file_path
stringlengths 11
79
| full_name
stringlengths 2
100
| traced_tactics
list | end
list | commit
stringclasses 4
values | url
stringclasses 4
values | start
list |
|---|---|---|---|---|---|---|
Mathlib/Order/SymmDiff.lean
|
bihimp_self
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.36681\nα : Type u_1\nβ : Type ?u.36687\nπ : ι → Type ?u.36692\ninst✝ : GeneralizedHeytingAlgebra α\na b c d : α\n⊢ a ⇔ a = ⊤",
"tactic": "rw [bihimp, inf_idem, himp_self]"
}
] |
[
246,
71
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
246,
1
] |
Mathlib/Order/Hom/Basic.lean
|
OrderEmbedding.strictMono
|
[] |
[
669,
72
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
669,
11
] |
Mathlib/RingTheory/Polynomial/Basic.lean
|
Polynomial.support_toSubring
|
[
{
"state_after": "case a\nR : Type u\nS : Type ?u.88744\ninst✝¹ : Ring R\ninst✝ : Semiring S\nf : R →+* S\nx : S\np : R[X]\nT : Subring R\nhp : ↑(frange p) ⊆ ↑T\ni : ℕ\n⊢ i ∈ support (toSubring p T hp) ↔ i ∈ support p",
"state_before": "R : Type u\nS : Type ?u.88744\ninst✝¹ : Ring R\ninst✝ : Semiring S\nf : R →+* S\nx : S\np : R[X]\nT : Subring R\nhp : ↑(frange p) ⊆ ↑T\n⊢ support (toSubring p T hp) = support p",
"tactic": "ext i"
},
{
"state_after": "case a\nR : Type u\nS : Type ?u.88744\ninst✝¹ : Ring R\ninst✝ : Semiring S\nf : R →+* S\nx : S\np : R[X]\nT : Subring R\nhp : ↑(frange p) ⊆ ↑T\ni : ℕ\n⊢ coeff (toSubring p T hp) i = 0 ↔ coeff p i = 0",
"state_before": "case a\nR : Type u\nS : Type ?u.88744\ninst✝¹ : Ring R\ninst✝ : Semiring S\nf : R →+* S\nx : S\np : R[X]\nT : Subring R\nhp : ↑(frange p) ⊆ ↑T\ni : ℕ\n⊢ i ∈ support (toSubring p T hp) ↔ i ∈ support p",
"tactic": "simp only [mem_support_iff, not_iff_not, Ne.def]"
},
{
"state_after": "case a\nR : Type u\nS : Type ?u.88744\ninst✝¹ : Ring R\ninst✝ : Semiring S\nf : R →+* S\nx : S\np : R[X]\nT : Subring R\nhp : ↑(frange p) ⊆ ↑T\ni : ℕ\n⊢ coeff (toSubring p T hp) i = 0 ↔ ↑(coeff (toSubring p T hp) i) = 0",
"state_before": "case a\nR : Type u\nS : Type ?u.88744\ninst✝¹ : Ring R\ninst✝ : Semiring S\nf : R →+* S\nx : S\np : R[X]\nT : Subring R\nhp : ↑(frange p) ⊆ ↑T\ni : ℕ\n⊢ coeff (toSubring p T hp) i = 0 ↔ coeff p i = 0",
"tactic": "conv_rhs => rw [← coeff_toSubring p T hp]"
},
{
"state_after": "no goals",
"state_before": "case a\nR : Type u\nS : Type ?u.88744\ninst✝¹ : Ring R\ninst✝ : Semiring S\nf : R →+* S\nx : S\np : R[X]\nT : Subring R\nhp : ↑(frange p) ⊆ ↑T\ni : ℕ\n⊢ coeff (toSubring p T hp) i = 0 ↔ ↑(coeff (toSubring p T hp) i) = 0",
"tactic": "exact ⟨fun H => by rw [H, ZeroMemClass.coe_zero], fun H => Subtype.coe_injective H⟩"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type ?u.88744\ninst✝¹ : Ring R\ninst✝ : Semiring S\nf : R →+* S\nx : S\np : R[X]\nT : Subring R\nhp : ↑(frange p) ⊆ ↑T\ni : ℕ\nH : coeff (toSubring p T hp) i = 0\n⊢ ↑(coeff (toSubring p T hp) i) = 0",
"tactic": "rw [H, ZeroMemClass.coe_zero]"
}
] |
[
375,
86
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
371,
1
] |
Mathlib/Logic/Equiv/Defs.lean
|
Equiv.arrowCongr_symm
|
[] |
[
538,
64
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
537,
9
] |
Mathlib/Data/List/Perm.lean
|
List.Perm.product_right
|
[] |
[
1108,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1106,
1
] |
Mathlib/FieldTheory/PerfectClosure.lean
|
PerfectClosure.frobenius_mk
|
[
{
"state_after": "K : Type u\ninst✝² : CommRing K\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP K p\nx : ℕ × K\n⊢ mk K p x ^ p = mk K p (x.fst, x.snd ^ p)",
"state_before": "K : Type u\ninst✝² : CommRing K\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP K p\nx : ℕ × K\n⊢ ↑(frobenius (PerfectClosure K p) p) (mk K p x) = mk K p (x.fst, x.snd ^ p)",
"tactic": "simp only [frobenius_def]"
},
{
"state_after": "case mk\nK : Type u\ninst✝² : CommRing K\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP K p\nn : ℕ\nx : K\n⊢ mk K p (n, x) ^ p = mk K p ((n, x).fst, (n, x).snd ^ p)",
"state_before": "K : Type u\ninst✝² : CommRing K\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP K p\nx : ℕ × K\n⊢ mk K p x ^ p = mk K p (x.fst, x.snd ^ p)",
"tactic": "cases' x with n x"
},
{
"state_after": "case mk\nK : Type u\ninst✝² : CommRing K\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP K p\nn : ℕ\nx : K\n⊢ mk K p (n, x) ^ p = mk K p (n, x ^ p)",
"state_before": "case mk\nK : Type u\ninst✝² : CommRing K\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP K p\nn : ℕ\nx : K\n⊢ mk K p (n, x) ^ p = mk K p ((n, x).fst, (n, x).snd ^ p)",
"tactic": "dsimp only"
},
{
"state_after": "case mk\nK : Type u\ninst✝² : CommRing K\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP K p\nn : ℕ\nx : K\n⊢ ∀ (p' : ℕ), mk K p (n, x) ^ p' = mk K p (n, x ^ p')",
"state_before": "case mk\nK : Type u\ninst✝² : CommRing K\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP K p\nn : ℕ\nx : K\n⊢ mk K p (n, x) ^ p = mk K p (n, x ^ p)",
"tactic": "suffices ∀ p' : ℕ, mk K p (n, x) ^ p' = mk K p (n, x ^ p') by apply this"
},
{
"state_after": "case mk\nK : Type u\ninst✝² : CommRing K\np✝ : ℕ\ninst✝¹ : Fact (Nat.Prime p✝)\ninst✝ : CharP K p✝\nn : ℕ\nx : K\np : ℕ\n⊢ mk K p✝ (n, x) ^ p = mk K p✝ (n, x ^ p)",
"state_before": "case mk\nK : Type u\ninst✝² : CommRing K\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP K p\nn : ℕ\nx : K\n⊢ ∀ (p' : ℕ), mk K p (n, x) ^ p' = mk K p (n, x ^ p')",
"tactic": "intro p"
},
{
"state_after": "case mk.zero\nK : Type u\ninst✝² : CommRing K\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP K p\nn : ℕ\nx : K\n⊢ mk K p (n, x) ^ Nat.zero = mk K p (n, x ^ Nat.zero)\n\ncase mk.succ\nK : Type u\ninst✝² : CommRing K\np✝ : ℕ\ninst✝¹ : Fact (Nat.Prime p✝)\ninst✝ : CharP K p✝\nn : ℕ\nx : K\np : ℕ\nih : mk K p✝ (n, x) ^ p = mk K p✝ (n, x ^ p)\n⊢ mk K p✝ (n, x) ^ Nat.succ p = mk K p✝ (n, x ^ Nat.succ p)",
"state_before": "case mk\nK : Type u\ninst✝² : CommRing K\np✝ : ℕ\ninst✝¹ : Fact (Nat.Prime p✝)\ninst✝ : CharP K p✝\nn : ℕ\nx : K\np : ℕ\n⊢ mk K p✝ (n, x) ^ p = mk K p✝ (n, x ^ p)",
"tactic": "induction' p with p ih"
},
{
"state_after": "case mk.succ\nK : Type u\ninst✝² : CommRing K\np✝ : ℕ\ninst✝¹ : Fact (Nat.Prime p✝)\ninst✝ : CharP K p✝\nn : ℕ\nx : K\np : ℕ\nih : mk K p✝ (n, x) ^ p = mk K p✝ (n, x ^ p)\n⊢ mk K p✝ (n, x) ^ Nat.succ p = mk K p✝ (n, x ^ Nat.succ p)",
"state_before": "case mk.zero\nK : Type u\ninst✝² : CommRing K\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP K p\nn : ℕ\nx : K\n⊢ mk K p (n, x) ^ Nat.zero = mk K p (n, x ^ Nat.zero)\n\ncase mk.succ\nK : Type u\ninst✝² : CommRing K\np✝ : ℕ\ninst✝¹ : Fact (Nat.Prime p✝)\ninst✝ : CharP K p✝\nn : ℕ\nx : K\np : ℕ\nih : mk K p✝ (n, x) ^ p = mk K p✝ (n, x ^ p)\n⊢ mk K p✝ (n, x) ^ Nat.succ p = mk K p✝ (n, x ^ Nat.succ p)",
"tactic": "case zero => apply R.sound; rw [(frobenius _ _).iterate_map_one, pow_zero]"
},
{
"state_after": "no goals",
"state_before": "case mk.succ\nK : Type u\ninst✝² : CommRing K\np✝ : ℕ\ninst✝¹ : Fact (Nat.Prime p✝)\ninst✝ : CharP K p✝\nn : ℕ\nx : K\np : ℕ\nih : mk K p✝ (n, x) ^ p = mk K p✝ (n, x ^ p)\n⊢ mk K p✝ (n, x) ^ Nat.succ p = mk K p✝ (n, x ^ Nat.succ p)",
"tactic": "case succ =>\n rw [pow_succ, ih]\n symm\n apply R.sound\n simp only [pow_succ, (frobenius _ _).iterate_map_mul]"
},
{
"state_after": "no goals",
"state_before": "K : Type u\ninst✝² : CommRing K\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP K p\nn : ℕ\nx : K\nthis : ∀ (p' : ℕ), mk K p (n, x) ^ p' = mk K p (n, x ^ p')\n⊢ mk K p (n, x) ^ p = mk K p (n, x ^ p)",
"tactic": "apply this"
},
{
"state_after": "case H\nK : Type u\ninst✝² : CommRing K\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP K p\nn : ℕ\nx : K\n⊢ (↑(frobenius K p)^[n]) 1 = x ^ Nat.zero",
"state_before": "K : Type u\ninst✝² : CommRing K\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP K p\nn : ℕ\nx : K\n⊢ mk K p (n, x) ^ Nat.zero = mk K p (n, x ^ Nat.zero)",
"tactic": "apply R.sound"
},
{
"state_after": "no goals",
"state_before": "case H\nK : Type u\ninst✝² : CommRing K\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP K p\nn : ℕ\nx : K\n⊢ (↑(frobenius K p)^[n]) 1 = x ^ Nat.zero",
"tactic": "rw [(frobenius _ _).iterate_map_one, pow_zero]"
},
{
"state_after": "K : Type u\ninst✝² : CommRing K\np✝ : ℕ\ninst✝¹ : Fact (Nat.Prime p✝)\ninst✝ : CharP K p✝\nn : ℕ\nx : K\np : ℕ\nih : mk K p✝ (n, x) ^ p = mk K p✝ (n, x ^ p)\n⊢ mk K p✝ (n, x) * mk K p✝ (n, x ^ p) = mk K p✝ (n, x ^ Nat.succ p)",
"state_before": "K : Type u\ninst✝² : CommRing K\np✝ : ℕ\ninst✝¹ : Fact (Nat.Prime p✝)\ninst✝ : CharP K p✝\nn : ℕ\nx : K\np : ℕ\nih : mk K p✝ (n, x) ^ p = mk K p✝ (n, x ^ p)\n⊢ mk K p✝ (n, x) ^ Nat.succ p = mk K p✝ (n, x ^ Nat.succ p)",
"tactic": "rw [pow_succ, ih]"
},
{
"state_after": "K : Type u\ninst✝² : CommRing K\np✝ : ℕ\ninst✝¹ : Fact (Nat.Prime p✝)\ninst✝ : CharP K p✝\nn : ℕ\nx : K\np : ℕ\nih : mk K p✝ (n, x) ^ p = mk K p✝ (n, x ^ p)\n⊢ mk K p✝ (n, x ^ Nat.succ p) = mk K p✝ (n, x) * mk K p✝ (n, x ^ p)",
"state_before": "K : Type u\ninst✝² : CommRing K\np✝ : ℕ\ninst✝¹ : Fact (Nat.Prime p✝)\ninst✝ : CharP K p✝\nn : ℕ\nx : K\np : ℕ\nih : mk K p✝ (n, x) ^ p = mk K p✝ (n, x ^ p)\n⊢ mk K p✝ (n, x) * mk K p✝ (n, x ^ p) = mk K p✝ (n, x ^ Nat.succ p)",
"tactic": "symm"
},
{
"state_after": "case H\nK : Type u\ninst✝² : CommRing K\np✝ : ℕ\ninst✝¹ : Fact (Nat.Prime p✝)\ninst✝ : CharP K p✝\nn : ℕ\nx : K\np : ℕ\nih : mk K p✝ (n, x) ^ p = mk K p✝ (n, x ^ p)\n⊢ (↑(frobenius K p✝)^[(n, x).fst]) (x ^ Nat.succ p) =\n (↑(frobenius K p✝)^[(n, x ^ p).fst]) (n, x).snd * (↑(frobenius K p✝)^[(n, x).fst]) (n, x ^ p).snd",
"state_before": "K : Type u\ninst✝² : CommRing K\np✝ : ℕ\ninst✝¹ : Fact (Nat.Prime p✝)\ninst✝ : CharP K p✝\nn : ℕ\nx : K\np : ℕ\nih : mk K p✝ (n, x) ^ p = mk K p✝ (n, x ^ p)\n⊢ mk K p✝ (n, x ^ Nat.succ p) = mk K p✝ (n, x) * mk K p✝ (n, x ^ p)",
"tactic": "apply R.sound"
},
{
"state_after": "no goals",
"state_before": "case H\nK : Type u\ninst✝² : CommRing K\np✝ : ℕ\ninst✝¹ : Fact (Nat.Prime p✝)\ninst✝ : CharP K p✝\nn : ℕ\nx : K\np : ℕ\nih : mk K p✝ (n, x) ^ p = mk K p✝ (n, x ^ p)\n⊢ (↑(frobenius K p✝)^[(n, x).fst]) (x ^ Nat.succ p) =\n (↑(frobenius K p✝)^[(n, x ^ p).fst]) (n, x).snd * (↑(frobenius K p✝)^[(n, x).fst]) (n, x ^ p).snd",
"tactic": "simp only [pow_succ, (frobenius _ _).iterate_map_mul]"
}
] |
[
463,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
449,
1
] |
Mathlib/Algebra/Module/Submodule/Basic.lean
|
Submodule.sum_mem
|
[] |
[
247,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
246,
11
] |
Mathlib/Algebra/Lie/Submodule.lean
|
LieIdeal.mem_map_of_surjective
|
[
{
"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'\ny : L'\nh₁ : Function.Surjective ↑f\nh₂ : ∃ y_1, y_1 ∈ ↑I ∧ ↑↑f y_1 = y\n⊢ ∃ x, ↑f ↑x = y",
"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'\ny : L'\nh₁ : Function.Surjective ↑f\nh₂ : y ∈ map f I\n⊢ ∃ x, ↑f ↑x = y",
"tactic": "rw [← LieSubmodule.mem_coeSubmodule, coe_map_of_surjective h₁, Submodule.mem_map] at h₂"
},
{
"state_after": "case intro.intro\nR : 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'\nh₁ : Function.Surjective ↑f\nx : L\nhx : x ∈ ↑I\n⊢ ∃ x_1, ↑f ↑x_1 = ↑↑f x",
"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'\ny : L'\nh₁ : Function.Surjective ↑f\nh₂ : ∃ y_1, y_1 ∈ ↑I ∧ ↑↑f y_1 = y\n⊢ ∃ x, ↑f ↑x = y",
"tactic": "obtain ⟨x, hx, rfl⟩ := h₂"
},
{
"state_after": "case intro.intro\nR : 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'\nh₁ : Function.Surjective ↑f\nx : L\nhx : x ∈ ↑I\n⊢ ↑f ↑{ val := x, property := hx } = ↑↑f x",
"state_before": "case intro.intro\nR : 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'\nh₁ : Function.Surjective ↑f\nx : L\nhx : x ∈ ↑I\n⊢ ∃ x_1, ↑f ↑x_1 = ↑↑f x",
"tactic": "use ⟨x, hx⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\nR : 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'\nh₁ : Function.Surjective ↑f\nx : L\nhx : x ∈ ↑I\n⊢ ↑f ↑{ val := x, property := hx } = ↑↑f x",
"tactic": "rfl"
}
] |
[
1042,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1037,
1
] |
Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean
|
MeasureTheory.AEStronglyMeasurable.iUnion
|
[] |
[
1706,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1703,
11
] |
Mathlib/Topology/Order.lean
|
isClosed_iSup_iff
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nι : Sort v\nt : ι → TopologicalSpace α\ns : Set α\n⊢ IsClosed s ↔ ∀ (i : ι), IsClosed s",
"tactic": "simp [← @isOpen_compl_iff _ (⨆ i, t i), ← @isOpen_compl_iff _ (t _), isOpen_iSup_iff]"
}
] |
[
1002,
88
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1001,
1
] |
Mathlib/Order/LocallyFinite.lean
|
Finset.coe_uIcc
|
[] |
[
509,
14
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
508,
1
] |
Mathlib/Data/Set/Image.lean
|
Set.preimage_singleton_eq_empty
|
[] |
[
1033,
71
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1032,
1
] |
Mathlib/MeasureTheory/Function/SimpleFunc.lean
|
MeasureTheory.SimpleFunc.measurableSet_fiber
|
[] |
[
85,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
84,
1
] |
Mathlib/Init/Algebra/Classes.lean
|
total_of
|
[] |
[
312,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
311,
1
] |
Mathlib/RingTheory/Coprime/Basic.lean
|
isCoprime_one_right
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\ninst✝ : CommSemiring R\nx y z : R\n⊢ 0 * x + 1 * 1 = 1",
"tactic": "rw [one_mul, zero_mul, zero_add]"
}
] |
[
87,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
86,
1
] |
Mathlib/Computability/Primrec.lean
|
Primrec.vector_cons
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.361685\nγ : Type ?u.361688\nσ : Type ?u.361691\ninst✝³ : Primcodable α\ninst✝² : Primcodable β\ninst✝¹ : Primcodable γ\ninst✝ : Primcodable σ\nn : ℕ\n⊢ Primrec fun a => a.fst :: Vector.toList a.snd",
"state_before": "α : Type u_1\nβ : Type ?u.361685\nγ : Type ?u.361688\nσ : Type ?u.361691\ninst✝³ : Primcodable α\ninst✝² : Primcodable β\ninst✝¹ : Primcodable γ\ninst✝ : Primcodable σ\nn : ℕ\n⊢ Primrec fun a => Vector.toList (a.fst ::ᵥ a.snd)",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.361685\nγ : Type ?u.361688\nσ : Type ?u.361691\ninst✝³ : Primcodable α\ninst✝² : Primcodable β\ninst✝¹ : Primcodable γ\ninst✝ : Primcodable σ\nn : ℕ\n⊢ Primrec fun a => a.fst :: Vector.toList a.snd",
"tactic": "exact list_cons.comp fst (vector_toList_iff.2 snd)"
}
] |
[
1293,
85
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1292,
1
] |
Mathlib/Analysis/NormedSpace/ContinuousLinearMap.lean
|
LinearMap.toContinuousLinearMap₁_apply
|
[] |
[
133,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
131,
1
] |
Mathlib/Data/Set/Intervals/Basic.lean
|
Set.Ioi_subset_Ioi_iff
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.72368\ninst✝ : LinearOrder α\na a₁ a₂ b b₁ b₂ c d : α\nh : Ioi b ⊆ Ioi a\n⊢ a ≤ b",
"state_before": "α : Type u_1\nβ : Type ?u.72368\ninst✝ : LinearOrder α\na a₁ a₂ b b₁ b₂ c d : α\n⊢ Ioi b ⊆ Ioi a ↔ a ≤ b",
"tactic": "refine' ⟨fun h => _, fun h => Ioi_subset_Ioi h⟩"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.72368\ninst✝ : LinearOrder α\na a₁ a₂ b b₁ b₂ c d : α\nh : Ioi b ⊆ Ioi a\nba : ¬a ≤ b\n⊢ False",
"state_before": "α : Type u_1\nβ : Type ?u.72368\ninst✝ : LinearOrder α\na a₁ a₂ b b₁ b₂ c d : α\nh : Ioi b ⊆ Ioi a\n⊢ a ≤ b",
"tactic": "by_contra ba"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.72368\ninst✝ : LinearOrder α\na a₁ a₂ b b₁ b₂ c d : α\nh : Ioi b ⊆ Ioi a\nba : ¬a ≤ b\n⊢ False",
"tactic": "exact lt_irrefl _ (h (not_le.mp ba))"
}
] |
[
1173,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1170,
1
] |
Mathlib/GroupTheory/GroupAction/Prod.lean
|
Prod.pow_swap
|
[] |
[
117,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
116,
1
] |
Mathlib/Combinatorics/Composition.lean
|
CompositionAsSet.toComposition_blocks
|
[] |
[
1043,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1041,
1
] |
Mathlib/Logic/Equiv/Fin.lean
|
coe_finRotate_of_ne_last
|
[
{
"state_after": "m n : ℕ\ni : Fin (Nat.succ n)\nh : i ≠ Fin.last n\n⊢ ↑(i + 1) = ↑i + 1",
"state_before": "m n : ℕ\ni : Fin (Nat.succ n)\nh : i ≠ Fin.last n\n⊢ ↑(↑(finRotate (n + 1)) i) = ↑i + 1",
"tactic": "rw [finRotate_succ_apply]"
},
{
"state_after": "m n : ℕ\ni : Fin (Nat.succ n)\nh : i ≠ Fin.last n\nthis : ↑i < n\n⊢ ↑(i + 1) = ↑i + 1",
"state_before": "m n : ℕ\ni : Fin (Nat.succ n)\nh : i ≠ Fin.last n\n⊢ ↑(i + 1) = ↑i + 1",
"tactic": "have : (i : ℕ) < n := Fin.val_lt_last h"
},
{
"state_after": "no goals",
"state_before": "m n : ℕ\ni : Fin (Nat.succ n)\nh : i ≠ Fin.last n\nthis : ↑i < n\n⊢ ↑(i + 1) = ↑i + 1",
"tactic": "exact Fin.val_add_one_of_lt this"
}
] |
[
454,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
450,
1
] |
Mathlib/Order/Cover.lean
|
Set.OrdConnected.apply_covby_apply_iff
|
[] |
[
355,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
353,
1
] |
Mathlib/Topology/ContinuousOn.lean
|
continuousOn_to_generateFrom_iff
|
[
{
"state_after": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.347950\nδ : Type ?u.347953\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\ninst✝¹ : TopologicalSpace γ\ninst✝ : TopologicalSpace δ\ns : Set α\nT : Set (Set β)\nf : α → β\nx : α\nx✝ : x ∈ s\n⊢ Tendsto f (𝓝[s] x) (𝓝 (f x)) ↔ ∀ (t : Set β), t ∈ T → f x ∈ t → f ⁻¹' t ∈ 𝓝[s] x",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.347950\nδ : Type ?u.347953\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\ninst✝¹ : TopologicalSpace γ\ninst✝ : TopologicalSpace δ\ns : Set α\nT : Set (Set β)\nf : α → β\nx : α\nx✝ : x ∈ s\n⊢ ContinuousWithinAt f s x ↔ ∀ (t : Set β), t ∈ T → f x ∈ t → f ⁻¹' t ∈ 𝓝[s] x",
"tactic": "delta ContinuousWithinAt"
},
{
"state_after": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.347950\nδ : Type ?u.347953\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\ninst✝¹ : TopologicalSpace γ\ninst✝ : TopologicalSpace δ\ns : Set α\nT : Set (Set β)\nf : α → β\nx : α\nx✝ : x ∈ s\n⊢ (∀ (i : Set β), f x ∈ i → i ∈ T → ∀ᶠ (a : α) in 𝓝[s] x, f a ∈ i) ↔ ∀ (t : Set β), t ∈ T → f x ∈ t → f ⁻¹' t ∈ 𝓝[s] x",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.347950\nδ : Type ?u.347953\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\ninst✝¹ : TopologicalSpace γ\ninst✝ : TopologicalSpace δ\ns : Set α\nT : Set (Set β)\nf : α → β\nx : α\nx✝ : x ∈ s\n⊢ Tendsto f (𝓝[s] x) (𝓝 (f x)) ↔ ∀ (t : Set β), t ∈ T → f x ∈ t → f ⁻¹' t ∈ 𝓝[s] x",
"tactic": "simp only [TopologicalSpace.nhds_generateFrom, tendsto_iInf, tendsto_principal, mem_setOf_eq,\n and_imp]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.347950\nδ : Type ?u.347953\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\ninst✝¹ : TopologicalSpace γ\ninst✝ : TopologicalSpace δ\ns : Set α\nT : Set (Set β)\nf : α → β\nx : α\nx✝ : x ∈ s\n⊢ (∀ (i : Set β), f x ∈ i → i ∈ T → ∀ᶠ (a : α) in 𝓝[s] x, f a ∈ i) ↔ ∀ (t : Set β), t ∈ T → f x ∈ t → f ⁻¹' t ∈ 𝓝[s] x",
"tactic": "exact forall_congr' fun t => forall_swap"
}
] |
[
1049,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1043,
1
] |
Mathlib/Data/List/AList.lean
|
AList.not_mem_empty
|
[] |
[
117,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
116,
1
] |
Mathlib/Data/Quot.lean
|
Quotient.choice_eq
|
[] |
[
428,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
426,
1
] |
Mathlib/Data/List/Rdrop.lean
|
List.rdropWhile_prefix
|
[
{
"state_after": "α : Type u_1\np : α → Bool\nl : List α\nn : ℕ\n⊢ dropWhile p (reverse l) <:+ reverse l",
"state_before": "α : Type u_1\np : α → Bool\nl : List α\nn : ℕ\n⊢ rdropWhile p l <+: l",
"tactic": "rw [← reverse_suffix, rdropWhile, reverse_reverse]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\np : α → Bool\nl : List α\nn : ℕ\n⊢ dropWhile p (reverse l) <:+ reverse l",
"tactic": "exact dropWhile_suffix _"
}
] |
[
135,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
133,
1
] |
Mathlib/MeasureTheory/Function/L1Space.lean
|
MeasureTheory.Integrable.neg
|
[] |
[
680,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
679,
1
] |
Mathlib/Algebra/Order/AbsoluteValue.lean
|
AbsoluteValue.coe_toMonoidWithZeroHom
|
[] |
[
195,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
194,
1
] |
Mathlib/Analysis/NormedSpace/LinearIsometry.lean
|
LinearIsometry.comp_continuous_iff
|
[] |
[
364,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
362,
1
] |
Mathlib/MeasureTheory/Function/L2Space.lean
|
MeasureTheory.L2.inner_def
|
[] |
[
157,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
156,
1
] |
Mathlib/Order/BooleanAlgebra.lean
|
compl_injective
|
[] |
[
658,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
657,
1
] |
Mathlib/LinearAlgebra/Eigenspace/Basic.lean
|
Module.End.eigenspace_le_generalizedEigenspace
|
[] |
[
345,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
343,
1
] |
Mathlib/GroupTheory/Perm/Cycle/Concrete.lean
|
Equiv.Perm.next_toList_eq_apply
|
[
{
"state_after": "α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\np✝ : Perm α\nx✝ : α\np : Perm α\nx y : α\nhy✝ : y ∈ toList p x\nhy : SameCycle p x y ∧ x ∈ support p\n⊢ next (toList p x) y hy✝ = ↑p y",
"state_before": "α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\np✝ : Perm α\nx✝ : α\np : Perm α\nx y : α\nhy : y ∈ toList p x\n⊢ next (toList p x) y hy = ↑p y",
"tactic": "rw [mem_toList_iff] at hy"
},
{
"state_after": "case intro.intro\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\np✝ : Perm α\nx✝ : α\np : Perm α\nx y : α\nhy✝ : y ∈ toList p x\nhy : SameCycle p x y ∧ x ∈ support p\nk : ℕ\nhk : k < Finset.card (support (cycleOf p x))\nhk' : ↑(p ^ k) x = y\n⊢ next (toList p x) y hy✝ = ↑p y",
"state_before": "α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\np✝ : Perm α\nx✝ : α\np : Perm α\nx y : α\nhy✝ : y ∈ toList p x\nhy : SameCycle p x y ∧ x ∈ support p\n⊢ next (toList p x) y hy✝ = ↑p y",
"tactic": "obtain ⟨k, hk, hk'⟩ := hy.left.exists_pow_eq_of_mem_support hy.right"
},
{
"state_after": "case intro.intro\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\np✝ : Perm α\nx✝ : α\np : Perm α\nx y : α\nhy✝ : y ∈ toList p x\nhy : SameCycle p x y ∧ x ∈ support p\nk : ℕ\nhk : k < Finset.card (support (cycleOf p x))\nhk'✝ : ↑(p ^ k) x = y\nhk' : nthLe (toList p x) k (_ : k < length (toList p x)) = y\n⊢ next (toList p x) y hy✝ = ↑p y",
"state_before": "case intro.intro\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\np✝ : Perm α\nx✝ : α\np : Perm α\nx y : α\nhy✝ : y ∈ toList p x\nhy : SameCycle p x y ∧ x ∈ support p\nk : ℕ\nhk : k < Finset.card (support (cycleOf p x))\nhk' : ↑(p ^ k) x = y\n⊢ next (toList p x) y hy✝ = ↑p y",
"tactic": "rw [← nthLe_toList p x k (by simpa using hk)] at hk'"
},
{
"state_after": "case intro.intro\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\np✝ : Perm α\nx✝ : α\np : Perm α\nx y : α\nhy✝ : y ∈ toList p x\nhy : SameCycle p x y ∧ x ∈ support p\nk : ℕ\nhk : k < Finset.card (support (cycleOf p x))\nhk'✝ : ↑(p ^ k) x = y\nhk' : nthLe (toList p x) k (_ : k < length (toList p x)) = y\n⊢ next (toList p x) (nthLe (toList p x) k (_ : k < length (toList p x)))\n (_ : nthLe (toList p x) k (_ : k < length (toList p x)) ∈ toList p x) =\n ↑p (nthLe (toList p x) k (_ : k < length (toList p x)))",
"state_before": "case intro.intro\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\np✝ : Perm α\nx✝ : α\np : Perm α\nx y : α\nhy✝ : y ∈ toList p x\nhy : SameCycle p x y ∧ x ∈ support p\nk : ℕ\nhk : k < Finset.card (support (cycleOf p x))\nhk'✝ : ↑(p ^ k) x = y\nhk' : nthLe (toList p x) k (_ : k < length (toList p x)) = y\n⊢ next (toList p x) y hy✝ = ↑p y",
"tactic": "simp_rw [← hk']"
},
{
"state_after": "case intro.intro\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\np✝ : Perm α\nx✝ : α\np : Perm α\nx y : α\nhy✝ : y ∈ toList p x\nhy : SameCycle p x y ∧ x ∈ support p\nk : ℕ\nhk : k < Finset.card (support (cycleOf p x))\nhk'✝ : ↑(p ^ k) x = y\nhk' : nthLe (toList p x) k (_ : k < length (toList p x)) = y\n⊢ IsCycle (cycleOf p x)",
"state_before": "case intro.intro\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\np✝ : Perm α\nx✝ : α\np : Perm α\nx y : α\nhy✝ : y ∈ toList p x\nhy : SameCycle p x y ∧ x ∈ support p\nk : ℕ\nhk : k < Finset.card (support (cycleOf p x))\nhk'✝ : ↑(p ^ k) x = y\nhk' : nthLe (toList p x) k (_ : k < length (toList p x)) = y\n⊢ next (toList p x) (nthLe (toList p x) k (_ : k < length (toList p x)))\n (_ : nthLe (toList p x) k (_ : k < length (toList p x)) ∈ toList p x) =\n ↑p (nthLe (toList p x) k (_ : k < length (toList p x)))",
"tactic": "rw [next_nthLe _ (nodup_toList _ _), nthLe_toList, nthLe_toList, ← mul_apply, ← pow_succ,\n length_toList, pow_apply_eq_pow_mod_orderOf_cycleOf_apply p (k + 1), IsCycle.orderOf]"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\np✝ : Perm α\nx✝ : α\np : Perm α\nx y : α\nhy✝ : y ∈ toList p x\nhy : SameCycle p x y ∧ x ∈ support p\nk : ℕ\nhk : k < Finset.card (support (cycleOf p x))\nhk'✝ : ↑(p ^ k) x = y\nhk' : nthLe (toList p x) k (_ : k < length (toList p x)) = y\n⊢ IsCycle (cycleOf p x)",
"tactic": "exact isCycle_cycleOf _ (mem_support.mp hy.right)"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\np✝ : Perm α\nx✝ : α\np : Perm α\nx y : α\nhy✝ : y ∈ toList p x\nhy : SameCycle p x y ∧ x ∈ support p\nk : ℕ\nhk : k < Finset.card (support (cycleOf p x))\nhk' : ↑(p ^ k) x = y\n⊢ k < length (toList p x)",
"tactic": "simpa using hk"
}
] |
[
314,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
306,
1
] |
Mathlib/Algebra/Homology/ShortExact/Preadditive.lean
|
CategoryTheory.Splitting.shortExact
|
[] |
[
328,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
325,
11
] |
Mathlib/Algebra/Parity.lean
|
Even.pow_pos
|
[
{
"state_after": "case intro\nF : Type ?u.107024\nα : Type ?u.107027\nβ : Type ?u.107030\nR : Type u_1\ninst✝ : LinearOrderedRing R\na : R\nn : ℕ\nha : a ≠ 0\nk : ℕ\nhk : n = k + k\n⊢ 0 < a ^ n",
"state_before": "F : Type ?u.107024\nα : Type ?u.107027\nβ : Type ?u.107030\nR : Type u_1\ninst✝ : LinearOrderedRing R\na : R\nn : ℕ\nhn : Even n\nha : a ≠ 0\n⊢ 0 < a ^ n",
"tactic": "cases' hn with k hk"
},
{
"state_after": "no goals",
"state_before": "case intro\nF : Type ?u.107024\nα : Type ?u.107027\nβ : Type ?u.107030\nR : Type u_1\ninst✝ : LinearOrderedRing R\na : R\nn : ℕ\nha : a ≠ 0\nk : ℕ\nhk : n = k + k\n⊢ 0 < a ^ n",
"tactic": "simpa only [hk, two_mul] using pow_bit0_pos ha k"
}
] |
[
475,
73
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
474,
1
] |
Mathlib/Algebra/CharP/Basic.lean
|
frobenius_one
|
[] |
[
364,
12
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
363,
1
] |
Mathlib/LinearAlgebra/QuadraticForm/Isometry.lean
|
QuadraticForm.Equivalent.trans
|
[] |
[
119,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
118,
1
] |
Mathlib/SetTheory/Cardinal/Basic.lean
|
Cardinal.toNat_mul
|
[
{
"state_after": "case inl\nα β : Type u\ny : Cardinal\n⊢ ↑toNat (0 * y) = ↑toNat 0 * ↑toNat y\n\ncase inr\nα β : Type u\nx y : Cardinal\nhx1 : x ≠ 0\n⊢ ↑toNat (x * y) = ↑toNat x * ↑toNat y",
"state_before": "α β : Type u\nx y : Cardinal\n⊢ ↑toNat (x * y) = ↑toNat x * ↑toNat y",
"tactic": "rcases eq_or_ne x 0 with (rfl | hx1)"
},
{
"state_after": "case inr.inl\nα β : Type u\nx : Cardinal\nhx1 : x ≠ 0\n⊢ ↑toNat (x * 0) = ↑toNat x * ↑toNat 0\n\ncase inr.inr\nα β : Type u\nx y : Cardinal\nhx1 : x ≠ 0\nhy1 : y ≠ 0\n⊢ ↑toNat (x * y) = ↑toNat x * ↑toNat y",
"state_before": "case inr\nα β : Type u\nx y : Cardinal\nhx1 : x ≠ 0\n⊢ ↑toNat (x * y) = ↑toNat x * ↑toNat y",
"tactic": "rcases eq_or_ne y 0 with (rfl | hy1)"
},
{
"state_after": "case inr.inr.inl\nα β : Type u\nx y : Cardinal\nhx1 : x ≠ 0\nhy1 : y ≠ 0\nhx2 : x < ℵ₀\n⊢ ↑toNat (x * y) = ↑toNat x * ↑toNat y\n\ncase inr.inr.inr\nα β : Type u\nx y : Cardinal\nhx1 : x ≠ 0\nhy1 : y ≠ 0\nhx2 : ℵ₀ ≤ x\n⊢ ↑toNat (x * y) = ↑toNat x * ↑toNat y",
"state_before": "case inr.inr\nα β : Type u\nx y : Cardinal\nhx1 : x ≠ 0\nhy1 : y ≠ 0\n⊢ ↑toNat (x * y) = ↑toNat x * ↑toNat y",
"tactic": "cases' lt_or_le x ℵ₀ with hx2 hx2"
},
{
"state_after": "no goals",
"state_before": "case inl\nα β : Type u\ny : Cardinal\n⊢ ↑toNat (0 * y) = ↑toNat 0 * ↑toNat y",
"tactic": "rw [zero_mul, zero_toNat, zero_mul]"
},
{
"state_after": "no goals",
"state_before": "case inr.inl\nα β : Type u\nx : Cardinal\nhx1 : x ≠ 0\n⊢ ↑toNat (x * 0) = ↑toNat x * ↑toNat 0",
"tactic": "rw [mul_zero, zero_toNat, mul_zero]"
},
{
"state_after": "case inr.inr.inl.inl\nα β : Type u\nx y : Cardinal\nhx1 : x ≠ 0\nhy1 : y ≠ 0\nhx2 : x < ℵ₀\nhy2 : y < ℵ₀\n⊢ ↑toNat (x * y) = ↑toNat x * ↑toNat y\n\ncase inr.inr.inl.inr\nα β : Type u\nx y : Cardinal\nhx1 : x ≠ 0\nhy1 : y ≠ 0\nhx2 : x < ℵ₀\nhy2 : ℵ₀ ≤ y\n⊢ ↑toNat (x * y) = ↑toNat x * ↑toNat y",
"state_before": "case inr.inr.inl\nα β : Type u\nx y : Cardinal\nhx1 : x ≠ 0\nhy1 : y ≠ 0\nhx2 : x < ℵ₀\n⊢ ↑toNat (x * y) = ↑toNat x * ↑toNat y",
"tactic": "cases' lt_or_le y ℵ₀ with hy2 hy2"
},
{
"state_after": "case inr.inr.inl.inl.intro\nα β : Type u\ny : Cardinal\nhy1 : y ≠ 0\nhy2 : y < ℵ₀\nx : ℕ\nhx1 : ↑x ≠ 0\n⊢ ↑toNat (↑x * y) = ↑toNat ↑x * ↑toNat y",
"state_before": "case inr.inr.inl.inl\nα β : Type u\nx y : Cardinal\nhx1 : x ≠ 0\nhy1 : y ≠ 0\nhx2 : x < ℵ₀\nhy2 : y < ℵ₀\n⊢ ↑toNat (x * y) = ↑toNat x * ↑toNat y",
"tactic": "lift x to ℕ using hx2"
},
{
"state_after": "case inr.inr.inl.inl.intro.intro\nα β : Type u\nx : ℕ\nhx1 : ↑x ≠ 0\ny : ℕ\nhy1 : ↑y ≠ 0\n⊢ ↑toNat (↑x * ↑y) = ↑toNat ↑x * ↑toNat ↑y",
"state_before": "case inr.inr.inl.inl.intro\nα β : Type u\ny : Cardinal\nhy1 : y ≠ 0\nhy2 : y < ℵ₀\nx : ℕ\nhx1 : ↑x ≠ 0\n⊢ ↑toNat (↑x * y) = ↑toNat ↑x * ↑toNat y",
"tactic": "lift y to ℕ using hy2"
},
{
"state_after": "no goals",
"state_before": "case inr.inr.inl.inl.intro.intro\nα β : Type u\nx : ℕ\nhx1 : ↑x ≠ 0\ny : ℕ\nhy1 : ↑y ≠ 0\n⊢ ↑toNat (↑x * ↑y) = ↑toNat ↑x * ↑toNat ↑y",
"tactic": "rw [← Nat.cast_mul, toNat_cast, toNat_cast, toNat_cast]"
},
{
"state_after": "case inr.inr.inl.inr\nα β : Type u\nx y : Cardinal\nhx1 : x ≠ 0\nhy1 : y ≠ 0\nhx2 : x < ℵ₀\nhy2 : ℵ₀ ≤ y\n⊢ ℵ₀ ≤ x * y",
"state_before": "case inr.inr.inl.inr\nα β : Type u\nx y : Cardinal\nhx1 : x ≠ 0\nhy1 : y ≠ 0\nhx2 : x < ℵ₀\nhy2 : ℵ₀ ≤ y\n⊢ ↑toNat (x * y) = ↑toNat x * ↑toNat y",
"tactic": "rw [toNat_apply_of_aleph0_le hy2, mul_zero, toNat_apply_of_aleph0_le]"
},
{
"state_after": "no goals",
"state_before": "case inr.inr.inl.inr\nα β : Type u\nx y : Cardinal\nhx1 : x ≠ 0\nhy1 : y ≠ 0\nhx2 : x < ℵ₀\nhy2 : ℵ₀ ≤ y\n⊢ ℵ₀ ≤ x * y",
"tactic": "exact aleph0_le_mul_iff'.2 (Or.inl ⟨hx1, hy2⟩)"
},
{
"state_after": "case inr.inr.inr\nα β : Type u\nx y : Cardinal\nhx1 : x ≠ 0\nhy1 : y ≠ 0\nhx2 : ℵ₀ ≤ x\n⊢ ℵ₀ ≤ x * y",
"state_before": "case inr.inr.inr\nα β : Type u\nx y : Cardinal\nhx1 : x ≠ 0\nhy1 : y ≠ 0\nhx2 : ℵ₀ ≤ x\n⊢ ↑toNat (x * y) = ↑toNat x * ↑toNat y",
"tactic": "rw [toNat_apply_of_aleph0_le hx2, zero_mul, toNat_apply_of_aleph0_le]"
},
{
"state_after": "no goals",
"state_before": "case inr.inr.inr\nα β : Type u\nx y : Cardinal\nhx1 : x ≠ 0\nhy1 : y ≠ 0\nhx2 : ℵ₀ ≤ x\n⊢ ℵ₀ ≤ x * y",
"tactic": "exact aleph0_le_mul_iff'.2 (Or.inr ⟨hx2, hy1⟩)"
}
] |
[
1805,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1792,
1
] |
Mathlib/Algebra/DirectSum/Internal.lean
|
DirectSum.coe_of_mul_apply_add
|
[] |
[
214,
86
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
212,
1
] |
Mathlib/Algebra/Order/Monoid/Lemmas.lean
|
StrictAnti.mul'
|
[] |
[
1458,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1455,
1
] |
Mathlib/Order/Filter/Extr.lean
|
IsMaxFilter.max
|
[] |
[
617,
68
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
614,
1
] |
Mathlib/Dynamics/FixedPoints/Basic.lean
|
Function.IsFixedPt.left_of_comp
|
[] |
[
75,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
72,
1
] |
Mathlib/NumberTheory/LucasLehmer.lean
|
LucasLehmer.mersenne_coe_X
|
[
{
"state_after": "case h₁\np : ℕ\n⊢ minFac (2 ^ p - 1) ∣ 2 ^ p - 1",
"state_before": "p : ℕ\n⊢ ↑(mersenne p) = 0",
"tactic": "ext <;> simp [mersenne, q, ZMod.nat_cast_zmod_eq_zero_iff_dvd, -pow_pos]"
},
{
"state_after": "no goals",
"state_before": "case h₁\np : ℕ\n⊢ minFac (2 ^ p - 1) ∣ 2 ^ p - 1",
"tactic": "apply Nat.minFac_dvd"
}
] |
[
438,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
436,
1
] |
Mathlib/Data/Set/Pointwise/SMul.lean
|
Set.singleton_vsub_singleton
|
[] |
[
660,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
659,
1
] |
Mathlib/RingTheory/Localization/Basic.lean
|
IsLocalization.map_nonZeroDivisors_le
|
[] |
[
839,
67
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
837,
1
] |
Mathlib/Order/Filter/Pointwise.lean
|
Filter.inv_le_self
|
[] |
[
268,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
267,
1
] |
Mathlib/RingTheory/Polynomial/Quotient.lean
|
Ideal.quotient_map_C_eq_zero
|
[
{
"state_after": "R : Type u_1\ninst✝ : CommRing R\nI : Ideal R\na : R\nha : a ∈ I\n⊢ ↑(RingHom.comp (Quotient.mk (map C I)) C) a = 0",
"state_before": "R : Type u_1\ninst✝ : CommRing R\nI : Ideal R\n⊢ ∀ (a : R), a ∈ I → ↑(RingHom.comp (Quotient.mk (map C I)) C) a = 0",
"tactic": "intro a ha"
},
{
"state_after": "R : Type u_1\ninst✝ : CommRing R\nI : Ideal R\na : R\nha : a ∈ I\n⊢ ↑C a ∈ map C I",
"state_before": "R : Type u_1\ninst✝ : CommRing R\nI : Ideal R\na : R\nha : a ∈ I\n⊢ ↑(RingHom.comp (Quotient.mk (map C I)) C) a = 0",
"tactic": "rw [RingHom.comp_apply, Quotient.eq_zero_iff_mem]"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝ : CommRing R\nI : Ideal R\na : R\nha : a ∈ I\n⊢ ↑C a ∈ map C I",
"tactic": "exact mem_map_of_mem _ ha"
}
] |
[
83,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
79,
1
] |
Mathlib/Order/Filter/Basic.lean
|
Filter.not_frequently
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nδ : Type ?u.161580\nι : Sort x\np : α → Prop\nf : Filter α\n⊢ (¬∃ᶠ (x : α) in f, p x) ↔ ∀ᶠ (x : α) in f, ¬p x",
"tactic": "simp only [Filter.Frequently, not_not]"
}
] |
[
1322,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1321,
1
] |
Mathlib/CategoryTheory/Generator.lean
|
CategoryTheory.isCoseparator_unop_iff
|
[
{
"state_after": "no goals",
"state_before": "C : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nG : Cᵒᵖ\n⊢ IsCoseparator G.unop ↔ IsSeparator G",
"tactic": "rw [IsSeparator, IsCoseparator, ← isCoseparating_unop_iff, Set.singleton_unop]"
}
] |
[
415,
81
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
414,
1
] |
Mathlib/GroupTheory/Submonoid/Membership.lean
|
Submonoid.mem_sup_left
|
[
{
"state_after": "M : Type u_1\nA : Type ?u.43722\nB : Type ?u.43725\ninst✝ : MulOneClass M\nS T : Submonoid M\n⊢ S ≤ S ⊔ T",
"state_before": "M : Type u_1\nA : Type ?u.43722\nB : Type ?u.43725\ninst✝ : MulOneClass M\nS T : Submonoid M\n⊢ ∀ {x : M}, x ∈ S → x ∈ S ⊔ T",
"tactic": "rw [←SetLike.le_def]"
},
{
"state_after": "no goals",
"state_before": "M : Type u_1\nA : Type ?u.43722\nB : Type ?u.43725\ninst✝ : MulOneClass M\nS T : Submonoid M\n⊢ S ≤ S ⊔ T",
"tactic": "exact le_sup_left"
}
] |
[
239,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
237,
1
] |
Mathlib/Order/Filter/Pointwise.lean
|
Filter.pure_mul_pure
|
[] |
[
366,
12
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
365,
1
] |
Mathlib/Topology/MetricSpace/Basic.lean
|
Filter.Tendsto.congr_dist
|
[] |
[
1471,
70
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1468,
1
] |
Mathlib/Combinatorics/SimpleGraph/Connectivity.lean
|
SimpleGraph.Walk.support_eq_cons
|
[
{
"state_after": "no goals",
"state_before": "V : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nu v : V\np : Walk G u v\n⊢ support p = u :: List.tail (support p)",
"tactic": "cases p <;> simp"
}
] |
[
571,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
570,
1
] |
Mathlib/Algebra/DirectLimit.lean
|
Ring.DirectLimit.Polynomial.exists_of
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\ninst✝⁴ : Ring R\nι : Type v\ndec_ι : DecidableEq ι\ninst✝³ : Preorder ι\nG : ι → Type w\ninst✝² : (i : ι) → CommRing (G i)\nf : (i j : ι) → i ≤ j → G i → G j\nf' : (i j : ι) → i ≤ j → G i →+* G j\ninst✝¹ : Nonempty ι\ninst✝ : IsDirected ι fun x x_1 => x ≤ x_1\nq : (DirectLimit G fun i j h => ↑(f' i j h))[X]\nz : DirectLimit G fun i j h => ↑(f' i j h)\ni : ι\nx : G i\nh : ↑(of G (fun i j h => ↑(f' i j h)) i) x = z\n⊢ Polynomial.map (of G (fun i j h => ↑(f' i j h)) i) (↑C x) = ↑C z",
"tactic": "rw [map_C, h]"
},
{
"state_after": "R : Type u\ninst✝⁴ : Ring R\nι : Type v\ndec_ι : DecidableEq ι\ninst✝³ : Preorder ι\nG : ι → Type w\ninst✝² : (i : ι) → CommRing (G i)\nf : (i j : ι) → i ≤ j → G i → G j\nf' : (i j : ι) → i ≤ j → G i →+* G j\ninst✝¹ : Nonempty ι\ninst✝ : IsDirected ι fun x x_1 => x ≤ x_1\nq : (DirectLimit G fun i j h => ↑(f' i j h))[X]\nq₁ q₂ : (DirectLimit G fun i j h => ↑(f' i j h))[X]\nx✝¹ : ∃ i p, Polynomial.map (of G (fun i j h => ↑(f' i j h)) i) p = q₁\nx✝ : ∃ i p, Polynomial.map (of G (fun i j h => ↑(f' i j h)) i) p = q₂\ni₁ : ι\np₁ : (G i₁)[X]\nih₁ : Polynomial.map (of G (fun i j h => ↑(f' i j h)) i₁) p₁ = q₁\ni₂ : ι\np₂ : (G i₂)[X]\nih₂ : Polynomial.map (of G (fun i j h => ↑(f' i j h)) i₂) p₂ = q₂\ni : ι\nh1 : i₁ ≤ i\nh2 : i₂ ≤ i\n⊢ Polynomial.map (RingHom.comp (of G (fun i j h => ↑(f' i j h)) i) (f' i₁ i h1)) p₁ +\n Polynomial.map (RingHom.comp (of G (fun i j h => ↑(f' i j h)) i) (f' i₂ i h2)) p₂ =\n Polynomial.map (of G (fun i j h => ↑(f' i j h)) i₁) p₁ + Polynomial.map (of G (fun i j h => ↑(f' i j h)) i₂) p₂",
"state_before": "R : Type u\ninst✝⁴ : Ring R\nι : Type v\ndec_ι : DecidableEq ι\ninst✝³ : Preorder ι\nG : ι → Type w\ninst✝² : (i : ι) → CommRing (G i)\nf : (i j : ι) → i ≤ j → G i → G j\nf' : (i j : ι) → i ≤ j → G i →+* G j\ninst✝¹ : Nonempty ι\ninst✝ : IsDirected ι fun x x_1 => x ≤ x_1\nq : (DirectLimit G fun i j h => ↑(f' i j h))[X]\nq₁ q₂ : (DirectLimit G fun i j h => ↑(f' i j h))[X]\nx✝¹ : ∃ i p, Polynomial.map (of G (fun i j h => ↑(f' i j h)) i) p = q₁\nx✝ : ∃ i p, Polynomial.map (of G (fun i j h => ↑(f' i j h)) i) p = q₂\ni₁ : ι\np₁ : (G i₁)[X]\nih₁ : Polynomial.map (of G (fun i j h => ↑(f' i j h)) i₁) p₁ = q₁\ni₂ : ι\np₂ : (G i₂)[X]\nih₂ : Polynomial.map (of G (fun i j h => ↑(f' i j h)) i₂) p₂ = q₂\ni : ι\nh1 : i₁ ≤ i\nh2 : i₂ ≤ i\n⊢ Polynomial.map (of G (fun i j h => ↑(f' i j h)) i) (Polynomial.map (f' i₁ i h1) p₁ + Polynomial.map (f' i₂ i h2) p₂) =\n q₁ + q₂",
"tactic": "rw [Polynomial.map_add, map_map, map_map, ← ih₁, ← ih₂]"
},
{
"state_after": "no goals",
"state_before": "R : Type u\ninst✝⁴ : Ring R\nι : Type v\ndec_ι : DecidableEq ι\ninst✝³ : Preorder ι\nG : ι → Type w\ninst✝² : (i : ι) → CommRing (G i)\nf : (i j : ι) → i ≤ j → G i → G j\nf' : (i j : ι) → i ≤ j → G i →+* G j\ninst✝¹ : Nonempty ι\ninst✝ : IsDirected ι fun x x_1 => x ≤ x_1\nq : (DirectLimit G fun i j h => ↑(f' i j h))[X]\nq₁ q₂ : (DirectLimit G fun i j h => ↑(f' i j h))[X]\nx✝¹ : ∃ i p, Polynomial.map (of G (fun i j h => ↑(f' i j h)) i) p = q₁\nx✝ : ∃ i p, Polynomial.map (of G (fun i j h => ↑(f' i j h)) i) p = q₂\ni₁ : ι\np₁ : (G i₁)[X]\nih₁ : Polynomial.map (of G (fun i j h => ↑(f' i j h)) i₁) p₁ = q₁\ni₂ : ι\np₂ : (G i₂)[X]\nih₂ : Polynomial.map (of G (fun i j h => ↑(f' i j h)) i₂) p₂ = q₂\ni : ι\nh1 : i₁ ≤ i\nh2 : i₂ ≤ i\n⊢ Polynomial.map (RingHom.comp (of G (fun i j h => ↑(f' i j h)) i) (f' i₁ i h1)) p₁ +\n Polynomial.map (RingHom.comp (of G (fun i j h => ↑(f' i j h)) i) (f' i₂ i h2)) p₂ =\n Polynomial.map (of G (fun i j h => ↑(f' i j h)) i₁) p₁ + Polynomial.map (of G (fun i j h => ↑(f' i j h)) i₂) p₂",
"tactic": "congr 2 <;> ext x <;> simp_rw [RingHom.comp_apply, of_f]"
},
{
"state_after": "no goals",
"state_before": "R : Type u\ninst✝⁴ : Ring R\nι : Type v\ndec_ι : DecidableEq ι\ninst✝³ : Preorder ι\nG : ι → Type w\ninst✝² : (i : ι) → CommRing (G i)\nf : (i j : ι) → i ≤ j → G i → G j\nf' : (i j : ι) → i ≤ j → G i →+* G j\ninst✝¹ : Nonempty ι\ninst✝ : IsDirected ι fun x x_1 => x ≤ x_1\nq : (DirectLimit G fun i j h => ↑(f' i j h))[X]\nn : ℕ\nz : DirectLimit G fun i j h => ↑(f' i j h)\nx✝ : ∃ i p, Polynomial.map (of G (fun i j h => ↑(f' i j h)) i) p = ↑C z * X ^ n\ni : ι\nx : G i\nh : ↑(of G (fun i j h => ↑(f' i j h)) i) x = z\n⊢ Polynomial.map (of G (fun i j h => ↑(f' i j h)) i) (↑C x * X ^ (n + 1)) = ↑C z * X ^ (n + 1)",
"tactic": "rw [Polynomial.map_mul, map_C, h, Polynomial.map_pow, map_X]"
}
] |
[
464,
92
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
450,
8
] |
Mathlib/LinearAlgebra/Trace.lean
|
LinearMap.traceAux_eq
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\ninst✝⁶ : CommSemiring R\nM : Type v\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\nι : Type w\ninst✝³ : DecidableEq ι\ninst✝² : Fintype ι\nκ : Type u_1\ninst✝¹ : DecidableEq κ\ninst✝ : Fintype κ\nb : Basis ι R M\nc : Basis κ R M\nf : M →ₗ[R] M\n⊢ trace (↑(toMatrix b b) f) = trace (↑(toMatrix b b) (comp (comp id f) id))",
"tactic": "rw [LinearMap.id_comp, LinearMap.comp_id]"
},
{
"state_after": "no goals",
"state_before": "R : Type u\ninst✝⁶ : CommSemiring R\nM : Type v\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\nι : Type w\ninst✝³ : DecidableEq ι\ninst✝² : Fintype ι\nκ : Type u_1\ninst✝¹ : DecidableEq κ\ninst✝ : Fintype κ\nb : Basis ι R M\nc : Basis κ R M\nf : M →ₗ[R] M\n⊢ trace (↑(toMatrix b b) (comp (comp id f) id)) = trace (↑(toMatrix c b) id ⬝ ↑(toMatrix c c) f ⬝ ↑(toMatrix b c) id)",
"tactic": "rw [LinearMap.toMatrix_comp _ c, LinearMap.toMatrix_comp _ c]"
},
{
"state_after": "no goals",
"state_before": "R : Type u\ninst✝⁶ : CommSemiring R\nM : Type v\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\nι : Type w\ninst✝³ : DecidableEq ι\ninst✝² : Fintype ι\nκ : Type u_1\ninst✝¹ : DecidableEq κ\ninst✝ : Fintype κ\nb : Basis ι R M\nc : Basis κ R M\nf : M →ₗ[R] M\n⊢ trace (↑(toMatrix c b) id ⬝ ↑(toMatrix c c) f ⬝ ↑(toMatrix b c) id) =\n trace (↑(toMatrix c c) f ⬝ ↑(toMatrix b c) id ⬝ ↑(toMatrix c b) id)",
"tactic": "rw [Matrix.mul_assoc, Matrix.trace_mul_comm]"
},
{
"state_after": "no goals",
"state_before": "R : Type u\ninst✝⁶ : CommSemiring R\nM : Type v\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\nι : Type w\ninst✝³ : DecidableEq ι\ninst✝² : Fintype ι\nκ : Type u_1\ninst✝¹ : DecidableEq κ\ninst✝ : Fintype κ\nb : Basis ι R M\nc : Basis κ R M\nf : M →ₗ[R] M\n⊢ trace (↑(toMatrix c c) f ⬝ ↑(toMatrix b c) id ⬝ ↑(toMatrix c b) id) = trace (↑(toMatrix c c) (comp (comp f id) id))",
"tactic": "rw [LinearMap.toMatrix_comp _ b, LinearMap.toMatrix_comp _ c]"
},
{
"state_after": "no goals",
"state_before": "R : Type u\ninst✝⁶ : CommSemiring R\nM : Type v\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\nι : Type w\ninst✝³ : DecidableEq ι\ninst✝² : Fintype ι\nκ : Type u_1\ninst✝¹ : DecidableEq κ\ninst✝ : Fintype κ\nb : Basis ι R M\nc : Basis κ R M\nf : M →ₗ[R] M\n⊢ trace (↑(toMatrix c c) (comp (comp f id) id)) = trace (↑(toMatrix c c) f)",
"tactic": "rw [LinearMap.comp_id, LinearMap.comp_id]"
}
] |
[
82,
98
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
68,
1
] |
Mathlib/LinearAlgebra/StdBasis.lean
|
LinearMap.iSup_range_stdBasis
|
[
{
"state_after": "case intro\nR : Type u_3\nι : Type u_1\ninst✝⁴ : Semiring R\nφ : ι → Type u_2\ninst✝³ : (i : ι) → AddCommMonoid (φ i)\ninst✝² : (i : ι) → Module R (φ i)\ninst✝¹ : DecidableEq ι\ninst✝ : Finite ι\nval✝ : Fintype ι\n⊢ (⨆ (i : ι), range (stdBasis R φ i)) = ⊤",
"state_before": "R : Type u_3\nι : Type u_1\ninst✝⁴ : Semiring R\nφ : ι → Type u_2\ninst✝³ : (i : ι) → AddCommMonoid (φ i)\ninst✝² : (i : ι) → Module R (φ i)\ninst✝¹ : DecidableEq ι\ninst✝ : Finite ι\n⊢ (⨆ (i : ι), range (stdBasis R φ i)) = ⊤",
"tactic": "cases nonempty_fintype ι"
},
{
"state_after": "case h.e'_2.h.e'_4.h\nR : Type u_3\nι : Type u_1\ninst✝⁴ : Semiring R\nφ : ι → Type u_2\ninst✝³ : (i : ι) → AddCommMonoid (φ i)\ninst✝² : (i : ι) → Module R (φ i)\ninst✝¹ : DecidableEq ι\ninst✝ : Finite ι\nval✝ : Fintype ι\nx✝ : ι\n⊢ range (stdBasis R φ x✝) = ⨆ (_ : x✝ ∈ ?intro.convert_1), range (stdBasis R φ x✝)\n\ncase intro.convert_1\nR : Type u_3\nι : Type u_1\ninst✝⁴ : Semiring R\nφ : ι → Type u_2\ninst✝³ : (i : ι) → AddCommMonoid (φ i)\ninst✝² : (i : ι) → Module R (φ i)\ninst✝¹ : DecidableEq ι\ninst✝ : Finite ι\nval✝ : Fintype ι\n⊢ Finset ι\n\ncase intro.convert_2\nR : Type u_3\nι : Type u_1\ninst✝⁴ : Semiring R\nφ : ι → Type u_2\ninst✝³ : (i : ι) → AddCommMonoid (φ i)\ninst✝² : (i : ι) → Module R (φ i)\ninst✝¹ : DecidableEq ι\ninst✝ : Finite ι\nval✝ : Fintype ι\n⊢ Set.univ ⊆ ↑?intro.convert_1 ∪ ∅",
"state_before": "case intro\nR : Type u_3\nι : Type u_1\ninst✝⁴ : Semiring R\nφ : ι → Type u_2\ninst✝³ : (i : ι) → AddCommMonoid (φ i)\ninst✝² : (i : ι) → Module R (φ i)\ninst✝¹ : DecidableEq ι\ninst✝ : Finite ι\nval✝ : Fintype ι\n⊢ (⨆ (i : ι), range (stdBasis R φ i)) = ⊤",
"tactic": "convert top_unique (iInf_emptyset.ge.trans <| iInf_ker_proj_le_iSup_range_stdBasis R φ _)"
},
{
"state_after": "case h.e'_2.h.e'_4.h\nR : Type u_3\nι : Type u_1\ninst✝⁴ : Semiring R\nφ : ι → Type u_2\ninst✝³ : (i : ι) → AddCommMonoid (φ i)\ninst✝² : (i : ι) → Module R (φ i)\ninst✝¹ : DecidableEq ι\ninst✝ : Finite ι\nval✝ : Fintype ι\ni : ι\n⊢ range (stdBasis R φ i) = ⨆ (_ : i ∈ ?intro.convert_1), range (stdBasis R φ i)",
"state_before": "case h.e'_2.h.e'_4.h\nR : Type u_3\nι : Type u_1\ninst✝⁴ : Semiring R\nφ : ι → Type u_2\ninst✝³ : (i : ι) → AddCommMonoid (φ i)\ninst✝² : (i : ι) → Module R (φ i)\ninst✝¹ : DecidableEq ι\ninst✝ : Finite ι\nval✝ : Fintype ι\nx✝ : ι\n⊢ range (stdBasis R φ x✝) = ⨆ (_ : x✝ ∈ ?intro.convert_1), range (stdBasis R φ x✝)",
"tactic": "rename_i i"
},
{
"state_after": "no goals",
"state_before": "case h.e'_2.h.e'_4.h\nR : Type u_3\nι : Type u_1\ninst✝⁴ : Semiring R\nφ : ι → Type u_2\ninst✝³ : (i : ι) → AddCommMonoid (φ i)\ninst✝² : (i : ι) → Module R (φ i)\ninst✝¹ : DecidableEq ι\ninst✝ : Finite ι\nval✝ : Fintype ι\ni : ι\n⊢ range (stdBasis R φ i) = ⨆ (_ : i ∈ ?intro.convert_1), range (stdBasis R φ i)",
"tactic": "exact ((@iSup_pos _ _ _ fun _ => range <| stdBasis R φ i) <| Finset.mem_univ i).symm"
},
{
"state_after": "no goals",
"state_before": "case intro.convert_2\nR : Type u_3\nι : Type u_1\ninst✝⁴ : Semiring R\nφ : ι → Type u_2\ninst✝³ : (i : ι) → AddCommMonoid (φ i)\ninst✝² : (i : ι) → Module R (φ i)\ninst✝¹ : DecidableEq ι\ninst✝ : Finite ι\nval✝ : Fintype ι\n⊢ Set.univ ⊆ ↑Finset.univ ∪ ∅",
"tactic": "rw [Finset.coe_univ, Set.union_empty]"
}
] |
[
142,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
137,
1
] |
Mathlib/Topology/UniformSpace/UniformConvergence.lean
|
TendstoUniformlyOn.uniformCauchySeqOn
|
[] |
[
437,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
434,
1
] |
Mathlib/Algebra/Star/Subalgebra.lean
|
StarSubalgebra.mem_centralizer_iff
|
[] |
[
312,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
310,
1
] |
Mathlib/Analysis/BoxIntegral/Partition/Basic.lean
|
BoxIntegral.Prepartition.mem_single
|
[] |
[
124,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
123,
1
] |
Mathlib/CategoryTheory/Abelian/RightDerived.lean
|
CategoryTheory.Abelian.Functor.rightDerivedZeroToSelf_natural
|
[
{
"state_after": "C : Type u\ninst✝⁵ : Category C\nD : Type u\ninst✝⁴ : Category D\nF : C ⥤ D\nX✝ Y✝ Z : C\nf✝ : X✝ ⟶ Y✝\ng : Y✝ ⟶ Z\ninst✝³ : Abelian C\ninst✝² : Abelian D\ninst✝¹ : Functor.Additive F\ninst✝ : EnoughInjectives C\nX Y : C\nf : X ⟶ Y\nP : InjectiveResolution X\nQ : InjectiveResolution Y\n⊢ F.map f ≫\n homology.lift (F.map (HomologicalComplex.d Q.cocomplex (ComplexShape.prev (ComplexShape.up ℕ) 0) 0))\n (F.map (HomologicalComplex.d Q.cocomplex 0 (ComplexShape.next (ComplexShape.up ℕ) 0)))\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj Q.cocomplex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj Q.cocomplex) 0 =\n 0)\n (F.map (HomologicalComplex.Hom.f Q.ι 0) ≫\n cokernel.π (F.map (HomologicalComplex.d Q.cocomplex (ComplexShape.prev (ComplexShape.up ℕ) 0) 0)))\n (_ :\n (F.map (HomologicalComplex.Hom.f Q.ι 0) ≫\n cokernel.π\n (HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj Q.cocomplex) 0)) ≫\n cokernel.desc (HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj Q.cocomplex) 0)\n (HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj Q.cocomplex) 0)\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj Q.cocomplex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj Q.cocomplex) 0 =\n 0) =\n 0) ≫\n (rightDerivedObjIso F 0 Q).inv =\n (homology.lift (F.map (HomologicalComplex.d P.cocomplex (ComplexShape.prev (ComplexShape.up ℕ) 0) 0))\n (F.map (HomologicalComplex.d P.cocomplex 0 (ComplexShape.next (ComplexShape.up ℕ) 0)))\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) 0 =\n 0)\n (F.map (HomologicalComplex.Hom.f P.ι 0) ≫\n cokernel.π (F.map (HomologicalComplex.d P.cocomplex (ComplexShape.prev (ComplexShape.up ℕ) 0) 0)))\n (_ :\n (F.map (HomologicalComplex.Hom.f P.ι 0) ≫\n cokernel.π\n (HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) 0)) ≫\n cokernel.desc (HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) 0)\n (HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) 0)\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) 0 =\n 0) =\n 0) ≫\n (rightDerivedObjIso F 0 P).inv) ≫\n (rightDerived F 0).map f",
"state_before": "C : Type u\ninst✝⁵ : Category C\nD : Type u\ninst✝⁴ : Category D\nF : C ⥤ D\nX✝ Y✝ Z : C\nf✝ : X✝ ⟶ Y✝\ng : Y✝ ⟶ Z\ninst✝³ : Abelian C\ninst✝² : Abelian D\ninst✝¹ : Functor.Additive F\ninst✝ : EnoughInjectives C\nX Y : C\nf : X ⟶ Y\nP : InjectiveResolution X\nQ : InjectiveResolution Y\n⊢ F.map f ≫ rightDerivedZeroToSelfAppInv F Q = rightDerivedZeroToSelfAppInv F P ≫ (rightDerived F 0).map f",
"tactic": "dsimp [rightDerivedZeroToSelfAppInv]"
},
{
"state_after": "C : Type u\ninst✝⁵ : Category C\nD : Type u\ninst✝⁴ : Category D\nF : C ⥤ D\nX✝ Y✝ Z : C\nf✝ : X✝ ⟶ Y✝\ng : Y✝ ⟶ Z\ninst✝³ : Abelian C\ninst✝² : Abelian D\ninst✝¹ : Functor.Additive F\ninst✝ : EnoughInjectives C\nX Y : C\nf : X ⟶ Y\nP : InjectiveResolution X\nQ : InjectiveResolution Y\n⊢ (F.map f ≫\n homology.lift (F.map (HomologicalComplex.d Q.cocomplex (ComplexShape.prev (ComplexShape.up ℕ) 0) 0))\n (F.map (HomologicalComplex.d Q.cocomplex 0 (ComplexShape.next (ComplexShape.up ℕ) 0)))\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj Q.cocomplex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj Q.cocomplex) 0 =\n 0)\n (F.map (HomologicalComplex.Hom.f Q.ι 0) ≫\n cokernel.π (F.map (HomologicalComplex.d Q.cocomplex (ComplexShape.prev (ComplexShape.up ℕ) 0) 0)))\n (_ :\n (F.map (HomologicalComplex.Hom.f Q.ι 0) ≫\n cokernel.π\n (HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj Q.cocomplex) 0)) ≫\n cokernel.desc (HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj Q.cocomplex) 0)\n (HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj Q.cocomplex) 0)\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj Q.cocomplex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj Q.cocomplex) 0 =\n 0) =\n 0)) ≫\n (rightDerivedObjIso F 0 Q).inv =\n (homology.lift (F.map (HomologicalComplex.d P.cocomplex (ComplexShape.prev (ComplexShape.up ℕ) 0) 0))\n (F.map (HomologicalComplex.d P.cocomplex 0 (ComplexShape.next (ComplexShape.up ℕ) 0)))\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) 0 =\n 0)\n (F.map (HomologicalComplex.Hom.f P.ι 0) ≫\n cokernel.π (F.map (HomologicalComplex.d P.cocomplex (ComplexShape.prev (ComplexShape.up ℕ) 0) 0)))\n (_ :\n (F.map (HomologicalComplex.Hom.f P.ι 0) ≫\n cokernel.π\n (HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) 0)) ≫\n cokernel.desc (HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) 0)\n (HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) 0)\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) 0 =\n 0) =\n 0) ≫\n (rightDerivedObjIso F 0 P).inv) ≫\n (rightDerived F 0).map f",
"state_before": "C : Type u\ninst✝⁵ : Category C\nD : Type u\ninst✝⁴ : Category D\nF : C ⥤ D\nX✝ Y✝ Z : C\nf✝ : X✝ ⟶ Y✝\ng : Y✝ ⟶ Z\ninst✝³ : Abelian C\ninst✝² : Abelian D\ninst✝¹ : Functor.Additive F\ninst✝ : EnoughInjectives C\nX Y : C\nf : X ⟶ Y\nP : InjectiveResolution X\nQ : InjectiveResolution Y\n⊢ F.map f ≫\n homology.lift (F.map (HomologicalComplex.d Q.cocomplex (ComplexShape.prev (ComplexShape.up ℕ) 0) 0))\n (F.map (HomologicalComplex.d Q.cocomplex 0 (ComplexShape.next (ComplexShape.up ℕ) 0)))\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj Q.cocomplex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj Q.cocomplex) 0 =\n 0)\n (F.map (HomologicalComplex.Hom.f Q.ι 0) ≫\n cokernel.π (F.map (HomologicalComplex.d Q.cocomplex (ComplexShape.prev (ComplexShape.up ℕ) 0) 0)))\n (_ :\n (F.map (HomologicalComplex.Hom.f Q.ι 0) ≫\n cokernel.π\n (HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj Q.cocomplex) 0)) ≫\n cokernel.desc (HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj Q.cocomplex) 0)\n (HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj Q.cocomplex) 0)\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj Q.cocomplex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj Q.cocomplex) 0 =\n 0) =\n 0) ≫\n (rightDerivedObjIso F 0 Q).inv =\n (homology.lift (F.map (HomologicalComplex.d P.cocomplex (ComplexShape.prev (ComplexShape.up ℕ) 0) 0))\n (F.map (HomologicalComplex.d P.cocomplex 0 (ComplexShape.next (ComplexShape.up ℕ) 0)))\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) 0 =\n 0)\n (F.map (HomologicalComplex.Hom.f P.ι 0) ≫\n cokernel.π (F.map (HomologicalComplex.d P.cocomplex (ComplexShape.prev (ComplexShape.up ℕ) 0) 0)))\n (_ :\n (F.map (HomologicalComplex.Hom.f P.ι 0) ≫\n cokernel.π\n (HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) 0)) ≫\n cokernel.desc (HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) 0)\n (HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) 0)\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) 0 =\n 0) =\n 0) ≫\n (rightDerivedObjIso F 0 P).inv) ≫\n (rightDerived F 0).map f",
"tactic": "simp only [CategoryTheory.Functor.map_id, Category.id_comp, ← Category.assoc]"
},
{
"state_after": "C : Type u\ninst✝⁵ : Category C\nD : Type u\ninst✝⁴ : Category D\nF : C ⥤ D\nX✝ Y✝ Z : C\nf✝ : X✝ ⟶ Y✝\ng : Y✝ ⟶ Z\ninst✝³ : Abelian C\ninst✝² : Abelian D\ninst✝¹ : Functor.Additive F\ninst✝ : EnoughInjectives C\nX Y : C\nf : X ⟶ Y\nP : InjectiveResolution X\nQ : InjectiveResolution Y\n⊢ F.map f ≫\n homology.lift (F.map (HomologicalComplex.d Q.cocomplex (ComplexShape.prev (ComplexShape.up ℕ) 0) 0))\n (F.map (HomologicalComplex.d Q.cocomplex 0 (ComplexShape.next (ComplexShape.up ℕ) 0)))\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj Q.cocomplex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj Q.cocomplex) 0 =\n 0)\n (F.map (HomologicalComplex.Hom.f Q.ι 0) ≫\n cokernel.π (F.map (HomologicalComplex.d Q.cocomplex (ComplexShape.prev (ComplexShape.up ℕ) 0) 0)))\n (_ :\n (F.map (HomologicalComplex.Hom.f Q.ι 0) ≫\n cokernel.π (HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj Q.cocomplex) 0)) ≫\n cokernel.desc (HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj Q.cocomplex) 0)\n (HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj Q.cocomplex) 0)\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj Q.cocomplex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj Q.cocomplex) 0 =\n 0) =\n 0) =\n homology.lift (F.map (HomologicalComplex.d P.cocomplex (ComplexShape.prev (ComplexShape.up ℕ) 0) 0))\n (F.map (HomologicalComplex.d P.cocomplex 0 (ComplexShape.next (ComplexShape.up ℕ) 0)))\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) 0 =\n 0)\n (F.map (HomologicalComplex.Hom.f P.ι 0) ≫\n cokernel.π (F.map (HomologicalComplex.d P.cocomplex (ComplexShape.prev (ComplexShape.up ℕ) 0) 0)))\n (_ :\n (F.map (HomologicalComplex.Hom.f P.ι 0) ≫\n cokernel.π (HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) 0)) ≫\n cokernel.desc (HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) 0)\n (HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) 0)\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) 0 =\n 0) =\n 0) ≫\n (homologyFunctor D (ComplexShape.up ℕ) 0).map\n ((mapHomologicalComplex F (ComplexShape.up ℕ)).map (InjectiveResolution.desc f Q P))",
"state_before": "C : Type u\ninst✝⁵ : Category C\nD : Type u\ninst✝⁴ : Category D\nF : C ⥤ D\nX✝ Y✝ Z : C\nf✝ : X✝ ⟶ Y✝\ng : Y✝ ⟶ Z\ninst✝³ : Abelian C\ninst✝² : Abelian D\ninst✝¹ : Functor.Additive F\ninst✝ : EnoughInjectives C\nX Y : C\nf : X ⟶ Y\nP : InjectiveResolution X\nQ : InjectiveResolution Y\n⊢ (F.map f ≫\n homology.lift (F.map (HomologicalComplex.d Q.cocomplex (ComplexShape.prev (ComplexShape.up ℕ) 0) 0))\n (F.map (HomologicalComplex.d Q.cocomplex 0 (ComplexShape.next (ComplexShape.up ℕ) 0)))\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj Q.cocomplex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj Q.cocomplex) 0 =\n 0)\n (F.map (HomologicalComplex.Hom.f Q.ι 0) ≫\n cokernel.π (F.map (HomologicalComplex.d Q.cocomplex (ComplexShape.prev (ComplexShape.up ℕ) 0) 0)))\n (_ :\n (F.map (HomologicalComplex.Hom.f Q.ι 0) ≫\n cokernel.π\n (HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj Q.cocomplex) 0)) ≫\n cokernel.desc (HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj Q.cocomplex) 0)\n (HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj Q.cocomplex) 0)\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj Q.cocomplex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj Q.cocomplex) 0 =\n 0) =\n 0)) ≫\n (rightDerivedObjIso F 0 Q).inv =\n (homology.lift (F.map (HomologicalComplex.d P.cocomplex (ComplexShape.prev (ComplexShape.up ℕ) 0) 0))\n (F.map (HomologicalComplex.d P.cocomplex 0 (ComplexShape.next (ComplexShape.up ℕ) 0)))\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) 0 =\n 0)\n (F.map (HomologicalComplex.Hom.f P.ι 0) ≫\n cokernel.π (F.map (HomologicalComplex.d P.cocomplex (ComplexShape.prev (ComplexShape.up ℕ) 0) 0)))\n (_ :\n (F.map (HomologicalComplex.Hom.f P.ι 0) ≫\n cokernel.π\n (HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) 0)) ≫\n cokernel.desc (HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) 0)\n (HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) 0)\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) 0 =\n 0) =\n 0) ≫\n (rightDerivedObjIso F 0 P).inv) ≫\n (rightDerived F 0).map f",
"tactic": "rw [Iso.comp_inv_eq, rightDerived_map_eq F 0 f (InjectiveResolution.desc f Q P) (by simp),\n Category.assoc, Category.assoc, Category.assoc, Category.assoc, Iso.inv_hom_id,\n Category.comp_id, ← Category.assoc (F.rightDerivedObjIso 0 P).inv, Iso.inv_hom_id,\n Category.id_comp]"
},
{
"state_after": "C : Type u\ninst✝⁵ : Category C\nD : Type u\ninst✝⁴ : Category D\nF : C ⥤ D\nX✝ Y✝ Z : C\nf✝ : X✝ ⟶ Y✝\ng : Y✝ ⟶ Z\ninst✝³ : Abelian C\ninst✝² : Abelian D\ninst✝¹ : Functor.Additive F\ninst✝ : EnoughInjectives C\nX Y : C\nf : X ⟶ Y\nP : InjectiveResolution X\nQ : InjectiveResolution Y\n⊢ F.map f ≫\n homology.lift (F.map (HomologicalComplex.d Q.cocomplex (ComplexShape.prev (ComplexShape.up ℕ) 0) 0))\n (F.map (HomologicalComplex.d Q.cocomplex 0 (ComplexShape.next (ComplexShape.up ℕ) 0)))\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj Q.cocomplex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj Q.cocomplex) 0 =\n 0)\n (F.map (HomologicalComplex.Hom.f Q.ι 0) ≫\n cokernel.π (F.map (HomologicalComplex.d Q.cocomplex (ComplexShape.prev (ComplexShape.up ℕ) 0) 0)))\n (_ :\n (F.map (HomologicalComplex.Hom.f Q.ι 0) ≫\n cokernel.π (HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj Q.cocomplex) 0)) ≫\n cokernel.desc (HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj Q.cocomplex) 0)\n (HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj Q.cocomplex) 0)\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj Q.cocomplex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj Q.cocomplex) 0 =\n 0) =\n 0) =\n homology.lift (F.map (HomologicalComplex.d P.cocomplex (ComplexShape.prev (ComplexShape.up ℕ) 0) 0))\n (F.map (HomologicalComplex.d P.cocomplex 0 (ComplexShape.next (ComplexShape.up ℕ) 0)))\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) 0 =\n 0)\n (F.map (HomologicalComplex.Hom.f P.ι 0) ≫\n cokernel.π (F.map (HomologicalComplex.d P.cocomplex (ComplexShape.prev (ComplexShape.up ℕ) 0) 0)))\n (_ :\n (F.map (HomologicalComplex.Hom.f P.ι 0) ≫\n cokernel.π (HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) 0)) ≫\n cokernel.desc (HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) 0)\n (HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) 0)\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) 0 =\n 0) =\n 0) ≫\n homology.map\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) 0 =\n 0)\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj Q.cocomplex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj Q.cocomplex) 0 =\n 0)\n (HomologicalComplex.Hom.sqTo\n ((mapHomologicalComplex F (ComplexShape.up ℕ)).map (InjectiveResolution.desc f Q P)) 0)\n (HomologicalComplex.Hom.sqFrom\n ((mapHomologicalComplex F (ComplexShape.up ℕ)).map (InjectiveResolution.desc f Q P)) 0)\n (_ :\n (HomologicalComplex.Hom.sqTo\n ((mapHomologicalComplex F (ComplexShape.up ℕ)).map (InjectiveResolution.desc f Q P)) 0).right =\n (HomologicalComplex.Hom.sqTo\n ((mapHomologicalComplex F (ComplexShape.up ℕ)).map (InjectiveResolution.desc f Q P)) 0).right)",
"state_before": "C : Type u\ninst✝⁵ : Category C\nD : Type u\ninst✝⁴ : Category D\nF : C ⥤ D\nX✝ Y✝ Z : C\nf✝ : X✝ ⟶ Y✝\ng : Y✝ ⟶ Z\ninst✝³ : Abelian C\ninst✝² : Abelian D\ninst✝¹ : Functor.Additive F\ninst✝ : EnoughInjectives C\nX Y : C\nf : X ⟶ Y\nP : InjectiveResolution X\nQ : InjectiveResolution Y\n⊢ F.map f ≫\n homology.lift (F.map (HomologicalComplex.d Q.cocomplex (ComplexShape.prev (ComplexShape.up ℕ) 0) 0))\n (F.map (HomologicalComplex.d Q.cocomplex 0 (ComplexShape.next (ComplexShape.up ℕ) 0)))\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj Q.cocomplex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj Q.cocomplex) 0 =\n 0)\n (F.map (HomologicalComplex.Hom.f Q.ι 0) ≫\n cokernel.π (F.map (HomologicalComplex.d Q.cocomplex (ComplexShape.prev (ComplexShape.up ℕ) 0) 0)))\n (_ :\n (F.map (HomologicalComplex.Hom.f Q.ι 0) ≫\n cokernel.π (HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj Q.cocomplex) 0)) ≫\n cokernel.desc (HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj Q.cocomplex) 0)\n (HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj Q.cocomplex) 0)\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj Q.cocomplex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj Q.cocomplex) 0 =\n 0) =\n 0) =\n homology.lift (F.map (HomologicalComplex.d P.cocomplex (ComplexShape.prev (ComplexShape.up ℕ) 0) 0))\n (F.map (HomologicalComplex.d P.cocomplex 0 (ComplexShape.next (ComplexShape.up ℕ) 0)))\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) 0 =\n 0)\n (F.map (HomologicalComplex.Hom.f P.ι 0) ≫\n cokernel.π (F.map (HomologicalComplex.d P.cocomplex (ComplexShape.prev (ComplexShape.up ℕ) 0) 0)))\n (_ :\n (F.map (HomologicalComplex.Hom.f P.ι 0) ≫\n cokernel.π (HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) 0)) ≫\n cokernel.desc (HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) 0)\n (HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) 0)\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) 0 =\n 0) =\n 0) ≫\n (homologyFunctor D (ComplexShape.up ℕ) 0).map\n ((mapHomologicalComplex F (ComplexShape.up ℕ)).map (InjectiveResolution.desc f Q P))",
"tactic": "dsimp only [homologyFunctor_map]"
},
{
"state_after": "case h\nC : Type u\ninst✝⁵ : Category C\nD : Type u\ninst✝⁴ : Category D\nF : C ⥤ D\nX✝ Y✝ Z : C\nf✝ : X✝ ⟶ Y✝\ng : Y✝ ⟶ Z\ninst✝³ : Abelian C\ninst✝² : Abelian D\ninst✝¹ : Functor.Additive F\ninst✝ : EnoughInjectives C\nX Y : C\nf : X ⟶ Y\nP : InjectiveResolution X\nQ : InjectiveResolution Y\n⊢ (F.map f ≫\n homology.lift (F.map (HomologicalComplex.d Q.cocomplex (ComplexShape.prev (ComplexShape.up ℕ) 0) 0))\n (F.map (HomologicalComplex.d Q.cocomplex 0 (ComplexShape.next (ComplexShape.up ℕ) 0)))\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj Q.cocomplex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj Q.cocomplex) 0 =\n 0)\n (F.map (HomologicalComplex.Hom.f Q.ι 0) ≫\n cokernel.π (F.map (HomologicalComplex.d Q.cocomplex (ComplexShape.prev (ComplexShape.up ℕ) 0) 0)))\n (_ :\n (F.map (HomologicalComplex.Hom.f Q.ι 0) ≫\n cokernel.π\n (HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj Q.cocomplex) 0)) ≫\n cokernel.desc (HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj Q.cocomplex) 0)\n (HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj Q.cocomplex) 0)\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj Q.cocomplex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj Q.cocomplex) 0 =\n 0) =\n 0)) ≫\n homology.ι (F.map (HomologicalComplex.d Q.cocomplex (ComplexShape.prev (ComplexShape.up ℕ) 0) 0))\n (F.map (HomologicalComplex.d Q.cocomplex 0 (ComplexShape.next (ComplexShape.up ℕ) 0)))\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj Q.cocomplex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj Q.cocomplex) 0 =\n 0) =\n (homology.lift (F.map (HomologicalComplex.d P.cocomplex (ComplexShape.prev (ComplexShape.up ℕ) 0) 0))\n (F.map (HomologicalComplex.d P.cocomplex 0 (ComplexShape.next (ComplexShape.up ℕ) 0)))\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) 0 =\n 0)\n (F.map (HomologicalComplex.Hom.f P.ι 0) ≫\n cokernel.π (F.map (HomologicalComplex.d P.cocomplex (ComplexShape.prev (ComplexShape.up ℕ) 0) 0)))\n (_ :\n (F.map (HomologicalComplex.Hom.f P.ι 0) ≫\n cokernel.π\n (HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) 0)) ≫\n cokernel.desc (HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) 0)\n (HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) 0)\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) 0 =\n 0) =\n 0) ≫\n homology.map\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) 0 =\n 0)\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj Q.cocomplex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj Q.cocomplex) 0 =\n 0)\n (HomologicalComplex.Hom.sqTo\n ((mapHomologicalComplex F (ComplexShape.up ℕ)).map (InjectiveResolution.desc f Q P)) 0)\n (HomologicalComplex.Hom.sqFrom\n ((mapHomologicalComplex F (ComplexShape.up ℕ)).map (InjectiveResolution.desc f Q P)) 0)\n (_ :\n (HomologicalComplex.Hom.sqTo\n ((mapHomologicalComplex F (ComplexShape.up ℕ)).map (InjectiveResolution.desc f Q P)) 0).right =\n (HomologicalComplex.Hom.sqTo\n ((mapHomologicalComplex F (ComplexShape.up ℕ)).map (InjectiveResolution.desc f Q P)) 0).right)) ≫\n homology.ι (F.map (HomologicalComplex.d Q.cocomplex (ComplexShape.prev (ComplexShape.up ℕ) 0) 0))\n (F.map (HomologicalComplex.d Q.cocomplex 0 (ComplexShape.next (ComplexShape.up ℕ) 0)))\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj Q.cocomplex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj Q.cocomplex) 0 =\n 0)",
"state_before": "C : Type u\ninst✝⁵ : Category C\nD : Type u\ninst✝⁴ : Category D\nF : C ⥤ D\nX✝ Y✝ Z : C\nf✝ : X✝ ⟶ Y✝\ng : Y✝ ⟶ Z\ninst✝³ : Abelian C\ninst✝² : Abelian D\ninst✝¹ : Functor.Additive F\ninst✝ : EnoughInjectives C\nX Y : C\nf : X ⟶ Y\nP : InjectiveResolution X\nQ : InjectiveResolution Y\n⊢ F.map f ≫\n homology.lift (F.map (HomologicalComplex.d Q.cocomplex (ComplexShape.prev (ComplexShape.up ℕ) 0) 0))\n (F.map (HomologicalComplex.d Q.cocomplex 0 (ComplexShape.next (ComplexShape.up ℕ) 0)))\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj Q.cocomplex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj Q.cocomplex) 0 =\n 0)\n (F.map (HomologicalComplex.Hom.f Q.ι 0) ≫\n cokernel.π (F.map (HomologicalComplex.d Q.cocomplex (ComplexShape.prev (ComplexShape.up ℕ) 0) 0)))\n (_ :\n (F.map (HomologicalComplex.Hom.f Q.ι 0) ≫\n cokernel.π (HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj Q.cocomplex) 0)) ≫\n cokernel.desc (HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj Q.cocomplex) 0)\n (HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj Q.cocomplex) 0)\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj Q.cocomplex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj Q.cocomplex) 0 =\n 0) =\n 0) =\n homology.lift (F.map (HomologicalComplex.d P.cocomplex (ComplexShape.prev (ComplexShape.up ℕ) 0) 0))\n (F.map (HomologicalComplex.d P.cocomplex 0 (ComplexShape.next (ComplexShape.up ℕ) 0)))\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) 0 =\n 0)\n (F.map (HomologicalComplex.Hom.f P.ι 0) ≫\n cokernel.π (F.map (HomologicalComplex.d P.cocomplex (ComplexShape.prev (ComplexShape.up ℕ) 0) 0)))\n (_ :\n (F.map (HomologicalComplex.Hom.f P.ι 0) ≫\n cokernel.π (HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) 0)) ≫\n cokernel.desc (HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) 0)\n (HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) 0)\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) 0 =\n 0) =\n 0) ≫\n homology.map\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) 0 =\n 0)\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj Q.cocomplex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj Q.cocomplex) 0 =\n 0)\n (HomologicalComplex.Hom.sqTo\n ((mapHomologicalComplex F (ComplexShape.up ℕ)).map (InjectiveResolution.desc f Q P)) 0)\n (HomologicalComplex.Hom.sqFrom\n ((mapHomologicalComplex F (ComplexShape.up ℕ)).map (InjectiveResolution.desc f Q P)) 0)\n (_ :\n (HomologicalComplex.Hom.sqTo\n ((mapHomologicalComplex F (ComplexShape.up ℕ)).map (InjectiveResolution.desc f Q P)) 0).right =\n (HomologicalComplex.Hom.sqTo\n ((mapHomologicalComplex F (ComplexShape.up ℕ)).map (InjectiveResolution.desc f Q P)) 0).right)",
"tactic": "apply homology.hom_to_ext"
},
{
"state_after": "case h\nC : Type u\ninst✝⁵ : Category C\nD : Type u\ninst✝⁴ : Category D\nF : C ⥤ D\nX✝ Y✝ Z : C\nf✝ : X✝ ⟶ Y✝\ng : Y✝ ⟶ Z\ninst✝³ : Abelian C\ninst✝² : Abelian D\ninst✝¹ : Functor.Additive F\ninst✝ : EnoughInjectives C\nX Y : C\nf : X ⟶ Y\nP : InjectiveResolution X\nQ : InjectiveResolution Y\n⊢ F.map f ≫\n F.map (HomologicalComplex.Hom.f Q.ι 0) ≫\n cokernel.π (F.map (HomologicalComplex.d Q.cocomplex (ComplexShape.prev (ComplexShape.up ℕ) 0) 0)) =\n homology.lift (F.map (HomologicalComplex.d P.cocomplex (ComplexShape.prev (ComplexShape.up ℕ) 0) 0))\n (F.map (HomologicalComplex.d P.cocomplex 0 (ComplexShape.next (ComplexShape.up ℕ) 0)))\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) 0 =\n 0)\n (F.map (HomologicalComplex.Hom.f P.ι 0) ≫\n cokernel.π (F.map (HomologicalComplex.d P.cocomplex (ComplexShape.prev (ComplexShape.up ℕ) 0) 0)))\n (_ :\n (F.map (HomologicalComplex.Hom.f P.ι 0) ≫\n cokernel.π (HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) 0)) ≫\n cokernel.desc (HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) 0)\n (HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) 0)\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) 0 =\n 0) =\n 0) ≫\n homology.map\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) 0 =\n 0)\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj Q.cocomplex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj Q.cocomplex) 0 =\n 0)\n (HomologicalComplex.Hom.sqTo\n ((mapHomologicalComplex F (ComplexShape.up ℕ)).map (InjectiveResolution.desc f Q P)) 0)\n (HomologicalComplex.Hom.sqFrom\n ((mapHomologicalComplex F (ComplexShape.up ℕ)).map (InjectiveResolution.desc f Q P)) 0)\n (_ :\n (HomologicalComplex.Hom.sqTo\n ((mapHomologicalComplex F (ComplexShape.up ℕ)).map (InjectiveResolution.desc f Q P)) 0).right =\n (HomologicalComplex.Hom.sqTo\n ((mapHomologicalComplex F (ComplexShape.up ℕ)).map (InjectiveResolution.desc f Q P)) 0).right) ≫\n homology.ι (F.map (HomologicalComplex.d Q.cocomplex (ComplexShape.prev (ComplexShape.up ℕ) 0) 0))\n (F.map (HomologicalComplex.d Q.cocomplex 0 (ComplexShape.next (ComplexShape.up ℕ) 0)))\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj Q.cocomplex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj Q.cocomplex) 0 =\n 0)",
"state_before": "case h\nC : Type u\ninst✝⁵ : Category C\nD : Type u\ninst✝⁴ : Category D\nF : C ⥤ D\nX✝ Y✝ Z : C\nf✝ : X✝ ⟶ Y✝\ng : Y✝ ⟶ Z\ninst✝³ : Abelian C\ninst✝² : Abelian D\ninst✝¹ : Functor.Additive F\ninst✝ : EnoughInjectives C\nX Y : C\nf : X ⟶ Y\nP : InjectiveResolution X\nQ : InjectiveResolution Y\n⊢ (F.map f ≫\n homology.lift (F.map (HomologicalComplex.d Q.cocomplex (ComplexShape.prev (ComplexShape.up ℕ) 0) 0))\n (F.map (HomologicalComplex.d Q.cocomplex 0 (ComplexShape.next (ComplexShape.up ℕ) 0)))\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj Q.cocomplex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj Q.cocomplex) 0 =\n 0)\n (F.map (HomologicalComplex.Hom.f Q.ι 0) ≫\n cokernel.π (F.map (HomologicalComplex.d Q.cocomplex (ComplexShape.prev (ComplexShape.up ℕ) 0) 0)))\n (_ :\n (F.map (HomologicalComplex.Hom.f Q.ι 0) ≫\n cokernel.π\n (HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj Q.cocomplex) 0)) ≫\n cokernel.desc (HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj Q.cocomplex) 0)\n (HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj Q.cocomplex) 0)\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj Q.cocomplex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj Q.cocomplex) 0 =\n 0) =\n 0)) ≫\n homology.ι (F.map (HomologicalComplex.d Q.cocomplex (ComplexShape.prev (ComplexShape.up ℕ) 0) 0))\n (F.map (HomologicalComplex.d Q.cocomplex 0 (ComplexShape.next (ComplexShape.up ℕ) 0)))\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj Q.cocomplex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj Q.cocomplex) 0 =\n 0) =\n (homology.lift (F.map (HomologicalComplex.d P.cocomplex (ComplexShape.prev (ComplexShape.up ℕ) 0) 0))\n (F.map (HomologicalComplex.d P.cocomplex 0 (ComplexShape.next (ComplexShape.up ℕ) 0)))\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) 0 =\n 0)\n (F.map (HomologicalComplex.Hom.f P.ι 0) ≫\n cokernel.π (F.map (HomologicalComplex.d P.cocomplex (ComplexShape.prev (ComplexShape.up ℕ) 0) 0)))\n (_ :\n (F.map (HomologicalComplex.Hom.f P.ι 0) ≫\n cokernel.π\n (HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) 0)) ≫\n cokernel.desc (HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) 0)\n (HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) 0)\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) 0 =\n 0) =\n 0) ≫\n homology.map\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) 0 =\n 0)\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj Q.cocomplex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj Q.cocomplex) 0 =\n 0)\n (HomologicalComplex.Hom.sqTo\n ((mapHomologicalComplex F (ComplexShape.up ℕ)).map (InjectiveResolution.desc f Q P)) 0)\n (HomologicalComplex.Hom.sqFrom\n ((mapHomologicalComplex F (ComplexShape.up ℕ)).map (InjectiveResolution.desc f Q P)) 0)\n (_ :\n (HomologicalComplex.Hom.sqTo\n ((mapHomologicalComplex F (ComplexShape.up ℕ)).map (InjectiveResolution.desc f Q P)) 0).right =\n (HomologicalComplex.Hom.sqTo\n ((mapHomologicalComplex F (ComplexShape.up ℕ)).map (InjectiveResolution.desc f Q P)) 0).right)) ≫\n homology.ι (F.map (HomologicalComplex.d Q.cocomplex (ComplexShape.prev (ComplexShape.up ℕ) 0) 0))\n (F.map (HomologicalComplex.d Q.cocomplex 0 (ComplexShape.next (ComplexShape.up ℕ) 0)))\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj Q.cocomplex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj Q.cocomplex) 0 =\n 0)",
"tactic": "rw [Category.assoc, homology.lift_ι, Category.assoc]"
},
{
"state_after": "case h\nC : Type u\ninst✝⁵ : Category C\nD : Type u\ninst✝⁴ : Category D\nF : C ⥤ D\nX✝ Y✝ Z : C\nf✝ : X✝ ⟶ Y✝\ng : Y✝ ⟶ Z\ninst✝³ : Abelian C\ninst✝² : Abelian D\ninst✝¹ : Functor.Additive F\ninst✝ : EnoughInjectives C\nX Y : C\nf : X ⟶ Y\nP : InjectiveResolution X\nQ : InjectiveResolution Y\n⊢ F.map f ≫\n F.map (HomologicalComplex.Hom.f Q.ι 0) ≫\n cokernel.π (F.map (HomologicalComplex.d Q.cocomplex (ComplexShape.prev (ComplexShape.up ℕ) 0) 0)) =\n homology.lift (F.map (HomologicalComplex.d P.cocomplex (ComplexShape.prev (ComplexShape.up ℕ) 0) 0))\n (F.map (HomologicalComplex.d P.cocomplex 0 (ComplexShape.next (ComplexShape.up ℕ) 0)))\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) 0 =\n 0)\n (F.map (HomologicalComplex.Hom.f P.ι 0) ≫\n cokernel.π (F.map (HomologicalComplex.d P.cocomplex (ComplexShape.prev (ComplexShape.up ℕ) 0) 0)))\n (_ :\n (F.map (HomologicalComplex.Hom.f P.ι 0) ≫\n cokernel.π (HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) 0)) ≫\n cokernel.desc (HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) 0)\n (HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) 0)\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) 0 =\n 0) =\n 0) ≫\n homology.ι (HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) 0)\n (HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) 0)\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) 0 =\n 0) ≫\n cokernel.map (HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) 0)\n (HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj Q.cocomplex) 0)\n (HomologicalComplex.Hom.sqTo\n ((mapHomologicalComplex F (ComplexShape.up ℕ)).map (InjectiveResolution.desc f Q P)) 0).left\n (HomologicalComplex.Hom.sqFrom\n ((mapHomologicalComplex F (ComplexShape.up ℕ)).map (InjectiveResolution.desc f Q P)) 0).left\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) 0 ≫\n (HomologicalComplex.Hom.sqFrom\n ((mapHomologicalComplex F (ComplexShape.up ℕ)).map (InjectiveResolution.desc f Q P)) 0).left =\n (HomologicalComplex.Hom.sqTo\n ((mapHomologicalComplex F (ComplexShape.up ℕ)).map (InjectiveResolution.desc f Q P)) 0).left ≫\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj Q.cocomplex) 0)",
"state_before": "case h\nC : Type u\ninst✝⁵ : Category C\nD : Type u\ninst✝⁴ : Category D\nF : C ⥤ D\nX✝ Y✝ Z : C\nf✝ : X✝ ⟶ Y✝\ng : Y✝ ⟶ Z\ninst✝³ : Abelian C\ninst✝² : Abelian D\ninst✝¹ : Functor.Additive F\ninst✝ : EnoughInjectives C\nX Y : C\nf : X ⟶ Y\nP : InjectiveResolution X\nQ : InjectiveResolution Y\n⊢ F.map f ≫\n F.map (HomologicalComplex.Hom.f Q.ι 0) ≫\n cokernel.π (F.map (HomologicalComplex.d Q.cocomplex (ComplexShape.prev (ComplexShape.up ℕ) 0) 0)) =\n homology.lift (F.map (HomologicalComplex.d P.cocomplex (ComplexShape.prev (ComplexShape.up ℕ) 0) 0))\n (F.map (HomologicalComplex.d P.cocomplex 0 (ComplexShape.next (ComplexShape.up ℕ) 0)))\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) 0 =\n 0)\n (F.map (HomologicalComplex.Hom.f P.ι 0) ≫\n cokernel.π (F.map (HomologicalComplex.d P.cocomplex (ComplexShape.prev (ComplexShape.up ℕ) 0) 0)))\n (_ :\n (F.map (HomologicalComplex.Hom.f P.ι 0) ≫\n cokernel.π (HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) 0)) ≫\n cokernel.desc (HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) 0)\n (HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) 0)\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) 0 =\n 0) =\n 0) ≫\n homology.map\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) 0 =\n 0)\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj Q.cocomplex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj Q.cocomplex) 0 =\n 0)\n (HomologicalComplex.Hom.sqTo\n ((mapHomologicalComplex F (ComplexShape.up ℕ)).map (InjectiveResolution.desc f Q P)) 0)\n (HomologicalComplex.Hom.sqFrom\n ((mapHomologicalComplex F (ComplexShape.up ℕ)).map (InjectiveResolution.desc f Q P)) 0)\n (_ :\n (HomologicalComplex.Hom.sqTo\n ((mapHomologicalComplex F (ComplexShape.up ℕ)).map (InjectiveResolution.desc f Q P)) 0).right =\n (HomologicalComplex.Hom.sqTo\n ((mapHomologicalComplex F (ComplexShape.up ℕ)).map (InjectiveResolution.desc f Q P)) 0).right) ≫\n homology.ι (F.map (HomologicalComplex.d Q.cocomplex (ComplexShape.prev (ComplexShape.up ℕ) 0) 0))\n (F.map (HomologicalComplex.d Q.cocomplex 0 (ComplexShape.next (ComplexShape.up ℕ) 0)))\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj Q.cocomplex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj Q.cocomplex) 0 =\n 0)",
"tactic": "erw [homology.map_ι]"
},
{
"state_after": "case h\nC : Type u\ninst✝⁵ : Category C\nD : Type u\ninst✝⁴ : Category D\nF : C ⥤ D\nX✝ Y✝ Z : C\nf✝ : X✝ ⟶ Y✝\ng : Y✝ ⟶ Z\ninst✝³ : Abelian C\ninst✝² : Abelian D\ninst✝¹ : Functor.Additive F\ninst✝ : EnoughInjectives C\nX Y : C\nf : X ⟶ Y\nP : InjectiveResolution X\nQ : InjectiveResolution Y\n⊢ F.map f ≫\n F.map (HomologicalComplex.Hom.f Q.ι 0) ≫\n cokernel.π (F.map (HomologicalComplex.d Q.cocomplex (ComplexShape.prev (ComplexShape.up ℕ) 0) 0)) =\n (homology.lift (F.map (HomologicalComplex.d P.cocomplex (ComplexShape.prev (ComplexShape.up ℕ) 0) 0))\n (F.map (HomologicalComplex.d P.cocomplex 0 (ComplexShape.next (ComplexShape.up ℕ) 0)))\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) 0 =\n 0)\n (F.map (HomologicalComplex.Hom.f P.ι 0) ≫\n cokernel.π (F.map (HomologicalComplex.d P.cocomplex (ComplexShape.prev (ComplexShape.up ℕ) 0) 0)))\n (_ :\n (F.map (HomologicalComplex.Hom.f P.ι 0) ≫\n cokernel.π\n (HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) 0)) ≫\n cokernel.desc (HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) 0)\n (HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) 0)\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) 0 =\n 0) =\n 0) ≫\n homology.ι (HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) 0)\n (HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) 0)\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) 0 =\n 0)) ≫\n cokernel.map (HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) 0)\n (HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj Q.cocomplex) 0)\n (HomologicalComplex.Hom.sqTo\n ((mapHomologicalComplex F (ComplexShape.up ℕ)).map (InjectiveResolution.desc f Q P)) 0).left\n (HomologicalComplex.Hom.sqFrom\n ((mapHomologicalComplex F (ComplexShape.up ℕ)).map (InjectiveResolution.desc f Q P)) 0).left\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) 0 ≫\n (HomologicalComplex.Hom.sqFrom\n ((mapHomologicalComplex F (ComplexShape.up ℕ)).map (InjectiveResolution.desc f Q P)) 0).left =\n (HomologicalComplex.Hom.sqTo\n ((mapHomologicalComplex F (ComplexShape.up ℕ)).map (InjectiveResolution.desc f Q P)) 0).left ≫\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj Q.cocomplex) 0)",
"state_before": "case h\nC : Type u\ninst✝⁵ : Category C\nD : Type u\ninst✝⁴ : Category D\nF : C ⥤ D\nX✝ Y✝ Z : C\nf✝ : X✝ ⟶ Y✝\ng : Y✝ ⟶ Z\ninst✝³ : Abelian C\ninst✝² : Abelian D\ninst✝¹ : Functor.Additive F\ninst✝ : EnoughInjectives C\nX Y : C\nf : X ⟶ Y\nP : InjectiveResolution X\nQ : InjectiveResolution Y\n⊢ F.map f ≫\n F.map (HomologicalComplex.Hom.f Q.ι 0) ≫\n cokernel.π (F.map (HomologicalComplex.d Q.cocomplex (ComplexShape.prev (ComplexShape.up ℕ) 0) 0)) =\n homology.lift (F.map (HomologicalComplex.d P.cocomplex (ComplexShape.prev (ComplexShape.up ℕ) 0) 0))\n (F.map (HomologicalComplex.d P.cocomplex 0 (ComplexShape.next (ComplexShape.up ℕ) 0)))\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) 0 =\n 0)\n (F.map (HomologicalComplex.Hom.f P.ι 0) ≫\n cokernel.π (F.map (HomologicalComplex.d P.cocomplex (ComplexShape.prev (ComplexShape.up ℕ) 0) 0)))\n (_ :\n (F.map (HomologicalComplex.Hom.f P.ι 0) ≫\n cokernel.π (HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) 0)) ≫\n cokernel.desc (HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) 0)\n (HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) 0)\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) 0 =\n 0) =\n 0) ≫\n homology.ι (HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) 0)\n (HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) 0)\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) 0 =\n 0) ≫\n cokernel.map (HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) 0)\n (HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj Q.cocomplex) 0)\n (HomologicalComplex.Hom.sqTo\n ((mapHomologicalComplex F (ComplexShape.up ℕ)).map (InjectiveResolution.desc f Q P)) 0).left\n (HomologicalComplex.Hom.sqFrom\n ((mapHomologicalComplex F (ComplexShape.up ℕ)).map (InjectiveResolution.desc f Q P)) 0).left\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) 0 ≫\n (HomologicalComplex.Hom.sqFrom\n ((mapHomologicalComplex F (ComplexShape.up ℕ)).map (InjectiveResolution.desc f Q P)) 0).left =\n (HomologicalComplex.Hom.sqTo\n ((mapHomologicalComplex F (ComplexShape.up ℕ)).map (InjectiveResolution.desc f Q P)) 0).left ≫\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj Q.cocomplex) 0)",
"tactic": "rw [←Category.assoc (homology.lift _ _ _ _ _) _ _]"
},
{
"state_after": "case h\nC : Type u\ninst✝⁵ : Category C\nD : Type u\ninst✝⁴ : Category D\nF : C ⥤ D\nX✝ Y✝ Z : C\nf✝ : X✝ ⟶ Y✝\ng : Y✝ ⟶ Z\ninst✝³ : Abelian C\ninst✝² : Abelian D\ninst✝¹ : Functor.Additive F\ninst✝ : EnoughInjectives C\nX Y : C\nf : X ⟶ Y\nP : InjectiveResolution X\nQ : InjectiveResolution Y\n⊢ F.map f ≫\n F.map (HomologicalComplex.Hom.f Q.ι 0) ≫\n cokernel.π (F.map (HomologicalComplex.d Q.cocomplex (ComplexShape.prev (ComplexShape.up ℕ) 0) 0)) =\n (F.map (HomologicalComplex.Hom.f P.ι 0) ≫\n cokernel.π (F.map (HomologicalComplex.d P.cocomplex (ComplexShape.prev (ComplexShape.up ℕ) 0) 0))) ≫\n cokernel.map (HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) 0)\n (HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj Q.cocomplex) 0)\n (HomologicalComplex.Hom.sqTo\n ((mapHomologicalComplex F (ComplexShape.up ℕ)).map (InjectiveResolution.desc f Q P)) 0).left\n (HomologicalComplex.Hom.sqFrom\n ((mapHomologicalComplex F (ComplexShape.up ℕ)).map (InjectiveResolution.desc f Q P)) 0).left\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) 0 ≫\n (HomologicalComplex.Hom.sqFrom\n ((mapHomologicalComplex F (ComplexShape.up ℕ)).map (InjectiveResolution.desc f Q P)) 0).left =\n (HomologicalComplex.Hom.sqTo\n ((mapHomologicalComplex F (ComplexShape.up ℕ)).map (InjectiveResolution.desc f Q P)) 0).left ≫\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj Q.cocomplex) 0)",
"state_before": "case h\nC : Type u\ninst✝⁵ : Category C\nD : Type u\ninst✝⁴ : Category D\nF : C ⥤ D\nX✝ Y✝ Z : C\nf✝ : X✝ ⟶ Y✝\ng : Y✝ ⟶ Z\ninst✝³ : Abelian C\ninst✝² : Abelian D\ninst✝¹ : Functor.Additive F\ninst✝ : EnoughInjectives C\nX Y : C\nf : X ⟶ Y\nP : InjectiveResolution X\nQ : InjectiveResolution Y\n⊢ F.map f ≫\n F.map (HomologicalComplex.Hom.f Q.ι 0) ≫\n cokernel.π (F.map (HomologicalComplex.d Q.cocomplex (ComplexShape.prev (ComplexShape.up ℕ) 0) 0)) =\n (homology.lift (F.map (HomologicalComplex.d P.cocomplex (ComplexShape.prev (ComplexShape.up ℕ) 0) 0))\n (F.map (HomologicalComplex.d P.cocomplex 0 (ComplexShape.next (ComplexShape.up ℕ) 0)))\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) 0 =\n 0)\n (F.map (HomologicalComplex.Hom.f P.ι 0) ≫\n cokernel.π (F.map (HomologicalComplex.d P.cocomplex (ComplexShape.prev (ComplexShape.up ℕ) 0) 0)))\n (_ :\n (F.map (HomologicalComplex.Hom.f P.ι 0) ≫\n cokernel.π\n (HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) 0)) ≫\n cokernel.desc (HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) 0)\n (HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) 0)\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) 0 =\n 0) =\n 0) ≫\n homology.ι (HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) 0)\n (HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) 0)\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) 0 =\n 0)) ≫\n cokernel.map (HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) 0)\n (HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj Q.cocomplex) 0)\n (HomologicalComplex.Hom.sqTo\n ((mapHomologicalComplex F (ComplexShape.up ℕ)).map (InjectiveResolution.desc f Q P)) 0).left\n (HomologicalComplex.Hom.sqFrom\n ((mapHomologicalComplex F (ComplexShape.up ℕ)).map (InjectiveResolution.desc f Q P)) 0).left\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) 0 ≫\n (HomologicalComplex.Hom.sqFrom\n ((mapHomologicalComplex F (ComplexShape.up ℕ)).map (InjectiveResolution.desc f Q P)) 0).left =\n (HomologicalComplex.Hom.sqTo\n ((mapHomologicalComplex F (ComplexShape.up ℕ)).map (InjectiveResolution.desc f Q P)) 0).left ≫\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj Q.cocomplex) 0)",
"tactic": "erw [homology.lift_ι]"
},
{
"state_after": "case h\nC : Type u\ninst✝⁵ : Category C\nD : Type u\ninst✝⁴ : Category D\nF : C ⥤ D\nX✝ Y✝ Z : C\nf✝ : X✝ ⟶ Y✝\ng : Y✝ ⟶ Z\ninst✝³ : Abelian C\ninst✝² : Abelian D\ninst✝¹ : Functor.Additive F\ninst✝ : EnoughInjectives C\nX Y : C\nf : X ⟶ Y\nP : InjectiveResolution X\nQ : InjectiveResolution Y\n⊢ F.map f ≫\n F.map (HomologicalComplex.Hom.f Q.ι 0) ≫\n cokernel.π (F.map (HomologicalComplex.d Q.cocomplex (ComplexShape.prev (ComplexShape.up ℕ) 0) 0)) =\n F.map (HomologicalComplex.Hom.f P.ι 0) ≫\n cokernel.π (F.map (HomologicalComplex.d P.cocomplex (ComplexShape.prev (ComplexShape.up ℕ) 0) 0)) ≫\n cokernel.map (HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) 0)\n (HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj Q.cocomplex) 0)\n (HomologicalComplex.Hom.sqTo\n ((mapHomologicalComplex F (ComplexShape.up ℕ)).map (InjectiveResolution.desc f Q P)) 0).left\n (HomologicalComplex.Hom.sqFrom\n ((mapHomologicalComplex F (ComplexShape.up ℕ)).map (InjectiveResolution.desc f Q P)) 0).left\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) 0 ≫\n (HomologicalComplex.Hom.sqFrom\n ((mapHomologicalComplex F (ComplexShape.up ℕ)).map (InjectiveResolution.desc f Q P)) 0).left =\n (HomologicalComplex.Hom.sqTo\n ((mapHomologicalComplex F (ComplexShape.up ℕ)).map (InjectiveResolution.desc f Q P)) 0).left ≫\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj Q.cocomplex) 0)",
"state_before": "case h\nC : Type u\ninst✝⁵ : Category C\nD : Type u\ninst✝⁴ : Category D\nF : C ⥤ D\nX✝ Y✝ Z : C\nf✝ : X✝ ⟶ Y✝\ng : Y✝ ⟶ Z\ninst✝³ : Abelian C\ninst✝² : Abelian D\ninst✝¹ : Functor.Additive F\ninst✝ : EnoughInjectives C\nX Y : C\nf : X ⟶ Y\nP : InjectiveResolution X\nQ : InjectiveResolution Y\n⊢ F.map f ≫\n F.map (HomologicalComplex.Hom.f Q.ι 0) ≫\n cokernel.π (F.map (HomologicalComplex.d Q.cocomplex (ComplexShape.prev (ComplexShape.up ℕ) 0) 0)) =\n (F.map (HomologicalComplex.Hom.f P.ι 0) ≫\n cokernel.π (F.map (HomologicalComplex.d P.cocomplex (ComplexShape.prev (ComplexShape.up ℕ) 0) 0))) ≫\n cokernel.map (HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) 0)\n (HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj Q.cocomplex) 0)\n (HomologicalComplex.Hom.sqTo\n ((mapHomologicalComplex F (ComplexShape.up ℕ)).map (InjectiveResolution.desc f Q P)) 0).left\n (HomologicalComplex.Hom.sqFrom\n ((mapHomologicalComplex F (ComplexShape.up ℕ)).map (InjectiveResolution.desc f Q P)) 0).left\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) 0 ≫\n (HomologicalComplex.Hom.sqFrom\n ((mapHomologicalComplex F (ComplexShape.up ℕ)).map (InjectiveResolution.desc f Q P)) 0).left =\n (HomologicalComplex.Hom.sqTo\n ((mapHomologicalComplex F (ComplexShape.up ℕ)).map (InjectiveResolution.desc f Q P)) 0).left ≫\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj Q.cocomplex) 0)",
"tactic": "rw [Category.assoc]"
},
{
"state_after": "case h\nC : Type u\ninst✝⁵ : Category C\nD : Type u\ninst✝⁴ : Category D\nF : C ⥤ D\nX✝ Y✝ Z : C\nf✝ : X✝ ⟶ Y✝\ng : Y✝ ⟶ Z\ninst✝³ : Abelian C\ninst✝² : Abelian D\ninst✝¹ : Functor.Additive F\ninst✝ : EnoughInjectives C\nX Y : C\nf : X ⟶ Y\nP : InjectiveResolution X\nQ : InjectiveResolution Y\n⊢ F.map f ≫\n F.map (HomologicalComplex.Hom.f Q.ι 0) ≫\n cokernel.π (F.map (HomologicalComplex.d Q.cocomplex (ComplexShape.prev (ComplexShape.up ℕ) 0) 0)) =\n F.map (HomologicalComplex.Hom.f P.ι 0) ≫\n (HomologicalComplex.Hom.sqFrom\n ((mapHomologicalComplex F (ComplexShape.up ℕ)).map (InjectiveResolution.desc f Q P)) 0).left ≫\n cokernel.π (HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj Q.cocomplex) 0)",
"state_before": "case h\nC : Type u\ninst✝⁵ : Category C\nD : Type u\ninst✝⁴ : Category D\nF : C ⥤ D\nX✝ Y✝ Z : C\nf✝ : X✝ ⟶ Y✝\ng : Y✝ ⟶ Z\ninst✝³ : Abelian C\ninst✝² : Abelian D\ninst✝¹ : Functor.Additive F\ninst✝ : EnoughInjectives C\nX Y : C\nf : X ⟶ Y\nP : InjectiveResolution X\nQ : InjectiveResolution Y\n⊢ F.map f ≫\n F.map (HomologicalComplex.Hom.f Q.ι 0) ≫\n cokernel.π (F.map (HomologicalComplex.d Q.cocomplex (ComplexShape.prev (ComplexShape.up ℕ) 0) 0)) =\n F.map (HomologicalComplex.Hom.f P.ι 0) ≫\n cokernel.π (F.map (HomologicalComplex.d P.cocomplex (ComplexShape.prev (ComplexShape.up ℕ) 0) 0)) ≫\n cokernel.map (HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) 0)\n (HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj Q.cocomplex) 0)\n (HomologicalComplex.Hom.sqTo\n ((mapHomologicalComplex F (ComplexShape.up ℕ)).map (InjectiveResolution.desc f Q P)) 0).left\n (HomologicalComplex.Hom.sqFrom\n ((mapHomologicalComplex F (ComplexShape.up ℕ)).map (InjectiveResolution.desc f Q P)) 0).left\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) 0 ≫\n (HomologicalComplex.Hom.sqFrom\n ((mapHomologicalComplex F (ComplexShape.up ℕ)).map (InjectiveResolution.desc f Q P)) 0).left =\n (HomologicalComplex.Hom.sqTo\n ((mapHomologicalComplex F (ComplexShape.up ℕ)).map (InjectiveResolution.desc f Q P)) 0).left ≫\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj Q.cocomplex) 0)",
"tactic": "erw [cokernel.π_desc]"
},
{
"state_after": "case h\nC : Type u\ninst✝⁵ : Category C\nD : Type u\ninst✝⁴ : Category D\nF : C ⥤ D\nX✝ Y✝ Z : C\nf✝ : X✝ ⟶ Y✝\ng : Y✝ ⟶ Z\ninst✝³ : Abelian C\ninst✝² : Abelian D\ninst✝¹ : Functor.Additive F\ninst✝ : EnoughInjectives C\nX Y : C\nf : X ⟶ Y\nP : InjectiveResolution X\nQ : InjectiveResolution Y\n⊢ F.map (HomologicalComplex.Hom.f P.ι 0 ≫ HomologicalComplex.Hom.f (InjectiveResolution.desc f Q P) 0) ≫\n cokernel.π (F.map (HomologicalComplex.d Q.cocomplex (ComplexShape.prev (ComplexShape.up ℕ) 0) 0)) =\n F.map (HomologicalComplex.Hom.f P.ι 0 ≫ HomologicalComplex.Hom.f (InjectiveResolution.desc f Q P) 0) ≫\n cokernel.π (HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj Q.cocomplex) 0)",
"state_before": "case h\nC : Type u\ninst✝⁵ : Category C\nD : Type u\ninst✝⁴ : Category D\nF : C ⥤ D\nX✝ Y✝ Z : C\nf✝ : X✝ ⟶ Y✝\ng : Y✝ ⟶ Z\ninst✝³ : Abelian C\ninst✝² : Abelian D\ninst✝¹ : Functor.Additive F\ninst✝ : EnoughInjectives C\nX Y : C\nf : X ⟶ Y\nP : InjectiveResolution X\nQ : InjectiveResolution Y\n⊢ F.map f ≫\n F.map (HomologicalComplex.Hom.f Q.ι 0) ≫\n cokernel.π (F.map (HomologicalComplex.d Q.cocomplex (ComplexShape.prev (ComplexShape.up ℕ) 0) 0)) =\n F.map (HomologicalComplex.Hom.f P.ι 0) ≫\n (HomologicalComplex.Hom.sqFrom\n ((mapHomologicalComplex F (ComplexShape.up ℕ)).map (InjectiveResolution.desc f Q P)) 0).left ≫\n cokernel.π (HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj Q.cocomplex) 0)",
"tactic": "rw [← Category.assoc, ← Functor.map_comp, ← Category.assoc,\n HomologicalComplex.Hom.sqFrom_left, mapHomologicalComplex_map_f, ← Functor.map_comp,\n show f ≫ Q.ι.f 0 = P.ι.f 0 ≫ (InjectiveResolution.desc f Q P).f 0 from\n HomologicalComplex.congr_hom (InjectiveResolution.desc_commutes f Q P).symm 0]"
},
{
"state_after": "no goals",
"state_before": "case h\nC : Type u\ninst✝⁵ : Category C\nD : Type u\ninst✝⁴ : Category D\nF : C ⥤ D\nX✝ Y✝ Z : C\nf✝ : X✝ ⟶ Y✝\ng : Y✝ ⟶ Z\ninst✝³ : Abelian C\ninst✝² : Abelian D\ninst✝¹ : Functor.Additive F\ninst✝ : EnoughInjectives C\nX Y : C\nf : X ⟶ Y\nP : InjectiveResolution X\nQ : InjectiveResolution Y\n⊢ F.map (HomologicalComplex.Hom.f P.ι 0 ≫ HomologicalComplex.Hom.f (InjectiveResolution.desc f Q P) 0) ≫\n cokernel.π (F.map (HomologicalComplex.d Q.cocomplex (ComplexShape.prev (ComplexShape.up ℕ) 0) 0)) =\n F.map (HomologicalComplex.Hom.f P.ι 0 ≫ HomologicalComplex.Hom.f (InjectiveResolution.desc f Q P) 0) ≫\n cokernel.π (HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj Q.cocomplex) 0)",
"tactic": "rfl"
},
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝⁵ : Category C\nD : Type u\ninst✝⁴ : Category D\nF : C ⥤ D\nX✝ Y✝ Z : C\nf✝ : X✝ ⟶ Y✝\ng : Y✝ ⟶ Z\ninst✝³ : Abelian C\ninst✝² : Abelian D\ninst✝¹ : Functor.Additive F\ninst✝ : EnoughInjectives C\nX Y : C\nf : X ⟶ Y\nP : InjectiveResolution X\nQ : InjectiveResolution Y\n⊢ P.ι ≫ InjectiveResolution.desc f Q P = (CochainComplex.single₀ C).map f ≫ Q.ι",
"tactic": "simp"
}
] |
[
323,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
300,
1
] |
Mathlib/Topology/Bases.lean
|
isTopologicalBasis_singletons
|
[] |
[
493,
64
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
490,
1
] |
Mathlib/Data/Num/Lemmas.lean
|
Num.mod_zero
|
[
{
"state_after": "case zero\n\n⊢ mod zero 0 = zero\n\ncase pos\na✝ : PosNum\n⊢ mod (pos a✝) 0 = pos a✝",
"state_before": "n : Num\n⊢ mod n 0 = n",
"tactic": "cases n"
},
{
"state_after": "no goals",
"state_before": "case zero\n\n⊢ mod zero 0 = zero",
"tactic": "rfl"
},
{
"state_after": "no goals",
"state_before": "case pos\na✝ : PosNum\n⊢ mod (pos a✝) 0 = pos a✝",
"tactic": "simp [Num.mod]"
}
] |
[
1663,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1659,
11
] |
Mathlib/Data/Set/Pointwise/Basic.lean
|
Set.mul_union
|
[] |
[
448,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
447,
1
] |
Mathlib/MeasureTheory/Function/SpecialFunctions/Basic.lean
|
Measurable.csin
|
[] |
[
196,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
195,
1
] |
Mathlib/Data/Finset/LocallyFinite.lean
|
Finset.Icc_eq_empty_iff
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.1120\nα : Type u_1\ninst✝¹ : Preorder α\ninst✝ : LocallyFiniteOrder α\na a₁ a₂ b b₁ b₂ c x : α\n⊢ Icc a b = ∅ ↔ ¬a ≤ b",
"tactic": "rw [← coe_eq_empty, coe_Icc, Set.Icc_eq_empty_iff]"
}
] |
[
70,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
69,
1
] |
Mathlib/Analysis/Calculus/Deriv/Basic.lean
|
derivWithin_id
|
[] |
[
667,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
666,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/Pullbacks.lean
|
CategoryTheory.Limits.pullback.lift_snd
|
[] |
[
1180,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1178,
1
] |
Mathlib/Analysis/InnerProductSpace/PiL2.lean
|
OrthonormalBasis.sum_inner_mul_inner
|
[
{
"state_after": "ι : Type u_1\nι' : Type ?u.827323\n𝕜 : Type u_2\ninst✝⁹ : IsROrC 𝕜\nE : Type u_3\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : InnerProductSpace 𝕜 E\nE' : Type ?u.827352\ninst✝⁶ : NormedAddCommGroup E'\ninst✝⁵ : InnerProductSpace 𝕜 E'\nF : Type ?u.827370\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : InnerProductSpace ℝ F\nF' : Type ?u.827390\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : InnerProductSpace ℝ F'\ninst✝ : Fintype ι\nb : OrthonormalBasis ι 𝕜 E\nx y : E\nthis : ↑(↑(innerSL 𝕜) x) (∑ i : ι, ↑b.repr y i • ↑b i) = ↑(↑(innerSL 𝕜) x) y\n⊢ ∑ i : ι, inner x (↑b i) * inner (↑b i) y = inner x y",
"state_before": "ι : Type u_1\nι' : Type ?u.827323\n𝕜 : Type u_2\ninst✝⁹ : IsROrC 𝕜\nE : Type u_3\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : InnerProductSpace 𝕜 E\nE' : Type ?u.827352\ninst✝⁶ : NormedAddCommGroup E'\ninst✝⁵ : InnerProductSpace 𝕜 E'\nF : Type ?u.827370\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : InnerProductSpace ℝ F\nF' : Type ?u.827390\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : InnerProductSpace ℝ F'\ninst✝ : Fintype ι\nb : OrthonormalBasis ι 𝕜 E\nx y : E\n⊢ ∑ i : ι, inner x (↑b i) * inner (↑b i) y = inner x y",
"tactic": "have := congr_arg (innerSL 𝕜 x) (b.sum_repr y)"
},
{
"state_after": "ι : Type u_1\nι' : Type ?u.827323\n𝕜 : Type u_2\ninst✝⁹ : IsROrC 𝕜\nE : Type u_3\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : InnerProductSpace 𝕜 E\nE' : Type ?u.827352\ninst✝⁶ : NormedAddCommGroup E'\ninst✝⁵ : InnerProductSpace 𝕜 E'\nF : Type ?u.827370\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : InnerProductSpace ℝ F\nF' : Type ?u.827390\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : InnerProductSpace ℝ F'\ninst✝ : Fintype ι\nb : OrthonormalBasis ι 𝕜 E\nx y : E\nthis : ∑ x_1 : ι, ↑(↑(innerSL 𝕜) x) (↑b.repr y x_1 • ↑b x_1) = ↑(↑(innerSL 𝕜) x) y\n⊢ ∑ i : ι, inner x (↑b i) * inner (↑b i) y = inner x y",
"state_before": "ι : Type u_1\nι' : Type ?u.827323\n𝕜 : Type u_2\ninst✝⁹ : IsROrC 𝕜\nE : Type u_3\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : InnerProductSpace 𝕜 E\nE' : Type ?u.827352\ninst✝⁶ : NormedAddCommGroup E'\ninst✝⁵ : InnerProductSpace 𝕜 E'\nF : Type ?u.827370\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : InnerProductSpace ℝ F\nF' : Type ?u.827390\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : InnerProductSpace ℝ F'\ninst✝ : Fintype ι\nb : OrthonormalBasis ι 𝕜 E\nx y : E\nthis : ↑(↑(innerSL 𝕜) x) (∑ i : ι, ↑b.repr y i • ↑b i) = ↑(↑(innerSL 𝕜) x) y\n⊢ ∑ i : ι, inner x (↑b i) * inner (↑b i) y = inner x y",
"tactic": "rw [map_sum] at this"
},
{
"state_after": "case h.e'_2.a\nι : Type u_1\nι' : Type ?u.827323\n𝕜 : Type u_2\ninst✝⁹ : IsROrC 𝕜\nE : Type u_3\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : InnerProductSpace 𝕜 E\nE' : Type ?u.827352\ninst✝⁶ : NormedAddCommGroup E'\ninst✝⁵ : InnerProductSpace 𝕜 E'\nF : Type ?u.827370\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : InnerProductSpace ℝ F\nF' : Type ?u.827390\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : InnerProductSpace ℝ F'\ninst✝ : Fintype ι\nb : OrthonormalBasis ι 𝕜 E\nx y : E\nthis : ∑ x_1 : ι, ↑(↑(innerSL 𝕜) x) (↑b.repr y x_1 • ↑b x_1) = ↑(↑(innerSL 𝕜) x) y\nx✝ : ι\na✝ : x✝ ∈ Finset.univ\n⊢ inner x (↑b x✝) * inner (↑b x✝) y = ↑(↑(innerSL 𝕜) x) (↑b.repr y x✝ • ↑b x✝)",
"state_before": "ι : Type u_1\nι' : Type ?u.827323\n𝕜 : Type u_2\ninst✝⁹ : IsROrC 𝕜\nE : Type u_3\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : InnerProductSpace 𝕜 E\nE' : Type ?u.827352\ninst✝⁶ : NormedAddCommGroup E'\ninst✝⁵ : InnerProductSpace 𝕜 E'\nF : Type ?u.827370\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : InnerProductSpace ℝ F\nF' : Type ?u.827390\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : InnerProductSpace ℝ F'\ninst✝ : Fintype ι\nb : OrthonormalBasis ι 𝕜 E\nx y : E\nthis : ∑ x_1 : ι, ↑(↑(innerSL 𝕜) x) (↑b.repr y x_1 • ↑b x_1) = ↑(↑(innerSL 𝕜) x) y\n⊢ ∑ i : ι, inner x (↑b i) * inner (↑b i) y = inner x y",
"tactic": "convert this"
},
{
"state_after": "case h.e'_2.a\nι : Type u_1\nι' : Type ?u.827323\n𝕜 : Type u_2\ninst✝⁹ : IsROrC 𝕜\nE : Type u_3\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : InnerProductSpace 𝕜 E\nE' : Type ?u.827352\ninst✝⁶ : NormedAddCommGroup E'\ninst✝⁵ : InnerProductSpace 𝕜 E'\nF : Type ?u.827370\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : InnerProductSpace ℝ F\nF' : Type ?u.827390\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : InnerProductSpace ℝ F'\ninst✝ : Fintype ι\nb : OrthonormalBasis ι 𝕜 E\nx y : E\nthis : ∑ x_1 : ι, ↑(↑(innerSL 𝕜) x) (↑b.repr y x_1 • ↑b x_1) = ↑(↑(innerSL 𝕜) x) y\nx✝ : ι\na✝ : x✝ ∈ Finset.univ\n⊢ inner (↑b x✝) y * inner x (↑b x✝) = inner (↑b x✝) y • ↑(↑(innerSL 𝕜) x) (↑b x✝)",
"state_before": "case h.e'_2.a\nι : Type u_1\nι' : Type ?u.827323\n𝕜 : Type u_2\ninst✝⁹ : IsROrC 𝕜\nE : Type u_3\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : InnerProductSpace 𝕜 E\nE' : Type ?u.827352\ninst✝⁶ : NormedAddCommGroup E'\ninst✝⁵ : InnerProductSpace 𝕜 E'\nF : Type ?u.827370\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : InnerProductSpace ℝ F\nF' : Type ?u.827390\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : InnerProductSpace ℝ F'\ninst✝ : Fintype ι\nb : OrthonormalBasis ι 𝕜 E\nx y : E\nthis : ∑ x_1 : ι, ↑(↑(innerSL 𝕜) x) (↑b.repr y x_1 • ↑b x_1) = ↑(↑(innerSL 𝕜) x) y\nx✝ : ι\na✝ : x✝ ∈ Finset.univ\n⊢ inner x (↑b x✝) * inner (↑b x✝) y = ↑(↑(innerSL 𝕜) x) (↑b.repr y x✝ • ↑b x✝)",
"tactic": "rw [SMulHomClass.map_smul, b.repr_apply_apply, mul_comm]"
},
{
"state_after": "no goals",
"state_before": "case h.e'_2.a\nι : Type u_1\nι' : Type ?u.827323\n𝕜 : Type u_2\ninst✝⁹ : IsROrC 𝕜\nE : Type u_3\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : InnerProductSpace 𝕜 E\nE' : Type ?u.827352\ninst✝⁶ : NormedAddCommGroup E'\ninst✝⁵ : InnerProductSpace 𝕜 E'\nF : Type ?u.827370\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : InnerProductSpace ℝ F\nF' : Type ?u.827390\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : InnerProductSpace ℝ F'\ninst✝ : Fintype ι\nb : OrthonormalBasis ι 𝕜 E\nx y : E\nthis : ∑ x_1 : ι, ↑(↑(innerSL 𝕜) x) (↑b.repr y x_1 • ↑b x_1) = ↑(↑(innerSL 𝕜) x) y\nx✝ : ι\na✝ : x✝ ∈ Finset.univ\n⊢ inner (↑b x✝) y * inner x (↑b x✝) = inner (↑b x✝) y • ↑(↑(innerSL 𝕜) x) (↑b x✝)",
"tactic": "simp only [innerSL_apply, smul_eq_mul]"
}
] |
[
454,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
448,
11
] |
Mathlib/GroupTheory/Nilpotent.lean
|
nilpotencyClass_zero_iff_subsingleton
|
[
{
"state_after": "no goals",
"state_before": "G : Type u_1\ninst✝² : Group G\nH : Subgroup G\ninst✝¹ : Normal H\ninst✝ : Group.IsNilpotent G\n⊢ nilpotencyClass G = 0 ↔ Subsingleton G",
"tactic": "rw [Group.nilpotencyClass, @Nat.find_eq_zero _ (Classical.decPred _), upperCentralSeries_zero,\n subsingleton_iff_bot_eq_top, Subgroup.subsingleton_iff]"
}
] |
[
618,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
614,
1
] |
Mathlib/Data/List/Sigma.lean
|
List.Perm.kreplace
|
[
{
"state_after": "α : Type u\nβ : α → Type v\nl l₁✝ l₂✝ : List (Sigma β)\ninst✝ : DecidableEq α\na : α\nb : β a\nl₁ l₂ : List (Sigma β)\nnd : NodupKeys l₁\n⊢ ∀ {a_1 b_1 : Sigma β},\n a_1.fst ≠ b_1.fst →\n ∀ (c : Sigma β),\n (c ∈ if a = a_1.fst then some { fst := a, snd := b } else none) →\n ∀ (d : Sigma β), (d ∈ if a = b_1.fst then some { fst := a, snd := b } else none) → a_1 = b_1 ∧ c = d",
"state_before": "α : Type u\nβ : α → Type v\nl l₁✝ l₂✝ : List (Sigma β)\ninst✝ : DecidableEq α\na : α\nb : β a\nl₁ l₂ : List (Sigma β)\nnd : NodupKeys l₁\n⊢ Pairwise\n (fun a_1 b_1 =>\n ∀ (c : Sigma β),\n (c ∈ if a = a_1.fst then some { fst := a, snd := b } else none) →\n ∀ (d : Sigma β), (d ∈ if a = b_1.fst then some { fst := a, snd := b } else none) → a_1 = b_1 ∧ c = d)\n l₁",
"tactic": "refine' nd.pairwise_ne.imp _"
},
{
"state_after": "α : Type u\nβ : α → Type v\nl l₁✝ l₂✝ : List (Sigma β)\ninst✝ : DecidableEq α\na : α\nb : β a\nl₁ l₂ : List (Sigma β)\nnd : NodupKeys l₁\nx y : Sigma β\nh : x.fst ≠ y.fst\nz : Sigma β\nh₁ : z ∈ if a = x.fst then some { fst := a, snd := b } else none\nw : Sigma β\nh₂ : w ∈ if a = y.fst then some { fst := a, snd := b } else none\n⊢ x = y ∧ z = w",
"state_before": "α : Type u\nβ : α → Type v\nl l₁✝ l₂✝ : List (Sigma β)\ninst✝ : DecidableEq α\na : α\nb : β a\nl₁ l₂ : List (Sigma β)\nnd : NodupKeys l₁\n⊢ ∀ {a_1 b_1 : Sigma β},\n a_1.fst ≠ b_1.fst →\n ∀ (c : Sigma β),\n (c ∈ if a = a_1.fst then some { fst := a, snd := b } else none) →\n ∀ (d : Sigma β), (d ∈ if a = b_1.fst then some { fst := a, snd := b } else none) → a_1 = b_1 ∧ c = d",
"tactic": "intro x y h z h₁ w h₂"
},
{
"state_after": "case inl.inl.refl.refl\nα : Type u\nβ : α → Type v\nl l₁✝ l₂✝ : List (Sigma β)\ninst✝ : DecidableEq α\na : α\nb : β a\nl₁ l₂ : List (Sigma β)\nnd : NodupKeys l₁\nx y : Sigma β\nh : x.fst ≠ y.fst\nh_2 : a = x.fst\nh_1 : a = y.fst\n⊢ x = y ∧ { fst := a, snd := b } = { fst := a, snd := b }",
"state_before": "α : Type u\nβ : α → Type v\nl l₁✝ l₂✝ : List (Sigma β)\ninst✝ : DecidableEq α\na : α\nb : β a\nl₁ l₂ : List (Sigma β)\nnd : NodupKeys l₁\nx y : Sigma β\nh : x.fst ≠ y.fst\nz : Sigma β\nh₁ : z ∈ if a = x.fst then some { fst := a, snd := b } else none\nw : Sigma β\nh₂ : w ∈ if a = y.fst then some { fst := a, snd := b } else none\n⊢ x = y ∧ z = w",
"tactic": "split_ifs at h₁ h₂ with h_2 h_1 <;> cases h₁ <;> cases h₂"
},
{
"state_after": "no goals",
"state_before": "case inl.inl.refl.refl\nα : Type u\nβ : α → Type v\nl l₁✝ l₂✝ : List (Sigma β)\ninst✝ : DecidableEq α\na : α\nb : β a\nl₁ l₂ : List (Sigma β)\nnd : NodupKeys l₁\nx y : Sigma β\nh : x.fst ≠ y.fst\nh_2 : a = x.fst\nh_1 : a = y.fst\n⊢ x = y ∧ { fst := a, snd := b } = { fst := a, snd := b }",
"tactic": "exact (h (h_2.symm.trans h_1)).elim"
}
] |
[
387,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
381,
1
] |
Std/Data/List/Lemmas.lean
|
List.Sublist.append_right
|
[] |
[
371,
48
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
368,
1
] |
Mathlib/Analysis/NormedSpace/lpSpace.lean
|
lp.monotone
|
[] |
[
351,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
350,
11
] |
Mathlib/Analysis/NormedSpace/Complemented.lean
|
Subspace.closedComplemented_of_closed_compl
|
[] |
[
130,
84
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
128,
1
] |
Mathlib/Topology/Instances/ENNReal.lean
|
ENNReal.nhdsWithin_Ioi_one_neBot
|
[] |
[
220,
87
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
220,
1
] |
Mathlib/MeasureTheory/Measure/Haar/NormedSpace.lean
|
MeasureTheory.Measure.integral_comp_div
|
[] |
[
131,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
130,
1
] |
Mathlib/LinearAlgebra/Span.lean
|
LinearEquiv.toSpanNonzeroSingleton_one
|
[
{
"state_after": "R : Type u_2\nR₂ : Type ?u.381021\nK : Type ?u.381024\nM : Type u_1\nM₂ : Type ?u.381030\nV : Type ?u.381033\nS : Type ?u.381036\ninst✝³ : Ring R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : NoZeroSMulDivisors R M\nx : M\nh : x ≠ 0\n⊢ ↑(↑(toSpanNonzeroSingleton R M x h) 1) = ↑{ val := x, property := (_ : x ∈ Submodule.span R {x}) }",
"state_before": "R : Type u_2\nR₂ : Type ?u.381021\nK : Type ?u.381024\nM : Type u_1\nM₂ : Type ?u.381030\nV : Type ?u.381033\nS : Type ?u.381036\ninst✝³ : Ring R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : NoZeroSMulDivisors R M\nx : M\nh : x ≠ 0\n⊢ ↑(toSpanNonzeroSingleton R M x h) 1 = { val := x, property := (_ : x ∈ Submodule.span R {x}) }",
"tactic": "apply SetLike.coe_eq_coe.mp"
},
{
"state_after": "R : Type u_2\nR₂ : Type ?u.381021\nK : Type ?u.381024\nM : Type u_1\nM₂ : Type ?u.381030\nV : Type ?u.381033\nS : Type ?u.381036\ninst✝³ : Ring R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : NoZeroSMulDivisors R M\nx : M\nh : x ≠ 0\nthis : ↑(↑(toSpanNonzeroSingleton R M x h) 1) = ↑(toSpanSingleton R M x) 1\n⊢ ↑(↑(toSpanNonzeroSingleton R M x h) 1) = ↑{ val := x, property := (_ : x ∈ Submodule.span R {x}) }",
"state_before": "R : Type u_2\nR₂ : Type ?u.381021\nK : Type ?u.381024\nM : Type u_1\nM₂ : Type ?u.381030\nV : Type ?u.381033\nS : Type ?u.381036\ninst✝³ : Ring R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : NoZeroSMulDivisors R M\nx : M\nh : x ≠ 0\n⊢ ↑(↑(toSpanNonzeroSingleton R M x h) 1) = ↑{ val := x, property := (_ : x ∈ Submodule.span R {x}) }",
"tactic": "have : ↑(toSpanNonzeroSingleton R M x h 1) = toSpanSingleton R M x 1 := rfl"
},
{
"state_after": "no goals",
"state_before": "R : Type u_2\nR₂ : Type ?u.381021\nK : Type ?u.381024\nM : Type u_1\nM₂ : Type ?u.381030\nV : Type ?u.381033\nS : Type ?u.381036\ninst✝³ : Ring R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : NoZeroSMulDivisors R M\nx : M\nh : x ≠ 0\nthis : ↑(↑(toSpanNonzeroSingleton R M x h) 1) = ↑(toSpanSingleton R M x) 1\n⊢ ↑(↑(toSpanNonzeroSingleton R M x h) 1) = ↑{ val := x, property := (_ : x ∈ Submodule.span R {x}) }",
"tactic": "rw [this, toSpanSingleton_one, Submodule.coe_mk]"
}
] |
[
1031,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1026,
1
] |
Mathlib/Data/Sum/Order.lean
|
OrderIso.sumAssoc_symm_apply_inr_inr
|
[] |
[
599,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
598,
1
] |
Mathlib/Algebra/MonoidAlgebra/Division.lean
|
AddMonoidAlgebra.modOf_apply_of_not_exists_add
|
[] |
[
138,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
136,
1
] |
Mathlib/Algebra/Order/Group/Defs.lean
|
mul_inv_le_inv_mul_iff
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\ninst✝³ : Group α\ninst✝² : LE α\ninst✝¹ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1\ninst✝ : CovariantClass α α (swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1\na b c d : α\n⊢ a * b⁻¹ ≤ d⁻¹ * c ↔ d * a ≤ c * b",
"tactic": "rw [← mul_le_mul_iff_left d, ← mul_le_mul_iff_right b, mul_inv_cancel_left, mul_assoc,\n inv_mul_cancel_right]"
}
] |
[
358,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
356,
1
] |
Mathlib/Data/Fin/Tuple/Sort.lean
|
Tuple.eq_sort_iff'
|
[
{
"state_after": "case mp\nn : ℕ\nα : Type u_1\ninst✝ : LinearOrder α\nf : Fin n → α\nσ : Equiv.Perm (Fin n)\nh : σ = sort f\n⊢ StrictMono ↑(σ.trans (graphEquiv₁ f))\n\ncase mpr\nn : ℕ\nα : Type u_1\ninst✝ : LinearOrder α\nf : Fin n → α\nσ : Equiv.Perm (Fin n)\nh : StrictMono ↑(σ.trans (graphEquiv₁ f))\n⊢ σ = sort f",
"state_before": "n : ℕ\nα : Type u_1\ninst✝ : LinearOrder α\nf : Fin n → α\nσ : Equiv.Perm (Fin n)\n⊢ σ = sort f ↔ StrictMono ↑(σ.trans (graphEquiv₁ f))",
"tactic": "constructor <;> intro h"
},
{
"state_after": "case mp\nn : ℕ\nα : Type u_1\ninst✝ : LinearOrder α\nf : Fin n → α\nσ : Equiv.Perm (Fin n)\nh : σ = sort f\n⊢ StrictMono ↑((graphEquiv₂ f).trans (Equiv.refl { x // x ∈ graph f }))",
"state_before": "case mp\nn : ℕ\nα : Type u_1\ninst✝ : LinearOrder α\nf : Fin n → α\nσ : Equiv.Perm (Fin n)\nh : σ = sort f\n⊢ StrictMono ↑(σ.trans (graphEquiv₁ f))",
"tactic": "rw [h, sort, Equiv.trans_assoc, Equiv.symm_trans_self]"
},
{
"state_after": "no goals",
"state_before": "case mp\nn : ℕ\nα : Type u_1\ninst✝ : LinearOrder α\nf : Fin n → α\nσ : Equiv.Perm (Fin n)\nh : σ = sort f\n⊢ StrictMono ↑((graphEquiv₂ f).trans (Equiv.refl { x // x ∈ graph f }))",
"tactic": "exact (graphEquiv₂ f).strictMono"
},
{
"state_after": "case mpr\nn : ℕ\nα : Type u_1\ninst✝ : LinearOrder α\nf : Fin n → α\nσ : Equiv.Perm (Fin n)\nh : StrictMono ↑(σ.trans (graphEquiv₁ f))\nthis :\n graphEquiv₂ f =\n StrictMono.orderIsoOfSurjective (↑(σ.trans (graphEquiv₁ f))) h (_ : Function.Surjective ↑(σ.trans (graphEquiv₁ f)))\n⊢ σ = sort f",
"state_before": "case mpr\nn : ℕ\nα : Type u_1\ninst✝ : LinearOrder α\nf : Fin n → α\nσ : Equiv.Perm (Fin n)\nh : StrictMono ↑(σ.trans (graphEquiv₁ f))\n⊢ σ = sort f",
"tactic": "have := Subsingleton.elim (graphEquiv₂ f) (h.orderIsoOfSurjective _ <| Equiv.surjective _)"
},
{
"state_after": "case mpr.H\nn : ℕ\nα : Type u_1\ninst✝ : LinearOrder α\nf : Fin n → α\nσ : Equiv.Perm (Fin n)\nh : StrictMono ↑(σ.trans (graphEquiv₁ f))\nthis :\n graphEquiv₂ f =\n StrictMono.orderIsoOfSurjective (↑(σ.trans (graphEquiv₁ f))) h (_ : Function.Surjective ↑(σ.trans (graphEquiv₁ f)))\nx : Fin n\n⊢ ↑σ x = ↑(sort f) x",
"state_before": "case mpr\nn : ℕ\nα : Type u_1\ninst✝ : LinearOrder α\nf : Fin n → α\nσ : Equiv.Perm (Fin n)\nh : StrictMono ↑(σ.trans (graphEquiv₁ f))\nthis :\n graphEquiv₂ f =\n StrictMono.orderIsoOfSurjective (↑(σ.trans (graphEquiv₁ f))) h (_ : Function.Surjective ↑(σ.trans (graphEquiv₁ f)))\n⊢ σ = sort f",
"tactic": "ext1 x"
},
{
"state_after": "no goals",
"state_before": "case mpr.H\nn : ℕ\nα : Type u_1\ninst✝ : LinearOrder α\nf : Fin n → α\nσ : Equiv.Perm (Fin n)\nh : StrictMono ↑(σ.trans (graphEquiv₁ f))\nthis :\n graphEquiv₂ f =\n StrictMono.orderIsoOfSurjective (↑(σ.trans (graphEquiv₁ f))) h (_ : Function.Surjective ↑(σ.trans (graphEquiv₁ f)))\nx : Fin n\n⊢ ↑σ x = ↑(sort f) x",
"tactic": "exact (graphEquiv₁ f).apply_eq_iff_eq_symm_apply.1 (FunLike.congr_fun this x).symm"
}
] |
[
139,
87
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
133,
1
] |
Mathlib/Combinatorics/SimpleGraph/DegreeSum.lean
|
SimpleGraph.Dart.edge_fiber
|
[
{
"state_after": "no goals",
"state_before": "V : Type u\nG : SimpleGraph V\ninst✝³ : Fintype V\ninst✝² : DecidableRel G.Adj\ninst✝¹ : Fintype (Sym2 V)\ninst✝ : DecidableEq V\nd d' : Dart G\n⊢ d' ∈ filter (fun d' => edge d' = edge d) univ ↔ d' ∈ {d, symm d}",
"tactic": "simpa using dart_edge_eq_iff d' d"
}
] |
[
88,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
86,
1
] |
Mathlib/Data/Set/Intervals/Basic.lean
|
Set.not_mem_Iio_self
|
[] |
[
737,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
737,
1
] |
Mathlib/Topology/FiberBundle/Trivialization.lean
|
Pretrivialization.apply_symm_apply
|
[] |
[
165,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
164,
1
] |
Mathlib/LinearAlgebra/BilinearForm.lean
|
BilinForm.neg_apply
|
[] |
[
248,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
247,
1
] |
Mathlib/Data/Multiset/Functor.lean
|
Multiset.id_traverse
|
[
{
"state_after": "F : Type u → Type u\ninst✝¹ : Applicative F\ninst✝ : CommApplicative F\nα' β' : Type u\nf : α' → F β'\nα : Type u_1\nx : Multiset α\n⊢ ∀ (a : List α), traverse pure (Quotient.mk (isSetoid α) a) = Quotient.mk (isSetoid α) a",
"state_before": "F : Type u → Type u\ninst✝¹ : Applicative F\ninst✝ : CommApplicative F\nα' β' : Type u\nf : α' → F β'\nα : Type u_1\nx : Multiset α\n⊢ traverse pure x = x",
"tactic": "refine' Quotient.inductionOn x _"
},
{
"state_after": "F : Type u → Type u\ninst✝¹ : Applicative F\ninst✝ : CommApplicative F\nα' β' : Type u\nf : α' → F β'\nα : Type u_1\nx : Multiset α\na✝ : List α\n⊢ traverse pure (Quotient.mk (isSetoid α) a✝) = Quotient.mk (isSetoid α) a✝",
"state_before": "F : Type u → Type u\ninst✝¹ : Applicative F\ninst✝ : CommApplicative F\nα' β' : Type u\nf : α' → F β'\nα : Type u_1\nx : Multiset α\n⊢ ∀ (a : List α), traverse pure (Quotient.mk (isSetoid α) a) = Quotient.mk (isSetoid α) a",
"tactic": "intro"
},
{
"state_after": "no goals",
"state_before": "F : Type u → Type u\ninst✝¹ : Applicative F\ninst✝ : CommApplicative F\nα' β' : Type u\nf : α' → F β'\nα : Type u_1\nx : Multiset α\na✝ : List α\n⊢ traverse pure (Quotient.mk (isSetoid α) a✝) = Quotient.mk (isSetoid α) a✝",
"tactic": "simp [traverse, Coe.coe]"
}
] |
[
106,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
103,
1
] |
Mathlib/Data/PNat/Prime.lean
|
PNat.dvd_lcm_left
|
[] |
[
91,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
90,
1
] |
Mathlib/Data/Real/Hyperreal.lean
|
Hyperreal.epsilon_ne_zero
|
[] |
[
190,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
189,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/ArctanDeriv.lean
|
HasDerivWithinAt.arctan
|
[] |
[
146,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
144,
1
] |
Mathlib/Combinatorics/Composition.lean
|
Composition.orderEmbOfFin_boundaries
|
[
{
"state_after": "n : ℕ\nc : Composition n\n⊢ ∀ (x : Fin (length c + 1)), ↑(boundary c) x ∈ boundaries c",
"state_before": "n : ℕ\nc : Composition n\n⊢ Finset.orderEmbOfFin (boundaries c) (_ : Finset.card (boundaries c) = length c + 1) = boundary c",
"tactic": "refine' (Finset.orderEmbOfFin_unique' _ _).symm"
},
{
"state_after": "no goals",
"state_before": "n : ℕ\nc : Composition n\n⊢ ∀ (x : Fin (length c + 1)), ↑(boundary c) x ∈ boundaries c",
"tactic": "exact fun i => (Finset.mem_map' _).2 (Finset.mem_univ _)"
}
] |
[
302,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
299,
1
] |
Mathlib/Data/Finsupp/Basic.lean
|
Finsupp.mk_mem_graph_iff
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.366\nγ : Type ?u.369\nι : Type ?u.372\nM : Type u_2\nM' : Type ?u.378\nN : Type ?u.381\nP : Type ?u.384\nG : Type ?u.387\nH : Type ?u.390\nR : Type ?u.393\nS : Type ?u.396\ninst✝ : Zero M\na : α\nm : M\nf : α →₀ M\n⊢ (∃ a_1,\n ↑f a_1 ≠ 0 ∧\n ↑{ toFun := fun a => (a, ↑f a),\n inj' := (_ : ∀ (x x_1 : α), (fun a => (a, ↑f a)) x = (fun a => (a, ↑f a)) x_1 → x = x_1) }\n a_1 =\n (a, m)) ↔\n ↑f a = m ∧ m ≠ 0",
"state_before": "α : Type u_1\nβ : Type ?u.366\nγ : Type ?u.369\nι : Type ?u.372\nM : Type u_2\nM' : Type ?u.378\nN : Type ?u.381\nP : Type ?u.384\nG : Type ?u.387\nH : Type ?u.390\nR : Type ?u.393\nS : Type ?u.396\ninst✝ : Zero M\na : α\nm : M\nf : α →₀ M\n⊢ (a, m) ∈ graph f ↔ ↑f a = m ∧ m ≠ 0",
"tactic": "simp_rw [graph, mem_map, mem_support_iff]"
},
{
"state_after": "case mp\nα : Type u_1\nβ : Type ?u.366\nγ : Type ?u.369\nι : Type ?u.372\nM : Type u_2\nM' : Type ?u.378\nN : Type ?u.381\nP : Type ?u.384\nG : Type ?u.387\nH : Type ?u.390\nR : Type ?u.393\nS : Type ?u.396\ninst✝ : Zero M\na : α\nm : M\nf : α →₀ M\n⊢ (∃ a_1,\n ↑f a_1 ≠ 0 ∧\n ↑{ toFun := fun a => (a, ↑f a),\n inj' := (_ : ∀ (x x_1 : α), (fun a => (a, ↑f a)) x = (fun a => (a, ↑f a)) x_1 → x = x_1) }\n a_1 =\n (a, m)) →\n ↑f a = m ∧ m ≠ 0\n\ncase mpr\nα : Type u_1\nβ : Type ?u.366\nγ : Type ?u.369\nι : Type ?u.372\nM : Type u_2\nM' : Type ?u.378\nN : Type ?u.381\nP : Type ?u.384\nG : Type ?u.387\nH : Type ?u.390\nR : Type ?u.393\nS : Type ?u.396\ninst✝ : Zero M\na : α\nm : M\nf : α →₀ M\n⊢ ↑f a = m ∧ m ≠ 0 →\n ∃ a_2,\n ↑f a_2 ≠ 0 ∧\n ↑{ toFun := fun a => (a, ↑f a),\n inj' := (_ : ∀ (x x_1 : α), (fun a => (a, ↑f a)) x = (fun a => (a, ↑f a)) x_1 → x = x_1) }\n a_2 =\n (a, m)",
"state_before": "α : Type u_1\nβ : Type ?u.366\nγ : Type ?u.369\nι : Type ?u.372\nM : Type u_2\nM' : Type ?u.378\nN : Type ?u.381\nP : Type ?u.384\nG : Type ?u.387\nH : Type ?u.390\nR : Type ?u.393\nS : Type ?u.396\ninst✝ : Zero M\na : α\nm : M\nf : α →₀ M\n⊢ (∃ a_1,\n ↑f a_1 ≠ 0 ∧\n ↑{ toFun := fun a => (a, ↑f a),\n inj' := (_ : ∀ (x x_1 : α), (fun a => (a, ↑f a)) x = (fun a => (a, ↑f a)) x_1 → x = x_1) }\n a_1 =\n (a, m)) ↔\n ↑f a = m ∧ m ≠ 0",
"tactic": "constructor"
},
{
"state_after": "case mp.intro.intro.refl\nα : Type u_1\nβ : Type ?u.366\nγ : Type ?u.369\nι : Type ?u.372\nM : Type u_2\nM' : Type ?u.378\nN : Type ?u.381\nP : Type ?u.384\nG : Type ?u.387\nH : Type ?u.390\nR : Type ?u.393\nS : Type ?u.396\ninst✝ : Zero M\na : α\nf : α →₀ M\nha : ↑f a ≠ 0\n⊢ ↑f a = ↑f a ∧ ↑f a ≠ 0",
"state_before": "case mp\nα : Type u_1\nβ : Type ?u.366\nγ : Type ?u.369\nι : Type ?u.372\nM : Type u_2\nM' : Type ?u.378\nN : Type ?u.381\nP : Type ?u.384\nG : Type ?u.387\nH : Type ?u.390\nR : Type ?u.393\nS : Type ?u.396\ninst✝ : Zero M\na : α\nm : M\nf : α →₀ M\n⊢ (∃ a_1,\n ↑f a_1 ≠ 0 ∧\n ↑{ toFun := fun a => (a, ↑f a),\n inj' := (_ : ∀ (x x_1 : α), (fun a => (a, ↑f a)) x = (fun a => (a, ↑f a)) x_1 → x = x_1) }\n a_1 =\n (a, m)) →\n ↑f a = m ∧ m ≠ 0",
"tactic": "rintro ⟨b, ha, rfl, -⟩"
},
{
"state_after": "no goals",
"state_before": "case mp.intro.intro.refl\nα : Type u_1\nβ : Type ?u.366\nγ : Type ?u.369\nι : Type ?u.372\nM : Type u_2\nM' : Type ?u.378\nN : Type ?u.381\nP : Type ?u.384\nG : Type ?u.387\nH : Type ?u.390\nR : Type ?u.393\nS : Type ?u.396\ninst✝ : Zero M\na : α\nf : α →₀ M\nha : ↑f a ≠ 0\n⊢ ↑f a = ↑f a ∧ ↑f a ≠ 0",
"tactic": "exact ⟨rfl, ha⟩"
},
{
"state_after": "case mpr.intro\nα : Type u_1\nβ : Type ?u.366\nγ : Type ?u.369\nι : Type ?u.372\nM : Type u_2\nM' : Type ?u.378\nN : Type ?u.381\nP : Type ?u.384\nG : Type ?u.387\nH : Type ?u.390\nR : Type ?u.393\nS : Type ?u.396\ninst✝ : Zero M\na : α\nf : α →₀ M\nha : ↑f a ≠ 0\n⊢ ∃ a_1,\n ↑f a_1 ≠ 0 ∧\n ↑{ toFun := fun a => (a, ↑f a),\n inj' := (_ : ∀ (x x_1 : α), (fun a => (a, ↑f a)) x = (fun a => (a, ↑f a)) x_1 → x = x_1) }\n a_1 =\n (a, ↑f a)",
"state_before": "case mpr\nα : Type u_1\nβ : Type ?u.366\nγ : Type ?u.369\nι : Type ?u.372\nM : Type u_2\nM' : Type ?u.378\nN : Type ?u.381\nP : Type ?u.384\nG : Type ?u.387\nH : Type ?u.390\nR : Type ?u.393\nS : Type ?u.396\ninst✝ : Zero M\na : α\nm : M\nf : α →₀ M\n⊢ ↑f a = m ∧ m ≠ 0 →\n ∃ a_2,\n ↑f a_2 ≠ 0 ∧\n ↑{ toFun := fun a => (a, ↑f a),\n inj' := (_ : ∀ (x x_1 : α), (fun a => (a, ↑f a)) x = (fun a => (a, ↑f a)) x_1 → x = x_1) }\n a_2 =\n (a, m)",
"tactic": "rintro ⟨rfl, ha⟩"
},
{
"state_after": "no goals",
"state_before": "case mpr.intro\nα : Type u_1\nβ : Type ?u.366\nγ : Type ?u.369\nι : Type ?u.372\nM : Type u_2\nM' : Type ?u.378\nN : Type ?u.381\nP : Type ?u.384\nG : Type ?u.387\nH : Type ?u.390\nR : Type ?u.393\nS : Type ?u.396\ninst✝ : Zero M\na : α\nf : α →₀ M\nha : ↑f a ≠ 0\n⊢ ∃ a_1,\n ↑f a_1 ≠ 0 ∧\n ↑{ toFun := fun a => (a, ↑f a),\n inj' := (_ : ∀ (x x_1 : α), (fun a => (a, ↑f a)) x = (fun a => (a, ↑f a)) x_1 → x = x_1) }\n a_1 =\n (a, ↑f a)",
"tactic": "exact ⟨a, ha, rfl⟩"
}
] |
[
77,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
71,
1
] |
Mathlib/Data/Set/Function.lean
|
Set.range_restrictPreimage
|
[
{
"state_after": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.25203\nι : Sort ?u.25206\nπ : α → Type ?u.25211\ns s₁ s₂ : Set α\nt t₁ t₂ : Set β\np : Set γ\nf f₁ f₂ f₃ : α → β\ng g₁ g₂ : β → γ\nf' f₁' f₂' : β → α\ng' : γ → β\na : α\nb : β\n⊢ range (MapsTo.restrict f (f ⁻¹' t) t (_ : MapsTo f (f ⁻¹' t) t)) = Subtype.val ⁻¹' range f",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.25203\nι : Sort ?u.25206\nπ : α → Type ?u.25211\ns s₁ s₂ : Set α\nt t₁ t₂ : Set β\np : Set γ\nf f₁ f₂ f₃ : α → β\ng g₁ g₂ : β → γ\nf' f₁' f₂' : β → α\ng' : γ → β\na : α\nb : β\n⊢ range (restrictPreimage t f) = Subtype.val ⁻¹' range f",
"tactic": "delta Set.restrictPreimage"
},
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.25203\nι : Sort ?u.25206\nπ : α → Type ?u.25211\ns s₁ s₂ : Set α\nt t₁ t₂ : Set β\np : Set γ\nf f₁ f₂ f₃ : α → β\ng g₁ g₂ : β → γ\nf' f₁' f₂' : β → α\ng' : γ → β\na : α\nb : β\n⊢ range (MapsTo.restrict f (f ⁻¹' t) t (_ : MapsTo f (f ⁻¹' t) t)) = Subtype.val ⁻¹' range f",
"tactic": "rw [MapsTo.range_restrict, Set.image_preimage_eq_inter_range, Set.preimage_inter,\n Subtype.coe_preimage_self, Set.univ_inter]"
}
] |
[
562,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
559,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean
|
Real.tan_sub_pi
|
[] |
[
1015,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1014,
1
] |
Mathlib/Order/Filter/SmallSets.lean
|
Filter.monotone_smallSets
|
[] |
[
94,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
93,
1
] |
Mathlib/Order/ConditionallyCompleteLattice/Basic.lean
|
isGLB_ciInf_set
|
[] |
[
509,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
507,
1
] |
Mathlib/Algebra/Hom/Freiman.lean
|
FreimanHom.one_apply
|
[] |
[
318,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
317,
1
] |
Mathlib/Data/FunLike/Equiv.lean
|
EquivLike.surjective_comp
|
[] |
[
188,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
187,
1
] |
Mathlib/Combinatorics/SimpleGraph/Density.lean
|
SimpleGraph.edgeDensity_comm
|
[] |
[
430,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
429,
1
] |
Mathlib/Analysis/NormedSpace/Multilinear.lean
|
ContinuousMultilinearMap.mkPiField_eq_iff
|
[
{
"state_after": "𝕜 : Type u\nι : Type v\nι' : Type v'\nn : ℕ\nE : ι → Type wE\nE₁ : ι → Type wE₁\nE' : ι' → Type wE'\nEi : Fin (Nat.succ n) → Type wEi\nG : Type wG\nG' : Type wG'\ninst✝¹⁴ : Fintype ι\ninst✝¹³ : Fintype ι'\ninst✝¹² : NontriviallyNormedField 𝕜\ninst✝¹¹ : (i : ι) → NormedAddCommGroup (E i)\ninst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i)\ninst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i)\ninst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i)\ninst✝⁷ : (i : ι') → NormedAddCommGroup (E' i)\ninst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i)\ninst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i)\ninst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i)\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\ninst✝¹ : NormedAddCommGroup G'\ninst✝ : NormedSpace 𝕜 G'\nz₁ z₂ : G\n⊢ (ContinuousMultilinearMap.mkPiField 𝕜 ι z₁).toMultilinearMap =\n (ContinuousMultilinearMap.mkPiField 𝕜 ι z₂).toMultilinearMap ↔\n z₁ = z₂",
"state_before": "𝕜 : Type u\nι : Type v\nι' : Type v'\nn : ℕ\nE : ι → Type wE\nE₁ : ι → Type wE₁\nE' : ι' → Type wE'\nEi : Fin (Nat.succ n) → Type wEi\nG : Type wG\nG' : Type wG'\ninst✝¹⁴ : Fintype ι\ninst✝¹³ : Fintype ι'\ninst✝¹² : NontriviallyNormedField 𝕜\ninst✝¹¹ : (i : ι) → NormedAddCommGroup (E i)\ninst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i)\ninst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i)\ninst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i)\ninst✝⁷ : (i : ι') → NormedAddCommGroup (E' i)\ninst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i)\ninst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i)\ninst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i)\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\ninst✝¹ : NormedAddCommGroup G'\ninst✝ : NormedSpace 𝕜 G'\nz₁ z₂ : G\n⊢ ContinuousMultilinearMap.mkPiField 𝕜 ι z₁ = ContinuousMultilinearMap.mkPiField 𝕜 ι z₂ ↔ z₁ = z₂",
"tactic": "rw [← toMultilinearMap_injective.eq_iff]"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u\nι : Type v\nι' : Type v'\nn : ℕ\nE : ι → Type wE\nE₁ : ι → Type wE₁\nE' : ι' → Type wE'\nEi : Fin (Nat.succ n) → Type wEi\nG : Type wG\nG' : Type wG'\ninst✝¹⁴ : Fintype ι\ninst✝¹³ : Fintype ι'\ninst✝¹² : NontriviallyNormedField 𝕜\ninst✝¹¹ : (i : ι) → NormedAddCommGroup (E i)\ninst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i)\ninst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i)\ninst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i)\ninst✝⁷ : (i : ι') → NormedAddCommGroup (E' i)\ninst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i)\ninst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i)\ninst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i)\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\ninst✝¹ : NormedAddCommGroup G'\ninst✝ : NormedSpace 𝕜 G'\nz₁ z₂ : G\n⊢ (ContinuousMultilinearMap.mkPiField 𝕜 ι z₁).toMultilinearMap =\n (ContinuousMultilinearMap.mkPiField 𝕜 ι z₂).toMultilinearMap ↔\n z₁ = z₂",
"tactic": "exact MultilinearMap.mkPiRing_eq_iff"
}
] |
[
915,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
911,
1
] |
Mathlib/Analysis/Calculus/UniformLimitsDeriv.lean
|
hasDerivAt_of_tendstoUniformlyOnFilter
|
[
{
"state_after": "ι : Type u_1\nl : Filter ι\n𝕜 : Type u_2\ninst✝³ : IsROrC 𝕜\nG : Type u_3\ninst✝² : NormedAddCommGroup G\ninst✝¹ : NormedSpace 𝕜 G\nf : ι → 𝕜 → G\ng : 𝕜 → G\nf' : ι → 𝕜 → G\ng' : 𝕜 → G\nx : 𝕜\ninst✝ : NeBot l\nhf' : TendstoUniformlyOnFilter f' g' l (𝓝 x)\nhf : ∀ᶠ (n : ι × 𝕜) in l ×ˢ 𝓝 x, HasDerivAt (f n.fst) (f' n.fst n.snd) n.snd\nhfg : ∀ᶠ (y : 𝕜) in 𝓝 x, Tendsto (fun n => f n y) l (𝓝 (g y))\nF' : ι → 𝕜 → 𝕜 →L[𝕜] G := fun n z => ContinuousLinearMap.smulRight 1 (f' n z)\n⊢ HasDerivAt g (g' x) x",
"state_before": "ι : Type u_1\nl : Filter ι\n𝕜 : Type u_2\ninst✝³ : IsROrC 𝕜\nG : Type u_3\ninst✝² : NormedAddCommGroup G\ninst✝¹ : NormedSpace 𝕜 G\nf : ι → 𝕜 → G\ng : 𝕜 → G\nf' : ι → 𝕜 → G\ng' : 𝕜 → G\nx : 𝕜\ninst✝ : NeBot l\nhf' : TendstoUniformlyOnFilter f' g' l (𝓝 x)\nhf : ∀ᶠ (n : ι × 𝕜) in l ×ˢ 𝓝 x, HasDerivAt (f n.fst) (f' n.fst n.snd) n.snd\nhfg : ∀ᶠ (y : 𝕜) in 𝓝 x, Tendsto (fun n => f n y) l (𝓝 (g y))\n⊢ HasDerivAt g (g' x) x",
"tactic": "let F' n z := (1 : 𝕜 →L[𝕜] 𝕜).smulRight (f' n z)"
},
{
"state_after": "ι : Type u_1\nl : Filter ι\n𝕜 : Type u_2\ninst✝³ : IsROrC 𝕜\nG : Type u_3\ninst✝² : NormedAddCommGroup G\ninst✝¹ : NormedSpace 𝕜 G\nf : ι → 𝕜 → G\ng : 𝕜 → G\nf' : ι → 𝕜 → G\ng' : 𝕜 → G\nx : 𝕜\ninst✝ : NeBot l\nhf' : TendstoUniformlyOnFilter f' g' l (𝓝 x)\nhf : ∀ᶠ (n : ι × 𝕜) in l ×ˢ 𝓝 x, HasDerivAt (f n.fst) (f' n.fst n.snd) n.snd\nhfg : ∀ᶠ (y : 𝕜) in 𝓝 x, Tendsto (fun n => f n y) l (𝓝 (g y))\nF' : ι → 𝕜 → 𝕜 →L[𝕜] G := fun n z => ContinuousLinearMap.smulRight 1 (f' n z)\nG' : 𝕜 → 𝕜 →L[𝕜] G := fun z => ContinuousLinearMap.smulRight 1 (g' z)\n⊢ HasDerivAt g (g' x) x",
"state_before": "ι : Type u_1\nl : Filter ι\n𝕜 : Type u_2\ninst✝³ : IsROrC 𝕜\nG : Type u_3\ninst✝² : NormedAddCommGroup G\ninst✝¹ : NormedSpace 𝕜 G\nf : ι → 𝕜 → G\ng : 𝕜 → G\nf' : ι → 𝕜 → G\ng' : 𝕜 → G\nx : 𝕜\ninst✝ : NeBot l\nhf' : TendstoUniformlyOnFilter f' g' l (𝓝 x)\nhf : ∀ᶠ (n : ι × 𝕜) in l ×ˢ 𝓝 x, HasDerivAt (f n.fst) (f' n.fst n.snd) n.snd\nhfg : ∀ᶠ (y : 𝕜) in 𝓝 x, Tendsto (fun n => f n y) l (𝓝 (g y))\nF' : ι → 𝕜 → 𝕜 →L[𝕜] G := fun n z => ContinuousLinearMap.smulRight 1 (f' n z)\n⊢ HasDerivAt g (g' x) x",
"tactic": "let G' z := (1 : 𝕜 →L[𝕜] 𝕜).smulRight (g' z)"
},
{
"state_after": "ι : Type u_1\nl : Filter ι\n𝕜 : Type u_2\ninst✝³ : IsROrC 𝕜\nG : Type u_3\ninst✝² : NormedAddCommGroup G\ninst✝¹ : NormedSpace 𝕜 G\nf : ι → 𝕜 → G\ng : 𝕜 → G\nf' : ι → 𝕜 → G\ng' : 𝕜 → G\nx : 𝕜\ninst✝ : NeBot l\nhf' : TendstoUniformlyOnFilter f' g' l (𝓝 x)\nhfg : ∀ᶠ (y : 𝕜) in 𝓝 x, Tendsto (fun n => f n y) l (𝓝 (g y))\nF' : ι → 𝕜 → 𝕜 →L[𝕜] G := fun n z => ContinuousLinearMap.smulRight 1 (f' n z)\nG' : 𝕜 → 𝕜 →L[𝕜] G := fun z => ContinuousLinearMap.smulRight 1 (g' z)\nhf : ∀ᶠ (n : ι × 𝕜) in l ×ˢ 𝓝 x, HasFDerivAt (f n.fst) (ContinuousLinearMap.smulRight 1 (f' n.fst n.snd)) n.snd\n⊢ HasFDerivAt g (ContinuousLinearMap.smulRight 1 (g' x)) x",
"state_before": "ι : Type u_1\nl : Filter ι\n𝕜 : Type u_2\ninst✝³ : IsROrC 𝕜\nG : Type u_3\ninst✝² : NormedAddCommGroup G\ninst✝¹ : NormedSpace 𝕜 G\nf : ι → 𝕜 → G\ng : 𝕜 → G\nf' : ι → 𝕜 → G\ng' : 𝕜 → G\nx : 𝕜\ninst✝ : NeBot l\nhf' : TendstoUniformlyOnFilter f' g' l (𝓝 x)\nhf : ∀ᶠ (n : ι × 𝕜) in l ×ˢ 𝓝 x, HasDerivAt (f n.fst) (f' n.fst n.snd) n.snd\nhfg : ∀ᶠ (y : 𝕜) in 𝓝 x, Tendsto (fun n => f n y) l (𝓝 (g y))\nF' : ι → 𝕜 → 𝕜 →L[𝕜] G := fun n z => ContinuousLinearMap.smulRight 1 (f' n z)\nG' : 𝕜 → 𝕜 →L[𝕜] G := fun z => ContinuousLinearMap.smulRight 1 (g' z)\n⊢ HasDerivAt g (g' x) x",
"tactic": "simp_rw [hasDerivAt_iff_hasFDerivAt] at hf ⊢"
},
{
"state_after": "ι : Type u_1\nl : Filter ι\n𝕜 : Type u_2\ninst✝³ : IsROrC 𝕜\nG : Type u_3\ninst✝² : NormedAddCommGroup G\ninst✝¹ : NormedSpace 𝕜 G\nf : ι → 𝕜 → G\ng : 𝕜 → G\nf' : ι → 𝕜 → G\ng' : 𝕜 → G\nx : 𝕜\ninst✝ : NeBot l\nhf'✝ : TendstoUniformlyOnFilter f' g' l (𝓝 x)\nhfg : ∀ᶠ (y : 𝕜) in 𝓝 x, Tendsto (fun n => f n y) l (𝓝 (g y))\nF' : ι → 𝕜 → 𝕜 →L[𝕜] G := fun n z => ContinuousLinearMap.smulRight 1 (f' n z)\nG' : 𝕜 → 𝕜 →L[𝕜] G := fun z => ContinuousLinearMap.smulRight 1 (g' z)\nhf : ∀ᶠ (n : ι × 𝕜) in l ×ˢ 𝓝 x, HasFDerivAt (f n.fst) (ContinuousLinearMap.smulRight 1 (f' n.fst n.snd)) n.snd\nhf' : TendstoUniformlyOnFilter F' G' l (𝓝 x)\n⊢ HasFDerivAt g (ContinuousLinearMap.smulRight 1 (g' x)) x",
"state_before": "ι : Type u_1\nl : Filter ι\n𝕜 : Type u_2\ninst✝³ : IsROrC 𝕜\nG : Type u_3\ninst✝² : NormedAddCommGroup G\ninst✝¹ : NormedSpace 𝕜 G\nf : ι → 𝕜 → G\ng : 𝕜 → G\nf' : ι → 𝕜 → G\ng' : 𝕜 → G\nx : 𝕜\ninst✝ : NeBot l\nhf' : TendstoUniformlyOnFilter f' g' l (𝓝 x)\nhfg : ∀ᶠ (y : 𝕜) in 𝓝 x, Tendsto (fun n => f n y) l (𝓝 (g y))\nF' : ι → 𝕜 → 𝕜 →L[𝕜] G := fun n z => ContinuousLinearMap.smulRight 1 (f' n z)\nG' : 𝕜 → 𝕜 →L[𝕜] G := fun z => ContinuousLinearMap.smulRight 1 (g' z)\nhf : ∀ᶠ (n : ι × 𝕜) in l ×ˢ 𝓝 x, HasFDerivAt (f n.fst) (ContinuousLinearMap.smulRight 1 (f' n.fst n.snd)) n.snd\n⊢ HasFDerivAt g (ContinuousLinearMap.smulRight 1 (g' x)) x",
"tactic": "have hf' : TendstoUniformlyOnFilter F' G' l (𝓝 x) := by\n rw [Metric.tendstoUniformlyOnFilter_iff] at hf' ⊢\n intro ε hε\n obtain ⟨q, hq, hq'⟩ := exists_between hε.lt\n apply (hf' q hq).mono\n intro n hn\n refine' lt_of_le_of_lt _ hq'\n simp only [dist_eq_norm] at hn ⊢\n refine' ContinuousLinearMap.op_norm_le_bound _ hq.le _\n intro z\n simp only [ContinuousLinearMap.coe_sub', Pi.sub_apply, ContinuousLinearMap.smulRight_apply,\n ContinuousLinearMap.one_apply]\n rw [← smul_sub, norm_smul, mul_comm]\n gcongr"
},
{
"state_after": "no goals",
"state_before": "ι : Type u_1\nl : Filter ι\n𝕜 : Type u_2\ninst✝³ : IsROrC 𝕜\nG : Type u_3\ninst✝² : NormedAddCommGroup G\ninst✝¹ : NormedSpace 𝕜 G\nf : ι → 𝕜 → G\ng : 𝕜 → G\nf' : ι → 𝕜 → G\ng' : 𝕜 → G\nx : 𝕜\ninst✝ : NeBot l\nhf'✝ : TendstoUniformlyOnFilter f' g' l (𝓝 x)\nhfg : ∀ᶠ (y : 𝕜) in 𝓝 x, Tendsto (fun n => f n y) l (𝓝 (g y))\nF' : ι → 𝕜 → 𝕜 →L[𝕜] G := fun n z => ContinuousLinearMap.smulRight 1 (f' n z)\nG' : 𝕜 → 𝕜 →L[𝕜] G := fun z => ContinuousLinearMap.smulRight 1 (g' z)\nhf : ∀ᶠ (n : ι × 𝕜) in l ×ˢ 𝓝 x, HasFDerivAt (f n.fst) (ContinuousLinearMap.smulRight 1 (f' n.fst n.snd)) n.snd\nhf' : TendstoUniformlyOnFilter F' G' l (𝓝 x)\n⊢ HasFDerivAt g (ContinuousLinearMap.smulRight 1 (g' x)) x",
"tactic": "exact hasFDerivAt_of_tendstoUniformlyOnFilter hf' hf hfg"
},
{
"state_after": "ι : Type u_1\nl : Filter ι\n𝕜 : Type u_2\ninst✝³ : IsROrC 𝕜\nG : Type u_3\ninst✝² : NormedAddCommGroup G\ninst✝¹ : NormedSpace 𝕜 G\nf : ι → 𝕜 → G\ng : 𝕜 → G\nf' : ι → 𝕜 → G\ng' : 𝕜 → G\nx : 𝕜\ninst✝ : NeBot l\nhf' : ∀ (ε : ℝ), ε > 0 → ∀ᶠ (n : ι × 𝕜) in l ×ˢ 𝓝 x, dist (g' n.snd) (f' n.fst n.snd) < ε\nhfg : ∀ᶠ (y : 𝕜) in 𝓝 x, Tendsto (fun n => f n y) l (𝓝 (g y))\nF' : ι → 𝕜 → 𝕜 →L[𝕜] G := fun n z => ContinuousLinearMap.smulRight 1 (f' n z)\nG' : 𝕜 → 𝕜 →L[𝕜] G := fun z => ContinuousLinearMap.smulRight 1 (g' z)\nhf : ∀ᶠ (n : ι × 𝕜) in l ×ˢ 𝓝 x, HasFDerivAt (f n.fst) (ContinuousLinearMap.smulRight 1 (f' n.fst n.snd)) n.snd\n⊢ ∀ (ε : ℝ), ε > 0 → ∀ᶠ (n : ι × 𝕜) in l ×ˢ 𝓝 x, dist (G' n.snd) (F' n.fst n.snd) < ε",
"state_before": "ι : Type u_1\nl : Filter ι\n𝕜 : Type u_2\ninst✝³ : IsROrC 𝕜\nG : Type u_3\ninst✝² : NormedAddCommGroup G\ninst✝¹ : NormedSpace 𝕜 G\nf : ι → 𝕜 → G\ng : 𝕜 → G\nf' : ι → 𝕜 → G\ng' : 𝕜 → G\nx : 𝕜\ninst✝ : NeBot l\nhf' : TendstoUniformlyOnFilter f' g' l (𝓝 x)\nhfg : ∀ᶠ (y : 𝕜) in 𝓝 x, Tendsto (fun n => f n y) l (𝓝 (g y))\nF' : ι → 𝕜 → 𝕜 →L[𝕜] G := fun n z => ContinuousLinearMap.smulRight 1 (f' n z)\nG' : 𝕜 → 𝕜 →L[𝕜] G := fun z => ContinuousLinearMap.smulRight 1 (g' z)\nhf : ∀ᶠ (n : ι × 𝕜) in l ×ˢ 𝓝 x, HasFDerivAt (f n.fst) (ContinuousLinearMap.smulRight 1 (f' n.fst n.snd)) n.snd\n⊢ TendstoUniformlyOnFilter F' G' l (𝓝 x)",
"tactic": "rw [Metric.tendstoUniformlyOnFilter_iff] at hf' ⊢"
},
{
"state_after": "ι : Type u_1\nl : Filter ι\n𝕜 : Type u_2\ninst✝³ : IsROrC 𝕜\nG : Type u_3\ninst✝² : NormedAddCommGroup G\ninst✝¹ : NormedSpace 𝕜 G\nf : ι → 𝕜 → G\ng : 𝕜 → G\nf' : ι → 𝕜 → G\ng' : 𝕜 → G\nx : 𝕜\ninst✝ : NeBot l\nhf' : ∀ (ε : ℝ), ε > 0 → ∀ᶠ (n : ι × 𝕜) in l ×ˢ 𝓝 x, dist (g' n.snd) (f' n.fst n.snd) < ε\nhfg : ∀ᶠ (y : 𝕜) in 𝓝 x, Tendsto (fun n => f n y) l (𝓝 (g y))\nF' : ι → 𝕜 → 𝕜 →L[𝕜] G := fun n z => ContinuousLinearMap.smulRight 1 (f' n z)\nG' : 𝕜 → 𝕜 →L[𝕜] G := fun z => ContinuousLinearMap.smulRight 1 (g' z)\nhf : ∀ᶠ (n : ι × 𝕜) in l ×ˢ 𝓝 x, HasFDerivAt (f n.fst) (ContinuousLinearMap.smulRight 1 (f' n.fst n.snd)) n.snd\nε : ℝ\nhε : ε > 0\n⊢ ∀ᶠ (n : ι × 𝕜) in l ×ˢ 𝓝 x, dist (G' n.snd) (F' n.fst n.snd) < ε",
"state_before": "ι : Type u_1\nl : Filter ι\n𝕜 : Type u_2\ninst✝³ : IsROrC 𝕜\nG : Type u_3\ninst✝² : NormedAddCommGroup G\ninst✝¹ : NormedSpace 𝕜 G\nf : ι → 𝕜 → G\ng : 𝕜 → G\nf' : ι → 𝕜 → G\ng' : 𝕜 → G\nx : 𝕜\ninst✝ : NeBot l\nhf' : ∀ (ε : ℝ), ε > 0 → ∀ᶠ (n : ι × 𝕜) in l ×ˢ 𝓝 x, dist (g' n.snd) (f' n.fst n.snd) < ε\nhfg : ∀ᶠ (y : 𝕜) in 𝓝 x, Tendsto (fun n => f n y) l (𝓝 (g y))\nF' : ι → 𝕜 → 𝕜 →L[𝕜] G := fun n z => ContinuousLinearMap.smulRight 1 (f' n z)\nG' : 𝕜 → 𝕜 →L[𝕜] G := fun z => ContinuousLinearMap.smulRight 1 (g' z)\nhf : ∀ᶠ (n : ι × 𝕜) in l ×ˢ 𝓝 x, HasFDerivAt (f n.fst) (ContinuousLinearMap.smulRight 1 (f' n.fst n.snd)) n.snd\n⊢ ∀ (ε : ℝ), ε > 0 → ∀ᶠ (n : ι × 𝕜) in l ×ˢ 𝓝 x, dist (G' n.snd) (F' n.fst n.snd) < ε",
"tactic": "intro ε hε"
},
{
"state_after": "case intro.intro\nι : Type u_1\nl : Filter ι\n𝕜 : Type u_2\ninst✝³ : IsROrC 𝕜\nG : Type u_3\ninst✝² : NormedAddCommGroup G\ninst✝¹ : NormedSpace 𝕜 G\nf : ι → 𝕜 → G\ng : 𝕜 → G\nf' : ι → 𝕜 → G\ng' : 𝕜 → G\nx : 𝕜\ninst✝ : NeBot l\nhf' : ∀ (ε : ℝ), ε > 0 → ∀ᶠ (n : ι × 𝕜) in l ×ˢ 𝓝 x, dist (g' n.snd) (f' n.fst n.snd) < ε\nhfg : ∀ᶠ (y : 𝕜) in 𝓝 x, Tendsto (fun n => f n y) l (𝓝 (g y))\nF' : ι → 𝕜 → 𝕜 →L[𝕜] G := fun n z => ContinuousLinearMap.smulRight 1 (f' n z)\nG' : 𝕜 → 𝕜 →L[𝕜] G := fun z => ContinuousLinearMap.smulRight 1 (g' z)\nhf : ∀ᶠ (n : ι × 𝕜) in l ×ˢ 𝓝 x, HasFDerivAt (f n.fst) (ContinuousLinearMap.smulRight 1 (f' n.fst n.snd)) n.snd\nε : ℝ\nhε : ε > 0\nq : ℝ\nhq : 0 < q\nhq' : q < ε\n⊢ ∀ᶠ (n : ι × 𝕜) in l ×ˢ 𝓝 x, dist (G' n.snd) (F' n.fst n.snd) < ε",
"state_before": "ι : Type u_1\nl : Filter ι\n𝕜 : Type u_2\ninst✝³ : IsROrC 𝕜\nG : Type u_3\ninst✝² : NormedAddCommGroup G\ninst✝¹ : NormedSpace 𝕜 G\nf : ι → 𝕜 → G\ng : 𝕜 → G\nf' : ι → 𝕜 → G\ng' : 𝕜 → G\nx : 𝕜\ninst✝ : NeBot l\nhf' : ∀ (ε : ℝ), ε > 0 → ∀ᶠ (n : ι × 𝕜) in l ×ˢ 𝓝 x, dist (g' n.snd) (f' n.fst n.snd) < ε\nhfg : ∀ᶠ (y : 𝕜) in 𝓝 x, Tendsto (fun n => f n y) l (𝓝 (g y))\nF' : ι → 𝕜 → 𝕜 →L[𝕜] G := fun n z => ContinuousLinearMap.smulRight 1 (f' n z)\nG' : 𝕜 → 𝕜 →L[𝕜] G := fun z => ContinuousLinearMap.smulRight 1 (g' z)\nhf : ∀ᶠ (n : ι × 𝕜) in l ×ˢ 𝓝 x, HasFDerivAt (f n.fst) (ContinuousLinearMap.smulRight 1 (f' n.fst n.snd)) n.snd\nε : ℝ\nhε : ε > 0\n⊢ ∀ᶠ (n : ι × 𝕜) in l ×ˢ 𝓝 x, dist (G' n.snd) (F' n.fst n.snd) < ε",
"tactic": "obtain ⟨q, hq, hq'⟩ := exists_between hε.lt"
},
{
"state_after": "case intro.intro\nι : Type u_1\nl : Filter ι\n𝕜 : Type u_2\ninst✝³ : IsROrC 𝕜\nG : Type u_3\ninst✝² : NormedAddCommGroup G\ninst✝¹ : NormedSpace 𝕜 G\nf : ι → 𝕜 → G\ng : 𝕜 → G\nf' : ι → 𝕜 → G\ng' : 𝕜 → G\nx : 𝕜\ninst✝ : NeBot l\nhf' : ∀ (ε : ℝ), ε > 0 → ∀ᶠ (n : ι × 𝕜) in l ×ˢ 𝓝 x, dist (g' n.snd) (f' n.fst n.snd) < ε\nhfg : ∀ᶠ (y : 𝕜) in 𝓝 x, Tendsto (fun n => f n y) l (𝓝 (g y))\nF' : ι → 𝕜 → 𝕜 →L[𝕜] G := fun n z => ContinuousLinearMap.smulRight 1 (f' n z)\nG' : 𝕜 → 𝕜 →L[𝕜] G := fun z => ContinuousLinearMap.smulRight 1 (g' z)\nhf : ∀ᶠ (n : ι × 𝕜) in l ×ˢ 𝓝 x, HasFDerivAt (f n.fst) (ContinuousLinearMap.smulRight 1 (f' n.fst n.snd)) n.snd\nε : ℝ\nhε : ε > 0\nq : ℝ\nhq : 0 < q\nhq' : q < ε\n⊢ ∀ (x : ι × 𝕜), dist (g' x.snd) (f' x.fst x.snd) < q → dist (G' x.snd) (F' x.fst x.snd) < ε",
"state_before": "case intro.intro\nι : Type u_1\nl : Filter ι\n𝕜 : Type u_2\ninst✝³ : IsROrC 𝕜\nG : Type u_3\ninst✝² : NormedAddCommGroup G\ninst✝¹ : NormedSpace 𝕜 G\nf : ι → 𝕜 → G\ng : 𝕜 → G\nf' : ι → 𝕜 → G\ng' : 𝕜 → G\nx : 𝕜\ninst✝ : NeBot l\nhf' : ∀ (ε : ℝ), ε > 0 → ∀ᶠ (n : ι × 𝕜) in l ×ˢ 𝓝 x, dist (g' n.snd) (f' n.fst n.snd) < ε\nhfg : ∀ᶠ (y : 𝕜) in 𝓝 x, Tendsto (fun n => f n y) l (𝓝 (g y))\nF' : ι → 𝕜 → 𝕜 →L[𝕜] G := fun n z => ContinuousLinearMap.smulRight 1 (f' n z)\nG' : 𝕜 → 𝕜 →L[𝕜] G := fun z => ContinuousLinearMap.smulRight 1 (g' z)\nhf : ∀ᶠ (n : ι × 𝕜) in l ×ˢ 𝓝 x, HasFDerivAt (f n.fst) (ContinuousLinearMap.smulRight 1 (f' n.fst n.snd)) n.snd\nε : ℝ\nhε : ε > 0\nq : ℝ\nhq : 0 < q\nhq' : q < ε\n⊢ ∀ᶠ (n : ι × 𝕜) in l ×ˢ 𝓝 x, dist (G' n.snd) (F' n.fst n.snd) < ε",
"tactic": "apply (hf' q hq).mono"
},
{
"state_after": "case intro.intro\nι : Type u_1\nl : Filter ι\n𝕜 : Type u_2\ninst✝³ : IsROrC 𝕜\nG : Type u_3\ninst✝² : NormedAddCommGroup G\ninst✝¹ : NormedSpace 𝕜 G\nf : ι → 𝕜 → G\ng : 𝕜 → G\nf' : ι → 𝕜 → G\ng' : 𝕜 → G\nx : 𝕜\ninst✝ : NeBot l\nhf' : ∀ (ε : ℝ), ε > 0 → ∀ᶠ (n : ι × 𝕜) in l ×ˢ 𝓝 x, dist (g' n.snd) (f' n.fst n.snd) < ε\nhfg : ∀ᶠ (y : 𝕜) in 𝓝 x, Tendsto (fun n => f n y) l (𝓝 (g y))\nF' : ι → 𝕜 → 𝕜 →L[𝕜] G := fun n z => ContinuousLinearMap.smulRight 1 (f' n z)\nG' : 𝕜 → 𝕜 →L[𝕜] G := fun z => ContinuousLinearMap.smulRight 1 (g' z)\nhf : ∀ᶠ (n : ι × 𝕜) in l ×ˢ 𝓝 x, HasFDerivAt (f n.fst) (ContinuousLinearMap.smulRight 1 (f' n.fst n.snd)) n.snd\nε : ℝ\nhε : ε > 0\nq : ℝ\nhq : 0 < q\nhq' : q < ε\nn : ι × 𝕜\nhn : dist (g' n.snd) (f' n.fst n.snd) < q\n⊢ dist (G' n.snd) (F' n.fst n.snd) < ε",
"state_before": "case intro.intro\nι : Type u_1\nl : Filter ι\n𝕜 : Type u_2\ninst✝³ : IsROrC 𝕜\nG : Type u_3\ninst✝² : NormedAddCommGroup G\ninst✝¹ : NormedSpace 𝕜 G\nf : ι → 𝕜 → G\ng : 𝕜 → G\nf' : ι → 𝕜 → G\ng' : 𝕜 → G\nx : 𝕜\ninst✝ : NeBot l\nhf' : ∀ (ε : ℝ), ε > 0 → ∀ᶠ (n : ι × 𝕜) in l ×ˢ 𝓝 x, dist (g' n.snd) (f' n.fst n.snd) < ε\nhfg : ∀ᶠ (y : 𝕜) in 𝓝 x, Tendsto (fun n => f n y) l (𝓝 (g y))\nF' : ι → 𝕜 → 𝕜 →L[𝕜] G := fun n z => ContinuousLinearMap.smulRight 1 (f' n z)\nG' : 𝕜 → 𝕜 →L[𝕜] G := fun z => ContinuousLinearMap.smulRight 1 (g' z)\nhf : ∀ᶠ (n : ι × 𝕜) in l ×ˢ 𝓝 x, HasFDerivAt (f n.fst) (ContinuousLinearMap.smulRight 1 (f' n.fst n.snd)) n.snd\nε : ℝ\nhε : ε > 0\nq : ℝ\nhq : 0 < q\nhq' : q < ε\n⊢ ∀ (x : ι × 𝕜), dist (g' x.snd) (f' x.fst x.snd) < q → dist (G' x.snd) (F' x.fst x.snd) < ε",
"tactic": "intro n hn"
},
{
"state_after": "case intro.intro\nι : Type u_1\nl : Filter ι\n𝕜 : Type u_2\ninst✝³ : IsROrC 𝕜\nG : Type u_3\ninst✝² : NormedAddCommGroup G\ninst✝¹ : NormedSpace 𝕜 G\nf : ι → 𝕜 → G\ng : 𝕜 → G\nf' : ι → 𝕜 → G\ng' : 𝕜 → G\nx : 𝕜\ninst✝ : NeBot l\nhf' : ∀ (ε : ℝ), ε > 0 → ∀ᶠ (n : ι × 𝕜) in l ×ˢ 𝓝 x, dist (g' n.snd) (f' n.fst n.snd) < ε\nhfg : ∀ᶠ (y : 𝕜) in 𝓝 x, Tendsto (fun n => f n y) l (𝓝 (g y))\nF' : ι → 𝕜 → 𝕜 →L[𝕜] G := fun n z => ContinuousLinearMap.smulRight 1 (f' n z)\nG' : 𝕜 → 𝕜 →L[𝕜] G := fun z => ContinuousLinearMap.smulRight 1 (g' z)\nhf : ∀ᶠ (n : ι × 𝕜) in l ×ˢ 𝓝 x, HasFDerivAt (f n.fst) (ContinuousLinearMap.smulRight 1 (f' n.fst n.snd)) n.snd\nε : ℝ\nhε : ε > 0\nq : ℝ\nhq : 0 < q\nhq' : q < ε\nn : ι × 𝕜\nhn : dist (g' n.snd) (f' n.fst n.snd) < q\n⊢ dist (G' n.snd) (F' n.fst n.snd) ≤ q",
"state_before": "case intro.intro\nι : Type u_1\nl : Filter ι\n𝕜 : Type u_2\ninst✝³ : IsROrC 𝕜\nG : Type u_3\ninst✝² : NormedAddCommGroup G\ninst✝¹ : NormedSpace 𝕜 G\nf : ι → 𝕜 → G\ng : 𝕜 → G\nf' : ι → 𝕜 → G\ng' : 𝕜 → G\nx : 𝕜\ninst✝ : NeBot l\nhf' : ∀ (ε : ℝ), ε > 0 → ∀ᶠ (n : ι × 𝕜) in l ×ˢ 𝓝 x, dist (g' n.snd) (f' n.fst n.snd) < ε\nhfg : ∀ᶠ (y : 𝕜) in 𝓝 x, Tendsto (fun n => f n y) l (𝓝 (g y))\nF' : ι → 𝕜 → 𝕜 →L[𝕜] G := fun n z => ContinuousLinearMap.smulRight 1 (f' n z)\nG' : 𝕜 → 𝕜 →L[𝕜] G := fun z => ContinuousLinearMap.smulRight 1 (g' z)\nhf : ∀ᶠ (n : ι × 𝕜) in l ×ˢ 𝓝 x, HasFDerivAt (f n.fst) (ContinuousLinearMap.smulRight 1 (f' n.fst n.snd)) n.snd\nε : ℝ\nhε : ε > 0\nq : ℝ\nhq : 0 < q\nhq' : q < ε\nn : ι × 𝕜\nhn : dist (g' n.snd) (f' n.fst n.snd) < q\n⊢ dist (G' n.snd) (F' n.fst n.snd) < ε",
"tactic": "refine' lt_of_le_of_lt _ hq'"
},
{
"state_after": "case intro.intro\nι : Type u_1\nl : Filter ι\n𝕜 : Type u_2\ninst✝³ : IsROrC 𝕜\nG : Type u_3\ninst✝² : NormedAddCommGroup G\ninst✝¹ : NormedSpace 𝕜 G\nf : ι → 𝕜 → G\ng : 𝕜 → G\nf' : ι → 𝕜 → G\ng' : 𝕜 → G\nx : 𝕜\ninst✝ : NeBot l\nhf' : ∀ (ε : ℝ), ε > 0 → ∀ᶠ (n : ι × 𝕜) in l ×ˢ 𝓝 x, dist (g' n.snd) (f' n.fst n.snd) < ε\nhfg : ∀ᶠ (y : 𝕜) in 𝓝 x, Tendsto (fun n => f n y) l (𝓝 (g y))\nF' : ι → 𝕜 → 𝕜 →L[𝕜] G := fun n z => ContinuousLinearMap.smulRight 1 (f' n z)\nG' : 𝕜 → 𝕜 →L[𝕜] G := fun z => ContinuousLinearMap.smulRight 1 (g' z)\nhf : ∀ᶠ (n : ι × 𝕜) in l ×ˢ 𝓝 x, HasFDerivAt (f n.fst) (ContinuousLinearMap.smulRight 1 (f' n.fst n.snd)) n.snd\nε : ℝ\nhε : ε > 0\nq : ℝ\nhq : 0 < q\nhq' : q < ε\nn : ι × 𝕜\nhn : ‖g' n.snd - f' n.fst n.snd‖ < q\n⊢ ‖ContinuousLinearMap.smulRight 1 (g' n.snd) - ContinuousLinearMap.smulRight 1 (f' n.fst n.snd)‖ ≤ q",
"state_before": "case intro.intro\nι : Type u_1\nl : Filter ι\n𝕜 : Type u_2\ninst✝³ : IsROrC 𝕜\nG : Type u_3\ninst✝² : NormedAddCommGroup G\ninst✝¹ : NormedSpace 𝕜 G\nf : ι → 𝕜 → G\ng : 𝕜 → G\nf' : ι → 𝕜 → G\ng' : 𝕜 → G\nx : 𝕜\ninst✝ : NeBot l\nhf' : ∀ (ε : ℝ), ε > 0 → ∀ᶠ (n : ι × 𝕜) in l ×ˢ 𝓝 x, dist (g' n.snd) (f' n.fst n.snd) < ε\nhfg : ∀ᶠ (y : 𝕜) in 𝓝 x, Tendsto (fun n => f n y) l (𝓝 (g y))\nF' : ι → 𝕜 → 𝕜 →L[𝕜] G := fun n z => ContinuousLinearMap.smulRight 1 (f' n z)\nG' : 𝕜 → 𝕜 →L[𝕜] G := fun z => ContinuousLinearMap.smulRight 1 (g' z)\nhf : ∀ᶠ (n : ι × 𝕜) in l ×ˢ 𝓝 x, HasFDerivAt (f n.fst) (ContinuousLinearMap.smulRight 1 (f' n.fst n.snd)) n.snd\nε : ℝ\nhε : ε > 0\nq : ℝ\nhq : 0 < q\nhq' : q < ε\nn : ι × 𝕜\nhn : dist (g' n.snd) (f' n.fst n.snd) < q\n⊢ dist (G' n.snd) (F' n.fst n.snd) ≤ q",
"tactic": "simp only [dist_eq_norm] at hn ⊢"
},
{
"state_after": "case intro.intro\nι : Type u_1\nl : Filter ι\n𝕜 : Type u_2\ninst✝³ : IsROrC 𝕜\nG : Type u_3\ninst✝² : NormedAddCommGroup G\ninst✝¹ : NormedSpace 𝕜 G\nf : ι → 𝕜 → G\ng : 𝕜 → G\nf' : ι → 𝕜 → G\ng' : 𝕜 → G\nx : 𝕜\ninst✝ : NeBot l\nhf' : ∀ (ε : ℝ), ε > 0 → ∀ᶠ (n : ι × 𝕜) in l ×ˢ 𝓝 x, dist (g' n.snd) (f' n.fst n.snd) < ε\nhfg : ∀ᶠ (y : 𝕜) in 𝓝 x, Tendsto (fun n => f n y) l (𝓝 (g y))\nF' : ι → 𝕜 → 𝕜 →L[𝕜] G := fun n z => ContinuousLinearMap.smulRight 1 (f' n z)\nG' : 𝕜 → 𝕜 →L[𝕜] G := fun z => ContinuousLinearMap.smulRight 1 (g' z)\nhf : ∀ᶠ (n : ι × 𝕜) in l ×ˢ 𝓝 x, HasFDerivAt (f n.fst) (ContinuousLinearMap.smulRight 1 (f' n.fst n.snd)) n.snd\nε : ℝ\nhε : ε > 0\nq : ℝ\nhq : 0 < q\nhq' : q < ε\nn : ι × 𝕜\nhn : ‖g' n.snd - f' n.fst n.snd‖ < q\n⊢ ∀ (x : 𝕜),\n ‖↑(ContinuousLinearMap.smulRight 1 (g' n.snd) - ContinuousLinearMap.smulRight 1 (f' n.fst n.snd)) x‖ ≤ q * ‖x‖",
"state_before": "case intro.intro\nι : Type u_1\nl : Filter ι\n𝕜 : Type u_2\ninst✝³ : IsROrC 𝕜\nG : Type u_3\ninst✝² : NormedAddCommGroup G\ninst✝¹ : NormedSpace 𝕜 G\nf : ι → 𝕜 → G\ng : 𝕜 → G\nf' : ι → 𝕜 → G\ng' : 𝕜 → G\nx : 𝕜\ninst✝ : NeBot l\nhf' : ∀ (ε : ℝ), ε > 0 → ∀ᶠ (n : ι × 𝕜) in l ×ˢ 𝓝 x, dist (g' n.snd) (f' n.fst n.snd) < ε\nhfg : ∀ᶠ (y : 𝕜) in 𝓝 x, Tendsto (fun n => f n y) l (𝓝 (g y))\nF' : ι → 𝕜 → 𝕜 →L[𝕜] G := fun n z => ContinuousLinearMap.smulRight 1 (f' n z)\nG' : 𝕜 → 𝕜 →L[𝕜] G := fun z => ContinuousLinearMap.smulRight 1 (g' z)\nhf : ∀ᶠ (n : ι × 𝕜) in l ×ˢ 𝓝 x, HasFDerivAt (f n.fst) (ContinuousLinearMap.smulRight 1 (f' n.fst n.snd)) n.snd\nε : ℝ\nhε : ε > 0\nq : ℝ\nhq : 0 < q\nhq' : q < ε\nn : ι × 𝕜\nhn : ‖g' n.snd - f' n.fst n.snd‖ < q\n⊢ ‖ContinuousLinearMap.smulRight 1 (g' n.snd) - ContinuousLinearMap.smulRight 1 (f' n.fst n.snd)‖ ≤ q",
"tactic": "refine' ContinuousLinearMap.op_norm_le_bound _ hq.le _"
},
{
"state_after": "case intro.intro\nι : Type u_1\nl : Filter ι\n𝕜 : Type u_2\ninst✝³ : IsROrC 𝕜\nG : Type u_3\ninst✝² : NormedAddCommGroup G\ninst✝¹ : NormedSpace 𝕜 G\nf : ι → 𝕜 → G\ng : 𝕜 → G\nf' : ι → 𝕜 → G\ng' : 𝕜 → G\nx : 𝕜\ninst✝ : NeBot l\nhf' : ∀ (ε : ℝ), ε > 0 → ∀ᶠ (n : ι × 𝕜) in l ×ˢ 𝓝 x, dist (g' n.snd) (f' n.fst n.snd) < ε\nhfg : ∀ᶠ (y : 𝕜) in 𝓝 x, Tendsto (fun n => f n y) l (𝓝 (g y))\nF' : ι → 𝕜 → 𝕜 →L[𝕜] G := fun n z => ContinuousLinearMap.smulRight 1 (f' n z)\nG' : 𝕜 → 𝕜 →L[𝕜] G := fun z => ContinuousLinearMap.smulRight 1 (g' z)\nhf : ∀ᶠ (n : ι × 𝕜) in l ×ˢ 𝓝 x, HasFDerivAt (f n.fst) (ContinuousLinearMap.smulRight 1 (f' n.fst n.snd)) n.snd\nε : ℝ\nhε : ε > 0\nq : ℝ\nhq : 0 < q\nhq' : q < ε\nn : ι × 𝕜\nhn : ‖g' n.snd - f' n.fst n.snd‖ < q\nz : 𝕜\n⊢ ‖↑(ContinuousLinearMap.smulRight 1 (g' n.snd) - ContinuousLinearMap.smulRight 1 (f' n.fst n.snd)) z‖ ≤ q * ‖z‖",
"state_before": "case intro.intro\nι : Type u_1\nl : Filter ι\n𝕜 : Type u_2\ninst✝³ : IsROrC 𝕜\nG : Type u_3\ninst✝² : NormedAddCommGroup G\ninst✝¹ : NormedSpace 𝕜 G\nf : ι → 𝕜 → G\ng : 𝕜 → G\nf' : ι → 𝕜 → G\ng' : 𝕜 → G\nx : 𝕜\ninst✝ : NeBot l\nhf' : ∀ (ε : ℝ), ε > 0 → ∀ᶠ (n : ι × 𝕜) in l ×ˢ 𝓝 x, dist (g' n.snd) (f' n.fst n.snd) < ε\nhfg : ∀ᶠ (y : 𝕜) in 𝓝 x, Tendsto (fun n => f n y) l (𝓝 (g y))\nF' : ι → 𝕜 → 𝕜 →L[𝕜] G := fun n z => ContinuousLinearMap.smulRight 1 (f' n z)\nG' : 𝕜 → 𝕜 →L[𝕜] G := fun z => ContinuousLinearMap.smulRight 1 (g' z)\nhf : ∀ᶠ (n : ι × 𝕜) in l ×ˢ 𝓝 x, HasFDerivAt (f n.fst) (ContinuousLinearMap.smulRight 1 (f' n.fst n.snd)) n.snd\nε : ℝ\nhε : ε > 0\nq : ℝ\nhq : 0 < q\nhq' : q < ε\nn : ι × 𝕜\nhn : ‖g' n.snd - f' n.fst n.snd‖ < q\n⊢ ∀ (x : 𝕜),\n ‖↑(ContinuousLinearMap.smulRight 1 (g' n.snd) - ContinuousLinearMap.smulRight 1 (f' n.fst n.snd)) x‖ ≤ q * ‖x‖",
"tactic": "intro z"
},
{
"state_after": "case intro.intro\nι : Type u_1\nl : Filter ι\n𝕜 : Type u_2\ninst✝³ : IsROrC 𝕜\nG : Type u_3\ninst✝² : NormedAddCommGroup G\ninst✝¹ : NormedSpace 𝕜 G\nf : ι → 𝕜 → G\ng : 𝕜 → G\nf' : ι → 𝕜 → G\ng' : 𝕜 → G\nx : 𝕜\ninst✝ : NeBot l\nhf' : ∀ (ε : ℝ), ε > 0 → ∀ᶠ (n : ι × 𝕜) in l ×ˢ 𝓝 x, dist (g' n.snd) (f' n.fst n.snd) < ε\nhfg : ∀ᶠ (y : 𝕜) in 𝓝 x, Tendsto (fun n => f n y) l (𝓝 (g y))\nF' : ι → 𝕜 → 𝕜 →L[𝕜] G := fun n z => ContinuousLinearMap.smulRight 1 (f' n z)\nG' : 𝕜 → 𝕜 →L[𝕜] G := fun z => ContinuousLinearMap.smulRight 1 (g' z)\nhf : ∀ᶠ (n : ι × 𝕜) in l ×ˢ 𝓝 x, HasFDerivAt (f n.fst) (ContinuousLinearMap.smulRight 1 (f' n.fst n.snd)) n.snd\nε : ℝ\nhε : ε > 0\nq : ℝ\nhq : 0 < q\nhq' : q < ε\nn : ι × 𝕜\nhn : ‖g' n.snd - f' n.fst n.snd‖ < q\nz : 𝕜\n⊢ ‖z • g' n.snd - z • f' n.fst n.snd‖ ≤ q * ‖z‖",
"state_before": "case intro.intro\nι : Type u_1\nl : Filter ι\n𝕜 : Type u_2\ninst✝³ : IsROrC 𝕜\nG : Type u_3\ninst✝² : NormedAddCommGroup G\ninst✝¹ : NormedSpace 𝕜 G\nf : ι → 𝕜 → G\ng : 𝕜 → G\nf' : ι → 𝕜 → G\ng' : 𝕜 → G\nx : 𝕜\ninst✝ : NeBot l\nhf' : ∀ (ε : ℝ), ε > 0 → ∀ᶠ (n : ι × 𝕜) in l ×ˢ 𝓝 x, dist (g' n.snd) (f' n.fst n.snd) < ε\nhfg : ∀ᶠ (y : 𝕜) in 𝓝 x, Tendsto (fun n => f n y) l (𝓝 (g y))\nF' : ι → 𝕜 → 𝕜 →L[𝕜] G := fun n z => ContinuousLinearMap.smulRight 1 (f' n z)\nG' : 𝕜 → 𝕜 →L[𝕜] G := fun z => ContinuousLinearMap.smulRight 1 (g' z)\nhf : ∀ᶠ (n : ι × 𝕜) in l ×ˢ 𝓝 x, HasFDerivAt (f n.fst) (ContinuousLinearMap.smulRight 1 (f' n.fst n.snd)) n.snd\nε : ℝ\nhε : ε > 0\nq : ℝ\nhq : 0 < q\nhq' : q < ε\nn : ι × 𝕜\nhn : ‖g' n.snd - f' n.fst n.snd‖ < q\nz : 𝕜\n⊢ ‖↑(ContinuousLinearMap.smulRight 1 (g' n.snd) - ContinuousLinearMap.smulRight 1 (f' n.fst n.snd)) z‖ ≤ q * ‖z‖",
"tactic": "simp only [ContinuousLinearMap.coe_sub', Pi.sub_apply, ContinuousLinearMap.smulRight_apply,\n ContinuousLinearMap.one_apply]"
},
{
"state_after": "case intro.intro\nι : Type u_1\nl : Filter ι\n𝕜 : Type u_2\ninst✝³ : IsROrC 𝕜\nG : Type u_3\ninst✝² : NormedAddCommGroup G\ninst✝¹ : NormedSpace 𝕜 G\nf : ι → 𝕜 → G\ng : 𝕜 → G\nf' : ι → 𝕜 → G\ng' : 𝕜 → G\nx : 𝕜\ninst✝ : NeBot l\nhf' : ∀ (ε : ℝ), ε > 0 → ∀ᶠ (n : ι × 𝕜) in l ×ˢ 𝓝 x, dist (g' n.snd) (f' n.fst n.snd) < ε\nhfg : ∀ᶠ (y : 𝕜) in 𝓝 x, Tendsto (fun n => f n y) l (𝓝 (g y))\nF' : ι → 𝕜 → 𝕜 →L[𝕜] G := fun n z => ContinuousLinearMap.smulRight 1 (f' n z)\nG' : 𝕜 → 𝕜 →L[𝕜] G := fun z => ContinuousLinearMap.smulRight 1 (g' z)\nhf : ∀ᶠ (n : ι × 𝕜) in l ×ˢ 𝓝 x, HasFDerivAt (f n.fst) (ContinuousLinearMap.smulRight 1 (f' n.fst n.snd)) n.snd\nε : ℝ\nhε : ε > 0\nq : ℝ\nhq : 0 < q\nhq' : q < ε\nn : ι × 𝕜\nhn : ‖g' n.snd - f' n.fst n.snd‖ < q\nz : 𝕜\n⊢ ‖g' n.snd - f' n.fst n.snd‖ * ‖z‖ ≤ q * ‖z‖",
"state_before": "case intro.intro\nι : Type u_1\nl : Filter ι\n𝕜 : Type u_2\ninst✝³ : IsROrC 𝕜\nG : Type u_3\ninst✝² : NormedAddCommGroup G\ninst✝¹ : NormedSpace 𝕜 G\nf : ι → 𝕜 → G\ng : 𝕜 → G\nf' : ι → 𝕜 → G\ng' : 𝕜 → G\nx : 𝕜\ninst✝ : NeBot l\nhf' : ∀ (ε : ℝ), ε > 0 → ∀ᶠ (n : ι × 𝕜) in l ×ˢ 𝓝 x, dist (g' n.snd) (f' n.fst n.snd) < ε\nhfg : ∀ᶠ (y : 𝕜) in 𝓝 x, Tendsto (fun n => f n y) l (𝓝 (g y))\nF' : ι → 𝕜 → 𝕜 →L[𝕜] G := fun n z => ContinuousLinearMap.smulRight 1 (f' n z)\nG' : 𝕜 → 𝕜 →L[𝕜] G := fun z => ContinuousLinearMap.smulRight 1 (g' z)\nhf : ∀ᶠ (n : ι × 𝕜) in l ×ˢ 𝓝 x, HasFDerivAt (f n.fst) (ContinuousLinearMap.smulRight 1 (f' n.fst n.snd)) n.snd\nε : ℝ\nhε : ε > 0\nq : ℝ\nhq : 0 < q\nhq' : q < ε\nn : ι × 𝕜\nhn : ‖g' n.snd - f' n.fst n.snd‖ < q\nz : 𝕜\n⊢ ‖z • g' n.snd - z • f' n.fst n.snd‖ ≤ q * ‖z‖",
"tactic": "rw [← smul_sub, norm_smul, mul_comm]"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\nι : Type u_1\nl : Filter ι\n𝕜 : Type u_2\ninst✝³ : IsROrC 𝕜\nG : Type u_3\ninst✝² : NormedAddCommGroup G\ninst✝¹ : NormedSpace 𝕜 G\nf : ι → 𝕜 → G\ng : 𝕜 → G\nf' : ι → 𝕜 → G\ng' : 𝕜 → G\nx : 𝕜\ninst✝ : NeBot l\nhf' : ∀ (ε : ℝ), ε > 0 → ∀ᶠ (n : ι × 𝕜) in l ×ˢ 𝓝 x, dist (g' n.snd) (f' n.fst n.snd) < ε\nhfg : ∀ᶠ (y : 𝕜) in 𝓝 x, Tendsto (fun n => f n y) l (𝓝 (g y))\nF' : ι → 𝕜 → 𝕜 →L[𝕜] G := fun n z => ContinuousLinearMap.smulRight 1 (f' n z)\nG' : 𝕜 → 𝕜 →L[𝕜] G := fun z => ContinuousLinearMap.smulRight 1 (g' z)\nhf : ∀ᶠ (n : ι × 𝕜) in l ×ˢ 𝓝 x, HasFDerivAt (f n.fst) (ContinuousLinearMap.smulRight 1 (f' n.fst n.snd)) n.snd\nε : ℝ\nhε : ε > 0\nq : ℝ\nhq : 0 < q\nhq' : q < ε\nn : ι × 𝕜\nhn : ‖g' n.snd - f' n.fst n.snd‖ < q\nz : 𝕜\n⊢ ‖g' n.snd - f' n.fst n.snd‖ * ‖z‖ ≤ q * ‖z‖",
"tactic": "gcongr"
}
] |
[
530,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
504,
1
] |
Mathlib/Data/Finset/Image.lean
|
Finset.image_erase
|
[
{
"state_after": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.92014\ninst✝¹ : DecidableEq β\nf✝ g : α → β\ns✝ : Finset α\nt : Finset β\na✝ : α\nb✝ c : β\ninst✝ : DecidableEq α\nf : α → β\nhf : Injective f\ns : Finset α\na : α\nb : β\n⊢ b ∈ image f (erase s a) → b ∈ erase (image f s) (f a)",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.92014\ninst✝¹ : DecidableEq β\nf✝ g : α → β\ns✝ : Finset α\nt : Finset β\na✝ : α\nb c : β\ninst✝ : DecidableEq α\nf : α → β\nhf : Injective f\ns : Finset α\na : α\n⊢ image f (erase s a) = erase (image f s) (f a)",
"tactic": "refine' (erase_image_subset_image_erase _ _ _).antisymm' fun b => _"
},
{
"state_after": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.92014\ninst✝¹ : DecidableEq β\nf✝ g : α → β\ns✝ : Finset α\nt : Finset β\na✝ : α\nb✝ c : β\ninst✝ : DecidableEq α\nf : α → β\nhf : Injective f\ns : Finset α\na : α\nb : β\n⊢ (∃ a_1, (a_1 ≠ a ∧ a_1 ∈ s) ∧ f a_1 = b) → b ≠ f a ∧ ∃ a, a ∈ s ∧ f a = b",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.92014\ninst✝¹ : DecidableEq β\nf✝ g : α → β\ns✝ : Finset α\nt : Finset β\na✝ : α\nb✝ c : β\ninst✝ : DecidableEq α\nf : α → β\nhf : Injective f\ns : Finset α\na : α\nb : β\n⊢ b ∈ image f (erase s a) → b ∈ erase (image f s) (f a)",
"tactic": "simp only [mem_image, exists_prop, mem_erase]"
},
{
"state_after": "case intro.intro.intro\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.92014\ninst✝¹ : DecidableEq β\nf✝ g : α → β\ns✝ : Finset α\nt : Finset β\na✝ : α\nb c : β\ninst✝ : DecidableEq α\nf : α → β\nhf : Injective f\ns : Finset α\na a' : α\nhaa' : a' ≠ a\nha' : a' ∈ s\n⊢ f a' ≠ f a ∧ ∃ a, a ∈ s ∧ f a = f a'",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.92014\ninst✝¹ : DecidableEq β\nf✝ g : α → β\ns✝ : Finset α\nt : Finset β\na✝ : α\nb✝ c : β\ninst✝ : DecidableEq α\nf : α → β\nhf : Injective f\ns : Finset α\na : α\nb : β\n⊢ (∃ a_1, (a_1 ≠ a ∧ a_1 ∈ s) ∧ f a_1 = b) → b ≠ f a ∧ ∃ a, a ∈ s ∧ f a = b",
"tactic": "rintro ⟨a', ⟨haa', ha'⟩, rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.92014\ninst✝¹ : DecidableEq β\nf✝ g : α → β\ns✝ : Finset α\nt : Finset β\na✝ : α\nb c : β\ninst✝ : DecidableEq α\nf : α → β\nhf : Injective f\ns : Finset α\na a' : α\nhaa' : a' ≠ a\nha' : a' ∈ s\n⊢ f a' ≠ f a ∧ ∃ a, a ∈ s ∧ f a = f a'",
"tactic": "exact ⟨hf.ne haa', a', ha', rfl⟩"
}
] |
[
522,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
517,
1
] |
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