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 |
|---|---|---|---|---|---|---|
Std/Data/Nat/Lemmas.lean
|
Nat.div_div_eq_div_mul
|
[
{
"state_after": "case inr\nm n k : Nat\nkpos : k > 0\n⊢ m / n / k = m / (n * k)",
"state_before": "m n k : Nat\n⊢ m / n / k = m / (n * k)",
"tactic": "cases eq_zero_or_pos k with\n| inl k0 => rw [k0, Nat.mul_zero, Nat.div_zero, Nat.div_zero] | inr kpos => ?_"
},
{
"state_after": "case inr.inr\nm n k : Nat\nkpos : k > 0\nnpos : n > 0\n⊢ m / n / k = m / (n * k)",
"state_before": "case inr\nm n k : Nat\nkpos : k > 0\n⊢ m / n / k = m / (n * k)",
"tactic": "cases eq_zero_or_pos n with\n| inl n0 => rw [n0, Nat.zero_mul, Nat.div_zero, Nat.zero_div] | inr npos => ?_"
},
{
"state_after": "case inr.inr.h₁\nm n k : Nat\nkpos : k > 0\nnpos : n > 0\n⊢ m / n / k ≤ m / (n * k)\n\ncase inr.inr.h₂\nm n k : Nat\nkpos : k > 0\nnpos : n > 0\n⊢ m / (n * k) ≤ m / n / k",
"state_before": "case inr.inr\nm n k : Nat\nkpos : k > 0\nnpos : n > 0\n⊢ m / n / k = m / (n * k)",
"tactic": "apply Nat.le_antisymm"
},
{
"state_after": "no goals",
"state_before": "case inl\nm n k : Nat\nk0 : k = 0\n⊢ m / n / k = m / (n * k)",
"tactic": "rw [k0, Nat.mul_zero, Nat.div_zero, Nat.div_zero]"
},
{
"state_after": "no goals",
"state_before": "case inr.inl\nm n k : Nat\nkpos : k > 0\nn0 : n = 0\n⊢ m / n / k = m / (n * k)",
"tactic": "rw [n0, Nat.zero_mul, Nat.div_zero, Nat.zero_div]"
},
{
"state_after": "case inr.inr.h₁\nm n k : Nat\nkpos : k > 0\nnpos : n > 0\n⊢ m / n / k * (n * k) ≤ m",
"state_before": "case inr.inr.h₁\nm n k : Nat\nkpos : k > 0\nnpos : n > 0\n⊢ m / n / k ≤ m / (n * k)",
"tactic": "apply (le_div_iff_mul_le (Nat.mul_pos npos kpos)).2"
},
{
"state_after": "case inr.inr.h₁\nm n k : Nat\nkpos : k > 0\nnpos : n > 0\n⊢ m / n / k * k * n ≤ m",
"state_before": "case inr.inr.h₁\nm n k : Nat\nkpos : k > 0\nnpos : n > 0\n⊢ m / n / k * (n * k) ≤ m",
"tactic": "rw [Nat.mul_comm n k, ← Nat.mul_assoc]"
},
{
"state_after": "case inr.inr.h₁\nm n k : Nat\nkpos : k > 0\nnpos : n > 0\n⊢ m / n / k * k ≤ m / n",
"state_before": "case inr.inr.h₁\nm n k : Nat\nkpos : k > 0\nnpos : n > 0\n⊢ m / n / k * k * n ≤ m",
"tactic": "apply (le_div_iff_mul_le npos).1"
},
{
"state_after": "case inr.inr.h₁\nm n k : Nat\nkpos : k > 0\nnpos : n > 0\n⊢ m / n / k ≤ m / n / k",
"state_before": "case inr.inr.h₁\nm n k : Nat\nkpos : k > 0\nnpos : n > 0\n⊢ m / n / k * k ≤ m / n",
"tactic": "apply (le_div_iff_mul_le kpos).1"
},
{
"state_after": "no goals",
"state_before": "case inr.inr.h₁\nm n k : Nat\nkpos : k > 0\nnpos : n > 0\n⊢ m / n / k ≤ m / n / k",
"tactic": "(apply Nat.le_refl)"
},
{
"state_after": "no goals",
"state_before": "case inr.inr.h₁\nm n k : Nat\nkpos : k > 0\nnpos : n > 0\n⊢ m / n / k ≤ m / n / k",
"tactic": "apply Nat.le_refl"
},
{
"state_after": "case inr.inr.h₂\nm n k : Nat\nkpos : k > 0\nnpos : n > 0\n⊢ m / (n * k) * k ≤ m / n",
"state_before": "case inr.inr.h₂\nm n k : Nat\nkpos : k > 0\nnpos : n > 0\n⊢ m / (n * k) ≤ m / n / k",
"tactic": "apply (le_div_iff_mul_le kpos).2"
},
{
"state_after": "case inr.inr.h₂\nm n k : Nat\nkpos : k > 0\nnpos : n > 0\n⊢ m / (n * k) * k * n ≤ m",
"state_before": "case inr.inr.h₂\nm n k : Nat\nkpos : k > 0\nnpos : n > 0\n⊢ m / (n * k) * k ≤ m / n",
"tactic": "apply (le_div_iff_mul_le npos).2"
},
{
"state_after": "case inr.inr.h₂\nm n k : Nat\nkpos : k > 0\nnpos : n > 0\n⊢ m / (k * n) * (k * n) ≤ m",
"state_before": "case inr.inr.h₂\nm n k : Nat\nkpos : k > 0\nnpos : n > 0\n⊢ m / (n * k) * k * n ≤ m",
"tactic": "rw [Nat.mul_assoc, Nat.mul_comm n k]"
},
{
"state_after": "case inr.inr.h₂\nm n k : Nat\nkpos : k > 0\nnpos : n > 0\n⊢ m / (k * n) ≤ m / (k * n)",
"state_before": "case inr.inr.h₂\nm n k : Nat\nkpos : k > 0\nnpos : n > 0\n⊢ m / (k * n) * (k * n) ≤ m",
"tactic": "apply (le_div_iff_mul_le (Nat.mul_pos kpos npos)).1"
},
{
"state_after": "no goals",
"state_before": "case inr.inr.h₂\nm n k : Nat\nkpos : k > 0\nnpos : n > 0\n⊢ m / (k * n) ≤ m / (k * n)",
"tactic": "apply Nat.le_refl"
}
] |
[
636,
22
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
621,
11
] |
Mathlib/Order/Disjoint.lean
|
symmetric_codisjoint
|
[] |
[
241,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
240,
1
] |
Mathlib/RingTheory/Subring/Basic.lean
|
Subring.coe_centralizer
|
[] |
[
852,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
851,
1
] |
Mathlib/Topology/UniformSpace/UniformConvergence.lean
|
TendstoUniformlyOnFilter.congr
|
[
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝ : UniformSpace β\nF : ι → α → β\nf : α → β\ns s' : Set α\nx : α\np : Filter ι\np' : Filter α\ng : ι → α\nF' : ι → α → β\nhf : TendstoUniformlyOnFilter F f p p'\nhff' : ∀ᶠ (n : ι × α) in p ×ˢ p', F n.fst n.snd = F' n.fst n.snd\nu : Set (β × β)\nhu : u ∈ 𝓤 β\nn : ι × α\nh : (f n.snd, F n.fst n.snd) ∈ u ∧ F n.fst n.snd = F' n.fst n.snd\n⊢ (f n.snd, F' n.fst n.snd) ∈ u",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝ : UniformSpace β\nF : ι → α → β\nf : α → β\ns s' : Set α\nx : α\np : Filter ι\np' : Filter α\ng : ι → α\nF' : ι → α → β\nhf : TendstoUniformlyOnFilter F f p p'\nhff' : ∀ᶠ (n : ι × α) in p ×ˢ p', F n.fst n.snd = F' n.fst n.snd\n⊢ TendstoUniformlyOnFilter F' f p p'",
"tactic": "refine' fun u hu => ((hf u hu).and hff').mono fun n h => _"
},
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝ : UniformSpace β\nF : ι → α → β\nf : α → β\ns s' : Set α\nx : α\np : Filter ι\np' : Filter α\ng : ι → α\nF' : ι → α → β\nhf : TendstoUniformlyOnFilter F f p p'\nhff' : ∀ᶠ (n : ι × α) in p ×ˢ p', F n.fst n.snd = F' n.fst n.snd\nu : Set (β × β)\nhu : u ∈ 𝓤 β\nn : ι × α\nh : (f n.snd, F n.fst n.snd) ∈ u ∧ F n.fst n.snd = F' n.fst n.snd\n⊢ (f n.snd, F n.fst n.snd) ∈ u",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝ : UniformSpace β\nF : ι → α → β\nf : α → β\ns s' : Set α\nx : α\np : Filter ι\np' : Filter α\ng : ι → α\nF' : ι → α → β\nhf : TendstoUniformlyOnFilter F f p p'\nhff' : ∀ᶠ (n : ι × α) in p ×ˢ p', F n.fst n.snd = F' n.fst n.snd\nu : Set (β × β)\nhu : u ∈ 𝓤 β\nn : ι × α\nh : (f n.snd, F n.fst n.snd) ∈ u ∧ F n.fst n.snd = F' n.fst n.snd\n⊢ (f n.snd, F' n.fst n.snd) ∈ u",
"tactic": "rw [← h.right]"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝ : UniformSpace β\nF : ι → α → β\nf : α → β\ns s' : Set α\nx : α\np : Filter ι\np' : Filter α\ng : ι → α\nF' : ι → α → β\nhf : TendstoUniformlyOnFilter F f p p'\nhff' : ∀ᶠ (n : ι × α) in p ×ˢ p', F n.fst n.snd = F' n.fst n.snd\nu : Set (β × β)\nhu : u ∈ 𝓤 β\nn : ι × α\nh : (f n.snd, F n.fst n.snd) ∈ u ∧ F n.fst n.snd = F' n.fst n.snd\n⊢ (f n.snd, F n.fst n.snd) ∈ u",
"tactic": "exact h.left"
}
] |
[
212,
15
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
207,
1
] |
Mathlib/MeasureTheory/Integral/IntegralEqImproper.lean
|
MeasureTheory.AECover.restrict
|
[] |
[
281,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
279,
11
] |
Mathlib/Topology/LocalAtTarget.lean
|
isClosed_iff_coe_preimage_of_iSup_eq_top
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.18470\nβ : Type u_1\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → β\ns✝ : Set β\nι : Type ?u.18489\nU : ι → Opens β\nhU : iSup U = ⊤\ns : Set β\n⊢ IsClosed s ↔ ∀ (i : ι), IsClosed (Subtype.val ⁻¹' s)",
"tactic": "simpa using isOpen_iff_coe_preimage_of_iSup_eq_top hU (sᶜ)"
}
] |
[
105,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
103,
1
] |
Mathlib/RingTheory/AdjoinRoot.lean
|
PowerBasis.quotientEquivQuotientMinpolyMap_apply_mk
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\nK : Type w\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\npb : PowerBasis R S\nI : Ideal R\ng : R[X]\n⊢ ↑(quotientEquivQuotientMinpolyMap pb I) (↑(Ideal.Quotient.mk (Ideal.map (algebraMap R S) I)) (↑(aeval pb.gen) g)) =\n ↑(Ideal.Quotient.mk (span {Polynomial.map (Ideal.Quotient.mk I) (minpoly R pb.gen)}))\n (Polynomial.map (Ideal.Quotient.mk I) g)",
"tactic": "rw [PowerBasis.quotientEquivQuotientMinpolyMap, AlgEquiv.trans_apply, AlgEquiv.ofRingEquiv_apply,\n quotientEquiv_mk, AlgEquiv.coe_ringEquiv', AdjoinRoot.equiv'_symm_apply, PowerBasis.lift_aeval,\n AdjoinRoot.aeval_eq, AdjoinRoot.quotEquivQuotMap_apply_mk]"
}
] |
[
922,
63
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
915,
1
] |
Mathlib/Order/Bounds/Basic.lean
|
IsLUB.insert
|
[
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\nι : Sort x\ninst✝² : Preorder α\ninst✝¹ : Preorder β\ns✝ t : Set α\na✝ b✝ : α\ninst✝ : SemilatticeSup γ\na b : γ\ns : Set γ\nhs : IsLUB s b\n⊢ IsLUB ({a} ∪ s) (a ⊔ b)",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nι : Sort x\ninst✝² : Preorder α\ninst✝¹ : Preorder β\ns✝ t : Set α\na✝ b✝ : α\ninst✝ : SemilatticeSup γ\na b : γ\ns : Set γ\nhs : IsLUB s b\n⊢ IsLUB (Insert.insert a s) (a ⊔ b)",
"tactic": "rw [insert_eq]"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nι : Sort x\ninst✝² : Preorder α\ninst✝¹ : Preorder β\ns✝ t : Set α\na✝ b✝ : α\ninst✝ : SemilatticeSup γ\na b : γ\ns : Set γ\nhs : IsLUB s b\n⊢ IsLUB ({a} ∪ s) (a ⊔ b)",
"tactic": "exact isLUB_singleton.union hs"
}
] |
[
949,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
946,
1
] |
Mathlib/Topology/Constructions.lean
|
IsOpen.openEmbedding_subtype_val
|
[] |
[
1024,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1022,
1
] |
Mathlib/Data/List/Chain.lean
|
List.chain'_pair
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nR r : α → α → Prop\nl l₁ l₂ : List α\na b x y : α\n⊢ Chain' R [x, y] ↔ R x y",
"tactic": "simp only [chain'_singleton, chain'_cons, and_true_iff]"
}
] |
[
335,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
334,
1
] |
Mathlib/Data/Fin/Basic.lean
|
Fin.succ_last
|
[] |
[
1239,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1238,
1
] |
Mathlib/Analysis/Normed/Group/Hom.lean
|
NormedAddGroupHom.ker.incl_comp_lift
|
[
{
"state_after": "case H\nV : Type ?u.485294\nW : Type ?u.485297\nV₁ : Type u_1\nV₂ : Type u_3\nV₃ : Type u_2\ninst✝⁴ : SeminormedAddCommGroup V\ninst✝³ : SeminormedAddCommGroup W\ninst✝² : SeminormedAddCommGroup V₁\ninst✝¹ : SeminormedAddCommGroup V₂\ninst✝ : SeminormedAddCommGroup V₃\nf : NormedAddGroupHom V₁ V₂\ng : NormedAddGroupHom V₂ V₃\nh : NormedAddGroupHom.comp g f = 0\nx✝ : V₁\n⊢ ↑(NormedAddGroupHom.comp (incl (ker g)) (lift f g h)) x✝ = ↑f x✝",
"state_before": "V : Type ?u.485294\nW : Type ?u.485297\nV₁ : Type u_1\nV₂ : Type u_3\nV₃ : Type u_2\ninst✝⁴ : SeminormedAddCommGroup V\ninst✝³ : SeminormedAddCommGroup W\ninst✝² : SeminormedAddCommGroup V₁\ninst✝¹ : SeminormedAddCommGroup V₂\ninst✝ : SeminormedAddCommGroup V₃\nf : NormedAddGroupHom V₁ V₂\ng : NormedAddGroupHom V₂ V₃\nh : NormedAddGroupHom.comp g f = 0\n⊢ NormedAddGroupHom.comp (incl (ker g)) (lift f g h) = f",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case H\nV : Type ?u.485294\nW : Type ?u.485297\nV₁ : Type u_1\nV₂ : Type u_3\nV₃ : Type u_2\ninst✝⁴ : SeminormedAddCommGroup V\ninst✝³ : SeminormedAddCommGroup W\ninst✝² : SeminormedAddCommGroup V₁\ninst✝¹ : SeminormedAddCommGroup V₂\ninst✝ : SeminormedAddCommGroup V₃\nf : NormedAddGroupHom V₁ V₂\ng : NormedAddGroupHom V₂ V₃\nh : NormedAddGroupHom.comp g f = 0\nx✝ : V₁\n⊢ ↑(NormedAddGroupHom.comp (incl (ker g)) (lift f g h)) x✝ = ↑f x✝",
"tactic": "rfl"
}
] |
[
761,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
759,
1
] |
Mathlib/RingTheory/FreeCommRing.lean
|
FreeCommRing.isSupported_mul
|
[] |
[
202,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
201,
1
] |
Mathlib/Algebra/Order/ToIntervalMod.lean
|
toIcoDiv_zsmul_sub_self
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : LinearOrderedAddCommGroup α\nhα : Archimedean α\np : α\nhp : 0 < p\na✝ b✝ c : α\nn : ℤ\na b : α\n⊢ toIcoDiv hp a b • p - b = -toIcoMod hp a b",
"tactic": "rw [toIcoMod, neg_sub]"
}
] |
[
126,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
125,
1
] |
Mathlib/Order/Hom/Bounded.lean
|
BotHom.ext
|
[] |
[
401,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
400,
1
] |
Mathlib/LinearAlgebra/Ray.lean
|
SameRay.pos_smul_left
|
[] |
[
161,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
160,
1
] |
Mathlib/Algebra/Algebra/Hom.lean
|
AlgHom.comp_apply
|
[] |
[
339,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
338,
1
] |
Mathlib/Combinatorics/Young/YoungDiagram.lean
|
YoungDiagram.pos_of_mem_rowLens
|
[
{
"state_after": "μ : YoungDiagram\nx : ℕ\nhx : ∃ a, a ∈ List.range (colLen μ 0) ∧ rowLen μ a = x\n⊢ 0 < x",
"state_before": "μ : YoungDiagram\nx : ℕ\nhx : x ∈ rowLens μ\n⊢ 0 < x",
"tactic": "rw [rowLens, List.mem_map] at hx"
},
{
"state_after": "case intro.intro\nμ : YoungDiagram\ni : ℕ\nhi : i ∈ List.range (colLen μ 0)\n⊢ 0 < rowLen μ i",
"state_before": "μ : YoungDiagram\nx : ℕ\nhx : ∃ a, a ∈ List.range (colLen μ 0) ∧ rowLen μ a = x\n⊢ 0 < x",
"tactic": "obtain ⟨i, hi, rfl : μ.rowLen i = x⟩ := hx"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\nμ : YoungDiagram\ni : ℕ\nhi : i ∈ List.range (colLen μ 0)\n⊢ 0 < rowLen μ i",
"tactic": "rwa [List.mem_range, ← mem_iff_lt_colLen, mem_iff_lt_rowLen] at hi"
}
] |
[
436,
69
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
433,
1
] |
Mathlib/Analysis/SpecialFunctions/Exponential.lean
|
hasDerivAt_exp_zero_of_radius_pos
|
[] |
[
154,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
152,
1
] |
Mathlib/Topology/MetricSpace/Holder.lean
|
HolderWith.edist_le_of_le
|
[] |
[
201,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
199,
1
] |
Mathlib/Algebra/Group/Basic.lean
|
inv_mul_eq_of_eq_mul
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.50703\nβ : Type ?u.50706\nG : Type u_1\ninst✝ : Group G\na b c d : G\nh : b = a * c\n⊢ a⁻¹ * b = c",
"tactic": "simp [h]"
}
] |
[
638,
74
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
638,
1
] |
Mathlib/Topology/Maps.lean
|
ClosedEmbedding.closure_image_eq
|
[] |
[
700,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
697,
1
] |
Mathlib/Data/Real/Hyperreal.lean
|
Hyperreal.not_infinite_iff_exist_lt_gt
|
[] |
[
615,
93
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
613,
1
] |
Mathlib/Algebra/Module/Zlattice.lean
|
Zspan.repr_floor_apply
|
[
{
"state_after": "no goals",
"state_before": "E : Type u_3\nι : Type u_2\nK : Type u_1\ninst✝⁴ : NormedLinearOrderedField K\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace K E\nb : Basis ι K E\ninst✝¹ : FloorRing K\ninst✝ : Fintype ι\nm : E\ni : ι\n⊢ ↑(↑b.repr ↑(floor b m)) i = ↑⌊↑(↑b.repr m) i⌋",
"tactic": "classical simp only [floor, zsmul_eq_smul_cast K, b.repr.map_smul, Finsupp.single_apply,\n Finset.sum_apply', Basis.repr_self, Finsupp.smul_single', mul_one, Finset.sum_ite_eq', coe_sum,\n Finset.mem_univ, if_true, coe_smul_of_tower, Basis.restrictScalars_apply, LinearEquiv.map_sum]"
},
{
"state_after": "no goals",
"state_before": "E : Type u_3\nι : Type u_2\nK : Type u_1\ninst✝⁴ : NormedLinearOrderedField K\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace K E\nb : Basis ι K E\ninst✝¹ : FloorRing K\ninst✝ : Fintype ι\nm : E\ni : ι\n⊢ ↑(↑b.repr ↑(floor b m)) i = ↑⌊↑(↑b.repr m) i⌋",
"tactic": "simp only [floor, zsmul_eq_smul_cast K, b.repr.map_smul, Finsupp.single_apply,\nFinset.sum_apply', Basis.repr_self, Finsupp.smul_single', mul_one, Finset.sum_ite_eq', coe_sum,\nFinset.mem_univ, if_true, coe_smul_of_tower, Basis.restrictScalars_apply, LinearEquiv.map_sum]"
}
] |
[
82,
99
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
79,
1
] |
Mathlib/LinearAlgebra/Basic.lean
|
LinearMap.finsupp_sum_apply
|
[] |
[
1130,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1128,
1
] |
Mathlib/Analysis/InnerProductSpace/Orientation.lean
|
OrthonormalBasis.abs_det_adjustToOrientation
|
[
{
"state_after": "no goals",
"state_before": "E : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace ℝ E\nι : Type u_2\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\nne : Nonempty ι\ne f : OrthonormalBasis ι ℝ E\nx : Orientation ℝ E ι\nv : ι → E\n⊢ abs (↑(Basis.det (OrthonormalBasis.toBasis (adjustToOrientation e x))) v) =\n abs (↑(Basis.det (OrthonormalBasis.toBasis e)) v)",
"tactic": "simp [toBasis_adjustToOrientation]"
}
] |
[
146,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
144,
1
] |
Mathlib/Order/Heyting/Hom.lean
|
CoheytingHom.id_apply
|
[] |
[
438,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
437,
1
] |
Mathlib/Analysis/LocallyConvex/Basic.lean
|
Balanced.union
|
[] |
[
183,
73
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
182,
1
] |
Mathlib/Combinatorics/SimpleGraph/Regularity/Uniform.lean
|
Finpartition.isUniformOfEmpty
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\n𝕜 : Type u_2\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : DecidableEq α\nA : Finset α\nP : Finpartition A\nG : SimpleGraph α\ninst✝ : DecidableRel G.Adj\nε : 𝕜\nhP : P.parts = ∅\n⊢ IsUniform P G ε",
"tactic": "simp [IsUniform, hP, nonUniforms]"
}
] |
[
256,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
255,
1
] |
Mathlib/Data/Multiset/Fold.lean
|
Multiset.fold_eq_foldr
|
[] |
[
40,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
38,
1
] |
Mathlib/AlgebraicGeometry/StructureSheaf.lean
|
AlgebraicGeometry.StructureSheaf.localizationToStalk_stalkSpecializes
|
[
{
"state_after": "case h\nR✝ : Type u\ninst✝¹ : CommRing R✝\nR : Type u_1\ninst✝ : CommRing R\nx y : PrimeSpectrum R\nh : x ⤳ y\n⊢ RingHom.comp (localizationToStalk R y ≫ stalkSpecializes (Sheaf.presheaf (structureSheaf R)) h)\n (algebraMap R (Localization.AtPrime y.asIdeal)) =\n RingHom.comp (CommRingCat.ofHom (PrimeSpectrum.localizationMapOfSpecializes h) ≫ localizationToStalk R x)\n (algebraMap R (Localization.AtPrime y.asIdeal))",
"state_before": "R✝ : Type u\ninst✝¹ : CommRing R✝\nR : Type u_1\ninst✝ : CommRing R\nx y : PrimeSpectrum R\nh : x ⤳ y\n⊢ localizationToStalk R y ≫ stalkSpecializes (Sheaf.presheaf (structureSheaf R)) h =\n CommRingCat.ofHom (PrimeSpectrum.localizationMapOfSpecializes h) ≫ localizationToStalk R x",
"tactic": "apply IsLocalization.ringHom_ext (S := Localization.AtPrime y.asIdeal) y.asIdeal.primeCompl"
},
{
"state_after": "case h\nR✝ : Type u\ninst✝¹ : CommRing R✝\nR : Type u_1\ninst✝ : CommRing R\nx y : PrimeSpectrum R\nh : x ⤳ y\n⊢ RingHom.comp (stalkSpecializes (Sheaf.presheaf (structureSheaf R)) h)\n (RingHom.comp (localizationToStalk R y) (algebraMap R (Localization.AtPrime y.asIdeal))) =\n RingHom.comp (CommRingCat.ofHom (PrimeSpectrum.localizationMapOfSpecializes h) ≫ localizationToStalk R x)\n (algebraMap R (Localization.AtPrime y.asIdeal))",
"state_before": "case h\nR✝ : Type u\ninst✝¹ : CommRing R✝\nR : Type u_1\ninst✝ : CommRing R\nx y : PrimeSpectrum R\nh : x ⤳ y\n⊢ RingHom.comp (localizationToStalk R y ≫ stalkSpecializes (Sheaf.presheaf (structureSheaf R)) h)\n (algebraMap R (Localization.AtPrime y.asIdeal)) =\n RingHom.comp (CommRingCat.ofHom (PrimeSpectrum.localizationMapOfSpecializes h) ≫ localizationToStalk R x)\n (algebraMap R (Localization.AtPrime y.asIdeal))",
"tactic": "erw [RingHom.comp_assoc]"
},
{
"state_after": "case h\nR✝ : Type u\ninst✝¹ : CommRing R✝\nR : Type u_1\ninst✝ : CommRing R\nx y : PrimeSpectrum R\nh : x ⤳ y\n⊢ RingHom.comp (stalkSpecializes (Sheaf.presheaf (structureSheaf R)) h)\n (RingHom.comp (localizationToStalk R y) (algebraMap R (Localization.AtPrime y.asIdeal))) =\n RingHom.comp (localizationToStalk R x)\n (RingHom.comp (CommRingCat.ofHom (PrimeSpectrum.localizationMapOfSpecializes h))\n (algebraMap R (Localization.AtPrime y.asIdeal)))",
"state_before": "case h\nR✝ : Type u\ninst✝¹ : CommRing R✝\nR : Type u_1\ninst✝ : CommRing R\nx y : PrimeSpectrum R\nh : x ⤳ y\n⊢ RingHom.comp (stalkSpecializes (Sheaf.presheaf (structureSheaf R)) h)\n (RingHom.comp (localizationToStalk R y) (algebraMap R (Localization.AtPrime y.asIdeal))) =\n RingHom.comp (CommRingCat.ofHom (PrimeSpectrum.localizationMapOfSpecializes h) ≫ localizationToStalk R x)\n (algebraMap R (Localization.AtPrime y.asIdeal))",
"tactic": "conv_rhs => erw [RingHom.comp_assoc]"
},
{
"state_after": "case h\nR✝ : Type u\ninst✝¹ : CommRing R✝\nR : Type u_1\ninst✝ : CommRing R\nx y : PrimeSpectrum R\nh : x ⤳ y\n⊢ RingHom.comp (stalkSpecializes (Sheaf.presheaf (structureSheaf R)) h)\n (RingHom.comp\n (IsLocalization.lift (_ : ∀ (f : { x // x ∈ Ideal.primeCompl y.asIdeal }), IsUnit (↑(toStalk R y) ↑f)))\n (algebraMap R (Localization.AtPrime y.asIdeal))) =\n RingHom.comp\n (IsLocalization.lift (_ : ∀ (f : { x_1 // x_1 ∈ Ideal.primeCompl x.asIdeal }), IsUnit (↑(toStalk R x) ↑f)))\n (RingHom.comp\n (IsLocalization.lift\n (_ :\n ∀ (y_1 : { x // x ∈ Ideal.primeCompl y.asIdeal }),\n IsUnit (↑(algebraMap R (Localization.AtPrime x.asIdeal)) ↑y_1)))\n (algebraMap R (Localization.AtPrime y.asIdeal)))",
"state_before": "case h\nR✝ : Type u\ninst✝¹ : CommRing R✝\nR : Type u_1\ninst✝ : CommRing R\nx y : PrimeSpectrum R\nh : x ⤳ y\n⊢ RingHom.comp (stalkSpecializes (Sheaf.presheaf (structureSheaf R)) h)\n (RingHom.comp (localizationToStalk R y) (algebraMap R (Localization.AtPrime y.asIdeal))) =\n RingHom.comp (localizationToStalk R x)\n (RingHom.comp (CommRingCat.ofHom (PrimeSpectrum.localizationMapOfSpecializes h))\n (algebraMap R (Localization.AtPrime y.asIdeal)))",
"tactic": "dsimp [CommRingCat.ofHom, localizationToStalk, PrimeSpectrum.localizationMapOfSpecializes]"
},
{
"state_after": "case h\nR✝ : Type u\ninst✝¹ : CommRing R✝\nR : Type u_1\ninst✝ : CommRing R\nx y : PrimeSpectrum R\nh : x ⤳ y\n⊢ RingHom.comp (stalkSpecializes (Sheaf.presheaf (structureSheaf R)) h) (toStalk R y) = toStalk R x",
"state_before": "case h\nR✝ : Type u\ninst✝¹ : CommRing R✝\nR : Type u_1\ninst✝ : CommRing R\nx y : PrimeSpectrum R\nh : x ⤳ y\n⊢ RingHom.comp (stalkSpecializes (Sheaf.presheaf (structureSheaf R)) h)\n (RingHom.comp\n (IsLocalization.lift (_ : ∀ (f : { x // x ∈ Ideal.primeCompl y.asIdeal }), IsUnit (↑(toStalk R y) ↑f)))\n (algebraMap R (Localization.AtPrime y.asIdeal))) =\n RingHom.comp\n (IsLocalization.lift (_ : ∀ (f : { x_1 // x_1 ∈ Ideal.primeCompl x.asIdeal }), IsUnit (↑(toStalk R x) ↑f)))\n (RingHom.comp\n (IsLocalization.lift\n (_ :\n ∀ (y_1 : { x // x ∈ Ideal.primeCompl y.asIdeal }),\n IsUnit (↑(algebraMap R (Localization.AtPrime x.asIdeal)) ↑y_1)))\n (algebraMap R (Localization.AtPrime y.asIdeal)))",
"tactic": "rw [IsLocalization.lift_comp, IsLocalization.lift_comp, IsLocalization.lift_comp]"
},
{
"state_after": "no goals",
"state_before": "case h\nR✝ : Type u\ninst✝¹ : CommRing R✝\nR : Type u_1\ninst✝ : CommRing R\nx y : PrimeSpectrum R\nh : x ⤳ y\n⊢ RingHom.comp (stalkSpecializes (Sheaf.presheaf (structureSheaf R)) h) (toStalk R y) = toStalk R x",
"tactic": "exact toStalk_stalkSpecializes h"
}
] |
[
1036,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1026,
1
] |
Mathlib/Data/Finsupp/Basic.lean
|
Finsupp.erase_neg
|
[] |
[
1157,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1156,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Deriv.lean
|
deriv_cos
|
[] |
[
820,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
818,
1
] |
Mathlib/Analysis/Calculus/FDeriv/Comp.lean
|
fderiv.comp_fderivWithin
|
[] |
[
178,
82
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
175,
1
] |
Mathlib/Data/Real/CauSeq.lean
|
CauSeq.const_le
|
[
{
"state_after": "α : Type u_1\ninst✝ : LinearOrderedField α\nx y : α\n⊢ const x ≤ const y ↔ x < y ∨ x = y",
"state_before": "α : Type u_1\ninst✝ : LinearOrderedField α\nx y : α\n⊢ const x ≤ const y ↔ x ≤ y",
"tactic": "rw [le_iff_lt_or_eq]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : LinearOrderedField α\nx y : α\n⊢ const x ≤ const y ↔ x < y ∨ x = y",
"tactic": "exact or_congr const_lt const_equiv"
}
] |
[
788,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
787,
1
] |
Mathlib/RingTheory/Localization/Ideal.lean
|
IsLocalization.surjective_quotientMap_of_maximal_of_localization
|
[
{
"state_after": "R : Type u_2\ninst✝⁴ : CommRing R\nM : Submonoid R\nS : Type u_1\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : IsLocalization M S\nI : Ideal S\ninst✝ : Ideal.IsPrime I\nJ : Ideal R\nH : J ≤ Ideal.comap (algebraMap R S) I\nhI : Ideal.IsMaximal (Ideal.comap (algebraMap R S) I)\ns : S ⧸ I\n⊢ ∃ a, ↑(Ideal.quotientMap I (algebraMap R S) H) a = s",
"state_before": "R : Type u_2\ninst✝⁴ : CommRing R\nM : Submonoid R\nS : Type u_1\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : IsLocalization M S\nI : Ideal S\ninst✝ : Ideal.IsPrime I\nJ : Ideal R\nH : J ≤ Ideal.comap (algebraMap R S) I\nhI : Ideal.IsMaximal (Ideal.comap (algebraMap R S) I)\n⊢ Function.Surjective ↑(Ideal.quotientMap I (algebraMap R S) H)",
"tactic": "intro s"
},
{
"state_after": "case intro\nR : Type u_2\ninst✝⁴ : CommRing R\nM : Submonoid R\nS : Type u_1\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : IsLocalization M S\nI : Ideal S\ninst✝ : Ideal.IsPrime I\nJ : Ideal R\nH : J ≤ Ideal.comap (algebraMap R S) I\nhI : Ideal.IsMaximal (Ideal.comap (algebraMap R S) I)\ns : S\n⊢ ∃ a, ↑(Ideal.quotientMap I (algebraMap R S) H) a = ↑(Ideal.Quotient.mk I) s",
"state_before": "R : Type u_2\ninst✝⁴ : CommRing R\nM : Submonoid R\nS : Type u_1\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : IsLocalization M S\nI : Ideal S\ninst✝ : Ideal.IsPrime I\nJ : Ideal R\nH : J ≤ Ideal.comap (algebraMap R S) I\nhI : Ideal.IsMaximal (Ideal.comap (algebraMap R S) I)\ns : S ⧸ I\n⊢ ∃ a, ↑(Ideal.quotientMap I (algebraMap R S) H) a = s",
"tactic": "obtain ⟨s, rfl⟩ := Ideal.Quotient.mk_surjective s"
},
{
"state_after": "case intro.intro.intro.mk\nR : Type u_2\ninst✝⁴ : CommRing R\nM : Submonoid R\nS : Type u_1\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : IsLocalization M S\nI : Ideal S\ninst✝ : Ideal.IsPrime I\nJ : Ideal R\nH : J ≤ Ideal.comap (algebraMap R S) I\nhI : Ideal.IsMaximal (Ideal.comap (algebraMap R S) I)\nr m : R\nhm : m ∈ M\n⊢ ∃ a, ↑(Ideal.quotientMap I (algebraMap R S) H) a = ↑(Ideal.Quotient.mk I) (mk' S r { val := m, property := hm })",
"state_before": "case intro\nR : Type u_2\ninst✝⁴ : CommRing R\nM : Submonoid R\nS : Type u_1\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : IsLocalization M S\nI : Ideal S\ninst✝ : Ideal.IsPrime I\nJ : Ideal R\nH : J ≤ Ideal.comap (algebraMap R S) I\nhI : Ideal.IsMaximal (Ideal.comap (algebraMap R S) I)\ns : S\n⊢ ∃ a, ↑(Ideal.quotientMap I (algebraMap R S) H) a = ↑(Ideal.Quotient.mk I) s",
"tactic": "obtain ⟨r, ⟨m, hm⟩, rfl⟩ := mk'_surjective M s"
},
{
"state_after": "case pos\nR : Type u_2\ninst✝⁴ : CommRing R\nM : Submonoid R\nS : Type u_1\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : IsLocalization M S\nI : Ideal S\ninst✝ : Ideal.IsPrime I\nJ : Ideal R\nH : J ≤ Ideal.comap (algebraMap R S) I\nhI : Ideal.IsMaximal (Ideal.comap (algebraMap R S) I)\nr m : R\nhm : m ∈ M\nhM : ↑(Ideal.Quotient.mk (Ideal.comap (algebraMap R S) I)) m = 0\n⊢ ∃ a, ↑(Ideal.quotientMap I (algebraMap R S) H) a = ↑(Ideal.Quotient.mk I) (mk' S r { val := m, property := hm })\n\ncase neg\nR : Type u_2\ninst✝⁴ : CommRing R\nM : Submonoid R\nS : Type u_1\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : IsLocalization M S\nI : Ideal S\ninst✝ : Ideal.IsPrime I\nJ : Ideal R\nH : J ≤ Ideal.comap (algebraMap R S) I\nhI : Ideal.IsMaximal (Ideal.comap (algebraMap R S) I)\nr m : R\nhm : m ∈ M\nhM : ¬↑(Ideal.Quotient.mk (Ideal.comap (algebraMap R S) I)) m = 0\n⊢ ∃ a, ↑(Ideal.quotientMap I (algebraMap R S) H) a = ↑(Ideal.Quotient.mk I) (mk' S r { val := m, property := hm })",
"state_before": "case intro.intro.intro.mk\nR : Type u_2\ninst✝⁴ : CommRing R\nM : Submonoid R\nS : Type u_1\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : IsLocalization M S\nI : Ideal S\ninst✝ : Ideal.IsPrime I\nJ : Ideal R\nH : J ≤ Ideal.comap (algebraMap R S) I\nhI : Ideal.IsMaximal (Ideal.comap (algebraMap R S) I)\nr m : R\nhm : m ∈ M\n⊢ ∃ a, ↑(Ideal.quotientMap I (algebraMap R S) H) a = ↑(Ideal.Quotient.mk I) (mk' S r { val := m, property := hm })",
"tactic": "by_cases hM : (Ideal.Quotient.mk (I.comap (algebraMap R S))) m = 0"
},
{
"state_after": "case pos\nR : Type u_2\ninst✝⁴ : CommRing R\nM : Submonoid R\nS : Type u_1\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : IsLocalization M S\nI : Ideal S\ninst✝ : Ideal.IsPrime I\nJ : Ideal R\nH : J ≤ Ideal.comap (algebraMap R S) I\nhI : Ideal.IsMaximal (Ideal.comap (algebraMap R S) I)\nr m : R\nhm : m ∈ M\nhM : ↑(Ideal.Quotient.mk (Ideal.comap (algebraMap R S) I)) m = 0\nthis : I = ⊤\n⊢ ∃ a, ↑(Ideal.quotientMap I (algebraMap R S) H) a = ↑(Ideal.Quotient.mk I) (mk' S r { val := m, property := hm })",
"state_before": "case pos\nR : Type u_2\ninst✝⁴ : CommRing R\nM : Submonoid R\nS : Type u_1\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : IsLocalization M S\nI : Ideal S\ninst✝ : Ideal.IsPrime I\nJ : Ideal R\nH : J ≤ Ideal.comap (algebraMap R S) I\nhI : Ideal.IsMaximal (Ideal.comap (algebraMap R S) I)\nr m : R\nhm : m ∈ M\nhM : ↑(Ideal.Quotient.mk (Ideal.comap (algebraMap R S) I)) m = 0\n⊢ ∃ a, ↑(Ideal.quotientMap I (algebraMap R S) H) a = ↑(Ideal.Quotient.mk I) (mk' S r { val := m, property := hm })",
"tactic": "have : I = ⊤ := by\n rw [Ideal.eq_top_iff_one]\n rw [Ideal.Quotient.eq_zero_iff_mem, Ideal.mem_comap] at hM\n convert I.mul_mem_right (mk' S (1 : R) ⟨m, hm⟩) hM\n rw [← mk'_eq_mul_mk'_one, mk'_self]"
},
{
"state_after": "no goals",
"state_before": "case pos\nR : Type u_2\ninst✝⁴ : CommRing R\nM : Submonoid R\nS : Type u_1\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : IsLocalization M S\nI : Ideal S\ninst✝ : Ideal.IsPrime I\nJ : Ideal R\nH : J ≤ Ideal.comap (algebraMap R S) I\nhI : Ideal.IsMaximal (Ideal.comap (algebraMap R S) I)\nr m : R\nhm : m ∈ M\nhM : ↑(Ideal.Quotient.mk (Ideal.comap (algebraMap R S) I)) m = 0\nthis : I = ⊤\n⊢ ∃ a, ↑(Ideal.quotientMap I (algebraMap R S) H) a = ↑(Ideal.Quotient.mk I) (mk' S r { val := m, property := hm })",
"tactic": "exact ⟨0, eq_comm.1 (by simp [Ideal.Quotient.eq_zero_iff_mem, this])⟩"
},
{
"state_after": "R : Type u_2\ninst✝⁴ : CommRing R\nM : Submonoid R\nS : Type u_1\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : IsLocalization M S\nI : Ideal S\ninst✝ : Ideal.IsPrime I\nJ : Ideal R\nH : J ≤ Ideal.comap (algebraMap R S) I\nhI : Ideal.IsMaximal (Ideal.comap (algebraMap R S) I)\nr m : R\nhm : m ∈ M\nhM : ↑(Ideal.Quotient.mk (Ideal.comap (algebraMap R S) I)) m = 0\n⊢ 1 ∈ I",
"state_before": "R : Type u_2\ninst✝⁴ : CommRing R\nM : Submonoid R\nS : Type u_1\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : IsLocalization M S\nI : Ideal S\ninst✝ : Ideal.IsPrime I\nJ : Ideal R\nH : J ≤ Ideal.comap (algebraMap R S) I\nhI : Ideal.IsMaximal (Ideal.comap (algebraMap R S) I)\nr m : R\nhm : m ∈ M\nhM : ↑(Ideal.Quotient.mk (Ideal.comap (algebraMap R S) I)) m = 0\n⊢ I = ⊤",
"tactic": "rw [Ideal.eq_top_iff_one]"
},
{
"state_after": "R : Type u_2\ninst✝⁴ : CommRing R\nM : Submonoid R\nS : Type u_1\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : IsLocalization M S\nI : Ideal S\ninst✝ : Ideal.IsPrime I\nJ : Ideal R\nH : J ≤ Ideal.comap (algebraMap R S) I\nhI : Ideal.IsMaximal (Ideal.comap (algebraMap R S) I)\nr m : R\nhm : m ∈ M\nhM : ↑(algebraMap R S) m ∈ I\n⊢ 1 ∈ I",
"state_before": "R : Type u_2\ninst✝⁴ : CommRing R\nM : Submonoid R\nS : Type u_1\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : IsLocalization M S\nI : Ideal S\ninst✝ : Ideal.IsPrime I\nJ : Ideal R\nH : J ≤ Ideal.comap (algebraMap R S) I\nhI : Ideal.IsMaximal (Ideal.comap (algebraMap R S) I)\nr m : R\nhm : m ∈ M\nhM : ↑(Ideal.Quotient.mk (Ideal.comap (algebraMap R S) I)) m = 0\n⊢ 1 ∈ I",
"tactic": "rw [Ideal.Quotient.eq_zero_iff_mem, Ideal.mem_comap] at hM"
},
{
"state_after": "case h.e'_4\nR : Type u_2\ninst✝⁴ : CommRing R\nM : Submonoid R\nS : Type u_1\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : IsLocalization M S\nI : Ideal S\ninst✝ : Ideal.IsPrime I\nJ : Ideal R\nH : J ≤ Ideal.comap (algebraMap R S) I\nhI : Ideal.IsMaximal (Ideal.comap (algebraMap R S) I)\nr m : R\nhm : m ∈ M\nhM : ↑(algebraMap R S) m ∈ I\n⊢ 1 = ↑(algebraMap R S) m * mk' S 1 { val := m, property := hm }",
"state_before": "R : Type u_2\ninst✝⁴ : CommRing R\nM : Submonoid R\nS : Type u_1\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : IsLocalization M S\nI : Ideal S\ninst✝ : Ideal.IsPrime I\nJ : Ideal R\nH : J ≤ Ideal.comap (algebraMap R S) I\nhI : Ideal.IsMaximal (Ideal.comap (algebraMap R S) I)\nr m : R\nhm : m ∈ M\nhM : ↑(algebraMap R S) m ∈ I\n⊢ 1 ∈ I",
"tactic": "convert I.mul_mem_right (mk' S (1 : R) ⟨m, hm⟩) hM"
},
{
"state_after": "no goals",
"state_before": "case h.e'_4\nR : Type u_2\ninst✝⁴ : CommRing R\nM : Submonoid R\nS : Type u_1\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : IsLocalization M S\nI : Ideal S\ninst✝ : Ideal.IsPrime I\nJ : Ideal R\nH : J ≤ Ideal.comap (algebraMap R S) I\nhI : Ideal.IsMaximal (Ideal.comap (algebraMap R S) I)\nr m : R\nhm : m ∈ M\nhM : ↑(algebraMap R S) m ∈ I\n⊢ 1 = ↑(algebraMap R S) m * mk' S 1 { val := m, property := hm }",
"tactic": "rw [← mk'_eq_mul_mk'_one, mk'_self]"
},
{
"state_after": "no goals",
"state_before": "R : Type u_2\ninst✝⁴ : CommRing R\nM : Submonoid R\nS : Type u_1\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : IsLocalization M S\nI : Ideal S\ninst✝ : Ideal.IsPrime I\nJ : Ideal R\nH : J ≤ Ideal.comap (algebraMap R S) I\nhI : Ideal.IsMaximal (Ideal.comap (algebraMap R S) I)\nr m : R\nhm : m ∈ M\nhM : ↑(Ideal.Quotient.mk (Ideal.comap (algebraMap R S) I)) m = 0\nthis : I = ⊤\n⊢ ↑(Ideal.Quotient.mk I) (mk' S r { val := m, property := hm }) = ↑(Ideal.quotientMap I (algebraMap R S) H) 0",
"tactic": "simp [Ideal.Quotient.eq_zero_iff_mem, this]"
},
{
"state_after": "case neg\nR : Type u_2\ninst✝⁴ : CommRing R\nM : Submonoid R\nS : Type u_1\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : IsLocalization M S\nI : Ideal S\ninst✝ : Ideal.IsPrime I\nJ : Ideal R\nH : J ≤ Ideal.comap (algebraMap R S) I\nhI : IsField (R ⧸ Ideal.comap (algebraMap R S) I)\nr m : R\nhm : m ∈ M\nhM : ¬↑(Ideal.Quotient.mk (Ideal.comap (algebraMap R S) I)) m = 0\n⊢ ∃ a, ↑(Ideal.quotientMap I (algebraMap R S) H) a = ↑(Ideal.Quotient.mk I) (mk' S r { val := m, property := hm })",
"state_before": "case neg\nR : Type u_2\ninst✝⁴ : CommRing R\nM : Submonoid R\nS : Type u_1\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : IsLocalization M S\nI : Ideal S\ninst✝ : Ideal.IsPrime I\nJ : Ideal R\nH : J ≤ Ideal.comap (algebraMap R S) I\nhI : Ideal.IsMaximal (Ideal.comap (algebraMap R S) I)\nr m : R\nhm : m ∈ M\nhM : ¬↑(Ideal.Quotient.mk (Ideal.comap (algebraMap R S) I)) m = 0\n⊢ ∃ a, ↑(Ideal.quotientMap I (algebraMap R S) H) a = ↑(Ideal.Quotient.mk I) (mk' S r { val := m, property := hm })",
"tactic": "rw [Ideal.Quotient.maximal_ideal_iff_isField_quotient] at hI"
},
{
"state_after": "case neg.intro\nR : Type u_2\ninst✝⁴ : CommRing R\nM : Submonoid R\nS : Type u_1\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : IsLocalization M S\nI : Ideal S\ninst✝ : Ideal.IsPrime I\nJ : Ideal R\nH : J ≤ Ideal.comap (algebraMap R S) I\nhI : IsField (R ⧸ Ideal.comap (algebraMap R S) I)\nr m : R\nhm : m ∈ M\nhM : ¬↑(Ideal.Quotient.mk (Ideal.comap (algebraMap R S) I)) m = 0\nn : R ⧸ Ideal.comap (algebraMap R S) I\nhn : ↑(Ideal.Quotient.mk (Ideal.comap (algebraMap R S) I)) m * n = 1\n⊢ ∃ a, ↑(Ideal.quotientMap I (algebraMap R S) H) a = ↑(Ideal.Quotient.mk I) (mk' S r { val := m, property := hm })",
"state_before": "case neg\nR : Type u_2\ninst✝⁴ : CommRing R\nM : Submonoid R\nS : Type u_1\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : IsLocalization M S\nI : Ideal S\ninst✝ : Ideal.IsPrime I\nJ : Ideal R\nH : J ≤ Ideal.comap (algebraMap R S) I\nhI : IsField (R ⧸ Ideal.comap (algebraMap R S) I)\nr m : R\nhm : m ∈ M\nhM : ¬↑(Ideal.Quotient.mk (Ideal.comap (algebraMap R S) I)) m = 0\n⊢ ∃ a, ↑(Ideal.quotientMap I (algebraMap R S) H) a = ↑(Ideal.Quotient.mk I) (mk' S r { val := m, property := hm })",
"tactic": "obtain ⟨n, hn⟩ := hI.3 hM"
},
{
"state_after": "case neg.intro.intro\nR : Type u_2\ninst✝⁴ : CommRing R\nM : Submonoid R\nS : Type u_1\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : IsLocalization M S\nI : Ideal S\ninst✝ : Ideal.IsPrime I\nJ : Ideal R\nH : J ≤ Ideal.comap (algebraMap R S) I\nhI : IsField (R ⧸ Ideal.comap (algebraMap R S) I)\nr m : R\nhm : m ∈ M\nhM : ¬↑(Ideal.Quotient.mk (Ideal.comap (algebraMap R S) I)) m = 0\nrn : R\nhn :\n ↑(Ideal.Quotient.mk (Ideal.comap (algebraMap R S) I)) m * ↑(Ideal.Quotient.mk (Ideal.comap (algebraMap R S) I)) rn = 1\n⊢ ∃ a, ↑(Ideal.quotientMap I (algebraMap R S) H) a = ↑(Ideal.Quotient.mk I) (mk' S r { val := m, property := hm })",
"state_before": "case neg.intro\nR : Type u_2\ninst✝⁴ : CommRing R\nM : Submonoid R\nS : Type u_1\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : IsLocalization M S\nI : Ideal S\ninst✝ : Ideal.IsPrime I\nJ : Ideal R\nH : J ≤ Ideal.comap (algebraMap R S) I\nhI : IsField (R ⧸ Ideal.comap (algebraMap R S) I)\nr m : R\nhm : m ∈ M\nhM : ¬↑(Ideal.Quotient.mk (Ideal.comap (algebraMap R S) I)) m = 0\nn : R ⧸ Ideal.comap (algebraMap R S) I\nhn : ↑(Ideal.Quotient.mk (Ideal.comap (algebraMap R S) I)) m * n = 1\n⊢ ∃ a, ↑(Ideal.quotientMap I (algebraMap R S) H) a = ↑(Ideal.Quotient.mk I) (mk' S r { val := m, property := hm })",
"tactic": "obtain ⟨rn, rfl⟩ := Ideal.Quotient.mk_surjective n"
},
{
"state_after": "case neg.intro.intro\nR : Type u_2\ninst✝⁴ : CommRing R\nM : Submonoid R\nS : Type u_1\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : IsLocalization M S\nI : Ideal S\ninst✝ : Ideal.IsPrime I\nJ : Ideal R\nH : J ≤ Ideal.comap (algebraMap R S) I\nhI : IsField (R ⧸ Ideal.comap (algebraMap R S) I)\nr m : R\nhm : m ∈ M\nhM : ¬↑(Ideal.Quotient.mk (Ideal.comap (algebraMap R S) I)) m = 0\nrn : R\nhn :\n ↑(Ideal.Quotient.mk (Ideal.comap (algebraMap R S) I)) m * ↑(Ideal.Quotient.mk (Ideal.comap (algebraMap R S) I)) rn = 1\n⊢ ↑(Ideal.quotientMap I (algebraMap R S) H) (↑(Ideal.Quotient.mk J) (r * rn)) =\n ↑(Ideal.Quotient.mk I) (mk' S r { val := m, property := hm })",
"state_before": "case neg.intro.intro\nR : Type u_2\ninst✝⁴ : CommRing R\nM : Submonoid R\nS : Type u_1\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : IsLocalization M S\nI : Ideal S\ninst✝ : Ideal.IsPrime I\nJ : Ideal R\nH : J ≤ Ideal.comap (algebraMap R S) I\nhI : IsField (R ⧸ Ideal.comap (algebraMap R S) I)\nr m : R\nhm : m ∈ M\nhM : ¬↑(Ideal.Quotient.mk (Ideal.comap (algebraMap R S) I)) m = 0\nrn : R\nhn :\n ↑(Ideal.Quotient.mk (Ideal.comap (algebraMap R S) I)) m * ↑(Ideal.Quotient.mk (Ideal.comap (algebraMap R S) I)) rn = 1\n⊢ ∃ a, ↑(Ideal.quotientMap I (algebraMap R S) H) a = ↑(Ideal.Quotient.mk I) (mk' S r { val := m, property := hm })",
"tactic": "refine' ⟨(Ideal.Quotient.mk J) (r * rn), _⟩"
},
{
"state_after": "case neg.intro.intro\nR : Type u_2\ninst✝⁴ : CommRing R\nM : Submonoid R\nS : Type u_1\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : IsLocalization M S\nI : Ideal S\ninst✝ : Ideal.IsPrime I\nJ : Ideal R\nH : J ≤ Ideal.comap (algebraMap R S) I\nhI : IsField (R ⧸ Ideal.comap (algebraMap R S) I)\nr m : R\nhm : m ∈ M\nhM : ¬↑(Ideal.Quotient.mk (Ideal.comap (algebraMap R S) I)) m = 0\nrn : R\nhn :\n ↑(Ideal.quotientMap I (algebraMap R S) (_ : Ideal.comap (algebraMap R S) I ≤ Ideal.comap (algebraMap R S) I))\n (↑(Ideal.Quotient.mk (Ideal.comap (algebraMap R S) I)) m *\n ↑(Ideal.Quotient.mk (Ideal.comap (algebraMap R S) I)) rn) =\n ↑(Ideal.quotientMap I (algebraMap R S) (_ : Ideal.comap (algebraMap R S) I ≤ Ideal.comap (algebraMap R S) I)) 1\n⊢ ↑(Ideal.quotientMap I (algebraMap R S) H) (↑(Ideal.Quotient.mk J) (r * rn)) =\n ↑(Ideal.Quotient.mk I) (mk' S r { val := m, property := hm })",
"state_before": "case neg.intro.intro\nR : Type u_2\ninst✝⁴ : CommRing R\nM : Submonoid R\nS : Type u_1\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : IsLocalization M S\nI : Ideal S\ninst✝ : Ideal.IsPrime I\nJ : Ideal R\nH : J ≤ Ideal.comap (algebraMap R S) I\nhI : IsField (R ⧸ Ideal.comap (algebraMap R S) I)\nr m : R\nhm : m ∈ M\nhM : ¬↑(Ideal.Quotient.mk (Ideal.comap (algebraMap R S) I)) m = 0\nrn : R\nhn :\n ↑(Ideal.Quotient.mk (Ideal.comap (algebraMap R S) I)) m * ↑(Ideal.Quotient.mk (Ideal.comap (algebraMap R S) I)) rn = 1\n⊢ ↑(Ideal.quotientMap I (algebraMap R S) H) (↑(Ideal.Quotient.mk J) (r * rn)) =\n ↑(Ideal.Quotient.mk I) (mk' S r { val := m, property := hm })",
"tactic": "replace hn := congr_arg (Ideal.quotientMap I (algebraMap R S) le_rfl) hn"
},
{
"state_after": "case neg.intro.intro\nR : Type u_2\ninst✝⁴ : CommRing R\nM : Submonoid R\nS : Type u_1\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : IsLocalization M S\nI : Ideal S\ninst✝ : Ideal.IsPrime I\nJ : Ideal R\nH : J ≤ Ideal.comap (algebraMap R S) I\nhI : IsField (R ⧸ Ideal.comap (algebraMap R S) I)\nr m : R\nhm : m ∈ M\nhM : ¬↑(Ideal.Quotient.mk (Ideal.comap (algebraMap R S) I)) m = 0\nrn : R\nhn :\n ↑(Ideal.quotientMap I (algebraMap R S) (_ : Ideal.comap (algebraMap R S) I ≤ Ideal.comap (algebraMap R S) I))\n (↑(Ideal.Quotient.mk (Ideal.comap (algebraMap R S) I)) m) *\n ↑(Ideal.quotientMap I (algebraMap R S) (_ : Ideal.comap (algebraMap R S) I ≤ Ideal.comap (algebraMap R S) I))\n (↑(Ideal.Quotient.mk (Ideal.comap (algebraMap R S) I)) rn) =\n 1\n⊢ ↑(Ideal.quotientMap I (algebraMap R S) H) (↑(Ideal.Quotient.mk J) (r * rn)) =\n ↑(Ideal.Quotient.mk I) (mk' S r { val := m, property := hm })",
"state_before": "case neg.intro.intro\nR : Type u_2\ninst✝⁴ : CommRing R\nM : Submonoid R\nS : Type u_1\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : IsLocalization M S\nI : Ideal S\ninst✝ : Ideal.IsPrime I\nJ : Ideal R\nH : J ≤ Ideal.comap (algebraMap R S) I\nhI : IsField (R ⧸ Ideal.comap (algebraMap R S) I)\nr m : R\nhm : m ∈ M\nhM : ¬↑(Ideal.Quotient.mk (Ideal.comap (algebraMap R S) I)) m = 0\nrn : R\nhn :\n ↑(Ideal.quotientMap I (algebraMap R S) (_ : Ideal.comap (algebraMap R S) I ≤ Ideal.comap (algebraMap R S) I))\n (↑(Ideal.Quotient.mk (Ideal.comap (algebraMap R S) I)) m *\n ↑(Ideal.Quotient.mk (Ideal.comap (algebraMap R S) I)) rn) =\n ↑(Ideal.quotientMap I (algebraMap R S) (_ : Ideal.comap (algebraMap R S) I ≤ Ideal.comap (algebraMap R S) I)) 1\n⊢ ↑(Ideal.quotientMap I (algebraMap R S) H) (↑(Ideal.Quotient.mk J) (r * rn)) =\n ↑(Ideal.Quotient.mk I) (mk' S r { val := m, property := hm })",
"tactic": "rw [RingHom.map_one, RingHom.map_mul] at hn"
},
{
"state_after": "case neg.intro.intro\nR : Type u_2\ninst✝⁴ : CommRing R\nM : Submonoid R\nS : Type u_1\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : IsLocalization M S\nI : Ideal S\ninst✝ : Ideal.IsPrime I\nJ : Ideal R\nH : J ≤ Ideal.comap (algebraMap R S) I\nhI : IsField (R ⧸ Ideal.comap (algebraMap R S) I)\nr m : R\nhm : m ∈ M\nhM : ¬↑(Ideal.Quotient.mk (Ideal.comap (algebraMap R S) I)) m = 0\nrn : R\nhn :\n ↑(Ideal.quotientMap I (algebraMap R S) (_ : Ideal.comap (algebraMap R S) I ≤ Ideal.comap (algebraMap R S) I))\n (↑(Ideal.Quotient.mk (Ideal.comap (algebraMap R S) I)) m) *\n ↑(Ideal.quotientMap I (algebraMap R S) (_ : Ideal.comap (algebraMap R S) I ≤ Ideal.comap (algebraMap R S) I))\n (↑(Ideal.Quotient.mk (Ideal.comap (algebraMap R S) I)) rn) =\n 1\n⊢ ↑(Ideal.Quotient.mk I) (↑(algebraMap R S) (r * rn)) =\n ↑(Ideal.Quotient.mk I) (↑(algebraMap R S) r * mk' S 1 { val := m, property := hm })",
"state_before": "case neg.intro.intro\nR : Type u_2\ninst✝⁴ : CommRing R\nM : Submonoid R\nS : Type u_1\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : IsLocalization M S\nI : Ideal S\ninst✝ : Ideal.IsPrime I\nJ : Ideal R\nH : J ≤ Ideal.comap (algebraMap R S) I\nhI : IsField (R ⧸ Ideal.comap (algebraMap R S) I)\nr m : R\nhm : m ∈ M\nhM : ¬↑(Ideal.Quotient.mk (Ideal.comap (algebraMap R S) I)) m = 0\nrn : R\nhn :\n ↑(Ideal.quotientMap I (algebraMap R S) (_ : Ideal.comap (algebraMap R S) I ≤ Ideal.comap (algebraMap R S) I))\n (↑(Ideal.Quotient.mk (Ideal.comap (algebraMap R S) I)) m) *\n ↑(Ideal.quotientMap I (algebraMap R S) (_ : Ideal.comap (algebraMap R S) I ≤ Ideal.comap (algebraMap R S) I))\n (↑(Ideal.Quotient.mk (Ideal.comap (algebraMap R S) I)) rn) =\n 1\n⊢ ↑(Ideal.quotientMap I (algebraMap R S) H) (↑(Ideal.Quotient.mk J) (r * rn)) =\n ↑(Ideal.Quotient.mk I) (mk' S r { val := m, property := hm })",
"tactic": "rw [Ideal.quotientMap_mk, ← sub_eq_zero, ← RingHom.map_sub, Ideal.Quotient.eq_zero_iff_mem, ←\n Ideal.Quotient.eq_zero_iff_mem, RingHom.map_sub, sub_eq_zero, mk'_eq_mul_mk'_one]"
},
{
"state_after": "case neg.intro.intro\nR : Type u_2\ninst✝⁴ : CommRing R\nM : Submonoid R\nS : Type u_1\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : IsLocalization M S\nI : Ideal S\ninst✝ : Ideal.IsPrime I\nJ : Ideal R\nH : J ≤ Ideal.comap (algebraMap R S) I\nhI : IsField (R ⧸ Ideal.comap (algebraMap R S) I)\nr m : R\nhm : m ∈ M\nhM : ¬↑(Ideal.Quotient.mk (Ideal.comap (algebraMap R S) I)) m = 0\nrn : R\nhn :\n ↑(Ideal.quotientMap I (algebraMap R S) (_ : Ideal.comap (algebraMap R S) I ≤ Ideal.comap (algebraMap R S) I))\n (↑(Ideal.Quotient.mk (Ideal.comap (algebraMap R S) I)) m) *\n ↑(Ideal.quotientMap I (algebraMap R S) (_ : Ideal.comap (algebraMap R S) I ≤ Ideal.comap (algebraMap R S) I))\n (↑(Ideal.Quotient.mk (Ideal.comap (algebraMap R S) I)) rn) =\n 1\n⊢ ↑(Ideal.Quotient.mk I) (↑(algebraMap R S) rn) = ↑(Ideal.Quotient.mk I) (mk' S 1 { val := m, property := hm }) ∨\n ↑(Ideal.Quotient.mk I) (↑(algebraMap R S) r) = 0",
"state_before": "case neg.intro.intro\nR : Type u_2\ninst✝⁴ : CommRing R\nM : Submonoid R\nS : Type u_1\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : IsLocalization M S\nI : Ideal S\ninst✝ : Ideal.IsPrime I\nJ : Ideal R\nH : J ≤ Ideal.comap (algebraMap R S) I\nhI : IsField (R ⧸ Ideal.comap (algebraMap R S) I)\nr m : R\nhm : m ∈ M\nhM : ¬↑(Ideal.Quotient.mk (Ideal.comap (algebraMap R S) I)) m = 0\nrn : R\nhn :\n ↑(Ideal.quotientMap I (algebraMap R S) (_ : Ideal.comap (algebraMap R S) I ≤ Ideal.comap (algebraMap R S) I))\n (↑(Ideal.Quotient.mk (Ideal.comap (algebraMap R S) I)) m) *\n ↑(Ideal.quotientMap I (algebraMap R S) (_ : Ideal.comap (algebraMap R S) I ≤ Ideal.comap (algebraMap R S) I))\n (↑(Ideal.Quotient.mk (Ideal.comap (algebraMap R S) I)) rn) =\n 1\n⊢ ↑(Ideal.Quotient.mk I) (↑(algebraMap R S) (r * rn)) =\n ↑(Ideal.Quotient.mk I) (↑(algebraMap R S) r * mk' S 1 { val := m, property := hm })",
"tactic": "simp only [mul_eq_mul_left_iff, RingHom.map_mul]"
},
{
"state_after": "case neg.intro.intro\nR : Type u_2\ninst✝⁴ : CommRing R\nM : Submonoid R\nS : Type u_1\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : IsLocalization M S\nI : Ideal S\ninst✝ : Ideal.IsPrime I\nJ : Ideal R\nH : J ≤ Ideal.comap (algebraMap R S) I\nhI : IsField (R ⧸ Ideal.comap (algebraMap R S) I)\nr m : R\nhm : m ∈ M\nhM : ¬↑(Ideal.Quotient.mk (Ideal.comap (algebraMap R S) I)) m = 0\nrn : R\nhn :\n ↑(Ideal.quotientMap I (algebraMap R S) (_ : Ideal.comap (algebraMap R S) I ≤ Ideal.comap (algebraMap R S) I))\n (↑(Ideal.Quotient.mk (Ideal.comap (algebraMap R S) I)) m) *\n ↑(Ideal.quotientMap I (algebraMap R S) (_ : Ideal.comap (algebraMap R S) I ≤ Ideal.comap (algebraMap R S) I))\n (↑(Ideal.Quotient.mk (Ideal.comap (algebraMap R S) I)) rn) =\n 1\n⊢ 1 = ↑(Ideal.Quotient.mk I) (↑(algebraMap R S) m) * ↑(Ideal.Quotient.mk I) (mk' S 1 { val := m, property := hm })",
"state_before": "case neg.intro.intro\nR : Type u_2\ninst✝⁴ : CommRing R\nM : Submonoid R\nS : Type u_1\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : IsLocalization M S\nI : Ideal S\ninst✝ : Ideal.IsPrime I\nJ : Ideal R\nH : J ≤ Ideal.comap (algebraMap R S) I\nhI : IsField (R ⧸ Ideal.comap (algebraMap R S) I)\nr m : R\nhm : m ∈ M\nhM : ¬↑(Ideal.Quotient.mk (Ideal.comap (algebraMap R S) I)) m = 0\nrn : R\nhn :\n ↑(Ideal.quotientMap I (algebraMap R S) (_ : Ideal.comap (algebraMap R S) I ≤ Ideal.comap (algebraMap R S) I))\n (↑(Ideal.Quotient.mk (Ideal.comap (algebraMap R S) I)) m) *\n ↑(Ideal.quotientMap I (algebraMap R S) (_ : Ideal.comap (algebraMap R S) I ≤ Ideal.comap (algebraMap R S) I))\n (↑(Ideal.Quotient.mk (Ideal.comap (algebraMap R S) I)) rn) =\n 1\n⊢ ↑(Ideal.Quotient.mk I) (↑(algebraMap R S) rn) = ↑(Ideal.Quotient.mk I) (mk' S 1 { val := m, property := hm }) ∨\n ↑(Ideal.Quotient.mk I) (↑(algebraMap R S) r) = 0",
"tactic": "refine\n Or.inl\n (mul_left_cancel₀ (M₀ := S ⧸ I)\n (fun hn =>\n hM\n (Ideal.Quotient.eq_zero_iff_mem.2\n (Ideal.mem_comap.2 (Ideal.Quotient.eq_zero_iff_mem.1 hn))))\n (_root_.trans hn ?_))"
},
{
"state_after": "case neg.intro.intro\nR : Type u_2\ninst✝⁴ : CommRing R\nM : Submonoid R\nS : Type u_1\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : IsLocalization M S\nI : Ideal S\ninst✝ : Ideal.IsPrime I\nJ : Ideal R\nH : J ≤ Ideal.comap (algebraMap R S) I\nhI : IsField (R ⧸ Ideal.comap (algebraMap R S) I)\nr m : R\nhm : m ∈ M\nhM : ¬↑(Ideal.Quotient.mk (Ideal.comap (algebraMap R S) I)) m = 0\nrn : R\nhn :\n ↑(Ideal.quotientMap I (algebraMap R S) (_ : Ideal.comap (algebraMap R S) I ≤ Ideal.comap (algebraMap R S) I))\n (↑(Ideal.Quotient.mk (Ideal.comap (algebraMap R S) I)) m) *\n ↑(Ideal.quotientMap I (algebraMap R S) (_ : Ideal.comap (algebraMap R S) I ≤ Ideal.comap (algebraMap R S) I))\n (↑(Ideal.Quotient.mk (Ideal.comap (algebraMap R S) I)) rn) =\n 1\n⊢ 1 = ↑(Ideal.Quotient.mk I) (↑(algebraMap R S) m * mk' S 1 { val := m, property := hm })",
"state_before": "case neg.intro.intro\nR : Type u_2\ninst✝⁴ : CommRing R\nM : Submonoid R\nS : Type u_1\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : IsLocalization M S\nI : Ideal S\ninst✝ : Ideal.IsPrime I\nJ : Ideal R\nH : J ≤ Ideal.comap (algebraMap R S) I\nhI : IsField (R ⧸ Ideal.comap (algebraMap R S) I)\nr m : R\nhm : m ∈ M\nhM : ¬↑(Ideal.Quotient.mk (Ideal.comap (algebraMap R S) I)) m = 0\nrn : R\nhn :\n ↑(Ideal.quotientMap I (algebraMap R S) (_ : Ideal.comap (algebraMap R S) I ≤ Ideal.comap (algebraMap R S) I))\n (↑(Ideal.Quotient.mk (Ideal.comap (algebraMap R S) I)) m) *\n ↑(Ideal.quotientMap I (algebraMap R S) (_ : Ideal.comap (algebraMap R S) I ≤ Ideal.comap (algebraMap R S) I))\n (↑(Ideal.Quotient.mk (Ideal.comap (algebraMap R S) I)) rn) =\n 1\n⊢ 1 = ↑(Ideal.Quotient.mk I) (↑(algebraMap R S) m) * ↑(Ideal.Quotient.mk I) (mk' S 1 { val := m, property := hm })",
"tactic": "refine Eq.trans ?_ (RingHom.map_mul (Ideal.Quotient.mk I) (algebraMap R S m) (mk' S 1 ⟨m, hm⟩))"
},
{
"state_after": "no goals",
"state_before": "case neg.intro.intro\nR : Type u_2\ninst✝⁴ : CommRing R\nM : Submonoid R\nS : Type u_1\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : IsLocalization M S\nI : Ideal S\ninst✝ : Ideal.IsPrime I\nJ : Ideal R\nH : J ≤ Ideal.comap (algebraMap R S) I\nhI : IsField (R ⧸ Ideal.comap (algebraMap R S) I)\nr m : R\nhm : m ∈ M\nhM : ¬↑(Ideal.Quotient.mk (Ideal.comap (algebraMap R S) I)) m = 0\nrn : R\nhn :\n ↑(Ideal.quotientMap I (algebraMap R S) (_ : Ideal.comap (algebraMap R S) I ≤ Ideal.comap (algebraMap R S) I))\n (↑(Ideal.Quotient.mk (Ideal.comap (algebraMap R S) I)) m) *\n ↑(Ideal.quotientMap I (algebraMap R S) (_ : Ideal.comap (algebraMap R S) I ≤ Ideal.comap (algebraMap R S) I))\n (↑(Ideal.Quotient.mk (Ideal.comap (algebraMap R S) I)) rn) =\n 1\n⊢ 1 = ↑(Ideal.Quotient.mk I) (↑(algebraMap R S) m * mk' S 1 { val := m, property := hm })",
"tactic": "rw [← mk'_eq_mul_mk'_one, mk'_self, RingHom.map_one]"
}
] |
[
208,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
175,
1
] |
Mathlib/Order/SuccPred/Basic.lean
|
Pred.rec_top
|
[] |
[
1513,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1512,
1
] |
Mathlib/ModelTheory/Satisfiability.lean
|
FirstOrder.Language.Theory.isSatisfiable_union_distinctConstantsTheory_of_card_le
|
[
{
"state_after": "L : Language\nT✝ : Theory L\nα : Type w\nn : ℕ\nT' T : Theory L\ns : Set α\nM : Type w'\ninst✝² : Nonempty M\ninst✝¹ : Structure L M\ninst✝ : M ⊨ T\nh : lift (#↑s) ≤ lift (#M)\nthis : Inhabited M\n⊢ IsSatisfiable (LHom.onTheory (lhomWithConstants L α) T ∪ distinctConstantsTheory L s)",
"state_before": "L : Language\nT✝ : Theory L\nα : Type w\nn : ℕ\nT' T : Theory L\ns : Set α\nM : Type w'\ninst✝² : Nonempty M\ninst✝¹ : Structure L M\ninst✝ : M ⊨ T\nh : lift (#↑s) ≤ lift (#M)\n⊢ IsSatisfiable (LHom.onTheory (lhomWithConstants L α) T ∪ distinctConstantsTheory L s)",
"tactic": "haveI : Inhabited M := Classical.inhabited_of_nonempty inferInstance"
},
{
"state_after": "L : Language\nT✝ : Theory L\nα : Type w\nn : ℕ\nT' T : Theory L\ns : Set α\nM : Type w'\ninst✝² : Nonempty M\ninst✝¹ : Structure L M\ninst✝ : M ⊨ T\nh : Nonempty (↑s ↪ M)\nthis : Inhabited M\n⊢ IsSatisfiable (LHom.onTheory (lhomWithConstants L α) T ∪ distinctConstantsTheory L s)",
"state_before": "L : Language\nT✝ : Theory L\nα : Type w\nn : ℕ\nT' T : Theory L\ns : Set α\nM : Type w'\ninst✝² : Nonempty M\ninst✝¹ : Structure L M\ninst✝ : M ⊨ T\nh : lift (#↑s) ≤ lift (#M)\nthis : Inhabited M\n⊢ IsSatisfiable (LHom.onTheory (lhomWithConstants L α) T ∪ distinctConstantsTheory L s)",
"tactic": "rw [Cardinal.lift_mk_le'] at h"
},
{
"state_after": "L : Language\nT✝ : Theory L\nα : Type w\nn : ℕ\nT' T : Theory L\ns : Set α\nM : Type w'\ninst✝² : Nonempty M\ninst✝¹ : Structure L M\ninst✝ : M ⊨ T\nh : Nonempty (↑s ↪ M)\nthis✝ : Inhabited M\nthis : Structure (constantsOn α) M := constantsOn.structure (Function.extend Subtype.val (↑(Nonempty.some h)) default)\n⊢ IsSatisfiable (LHom.onTheory (lhomWithConstants L α) T ∪ distinctConstantsTheory L s)",
"state_before": "L : Language\nT✝ : Theory L\nα : Type w\nn : ℕ\nT' T : Theory L\ns : Set α\nM : Type w'\ninst✝² : Nonempty M\ninst✝¹ : Structure L M\ninst✝ : M ⊨ T\nh : Nonempty (↑s ↪ M)\nthis : Inhabited M\n⊢ IsSatisfiable (LHom.onTheory (lhomWithConstants L α) T ∪ distinctConstantsTheory L s)",
"tactic": "letI : (constantsOn α).Structure M := constantsOn.structure (Function.extend (↑) h.some default)"
},
{
"state_after": "L : Language\nT✝ : Theory L\nα : Type w\nn : ℕ\nT' T : Theory L\ns : Set α\nM : Type w'\ninst✝² : Nonempty M\ninst✝¹ : Structure L M\ninst✝ : M ⊨ T\nh : Nonempty (↑s ↪ M)\nthis✝¹ : Inhabited M\nthis✝ : Structure (constantsOn α) M := constantsOn.structure (Function.extend Subtype.val (↑(Nonempty.some h)) default)\nthis : M ⊨ LHom.onTheory (lhomWithConstants L α) T ∪ distinctConstantsTheory L s\n⊢ IsSatisfiable (LHom.onTheory (lhomWithConstants L α) T ∪ distinctConstantsTheory L s)",
"state_before": "L : Language\nT✝ : Theory L\nα : Type w\nn : ℕ\nT' T : Theory L\ns : Set α\nM : Type w'\ninst✝² : Nonempty M\ninst✝¹ : Structure L M\ninst✝ : M ⊨ T\nh : Nonempty (↑s ↪ M)\nthis✝ : Inhabited M\nthis : Structure (constantsOn α) M := constantsOn.structure (Function.extend Subtype.val (↑(Nonempty.some h)) default)\n⊢ IsSatisfiable (LHom.onTheory (lhomWithConstants L α) T ∪ distinctConstantsTheory L s)",
"tactic": "have : M ⊨ (L.lhomWithConstants α).onTheory T ∪ L.distinctConstantsTheory s := by\n refine' ((LHom.onTheory_model _ _).2 inferInstance).union _\n rw [model_distinctConstantsTheory]\n refine' fun a as b bs ab => _\n rw [← Subtype.coe_mk a as, ← Subtype.coe_mk b bs, ← Subtype.ext_iff]\n exact\n h.some.injective\n ((Subtype.coe_injective.extend_apply h.some default ⟨a, as⟩).symm.trans\n (ab.trans (Subtype.coe_injective.extend_apply h.some default ⟨b, bs⟩)))"
},
{
"state_after": "no goals",
"state_before": "L : Language\nT✝ : Theory L\nα : Type w\nn : ℕ\nT' T : Theory L\ns : Set α\nM : Type w'\ninst✝² : Nonempty M\ninst✝¹ : Structure L M\ninst✝ : M ⊨ T\nh : Nonempty (↑s ↪ M)\nthis✝¹ : Inhabited M\nthis✝ : Structure (constantsOn α) M := constantsOn.structure (Function.extend Subtype.val (↑(Nonempty.some h)) default)\nthis : M ⊨ LHom.onTheory (lhomWithConstants L α) T ∪ distinctConstantsTheory L s\n⊢ IsSatisfiable (LHom.onTheory (lhomWithConstants L α) T ∪ distinctConstantsTheory L s)",
"tactic": "exact Model.isSatisfiable M"
},
{
"state_after": "L : Language\nT✝ : Theory L\nα : Type w\nn : ℕ\nT' T : Theory L\ns : Set α\nM : Type w'\ninst✝² : Nonempty M\ninst✝¹ : Structure L M\ninst✝ : M ⊨ T\nh : Nonempty (↑s ↪ M)\nthis✝ : Inhabited M\nthis : Structure (constantsOn α) M := constantsOn.structure (Function.extend Subtype.val (↑(Nonempty.some h)) default)\n⊢ M ⊨ distinctConstantsTheory L s",
"state_before": "L : Language\nT✝ : Theory L\nα : Type w\nn : ℕ\nT' T : Theory L\ns : Set α\nM : Type w'\ninst✝² : Nonempty M\ninst✝¹ : Structure L M\ninst✝ : M ⊨ T\nh : Nonempty (↑s ↪ M)\nthis✝ : Inhabited M\nthis : Structure (constantsOn α) M := constantsOn.structure (Function.extend Subtype.val (↑(Nonempty.some h)) default)\n⊢ M ⊨ LHom.onTheory (lhomWithConstants L α) T ∪ distinctConstantsTheory L s",
"tactic": "refine' ((LHom.onTheory_model _ _).2 inferInstance).union _"
},
{
"state_after": "L : Language\nT✝ : Theory L\nα : Type w\nn : ℕ\nT' T : Theory L\ns : Set α\nM : Type w'\ninst✝² : Nonempty M\ninst✝¹ : Structure L M\ninst✝ : M ⊨ T\nh : Nonempty (↑s ↪ M)\nthis✝ : Inhabited M\nthis : Structure (constantsOn α) M := constantsOn.structure (Function.extend Subtype.val (↑(Nonempty.some h)) default)\n⊢ Set.InjOn (fun i => ↑(Language.con L i)) s",
"state_before": "L : Language\nT✝ : Theory L\nα : Type w\nn : ℕ\nT' T : Theory L\ns : Set α\nM : Type w'\ninst✝² : Nonempty M\ninst✝¹ : Structure L M\ninst✝ : M ⊨ T\nh : Nonempty (↑s ↪ M)\nthis✝ : Inhabited M\nthis : Structure (constantsOn α) M := constantsOn.structure (Function.extend Subtype.val (↑(Nonempty.some h)) default)\n⊢ M ⊨ distinctConstantsTheory L s",
"tactic": "rw [model_distinctConstantsTheory]"
},
{
"state_after": "L : Language\nT✝ : Theory L\nα : Type w\nn : ℕ\nT' T : Theory L\ns : Set α\nM : Type w'\ninst✝² : Nonempty M\ninst✝¹ : Structure L M\ninst✝ : M ⊨ T\nh : Nonempty (↑s ↪ M)\nthis✝ : Inhabited M\nthis : Structure (constantsOn α) M := constantsOn.structure (Function.extend Subtype.val (↑(Nonempty.some h)) default)\na : α\nas : a ∈ s\nb : α\nbs : b ∈ s\nab : (fun i => ↑(Language.con L i)) a = (fun i => ↑(Language.con L i)) b\n⊢ a = b",
"state_before": "L : Language\nT✝ : Theory L\nα : Type w\nn : ℕ\nT' T : Theory L\ns : Set α\nM : Type w'\ninst✝² : Nonempty M\ninst✝¹ : Structure L M\ninst✝ : M ⊨ T\nh : Nonempty (↑s ↪ M)\nthis✝ : Inhabited M\nthis : Structure (constantsOn α) M := constantsOn.structure (Function.extend Subtype.val (↑(Nonempty.some h)) default)\n⊢ Set.InjOn (fun i => ↑(Language.con L i)) s",
"tactic": "refine' fun a as b bs ab => _"
},
{
"state_after": "L : Language\nT✝ : Theory L\nα : Type w\nn : ℕ\nT' T : Theory L\ns : Set α\nM : Type w'\ninst✝² : Nonempty M\ninst✝¹ : Structure L M\ninst✝ : M ⊨ T\nh : Nonempty (↑s ↪ M)\nthis✝ : Inhabited M\nthis : Structure (constantsOn α) M := constantsOn.structure (Function.extend Subtype.val (↑(Nonempty.some h)) default)\na : α\nas : a ∈ s\nb : α\nbs : b ∈ s\nab : (fun i => ↑(Language.con L i)) a = (fun i => ↑(Language.con L i)) b\n⊢ { val := a, property := as } = { val := b, property := bs }",
"state_before": "L : Language\nT✝ : Theory L\nα : Type w\nn : ℕ\nT' T : Theory L\ns : Set α\nM : Type w'\ninst✝² : Nonempty M\ninst✝¹ : Structure L M\ninst✝ : M ⊨ T\nh : Nonempty (↑s ↪ M)\nthis✝ : Inhabited M\nthis : Structure (constantsOn α) M := constantsOn.structure (Function.extend Subtype.val (↑(Nonempty.some h)) default)\na : α\nas : a ∈ s\nb : α\nbs : b ∈ s\nab : (fun i => ↑(Language.con L i)) a = (fun i => ↑(Language.con L i)) b\n⊢ a = b",
"tactic": "rw [← Subtype.coe_mk a as, ← Subtype.coe_mk b bs, ← Subtype.ext_iff]"
},
{
"state_after": "no goals",
"state_before": "L : Language\nT✝ : Theory L\nα : Type w\nn : ℕ\nT' T : Theory L\ns : Set α\nM : Type w'\ninst✝² : Nonempty M\ninst✝¹ : Structure L M\ninst✝ : M ⊨ T\nh : Nonempty (↑s ↪ M)\nthis✝ : Inhabited M\nthis : Structure (constantsOn α) M := constantsOn.structure (Function.extend Subtype.val (↑(Nonempty.some h)) default)\na : α\nas : a ∈ s\nb : α\nbs : b ∈ s\nab : (fun i => ↑(Language.con L i)) a = (fun i => ↑(Language.con L i)) b\n⊢ { val := a, property := as } = { val := b, property := bs }",
"tactic": "exact\n h.some.injective\n ((Subtype.coe_injective.extend_apply h.some default ⟨a, as⟩).symm.trans\n (ab.trans (Subtype.coe_injective.extend_apply h.some default ⟨b, bs⟩)))"
}
] |
[
157,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
141,
1
] |
Mathlib/Topology/Sheaves/Sheaf.lean
|
TopCat.Presheaf.isSheaf_of_iso
|
[] |
[
107,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
106,
1
] |
Mathlib/Analysis/Normed/Group/Pointwise.lean
|
smul_ball_one
|
[
{
"state_after": "no goals",
"state_before": "E : Type u_1\ninst✝ : SeminormedCommGroup E\nε δ : ℝ\ns t : Set E\nx y : E\n⊢ x • ball 1 δ = ball x δ",
"tactic": "rw [smul_ball, smul_eq_mul, mul_one]"
}
] |
[
152,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
151,
1
] |
Std/Data/Rat/Lemmas.lean
|
Rat.normalize_eq_mkRat
|
[
{
"state_after": "no goals",
"state_before": "num : Int\nden : Nat\nden_nz : den ≠ 0\n⊢ normalize num den = mkRat num den",
"tactic": "simp [mkRat, den_nz]"
}
] |
[
83,
23
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
82,
1
] |
Mathlib/Data/Fin/Tuple/Basic.lean
|
Fin.repeat_succ
|
[
{
"state_after": "m✝ n : ℕ\nα✝ : Fin (n + 1) → Type u\nx : α✝ 0\nq : (i : Fin (n + 1)) → α✝ i\np : (i : Fin n) → α✝ (succ i)\ni : Fin n\ny : α✝ (succ i)\nz : α✝ 0\nα : Type u_1\na : Fin n → α\nm : ℕ\nh : Nat.succ m * n = n + m * n\n⊢ repeat (Nat.succ m) a = append a (repeat m a) ∘ ↑(cast h)",
"state_before": "m✝ n : ℕ\nα✝ : Fin (n + 1) → Type u\nx : α✝ 0\nq : (i : Fin (n + 1)) → α✝ i\np : (i : Fin n) → α✝ (succ i)\ni : Fin n\ny : α✝ (succ i)\nz : α✝ 0\nα : Type u_1\na : Fin n → α\nm : ℕ\n⊢ repeat (Nat.succ m) a = append a (repeat m a) ∘ ↑(cast (_ : Nat.succ m * n = n + m * n))",
"tactic": "generalize_proofs h"
},
{
"state_after": "case h\nm✝ n : ℕ\nα✝ : Fin (n + 1) → Type u\nx : α✝ 0\nq : (i : Fin (n + 1)) → α✝ i\np : (i : Fin n) → α✝ (succ i)\ni : Fin n\ny : α✝ (succ i)\nz : α✝ 0\nα : Type u_1\na : Fin n → α\nm : ℕ\nh : Nat.succ m * n = n + m * n\n⊢ ∀ (x : Fin (Nat.succ m * n)), repeat (Nat.succ m) a x = (append a (repeat m a) ∘ ↑(cast h)) x",
"state_before": "m✝ n : ℕ\nα✝ : Fin (n + 1) → Type u\nx : α✝ 0\nq : (i : Fin (n + 1)) → α✝ i\np : (i : Fin n) → α✝ (succ i)\ni : Fin n\ny : α✝ (succ i)\nz : α✝ 0\nα : Type u_1\na : Fin n → α\nm : ℕ\nh : Nat.succ m * n = n + m * n\n⊢ repeat (Nat.succ m) a = append a (repeat m a) ∘ ↑(cast h)",
"tactic": "apply funext"
},
{
"state_after": "case h\nm✝ n : ℕ\nα✝ : Fin (n + 1) → Type u\nx : α✝ 0\nq : (i : Fin (n + 1)) → α✝ i\np : (i : Fin n) → α✝ (succ i)\ni : Fin n\ny : α✝ (succ i)\nz : α✝ 0\nα : Type u_1\na : Fin n → α\nm : ℕ\nh : Nat.succ m * n = n + m * n\n⊢ ∀ (x : Fin (n + m * n)),\n repeat (Nat.succ m) a (↑(cast (_ : n + m * n = Nat.succ m * n)) x) =\n (append a (repeat m a) ∘ ↑(cast h)) (↑(cast (_ : n + m * n = Nat.succ m * n)) x)",
"state_before": "case h\nm✝ n : ℕ\nα✝ : Fin (n + 1) → Type u\nx : α✝ 0\nq : (i : Fin (n + 1)) → α✝ i\np : (i : Fin n) → α✝ (succ i)\ni : Fin n\ny : α✝ (succ i)\nz : α✝ 0\nα : Type u_1\na : Fin n → α\nm : ℕ\nh : Nat.succ m * n = n + m * n\n⊢ ∀ (x : Fin (Nat.succ m * n)), repeat (Nat.succ m) a x = (append a (repeat m a) ∘ ↑(cast h)) x",
"tactic": "rw [(Fin.cast h.symm).surjective.forall]"
},
{
"state_after": "case h.refine'_1\nm✝ n : ℕ\nα✝ : Fin (n + 1) → Type u\nx : α✝ 0\nq : (i : Fin (n + 1)) → α✝ i\np : (i : Fin n) → α✝ (succ i)\ni : Fin n\ny : α✝ (succ i)\nz : α✝ 0\nα : Type u_1\na : Fin n → α\nm : ℕ\nh : Nat.succ m * n = n + m * n\nl : Fin n\n⊢ repeat (Nat.succ m) a (↑(cast (_ : n + m * n = Nat.succ m * n)) (↑(castAdd (m * n)) l)) =\n (append a (repeat m a) ∘ ↑(cast h)) (↑(cast (_ : n + m * n = Nat.succ m * n)) (↑(castAdd (m * n)) l))\n\ncase h.refine'_2\nm✝ n : ℕ\nα✝ : Fin (n + 1) → Type u\nx : α✝ 0\nq : (i : Fin (n + 1)) → α✝ i\np : (i : Fin n) → α✝ (succ i)\ni : Fin n\ny : α✝ (succ i)\nz : α✝ 0\nα : Type u_1\na : Fin n → α\nm : ℕ\nh : Nat.succ m * n = n + m * n\nr : Fin (m * n)\n⊢ repeat (Nat.succ m) a (↑(cast (_ : n + m * n = Nat.succ m * n)) (↑(natAdd n) r)) =\n (append a (repeat m a) ∘ ↑(cast h)) (↑(cast (_ : n + m * n = Nat.succ m * n)) (↑(natAdd n) r))",
"state_before": "case h\nm✝ n : ℕ\nα✝ : Fin (n + 1) → Type u\nx : α✝ 0\nq : (i : Fin (n + 1)) → α✝ i\np : (i : Fin n) → α✝ (succ i)\ni : Fin n\ny : α✝ (succ i)\nz : α✝ 0\nα : Type u_1\na : Fin n → α\nm : ℕ\nh : Nat.succ m * n = n + m * n\n⊢ ∀ (x : Fin (n + m * n)),\n repeat (Nat.succ m) a (↑(cast (_ : n + m * n = Nat.succ m * n)) x) =\n (append a (repeat m a) ∘ ↑(cast h)) (↑(cast (_ : n + m * n = Nat.succ m * n)) x)",
"tactic": "refine' Fin.addCases (fun l => _) fun r => _"
},
{
"state_after": "no goals",
"state_before": "case h.refine'_1\nm✝ n : ℕ\nα✝ : Fin (n + 1) → Type u\nx : α✝ 0\nq : (i : Fin (n + 1)) → α✝ i\np : (i : Fin n) → α✝ (succ i)\ni : Fin n\ny : α✝ (succ i)\nz : α✝ 0\nα : Type u_1\na : Fin n → α\nm : ℕ\nh : Nat.succ m * n = n + m * n\nl : Fin n\n⊢ repeat (Nat.succ m) a (↑(cast (_ : n + m * n = Nat.succ m * n)) (↑(castAdd (m * n)) l)) =\n (append a (repeat m a) ∘ ↑(cast h)) (↑(cast (_ : n + m * n = Nat.succ m * n)) (↑(castAdd (m * n)) l))",
"tactic": "simp [modNat, Nat.mod_eq_of_lt l.is_lt]"
},
{
"state_after": "no goals",
"state_before": "case h.refine'_2\nm✝ n : ℕ\nα✝ : Fin (n + 1) → Type u\nx : α✝ 0\nq : (i : Fin (n + 1)) → α✝ i\np : (i : Fin n) → α✝ (succ i)\ni : Fin n\ny : α✝ (succ i)\nz : α✝ 0\nα : Type u_1\na : Fin n → α\nm : ℕ\nh : Nat.succ m * n = n + m * n\nr : Fin (m * n)\n⊢ repeat (Nat.succ m) a (↑(cast (_ : n + m * n = Nat.succ m * n)) (↑(natAdd n) r)) =\n (append a (repeat m a) ∘ ↑(cast h)) (↑(cast (_ : n + m * n = Nat.succ m * n)) (↑(natAdd n) r))",
"tactic": "simp [modNat]"
}
] |
[
404,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
396,
1
] |
Mathlib/Geometry/Manifold/SmoothManifoldWithCorners.lean
|
map_extChartAt_nhds
|
[] |
[
1104,
54
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1103,
1
] |
Mathlib/Data/Finset/Basic.lean
|
Finset.mem_of_mem_filter
|
[] |
[
2664,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2663,
1
] |
Mathlib/Combinatorics/SimpleGraph/Connectivity.lean
|
SimpleGraph.Walk.support_map
|
[
{
"state_after": "no goals",
"state_before": "V : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nf : G →g G'\nf' : G' →g G''\nu v u' v' : V\np : Walk G u v\n⊢ support (Walk.map f p) = List.map (↑f) (support p)",
"tactic": "induction p <;> simp [*]"
}
] |
[
1511,
89
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1511,
1
] |
Std/Logic.lean
|
ne_self_iff_false
|
[] |
[
84,
77
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
84,
1
] |
Mathlib/Geometry/Euclidean/Inversion.lean
|
EuclideanGeometry.inversion_self
|
[
{
"state_after": "no goals",
"state_before": "V : Type u_2\nP : Type u_1\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\na b c✝ d x y z : P\nR✝ : ℝ\nc : P\nR : ℝ\n⊢ inversion c R c = c",
"tactic": "simp [inversion]"
}
] |
[
51,
84
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
51,
1
] |
Std/Classes/LawfulMonad.lean
|
SatisfiesM_ReaderT_eq
|
[
{
"state_after": "no goals",
"state_before": "m : Type u_1 → Type u_2\nρ α✝ : Type u_1\np : α✝ → Prop\nx : ReaderT ρ m α✝\ninst✝ : Monad m\na : ReaderT ρ m { a // p a }\n⊢ Subtype.val <$> a = x ↔ ∀ (x_1 : ρ), Subtype.val <$> a x_1 = x x_1",
"tactic": "exact ⟨fun eq _ => eq ▸ rfl, funext⟩"
}
] |
[
197,
94
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
195,
9
] |
Mathlib/CategoryTheory/Limits/Shapes/BinaryProducts.lean
|
CategoryTheory.Limits.coprodComparison_inl
|
[] |
[
1345,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1344,
1
] |
Mathlib/Analysis/Convex/Contractible.lean
|
Convex.contractibleSpace
|
[] |
[
48,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
45,
11
] |
Mathlib/Analysis/Asymptotics/Theta.lean
|
Asymptotics.IsTheta.trans_isLittleO
|
[] |
[
122,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
120,
1
] |
Mathlib/Combinatorics/SimpleGraph/Basic.lean
|
SimpleGraph.fromEdgeSet_adj
|
[] |
[
619,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
618,
1
] |
Mathlib/RingTheory/FractionalIdeal.lean
|
FractionalIdeal.map_symm_map
|
[
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝⁶ : CommRing R\nS : Submonoid R\nP : Type u_3\ninst✝⁵ : CommRing P\ninst✝⁴ : Algebra R P\nloc : IsLocalization S P\nP' : Type u_2\ninst✝³ : CommRing P'\ninst✝² : Algebra R P'\nloc' : IsLocalization S P'\nP'' : Type ?u.574326\ninst✝¹ : CommRing P''\ninst✝ : Algebra R P''\nloc'' : IsLocalization S P''\nI✝ J : FractionalIdeal S P\ng✝ : P →ₐ[R] P'\nI : FractionalIdeal S P'\ng : P ≃ₐ[R] P'\n⊢ map (↑g) (map (↑(AlgEquiv.symm g)) I) = I",
"tactic": "rw [← map_comp, g.comp_symm, map_id]"
}
] |
[
797,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
795,
1
] |
Mathlib/Order/SupIndep.lean
|
CompleteLattice.independent_empty
|
[] |
[
272,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
271,
1
] |
Mathlib/Algebra/Lie/Basic.lean
|
LieModuleHom.coe_neg
|
[] |
[
897,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
896,
1
] |
Mathlib/Analysis/Calculus/FDeriv/Prod.lean
|
fderiv.fst
|
[] |
[
220,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
218,
1
] |
Mathlib/Algebra/Hom/Units.lean
|
IsUnit.mul_one_div_cancel
|
[
{
"state_after": "no goals",
"state_before": "F : Type ?u.54504\nG : Type ?u.54507\nα : Type u_1\nM : Type ?u.54513\nN : Type ?u.54516\ninst✝ : DivisionMonoid α\na b c : α\nh : IsUnit a\n⊢ a * (1 / a) = 1",
"tactic": "simp [h]"
}
] |
[
379,
85
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
379,
11
] |
Mathlib/Data/Seq/WSeq.lean
|
Stream'.WSeq.destruct_tail
|
[
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\ns : WSeq α\n⊢ (fun o => Option.rec nil Prod.snd o) <$> destruct s >>= destruct = do\n let x ← destruct s\n match x with\n | none => Computation.pure none\n | some (fst, s) => destruct s",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ns : WSeq α\n⊢ destruct (tail s) = destruct s >>= tail.aux",
"tactic": "simp [tail]"
},
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\ns : WSeq α\n⊢ (destruct s >>= fun x => Pure.pure (Option.rec nil Prod.snd x) >>= destruct) = do\n let x ← destruct s\n match x with\n | none => Computation.pure none\n | some (fst, s) => destruct s",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ns : WSeq α\n⊢ (fun o => Option.rec nil Prod.snd o) <$> destruct s >>= destruct = do\n let x ← destruct s\n match x with\n | none => Computation.pure none\n | some (fst, s) => destruct s",
"tactic": "rw [← bind_pure_comp, LawfulMonad.bind_assoc]"
},
{
"state_after": "case h\nα : Type u\nβ : Type v\nγ : Type w\ns : WSeq α\n⊢ (fun x => Pure.pure (Option.rec nil Prod.snd x) >>= destruct) = fun x =>\n match x with\n | none => Computation.pure none\n | some (fst, s) => destruct s",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ns : WSeq α\n⊢ (destruct s >>= fun x => Pure.pure (Option.rec nil Prod.snd x) >>= destruct) = do\n let x ← destruct s\n match x with\n | none => Computation.pure none\n | some (fst, s) => destruct s",
"tactic": "apply congr_arg"
},
{
"state_after": "no goals",
"state_before": "case h\nα : Type u\nβ : Type v\nγ : Type w\ns : WSeq α\n⊢ (fun x => Pure.pure (Option.rec nil Prod.snd x) >>= destruct) = fun x =>\n match x with\n | none => Computation.pure none\n | some (fst, s) => destruct s",
"tactic": "ext1 (_ | ⟨a, s⟩) <;> apply (@pure_bind Computation _ _ _ _ _ _).trans _ <;> simp"
}
] |
[
807,
101
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
805,
1
] |
Mathlib/Data/Nat/Digits.lean
|
Nat.NormDigits.digits_one
|
[
{
"state_after": "n✝ b n : ℕ\nn0 : 0 < n\nnb : n < b\nb2 : 1 < b\n⊢ digits b n = [n] ∧ 1 < b ∧ 0 < n",
"state_before": "n✝ b n : ℕ\nn0 : 0 < n\nnb : n < b\n⊢ digits b n = [n] ∧ 1 < b ∧ 0 < n",
"tactic": "have b2 : 1 < b :=\n lt_iff_add_one_le.mpr (le_trans (add_le_add_right (lt_iff_add_one_le.mp n0) 1) nb)"
},
{
"state_after": "n✝ b n : ℕ\nn0 : 0 < n\nnb : n < b\nb2 : 1 < b\n⊢ digits b n = [n]",
"state_before": "n✝ b n : ℕ\nn0 : 0 < n\nnb : n < b\nb2 : 1 < b\n⊢ digits b n = [n] ∧ 1 < b ∧ 0 < n",
"tactic": "refine' ⟨_, b2, n0⟩"
},
{
"state_after": "no goals",
"state_before": "n✝ b n : ℕ\nn0 : 0 < n\nnb : n < b\nb2 : 1 < b\n⊢ digits b n = [n]",
"tactic": "rw [Nat.digits_def' b2 n0, Nat.mod_eq_of_lt nb,\n (Nat.div_eq_zero_iff ((zero_le n).trans_lt nb)).2 nb, Nat.digits_zero]"
}
] |
[
647,
75
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
642,
1
] |
Mathlib/Order/Atoms.lean
|
IsCoatom.le_iff
|
[] |
[
163,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
162,
1
] |
Mathlib/Data/Real/ENNReal.lean
|
ENNReal.coe_lt_coe_nat
|
[] |
[
850,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
849,
1
] |
Mathlib/Algebra/Algebra/Basic.lean
|
LinearMap.map_mul_algebraMap
|
[
{
"state_after": "no goals",
"state_before": "R : Type u_1\nA : Type u_2\nB : Type u_3\ninst✝⁴ : CommSemiring R\ninst✝³ : Semiring A\ninst✝² : Semiring B\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : A →ₗ[R] B\na : A\nr : R\n⊢ ↑f (a * ↑(algebraMap R A) r) = ↑f a * ↑(algebraMap R B) r",
"tactic": "rw [← Algebra.commutes, ← Algebra.commutes, map_algebraMap_mul]"
}
] |
[
704,
66
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
702,
1
] |
Mathlib/Data/Polynomial/Degree/TrailingDegree.lean
|
Polynomial.trailingDegree_ne_of_natTrailingDegree_ne
|
[
{
"state_after": "R : Type u\nS : Type v\na b : R\nn✝ m : ℕ\ninst✝ : Semiring R\np q r : R[X]\nn : ℕ\nthis : ↑n = ↑n\n⊢ natTrailingDegree p ≠ n → trailingDegree p ≠ ↑n",
"state_before": "R : Type u\nS : Type v\na b : R\nn✝ m : ℕ\ninst✝ : Semiring R\np q r : R[X]\nn : ℕ\n⊢ natTrailingDegree p ≠ n → trailingDegree p ≠ ↑n",
"tactic": "have : Nat.cast n = WithTop.some n := rfl"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\na b : R\nn✝ m : ℕ\ninst✝ : Semiring R\np q r : R[X]\nn : ℕ\nthis : ↑n = ↑n\n⊢ natTrailingDegree p ≠ n → trailingDegree p ≠ ↑n",
"tactic": "exact mt fun h => by rw [natTrailingDegree, h, this, ←WithTop.some_eq_coe, Option.getD_coe]"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\na b : R\nn✝ m : ℕ\ninst✝ : Semiring R\np q r : R[X]\nn : ℕ\nthis : ↑n = ↑n\nh : trailingDegree p = ↑n\n⊢ natTrailingDegree p = n",
"tactic": "rw [natTrailingDegree, h, this, ←WithTop.some_eq_coe, Option.getD_coe]"
}
] |
[
183,
94
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
179,
1
] |
Mathlib/Analysis/Convex/Function.lean
|
StrictConcaveOn.inf
|
[] |
[
630,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
628,
1
] |
Mathlib/Algebra/Ring/Idempotents.lean
|
IsIdempotentElem.zero
|
[] |
[
61,
13
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
60,
1
] |
Mathlib/MeasureTheory/Function/StronglyMeasurable/Inner.lean
|
MeasureTheory.AEStronglyMeasurable.im
|
[] |
[
51,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
49,
11
] |
Mathlib/Analysis/BoxIntegral/Partition/Filter.lean
|
BoxIntegral.IntegrationParams.MemBaseSet.unionComplToSubordinate
|
[
{
"state_after": "no goals",
"state_before": "ι : Type u_1\ninst✝ : Fintype ι\nI J : Box ι\nc c₁ c₂ : ℝ≥0\nr r₁ r₂ : (ι → ℝ) → ↑(Set.Ioi 0)\nπ π₁ π₂✝ : TaggedPrepartition I\nl l₁ l₂ : IntegrationParams\nhπ₁ : MemBaseSet l I c r₁ π₁\nhle : ∀ (x : ι → ℝ), x ∈ ↑Box.Icc I → r₂ x ≤ r₁ x\nπ₂ : Prepartition I\nhU : Prepartition.iUnion π₂ = ↑I \\ TaggedPrepartition.iUnion π₁\nhc : l.bDistortion = true → Prepartition.distortion π₂ ≤ c\nx✝ : l.bDistortion = true\n⊢ Prepartition.iUnion ⊥ = ↑I \\ TaggedPrepartition.iUnion (unionComplToSubordinate π₁ π₂ hU r₂) ∧\n Prepartition.distortion ⊥ ≤ c",
"tactic": "simp"
}
] |
[
395,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
388,
11
] |
Mathlib/Combinatorics/SimpleGraph/Connectivity.lean
|
SimpleGraph.Walk.length_bypass_le
|
[
{
"state_after": "no goals",
"state_before": "case nil\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\ninst✝ : DecidableEq V\nu v u✝ : V\n⊢ length (bypass nil) ≤ length nil",
"tactic": "rfl"
},
{
"state_after": "case cons\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\ninst✝ : DecidableEq V\nu v u✝ v✝ w✝ : V\nh✝ : Adj G u✝ v✝\np✝ : Walk G v✝ w✝\nih : length (bypass p✝) ≤ length p✝\n⊢ length (if hs : u✝ ∈ support (bypass p✝) then dropUntil (bypass p✝) u✝ hs else cons h✝ (bypass p✝)) ≤\n length (cons h✝ p✝)",
"state_before": "case cons\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\ninst✝ : DecidableEq V\nu v u✝ v✝ w✝ : V\nh✝ : Adj G u✝ v✝\np✝ : Walk G v✝ w✝\nih : length (bypass p✝) ≤ length p✝\n⊢ length (bypass (cons h✝ p✝)) ≤ length (cons h✝ p✝)",
"tactic": "simp only [bypass]"
},
{
"state_after": "case cons.inl\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\ninst✝ : DecidableEq V\nu v u✝ v✝ w✝ : V\nh✝¹ : Adj G u✝ v✝\np✝ : Walk G v✝ w✝\nih : length (bypass p✝) ≤ length p✝\nh✝ : u✝ ∈ support (bypass p✝)\n⊢ length (dropUntil (bypass p✝) u✝ h✝) ≤ length (cons h✝¹ p✝)\n\ncase cons.inr\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\ninst✝ : DecidableEq V\nu v u✝ v✝ w✝ : V\nh✝¹ : Adj G u✝ v✝\np✝ : Walk G v✝ w✝\nih : length (bypass p✝) ≤ length p✝\nh✝ : ¬u✝ ∈ support (bypass p✝)\n⊢ length (cons h✝¹ (bypass p✝)) ≤ length (cons h✝¹ p✝)",
"state_before": "case cons\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\ninst✝ : DecidableEq V\nu v u✝ v✝ w✝ : V\nh✝ : Adj G u✝ v✝\np✝ : Walk G v✝ w✝\nih : length (bypass p✝) ≤ length p✝\n⊢ length (if hs : u✝ ∈ support (bypass p✝) then dropUntil (bypass p✝) u✝ hs else cons h✝ (bypass p✝)) ≤\n length (cons h✝ p✝)",
"tactic": "split_ifs"
},
{
"state_after": "V : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\ninst✝ : DecidableEq V\nu v u✝ v✝ w✝ : V\nh✝¹ : Adj G u✝ v✝\np✝ : Walk G v✝ w✝\nih : length (bypass p✝) ≤ length p✝\nh✝ : u✝ ∈ support (bypass p✝)\n⊢ length (dropUntil (bypass p✝) u✝ h✝) ≤ ?m.279260\n\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\ninst✝ : DecidableEq V\nu v u✝ v✝ w✝ : V\nh✝¹ : Adj G u✝ v✝\np✝ : Walk G v✝ w✝\nih : length (bypass p✝) ≤ length p✝\nh✝ : u✝ ∈ support (bypass p✝)\n⊢ ?m.279260 ≤ length (cons h✝¹ p✝)\n\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\ninst✝ : DecidableEq V\nu v u✝ v✝ w✝ : V\nh✝¹ : Adj G u✝ v✝\np✝ : Walk G v✝ w✝\nih : length (bypass p✝) ≤ length p✝\nh✝ : u✝ ∈ support (bypass p✝)\n⊢ ℕ",
"state_before": "case cons.inl\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\ninst✝ : DecidableEq V\nu v u✝ v✝ w✝ : V\nh✝¹ : Adj G u✝ v✝\np✝ : Walk G v✝ w✝\nih : length (bypass p✝) ≤ length p✝\nh✝ : u✝ ∈ support (bypass p✝)\n⊢ length (dropUntil (bypass p✝) u✝ h✝) ≤ length (cons h✝¹ p✝)",
"tactic": "trans"
},
{
"state_after": "V : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\ninst✝ : DecidableEq V\nu v u✝ v✝ w✝ : V\nh✝¹ : Adj G u✝ v✝\np✝ : Walk G v✝ w✝\nih : length (bypass p✝) ≤ length p✝\nh✝ : u✝ ∈ support (bypass p✝)\n⊢ length (bypass p✝) ≤ length (cons h✝¹ p✝)",
"state_before": "V : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\ninst✝ : DecidableEq V\nu v u✝ v✝ w✝ : V\nh✝¹ : Adj G u✝ v✝\np✝ : Walk G v✝ w✝\nih : length (bypass p✝) ≤ length p✝\nh✝ : u✝ ∈ support (bypass p✝)\n⊢ length (dropUntil (bypass p✝) u✝ h✝) ≤ ?m.279260\n\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\ninst✝ : DecidableEq V\nu v u✝ v✝ w✝ : V\nh✝¹ : Adj G u✝ v✝\np✝ : Walk G v✝ w✝\nih : length (bypass p✝) ≤ length p✝\nh✝ : u✝ ∈ support (bypass p✝)\n⊢ ?m.279260 ≤ length (cons h✝¹ p✝)\n\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\ninst✝ : DecidableEq V\nu v u✝ v✝ w✝ : V\nh✝¹ : Adj G u✝ v✝\np✝ : Walk G v✝ w✝\nih : length (bypass p✝) ≤ length p✝\nh✝ : u✝ ∈ support (bypass p✝)\n⊢ ℕ",
"tactic": "apply length_dropUntil_le"
},
{
"state_after": "V : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\ninst✝ : DecidableEq V\nu v u✝ v✝ w✝ : V\nh✝¹ : Adj G u✝ v✝\np✝ : Walk G v✝ w✝\nih : length (bypass p✝) ≤ length p✝\nh✝ : u✝ ∈ support (bypass p✝)\n⊢ length (bypass p✝) ≤ length p✝ + 1",
"state_before": "V : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\ninst✝ : DecidableEq V\nu v u✝ v✝ w✝ : V\nh✝¹ : Adj G u✝ v✝\np✝ : Walk G v✝ w✝\nih : length (bypass p✝) ≤ length p✝\nh✝ : u✝ ∈ support (bypass p✝)\n⊢ length (bypass p✝) ≤ length (cons h✝¹ p✝)",
"tactic": "rw [length_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''\ninst✝ : DecidableEq V\nu v u✝ v✝ w✝ : V\nh✝¹ : Adj G u✝ v✝\np✝ : Walk G v✝ w✝\nih : length (bypass p✝) ≤ length p✝\nh✝ : u✝ ∈ support (bypass p✝)\n⊢ length (bypass p✝) ≤ length p✝ + 1",
"tactic": "exact le_add_right ih"
},
{
"state_after": "case cons.inr\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\ninst✝ : DecidableEq V\nu v u✝ v✝ w✝ : V\nh✝¹ : Adj G u✝ v✝\np✝ : Walk G v✝ w✝\nih : length (bypass p✝) ≤ length p✝\nh✝ : ¬u✝ ∈ support (bypass p✝)\n⊢ length (bypass p✝) + 1 ≤ length p✝ + 1",
"state_before": "case cons.inr\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\ninst✝ : DecidableEq V\nu v u✝ v✝ w✝ : V\nh✝¹ : Adj G u✝ v✝\np✝ : Walk G v✝ w✝\nih : length (bypass p✝) ≤ length p✝\nh✝ : ¬u✝ ∈ support (bypass p✝)\n⊢ length (cons h✝¹ (bypass p✝)) ≤ length (cons h✝¹ p✝)",
"tactic": "rw [length_cons, length_cons]"
},
{
"state_after": "no goals",
"state_before": "case cons.inr\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\ninst✝ : DecidableEq V\nu v u✝ v✝ w✝ : V\nh✝¹ : Adj G u✝ v✝\np✝ : Walk G v✝ w✝\nih : length (bypass p✝) ≤ length p✝\nh✝ : ¬u✝ ∈ support (bypass p✝)\n⊢ length (bypass p✝) + 1 ≤ length p✝ + 1",
"tactic": "exact add_le_add_right ih 1"
}
] |
[
1389,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1378,
1
] |
Mathlib/Topology/UniformSpace/Equicontinuity.lean
|
Set.UniformEquicontinuous.uniformContinuous_of_mem
|
[] |
[
184,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
182,
11
] |
Mathlib/Order/Monotone/Monovary.lean
|
AntivaryOn.comp_antitone_on_left
|
[] |
[
192,
70
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
191,
1
] |
Mathlib/LinearAlgebra/Basic.lean
|
Finsupp.LinearEquiv.finsuppUnique_symm_apply
|
[
{
"state_after": "case h\nR : Type u_3\nR₁ : Type ?u.31136\nR₂ : Type ?u.31139\nR₃ : Type ?u.31142\nR₄ : Type ?u.31145\nS : Type ?u.31148\nK : Type ?u.31151\nK₂ : Type ?u.31154\nM : Type u_2\nM' : Type ?u.31160\nM₁ : Type ?u.31163\nM₂ : Type ?u.31166\nM₃ : Type ?u.31169\nM₄ : Type ?u.31172\nN : Type ?u.31175\nN₂ : Type ?u.31178\nι : Type ?u.31181\nV : Type ?u.31184\nV₂ : Type ?u.31187\nα✝ : Type ?u.31190\ninst✝⁴ : Finite α✝\ninst✝³ : AddCommMonoid M\ninst✝² : Semiring R\ninst✝¹ : Module R M\nα : Type u_1\ninst✝ : Unique α\nm : M\n⊢ ↑(↑(LinearEquiv.symm (finsuppUnique R M α)) m) default = ↑(single default m) default",
"state_before": "R : Type u_3\nR₁ : Type ?u.31136\nR₂ : Type ?u.31139\nR₃ : Type ?u.31142\nR₄ : Type ?u.31145\nS : Type ?u.31148\nK : Type ?u.31151\nK₂ : Type ?u.31154\nM : Type u_2\nM' : Type ?u.31160\nM₁ : Type ?u.31163\nM₂ : Type ?u.31166\nM₃ : Type ?u.31169\nM₄ : Type ?u.31172\nN : Type ?u.31175\nN₂ : Type ?u.31178\nι : Type ?u.31181\nV : Type ?u.31184\nV₂ : Type ?u.31187\nα✝ : Type ?u.31190\ninst✝⁴ : Finite α✝\ninst✝³ : AddCommMonoid M\ninst✝² : Semiring R\ninst✝¹ : Module R M\nα : Type u_1\ninst✝ : Unique α\nm : M\n⊢ ↑(LinearEquiv.symm (finsuppUnique R M α)) m = single default m",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case h\nR : Type u_3\nR₁ : Type ?u.31136\nR₂ : Type ?u.31139\nR₃ : Type ?u.31142\nR₄ : Type ?u.31145\nS : Type ?u.31148\nK : Type ?u.31151\nK₂ : Type ?u.31154\nM : Type u_2\nM' : Type ?u.31160\nM₁ : Type ?u.31163\nM₂ : Type ?u.31166\nM₃ : Type ?u.31169\nM₄ : Type ?u.31172\nN : Type ?u.31175\nN₂ : Type ?u.31178\nι : Type ?u.31181\nV : Type ?u.31184\nV₂ : Type ?u.31187\nα✝ : Type ?u.31190\ninst✝⁴ : Finite α✝\ninst✝³ : AddCommMonoid M\ninst✝² : Semiring R\ninst✝¹ : Module R M\nα : Type u_1\ninst✝ : Unique α\nm : M\n⊢ ↑(↑(LinearEquiv.symm (finsuppUnique R M α)) m) default = ↑(single default m) default",
"tactic": "simp [LinearEquiv.finsuppUnique, Equiv.funUnique, single, Pi.single,\nequivFunOnFinite, Function.update]"
}
] |
[
148,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
145,
1
] |
Mathlib/Data/Multiset/Basic.lean
|
Multiset.coe_eq_zero
|
[] |
[
111,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
110,
1
] |
Mathlib/LinearAlgebra/Finsupp.lean
|
Finsupp.span_single_image
|
[
{
"state_after": "α : Type u_2\nM : Type u_1\nN : Type ?u.43675\nP : Type ?u.43678\nR : Type u_3\nS : Type ?u.43684\ninst✝⁷ : Semiring R\ninst✝⁶ : Semiring S\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\ninst✝³ : AddCommMonoid N\ninst✝² : Module R N\ninst✝¹ : AddCommMonoid P\ninst✝ : Module R P\ns : Set M\na : α\n⊢ span R (single a '' s) = span R (↑(lsingle a) '' s)",
"state_before": "α : Type u_2\nM : Type u_1\nN : Type ?u.43675\nP : Type ?u.43678\nR : Type u_3\nS : Type ?u.43684\ninst✝⁷ : Semiring R\ninst✝⁶ : Semiring S\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\ninst✝³ : AddCommMonoid N\ninst✝² : Module R N\ninst✝¹ : AddCommMonoid P\ninst✝ : Module R P\ns : Set M\na : α\n⊢ span R (single a '' s) = map (lsingle a) (span R s)",
"tactic": "rw [← span_image]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_2\nM : Type u_1\nN : Type ?u.43675\nP : Type ?u.43678\nR : Type u_3\nS : Type ?u.43684\ninst✝⁷ : Semiring R\ninst✝⁶ : Semiring S\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\ninst✝³ : AddCommMonoid N\ninst✝² : Module R N\ninst✝¹ : AddCommMonoid P\ninst✝ : Module R P\ns : Set M\na : α\n⊢ span R (single a '' s) = span R (↑(lsingle a) '' s)",
"tactic": "rfl"
}
] |
[
180,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
178,
1
] |
src/lean/Init/SizeOf.lean
|
Bool.sizeOf_eq_one
|
[
{
"state_after": "no goals",
"state_before": "b : Bool\n⊢ sizeOf b = 1",
"tactic": "cases b <;> rfl"
}
] |
[
86,
83
] |
d5348dfac847a56a4595fb6230fd0708dcb4e7e9
|
https://github.com/leanprover/lean4
|
[
86,
9
] |
Mathlib/Dynamics/PeriodicPts.lean
|
Function.periodicOrbit_length
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.29967\nf fa : α → α\nfb : β → β\nx y : α\nm n : ℕ\n⊢ Cycle.length (periodicOrbit f x) = minimalPeriod f x",
"tactic": "rw [periodicOrbit, Cycle.length_coe, List.length_map, List.length_range]"
}
] |
[
502,
75
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
501,
1
] |
Mathlib/Data/Complex/Exponential.lean
|
Real.add_one_le_exp_of_nonneg
|
[
{
"state_after": "case inl\nx y : ℝ\nhx : 0 ≤ 0\n⊢ 0 + 1 ≤ exp 0\n\ncase inr\nx✝ y x : ℝ\nhx : 0 ≤ x\nh : 0 < x\n⊢ x + 1 ≤ exp x",
"state_before": "x✝ y x : ℝ\nhx : 0 ≤ x\n⊢ x + 1 ≤ exp x",
"tactic": "rcases eq_or_lt_of_le hx with (rfl | h)"
},
{
"state_after": "no goals",
"state_before": "case inr\nx✝ y x : ℝ\nhx : 0 ≤ x\nh : 0 < x\n⊢ x + 1 ≤ exp x",
"tactic": "exact (add_one_lt_exp_of_pos h).le"
},
{
"state_after": "no goals",
"state_before": "case inl\nx y : ℝ\nhx : 0 ≤ 0\n⊢ 0 + 1 ≤ exp 0",
"tactic": "simp"
}
] |
[
1489,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1486,
1
] |
Mathlib/Topology/Sets/Compacts.lean
|
TopologicalSpace.PositiveCompacts.interior_nonempty
|
[] |
[
337,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
336,
1
] |
Mathlib/Data/Nat/Choose/Central.lean
|
Nat.centralBinom_zero
|
[] |
[
54,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
53,
1
] |
Mathlib/RingTheory/IntegralClosure.lean
|
IsIntegralClosure.mk'_one
|
[
{
"state_after": "no goals",
"state_before": "R : Type u_1\nA : Type u_3\nB : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : Algebra R B\ninst✝¹ : Algebra A B\ninst✝ : IsIntegralClosure A R B\nh : optParam (IsIntegral R 1) (_ : IsIntegral R 1)\n⊢ ↑(algebraMap A B) (mk' A 1 h) = ↑(algebraMap A B) 1",
"tactic": "rw [algebraMap_mk', RingHom.map_one]"
}
] |
[
873,
72
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
872,
1
] |
Std/Data/Nat/Lemmas.lean
|
Nat.lt_of_add_lt_add_left
|
[
{
"state_after": "no goals",
"state_before": "k n m : Nat\nh : k + n < k + m\nheq : n = m\n⊢ k + m < k + m",
"tactic": "rwa [heq] at h"
}
] |
[
75,
47
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
73,
11
] |
Mathlib/Data/Stream/Init.lean
|
Stream'.mem_map
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nδ : Type w\nf : α → β\na : α\ns : Stream' α\nx✝ : a ∈ s\nn : ℕ\nh : (fun b => a = b) (nth s n)\n⊢ (fun b => f a = b) (nth (map f s) n)",
"tactic": "rw [nth_map, h]"
}
] |
[
188,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
187,
1
] |
Mathlib/Topology/Category/TopCat/Opens.lean
|
TopologicalSpace.Opens.inclusion_top_functor
|
[
{
"state_after": "case refine'_1\nX : TopCat\n⊢ ∀ (X_1 : Opens ↑((toTopCat X).obj ⊤)),\n (IsOpenMap.functor (_ : IsOpenMap ((forget TopCat).map (inclusion ⊤)))).obj X_1 =\n (map (inclusionTopIso X).inv).obj X_1\n\ncase refine'_2\nX : TopCat\n⊢ ∀ (X_1 Y : Opens ↑((toTopCat X).obj ⊤)) (f : X_1 ⟶ Y),\n (IsOpenMap.functor (_ : IsOpenMap ((forget TopCat).map (inclusion ⊤)))).map f =\n eqToHom (_ : ?m.132552.obj X_1 = ?m.132553.obj X_1) ≫\n (map (inclusionTopIso X).inv).map f ≫\n eqToHom\n (_ :\n (map (inclusionTopIso X).inv).obj Y =\n (IsOpenMap.functor (_ : IsOpenMap ((forget TopCat).map (inclusion ⊤)))).obj Y)",
"state_before": "X : TopCat\n⊢ IsOpenMap.functor (_ : IsOpenMap ((forget TopCat).map (inclusion ⊤))) = map (inclusionTopIso X).inv",
"tactic": "refine' CategoryTheory.Functor.ext _ _"
},
{
"state_after": "case refine'_2\nX : TopCat\n⊢ ∀ (X_1 Y : Opens ↑((toTopCat X).obj ⊤)) (f : X_1 ⟶ Y),\n (IsOpenMap.functor (_ : IsOpenMap ((forget TopCat).map (inclusion ⊤)))).map f =\n eqToHom\n (_ :\n (IsOpenMap.functor (_ : IsOpenMap ((forget TopCat).map (inclusion ⊤)))).obj X_1 =\n (map (inclusionTopIso X).inv).obj X_1) ≫\n (map (inclusionTopIso X).inv).map f ≫\n eqToHom\n (_ :\n (map (inclusionTopIso X).inv).obj Y =\n (IsOpenMap.functor (_ : IsOpenMap ((forget TopCat).map (inclusion ⊤)))).obj Y)",
"state_before": "case refine'_1\nX : TopCat\n⊢ ∀ (X_1 : Opens ↑((toTopCat X).obj ⊤)),\n (IsOpenMap.functor (_ : IsOpenMap ((forget TopCat).map (inclusion ⊤)))).obj X_1 =\n (map (inclusionTopIso X).inv).obj X_1\n\ncase refine'_2\nX : TopCat\n⊢ ∀ (X_1 Y : Opens ↑((toTopCat X).obj ⊤)) (f : X_1 ⟶ Y),\n (IsOpenMap.functor (_ : IsOpenMap ((forget TopCat).map (inclusion ⊤)))).map f =\n eqToHom (_ : ?m.132552.obj X_1 = ?m.132553.obj X_1) ≫\n (map (inclusionTopIso X).inv).map f ≫\n eqToHom\n (_ :\n (map (inclusionTopIso X).inv).obj Y =\n (IsOpenMap.functor (_ : IsOpenMap ((forget TopCat).map (inclusion ⊤)))).obj Y)",
"tactic": ". intro U\n ext x\n exact ⟨fun ⟨⟨_, _⟩, h, rfl⟩ => h, fun h => ⟨⟨x, trivial⟩, h, rfl⟩⟩"
},
{
"state_after": "no goals",
"state_before": "case refine'_2\nX : TopCat\n⊢ ∀ (X_1 Y : Opens ↑((toTopCat X).obj ⊤)) (f : X_1 ⟶ Y),\n (IsOpenMap.functor (_ : IsOpenMap ((forget TopCat).map (inclusion ⊤)))).map f =\n eqToHom\n (_ :\n (IsOpenMap.functor (_ : IsOpenMap ((forget TopCat).map (inclusion ⊤)))).obj X_1 =\n (map (inclusionTopIso X).inv).obj X_1) ≫\n (map (inclusionTopIso X).inv).map f ≫\n eqToHom\n (_ :\n (map (inclusionTopIso X).inv).obj Y =\n (IsOpenMap.functor (_ : IsOpenMap ((forget TopCat).map (inclusion ⊤)))).obj Y)",
"tactic": ". intros U V f\n apply Subsingleton.elim"
},
{
"state_after": "case refine'_1\nX : TopCat\nU : Opens ↑((toTopCat X).obj ⊤)\n⊢ (IsOpenMap.functor (_ : IsOpenMap ((forget TopCat).map (inclusion ⊤)))).obj U = (map (inclusionTopIso X).inv).obj U",
"state_before": "case refine'_1\nX : TopCat\n⊢ ∀ (X_1 : Opens ↑((toTopCat X).obj ⊤)),\n (IsOpenMap.functor (_ : IsOpenMap ((forget TopCat).map (inclusion ⊤)))).obj X_1 =\n (map (inclusionTopIso X).inv).obj X_1",
"tactic": "intro U"
},
{
"state_after": "case refine'_1.h.h\nX : TopCat\nU : Opens ↑((toTopCat X).obj ⊤)\nx : ↑X\n⊢ x ∈ ↑((IsOpenMap.functor (_ : IsOpenMap ((forget TopCat).map (inclusion ⊤)))).obj U) ↔\n x ∈ ↑((map (inclusionTopIso X).inv).obj U)",
"state_before": "case refine'_1\nX : TopCat\nU : Opens ↑((toTopCat X).obj ⊤)\n⊢ (IsOpenMap.functor (_ : IsOpenMap ((forget TopCat).map (inclusion ⊤)))).obj U = (map (inclusionTopIso X).inv).obj U",
"tactic": "ext x"
},
{
"state_after": "no goals",
"state_before": "case refine'_1.h.h\nX : TopCat\nU : Opens ↑((toTopCat X).obj ⊤)\nx : ↑X\n⊢ x ∈ ↑((IsOpenMap.functor (_ : IsOpenMap ((forget TopCat).map (inclusion ⊤)))).obj U) ↔\n x ∈ ↑((map (inclusionTopIso X).inv).obj U)",
"tactic": "exact ⟨fun ⟨⟨_, _⟩, h, rfl⟩ => h, fun h => ⟨⟨x, trivial⟩, h, rfl⟩⟩"
},
{
"state_after": "case refine'_2\nX : TopCat\nU V : Opens ↑((toTopCat X).obj ⊤)\nf : U ⟶ V\n⊢ (IsOpenMap.functor (_ : IsOpenMap ((forget TopCat).map (inclusion ⊤)))).map f =\n eqToHom\n (_ :\n (IsOpenMap.functor (_ : IsOpenMap ((forget TopCat).map (inclusion ⊤)))).obj U =\n (map (inclusionTopIso X).inv).obj U) ≫\n (map (inclusionTopIso X).inv).map f ≫\n eqToHom\n (_ :\n (map (inclusionTopIso X).inv).obj V =\n (IsOpenMap.functor (_ : IsOpenMap ((forget TopCat).map (inclusion ⊤)))).obj V)",
"state_before": "case refine'_2\nX : TopCat\n⊢ ∀ (X_1 Y : Opens ↑((toTopCat X).obj ⊤)) (f : X_1 ⟶ Y),\n (IsOpenMap.functor (_ : IsOpenMap ((forget TopCat).map (inclusion ⊤)))).map f =\n eqToHom\n (_ :\n (IsOpenMap.functor (_ : IsOpenMap ((forget TopCat).map (inclusion ⊤)))).obj X_1 =\n (map (inclusionTopIso X).inv).obj X_1) ≫\n (map (inclusionTopIso X).inv).map f ≫\n eqToHom\n (_ :\n (map (inclusionTopIso X).inv).obj Y =\n (IsOpenMap.functor (_ : IsOpenMap ((forget TopCat).map (inclusion ⊤)))).obj Y)",
"tactic": "intros U V f"
},
{
"state_after": "no goals",
"state_before": "case refine'_2\nX : TopCat\nU V : Opens ↑((toTopCat X).obj ⊤)\nf : U ⟶ V\n⊢ (IsOpenMap.functor (_ : IsOpenMap ((forget TopCat).map (inclusion ⊤)))).map f =\n eqToHom\n (_ :\n (IsOpenMap.functor (_ : IsOpenMap ((forget TopCat).map (inclusion ⊤)))).obj U =\n (map (inclusionTopIso X).inv).obj U) ≫\n (map (inclusionTopIso X).inv).map f ≫\n eqToHom\n (_ :\n (map (inclusionTopIso X).inv).obj V =\n (IsOpenMap.functor (_ : IsOpenMap ((forget TopCat).map (inclusion ⊤)))).obj V)",
"tactic": "apply Subsingleton.elim"
}
] |
[
362,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
355,
1
] |
Mathlib/Data/Finset/Lattice.lean
|
Finset.sup_image
|
[] |
[
66,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
64,
1
] |
Mathlib/Data/Set/Intervals/Basic.lean
|
Set.left_mem_Ico
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.3745\ninst✝ : Preorder α\na a₁ a₂ b b₁ b₂ c x : α\n⊢ a ∈ Ico a b ↔ a < b",
"tactic": "simp [le_refl]"
}
] |
[
190,
64
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
190,
1
] |
Mathlib/Analysis/Calculus/ContDiff.lean
|
ContDiffAt.prod_map'
|
[
{
"state_after": "case mk\n𝕜 : Type u_2\ninst✝¹⁴ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝¹³ : NormedAddCommGroup D\ninst✝¹² : NormedSpace 𝕜 D\nE : Type uE\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedSpace 𝕜 F\nG : Type uG\ninst✝⁷ : NormedAddCommGroup G\ninst✝⁶ : NormedSpace 𝕜 G\nX : Type ?u.2192953\ninst✝⁵ : NormedAddCommGroup X\ninst✝⁴ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf✝ f₁ : E → F\ng✝ : F → G\nx x₀ : E\nc : F\nb : E × F → G\nm n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nE' : Type u_1\ninst✝³ : NormedAddCommGroup E'\ninst✝² : NormedSpace 𝕜 E'\nF' : Type u_3\ninst✝¹ : NormedAddCommGroup F'\ninst✝ : NormedSpace 𝕜 F'\nf : E → F\ng : E' → F'\nfst✝ : E\nsnd✝ : E'\nhf : ContDiffAt 𝕜 n f (fst✝, snd✝).fst\nhg : ContDiffAt 𝕜 n g (fst✝, snd✝).snd\n⊢ ContDiffAt 𝕜 n (Prod.map f g) (fst✝, snd✝)",
"state_before": "𝕜 : Type u_2\ninst✝¹⁴ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝¹³ : NormedAddCommGroup D\ninst✝¹² : NormedSpace 𝕜 D\nE : Type uE\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedSpace 𝕜 F\nG : Type uG\ninst✝⁷ : NormedAddCommGroup G\ninst✝⁶ : NormedSpace 𝕜 G\nX : Type ?u.2192953\ninst✝⁵ : NormedAddCommGroup X\ninst✝⁴ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf✝ f₁ : E → F\ng✝ : F → G\nx x₀ : E\nc : F\nb : E × F → G\nm n : ℕ∞\np✝ : E → FormalMultilinearSeries 𝕜 E F\nE' : Type u_1\ninst✝³ : NormedAddCommGroup E'\ninst✝² : NormedSpace 𝕜 E'\nF' : Type u_3\ninst✝¹ : NormedAddCommGroup F'\ninst✝ : NormedSpace 𝕜 F'\nf : E → F\ng : E' → F'\np : E × E'\nhf : ContDiffAt 𝕜 n f p.fst\nhg : ContDiffAt 𝕜 n g p.snd\n⊢ ContDiffAt 𝕜 n (Prod.map f g) p",
"tactic": "rcases p with ⟨⟩"
},
{
"state_after": "no goals",
"state_before": "case mk\n𝕜 : Type u_2\ninst✝¹⁴ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝¹³ : NormedAddCommGroup D\ninst✝¹² : NormedSpace 𝕜 D\nE : Type uE\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedSpace 𝕜 F\nG : Type uG\ninst✝⁷ : NormedAddCommGroup G\ninst✝⁶ : NormedSpace 𝕜 G\nX : Type ?u.2192953\ninst✝⁵ : NormedAddCommGroup X\ninst✝⁴ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf✝ f₁ : E → F\ng✝ : F → G\nx x₀ : E\nc : F\nb : E × F → G\nm n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nE' : Type u_1\ninst✝³ : NormedAddCommGroup E'\ninst✝² : NormedSpace 𝕜 E'\nF' : Type u_3\ninst✝¹ : NormedAddCommGroup F'\ninst✝ : NormedSpace 𝕜 F'\nf : E → F\ng : E' → F'\nfst✝ : E\nsnd✝ : E'\nhf : ContDiffAt 𝕜 n f (fst✝, snd✝).fst\nhg : ContDiffAt 𝕜 n g (fst✝, snd✝).snd\n⊢ ContDiffAt 𝕜 n (Prod.map f g) (fst✝, snd✝)",
"tactic": "exact ContDiffAt.prod_map hf hg"
}
] |
[
1628,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1625,
1
] |
Mathlib/ModelTheory/Definability.lean
|
Set.definable_finset_sup
|
[
{
"state_after": "no goals",
"state_before": "M : Type w\nA : Set M\nL : Language\ninst✝ : Structure L M\nα : Type u₁\nβ : Type ?u.14150\nB : Set M\ns✝ : Set (α → M)\nι : Type u_1\nf : ι → Set (α → M)\nhf : ∀ (i : ι), Definable A L (f i)\ns : Finset ι\n⊢ Definable A L (Finset.sup s f)",
"tactic": "classical\n refine' Finset.induction definable_empty (fun i s _ h => _) s\n rw [Finset.sup_insert]\n exact (hf i).union h"
},
{
"state_after": "M : Type w\nA : Set M\nL : Language\ninst✝ : Structure L M\nα : Type u₁\nβ : Type ?u.14150\nB : Set M\ns✝¹ : Set (α → M)\nι : Type u_1\nf : ι → Set (α → M)\nhf : ∀ (i : ι), Definable A L (f i)\ns✝ : Finset ι\ni : ι\ns : Finset ι\nx✝ : ¬i ∈ s\nh : Definable A L (Finset.sup s f)\n⊢ Definable A L (Finset.sup (insert i s) f)",
"state_before": "M : Type w\nA : Set M\nL : Language\ninst✝ : Structure L M\nα : Type u₁\nβ : Type ?u.14150\nB : Set M\ns✝ : Set (α → M)\nι : Type u_1\nf : ι → Set (α → M)\nhf : ∀ (i : ι), Definable A L (f i)\ns : Finset ι\n⊢ Definable A L (Finset.sup s f)",
"tactic": "refine' Finset.induction definable_empty (fun i s _ h => _) s"
},
{
"state_after": "M : Type w\nA : Set M\nL : Language\ninst✝ : Structure L M\nα : Type u₁\nβ : Type ?u.14150\nB : Set M\ns✝¹ : Set (α → M)\nι : Type u_1\nf : ι → Set (α → M)\nhf : ∀ (i : ι), Definable A L (f i)\ns✝ : Finset ι\ni : ι\ns : Finset ι\nx✝ : ¬i ∈ s\nh : Definable A L (Finset.sup s f)\n⊢ Definable A L (f i ⊔ Finset.sup s f)",
"state_before": "M : Type w\nA : Set M\nL : Language\ninst✝ : Structure L M\nα : Type u₁\nβ : Type ?u.14150\nB : Set M\ns✝¹ : Set (α → M)\nι : Type u_1\nf : ι → Set (α → M)\nhf : ∀ (i : ι), Definable A L (f i)\ns✝ : Finset ι\ni : ι\ns : Finset ι\nx✝ : ¬i ∈ s\nh : Definable A L (Finset.sup s f)\n⊢ Definable A L (Finset.sup (insert i s) f)",
"tactic": "rw [Finset.sup_insert]"
},
{
"state_after": "no goals",
"state_before": "M : Type w\nA : Set M\nL : Language\ninst✝ : Structure L M\nα : Type u₁\nβ : Type ?u.14150\nB : Set M\ns✝¹ : Set (α → M)\nι : Type u_1\nf : ι → Set (α → M)\nhf : ∀ (i : ι), Definable A L (f i)\ns✝ : Finset ι\ni : ι\ns : Finset ι\nx✝ : ¬i ∈ s\nh : Definable A L (Finset.sup s f)\n⊢ Definable A L (f i ⊔ Finset.sup s f)",
"tactic": "exact (hf i).union h"
}
] |
[
125,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
120,
1
] |
Mathlib/Geometry/Manifold/ChartedSpace.lean
|
mem_pregroupoid_of_eq_on_source
|
[
{
"state_after": "H : Type u\nH' : Type ?u.33053\nM : Type ?u.33056\nM' : Type ?u.33059\nM'' : Type ?u.33062\ninst✝ : TopologicalSpace H\nPG : Pregroupoid H\ne e' : LocalHomeomorph H H\nhe' : e ≈ e'\nhe : Pregroupoid.property PG (↑e) e.source\n⊢ Pregroupoid.property PG (↑e') e.source",
"state_before": "H : Type u\nH' : Type ?u.33053\nM : Type ?u.33056\nM' : Type ?u.33059\nM'' : Type ?u.33062\ninst✝ : TopologicalSpace H\nPG : Pregroupoid H\ne e' : LocalHomeomorph H H\nhe' : e ≈ e'\nhe : Pregroupoid.property PG (↑e) e.source\n⊢ Pregroupoid.property PG (↑e') e'.source",
"tactic": "rw [← he'.1]"
},
{
"state_after": "no goals",
"state_before": "H : Type u\nH' : Type ?u.33053\nM : Type ?u.33056\nM' : Type ?u.33059\nM'' : Type ?u.33062\ninst✝ : TopologicalSpace H\nPG : Pregroupoid H\ne e' : LocalHomeomorph H H\nhe' : e ≈ e'\nhe : Pregroupoid.property PG (↑e) e.source\n⊢ Pregroupoid.property PG (↑e') e.source",
"tactic": "exact PG.congr e.open_source he'.eqOn.symm he"
}
] |
[
362,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
359,
1
] |
Mathlib/Algebra/Parity.lean
|
odd_neg_one
|
[
{
"state_after": "no goals",
"state_before": "F : Type ?u.103565\nα : Type u_1\nβ : Type ?u.103571\nR : Type ?u.103574\ninst✝ : Ring α\na b : α\nn : ℕ\n⊢ Odd (-1)",
"tactic": "simp"
}
] |
[
440,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
440,
1
] |
Mathlib/RingTheory/PolynomialAlgebra.lean
|
PolyEquivTensor.invFun_monomial
|
[] |
[
151,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
149,
1
] |
Mathlib/Combinatorics/Catalan.lean
|
gosper_catalan_sub_eq_central_binom_div
|
[
{
"state_after": "n : ℕ\nthis : ↑n + 1 ≠ 0\n⊢ gosperCatalan (n + 1) (n + 1) - gosperCatalan (n + 1) 0 = ↑(Nat.centralBinom (n + 1)) / (↑n + 2)",
"state_before": "n : ℕ\n⊢ gosperCatalan (n + 1) (n + 1) - gosperCatalan (n + 1) 0 = ↑(Nat.centralBinom (n + 1)) / (↑n + 2)",
"tactic": "have : (n : ℚ) + 1 ≠ 0 := by norm_cast; exact n.succ_ne_zero"
},
{
"state_after": "n : ℕ\nthis✝ : ↑n + 1 ≠ 0\nthis : ↑n + 1 + 1 ≠ 0\n⊢ gosperCatalan (n + 1) (n + 1) - gosperCatalan (n + 1) 0 = ↑(Nat.centralBinom (n + 1)) / (↑n + 2)",
"state_before": "n : ℕ\nthis : ↑n + 1 ≠ 0\n⊢ gosperCatalan (n + 1) (n + 1) - gosperCatalan (n + 1) 0 = ↑(Nat.centralBinom (n + 1)) / (↑n + 2)",
"tactic": "have : (n : ℚ) + 1 + 1 ≠ 0 := by norm_cast; exact (n + 1).succ_ne_zero"
},
{
"state_after": "n : ℕ\nthis✝ : ↑n + 1 ≠ 0\nthis : ↑n + 1 + 1 ≠ 0\nh : ↑n + 2 ≠ 0\n⊢ gosperCatalan (n + 1) (n + 1) - gosperCatalan (n + 1) 0 = ↑(Nat.centralBinom (n + 1)) / (↑n + 2)",
"state_before": "n : ℕ\nthis✝ : ↑n + 1 ≠ 0\nthis : ↑n + 1 + 1 ≠ 0\n⊢ gosperCatalan (n + 1) (n + 1) - gosperCatalan (n + 1) 0 = ↑(Nat.centralBinom (n + 1)) / (↑n + 2)",
"tactic": "have h : (n : ℚ) + 2 ≠ 0 := by norm_cast; exact (n + 1).succ_ne_zero"
},
{
"state_after": "n : ℕ\nthis✝ : ↑n + 1 ≠ 0\nthis : ↑n + 1 + 1 ≠ 0\nh : ↑n + 2 ≠ 0\n⊢ ↑(Nat.centralBinom (n + 1)) * ↑1 * (2 * ↑(n + 1) - ↑(n + 1)) / (2 * ↑(n + 1) * (↑(n + 1) + 1)) -\n ↑1 * ↑(Nat.centralBinom (n + 1)) * (2 * ↑0 - ↑(n + 1)) / (2 * ↑(n + 1) * (↑(n + 1) + 1)) =\n ↑(Nat.centralBinom (n + 1)) / (↑n + 2)",
"state_before": "n : ℕ\nthis✝ : ↑n + 1 ≠ 0\nthis : ↑n + 1 + 1 ≠ 0\nh : ↑n + 2 ≠ 0\n⊢ gosperCatalan (n + 1) (n + 1) - gosperCatalan (n + 1) 0 = ↑(Nat.centralBinom (n + 1)) / (↑n + 2)",
"tactic": "simp only [gosperCatalan, Nat.sub_zero, Nat.centralBinom_zero, Nat.sub_self]"
},
{
"state_after": "n : ℕ\nthis✝ : ↑n + 1 ≠ 0\nthis : ↑n + 1 + 1 ≠ 0\nh : ↑n + 2 ≠ 0\n⊢ (↑(Nat.centralBinom (n + 1)) * (2 * (↑n + 1) - (↑n + 1)) - ↑(Nat.centralBinom (n + 1)) * (-1 + -↑n)) * (↑n + 2) =\n ↑(Nat.centralBinom (n + 1)) * (2 * (↑n + 1) * (↑n + 1 + 1))",
"state_before": "n : ℕ\nthis✝ : ↑n + 1 ≠ 0\nthis : ↑n + 1 + 1 ≠ 0\nh : ↑n + 2 ≠ 0\n⊢ ↑(Nat.centralBinom (n + 1)) * ↑1 * (2 * ↑(n + 1) - ↑(n + 1)) / (2 * ↑(n + 1) * (↑(n + 1) + 1)) -\n ↑1 * ↑(Nat.centralBinom (n + 1)) * (2 * ↑0 - ↑(n + 1)) / (2 * ↑(n + 1) * (↑(n + 1) + 1)) =\n ↑(Nat.centralBinom (n + 1)) / (↑n + 2)",
"tactic": "field_simp"
},
{
"state_after": "no goals",
"state_before": "n : ℕ\nthis✝ : ↑n + 1 ≠ 0\nthis : ↑n + 1 + 1 ≠ 0\nh : ↑n + 2 ≠ 0\n⊢ (↑(Nat.centralBinom (n + 1)) * (2 * (↑n + 1) - (↑n + 1)) - ↑(Nat.centralBinom (n + 1)) * (-1 + -↑n)) * (↑n + 2) =\n ↑(Nat.centralBinom (n + 1)) * (2 * (↑n + 1) * (↑n + 1 + 1))",
"tactic": "ring"
},
{
"state_after": "n : ℕ\n⊢ ¬n + 1 = 0",
"state_before": "n : ℕ\n⊢ ↑n + 1 ≠ 0",
"tactic": "norm_cast"
},
{
"state_after": "no goals",
"state_before": "n : ℕ\n⊢ ¬n + 1 = 0",
"tactic": "exact n.succ_ne_zero"
},
{
"state_after": "n : ℕ\nthis : ↑n + 1 ≠ 0\n⊢ ¬n + 1 + 1 = 0",
"state_before": "n : ℕ\nthis : ↑n + 1 ≠ 0\n⊢ ↑n + 1 + 1 ≠ 0",
"tactic": "norm_cast"
},
{
"state_after": "no goals",
"state_before": "n : ℕ\nthis : ↑n + 1 ≠ 0\n⊢ ¬n + 1 + 1 = 0",
"tactic": "exact (n + 1).succ_ne_zero"
},
{
"state_after": "n : ℕ\nthis✝ : ↑n + 1 ≠ 0\nthis : ↑n + 1 + 1 ≠ 0\n⊢ ¬n + 2 = 0",
"state_before": "n : ℕ\nthis✝ : ↑n + 1 ≠ 0\nthis : ↑n + 1 + 1 ≠ 0\n⊢ ↑n + 2 ≠ 0",
"tactic": "norm_cast"
},
{
"state_after": "no goals",
"state_before": "n : ℕ\nthis✝ : ↑n + 1 ≠ 0\nthis : ↑n + 1 + 1 ≠ 0\n⊢ ¬n + 2 = 0",
"tactic": "exact (n + 1).succ_ne_zero"
}
] |
[
118,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
111,
9
] |
Mathlib/Data/Polynomial/Basic.lean
|
Polynomial.toFinsupp_add
|
[
{
"state_after": "case ofFinsupp\nR : Type u\na b✝ : R\nm n : ℕ\ninst✝ : Semiring R\np q b : R[X]\ntoFinsupp✝ : AddMonoidAlgebra R ℕ\n⊢ ({ toFinsupp := toFinsupp✝ } + b).toFinsupp = { toFinsupp := toFinsupp✝ }.toFinsupp + b.toFinsupp",
"state_before": "R : Type u\na✝ b✝ : R\nm n : ℕ\ninst✝ : Semiring R\np q a b : R[X]\n⊢ (a + b).toFinsupp = a.toFinsupp + b.toFinsupp",
"tactic": "cases a"
},
{
"state_after": "case ofFinsupp.ofFinsupp\nR : Type u\na b : R\nm n : ℕ\ninst✝ : Semiring R\np q : R[X]\ntoFinsupp✝¹ toFinsupp✝ : AddMonoidAlgebra R ℕ\n⊢ ({ toFinsupp := toFinsupp✝¹ } + { toFinsupp := toFinsupp✝ }).toFinsupp =\n { toFinsupp := toFinsupp✝¹ }.toFinsupp + { toFinsupp := toFinsupp✝ }.toFinsupp",
"state_before": "case ofFinsupp\nR : Type u\na b✝ : R\nm n : ℕ\ninst✝ : Semiring R\np q b : R[X]\ntoFinsupp✝ : AddMonoidAlgebra R ℕ\n⊢ ({ toFinsupp := toFinsupp✝ } + b).toFinsupp = { toFinsupp := toFinsupp✝ }.toFinsupp + b.toFinsupp",
"tactic": "cases b"
},
{
"state_after": "no goals",
"state_before": "case ofFinsupp.ofFinsupp\nR : Type u\na b : R\nm n : ℕ\ninst✝ : Semiring R\np q : R[X]\ntoFinsupp✝¹ toFinsupp✝ : AddMonoidAlgebra R ℕ\n⊢ ({ toFinsupp := toFinsupp✝¹ } + { toFinsupp := toFinsupp✝ }).toFinsupp =\n { toFinsupp := toFinsupp✝¹ }.toFinsupp + { toFinsupp := toFinsupp✝ }.toFinsupp",
"tactic": "rw [← ofFinsupp_add]"
}
] |
[
215,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
212,
1
] |
Mathlib/Algebra/Lie/Basic.lean
|
lie_jacobi
|
[
{
"state_after": "R : Type u\nL : Type v\nM : Type w\nN : Type w₁\ninst✝¹⁰ : CommRing R\ninst✝⁹ : LieRing L\ninst✝⁸ : LieAlgebra R L\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\ninst✝³ : AddCommGroup N\ninst✝² : Module R N\ninst✝¹ : LieRingModule L N\ninst✝ : LieModule R L N\nt : R\nx y z : L\nm n : M\n⊢ -(⁅y, ⁅z, x⁆⁆ - ⁅z, ⁅y, x⁆⁆) + ⁅y, ⁅z, x⁆⁆ + -⁅z, ⁅y, x⁆⁆ = 0",
"state_before": "R : Type u\nL : Type v\nM : Type w\nN : Type w₁\ninst✝¹⁰ : CommRing R\ninst✝⁹ : LieRing L\ninst✝⁸ : LieAlgebra R L\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\ninst✝³ : AddCommGroup N\ninst✝² : Module R N\ninst✝¹ : LieRingModule L N\ninst✝ : LieModule R L N\nt : R\nx y z : L\nm n : M\n⊢ ⁅x, ⁅y, z⁆⁆ + ⁅y, ⁅z, x⁆⁆ + ⁅z, ⁅x, y⁆⁆ = 0",
"tactic": "rw [← neg_neg ⁅x, y⁆, lie_neg z, lie_skew y x, ← lie_skew, lie_lie]"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nL : Type v\nM : Type w\nN : Type w₁\ninst✝¹⁰ : CommRing R\ninst✝⁹ : LieRing L\ninst✝⁸ : LieAlgebra R L\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\ninst✝³ : AddCommGroup N\ninst✝² : Module R N\ninst✝¹ : LieRingModule L N\ninst✝ : LieModule R L N\nt : R\nx y z : L\nm n : M\n⊢ -(⁅y, ⁅z, x⁆⁆ - ⁅z, ⁅y, x⁆⁆) + ⁅y, ⁅z, x⁆⁆ + -⁅z, ⁅y, x⁆⁆ = 0",
"tactic": "abel"
}
] |
[
217,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
215,
1
] |
Mathlib/Topology/Order/Hom/Esakia.lean
|
EsakiaHom.id_comp
|
[] |
[
347,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
346,
1
] |
Mathlib/Topology/CompactOpen.lean
|
QuotientMap.continuous_lift_prod_left
|
[
{
"state_after": "X₀ : Type u_1\nX : Type u_2\nY : Type u_3\nZ : Type u_4\ninst✝⁴ : TopologicalSpace X₀\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace Y\ninst✝¹ : TopologicalSpace Z\ninst✝ : LocallyCompactSpace Y\nf : X₀ → X\nhf : QuotientMap f\ng : X × Y → Z\nhg : Continuous fun p => g (f p.fst, p.snd)\nGf : C(X₀, C(Y, Z)) := curry (mk fun p => g (f p.fst, p.snd))\n⊢ Continuous g",
"state_before": "X₀ : Type u_1\nX : Type u_2\nY : Type u_3\nZ : Type u_4\ninst✝⁴ : TopologicalSpace X₀\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace Y\ninst✝¹ : TopologicalSpace Z\ninst✝ : LocallyCompactSpace Y\nf : X₀ → X\nhf : QuotientMap f\ng : X × Y → Z\nhg : Continuous fun p => g (f p.fst, p.snd)\n⊢ Continuous g",
"tactic": "let Gf : C(X₀, C(Y, Z)) := ContinuousMap.curry ⟨_, hg⟩"
},
{
"state_after": "X₀ : Type u_1\nX : Type u_2\nY : Type u_3\nZ : Type u_4\ninst✝⁴ : TopologicalSpace X₀\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace Y\ninst✝¹ : TopologicalSpace Z\ninst✝ : LocallyCompactSpace Y\nf : X₀ → X\nhf : QuotientMap f\ng : X × Y → Z\nhg : Continuous fun p => g (f p.fst, p.snd)\nGf : C(X₀, C(Y, Z)) := curry (mk fun p => g (f p.fst, p.snd))\nh : ∀ (x : X), Continuous fun y => g (x, y)\n⊢ Continuous g",
"state_before": "X₀ : Type u_1\nX : Type u_2\nY : Type u_3\nZ : Type u_4\ninst✝⁴ : TopologicalSpace X₀\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace Y\ninst✝¹ : TopologicalSpace Z\ninst✝ : LocallyCompactSpace Y\nf : X₀ → X\nhf : QuotientMap f\ng : X × Y → Z\nhg : Continuous fun p => g (f p.fst, p.snd)\nGf : C(X₀, C(Y, Z)) := curry (mk fun p => g (f p.fst, p.snd))\n⊢ Continuous g",
"tactic": "have h : ∀ x : X, Continuous fun y => g (x, y) := by\n intro x\n obtain ⟨x₀, rfl⟩ := hf.surjective x\n exact (Gf x₀).continuous"
},
{
"state_after": "X₀ : Type u_1\nX : Type u_2\nY : Type u_3\nZ : Type u_4\ninst✝⁴ : TopologicalSpace X₀\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace Y\ninst✝¹ : TopologicalSpace Z\ninst✝ : LocallyCompactSpace Y\nf : X₀ → X\nhf : QuotientMap f\ng : X × Y → Z\nhg : Continuous fun p => g (f p.fst, p.snd)\nGf : C(X₀, C(Y, Z)) := curry (mk fun p => g (f p.fst, p.snd))\nh : ∀ (x : X), Continuous fun y => g (x, y)\nG : X → C(Y, Z) := fun x => mk fun y => g (x, y)\n⊢ Continuous g",
"state_before": "X₀ : Type u_1\nX : Type u_2\nY : Type u_3\nZ : Type u_4\ninst✝⁴ : TopologicalSpace X₀\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace Y\ninst✝¹ : TopologicalSpace Z\ninst✝ : LocallyCompactSpace Y\nf : X₀ → X\nhf : QuotientMap f\ng : X × Y → Z\nhg : Continuous fun p => g (f p.fst, p.snd)\nGf : C(X₀, C(Y, Z)) := curry (mk fun p => g (f p.fst, p.snd))\nh : ∀ (x : X), Continuous fun y => g (x, y)\n⊢ Continuous g",
"tactic": "let G : X → C(Y, Z) := fun x => ⟨_, h x⟩"
},
{
"state_after": "X₀ : Type u_1\nX : Type u_2\nY : Type u_3\nZ : Type u_4\ninst✝⁴ : TopologicalSpace X₀\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace Y\ninst✝¹ : TopologicalSpace Z\ninst✝ : LocallyCompactSpace Y\nf : X₀ → X\nhf : QuotientMap f\ng : X × Y → Z\nhg : Continuous fun p => g (f p.fst, p.snd)\nGf : C(X₀, C(Y, Z)) := curry (mk fun p => g (f p.fst, p.snd))\nh : ∀ (x : X), Continuous fun y => g (x, y)\nG : X → C(Y, Z) := fun x => mk fun y => g (x, y)\nthis : Continuous G\n⊢ Continuous g",
"state_before": "X₀ : Type u_1\nX : Type u_2\nY : Type u_3\nZ : Type u_4\ninst✝⁴ : TopologicalSpace X₀\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace Y\ninst✝¹ : TopologicalSpace Z\ninst✝ : LocallyCompactSpace Y\nf : X₀ → X\nhf : QuotientMap f\ng : X × Y → Z\nhg : Continuous fun p => g (f p.fst, p.snd)\nGf : C(X₀, C(Y, Z)) := curry (mk fun p => g (f p.fst, p.snd))\nh : ∀ (x : X), Continuous fun y => g (x, y)\nG : X → C(Y, Z) := fun x => mk fun y => g (x, y)\n⊢ Continuous g",
"tactic": "have : Continuous G := by\n rw [hf.continuous_iff]\n exact Gf.continuous"
},
{
"state_after": "no goals",
"state_before": "X₀ : Type u_1\nX : Type u_2\nY : Type u_3\nZ : Type u_4\ninst✝⁴ : TopologicalSpace X₀\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace Y\ninst✝¹ : TopologicalSpace Z\ninst✝ : LocallyCompactSpace Y\nf : X₀ → X\nhf : QuotientMap f\ng : X × Y → Z\nhg : Continuous fun p => g (f p.fst, p.snd)\nGf : C(X₀, C(Y, Z)) := curry (mk fun p => g (f p.fst, p.snd))\nh : ∀ (x : X), Continuous fun y => g (x, y)\nG : X → C(Y, Z) := fun x => mk fun y => g (x, y)\nthis : Continuous G\n⊢ Continuous g",
"tactic": "exact ContinuousMap.continuous_uncurry_of_continuous ⟨G, this⟩"
},
{
"state_after": "X₀ : Type u_1\nX : Type u_2\nY : Type u_3\nZ : Type u_4\ninst✝⁴ : TopologicalSpace X₀\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace Y\ninst✝¹ : TopologicalSpace Z\ninst✝ : LocallyCompactSpace Y\nf : X₀ → X\nhf : QuotientMap f\ng : X × Y → Z\nhg : Continuous fun p => g (f p.fst, p.snd)\nGf : C(X₀, C(Y, Z)) := curry (mk fun p => g (f p.fst, p.snd))\nx : X\n⊢ Continuous fun y => g (x, y)",
"state_before": "X₀ : Type u_1\nX : Type u_2\nY : Type u_3\nZ : Type u_4\ninst✝⁴ : TopologicalSpace X₀\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace Y\ninst✝¹ : TopologicalSpace Z\ninst✝ : LocallyCompactSpace Y\nf : X₀ → X\nhf : QuotientMap f\ng : X × Y → Z\nhg : Continuous fun p => g (f p.fst, p.snd)\nGf : C(X₀, C(Y, Z)) := curry (mk fun p => g (f p.fst, p.snd))\n⊢ ∀ (x : X), Continuous fun y => g (x, y)",
"tactic": "intro x"
},
{
"state_after": "case intro\nX₀ : Type u_1\nX : Type u_2\nY : Type u_3\nZ : Type u_4\ninst✝⁴ : TopologicalSpace X₀\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace Y\ninst✝¹ : TopologicalSpace Z\ninst✝ : LocallyCompactSpace Y\nf : X₀ → X\nhf : QuotientMap f\ng : X × Y → Z\nhg : Continuous fun p => g (f p.fst, p.snd)\nGf : C(X₀, C(Y, Z)) := curry (mk fun p => g (f p.fst, p.snd))\nx₀ : X₀\n⊢ Continuous fun y => g (f x₀, y)",
"state_before": "X₀ : Type u_1\nX : Type u_2\nY : Type u_3\nZ : Type u_4\ninst✝⁴ : TopologicalSpace X₀\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace Y\ninst✝¹ : TopologicalSpace Z\ninst✝ : LocallyCompactSpace Y\nf : X₀ → X\nhf : QuotientMap f\ng : X × Y → Z\nhg : Continuous fun p => g (f p.fst, p.snd)\nGf : C(X₀, C(Y, Z)) := curry (mk fun p => g (f p.fst, p.snd))\nx : X\n⊢ Continuous fun y => g (x, y)",
"tactic": "obtain ⟨x₀, rfl⟩ := hf.surjective x"
},
{
"state_after": "no goals",
"state_before": "case intro\nX₀ : Type u_1\nX : Type u_2\nY : Type u_3\nZ : Type u_4\ninst✝⁴ : TopologicalSpace X₀\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace Y\ninst✝¹ : TopologicalSpace Z\ninst✝ : LocallyCompactSpace Y\nf : X₀ → X\nhf : QuotientMap f\ng : X × Y → Z\nhg : Continuous fun p => g (f p.fst, p.snd)\nGf : C(X₀, C(Y, Z)) := curry (mk fun p => g (f p.fst, p.snd))\nx₀ : X₀\n⊢ Continuous fun y => g (f x₀, y)",
"tactic": "exact (Gf x₀).continuous"
},
{
"state_after": "X₀ : Type u_1\nX : Type u_2\nY : Type u_3\nZ : Type u_4\ninst✝⁴ : TopologicalSpace X₀\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace Y\ninst✝¹ : TopologicalSpace Z\ninst✝ : LocallyCompactSpace Y\nf : X₀ → X\nhf : QuotientMap f\ng : X × Y → Z\nhg : Continuous fun p => g (f p.fst, p.snd)\nGf : C(X₀, C(Y, Z)) := curry (mk fun p => g (f p.fst, p.snd))\nh : ∀ (x : X), Continuous fun y => g (x, y)\nG : X → C(Y, Z) := fun x => mk fun y => g (x, y)\n⊢ Continuous (G ∘ f)",
"state_before": "X₀ : Type u_1\nX : Type u_2\nY : Type u_3\nZ : Type u_4\ninst✝⁴ : TopologicalSpace X₀\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace Y\ninst✝¹ : TopologicalSpace Z\ninst✝ : LocallyCompactSpace Y\nf : X₀ → X\nhf : QuotientMap f\ng : X × Y → Z\nhg : Continuous fun p => g (f p.fst, p.snd)\nGf : C(X₀, C(Y, Z)) := curry (mk fun p => g (f p.fst, p.snd))\nh : ∀ (x : X), Continuous fun y => g (x, y)\nG : X → C(Y, Z) := fun x => mk fun y => g (x, y)\n⊢ Continuous G",
"tactic": "rw [hf.continuous_iff]"
},
{
"state_after": "no goals",
"state_before": "X₀ : Type u_1\nX : Type u_2\nY : Type u_3\nZ : Type u_4\ninst✝⁴ : TopologicalSpace X₀\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace Y\ninst✝¹ : TopologicalSpace Z\ninst✝ : LocallyCompactSpace Y\nf : X₀ → X\nhf : QuotientMap f\ng : X × Y → Z\nhg : Continuous fun p => g (f p.fst, p.snd)\nGf : C(X₀, C(Y, Z)) := curry (mk fun p => g (f p.fst, p.snd))\nh : ∀ (x : X), Continuous fun y => g (x, y)\nG : X → C(Y, Z) := fun x => mk fun y => g (x, y)\n⊢ Continuous (G ∘ f)",
"tactic": "exact Gf.continuous"
}
] |
[
489,
65
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
478,
1
] |
Mathlib/Data/List/BigOperators/Basic.lean
|
List.one_le_prod_of_one_le
|
[
{
"state_after": "case nil\nι : Type ?u.105808\nα : Type ?u.105811\nM : Type u_1\nN : Type ?u.105817\nP : Type ?u.105820\nM₀ : Type ?u.105823\nG : Type ?u.105826\nR : Type ?u.105829\ninst✝⁴ : Monoid M\ninst✝³ : Monoid N\ninst✝² : Monoid P\nl✝ l₁ l₂ : List M\na : M\ninst✝¹ : Preorder M\ninst✝ : CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1\nl : List M\nhl₁✝ : ∀ (x : M), x ∈ l → 1 ≤ x\nhl₁ : ∀ (x : M), x ∈ [] → 1 ≤ x\n⊢ 1 ≤ prod []\n\ncase cons\nι : Type ?u.105808\nα : Type ?u.105811\nM : Type u_1\nN : Type ?u.105817\nP : Type ?u.105820\nM₀ : Type ?u.105823\nG : Type ?u.105826\nR : Type ?u.105829\ninst✝⁴ : Monoid M\ninst✝³ : Monoid N\ninst✝² : Monoid P\nl✝ l₁ l₂ : List M\na : M\ninst✝¹ : Preorder M\ninst✝ : CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1\nl : List M\nhl₁✝ : ∀ (x : M), x ∈ l → 1 ≤ x\nhd : M\ntl : List M\nih : (∀ (x : M), x ∈ tl → 1 ≤ x) → 1 ≤ prod tl\nhl₁ : ∀ (x : M), x ∈ hd :: tl → 1 ≤ x\n⊢ 1 ≤ prod (hd :: tl)",
"state_before": "ι : Type ?u.105808\nα : Type ?u.105811\nM : Type u_1\nN : Type ?u.105817\nP : Type ?u.105820\nM₀ : Type ?u.105823\nG : Type ?u.105826\nR : Type ?u.105829\ninst✝⁴ : Monoid M\ninst✝³ : Monoid N\ninst✝² : Monoid P\nl✝ l₁ l₂ : List M\na : M\ninst✝¹ : Preorder M\ninst✝ : CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1\nl : List M\nhl₁ : ∀ (x : M), x ∈ l → 1 ≤ x\n⊢ 1 ≤ prod l",
"tactic": "induction' l with hd tl ih"
},
{
"state_after": "case cons\nι : Type ?u.105808\nα : Type ?u.105811\nM : Type u_1\nN : Type ?u.105817\nP : Type ?u.105820\nM₀ : Type ?u.105823\nG : Type ?u.105826\nR : Type ?u.105829\ninst✝⁴ : Monoid M\ninst✝³ : Monoid N\ninst✝² : Monoid P\nl✝ l₁ l₂ : List M\na : M\ninst✝¹ : Preorder M\ninst✝ : CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1\nl : List M\nhl₁✝ : ∀ (x : M), x ∈ l → 1 ≤ x\nhd : M\ntl : List M\nih : (∀ (x : M), x ∈ tl → 1 ≤ x) → 1 ≤ prod tl\nhl₁ : ∀ (x : M), x ∈ hd :: tl → 1 ≤ x\n⊢ 1 ≤ hd * prod tl",
"state_before": "case cons\nι : Type ?u.105808\nα : Type ?u.105811\nM : Type u_1\nN : Type ?u.105817\nP : Type ?u.105820\nM₀ : Type ?u.105823\nG : Type ?u.105826\nR : Type ?u.105829\ninst✝⁴ : Monoid M\ninst✝³ : Monoid N\ninst✝² : Monoid P\nl✝ l₁ l₂ : List M\na : M\ninst✝¹ : Preorder M\ninst✝ : CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1\nl : List M\nhl₁✝ : ∀ (x : M), x ∈ l → 1 ≤ x\nhd : M\ntl : List M\nih : (∀ (x : M), x ∈ tl → 1 ≤ x) → 1 ≤ prod tl\nhl₁ : ∀ (x : M), x ∈ hd :: tl → 1 ≤ x\n⊢ 1 ≤ prod (hd :: tl)",
"tactic": "rw [prod_cons]"
},
{
"state_after": "no goals",
"state_before": "case cons\nι : Type ?u.105808\nα : Type ?u.105811\nM : Type u_1\nN : Type ?u.105817\nP : Type ?u.105820\nM₀ : Type ?u.105823\nG : Type ?u.105826\nR : Type ?u.105829\ninst✝⁴ : Monoid M\ninst✝³ : Monoid N\ninst✝² : Monoid P\nl✝ l₁ l₂ : List M\na : M\ninst✝¹ : Preorder M\ninst✝ : CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1\nl : List M\nhl₁✝ : ∀ (x : M), x ∈ l → 1 ≤ x\nhd : M\ntl : List M\nih : (∀ (x : M), x ∈ tl → 1 ≤ x) → 1 ≤ prod tl\nhl₁ : ∀ (x : M), x ∈ hd :: tl → 1 ≤ x\n⊢ 1 ≤ hd * prod tl",
"tactic": "exact one_le_mul (hl₁ hd (mem_cons_self hd tl)) (ih fun x h => hl₁ x (mem_cons_of_mem hd h))"
},
{
"state_after": "no goals",
"state_before": "case nil\nι : Type ?u.105808\nα : Type ?u.105811\nM : Type u_1\nN : Type ?u.105817\nP : Type ?u.105820\nM₀ : Type ?u.105823\nG : Type ?u.105826\nR : Type ?u.105829\ninst✝⁴ : Monoid M\ninst✝³ : Monoid N\ninst✝² : Monoid P\nl✝ l₁ l₂ : List M\na : M\ninst✝¹ : Preorder M\ninst✝ : CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1\nl : List M\nhl₁✝ : ∀ (x : M), x ∈ l → 1 ≤ x\nhl₁ : ∀ (x : M), x ∈ [] → 1 ≤ x\n⊢ 1 ≤ prod []",
"tactic": "rfl"
}
] |
[
371,
95
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
364,
1
] |
Mathlib/Data/List/MinMax.lean
|
List.maximum_nil
|
[] |
[
287,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
286,
1
] |
Mathlib/Algebra/Hom/NonUnitalAlg.lean
|
NonUnitalAlgHom.toFun_eq_coe
|
[] |
[
128,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
127,
1
] |
Mathlib/Algebra/Tropical/Basic.lean
|
Tropical.untrop_max
|
[] |
[
307,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
306,
1
] |
Mathlib/Topology/Algebra/InfiniteSum/Basic.lean
|
tsum_eq_tsum_of_ne_zero_bij
|
[] |
[
580,
83
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
577,
1
] |
Mathlib/Analysis/Normed/Group/Pointwise.lean
|
ball_mul
|
[
{
"state_after": "no goals",
"state_before": "E : Type u_1\ninst✝ : SeminormedCommGroup E\nε δ : ℝ\ns t : Set E\nx y : E\n⊢ ball x δ * s = x • thickening δ s",
"tactic": "rw [mul_comm, mul_ball]"
}
] |
[
243,
83
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
243,
1
] |
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