂)",
"tactic": "rw [union_eq_iUnion, union_eq_iUnion]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.36188\nγ : Type ?u.36191\nδ : Type ?u.36194\nι : Type ?u.36197\nR : Type ?u.36200\nR' : Type ?u.36203\nm : MeasurableSpace α\nμ μ₁ μ₂ : Measure α\ns s₁ s₂ t t₁ t₂ : Set α\nhs : s₁ ⊆ s₂\nhsμ : ↑↑μ s₂ ≤ ↑↑μ s₁\nht : t₁ ⊆ t₂\nhtμ : ↑↑μ t₂ ≤ ↑↑μ t₁\n⊢ ↑↑μ (⋃ (b : Bool), bif b then s₁ else t₁) = ↑↑μ (⋃ (b : Bool), bif b then s₂ else t₂)",
"tactic": "exact measure_iUnion_congr_of_subset (Bool.forall_bool.2 ⟨ht, hs⟩) (Bool.forall_bool.2 ⟨htμ, hsμ⟩)"
}
] |
[
356,
101
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
353,
1
] |
Mathlib/NumberTheory/Padics/RingHoms.lean
|
PadicInt.nthHomSeq_mul
|
[
{
"state_after": "p : ℕ\nhp_prime : Fact (Nat.Prime p)\nR : Type u_1\ninst✝ : NonAssocSemiring R\nf : (k : ℕ) → R →+* ZMod (p ^ k)\nf_compat : ∀ (k1 k2 : ℕ) (hk : k1 ≤ k2), RingHom.comp (ZMod.castHom (_ : p ^ k1 ∣ p ^ k2) (ZMod (p ^ k1))) (f k2) = f k1\nr s : R\nε : ℚ\nhε : ε > 0\n⊢ ∃ i,\n ∀ (j : ℕ), j ≥ i → padicNorm p (↑(nthHomSeq f_compat (r * s) - nthHomSeq f_compat r * nthHomSeq f_compat s) j) < ε",
"state_before": "p : ℕ\nhp_prime : Fact (Nat.Prime p)\nR : Type u_1\ninst✝ : NonAssocSemiring R\nf : (k : ℕ) → R →+* ZMod (p ^ k)\nf_compat : ∀ (k1 k2 : ℕ) (hk : k1 ≤ k2), RingHom.comp (ZMod.castHom (_ : p ^ k1 ∣ p ^ k2) (ZMod (p ^ k1))) (f k2) = f k1\nr s : R\n⊢ nthHomSeq f_compat (r * s) ≈ nthHomSeq f_compat r * nthHomSeq f_compat s",
"tactic": "intro ε hε"
},
{
"state_after": "case intro\np : ℕ\nhp_prime : Fact (Nat.Prime p)\nR : Type u_1\ninst✝ : NonAssocSemiring R\nf : (k : ℕ) → R →+* ZMod (p ^ k)\nf_compat : ∀ (k1 k2 : ℕ) (hk : k1 ≤ k2), RingHom.comp (ZMod.castHom (_ : p ^ k1 ∣ p ^ k2) (ZMod (p ^ k1))) (f k2) = f k1\nr s : R\nε : ℚ\nhε : ε > 0\nn : ℕ\nhn : ↑p ^ (-↑n) < ε\n⊢ ∃ i,\n ∀ (j : ℕ), j ≥ i → padicNorm p (↑(nthHomSeq f_compat (r * s) - nthHomSeq f_compat r * nthHomSeq f_compat s) j) < ε",
"state_before": "p : ℕ\nhp_prime : Fact (Nat.Prime p)\nR : Type u_1\ninst✝ : NonAssocSemiring R\nf : (k : ℕ) → R →+* ZMod (p ^ k)\nf_compat : ∀ (k1 k2 : ℕ) (hk : k1 ≤ k2), RingHom.comp (ZMod.castHom (_ : p ^ k1 ∣ p ^ k2) (ZMod (p ^ k1))) (f k2) = f k1\nr s : R\nε : ℚ\nhε : ε > 0\n⊢ ∃ i,\n ∀ (j : ℕ), j ≥ i → padicNorm p (↑(nthHomSeq f_compat (r * s) - nthHomSeq f_compat r * nthHomSeq f_compat s) j) < ε",
"tactic": "obtain ⟨n, hn⟩ := exists_pow_neg_lt_rat p hε"
},
{
"state_after": "case intro\np : ℕ\nhp_prime : Fact (Nat.Prime p)\nR : Type u_1\ninst✝ : NonAssocSemiring R\nf : (k : ℕ) → R →+* ZMod (p ^ k)\nf_compat : ∀ (k1 k2 : ℕ) (hk : k1 ≤ k2), RingHom.comp (ZMod.castHom (_ : p ^ k1 ∣ p ^ k2) (ZMod (p ^ k1))) (f k2) = f k1\nr s : R\nε : ℚ\nhε : ε > 0\nn : ℕ\nhn : ↑p ^ (-↑n) < ε\n⊢ ∀ (j : ℕ), j ≥ n → padicNorm p (↑(nthHomSeq f_compat (r * s) - nthHomSeq f_compat r * nthHomSeq f_compat s) j) < ε",
"state_before": "case intro\np : ℕ\nhp_prime : Fact (Nat.Prime p)\nR : Type u_1\ninst✝ : NonAssocSemiring R\nf : (k : ℕ) → R →+* ZMod (p ^ k)\nf_compat : ∀ (k1 k2 : ℕ) (hk : k1 ≤ k2), RingHom.comp (ZMod.castHom (_ : p ^ k1 ∣ p ^ k2) (ZMod (p ^ k1))) (f k2) = f k1\nr s : R\nε : ℚ\nhε : ε > 0\nn : ℕ\nhn : ↑p ^ (-↑n) < ε\n⊢ ∃ i,\n ∀ (j : ℕ), j ≥ i → padicNorm p (↑(nthHomSeq f_compat (r * s) - nthHomSeq f_compat r * nthHomSeq f_compat s) j) < ε",
"tactic": "use n"
},
{
"state_after": "case intro\np : ℕ\nhp_prime : Fact (Nat.Prime p)\nR : Type u_1\ninst✝ : NonAssocSemiring R\nf : (k : ℕ) → R →+* ZMod (p ^ k)\nf_compat : ∀ (k1 k2 : ℕ) (hk : k1 ≤ k2), RingHom.comp (ZMod.castHom (_ : p ^ k1 ∣ p ^ k2) (ZMod (p ^ k1))) (f k2) = f k1\nr s : R\nε : ℚ\nhε : ε > 0\nn : ℕ\nhn : ↑p ^ (-↑n) < ε\nj : ℕ\nhj : j ≥ n\n⊢ padicNorm p (↑(nthHomSeq f_compat (r * s) - nthHomSeq f_compat r * nthHomSeq f_compat s) j) < ε",
"state_before": "case intro\np : ℕ\nhp_prime : Fact (Nat.Prime p)\nR : Type u_1\ninst✝ : NonAssocSemiring R\nf : (k : ℕ) → R →+* ZMod (p ^ k)\nf_compat : ∀ (k1 k2 : ℕ) (hk : k1 ≤ k2), RingHom.comp (ZMod.castHom (_ : p ^ k1 ∣ p ^ k2) (ZMod (p ^ k1))) (f k2) = f k1\nr s : R\nε : ℚ\nhε : ε > 0\nn : ℕ\nhn : ↑p ^ (-↑n) < ε\n⊢ ∀ (j : ℕ), j ≥ n → padicNorm p (↑(nthHomSeq f_compat (r * s) - nthHomSeq f_compat r * nthHomSeq f_compat s) j) < ε",
"tactic": "intro j hj"
},
{
"state_after": "case intro\np : ℕ\nhp_prime : Fact (Nat.Prime p)\nR : Type u_1\ninst✝ : NonAssocSemiring R\nf : (k : ℕ) → R →+* ZMod (p ^ k)\nf_compat : ∀ (k1 k2 : ℕ) (hk : k1 ≤ k2), RingHom.comp (ZMod.castHom (_ : p ^ k1 ∣ p ^ k2) (ZMod (p ^ k1))) (f k2) = f k1\nr s : R\nε : ℚ\nhε : ε > 0\nn : ℕ\nhn : ↑p ^ (-↑n) < ε\nj : ℕ\nhj : j ≥ n\n⊢ padicNorm p (↑(nthHom (fun k2 => f k2) (r * s) j) - ↑(nthHom (fun k2 => f k2) r j) * ↑(nthHom (fun k2 => f k2) s j)) <\n ε",
"state_before": "case intro\np : ℕ\nhp_prime : Fact (Nat.Prime p)\nR : Type u_1\ninst✝ : NonAssocSemiring R\nf : (k : ℕ) → R →+* ZMod (p ^ k)\nf_compat : ∀ (k1 k2 : ℕ) (hk : k1 ≤ k2), RingHom.comp (ZMod.castHom (_ : p ^ k1 ∣ p ^ k2) (ZMod (p ^ k1))) (f k2) = f k1\nr s : R\nε : ℚ\nhε : ε > 0\nn : ℕ\nhn : ↑p ^ (-↑n) < ε\nj : ℕ\nhj : j ≥ n\n⊢ padicNorm p (↑(nthHomSeq f_compat (r * s) - nthHomSeq f_compat r * nthHomSeq f_compat s) j) < ε",
"tactic": "dsimp [nthHomSeq]"
},
{
"state_after": "p : ℕ\nhp_prime : Fact (Nat.Prime p)\nR : Type u_1\ninst✝ : NonAssocSemiring R\nf : (k : ℕ) → R →+* ZMod (p ^ k)\nf_compat : ∀ (k1 k2 : ℕ) (hk : k1 ≤ k2), RingHom.comp (ZMod.castHom (_ : p ^ k1 ∣ p ^ k2) (ZMod (p ^ k1))) (f k2) = f k1\nr s : R\nε : ℚ\nhε : ε > 0\nn : ℕ\nhn : ↑p ^ (-↑n) < ε\nj : ℕ\nhj : j ≥ n\n⊢ padicNorm p (↑(nthHom (fun k2 => f k2) (r * s) j) - ↑(nthHom (fun k2 => f k2) r j) * ↑(nthHom (fun k2 => f k2) s j)) ≤\n ↑p ^ (-↑n)",
"state_before": "case intro\np : ℕ\nhp_prime : Fact (Nat.Prime p)\nR : Type u_1\ninst✝ : NonAssocSemiring R\nf : (k : ℕ) → R →+* ZMod (p ^ k)\nf_compat : ∀ (k1 k2 : ℕ) (hk : k1 ≤ k2), RingHom.comp (ZMod.castHom (_ : p ^ k1 ∣ p ^ k2) (ZMod (p ^ k1))) (f k2) = f k1\nr s : R\nε : ℚ\nhε : ε > 0\nn : ℕ\nhn : ↑p ^ (-↑n) < ε\nj : ℕ\nhj : j ≥ n\n⊢ padicNorm p (↑(nthHom (fun k2 => f k2) (r * s) j) - ↑(nthHom (fun k2 => f k2) r j) * ↑(nthHom (fun k2 => f k2) s j)) <\n ε",
"tactic": "apply lt_of_le_of_lt _ hn"
},
{
"state_after": "p : ℕ\nhp_prime : Fact (Nat.Prime p)\nR : Type u_1\ninst✝ : NonAssocSemiring R\nf : (k : ℕ) → R →+* ZMod (p ^ k)\nf_compat : ∀ (k1 k2 : ℕ) (hk : k1 ≤ k2), RingHom.comp (ZMod.castHom (_ : p ^ k1 ∣ p ^ k2) (ZMod (p ^ k1))) (f k2) = f k1\nr s : R\nε : ℚ\nhε : ε > 0\nn : ℕ\nhn : ↑p ^ (-↑n) < ε\nj : ℕ\nhj : j ≥ n\n⊢ ↑(nthHom (fun k2 => f k2) (r * s) j - nthHom (fun k2 => f k2) r j * nthHom (fun k2 => f k2) s j) = 0",
"state_before": "p : ℕ\nhp_prime : Fact (Nat.Prime p)\nR : Type u_1\ninst✝ : NonAssocSemiring R\nf : (k : ℕ) → R →+* ZMod (p ^ k)\nf_compat : ∀ (k1 k2 : ℕ) (hk : k1 ≤ k2), RingHom.comp (ZMod.castHom (_ : p ^ k1 ∣ p ^ k2) (ZMod (p ^ k1))) (f k2) = f k1\nr s : R\nε : ℚ\nhε : ε > 0\nn : ℕ\nhn : ↑p ^ (-↑n) < ε\nj : ℕ\nhj : j ≥ n\n⊢ padicNorm p (↑(nthHom (fun k2 => f k2) (r * s) j) - ↑(nthHom (fun k2 => f k2) r j) * ↑(nthHom (fun k2 => f k2) s j)) ≤\n ↑p ^ (-↑n)",
"tactic": "rw [← Int.cast_mul, ← Int.cast_sub, ← padicNorm.dvd_iff_norm_le, ←\n ZMod.int_cast_zmod_eq_zero_iff_dvd]"
},
{
"state_after": "p : ℕ\nhp_prime : Fact (Nat.Prime p)\nR : Type u_1\ninst✝ : NonAssocSemiring R\nf : (k : ℕ) → R →+* ZMod (p ^ k)\nf_compat : ∀ (k1 k2 : ℕ) (hk : k1 ≤ k2), RingHom.comp (ZMod.castHom (_ : p ^ k1 ∣ p ^ k2) (ZMod (p ^ k1))) (f k2) = f k1\nr s : R\nε : ℚ\nhε : ε > 0\nn : ℕ\nhn : ↑p ^ (-↑n) < ε\nj : ℕ\nhj : j ≥ n\n⊢ ↑(↑(ZMod.val (↑(f j) (r * s))) - ↑(ZMod.val (↑(f j) r)) * ↑(ZMod.val (↑(f j) s))) = 0",
"state_before": "p : ℕ\nhp_prime : Fact (Nat.Prime p)\nR : Type u_1\ninst✝ : NonAssocSemiring R\nf : (k : ℕ) → R →+* ZMod (p ^ k)\nf_compat : ∀ (k1 k2 : ℕ) (hk : k1 ≤ k2), RingHom.comp (ZMod.castHom (_ : p ^ k1 ∣ p ^ k2) (ZMod (p ^ k1))) (f k2) = f k1\nr s : R\nε : ℚ\nhε : ε > 0\nn : ℕ\nhn : ↑p ^ (-↑n) < ε\nj : ℕ\nhj : j ≥ n\n⊢ ↑(nthHom (fun k2 => f k2) (r * s) j - nthHom (fun k2 => f k2) r j * nthHom (fun k2 => f k2) s j) = 0",
"tactic": "dsimp [nthHom]"
},
{
"state_after": "p : ℕ\nhp_prime : Fact (Nat.Prime p)\nR : Type u_1\ninst✝ : NonAssocSemiring R\nf : (k : ℕ) → R →+* ZMod (p ^ k)\nf_compat : ∀ (k1 k2 : ℕ) (hk : k1 ≤ k2), RingHom.comp (ZMod.castHom (_ : p ^ k1 ∣ p ^ k2) (ZMod (p ^ k1))) (f k2) = f k1\nr s : R\nε : ℚ\nhε : ε > 0\nn : ℕ\nhn : ↑p ^ (-↑n) < ε\nj : ℕ\nhj : j ≥ n\n⊢ ↑(↑(f j) r * ↑(f j) s) - ↑(↑(f j) r) * ↑(↑(f j) s) = 0",
"state_before": "p : ℕ\nhp_prime : Fact (Nat.Prime p)\nR : Type u_1\ninst✝ : NonAssocSemiring R\nf : (k : ℕ) → R →+* ZMod (p ^ k)\nf_compat : ∀ (k1 k2 : ℕ) (hk : k1 ≤ k2), RingHom.comp (ZMod.castHom (_ : p ^ k1 ∣ p ^ k2) (ZMod (p ^ k1))) (f k2) = f k1\nr s : R\nε : ℚ\nhε : ε > 0\nn : ℕ\nhn : ↑p ^ (-↑n) < ε\nj : ℕ\nhj : j ≥ n\n⊢ ↑(↑(ZMod.val (↑(f j) (r * s))) - ↑(ZMod.val (↑(f j) r)) * ↑(ZMod.val (↑(f j) s))) = 0",
"tactic": "simp only [ZMod.nat_cast_val, RingHom.map_mul, Int.cast_sub, ZMod.int_cast_cast, Int.cast_mul]"
},
{
"state_after": "no goals",
"state_before": "p : ℕ\nhp_prime : Fact (Nat.Prime p)\nR : Type u_1\ninst✝ : NonAssocSemiring R\nf : (k : ℕ) → R →+* ZMod (p ^ k)\nf_compat : ∀ (k1 k2 : ℕ) (hk : k1 ≤ k2), RingHom.comp (ZMod.castHom (_ : p ^ k1 ∣ p ^ k2) (ZMod (p ^ k1))) (f k2) = f k1\nr s : R\nε : ℚ\nhε : ε > 0\nn : ℕ\nhn : ↑p ^ (-↑n) < ε\nj : ℕ\nhj : j ≥ n\n⊢ ↑(↑(f j) r * ↑(f j) s) - ↑(↑(f j) r) * ↑(↑(f j) s) = 0",
"tactic": "rw [ZMod.cast_mul (show p ^ n ∣ p ^ j from pow_dvd_pow _ hj), sub_self]"
}
] |
[
567,
74
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
555,
1
] |
Mathlib/Topology/MetricSpace/Basic.lean
|
tendsto_uniformity_iff_dist_tendsto_zero
|
[
{
"state_after": "α : Type u\nβ : Type v\nX : Type ?u.214516\nι✝ : Type ?u.214519\ninst✝ : PseudoMetricSpace α\nι : Type u_1\nf : ι → α × α\np : Filter ι\n⊢ Tendsto ((fun p => dist p.fst p.snd) ∘ f) p (𝓝 0) ↔ Tendsto (fun x => dist (f x).fst (f x).snd) p (𝓝 0)",
"state_before": "α : Type u\nβ : Type v\nX : Type ?u.214516\nι✝ : Type ?u.214519\ninst✝ : PseudoMetricSpace α\nι : Type u_1\nf : ι → α × α\np : Filter ι\n⊢ Tendsto f p (𝓤 α) ↔ Tendsto (fun x => dist (f x).fst (f x).snd) p (𝓝 0)",
"tactic": "rw [Metric.uniformity_eq_comap_nhds_zero, tendsto_comap_iff]"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nX : Type ?u.214516\nι✝ : Type ?u.214519\ninst✝ : PseudoMetricSpace α\nι : Type u_1\nf : ι → α × α\np : Filter ι\n⊢ Tendsto ((fun p => dist p.fst p.snd) ∘ f) p (𝓝 0) ↔ Tendsto (fun x => dist (f x).fst (f x).snd) p (𝓝 0)",
"tactic": "rfl"
}
] |
[
1465,
68
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1463,
1
] |
Mathlib/Topology/Order.lean
|
TopologicalSpace.generateFrom_setOf_isOpen
|
[] |
[
220,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
218,
1
] |
Mathlib/Analysis/SpecialFunctions/Pow/Continuity.lean
|
Filter.Tendsto.nnrpow
|
[] |
[
469,
65
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
466,
1
] |
Mathlib/Topology/Algebra/Ring/Basic.lean
|
RingTopology.ext
|
[] |
[
310,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
309,
1
] |
Mathlib/Analysis/Calculus/Deriv/Basic.lean
|
Filter.EventuallyEq.derivWithin_eq
|
[
{
"state_after": "𝕜 : Type u\ninst✝⁴ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nE : Type w\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf f₀ f₁ g : 𝕜 → F\nf' f₀' f₁' g' : F\nx : 𝕜\ns t : Set 𝕜\nL L₁ L₂ : Filter 𝕜\nhs : f₁ =ᶠ[𝓝[s] x] f\nhx : f₁ x = f x\n⊢ ↑(fderivWithin 𝕜 f₁ s x) 1 = ↑(fderivWithin 𝕜 f s x) 1",
"state_before": "𝕜 : Type u\ninst✝⁴ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nE : Type w\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf f₀ f₁ g : 𝕜 → F\nf' f₀' f₁' g' : F\nx : 𝕜\ns t : Set 𝕜\nL L₁ L₂ : Filter 𝕜\nhs : f₁ =ᶠ[𝓝[s] x] f\nhx : f₁ x = f x\n⊢ derivWithin f₁ s x = derivWithin f s x",
"tactic": "unfold derivWithin"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u\ninst✝⁴ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nE : Type w\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf f₀ f₁ g : 𝕜 → F\nf' f₀' f₁' g' : F\nx : 𝕜\ns t : Set 𝕜\nL L₁ L₂ : Filter 𝕜\nhs : f₁ =ᶠ[𝓝[s] x] f\nhx : f₁ x = f x\n⊢ ↑(fderivWithin 𝕜 f₁ s x) 1 = ↑(fderivWithin 𝕜 f s x) 1",
"tactic": "rw [hs.fderivWithin_eq hx]"
}
] |
[
606,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
603,
1
] |
Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean
|
MeasureTheory.AEStronglyMeasurable.congr
|
[] |
[
1222,
64
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1221,
1
] |
Mathlib/LinearAlgebra/Alternating.lean
|
Basis.ext_alternating
|
[
{
"state_after": "R : Type ?u.1290808\ninst✝¹⁶ : Semiring R\nM : Type ?u.1290814\ninst✝¹⁵ : AddCommMonoid M\ninst✝¹⁴ : Module R M\nN : Type ?u.1290846\ninst✝¹³ : AddCommMonoid N\ninst✝¹² : Module R N\nP : Type ?u.1290876\ninst✝¹¹ : AddCommMonoid P\ninst✝¹⁰ : Module R P\nM' : Type ?u.1290906\ninst✝⁹ : AddCommGroup M'\ninst✝⁸ : Module R M'\nN' : Type ?u.1291294\ninst✝⁷ : AddCommGroup N'\ninst✝⁶ : Module R N'\nι : Type u_4\nι' : Type ?u.1291685\nι'' : Type ?u.1291688\nι₁ : Type u_5\ninst✝⁵ : Finite ι\nR' : Type u_1\nN₁ : Type u_2\nN₂ : Type u_3\ninst✝⁴ : CommSemiring R'\ninst✝³ : AddCommMonoid N₁\ninst✝² : AddCommMonoid N₂\ninst✝¹ : Module R' N₁\ninst✝ : Module R' N₂\nf g : AlternatingMap R' N₁ N₂ ι\ne : Basis ι₁ R' N₁\nh : ∀ (v : ι → ι₁), Function.Injective v → (↑f fun i => ↑e (v i)) = ↑g fun i => ↑e (v i)\nv : ι → ι₁\n⊢ (↑↑f fun i => ↑e (v i)) = ↑↑g fun i => ↑e (v i)",
"state_before": "R : Type ?u.1290808\ninst✝¹⁶ : Semiring R\nM : Type ?u.1290814\ninst✝¹⁵ : AddCommMonoid M\ninst✝¹⁴ : Module R M\nN : Type ?u.1290846\ninst✝¹³ : AddCommMonoid N\ninst✝¹² : Module R N\nP : Type ?u.1290876\ninst✝¹¹ : AddCommMonoid P\ninst✝¹⁰ : Module R P\nM' : Type ?u.1290906\ninst✝⁹ : AddCommGroup M'\ninst✝⁸ : Module R M'\nN' : Type ?u.1291294\ninst✝⁷ : AddCommGroup N'\ninst✝⁶ : Module R N'\nι : Type u_4\nι' : Type ?u.1291685\nι'' : Type ?u.1291688\nι₁ : Type u_5\ninst✝⁵ : Finite ι\nR' : Type u_1\nN₁ : Type u_2\nN₂ : Type u_3\ninst✝⁴ : CommSemiring R'\ninst✝³ : AddCommMonoid N₁\ninst✝² : AddCommMonoid N₂\ninst✝¹ : Module R' N₁\ninst✝ : Module R' N₂\nf g : AlternatingMap R' N₁ N₂ ι\ne : Basis ι₁ R' N₁\nh : ∀ (v : ι → ι₁), Function.Injective v → (↑f fun i => ↑e (v i)) = ↑g fun i => ↑e (v i)\n⊢ f = g",
"tactic": "refine' AlternatingMap.coe_multilinearMap_injective (Basis.ext_multilinear e fun v => _)"
},
{
"state_after": "case pos\nR : Type ?u.1290808\ninst✝¹⁶ : Semiring R\nM : Type ?u.1290814\ninst✝¹⁵ : AddCommMonoid M\ninst✝¹⁴ : Module R M\nN : Type ?u.1290846\ninst✝¹³ : AddCommMonoid N\ninst✝¹² : Module R N\nP : Type ?u.1290876\ninst✝¹¹ : AddCommMonoid P\ninst✝¹⁰ : Module R P\nM' : Type ?u.1290906\ninst✝⁹ : AddCommGroup M'\ninst✝⁸ : Module R M'\nN' : Type ?u.1291294\ninst✝⁷ : AddCommGroup N'\ninst✝⁶ : Module R N'\nι : Type u_4\nι' : Type ?u.1291685\nι'' : Type ?u.1291688\nι₁ : Type u_5\ninst✝⁵ : Finite ι\nR' : Type u_1\nN₁ : Type u_2\nN₂ : Type u_3\ninst✝⁴ : CommSemiring R'\ninst✝³ : AddCommMonoid N₁\ninst✝² : AddCommMonoid N₂\ninst✝¹ : Module R' N₁\ninst✝ : Module R' N₂\nf g : AlternatingMap R' N₁ N₂ ι\ne : Basis ι₁ R' N₁\nh : ∀ (v : ι → ι₁), Function.Injective v → (↑f fun i => ↑e (v i)) = ↑g fun i => ↑e (v i)\nv : ι → ι₁\nhi : Function.Injective v\n⊢ (↑↑f fun i => ↑e (v i)) = ↑↑g fun i => ↑e (v i)\n\ncase neg\nR : Type ?u.1290808\ninst✝¹⁶ : Semiring R\nM : Type ?u.1290814\ninst✝¹⁵ : AddCommMonoid M\ninst✝¹⁴ : Module R M\nN : Type ?u.1290846\ninst✝¹³ : AddCommMonoid N\ninst✝¹² : Module R N\nP : Type ?u.1290876\ninst✝¹¹ : AddCommMonoid P\ninst✝¹⁰ : Module R P\nM' : Type ?u.1290906\ninst✝⁹ : AddCommGroup M'\ninst✝⁸ : Module R M'\nN' : Type ?u.1291294\ninst✝⁷ : AddCommGroup N'\ninst✝⁶ : Module R N'\nι : Type u_4\nι' : Type ?u.1291685\nι'' : Type ?u.1291688\nι₁ : Type u_5\ninst✝⁵ : Finite ι\nR' : Type u_1\nN₁ : Type u_2\nN₂ : Type u_3\ninst✝⁴ : CommSemiring R'\ninst✝³ : AddCommMonoid N₁\ninst✝² : AddCommMonoid N₂\ninst✝¹ : Module R' N₁\ninst✝ : Module R' N₂\nf g : AlternatingMap R' N₁ N₂ ι\ne : Basis ι₁ R' N₁\nh : ∀ (v : ι → ι₁), Function.Injective v → (↑f fun i => ↑e (v i)) = ↑g fun i => ↑e (v i)\nv : ι → ι₁\nhi : ¬Function.Injective v\n⊢ (↑↑f fun i => ↑e (v i)) = ↑↑g fun i => ↑e (v i)",
"state_before": "R : Type ?u.1290808\ninst✝¹⁶ : Semiring R\nM : Type ?u.1290814\ninst✝¹⁵ : AddCommMonoid M\ninst✝¹⁴ : Module R M\nN : Type ?u.1290846\ninst✝¹³ : AddCommMonoid N\ninst✝¹² : Module R N\nP : Type ?u.1290876\ninst✝¹¹ : AddCommMonoid P\ninst✝¹⁰ : Module R P\nM' : Type ?u.1290906\ninst✝⁹ : AddCommGroup M'\ninst✝⁸ : Module R M'\nN' : Type ?u.1291294\ninst✝⁷ : AddCommGroup N'\ninst✝⁶ : Module R N'\nι : Type u_4\nι' : Type ?u.1291685\nι'' : Type ?u.1291688\nι₁ : Type u_5\ninst✝⁵ : Finite ι\nR' : Type u_1\nN₁ : Type u_2\nN₂ : Type u_3\ninst✝⁴ : CommSemiring R'\ninst✝³ : AddCommMonoid N₁\ninst✝² : AddCommMonoid N₂\ninst✝¹ : Module R' N₁\ninst✝ : Module R' N₂\nf g : AlternatingMap R' N₁ N₂ ι\ne : Basis ι₁ R' N₁\nh : ∀ (v : ι → ι₁), Function.Injective v → (↑f fun i => ↑e (v i)) = ↑g fun i => ↑e (v i)\nv : ι → ι₁\n⊢ (↑↑f fun i => ↑e (v i)) = ↑↑g fun i => ↑e (v i)",
"tactic": "by_cases hi : Function.Injective v"
},
{
"state_after": "no goals",
"state_before": "case pos\nR : Type ?u.1290808\ninst✝¹⁶ : Semiring R\nM : Type ?u.1290814\ninst✝¹⁵ : AddCommMonoid M\ninst✝¹⁴ : Module R M\nN : Type ?u.1290846\ninst✝¹³ : AddCommMonoid N\ninst✝¹² : Module R N\nP : Type ?u.1290876\ninst✝¹¹ : AddCommMonoid P\ninst✝¹⁰ : Module R P\nM' : Type ?u.1290906\ninst✝⁹ : AddCommGroup M'\ninst✝⁸ : Module R M'\nN' : Type ?u.1291294\ninst✝⁷ : AddCommGroup N'\ninst✝⁶ : Module R N'\nι : Type u_4\nι' : Type ?u.1291685\nι'' : Type ?u.1291688\nι₁ : Type u_5\ninst✝⁵ : Finite ι\nR' : Type u_1\nN₁ : Type u_2\nN₂ : Type u_3\ninst✝⁴ : CommSemiring R'\ninst✝³ : AddCommMonoid N₁\ninst✝² : AddCommMonoid N₂\ninst✝¹ : Module R' N₁\ninst✝ : Module R' N₂\nf g : AlternatingMap R' N₁ N₂ ι\ne : Basis ι₁ R' N₁\nh : ∀ (v : ι → ι₁), Function.Injective v → (↑f fun i => ↑e (v i)) = ↑g fun i => ↑e (v i)\nv : ι → ι₁\nhi : Function.Injective v\n⊢ (↑↑f fun i => ↑e (v i)) = ↑↑g fun i => ↑e (v i)",
"tactic": "exact h v hi"
},
{
"state_after": "case neg\nR : Type ?u.1290808\ninst✝¹⁶ : Semiring R\nM : Type ?u.1290814\ninst✝¹⁵ : AddCommMonoid M\ninst✝¹⁴ : Module R M\nN : Type ?u.1290846\ninst✝¹³ : AddCommMonoid N\ninst✝¹² : Module R N\nP : Type ?u.1290876\ninst✝¹¹ : AddCommMonoid P\ninst✝¹⁰ : Module R P\nM' : Type ?u.1290906\ninst✝⁹ : AddCommGroup M'\ninst✝⁸ : Module R M'\nN' : Type ?u.1291294\ninst✝⁷ : AddCommGroup N'\ninst✝⁶ : Module R N'\nι : Type u_4\nι' : Type ?u.1291685\nι'' : Type ?u.1291688\nι₁ : Type u_5\ninst✝⁵ : Finite ι\nR' : Type u_1\nN₁ : Type u_2\nN₂ : Type u_3\ninst✝⁴ : CommSemiring R'\ninst✝³ : AddCommMonoid N₁\ninst✝² : AddCommMonoid N₂\ninst✝¹ : Module R' N₁\ninst✝ : Module R' N₂\nf g : AlternatingMap R' N₁ N₂ ι\ne : Basis ι₁ R' N₁\nh : ∀ (v : ι → ι₁), Function.Injective v → (↑f fun i => ↑e (v i)) = ↑g fun i => ↑e (v i)\nv : ι → ι₁\nhi : ¬Function.Injective v\nthis : ¬Function.Injective fun i => ↑e (v i)\n⊢ (↑↑f fun i => ↑e (v i)) = ↑↑g fun i => ↑e (v i)",
"state_before": "case neg\nR : Type ?u.1290808\ninst✝¹⁶ : Semiring R\nM : Type ?u.1290814\ninst✝¹⁵ : AddCommMonoid M\ninst✝¹⁴ : Module R M\nN : Type ?u.1290846\ninst✝¹³ : AddCommMonoid N\ninst✝¹² : Module R N\nP : Type ?u.1290876\ninst✝¹¹ : AddCommMonoid P\ninst✝¹⁰ : Module R P\nM' : Type ?u.1290906\ninst✝⁹ : AddCommGroup M'\ninst✝⁸ : Module R M'\nN' : Type ?u.1291294\ninst✝⁷ : AddCommGroup N'\ninst✝⁶ : Module R N'\nι : Type u_4\nι' : Type ?u.1291685\nι'' : Type ?u.1291688\nι₁ : Type u_5\ninst✝⁵ : Finite ι\nR' : Type u_1\nN₁ : Type u_2\nN₂ : Type u_3\ninst✝⁴ : CommSemiring R'\ninst✝³ : AddCommMonoid N₁\ninst✝² : AddCommMonoid N₂\ninst✝¹ : Module R' N₁\ninst✝ : Module R' N₂\nf g : AlternatingMap R' N₁ N₂ ι\ne : Basis ι₁ R' N₁\nh : ∀ (v : ι → ι₁), Function.Injective v → (↑f fun i => ↑e (v i)) = ↑g fun i => ↑e (v i)\nv : ι → ι₁\nhi : ¬Function.Injective v\n⊢ (↑↑f fun i => ↑e (v i)) = ↑↑g fun i => ↑e (v i)",
"tactic": "have : ¬Function.Injective fun i => e (v i) := hi.imp Function.Injective.of_comp"
},
{
"state_after": "no goals",
"state_before": "case neg\nR : Type ?u.1290808\ninst✝¹⁶ : Semiring R\nM : Type ?u.1290814\ninst✝¹⁵ : AddCommMonoid M\ninst✝¹⁴ : Module R M\nN : Type ?u.1290846\ninst✝¹³ : AddCommMonoid N\ninst✝¹² : Module R N\nP : Type ?u.1290876\ninst✝¹¹ : AddCommMonoid P\ninst✝¹⁰ : Module R P\nM' : Type ?u.1290906\ninst✝⁹ : AddCommGroup M'\ninst✝⁸ : Module R M'\nN' : Type ?u.1291294\ninst✝⁷ : AddCommGroup N'\ninst✝⁶ : Module R N'\nι : Type u_4\nι' : Type ?u.1291685\nι'' : Type ?u.1291688\nι₁ : Type u_5\ninst✝⁵ : Finite ι\nR' : Type u_1\nN₁ : Type u_2\nN₂ : Type u_3\ninst✝⁴ : CommSemiring R'\ninst✝³ : AddCommMonoid N₁\ninst✝² : AddCommMonoid N₂\ninst✝¹ : Module R' N₁\ninst✝ : Module R' N₂\nf g : AlternatingMap R' N₁ N₂ ι\ne : Basis ι₁ R' N₁\nh : ∀ (v : ι → ι₁), Function.Injective v → (↑f fun i => ↑e (v i)) = ↑g fun i => ↑e (v i)\nv : ι → ι₁\nhi : ¬Function.Injective v\nthis : ¬Function.Injective fun i => ↑e (v i)\n⊢ (↑↑f fun i => ↑e (v i)) = ↑↑g fun i => ↑e (v i)",
"tactic": "rw [coe_multilinearMap, coe_multilinearMap, f.map_eq_zero_of_not_injective _ this,\n g.map_eq_zero_of_not_injective _ this]"
}
] |
[
1222,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1213,
1
] |
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