module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.RingTheory.TwoSidedIdeal.Kernel | {
"line": 63,
"column": 19
} | {
"line": 63,
"column": 30
} | {
"line": 63,
"column": 31
} | [
{
"pp": "R : Type u_4\ninst✝ : NonAssocRing R\nI : TwoSidedIdeal R\nx✝ : R\nh : x✝ ∈ ker I.ringCon.mk'\n⊢ x✝ ∈ I",
"ppTerm": "?m.42",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u_4\ninst✝ : NonAssocRing R\nI : TwoSidedIdeal R\nx✝ : R\nh : x✝ ∈ ker I.ringCon.mk'\n⊢ x✝ ∈ I"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.TwoSidedIdeal.Kernel | {
"line": 64,
"column": 58
} | {
"line": 64,
"column": 69
} | {
"line": 64,
"column": 70
} | [
{
"pp": "R : Type u_4\ninst✝ : NonAssocRing R\nI : TwoSidedIdeal R\nx✝ : R\nh : x✝ ∈ I\n⊢ x✝ - 0 ∈ I",
"ppTerm": "?m.40",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"TwoSidedIdeal",
"sub_zero",
"NonUnitalNonAssocRing.toAddCommGroup",
"HSub.hSub",
... | [
"R : Type u_4\ninst✝ : NonAssocRing R\nI : TwoSidedIdeal R\nx✝ : R\nh : x✝ ∈ I\n⊢ x✝ ∈ I"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Dimension.Free | {
"line": 362,
"column": 4
} | {
"line": 362,
"column": 15
} | {
"line": 362,
"column": 16
} | [
{
"pp": "case h\nR : Type u_2\ninst✝⁴ : CommSemiring R\ninst✝³ : StrongRankCondition R\nM : Type u_3\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ninst✝ : Free R M\nd1 : finrank R M = 1\nu : M →ₗ[R] M\ne : R ≃ₗ[R] M := ⋯.some\nc : R := e.symm (u (e 1))\nhc : c = e.symm (u (e 1))\nthis : u = c • LinearMap.id\n... | [
"case h\nR : Type u_2\ninst✝⁴ : CommSemiring R\ninst✝³ : StrongRankCondition R\nM : Type u_3\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ninst✝ : Free R M\nd1 : finrank R M = 1\nu : M →ₗ[R] M\ne : R ≃ₗ[R] M := ⋯.some\nc : R := e.symm (u (e 1))\nhc : c = e.symm (u (e 1))\nthis : u = c • LinearMap.id\nd : R\nhcd :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.SimpleRing.Basic | {
"line": 55,
"column": 4
} | {
"line": 55,
"column": 37
} | {
"line": 55,
"column": 38
} | [
{
"pp": "R : Type u_1\ninst✝¹ : NonUnitalNonAssocRing R\nA : Type u_2\ninst✝ : DivisionRing A\nI : TwoSidedIdeal A\nH : ¬I = ⊥\nx : A\nhx1 : x ∈ I\nhx2 : x ≠ 0\n⊢ 1 ∈ I",
"ppTerm": "?m.53",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u_1\ninst✝¹ : NonUnitalNonAssocRing R\nA : Type u_2\ninst✝ : DivisionRing A\nI : TwoSidedIdeal A\nH : ¬I = ⊥\nx : A\nhx1 : x ∈ I\nhx2 : x ≠ 0\n⊢ 1 ∈ I"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.TwoSidedIdeal.Lattice | {
"line": 141,
"column": 39
} | {
"line": 141,
"column": 50
} | {
"line": 141,
"column": 51
} | [
{
"pp": "R : Type u_2\ninst✝ : NonAssocRing R\nI : TwoSidedIdeal R\nh : 1 ∈ I\nx : R\nx✝ : x ∈ ⊤\n⊢ x ∈ I",
"ppTerm": "?m.21",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u_2\ninst✝ : NonAssocRing R\nI : TwoSidedIdeal R\nh : 1 ∈ I\nx : R\nx✝ : x ∈ ⊤\n⊢ x ∈ I"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Algebra.Subalgebra.Lattice | {
"line": 180,
"column": 2
} | {
"line": 180,
"column": 13
} | {
"line": 180,
"column": 14
} | [
{
"pp": "R : Type u\nA : Type v\nB : Type w\ninst✝⁵ : CommSemiring R\ninst✝⁴ : Semiring A\ninst✝³ : Algebra R A\ninst✝² : Semiring B\ninst✝¹ : Algebra R B\nι : Sort u_1\ninst✝ : Nonempty ι\nf : A →ₐ[R] B\nhf : Function.Injective ⇑f\ns : ι → Subalgebra R A\n⊢ ↑(Subalgebra.map f (iInf s)) = ↑(⨅ i, Subalgebra.map ... | [
"R : Type u\nA : Type v\nB : Type w\ninst✝⁵ : CommSemiring R\ninst✝⁴ : Semiring A\ninst✝³ : Algebra R A\ninst✝² : Semiring B\ninst✝¹ : Algebra R B\nι : Sort u_1\ninst✝ : Nonempty ι\nf : A →ₐ[R] B\nhf : Function.Injective ⇑f\ns : ι → Subalgebra R A\n⊢ ⇑f '' ⋂ i, ↑(s i) = ⋂ i, ⇑f '' ↑(s i)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Algebra.Subalgebra.Basic | {
"line": 82,
"column": 23
} | {
"line": 82,
"column": 34
} | {
"line": 82,
"column": 35
} | [
{
"pp": "R' : Type u'\nR : Type u\nA : Type v\nB : Type w\nC : Type w'\ninst✝⁶ : CommSemiring R\ninst✝⁵ : Semiring A\ninst✝⁴ : Algebra R A\ninst✝³ : Semiring B\ninst✝² : Algebra R B\ninst✝¹ : Semiring C\ninst✝ : Algebra R C\ns : Set A\nh :\n (∀ {x y : A}, x ∈ s → y ∈ s → x + y ∈ s) ∧\n (∀ {x y : A}, x ∈ s →... | [
"R' : Type u'\nR : Type u\nA : Type v\nB : Type w\nC : Type w'\ninst✝⁶ : CommSemiring R\ninst✝⁵ : Semiring A\ninst✝⁴ : Algebra R A\ninst✝³ : Semiring B\ninst✝² : Algebra R B\ninst✝¹ : Semiring C\ninst✝ : Algebra R C\ns : Set A\nh :\n (∀ {x y : A}, x ∈ s → y ∈ s → x + y ∈ s) ∧\n (∀ {x y : A}, x ∈ s → y ∈ s → x *... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Algebra.Subalgebra.Basic | {
"line": 80,
"column": 24
} | {
"line": 80,
"column": 35
} | {
"line": 80,
"column": 36
} | [
{
"pp": "R' : Type u'\nR : Type u\nA : Type v\nB : Type w\nC : Type w'\ninst✝⁶ : CommSemiring R\ninst✝⁵ : Semiring A\ninst✝⁴ : Algebra R A\ninst✝³ : Semiring B\ninst✝² : Algebra R B\ninst✝¹ : Semiring C\ninst✝ : Algebra R C\ns : Set A\nh :\n (∀ {x y : A}, x ∈ s → y ∈ s → x + y ∈ s) ∧\n (∀ {x y : A}, x ∈ s →... | [
"R' : Type u'\nR : Type u\nA : Type v\nB : Type w\nC : Type w'\ninst✝⁶ : CommSemiring R\ninst✝⁵ : Semiring A\ninst✝⁴ : Algebra R A\ninst✝³ : Semiring B\ninst✝² : Algebra R B\ninst✝¹ : Semiring C\ninst✝ : Algebra R C\ns : Set A\nh :\n (∀ {x y : A}, x ∈ s → y ∈ s → x + y ∈ s) ∧\n (∀ {x y : A}, x ∈ s → y ∈ s → x *... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Algebra.Subalgebra.Lattice | {
"line": 269,
"column": 4
} | {
"line": 269,
"column": 66
} | {
"line": 269,
"column": 67
} | [
{
"pp": "R : Type u\nA : Type v\nB : Type w\ninst✝⁴ : CommSemiring R\ninst✝³ : Semiring A\ninst✝² : Algebra R A\ninst✝¹ : Semiring B\ninst✝ : Algebra R B\nf : A →ₐ[R] B\n⊢ Subalgebra.toSubmodule (Subalgebra.map f ⊥) = Subalgebra.toSubmodule ⊥",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [... | [
"R : Type u\nA : Type v\nB : Type w\ninst✝⁴ : CommSemiring R\ninst✝³ : Semiring A\ninst✝² : Algebra R A\ninst✝¹ : Semiring B\ninst✝ : Algebra R B\nf : A →ₐ[R] B\n⊢ Submodule.map f.toLinearMap 1 = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Algebra.Subalgebra.Lattice | {
"line": 406,
"column": 2
} | {
"line": 406,
"column": 37
} | {
"line": 406,
"column": 38
} | [
{
"pp": "R : Type u_1\nA : Type u_2\nB : Type u_3\ninst✝⁴ : CommSemiring R\ninst✝³ : Semiring A\ninst✝² : Algebra R A\ninst✝¹ : Semiring B\ninst✝ : Algebra R B\nf : A →ₐ[R] B\nS : Subalgebra R B\nh : S ≤ f.range\n⊢ map f (comap f S) = S",
"ppTerm": "?m.40",
"assigned": false,
"usedConstants": [],
... | [
"R : Type u_1\nA : Type u_2\nB : Type u_3\ninst✝⁴ : CommSemiring R\ninst✝³ : Semiring A\ninst✝² : Algebra R A\ninst✝¹ : Semiring B\ninst✝ : Algebra R B\nf : A →ₐ[R] B\nS : Subalgebra R B\nh : S ≤ f.range\n⊢ map f (comap f S) = S"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Algebra.Subalgebra.Basic | {
"line": 893,
"column": 36
} | {
"line": 893,
"column": 71
} | {
"line": 893,
"column": 72
} | [
{
"pp": "R : Type u\nA : Type v\nB : Type w\ninst✝⁵ : CommSemiring R\ninst✝⁴ : Semiring A\ninst✝³ : Algebra R A\ninst✝² : Semiring B\ninst✝¹ : Algebra R B\nS✝ T U : Subalgebra R A\nh : S✝ ≤ T\nα : Type u_1\nβ : Type u_2\ninst✝ : IsDomain A\nS : Subalgebra R A\nr : ↥S\nhr : IsRegular r\n⊢ IsRegular ↑r",
"ppT... | [
"R : Type u\nA : Type v\nB : Type w\ninst✝⁵ : CommSemiring R\ninst✝⁴ : Semiring A\ninst✝³ : Algebra R A\ninst✝² : Semiring B\ninst✝¹ : Algebra R B\nS✝ T U : Subalgebra R A\nh : S✝ ≤ T\nα : Type u_1\nβ : Type u_2\ninst✝ : IsDomain A\nS : Subalgebra R A\nr : ↥S\nhr : IsRegular r\n⊢ ¬r = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Algebra.Subalgebra.Basic | {
"line": 999,
"column": 24
} | {
"line": 999,
"column": 35
} | {
"line": 999,
"column": 36
} | [
{
"pp": "R : Type u_1\ninst✝ : Ring R\nS : Subring R\ni✝ : ℤ\ni : ℕ\nih : (algebraMap ℤ R) ↑i ∈ S.carrier\n⊢ (algebraMap ℤ R) (↑i + 1) ∈ S.carrier",
"ppTerm": "?m.40",
"assigned": true,
"usedConstants": [
"Int.cast",
"Eq.mpr",
"Int.cast_natCast",
"RingHom.instRingHomClass",
... | [
"R : Type u_1\ninst✝ : Ring R\nS : Subring R\ni✝ : ℤ\ni : ℕ\nih : (algebraMap ℤ R) ↑i ∈ S.carrier\n⊢ ↑i + 1 ∈ S"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.DirectSum.Module | {
"line": 526,
"column": 42
} | {
"line": 526,
"column": 53
} | {
"line": 526,
"column": 54
} | [
{
"pp": "R : Type u\ninst✝² : Ring R\nι : Type v\ndec_ι : DecidableEq ι\nM : Type u_1\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nA : ι → Submodule R M\ni j : ι\nhij : i ≠ j\nh : Set.univ = {i, j}\nk : ι\n⊢ k = i ∨ k = j",
"ppTerm": "?m.32",
"assigned": false,
"usedConstants": [],
"usedFVars":... | [
"R : Type u\ninst✝² : Ring R\nι : Type v\ndec_ι : DecidableEq ι\nM : Type u_1\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nA : ι → Submodule R M\ni j : ι\nhij : i ≠ j\nh : Set.univ = {i, j}\nk : ι\n⊢ k = i ∨ k = j"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Algebra.Subalgebra.Lattice | {
"line": 650,
"column": 2
} | {
"line": 650,
"column": 13
} | {
"line": 650,
"column": 14
} | [
{
"pp": "R : Type uR\nA : Type uA\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nn : ℕ\n⊢ R[↑n] = ⊥",
"ppTerm": "?m.22",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type uR\nA : Type uA\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nn : ℕ\n⊢ R[↑n] = ⊥"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Algebra.Subalgebra.Lattice | {
"line": 746,
"column": 25
} | {
"line": 746,
"column": 45
} | {
"line": 746,
"column": 46
} | [
{
"pp": "case add\nR : Type uR\nA : Type uA\nB : Type uB\ninst✝⁴ : CommSemiring R\ninst✝³ : Semiring A\ninst✝² : Semiring B\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\ns : Set A\nx : A\nf : A →ₗ[R] B\nhf : ∀ (a₁ a₂ : A), f (a₁ * a₂) = f a₁ * f a₂\ny z : A\nhx✝ : y ∈ adjoin R s\nhy✝ : z ∈ adjoin R s\nhy : f y ∈ ... | [
"case add\nR : Type uR\nA : Type uA\nB : Type uB\ninst✝⁴ : CommSemiring R\ninst✝³ : Semiring A\ninst✝² : Semiring B\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\ns : Set A\nx : A\nf : A →ₗ[R] B\nhf : ∀ (a₁ a₂ : A), f (a₁ * a₂) = f a₁ * f a₂\ny z : A\nhx✝ : y ∈ adjoin R s\nhy✝ : z ∈ adjoin R s\nhy : f y ∈ adjoin R (⇑f... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Algebra.Subalgebra.Lattice | {
"line": 747,
"column": 25
} | {
"line": 747,
"column": 49
} | {
"line": 747,
"column": 50
} | [
{
"pp": "case mul\nR : Type uR\nA : Type uA\nB : Type uB\ninst✝⁴ : CommSemiring R\ninst✝³ : Semiring A\ninst✝² : Semiring B\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\ns : Set A\nx : A\nf : A →ₗ[R] B\nhf : ∀ (a₁ a₂ : A), f (a₁ * a₂) = f a₁ * f a₂\ny z : A\nhx✝ : y ∈ adjoin R s\nhy✝ : z ∈ adjoin R s\nhy : f y ∈ ... | [
"case mul\nR : Type uR\nA : Type uA\nB : Type uB\ninst✝⁴ : CommSemiring R\ninst✝³ : Semiring A\ninst✝² : Semiring B\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\ns : Set A\nx : A\nf : A →ₗ[R] B\nhf : ∀ (a₁ a₂ : A), f (a₁ * a₂) = f a₁ * f a₂\ny z : A\nhx✝ : y ∈ adjoin R s\nhy✝ : z ∈ adjoin R s\nhy : f y ∈ adjoin R (⇑f... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Algebra.Subalgebra.Lattice | {
"line": 828,
"column": 2
} | {
"line": 828,
"column": 13
} | {
"line": 828,
"column": 14
} | [
{
"pp": "R : Type uR\nA : Type uA\ninst✝² : CommRing R\ninst✝¹ : Ring A\ninst✝ : Algebra R A\nn : ℤ\n⊢ R[↑n] = ⊥",
"ppTerm": "?m.13",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type uR\nA : Type uA\ninst✝² : CommRing R\ninst✝¹ : Ring A\ninst✝ : Algebra R A\nn : ℤ\n⊢ R[↑n] = ⊥"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Algebra.Subalgebra.Lattice | {
"line": 832,
"column": 2
} | {
"line": 832,
"column": 13
} | {
"line": 832,
"column": 14
} | [
{
"pp": "R : Type uR\nA : Type uA\ninst✝² : CommRing R\ninst✝¹ : Ring A\ninst✝ : Algebra R A\nn : ℤ\ns : Set A\n⊢ adjoin R (insert (↑n) s) = adjoin R s",
"ppTerm": "?m.15",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type uR\nA : Type uA\ninst✝² : CommRing R\ninst✝¹ : Ring A\ninst✝ : Algebra R A\nn : ℤ\ns : Set A\n⊢ adjoin R (insert (↑n) s) = adjoin R s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.DirectSum.TensorProduct | {
"line": 52,
"column": 2
} | {
"line": 64,
"column": 50
} | {
"line": 66,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝¹⁶ : CommSemiring R\nS : Type ?u.6\ninst✝¹⁵ : Semiring S\ninst✝¹⁴ : Algebra R S\nι₁ : Type v₁\nι₂ : Type v₂\ninst✝¹³ : DecidableEq ι₁\ninst✝¹² : DecidableEq ι₂\nM₁ : ι₁ → Type w₁\nM₁' : Type w₁'\nM₂ : ι₂ → Type w₂\nM₂' : Type w₂'\ninst✝¹¹ : (i₁ : ι₁) → AddCommMonoid (M₁ i₁)\ninst✝¹⁰ : ... | [] | refine LinearEquiv.ofLinear ?toFun ?invFun ?left ?right
· exact AlgebraTensorModule.lift <|
toModule S _ _ fun i₁ => flip <| toModule R _ _ fun i₂ => flip <| AlgebraTensorModule.curry <|
DirectSum.lof S (ι₁ × ι₂) (fun i => M₁ i.1 ⊗[R] M₂ i.2) (i₁, i₂)
· exact toModule S _ _ fun i => AlgebraTensorModule.... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.DirectSum.TensorProduct | {
"line": 52,
"column": 2
} | {
"line": 64,
"column": 50
} | {
"line": 66,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝¹⁶ : CommSemiring R\nS : Type ?u.6\ninst✝¹⁵ : Semiring S\ninst✝¹⁴ : Algebra R S\nι₁ : Type v₁\nι₂ : Type v₂\ninst✝¹³ : DecidableEq ι₁\ninst✝¹² : DecidableEq ι₂\nM₁ : ι₁ → Type w₁\nM₁' : Type w₁'\nM₂ : ι₂ → Type w₂\nM₂' : Type w₂'\ninst✝¹¹ : (i₁ : ι₁) → AddCommMonoid (M₁ i₁)\ninst✝¹⁰ : ... | [] | refine LinearEquiv.ofLinear ?toFun ?invFun ?left ?right
· exact AlgebraTensorModule.lift <|
toModule S _ _ fun i₁ => flip <| toModule R _ _ fun i₂ => flip <| AlgebraTensorModule.curry <|
DirectSum.lof S (ι₁ × ι₂) (fun i => M₁ i.1 ⊗[R] M₂ i.2) (i₁, i₂)
· exact toModule S _ _ fun i => AlgebraTensorModule.... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Algebra.NonUnitalSubalgebra | {
"line": 672,
"column": 56
} | {
"line": 672,
"column": 77
} | {
"line": 672,
"column": 78
} | [
{
"pp": "R : Type u\nA : Type v\ninst✝⁴ : CommSemiring R\ninst✝³ : NonUnitalNonAssocSemiring A\ninst✝² : Module R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\ns : Set A\nx✝⁴ x✝³ y✝ : A\nhx✝ : x✝³ ∈ adjoin R s\nhy✝ : y✝ ∈ adjoin R s\nhpx : x✝³ ∈ span R ↑(Subsemigroup.closure s)\nhpy : y✝ ∈ span ... | [
"R : Type u\nA : Type v\ninst✝⁴ : CommSemiring R\ninst✝³ : NonUnitalNonAssocSemiring A\ninst✝² : Module R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\ns : Set A\nx✝⁴ x✝³ y✝ : A\nhx✝ : x✝³ ∈ adjoin R s\nhy✝ : y✝ ∈ adjoin R s\nhpx : x✝³ ∈ span R ↑(Subsemigroup.closure s)\nhpy : y✝ ∈ span R ↑(Subsemig... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Algebra.NonUnitalSubalgebra | {
"line": 673,
"column": 56
} | {
"line": 673,
"column": 77
} | {
"line": 673,
"column": 78
} | [
{
"pp": "R : Type u\nA : Type v\ninst✝⁴ : CommSemiring R\ninst✝³ : NonUnitalNonAssocSemiring A\ninst✝² : Module R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\ns : Set A\nx✝⁴ x✝³ y✝ : A\nhx✝ : x✝³ ∈ adjoin R s\nhy✝ : y✝ ∈ adjoin R s\nhpx : x✝³ ∈ span R ↑(Subsemigroup.closure s)\nhpy : y✝ ∈ span ... | [
"R : Type u\nA : Type v\ninst✝⁴ : CommSemiring R\ninst✝³ : NonUnitalNonAssocSemiring A\ninst✝² : Module R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\ns : Set A\nx✝⁴ x✝³ y✝ : A\nhx✝ : x✝³ ∈ adjoin R s\nhy✝ : y✝ ∈ adjoin R s\nhpx : x✝³ ∈ span R ↑(Subsemigroup.closure s)\nhpy : y✝ ∈ span R ↑(Subsemig... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Algebra.NonUnitalSubalgebra | {
"line": 674,
"column": 51
} | {
"line": 674,
"column": 79
} | {
"line": 674,
"column": 80
} | [
{
"pp": "R : Type u\nA : Type v\ninst✝⁴ : CommSemiring R\ninst✝³ : NonUnitalNonAssocSemiring A\ninst✝² : Module R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\ns : Set A\nx✝³ x✝² y✝ : A\nhx✝ : x✝² ∈ adjoin R s\nhy✝ : y✝ ∈ adjoin R s\nhpx : x✝² ∈ span R ↑(Subsemigroup.closure s)\nhpy : y✝ ∈ span ... | [
"R : Type u\nA : Type v\ninst✝⁴ : CommSemiring R\ninst✝³ : NonUnitalNonAssocSemiring A\ninst✝² : Module R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\ns : Set A\nx✝³ x✝² y✝ : A\nhx✝ : x✝² ∈ adjoin R s\nhy✝ : y✝ ∈ adjoin R s\nhpx : x✝² ∈ span R ↑(Subsemigroup.closure s)\nhpy : y✝ ∈ span R ↑(Subsemig... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Algebra.NonUnitalSubalgebra | {
"line": 675,
"column": 51
} | {
"line": 675,
"column": 78
} | {
"line": 675,
"column": 79
} | [
{
"pp": "R : Type u\nA : Type v\ninst✝⁴ : CommSemiring R\ninst✝³ : NonUnitalNonAssocSemiring A\ninst✝² : Module R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\ns : Set A\nx✝³ x✝² y✝ : A\nhx✝ : x✝² ∈ adjoin R s\nhy✝ : y✝ ∈ adjoin R s\nhpx : x✝² ∈ span R ↑(Subsemigroup.closure s)\nhpy : y✝ ∈ span ... | [
"R : Type u\nA : Type v\ninst✝⁴ : CommSemiring R\ninst✝³ : NonUnitalNonAssocSemiring A\ninst✝² : Module R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\ns : Set A\nx✝³ x✝² y✝ : A\nhx✝ : x✝² ∈ adjoin R s\nhy✝ : y✝ ∈ adjoin R s\nhpx : x✝² ∈ span R ↑(Subsemigroup.closure s)\nhpy : y✝ ∈ span R ↑(Subsemig... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Algebra.Subalgebra.Lattice | {
"line": 1000,
"column": 4
} | {
"line": 1003,
"column": 97
} | {
"line": 1005,
"column": 0
} | [
{
"pp": "case a\nR : Type uR\nA : Type uA\nB : Type uB\ninst✝⁴ : CommSemiring R\ninst✝³ : Ring A\ninst✝² : Algebra R A\ninst✝¹ : Ring B\ninst✝ : Algebra R B\nf : A →ₐ[R] B\nS : Subalgebra R A\n⊢ S ⊔ adjoin R (⇑f ⁻¹' {0}) ≤ comap f (map f S)",
"ppTerm": "?a✝",
"assigned": true,
"usedConstants": [
... | [] | rw [← map_le, Algebra.map_sup, f.map_adjoin]
apply le_of_eq
rw [sup_eq_left, Algebra.adjoin_le_iff]
exact (Set.image_preimage_subset f {0}).trans (Set.singleton_subset_iff.2 (S.map f).zero_mem) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Algebra.Subalgebra.Lattice | {
"line": 1000,
"column": 4
} | {
"line": 1003,
"column": 97
} | {
"line": 1005,
"column": 0
} | [
{
"pp": "case a\nR : Type uR\nA : Type uA\nB : Type uB\ninst✝⁴ : CommSemiring R\ninst✝³ : Ring A\ninst✝² : Algebra R A\ninst✝¹ : Ring B\ninst✝ : Algebra R B\nf : A →ₐ[R] B\nS : Subalgebra R A\n⊢ S ⊔ adjoin R (⇑f ⁻¹' {0}) ≤ comap f (map f S)",
"ppTerm": "?a✝",
"assigned": true,
"usedConstants": [
... | [] | rw [← map_le, Algebra.map_sup, f.map_adjoin]
apply le_of_eq
rw [sup_eq_left, Algebra.adjoin_le_iff]
exact (Set.image_preimage_subset f {0}).trans (Set.singleton_subset_iff.2 (S.map f).zero_mem) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Algebra.NonUnitalSubalgebra | {
"line": 813,
"column": 2
} | {
"line": 813,
"column": 13
} | {
"line": 813,
"column": 14
} | [
{
"pp": "F : Type u_1\nR : Type u\nA : Type v\nB : Type w\ninst✝¹¹ : CommSemiring R\ninst✝¹⁰ : NonUnitalNonAssocSemiring A\ninst✝⁹ : Module R A\ninst✝⁸ : NonUnitalNonAssocSemiring B\ninst✝⁷ : Module R B\ninst✝⁶ : FunLike F A B\ninst✝⁵ : NonUnitalAlgHomClass F R A B\ninst✝⁴ : IsScalarTower R A A\ninst✝³ : SMulCo... | [
"F : Type u_1\nR : Type u\nA : Type v\nB : Type w\ninst✝¹¹ : CommSemiring R\ninst✝¹⁰ : NonUnitalNonAssocSemiring A\ninst✝⁹ : Module R A\ninst✝⁸ : NonUnitalNonAssocSemiring B\ninst✝⁷ : Module R B\ninst✝⁶ : FunLike F A B\ninst✝⁵ : NonUnitalAlgHomClass F R A B\ninst✝⁴ : IsScalarTower R A A\ninst✝³ : SMulCommClass R A ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.TensorProduct.Basis | {
"line": 172,
"column": 12
} | {
"line": 172,
"column": 23
} | {
"line": 172,
"column": 24
} | [
{
"pp": "R : Type u_1\nM : Type u_3\nN : Type u_4\nκ : Type u_6\ninst✝⁴ : CommSemiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\ninst✝¹ : AddCommMonoid N\ninst✝ : Module R N\n𝒞 : Basis κ R N\nx : M ⊗[R] N\n⊢ ∃ b, (b.sum fun i m ↦ m ⊗ₜ[R] 𝒞 i) = x",
"ppTerm": "?m.50",
"assigned": false,
"us... | [
"R : Type u_1\nM : Type u_3\nN : Type u_4\nκ : Type u_6\ninst✝⁴ : CommSemiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\ninst✝¹ : AddCommMonoid N\ninst✝ : Module R N\n𝒞 : Basis κ R N\nx : M ⊗[R] N\n⊢ ∃ b, (b.sum fun i m ↦ m ⊗ₜ[R] 𝒞 i) = x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Star.Basic | {
"line": 516,
"column": 42
} | {
"line": 516,
"column": 53
} | {
"line": 516,
"column": 54
} | [
{
"pp": "R : Type u\ninst✝¹ : Mul R\ninst✝ : StarMul R\nx : R\nhx : IsLeftRegular x\na b : R\nh : (fun x_1 ↦ x_1 * star x) a = (fun x_1 ↦ x_1 * star x) b\n⊢ (fun x_1 ↦ x * x_1) (star a) = (fun x_1 ↦ x * x_1) (star b)",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"HMul.hMul",
... | [
"R : Type u\ninst✝¹ : Mul R\ninst✝ : StarMul R\nx : R\nhx : IsLeftRegular x\na b : R\nh : (fun x_1 ↦ x_1 * star x) a = (fun x_1 ↦ x_1 * star x) b\n⊢ x * star a = x * star b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Star.Basic | {
"line": 520,
"column": 42
} | {
"line": 520,
"column": 53
} | {
"line": 520,
"column": 54
} | [
{
"pp": "R : Type u\ninst✝¹ : Mul R\ninst✝ : StarMul R\nx : R\nhx : IsRightRegular x\na b : R\nh : (fun x_1 ↦ star x * x_1) a = (fun x_1 ↦ star x * x_1) b\n⊢ (fun x_1 ↦ x_1 * x) (star a) = (fun x_1 ↦ x_1 * x) (star b)",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"HMul.hMul",
... | [
"R : Type u\ninst✝¹ : Mul R\ninst✝ : StarMul R\nx : R\nhx : IsRightRegular x\na b : R\nh : (fun x_1 ↦ star x * x_1) a = (fun x_1 ↦ star x * x_1) b\n⊢ star a * x = star b * x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Star.BigOperators | {
"line": 33,
"column": 2
} | {
"line": 33,
"column": 39
} | {
"line": 33,
"column": 40
} | [
{
"pp": "R : Type u_1\nι : Type u_2\ninst✝¹ : AddCommMonoid R\ninst✝ : StarAddMonoid R\ns : Finset ι\nx : ι → R\nh : ∀ i ∈ s, IsSelfAdjoint (x i)\n⊢ IsSelfAdjoint (∑ i ∈ s, x i)",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"IsSelfAdjoint",
"congrArg",
"... | [
"R : Type u_1\nι : Type u_2\ninst✝¹ : AddCommMonoid R\ninst✝ : StarAddMonoid R\ns : Finset ι\nx : ι → R\nh : ∀ i ∈ s, IsSelfAdjoint (x i)\n⊢ ∑ x_1 ∈ s, star (x x_1) = ∑ i ∈ s, x i"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Module.Rat | {
"line": 117,
"column": 4
} | {
"line": 117,
"column": 51
} | {
"line": 117,
"column": 52
} | [
{
"pp": "M : Type u_1\ninst✝¹ : AddCommMonoid M\ninst✝ : Module ℚ≥0 M\nn : ℕ\nhn : n ≠ 0\nx y : M\nhxy : (fun a ↦ n • a) x = (fun a ↦ n • a) y\n⊢ x = y",
"ppTerm": "?m.11",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"M : Type u_1\ninst✝¹ : AddCommMonoid M\ninst✝ : Module ℚ≥0 M\nn : ℕ\nhn : n ≠ 0\nx y : M\nhxy : (fun a ↦ n • a) x = (fun a ↦ n • a) y\n⊢ x = y"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Module.Rat | {
"line": 125,
"column": 4
} | {
"line": 125,
"column": 49
} | {
"line": 125,
"column": 50
} | [
{
"pp": "M : Type u_1\ninst✝¹ : AddCommGroup M\ninst✝ : Module ℚ M\nn : ℕ\nhn : n ≠ 0\nx y : M\nhxy : (fun a ↦ n • a) x = (fun a ↦ n • a) y\n⊢ x = y",
"ppTerm": "?m.11",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"M : Type u_1\ninst✝¹ : AddCommGroup M\ninst✝ : Module ℚ M\nn : ℕ\nhn : n ≠ 0\nx y : M\nhxy : (fun a ↦ n • a) x = (fun a ↦ n • a) y\n⊢ x = y"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Star.SelfAdjoint | {
"line": 83,
"column": 2
} | {
"line": 83,
"column": 45
} | {
"line": 83,
"column": 46
} | [
{
"pp": "R : Type u_1\ninst✝ : InvolutiveStar R\nx : R\n⊢ IsSelfAdjoint (star x) ↔ IsSelfAdjoint x",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"IsSelfAdjoint",
"star_star",
"congrArg",
"id",
"Iff",
"InvolutiveStar.toStar",
"congr... | [
"R : Type u_1\ninst✝ : InvolutiveStar R\nx : R\n⊢ x = star x ↔ star x = x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Star.SelfAdjoint | {
"line": 91,
"column": 2
} | {
"line": 91,
"column": 30
} | {
"line": 91,
"column": 31
} | [
{
"pp": "R : Type u_1\ninst✝¹ : Mul R\ninst✝ : StarMul R\nx : R\n⊢ IsSelfAdjoint (x * star x)",
"ppTerm": "?m.10",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u_1\ninst✝¹ : Mul R\ninst✝ : StarMul R\nx : R\n⊢ IsSelfAdjoint (x * star x)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Star.SelfAdjoint | {
"line": 212,
"column": 2
} | {
"line": 212,
"column": 13
} | {
"line": 212,
"column": 14
} | [
{
"pp": "R : Type u_1\ninst✝¹ : Monoid R\ninst✝ : StarMul R\na u : R\nhu : IsUnit u\n⊢ IsSelfAdjoint (star u * a * u) ↔ IsSelfAdjoint a",
"ppTerm": "?m.18",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u_1\ninst✝¹ : Monoid R\ninst✝ : StarMul R\na u : R\nhu : IsUnit u\n⊢ IsSelfAdjoint (star u * a * u) ↔ IsSelfAdjoint a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Star.SelfAdjoint | {
"line": 627,
"column": 23
} | {
"line": 627,
"column": 81
} | {
"line": 627,
"column": 82
} | [
{
"pp": "R : Type u_1\nA : Type u_2\ninst✝² : Monoid R\ninst✝¹ : StarMul R\nx : Rˣ\ninst✝ : IsStarNormal ↑x\n⊢ Commute (star ↑x⁻¹) ↑x⁻¹",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Units.val",
"Eq.mpr",
"Monoid.toMulOneClass",
"Commute",
"Units",
"id... | [
"R : Type u_1\nA : Type u_2\ninst✝² : Monoid R\ninst✝¹ : StarMul R\nx : Rˣ\ninst✝ : IsStarNormal ↑x\n⊢ Commute (star ↑x) ↑x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Star.SelfAdjoint | {
"line": 673,
"column": 25
} | {
"line": 673,
"column": 61
} | {
"line": 673,
"column": 62
} | [
{
"pp": "R : Type u_1\ninst✝¹ : NonUnitalNonAssocRing R\ninst✝ : StarRing R\na b : R\nha : IsSelfAdjoint a\nhb : IsSelfAdjoint b\nhab : a * b = 0\n⊢ b * a = 0",
"ppTerm": "?m.23",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u_1\ninst✝¹ : NonUnitalNonAssocRing R\ninst✝ : StarRing R\na b : R\nha : IsSelfAdjoint a\nhb : IsSelfAdjoint b\nhab : a * b = 0\n⊢ b * a = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.ConjTranspose | {
"line": 195,
"column": 52
} | {
"line": 195,
"column": 74
} | {
"line": 197,
"column": 0
} | [
{
"pp": "n : Type u_3\nα : Type v\ninst✝² : DecidableEq n\ninst✝¹ : NonAssocSemiring α\ninst✝ : StarRing α\nM : Matrix n n α\n⊢ M = 1ᴴ ↔ M = 1",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"congrArg",
"Matrix",
... | [] | rw [conjTranspose_one] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.Matrix.ConjTranspose | {
"line": 195,
"column": 52
} | {
"line": 195,
"column": 74
} | {
"line": 197,
"column": 0
} | [
{
"pp": "n : Type u_3\nα : Type v\ninst✝² : DecidableEq n\ninst✝¹ : NonAssocSemiring α\ninst✝ : StarRing α\nM : Matrix n n α\n⊢ M = 1ᴴ ↔ M = 1",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"congrArg",
"Matrix",
... | [] | rw [conjTranspose_one] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Matrix.ConjTranspose | {
"line": 195,
"column": 52
} | {
"line": 195,
"column": 74
} | {
"line": 197,
"column": 0
} | [
{
"pp": "n : Type u_3\nα : Type v\ninst✝² : DecidableEq n\ninst✝¹ : NonAssocSemiring α\ninst✝ : StarRing α\nM : Matrix n n α\n⊢ M = 1ᴴ ↔ M = 1",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"congrArg",
"Matrix",
... | [] | rw [conjTranspose_one] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Basis.Fin | {
"line": 55,
"column": 18
} | {
"line": 55,
"column": 66
} | {
"line": 55,
"column": 67
} | [
{
"pp": "ι : Type u_1\nι' : Type u_2\nR : Type u_3\nR₂ : Type u_4\nM : Type u_5\nM' : Type u_6\nv : ι → M\ninst✝⁴ : Ring R\ninst✝³ : CommRing R₂\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : Module R₂ M\nx✝¹ y✝ : M\nb✝ : Basis ι R M\nn : ℕ\nN : Submodule R M\ny : M\nb : Basis (Fin n) R ↥N\nhli : ∀ (c :... | [
"ι : Type u_1\nι' : Type u_2\nR : Type u_3\nR₂ : Type u_4\nM : Type u_5\nM' : Type u_6\nv : ι → M\ninst✝⁴ : Ring R\ninst✝³ : CommRing R₂\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : Module R₂ M\nx✝¹ y✝ : M\nb✝ : Basis ι R M\nn : ℕ\nN : Submodule R M\ny : M\nb : Basis (Fin n) R ↥N\nhli : ∀ (c : R), ∀ x ∈ N... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Basis.Fin | {
"line": 96,
"column": 17
} | {
"line": 96,
"column": 65
} | {
"line": 96,
"column": 66
} | [
{
"pp": "ι : Type u_1\nι' : Type u_2\nR : Type u_3\nR₂ : Type u_4\nM : Type u_5\nM' : Type u_6\nv : ι → M\ninst✝⁴ : Ring R\ninst✝³ : CommRing R₂\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : Module R₂ M\nx✝¹ y✝ : M\nb✝ : Basis ι R M\nn : ℕ\nN : Submodule R M\nb : Basis (Fin n) R ↥N\ny : M\nhli : ∀ (c :... | [
"ι : Type u_1\nι' : Type u_2\nR : Type u_3\nR₂ : Type u_4\nM : Type u_5\nM' : Type u_6\nv : ι → M\ninst✝⁴ : Ring R\ninst✝³ : CommRing R₂\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : Module R₂ M\nx✝¹ y✝ : M\nb✝ : Basis ι R M\nn : ℕ\nN : Submodule R M\nb : Basis (Fin n) R ↥N\ny : M\nhli : ∀ (c : R), ∀ x ∈ N... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Matrix.Block | {
"line": 201,
"column": 2
} | {
"line": 201,
"column": 43
} | {
"line": 203,
"column": 0
} | [
{
"pp": "l : Type u_1\nm : Type u_2\nn : Type u_3\no : Type u_4\nR : Type u_10\ninst✝ : Neg R\nA : Matrix n l R\nB : Matrix n m R\nC : Matrix o l R\nD : Matrix o m R\ni : n ⊕ o\nj : l ⊕ m\n⊢ (-fromBlocks A B C D) i j = fromBlocks (-A) (-B) (-C) (-D) i j",
"ppTerm": "?m.31",
"assigned": true,
"usedCo... | [] | cases i <;> cases j <;> simp [fromBlocks] | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Data.Matrix.Block | {
"line": 378,
"column": 2
} | {
"line": 381,
"column": 15
} | {
"line": 383,
"column": 0
} | [
{
"pp": "m : Type u_2\no : Type u_4\nα : Type u_12\ninst✝² : DecidableEq o\ninst✝¹ : Zero α\ninst✝ : DecidableEq m\nd : o → m → α\n⊢ (blockDiagonal fun k ↦ diagonal (d k)) = diagonal fun ik ↦ d ik.2 ik.1",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instSubsingleto... | [] | ext ⟨i, k⟩ ⟨j, k'⟩
simp only [blockDiagonal_apply, diagonal_apply, Prod.mk_inj, ← ite_and]
congr 1
rw [and_comm] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Matrix.Block | {
"line": 378,
"column": 2
} | {
"line": 381,
"column": 15
} | {
"line": 383,
"column": 0
} | [
{
"pp": "m : Type u_2\no : Type u_4\nα : Type u_12\ninst✝² : DecidableEq o\ninst✝¹ : Zero α\ninst✝ : DecidableEq m\nd : o → m → α\n⊢ (blockDiagonal fun k ↦ diagonal (d k)) = diagonal fun ik ↦ d ik.2 ik.1",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instSubsingleto... | [] | ext ⟨i, k⟩ ⟨j, k'⟩
simp only [blockDiagonal_apply, diagonal_apply, Prod.mk_inj, ← ite_and]
congr 1
rw [and_comm] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Matrix.RowCol | {
"line": 319,
"column": 76
} | {
"line": 319,
"column": 94
} | {
"line": 319,
"column": 94
} | [
{
"pp": "n : Type u_3\nα : Type v\ninst✝¹ : DecidableEq n\ninst✝ : Zero α\nv : n → α\ni : n\nx : α\n⊢ (diagonal (Function.update v i x))ᵀ = diagonal (Function.update v i x)",
"ppTerm": "?m.44",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Function.update",
"congrArg",
"Mat... | [
"n : Type u_3\nα : Type v\ninst✝¹ : DecidableEq n\ninst✝ : Zero α\nv : n → α\ni : n\nx : α\n⊢ diagonal (Function.update v i x) = diagonal (Function.update v i x)"
] | diagonal_transpose | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Matrix.RowCol | {
"line": 392,
"column": 2
} | {
"line": 392,
"column": 37
} | {
"line": 392,
"column": 38
} | [
{
"pp": "m : Type u_2\nn : Type u_3\nα : Type v\ninst✝² : DecidableEq m\ninst✝¹ : DecidableEq n\ninst✝ : Zero α\ni : m\nj : n\nr : α\n⊢ single i j r = updateCol 0 j (Pi.single i r)",
"ppTerm": "?m.19",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"m : Type u_2\nn : Type u_3\nα : Type v\ninst✝² : DecidableEq m\ninst✝¹ : DecidableEq n\ninst✝ : Zero α\ni : m\nj : n\nr : α\n⊢ single i j r = updateCol 0 j (Pi.single i r)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.RowCol | {
"line": 429,
"column": 2
} | {
"line": 432,
"column": 24
} | {
"line": 434,
"column": 0
} | [
{
"pp": "l : Type u_1\nm : Type u_2\nn : Type u_3\nα : Type v\ninst✝² : DecidableEq l\ninst✝¹ : Fintype m\ninst✝ : NonUnitalNonAssocSemiring α\nA : Matrix l m α\ni : l\nr : m → α\nB : Matrix m n α\n⊢ A.updateRow i r * B = (A * B).updateRow i (r ᵥ* B)",
"ppTerm": "?m.23",
"assigned": true,
"usedConst... | [] | ext i' j'
obtain rfl | hi := eq_or_ne i' i
· simp [mul_apply, vecMul, dotProduct]
· simp [mul_apply, hi] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Matrix.RowCol | {
"line": 429,
"column": 2
} | {
"line": 432,
"column": 24
} | {
"line": 434,
"column": 0
} | [
{
"pp": "l : Type u_1\nm : Type u_2\nn : Type u_3\nα : Type v\ninst✝² : DecidableEq l\ninst✝¹ : Fintype m\ninst✝ : NonUnitalNonAssocSemiring α\nA : Matrix l m α\ni : l\nr : m → α\nB : Matrix m n α\n⊢ A.updateRow i r * B = (A * B).updateRow i (r ᵥ* B)",
"ppTerm": "?m.23",
"assigned": true,
"usedConst... | [] | ext i' j'
obtain rfl | hi := eq_or_ne i' i
· simp [mul_apply, vecMul, dotProduct]
· simp [mul_apply, hi] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.TensorProduct.Tower | {
"line": 672,
"column": 2
} | {
"line": 673,
"column": 48
} | {
"line": 675,
"column": 0
} | [
{
"pp": "R : Type u_1\nA : Type u_2\nM : Type u_4\nN : Type u_5\ninst✝⁶ : CommSemiring R\ninst✝⁵ : Semiring A\ninst✝⁴ : Algebra R A\ninst✝³ : AddCommMonoid M\ninst✝² : AddCommMonoid N\ninst✝¹ : Module R M\ninst✝ : Module R N\nf g : M →ₗ[R] N\n⊢ baseChange A (f + g) = baseChange A f + baseChange A g",
"ppTer... | [] | ext
simp [baseChange_eq_ltensor, -baseChange_tmul] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.TensorProduct.Tower | {
"line": 672,
"column": 2
} | {
"line": 673,
"column": 48
} | {
"line": 675,
"column": 0
} | [
{
"pp": "R : Type u_1\nA : Type u_2\nM : Type u_4\nN : Type u_5\ninst✝⁶ : CommSemiring R\ninst✝⁵ : Semiring A\ninst✝⁴ : Algebra R A\ninst✝³ : AddCommMonoid M\ninst✝² : AddCommMonoid N\ninst✝¹ : Module R M\ninst✝ : Module R N\nf g : M →ₗ[R] N\n⊢ baseChange A (f + g) = baseChange A f + baseChange A g",
"ppTer... | [] | ext
simp [baseChange_eq_ltensor, -baseChange_tmul] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Matrix.Notation | {
"line": 546,
"column": 4
} | {
"line": 546,
"column": 15
} | {
"line": 546,
"column": 16
} | [
{
"pp": "case refine_1\nα : Type u_1\nx y : α\nh : Function.Injective ![x, y]\n⊢ x ≠ y",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"id",
"Ne"
],
"usedFVars": [
"α",
"x",
"y"
],
"usedGoals": [
{
"new": true,
"index"... | [
"case refine_1\nα : Type u_1\nx y : α\nh : Function.Injective ![x, y]\n⊢ ¬x = y"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.AffineMonoid.Irreducible | {
"line": 41,
"column": 2
} | {
"line": 41,
"column": 18
} | {
"line": 41,
"column": 19
} | [
{
"pp": "M : Type u_1\ninst✝¹ : CommMonoid M\ninst✝ : Subsingleton Mˣ\nS : Set M\nhS : Submonoid.closure S = ⊤\n⊢ {p | Irreducible p} ⊆ S",
"ppTerm": "?m.14",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"M : Type u_1\ninst✝¹ : CommMonoid M\ninst✝ : Subsingleton Mˣ\nS : Set M\nhS : Submonoid.closure S = ⊤\n⊢ {p | Irreducible p} ⊆ S"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.AffineMonoid.Irreducible | {
"line": 59,
"column": 2
} | {
"line": 59,
"column": 13
} | {
"line": 59,
"column": 14
} | [
{
"pp": "M : Type u_1\ninst✝² : CommMonoid M\ninst✝¹ : Subsingleton Mˣ\ninst✝ : Monoid.FG M\n⊢ {p | Irreducible p}.Finite",
"ppTerm": "?m.7",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"M : Type u_1\ninst✝² : CommMonoid M\ninst✝¹ : Subsingleton Mˣ\ninst✝ : Monoid.FG M\n⊢ {p | Irreducible p}.Finite"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.AffineMonoid.Irreducible | {
"line": 84,
"column": 4
} | {
"line": 84,
"column": 15
} | {
"line": 84,
"column": 16
} | [
{
"pp": "case inl\nM : Type u_1\ninst✝² : CancelCommMonoid M\ninst✝¹ : Subsingleton Mˣ\ninst✝ : Monoid.FG M\nS : Finset M\nhSgen : closure ↑S = ⊤\nhSmax : ∀ ⦃y : Finset M⦄, (fun S ↦ closure ↑S = ⊤) y → y ⊆ S → S ⊆ y\nhrS : 1 ∈ ↑S\nhrirred : 1 ∉ {p | Irreducible p}\n⊢ False",
"ppTerm": "?inl",
"assigned"... | [
"case inl\nM : Type u_1\ninst✝² : CancelCommMonoid M\ninst✝¹ : Subsingleton Mˣ\ninst✝ : Monoid.FG M\nS : Finset M\nhSgen : closure ↑S = ⊤\nhSmax : ∀ ⦃y : Finset M⦄, (fun S ↦ closure ↑S = ⊤) y → y ⊆ S → S ⊆ y\nhrS : 1 ∈ ↑S\nhrirred : 1 ∉ {p | Irreducible p}\n⊢ False"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.TensorProduct.Tower | {
"line": 843,
"column": 68
} | {
"line": 843,
"column": 80
} | {
"line": 843,
"column": 81
} | [
{
"pp": "case refine_1\nR : Type u_1\nM : Type u_2\nA : Type u_3\ninst✝⁴ : CommSemiring R\ninst✝³ : Semiring A\ninst✝² : Algebra R A\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\np : Submodule R M\nx✝¹ : A ⊗[R] ↥p\nx✝ : x✝¹ ∈ {t | ∃ m n, m ⊗ₜ[R] n = t}\na : A\nm : ↥p\nh : a ⊗ₜ[R] m = x✝¹\n⊢ a ⊗ₜ[R] p.subtype m... | [
"case refine_1\nR : Type u_1\nM : Type u_2\nA : Type u_3\ninst✝⁴ : CommSemiring R\ninst✝³ : Semiring A\ninst✝² : Algebra R A\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\np : Submodule R M\nx✝¹ : A ⊗[R] ↥p\nx✝ : x✝¹ ∈ {t | ∃ m n, m ⊗ₜ[R] n = t}\na : A\nm : ↥p\nh : a ⊗ₜ[R] m = x✝¹\n⊢ (a * 1) ⊗ₜ[R] p.subtype m ∈ spa... | ← mul_one a, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.AffineMonoid.Irreducible | {
"line": 95,
"column": 4
} | {
"line": 95,
"column": 79
} | {
"line": 98,
"column": 2
} | [
{
"pp": "M : Type u_1\ninst✝² : CancelCommMonoid M\ninst✝¹ : Subsingleton Mˣ\ninst✝ : Monoid.FG M\nS : Finset M\nhSgen : closure ↑S = ⊤\nhSmax : ∀ ⦃y : Finset M⦄, (fun S ↦ closure ↑S = ⊤) y → y ⊆ S → S ⊆ y\nr : M\nhrS : r ∈ ↑S\nhr₀ : r ≠ 1\nhrirred : ¬IsUnit r → ∃ x x_1, ∃ (_ : r = x * x_1), ¬IsUnit x ∧ ¬IsUnit... | [] | rw [← hn, Finset.sdiff_singleton_eq_erase, ← Finset.mul_prod_erase _ _ hrS] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.AffineMonoid.Irreducible | {
"line": 110,
"column": 23
} | {
"line": 110,
"column": 34
} | {
"line": 110,
"column": 35
} | [
{
"pp": "case refine_1\nM : Type u_1\ninst✝² : CancelCommMonoid M\ninst✝¹ : Subsingleton Mˣ\ninst✝ : Monoid.FG M\nS : Finset M\nhSgen : closure ↑S = ⊤\nhSmax : ∀ ⦃y : Finset M⦄, (fun S ↦ closure ↑S = ⊤) y → y ⊆ S → S ⊆ y\nr : M\nhrS : r ∈ ↑S\nhr₀ : r ≠ 1\nhrirred : ¬IsUnit r → ∃ x x_1, ∃ (_ : r = x * x_1), ¬IsU... | [
"case refine_1\nM : Type u_1\ninst✝² : CancelCommMonoid M\ninst✝¹ : Subsingleton Mˣ\ninst✝ : Monoid.FG M\nS : Finset M\nhSgen : closure ↑S = ⊤\nhSmax : ∀ ⦃y : Finset M⦄, (fun S ↦ closure ↑S = ⊤) y → y ⊆ S → S ⊆ y\nr : M\nhrS : r ∈ ↑S\nhr₀ : r ≠ 1\nhrirred : ¬IsUnit r → ∃ x x_1, ∃ (_ : r = x * x_1), ¬IsUnit x ∧ ¬IsU... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.AffineMonoid.Irreducible | {
"line": 110,
"column": 23
} | {
"line": 110,
"column": 34
} | {
"line": 110,
"column": 35
} | [
{
"pp": "case refine_2\nM : Type u_1\ninst✝² : CancelCommMonoid M\ninst✝¹ : Subsingleton Mˣ\ninst✝ : Monoid.FG M\nS : Finset M\nhSgen : closure ↑S = ⊤\nhSmax : ∀ ⦃y : Finset M⦄, (fun S ↦ closure ↑S = ⊤) y → y ⊆ S → S ⊆ y\nr : M\nhrS : r ∈ ↑S\nhr₀ : r ≠ 1\nhrirred : ¬IsUnit r → ∃ x x_1, ∃ (_ : r = x * x_1), ¬IsU... | [
"case refine_2\nM : Type u_1\ninst✝² : CancelCommMonoid M\ninst✝¹ : Subsingleton Mˣ\ninst✝ : Monoid.FG M\nS : Finset M\nhSgen : closure ↑S = ⊤\nhSmax : ∀ ⦃y : Finset M⦄, (fun S ↦ closure ↑S = ⊤) y → y ⊆ S → S ⊆ y\nr : M\nhrS : r ∈ ↑S\nhr₀ : r ≠ 1\nhrirred : ¬IsUnit r → ∃ x x_1, ∃ (_ : r = x * x_1), ¬IsUnit x ∧ ¬IsU... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.AffineMonoid.Irreducible | {
"line": 113,
"column": 4
} | {
"line": 113,
"column": 15
} | {
"line": 113,
"column": 16
} | [
{
"pp": "M : Type u_1\ninst✝² : CancelCommMonoid M\ninst✝¹ : Subsingleton Mˣ\ninst✝ : Monoid.FG M\nS : Finset M\nhSgen : closure ↑S = ⊤\nr : M\nhSmax : S ⊆ S \\ {r}\nhrS : r ∈ ↑S\nhr₀ : r ≠ 1\nhrirred : ¬IsUnit r → ∃ x x_1, ∃ (_ : r = x * x_1), ¬IsUnit x ∧ ¬IsUnit x_1\na b : M\nhr✝ : r = a * b\nha : ¬IsUnit a\n... | [
"M : Type u_1\ninst✝² : CancelCommMonoid M\ninst✝¹ : Subsingleton Mˣ\ninst✝ : Monoid.FG M\nS : Finset M\nhSgen : closure ↑S = ⊤\nr : M\nhSmax : S ⊆ S \\ {r}\nhrS : r ∈ ↑S\nhr₀ : r ≠ 1\nhrirred : ¬IsUnit r → ∃ x x_1, ∃ (_ : r = x * x_1), ¬IsUnit x ∧ ¬IsUnit x_1\na b : M\nhr✝ : r = a * b\nha : ¬IsUnit a\nhb : ¬IsUnit... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.AffineMonoid.Irreducible | {
"line": 124,
"column": 31
} | {
"line": 124,
"column": 72
} | {
"line": 124,
"column": 73
} | [
{
"pp": "M : Type u_1\ninst✝² : CancelCommMonoid M\ninst✝¹ : Subsingleton Mˣ\ninst✝ : Monoid.FG M\nS : Finset M\nhSgen : closure ↑S = ⊤\nhSmax : ∀ ⦃y : Finset M⦄, (fun S ↦ closure ↑S = ⊤) y → y ⊆ S → S ⊆ y\nr : M\nhrS : r ∈ ↑S\nhr₀ : r ≠ 1\nhrirred : ¬IsUnit r → ∃ x x_1, ∃ (_ : r = x * x_1), ¬IsUnit x ∧ ¬IsUnit... | [
"M : Type u_1\ninst✝² : CancelCommMonoid M\ninst✝¹ : Subsingleton Mˣ\ninst✝ : Monoid.FG M\nS : Finset M\nhSgen : closure ↑S = ⊤\nhSmax : ∀ ⦃y : Finset M⦄, (fun S ↦ closure ↑S = ⊤) y → y ⊆ S → S ⊆ y\nr : M\nhrS : r ∈ ↑S\nhr₀ : r ≠ 1\nhrirred : ¬IsUnit r → ∃ x x_1, ∃ (_ : r = x * x_1), ¬IsUnit x ∧ ¬IsUnit x_1\na b : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.AffineMonoid.Irreducible | {
"line": 126,
"column": 31
} | {
"line": 126,
"column": 72
} | {
"line": 126,
"column": 73
} | [
{
"pp": "M : Type u_1\ninst✝² : CancelCommMonoid M\ninst✝¹ : Subsingleton Mˣ\ninst✝ : Monoid.FG M\nS : Finset M\nhSgen : closure ↑S = ⊤\nhSmax : ∀ ⦃y : Finset M⦄, (fun S ↦ closure ↑S = ⊤) y → y ⊆ S → S ⊆ y\nr : M\nhrS : r ∈ ↑S\nhr₀ : r ≠ 1\nhrirred : ¬IsUnit r → ∃ x x_1, ∃ (_ : r = x * x_1), ¬IsUnit x ∧ ¬IsUnit... | [
"M : Type u_1\ninst✝² : CancelCommMonoid M\ninst✝¹ : Subsingleton Mˣ\ninst✝ : Monoid.FG M\nS : Finset M\nhSgen : closure ↑S = ⊤\nhSmax : ∀ ⦃y : Finset M⦄, (fun S ↦ closure ↑S = ⊤) y → y ⊆ S → S ⊆ y\nr : M\nhrS : r ∈ ↑S\nhr₀ : r ≠ 1\nhrirred : ¬IsUnit r → ∃ x x_1, ∃ (_ : r = x * x_1), ¬IsUnit x ∧ ¬IsUnit x_1\na b : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.FreeModule.PID | {
"line": 213,
"column": 43
} | {
"line": 213,
"column": 68
} | {
"line": 213,
"column": 69
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : IsPrincipalIdealRing R\ninst✝³ : IsDomain R\ninst✝² : Finite ι\nO : Type u_4\ninst✝¹ : AddCommGroup O\ninst✝ : Module R O\nM N : Submodule R O\nb'M : Basis ι R ↥M\nN_bot : N ≠ ⊥\nN_le_M : N ≤ M\nthis : ∃ ϕ, ∀ (ψ : ↥M →ₗ[R] R), ¬ϕ.submoduleImage ... | [
"ι : Type u_1\nR : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : IsPrincipalIdealRing R\ninst✝³ : IsDomain R\ninst✝² : Finite ι\nO : Type u_4\ninst✝¹ : AddCommGroup O\ninst✝ : Module R O\nM N : Submodule R O\nb'M : Basis ι R ↥M\nN_bot : N ≠ ⊥\nN_le_M : N ≤ M\nthis : ∃ ϕ, ∀ (ψ : ↥M →ₗ[R] R), ¬ϕ.submoduleImage N < ψ.submod... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Field.Subfield.Defs | {
"line": 77,
"column": 2
} | {
"line": 77,
"column": 35
} | {
"line": 77,
"column": 36
} | [
{
"pp": "K : Type u\ninst✝¹ : DivisionRing K\nS : Type u_1\ninst✝ : SetLike S K\nh : SubfieldClass S K\ns : S\nq : ℚ≥0\n⊢ ↑q ∈ s",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"instHDiv",
"GroupWithZero.toDivInvMo... | [
"K : Type u\ninst✝¹ : DivisionRing K\nS : Type u_1\ninst✝ : SetLike S K\nh : SubfieldClass S K\ns : S\nq : ℚ≥0\n⊢ ↑q.num / ↑q.den ∈ s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Field.Subfield.Defs | {
"line": 81,
"column": 2
} | {
"line": 81,
"column": 33
} | {
"line": 81,
"column": 34
} | [
{
"pp": "K : Type u\ninst✝¹ : DivisionRing K\nS : Type u_1\ninst✝ : SetLike S K\nh : SubfieldClass S K\ns : S\nq : ℚ\n⊢ ↑q ∈ s",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"Int.cast",
"Eq.mpr",
"Rat.num",
"instHDiv",
"DivisionRing.toRatCast",
"congrArg... | [
"K : Type u\ninst✝¹ : DivisionRing K\nS : Type u_1\ninst✝ : SetLike S K\nh : SubfieldClass S K\ns : S\nq : ℚ\n⊢ ↑q.num / ↑q.den ∈ s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Field.Subfield.Defs | {
"line": 91,
"column": 2
} | {
"line": 91,
"column": 35
} | {
"line": 91,
"column": 36
} | [
{
"pp": "K : Type u\ninst✝¹ : DivisionRing K\nS : Type u_1\ninst✝ : SetLike S K\nh : SubfieldClass S K\nx : K\ns : S\nq : ℚ≥0\nhx : x ∈ s\n⊢ q • x ∈ s",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instHSMul",
"HMul.hMul",
"congrArg",
"Membership.m... | [
"K : Type u\ninst✝¹ : DivisionRing K\nS : Type u_1\ninst✝ : SetLike S K\nh : SubfieldClass S K\nx : K\ns : S\nq : ℚ≥0\nhx : x ∈ s\n⊢ ↑q * x ∈ s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Field.Subfield.Defs | {
"line": 95,
"column": 2
} | {
"line": 95,
"column": 33
} | {
"line": 95,
"column": 34
} | [
{
"pp": "K : Type u\ninst✝¹ : DivisionRing K\nS : Type u_1\ninst✝ : SetLike S K\nh : SubfieldClass S K\nx : K\ns : S\nq : ℚ\nhx : x ∈ s\n⊢ q • x ∈ s",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instHSMul",
"HMul.hMul",
"DivisionRing.toRatCast",
"... | [
"K : Type u\ninst✝¹ : DivisionRing K\nS : Type u_1\ninst✝ : SetLike S K\nh : SubfieldClass S K\nx : K\ns : S\nq : ℚ\nhx : x ∈ s\n⊢ ↑q * x ∈ s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.FreeModule.PID | {
"line": 232,
"column": 4
} | {
"line": 233,
"column": 21
} | {
"line": 233,
"column": 22
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : IsPrincipalIdealRing R\ninst✝³ : IsDomain R\ninst✝² : Finite ι\nO : Type u_4\ninst✝¹ : AddCommGroup O\ninst✝ : Module R O\nM N : Submodule R O\nb'M : Basis ι R ↥M\nN_bot : N ≠ ⊥\nN_le_M : N ≤ M\nthis : ∃ ϕ, ∀ (ψ : ↥M →ₗ[R] R), ¬ϕ.submoduleImage ... | [
"ι : Type u_1\nR : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : IsPrincipalIdealRing R\ninst✝³ : IsDomain R\ninst✝² : Finite ι\nO : Type u_4\ninst✝¹ : AddCommGroup O\ninst✝ : Module R O\nM N : Submodule R O\nb'M : Basis ι R ↥M\nN_bot : N ≠ ⊥\nN_le_M : N ≤ M\nthis : ∃ ϕ, ∀ (ψ : ↥M →ₗ[R] R), ¬ϕ.submoduleImage N < ψ.submod... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Field.Subfield.Basic | {
"line": 206,
"column": 2
} | {
"line": 206,
"column": 38
} | {
"line": 206,
"column": 39
} | [
{
"pp": "K : Type u\nL : Type v\nM : Type w\ninst✝² : DivisionRing K\ninst✝¹ : DivisionRing L\ninst✝ : DivisionRing M\ng : L →+* M\nf : K →+* L\n⊢ Subfield.map g f.fieldRange = (g.comp f).fieldRange",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Di... | [
"K : Type u\nL : Type v\nM : Type w\ninst✝² : DivisionRing K\ninst✝¹ : DivisionRing L\ninst✝ : DivisionRing M\ng : L →+* M\nf : K →+* L\n⊢ Subfield.map g (Subfield.map f ⊤) = Subfield.map (g.comp f) ⊤"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Field.Subfield.Basic | {
"line": 275,
"column": 64
} | {
"line": 279,
"column": 18
} | {
"line": 281,
"column": 0
} | [
{
"pp": "K : Type u\ninst✝ : DivisionRing K\nS : Set (Subfield K)\n⊢ IsGLB S (sInf S)",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"IsGLB.of_image",
"iInf",
"SetLike.coe_subset_coe._simp_1",
"congrArg",
"HEq.refl",
"PartialOrder.toPreor... | [] | by
have : ∀ {s t : Subfield K}, (s : Set K) ≤ t ↔ s ≤ t := by simp [SetLike.coe_subset_coe]
refine IsGLB.of_image this ?_
convert! isGLB_biInf (s := S) (f := SetLike.coe)
exact coe_sInf _ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Field.Subfield.Basic | {
"line": 391,
"column": 2
} | {
"line": 391,
"column": 13
} | {
"line": 391,
"column": 14
} | [
{
"pp": "K : Type u\nL : Type v\ninst✝² : DivisionRing K\ninst✝¹ : DivisionRing L\nι : Sort u_1\ninst✝ : Nonempty ι\nf : K →+* L\ns : ι → Subfield K\n⊢ ↑(map f (iInf s)) = ↑(⨅ i, map f (s i))",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"iInf",
"congrArg",
... | [
"K : Type u\nL : Type v\ninst✝² : DivisionRing K\ninst✝¹ : DivisionRing L\nι : Sort u_1\ninst✝ : Nonempty ι\nf : K →+* L\ns : ι → Subfield K\n⊢ ⇑f '' ⋂ i, ↑(s i) = ⋂ i, ⇑f '' ↑(s i)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.OreLocalization.Basic | {
"line": 185,
"column": 60
} | {
"line": 185,
"column": 68
} | {
"line": 185,
"column": 68
} | [
{
"pp": "case c.c.c\nR : Type u_1\ninst✝³ : Monoid R\nS : Submonoid R\ninst✝² : OreSet S\nX : Type u_2\ninst✝¹ : AddMonoid X\ninst✝ : DistribMulAction R X\nr₁ : X\ns₁ : ↥S\nr₂ : X\ns₂ : ↥S\nr₃ : X\ns₃ : ↥S\nra : R\nsa : ↥S\nha : ↑sa * ↑s₁ = ra * ↑s₂\nha' : r₁ /ₒ s₁ + r₂ /ₒ s₂ = (sa • r₁ + ra • r₂) /ₒ (sa * s₁)\... | [
"case c.c.c\nR : Type u_1\ninst✝³ : Monoid R\nS : Submonoid R\ninst✝² : OreSet S\nX : Type u_2\ninst✝¹ : AddMonoid X\ninst✝ : DistribMulAction R X\nr₁ : X\ns₁ : ↥S\nr₂ : X\ns₂ : ↥S\nr₃ : X\ns₃ : ↥S\nra : R\nsa : ↥S\nha : ↑sa * ↑s₁ = ra * ↑s₂\nha' : r₁ /ₒ s₁ + r₂ /ₒ s₂ = (sa • r₁ + ra • r₂) /ₒ (sa * s₁)\n⊢ (sa • r₁ ... | rw [ha'] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Field.Subfield.Basic | {
"line": 575,
"column": 4
} | {
"line": 581,
"column": 77
} | {
"line": 582,
"column": 2
} | [
{
"pp": "K✝ : Type u\nL : Type v\nM : Type w\ninst✝³ : DivisionRing K✝\ninst✝² : DivisionRing L\ninst✝¹ : DivisionRing M\ns✝¹ : Set K✝\nK : Type u\ninst✝ : Field K\ns✝ : Subfield K\ns : Set K\na✝ b✝ : K\nx_mem : a✝ ∈ {z | ∃ x ∈ Subring.closure s, ∃ y ∈ Subring.closure s, x / y = z}\ny_mem : b✝ ∈ {z | ∃ x ∈ Subr... | [] | obtain ⟨nx, hnx, dx, hdx, rfl⟩ := id x_mem
obtain ⟨ny, hny, dy, hdy, rfl⟩ := id y_mem
by_cases hx0 : dx = 0; · rwa [hx0, div_zero, zero_add]
by_cases hy0 : dy = 0; · rwa [hy0, div_zero, add_zero]
exact
⟨nx * dy + dx * ny, Subring.add_mem _ (Subring.mul_mem _ hnx hdy) (Subring.mul_mem _ hdx hny),
... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Field.Subfield.Basic | {
"line": 575,
"column": 4
} | {
"line": 581,
"column": 77
} | {
"line": 582,
"column": 2
} | [
{
"pp": "K✝ : Type u\nL : Type v\nM : Type w\ninst✝³ : DivisionRing K✝\ninst✝² : DivisionRing L\ninst✝¹ : DivisionRing M\ns✝¹ : Set K✝\nK : Type u\ninst✝ : Field K\ns✝ : Subfield K\ns : Set K\na✝ b✝ : K\nx_mem : a✝ ∈ {z | ∃ x ∈ Subring.closure s, ∃ y ∈ Subring.closure s, x / y = z}\ny_mem : b✝ ∈ {z | ∃ x ∈ Subr... | [] | obtain ⟨nx, hnx, dx, hdx, rfl⟩ := id x_mem
obtain ⟨ny, hny, dy, hdy, rfl⟩ := id y_mem
by_cases hx0 : dx = 0; · rwa [hx0, div_zero, zero_add]
by_cases hy0 : dy = 0; · rwa [hy0, div_zero, add_zero]
exact
⟨nx * dy + dx * ny, Subring.add_mem _ (Subring.mul_mem _ hnx hdy) (Subring.mul_mem _ hdx hny),
... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Field.Subfield.Basic | {
"line": 607,
"column": 2
} | {
"line": 607,
"column": 37
} | {
"line": 607,
"column": 38
} | [
{
"pp": "K : Type u\nL : Type v\ninst✝¹ : DivisionRing K\ninst✝ : DivisionRing L\nf : K →+* L\ns : Subfield L\nh : s ≤ f.fieldRange\n⊢ map f (comap f s) = s",
"ppTerm": "?m.19",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"K : Type u\nL : Type v\ninst✝¹ : DivisionRing K\ninst✝ : DivisionRing L\nf : K →+* L\ns : Subfield L\nh : s ≤ f.fieldRange\n⊢ map f (comap f s) = s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.OreLocalization.Basic | {
"line": 227,
"column": 60
} | {
"line": 227,
"column": 68
} | {
"line": 227,
"column": 68
} | [
{
"pp": "case c.c.c\nR : Type u_1\ninst✝³ : Monoid R\nS : Submonoid R\ninst✝² : OreSet S\nX : Type u_2\ninst✝¹ : AddMonoid X\ninst✝ : DistribMulAction R X\nr₁ : X\ns₁ : ↥S\nr₂ : X\ns₂ : ↥S\nr₃ : R\ns₃ : ↥S\nra : R\nsa : ↥S\nha : ↑sa * ↑s₁ = ra * ↑s₂\nha' : r₁ /ₒ s₁ + r₂ /ₒ s₂ = (sa • r₁ + ra • r₂) /ₒ (sa * s₁)\... | [
"case c.c.c\nR : Type u_1\ninst✝³ : Monoid R\nS : Submonoid R\ninst✝² : OreSet S\nX : Type u_2\ninst✝¹ : AddMonoid X\ninst✝ : DistribMulAction R X\nr₁ : X\ns₁ : ↥S\nr₂ : X\ns₂ : ↥S\nr₃ : R\ns₃ : ↥S\nra : R\nsa : ↥S\nha : ↑sa * ↑s₁ = ra * ↑s₂\nha' : r₁ /ₒ s₁ + r₂ /ₒ s₂ = (sa • r₁ + ra • r₂) /ₒ (sa * s₁)\n⊢ (r₃ /ₒ s₃... | rw [ha'] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.GroupTheory.MonoidLocalization.MonoidWithZero | {
"line": 40,
"column": 37
} | {
"line": 40,
"column": 51
} | {
"line": 40,
"column": 51
} | [
{
"pp": "M : Type u_1\ninst✝¹ : CommMonoidWithZero M\nS : Submonoid M\nN : Type u_2\ninst✝ : CommMonoidWithZero N\nf : S.LocalizationMap N\nx✝ : Subsingleton N\nc : ↥S\neq : 0 = ↑c\n⊢ 0 ∈ S",
"ppTerm": "?m.63",
"assigned": true,
"usedConstants": [
"CommMonoidWithZero.toCommMonoid",
"Mono... | [] | exact eq ▸ c.2 | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.OreLocalization.NonZeroDivisors | {
"line": 76,
"column": 2
} | {
"line": 76,
"column": 55
} | {
"line": 77,
"column": 2
} | [
{
"pp": "case c\nR : Type u_1\ninst✝³ : MonoidWithZero R\ninst✝² : Nontrivial R\ninst✝¹ : OreSet R⁰\ninst✝ : NoZeroDivisors R\nr : R\ns : ↥R⁰\nh : r /ₒ s ≠ 0\n⊢ r /ₒ s * (r /ₒ s)⁻¹ = 1",
"ppTerm": "?c",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"HMul.hMul",
... | [
"case c\nR : Type u_1\ninst✝³ : MonoidWithZero R\ninst✝² : Nontrivial R\ninst✝¹ : OreSet R⁰\ninst✝ : NoZeroDivisors R\nr : R\ns : ↥R⁰\nh : r /ₒ s ≠ 0\n⊢ (r /ₒ s * if hr : r = 0 then 0 else ↑s /ₒ ⟨r, ⋯⟩) = 1 /ₒ 1"
] | rw [OreLocalization.inv_def, OreLocalization.one_def] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.FreeModule.PID | {
"line": 307,
"column": 27
} | {
"line": 307,
"column": 38
} | {
"line": 307,
"column": 39
} | [
{
"pp": "R : Type u_2\ninst✝⁵ : CommRing R\nM : Type u_3\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : IsPrincipalIdealRing R\ninst✝¹ : IsDomain R\nι : Type u_4\ninst✝ : Finite ι\nb : Basis ι R M\nn : ℕ\nb' : Basis (Fin n) R ↥⊥\ne : Fin n ≃ Fin 0 := b'.indexEquiv (Basis.empty ↥⊥)\n⊢ n = 0",
"ppTer... | [
"R : Type u_2\ninst✝⁵ : CommRing R\nM : Type u_3\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : IsPrincipalIdealRing R\ninst✝¹ : IsDomain R\nι : Type u_4\ninst✝ : Finite ι\nb : Basis ι R M\nn : ℕ\nb' : Basis (Fin n) R ↥⊥\ne : Fin n ≃ Fin 0 := b'.indexEquiv (Basis.empty ↥⊥)\n⊢ n = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.ToLin | {
"line": 653,
"column": 40
} | {
"line": 653,
"column": 83
} | {
"line": 653,
"column": 83
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommSemiring R\nn : Type u_4\ninst✝³ : Fintype n\ninst✝² : DecidableEq n\nM₁ : Type u_5\ninst✝¹ : AddCommMonoid M₁\ninst✝ : Module R M₁\nv₁ : Basis n R M₁\nr : R\n⊢ (toMatrix v₁ v₁) ((toLin v₁ v₁) ((scalar n) r)) = (toMatrix v₁ v₁) (r • LinearMap.id)",
"ppTerm": "?m.72",
... | [] | by simp [toMatrix_id, smul_one_eq_diagonal] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.FreeModule.PID | {
"line": 369,
"column": 32
} | {
"line": 369,
"column": 82
} | {
"line": 371,
"column": 4
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\ninst✝⁶ : CommRing R\nM : Type u_3\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\nb : ι → M\ninst✝³ : IsPrincipalIdealRing R\ninst✝² : IsDomain R\ninst✝¹ : Fintype ι\ns : ι → M\nhs : span R (range s) = ⊤\ninst✝ : IsTorsionFree R M\nthis✝ : ∃ s_1, LinearIndepOn R s s_1 ∧ ∀ i ∉... | [] | by rw [Fintype.prod_eq_prod_compl_mul i, mul_smul] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.FreeModule.PID | {
"line": 428,
"column": 38
} | {
"line": 428,
"column": 49
} | {
"line": 428,
"column": 50
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\ninst✝² : CommRing R\nM : Type u_3\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nn : ℕ\nN : Submodule R M\nsnf : SmithNormalForm N ι n\ni : ι\nhi : i ∉ range ⇑snf.f\nc : Fin n →₀ R\nhm : (c.sum fun i x ↦ x • ↑(snf.bN i)) ∈ N\n⊢ ∀ (j : Fin n), snf.f j ≠ i",
"ppTerm": "?m.1... | [
"ι : Type u_1\nR : Type u_2\ninst✝² : CommRing R\nM : Type u_3\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nn : ℕ\nN : Submodule R M\nsnf : SmithNormalForm N ι n\ni : ι\nhi : i ∉ range ⇑snf.f\nc : Fin n →₀ R\nhm : (c.sum fun i x ↦ x • ↑(snf.bN i)) ∈ N\n⊢ ∀ (j : Fin n), ¬snf.f j = i"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.ToLin | {
"line": 725,
"column": 2
} | {
"line": 725,
"column": 47
} | {
"line": 725,
"column": 48
} | [
{
"pp": "R : Type u_1\ninst✝⁷ : CommSemiring R\nm : Type u_3\nn : Type u_4\ninst✝⁶ : Fintype n\ninst✝⁵ : DecidableEq n\nM₁ : Type u_5\nM₂ : Type u_6\ninst✝⁴ : AddCommMonoid M₁\ninst✝³ : AddCommMonoid M₂\ninst✝² : Module R M₁\ninst✝¹ : Module R M₂\nv₁ : Basis n R M₁\nv₂ : Basis m R M₂\ninst✝ : Finite m\nf : M₁ →... | [
"R : Type u_1\ninst✝⁷ : CommSemiring R\nm : Type u_3\nn : Type u_4\ninst✝⁶ : Fintype n\ninst✝⁵ : DecidableEq n\nM₁ : Type u_5\nM₂ : Type u_6\ninst✝⁴ : AddCommMonoid M₁\ninst✝³ : AddCommMonoid M₂\ninst✝² : Module R M₁\ninst✝¹ : Module R M₂\nv₁ : Basis n R M₁\nv₂ : Basis m R M₂\ninst✝ : Finite m\nf : M₁ →ₗ[R] R\nx : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.ToLin | {
"line": 764,
"column": 6
} | {
"line": 764,
"column": 29
} | {
"line": 764,
"column": 30
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommSemiring R\nn : Type u_4\ninst✝³ : Fintype n\ninst✝² : DecidableEq n\nM₁ : Type u_5\ninst✝¹ : AddCommMonoid M₁\ninst✝ : Module R M₁\nv₁ : Basis n R M₁\nf g : M₁ →ₗ[R] M₁\n⊢ (toMatrix v₁ v₁) (f * g) = (toMatrix v₁ v₁) f * (toMatrix v₁ v₁) g",
"ppTerm": "?m.85",
"assign... | [
"R : Type u_1\ninst✝⁴ : CommSemiring R\nn : Type u_4\ninst✝³ : Fintype n\ninst✝² : DecidableEq n\nM₁ : Type u_5\ninst✝¹ : AddCommMonoid M₁\ninst✝ : Module R M₁\nv₁ : Basis n R M₁\nf g : M₁ →ₗ[R] M₁\n⊢ (toMatrix v₁ v₁) (f ∘ₗ g) = (toMatrix v₁ v₁) f * (toMatrix v₁ v₁) g"
] | Module.End.mul_eq_comp, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Matrix.ToLin | {
"line": 789,
"column": 6
} | {
"line": 789,
"column": 29
} | {
"line": 789,
"column": 30
} | [
{
"pp": "R : Type u_1\ninst✝¹¹ : CommSemiring R\nl : Type u_2\nm : Type u_3\nn : Type u_4\ninst✝¹⁰ : Fintype n\ninst✝⁹ : DecidableEq n\nM₁ : Type u_5\nM₂ : Type u_6\ninst✝⁸ : AddCommMonoid M₁\ninst✝⁷ : AddCommMonoid M₂\ninst✝⁶ : Module R M₁\ninst✝⁵ : Module R M₂\nv₁ : Basis n R M₁\nv₂ : Basis m R M₂\ninst✝⁴ : F... | [
"R : Type u_1\ninst✝¹¹ : CommSemiring R\nl : Type u_2\nm : Type u_3\nn : Type u_4\ninst✝¹⁰ : Fintype n\ninst✝⁹ : DecidableEq n\nM₁ : Type u_5\nM₂ : Type u_6\ninst✝⁸ : AddCommMonoid M₁\ninst✝⁷ : AddCommMonoid M₂\ninst✝⁶ : Module R M₁\ninst✝⁵ : Module R M₂\nv₁ : Basis n R M₁\nv₂ : Basis m R M₂\ninst✝⁴ : Fintype m\nM₃... | Matrix.toLin_mul v₁ v₂, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Matrix.ToLin | {
"line": 880,
"column": 6
} | {
"line": 880,
"column": 29
} | {
"line": 880,
"column": 30
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommSemiring R\nn : Type u_4\ninst✝³ : Fintype n\ninst✝² : DecidableEq n\nM₁ : Type u_5\ninst✝¹ : AddCommMonoid M₁\ninst✝ : Module R M₁\nv₁ : Basis n R M₁\nf g : M₁ →ₗ[R] M₁\n⊢ (toMatrixAlgEquiv v₁) (f * g) = (toMatrixAlgEquiv v₁) f * (toMatrixAlgEquiv v₁) g",
"ppTerm": "?m.7... | [
"R : Type u_1\ninst✝⁴ : CommSemiring R\nn : Type u_4\ninst✝³ : Fintype n\ninst✝² : DecidableEq n\nM₁ : Type u_5\ninst✝¹ : AddCommMonoid M₁\ninst✝ : Module R M₁\nv₁ : Basis n R M₁\nf g : M₁ →ₗ[R] M₁\n⊢ (toMatrixAlgEquiv v₁) (f ∘ₗ g) = (toMatrixAlgEquiv v₁) f * (toMatrixAlgEquiv v₁) g"
] | Module.End.mul_eq_comp, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Localization.FractionRing | {
"line": 69,
"column": 4
} | {
"line": 69,
"column": 91
} | {
"line": 69,
"column": 92
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommRing R\nM : Submonoid R\nS : Type u_2\ninst✝³ : CommRing S\ninst✝² : Algebra R S\nP : Type u_3\ninst✝¹ : CommRing P\nA : Type u_4\ninst✝ : CommRing A\nK : Type u_5\nx : ℤ\nhx✝ : x ∈ ℤ⁰\nhx : x ≠ 0\n⊢ IsUnit ((algebraMap ℤ ℚ) ↑⟨x, hx✝⟩)",
"ppTerm": "?m.34",
"assigned":... | [
"R : Type u_1\ninst✝⁴ : CommRing R\nM : Submonoid R\nS : Type u_2\ninst✝³ : CommRing S\ninst✝² : Algebra R S\nP : Type u_3\ninst✝¹ : CommRing P\nA : Type u_4\ninst✝ : CommRing A\nK : Type u_5\nx : ℤ\nhx✝ : x ∈ ℤ⁰\nhx : x ≠ 0\n⊢ ¬x = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Localization.FractionRing | {
"line": 82,
"column": 20
} | {
"line": 82,
"column": 31
} | {
"line": 82,
"column": 32
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommRing R\nM : Submonoid R\nS : Type u_2\ninst✝³ : CommRing S\ninst✝² : Algebra R S\nP : Type u_3\ninst✝¹ : CommRing P\nA : Type u_4\ninst✝ : CommRing A\nK : Type u_5\ny : ↥(Submonoid.pos ℤ)\n⊢ IsUnit ((algebraMap ℤ ℚ) ↑y)",
"ppTerm": "?m.20",
"assigned": true,
"used... | [
"R : Type u_1\ninst✝⁴ : CommRing R\nM : Submonoid R\nS : Type u_2\ninst✝³ : CommRing S\ninst✝² : Algebra R S\nP : Type u_3\ninst✝¹ : CommRing P\nA : Type u_4\ninst✝ : CommRing A\nK : Type u_5\ny : ↥(Submonoid.pos ℤ)\n⊢ ¬↑y = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Localization.FractionRing | {
"line": 86,
"column": 32
} | {
"line": 86,
"column": 43
} | {
"line": 86,
"column": 44
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommRing R\nM : Submonoid R\nS : Type u_2\ninst✝³ : CommRing S\ninst✝² : Algebra R S\nP : Type u_3\ninst✝¹ : CommRing P\nA : Type u_4\ninst✝ : CommRing A\nK : Type u_5\nz : ℚ\nx1 : ℤ\nx2 : ↥ℤ⁰\nhx : z * (algebraMap ℤ ℚ) ↑(x1, x2).2 = (algebraMap ℤ ℚ) (x1, x2).1\nhx2 : ↑x2 < 0\n⊢ ... | [
"R : Type u_1\ninst✝⁴ : CommRing R\nM : Submonoid R\nS : Type u_2\ninst✝³ : CommRing S\ninst✝² : Algebra R S\nP : Type u_3\ninst✝¹ : CommRing P\nA : Type u_4\ninst✝ : CommRing A\nK : Type u_5\nz : ℚ\nx1 : ℤ\nx2 : ↥ℤ⁰\nhx : z * (algebraMap ℤ ℚ) ↑(x1, x2).2 = (algebraMap ℤ ℚ) (x1, x2).1\nhx2 : ↑x2 < 0\n⊢ ↑x2 < 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Localization.FractionRing | {
"line": 86,
"column": 54
} | {
"line": 86,
"column": 65
} | {
"line": 86,
"column": 66
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommRing R\nM : Submonoid R\nS : Type u_2\ninst✝³ : CommRing S\ninst✝² : Algebra R S\nP : Type u_3\ninst✝¹ : CommRing P\nA : Type u_4\ninst✝ : CommRing A\nK : Type u_5\nz : ℚ\nx1 : ℤ\nx2 : ↥ℤ⁰\nhx : z * (algebraMap ℤ ℚ) ↑(x1, x2).2 = (algebraMap ℤ ℚ) (x1, x2).1\nhx2 : ↑x2 < 0\n⊢ ... | [
"R : Type u_1\ninst✝⁴ : CommRing R\nM : Submonoid R\nS : Type u_2\ninst✝³ : CommRing S\ninst✝² : Algebra R S\nP : Type u_3\ninst✝¹ : CommRing P\nA : Type u_4\ninst✝ : CommRing A\nK : Type u_5\nz : ℚ\nx1 : ℤ\nx2 : ↥ℤ⁰\nhx : z * (algebraMap ℤ ℚ) ↑(x1, x2).2 = (algebraMap ℤ ℚ) (x1, x2).1\nhx2 : ↑x2 < 0\n⊢ z * ↑↑x2 = ↑... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Localization.Basic | {
"line": 103,
"column": 6
} | {
"line": 103,
"column": 17
} | {
"line": 103,
"column": 18
} | [
{
"pp": "case refine_1.inl\nι : Type u_1\nR : ι → Type u_2\ninst✝ : (i : ι) → CommSemiring (R i)\nr₁ r₂ : (i : ι) → R i\nj : ι\nS : Submonoid (R j)\nT : Submonoid ((i : ι) → R i) := Submonoid.comap (Pi.evalRingHom R j) S\ns₁ s₂ : ↥T\ns : ↥S\nhs :\n ↑s * (↑⟨(Pi.evalRingHom R j) ↑s₂, ⋯⟩ * (Pi.evalRingHom R j) r₁... | [
"case refine_1.inl\nι : Type u_1\nR : ι → Type u_2\ninst✝ : (i : ι) → CommSemiring (R i)\nr₁ r₂ : (i : ι) → R i\nj : ι\nS : Submonoid (R j)\nT : Submonoid ((i : ι) → R i) := Submonoid.comap (Pi.evalRingHom R j) S\ns₁ s₂ : ↥T\ns : ↥S\nhs :\n ↑s * (↑⟨(Pi.evalRingHom R j) ↑s₂, ⋯⟩ * (Pi.evalRingHom R j) r₁) =\n ↑s ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Localization.FractionRing | {
"line": 88,
"column": 33
} | {
"line": 88,
"column": 44
} | {
"line": 88,
"column": 45
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommRing R\nM : Submonoid R\nS : Type u_2\ninst✝³ : CommRing S\ninst✝² : Algebra R S\nP : Type u_3\ninst✝¹ : CommRing P\nA : Type u_4\ninst✝ : CommRing A\nK : Type u_5\nx y : ℤ\nh : (algebraMap ℤ ℚ) x = (algebraMap ℤ ℚ) y\n⊢ ↑1 * x = ↑1 * y",
"ppTerm": "?m.32",
"assigned"... | [
"R : Type u_1\ninst✝⁴ : CommRing R\nM : Submonoid R\nS : Type u_2\ninst✝³ : CommRing S\ninst✝² : Algebra R S\nP : Type u_3\ninst✝¹ : CommRing P\nA : Type u_4\ninst✝ : CommRing A\nK : Type u_5\nx y : ℤ\nh : (algebraMap ℤ ℚ) x = (algebraMap ℤ ℚ) y\n⊢ x = y"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Localization.FractionRing | {
"line": 94,
"column": 33
} | {
"line": 94,
"column": 44
} | {
"line": 94,
"column": 45
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommRing R\nM : Submonoid R\nS : Type u_2\ninst✝³ : CommRing S\ninst✝² : Algebra R S\nP : Type u_3\ninst✝¹ : CommRing P\nA : Type u_4\ninst✝ : CommRing A\nK : Type u_5\nx y : ℕ\nh : (algebraMap ℕ ℚ≥0) x = (algebraMap ℕ ℚ≥0) y\n⊢ ↑1 * x = ↑1 * y",
"ppTerm": "?m.35",
"assig... | [
"R : Type u_1\ninst✝⁴ : CommRing R\nM : Submonoid R\nS : Type u_2\ninst✝³ : CommRing S\ninst✝² : Algebra R S\nP : Type u_3\ninst✝¹ : CommRing P\nA : Type u_4\ninst✝ : CommRing A\nK : Type u_5\nx y : ℕ\nh : (algebraMap ℕ ℚ≥0) x = (algebraMap ℕ ℚ≥0) y\n⊢ x = y"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Localization.FractionRing | {
"line": 111,
"column": 31
} | {
"line": 111,
"column": 42
} | {
"line": 111,
"column": 43
} | [
{
"pp": "R : Type u_1\ninst✝³ : CommRing R\nK : Type u_5\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : FaithfulSMul R K\nsurj : ∀ (z : K), ∃ x y, z = (algebraMap R K) x / (algebraMap R K) y\ninj : Function.Injective ⇑(algebraMap R K)\nthis✝ : NoZeroDivisors R\nthis : Nontrivial R\nz : K\nx : R\neq : z = (alg... | [
"R : Type u_1\ninst✝³ : CommRing R\nK : Type u_5\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : FaithfulSMul R K\nsurj : ∀ (z : K), ∃ x y, z = (algebraMap R K) x / (algebraMap R K) y\ninj : Function.Injective ⇑(algebraMap R K)\nthis✝ : NoZeroDivisors R\nthis : Nontrivial R\nz : K\nx : R\neq : z = (algebraMap R K)... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Localization.Basic | {
"line": 126,
"column": 4
} | {
"line": 126,
"column": 15
} | {
"line": 126,
"column": 16
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝³ : CommSemiring S\ninst✝² : Algebra R S\ninst✝¹ : IsLocalization M S\ninst✝ : Finite R\nx : S\n⊢ ∃ a, uncurry (mk' S) a = x",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",... | [
"R : Type u_1\ninst✝⁴ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝³ : CommSemiring S\ninst✝² : Algebra R S\ninst✝¹ : IsLocalization M S\ninst✝ : Finite R\nx : S\n⊢ ∃ a a_1, ∃ (b : a_1 ∈ M), mk' S a ⟨a_1, b⟩ = x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Localization.FractionRing | {
"line": 115,
"column": 28
} | {
"line": 115,
"column": 39
} | {
"line": 115,
"column": 40
} | [
{
"pp": "R : Type u_1\ninst✝³ : CommRing R\nK : Type u_5\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : FaithfulSMul R K\nsurj : ∀ (z : K), ∃ x y, z = (algebraMap R K) x / (algebraMap R K) y\ninj : Function.Injective ⇑(algebraMap R K)\nthis✝ : NoZeroDivisors R\nthis : Nontrivial R\nx✝ y✝ : R\neq : (algebraMap... | [
"R : Type u_1\ninst✝³ : CommRing R\nK : Type u_5\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : FaithfulSMul R K\nsurj : ∀ (z : K), ∃ x y, z = (algebraMap R K) x / (algebraMap R K) y\ninj : Function.Injective ⇑(algebraMap R K)\nthis✝ : NoZeroDivisors R\nthis : Nontrivial R\nx✝ y✝ : R\neq : (algebraMap R K) x✝ = (... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.ToLin | {
"line": 1085,
"column": 2
} | {
"line": 1085,
"column": 18
} | {
"line": 1087,
"column": 0
} | [
{
"pp": "R : Type u_1\nM₁ : Type u_3\nM₂ : Type u_4\nι₁ : Type u_6\nι₂ : Type u_7\ninst✝⁷ : CommSemiring R\ninst✝⁶ : AddCommMonoid M₁\ninst✝⁵ : AddCommMonoid M₂\ninst✝⁴ : Module R M₁\ninst✝³ : Module R M₂\ninst✝² : Fintype ι₁\ninst✝¹ : Fintype ι₂\ninst✝ : DecidableEq ι₁\nb₁ : Basis ι₁ R M₁\nb₂ : Basis ι₂ R M₂\n... | [] | simp [linearMap] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.LinearAlgebra.Matrix.ToLin | {
"line": 1085,
"column": 2
} | {
"line": 1085,
"column": 18
} | {
"line": 1087,
"column": 0
} | [
{
"pp": "R : Type u_1\nM₁ : Type u_3\nM₂ : Type u_4\nι₁ : Type u_6\nι₂ : Type u_7\ninst✝⁷ : CommSemiring R\ninst✝⁶ : AddCommMonoid M₁\ninst✝⁵ : AddCommMonoid M₂\ninst✝⁴ : Module R M₁\ninst✝³ : Module R M₂\ninst✝² : Fintype ι₁\ninst✝¹ : Fintype ι₂\ninst✝ : DecidableEq ι₁\nb₁ : Basis ι₁ R M₁\nb₂ : Basis ι₂ R M₂\n... | [] | simp [linearMap] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Matrix.ToLin | {
"line": 1085,
"column": 2
} | {
"line": 1085,
"column": 18
} | {
"line": 1087,
"column": 0
} | [
{
"pp": "R : Type u_1\nM₁ : Type u_3\nM₂ : Type u_4\nι₁ : Type u_6\nι₂ : Type u_7\ninst✝⁷ : CommSemiring R\ninst✝⁶ : AddCommMonoid M₁\ninst✝⁵ : AddCommMonoid M₂\ninst✝⁴ : Module R M₁\ninst✝³ : Module R M₂\ninst✝² : Fintype ι₁\ninst✝¹ : Fintype ι₂\ninst✝ : DecidableEq ι₁\nb₁ : Basis ι₁ R M₁\nb₂ : Basis ι₂ R M₂\n... | [] | simp [linearMap] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
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