file_path
stringlengths 11
79
| full_name
stringlengths 2
100
| traced_tactics
list | end
list | commit
stringclasses 4
values | url
stringclasses 4
values | start
list |
|---|---|---|---|---|---|---|
Mathlib/Data/Set/Intervals/OrderIso.lean
|
OrderIso.preimage_Ici
|
[
{
"state_after": "case h\nα : Type u_1\nβ : Type u_2\ninst✝¹ : Preorder α\ninst✝ : Preorder β\ne : α ≃o β\nb : β\nx : α\n⊢ x ∈ ↑e ⁻¹' Ici b ↔ x ∈ Ici (↑(symm e) b)",
"state_before": "α : Type u_1\nβ : Type u_2\ninst✝¹ : Preorder α\ninst✝ : Preorder β\ne : α ≃o β\nb : β\n⊢ ↑e ⁻¹' Ici b = Ici (↑(symm e) b)",
"tactic": "ext x"
},
{
"state_after": "no goals",
"state_before": "case h\nα : Type u_1\nβ : Type u_2\ninst✝¹ : Preorder α\ninst✝ : Preorder β\ne : α ≃o β\nb : β\nx : α\n⊢ x ∈ ↑e ⁻¹' Ici b ↔ x ∈ Ici (↑(symm e) b)",
"tactic": "simp [← e.le_iff_le]"
}
] |
[
34,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
32,
1
] |
Mathlib/CategoryTheory/Preadditive/OfBiproducts.lean
|
CategoryTheory.SemiadditiveOfBinaryBiproducts.isUnital_leftAdd
|
[
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝² : Category C\ninst✝¹ : HasZeroMorphisms C\ninst✝ : HasBinaryBiproducts C\nX Y : C\nhr : ∀ (f : X ⟶ Y), biprod.lift 0 f = f ≫ biprod.inr\nhl : ∀ (f : X ⟶ Y), biprod.lift f 0 = f ≫ biprod.inl\n⊢ EckmannHilton.IsUnital (fun x x_1 => leftAdd X Y x x_1) 0",
"tactic": "exact ⟨⟨fun f => by simp [hr f, leftAdd, Category.assoc, Category.comp_id, biprod.inr_desc]⟩,\n ⟨fun f => by simp [hl f, leftAdd, Category.assoc, Category.comp_id, biprod.inl_desc]⟩⟩"
},
{
"state_after": "C : Type u\ninst✝² : Category C\ninst✝¹ : HasZeroMorphisms C\ninst✝ : HasBinaryBiproducts C\nX Y : C\nf : X ⟶ Y\n⊢ biprod.lift 0 f = f ≫ biprod.inr",
"state_before": "C : Type u\ninst✝² : Category C\ninst✝¹ : HasZeroMorphisms C\ninst✝ : HasBinaryBiproducts C\nX Y : C\n⊢ ∀ (f : X ⟶ Y), biprod.lift 0 f = f ≫ biprod.inr",
"tactic": "intro f"
},
{
"state_after": "case h₀\nC : Type u\ninst✝² : Category C\ninst✝¹ : HasZeroMorphisms C\ninst✝ : HasBinaryBiproducts C\nX Y : C\nf : X ⟶ Y\n⊢ biprod.lift 0 f ≫ biprod.fst = (f ≫ biprod.inr) ≫ biprod.fst\n\ncase h₁\nC : Type u\ninst✝² : Category C\ninst✝¹ : HasZeroMorphisms C\ninst✝ : HasBinaryBiproducts C\nX Y : C\nf : X ⟶ Y\n⊢ biprod.lift 0 f ≫ biprod.snd = (f ≫ biprod.inr) ≫ biprod.snd",
"state_before": "C : Type u\ninst✝² : Category C\ninst✝¹ : HasZeroMorphisms C\ninst✝ : HasBinaryBiproducts C\nX Y : C\nf : X ⟶ Y\n⊢ biprod.lift 0 f = f ≫ biprod.inr",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case h₀\nC : Type u\ninst✝² : Category C\ninst✝¹ : HasZeroMorphisms C\ninst✝ : HasBinaryBiproducts C\nX Y : C\nf : X ⟶ Y\n⊢ biprod.lift 0 f ≫ biprod.fst = (f ≫ biprod.inr) ≫ biprod.fst",
"tactic": "aesop_cat"
},
{
"state_after": "no goals",
"state_before": "case h₁\nC : Type u\ninst✝² : Category C\ninst✝¹ : HasZeroMorphisms C\ninst✝ : HasBinaryBiproducts C\nX Y : C\nf : X ⟶ Y\n⊢ biprod.lift 0 f ≫ biprod.snd = (f ≫ biprod.inr) ≫ biprod.snd",
"tactic": "simp [biprod.lift_fst, Category.assoc, biprod.inr_fst, comp_zero]"
},
{
"state_after": "C : Type u\ninst✝² : Category C\ninst✝¹ : HasZeroMorphisms C\ninst✝ : HasBinaryBiproducts C\nX Y : C\nhr : ∀ (f : X ⟶ Y), biprod.lift 0 f = f ≫ biprod.inr\nf : X ⟶ Y\n⊢ biprod.lift f 0 = f ≫ biprod.inl",
"state_before": "C : Type u\ninst✝² : Category C\ninst✝¹ : HasZeroMorphisms C\ninst✝ : HasBinaryBiproducts C\nX Y : C\nhr : ∀ (f : X ⟶ Y), biprod.lift 0 f = f ≫ biprod.inr\n⊢ ∀ (f : X ⟶ Y), biprod.lift f 0 = f ≫ biprod.inl",
"tactic": "intro f"
},
{
"state_after": "case h₀\nC : Type u\ninst✝² : Category C\ninst✝¹ : HasZeroMorphisms C\ninst✝ : HasBinaryBiproducts C\nX Y : C\nhr : ∀ (f : X ⟶ Y), biprod.lift 0 f = f ≫ biprod.inr\nf : X ⟶ Y\n⊢ biprod.lift f 0 ≫ biprod.fst = (f ≫ biprod.inl) ≫ biprod.fst\n\ncase h₁\nC : Type u\ninst✝² : Category C\ninst✝¹ : HasZeroMorphisms C\ninst✝ : HasBinaryBiproducts C\nX Y : C\nhr : ∀ (f : X ⟶ Y), biprod.lift 0 f = f ≫ biprod.inr\nf : X ⟶ Y\n⊢ biprod.lift f 0 ≫ biprod.snd = (f ≫ biprod.inl) ≫ biprod.snd",
"state_before": "C : Type u\ninst✝² : Category C\ninst✝¹ : HasZeroMorphisms C\ninst✝ : HasBinaryBiproducts C\nX Y : C\nhr : ∀ (f : X ⟶ Y), biprod.lift 0 f = f ≫ biprod.inr\nf : X ⟶ Y\n⊢ biprod.lift f 0 = f ≫ biprod.inl",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case h₀\nC : Type u\ninst✝² : Category C\ninst✝¹ : HasZeroMorphisms C\ninst✝ : HasBinaryBiproducts C\nX Y : C\nhr : ∀ (f : X ⟶ Y), biprod.lift 0 f = f ≫ biprod.inr\nf : X ⟶ Y\n⊢ biprod.lift f 0 ≫ biprod.fst = (f ≫ biprod.inl) ≫ biprod.fst",
"tactic": "aesop_cat"
},
{
"state_after": "no goals",
"state_before": "case h₁\nC : Type u\ninst✝² : Category C\ninst✝¹ : HasZeroMorphisms C\ninst✝ : HasBinaryBiproducts C\nX Y : C\nhr : ∀ (f : X ⟶ Y), biprod.lift 0 f = f ≫ biprod.inr\nf : X ⟶ Y\n⊢ biprod.lift f 0 ≫ biprod.snd = (f ≫ biprod.inl) ≫ biprod.snd",
"tactic": "simp [biprod.lift_snd, Category.assoc, biprod.inl_snd, comp_zero]"
},
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝² : Category C\ninst✝¹ : HasZeroMorphisms C\ninst✝ : HasBinaryBiproducts C\nX Y : C\nhr : ∀ (f : X ⟶ Y), biprod.lift 0 f = f ≫ biprod.inr\nhl : ∀ (f : X ⟶ Y), biprod.lift f 0 = f ≫ biprod.inl\nf : X ⟶ Y\n⊢ leftAdd X Y 0 f = f",
"tactic": "simp [hr f, leftAdd, Category.assoc, Category.comp_id, biprod.inr_desc]"
},
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝² : Category C\ninst✝¹ : HasZeroMorphisms C\ninst✝ : HasBinaryBiproducts C\nX Y : C\nhr : ∀ (f : X ⟶ Y), biprod.lift 0 f = f ≫ biprod.inr\nhl : ∀ (f : X ⟶ Y), biprod.lift f 0 = f ≫ biprod.inl\nf : X ⟶ Y\n⊢ leftAdd X Y f 0 = f",
"tactic": "simp [hl f, leftAdd, Category.assoc, Category.comp_id, biprod.inl_desc]"
}
] |
[
71,
91
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
59,
1
] |
Std/Data/String/Lemmas.lean
|
Substring.Valid.next
|
[
{
"state_after": "m₁ : List Char\nc : Char\nm₂ : List Char\nx✝ : Substring\nh✝ : Valid x✝\ne : (toString x✝).data = m₁ ++ c :: m₂\nl m r : List Char\nh : ValidFor l m r x✝\n⊢ Substring.next x✝ { byteIdx := utf8Len m₁ } = { byteIdx := utf8Len m₁ + csize c }",
"state_before": "m₁ : List Char\nc : Char\nm₂ : List Char\nx✝ : Substring\nh : Valid x✝\ne : (toString x✝).data = m₁ ++ c :: m₂\n⊢ Substring.next x✝ { byteIdx := utf8Len m₁ } = { byteIdx := utf8Len m₁ + csize c }",
"tactic": "let ⟨l, m, r, h⟩ := h.validFor"
},
{
"state_after": "m₁ : List Char\nc : Char\nm₂ : List Char\nx✝ : Substring\nh✝ : Valid x✝\nl m r : List Char\nh : ValidFor l m r x✝\ne : m = m₁ ++ c :: m₂\n⊢ Substring.next x✝ { byteIdx := utf8Len m₁ } = { byteIdx := utf8Len m₁ + csize c }",
"state_before": "m₁ : List Char\nc : Char\nm₂ : List Char\nx✝ : Substring\nh✝ : Valid x✝\ne : (toString x✝).data = m₁ ++ c :: m₂\nl m r : List Char\nh : ValidFor l m r x✝\n⊢ Substring.next x✝ { byteIdx := utf8Len m₁ } = { byteIdx := utf8Len m₁ + csize c }",
"tactic": "simp [h.toString] at e"
},
{
"state_after": "m₁ : List Char\nc : Char\nm₂ : List Char\nx✝ : Substring\nh✝ : Valid x✝\nl r : List Char\nh : ValidFor l (m₁ ++ c :: m₂) r x✝\n⊢ Substring.next x✝ { byteIdx := utf8Len m₁ } = { byteIdx := utf8Len m₁ + csize c }",
"state_before": "m₁ : List Char\nc : Char\nm₂ : List Char\nx✝ : Substring\nh✝ : Valid x✝\nl m r : List Char\nh : ValidFor l m r x✝\ne : m = m₁ ++ c :: m₂\n⊢ Substring.next x✝ { byteIdx := utf8Len m₁ } = { byteIdx := utf8Len m₁ + csize c }",
"tactic": "subst e"
},
{
"state_after": "no goals",
"state_before": "m₁ : List Char\nc : Char\nm₂ : List Char\nx✝ : Substring\nh✝ : Valid x✝\nl r : List Char\nh : ValidFor l (m₁ ++ c :: m₂) r x✝\n⊢ Substring.next x✝ { byteIdx := utf8Len m₁ } = { byteIdx := utf8Len m₁ + csize c }",
"tactic": "simp [h.next]"
}
] |
[
990,
51
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
986,
1
] |
Mathlib/Order/SuccPred/Basic.lean
|
Order.Ico_succ_right
|
[] |
[
377,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
376,
1
] |
Mathlib/Data/Real/CauSeq.lean
|
CauSeq.lt_irrefl
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : LinearOrderedField α\nf : CauSeq α abs\nx✝ : f < f\nh : f < f := x✝\n⊢ LimZero (f - f)",
"tactic": "simp [zero_limZero]"
}
] |
[
745,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
744,
1
] |
Mathlib/NumberTheory/Liouville/LiouvilleWith.lean
|
LiouvilleWith.nat_sub_iff
|
[
{
"state_after": "no goals",
"state_before": "p q x y : ℝ\nr : ℚ\nm : ℤ\nn : ℕ\n⊢ LiouvilleWith p (↑n - x) ↔ LiouvilleWith p x",
"tactic": "simp [sub_eq_add_neg]"
}
] |
[
307,
94
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
307,
1
] |
Mathlib/MeasureTheory/Function/AEEqFun.lean
|
MeasureTheory.AEEqFun.compMeasurable_toGerm
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_3\nβ : Type u_1\nγ : Type u_2\nδ : Type ?u.445345\ninst✝¹⁰ : MeasurableSpace α\nμ ν : Measure α\ninst✝⁹ : TopologicalSpace β\ninst✝⁸ : TopologicalSpace γ\ninst✝⁷ : TopologicalSpace δ\ninst✝⁶ : MeasurableSpace β\ninst✝⁵ : BorelSpace β\ninst✝⁴ : PseudoMetrizableSpace β\ninst✝³ : PseudoMetrizableSpace γ\ninst✝² : SecondCountableTopology γ\ninst✝¹ : MeasurableSpace γ\ninst✝ : OpensMeasurableSpace γ\ng : β → γ\nhg : Measurable g\nf✝ : α →ₘ[μ] β\nf : α → β\nx✝ : AEStronglyMeasurable f μ\n⊢ toGerm (compMeasurable g hg (mk f x✝)) = Germ.map g (toGerm (mk f x✝))",
"tactic": "simp"
}
] |
[
399,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
395,
1
] |
Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean
|
affineSpan_singleton_union_vadd_eq_top_of_span_eq_top
|
[
{
"state_after": "case convert_1\nk : Type u_2\nV : Type u_1\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type ?u.351531\ns : Set V\np : P\nh : Submodule.span k (range Subtype.val) = ⊤\n⊢ direction (affineSpan k ({p} ∪ (fun v => v +ᵥ p) '' s)) = direction ⊤",
"state_before": "k : Type u_2\nV : Type u_1\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type ?u.351531\ns : Set V\np : P\nh : Submodule.span k (range Subtype.val) = ⊤\n⊢ affineSpan k ({p} ∪ (fun v => v +ᵥ p) '' s) = ⊤",
"tactic": "convert ext_of_direction_eq _\n ⟨p, mem_affineSpan k (Set.mem_union_left _ (Set.mem_singleton _)), mem_top k V p⟩"
},
{
"state_after": "case convert_1\nk : Type u_2\nV : Type u_1\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type ?u.351531\ns : Set V\np : P\nh : Submodule.span k (range Subtype.val) = ⊤\n⊢ Submodule.span k (range Subtype.val) ≤ Submodule.span k ((fun x => x -ᵥ p) '' ({p} ∪ (fun v => v +ᵥ p) '' s))",
"state_before": "case convert_1\nk : Type u_2\nV : Type u_1\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type ?u.351531\ns : Set V\np : P\nh : Submodule.span k (range Subtype.val) = ⊤\n⊢ direction (affineSpan k ({p} ∪ (fun v => v +ᵥ p) '' s)) = direction ⊤",
"tactic": "rw [direction_affineSpan, direction_top,\n vectorSpan_eq_span_vsub_set_right k (Set.mem_union_left _ (Set.mem_singleton _) : p ∈ _),\n eq_top_iff, ← h]"
},
{
"state_after": "case convert_1.h\nk : Type u_2\nV : Type u_1\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type ?u.351531\ns : Set V\np : P\nh : Submodule.span k (range Subtype.val) = ⊤\n⊢ range Subtype.val ⊆ (fun x => x -ᵥ p) '' ({p} ∪ (fun v => v +ᵥ p) '' s)",
"state_before": "case convert_1\nk : Type u_2\nV : Type u_1\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type ?u.351531\ns : Set V\np : P\nh : Submodule.span k (range Subtype.val) = ⊤\n⊢ Submodule.span k (range Subtype.val) ≤ Submodule.span k ((fun x => x -ᵥ p) '' ({p} ∪ (fun v => v +ᵥ p) '' s))",
"tactic": "apply Submodule.span_mono"
},
{
"state_after": "case convert_1.h.intro\nk : Type u_2\nV : Type u_1\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type ?u.351531\ns : Set V\np : P\nh : Submodule.span k (range Subtype.val) = ⊤\nv' : { x // x ∈ s }\n⊢ ↑v' ∈ (fun x => x -ᵥ p) '' ({p} ∪ (fun v => v +ᵥ p) '' s)",
"state_before": "case convert_1.h\nk : Type u_2\nV : Type u_1\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type ?u.351531\ns : Set V\np : P\nh : Submodule.span k (range Subtype.val) = ⊤\n⊢ range Subtype.val ⊆ (fun x => x -ᵥ p) '' ({p} ∪ (fun v => v +ᵥ p) '' s)",
"tactic": "rintro v ⟨v', rfl⟩"
},
{
"state_after": "case convert_1.h.intro\nk : Type u_2\nV : Type u_1\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type ?u.351531\ns : Set V\np : P\nh : Submodule.span k (range Subtype.val) = ⊤\nv' : { x // x ∈ s }\n⊢ ↑v' +ᵥ p ∈ {p} ∪ (fun v => v +ᵥ p) '' s ∧ (fun x => x -ᵥ p) (↑v' +ᵥ p) = ↑v'",
"state_before": "case convert_1.h.intro\nk : Type u_2\nV : Type u_1\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type ?u.351531\ns : Set V\np : P\nh : Submodule.span k (range Subtype.val) = ⊤\nv' : { x // x ∈ s }\n⊢ ↑v' ∈ (fun x => x -ᵥ p) '' ({p} ∪ (fun v => v +ᵥ p) '' s)",
"tactic": "use (v' : V) +ᵥ p"
},
{
"state_after": "no goals",
"state_before": "case convert_1.h.intro\nk : Type u_2\nV : Type u_1\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type ?u.351531\ns : Set V\np : P\nh : Submodule.span k (range Subtype.val) = ⊤\nv' : { x // x ∈ s }\n⊢ ↑v' +ᵥ p ∈ {p} ∪ (fun v => v +ᵥ p) '' s ∧ (fun x => x -ᵥ p) (↑v' +ᵥ p) = ↑v'",
"tactic": "simp"
}
] |
[
1238,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1227,
1
] |
Mathlib/Data/Real/CauSeq.lean
|
CauSeq.const_limZero
|
[] |
[
464,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
458,
1
] |
Mathlib/Topology/MetricSpace/Kuratowski.lean
|
KuratowskiEmbedding.exists_isometric_embedding
|
[
{
"state_after": "case inl\nα✝ : Type u\nβ : Type v\nγ : Type w\nf g : { x // x ∈ lp (fun n => ℝ) ⊤ }\nn : ℕ\nC : ℝ\ninst✝² : MetricSpace α✝\nx : ℕ → α✝\na b : α✝\nα : Type u\ninst✝¹ : MetricSpace α\ninst✝ : SeparableSpace α\nh : univ = ∅\n⊢ ∃ f, Isometry f\n\ncase inr\nα✝ : Type u\nβ : Type v\nγ : Type w\nf g : { x // x ∈ lp (fun n => ℝ) ⊤ }\nn : ℕ\nC : ℝ\ninst✝² : MetricSpace α✝\nx : ℕ → α✝\na b : α✝\nα : Type u\ninst✝¹ : MetricSpace α\ninst✝ : SeparableSpace α\nh : Set.Nonempty univ\n⊢ ∃ f, Isometry f",
"state_before": "α✝ : Type u\nβ : Type v\nγ : Type w\nf g : { x // x ∈ lp (fun n => ℝ) ⊤ }\nn : ℕ\nC : ℝ\ninst✝² : MetricSpace α✝\nx : ℕ → α✝\na b : α✝\nα : Type u\ninst✝¹ : MetricSpace α\ninst✝ : SeparableSpace α\n⊢ ∃ f, Isometry f",
"tactic": "cases' (univ : Set α).eq_empty_or_nonempty with h h"
},
{
"state_after": "case inl\nα✝ : Type u\nβ : Type v\nγ : Type w\nf g : { x // x ∈ lp (fun n => ℝ) ⊤ }\nn : ℕ\nC : ℝ\ninst✝² : MetricSpace α✝\nx : ℕ → α✝\na b : α✝\nα : Type u\ninst✝¹ : MetricSpace α\ninst✝ : SeparableSpace α\nh : univ = ∅\n⊢ Isometry fun x => 0",
"state_before": "case inl\nα✝ : Type u\nβ : Type v\nγ : Type w\nf g : { x // x ∈ lp (fun n => ℝ) ⊤ }\nn : ℕ\nC : ℝ\ninst✝² : MetricSpace α✝\nx : ℕ → α✝\na b : α✝\nα : Type u\ninst✝¹ : MetricSpace α\ninst✝ : SeparableSpace α\nh : univ = ∅\n⊢ ∃ f, Isometry f",
"tactic": "use fun _ => 0"
},
{
"state_after": "case inl\nα✝ : Type u\nβ : Type v\nγ : Type w\nf g : { x // x ∈ lp (fun n => ℝ) ⊤ }\nn : ℕ\nC : ℝ\ninst✝² : MetricSpace α✝\nx✝ : ℕ → α✝\na b : α✝\nα : Type u\ninst✝¹ : MetricSpace α\ninst✝ : SeparableSpace α\nh : univ = ∅\nx : α\n⊢ ∀ (x2 : α), edist ((fun x => 0) x) ((fun x => 0) x2) = edist x x2",
"state_before": "case inl\nα✝ : Type u\nβ : Type v\nγ : Type w\nf g : { x // x ∈ lp (fun n => ℝ) ⊤ }\nn : ℕ\nC : ℝ\ninst✝² : MetricSpace α✝\nx : ℕ → α✝\na b : α✝\nα : Type u\ninst✝¹ : MetricSpace α\ninst✝ : SeparableSpace α\nh : univ = ∅\n⊢ Isometry fun x => 0",
"tactic": "intro x"
},
{
"state_after": "no goals",
"state_before": "case inl\nα✝ : Type u\nβ : Type v\nγ : Type w\nf g : { x // x ∈ lp (fun n => ℝ) ⊤ }\nn : ℕ\nC : ℝ\ninst✝² : MetricSpace α✝\nx✝ : ℕ → α✝\na b : α✝\nα : Type u\ninst✝¹ : MetricSpace α\ninst✝ : SeparableSpace α\nh : univ = ∅\nx : α\n⊢ ∀ (x2 : α), edist ((fun x => 0) x) ((fun x => 0) x2) = edist x x2",
"tactic": "exact absurd h (Nonempty.ne_empty ⟨x, mem_univ x⟩)"
},
{
"state_after": "case inr.intro\nα✝ : Type u\nβ : Type v\nγ : Type w\nf g : { x // x ∈ lp (fun n => ℝ) ⊤ }\nn : ℕ\nC : ℝ\ninst✝² : MetricSpace α✝\nx : ℕ → α✝\na b : α✝\nα : Type u\ninst✝¹ : MetricSpace α\ninst✝ : SeparableSpace α\nbasepoint : α\nh✝ : basepoint ∈ univ\n⊢ ∃ f, Isometry f",
"state_before": "case inr\nα✝ : Type u\nβ : Type v\nγ : Type w\nf g : { x // x ∈ lp (fun n => ℝ) ⊤ }\nn : ℕ\nC : ℝ\ninst✝² : MetricSpace α✝\nx : ℕ → α✝\na b : α✝\nα : Type u\ninst✝¹ : MetricSpace α\ninst✝ : SeparableSpace α\nh : Set.Nonempty univ\n⊢ ∃ f, Isometry f",
"tactic": "rcases h with ⟨basepoint⟩"
},
{
"state_after": "case inr.intro\nα✝ : Type u\nβ : Type v\nγ : Type w\nf g : { x // x ∈ lp (fun n => ℝ) ⊤ }\nn : ℕ\nC : ℝ\ninst✝² : MetricSpace α✝\nx : ℕ → α✝\na b : α✝\nα : Type u\ninst✝¹ : MetricSpace α\ninst✝ : SeparableSpace α\nbasepoint : α\nh✝ : basepoint ∈ univ\nthis : Inhabited α\n⊢ ∃ f, Isometry f",
"state_before": "case inr.intro\nα✝ : Type u\nβ : Type v\nγ : Type w\nf g : { x // x ∈ lp (fun n => ℝ) ⊤ }\nn : ℕ\nC : ℝ\ninst✝² : MetricSpace α✝\nx : ℕ → α✝\na b : α✝\nα : Type u\ninst✝¹ : MetricSpace α\ninst✝ : SeparableSpace α\nbasepoint : α\nh✝ : basepoint ∈ univ\n⊢ ∃ f, Isometry f",
"tactic": "haveI : Inhabited α := ⟨basepoint⟩"
},
{
"state_after": "case inr.intro\nα✝ : Type u\nβ : Type v\nγ : Type w\nf g : { x // x ∈ lp (fun n => ℝ) ⊤ }\nn : ℕ\nC : ℝ\ninst✝² : MetricSpace α✝\nx : ℕ → α✝\na b : α✝\nα : Type u\ninst✝¹ : MetricSpace α\ninst✝ : SeparableSpace α\nbasepoint : α\nh✝ : basepoint ∈ univ\nthis✝ : Inhabited α\nthis : ∃ s, Set.Countable s ∧ Dense s\n⊢ ∃ f, Isometry f",
"state_before": "case inr.intro\nα✝ : Type u\nβ : Type v\nγ : Type w\nf g : { x // x ∈ lp (fun n => ℝ) ⊤ }\nn : ℕ\nC : ℝ\ninst✝² : MetricSpace α✝\nx : ℕ → α✝\na b : α✝\nα : Type u\ninst✝¹ : MetricSpace α\ninst✝ : SeparableSpace α\nbasepoint : α\nh✝ : basepoint ∈ univ\nthis : Inhabited α\n⊢ ∃ f, Isometry f",
"tactic": "have : ∃ s : Set α, s.Countable ∧ Dense s := exists_countable_dense α"
},
{
"state_after": "case inr.intro.intro.intro\nα✝ : Type u\nβ : Type v\nγ : Type w\nf g : { x // x ∈ lp (fun n => ℝ) ⊤ }\nn : ℕ\nC : ℝ\ninst✝² : MetricSpace α✝\nx : ℕ → α✝\na b : α✝\nα : Type u\ninst✝¹ : MetricSpace α\ninst✝ : SeparableSpace α\nbasepoint : α\nh✝ : basepoint ∈ univ\nthis : Inhabited α\nS : Set α\nS_countable : Set.Countable S\nS_dense : Dense S\n⊢ ∃ f, Isometry f",
"state_before": "case inr.intro\nα✝ : Type u\nβ : Type v\nγ : Type w\nf g : { x // x ∈ lp (fun n => ℝ) ⊤ }\nn : ℕ\nC : ℝ\ninst✝² : MetricSpace α✝\nx : ℕ → α✝\na b : α✝\nα : Type u\ninst✝¹ : MetricSpace α\ninst✝ : SeparableSpace α\nbasepoint : α\nh✝ : basepoint ∈ univ\nthis✝ : Inhabited α\nthis : ∃ s, Set.Countable s ∧ Dense s\n⊢ ∃ f, Isometry f",
"tactic": "rcases this with ⟨S, ⟨S_countable, S_dense⟩⟩"
},
{
"state_after": "case inr.intro.intro.intro.intro\nα✝ : Type u\nβ : Type v\nγ : Type w\nf g : { x // x ∈ lp (fun n => ℝ) ⊤ }\nn : ℕ\nC : ℝ\ninst✝² : MetricSpace α✝\nx✝ : ℕ → α✝\na b : α✝\nα : Type u\ninst✝¹ : MetricSpace α\ninst✝ : SeparableSpace α\nbasepoint : α\nh✝ : basepoint ∈ univ\nthis : Inhabited α\nS : Set α\nS_countable : Set.Countable S\nS_dense : Dense S\nx : ℕ → α\nx_range : S ⊆ range x\n⊢ ∃ f, Isometry f",
"state_before": "case inr.intro.intro.intro\nα✝ : Type u\nβ : Type v\nγ : Type w\nf g : { x // x ∈ lp (fun n => ℝ) ⊤ }\nn : ℕ\nC : ℝ\ninst✝² : MetricSpace α✝\nx : ℕ → α✝\na b : α✝\nα : Type u\ninst✝¹ : MetricSpace α\ninst✝ : SeparableSpace α\nbasepoint : α\nh✝ : basepoint ∈ univ\nthis : Inhabited α\nS : Set α\nS_countable : Set.Countable S\nS_dense : Dense S\n⊢ ∃ f, Isometry f",
"tactic": "rcases Set.countable_iff_exists_subset_range.1 S_countable with ⟨x, x_range⟩"
},
{
"state_after": "no goals",
"state_before": "case inr.intro.intro.intro.intro\nα✝ : Type u\nβ : Type v\nγ : Type w\nf g : { x // x ∈ lp (fun n => ℝ) ⊤ }\nn : ℕ\nC : ℝ\ninst✝² : MetricSpace α✝\nx✝ : ℕ → α✝\na b : α✝\nα : Type u\ninst✝¹ : MetricSpace α\ninst✝ : SeparableSpace α\nbasepoint : α\nh✝ : basepoint ∈ univ\nthis : Inhabited α\nS : Set α\nS_countable : Set.Countable S\nS_dense : Dense S\nx : ℕ → α\nx_range : S ⊆ range x\n⊢ ∃ f, Isometry f",
"tactic": "exact ⟨embeddingOfSubset x, embeddingOfSubset_isometry x (S_dense.mono x_range)⟩"
}
] |
[
108,
85
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
97,
1
] |
Mathlib/Algebra/Module/Submodule/Lattice.lean
|
Submodule.toAddSubmonoid_toNatSubmodule
|
[] |
[
376,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
374,
1
] |
Std/Data/List/Lemmas.lean
|
List.get?_modifyNth_eq
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nf : α → α\nn : Nat\nl : List α\n⊢ get? (modifyNth f n l) n = f <$> get? l n",
"tactic": "simp only [get?_modifyNth, if_pos]"
}
] |
[
758,
37
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
756,
9
] |
Mathlib/Tactic/LinearCombination.lean
|
Mathlib.Tactic.LinearCombination.add_pf
|
[] |
[
39,
94
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
39,
1
] |
Mathlib/Data/Dfinsupp/Basic.lean
|
Dfinsupp.filter_sub
|
[] |
[
480,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
478,
1
] |
Mathlib/Analysis/Calculus/FDeriv/Comp.lean
|
Differentiable.comp
|
[] |
[
188,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
186,
1
] |
Mathlib/FieldTheory/RatFunc.lean
|
RatFunc.liftOn_ofFractionRing_mk
|
[
{
"state_after": "K : Type u\ninst✝ : CommRing K\nP : Sort v\nn : K[X]\nd : { x // x ∈ K[X]⁰ }\nf : K[X] → K[X] → P\nH : ∀ {p q p' q' : K[X]}, q ∈ K[X]⁰ → q' ∈ K[X]⁰ → q' * p = q * p' → f p q = f p' q'\n⊢ Localization.liftOn { toFractionRing := Localization.mk n d }.toFractionRing (fun p q => f p ↑q)\n (_ : ∀ {p p' : K[X]} {q q' : { x // x ∈ K[X]⁰ }}, ↑(Localization.r K[X]⁰) (p, q) (p', q') → f p ↑q = f p' ↑q') =\n f n ↑d",
"state_before": "K : Type u\ninst✝ : CommRing K\nP : Sort v\nn : K[X]\nd : { x // x ∈ K[X]⁰ }\nf : K[X] → K[X] → P\nH : ∀ {p q p' q' : K[X]}, q ∈ K[X]⁰ → q' ∈ K[X]⁰ → q' * p = q * p' → f p q = f p' q'\n⊢ RatFunc.liftOn { toFractionRing := Localization.mk n d } f H = f n ↑d",
"tactic": "rw [RatFunc.liftOn]"
},
{
"state_after": "no goals",
"state_before": "K : Type u\ninst✝ : CommRing K\nP : Sort v\nn : K[X]\nd : { x // x ∈ K[X]⁰ }\nf : K[X] → K[X] → P\nH : ∀ {p q p' q' : K[X]}, q ∈ K[X]⁰ → q' ∈ K[X]⁰ → q' * p = q * p' → f p q = f p' q'\n⊢ Localization.liftOn { toFractionRing := Localization.mk n d }.toFractionRing (fun p q => f p ↑q)\n (_ : ∀ {p p' : K[X]} {q q' : { x // x ∈ K[X]⁰ }}, ↑(Localization.r K[X]⁰) (p, q) (p', q') → f p ↑q = f p' ↑q') =\n f n ↑d",
"tactic": "exact Localization.liftOn_mk _ _ _ _"
}
] |
[
172,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
168,
1
] |
Mathlib/Logic/Function/Basic.lean
|
eq_rec_on_bijective
|
[] |
[
997,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
995,
1
] |
Mathlib/RingTheory/Ideal/Operations.lean
|
Ideal.map_of_equiv
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\nF : Type ?u.1386892\ninst✝² : Ring R\ninst✝¹ : Ring S\ninst✝ : RingHomClass F R S\nf✝ : F\nI✝ I : Ideal R\nf : R ≃+* S\n⊢ map (↑(RingEquiv.symm f)) (map (↑f) I) = I",
"tactic": "simp [← RingEquiv.toRingHom_eq_coe, map_map]"
}
] |
[
1713,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1711,
1
] |
Mathlib/Topology/MetricSpace/Lipschitz.lean
|
LipschitzWith.eval
|
[
{
"state_after": "no goals",
"state_before": "α✝ : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝⁴ : PseudoEMetricSpace α✝\ninst✝³ : PseudoEMetricSpace β\ninst✝² : PseudoEMetricSpace γ\nK : ℝ≥0\nf✝ : α✝ → β\nx y : α✝\nr : ℝ≥0∞\nα : ι → Type u\ninst✝¹ : (i : ι) → PseudoEMetricSpace (α i)\ninst✝ : Fintype ι\ni : ι\nf g : (x : ι) → α x\n⊢ edist (eval i f) (eval i g) ≤ edist f g",
"tactic": "convert edist_le_pi_edist f g i"
}
] |
[
219,
74
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
217,
11
] |
Mathlib/LinearAlgebra/Alternating.lean
|
AlternatingMap.map_swap_add
|
[
{
"state_after": "R : Type u_4\ninst✝¹⁵ : Semiring R\nM : Type u_3\ninst✝¹⁴ : AddCommMonoid M\ninst✝¹³ : Module R M\nN : Type u_2\ninst✝¹² : AddCommMonoid N\ninst✝¹¹ : Module R N\nP : Type ?u.417280\ninst✝¹⁰ : AddCommMonoid P\ninst✝⁹ : Module R P\nM' : Type ?u.417310\ninst✝⁸ : AddCommGroup M'\ninst✝⁷ : Module R M'\nN' : Type ?u.417698\ninst✝⁶ : AddCommGroup N'\ninst✝⁵ : Module R N'\nι : Type u_1\nι' : Type ?u.418089\nι'' : Type ?u.418092\nM₂ : Type ?u.418095\ninst✝⁴ : AddCommMonoid M₂\ninst✝³ : Module R M₂\nM₃ : Type ?u.418125\ninst✝² : AddCommMonoid M₃\ninst✝¹ : Module R M₃\nf f' : AlternatingMap R M N ι\ng g₂ : AlternatingMap R M N' ι\ng' : AlternatingMap R M' N' ι\nv : ι → M\nv' : ι → M'\ninst✝ : DecidableEq ι\ni j : ι\nhij : i ≠ j\n⊢ ↑f (update (update v j (v i)) i (v j)) + ↑f v = 0",
"state_before": "R : Type u_4\ninst✝¹⁵ : Semiring R\nM : Type u_3\ninst✝¹⁴ : AddCommMonoid M\ninst✝¹³ : Module R M\nN : Type u_2\ninst✝¹² : AddCommMonoid N\ninst✝¹¹ : Module R N\nP : Type ?u.417280\ninst✝¹⁰ : AddCommMonoid P\ninst✝⁹ : Module R P\nM' : Type ?u.417310\ninst✝⁸ : AddCommGroup M'\ninst✝⁷ : Module R M'\nN' : Type ?u.417698\ninst✝⁶ : AddCommGroup N'\ninst✝⁵ : Module R N'\nι : Type u_1\nι' : Type ?u.418089\nι'' : Type ?u.418092\nM₂ : Type ?u.418095\ninst✝⁴ : AddCommMonoid M₂\ninst✝³ : Module R M₂\nM₃ : Type ?u.418125\ninst✝² : AddCommMonoid M₃\ninst✝¹ : Module R M₃\nf f' : AlternatingMap R M N ι\ng g₂ : AlternatingMap R M N' ι\ng' : AlternatingMap R M' N' ι\nv : ι → M\nv' : ι → M'\ninst✝ : DecidableEq ι\ni j : ι\nhij : i ≠ j\n⊢ ↑f (v ∘ ↑(Equiv.swap i j)) + ↑f v = 0",
"tactic": "rw [Equiv.comp_swap_eq_update]"
},
{
"state_after": "case h.e'_2\nR : Type u_4\ninst✝¹⁵ : Semiring R\nM : Type u_3\ninst✝¹⁴ : AddCommMonoid M\ninst✝¹³ : Module R M\nN : Type u_2\ninst✝¹² : AddCommMonoid N\ninst✝¹¹ : Module R N\nP : Type ?u.417280\ninst✝¹⁰ : AddCommMonoid P\ninst✝⁹ : Module R P\nM' : Type ?u.417310\ninst✝⁸ : AddCommGroup M'\ninst✝⁷ : Module R M'\nN' : Type ?u.417698\ninst✝⁶ : AddCommGroup N'\ninst✝⁵ : Module R N'\nι : Type u_1\nι' : Type ?u.418089\nι'' : Type ?u.418092\nM₂ : Type ?u.418095\ninst✝⁴ : AddCommMonoid M₂\ninst✝³ : Module R M₂\nM₃ : Type ?u.418125\ninst✝² : AddCommMonoid M₃\ninst✝¹ : Module R M₃\nf f' : AlternatingMap R M N ι\ng g₂ : AlternatingMap R M N' ι\ng' : AlternatingMap R M' N' ι\nv : ι → M\nv' : ι → M'\ninst✝ : DecidableEq ι\ni j : ι\nhij : i ≠ j\n⊢ ↑f (update (update v j (v i)) i (v j)) + ↑f v = ↑f (update (update v i (v i + v j)) j (v i + v j))",
"state_before": "R : Type u_4\ninst✝¹⁵ : Semiring R\nM : Type u_3\ninst✝¹⁴ : AddCommMonoid M\ninst✝¹³ : Module R M\nN : Type u_2\ninst✝¹² : AddCommMonoid N\ninst✝¹¹ : Module R N\nP : Type ?u.417280\ninst✝¹⁰ : AddCommMonoid P\ninst✝⁹ : Module R P\nM' : Type ?u.417310\ninst✝⁸ : AddCommGroup M'\ninst✝⁷ : Module R M'\nN' : Type ?u.417698\ninst✝⁶ : AddCommGroup N'\ninst✝⁵ : Module R N'\nι : Type u_1\nι' : Type ?u.418089\nι'' : Type ?u.418092\nM₂ : Type ?u.418095\ninst✝⁴ : AddCommMonoid M₂\ninst✝³ : Module R M₂\nM₃ : Type ?u.418125\ninst✝² : AddCommMonoid M₃\ninst✝¹ : Module R M₃\nf f' : AlternatingMap R M N ι\ng g₂ : AlternatingMap R M N' ι\ng' : AlternatingMap R M' N' ι\nv : ι → M\nv' : ι → M'\ninst✝ : DecidableEq ι\ni j : ι\nhij : i ≠ j\n⊢ ↑f (update (update v j (v i)) i (v j)) + ↑f v = 0",
"tactic": "convert f.map_update_update v hij (v i + v j)"
},
{
"state_after": "no goals",
"state_before": "case h.e'_2\nR : Type u_4\ninst✝¹⁵ : Semiring R\nM : Type u_3\ninst✝¹⁴ : AddCommMonoid M\ninst✝¹³ : Module R M\nN : Type u_2\ninst✝¹² : AddCommMonoid N\ninst✝¹¹ : Module R N\nP : Type ?u.417280\ninst✝¹⁰ : AddCommMonoid P\ninst✝⁹ : Module R P\nM' : Type ?u.417310\ninst✝⁸ : AddCommGroup M'\ninst✝⁷ : Module R M'\nN' : Type ?u.417698\ninst✝⁶ : AddCommGroup N'\ninst✝⁵ : Module R N'\nι : Type u_1\nι' : Type ?u.418089\nι'' : Type ?u.418092\nM₂ : Type ?u.418095\ninst✝⁴ : AddCommMonoid M₂\ninst✝³ : Module R M₂\nM₃ : Type ?u.418125\ninst✝² : AddCommMonoid M₃\ninst✝¹ : Module R M₃\nf f' : AlternatingMap R M N ι\ng g₂ : AlternatingMap R M N' ι\ng' : AlternatingMap R M' N' ι\nv : ι → M\nv' : ι → M'\ninst✝ : DecidableEq ι\ni j : ι\nhij : i ≠ j\n⊢ ↑f (update (update v j (v i)) i (v j)) + ↑f v = ↑f (update (update v i (v i + v j)) j (v i + v j))",
"tactic": "simp [f.map_update_self _ hij, f.map_update_self _ hij.symm,\n Function.update_comm hij (v i + v j) (v _) v, Function.update_comm hij.symm (v i) (v i) v]"
}
] |
[
694,
95
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
689,
1
] |
Mathlib/Data/List/Intervals.lean
|
List.Ico.succ_singleton
|
[
{
"state_after": "n : ℕ\n⊢ range' n (n + 1 - n) = [n]",
"state_before": "n : ℕ\n⊢ Ico n (n + 1) = [n]",
"tactic": "dsimp [Ico]"
},
{
"state_after": "no goals",
"state_before": "n : ℕ\n⊢ range' n (n + 1 - n) = [n]",
"tactic": "simp [range', add_tsub_cancel_left]"
}
] |
[
124,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
122,
1
] |
Mathlib/LinearAlgebra/Determinant.lean
|
LinearMap.associated_det_of_eq_comp
|
[
{
"state_after": "R : Type u_1\ninst✝⁶ : CommRing R\nM : Type u_2\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\nM' : Type ?u.2104049\ninst✝³ : AddCommGroup M'\ninst✝² : Module R M'\nι : Type ?u.2104591\ninst✝¹ : DecidableEq ι\ninst✝ : Fintype ι\ne✝ : Basis ι R M\ne : M ≃ₗ[R] M\nf f' : M →ₗ[R] M\nh : ∀ (x : M), ↑f x = ↑f' (↑e x)\n⊢ Associated (↑LinearMap.det (comp f' ↑e)) (↑LinearMap.det f')",
"state_before": "R : Type u_1\ninst✝⁶ : CommRing R\nM : Type u_2\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\nM' : Type ?u.2104049\ninst✝³ : AddCommGroup M'\ninst✝² : Module R M'\nι : Type ?u.2104591\ninst✝¹ : DecidableEq ι\ninst✝ : Fintype ι\ne✝ : Basis ι R M\ne : M ≃ₗ[R] M\nf f' : M →ₗ[R] M\nh : ∀ (x : M), ↑f x = ↑f' (↑e x)\n⊢ Associated (↑LinearMap.det f) (↑LinearMap.det f')",
"tactic": "suffices Associated (LinearMap.det (f' ∘ₗ ↑e)) (LinearMap.det f') by\n convert this using 2\n ext x\n exact h x"
},
{
"state_after": "R : Type u_1\ninst✝⁶ : CommRing R\nM : Type u_2\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\nM' : Type ?u.2104049\ninst✝³ : AddCommGroup M'\ninst✝² : Module R M'\nι : Type ?u.2104591\ninst✝¹ : DecidableEq ι\ninst✝ : Fintype ι\ne✝ : Basis ι R M\ne : M ≃ₗ[R] M\nf f' : M →ₗ[R] M\nh : ∀ (x : M), ↑f x = ↑f' (↑e x)\n⊢ Associated (↑LinearMap.det f' * ↑LinearMap.det ↑e) (↑LinearMap.det f' * 1)",
"state_before": "R : Type u_1\ninst✝⁶ : CommRing R\nM : Type u_2\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\nM' : Type ?u.2104049\ninst✝³ : AddCommGroup M'\ninst✝² : Module R M'\nι : Type ?u.2104591\ninst✝¹ : DecidableEq ι\ninst✝ : Fintype ι\ne✝ : Basis ι R M\ne : M ≃ₗ[R] M\nf f' : M →ₗ[R] M\nh : ∀ (x : M), ↑f x = ↑f' (↑e x)\n⊢ Associated (↑LinearMap.det (comp f' ↑e)) (↑LinearMap.det f')",
"tactic": "rw [← mul_one (LinearMap.det f'), LinearMap.det_comp]"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝⁶ : CommRing R\nM : Type u_2\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\nM' : Type ?u.2104049\ninst✝³ : AddCommGroup M'\ninst✝² : Module R M'\nι : Type ?u.2104591\ninst✝¹ : DecidableEq ι\ninst✝ : Fintype ι\ne✝ : Basis ι R M\ne : M ≃ₗ[R] M\nf f' : M →ₗ[R] M\nh : ∀ (x : M), ↑f x = ↑f' (↑e x)\n⊢ Associated (↑LinearMap.det f' * ↑LinearMap.det ↑e) (↑LinearMap.det f' * 1)",
"tactic": "exact Associated.mul_left _ (associated_one_iff_isUnit.mpr e.isUnit_det')"
},
{
"state_after": "case h.e'_3.h.e'_6\nR : Type u_1\ninst✝⁶ : CommRing R\nM : Type u_2\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\nM' : Type ?u.2104049\ninst✝³ : AddCommGroup M'\ninst✝² : Module R M'\nι : Type ?u.2104591\ninst✝¹ : DecidableEq ι\ninst✝ : Fintype ι\ne✝ : Basis ι R M\ne : M ≃ₗ[R] M\nf f' : M →ₗ[R] M\nh : ∀ (x : M), ↑f x = ↑f' (↑e x)\nthis : Associated (↑LinearMap.det (comp f' ↑e)) (↑LinearMap.det f')\n⊢ f = comp f' ↑e",
"state_before": "R : Type u_1\ninst✝⁶ : CommRing R\nM : Type u_2\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\nM' : Type ?u.2104049\ninst✝³ : AddCommGroup M'\ninst✝² : Module R M'\nι : Type ?u.2104591\ninst✝¹ : DecidableEq ι\ninst✝ : Fintype ι\ne✝ : Basis ι R M\ne : M ≃ₗ[R] M\nf f' : M →ₗ[R] M\nh : ∀ (x : M), ↑f x = ↑f' (↑e x)\nthis : Associated (↑LinearMap.det (comp f' ↑e)) (↑LinearMap.det f')\n⊢ Associated (↑LinearMap.det f) (↑LinearMap.det f')",
"tactic": "convert this using 2"
},
{
"state_after": "case h.e'_3.h.e'_6.h\nR : Type u_1\ninst✝⁶ : CommRing R\nM : Type u_2\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\nM' : Type ?u.2104049\ninst✝³ : AddCommGroup M'\ninst✝² : Module R M'\nι : Type ?u.2104591\ninst✝¹ : DecidableEq ι\ninst✝ : Fintype ι\ne✝ : Basis ι R M\ne : M ≃ₗ[R] M\nf f' : M →ₗ[R] M\nh : ∀ (x : M), ↑f x = ↑f' (↑e x)\nthis : Associated (↑LinearMap.det (comp f' ↑e)) (↑LinearMap.det f')\nx : M\n⊢ ↑f x = ↑(comp f' ↑e) x",
"state_before": "case h.e'_3.h.e'_6\nR : Type u_1\ninst✝⁶ : CommRing R\nM : Type u_2\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\nM' : Type ?u.2104049\ninst✝³ : AddCommGroup M'\ninst✝² : Module R M'\nι : Type ?u.2104591\ninst✝¹ : DecidableEq ι\ninst✝ : Fintype ι\ne✝ : Basis ι R M\ne : M ≃ₗ[R] M\nf f' : M →ₗ[R] M\nh : ∀ (x : M), ↑f x = ↑f' (↑e x)\nthis : Associated (↑LinearMap.det (comp f' ↑e)) (↑LinearMap.det f')\n⊢ f = comp f' ↑e",
"tactic": "ext x"
},
{
"state_after": "no goals",
"state_before": "case h.e'_3.h.e'_6.h\nR : Type u_1\ninst✝⁶ : CommRing R\nM : Type u_2\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\nM' : Type ?u.2104049\ninst✝³ : AddCommGroup M'\ninst✝² : Module R M'\nι : Type ?u.2104591\ninst✝¹ : DecidableEq ι\ninst✝ : Fintype ι\ne✝ : Basis ι R M\ne : M ≃ₗ[R] M\nf f' : M →ₗ[R] M\nh : ∀ (x : M), ↑f x = ↑f' (↑e x)\nthis : Associated (↑LinearMap.det (comp f' ↑e)) (↑LinearMap.det f')\nx : M\n⊢ ↑f x = ↑(comp f' ↑e) x",
"tactic": "exact h x"
}
] |
[
496,
76
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
489,
1
] |
Mathlib/Algebra/GroupPower/Basic.lean
|
zpow_two
|
[
{
"state_after": "case h.e'_2\nα : Type ?u.55852\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u₁\nS : Type u₂\ninst✝ : DivInvMonoid G\na : G\n⊢ a ^ 2 = a ^ 2",
"state_before": "α : Type ?u.55852\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u₁\nS : Type u₂\ninst✝ : DivInvMonoid G\na : G\n⊢ a ^ 2 = a * a",
"tactic": "convert pow_two a using 1"
},
{
"state_after": "no goals",
"state_before": "case h.e'_2\nα : Type ?u.55852\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u₁\nS : Type u₂\ninst✝ : DivInvMonoid G\na : G\n⊢ a ^ 2 = a ^ 2",
"tactic": "exact zpow_ofNat a 2"
}
] |
[
295,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
293,
1
] |
Mathlib/GroupTheory/MonoidLocalization.lean
|
Submonoid.LocalizationMap.map_units
|
[] |
[
577,
8
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
576,
1
] |
Mathlib/Analysis/NormedSpace/Multilinear.lean
|
ContinuousMultilinearMap.uncurry_curryRight
|
[
{
"state_after": "case H\n𝕜 : Type u\nι : Type v\nι' : Type v'\nn : ℕ\nE : ι → Type wE\nE₁ : ι → Type wE₁\nE' : ι' → Type wE'\nEi : Fin (Nat.succ n) → Type wEi\nG : Type wG\nG' : Type wG'\ninst✝¹⁴ : Fintype ι\ninst✝¹³ : Fintype ι'\ninst✝¹² : NontriviallyNormedField 𝕜\ninst✝¹¹ : (i : ι) → NormedAddCommGroup (E i)\ninst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i)\ninst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i)\ninst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i)\ninst✝⁷ : (i : ι') → NormedAddCommGroup (E' i)\ninst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i)\ninst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i)\ninst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i)\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\ninst✝¹ : NormedAddCommGroup G'\ninst✝ : NormedSpace 𝕜 G'\nf : ContinuousMultilinearMap 𝕜 Ei G\nm : (i : Fin (Nat.succ n)) → Ei i\n⊢ ↑(uncurryRight (curryRight f)) m = ↑f m",
"state_before": "𝕜 : Type u\nι : Type v\nι' : Type v'\nn : ℕ\nE : ι → Type wE\nE₁ : ι → Type wE₁\nE' : ι' → Type wE'\nEi : Fin (Nat.succ n) → Type wEi\nG : Type wG\nG' : Type wG'\ninst✝¹⁴ : Fintype ι\ninst✝¹³ : Fintype ι'\ninst✝¹² : NontriviallyNormedField 𝕜\ninst✝¹¹ : (i : ι) → NormedAddCommGroup (E i)\ninst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i)\ninst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i)\ninst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i)\ninst✝⁷ : (i : ι') → NormedAddCommGroup (E' i)\ninst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i)\ninst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i)\ninst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i)\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\ninst✝¹ : NormedAddCommGroup G'\ninst✝ : NormedSpace 𝕜 G'\nf : ContinuousMultilinearMap 𝕜 Ei G\n⊢ uncurryRight (curryRight f) = f",
"tactic": "ext m"
},
{
"state_after": "no goals",
"state_before": "case H\n𝕜 : Type u\nι : Type v\nι' : Type v'\nn : ℕ\nE : ι → Type wE\nE₁ : ι → Type wE₁\nE' : ι' → Type wE'\nEi : Fin (Nat.succ n) → Type wEi\nG : Type wG\nG' : Type wG'\ninst✝¹⁴ : Fintype ι\ninst✝¹³ : Fintype ι'\ninst✝¹² : NontriviallyNormedField 𝕜\ninst✝¹¹ : (i : ι) → NormedAddCommGroup (E i)\ninst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i)\ninst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i)\ninst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i)\ninst✝⁷ : (i : ι') → NormedAddCommGroup (E' i)\ninst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i)\ninst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i)\ninst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i)\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\ninst✝¹ : NormedAddCommGroup G'\ninst✝ : NormedSpace 𝕜 G'\nf : ContinuousMultilinearMap 𝕜 Ei G\nm : (i : Fin (Nat.succ n)) → Ei i\n⊢ ↑(uncurryRight (curryRight f)) m = ↑f m",
"tactic": "simp"
}
] |
[
1511,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1508,
1
] |
Mathlib/Algebra/DirectSum/Module.lean
|
DirectSum.lequivCongrLeft_apply
|
[] |
[
245,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
243,
1
] |
Mathlib/Data/Finset/Pointwise.lean
|
Finset.inv_empty
|
[] |
[
229,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
228,
1
] |
Mathlib/Algebra/Hom/Iterate.lean
|
Commute.function_commute_mul_right
|
[] |
[
260,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
258,
1
] |
Mathlib/Topology/MetricSpace/HausdorffDistance.lean
|
Metric.thickening_thickening_subset
|
[
{
"state_after": "case inl\nι : Sort ?u.153954\nα : Type u\nβ : Type v\ninst✝ : PseudoEMetricSpace α\nδ✝ ε✝ : ℝ\ns✝ t : Set α\nx : α\nε δ : ℝ\ns : Set α\nhε : ε ≤ 0\n⊢ thickening ε (thickening δ s) ⊆ thickening (ε + δ) s\n\ncase inr\nι : Sort ?u.153954\nα : Type u\nβ : Type v\ninst✝ : PseudoEMetricSpace α\nδ✝ ε✝ : ℝ\ns✝ t : Set α\nx : α\nε δ : ℝ\ns : Set α\nhε : 0 ≤ ε\n⊢ thickening ε (thickening δ s) ⊆ thickening (ε + δ) s",
"state_before": "ι : Sort ?u.153954\nα : Type u\nβ : Type v\ninst✝ : PseudoEMetricSpace α\nδ✝ ε✝ : ℝ\ns✝ t : Set α\nx : α\nε δ : ℝ\ns : Set α\n⊢ thickening ε (thickening δ s) ⊆ thickening (ε + δ) s",
"tactic": "obtain hε | hε := le_total ε 0"
},
{
"state_after": "case inr.inl\nι : Sort ?u.153954\nα : Type u\nβ : Type v\ninst✝ : PseudoEMetricSpace α\nδ✝ ε✝ : ℝ\ns✝ t : Set α\nx : α\nε δ : ℝ\ns : Set α\nhε : 0 ≤ ε\nhδ : δ ≤ 0\n⊢ thickening ε (thickening δ s) ⊆ thickening (ε + δ) s\n\ncase inr.inr\nι : Sort ?u.153954\nα : Type u\nβ : Type v\ninst✝ : PseudoEMetricSpace α\nδ✝ ε✝ : ℝ\ns✝ t : Set α\nx : α\nε δ : ℝ\ns : Set α\nhε : 0 ≤ ε\nhδ : 0 ≤ δ\n⊢ thickening ε (thickening δ s) ⊆ thickening (ε + δ) s",
"state_before": "case inr\nι : Sort ?u.153954\nα : Type u\nβ : Type v\ninst✝ : PseudoEMetricSpace α\nδ✝ ε✝ : ℝ\ns✝ t : Set α\nx : α\nε δ : ℝ\ns : Set α\nhε : 0 ≤ ε\n⊢ thickening ε (thickening δ s) ⊆ thickening (ε + δ) s",
"tactic": "obtain hδ | hδ := le_total δ 0"
},
{
"state_after": "case inr.inr\nι : Sort ?u.153954\nα : Type u\nβ : Type v\ninst✝ : PseudoEMetricSpace α\nδ✝ ε✝ : ℝ\ns✝ t : Set α\nx✝ : α\nε δ : ℝ\ns : Set α\nhε : 0 ≤ ε\nhδ : 0 ≤ δ\nx : α\n⊢ x ∈ thickening ε (thickening δ s) → x ∈ thickening (ε + δ) s",
"state_before": "case inr.inr\nι : Sort ?u.153954\nα : Type u\nβ : Type v\ninst✝ : PseudoEMetricSpace α\nδ✝ ε✝ : ℝ\ns✝ t : Set α\nx : α\nε δ : ℝ\ns : Set α\nhε : 0 ≤ ε\nhδ : 0 ≤ δ\n⊢ thickening ε (thickening δ s) ⊆ thickening (ε + δ) s",
"tactic": "intro x"
},
{
"state_after": "case inr.inr\nι : Sort ?u.153954\nα : Type u\nβ : Type v\ninst✝ : PseudoEMetricSpace α\nδ✝ ε✝ : ℝ\ns✝ t : Set α\nx✝ : α\nε δ : ℝ\ns : Set α\nhε : 0 ≤ ε\nhδ : 0 ≤ δ\nx : α\n⊢ (∃ z, (∃ z_1, z_1 ∈ s ∧ edist z z_1 < ENNReal.ofReal δ) ∧ edist x z < ENNReal.ofReal ε) →\n ∃ z, z ∈ s ∧ edist x z < ENNReal.ofReal ε + ENNReal.ofReal δ",
"state_before": "case inr.inr\nι : Sort ?u.153954\nα : Type u\nβ : Type v\ninst✝ : PseudoEMetricSpace α\nδ✝ ε✝ : ℝ\ns✝ t : Set α\nx✝ : α\nε δ : ℝ\ns : Set α\nhε : 0 ≤ ε\nhδ : 0 ≤ δ\nx : α\n⊢ x ∈ thickening ε (thickening δ s) → x ∈ thickening (ε + δ) s",
"tactic": "simp_rw [mem_thickening_iff_exists_edist_lt, ENNReal.ofReal_add hε hδ]"
},
{
"state_after": "no goals",
"state_before": "case inr.inr\nι : Sort ?u.153954\nα : Type u\nβ : Type v\ninst✝ : PseudoEMetricSpace α\nδ✝ ε✝ : ℝ\ns✝ t : Set α\nx✝ : α\nε δ : ℝ\ns : Set α\nhε : 0 ≤ ε\nhδ : 0 ≤ δ\nx : α\n⊢ (∃ z, (∃ z_1, z_1 ∈ s ∧ edist z z_1 < ENNReal.ofReal δ) ∧ edist x z < ENNReal.ofReal ε) →\n ∃ z, z ∈ s ∧ edist x z < ENNReal.ofReal ε + ENNReal.ofReal δ",
"tactic": "exact fun ⟨y, ⟨z, hz, hy⟩, hx⟩ =>\n ⟨z, hz, (edist_triangle _ _ _).trans_lt <| ENNReal.add_lt_add hx hy⟩"
},
{
"state_after": "no goals",
"state_before": "case inl\nι : Sort ?u.153954\nα : Type u\nβ : Type v\ninst✝ : PseudoEMetricSpace α\nδ✝ ε✝ : ℝ\ns✝ t : Set α\nx : α\nε δ : ℝ\ns : Set α\nhε : ε ≤ 0\n⊢ thickening ε (thickening δ s) ⊆ thickening (ε + δ) s",
"tactic": "simp only [thickening_of_nonpos hε, empty_subset]"
},
{
"state_after": "no goals",
"state_before": "case inr.inl\nι : Sort ?u.153954\nα : Type u\nβ : Type v\ninst✝ : PseudoEMetricSpace α\nδ✝ ε✝ : ℝ\ns✝ t : Set α\nx : α\nε δ : ℝ\ns : Set α\nhε : 0 ≤ ε\nhδ : δ ≤ 0\n⊢ thickening ε (thickening δ s) ⊆ thickening (ε + δ) s",
"tactic": "simp only [thickening_of_nonpos hδ, thickening_empty, empty_subset]"
}
] |
[
1472,
73
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1463,
1
] |
Mathlib/Analysis/SpecificLimits/Normed.lean
|
NormedAddCommGroup.cauchy_series_of_le_geometric'
|
[] |
[
427,
79
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
425,
1
] |
Mathlib/Combinatorics/Composition.lean
|
Composition.eq_ones_iff_length
|
[
{
"state_after": "case mp\nn : ℕ\nc✝ c : Composition n\n⊢ c = ones n → length c = n\n\ncase mpr\nn : ℕ\nc✝ c : Composition n\n⊢ length c = n → c = ones n",
"state_before": "n : ℕ\nc✝ c : Composition n\n⊢ c = ones n ↔ length c = n",
"tactic": "constructor"
},
{
"state_after": "case mp\nn : ℕ\nc : Composition n\n⊢ length (ones n) = n",
"state_before": "case mp\nn : ℕ\nc✝ c : Composition n\n⊢ c = ones n → length c = n",
"tactic": "rintro rfl"
},
{
"state_after": "no goals",
"state_before": "case mp\nn : ℕ\nc : Composition n\n⊢ length (ones n) = n",
"tactic": "exact ones_length n"
},
{
"state_after": "case mpr\nn : ℕ\nc✝ c : Composition n\n⊢ ¬c = ones n → ¬length c = n",
"state_before": "case mpr\nn : ℕ\nc✝ c : Composition n\n⊢ length c = n → c = ones n",
"tactic": "contrapose"
},
{
"state_after": "case mpr\nn : ℕ\nc✝ c : Composition n\nH : ¬c = ones n\nlength_n : length c = n\n⊢ False",
"state_before": "case mpr\nn : ℕ\nc✝ c : Composition n\n⊢ ¬c = ones n → ¬length c = n",
"tactic": "intro H length_n"
},
{
"state_after": "case mpr\nn : ℕ\nc✝ c : Composition n\nH : ¬c = ones n\nlength_n : length c = n\n⊢ n < n",
"state_before": "case mpr\nn : ℕ\nc✝ c : Composition n\nH : ¬c = ones n\nlength_n : length c = n\n⊢ False",
"tactic": "apply lt_irrefl n"
},
{
"state_after": "no goals",
"state_before": "case mpr\nn : ℕ\nc✝ c : Composition n\nH : ¬c = ones n\nlength_n : length c = n\n⊢ n < n",
"tactic": "calc\n n = ∑ i : Fin c.length, 1 := by simp [length_n]\n _ < ∑ i : Fin c.length, c.blocksFun i := by\n {\n obtain ⟨i, hi, i_blocks⟩ : ∃ i ∈ c.blocks, 1 < i := ne_ones_iff.1 H\n rw [← ofFn_blocksFun, mem_ofFn c.blocksFun, Set.mem_range] at hi\n obtain ⟨j : Fin c.length, hj : c.blocksFun j = i⟩ := hi\n rw [← hj] at i_blocks\n exact Finset.sum_lt_sum (fun i _ => by simp [blocksFun]) ⟨j, Finset.mem_univ _, i_blocks⟩\n }\n _ = n := c.sum_blocksFun"
},
{
"state_after": "no goals",
"state_before": "n : ℕ\nc✝ c : Composition n\nH : ¬c = ones n\nlength_n : length c = n\n⊢ n = ∑ i : Fin (length c), 1",
"tactic": "simp [length_n]"
},
{
"state_after": "no goals",
"state_before": "n : ℕ\nc✝ c : Composition n\nH : ¬c = ones n\nlength_n : length c = n\ni✝ : ℕ\nj : Fin (length c)\ni_blocks : 1 < blocksFun c j\nhj : blocksFun c j = i✝\ni : Fin (length c)\nx✝ : i ∈ Finset.univ\n⊢ 1 ≤ blocksFun c i",
"tactic": "simp [blocksFun]"
}
] |
[
562,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
545,
1
] |
Mathlib/Topology/Basic.lean
|
IsOpen.preimage
|
[] |
[
1572,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1570,
1
] |
Mathlib/Data/Nat/Factorization/Basic.lean
|
Nat.pos_of_mem_factorization
|
[] |
[
145,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
144,
1
] |
Mathlib/Data/Real/ENNReal.lean
|
Set.OrdConnected.preimage_coe_nnreal_ennreal
|
[] |
[
2508,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2507,
1
] |
Mathlib/Data/Prod/Basic.lean
|
Prod.fst_eq_iff
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.5233\nδ : Type ?u.5236\na : α\nb : β\nx : α\n⊢ (a, b).fst = x ↔ (a, b) = (x, (a, b).snd)",
"tactic": "simp"
}
] |
[
218,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
217,
1
] |
Mathlib/Analysis/Convex/Star.lean
|
StarConvex.add_smul_mem
|
[
{
"state_after": "𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.100004\ninst✝⁴ : OrderedRing 𝕜\ninst✝³ : AddCommGroup E\ninst✝² : AddCommGroup F\ninst✝¹ : Module 𝕜 E\ninst✝ : Module 𝕜 F\nx y : E\ns t✝ : Set E\nhs : StarConvex 𝕜 x s\nhy : x + y ∈ s\nt : 𝕜\nht₀ : 0 ≤ t\nht₁ : t ≤ 1\nh : x + t • y = (1 - t) • x + t • (x + y)\n⊢ x + t • y ∈ s",
"state_before": "𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.100004\ninst✝⁴ : OrderedRing 𝕜\ninst✝³ : AddCommGroup E\ninst✝² : AddCommGroup F\ninst✝¹ : Module 𝕜 E\ninst✝ : Module 𝕜 F\nx y : E\ns t✝ : Set E\nhs : StarConvex 𝕜 x s\nhy : x + y ∈ s\nt : 𝕜\nht₀ : 0 ≤ t\nht₁ : t ≤ 1\n⊢ x + t • y ∈ s",
"tactic": "have h : x + t • y = (1 - t) • x + t • (x + y) := by\n rw [smul_add, ← add_assoc, ← add_smul, sub_add_cancel, one_smul]"
},
{
"state_after": "𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.100004\ninst✝⁴ : OrderedRing 𝕜\ninst✝³ : AddCommGroup E\ninst✝² : AddCommGroup F\ninst✝¹ : Module 𝕜 E\ninst✝ : Module 𝕜 F\nx y : E\ns t✝ : Set E\nhs : StarConvex 𝕜 x s\nhy : x + y ∈ s\nt : 𝕜\nht₀ : 0 ≤ t\nht₁ : t ≤ 1\nh : x + t • y = (1 - t) • x + t • (x + y)\n⊢ (1 - t) • x + t • (x + y) ∈ s",
"state_before": "𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.100004\ninst✝⁴ : OrderedRing 𝕜\ninst✝³ : AddCommGroup E\ninst✝² : AddCommGroup F\ninst✝¹ : Module 𝕜 E\ninst✝ : Module 𝕜 F\nx y : E\ns t✝ : Set E\nhs : StarConvex 𝕜 x s\nhy : x + y ∈ s\nt : 𝕜\nht₀ : 0 ≤ t\nht₁ : t ≤ 1\nh : x + t • y = (1 - t) • x + t • (x + y)\n⊢ x + t • y ∈ s",
"tactic": "rw [h]"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.100004\ninst✝⁴ : OrderedRing 𝕜\ninst✝³ : AddCommGroup E\ninst✝² : AddCommGroup F\ninst✝¹ : Module 𝕜 E\ninst✝ : Module 𝕜 F\nx y : E\ns t✝ : Set E\nhs : StarConvex 𝕜 x s\nhy : x + y ∈ s\nt : 𝕜\nht₀ : 0 ≤ t\nht₁ : t ≤ 1\nh : x + t • y = (1 - t) • x + t • (x + y)\n⊢ (1 - t) • x + t • (x + y) ∈ s",
"tactic": "exact hs hy (sub_nonneg_of_le ht₁) ht₀ (sub_add_cancel _ _)"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.100004\ninst✝⁴ : OrderedRing 𝕜\ninst✝³ : AddCommGroup E\ninst✝² : AddCommGroup F\ninst✝¹ : Module 𝕜 E\ninst✝ : Module 𝕜 F\nx y : E\ns t✝ : Set E\nhs : StarConvex 𝕜 x s\nhy : x + y ∈ s\nt : 𝕜\nht₀ : 0 ≤ t\nht₁ : t ≤ 1\n⊢ x + t • y = (1 - t) • x + t • (x + y)",
"tactic": "rw [smul_add, ← add_assoc, ← add_smul, sub_add_cancel, one_smul]"
}
] |
[
340,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
335,
1
] |
Mathlib/Data/Num/Lemmas.lean
|
PosNum.cast_to_num
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.355613\nn : PosNum\n⊢ ↑n = pos n",
"tactic": "rw [← cast_to_nat, ← of_to_nat n]"
}
] |
[
655,
90
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
655,
1
] |
Mathlib/RingTheory/Multiplicity.lean
|
multiplicity.ne_top_iff_finite
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝¹ : Monoid α\ninst✝ : DecidableRel fun x x_1 => x ∣ x_1\na b : α\n⊢ multiplicity a b ≠ ⊤ ↔ Finite a b",
"tactic": "rw [Ne.def, eq_top_iff_not_finite, Classical.not_not]"
}
] |
[
218,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
217,
1
] |
Mathlib/Data/List/Range.lean
|
List.length_finRange
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nn : ℕ\n⊢ length (finRange n) = n",
"tactic": "rw [finRange, length_pmap, length_range]"
}
] |
[
152,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
151,
1
] |
Mathlib/Analysis/InnerProductSpace/Basic.lean
|
norm_inner_eq_norm_tfae
|
[
{
"state_after": "case tfae_1_to_2\n𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.3166004\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\ndec_E : DecidableEq E\nx y : E\n⊢ ‖inner x y‖ = ‖x‖ * ‖y‖ → x = 0 ∨ y = (inner x y / inner x x) • x\n\n𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.3166004\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\ndec_E : DecidableEq E\nx y : E\ntfae_1_to_2 : ‖inner x y‖ = ‖x‖ * ‖y‖ → x = 0 ∨ y = (inner x y / inner x x) • x\n⊢ List.TFAE\n [‖inner x y‖ = ‖x‖ * ‖y‖, x = 0 ∨ y = (inner x y / inner x x) • x, x = 0 ∨ ∃ r, y = r • x,\n x = 0 ∨ y ∈ Submodule.span 𝕜 {x}]",
"state_before": "𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.3166004\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\ndec_E : DecidableEq E\nx y : E\n⊢ List.TFAE\n [‖inner x y‖ = ‖x‖ * ‖y‖, x = 0 ∨ y = (inner x y / inner x x) • x, x = 0 ∨ ∃ r, y = r • x,\n x = 0 ∨ y ∈ Submodule.span 𝕜 {x}]",
"tactic": "tfae_have 1 → 2"
},
{
"state_after": "case tfae_2_to_3\n𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.3166004\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\ndec_E : DecidableEq E\nx y : E\ntfae_1_to_2 : ‖inner x y‖ = ‖x‖ * ‖y‖ → x = 0 ∨ y = (inner x y / inner x x) • x\n⊢ x = 0 ∨ y = (inner x y / inner x x) • x → x = 0 ∨ ∃ r, y = r • x\n\n𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.3166004\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\ndec_E : DecidableEq E\nx y : E\ntfae_1_to_2 : ‖inner x y‖ = ‖x‖ * ‖y‖ → x = 0 ∨ y = (inner x y / inner x x) • x\ntfae_2_to_3 : x = 0 ∨ y = (inner x y / inner x x) • x → x = 0 ∨ ∃ r, y = r • x\n⊢ List.TFAE\n [‖inner x y‖ = ‖x‖ * ‖y‖, x = 0 ∨ y = (inner x y / inner x x) • x, x = 0 ∨ ∃ r, y = r • x,\n x = 0 ∨ y ∈ Submodule.span 𝕜 {x}]",
"state_before": "𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.3166004\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\ndec_E : DecidableEq E\nx y : E\ntfae_1_to_2 : ‖inner x y‖ = ‖x‖ * ‖y‖ → x = 0 ∨ y = (inner x y / inner x x) • x\n⊢ List.TFAE\n [‖inner x y‖ = ‖x‖ * ‖y‖, x = 0 ∨ y = (inner x y / inner x x) • x, x = 0 ∨ ∃ r, y = r • x,\n x = 0 ∨ y ∈ Submodule.span 𝕜 {x}]",
"tactic": "tfae_have 2 → 3"
},
{
"state_after": "𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.3166004\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\ndec_E : DecidableEq E\nx y : E\ntfae_1_to_2 : ‖inner x y‖ = ‖x‖ * ‖y‖ → x = 0 ∨ y = (inner x y / inner x x) • x\ntfae_2_to_3 : x = 0 ∨ y = (inner x y / inner x x) • x → x = 0 ∨ ∃ r, y = r • x\n⊢ List.TFAE\n [‖inner x y‖ = ‖x‖ * ‖y‖, x = 0 ∨ y = (inner x y / inner x x) • x, x = 0 ∨ ∃ r, y = r • x,\n x = 0 ∨ y ∈ Submodule.span 𝕜 {x}]",
"state_before": "case tfae_2_to_3\n𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.3166004\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\ndec_E : DecidableEq E\nx y : E\ntfae_1_to_2 : ‖inner x y‖ = ‖x‖ * ‖y‖ → x = 0 ∨ y = (inner x y / inner x x) • x\n⊢ x = 0 ∨ y = (inner x y / inner x x) • x → x = 0 ∨ ∃ r, y = r • x\n\n𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.3166004\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\ndec_E : DecidableEq E\nx y : E\ntfae_1_to_2 : ‖inner x y‖ = ‖x‖ * ‖y‖ → x = 0 ∨ y = (inner x y / inner x x) • x\ntfae_2_to_3 : x = 0 ∨ y = (inner x y / inner x x) • x → x = 0 ∨ ∃ r, y = r • x\n⊢ List.TFAE\n [‖inner x y‖ = ‖x‖ * ‖y‖, x = 0 ∨ y = (inner x y / inner x x) • x, x = 0 ∨ ∃ r, y = r • x,\n x = 0 ∨ y ∈ Submodule.span 𝕜 {x}]",
"tactic": "exact fun h => h.imp_right fun h' => ⟨_, h'⟩"
},
{
"state_after": "case tfae_3_to_1\n𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.3166004\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\ndec_E : DecidableEq E\nx y : E\ntfae_1_to_2 : ‖inner x y‖ = ‖x‖ * ‖y‖ → x = 0 ∨ y = (inner x y / inner x x) • x\ntfae_2_to_3 : x = 0 ∨ y = (inner x y / inner x x) • x → x = 0 ∨ ∃ r, y = r • x\n⊢ (x = 0 ∨ ∃ r, y = r • x) → ‖inner x y‖ = ‖x‖ * ‖y‖\n\n𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.3166004\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\ndec_E : DecidableEq E\nx y : E\ntfae_1_to_2 : ‖inner x y‖ = ‖x‖ * ‖y‖ → x = 0 ∨ y = (inner x y / inner x x) • x\ntfae_2_to_3 : x = 0 ∨ y = (inner x y / inner x x) • x → x = 0 ∨ ∃ r, y = r • x\ntfae_3_to_1 : (x = 0 ∨ ∃ r, y = r • x) → ‖inner x y‖ = ‖x‖ * ‖y‖\n⊢ List.TFAE\n [‖inner x y‖ = ‖x‖ * ‖y‖, x = 0 ∨ y = (inner x y / inner x x) • x, x = 0 ∨ ∃ r, y = r • x,\n x = 0 ∨ y ∈ Submodule.span 𝕜 {x}]",
"state_before": "𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.3166004\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\ndec_E : DecidableEq E\nx y : E\ntfae_1_to_2 : ‖inner x y‖ = ‖x‖ * ‖y‖ → x = 0 ∨ y = (inner x y / inner x x) • x\ntfae_2_to_3 : x = 0 ∨ y = (inner x y / inner x x) • x → x = 0 ∨ ∃ r, y = r • x\n⊢ List.TFAE\n [‖inner x y‖ = ‖x‖ * ‖y‖, x = 0 ∨ y = (inner x y / inner x x) • x, x = 0 ∨ ∃ r, y = r • x,\n x = 0 ∨ y ∈ Submodule.span 𝕜 {x}]",
"tactic": "tfae_have 3 → 1"
},
{
"state_after": "case tfae_3_iff_4\n𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.3166004\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\ndec_E : DecidableEq E\nx y : E\ntfae_1_to_2 : ‖inner x y‖ = ‖x‖ * ‖y‖ → x = 0 ∨ y = (inner x y / inner x x) • x\ntfae_2_to_3 : x = 0 ∨ y = (inner x y / inner x x) • x → x = 0 ∨ ∃ r, y = r • x\ntfae_3_to_1 : (x = 0 ∨ ∃ r, y = r • x) → ‖inner x y‖ = ‖x‖ * ‖y‖\n⊢ (x = 0 ∨ ∃ r, y = r • x) ↔ x = 0 ∨ y ∈ Submodule.span 𝕜 {x}\n\n𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.3166004\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\ndec_E : DecidableEq E\nx y : E\ntfae_1_to_2 : ‖inner x y‖ = ‖x‖ * ‖y‖ → x = 0 ∨ y = (inner x y / inner x x) • x\ntfae_2_to_3 : x = 0 ∨ y = (inner x y / inner x x) • x → x = 0 ∨ ∃ r, y = r • x\ntfae_3_to_1 : (x = 0 ∨ ∃ r, y = r • x) → ‖inner x y‖ = ‖x‖ * ‖y‖\ntfae_3_iff_4 : (x = 0 ∨ ∃ r, y = r • x) ↔ x = 0 ∨ y ∈ Submodule.span 𝕜 {x}\n⊢ List.TFAE\n [‖inner x y‖ = ‖x‖ * ‖y‖, x = 0 ∨ y = (inner x y / inner x x) • x, x = 0 ∨ ∃ r, y = r • x,\n x = 0 ∨ y ∈ Submodule.span 𝕜 {x}]",
"state_before": "𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.3166004\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\ndec_E : DecidableEq E\nx y : E\ntfae_1_to_2 : ‖inner x y‖ = ‖x‖ * ‖y‖ → x = 0 ∨ y = (inner x y / inner x x) • x\ntfae_2_to_3 : x = 0 ∨ y = (inner x y / inner x x) • x → x = 0 ∨ ∃ r, y = r • x\ntfae_3_to_1 : (x = 0 ∨ ∃ r, y = r • x) → ‖inner x y‖ = ‖x‖ * ‖y‖\n⊢ List.TFAE\n [‖inner x y‖ = ‖x‖ * ‖y‖, x = 0 ∨ y = (inner x y / inner x x) • x, x = 0 ∨ ∃ r, y = r • x,\n x = 0 ∨ y ∈ Submodule.span 𝕜 {x}]",
"tactic": "tfae_have 3 ↔ 4"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.3166004\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\ndec_E : DecidableEq E\nx y : E\ntfae_1_to_2 : ‖inner x y‖ = ‖x‖ * ‖y‖ → x = 0 ∨ y = (inner x y / inner x x) • x\ntfae_2_to_3 : x = 0 ∨ y = (inner x y / inner x x) • x → x = 0 ∨ ∃ r, y = r • x\ntfae_3_to_1 : (x = 0 ∨ ∃ r, y = r • x) → ‖inner x y‖ = ‖x‖ * ‖y‖\ntfae_3_iff_4 : (x = 0 ∨ ∃ r, y = r • x) ↔ x = 0 ∨ y ∈ Submodule.span 𝕜 {x}\n⊢ List.TFAE\n [‖inner x y‖ = ‖x‖ * ‖y‖, x = 0 ∨ y = (inner x y / inner x x) • x, x = 0 ∨ ∃ r, y = r • x,\n x = 0 ∨ y ∈ Submodule.span 𝕜 {x}]",
"tactic": "tfae_finish"
},
{
"state_after": "case tfae_1_to_2\n𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.3166004\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\ndec_E : DecidableEq E\nx y : E\nh : ‖inner x y‖ = ‖x‖ * ‖y‖\nhx₀ : ¬x = 0\n⊢ y = (inner x y / inner x x) • x",
"state_before": "case tfae_1_to_2\n𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.3166004\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\ndec_E : DecidableEq E\nx y : E\n⊢ ‖inner x y‖ = ‖x‖ * ‖y‖ → x = 0 ∨ y = (inner x y / inner x x) • x",
"tactic": "refine' fun h => or_iff_not_imp_left.2 fun hx₀ => _"
},
{
"state_after": "case tfae_1_to_2\n𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.3166004\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\ndec_E : DecidableEq E\nx y : E\nh : ‖inner x y‖ = ‖x‖ * ‖y‖\nhx₀ : ¬x = 0\nthis : ‖x‖ ^ 2 ≠ 0\n⊢ y = (inner x y / inner x x) • x",
"state_before": "case tfae_1_to_2\n𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.3166004\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\ndec_E : DecidableEq E\nx y : E\nh : ‖inner x y‖ = ‖x‖ * ‖y‖\nhx₀ : ¬x = 0\n⊢ y = (inner x y / inner x x) • x",
"tactic": "have : ‖x‖ ^ 2 ≠ 0 := pow_ne_zero _ (norm_ne_zero_iff.2 hx₀)"
},
{
"state_after": "case tfae_1_to_2\n𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.3166004\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\ndec_E : DecidableEq E\nx y : E\nh✝ : ‖x‖ ^ 2 * (‖x‖ ^ 2 * ‖y‖ ^ 2) = ‖x‖ ^ 2 * ‖inner x y‖ ^ 2\nh : ‖x‖ ^ 2 * (‖x‖ ^ 2 * ‖y‖ ^ 2 - ‖inner x y‖ ^ 2) = 0\nhx₀ : ¬x = 0\nthis : ‖x‖ ^ 2 ≠ 0\n⊢ y = (inner x y / inner x x) • x",
"state_before": "case tfae_1_to_2\n𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.3166004\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\ndec_E : DecidableEq E\nx y : E\nh : ‖inner x y‖ = ‖x‖ * ‖y‖\nhx₀ : ¬x = 0\nthis : ‖x‖ ^ 2 ≠ 0\n⊢ y = (inner x y / inner x x) • x",
"tactic": "rw [← sq_eq_sq (norm_nonneg _) (mul_nonneg (norm_nonneg _) (norm_nonneg _)), mul_pow, ←\n mul_right_inj' this, eq_comm, ← sub_eq_zero, ← mul_sub] at h"
},
{
"state_after": "case tfae_1_to_2\n𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.3166004\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\ndec_E : DecidableEq E\nx y : E\nh✝ : ‖x‖ ^ 2 * (‖x‖ ^ 2 * ‖y‖ ^ 2) = ‖x‖ ^ 2 * ‖inner x y‖ ^ 2\nhx₀ : ¬x = 0\nthis : ‖x‖ ^ 2 ≠ 0\nh : ↑re (inner x x) * (↑re (inner x x) * ↑re (inner y y) - ‖inner x y‖ ^ 2) = 0\n⊢ y = (inner x y / inner x x) • x",
"state_before": "case tfae_1_to_2\n𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.3166004\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\ndec_E : DecidableEq E\nx y : E\nh✝ : ‖x‖ ^ 2 * (‖x‖ ^ 2 * ‖y‖ ^ 2) = ‖x‖ ^ 2 * ‖inner x y‖ ^ 2\nh : ‖x‖ ^ 2 * (‖x‖ ^ 2 * ‖y‖ ^ 2 - ‖inner x y‖ ^ 2) = 0\nhx₀ : ¬x = 0\nthis : ‖x‖ ^ 2 ≠ 0\n⊢ y = (inner x y / inner x x) • x",
"tactic": "simp only [@norm_sq_eq_inner 𝕜] at h"
},
{
"state_after": "case tfae_1_to_2\n𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.3166004\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\ndec_E : DecidableEq E\nx y : E\nh✝ : ‖x‖ ^ 2 * (‖x‖ ^ 2 * ‖y‖ ^ 2) = ‖x‖ ^ 2 * ‖inner x y‖ ^ 2\nhx₀ : ¬x = 0\nthis✝ : ‖x‖ ^ 2 ≠ 0\nh : ↑re (inner x x) * (↑re (inner x x) * ↑re (inner y y) - ‖inner x y‖ ^ 2) = 0\nthis : InnerProductSpace.Core 𝕜 E := InnerProductSpace.toCore\n⊢ y = (inner x y / inner x x) • x",
"state_before": "case tfae_1_to_2\n𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.3166004\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\ndec_E : DecidableEq E\nx y : E\nh✝ : ‖x‖ ^ 2 * (‖x‖ ^ 2 * ‖y‖ ^ 2) = ‖x‖ ^ 2 * ‖inner x y‖ ^ 2\nhx₀ : ¬x = 0\nthis : ‖x‖ ^ 2 ≠ 0\nh : ↑re (inner x x) * (↑re (inner x x) * ↑re (inner y y) - ‖inner x y‖ ^ 2) = 0\n⊢ y = (inner x y / inner x x) • x",
"tactic": "letI : InnerProductSpace.Core 𝕜 E := InnerProductSpace.toCore"
},
{
"state_after": "case tfae_1_to_2\n𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.3166004\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\ndec_E : DecidableEq E\nx y : E\nh✝ : ‖x‖ ^ 2 * (‖x‖ ^ 2 * ‖y‖ ^ 2) = ‖x‖ ^ 2 * ‖inner x y‖ ^ 2\nhx₀ : ¬x = 0\nthis✝ : ‖x‖ ^ 2 ≠ 0\nthis : InnerProductSpace.Core 𝕜 E := InnerProductSpace.toCore\nh : inner x y • x = inner x x • y\n⊢ y = (inner x y / inner x x) • x",
"state_before": "case tfae_1_to_2\n𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.3166004\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\ndec_E : DecidableEq E\nx y : E\nh✝ : ‖x‖ ^ 2 * (‖x‖ ^ 2 * ‖y‖ ^ 2) = ‖x‖ ^ 2 * ‖inner x y‖ ^ 2\nhx₀ : ¬x = 0\nthis✝ : ‖x‖ ^ 2 ≠ 0\nh : ↑re (inner x x) * (↑re (inner x x) * ↑re (inner y y) - ‖inner x y‖ ^ 2) = 0\nthis : InnerProductSpace.Core 𝕜 E := InnerProductSpace.toCore\n⊢ y = (inner x y / inner x x) • x",
"tactic": "erw [← InnerProductSpace.Core.cauchy_schwarz_aux, InnerProductSpace.Core.normSq_eq_zero,\n sub_eq_zero] at h"
},
{
"state_after": "case tfae_1_to_2.hc\n𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.3166004\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\ndec_E : DecidableEq E\nx y : E\nh✝ : ‖x‖ ^ 2 * (‖x‖ ^ 2 * ‖y‖ ^ 2) = ‖x‖ ^ 2 * ‖inner x y‖ ^ 2\nhx₀ : ¬x = 0\nthis✝ : ‖x‖ ^ 2 ≠ 0\nthis : InnerProductSpace.Core 𝕜 E := InnerProductSpace.toCore\nh : inner x y • x = inner x x • y\n⊢ inner x x ≠ 0",
"state_before": "case tfae_1_to_2\n𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.3166004\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\ndec_E : DecidableEq E\nx y : E\nh✝ : ‖x‖ ^ 2 * (‖x‖ ^ 2 * ‖y‖ ^ 2) = ‖x‖ ^ 2 * ‖inner x y‖ ^ 2\nhx₀ : ¬x = 0\nthis✝ : ‖x‖ ^ 2 ≠ 0\nthis : InnerProductSpace.Core 𝕜 E := InnerProductSpace.toCore\nh : inner x y • x = inner x x • y\n⊢ y = (inner x y / inner x x) • x",
"tactic": "rw [div_eq_inv_mul, mul_smul, h, inv_smul_smul₀]"
},
{
"state_after": "no goals",
"state_before": "case tfae_1_to_2.hc\n𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.3166004\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\ndec_E : DecidableEq E\nx y : E\nh✝ : ‖x‖ ^ 2 * (‖x‖ ^ 2 * ‖y‖ ^ 2) = ‖x‖ ^ 2 * ‖inner x y‖ ^ 2\nhx₀ : ¬x = 0\nthis✝ : ‖x‖ ^ 2 ≠ 0\nthis : InnerProductSpace.Core 𝕜 E := InnerProductSpace.toCore\nh : inner x y • x = inner x x • y\n⊢ inner x x ≠ 0",
"tactic": "rwa [inner_self_ne_zero]"
},
{
"state_after": "no goals",
"state_before": "case tfae_3_to_1\n𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.3166004\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\ndec_E : DecidableEq E\nx y : E\ntfae_1_to_2 : ‖inner x y‖ = ‖x‖ * ‖y‖ → x = 0 ∨ y = (inner x y / inner x x) • x\ntfae_2_to_3 : x = 0 ∨ y = (inner x y / inner x x) • x → x = 0 ∨ ∃ r, y = r • x\n⊢ (x = 0 ∨ ∃ r, y = r • x) → ‖inner x y‖ = ‖x‖ * ‖y‖",
"tactic": "rintro (rfl | ⟨r, rfl⟩) <;>\nsimp [inner_smul_right, norm_smul, inner_self_eq_norm_sq_to_K, inner_self_eq_norm_mul_norm,\n sq, mul_left_comm]"
},
{
"state_after": "no goals",
"state_before": "case tfae_3_iff_4\n𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.3166004\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\ndec_E : DecidableEq E\nx y : E\ntfae_1_to_2 : ‖inner x y‖ = ‖x‖ * ‖y‖ → x = 0 ∨ y = (inner x y / inner x x) • x\ntfae_2_to_3 : x = 0 ∨ y = (inner x y / inner x x) • x → x = 0 ∨ ∃ r, y = r • x\ntfae_3_to_1 : (x = 0 ∨ ∃ r, y = r • x) → ‖inner x y‖ = ‖x‖ * ‖y‖\n⊢ (x = 0 ∨ ∃ r, y = r • x) ↔ x = 0 ∨ y ∈ Submodule.span 𝕜 {x}",
"tactic": "simp only [Submodule.mem_span_singleton, eq_comm]"
}
] |
[
1611,
14
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1589,
1
] |
Mathlib/Algebra/GroupWithZero/Basic.lean
|
mul_self_mul_inv
|
[
{
"state_after": "case pos\nα : Type ?u.24169\nM₀ : Type ?u.24172\nG₀ : Type u_1\nM₀' : Type ?u.24178\nG₀' : Type ?u.24181\nF : Type ?u.24184\nF' : Type ?u.24187\ninst✝ : GroupWithZero G₀\na✝ b c a : G₀\nh : a = 0\n⊢ a * a * a⁻¹ = a\n\ncase neg\nα : Type ?u.24169\nM₀ : Type ?u.24172\nG₀ : Type u_1\nM₀' : Type ?u.24178\nG₀' : Type ?u.24181\nF : Type ?u.24184\nF' : Type ?u.24187\ninst✝ : GroupWithZero G₀\na✝ b c a : G₀\nh : ¬a = 0\n⊢ a * a * a⁻¹ = a",
"state_before": "α : Type ?u.24169\nM₀ : Type ?u.24172\nG₀ : Type u_1\nM₀' : Type ?u.24178\nG₀' : Type ?u.24181\nF : Type ?u.24184\nF' : Type ?u.24187\ninst✝ : GroupWithZero G₀\na✝ b c a : G₀\n⊢ a * a * a⁻¹ = a",
"tactic": "by_cases h : a = 0"
},
{
"state_after": "no goals",
"state_before": "case pos\nα : Type ?u.24169\nM₀ : Type ?u.24172\nG₀ : Type u_1\nM₀' : Type ?u.24178\nG₀' : Type ?u.24181\nF : Type ?u.24184\nF' : Type ?u.24187\ninst✝ : GroupWithZero G₀\na✝ b c a : G₀\nh : a = 0\n⊢ a * a * a⁻¹ = a",
"tactic": "rw [h, inv_zero, mul_zero]"
},
{
"state_after": "no goals",
"state_before": "case neg\nα : Type ?u.24169\nM₀ : Type ?u.24172\nG₀ : Type u_1\nM₀' : Type ?u.24178\nG₀' : Type ?u.24181\nF : Type ?u.24184\nF' : Type ?u.24187\ninst✝ : GroupWithZero G₀\na✝ b c a : G₀\nh : ¬a = 0\n⊢ a * a * a⁻¹ = a",
"tactic": "rw [mul_assoc, mul_inv_cancel h, mul_one]"
}
] |
[
348,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
345,
1
] |
Mathlib/Topology/MetricSpace/HausdorffDistance.lean
|
Metric.hausdorffDist_empty'
|
[
{
"state_after": "no goals",
"state_before": "ι : Sort ?u.75311\nα : Type u\nβ : Type v\ninst✝¹ : PseudoMetricSpace α\ninst✝ : PseudoMetricSpace β\ns t u : Set α\nx y : α\nΦ : α → β\n⊢ hausdorffDist ∅ s = 0",
"tactic": "simp [hausdorffDist_comm]"
}
] |
[
740,
85
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
740,
1
] |
Mathlib/Order/Heyting/Basic.lean
|
inf_sdiff_left
|
[] |
[
549,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
548,
1
] |
Mathlib/MeasureTheory/Measure/MeasureSpace.lean
|
MeasureTheory.NullMeasurableSet.mono_ac
|
[] |
[
2665,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2663,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/Equalizers.lean
|
CategoryTheory.Limits.Cofork.ofCocone_ι
|
[
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝ : Category C\nX Y : C\nf g : X ⟶ Y\nF : WalkingParallelPair ⥤ C\nt : Cocone F\nj : WalkingParallelPair\n⊢ (parallelPair (F.map left) (F.map right)).obj j = F.obj j",
"tactic": "aesop"
}
] |
[
659,
72
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
658,
1
] |
Mathlib/Logic/Basic.lean
|
exists_unique_const
|
[
{
"state_after": "no goals",
"state_before": "ι : Sort ?u.15425\nα✝ : Sort ?u.15430\nκ : ι → Sort ?u.15427\np q : α✝ → Prop\nb : Prop\nα : Sort u_1\ni : Nonempty α\ninst✝ : Subsingleton α\n⊢ (∃! x, b) ↔ b",
"tactic": "simp"
}
] |
[
743,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
742,
1
] |
Std/Data/Nat/Lemmas.lean
|
Nat.dvd_refl
|
[
{
"state_after": "no goals",
"state_before": "a : Nat\n⊢ a = a * 1",
"tactic": "simp"
}
] |
[
651,
61
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
651,
11
] |
Mathlib/Computability/Partrec.lean
|
Computable.vector_head
|
[] |
[
382,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
381,
1
] |
Mathlib/Topology/MetricSpace/EMetricSpace.lean
|
EMetric.tendsto_nhds
|
[] |
[
725,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
723,
1
] |
Mathlib/Data/Num/Lemmas.lean
|
Num.mul_to_nat
|
[] |
[
321,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
317,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean
|
Real.Angle.sign_pi_sub
|
[
{
"state_after": "no goals",
"state_before": "θ : Angle\n⊢ sign (↑π - θ) = sign θ",
"tactic": "simp [sign_antiperiodic.sub_eq']"
}
] |
[
890,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
889,
1
] |
Mathlib/Topology/Separation.lean
|
t2_iff_ultrafilter
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\ninst✝ : TopologicalSpace α\n⊢ (∀ {x y : α}, NeBot (𝓝 x ⊓ 𝓝 y) → x = y) ↔ ∀ {x y : α} (f : Ultrafilter α), ↑f ≤ 𝓝 x → ↑f ≤ 𝓝 y → x = y",
"tactic": "simp only [← exists_ultrafilter_iff, and_imp, le_inf_iff, exists_imp]"
}
] |
[
948,
96
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
946,
1
] |
Mathlib/Algebra/BigOperators/Pi.lean
|
prod_mk_prod
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝¹ : CommMonoid α\ninst✝ : CommMonoid β\ns : Finset γ\nf : γ → α\ng : γ → β\nthis : DecidableEq γ\n⊢ ∀ ⦃a : γ⦄ {s : Finset γ},\n ¬a ∈ s →\n (∏ x in s, f x, ∏ x in s, g x) = ∏ x in s, (f x, g x) →\n (∏ x in insert a s, f x, ∏ x in insert a s, g x) = ∏ x in insert a s, (f x, g x)",
"tactic": "simp (config := { contextual := true }) [Prod.ext_iff]"
}
] |
[
70,
88
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
67,
1
] |
Std/Data/Int/Lemmas.lean
|
Int.le_of_neg_le_neg
|
[
{
"state_after": "a b : Int\nh : -b ≤ -a\nthis : a ≤ b\n⊢ a ≤ b",
"state_before": "a b : Int\nh : -b ≤ -a\nthis : - -a ≤ - -b\n⊢ a ≤ b",
"tactic": "simp [Int.neg_neg] at this"
},
{
"state_after": "no goals",
"state_before": "a b : Int\nh : -b ≤ -a\nthis : a ≤ b\n⊢ a ≤ b",
"tactic": "assumption"
}
] |
[
863,
19
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
861,
11
] |
Mathlib/Data/Real/ENNReal.lean
|
ENNReal.toNNReal_sInf
|
[
{
"state_after": "α : Type ?u.839566\nβ : Type ?u.839569\na b c d : ℝ≥0∞\nr p q : ℝ≥0\nι : Sort ?u.839586\nf g : ι → ℝ≥0∞\ns : Set ℝ≥0∞\nhs : ∀ (r : ℝ≥0∞), r ∈ s → r ≠ ⊤\nhf : ∀ (i : { x // x ∈ s }), ↑i ≠ ⊤\n⊢ ENNReal.toNNReal (sInf s) = sInf (ENNReal.toNNReal '' s)",
"state_before": "α : Type ?u.839566\nβ : Type ?u.839569\na b c d : ℝ≥0∞\nr p q : ℝ≥0\nι : Sort ?u.839586\nf g : ι → ℝ≥0∞\ns : Set ℝ≥0∞\nhs : ∀ (r : ℝ≥0∞), r ∈ s → r ≠ ⊤\n⊢ ENNReal.toNNReal (sInf s) = sInf (ENNReal.toNNReal '' s)",
"tactic": "have hf : ∀ i, ((↑) : s → ℝ≥0∞) i ≠ ∞ := fun ⟨r, rs⟩ => hs r rs"
},
{
"state_after": "no goals",
"state_before": "α : Type ?u.839566\nβ : Type ?u.839569\na b c d : ℝ≥0∞\nr p q : ℝ≥0\nι : Sort ?u.839586\nf g : ι → ℝ≥0∞\ns : Set ℝ≥0∞\nhs : ∀ (r : ℝ≥0∞), r ∈ s → r ≠ ⊤\nhf : ∀ (i : { x // x ∈ s }), ↑i ≠ ⊤\n⊢ ENNReal.toNNReal (sInf s) = sInf (ENNReal.toNNReal '' s)",
"tactic": "simpa only [← sInf_range, ← image_eq_range, Subtype.range_coe_subtype] using (toNNReal_iInf hf)"
}
] |
[
2360,
98
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2355,
1
] |
Mathlib/Order/ConditionallyCompleteLattice/Basic.lean
|
csSup_insert
|
[] |
[
708,
65
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
707,
1
] |
Mathlib/MeasureTheory/Constructions/Polish.lean
|
IsClosed.analyticSet
|
[
{
"state_after": "α : Type u_1\ninst✝¹ : TopologicalSpace α\nι : Type ?u.62127\ninst✝ : PolishSpace α\ns : Set α\nhs : IsClosed s\nthis : PolishSpace ↑s\n⊢ AnalyticSet s",
"state_before": "α : Type u_1\ninst✝¹ : TopologicalSpace α\nι : Type ?u.62127\ninst✝ : PolishSpace α\ns : Set α\nhs : IsClosed s\n⊢ AnalyticSet s",
"tactic": "haveI : PolishSpace s := hs.polishSpace"
},
{
"state_after": "α : Type u_1\ninst✝¹ : TopologicalSpace α\nι : Type ?u.62127\ninst✝ : PolishSpace α\ns : Set α\nhs : IsClosed s\nthis : PolishSpace ↑s\n⊢ AnalyticSet (range Subtype.val)",
"state_before": "α : Type u_1\ninst✝¹ : TopologicalSpace α\nι : Type ?u.62127\ninst✝ : PolishSpace α\ns : Set α\nhs : IsClosed s\nthis : PolishSpace ↑s\n⊢ AnalyticSet s",
"tactic": "rw [← @Subtype.range_val α s]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝¹ : TopologicalSpace α\nι : Type ?u.62127\ninst✝ : PolishSpace α\ns : Set α\nhs : IsClosed s\nthis : PolishSpace ↑s\n⊢ AnalyticSet (range Subtype.val)",
"tactic": "exact analyticSet_range_of_polishSpace continuous_subtype_val"
}
] |
[
216,
64
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
212,
1
] |
Mathlib/Algebra/Support.lean
|
Function.mulSupport_eq_preimage
|
[] |
[
52,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
51,
1
] |
Mathlib/Topology/Algebra/InfiniteSum/Basic.lean
|
tsum_neg
|
[
{
"state_after": "case pos\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.442177\nδ : Type ?u.442180\ninst✝³ : AddCommGroup α\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalAddGroup α\nf g : β → α\na a₁ a₂ : α\ninst✝ : T2Space α\nhf : Summable f\n⊢ (∑' (b : β), -f b) = -∑' (b : β), f b\n\ncase neg\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.442177\nδ : Type ?u.442180\ninst✝³ : AddCommGroup α\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalAddGroup α\nf g : β → α\na a₁ a₂ : α\ninst✝ : T2Space α\nhf : ¬Summable f\n⊢ (∑' (b : β), -f b) = -∑' (b : β), f b",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.442177\nδ : Type ?u.442180\ninst✝³ : AddCommGroup α\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalAddGroup α\nf g : β → α\na a₁ a₂ : α\ninst✝ : T2Space α\n⊢ (∑' (b : β), -f b) = -∑' (b : β), f b",
"tactic": "by_cases hf : Summable f"
},
{
"state_after": "no goals",
"state_before": "case pos\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.442177\nδ : Type ?u.442180\ninst✝³ : AddCommGroup α\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalAddGroup α\nf g : β → α\na a₁ a₂ : α\ninst✝ : T2Space α\nhf : Summable f\n⊢ (∑' (b : β), -f b) = -∑' (b : β), f b",
"tactic": "exact hf.hasSum.neg.tsum_eq"
},
{
"state_after": "no goals",
"state_before": "case neg\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.442177\nδ : Type ?u.442180\ninst✝³ : AddCommGroup α\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalAddGroup α\nf g : β → α\na a₁ a₂ : α\ninst✝ : T2Space α\nhf : ¬Summable f\n⊢ (∑' (b : β), -f b) = -∑' (b : β), f b",
"tactic": "simp [tsum_eq_zero_of_not_summable hf, tsum_eq_zero_of_not_summable (mt Summable.of_neg hf)]"
}
] |
[
907,
97
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
904,
1
] |
Mathlib/Data/Nat/Factorial/Basic.lean
|
Nat.pow_succ_le_ascFactorial
|
[
{
"state_after": "no goals",
"state_before": "n : ℕ\n⊢ (n + 1) ^ 0 ≤ ascFactorial n 0",
"tactic": "rw [ascFactorial_zero, pow_zero]"
},
{
"state_after": "n k : ℕ\n⊢ (n + 1) * (n + 1) ^ k ≤ ascFactorial n (k + 1)",
"state_before": "n k : ℕ\n⊢ (n + 1) ^ (k + 1) ≤ ascFactorial n (k + 1)",
"tactic": "rw [pow_succ, mul_comm]"
},
{
"state_after": "no goals",
"state_before": "n k : ℕ\n⊢ (n + 1) * (n + 1) ^ k ≤ ascFactorial n (k + 1)",
"tactic": "exact Nat.mul_le_mul (Nat.add_le_add_right le_self_add _) (pow_succ_le_ascFactorial _ k)"
}
] |
[
289,
93
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
285,
1
] |
Mathlib/RingTheory/Polynomial/Basic.lean
|
Polynomial.degreeLTEquiv_eq_zero_iff_eq_zero
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type ?u.28643\ninst✝ : Semiring R\nn : ℕ\np : R[X]\nhp : p ∈ degreeLT R n\n⊢ ↑(degreeLTEquiv R n) { val := p, property := hp } = 0 ↔ p = 0",
"tactic": "rw [LinearEquiv.map_eq_zero_iff, Submodule.mk_eq_zero]"
}
] |
[
177,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
175,
1
] |
Mathlib/Algebra/Invertible.lean
|
invOf_mul_eq_iff_eq_mul_left
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nc a b : α\ninst✝¹ : Monoid α\ninst✝ : Invertible c\n⊢ ⅟c * a = b ↔ a = c * b",
"tactic": "rw [← mul_left_inj_of_invertible (c := c), mul_invOf_self_assoc]"
}
] |
[
309,
67
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
307,
1
] |
Mathlib/SetTheory/Game/PGame.lean
|
PGame.lf_of_lt_of_lf
|
[] |
[
583,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
582,
1
] |
Mathlib/Data/List/Perm.lean
|
List.exists_perm_sublist
|
[
{
"state_after": "no goals",
"state_before": "case nil\nα : Type uu\nβ : Type vv\nl₁✝ l₂✝ l₂ l₂' l₁ : List α\ns : l₁ <+ []\n⊢ ∃ l₁' x, l₁' <+ []",
"tactic": "exact ⟨[], eq_nil_of_sublist_nil s ▸ Perm.refl _, nil_sublist _⟩"
},
{
"state_after": "case cons.cons\nα : Type uu\nβ : Type vv\nl₁✝¹ l₂✝¹ l₂ l₂' : List α\nx : α\nl₁✝ l₂✝ : List α\na✝ : l₁✝ ~ l₂✝\nIH : ∀ {l₁ : List α}, l₁ <+ l₁✝ → ∃ l₁' x, l₁' <+ l₂✝\nl₁ : List α\ns : l₁ <+ l₁✝\n⊢ ∃ l₁' x_1, l₁' <+ x :: l₂✝\n\ncase cons.cons₂\nα : Type uu\nβ : Type vv\nl₁✝¹ l₂✝¹ l₂ l₂' : List α\nx : α\nl₁✝ l₂✝ : List α\na✝ : l₁✝ ~ l₂✝\nIH : ∀ {l₁ : List α}, l₁ <+ l₁✝ → ∃ l₁' x, l₁' <+ l₂✝\nl₁ : List α\ns : l₁ <+ l₁✝\n⊢ ∃ l₁' x_1, l₁' <+ x :: l₂✝",
"state_before": "case cons\nα : Type uu\nβ : Type vv\nl₁✝¹ l₂✝¹ l₂ l₂' : List α\nx : α\nl₁✝ l₂✝ : List α\na✝ : l₁✝ ~ l₂✝\nIH : ∀ {l₁ : List α}, l₁ <+ l₁✝ → ∃ l₁' x, l₁' <+ l₂✝\nl₁ : List α\ns : l₁ <+ x :: l₁✝\n⊢ ∃ l₁' x_1, l₁' <+ x :: l₂✝",
"tactic": "cases' s with _ _ _ s l₁ _ _ s"
},
{
"state_after": "no goals",
"state_before": "case cons.cons\nα : Type uu\nβ : Type vv\nl₁✝¹ l₂✝¹ l₂ l₂' : List α\nx : α\nl₁✝ l₂✝ : List α\na✝ : l₁✝ ~ l₂✝\nIH : ∀ {l₁ : List α}, l₁ <+ l₁✝ → ∃ l₁' x, l₁' <+ l₂✝\nl₁ : List α\ns : l₁ <+ l₁✝\n⊢ ∃ l₁' x_1, l₁' <+ x :: l₂✝",
"tactic": "exact\n let ⟨l₁', p', s'⟩ := IH s\n ⟨l₁', p', s'.cons _⟩"
},
{
"state_after": "no goals",
"state_before": "case cons.cons₂\nα : Type uu\nβ : Type vv\nl₁✝¹ l₂✝¹ l₂ l₂' : List α\nx : α\nl₁✝ l₂✝ : List α\na✝ : l₁✝ ~ l₂✝\nIH : ∀ {l₁ : List α}, l₁ <+ l₁✝ → ∃ l₁' x, l₁' <+ l₂✝\nl₁ : List α\ns : l₁ <+ l₁✝\n⊢ ∃ l₁' x_1, l₁' <+ x :: l₂✝",
"tactic": "exact\n let ⟨l₁', p', s'⟩ := IH s\n ⟨x :: l₁', p'.cons x, s'.cons₂ _⟩"
},
{
"state_after": "case swap.cons.cons\nα : Type uu\nβ : Type vv\nl₁✝ l₂✝ l₂ l₂' : List α\nx y : α\nl✝ l₁ : List α\ns : l₁ <+ l✝\n⊢ ∃ l₁' x_1, l₁' <+ x :: y :: l✝\n\ncase swap.cons.cons₂\nα : Type uu\nβ : Type vv\nl₁✝ l₂✝ l₂ l₂' : List α\nx y : α\nl✝ l₁ : List α\ns : l₁ <+ l✝\n⊢ ∃ l₁' x_1, l₁' <+ x :: y :: l✝\n\ncase swap.cons₂.cons\nα : Type uu\nβ : Type vv\nl₁✝ l₂✝ l₂ l₂' : List α\nx y : α\nl✝ l₁ : List α\ns : l₁ <+ l✝\n⊢ ∃ l₁' x_1, l₁' <+ x :: y :: l✝\n\ncase swap.cons₂.cons₂\nα : Type uu\nβ : Type vv\nl₁✝ l₂✝ l₂ l₂' : List α\nx y : α\nl✝ l₁ : List α\ns : l₁ <+ l✝\n⊢ ∃ l₁' x_1, l₁' <+ x :: y :: l✝",
"state_before": "case swap\nα : Type uu\nβ : Type vv\nl₁✝ l₂✝ l₂ l₂' : List α\nx y : α\nl✝ l₁ : List α\ns : l₁ <+ y :: x :: l✝\n⊢ ∃ l₁' x_1, l₁' <+ x :: y :: l✝",
"tactic": "cases' s with _ _ _ s l₁ _ _ s <;> cases' s with _ _ _ s l₁ _ _ s"
},
{
"state_after": "no goals",
"state_before": "case swap.cons.cons\nα : Type uu\nβ : Type vv\nl₁✝ l₂✝ l₂ l₂' : List α\nx y : α\nl✝ l₁ : List α\ns : l₁ <+ l✝\n⊢ ∃ l₁' x_1, l₁' <+ x :: y :: l✝",
"tactic": "exact ⟨l₁, Perm.refl _, (s.cons _).cons _⟩"
},
{
"state_after": "no goals",
"state_before": "case swap.cons.cons₂\nα : Type uu\nβ : Type vv\nl₁✝ l₂✝ l₂ l₂' : List α\nx y : α\nl✝ l₁ : List α\ns : l₁ <+ l✝\n⊢ ∃ l₁' x_1, l₁' <+ x :: y :: l✝",
"tactic": "exact ⟨x :: l₁, Perm.refl _, (s.cons _).cons₂ _⟩"
},
{
"state_after": "no goals",
"state_before": "case swap.cons₂.cons\nα : Type uu\nβ : Type vv\nl₁✝ l₂✝ l₂ l₂' : List α\nx y : α\nl✝ l₁ : List α\ns : l₁ <+ l✝\n⊢ ∃ l₁' x_1, l₁' <+ x :: y :: l✝",
"tactic": "exact ⟨y :: l₁, Perm.refl _, (s.cons₂ _).cons _⟩"
},
{
"state_after": "no goals",
"state_before": "case swap.cons₂.cons₂\nα : Type uu\nβ : Type vv\nl₁✝ l₂✝ l₂ l₂' : List α\nx y : α\nl✝ l₁ : List α\ns : l₁ <+ l✝\n⊢ ∃ l₁' x_1, l₁' <+ x :: y :: l✝",
"tactic": "exact ⟨x :: y :: l₁, Perm.swap _ _ _, (s.cons₂ _).cons₂ _⟩"
},
{
"state_after": "no goals",
"state_before": "case trans\nα : Type uu\nβ : Type vv\nl₁✝¹ l₂✝¹ l₂ l₂' l₁✝ l₂✝ l₃✝ : List α\na✝¹ : l₁✝ ~ l₂✝\na✝ : l₂✝ ~ l₃✝\nIH₁ : ∀ {l₁ : List α}, l₁ <+ l₁✝ → ∃ l₁' x, l₁' <+ l₂✝\nIH₂ : ∀ {l₁ : List α}, l₁ <+ l₂✝ → ∃ l₁' x, l₁' <+ l₃✝\nl₁ : List α\ns : l₁ <+ l₁✝\n⊢ ∃ l₁' x, l₁' <+ l₃✝",
"tactic": "exact\n let ⟨m₁, pm, sm⟩ := IH₁ s\n let ⟨r₁, pr, sr⟩ := IH₂ sm\n ⟨r₁, pr.trans pm, sr⟩"
}
] |
[
313,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
290,
1
] |
Mathlib/Order/ConditionallyCompleteLattice/Basic.lean
|
WithBot.coe_sSup'
|
[] |
[
142,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
140,
1
] |
Mathlib/Topology/MetricSpace/Isometry.lean
|
IsometryEquiv.eq_symm_apply
|
[] |
[
441,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
440,
1
] |
Mathlib/Computability/TuringMachine.lean
|
Turing.TM2to1.tr_respects
|
[
{
"state_after": "K : Type u_4\ninst✝² : DecidableEq K\nΓ : K → Type u_3\nΛ : Type u_2\ninst✝¹ : Inhabited Λ\nσ : Type u_1\ninst✝ : Inhabited σ\nM : Λ → Stmt₂\nc₁ : Cfg₂\nc₂ : TM1.Cfg Γ' Λ' σ\nh : TrCfg c₁ c₂\n⊢ match TM2.step M c₁ with\n | some b₁ => ∃ b₂, TrCfg b₁ b₂ ∧ Reaches₁ (TM1.step (tr M)) c₂ b₂\n | none => TM1.step (tr M) c₂ = none",
"state_before": "K : Type u_4\ninst✝² : DecidableEq K\nΓ : K → Type u_3\nΛ : Type u_2\ninst✝¹ : Inhabited Λ\nσ : Type u_1\ninst✝ : Inhabited σ\nM : Λ → Stmt₂\n⊢ Respects (TM2.step M) (TM1.step (tr M)) TrCfg",
"tactic": "intro c₁ c₂ h"
},
{
"state_after": "case mk\nK : Type u_4\ninst✝² : DecidableEq K\nΓ : K → Type u_3\nΛ : Type u_2\ninst✝¹ : Inhabited Λ\nσ : Type u_1\ninst✝ : Inhabited σ\nM : Λ → Stmt₂\nl : Option Λ\nv : σ\nS : (k : K) → List (Γ k)\nL : ListBlank ((k : K) → Option (Γ k))\nhT : ∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))\n⊢ match TM2.step M { l := l, var := v, stk := S } with\n | some b₁ =>\n ∃ b₂,\n TrCfg b₁ b₂ ∧\n Reaches₁ (TM1.step (tr M)) { l := Option.map normal l, var := v, Tape := Tape.mk' ∅ (addBottom L) } b₂\n | none => TM1.step (tr M) { l := Option.map normal l, var := v, Tape := Tape.mk' ∅ (addBottom L) } = none",
"state_before": "K : Type u_4\ninst✝² : DecidableEq K\nΓ : K → Type u_3\nΛ : Type u_2\ninst✝¹ : Inhabited Λ\nσ : Type u_1\ninst✝ : Inhabited σ\nM : Λ → Stmt₂\nc₁ : Cfg₂\nc₂ : TM1.Cfg Γ' Λ' σ\nh : TrCfg c₁ c₂\n⊢ match TM2.step M c₁ with\n | some b₁ => ∃ b₂, TrCfg b₁ b₂ ∧ Reaches₁ (TM1.step (tr M)) c₂ b₂\n | none => TM1.step (tr M) c₂ = none",
"tactic": "cases' h with l v S L hT"
},
{
"state_after": "case mk.none\nK : Type u_4\ninst✝² : DecidableEq K\nΓ : K → Type u_3\nΛ : Type u_2\ninst✝¹ : Inhabited Λ\nσ : Type u_1\ninst✝ : Inhabited σ\nM : Λ → Stmt₂\nv : σ\nS : (k : K) → List (Γ k)\nL : ListBlank ((k : K) → Option (Γ k))\nhT : ∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))\n⊢ match TM2.step M { l := none, var := v, stk := S } with\n | some b₁ =>\n ∃ b₂,\n TrCfg b₁ b₂ ∧\n Reaches₁ (TM1.step (tr M)) { l := Option.map normal none, var := v, Tape := Tape.mk' ∅ (addBottom L) } b₂\n | none => TM1.step (tr M) { l := Option.map normal none, var := v, Tape := Tape.mk' ∅ (addBottom L) } = none\n\ncase mk.some\nK : Type u_4\ninst✝² : DecidableEq K\nΓ : K → Type u_3\nΛ : Type u_2\ninst✝¹ : Inhabited Λ\nσ : Type u_1\ninst✝ : Inhabited σ\nM : Λ → Stmt₂\nv : σ\nS : (k : K) → List (Γ k)\nL : ListBlank ((k : K) → Option (Γ k))\nhT : ∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))\nl : Λ\n⊢ match TM2.step M { l := some l, var := v, stk := S } with\n | some b₁ =>\n ∃ b₂,\n TrCfg b₁ b₂ ∧\n Reaches₁ (TM1.step (tr M)) { l := Option.map normal (some l), var := v, Tape := Tape.mk' ∅ (addBottom L) } b₂\n | none => TM1.step (tr M) { l := Option.map normal (some l), var := v, Tape := Tape.mk' ∅ (addBottom L) } = none",
"state_before": "case mk\nK : Type u_4\ninst✝² : DecidableEq K\nΓ : K → Type u_3\nΛ : Type u_2\ninst✝¹ : Inhabited Λ\nσ : Type u_1\ninst✝ : Inhabited σ\nM : Λ → Stmt₂\nl : Option Λ\nv : σ\nS : (k : K) → List (Γ k)\nL : ListBlank ((k : K) → Option (Γ k))\nhT : ∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))\n⊢ match TM2.step M { l := l, var := v, stk := S } with\n | some b₁ =>\n ∃ b₂,\n TrCfg b₁ b₂ ∧\n Reaches₁ (TM1.step (tr M)) { l := Option.map normal l, var := v, Tape := Tape.mk' ∅ (addBottom L) } b₂\n | none => TM1.step (tr M) { l := Option.map normal l, var := v, Tape := Tape.mk' ∅ (addBottom L) } = none",
"tactic": "cases' l with l"
},
{
"state_after": "case mk.some\nK : Type u_4\ninst✝² : DecidableEq K\nΓ : K → Type u_3\nΛ : Type u_2\ninst✝¹ : Inhabited Λ\nσ : Type u_1\ninst✝ : Inhabited σ\nM : Λ → Stmt₂\nv : σ\nS : (k : K) → List (Γ k)\nL : ListBlank ((k : K) → Option (Γ k))\nhT : ∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))\nl : Λ\n⊢ ∃ b₂,\n TrCfg (TM2.stepAux (M l) v S) b₂ ∧\n Reaches₁ (TM1.step (tr M)) { l := some (normal l), var := v, Tape := Tape.mk' ∅ (addBottom L) } b₂",
"state_before": "case mk.some\nK : Type u_4\ninst✝² : DecidableEq K\nΓ : K → Type u_3\nΛ : Type u_2\ninst✝¹ : Inhabited Λ\nσ : Type u_1\ninst✝ : Inhabited σ\nM : Λ → Stmt₂\nv : σ\nS : (k : K) → List (Γ k)\nL : ListBlank ((k : K) → Option (Γ k))\nhT : ∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))\nl : Λ\n⊢ match TM2.step M { l := some l, var := v, stk := S } with\n | some b₁ =>\n ∃ b₂,\n TrCfg b₁ b₂ ∧\n Reaches₁ (TM1.step (tr M)) { l := Option.map normal (some l), var := v, Tape := Tape.mk' ∅ (addBottom L) } b₂\n | none => TM1.step (tr M) { l := Option.map normal (some l), var := v, Tape := Tape.mk' ∅ (addBottom L) } = none",
"tactic": "simp only [TM2.step, Respects, Option.map_some']"
},
{
"state_after": "case mk.some.intro.intro\nK : Type u_4\ninst✝² : DecidableEq K\nΓ : K → Type u_3\nΛ : Type u_2\ninst✝¹ : Inhabited Λ\nσ : Type u_1\ninst✝ : Inhabited σ\nM : Λ → Stmt₂\nv : σ\nS : (k : K) → List (Γ k)\nL : ListBlank ((k : K) → Option (Γ k))\nhT : ∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))\nl : Λ\nb : ?m.734846\nc : ?m.735219 b\nr : Reaches (TM1.step (tr M)) (?m.735220 b) (?m.735221 b)\n⊢ ∃ b₂,\n TrCfg (TM2.stepAux (M l) v S) b₂ ∧\n Reaches₁ (TM1.step (tr M)) { l := some (normal l), var := v, Tape := Tape.mk' ∅ (addBottom L) } b₂\n\nK : Type u_4\ninst✝² : DecidableEq K\nΓ : K → Type u_3\nΛ : Type u_2\ninst✝¹ : Inhabited Λ\nσ : Type u_1\ninst✝ : Inhabited σ\nM : Λ → Stmt₂\nv : σ\nS : (k : K) → List (Γ k)\nL : ListBlank ((k : K) → Option (Γ k))\nhT : ∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))\nl : Λ\n⊢ ∃ b, ?m.735219 b ∧ Reaches (TM1.step (tr M)) (?m.735220 b) (?m.735221 b)",
"state_before": "case mk.some\nK : Type u_4\ninst✝² : DecidableEq K\nΓ : K → Type u_3\nΛ : Type u_2\ninst✝¹ : Inhabited Λ\nσ : Type u_1\ninst✝ : Inhabited σ\nM : Λ → Stmt₂\nv : σ\nS : (k : K) → List (Γ k)\nL : ListBlank ((k : K) → Option (Γ k))\nhT : ∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))\nl : Λ\n⊢ ∃ b₂,\n TrCfg (TM2.stepAux (M l) v S) b₂ ∧\n Reaches₁ (TM1.step (tr M)) { l := some (normal l), var := v, Tape := Tape.mk' ∅ (addBottom L) } b₂",
"tactic": "rsuffices ⟨b, c, r⟩ : ∃ b, _ ∧ Reaches (TM1.step (tr M)) _ _"
},
{
"state_after": "K : Type u_4\ninst✝² : DecidableEq K\nΓ : K → Type u_3\nΛ : Type u_2\ninst✝¹ : Inhabited Λ\nσ : Type u_1\ninst✝ : Inhabited σ\nM : Λ → Stmt₂\nv : σ\nS : (k : K) → List (Γ k)\nL : ListBlank ((k : K) → Option (Γ k))\nhT : ∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))\nl : Λ\n⊢ ∃ b,\n TrCfg (TM2.stepAux (M l) v S) b ∧\n Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal (M l)) v (Tape.mk' ∅ (addBottom L))) b",
"state_before": "K : Type u_4\ninst✝² : DecidableEq K\nΓ : K → Type u_3\nΛ : Type u_2\ninst✝¹ : Inhabited Λ\nσ : Type u_1\ninst✝ : Inhabited σ\nM : Λ → Stmt₂\nv : σ\nS : (k : K) → List (Γ k)\nL : ListBlank ((k : K) → Option (Γ k))\nhT : ∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))\nl : Λ\n⊢ ∃ b,\n TrCfg (TM2.stepAux (M l) v S) b ∧\n Reaches (TM1.step (tr M)) (TM1.stepAux (tr M (normal l)) v (Tape.mk' ∅ (addBottom L))) b",
"tactic": "simp only [tr]"
},
{
"state_after": "K : Type u_4\ninst✝² : DecidableEq K\nΓ : K → Type u_3\nΛ : Type u_2\ninst✝¹ : Inhabited Λ\nσ : Type u_1\ninst✝ : Inhabited σ\nM : Λ → Stmt₂\nv : σ\nS : (k : K) → List (Γ k)\nL : ListBlank ((k : K) → Option (Γ k))\nhT : ∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))\nl : Λ\nN : Stmt₂\n⊢ ∃ b, TrCfg (TM2.stepAux N v S) b ∧ Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal N) v (Tape.mk' ∅ (addBottom L))) b",
"state_before": "K : Type u_4\ninst✝² : DecidableEq K\nΓ : K → Type u_3\nΛ : Type u_2\ninst✝¹ : Inhabited Λ\nσ : Type u_1\ninst✝ : Inhabited σ\nM : Λ → Stmt₂\nv : σ\nS : (k : K) → List (Γ k)\nL : ListBlank ((k : K) → Option (Γ k))\nhT : ∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))\nl : Λ\n⊢ ∃ b,\n TrCfg (TM2.stepAux (M l) v S) b ∧\n Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal (M l)) v (Tape.mk' ∅ (addBottom L))) b",
"tactic": "generalize M l = N"
},
{
"state_after": "no goals",
"state_before": "K : Type u_4\ninst✝² : DecidableEq K\nΓ : K → Type u_3\nΛ : Type u_2\ninst✝¹ : Inhabited Λ\nσ : Type u_1\ninst✝ : Inhabited σ\nM : Λ → Stmt₂\nv : σ\nS : (k : K) → List (Γ k)\nL : ListBlank ((k : K) → Option (Γ k))\nhT : ∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))\nl : Λ\nN : Stmt₂\n⊢ ∃ b, TrCfg (TM2.stepAux N v S) b ∧ Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal N) v (Tape.mk' ∅ (addBottom L))) b",
"tactic": "induction N using stmtStRec generalizing v S L hT with\n| H₁ k s q IH => exact tr_respects_aux M hT s @IH\n| H₂ a _ IH => exact IH _ hT\n| H₃ p q₁ q₂ IH₁ IH₂ =>\n unfold TM2.stepAux trNormal TM1.stepAux\n simp only []\n cases p v <;> [exact IH₂ _ hT; exact IH₁ _ hT]\n| H₄ => exact ⟨_, ⟨_, hT⟩, ReflTransGen.refl⟩\n| H₅ => exact ⟨_, ⟨_, hT⟩, ReflTransGen.refl⟩"
},
{
"state_after": "no goals",
"state_before": "case mk.none\nK : Type u_4\ninst✝² : DecidableEq K\nΓ : K → Type u_3\nΛ : Type u_2\ninst✝¹ : Inhabited Λ\nσ : Type u_1\ninst✝ : Inhabited σ\nM : Λ → Stmt₂\nv : σ\nS : (k : K) → List (Γ k)\nL : ListBlank ((k : K) → Option (Γ k))\nhT : ∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))\n⊢ match TM2.step M { l := none, var := v, stk := S } with\n | some b₁ =>\n ∃ b₂,\n TrCfg b₁ b₂ ∧\n Reaches₁ (TM1.step (tr M)) { l := Option.map normal none, var := v, Tape := Tape.mk' ∅ (addBottom L) } b₂\n | none => TM1.step (tr M) { l := Option.map normal none, var := v, Tape := Tape.mk' ∅ (addBottom L) } = none",
"tactic": "constructor"
},
{
"state_after": "no goals",
"state_before": "case mk.some.intro.intro\nK : Type u_4\ninst✝² : DecidableEq K\nΓ : K → Type u_3\nΛ : Type u_2\ninst✝¹ : Inhabited Λ\nσ : Type u_1\ninst✝ : Inhabited σ\nM : Λ → Stmt₂\nv : σ\nS : (k : K) → List (Γ k)\nL : ListBlank ((k : K) → Option (Γ k))\nhT : ∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))\nl : Λ\nb : ?m.734846\nc : ?m.735219 b\nr : Reaches (TM1.step (tr M)) (?m.735220 b) (?m.735221 b)\n⊢ ∃ b₂,\n TrCfg (TM2.stepAux (M l) v S) b₂ ∧\n Reaches₁ (TM1.step (tr M)) { l := some (normal l), var := v, Tape := Tape.mk' ∅ (addBottom L) } b₂",
"tactic": "exact ⟨b, c, TransGen.head' rfl r⟩"
},
{
"state_after": "no goals",
"state_before": "case H₁\nK : Type u_4\ninst✝² : DecidableEq K\nΓ : K → Type u_3\nΛ : Type u_2\ninst✝¹ : Inhabited Λ\nσ : Type u_1\ninst✝ : Inhabited σ\nM : Λ → Stmt₂\nl : Λ\nk : K\ns : StAct k\nq : Stmt₂\nIH :\n ∀ {v : σ} {S : (k : K) → List (Γ k)} (L : ListBlank ((k : K) → Option (Γ k))),\n (∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))) →\n ∃ b,\n TrCfg (TM2.stepAux q v S) b ∧\n Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal q) v (Tape.mk' ∅ (addBottom L))) b\nv : σ\nS : (k : K) → List (Γ k)\nL : ListBlank ((k : K) → Option (Γ k))\nhT : ∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))\n⊢ ∃ b,\n TrCfg (TM2.stepAux (stRun s q) v S) b ∧\n Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal (stRun s q)) v (Tape.mk' ∅ (addBottom L))) b",
"tactic": "exact tr_respects_aux M hT s @IH"
},
{
"state_after": "no goals",
"state_before": "case H₂\nK : Type u_4\ninst✝² : DecidableEq K\nΓ : K → Type u_3\nΛ : Type u_2\ninst✝¹ : Inhabited Λ\nσ : Type u_1\ninst✝ : Inhabited σ\nM : Λ → Stmt₂\nl : Λ\na : σ → σ\nq✝ : Stmt₂\nIH :\n ∀ {v : σ} {S : (k : K) → List (Γ k)} (L : ListBlank ((k : K) → Option (Γ k))),\n (∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))) →\n ∃ b,\n TrCfg (TM2.stepAux q✝ v S) b ∧\n Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal q✝) v (Tape.mk' ∅ (addBottom L))) b\nv : σ\nS : (k : K) → List (Γ k)\nL : ListBlank ((k : K) → Option (Γ k))\nhT : ∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))\n⊢ ∃ b,\n TrCfg (TM2.stepAux (TM2.Stmt.load a q✝) v S) b ∧\n Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal (TM2.Stmt.load a q✝)) v (Tape.mk' ∅ (addBottom L))) b",
"tactic": "exact IH _ hT"
},
{
"state_after": "case H₃\nK : Type u_4\ninst✝² : DecidableEq K\nΓ : K → Type u_3\nΛ : Type u_2\ninst✝¹ : Inhabited Λ\nσ : Type u_1\ninst✝ : Inhabited σ\nM : Λ → Stmt₂\nl : Λ\np : σ → Bool\nq₁ q₂ : Stmt₂\nIH₁ :\n ∀ {v : σ} {S : (k : K) → List (Γ k)} (L : ListBlank ((k : K) → Option (Γ k))),\n (∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))) →\n ∃ b,\n TrCfg (TM2.stepAux q₁ v S) b ∧\n Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal q₁) v (Tape.mk' ∅ (addBottom L))) b\nIH₂ :\n ∀ {v : σ} {S : (k : K) → List (Γ k)} (L : ListBlank ((k : K) → Option (Γ k))),\n (∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))) →\n ∃ b,\n TrCfg (TM2.stepAux q₂ v S) b ∧\n Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal q₂) v (Tape.mk' ∅ (addBottom L))) b\nv : σ\nS : (k : K) → List (Γ k)\nL : ListBlank ((k : K) → Option (Γ k))\nhT : ∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))\n⊢ ∃ b,\n TrCfg (bif p v then TM2.stepAux q₁ v S else TM2.stepAux q₂ v S) b ∧\n Reaches (TM1.step (tr M))\n (bif (fun x => p) (Tape.mk' ∅ (addBottom L)).head v then TM1.stepAux (trNormal q₁) v (Tape.mk' ∅ (addBottom L))\n else TM1.stepAux (trNormal q₂) v (Tape.mk' ∅ (addBottom L)))\n b",
"state_before": "case H₃\nK : Type u_4\ninst✝² : DecidableEq K\nΓ : K → Type u_3\nΛ : Type u_2\ninst✝¹ : Inhabited Λ\nσ : Type u_1\ninst✝ : Inhabited σ\nM : Λ → Stmt₂\nl : Λ\np : σ → Bool\nq₁ q₂ : Stmt₂\nIH₁ :\n ∀ {v : σ} {S : (k : K) → List (Γ k)} (L : ListBlank ((k : K) → Option (Γ k))),\n (∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))) →\n ∃ b,\n TrCfg (TM2.stepAux q₁ v S) b ∧\n Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal q₁) v (Tape.mk' ∅ (addBottom L))) b\nIH₂ :\n ∀ {v : σ} {S : (k : K) → List (Γ k)} (L : ListBlank ((k : K) → Option (Γ k))),\n (∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))) →\n ∃ b,\n TrCfg (TM2.stepAux q₂ v S) b ∧\n Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal q₂) v (Tape.mk' ∅ (addBottom L))) b\nv : σ\nS : (k : K) → List (Γ k)\nL : ListBlank ((k : K) → Option (Γ k))\nhT : ∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))\n⊢ ∃ b,\n TrCfg (TM2.stepAux (TM2.Stmt.branch p q₁ q₂) v S) b ∧\n Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal (TM2.Stmt.branch p q₁ q₂)) v (Tape.mk' ∅ (addBottom L))) b",
"tactic": "unfold TM2.stepAux trNormal TM1.stepAux"
},
{
"state_after": "case H₃\nK : Type u_4\ninst✝² : DecidableEq K\nΓ : K → Type u_3\nΛ : Type u_2\ninst✝¹ : Inhabited Λ\nσ : Type u_1\ninst✝ : Inhabited σ\nM : Λ → Stmt₂\nl : Λ\np : σ → Bool\nq₁ q₂ : Stmt₂\nIH₁ :\n ∀ {v : σ} {S : (k : K) → List (Γ k)} (L : ListBlank ((k : K) → Option (Γ k))),\n (∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))) →\n ∃ b,\n TrCfg (TM2.stepAux q₁ v S) b ∧\n Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal q₁) v (Tape.mk' ∅ (addBottom L))) b\nIH₂ :\n ∀ {v : σ} {S : (k : K) → List (Γ k)} (L : ListBlank ((k : K) → Option (Γ k))),\n (∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))) →\n ∃ b,\n TrCfg (TM2.stepAux q₂ v S) b ∧\n Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal q₂) v (Tape.mk' ∅ (addBottom L))) b\nv : σ\nS : (k : K) → List (Γ k)\nL : ListBlank ((k : K) → Option (Γ k))\nhT : ∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))\n⊢ ∃ b,\n TrCfg (bif p v then TM2.stepAux q₁ v S else TM2.stepAux q₂ v S) b ∧\n Reaches (TM1.step (tr M))\n (bif p v then TM1.stepAux (trNormal q₁) v (Tape.mk' ∅ (addBottom L))\n else TM1.stepAux (trNormal q₂) v (Tape.mk' ∅ (addBottom L)))\n b",
"state_before": "case H₃\nK : Type u_4\ninst✝² : DecidableEq K\nΓ : K → Type u_3\nΛ : Type u_2\ninst✝¹ : Inhabited Λ\nσ : Type u_1\ninst✝ : Inhabited σ\nM : Λ → Stmt₂\nl : Λ\np : σ → Bool\nq₁ q₂ : Stmt₂\nIH₁ :\n ∀ {v : σ} {S : (k : K) → List (Γ k)} (L : ListBlank ((k : K) → Option (Γ k))),\n (∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))) →\n ∃ b,\n TrCfg (TM2.stepAux q₁ v S) b ∧\n Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal q₁) v (Tape.mk' ∅ (addBottom L))) b\nIH₂ :\n ∀ {v : σ} {S : (k : K) → List (Γ k)} (L : ListBlank ((k : K) → Option (Γ k))),\n (∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))) →\n ∃ b,\n TrCfg (TM2.stepAux q₂ v S) b ∧\n Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal q₂) v (Tape.mk' ∅ (addBottom L))) b\nv : σ\nS : (k : K) → List (Γ k)\nL : ListBlank ((k : K) → Option (Γ k))\nhT : ∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))\n⊢ ∃ b,\n TrCfg (bif p v then TM2.stepAux q₁ v S else TM2.stepAux q₂ v S) b ∧\n Reaches (TM1.step (tr M))\n (bif (fun x => p) (Tape.mk' ∅ (addBottom L)).head v then TM1.stepAux (trNormal q₁) v (Tape.mk' ∅ (addBottom L))\n else TM1.stepAux (trNormal q₂) v (Tape.mk' ∅ (addBottom L)))\n b",
"tactic": "simp only []"
},
{
"state_after": "no goals",
"state_before": "case H₃\nK : Type u_4\ninst✝² : DecidableEq K\nΓ : K → Type u_3\nΛ : Type u_2\ninst✝¹ : Inhabited Λ\nσ : Type u_1\ninst✝ : Inhabited σ\nM : Λ → Stmt₂\nl : Λ\np : σ → Bool\nq₁ q₂ : Stmt₂\nIH₁ :\n ∀ {v : σ} {S : (k : K) → List (Γ k)} (L : ListBlank ((k : K) → Option (Γ k))),\n (∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))) →\n ∃ b,\n TrCfg (TM2.stepAux q₁ v S) b ∧\n Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal q₁) v (Tape.mk' ∅ (addBottom L))) b\nIH₂ :\n ∀ {v : σ} {S : (k : K) → List (Γ k)} (L : ListBlank ((k : K) → Option (Γ k))),\n (∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))) →\n ∃ b,\n TrCfg (TM2.stepAux q₂ v S) b ∧\n Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal q₂) v (Tape.mk' ∅ (addBottom L))) b\nv : σ\nS : (k : K) → List (Γ k)\nL : ListBlank ((k : K) → Option (Γ k))\nhT : ∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))\n⊢ ∃ b,\n TrCfg (bif p v then TM2.stepAux q₁ v S else TM2.stepAux q₂ v S) b ∧\n Reaches (TM1.step (tr M))\n (bif p v then TM1.stepAux (trNormal q₁) v (Tape.mk' ∅ (addBottom L))\n else TM1.stepAux (trNormal q₂) v (Tape.mk' ∅ (addBottom L)))\n b",
"tactic": "cases p v <;> [exact IH₂ _ hT; exact IH₁ _ hT]"
},
{
"state_after": "no goals",
"state_before": "case H₄\nK : Type u_4\ninst✝² : DecidableEq K\nΓ : K → Type u_3\nΛ : Type u_2\ninst✝¹ : Inhabited Λ\nσ : Type u_1\ninst✝ : Inhabited σ\nM : Λ → Stmt₂\nl : Λ\nl✝ : σ → Λ\nv : σ\nS : (k : K) → List (Γ k)\nL : ListBlank ((k : K) → Option (Γ k))\nhT : ∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))\n⊢ ∃ b,\n TrCfg (TM2.stepAux (TM2.Stmt.goto l✝) v S) b ∧\n Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal (TM2.Stmt.goto l✝)) v (Tape.mk' ∅ (addBottom L))) b",
"tactic": "exact ⟨_, ⟨_, hT⟩, ReflTransGen.refl⟩"
},
{
"state_after": "no goals",
"state_before": "case H₅\nK : Type u_4\ninst✝² : DecidableEq K\nΓ : K → Type u_3\nΛ : Type u_2\ninst✝¹ : Inhabited Λ\nσ : Type u_1\ninst✝ : Inhabited σ\nM : Λ → Stmt₂\nl : Λ\nv : σ\nS : (k : K) → List (Γ k)\nL : ListBlank ((k : K) → Option (Γ k))\nhT : ∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))\n⊢ ∃ b,\n TrCfg (TM2.stepAux TM2.Stmt.halt v S) b ∧\n Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal TM2.Stmt.halt) v (Tape.mk' ∅ (addBottom L))) b",
"tactic": "exact ⟨_, ⟨_, hT⟩, ReflTransGen.refl⟩"
}
] |
[
2723,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2704,
1
] |
Mathlib/Topology/Order.lean
|
continuous_iSup_rng
|
[] |
[
769,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
767,
1
] |
Mathlib/MeasureTheory/Function/L2Space.lean
|
MeasureTheory.L2.inner_indicatorConstLp_eq_inner_set_integral
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_2\nE : Type u_1\nF : Type ?u.159069\n𝕜 : Type u_3\ninst✝⁶ : IsROrC 𝕜\ninst✝⁵ : MeasurableSpace α\nμ : Measure α\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : InnerProductSpace 𝕜 E\ninst✝² : NormedAddCommGroup F\ns : Set α\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\nhs : MeasurableSet s\nhμs : ↑↑μ s ≠ ⊤\nc : E\nf : { x // x ∈ Lp E 2 }\n⊢ inner (indicatorConstLp 2 hs hμs c) f = inner c (∫ (x : α) in s, ↑↑f x ∂μ)",
"tactic": "rw [← integral_inner (integrableOn_Lp_of_measure_ne_top f fact_one_le_two_ennreal.elim hμs),\n L2.inner_indicatorConstLp_eq_set_integral_inner]"
}
] |
[
270,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
266,
1
] |
Mathlib/SetTheory/Ordinal/Basic.lean
|
Ordinal.type_ne_zero_of_nonempty
|
[] |
[
239,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
238,
1
] |
Mathlib/Data/Set/Pointwise/Interval.lean
|
Set.preimage_mul_const_Icc
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : LinearOrderedField α\na✝ a b c : α\nh : 0 < c\n⊢ (fun x => x * c) ⁻¹' Icc a b = Icc (a / c) (b / c)",
"tactic": "simp [← Ici_inter_Iic, h]"
}
] |
[
551,
87
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
550,
1
] |
Mathlib/NumberTheory/Padics/RingHoms.lean
|
PadicInt.zmod_congr_of_sub_mem_span
|
[
{
"state_after": "no goals",
"state_before": "p : ℕ\nhp_prime : Fact (Nat.Prime p)\nr : ℚ\nn : ℕ\nx : ℤ_[p]\na b : ℕ\nha : x - ↑a ∈ Ideal.span {↑p ^ n}\nhb : x - ↑b ∈ Ideal.span {↑p ^ n}\n⊢ ↑a = ↑b",
"tactic": "simpa using zmod_congr_of_sub_mem_span_aux n x a b ha hb"
}
] |
[
158,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
155,
1
] |
Mathlib/Data/Set/Lattice.lean
|
Set.inter_empty_of_inter_sUnion_empty
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.148065\nγ : Type ?u.148068\nι : Sort ?u.148071\nι' : Sort ?u.148074\nι₂ : Sort ?u.148077\nκ : ι → Sort ?u.148082\nκ₁ : ι → Sort ?u.148087\nκ₂ : ι → Sort ?u.148092\nκ' : ι' → Sort ?u.148097\ns t : Set α\nS : Set (Set α)\nhs : t ∈ S\nh : s ∩ ⋃₀ S = ∅\n⊢ s ∩ t ⊆ s ∩ ⋃₀ S",
"state_before": "α : Type u_1\nβ : Type ?u.148065\nγ : Type ?u.148068\nι : Sort ?u.148071\nι' : Sort ?u.148074\nι₂ : Sort ?u.148077\nκ : ι → Sort ?u.148082\nκ₁ : ι → Sort ?u.148087\nκ₂ : ι → Sort ?u.148092\nκ' : ι' → Sort ?u.148097\ns t : Set α\nS : Set (Set α)\nhs : t ∈ S\nh : s ∩ ⋃₀ S = ∅\n⊢ s ∩ t ⊆ ∅",
"tactic": "rw [← h]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.148065\nγ : Type ?u.148068\nι : Sort ?u.148071\nι' : Sort ?u.148074\nι₂ : Sort ?u.148077\nκ : ι → Sort ?u.148082\nκ₁ : ι → Sort ?u.148087\nκ₂ : ι → Sort ?u.148092\nκ' : ι' → Sort ?u.148097\ns t : Set α\nS : Set (Set α)\nhs : t ∈ S\nh : s ∩ ⋃₀ S = ∅\n⊢ s ∩ t ⊆ s ∩ ⋃₀ S",
"tactic": "exact inter_subset_inter_right _ (subset_sUnion_of_mem hs)"
}
] |
[
1260,
73
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1257,
1
] |
Mathlib/Data/Real/Sqrt.lean
|
Real.sqrt_eq_zero'
|
[
{
"state_after": "no goals",
"state_before": "x y : ℝ\n⊢ sqrt x = 0 ↔ x ≤ 0",
"tactic": "rw [sqrt, NNReal.coe_eq_zero, NNReal.sqrt_eq_zero, Real.toNNReal_eq_zero]"
}
] |
[
341,
76
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
340,
1
] |
Mathlib/Data/Dfinsupp/Basic.lean
|
Dfinsupp.subtypeDomain_zero
|
[] |
[
498,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
496,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean
|
Real.sin_add_pi
|
[] |
[
239,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
238,
1
] |
Mathlib/Analysis/InnerProductSpace/l2Space.lean
|
Submodule.isHilbertSumOrthogonal
|
[
{
"state_after": "ι : Type ?u.670433\n𝕜 : Type u_1\ninst✝⁴ : IsROrC 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ncplt : CompleteSpace E\nG : ι → Type ?u.670584\ninst✝¹ : (i : ι) → NormedAddCommGroup (G i)\ninst✝ : (i : ι) → InnerProductSpace 𝕜 (G i)\nV : (i : ι) → G i →ₗᵢ[𝕜] E\nF : ι → Submodule 𝕜 E\nK : Submodule 𝕜 E\nhK : CompleteSpace { x // x ∈ K }\nthis : ∀ (b : Bool), CompleteSpace { x // x ∈ bif b then K else Kᗮ }\n⊢ IsHilbertSum 𝕜 (fun b => { x // x ∈ bif b then K else Kᗮ }) fun b => subtypeₗᵢ (bif b then K else Kᗮ)",
"state_before": "ι : Type ?u.670433\n𝕜 : Type u_1\ninst✝⁴ : IsROrC 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ncplt : CompleteSpace E\nG : ι → Type ?u.670584\ninst✝¹ : (i : ι) → NormedAddCommGroup (G i)\ninst✝ : (i : ι) → InnerProductSpace 𝕜 (G i)\nV : (i : ι) → G i →ₗᵢ[𝕜] E\nF : ι → Submodule 𝕜 E\nK : Submodule 𝕜 E\nhK : CompleteSpace { x // x ∈ K }\n⊢ IsHilbertSum 𝕜 (fun b => { x // x ∈ bif b then K else Kᗮ }) fun b => subtypeₗᵢ (bif b then K else Kᗮ)",
"tactic": "have : ∀ b, CompleteSpace (↥(cond b K Kᗮ)) := by\n intro b\n cases b <;> first | exact instOrthogonalCompleteSpace K | assumption"
},
{
"state_after": "ι : Type ?u.670433\n𝕜 : Type u_1\ninst✝⁴ : IsROrC 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ncplt : CompleteSpace E\nG : ι → Type ?u.670584\ninst✝¹ : (i : ι) → NormedAddCommGroup (G i)\ninst✝ : (i : ι) → InnerProductSpace 𝕜 (G i)\nV : (i : ι) → G i →ₗᵢ[𝕜] E\nF : ι → Submodule 𝕜 E\nK : Submodule 𝕜 E\nhK : CompleteSpace { x // x ∈ K }\nthis : ∀ (b : Bool), CompleteSpace { x // x ∈ bif b then K else Kᗮ }\n⊢ ⊤ ≤ topologicalClosure (⨆ (i : Bool), bif i then K else Kᗮ)",
"state_before": "ι : Type ?u.670433\n𝕜 : Type u_1\ninst✝⁴ : IsROrC 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ncplt : CompleteSpace E\nG : ι → Type ?u.670584\ninst✝¹ : (i : ι) → NormedAddCommGroup (G i)\ninst✝ : (i : ι) → InnerProductSpace 𝕜 (G i)\nV : (i : ι) → G i →ₗᵢ[𝕜] E\nF : ι → Submodule 𝕜 E\nK : Submodule 𝕜 E\nhK : CompleteSpace { x // x ∈ K }\nthis : ∀ (b : Bool), CompleteSpace { x // x ∈ bif b then K else Kᗮ }\n⊢ IsHilbertSum 𝕜 (fun b => { x // x ∈ bif b then K else Kᗮ }) fun b => subtypeₗᵢ (bif b then K else Kᗮ)",
"tactic": "refine' IsHilbertSum.mkInternal _ K.orthogonalFamily_self _"
},
{
"state_after": "ι : Type ?u.670433\n𝕜 : Type u_1\ninst✝⁴ : IsROrC 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ncplt : CompleteSpace E\nG : ι → Type ?u.670584\ninst✝¹ : (i : ι) → NormedAddCommGroup (G i)\ninst✝ : (i : ι) → InnerProductSpace 𝕜 (G i)\nV : (i : ι) → G i →ₗᵢ[𝕜] E\nF : ι → Submodule 𝕜 E\nK : Submodule 𝕜 E\nhK : CompleteSpace { x // x ∈ K }\nthis : ∀ (b : Bool), CompleteSpace { x // x ∈ bif b then K else Kᗮ }\n⊢ ⊤ ≤ ⨆ (i : Bool), bif i then K else Kᗮ",
"state_before": "ι : Type ?u.670433\n𝕜 : Type u_1\ninst✝⁴ : IsROrC 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ncplt : CompleteSpace E\nG : ι → Type ?u.670584\ninst✝¹ : (i : ι) → NormedAddCommGroup (G i)\ninst✝ : (i : ι) → InnerProductSpace 𝕜 (G i)\nV : (i : ι) → G i →ₗᵢ[𝕜] E\nF : ι → Submodule 𝕜 E\nK : Submodule 𝕜 E\nhK : CompleteSpace { x // x ∈ K }\nthis : ∀ (b : Bool), CompleteSpace { x // x ∈ bif b then K else Kᗮ }\n⊢ ⊤ ≤ topologicalClosure (⨆ (i : Bool), bif i then K else Kᗮ)",
"tactic": "refine' le_trans _ (Submodule.le_topologicalClosure _)"
},
{
"state_after": "ι : Type ?u.670433\n𝕜 : Type u_1\ninst✝⁴ : IsROrC 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ncplt : CompleteSpace E\nG : ι → Type ?u.670584\ninst✝¹ : (i : ι) → NormedAddCommGroup (G i)\ninst✝ : (i : ι) → InnerProductSpace 𝕜 (G i)\nV : (i : ι) → G i →ₗᵢ[𝕜] E\nF : ι → Submodule 𝕜 E\nK : Submodule 𝕜 E\nhK : CompleteSpace { x // x ∈ K }\nthis : ∀ (b : Bool), CompleteSpace { x // x ∈ bif b then K else Kᗮ }\n⊢ ⊤ ≤\n (match true with\n | true => K\n | false => Kᗮ) ⊔\n match false with\n | true => K\n | false => Kᗮ",
"state_before": "ι : Type ?u.670433\n𝕜 : Type u_1\ninst✝⁴ : IsROrC 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ncplt : CompleteSpace E\nG : ι → Type ?u.670584\ninst✝¹ : (i : ι) → NormedAddCommGroup (G i)\ninst✝ : (i : ι) → InnerProductSpace 𝕜 (G i)\nV : (i : ι) → G i →ₗᵢ[𝕜] E\nF : ι → Submodule 𝕜 E\nK : Submodule 𝕜 E\nhK : CompleteSpace { x // x ∈ K }\nthis : ∀ (b : Bool), CompleteSpace { x // x ∈ bif b then K else Kᗮ }\n⊢ ⊤ ≤ ⨆ (i : Bool), bif i then K else Kᗮ",
"tactic": "rw [iSup_bool_eq, cond, cond]"
},
{
"state_after": "ι : Type ?u.670433\n𝕜 : Type u_1\ninst✝⁴ : IsROrC 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ncplt : CompleteSpace E\nG : ι → Type ?u.670584\ninst✝¹ : (i : ι) → NormedAddCommGroup (G i)\ninst✝ : (i : ι) → InnerProductSpace 𝕜 (G i)\nV : (i : ι) → G i →ₗᵢ[𝕜] E\nF : ι → Submodule 𝕜 E\nK : Submodule 𝕜 E\nhK : CompleteSpace { x // x ∈ K }\nthis : ∀ (b : Bool), CompleteSpace { x // x ∈ bif b then K else Kᗮ }\n⊢ Codisjoint\n (match true with\n | true => K\n | false => Kᗮ)\n (match false with\n | true => K\n | false => Kᗮ)",
"state_before": "ι : Type ?u.670433\n𝕜 : Type u_1\ninst✝⁴ : IsROrC 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ncplt : CompleteSpace E\nG : ι → Type ?u.670584\ninst✝¹ : (i : ι) → NormedAddCommGroup (G i)\ninst✝ : (i : ι) → InnerProductSpace 𝕜 (G i)\nV : (i : ι) → G i →ₗᵢ[𝕜] E\nF : ι → Submodule 𝕜 E\nK : Submodule 𝕜 E\nhK : CompleteSpace { x // x ∈ K }\nthis : ∀ (b : Bool), CompleteSpace { x // x ∈ bif b then K else Kᗮ }\n⊢ ⊤ ≤\n (match true with\n | true => K\n | false => Kᗮ) ⊔\n match false with\n | true => K\n | false => Kᗮ",
"tactic": "refine' Codisjoint.top_le _"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.670433\n𝕜 : Type u_1\ninst✝⁴ : IsROrC 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ncplt : CompleteSpace E\nG : ι → Type ?u.670584\ninst✝¹ : (i : ι) → NormedAddCommGroup (G i)\ninst✝ : (i : ι) → InnerProductSpace 𝕜 (G i)\nV : (i : ι) → G i →ₗᵢ[𝕜] E\nF : ι → Submodule 𝕜 E\nK : Submodule 𝕜 E\nhK : CompleteSpace { x // x ∈ K }\nthis : ∀ (b : Bool), CompleteSpace { x // x ∈ bif b then K else Kᗮ }\n⊢ Codisjoint\n (match true with\n | true => K\n | false => Kᗮ)\n (match false with\n | true => K\n | false => Kᗮ)",
"tactic": "exact Submodule.isCompl_orthogonal_of_completeSpace.codisjoint"
},
{
"state_after": "ι : Type ?u.670433\n𝕜 : Type u_1\ninst✝⁴ : IsROrC 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ncplt : CompleteSpace E\nG : ι → Type ?u.670584\ninst✝¹ : (i : ι) → NormedAddCommGroup (G i)\ninst✝ : (i : ι) → InnerProductSpace 𝕜 (G i)\nV : (i : ι) → G i →ₗᵢ[𝕜] E\nF : ι → Submodule 𝕜 E\nK : Submodule 𝕜 E\nhK : CompleteSpace { x // x ∈ K }\nb : Bool\n⊢ CompleteSpace { x // x ∈ bif b then K else Kᗮ }",
"state_before": "ι : Type ?u.670433\n𝕜 : Type u_1\ninst✝⁴ : IsROrC 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ncplt : CompleteSpace E\nG : ι → Type ?u.670584\ninst✝¹ : (i : ι) → NormedAddCommGroup (G i)\ninst✝ : (i : ι) → InnerProductSpace 𝕜 (G i)\nV : (i : ι) → G i →ₗᵢ[𝕜] E\nF : ι → Submodule 𝕜 E\nK : Submodule 𝕜 E\nhK : CompleteSpace { x // x ∈ K }\n⊢ ∀ (b : Bool), CompleteSpace { x // x ∈ bif b then K else Kᗮ }",
"tactic": "intro b"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.670433\n𝕜 : Type u_1\ninst✝⁴ : IsROrC 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ncplt : CompleteSpace E\nG : ι → Type ?u.670584\ninst✝¹ : (i : ι) → NormedAddCommGroup (G i)\ninst✝ : (i : ι) → InnerProductSpace 𝕜 (G i)\nV : (i : ι) → G i →ₗᵢ[𝕜] E\nF : ι → Submodule 𝕜 E\nK : Submodule 𝕜 E\nhK : CompleteSpace { x // x ∈ K }\nb : Bool\n⊢ CompleteSpace { x // x ∈ bif b then K else Kᗮ }",
"tactic": "cases b <;> first | exact instOrthogonalCompleteSpace K | assumption"
},
{
"state_after": "case true\nι : Type ?u.670433\n𝕜 : Type u_1\ninst✝⁴ : IsROrC 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ncplt : CompleteSpace E\nG : ι → Type ?u.670584\ninst✝¹ : (i : ι) → NormedAddCommGroup (G i)\ninst✝ : (i : ι) → InnerProductSpace 𝕜 (G i)\nV : (i : ι) → G i →ₗᵢ[𝕜] E\nF : ι → Submodule 𝕜 E\nK : Submodule 𝕜 E\nhK : CompleteSpace { x // x ∈ K }\n⊢ CompleteSpace { x // x ∈ bif true then K else Kᗮ }",
"state_before": "case true\nι : Type ?u.670433\n𝕜 : Type u_1\ninst✝⁴ : IsROrC 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ncplt : CompleteSpace E\nG : ι → Type ?u.670584\ninst✝¹ : (i : ι) → NormedAddCommGroup (G i)\ninst✝ : (i : ι) → InnerProductSpace 𝕜 (G i)\nV : (i : ι) → G i →ₗᵢ[𝕜] E\nF : ι → Submodule 𝕜 E\nK : Submodule 𝕜 E\nhK : CompleteSpace { x // x ∈ K }\n⊢ CompleteSpace { x // x ∈ bif true then K else Kᗮ }",
"tactic": "exact instOrthogonalCompleteSpace K"
},
{
"state_after": "no goals",
"state_before": "case true\nι : Type ?u.670433\n𝕜 : Type u_1\ninst✝⁴ : IsROrC 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ncplt : CompleteSpace E\nG : ι → Type ?u.670584\ninst✝¹ : (i : ι) → NormedAddCommGroup (G i)\ninst✝ : (i : ι) → InnerProductSpace 𝕜 (G i)\nV : (i : ι) → G i →ₗᵢ[𝕜] E\nF : ι → Submodule 𝕜 E\nK : Submodule 𝕜 E\nhK : CompleteSpace { x // x ∈ K }\n⊢ CompleteSpace { x // x ∈ bif true then K else Kᗮ }",
"tactic": "assumption"
}
] |
[
399,
65
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
390,
1
] |
Mathlib/Algebra/GroupWithZero/Divisibility.lean
|
eq_of_forall_dvd'
|
[] |
[
140,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
139,
1
] |
Mathlib/Order/Hom/CompleteLattice.lean
|
sInfHom.top_apply
|
[] |
[
521,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
520,
1
] |
Mathlib/Algebra/Order/Sub/Canonical.lean
|
tsub_pos_iff_not_le
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝² : CanonicallyOrderedAddMonoid α\ninst✝¹ : Sub α\ninst✝ : OrderedSub α\na b c d : α\n⊢ 0 < a - b ↔ ¬a ≤ b",
"tactic": "rw [pos_iff_ne_zero, Ne.def, tsub_eq_zero_iff_le]"
}
] |
[
353,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
352,
1
] |
Mathlib/Topology/Order/Basic.lean
|
eventually_gt_of_tendsto_gt
|
[] |
[
367,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
365,
1
] |
Mathlib/RingTheory/IsTensorProduct.lean
|
IsTensorProduct.map_eq
|
[
{
"state_after": "R : Type u_1\ninst✝¹⁴ : CommRing R\nM₁ : Type u_2\nM₂ : Type u_3\nM : Type u_4\nM' : Type ?u.102460\ninst✝¹³ : AddCommMonoid M₁\ninst✝¹² : AddCommMonoid M₂\ninst✝¹¹ : AddCommMonoid M\ninst✝¹⁰ : AddCommMonoid M'\ninst✝⁹ : Module R M₁\ninst✝⁸ : Module R M₂\ninst✝⁷ : Module R M\ninst✝⁶ : Module R M'\nf : M₁ →ₗ[R] M₂ →ₗ[R] M\nN₁ : Type u_5\nN₂ : Type u_6\nN : Type u_7\ninst✝⁵ : AddCommMonoid N₁\ninst✝⁴ : AddCommMonoid N₂\ninst✝³ : AddCommMonoid N\ninst✝² : Module R N₁\ninst✝¹ : Module R N₂\ninst✝ : Module R N\ng : N₁ →ₗ[R] N₂ →ₗ[R] N\nhf : IsTensorProduct f\nhg : IsTensorProduct g\ni₁ : M₁ →ₗ[R] N₁\ni₂ : M₂ →ₗ[R] N₂\nx₁ : M₁\nx₂ : M₂\n⊢ ↑(LinearMap.comp (↑(equiv hg)) (LinearMap.comp (TensorProduct.map i₁ i₂) ↑(LinearEquiv.symm (equiv hf))))\n (↑(↑f x₁) x₂) =\n ↑(↑g (↑i₁ x₁)) (↑i₂ x₂)",
"state_before": "R : Type u_1\ninst✝¹⁴ : CommRing R\nM₁ : Type u_2\nM₂ : Type u_3\nM : Type u_4\nM' : Type ?u.102460\ninst✝¹³ : AddCommMonoid M₁\ninst✝¹² : AddCommMonoid M₂\ninst✝¹¹ : AddCommMonoid M\ninst✝¹⁰ : AddCommMonoid M'\ninst✝⁹ : Module R M₁\ninst✝⁸ : Module R M₂\ninst✝⁷ : Module R M\ninst✝⁶ : Module R M'\nf : M₁ →ₗ[R] M₂ →ₗ[R] M\nN₁ : Type u_5\nN₂ : Type u_6\nN : Type u_7\ninst✝⁵ : AddCommMonoid N₁\ninst✝⁴ : AddCommMonoid N₂\ninst✝³ : AddCommMonoid N\ninst✝² : Module R N₁\ninst✝¹ : Module R N₂\ninst✝ : Module R N\ng : N₁ →ₗ[R] N₂ →ₗ[R] N\nhf : IsTensorProduct f\nhg : IsTensorProduct g\ni₁ : M₁ →ₗ[R] N₁\ni₂ : M₂ →ₗ[R] N₂\nx₁ : M₁\nx₂ : M₂\n⊢ ↑(map hf hg i₁ i₂) (↑(↑f x₁) x₂) = ↑(↑g (↑i₁ x₁)) (↑i₂ x₂)",
"tactic": "delta IsTensorProduct.map"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝¹⁴ : CommRing R\nM₁ : Type u_2\nM₂ : Type u_3\nM : Type u_4\nM' : Type ?u.102460\ninst✝¹³ : AddCommMonoid M₁\ninst✝¹² : AddCommMonoid M₂\ninst✝¹¹ : AddCommMonoid M\ninst✝¹⁰ : AddCommMonoid M'\ninst✝⁹ : Module R M₁\ninst✝⁸ : Module R M₂\ninst✝⁷ : Module R M\ninst✝⁶ : Module R M'\nf : M₁ →ₗ[R] M₂ →ₗ[R] M\nN₁ : Type u_5\nN₂ : Type u_6\nN : Type u_7\ninst✝⁵ : AddCommMonoid N₁\ninst✝⁴ : AddCommMonoid N₂\ninst✝³ : AddCommMonoid N\ninst✝² : Module R N₁\ninst✝¹ : Module R N₂\ninst✝ : Module R N\ng : N₁ →ₗ[R] N₂ →ₗ[R] N\nhf : IsTensorProduct f\nhg : IsTensorProduct g\ni₁ : M₁ →ₗ[R] N₁\ni₂ : M₂ →ₗ[R] N₂\nx₁ : M₁\nx₂ : M₂\n⊢ ↑(LinearMap.comp (↑(equiv hg)) (LinearMap.comp (TensorProduct.map i₁ i₂) ↑(LinearEquiv.symm (equiv hf))))\n (↑(↑f x₁) x₂) =\n ↑(↑g (↑i₁ x₁)) (↑i₂ x₂)",
"tactic": "simp"
}
] |
[
122,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
119,
1
] |
Mathlib/RingTheory/Ideal/Basic.lean
|
Ideal.span_singleton_one
|
[] |
[
163,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
162,
1
] |
Mathlib/Data/Set/Function.lean
|
Set.piecewise_compl
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.83376\nγ : Type ?u.83379\nι : Sort ?u.83382\nπ : α → Type ?u.83387\nδ : α → Sort u_2\ns : Set α\nf g : (i : α) → δ i\ninst✝¹ : (j : α) → Decidable (j ∈ s)\ninst✝ : (i : α) → Decidable (i ∈ sᶜ)\nx : α\nhx : x ∈ s\n⊢ piecewise (sᶜ) f g x = piecewise s g f x",
"tactic": "simp [hx]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.83376\nγ : Type ?u.83379\nι : Sort ?u.83382\nπ : α → Type ?u.83387\nδ : α → Sort u_2\ns : Set α\nf g : (i : α) → δ i\ninst✝¹ : (j : α) → Decidable (j ∈ s)\ninst✝ : (i : α) → Decidable (i ∈ sᶜ)\nx : α\nhx : ¬x ∈ s\n⊢ piecewise (sᶜ) f g x = piecewise s g f x",
"tactic": "simp [hx]"
}
] |
[
1434,
68
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1433,
1
] |
Mathlib/Geometry/Euclidean/Angle/Unoriented/Basic.lean
|
InnerProductGeometry.angle_neg_right
|
[
{
"state_after": "V : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nx✝ y✝ x y : V\n⊢ arccos (inner x (-y) / (‖x‖ * ‖-y‖)) = π - arccos (inner x y / (‖x‖ * ‖y‖))",
"state_before": "V : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nx✝ y✝ x y : V\n⊢ angle x (-y) = π - angle x y",
"tactic": "unfold angle"
},
{
"state_after": "no goals",
"state_before": "V : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nx✝ y✝ x y : V\n⊢ arccos (inner x (-y) / (‖x‖ * ‖-y‖)) = π - arccos (inner x y / (‖x‖ * ‖y‖))",
"tactic": "rw [← Real.arccos_neg, norm_neg, inner_neg_right, neg_div]"
}
] |
[
110,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
108,
1
] |
Mathlib/Algebra/Order/Kleene.lean
|
Prod.kstar_def
|
[] |
[
312,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
311,
1
] |
Mathlib/MeasureTheory/Measure/NullMeasurable.lean
|
Measurable.nullMeasurable
|
[] |
[
422,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
421,
11
] |
Mathlib/LinearAlgebra/Matrix/Transvection.lean
|
Matrix.Pivot.listTransvecCol_mul_last_row_drop
|
[
{
"state_after": "case refine'_1\nn : Type ?u.140137\np : Type ?u.140140\nR : Type u₂\n𝕜 : Type u_1\ninst✝³ : Field 𝕜\ninst✝² : DecidableEq n\ninst✝¹ : DecidableEq p\ninst✝ : CommRing R\nr : ℕ\nM : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜\ni : Fin r ⊕ Unit\nk : ℕ\nhk : k ≤ r\n⊢ ∀ (k_1 : ℕ),\n k_1 < r →\n k ≤ k_1 →\n (List.prod (List.drop (k_1 + 1) (listTransvecCol M)) ⬝ M) (inr ()) i = M (inr ()) i →\n (List.prod (List.drop k_1 (listTransvecCol M)) ⬝ M) (inr ()) i = M (inr ()) i\n\ncase refine'_2\nn : Type ?u.140137\np : Type ?u.140140\nR : Type u₂\n𝕜 : Type u_1\ninst✝³ : Field 𝕜\ninst✝² : DecidableEq n\ninst✝¹ : DecidableEq p\ninst✝ : CommRing R\nr : ℕ\nM : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜\ni : Fin r ⊕ Unit\nk : ℕ\nhk : k ≤ r\n⊢ (List.prod (List.drop r (listTransvecCol M)) ⬝ M) (inr ()) i = M (inr ()) i",
"state_before": "n : Type ?u.140137\np : Type ?u.140140\nR : Type u₂\n𝕜 : Type u_1\ninst✝³ : Field 𝕜\ninst✝² : DecidableEq n\ninst✝¹ : DecidableEq p\ninst✝ : CommRing R\nr : ℕ\nM : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜\ni : Fin r ⊕ Unit\nk : ℕ\nhk : k ≤ r\n⊢ (List.prod (List.drop k (listTransvecCol M)) ⬝ M) (inr ()) i = M (inr ()) i",
"tactic": "refine' Nat.decreasingInduction' _ hk _"
},
{
"state_after": "case refine'_1\nn✝ : Type ?u.140137\np : Type ?u.140140\nR : Type u₂\n𝕜 : Type u_1\ninst✝³ : Field 𝕜\ninst✝² : DecidableEq n✝\ninst✝¹ : DecidableEq p\ninst✝ : CommRing R\nr : ℕ\nM : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜\ni : Fin r ⊕ Unit\nk : ℕ\nhk : k ≤ r\nn : ℕ\nhn : n < r\na✝ : k ≤ n\nIH : (List.prod (List.drop (n + 1) (listTransvecCol M)) ⬝ M) (inr ()) i = M (inr ()) i\n⊢ (List.prod (List.drop n (listTransvecCol M)) ⬝ M) (inr ()) i = M (inr ()) i",
"state_before": "case refine'_1\nn : Type ?u.140137\np : Type ?u.140140\nR : Type u₂\n𝕜 : Type u_1\ninst✝³ : Field 𝕜\ninst✝² : DecidableEq n\ninst✝¹ : DecidableEq p\ninst✝ : CommRing R\nr : ℕ\nM : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜\ni : Fin r ⊕ Unit\nk : ℕ\nhk : k ≤ r\n⊢ ∀ (k_1 : ℕ),\n k_1 < r →\n k ≤ k_1 →\n (List.prod (List.drop (k_1 + 1) (listTransvecCol M)) ⬝ M) (inr ()) i = M (inr ()) i →\n (List.prod (List.drop k_1 (listTransvecCol M)) ⬝ M) (inr ()) i = M (inr ()) i",
"tactic": "intro n hn _ IH"
},
{
"state_after": "case refine'_1\nn✝ : Type ?u.140137\np : Type ?u.140140\nR : Type u₂\n𝕜 : Type u_1\ninst✝³ : Field 𝕜\ninst✝² : DecidableEq n✝\ninst✝¹ : DecidableEq p\ninst✝ : CommRing R\nr : ℕ\nM : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜\ni : Fin r ⊕ Unit\nk : ℕ\nhk : k ≤ r\nn : ℕ\nhn : n < r\na✝ : k ≤ n\nIH : (List.prod (List.drop (n + 1) (listTransvecCol M)) ⬝ M) (inr ()) i = M (inr ()) i\nhn' : n < List.length (listTransvecCol M)\n⊢ (List.prod (List.drop n (listTransvecCol M)) ⬝ M) (inr ()) i = M (inr ()) i",
"state_before": "case refine'_1\nn✝ : Type ?u.140137\np : Type ?u.140140\nR : Type u₂\n𝕜 : Type u_1\ninst✝³ : Field 𝕜\ninst✝² : DecidableEq n✝\ninst✝¹ : DecidableEq p\ninst✝ : CommRing R\nr : ℕ\nM : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜\ni : Fin r ⊕ Unit\nk : ℕ\nhk : k ≤ r\nn : ℕ\nhn : n < r\na✝ : k ≤ n\nIH : (List.prod (List.drop (n + 1) (listTransvecCol M)) ⬝ M) (inr ()) i = M (inr ()) i\n⊢ (List.prod (List.drop n (listTransvecCol M)) ⬝ M) (inr ()) i = M (inr ()) i",
"tactic": "have hn' : n < (listTransvecCol M).length := by simpa [listTransvecCol] using hn"
},
{
"state_after": "case refine'_1\nn✝ : Type ?u.140137\np : Type ?u.140140\nR : Type u₂\n𝕜 : Type u_1\ninst✝³ : Field 𝕜\ninst✝² : DecidableEq n✝\ninst✝¹ : DecidableEq p\ninst✝ : CommRing R\nr : ℕ\nM : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜\ni : Fin r ⊕ Unit\nk : ℕ\nhk : k ≤ r\nn : ℕ\nhn : n < r\na✝ : k ≤ n\nIH : (List.prod (List.drop (n + 1) (listTransvecCol M)) ⬝ M) (inr ()) i = M (inr ()) i\nhn' : n < List.length (listTransvecCol M)\n⊢ (List.prod\n (List.get (listTransvecCol M) { val := n, isLt := hn' } ::\n List.drop (↑{ val := n, isLt := hn' } + 1) (listTransvecCol M)) ⬝\n M)\n (inr ()) i =\n M (inr ()) i",
"state_before": "case refine'_1\nn✝ : Type ?u.140137\np : Type ?u.140140\nR : Type u₂\n𝕜 : Type u_1\ninst✝³ : Field 𝕜\ninst✝² : DecidableEq n✝\ninst✝¹ : DecidableEq p\ninst✝ : CommRing R\nr : ℕ\nM : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜\ni : Fin r ⊕ Unit\nk : ℕ\nhk : k ≤ r\nn : ℕ\nhn : n < r\na✝ : k ≤ n\nIH : (List.prod (List.drop (n + 1) (listTransvecCol M)) ⬝ M) (inr ()) i = M (inr ()) i\nhn' : n < List.length (listTransvecCol M)\n⊢ (List.prod (List.drop n (listTransvecCol M)) ⬝ M) (inr ()) i = M (inr ()) i",
"tactic": "rw [← @List.cons_get_drop_succ _ _ ⟨n, hn'⟩]"
},
{
"state_after": "no goals",
"state_before": "case refine'_1\nn✝ : Type ?u.140137\np : Type ?u.140140\nR : Type u₂\n𝕜 : Type u_1\ninst✝³ : Field 𝕜\ninst✝² : DecidableEq n✝\ninst✝¹ : DecidableEq p\ninst✝ : CommRing R\nr : ℕ\nM : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜\ni : Fin r ⊕ Unit\nk : ℕ\nhk : k ≤ r\nn : ℕ\nhn : n < r\na✝ : k ≤ n\nIH : (List.prod (List.drop (n + 1) (listTransvecCol M)) ⬝ M) (inr ()) i = M (inr ()) i\nhn' : n < List.length (listTransvecCol M)\n⊢ (List.prod\n (List.get (listTransvecCol M) { val := n, isLt := hn' } ::\n List.drop (↑{ val := n, isLt := hn' } + 1) (listTransvecCol M)) ⬝\n M)\n (inr ()) i =\n M (inr ()) i",
"tactic": "simpa [listTransvecCol, Matrix.mul_assoc]"
},
{
"state_after": "no goals",
"state_before": "n✝ : Type ?u.140137\np : Type ?u.140140\nR : Type u₂\n𝕜 : Type u_1\ninst✝³ : Field 𝕜\ninst✝² : DecidableEq n✝\ninst✝¹ : DecidableEq p\ninst✝ : CommRing R\nr : ℕ\nM : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜\ni : Fin r ⊕ Unit\nk : ℕ\nhk : k ≤ r\nn : ℕ\nhn : n < r\na✝ : k ≤ n\nIH : (List.prod (List.drop (n + 1) (listTransvecCol M)) ⬝ M) (inr ()) i = M (inr ()) i\n⊢ n < List.length (listTransvecCol M)",
"tactic": "simpa [listTransvecCol] using hn"
},
{
"state_after": "no goals",
"state_before": "case refine'_2\nn : Type ?u.140137\np : Type ?u.140140\nR : Type u₂\n𝕜 : Type u_1\ninst✝³ : Field 𝕜\ninst✝² : DecidableEq n\ninst✝¹ : DecidableEq p\ninst✝ : CommRing R\nr : ℕ\nM : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜\ni : Fin r ⊕ Unit\nk : ℕ\nhk : k ≤ r\n⊢ (List.prod (List.drop r (listTransvecCol M)) ⬝ M) (inr ()) i = M (inr ()) i",
"tactic": "simp only [listTransvecCol, List.length_ofFn, le_refl, List.drop_eq_nil_of_le, List.prod_nil,\n Matrix.one_mul]"
}
] |
[
369,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
359,
1
] |
Mathlib/Order/BoundedOrder.lean
|
max_bot_right
|
[] |
[
855,
13
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
854,
1
] |
Mathlib/LinearAlgebra/Matrix/NonsingularInverse.lean
|
Matrix.isUnit_diagonal
|
[
{
"state_after": "no goals",
"state_before": "l : Type ?u.338640\nm : Type u\nn : Type u'\nα : Type v\ninst✝² : Fintype n\ninst✝¹ : DecidableEq n\ninst✝ : CommRing α\nA B : Matrix n n α\nv : n → α\n⊢ IsUnit (diagonal v) ↔ IsUnit v",
"tactic": "simp only [← nonempty_invertible_iff_isUnit,\n (diagonalInvertibleEquivInvertible v).nonempty_congr]"
}
] |
[
565,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
563,
1
] |
Mathlib/Algebra/Ring/Commute.lean
|
Commute.mul_self_sub_mul_self_eq'
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nR : Type x\ninst✝ : NonUnitalNonAssocRing R\na b : R\nh : Commute a b\n⊢ a * a - b * b = (a - b) * (a + b)",
"tactic": "rw [mul_add, sub_mul, sub_mul, h.eq, sub_add_sub_cancel]"
}
] |
[
81,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
79,
1
] |
Mathlib/Topology/Algebra/Monoid.lean
|
continuous_one
|
[] |
[
36,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
35,
1
] |
Mathlib/Data/Real/ENNReal.lean
|
ENNReal.eq_inv_of_mul_eq_one_left
|
[
{
"state_after": "α : Type ?u.297877\nβ : Type ?u.297880\na b c d : ℝ≥0∞\nr p q : ℝ≥0\nh : a * b = 1\n⊢ b ≠ ⊤",
"state_before": "α : Type ?u.297877\nβ : Type ?u.297880\na b c d : ℝ≥0∞\nr p q : ℝ≥0\nh : a * b = 1\n⊢ a = b⁻¹",
"tactic": "rw [← mul_one a, ← ENNReal.mul_inv_cancel (right_ne_zero_of_mul_eq_one h), ← mul_assoc, h,\n one_mul]"
},
{
"state_after": "α : Type ?u.297877\nβ : Type ?u.297880\na c d : ℝ≥0∞\nr p q : ℝ≥0\nh : a * ⊤ = 1\n⊢ False",
"state_before": "α : Type ?u.297877\nβ : Type ?u.297880\na b c d : ℝ≥0∞\nr p q : ℝ≥0\nh : a * b = 1\n⊢ b ≠ ⊤",
"tactic": "rintro rfl"
},
{
"state_after": "no goals",
"state_before": "α : Type ?u.297877\nβ : Type ?u.297880\na c d : ℝ≥0∞\nr p q : ℝ≥0\nh : a * ⊤ = 1\n⊢ False",
"tactic": "simp [left_ne_zero_of_mul_eq_one h] at h"
}
] |
[
1653,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1649,
11
] |
Mathlib/LinearAlgebra/Matrix/Transvection.lean
|
Matrix.TransvectionStruct.det_toMatrix_prod
|
[
{
"state_after": "case nil\nn : Type u_1\np : Type ?u.23948\nR : Type u₂\n𝕜 : Type u_2\ninst✝⁴ : Field 𝕜\ninst✝³ : DecidableEq n\ninst✝² : DecidableEq p\ninst✝¹ : CommRing R\ni j : n\ninst✝ : Fintype n\n⊢ det (List.prod (List.map toMatrix [])) = 1\n\ncase cons\nn : Type u_1\np : Type ?u.23948\nR : Type u₂\n𝕜 : Type u_2\ninst✝⁴ : Field 𝕜\ninst✝³ : DecidableEq n\ninst✝² : DecidableEq p\ninst✝¹ : CommRing R\ni j : n\ninst✝ : Fintype n\nt : TransvectionStruct n 𝕜\nL : List (TransvectionStruct n 𝕜)\nIH : det (List.prod (List.map toMatrix L)) = 1\n⊢ det (List.prod (List.map toMatrix (t :: L))) = 1",
"state_before": "n : Type u_1\np : Type ?u.23948\nR : Type u₂\n𝕜 : Type u_2\ninst✝⁴ : Field 𝕜\ninst✝³ : DecidableEq n\ninst✝² : DecidableEq p\ninst✝¹ : CommRing R\ni j : n\ninst✝ : Fintype n\nL : List (TransvectionStruct n 𝕜)\n⊢ det (List.prod (List.map toMatrix L)) = 1",
"tactic": "induction' L with t L IH"
},
{
"state_after": "no goals",
"state_before": "case nil\nn : Type u_1\np : Type ?u.23948\nR : Type u₂\n𝕜 : Type u_2\ninst✝⁴ : Field 𝕜\ninst✝³ : DecidableEq n\ninst✝² : DecidableEq p\ninst✝¹ : CommRing R\ni j : n\ninst✝ : Fintype n\n⊢ det (List.prod (List.map toMatrix [])) = 1",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "case cons\nn : Type u_1\np : Type ?u.23948\nR : Type u₂\n𝕜 : Type u_2\ninst✝⁴ : Field 𝕜\ninst✝³ : DecidableEq n\ninst✝² : DecidableEq p\ninst✝¹ : CommRing R\ni j : n\ninst✝ : Fintype n\nt : TransvectionStruct n 𝕜\nL : List (TransvectionStruct n 𝕜)\nIH : det (List.prod (List.map toMatrix L)) = 1\n⊢ det (List.prod (List.map toMatrix (t :: L))) = 1",
"tactic": "simp [IH]"
}
] |
[
192,
14
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
188,
1
] |
Mathlib/Tactic/Linarith/Lemmas.lean
|
Linarith.lt_of_eq_of_lt
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : OrderedSemiring α\na b : α\nha : a = 0\nhb : b < 0\n⊢ a + b < 0",
"tactic": "simp [*]"
}
] |
[
34,
11
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
33,
1
] |
Mathlib/Data/Seq/WSeq.lean
|
Stream'.WSeq.get?_add
|
[] |
[
747,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
746,
1
] |
Mathlib/Logic/Lemmas.lean
|
ite_dite_distrib_right
|
[] |
[
53,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
51,
1
] |
Mathlib/Topology/MetricSpace/Holder.lean
|
HolderWith.ediam_image_le
|
[] |
[
227,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
224,
1
] |
Mathlib/RingTheory/PowerSeries/Basic.lean
|
PowerSeries.X_eq
|
[] |
[
1425,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1424,
1
] |
Mathlib/LinearAlgebra/Matrix/NonsingularInverse.lean
|
Matrix.det_smul_inv_mulVec_eq_cramer
|
[
{
"state_after": "no goals",
"state_before": "l : Type ?u.360583\nm : Type u\nn : Type u'\nα : Type v\ninst✝² : Fintype n\ninst✝¹ : DecidableEq n\ninst✝ : CommRing α\nA✝ B A : Matrix n n α\nb : n → α\nh : IsUnit (det A)\n⊢ det A • mulVec A⁻¹ b = ↑(cramer A) b",
"tactic": "rw [cramer_eq_adjugate_mulVec, A.nonsing_inv_apply h, ← smul_mulVec_assoc, smul_smul,\n h.mul_val_inv, one_smul]"
}
] |
[
607,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
604,
1
] |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.