module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.Topology.Algebra.Module.Equiv | {
"line": 825,
"column": 10
} | {
"line": 825,
"column": 24
} | {
"line": 826,
"column": 10
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝³ : Semiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ninst✝ : TopologicalSpace M\nf : (M →L[R] M)ˣ\nx : M\n⊢ (↑f * f.inv) x = x",
"ppTerm": "?m.105",
"assigned": true,
"usedConstants": [
"Units.val",
"Eq.mpr",
"MulOne.toOne",
... | [
"R : Type u_1\nM : Type u_2\ninst✝³ : Semiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ninst✝ : TopologicalSpace M\nf : (M →L[R] M)ˣ\nx : M\n⊢ 1 x = x"
] | rw [f.val_inv] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.Algebra.Module.Equiv | {
"line": 1113,
"column": 2
} | {
"line": 1113,
"column": 25
} | {
"line": 1114,
"column": 2
} | [
{
"pp": "R : Type u_1\nM : Type u_2\nM₂ : Type u_3\nM₃ : Type u_4\ninst✝⁹ : TopologicalSpace M\ninst✝⁸ : TopologicalSpace M₂\ninst✝⁷ : TopologicalSpace M₃\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\ninst✝³ : AddCommMonoid M₂\ninst✝² : Module R M₂\ninst✝¹ : AddCommMonoid M₃\ninst✝ : Modu... | [
"R : Type u_1\nM : Type u_2\nM₂ : Type u_3\nM₃ : Type u_4\ninst✝⁹ : TopologicalSpace M\ninst✝⁸ : TopologicalSpace M₂\ninst✝⁷ : TopologicalSpace M₃\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\ninst✝³ : AddCommMonoid M₂\ninst✝² : Module R M₂\ninst✝¹ : AddCommMonoid M₃\ninst✝ : Module R M₃\nf :... | rcases hg with ⟨N, rfl⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Topology.Algebra.Module.Equiv | {
"line": 1169,
"column": 4
} | {
"line": 1169,
"column": 94
} | {
"line": 1171,
"column": 0
} | [
{
"pp": "case neg\nR : Type u_1\nM : Type u_2\nM₂ : Type u_3\nM₃ : Type u_4\ninst✝⁹ : TopologicalSpace M\ninst✝⁸ : TopologicalSpace M₂\ninst✝⁷ : TopologicalSpace M₃\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\ninst✝³ : AddCommMonoid M₂\ninst✝² : Module R M₂\ninst✝¹ : AddCommMonoid M₃\nin... | [] | rw [inverse_of_not_isInvertible (by simp [hf]), inverse_of_not_isInvertible hf, comp_zero] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.Algebra.Module.Equiv | {
"line": 1169,
"column": 4
} | {
"line": 1169,
"column": 94
} | {
"line": 1171,
"column": 0
} | [
{
"pp": "case neg\nR : Type u_1\nM : Type u_2\nM₂ : Type u_3\nM₃ : Type u_4\ninst✝⁹ : TopologicalSpace M\ninst✝⁸ : TopologicalSpace M₂\ninst✝⁷ : TopologicalSpace M₃\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\ninst✝³ : AddCommMonoid M₂\ninst✝² : Module R M₂\ninst✝¹ : AddCommMonoid M₃\nin... | [] | rw [inverse_of_not_isInvertible (by simp [hf]), inverse_of_not_isInvertible hf, comp_zero] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Algebra.Module.Equiv | {
"line": 1169,
"column": 4
} | {
"line": 1169,
"column": 94
} | {
"line": 1171,
"column": 0
} | [
{
"pp": "case neg\nR : Type u_1\nM : Type u_2\nM₂ : Type u_3\nM₃ : Type u_4\ninst✝⁹ : TopologicalSpace M\ninst✝⁸ : TopologicalSpace M₂\ninst✝⁷ : TopologicalSpace M₃\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\ninst✝³ : AddCommMonoid M₂\ninst✝² : Module R M₂\ninst✝¹ : AddCommMonoid M₃\nin... | [] | rw [inverse_of_not_isInvertible (by simp [hf]), inverse_of_not_isInvertible hf, comp_zero] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Algebra.Module.Equiv | {
"line": 1173,
"column": 2
} | {
"line": 1173,
"column": 25
} | {
"line": 1174,
"column": 2
} | [
{
"pp": "R : Type u_1\nM : Type u_2\nM₂ : Type u_3\nM₃ : Type u_4\ninst✝⁹ : TopologicalSpace M\ninst✝⁸ : TopologicalSpace M₂\ninst✝⁷ : TopologicalSpace M₃\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\ninst✝³ : AddCommMonoid M₂\ninst✝² : Module R M₂\ninst✝¹ : AddCommMonoid M₃\ninst✝ : Modu... | [
"R : Type u_1\nM : Type u_2\nM₂ : Type u_3\nM₃ : Type u_4\ninst✝⁹ : TopologicalSpace M\ninst✝⁸ : TopologicalSpace M₂\ninst✝⁷ : TopologicalSpace M₃\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\ninst✝³ : AddCommMonoid M₂\ninst✝² : Module R M₂\ninst✝¹ : AddCommMonoid M₃\ninst✝ : Module R M₃\nf :... | rcases hg with ⟨N, rfl⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Algebra.Category.Ring.Topology | {
"line": 88,
"column": 2
} | {
"line": 88,
"column": 88
} | {
"line": 89,
"column": 2
} | [
{
"pp": "R A B : CommRingCat\ninst✝ : TopologicalSpace ↑R\nf : A ⟶ B\nhf : Function.Surjective ⇑(ConcreteCategory.hom f)\n⊢ IsEmbedding fun x ↦ f ≫ x",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"CommRingCat.Hom.hom",
"CommRingCat.carrier",
"Pi.topologicalSpace",
... | [
"R A B : CommRingCat\ninst✝ : TopologicalSpace ↑R\nf : A ⟶ B\nhf : Function.Surjective ⇑(ConcreteCategory.hom f)\n⊢ IsEmbedding ((fun f ↦ ⇑(Hom.hom f)) ∘ fun x ↦ f ≫ x)"
] | refine IsEmbedding.of_comp (continuous_precomp _) (IsInducing.induced _).continuous ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.LinearAlgebra.TensorProduct.Pi | {
"line": 157,
"column": 69
} | {
"line": 158,
"column": 66
} | {
"line": 160,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝⁸ : CommSemiring R\nS : Type u_2\ninst✝⁷ : CommSemiring S\ninst✝⁶ : Algebra R S\nN : Type u_3\ninst✝⁵ : AddCommMonoid N\ninst✝⁴ : Module R N\ninst✝³ : Module S N\ninst✝² : IsScalarTower R S N\nι : Type u_4\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\nx : N\ni : ι\n⊢ (piScalarRightInv ... | [] | by
simp [piScalarRightInv, Pi.single_apply, TensorProduct.ite_tmul] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Limits.Constructions.Over.Products | {
"line": 168,
"column": 13
} | {
"line": 168,
"column": 60
} | {
"line": 168,
"column": 60
} | [
{
"pp": "J : Type w\nC : Type u\ninst✝ : Category.{v, u} C\nX Y Z : C\nf : X ⟶ Y\ng : X ⟶ Z\nc : BinaryCofan (Under.mk f) (Under.mk g)\nhc : IsColimit c\ns :\n PushoutCocone ((pair (Under.mk f) (Under.mk g)).obj { as := WalkingPair.left }).hom\n ((pair (Under.mk f) (Under.mk g)).obj { as := WalkingPair.righ... | [] | by simpa only using! congr($(hc.fac _ _).right) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Limits.Constructions.Over.Products | {
"line": 169,
"column": 13
} | {
"line": 169,
"column": 60
} | {
"line": 169,
"column": 60
} | [
{
"pp": "J : Type w\nC : Type u\ninst✝ : Category.{v, u} C\nX Y Z : C\nf : X ⟶ Y\ng : X ⟶ Z\nc : BinaryCofan (Under.mk f) (Under.mk g)\nhc : IsColimit c\ns :\n PushoutCocone ((pair (Under.mk f) (Under.mk g)).obj { as := WalkingPair.left }).hom\n ((pair (Under.mk f) (Under.mk g)).obj { as := WalkingPair.righ... | [] | by simpa only using! congr($(hc.fac _ _).right) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.MorphismProperty.OverAdjunction | {
"line": 284,
"column": 21
} | {
"line": 284,
"column": 56
} | {
"line": 286,
"column": 0
} | [
{
"pp": "T : Type u_1\ninst✝⁴ : Category.{v_1, u_1} T\nP Q : MorphismProperty T\ninst✝³ : Q.IsMultiplicative\nX Y Z : T\nf : X ⟶ Y\ninst✝² : P.HasPushoutsAlong f\ninst✝¹ : P.IsStableUnderCobaseChangeAlong f\ninst✝ : Q.IsStableUnderCobaseChange\nX✝ Y✝ Z✝ : P.Under Q X\nx✝¹ : X✝ ⟶ Y✝\nx✝ : Y✝ ⟶ Z✝\n⊢ Under.homMk ... | [] | ext; apply pushout.hom_ext <;> simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.MorphismProperty.OverAdjunction | {
"line": 284,
"column": 21
} | {
"line": 284,
"column": 56
} | {
"line": 286,
"column": 0
} | [
{
"pp": "T : Type u_1\ninst✝⁴ : Category.{v_1, u_1} T\nP Q : MorphismProperty T\ninst✝³ : Q.IsMultiplicative\nX Y Z : T\nf : X ⟶ Y\ninst✝² : P.HasPushoutsAlong f\ninst✝¹ : P.IsStableUnderCobaseChangeAlong f\ninst✝ : Q.IsStableUnderCobaseChange\nX✝ Y✝ Z✝ : P.Under Q X\nx✝¹ : X✝ ⟶ Y✝\nx✝ : Y✝ ⟶ Z✝\n⊢ Under.homMk ... | [] | ext; apply pushout.hom_ext <;> simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.CharP.Invertible | {
"line": 126,
"column": 36
} | {
"line": 126,
"column": 42
} | {
"line": 126,
"column": 43
} | [
{
"pp": "R : Type u_1\nK : Type u_2\ninst✝¹ : DivisionSemiring K\ninst✝ : CharZero K\n⊢ 2 ≠ 0",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"instDecidableNot",
"of_decide_eq_true",
"id",
"Ne",
"instOfNatNat",
"Bool.true",
"Nat",
"Bool",
... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.Algebra.CharP.Invertible | {
"line": 126,
"column": 36
} | {
"line": 126,
"column": 42
} | {
"line": 126,
"column": 43
} | [
{
"pp": "R : Type u_1\nK : Type u_2\ninst✝¹ : DivisionSemiring K\ninst✝ : CharZero K\n⊢ 2 ≠ 0",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"instDecidableNot",
"of_decide_eq_true",
"id",
"Ne",
"instOfNatNat",
"Bool.true",
"Nat",
"Bool",
... | [] | decide | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.CharP.Invertible | {
"line": 126,
"column": 36
} | {
"line": 126,
"column": 42
} | {
"line": 126,
"column": 43
} | [
{
"pp": "R : Type u_1\nK : Type u_2\ninst✝¹ : DivisionSemiring K\ninst✝ : CharZero K\n⊢ 2 ≠ 0",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"instDecidableNot",
"of_decide_eq_true",
"id",
"Ne",
"instOfNatNat",
"Bool.true",
"Nat",
"Bool",
... | [] | decide | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.CharP.Invertible | {
"line": 129,
"column": 36
} | {
"line": 129,
"column": 42
} | {
"line": 129,
"column": 43
} | [
{
"pp": "R : Type u_1\nK : Type u_2\ninst✝¹ : DivisionSemiring K\ninst✝ : CharZero K\n⊢ 3 ≠ 0",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"instDecidableNot",
"of_decide_eq_true",
"id",
"Ne",
"instOfNatNat",
"Bool.true",
"Nat",
"Bool",
... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.Algebra.CharP.Invertible | {
"line": 129,
"column": 36
} | {
"line": 129,
"column": 42
} | {
"line": 129,
"column": 43
} | [
{
"pp": "R : Type u_1\nK : Type u_2\ninst✝¹ : DivisionSemiring K\ninst✝ : CharZero K\n⊢ 3 ≠ 0",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"instDecidableNot",
"of_decide_eq_true",
"id",
"Ne",
"instOfNatNat",
"Bool.true",
"Nat",
"Bool",
... | [] | decide | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.CharP.Invertible | {
"line": 129,
"column": 36
} | {
"line": 129,
"column": 42
} | {
"line": 129,
"column": 43
} | [
{
"pp": "R : Type u_1\nK : Type u_2\ninst✝¹ : DivisionSemiring K\ninst✝ : CharZero K\n⊢ 3 ≠ 0",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"instDecidableNot",
"of_decide_eq_true",
"id",
"Ne",
"instOfNatNat",
"Bool.true",
"Nat",
"Bool",
... | [] | decide | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.MorphismProperty.OverAdjunction | {
"line": 356,
"column": 32
} | {
"line": 356,
"column": 67
} | {
"line": 358,
"column": 0
} | [
{
"pp": "T : Type u_1\ninst✝⁶ : Category.{v_1, u_1} T\nP Q : MorphismProperty T\ninst✝⁵ : Q.IsMultiplicative\nX Y Z : T\ninst✝⁴ : P.IsStableUnderComposition\ninst✝³ : Q.IsStableUnderCobaseChange\nf : X ⟶ Y\ninst✝² : P.HasPushoutsAlong f\ninst✝¹ : P.IsStableUnderCobaseChangeAlong f\ninst✝ : Q.HasOfPrecompPropert... | [] | ext; apply pushout.hom_ext <;> simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.MorphismProperty.OverAdjunction | {
"line": 356,
"column": 32
} | {
"line": 356,
"column": 67
} | {
"line": 358,
"column": 0
} | [
{
"pp": "T : Type u_1\ninst✝⁶ : Category.{v_1, u_1} T\nP Q : MorphismProperty T\ninst✝⁵ : Q.IsMultiplicative\nX Y Z : T\ninst✝⁴ : P.IsStableUnderComposition\ninst✝³ : Q.IsStableUnderCobaseChange\nf : X ⟶ Y\ninst✝² : P.HasPushoutsAlong f\ninst✝¹ : P.IsStableUnderCobaseChangeAlong f\ninst✝ : Q.HasOfPrecompPropert... | [] | ext; apply pushout.hom_ext <;> simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.CharP.MixedCharZero | {
"line": 133,
"column": 8
} | {
"line": 133,
"column": 12
} | {
"line": 134,
"column": 8
} | [
{
"pp": "case e'_3\nR : Type u_1\ninst✝ : CommRing R\np : ℕ\nhp : Nat.Prime p\nI : Ideal R\nhI_not_top : I ≠ ⊤\nright✝ : CharP (R ⧸ I) p\nM : Ideal R\nhM_max : M.IsMaximal\nhM_ge : I ≤ M\nr : ℕ\nhr : CharP (R ⧸ M) r\nr_dvd_p : r ∣ p\n⊢ p = r",
"ppTerm": "?e'_3✝",
"assigned": true,
"usedConstants": [... | [
"case e'_3\nR : Type u_1\ninst✝ : CommRing R\np : ℕ\nhp : Nat.Prime p\nI : Ideal R\nhI_not_top : I ≠ ⊤\nright✝ : CharP (R ⧸ I) p\nM : Ideal R\nhM_max : M.IsMaximal\nhM_ge : I ≤ M\nr : ℕ\nhr : CharP (R ⧸ M) r\nr_dvd_p : r ∣ p\n⊢ r = p"
] | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.Data.Stream.Init | {
"line": 118,
"column": 2
} | {
"line": 119,
"column": 11
} | {
"line": 120,
"column": 2
} | [
{
"pp": "case zero\nα : Type u\na b : α\ns : Stream' α\nx✝ : a ∈ b :: s\nh : a = (b :: s).get 0\n⊢ a = b ∨ a ∈ s",
"ppTerm": "?zero",
"assigned": true,
"usedConstants": [
"Membership.mem",
"Stream'",
"Stream'.instMembership",
"Or.inl",
"Eq"
],
"usedFVars": [
... | [
"case succ\nα : Type u\na b : α\ns : Stream' α\nx✝ : a ∈ b :: s\nn' : ℕ\nh : a = (b :: s).get (n' + 1)\n⊢ a = b ∨ a ∈ s"
] | · left
exact h | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Data.Seq.Basic | {
"line": 151,
"column": 49
} | {
"line": 152,
"column": 17
} | {
"line": 154,
"column": 0
} | [
{
"pp": "α : Type u\ns : Seq α\n⊢ take 0 s = []",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"Stream'.Seq",
"Stream'.Seq.take",
"instOfNatNat",
"Stream'.Seq.nil",
"Stream'.Seq.recOn",
"List",
"Nat",
"Stream'.Seq.cons",
"Eq.ndrec",
... | [] | by
cases s <;> rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.Seq.Basic | {
"line": 163,
"column": 19
} | {
"line": 174,
"column": 64
} | {
"line": 176,
"column": 0
} | [
{
"pp": "α : Type u\nn k : ℕ\ns : Seq α\n⊢ (take (k + 1) s)[n]? = if n < k + 1 then s.get? n else none",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"IsRightCancelAdd.addRightStrictMono_of_addRightMono",
"not_lt_zero._simp_1",
"Eq.mpr",
"Stream'.Seq",
"False... | [] | by
rw [take]
cases h : destruct s with
| none =>
simp [destruct_eq_none h]
| some a =>
match a with
| (x, r) =>
rw [destruct_eq_cons h]
match n with
| 0 => simp
| n + 1 => simp [List.getElem?_cons_succ, getElem?_take] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.Seq.Computation | {
"line": 615,
"column": 6
} | {
"line": 615,
"column": 35
} | {
"line": 615,
"column": 36
} | [
{
"pp": "α : Type u\nβ : Type v\na : α\nf : α → Computation β\nc₁ c₂ : Computation β\nh : c₁ = (pure a).bind f ∧ c₂ = f a ∨ c₁ = corec (Bind.f f) (Sum.inr c₂)\nc : Computation β\n⊢ match\n match Bind.g c.destruct with\n | Sum.inl a => Sum.inl a\n | Sum.inr b => Sum.inr (corec (Bind.f f) b),\n c.dest... | [
"case inl\nα : Type u\nβ : Type v\na : α\nf : α → Computation β\nc₁ c₂ : Computation β\nh : c₁ = (pure a).bind f ∧ c₂ = f a ∨ c₁ = corec (Bind.f f) (Sum.inr c₂)\nc : Computation β\nb : β\n⊢ match\n match Bind.g (Sum.inl b) with\n | Sum.inl a => Sum.inl a\n | Sum.inr b => Sum.inr (corec (Bind.f f) b),\n ... | rcases destruct c with b | cb | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Data.Seq.Basic | {
"line": 281,
"column": 53
} | {
"line": 287,
"column": 28
} | {
"line": 289,
"column": 0
} | [
{
"pp": "α : Type u\ns : Seq α\n⊢ s.append nil = s",
"ppTerm": "?m.4",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Stream'.Seq",
"Stream'.Seq.coinduction2",
"congrArg",
"Option.some",
"Exists",
"id",
"Prod.mk",
"Stream'.Seq.nil",
"Strea... | [] | by
apply coinduction2 s; intro s
cases s
· trivial
· rw [cons_append, destruct_cons, destruct_cons]
dsimp
exact ⟨rfl, _, rfl, rfl⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.Seq.Basic | {
"line": 493,
"column": 21
} | {
"line": 493,
"column": 25
} | {
"line": 493,
"column": 25
} | [
{
"pp": "α : Type u\ns : Seq α\nn : ℕ\n⊢ s.tail.drop n = s.drop (1 + n)",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Stream'.Seq",
"Stream'.Seq.drop",
"instOfNatNat",
"instHAdd",
"Stream'.Seq.tail",
"HAdd.hAdd",
"Nat",
"instAddNat",
... | [
"α : Type u\ns : Seq α\nn : ℕ\n⊢ s.drop (1 + n) = s.tail.drop n"
] | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.Data.Seq.Basic | {
"line": 707,
"column": 92
} | {
"line": 708,
"column": 28
} | {
"line": 710,
"column": 0
} | [
{
"pp": "α : Type u\nx : α\ns : Seq α\nm n : ℕ\nh : n ≠ m\n⊢ (s.set m x).get? n = s.get? n",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"eq_false",
"congrArg",
"Stream'.Seq.update",
"Option.map",
"Nat",
"True",
"eq_self",
"of_eq_true",
... | [] | by
simp [set, get?_update, h] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.Seq.Basic | {
"line": 808,
"column": 4
} | {
"line": 812,
"column": 49
} | {
"line": 814,
"column": 0
} | [
{
"pp": "case succ\nα : Type u\nR : α → α → Prop\nhd : α\ntl : Seq α\nh_hd : ∀ x ∈ tl, R hd x\nh_tl : ∀ (i j : ℕ), i < j → ∀ x ∈ tl.get? i, ∀ y ∈ tl.get? j, R x y\ni : ℕ\nx : α\nhx : x ∈ (Seq.cons hd tl).get? i\ny : α\nk : ℕ\nh_ij : i < k + 1\nhy : y ∈ tl.get? k\n⊢ R x y",
"ppTerm": "?succ",
"assigned":... | [] | cases i with
| zero =>
simp only [get?_cons_zero, Option.mem_def, Option.some.injEq] at hx
exact hx ▸ all_get h_hd hy
| succ n => exact h_tl n k (by lia) x hx y hy | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalCases | Lean.Parser.Tactic.cases |
Mathlib.Data.Seq.Basic | {
"line": 849,
"column": 10
} | {
"line": 849,
"column": 23
} | {
"line": 849,
"column": 24
} | [
{
"pp": "α : Type u\nR : α → α → Prop\ns : Seq α\nmotive : Seq α → Prop\nbase : motive s\nstep : ∀ (hd : α) (tl : Seq α), motive (Seq.cons hd tl) → (∀ x ∈ tl, R hd x) ∧ motive tl\ni j : ℕ\nh_ij : i < j\nx : α\nhx : x ∈ (s.drop i).head\ny : α\nk : ℕ\nhy : y ∈ s.get? (i + k + 1)\nhj : j = i + k + 1\n⊢ R x y",
... | [
"α : Type u\nR : α → α → Prop\ns : Seq α\nmotive : Seq α → Prop\nbase : motive s\nstep : ∀ (hd : α) (tl : Seq α), motive (Seq.cons hd tl) → (∀ x ∈ tl, R hd x) ∧ motive tl\ni j : ℕ\nh_ij : i < j\nx : α\nhx : x ∈ (s.drop i).head\ny : α\nk : ℕ\nhy : y ∈ (s.drop (i + k + 1)).head\nhj : j = i + k + 1\n⊢ R x y"
] | ← head_dropn, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.Nat.Fib.Basic | {
"line": 123,
"column": 55
} | {
"line": 123,
"column": 61
} | {
"line": 123,
"column": 61
} | [
{
"pp": "n✝ n : ℕ\nfive_le_n : Nat.le 5 n\nIH : n ≤ fib n\n⊢ 2 ≤ 5",
"ppTerm": "?m.45",
"assigned": true,
"usedConstants": [
"of_decide_eq_true",
"Preorder.toLE",
"id",
"instOfNatNat",
"LE.le",
"Bool.true",
"Nat.instPreorder",
"Nat",
"Bool",
... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.Data.Nat.Fib.Basic | {
"line": 123,
"column": 55
} | {
"line": 123,
"column": 61
} | {
"line": 123,
"column": 61
} | [
{
"pp": "n✝ n : ℕ\nfive_le_n : Nat.le 5 n\nIH : n ≤ fib n\n⊢ 2 ≤ 5",
"ppTerm": "?m.45",
"assigned": true,
"usedConstants": [
"of_decide_eq_true",
"Preorder.toLE",
"id",
"instOfNatNat",
"LE.le",
"Bool.true",
"Nat.instPreorder",
"Nat",
"Bool",
... | [] | decide | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Nat.Fib.Basic | {
"line": 123,
"column": 55
} | {
"line": 123,
"column": 61
} | {
"line": 123,
"column": 61
} | [
{
"pp": "n✝ n : ℕ\nfive_le_n : Nat.le 5 n\nIH : n ≤ fib n\n⊢ 2 ≤ 5",
"ppTerm": "?m.45",
"assigned": true,
"usedConstants": [
"of_decide_eq_true",
"Preorder.toLE",
"id",
"instOfNatNat",
"LE.le",
"Bool.true",
"Nat.instPreorder",
"Nat",
"Bool",
... | [] | decide | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Nat.Fib.Basic | {
"line": 153,
"column": 4
} | {
"line": 155,
"column": 8
} | {
"line": 157,
"column": 0
} | [
{
"pp": "case succ\nn✝ : ℕ\n⊢ fib (2 * (n✝ + 1)) = fib (n✝ + 1) * (2 * fib (n✝ + 1 + 1) - fib (n✝ + 1))",
"ppTerm": "?succ",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Mathlib.Meta.NormN... | [] | rw [two_mul, ← add_assoc, fib_add, fib_add_two, two_mul]
simp only [← add_assoc, add_tsub_cancel_right]
ring | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Nat.Fib.Basic | {
"line": 153,
"column": 4
} | {
"line": 155,
"column": 8
} | {
"line": 157,
"column": 0
} | [
{
"pp": "case succ\nn✝ : ℕ\n⊢ fib (2 * (n✝ + 1)) = fib (n✝ + 1) * (2 * fib (n✝ + 1 + 1) - fib (n✝ + 1))",
"ppTerm": "?succ",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Mathlib.Meta.NormN... | [] | rw [two_mul, ← add_assoc, fib_add, fib_add_two, two_mul]
simp only [← add_assoc, add_tsub_cancel_right]
ring | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.ContinuedFractions.Computation.CorrectnessTerminating | {
"line": 136,
"column": 6
} | {
"line": 147,
"column": 28
} | {
"line": 148,
"column": 4
} | [
{
"pp": "case succ.inl\nK : Type u_1\ninst✝² : Field K\ninst✝¹ : LinearOrder K\nv : K\nn✝ : ℕ\ninst✝ : FloorRing K\ng : GenContFract K := of v\nn : ℕ\nifp_succ_n : IntFractPair K\nsucc_nth_stream_eq : IntFractPair.stream v (n + 1) = some ifp_succ_n\nifp_n : IntFractPair K\nnth_stream_eq : IntFractPair.stream v ... | [] | suffices v = conts.a / conts.b by simpa [compExactValue, ifp_succ_n_fr_eq_zero]
-- use the IH and the fact that ifp_n.fr⁻¹ = ⌊ifp_n.fr⁻¹⌋ to prove this case
obtain ⟨ifp_n', nth_stream_eq', ifp_n_fract_inv_eq_floor⟩ :
∃ ifp_n, IntFractPair.stream v n = some ifp_n ∧ ifp_n.fr⁻¹ = ⌊ifp_n.fr⁻¹⌋ :=
... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.ContinuedFractions.Computation.CorrectnessTerminating | {
"line": 136,
"column": 6
} | {
"line": 147,
"column": 28
} | {
"line": 148,
"column": 4
} | [
{
"pp": "case succ.inl\nK : Type u_1\ninst✝² : Field K\ninst✝¹ : LinearOrder K\nv : K\nn✝ : ℕ\ninst✝ : FloorRing K\ng : GenContFract K := of v\nn : ℕ\nifp_succ_n : IntFractPair K\nsucc_nth_stream_eq : IntFractPair.stream v (n + 1) = some ifp_succ_n\nifp_n : IntFractPair K\nnth_stream_eq : IntFractPair.stream v ... | [] | suffices v = conts.a / conts.b by simpa [compExactValue, ifp_succ_n_fr_eq_zero]
-- use the IH and the fact that ifp_n.fr⁻¹ = ⌊ifp_n.fr⁻¹⌋ to prove this case
obtain ⟨ifp_n', nth_stream_eq', ifp_n_fract_inv_eq_floor⟩ :
∃ ifp_n, IntFractPair.stream v n = some ifp_n ∧ ifp_n.fr⁻¹ = ⌊ifp_n.fr⁻¹⌋ :=
... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.ContinuedFractions.Computation.CorrectnessTerminating | {
"line": 106,
"column": 2
} | {
"line": 184,
"column": 11
} | {
"line": 186,
"column": 0
} | [
{
"pp": "K : Type u_1\ninst✝² : Field K\ninst✝¹ : LinearOrder K\nv : K\nn : ℕ\ninst✝ : FloorRing K\ng : GenContFract K := of v\n⊢ ∀ {ifp_n : IntFractPair K},\n IntFractPair.stream v n = some ifp_n → v = compExactValue ((of v).contsAux n) ((of v).contsAux (n + 1)) ifp_n.fr",
"ppTerm": "?m.38",
"assign... | [] | induction n with
| zero =>
intro ifp_zero stream_zero_eq
obtain rfl : IntFractPair.of v = ifp_zero := by
simpa only [IntFractPair.stream, Option.some.injEq] using stream_zero_eq
cases eq_or_ne (Int.fract v) 0 with
| inl fract_eq_zero =>
-- Int.fract v = 0; we must then have `v = ⌊v⌋`
... | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.Algebra.ContinuedFractions.Computation.CorrectnessTerminating | {
"line": 234,
"column": 41
} | {
"line": 240,
"column": 53
} | {
"line": 242,
"column": 0
} | [
{
"pp": "K : Type u_1\ninst✝² : Field K\ninst✝¹ : LinearOrder K\nv : K\ninst✝ : FloorRing K\nterminates : (of v).Terminates\n⊢ ∀ᶠ (n : ℕ) in atTop, v = (of v).convs n",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"GenContFract.convs",
"Lattice.toSemilatticeSup... | [] | by
rw [eventually_atTop]
obtain ⟨n, terminatedAt_n⟩ : ∃ n, (of v).TerminatedAt n := terminates
use n
intro m m_geq_n
rw [convs_stable_of_terminated m_geq_n terminatedAt_n]
exact of_correctness_of_terminatedAt terminatedAt_n | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.CubicDiscriminant | {
"line": 199,
"column": 29
} | {
"line": 199,
"column": 43
} | {
"line": 199,
"column": 43
} | [
{
"pp": "R : Type u_1\nP : Cubic R\ninst✝ : Semiring R\nha : P.a = 0\nhb : P.b = 0\nhc : P.c = 0\n⊢ (C P.d).leadingCoeff = P.d",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Polynomial.C",
"congrArg",
"RingHom",
"id",
"Polynomial.leadingCoeff... | [
"R : Type u_1\nP : Cubic R\ninst✝ : Semiring R\nha : P.a = 0\nhb : P.b = 0\nhc : P.c = 0\n⊢ P.d = P.d"
] | leadingCoeff_C | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.ContinuedFractions.ConvergentsEquiv | {
"line": 307,
"column": 8
} | {
"line": 307,
"column": 12
} | {
"line": 308,
"column": 8
} | [
{
"pp": "K : Type u_1\ng : GenContFract K\ninst✝ : Field K\na b : K\nb_ne_zero : b ≠ 0\nn' : ℕ\nnth_partDen_ne_zero : ∀ {b : K}, g.partDens.get? (n' + 1) = some b → b ≠ 0\nnot_terminatedAt_n : ¬g.TerminatedAt (n' + 1)\ns_nth_eq : g.s.get? (n' + 1) = some { a := a, b := b }\npa pb : K\ns_n'th_eq : g.s.get? n' = ... | [
"K : Type u_1\ng : GenContFract K\ninst✝ : Field K\na b : K\nb_ne_zero : b ≠ 0\nn' : ℕ\nnth_partDen_ne_zero : ∀ {b : K}, g.partDens.get? (n' + 1) = some b → b ≠ 0\nnot_terminatedAt_n : ¬g.TerminatedAt (n' + 1)\ns_nth_eq : g.s.get? (n' + 1) = some { a := a, b := b }\npa pb : K\ns_n'th_eq : g.s.get? n' = some { a := ... | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.Algebra.CubicDiscriminant | {
"line": 423,
"column": 2
} | {
"line": 423,
"column": 46
} | {
"line": 425,
"column": 0
} | [
{
"pp": "F : Type u_3\nK : Type u_4\nP : Cubic F\ninst✝¹ : Field F\ninst✝ : Field K\nφ : F →+* K\nha : P.a ≠ 0\n⊢ (Polynomial.map φ P.toPoly).Splits ↔ ∃ x y z, (map φ P).roots = {x, y, z}",
"ppTerm": "?m.37",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Cubic.roots",
"Multiset.c... | [] | rw [splits_iff_card_roots ha, card_eq_three] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.CubicDiscriminant | {
"line": 423,
"column": 2
} | {
"line": 423,
"column": 46
} | {
"line": 425,
"column": 0
} | [
{
"pp": "F : Type u_3\nK : Type u_4\nP : Cubic F\ninst✝¹ : Field F\ninst✝ : Field K\nφ : F →+* K\nha : P.a ≠ 0\n⊢ (Polynomial.map φ P.toPoly).Splits ↔ ∃ x y z, (map φ P).roots = {x, y, z}",
"ppTerm": "?m.37",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Cubic.roots",
"Multiset.c... | [] | rw [splits_iff_card_roots ha, card_eq_three] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.CubicDiscriminant | {
"line": 423,
"column": 2
} | {
"line": 423,
"column": 46
} | {
"line": 425,
"column": 0
} | [
{
"pp": "F : Type u_3\nK : Type u_4\nP : Cubic F\ninst✝¹ : Field F\ninst✝ : Field K\nφ : F →+* K\nha : P.a ≠ 0\n⊢ (Polynomial.map φ P.toPoly).Splits ↔ ∃ x y z, (map φ P).roots = {x, y, z}",
"ppTerm": "?m.37",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Cubic.roots",
"Multiset.c... | [] | rw [splits_iff_card_roots ha, card_eq_three] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Matrix.Ideal | {
"line": 91,
"column": 10
} | {
"line": 91,
"column": 13
} | {
"line": 91,
"column": 14
} | [
{
"pp": "R : Type u_1\ninst✝² : Ring R\nn : Type u_2\ninst✝¹ : Fintype n\ninst✝ : DecidableEq n\nI : Ideal R\nx : R\n⊢ (∀ (y : R), ∃ z, z * y * x + z - 1 ∈ I) →\n ∀ (i j : n) (y : Matrix n n R), ∃ z, z * y * single i j x + z - 1 ∈ matrix n I",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants":... | [
"R : Type u_1\ninst✝² : Ring R\nn : Type u_2\ninst✝¹ : Fintype n\ninst✝ : DecidableEq n\nI : Ideal R\nx : R\nxIJ : ∀ (y : R), ∃ z, z * y * x + z - 1 ∈ I\n⊢ ∀ (i j : n) (y : Matrix n n R), ∃ z, z * y * single i j x + z - 1 ∈ matrix n I"
] | xIJ | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.RingTheory.Idempotents | {
"line": 308,
"column": 2
} | {
"line": 322,
"column": 25
} | {
"line": 324,
"column": 0
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝¹ : Ring R\ninst✝ : Ring S\nf : R →+* S\nh : ∀ x ∈ RingHom.ker f, IsNilpotent x\nn : ℕ\ne : Fin n → S\nhe : CompleteOrthogonalIdempotents e\nhe' : ∀ (i : Fin n), e i ∈ f.range\n⊢ ∃ e', CompleteOrthogonalIdempotents e' ∧ ⇑f ∘ e' = e",
"ppTerm": "?m.40",
"assigned... | [] | cases subsingleton_or_nontrivial R
· choose e' he' using he'
exact ⟨e', .of_subsingleton, funext he'⟩
cases subsingleton_or_nontrivial S
· obtain ⟨n, hn⟩ := h 1 (Subsingleton.elim _ _)
simp at hn
rcases n with - | n
· simpa using he.complete
obtain ⟨e', h₁, h₂⟩ := OrthogonalIdempotents.lift_of_isNil... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Idempotents | {
"line": 308,
"column": 2
} | {
"line": 322,
"column": 25
} | {
"line": 324,
"column": 0
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝¹ : Ring R\ninst✝ : Ring S\nf : R →+* S\nh : ∀ x ∈ RingHom.ker f, IsNilpotent x\nn : ℕ\ne : Fin n → S\nhe : CompleteOrthogonalIdempotents e\nhe' : ∀ (i : Fin n), e i ∈ f.range\n⊢ ∃ e', CompleteOrthogonalIdempotents e' ∧ ⇑f ∘ e' = e",
"ppTerm": "?m.40",
"assigned... | [] | cases subsingleton_or_nontrivial R
· choose e' he' using he'
exact ⟨e', .of_subsingleton, funext he'⟩
cases subsingleton_or_nontrivial S
· obtain ⟨n, hn⟩ := h 1 (Subsingleton.elim _ _)
simp at hn
rcases n with - | n
· simpa using he.complete
obtain ⟨e', h₁, h₂⟩ := OrthogonalIdempotents.lift_of_isNil... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Matrix.Ideal | {
"line": 341,
"column": 51
} | {
"line": 353,
"column": 52
} | {
"line": 355,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝² : Ring R\nn : Type u_2\ninst✝¹ : Fintype n\ninst✝ : DecidableEq n\nI : TwoSidedIdeal R\n⊢ (matrix n I).jacobson ≤ matrix n I.jacobson",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"TwoSidedIdeal.jacobson",
"Eq.mpr",
"Matrix.smul",
"No... | [] | by
-- Proof generalized from example 8 in
-- https://ysharifi.wordpress.com/2022/08/16/the-jacobson-radical-basic-examples/
intro M Mmem p q
simp only [zero_apply, ← mem_iff]
rw [mem_jacobson_iff]
replace Mmem := mul_mem_right _ _ (single q p 1) Mmem
rw [mem_jacobson_iff] at Mmem
intro y
specialize Mm... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Idempotents | {
"line": 539,
"column": 2
} | {
"line": 539,
"column": 47
} | {
"line": 541,
"column": 0
} | [
{
"pp": "R : Type u_1\ne : R\ninst✝ : Semigroup R\nidem : IsIdempotentElem e\nhc : IsMulCentral e\nr : R\n⊢ r ∈ corner e ↔ r * e = r",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Semigroup.toMul",
"HMul.hMul",
"congrArg",
"Iff.rfl",
"Members... | [] | rw [mem_corner_iff_mul_left idem hc, hc.comm] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Idempotents | {
"line": 539,
"column": 2
} | {
"line": 539,
"column": 47
} | {
"line": 541,
"column": 0
} | [
{
"pp": "R : Type u_1\ne : R\ninst✝ : Semigroup R\nidem : IsIdempotentElem e\nhc : IsMulCentral e\nr : R\n⊢ r ∈ corner e ↔ r * e = r",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Semigroup.toMul",
"HMul.hMul",
"congrArg",
"Iff.rfl",
"Members... | [] | rw [mem_corner_iff_mul_left idem hc, hc.comm] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Idempotents | {
"line": 539,
"column": 2
} | {
"line": 539,
"column": 47
} | {
"line": 541,
"column": 0
} | [
{
"pp": "R : Type u_1\ne : R\ninst✝ : Semigroup R\nidem : IsIdempotentElem e\nhc : IsMulCentral e\nr : R\n⊢ r ∈ corner e ↔ r * e = r",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Semigroup.toMul",
"HMul.hMul",
"congrArg",
"Iff.rfl",
"Members... | [] | rw [mem_corner_iff_mul_left idem hc, hc.comm] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Idempotents | {
"line": 538,
"column": 91
} | {
"line": 539,
"column": 47
} | {
"line": 541,
"column": 0
} | [
{
"pp": "R : Type u_1\ne : R\ninst✝ : Semigroup R\nidem : IsIdempotentElem e\nhc : IsMulCentral e\nr : R\n⊢ r ∈ corner e ↔ r * e = r",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Semigroup.toMul",
"HMul.hMul",
"congrArg",
"Iff.rfl",
"Members... | [] | by
rw [mem_corner_iff_mul_left idem hc, hc.comm] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Divisibility.Prod | {
"line": 35,
"column": 4
} | {
"line": 35,
"column": 72
} | {
"line": 36,
"column": 4
} | [
{
"pp": "ι : Type u_1\nG₁ : Type u_2\nG₂ : Type u_3\nG : ι → Type u_4\ninst✝⁴ : Semigroup G₁\ninst✝³ : Semigroup G₂\ninst✝² : (i : ι) → Semigroup (G i)\ninst✝¹ : DecompositionMonoid G₁\ninst✝ : DecompositionMonoid G₂\na b c : G₁ × G₂\nh : a.1 ∣ (b * c).1 ∧ a.2 ∣ (b * c).2\n⊢ ∃ a₁ a₂, (a₁.1 ∣ b.1 ∧ a₁.2 ∣ b.2) ∧... | [
"ι : Type u_1\nG₁ : Type u_2\nG₂ : Type u_3\nG : ι → Type u_4\ninst✝⁴ : Semigroup G₁\ninst✝³ : Semigroup G₂\ninst✝² : (i : ι) → Semigroup (G i)\ninst✝¹ : DecompositionMonoid G₁\ninst✝ : DecompositionMonoid G₂\na b c : G₁ × G₂\nh : a.1 ∣ (b * c).1 ∧ a.2 ∣ (b * c).2\na₁ a₁' : G₁\nh₁ : a₁ ∣ b.1\nh₁' : a₁' ∣ c.1\neq₁ :... | obtain ⟨a₁, a₁', h₁, h₁', eq₁⟩ := DecompositionMonoid.primal a.1 h.1 | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.LinearAlgebra.Trace | {
"line": 179,
"column": 2
} | {
"line": 179,
"column": 38
} | {
"line": 181,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommRing R\nM : Type u_2\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : Free R M\ninst✝ : Module.Finite R M\nx : Dual R M ⊗[R] M\n⊢ (trace R M) ((dualTensorHom R M M) x) = (contractLeft R M) x",
"ppTerm": "?m.42",
"assigned": true,
"usedConstants": [
"L... | [] | rw [← comp_apply, trace_eq_contract] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.Trace | {
"line": 179,
"column": 2
} | {
"line": 179,
"column": 38
} | {
"line": 181,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommRing R\nM : Type u_2\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : Free R M\ninst✝ : Module.Finite R M\nx : Dual R M ⊗[R] M\n⊢ (trace R M) ((dualTensorHom R M M) x) = (contractLeft R M) x",
"ppTerm": "?m.42",
"assigned": true,
"usedConstants": [
"L... | [] | rw [← comp_apply, trace_eq_contract] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Trace | {
"line": 179,
"column": 2
} | {
"line": 179,
"column": 38
} | {
"line": 181,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommRing R\nM : Type u_2\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : Free R M\ninst✝ : Module.Finite R M\nx : Dual R M ⊗[R] M\n⊢ (trace R M) ((dualTensorHom R M M) x) = (contractLeft R M) x",
"ppTerm": "?m.42",
"assigned": true,
"usedConstants": [
"L... | [] | rw [← comp_apply, trace_eq_contract] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Trace | {
"line": 240,
"column": 2
} | {
"line": 247,
"column": 41
} | {
"line": 249,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝⁸ : CommRing R\nM : Type u_2\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\nN : Type u_3\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\ninst✝³ : Free R M\ninst✝² : Module.Finite R M\ninst✝¹ : Free R N\ninst✝ : Module.Finite R N\n⊢ (mapBilinear (RingHom.id R) M N M N).compr₂ (trace R ... | [] | apply
(compl₁₂_inj (show Surjective (dualTensorHom R M M) from (dualTensorHomEquiv R M M).surjective)
(show Surjective (dualTensorHom R N N) from (dualTensorHomEquiv R N N).surjective)).1
ext f m g n
simp only [AlgebraTensorModule.curry_apply, TensorProduct.curry_apply,
coe_restrictScalars, compl₁₂_... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Trace | {
"line": 240,
"column": 2
} | {
"line": 247,
"column": 41
} | {
"line": 249,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝⁸ : CommRing R\nM : Type u_2\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\nN : Type u_3\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\ninst✝³ : Free R M\ninst✝² : Module.Finite R M\ninst✝¹ : Free R N\ninst✝ : Module.Finite R N\n⊢ (mapBilinear (RingHom.id R) M N M N).compr₂ (trace R ... | [] | apply
(compl₁₂_inj (show Surjective (dualTensorHom R M M) from (dualTensorHomEquiv R M M).surjective)
(show Surjective (dualTensorHom R N N) from (dualTensorHomEquiv R N N).surjective)).1
ext f m g n
simp only [AlgebraTensorModule.curry_apply, TensorProduct.curry_apply,
coe_restrictScalars, compl₁₂_... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Eigenspace.Basic | {
"line": 826,
"column": 12
} | {
"line": 826,
"column": 51
} | {
"line": 826,
"column": 51
} | [
{
"pp": "K : Type v\nV : Type w\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : End K V\nμ : K\nn : ℕ\n⊢ (f * (f - (algebraMap K (End K V)) μ) ^ n).range = ((f - (algebraMap K (End K V)) μ) ^ n * f).range",
"ppTerm": "?m.114",
"assigned": true,
"usedConstants": [
"Module.En... | [
"K : Type v\nV : Type w\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : End K V\nμ : K\nn : ℕ\n⊢ LinearMap.range ((f - (algebraMap K (End K V)) μ) ^ n * f) = ((f - (algebraMap K (End K V)) μ) ^ n * f).range"
] | Algebra.mul_sub_algebraMap_pow_commutes | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Eigenspace.Basic | {
"line": 832,
"column": 2
} | {
"line": 832,
"column": 12
} | {
"line": 833,
"column": 2
} | [
{
"pp": "R : Type v\nM : Type w\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf : End R M\nμ t : R\nk : ℕ∞\n⊢ (f.genEigenspace μ) k ≤ ((t • f).genEigenspace (t * μ)) k",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Submodule",
"CommSemiring.toSemiring",
... | [
"R : Type v\nM : Type w\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf : End R M\nμ t : R\nk : ℕ∞\nm : M\nhm : m ∈ (f.genEigenspace μ) k\n⊢ m ∈ ((t • f).genEigenspace (t * μ)) k"
] | intro m hm | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.LinearAlgebra.Eigenspace.Basic | {
"line": 841,
"column": 2
} | {
"line": 841,
"column": 12
} | {
"line": 842,
"column": 2
} | [
{
"pp": "R : Type v\nM : Type w\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf₁ f₂ : End R M\nμ₁ μ₂ : R\nk₁ k₂ : ℕ∞\nh : Commute f₁ f₂\n⊢ (f₁.genEigenspace μ₁) k₁ ⊓ (f₂.genEigenspace μ₂) k₂ ≤ ((f₁ + f₂).genEigenspace (μ₁ + μ₂)) (k₁ + k₂)",
"ppTerm": "?m.47",
"assigned": true,
"... | [
"R : Type v\nM : Type w\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf₁ f₂ : End R M\nμ₁ μ₂ : R\nk₁ k₂ : ℕ∞\nh : Commute f₁ f₂\nm : M\nhm : m ∈ (f₁.genEigenspace μ₁) k₁ ⊓ (f₂.genEigenspace μ₂) k₂\n⊢ m ∈ ((f₁ + f₂).genEigenspace (μ₁ + μ₂)) (k₁ + k₂)"
] | intro m hm | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Algebra.DualQuaternion | {
"line": 43,
"column": 14
} | {
"line": 48,
"column": 31
} | {
"line": 49,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\n⊢ ∀ (x y : ℍ[DualNumber R]),\n ({ re := TrivSqZeroExt.fst (x * y).re, imI := TrivSqZeroExt.fst (x * y).imI, imJ := TrivSqZeroExt.fst (x * y).imJ,\n imK := TrivSqZeroExt.fst (x * y).imK },\n { re := TrivSqZeroExt.snd (x * y).re, imI := TrivSqZeroExt.s... | [] | by
intros
ext : 1
· rfl
· dsimp
congr 1 <;> simp <;> ring | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Field.GeomSum | {
"line": 62,
"column": 2
} | {
"line": 73,
"column": 50
} | {
"line": 75,
"column": 0
} | [
{
"pp": "K : Type u_2\ninst✝ : DivisionRing K\nx : K\nhx1 : x ≠ 1\nhx0 : x ≠ 0\nn : ℕ\n⊢ ∑ i ∈ range n, x⁻¹ ^ i = (x - 1)⁻¹ * (x - x⁻¹ ^ n * x)",
"ppTerm": "?m.45",
"assigned": true,
"usedConstants": [
"neg_add_rev",
"add_mul",
"AddGroup.toSubtractionMonoid",
"Distrib.leftDis... | [] | have h₁ : x⁻¹ ≠ 1 := by rwa [inv_eq_one_div, Ne, div_eq_iff_mul_eq hx0, one_mul]
have h₂ : x⁻¹ - 1 ≠ 0 := mt sub_eq_zero.1 h₁
have h₃ : x - 1 ≠ 0 := mt sub_eq_zero.1 hx1
have h₄ : x * (x ^ n)⁻¹ = (x ^ n)⁻¹ * x :=
Nat.recOn n (by simp) fun n h => by
rw [pow_succ', mul_inv_rev, ← mul_assoc, h, mul_assoc, ... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Field.GeomSum | {
"line": 62,
"column": 2
} | {
"line": 73,
"column": 50
} | {
"line": 75,
"column": 0
} | [
{
"pp": "K : Type u_2\ninst✝ : DivisionRing K\nx : K\nhx1 : x ≠ 1\nhx0 : x ≠ 0\nn : ℕ\n⊢ ∑ i ∈ range n, x⁻¹ ^ i = (x - 1)⁻¹ * (x - x⁻¹ ^ n * x)",
"ppTerm": "?m.45",
"assigned": true,
"usedConstants": [
"neg_add_rev",
"add_mul",
"AddGroup.toSubtractionMonoid",
"Distrib.leftDis... | [] | have h₁ : x⁻¹ ≠ 1 := by rwa [inv_eq_one_div, Ne, div_eq_iff_mul_eq hx0, one_mul]
have h₂ : x⁻¹ - 1 ≠ 0 := mt sub_eq_zero.1 h₁
have h₃ : x - 1 ≠ 0 := mt sub_eq_zero.1 hx1
have h₄ : x * (x ^ n)⁻¹ = (x ^ n)⁻¹ * x :=
Nat.recOn n (by simp) fun n h => by
rw [pow_succ', mul_inv_rev, ← mul_assoc, h, mul_assoc, ... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.FiveLemma | {
"line": 100,
"column": 2
} | {
"line": 100,
"column": 12
} | {
"line": 101,
"column": 2
} | [
{
"pp": "M₁ : Type u_1\nM₂ : Type u_2\nM₃ : Type u_3\nM₄ : Type u_4\nN₁ : Type u_6\nN₂ : Type u_7\nN₃ : Type u_8\nN₄ : Type u_9\ninst✝⁷ : Group M₁\ninst✝⁶ : Group M₂\ninst✝⁵ : Group M₃\ninst✝⁴ : Group M₄\ninst✝³ : Group N₁\ninst✝² : Group N₂\ninst✝¹ : Group N₃\ninst✝ : Group N₄\nf₁ : M₁ →* M₂\nf₂ : M₂ →* M₃\nf₃... | [
"M₁ : Type u_1\nM₂ : Type u_2\nM₃ : Type u_3\nM₄ : Type u_4\nN₁ : Type u_6\nN₂ : Type u_7\nN₃ : Type u_8\nN₄ : Type u_9\ninst✝⁷ : Group M₁\ninst✝⁶ : Group M₂\ninst✝⁵ : Group M₃\ninst✝⁴ : Group M₄\ninst✝³ : Group N₁\ninst✝² : Group N₂\ninst✝¹ : Group N₃\ninst✝ : Group N₄\nf₁ : M₁ →* M₂\nf₂ : M₂ →* M₃\nf₃ : M₃ →* M₄\... | intro m hm | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.RingTheory.Polynomial.Eisenstein.Criterion | {
"line": 107,
"column": 9
} | {
"line": 107,
"column": 23
} | {
"line": 107,
"column": 23
} | [
{
"pp": "R : Type u_1\ninst✝³ : CommRing R\ninst✝² : IsDomain R\nK : Type u_2\ninst✝¹ : Field K\ninst✝ : Algebra R K\nq f : R[X]\np : ℕ\nhq_irr : Irreducible (map (algebraMap R K) q)\nhq_monic : q.Monic\nhf_lC : (algebraMap R K) f.leadingCoeff ≠ 0\nhf_prim : ∀ (r : R), C r ∣ f → IsUnit r\nhfmodP : map (algebraM... | [
"R : Type u_1\ninst✝³ : CommRing R\ninst✝² : IsDomain R\nK : Type u_2\ninst✝¹ : Field K\ninst✝ : Algebra R K\nq f : R[X]\np : ℕ\nhq_irr : Irreducible (map (algebraMap R K) q)\nhq_monic : q.Monic\nhf_lC : (algebraMap R K) f.leadingCoeff ≠ 0\nhf_prim : ∀ (r : R), C r ∣ f → IsUnit r\nhfmodP : map (algebraMap R K) f = ... | leadingCoeff_C | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Polynomial.Eisenstein.Basic | {
"line": 116,
"column": 2
} | {
"line": 116,
"column": 92
} | {
"line": 117,
"column": 2
} | [
{
"pp": "R : Type u\ninst✝² : CommRing R\nf : R[X]\nS : Type v\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\np : R\nx : S\nhmo : f.Monic\nhf : f.IsWeaklyEisensteinAt (R ∙ p)\nφ : ℕ → R\nhx : x ^ (Polynomial.map (algebraMap R S) f).natDegree = -∑ i, (algebraMap R S) p * ((algebraMap R S) (φ ↑i) * x ^ ↑i)\nhφ : ∀ n ... | [
"R : Type u\ninst✝² : CommRing R\nf : R[X]\nS : Type v\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\np : R\nx : S\nhmo : f.Monic\nhf : f.IsWeaklyEisensteinAt (R ∙ p)\nφ : ℕ → R\nhx : x ^ (Polynomial.map (algebraMap R S) f).natDegree = -∑ i, (algebraMap R S) p * ((algebraMap R S) (φ ↑i) * x ^ ↑i)\nhφ : ∀ n < f.natDegre... | rw [hx, ← mul_sum, neg_eq_neg_one_mul, ← mul_assoc (-1 : S), mul_comm (-1 : S), mul_assoc] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Group.Ideal | {
"line": 80,
"column": 23
} | {
"line": 83,
"column": 71
} | {
"line": 83,
"column": 72
} | [
{
"pp": "M : Type u_1\ninst✝ : Semigroup M\ns : Set M\nx y : M\n⊢ y ∈ s ∪ univ * s → x • y ∈ s ∪ univ * s",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"_private.Mathlib.Algebra.Group.Ideal.0.SemigroupIdeal.coe_closure._simp_1_1",
"Semigroup.toMul",
"ins... | [] | by
rintro (hy | ⟨y, -, z, hz, rfl⟩)
· exact .inr <| mul_mem_mul (mem_univ _) hy
· simpa [← mul_assoc] using .inr <| mul_mem_mul (mem_univ _) hz | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Quaternion | {
"line": 1201,
"column": 6
} | {
"line": 1202,
"column": 55
} | {
"line": 1202,
"column": 56
} | [
{
"pp": "R : Type u_1\ninst✝² : Field R\na✝¹ b✝ : ℍ[R]\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\na✝ b a : ℍ[R]\nha : a ≠ 0\n⊢ a * a⁻¹ = 1",
"ppTerm": "?m.68",
"assigned": true,
"usedConstants": [
"Quaternion.coe",
"Iff.mpr",
"Eq.mpr",
"GroupWithZero.toMonoidWith... | [] | rw [inv_def, Algebra.mul_smul_comm (normSq a)⁻¹ a (star a), self_mul_star, smul_coe,
inv_mul_cancel₀ (normSq_ne_zero.2 ha), coe_one] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Quaternion | {
"line": 1201,
"column": 6
} | {
"line": 1202,
"column": 55
} | {
"line": 1202,
"column": 56
} | [
{
"pp": "R : Type u_1\ninst✝² : Field R\na✝¹ b✝ : ℍ[R]\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\na✝ b a : ℍ[R]\nha : a ≠ 0\n⊢ a * a⁻¹ = 1",
"ppTerm": "?m.68",
"assigned": true,
"usedConstants": [
"Quaternion.coe",
"Iff.mpr",
"Eq.mpr",
"GroupWithZero.toMonoidWith... | [] | rw [inv_def, Algebra.mul_smul_comm (normSq a)⁻¹ a (star a), self_mul_star, smul_coe,
inv_mul_cancel₀ (normSq_ne_zero.2 ha), coe_one] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Quaternion | {
"line": 1201,
"column": 6
} | {
"line": 1202,
"column": 55
} | {
"line": 1202,
"column": 56
} | [
{
"pp": "R : Type u_1\ninst✝² : Field R\na✝¹ b✝ : ℍ[R]\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\na✝ b a : ℍ[R]\nha : a ≠ 0\n⊢ a * a⁻¹ = 1",
"ppTerm": "?m.68",
"assigned": true,
"usedConstants": [
"Quaternion.coe",
"Iff.mpr",
"Eq.mpr",
"GroupWithZero.toMonoidWith... | [] | rw [inv_def, Algebra.mul_smul_comm (normSq a)⁻¹ a (star a), self_mul_star, smul_coe,
inv_mul_cancel₀ (normSq_ne_zero.2 ha), coe_one] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Quaternion | {
"line": 1252,
"column": 38
} | {
"line": 1252,
"column": 44
} | {
"line": 1254,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : Infinite R\n⊢ 1 ≤ 4",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"of_decide_eq_true",
"id",
"instOfNatNat",
"LE.le",
"instLENat",
"Bool.true",
"Nat",
"Bool",
"Eq.refl",
"OfNat.ofNat",
"Decid... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.Algebra.Quaternion | {
"line": 1252,
"column": 38
} | {
"line": 1252,
"column": 44
} | {
"line": 1254,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : Infinite R\n⊢ 1 ≤ 4",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"of_decide_eq_true",
"id",
"instOfNatNat",
"LE.le",
"instLENat",
"Bool.true",
"Nat",
"Bool",
"Eq.refl",
"OfNat.ofNat",
"Decid... | [] | decide | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Quaternion | {
"line": 1252,
"column": 38
} | {
"line": 1252,
"column": 44
} | {
"line": 1254,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : Infinite R\n⊢ 1 ≤ 4",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"of_decide_eq_true",
"id",
"instOfNatNat",
"LE.le",
"instLENat",
"Bool.true",
"Nat",
"Bool",
"Eq.refl",
"OfNat.ofNat",
"Decid... | [] | decide | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Group.Subgroup.ZPowers.Lemmas | {
"line": 79,
"column": 2
} | {
"line": 79,
"column": 96
} | {
"line": 81,
"column": 0
} | [
{
"pp": "a b : ℤ\n⊢ zmultiples a ⊔ zmultiples b = zmultiples ↑(a.gcd b)",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Int.gcd",
"Eq.mpr",
"AddSubgroup.instCompleteLattice",
"Lattice.toSemilatticeSup",
"CompleteLattice.toLattice",
"congrArg",
"Ad... | [] | simp_rw [← closure_eq_zmultiples, zmultiples_eq_closure, ← closure_union, Set.singleton_union] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Algebra.Group.Subgroup.ZPowers.Lemmas | {
"line": 79,
"column": 2
} | {
"line": 79,
"column": 96
} | {
"line": 81,
"column": 0
} | [
{
"pp": "a b : ℤ\n⊢ zmultiples a ⊔ zmultiples b = zmultiples ↑(a.gcd b)",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Int.gcd",
"Eq.mpr",
"AddSubgroup.instCompleteLattice",
"Lattice.toSemilatticeSup",
"CompleteLattice.toLattice",
"congrArg",
"Ad... | [] | simp_rw [← closure_eq_zmultiples, zmultiples_eq_closure, ← closure_union, Set.singleton_union] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Group.Subgroup.ZPowers.Lemmas | {
"line": 79,
"column": 2
} | {
"line": 79,
"column": 96
} | {
"line": 81,
"column": 0
} | [
{
"pp": "a b : ℤ\n⊢ zmultiples a ⊔ zmultiples b = zmultiples ↑(a.gcd b)",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Int.gcd",
"Eq.mpr",
"AddSubgroup.instCompleteLattice",
"Lattice.toSemilatticeSup",
"CompleteLattice.toLattice",
"congrArg",
"Ad... | [] | simp_rw [← closure_eq_zmultiples, zmultiples_eq_closure, ← closure_union, Set.singleton_union] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Group.Submonoid.Finsupp | {
"line": 32,
"column": 6
} | {
"line": 32,
"column": 20
} | {
"line": 34,
"column": 0
} | [
{
"pp": "case h.h_add\nM : Type u_1\ninst✝ : CommMonoid M\nι : Type u_2\nf : ι → M\nx : M\nv : ι →₀ ℕ\nhx : (v.prod fun x1 x2 ↦ f x1 ^ x2) ∈ closure (Set.range f)\nhx' : ∃ a, (v.prod fun x1 x2 ↦ f x1 ^ x2) = a.prod fun x1 x2 ↦ f x1 ^ x2\nw : ι →₀ ℕ\nhy : (w.prod fun x1 x2 ↦ f x1 ^ x2) ∈ closure (Set.range f)\nh... | [] | simp [pow_add] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Group.Submonoid.Finsupp | {
"line": 32,
"column": 6
} | {
"line": 32,
"column": 20
} | {
"line": 34,
"column": 0
} | [
{
"pp": "case h.h_add\nM : Type u_1\ninst✝ : CommMonoid M\nι : Type u_2\nf : ι → M\nx : M\nv : ι →₀ ℕ\nhx : (v.prod fun x1 x2 ↦ f x1 ^ x2) ∈ closure (Set.range f)\nhx' : ∃ a, (v.prod fun x1 x2 ↦ f x1 ^ x2) = a.prod fun x1 x2 ↦ f x1 ^ x2\nw : ι →₀ ℕ\nhy : (w.prod fun x1 x2 ↦ f x1 ^ x2) ∈ closure (Set.range f)\nh... | [] | simp [pow_add] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Group.Submonoid.Finsupp | {
"line": 32,
"column": 6
} | {
"line": 32,
"column": 20
} | {
"line": 34,
"column": 0
} | [
{
"pp": "case h.h_add\nM : Type u_1\ninst✝ : CommMonoid M\nι : Type u_2\nf : ι → M\nx : M\nv : ι →₀ ℕ\nhx : (v.prod fun x1 x2 ↦ f x1 ^ x2) ∈ closure (Set.range f)\nhx' : ∃ a, (v.prod fun x1 x2 ↦ f x1 ^ x2) = a.prod fun x1 x2 ↦ f x1 ^ x2\nw : ι →₀ ℕ\nhy : (w.prod fun x1 x2 ↦ f x1 ^ x2) ∈ closure (Set.range f)\nh... | [] | simp [pow_add] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.GroupWithZero.Pointwise.Finset | {
"line": 65,
"column": 52
} | {
"line": 65,
"column": 81
} | {
"line": 67,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝¹ : GroupWithZero α\ninst✝ : DecidableEq α\ns : Finset α\n⊢ s / 0 ⊆ 0",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Finset.mem_zero._simp_1",
"GroupWithZero.toMonoidWithZero",
"instHDiv",
"GroupWithZero.toDivInvMonoid",
"congrArg... | [] | by simp [subset_iff, mem_div] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.GroupWithZero.Pointwise.Finset | {
"line": 67,
"column": 52
} | {
"line": 67,
"column": 81
} | {
"line": 69,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝¹ : GroupWithZero α\ninst✝ : DecidableEq α\ns : Finset α\n⊢ 0 / s ⊆ 0",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Finset.mem_zero._simp_1",
"GroupWithZero.toMonoidWithZero",
"instHDiv",
"GroupWithZero.toDivInvMonoid",
"congrArg... | [] | by simp [subset_iff, mem_div] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.GroupWithZero.ProdHom | {
"line": 155,
"column": 58
} | {
"line": 157,
"column": 69
} | {
"line": 159,
"column": 0
} | [
{
"pp": "G₀ : Type u_1\nH₀ : Type u_2\ninst✝¹ : GroupWithZero G₀\ninst✝ : GroupWithZero H₀\n⊢ Function.Surjective ⇑(snd G₀ H₀)",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"GroupWithZero.toMonoidWithZero",
"congrArg",
"MonoidWithZeroHom.snd",
"Function.HasRightInv... | [] | by
classical
exact Function.HasRightInverse.surjective ⟨inr .., fun _ ↦ by simp⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.GroupWithZero.Range | {
"line": 304,
"column": 8
} | {
"line": 304,
"column": 35
} | {
"line": 304,
"column": 35
} | [
{
"pp": "case some\nA : Type u_1\nB : Type u_2\ninst✝¹ : MonoidWithZero A\ninst✝ : CommGroupWithZero B\nf : A →*₀ B\nx : Bˣ\nhx : x ∈ f.valueGroup\n⊢ some ⟨x, hx⟩ = 0 ∨ ∃ r s, ∃ (hr : f r ≠ 0) (hs : f s ≠ 0), some ⟨x, hx⟩ = ↑(valueGroup.mk f r s hr hs)",
"ppTerm": "?some",
"assigned": true,
"usedCon... | [
"case some\nA : Type u_1\nB : Type u_2\ninst✝¹ : MonoidWithZero A\ninst✝ : CommGroupWithZero B\nf : A →*₀ B\nx : Bˣ\nhx✝ : x ∈ f.valueGroup\nhx : ∃ a, f a ≠ 0 ∧ ∃ x_1, f x_1 ≠ 0 ∧ f a * ↑x = f x_1\n⊢ some ⟨x, hx✝⟩ = 0 ∨ ∃ r s, ∃ (hr : f r ≠ 0) (hs : f s ≠ 0), some ⟨x, hx✝⟩ = ↑(valueGroup.mk f r s hr hs)"
] | mem_valueGroup_iff_of_comm' | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Idempotents.Basic | {
"line": 79,
"column": 16
} | {
"line": 79,
"column": 26
} | {
"line": 80,
"column": 16
} | [
{
"pp": "case right\nC : Type u_1\ninst✝ : Category.{v_1, u_1} C\na✝ : IsIdempotentComplete C\nX : C\np : X ⟶ X\nhp : p ≫ p = p\nY : C\ni : Y ⟶ X\ne : X ⟶ Y\nh₁ : i ≫ e = 𝟙 Y\nh₂ : e ≫ i = p\ns : Fork (𝟙 X) p\n⊢ ∀ {m : s.pt ⟶ (Fork.ofι i ⋯).pt}, m ≫ (Fork.ofι i ⋯).ι = s.ι → m = s.ι ≫ e",
"ppTerm": "?right... | [
"case right\nC : Type u_1\ninst✝ : Category.{v_1, u_1} C\na✝ : IsIdempotentComplete C\nX : C\np : X ⟶ X\nhp : p ≫ p = p\nY : C\ni : Y ⟶ X\ne : X ⟶ Y\nh₁ : i ≫ e = 𝟙 Y\nh₂ : e ≫ i = p\ns : Fork (𝟙 X) p\nm : s.pt ⟶ (Fork.ofι i ⋯).pt\nhm : m ≫ (Fork.ofι i ⋯).ι = s.ι\n⊢ m = s.ι ≫ e"
] | intro m hm | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.CategoryTheory.Idempotents.Karoubi | {
"line": 159,
"column": 18
} | {
"line": 159,
"column": 74
} | {
"line": 159,
"column": 74
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Preadditive C\nP Q : Karoubi C\nf : P ⟶ Q\n⊢ P.p ≫ (-f.f) ≫ Q.p = -f.f",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NegZeroClass.toNeg",
"CategoryTheory.Idempotents.Karoubi.Hom.f",
"Subtrac... | [] | by simpa only [neg_comp, comp_neg, neg_inj] using f.comm | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Idempotents.FunctorCategories | {
"line": 83,
"column": 4
} | {
"line": 83,
"column": 73
} | {
"line": 84,
"column": 2
} | [
{
"pp": "case h.left\nJ : Type u_1\nC : Type u_2\ninst✝² : Category.{v_1, u_1} J\ninst✝¹ : Category.{v_2, u_2} C\nP Q : Karoubi (J ⥤ C)\nf : P ⟶ Q\nX : J\ninst✝ : IsIdempotentComplete C\nF : J ⥤ C\np : F ⟶ F\nhp : p ≫ p = p\nhC : ∀ (X : C) (p : X ⟶ X), p ≫ p = p → HasEqualizer (𝟙 X) p\nthis : ∀ (j : J), HasEqu... | [] | rw [assoc, equalizer.lift_ι, ← equalizer.condition, id_comp, comp_id] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.AlgebraicTopology.SimplexCategory.Basic | {
"line": 106,
"column": 38
} | {
"line": 106,
"column": 44
} | {
"line": 106,
"column": 45
} | [
{
"pp": "f : ⦋1⦌ ⟶ ⦋1⦌\ne0 : (Hom.toOrderHom f) 0 = 1\ne1 : (Hom.toOrderHom f) 1 = 0\n⊢ 0 ≤ 1",
"ppTerm": "?m.1280",
"assigned": true,
"usedConstants": [
"of_decide_eq_true",
"id",
"Fin.instOfNat",
"instOfNatNat",
"LE.le",
"instLEFin",
"Bool.true",
"Na... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.AlgebraicTopology.SimplexCategory.Basic | {
"line": 106,
"column": 38
} | {
"line": 106,
"column": 44
} | {
"line": 106,
"column": 45
} | [
{
"pp": "f : ⦋1⦌ ⟶ ⦋1⦌\ne0 : (Hom.toOrderHom f) 0 = 1\ne1 : (Hom.toOrderHom f) 1 = 0\n⊢ 0 ≤ 1",
"ppTerm": "?m.1280",
"assigned": true,
"usedConstants": [
"of_decide_eq_true",
"id",
"Fin.instOfNat",
"instOfNatNat",
"LE.le",
"instLEFin",
"Bool.true",
"Na... | [] | decide | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicTopology.SimplexCategory.Basic | {
"line": 106,
"column": 38
} | {
"line": 106,
"column": 44
} | {
"line": 106,
"column": 45
} | [
{
"pp": "f : ⦋1⦌ ⟶ ⦋1⦌\ne0 : (Hom.toOrderHom f) 0 = 1\ne1 : (Hom.toOrderHom f) 1 = 0\n⊢ 0 ≤ 1",
"ppTerm": "?m.1280",
"assigned": true,
"usedConstants": [
"of_decide_eq_true",
"id",
"Fin.instOfNat",
"instOfNatNat",
"LE.le",
"instLEFin",
"Bool.true",
"Na... | [] | decide | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Idempotents.FunctorCategories | {
"line": 99,
"column": 8
} | {
"line": 99,
"column": 46
} | {
"line": 100,
"column": 8
} | [
{
"pp": "J : Type u_1\nC : Type u_2\ninst✝¹ : Category.{v_1, u_1} J\ninst✝ : Category.{v_2, u_2} C\nP✝ Q : Karoubi (J ⥤ C)\nf : P✝ ⟶ Q\nX : J\nP : Karoubi (J ⥤ C)\nj j' : J\nφ : j ⟶ j'\n⊢ { X := P.X.obj j, p := P.p.app j, idem := ⋯ }.p ≫\n (P.p.app j ≫ P.X.map φ) ≫ { X := P.X.obj j', p := P.p.app j', idem ... | [
"J : Type u_1\nC : Type u_2\ninst✝¹ : Category.{v_1, u_1} J\ninst✝ : Category.{v_2, u_2} C\nP✝ Q : Karoubi (J ⥤ C)\nf : P✝ ⟶ Q\nX : J\nP : Karoubi (J ⥤ C)\nj j' : J\nφ : j ⟶ j'\n⊢ P.p.app j ≫ P.p.app j ≫ P.p.app j ≫ P.X.map φ = P.p.app j ≫ P.X.map φ"
] | simp only [NatTrans.naturality, assoc] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.AlgebraicTopology.SimplexCategory.Basic | {
"line": 108,
"column": 23
} | {
"line": 108,
"column": 29
} | {
"line": 108,
"column": 29
} | [
{
"pp": "f : ⦋1⦌ ⟶ ⦋1⦌\ne0 : (Hom.toOrderHom f) 0 = 1\ne1 : (Hom.toOrderHom f) 1 = 0\nthis : 1 ≤ 0\n⊢ ¬1 ≤ 0",
"ppTerm": "?m.1293",
"assigned": true,
"usedConstants": [
"instDecidableNot",
"of_decide_eq_true",
"PartialOrder.toPreorder",
"Preorder.toLE",
"id",
"Fin... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.AlgebraicTopology.SimplexCategory.Basic | {
"line": 108,
"column": 23
} | {
"line": 108,
"column": 29
} | {
"line": 108,
"column": 29
} | [
{
"pp": "f : ⦋1⦌ ⟶ ⦋1⦌\ne0 : (Hom.toOrderHom f) 0 = 1\ne1 : (Hom.toOrderHom f) 1 = 0\nthis : 1 ≤ 0\n⊢ ¬1 ≤ 0",
"ppTerm": "?m.1293",
"assigned": true,
"usedConstants": [
"instDecidableNot",
"of_decide_eq_true",
"PartialOrder.toPreorder",
"Preorder.toLE",
"id",
"Fin... | [] | decide | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicTopology.SimplexCategory.Basic | {
"line": 108,
"column": 23
} | {
"line": 108,
"column": 29
} | {
"line": 108,
"column": 29
} | [
{
"pp": "f : ⦋1⦌ ⟶ ⦋1⦌\ne0 : (Hom.toOrderHom f) 0 = 1\ne1 : (Hom.toOrderHom f) 1 = 0\nthis : 1 ≤ 0\n⊢ ¬1 ≤ 0",
"ppTerm": "?m.1293",
"assigned": true,
"usedConstants": [
"instDecidableNot",
"of_decide_eq_true",
"PartialOrder.toPreorder",
"Preorder.toLE",
"id",
"Fin... | [] | decide | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicTopology.SimplexCategory.Basic | {
"line": 157,
"column": 56
} | {
"line": 157,
"column": 62
} | {
"line": 159,
"column": 0
} | [
{
"pp": "⊢ mkOfSucc 0 = 𝟙 ⦋1⦌",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"of_decide_eq_true",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"SimplexCategory.instDecidableEqHom",
"CategoryTheory.CategoryStruct.id",
"id",
"Fin.instOfN... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.AlgebraicTopology.SimplexCategory.Basic | {
"line": 157,
"column": 56
} | {
"line": 157,
"column": 62
} | {
"line": 159,
"column": 0
} | [
{
"pp": "⊢ mkOfSucc 0 = 𝟙 ⦋1⦌",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"of_decide_eq_true",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"SimplexCategory.instDecidableEqHom",
"CategoryTheory.CategoryStruct.id",
"id",
"Fin.instOfN... | [] | decide | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicTopology.SimplexCategory.Basic | {
"line": 157,
"column": 56
} | {
"line": 157,
"column": 62
} | {
"line": 159,
"column": 0
} | [
{
"pp": "⊢ mkOfSucc 0 = 𝟙 ⦋1⦌",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"of_decide_eq_true",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"SimplexCategory.instDecidableEqHom",
"CategoryTheory.CategoryStruct.id",
"id",
"Fin.instOfN... | [] | decide | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Idempotents.FunctorCategories | {
"line": 149,
"column": 4
} | {
"line": 151,
"column": 8
} | {
"line": 152,
"column": 2
} | [
{
"pp": "case h_map\nJ : Type u_1\nC : Type u_2\ninst✝¹ : Category.{v_1, u_1} J\ninst✝ : Category.{v_2, u_2} C\n⊢ autoParam\n (∀ (X Y : J ⥤ C) (f : X ⟶ Y),\n (toKaroubi (J ⥤ C) ⋙ karoubiFunctorCategoryEmbedding J C).map f =\n eqToHom ⋯ ≫ ((Functor.whiskeringRight J C (Karoubi C)).obj (toKaroubi C... | [] | intro X Y f
ext j
simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Idempotents.FunctorCategories | {
"line": 149,
"column": 4
} | {
"line": 151,
"column": 8
} | {
"line": 152,
"column": 2
} | [
{
"pp": "case h_map\nJ : Type u_1\nC : Type u_2\ninst✝¹ : Category.{v_1, u_1} J\ninst✝ : Category.{v_2, u_2} C\n⊢ autoParam\n (∀ (X Y : J ⥤ C) (f : X ⟶ Y),\n (toKaroubi (J ⥤ C) ⋙ karoubiFunctorCategoryEmbedding J C).map f =\n eqToHom ⋯ ≫ ((Functor.whiskeringRight J C (Karoubi C)).obj (toKaroubi C... | [] | intro X Y f
ext j
simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicTopology.AlternatingFaceMapComplex | {
"line": 146,
"column": 4
} | {
"line": 146,
"column": 8
} | {
"line": 147,
"column": 4
} | [
{
"pp": "case e_a\nC : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Preadditive C\nX Y : SimplicialObject C\nf : X ⟶ Y\nn : ℕ\nx✝¹ : Fin (n + 2)\nx✝ : x✝¹ ∈ Finset.univ\n⊢ f.app (op ⦋n + 1⦌) ≫ Y.δ x✝¹ = X.δ x✝¹ ≫ f.app (op ⦋n⦌)",
"ppTerm": "?e_a✝",
"assigned": true,
"usedConstants": [
"Op... | [
"case e_a\nC : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Preadditive C\nX Y : SimplicialObject C\nf : X ⟶ Y\nn : ℕ\nx✝¹ : Fin (n + 2)\nx✝ : x✝¹ ∈ Finset.univ\n⊢ X.δ x✝¹ ≫ f.app (op ⦋n⦌) = f.app (op ⦋n + 1⦌) ≫ Y.δ x✝¹"
] | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
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