module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.Algebra.Algebra.Epi | {
"line": 140,
"column": 58
} | {
"line": 147,
"column": 61
} | {
"line": 148,
"column": 2
} | [
{
"pp": "R : Type u_1\nA : Type u_2\ninst✝⁷ : CommSemiring R\ninst✝⁶ : CommSemiring A\ninst✝⁵ : Algebra R A\ninst✝⁴ : Algebra.IsEpi R A\nM : Type u_3\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\ninst✝¹ : Module A M\ninst✝ : IsScalarTower R A M\nthis : ∀ (a : A) (m : M), 1 ⊗ₜ[R] (a • m) = a ⊗ₜ[R] m\n⊢ Injecti... | [] | by
let f : M →ₗ[R] A ⊗[R] M :=
{ toFun m := 1 ⊗ₜ m
map_add' m n := tmul_add _ _ _
map_smul' r m := tmul_smul _ _ _ }
have aux : f ∘ₗ (lift <| LinearMap.restrictScalars₁₂ R R (LinearMap.lsmul A M)) = .id := by
ext a m; simpa using! this a m
exact HasLeftInverse.injective ⟨f, fun x... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Star.Pointwise | {
"line": 109,
"column": 2
} | {
"line": 110,
"column": 13
} | {
"line": 112,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝¹ : AddMonoid α\ninst✝ : StarAddMonoid α\ns t : Set α\n⊢ (s + t)⋆ = s⋆ + t⋆",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Set.image2_image_right",
"Set.star",
"AddMonoid.toAddSemigroup",
"congrArg",
"StarAddMonoid... | [] | simp_rw [← image_star, ← image2_add, image_image2, image2_image_left, image2_image_right,
star_add] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Algebra.Star.Pointwise | {
"line": 109,
"column": 2
} | {
"line": 110,
"column": 13
} | {
"line": 112,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝¹ : AddMonoid α\ninst✝ : StarAddMonoid α\ns t : Set α\n⊢ (s + t)⋆ = s⋆ + t⋆",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Set.image2_image_right",
"Set.star",
"AddMonoid.toAddSemigroup",
"congrArg",
"StarAddMonoid... | [] | simp_rw [← image_star, ← image2_add, image_image2, image2_image_left, image2_image_right,
star_add] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Star.Pointwise | {
"line": 109,
"column": 2
} | {
"line": 110,
"column": 13
} | {
"line": 112,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝¹ : AddMonoid α\ninst✝ : StarAddMonoid α\ns t : Set α\n⊢ (s + t)⋆ = s⋆ + t⋆",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Set.image2_image_right",
"Set.star",
"AddMonoid.toAddSemigroup",
"congrArg",
"StarAddMonoid... | [] | simp_rw [← image_star, ← image2_add, image_image2, image2_image_left, image2_image_right,
star_add] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Star.StarProjection | {
"line": 127,
"column": 2
} | {
"line": 127,
"column": 31
} | {
"line": 128,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝² : NonUnitalRing R\ninst✝¹ : StarRing R\ninst✝ : IsAddTorsionFree R\np q : R\nhp : IsStarProjection p\nhq : IsStarProjection q\n⊢ p * q = p ∧ q * star p = p ↔ p * q = p",
"ppTerm": "?m.47",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"co... | [
"R : Type u_1\ninst✝² : NonUnitalRing R\ninst✝¹ : StarRing R\ninst✝ : IsAddTorsionFree R\np q : R\nhp : IsStarProjection p\nhq : IsStarProjection q\n⊢ p * q = p ∧ star q * star p = p ↔ p * q = p"
] | nth_rw 2 [← hq.isSelfAdjoint] | Mathlib.Tactic._aux_Mathlib_Tactic_NthRewrite___macroRules_Mathlib_Tactic_tacticNth_rw______1 | Mathlib.Tactic.tacticNth_rw_____ |
Mathlib.Algebra.Algebra.Spectrum.Basic | {
"line": 164,
"column": 71
} | {
"line": 165,
"column": 61
} | {
"line": 167,
"column": 0
} | [
{
"pp": "R : Type u\nA : Type v\ninst✝³ : CommSemiring R\ninst✝² : Ring A\ninst✝¹ : Algebra R A\ninst✝ : Subsingleton A\na : A\n⊢ σ a = ∅",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"spectrum.eq_1",
"spectrum",
"congrArg",
"Compl.compl",
"S... | [] | by
rw [spectrum, resolventSet_of_subsingleton, Set.compl_univ] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.TensorProduct.Basic | {
"line": 623,
"column": 23
} | {
"line": 623,
"column": 72
} | {
"line": 624,
"column": 2
} | [
{
"pp": "R : Type u_1\nA : Type u_2\nB : Type u_3\nM : Type u_4\ninst✝¹¹ : CommSemiring R\ninst✝¹⁰ : AddCommMonoid M\ninst✝⁹ : Module R M\ninst✝⁸ : Semiring A\ninst✝⁷ : Semiring B\ninst✝⁶ : Module A M\ninst✝⁵ : Module B M\ninst✝⁴ : Algebra R A\ninst✝³ : Algebra R B\ninst✝² : IsScalarTower R A M\ninst✝¹ : IsScal... | [] | simp only [(· • ·), map_add, LinearMap.add_apply] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.TensorProduct.Basic | {
"line": 623,
"column": 23
} | {
"line": 623,
"column": 72
} | {
"line": 624,
"column": 2
} | [
{
"pp": "R : Type u_1\nA : Type u_2\nB : Type u_3\nM : Type u_4\ninst✝¹¹ : CommSemiring R\ninst✝¹⁰ : AddCommMonoid M\ninst✝⁹ : Module R M\ninst✝⁸ : Semiring A\ninst✝⁷ : Semiring B\ninst✝⁶ : Module A M\ninst✝⁵ : Module B M\ninst✝⁴ : Algebra R A\ninst✝³ : Algebra R B\ninst✝² : IsScalarTower R A M\ninst✝¹ : IsScal... | [] | simp only [(· • ·), map_add, LinearMap.add_apply] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.TensorProduct.Basic | {
"line": 623,
"column": 23
} | {
"line": 623,
"column": 72
} | {
"line": 624,
"column": 2
} | [
{
"pp": "R : Type u_1\nA : Type u_2\nB : Type u_3\nM : Type u_4\ninst✝¹¹ : CommSemiring R\ninst✝¹⁰ : AddCommMonoid M\ninst✝⁹ : Module R M\ninst✝⁸ : Semiring A\ninst✝⁷ : Semiring B\ninst✝⁶ : Module A M\ninst✝⁵ : Module B M\ninst✝⁴ : Algebra R A\ninst✝³ : Algebra R B\ninst✝² : IsScalarTower R A M\ninst✝¹ : IsScal... | [] | simp only [(· • ·), map_add, LinearMap.add_apply] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Algebra.Unitization | {
"line": 745,
"column": 8
} | {
"line": 749,
"column": 75
} | {
"line": 750,
"column": 2
} | [
{
"pp": "case inl_add_inr.inl_add_inr\nS : Type u_1\nR : Type u_2\nA : Type u_3\ninst✝¹² : CommSemiring S\ninst✝¹¹ : CommSemiring R\ninst✝¹⁰ : NonUnitalSemiring A\ninst✝⁹ : Module R A\ninst✝⁸ : SMulCommClass R A A\ninst✝⁷ : IsScalarTower R A A\nB : Type u_4\ninst✝⁶ : Semiring B\ninst✝⁵ : Algebra S B\ninst✝⁴ : A... | [] | simp only [fst_mul, fst_add, fst_inl, fst_inr, snd_mul, snd_add, snd_inl, snd_inr, add_zero,
map_mul, zero_add, map_add, map_smul φ]
rw [add_mul, mul_add, mul_add]
rw [← Algebra.commutes _ (φ x_a)]
simp only [Algebra.algebraMap_eq_smul_one, smul_one_mul, add_assoc] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Algebra.Unitization | {
"line": 745,
"column": 8
} | {
"line": 749,
"column": 75
} | {
"line": 750,
"column": 2
} | [
{
"pp": "case inl_add_inr.inl_add_inr\nS : Type u_1\nR : Type u_2\nA : Type u_3\ninst✝¹² : CommSemiring S\ninst✝¹¹ : CommSemiring R\ninst✝¹⁰ : NonUnitalSemiring A\ninst✝⁹ : Module R A\ninst✝⁸ : SMulCommClass R A A\ninst✝⁷ : IsScalarTower R A A\nB : Type u_4\ninst✝⁶ : Semiring B\ninst✝⁵ : Algebra S B\ninst✝⁴ : A... | [] | simp only [fst_mul, fst_add, fst_inl, fst_inr, snd_mul, snd_add, snd_inl, snd_inr, add_zero,
map_mul, zero_add, map_add, map_smul φ]
rw [add_mul, mul_add, mul_add]
rw [← Algebra.commutes _ (φ x_a)]
simp only [Algebra.algebraMap_eq_smul_one, smul_one_mul, add_assoc] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Algebra.Spectrum.Quasispectrum | {
"line": 211,
"column": 2
} | {
"line": 211,
"column": 33
} | {
"line": 212,
"column": 2
} | [
{
"pp": "F : Type u_1\nR : Type u_2\nS : Type u_3\ninst✝³ : NonUnitalSemiring R\ninst✝² : NonUnitalSemiring S\ninst✝¹ : FunLike F R S\ninst✝ : NonUnitalRingHomClass F R S\nf : F\nx : R\nhx : IsQuasiregular x\n⊢ IsQuasiregular (f x)",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Eq.m... | [
"F : Type u_1\nR : Type u_2\nS : Type u_3\ninst✝³ : NonUnitalSemiring R\ninst✝² : NonUnitalSemiring S\ninst✝¹ : FunLike F R S\ninst✝ : NonUnitalRingHomClass F R S\nf : F\nx : R\nhx : ∃ y, y + x + x * y = 0 ∧ x + y + y * x = 0\n⊢ ∃ y, y + f x + f x * y = 0 ∧ f x + y + y * f x = 0"
] | rw [isQuasiregular_iff] at hx ⊢ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Algebra.Spectrum.Quasispectrum | {
"line": 454,
"column": 61
} | {
"line": 454,
"column": 88
} | {
"line": 454,
"column": 88
} | [
{
"pp": "S : Type u_3\nR : Type u_4\nA : Type u_5\ninst✝⁸ : Semifield R\ninst✝⁷ : Field S\ninst✝⁶ : NonUnitalRing A\ninst✝⁵ : Algebra R S\ninst✝⁴ : Module S A\ninst✝³ : IsScalarTower S A A\ninst✝² : SMulCommClass S A A\ninst✝¹ : Module R A\ninst✝ : IsScalarTower R S A\na : A\nr : R\n⊢ (algebraMap R S) r ∈ spect... | [] | spectrum.algebraMap_mem_iff | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Algebra.Order.Star.Basic | {
"line": 235,
"column": 5
} | {
"line": 235,
"column": 31
} | {
"line": 235,
"column": 31
} | [
{
"pp": "R : Type u_1\ninst✝³ : NonUnitalSemiring R\ninst✝² : PartialOrder R\ninst✝¹ : StarRing R\ninst✝ : StarOrderedRing R\na : R\nha : a ∈ AddSubmonoid.closure (range fun s ↦ star s * s)\nc : R\n⊢ 0 ≤ star c * 0 * c",
"ppTerm": "?m.49",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"... | [] | by rw [mul_zero, zero_mul] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Order.Star.Basic | {
"line": 382,
"column": 2
} | {
"line": 382,
"column": 38
} | {
"line": 383,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝³ : Semiring R\ninst✝² : PartialOrder R\ninst✝¹ : StarRing R\ninst✝ : StarOrderedRing R\nu x : R\nhu : IsUnit u\nh : 0 ≤ u * x * star u\n⊢ 0 ≤ x",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"MulOne.toOne",
"HMul.hMul",
"Monoid.toMulOneClass"... | [
"R : Type u_1\ninst✝³ : Semiring R\ninst✝² : PartialOrder R\ninst✝¹ : StarRing R\ninst✝ : StarOrderedRing R\nu x : R\nhu : IsUnit u\nh : 0 ≤ u * x * star u\nv : R\nhv : v * u = 1\n⊢ 0 ≤ x"
] | obtain ⟨v, hv⟩ := hu.exists_left_inv | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.RingTheory.TensorProduct.Maps | {
"line": 665,
"column": 4
} | {
"line": 665,
"column": 69
} | {
"line": 667,
"column": 0
} | [
{
"pp": "R : Type uR\nS : Type uS\nA : Type uA\nB : Type uB\ninst✝⁸ : CommSemiring R\ninst✝⁷ : CommSemiring S\ninst✝⁶ : Algebra R S\ninst✝⁵ : Semiring A\ninst✝⁴ : Algebra R A\ninst✝³ : Algebra S A\ninst✝² : IsScalarTower R S A\ninst✝¹ : Semiring B\ninst✝ : Algebra R B\nh : Function.Bijective ⇑(algebraMap R B)\n... | [] | exact Algebra.TensorProduct.map_bijective Function.bijective_id h | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.TensorProduct.Maps | {
"line": 665,
"column": 4
} | {
"line": 665,
"column": 69
} | {
"line": 667,
"column": 0
} | [
{
"pp": "R : Type uR\nS : Type uS\nA : Type uA\nB : Type uB\ninst✝⁸ : CommSemiring R\ninst✝⁷ : CommSemiring S\ninst✝⁶ : Algebra R S\ninst✝⁵ : Semiring A\ninst✝⁴ : Algebra R A\ninst✝³ : Algebra S A\ninst✝² : IsScalarTower R S A\ninst✝¹ : Semiring B\ninst✝ : Algebra R B\nh : Function.Bijective ⇑(algebraMap R B)\n... | [] | exact Algebra.TensorProduct.map_bijective Function.bijective_id h | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.TensorProduct.Maps | {
"line": 665,
"column": 4
} | {
"line": 665,
"column": 69
} | {
"line": 667,
"column": 0
} | [
{
"pp": "R : Type uR\nS : Type uS\nA : Type uA\nB : Type uB\ninst✝⁸ : CommSemiring R\ninst✝⁷ : CommSemiring S\ninst✝⁶ : Algebra R S\ninst✝⁵ : Semiring A\ninst✝⁴ : Algebra R A\ninst✝³ : Algebra S A\ninst✝² : IsScalarTower R S A\ninst✝¹ : Semiring B\ninst✝ : Algebra R B\nh : Function.Bijective ⇑(algebraMap R B)\n... | [] | exact Algebra.TensorProduct.map_bijective Function.bijective_id h | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Dimension.Constructions | {
"line": 196,
"column": 2
} | {
"line": 198,
"column": 54
} | {
"line": 200,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝⁴ : Semiring R\ninst✝³ : StrongRankCondition R\nι : Type v\nM : ι → Type w\ninst✝² : (i : ι) → AddCommMonoid (M i)\ninst✝¹ : (i : ι) → Module R (M i)\ninst✝ : ∀ (i : ι), Free R (M i)\n⊢ Module.rank R (⨁ (i : ι), M i) = sum fun i ↦ Module.rank R (M i)",
"ppTerm": "?m.29",
"assig... | [] | let B i := chooseBasis R (M i)
let b : Basis _ R (⨁ i, M i) := DFinsupp.basis fun i => B i
simp [← b.mk_eq_rank'', fun i => (B i).mk_eq_rank''] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Dimension.Constructions | {
"line": 196,
"column": 2
} | {
"line": 198,
"column": 54
} | {
"line": 200,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝⁴ : Semiring R\ninst✝³ : StrongRankCondition R\nι : Type v\nM : ι → Type w\ninst✝² : (i : ι) → AddCommMonoid (M i)\ninst✝¹ : (i : ι) → Module R (M i)\ninst✝ : ∀ (i : ι), Free R (M i)\n⊢ Module.rank R (⨁ (i : ι), M i) = sum fun i ↦ Module.rank R (M i)",
"ppTerm": "?m.29",
"assig... | [] | let B i := chooseBasis R (M i)
let b : Basis _ R (⨁ i, M i) := DFinsupp.basis fun i => B i
simp [← b.mk_eq_rank'', fun i => (B i).mk_eq_rank''] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Dimension.Constructions | {
"line": 433,
"column": 2
} | {
"line": 433,
"column": 38
} | {
"line": 435,
"column": 0
} | [
{
"pp": "R : Type u\nM : Type v\ninst✝³ : Semiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ninst✝ : StrongRankCondition R\ns : Finset M\n⊢ Module.rank R ↥(span R ↑s) ≤ ↑s.card",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Submodule",
"Cardinal",
"congrArg",
... | [] | simpa using rank_span_le (s : Set M) | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.LinearAlgebra.Dimension.Constructions | {
"line": 433,
"column": 2
} | {
"line": 433,
"column": 38
} | {
"line": 435,
"column": 0
} | [
{
"pp": "R : Type u\nM : Type v\ninst✝³ : Semiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ninst✝ : StrongRankCondition R\ns : Finset M\n⊢ Module.rank R ↥(span R ↑s) ≤ ↑s.card",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Submodule",
"Cardinal",
"congrArg",
... | [] | simpa using rank_span_le (s : Set M) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Dimension.Constructions | {
"line": 433,
"column": 2
} | {
"line": 433,
"column": 38
} | {
"line": 435,
"column": 0
} | [
{
"pp": "R : Type u\nM : Type v\ninst✝³ : Semiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ninst✝ : StrongRankCondition R\ns : Finset M\n⊢ Module.rank R ↥(span R ↑s) ≤ ↑s.card",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Submodule",
"Cardinal",
"congrArg",
... | [] | simpa using rank_span_le (s : Set M) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Dimension.Constructions | {
"line": 537,
"column": 2
} | {
"line": 537,
"column": 48
} | {
"line": 538,
"column": 2
} | [
{
"pp": "F : Type u_2\nE : Type u_3\ninst✝² : CommSemiring F\ninst✝¹ : Semiring E\ninst✝ : Algebra F E\n⊢ Module.rank F ↥⊤ = Module.rank F E",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Subalgebra.instSetLike",
"Eq.mpr",
"Submodule",
"Lattice.toSemilatticeSup",
... | [
"F : Type u_2\nE : Type u_3\ninst✝² : CommSemiring F\ninst✝¹ : Semiring E\ninst✝ : Algebra F E\n⊢ Module.rank F ↥⊤ = Module.rank F E"
] | rw [subalgebra_top_rank_eq_submodule_top_rank] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.Basis.VectorSpace | {
"line": 294,
"column": 2
} | {
"line": 294,
"column": 85
} | {
"line": 295,
"column": 2
} | [
{
"pp": "K : Type u_3\nV : Type u_4\nV' : Type u_5\ninst✝⁴ : DivisionRing K\ninst✝³ : AddCommGroup V\ninst✝² : AddCommGroup V'\ninst✝¹ : Module K V\ninst✝ : Module K V'\np : Submodule K V\nv : V\nf : ↥p →ₗ[K] V'\nhv : v ∉ p\ny : V'\n⊢ ∃ g, g ∘ₗ p.subtype = f ∧ g v = y",
"ppTerm": "?m.57",
"assigned": tr... | [
"K : Type u_3\nV : Type u_4\nV' : Type u_5\ninst✝⁴ : DivisionRing K\ninst✝³ : AddCommGroup V\ninst✝² : AddCommGroup V'\ninst✝¹ : Module K V\ninst✝ : Module K V'\np : Submodule K V\nv : V\nf : ↥p →ₗ[K] V'\nhv : v ∉ p\ny : V'\ng : V →ₗ[K] V'\nhg :\n g ∘ₗ ({ domain := p, toFun := f }.supSpanSingleton v y hv).domain.s... | rcases (LinearPMap.supSpanSingleton ⟨p, f⟩ v y hv).toFun.exists_extend with ⟨g, hg⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Algebra.FreeAlgebra | {
"line": 256,
"column": 18
} | {
"line": 256,
"column": 30
} | {
"line": 257,
"column": 2
} | [
{
"pp": "R : Type u_1\nX : Type u_2\ninst✝ : CommSemiring R\n⊢ Quot.mk (Rel R X) (Pre.ofScalar ↑0) = 0",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"congrArg",
"CommSemiring.toSemiring",
"AddMonoid.toAddZ... | [] | by simp; rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Polynomial.Eval.Defs | {
"line": 161,
"column": 6
} | {
"line": 161,
"column": 24
} | {
"line": 161,
"column": 25
} | [
{
"pp": "R : Type u\nS : Type v\na : R\ninst✝¹ : Semiring R\np : R[X]\ninst✝ : Semiring S\nf : R →+* S\nx : S\nh : Commute (f a) x\n⊢ eval₂ f x (p * C a) = eval₂ f x p * f a",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Polynomial.C",
"HMul.hMul",
"cong... | [
"R : Type u\nS : Type v\na : R\ninst✝¹ : Semiring R\np : R[X]\ninst✝ : Semiring S\nf : R →+* S\nx : S\nh : Commute (f a) x\n⊢ eval₂ f x p * eval₂ f x (C a) = eval₂ f x p * f a",
"case hf\nR : Type u\nS : Type v\na : R\ninst✝¹ : Semiring R\np : R[X]\ninst✝ : Semiring S\nf : R →+* S\nx : S\nh : Commute (f a) x\n⊢ ∀... | eval₂_mul_noncomm, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.Eval.Defs | {
"line": 413,
"column": 48
} | {
"line": 413,
"column": 66
} | {
"line": 415,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝ : Semiring R\np : R[X]\n⊢ comp 0 p = 0",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Polynomial.C",
"congrArg",
"Polynomial.C_0",
"RingHom",
"id",
"Polynomial",
"RingHom.instFunLike",
"Polynomial.... | [] | rw [← C_0, C_comp] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Polynomial.Eval.Defs | {
"line": 413,
"column": 48
} | {
"line": 413,
"column": 66
} | {
"line": 415,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝ : Semiring R\np : R[X]\n⊢ comp 0 p = 0",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Polynomial.C",
"congrArg",
"Polynomial.C_0",
"RingHom",
"id",
"Polynomial",
"RingHom.instFunLike",
"Polynomial.... | [] | rw [← C_0, C_comp] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Polynomial.Eval.Defs | {
"line": 413,
"column": 48
} | {
"line": 413,
"column": 66
} | {
"line": 415,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝ : Semiring R\np : R[X]\n⊢ comp 0 p = 0",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Polynomial.C",
"congrArg",
"Polynomial.C_0",
"RingHom",
"id",
"Polynomial",
"RingHom.instFunLike",
"Polynomial.... | [] | rw [← C_0, C_comp] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.MonoidAlgebra.Support | {
"line": 71,
"column": 2
} | {
"line": 73,
"column": 56
} | {
"line": 75,
"column": 0
} | [
{
"pp": "k : Type u₁\nG : Type u₂\ninst✝² : Semiring k\ninst✝¹ : Mul G\ninst✝ : DecidableEq G\nf : k[G]\nr : k\nhr : ∀ (y : k), y * r = 0 ↔ y = 0\nx : G\nrx : IsRightRegular x\n⊢ (f * single x r).coeff.support = image (fun x_1 ↦ x_1 * x) f.coeff.support",
"ppTerm": "?m.43",
"assigned": true,
"usedCo... | [] | refine subset_antisymm (support_coeff_mul_single_subset f _ _) fun y hy => ?_
obtain ⟨y, yf, rfl⟩ : ∃ a : G, a ∈ f.coeff.support ∧ a * x = y := by grind
simp [coeff_mul, mem_support_iff.mp yf, hr, rx.eq_iff] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.MonoidAlgebra.Support | {
"line": 71,
"column": 2
} | {
"line": 73,
"column": 56
} | {
"line": 75,
"column": 0
} | [
{
"pp": "k : Type u₁\nG : Type u₂\ninst✝² : Semiring k\ninst✝¹ : Mul G\ninst✝ : DecidableEq G\nf : k[G]\nr : k\nhr : ∀ (y : k), y * r = 0 ↔ y = 0\nx : G\nrx : IsRightRegular x\n⊢ (f * single x r).coeff.support = image (fun x_1 ↦ x_1 * x) f.coeff.support",
"ppTerm": "?m.43",
"assigned": true,
"usedCo... | [] | refine subset_antisymm (support_coeff_mul_single_subset f _ _) fun y hy => ?_
obtain ⟨y, yf, rfl⟩ : ∃ a : G, a ∈ f.coeff.support ∧ a * x = y := by grind
simp [coeff_mul, mem_support_iff.mp yf, hr, rx.eq_iff] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Polynomial.Eval.Defs | {
"line": 458,
"column": 58
} | {
"line": 459,
"column": 95
} | {
"line": 461,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝ : Semiring R\np q : R[X]\nn : ℕ\n⊢ (p * (X + ↑n)).comp q = p.comp q * (q + ↑n)",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"Nat.cast_comm",
"Distrib.leftDistribClass",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"HMul.hM... | [] | by
rw [mul_add, add_comp, mul_X_comp, ← Nat.cast_comm, natCast_mul_comp, Nat.cast_comm, mul_add] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.Nat.Prime.Defs | {
"line": 230,
"column": 37
} | {
"line": 231,
"column": 15
} | {
"line": 233,
"column": 0
} | [
{
"pp": "⊢ minFac 2 = 2",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"Dvd.dvd",
"congrArg",
"Nat.decidable_dvd",
"Nat.instMonoid",
"Nat.minFac",
"dvd_refl._simp_1",
"instOfNatNat",
"ite_cond_eq_true",
"Nat.minFacAux",
"Nat.instD... | [] | by
simp [minFac] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.LinearPMap | {
"line": 444,
"column": 4
} | {
"line": 444,
"column": 22
} | {
"line": 445,
"column": 2
} | [
{
"pp": "case h'\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝⁸ : Ring R\ninst✝⁷ : Ring S\ninst✝⁶ : Ring T\nσ : R →+* S\nτ : S →+* T\nE : Type u_4\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : Module R E\nF : Type u_5\ninst✝³ : AddCommGroup F\ninst✝² : Module S F\nG : Type u_6\ninst✝¹ : AddCommGroup G\ninst✝ : Module T... | [] | · simp [add_apply] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.LinearAlgebra.LinearPMap | {
"line": 448,
"column": 4
} | {
"line": 448,
"column": 22
} | {
"line": 450,
"column": 0
} | [
{
"pp": "case h'\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝⁸ : Ring R\ninst✝⁷ : Ring S\ninst✝⁶ : Ring T\nσ : R →+* S\nτ : S →+* T\nE : Type u_4\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : Module R E\nF : Type u_5\ninst✝³ : AddCommGroup F\ninst✝² : Module S F\nG : Type u_6\ninst✝¹ : AddCommGroup G\ninst✝ : Module T... | [] | · simp [add_apply] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.LinearAlgebra.LinearPMap | {
"line": 515,
"column": 26
} | {
"line": 525,
"column": 12
} | {
"line": 526,
"column": 2
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝⁸ : Ring R\ninst✝⁷ : Ring S\ninst✝⁶ : Ring T\nσ : R →+* S\nτ : S →+* T\nE : Type u_4\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : Module R E\nF : Type u_5\ninst✝³ : AddCommGroup F\ninst✝² : Module S F\nG : Type u_6\ninst✝¹ : AddCommGroup G\ninst✝ : Module T G\nf g :... | [] | by
ext x hf hg
· have : (0 : E →ₛₗ.[σ] F).domain = ⊤ := zero_domain
simp only [← h', add_domain, inf_eq_top_iff] at this
rw [neg_domain, this.1, this.2]
simp only [neg_domain, neg_apply, neg_eq_iff_add_eq_zero]
rw [ext_iff] at h'
rcases h' with ⟨hdom, h'⟩
rw [zero_domain] at hdom
... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.CharP.Defs | {
"line": 136,
"column": 22
} | {
"line": 136,
"column": 35
} | {
"line": 137,
"column": 22
} | [
{
"pp": "R : Type u_1\ninst✝ : NonAssocSemiring R\nthis : DecidableEq R := Classical.decEq R\nH : ¬∀ (p : ℕ), ↑p = 0 → p = 0\nx : ℕ\nH1 : ↑(x % Nat.find ⋯) + ↑(Nat.find ⋯ * (x / Nat.find ⋯)) = 0\nH2 : ¬x % Nat.find ⋯ = 0\n⊢ ↑(x % Nat.find ⋯) = 0",
"ppTerm": "?m.116",
"assigned": true,
"usedConstants... | [
"R : Type u_1\ninst✝ : NonAssocSemiring R\nthis : DecidableEq R := Classical.decEq R\nH : ¬∀ (p : ℕ), ↑p = 0 → p = 0\nx : ℕ\nH1 : ↑(x % Nat.find ⋯) + ↑(Nat.find ⋯) * ↑(x / Nat.find ⋯) = 0\nH2 : ¬x % Nat.find ⋯ = 0\n⊢ ↑(x % Nat.find ⋯) = 0"
] | Nat.cast_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.CharP.Defs | {
"line": 141,
"column": 40
} | {
"line": 141,
"column": 53
} | {
"line": 142,
"column": 12
} | [
{
"pp": "R : Type u_1\ninst✝ : NonAssocSemiring R\nthis : DecidableEq R := Classical.decEq R\nH : ¬∀ (p : ℕ), ↑p = 0 → p = 0\nx : ℕ\nH1 : Nat.find ⋯ ∣ x\n⊢ ↑(Nat.find ⋯ * (x / Nat.find ⋯)) = 0",
"ppTerm": "?m.166",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instDecidableNot",
... | [
"R : Type u_1\ninst✝ : NonAssocSemiring R\nthis : DecidableEq R := Classical.decEq R\nH : ¬∀ (p : ℕ), ↑p = 0 → p = 0\nx : ℕ\nH1 : Nat.find ⋯ ∣ x\n⊢ ↑(Nat.find ⋯) * ↑(x / Nat.find ⋯) = 0"
] | Nat.cast_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.LinearPMap | {
"line": 603,
"column": 56
} | {
"line": 603,
"column": 92
} | {
"line": 604,
"column": 6
} | [
{
"pp": "case w.refine_1\nR : Type u_1\nS : Type u_2\ninst✝⁵ : Ring R\ninst✝⁴ : Ring S\nσ : R →+* S\nE : Type u_4\ninst✝³ : AddCommGroup E\ninst✝² : Module R E\nF : Type u_5\ninst✝¹ : AddCommGroup F\ninst✝ : Module S F\nc : Set (E →ₛₗ.[σ] F)\nhc : DirectedOn (fun x1 x2 ↦ x1 ≤ x2) c\ncne : c.Nonempty\nhdir : Dir... | [
"case w.refine_1\nR : Type u_1\nS : Type u_2\ninst✝⁵ : Ring R\ninst✝⁴ : Ring S\nσ : R →+* S\nE : Type u_4\ninst✝³ : AddCommGroup E\ninst✝² : Module R E\nF : Type u_5\ninst✝¹ : AddCommGroup F\ninst✝ : Module S F\nc : Set (E →ₛₗ.[σ] F)\nhc : DirectedOn (fun x1 x2 ↦ x1 ≤ x2) c\ncne : c.Nonempty\nhdir : DirectedOn (fun... | f_eq ⟨p, hpc⟩ (x + y) (x' + y') rfl, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Order.Filter.Extr | {
"line": 724,
"column": 6
} | {
"line": 724,
"column": 18
} | {
"line": 724,
"column": 18
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝ : Preorder β\nf g : α → β\na : α\nl : Filter α\nh : IsExtrFilter f l a\nheq : f =ᶠ[l] g\nhfga : f a = g a\n⊢ IsExtrFilter g l a",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"IsMinFilter",
"Eq.mp",
... | [
"α : Type u_1\nβ : Type u_2\ninst✝ : Preorder β\nf g : α → β\na : α\nl : Filter α\nh : IsMinFilter f l a ∨ IsMaxFilter f l a\nheq : f =ᶠ[l] g\nhfga : f a = g a\n⊢ IsMinFilter g l a ∨ IsMaxFilter g l a"
] | IsExtrFilter | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.Coeff | {
"line": 376,
"column": 54
} | {
"line": 376,
"column": 66
} | {
"line": 376,
"column": 67
} | [
{
"pp": "R : Type u\nS : Type v\na b : R\nn✝ m : ℕ\ninst✝¹ : Semiring R\np✝ q r : R[X]\np : ℕ\ninst✝ : CharP R p\nn : ℕ\n⊢ ↑n = 0 ↔ C ↑n = C 0",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Polynomial.C",
"NonAssocSemiring.toAddCommMonoidWithOne",
"RingH... | [
"R : Type u\nS : Type v\na b : R\nn✝ m : ℕ\ninst✝¹ : Semiring R\np✝ q r : R[X]\np : ℕ\ninst✝ : CharP R p\nn : ℕ\n⊢ ↑n = 0 ↔ ↑n = C 0"
] | map_natCast, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.Degree.Defs | {
"line": 339,
"column": 57
} | {
"line": 339,
"column": 88
} | {
"line": 341,
"column": 0
} | [
{
"pp": "case inl\nR : Type u\ninst✝ : Semiring R\np q : R[X]\nh : (p + q).degree ≤ p.degree\n⊢ (p + q).natDegree ≤ max p.natDegree q.natDegree",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Polynomial.natDegree_le_natDegree",
"Lattice.toSemilatticeSup",
"congrArg",
... | [] | simp [natDegree_le_natDegree h] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Polynomial.Degree.Defs | {
"line": 339,
"column": 57
} | {
"line": 339,
"column": 88
} | {
"line": 341,
"column": 0
} | [
{
"pp": "case inr\nR : Type u\ninst✝ : Semiring R\np q : R[X]\nh : (p + q).degree ≤ q.degree\n⊢ (p + q).natDegree ≤ max p.natDegree q.natDegree",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Polynomial.natDegree_le_natDegree",
"Lattice.toSemilatticeSup",
"congrArg",
... | [] | simp [natDegree_le_natDegree h] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.LinearAlgebra.LinearPMap | {
"line": 882,
"column": 4
} | {
"line": 883,
"column": 38
} | {
"line": 884,
"column": 2
} | [
{
"pp": "case h\nR : Type u_1\ninst✝⁴ : Ring R\nE : Type u_4\ninst✝³ : AddCommGroup E\ninst✝² : Module R E\nF : Type u_5\ninst✝¹ : AddCommGroup F\ninst✝ : Module R F\nf g : E →ₗ.[R] F\nh : f.graph = g.graph\n⊢ f.domain = g.domain",
"ppTerm": "?h",
"assigned": true,
"usedConstants": [
"AddCommG... | [] | ext
exact mem_domain_iff_of_eq_graph h | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.LinearPMap | {
"line": 882,
"column": 4
} | {
"line": 883,
"column": 38
} | {
"line": 884,
"column": 2
} | [
{
"pp": "case h\nR : Type u_1\ninst✝⁴ : Ring R\nE : Type u_4\ninst✝³ : AddCommGroup E\ninst✝² : Module R E\nF : Type u_5\ninst✝¹ : AddCommGroup F\ninst✝ : Module R F\nf g : E →ₗ.[R] F\nh : f.graph = g.graph\n⊢ f.domain = g.domain",
"ppTerm": "?h",
"assigned": true,
"usedConstants": [
"AddCommG... | [] | ext
exact mem_domain_iff_of_eq_graph h | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Polynomial.Degree.Operations | {
"line": 350,
"column": 6
} | {
"line": 350,
"column": 49
} | {
"line": 351,
"column": 4
} | [
{
"pp": "R : Type u\ninst✝ : Semiring R\np : R[X]\nn : ℕ\nh : p.leadingCoeff ^ (n + 1) ≠ 0\nh₁ : p.leadingCoeff ^ n ≠ 0\n⊢ (p ^ n).leadingCoeff * p.leadingCoeff ≠ 0",
"ppTerm": "?m.114",
"assigned": true,
"usedConstants": [
"HMul.hMul",
"Monoid.toMulOneClass",
"congrArg",
"po... | [] | rwa [pow_succ, ← leadingCoeff_pow' h₁] at h | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.Algebra.Polynomial.Degree.SmallDegree | {
"line": 83,
"column": 23
} | {
"line": 83,
"column": 45
} | {
"line": 83,
"column": 45
} | [
{
"pp": "R : Type u\na b : R\ninst✝ : Semiring R\nha : a ≠ 0\n⊢ (C a * X).natDegree = 1",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Polynomial.C",
"HMul.hMul",
"congrArg",
"RingHom",
"Polynomial.natDegree_C_mul_X",
"id",
"instO... | [
"R : Type u\na b : R\ninst✝ : Semiring R\nha : a ≠ 0\n⊢ 1 = 1"
] | natDegree_C_mul_X a ha | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.Eval.Degree | {
"line": 83,
"column": 75
} | {
"line": 83,
"column": 88
} | {
"line": 84,
"column": 4
} | [
{
"pp": "S : Type v\ninst✝ : CommRing S\nd : ℕ\ny✝ : S\ncast_succ : ↑d + 1 = ↑d.succ\ny : ℕ\n_hy : y ∈ range d\n⊢ ↑((d + 1).choose (y + 1) * (y + 1)) * y✝ ^ y = ↑((d + 1).choose (y + 1)) * ↑(y + 1) * y✝ ^ (y + 1 - 1)",
"ppTerm": "?m.202",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"N... | [
"S : Type v\ninst✝ : CommRing S\nd : ℕ\ny✝ : S\ncast_succ : ↑d + 1 = ↑d.succ\ny : ℕ\n_hy : y ∈ range d\n⊢ ↑((d + 1).choose (y + 1)) * ↑(y + 1) * y✝ ^ y = ↑((d + 1).choose (y + 1)) * ↑(y + 1) * y✝ ^ (y + 1 - 1)"
] | Nat.cast_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.Eval.Degree | {
"line": 187,
"column": 2
} | {
"line": 187,
"column": 60
} | {
"line": 189,
"column": 0
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝¹ : Semiring R\ninst✝ : Semiring S\np : R[X]\nf : R →+* S\nhf : f p.leadingCoeff ≠ 0\n⊢ (map f p).coeff (map f p).natDegree = f (p.coeff p.natDegree)",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Polynomial.coeff_map",
"c... | [] | rw [coeff_map, natDegree_map_of_leadingCoeff_ne_zero f hf] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Polynomial.AlgebraMap | {
"line": 378,
"column": 61
} | {
"line": 379,
"column": 27
} | {
"line": 381,
"column": 0
} | [
{
"pp": "R : Type u_3\ninst✝ : CommRing R\nt t' : R\n⊢ algEquivAevalXAddC t = algEquivAevalXAddC t' ↔ t = t'",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Polynomial.C",
"AddLeftCancelSemigroup.toIsLeftCancelAdd",
"congrArg",
"CommSemiring.toSemiring",
"Pol... | [] | by
simp [algEquivAevalXAddC] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.MvPolynomial.Degrees | {
"line": 184,
"column": 6
} | {
"line": 184,
"column": 18
} | {
"line": 184,
"column": 19
} | [
{
"pp": "R : Type u\nσ : Type u_1\nτ : Type u_2\ninst✝ : CommSemiring R\nf : σ → τ\nφ : MvPolynomial σ R\ni : τ\n⊢ i ∈ ((rename f) φ).degrees → i ∈ Multiset.map f φ.degrees",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Nat.instMulZeroClass",
"AddMonoidAlgebra... | [
"R : Type u\nσ : Type u_1\nτ : Type u_2\ninst✝ : CommSemiring R\nf : σ → τ\nφ : MvPolynomial σ R\ni : τ\n⊢ (∃ d, coeff d ((rename f) φ) ≠ 0 ∧ i ∈ d.support) → i ∈ Multiset.map f φ.degrees"
] | mem_degrees, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.MvPolynomial.Eval | {
"line": 821,
"column": 4
} | {
"line": 821,
"column": 88
} | {
"line": 822,
"column": 4
} | [
{
"pp": "case monomial_add\nR : Type u\nσ : Type u_1\ninst✝³ : CommSemiring R\nS : Type u_2\nsubS : Type u_3\ninst✝² : CommSemiring S\ninst✝¹ : SetLike subS S\ninst✝ : SubsemiringClass subS S\nf✝ : R →+* S\ns : subS\nv : σ → S\nhv : ∀ (i : σ), v i ∈ s\na : σ →₀ ℕ\nb : R\nf : MvPolynomial σ R\nha : a ∉ f.coeff.s... | [
"case monomial_add.refine_1\nR : Type u\nσ : Type u_1\ninst✝³ : CommSemiring R\nS : Type u_2\nsubS : Type u_3\ninst✝² : CommSemiring S\ninst✝¹ : SetLike subS S\ninst✝ : SubsemiringClass subS S\nf✝ : R →+* S\ns : subS\nv : σ → S\nhv : ∀ (i : σ), v i ∈ s\na : σ →₀ ℕ\nb : R\nf : MvPolynomial σ R\nha : a ∉ f.coeff.supp... | refine add_mem (mul_mem ?_ <| prod_mem fun i _ => pow_mem (hv _) _) (ih fun i => ?_) | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Algebra.Polynomial.EraseLead | {
"line": 114,
"column": 4
} | {
"line": 114,
"column": 53
} | {
"line": 115,
"column": 2
} | [
{
"pp": "case pos\nR : Type u_1\ninst✝ : Semiring R\nf : R[X]\nhf : f = 0\n⊢ #f.eraseLead.support = #f.support - 1",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Finset",
"HSub.hSub",
"id",
"instSubNat",
"instOfNatNat",
... | [] | rw [hf, eraseLead_zero, support_zero, card_empty] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Polynomial.EraseLead | {
"line": 114,
"column": 4
} | {
"line": 114,
"column": 53
} | {
"line": 115,
"column": 2
} | [
{
"pp": "case pos\nR : Type u_1\ninst✝ : Semiring R\nf : R[X]\nhf : f = 0\n⊢ #f.eraseLead.support = #f.support - 1",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Finset",
"HSub.hSub",
"id",
"instSubNat",
"instOfNatNat",
... | [] | rw [hf, eraseLead_zero, support_zero, card_empty] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Polynomial.EraseLead | {
"line": 114,
"column": 4
} | {
"line": 114,
"column": 53
} | {
"line": 115,
"column": 2
} | [
{
"pp": "case pos\nR : Type u_1\ninst✝ : Semiring R\nf : R[X]\nhf : f = 0\n⊢ #f.eraseLead.support = #f.support - 1",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Finset",
"HSub.hSub",
"id",
"instSubNat",
"instOfNatNat",
... | [] | rw [hf, eraseLead_zero, support_zero, card_empty] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Polynomial.Degree.Lemmas | {
"line": 497,
"column": 59
} | {
"line": 498,
"column": 53
} | {
"line": 500,
"column": 0
} | [
{
"pp": "K : Type u_1\ninst✝ : DivisionRing K\np : K[X]\n⊢ p.degree + (C p.leadingCoeff⁻¹).degree = p.degree",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Polynomial.C",
"WithBot",
"IsDomain.to_noZeroDivisors",
"GroupWithZero.toDivisionMonoid",
... | [] | by
rw [← degree_mul, degree_mul_leadingCoeff_self_inv] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.MvPolynomial.Equiv | {
"line": 555,
"column": 82
} | {
"line": 563,
"column": 21
} | {
"line": 565,
"column": 0
} | [
{
"pp": "R : Type u\nσ : Type u_1\ninst✝ : CommSemiring R\np : MvPolynomial (Option σ) R\n⊢ ((optionEquivLeft R σ) p).support = Finset.image (fun m ↦ m none) p.support",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Finsupp.instFunLike",
"Eq.mpr",
"Nat.instMulZeroClass",... | [] | by
ext i
simp only [Polynomial.mem_support_iff, ne_eq, MvPolynomial.ext_iff, coeff_zero, not_forall,
Finset.mem_image, mem_support_iff, ← optionEquivLeft_coeff_some_coeff_none]
constructor
· rintro ⟨m, hm⟩
exact ⟨optionElim i m, by simpa using! hm, optionElim_apply_none _ _⟩
· rintro ⟨m, h, rfl⟩
e... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Polynomial.EraseLead | {
"line": 398,
"column": 4
} | {
"line": 401,
"column": 58
} | {
"line": 403,
"column": 0
} | [
{
"pp": "case refine_2\nR : Type u_1\ninst✝ : Semiring R\nn : ℕ\nk : Fin n → ℕ\nx : Fin n → R\nhk : Function.Injective k\nhx : ∀ (i : Fin n), x i ≠ 0\ni : ℕ\n⊢ (∃ a, k a = i) → (∑ x_1, if i = k x_1 then x x_1 else 0) ≠ 0",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"Finset.mem... | [] | rintro ⟨j, _, rfl⟩
rw [sum_eq_single_of_mem j (mem_univ j), if_pos rfl]
· exact hx j
· exact fun m _ hmj => if_neg fun h => hmj.symm (hk h) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Polynomial.EraseLead | {
"line": 398,
"column": 4
} | {
"line": 401,
"column": 58
} | {
"line": 403,
"column": 0
} | [
{
"pp": "case refine_2\nR : Type u_1\ninst✝ : Semiring R\nn : ℕ\nk : Fin n → ℕ\nx : Fin n → R\nhk : Function.Injective k\nhx : ∀ (i : Fin n), x i ≠ 0\ni : ℕ\n⊢ (∃ a, k a = i) → (∑ x_1, if i = k x_1 then x x_1 else 0) ≠ 0",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"Finset.mem... | [] | rintro ⟨j, _, rfl⟩
rw [sum_eq_single_of_mem j (mem_univ j), if_pos rfl]
· exact hx j
· exact fun m _ hmj => if_neg fun h => hmj.symm (hk h) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Polynomial.Basic | {
"line": 135,
"column": 20
} | {
"line": 135,
"column": 41
} | {
"line": 135,
"column": 42
} | [
{
"pp": "R✝ : Type u\nS : Type u_1\ninst✝¹ : Semiring R✝\nR : Type ?u.10\ninst✝ : Semiring R\nn : ℕ\np : R[X]\nhp : p.natDegree < n\nhp0 : ¬p = 0\n| p",
"ppTerm": "?m.196",
"assigned": true,
"usedConstants": [
"Semiring.toModule",
"congrArg",
"LinearMap.instFunLike",
"Polynom... | [
"R✝ : Type u\nS : Type u_1\ninst✝¹ : Semiring R✝\nR : Type ?u.10\ninst✝ : Semiring R\nn : ℕ\np : R[X]\nhp : p.natDegree < n\nhp0 : ¬p = 0\n| ∑ i ∈ range n, (monomial i) (p.coeff i)"
] | p.as_sum_range' n hp, | Lean.Elab.Tactic.Conv.evalRewrite | null |
Mathlib.RingTheory.Polynomial.Basic | {
"line": 165,
"column": 29
} | {
"line": 169,
"column": 88
} | {
"line": 169,
"column": 88
} | [
{
"pp": "R : Type u\nS : Type u_1\ninst✝¹ : Semiring R\ninst✝ : Nontrivial R\nn : ℕ\np : { p // p.Monic ∧ p.natDegree = n }\n⊢ (↑p).eraseLead ∈ degreeLT R n",
"ppTerm": "?m.53",
"assigned": true,
"usedConstants": [
"Polynomial.Monic.ne_zero_of_polynomial_ne",
"WithBot.instPreorder",
... | [] | by
rcases p with ⟨p, hp, rfl⟩
simp only [mem_degreeLT]
refine lt_of_lt_of_le ?_ degree_le_natDegree
exact degree_eraseLead_lt (Polynomial.Monic.ne_zero_of_polynomial_ne hp one_ne_zero) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Polynomial.Basic | {
"line": 276,
"column": 4
} | {
"line": 276,
"column": 28
} | {
"line": 278,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝ : Semiring R\nP : R[X]\nhP : P.Monic\nhdeg : 0 < P.natDegree\na✝ : Nontrivial R\nn : ℕ\nhn : n.succ ≠ 0\n⊢ ¬P ^ n = 0",
"ppTerm": "?m.107",
"assigned": true,
"usedConstants": [
"Polynomial",
"NPow.toPow",
"HPow.hPow",
"Polynomial.semiring",
"N... | [] | exact (hP.pow _).ne_zero | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.Polynomial.Basic | {
"line": 348,
"column": 57
} | {
"line": 348,
"column": 67
} | {
"line": 348,
"column": 68
} | [
{
"pp": "R : Type u\ninst✝ : Ring R\ni : ℕ\n⊢ coeff 1 i = ↑(coeff 1 i)",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Polynomial.coeff_one",
"Polynomial.instOne",
"Subring.instSetLike",
"Ring.toNonAs... | [
"R : Type u\ninst✝ : Ring R\ni : ℕ\n⊢ (if i = 0 then 1 else 0) = ↑(coeff 1 i)"
] | coeff_one, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Polynomial.Basic | {
"line": 547,
"column": 33
} | {
"line": 547,
"column": 43
} | {
"line": 547,
"column": 44
} | [
{
"pp": "case empty\nR : Type u\ninst✝ : CommSemiring R\nι : Type u_2\nf : ι → R[X]\nI : Ideal R\nn : ι → ℕ\nh : ∀ i ∈ ∅, ∀ (k : ℕ), (f i).coeff k ∈ I ^ (n i - k)\nk : ℕ\n⊢ coeff 1 k ∈ I ^ (0 - k)",
"ppTerm": "?empty",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAd... | [
"case empty\nR : Type u\ninst✝ : CommSemiring R\nι : Type u_2\nf : ι → R[X]\nI : Ideal R\nn : ι → ℕ\nh : ∀ i ∈ ∅, ∀ (k : ℕ), (f i).coeff k ∈ I ^ (n i - k)\nk : ℕ\n⊢ (if k = 0 then 1 else 0) ∈ I ^ (0 - k)"
] | coeff_one, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Polynomial.Basic | {
"line": 550,
"column": 10
} | {
"line": 550,
"column": 24
} | {
"line": 550,
"column": 25
} | [
{
"pp": "case insert\nR : Type u\ninst✝ : CommSemiring R\nι : Type u_2\nf : ι → R[X]\nI : Ideal R\nn : ι → ℕ\na : ι\ns : Finset ι\nha : a ∉ s\nhs : (∀ i ∈ s, ∀ (k : ℕ), (f i).coeff k ∈ I ^ (n i - k)) → ∀ (k : ℕ), (s.prod f).coeff k ∈ I ^ (s.sum n - k)\nh : ∀ i ∈ insert a s, ∀ (k : ℕ), (f i).coeff k ∈ I ^ (n i -... | [
"case insert\nR : Type u\ninst✝ : CommSemiring R\nι : Type u_2\nf : ι → R[X]\nI : Ideal R\nn : ι → ℕ\na : ι\ns : Finset ι\nha : a ∉ s\nhs : (∀ i ∈ s, ∀ (k : ℕ), (f i).coeff k ∈ I ^ (n i - k)) → ∀ (k : ℕ), (s.prod f).coeff k ∈ I ^ (s.sum n - k)\nh : ∀ i ∈ insert a s, ∀ (k : ℕ), (f i).coeff k ∈ I ^ (n i - k)\nk : ℕ\n... | sum_insert ha, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Ideal.Quotient.Operations | {
"line": 279,
"column": 36
} | {
"line": 279,
"column": 53
} | {
"line": 280,
"column": 4
} | [
{
"pp": "case «0».«0»\nR✝ : Type u\nS : Type v\nF : Type w\ninst✝³ : Ring R✝\ninst✝² : Semiring S\nι✝ : Type u_1\nR : Type u_2\ninst✝¹ : CommRing R\nι : Type u_3\ninst✝ : Finite ι\nI J : Ideal R\ncoprime : IsCoprime I J\nf : Fin 2 → Ideal R := ![I, J]\nh : (fun i ↦ i) ⟨0, ⋯⟩ ≠ (fun i ↦ i) ⟨0, ⋯⟩\n⊢ (IsCoprime o... | [] | try contradiction | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticTry__1 | Lean.Parser.Tactic.tacticTry_ |
Mathlib.RingTheory.Ideal.Quotient.Operations | {
"line": 279,
"column": 36
} | {
"line": 279,
"column": 53
} | {
"line": 280,
"column": 4
} | [
{
"pp": "case «0».«1»\nR✝ : Type u\nS : Type v\nF : Type w\ninst✝³ : Ring R✝\ninst✝² : Semiring S\nι✝ : Type u_1\nR : Type u_2\ninst✝¹ : CommRing R\nι : Type u_3\ninst✝ : Finite ι\nI J : Ideal R\ncoprime : IsCoprime I J\nf : Fin 2 → Ideal R := ![I, J]\nh : (fun i ↦ i) ⟨0, ⋯⟩ ≠ (fun i ↦ i) ⟨1, ⋯⟩\n⊢ (IsCoprime o... | [
"case «0».«1»\nR✝ : Type u\nS : Type v\nF : Type w\ninst✝³ : Ring R✝\ninst✝² : Semiring S\nι✝ : Type u_1\nR : Type u_2\ninst✝¹ : CommRing R\nι : Type u_3\ninst✝ : Finite ι\nI J : Ideal R\ncoprime : IsCoprime I J\nf : Fin 2 → Ideal R := ![I, J]\nh : (fun i ↦ i) ⟨0, ⋯⟩ ≠ (fun i ↦ i) ⟨1, ⋯⟩\n⊢ (IsCoprime on f) ((fun i... | try contradiction | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticTry__1 | Lean.Parser.Tactic.tacticTry_ |
Mathlib.RingTheory.Ideal.Quotient.Operations | {
"line": 279,
"column": 36
} | {
"line": 279,
"column": 53
} | {
"line": 280,
"column": 4
} | [
{
"pp": "case «1».«0»\nR✝ : Type u\nS : Type v\nF : Type w\ninst✝³ : Ring R✝\ninst✝² : Semiring S\nι✝ : Type u_1\nR : Type u_2\ninst✝¹ : CommRing R\nι : Type u_3\ninst✝ : Finite ι\nI J : Ideal R\ncoprime : IsCoprime I J\nf : Fin 2 → Ideal R := ![I, J]\nh : (fun i ↦ i) ⟨1, ⋯⟩ ≠ (fun i ↦ i) ⟨0, ⋯⟩\n⊢ (IsCoprime o... | [
"case «1».«0»\nR✝ : Type u\nS : Type v\nF : Type w\ninst✝³ : Ring R✝\ninst✝² : Semiring S\nι✝ : Type u_1\nR : Type u_2\ninst✝¹ : CommRing R\nι : Type u_3\ninst✝ : Finite ι\nI J : Ideal R\ncoprime : IsCoprime I J\nf : Fin 2 → Ideal R := ![I, J]\nh : (fun i ↦ i) ⟨1, ⋯⟩ ≠ (fun i ↦ i) ⟨0, ⋯⟩\n⊢ (IsCoprime on f) ((fun i... | try contradiction | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticTry__1 | Lean.Parser.Tactic.tacticTry_ |
Mathlib.RingTheory.Ideal.Quotient.Operations | {
"line": 279,
"column": 36
} | {
"line": 279,
"column": 53
} | {
"line": 280,
"column": 4
} | [
{
"pp": "case «1».«1»\nR✝ : Type u\nS : Type v\nF : Type w\ninst✝³ : Ring R✝\ninst✝² : Semiring S\nι✝ : Type u_1\nR : Type u_2\ninst✝¹ : CommRing R\nι : Type u_3\ninst✝ : Finite ι\nI J : Ideal R\ncoprime : IsCoprime I J\nf : Fin 2 → Ideal R := ![I, J]\nh : (fun i ↦ i) ⟨1, ⋯⟩ ≠ (fun i ↦ i) ⟨1, ⋯⟩\n⊢ (IsCoprime o... | [] | try contradiction | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticTry__1 | Lean.Parser.Tactic.tacticTry_ |
Mathlib.Algebra.GroupWithZero.Units.Fintype | {
"line": 46,
"column": 4
} | {
"line": 46,
"column": 78
} | {
"line": 48,
"column": 0
} | [
{
"pp": "case inr\nα : Type u_1\ninst✝ : GroupWithZero α\nh✝ : Infinite { a // a ≠ 0 }\n⊢ Nat.card ({ a // a = 0 } ⊕ { a // ¬a = 0 }) - 1 = Nat.card { a // a ≠ 0 }",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"Nat.instCanonica... | [] | rw [Nat.card_eq_zero_of_infinite, Nat.card_eq_zero_of_infinite, zero_tsub] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.GroupWithZero.Units.Fintype | {
"line": 46,
"column": 4
} | {
"line": 46,
"column": 78
} | {
"line": 48,
"column": 0
} | [
{
"pp": "case inr\nα : Type u_1\ninst✝ : GroupWithZero α\nh✝ : Infinite { a // a ≠ 0 }\n⊢ Nat.card ({ a // a = 0 } ⊕ { a // ¬a = 0 }) - 1 = Nat.card { a // a ≠ 0 }",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"Nat.instCanonica... | [] | rw [Nat.card_eq_zero_of_infinite, Nat.card_eq_zero_of_infinite, zero_tsub] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.GroupWithZero.Units.Fintype | {
"line": 46,
"column": 4
} | {
"line": 46,
"column": 78
} | {
"line": 48,
"column": 0
} | [
{
"pp": "case inr\nα : Type u_1\ninst✝ : GroupWithZero α\nh✝ : Infinite { a // a ≠ 0 }\n⊢ Nat.card ({ a // a = 0 } ⊕ { a // ¬a = 0 }) - 1 = Nat.card { a // a ≠ 0 }",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"Nat.instCanonica... | [] | rw [Nat.card_eq_zero_of_infinite, Nat.card_eq_zero_of_infinite, zero_tsub] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.CharP.Two | {
"line": 56,
"column": 6
} | {
"line": 56,
"column": 28
} | {
"line": 56,
"column": 29
} | [
{
"pp": "R : Type u_1\ninst✝¹ : AddMonoidWithOne R\ninst✝ : CharP R 2\n⊢ Set.range Nat.cast = {0, 1}",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"AddMonoid.toAddZeroClass",
"AddZeroClass.toAddZero",
"Set.instSingletonSet",
"id",... | [
"R : Type u_1\ninst✝¹ : AddMonoidWithOne R\ninst✝ : CharP R 2\n⊢ (Set.range fun x ↦ if Even x then 0 else 1) = {0, 1}"
] | funext natCast_eq_ite, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Ideal.Quotient.Operations | {
"line": 1134,
"column": 2
} | {
"line": 1134,
"column": 51
} | {
"line": 1135,
"column": 2
} | [
{
"pp": "R : Type u\nA : Type u_1\ninst✝² : CommSemiring R\ninst✝¹ : CommRing A\ninst✝ : Algebra R A\nI J : Ideal A\nh : I ≤ J\nx : A ⧸ I\n⊢ ((↑(quotQuotEquivQuotOfLEₐ R h)).comp (Quotient.mkₐ R (map (Quotient.mkₐ R I) J))) x = (Quotient.factorₐ R h) x",
"ppTerm": "?m.61",
"assigned": true,
"usedCon... | [
"R : Type u\nA : Type u_1\ninst✝² : CommSemiring R\ninst✝¹ : CommRing A\ninst✝ : Algebra R A\nI J : Ideal A\nh : I ≤ J\nx : A\n⊢ ((↑(quotQuotEquivQuotOfLEₐ R h)).comp (Quotient.mkₐ R (map (Quotient.mkₐ R I) J))) ((Ideal.Quotient.mk I) x) =\n (Quotient.factorₐ R h) ((Ideal.Quotient.mk I) x)"
] | obtain ⟨x, rfl⟩ := Ideal.Quotient.mk_surjective x | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Data.List.Permutation | {
"line": 152,
"column": 68
} | {
"line": 152,
"column": 77
} | {
"line": 152,
"column": 78
} | [
{
"pp": "case cons\nα : Type u_1\nt : α\nts l' : List α\ny : α\nys : List α\nih :\n ∀ {l : List α},\n l' ∈ (permutationsAux2 t ts [] ys fun x ↦ l ++ x).snd ↔\n ∃ l₁ l₂, l₂ ≠ [] ∧ ys = l₁ ++ l₂ ∧ l' = l ++ l₁ ++ t :: l₂ ++ ts\nl : List α\n⊢ l' ∈ (l ++ (t :: y :: ys ++ ts)) :: (permutationsAux2 t ts [] y... | [
"case cons\nα : Type u_1\nt : α\nts l' : List α\ny : α\nys : List α\nih :\n ∀ {l : List α},\n l' ∈ (permutationsAux2 t ts [] ys fun x ↦ l ++ x).snd ↔\n ∃ l₁ l₂, l₂ ≠ [] ∧ ys = l₁ ++ l₂ ∧ l' = l ++ l₁ ++ t :: l₂ ++ ts\nl : List α\n⊢ l' = l ++ (t :: y :: ys ++ ts) ∨ l' ∈ (permutationsAux2 t ts [] ys fun x ↦ ... | mem_cons, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Dynamics.PeriodicPts.Defs | {
"line": 223,
"column": 69
} | {
"line": 223,
"column": 87
} | {
"line": 223,
"column": 87
} | [
{
"pp": "case e'_2\nα : Type u_1\nf : α → α\nx : α\nm n : ℕ\nhm : IsPeriodicPt f m (f^[n] x)\nr : ℕ\nhr : r > 0\nhr' : IsPeriodicPt f r x\nthis : n ≤ (n / r + 1) * r\n⊢ x = f^[n / r + 1]^[r] x",
"ppTerm": "?e'_2",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instHDiv",
"congrArg... | [
"case e'_2\nα : Type u_1\nf : α → α\nx : α\nm n : ℕ\nhm : IsPeriodicPt f m (f^[n] x)\nr : ℕ\nhr : r > 0\nhr' : IsPeriodicPt f r x\nthis : n ≤ (n / r + 1) * r\n⊢ x = x"
] | (hr'.iterate _).eq | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Dynamics.PeriodicPts.Defs | {
"line": 223,
"column": 69
} | {
"line": 223,
"column": 87
} | {
"line": 223,
"column": 87
} | [
{
"pp": "case e'_3\nα : Type u_1\nf : α → α\nx : α\nm n : ℕ\nhm : IsPeriodicPt f m (f^[n] x)\nr : ℕ\nhr : r > 0\nhr' : IsPeriodicPt f r x\nthis : n ≤ (n / r + 1) * r\n⊢ x = f^[n / r + 1]^[r] x",
"ppTerm": "?e'_3",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instHDiv",
"congrArg... | [
"case e'_3\nα : Type u_1\nf : α → α\nx : α\nm n : ℕ\nhm : IsPeriodicPt f m (f^[n] x)\nr : ℕ\nhr : r > 0\nhr' : IsPeriodicPt f r x\nthis : n ≤ (n / r + 1) * r\n⊢ x = x"
] | (hr'.iterate _).eq | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.List.Permutation | {
"line": 177,
"column": 28
} | {
"line": 177,
"column": 39
} | {
"line": 177,
"column": 40
} | [
{
"pp": "case cons\nα : Type u_1\nt : α\nts : List α\nr : List (List α)\nl : List α\nL : List (List α)\nih :\n foldr (fun y r ↦ (permutationsAux2 t ts r y id).snd) r L =\n flatMap (fun y ↦ (permutationsAux2 t ts [] y id).snd) L ++ r\n⊢ foldr (fun y r ↦ (permutationsAux2 t ts r y id).snd) r (l :: L) =\n f... | [
"case cons\nα : Type u_1\nt : α\nts : List α\nr : List (List α)\nl : List α\nL : List (List α)\nih :\n foldr (fun y r ↦ (permutationsAux2 t ts r y id).snd) r L =\n flatMap (fun y ↦ (permutationsAux2 t ts [] y id).snd) L ++ r\n⊢ (permutationsAux2 t ts (foldr (fun y r ↦ (permutationsAux2 t ts r y id).snd) r L) l ... | foldr_cons, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Data.List.Cycle | {
"line": 161,
"column": 2
} | {
"line": 161,
"column": 82
} | {
"line": 163,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝ : DecidableEq α\nl : List α\nx : α\nh : x ∈ l.dropLast\ny : α\nhy : x ≠ y\n⊢ (y :: l).next x ⋯ = l.next x ⋯",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"List.nextOr",
"List.next.eq_1",
"Eq.mpr",
"List.nextOr_eq_nextOr_of_mem_dropLast... | [] | rwa [next, next, nextOr_cons_of_ne _ _ _ _ hy, nextOr_eq_nextOr_of_mem_dropLast] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.Data.List.Cycle | {
"line": 161,
"column": 2
} | {
"line": 161,
"column": 82
} | {
"line": 163,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝ : DecidableEq α\nl : List α\nx : α\nh : x ∈ l.dropLast\ny : α\nhy : x ≠ y\n⊢ (y :: l).next x ⋯ = l.next x ⋯",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"List.nextOr",
"List.next.eq_1",
"Eq.mpr",
"List.nextOr_eq_nextOr_of_mem_dropLast... | [] | rwa [next, next, nextOr_cons_of_ne _ _ _ _ hy, nextOr_eq_nextOr_of_mem_dropLast] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.List.Cycle | {
"line": 161,
"column": 2
} | {
"line": 161,
"column": 82
} | {
"line": 163,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝ : DecidableEq α\nl : List α\nx : α\nh : x ∈ l.dropLast\ny : α\nhy : x ≠ y\n⊢ (y :: l).next x ⋯ = l.next x ⋯",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"List.nextOr",
"List.next.eq_1",
"Eq.mpr",
"List.nextOr_eq_nextOr_of_mem_dropLast... | [] | rwa [next, next, nextOr_cons_of_ne _ _ _ _ hy, nextOr_eq_nextOr_of_mem_dropLast] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.IsPrimePow | {
"line": 108,
"column": 4
} | {
"line": 108,
"column": 42
} | {
"line": 109,
"column": 2
} | [
{
"pp": "case inr.mp\nn : ℕ\nh : n ≠ 1\nk : ℕ\nhkle : k ≤ Nat.log 2 n\nhk_pos : 0 < k\np : ℕ\nhle : p ≤ n\nheq : n = p ^ k\nhprime : Nat.Prime p\n⊢ n = n.minFac ^ k",
"ppTerm": "?inr.mp",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Nat.instMonoid",
"id",
... | [] | rw [heq, hprime.pow_minFac hk_pos.ne'] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Data.List.Cycle | {
"line": 601,
"column": 2
} | {
"line": 601,
"column": 26
} | {
"line": 602,
"column": 2
} | [
{
"pp": "case h\nα : Type u_1\nl : List α\nh : Subsingleton (Quot.mk (⇑(IsRotated.setoid α)) l)\n⊢ Nodup (Quot.mk (⇑(IsRotated.setoid α)) l)",
"ppTerm": "?h",
"assigned": true,
"usedConstants": [
"List.IsRotated.setoid",
"Cycle.Subsingleton",
"List.cons",
"List",
"List.... | [
"case h.nil\nα : Type u_1\nh : Subsingleton (Quot.mk ⇑(IsRotated.setoid α) [])\n⊢ Nodup (Quot.mk ⇑(IsRotated.setoid α) [])",
"case h.cons\nα : Type u_1\nhd : α\ntl : List α\nh : Subsingleton (Quot.mk (⇑(IsRotated.setoid α)) (hd :: tl))\n⊢ Nodup (Quot.mk (⇑(IsRotated.setoid α)) (hd :: tl))"
] | obtain - | ⟨hd, tl⟩ := l | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.GroupTheory.GroupAction.Quotient | {
"line": 288,
"column": 58
} | {
"line": 300,
"column": 70
} | {
"line": 302,
"column": 0
} | [
{
"pp": "G : Type u\nX : Type v\ninst✝³ : Group G\ninst✝² : MulAction G X\nx : X\ninst✝¹ : IsPretransitive G X\nH : Subgroup G\ninst✝ : Finite (G ⧸ H)\n⊢ Finite (orbitRel.Quotient (↥H) X)",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"Quotient.finite... | [] | by
rcases isEmpty_or_nonempty X with he | ⟨⟨b⟩⟩
· exact Quotient.finite _
· have h' : Finite (Quotient (rightRel H)) :=
Finite.of_equiv _ (quotientRightRelEquivQuotientLeftRel _).symm
let f : Quotient (rightRel H) → orbitRel.Quotient H X :=
fun a ↦ Quotient.liftOn' a (fun g ↦ ⟦g • b⟧) fun g₁ g₂ r ... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.Nat.PrimeFin | {
"line": 40,
"column": 11
} | {
"line": 40,
"column": 30
} | {
"line": 40,
"column": 31
} | [
{
"pp": "n p : ℕ\n⊢ p ∈ n.primeFactors ↔ Prime p ∧ p ∣ n ∧ n ≠ 0",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Nat.Prime",
"Dvd.dvd",
"Finset",
"Membership.mem",
"id",
"Ne",
"instOfNatNat",
"And",
"Iff",
"Nat.instDvd",
"Fi... | [
"n p : ℕ\n⊢ p ∈ n.primeFactorsList.toFinset ↔ Prime p ∧ p ∣ n ∧ n ≠ 0"
] | ← toFinset_factors, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Data.Nat.PrimeFin | {
"line": 93,
"column": 2
} | {
"line": 93,
"column": 52
} | {
"line": 95,
"column": 0
} | [
{
"pp": "p : ℕ\nhp : Prime p\n⊢ p.primeFactors = {p}",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"congrArg",
"Finset",
"Nat.primeFactorsList_prime",
"Insert.insert",
"List.toFinset",
"Finset.instEmptyCollection",
"Finset.instInsert",
"List... | [] | simp [Nat.primeFactors, primeFactorsList_prime hp] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Data.Nat.PrimeFin | {
"line": 93,
"column": 2
} | {
"line": 93,
"column": 52
} | {
"line": 95,
"column": 0
} | [
{
"pp": "p : ℕ\nhp : Prime p\n⊢ p.primeFactors = {p}",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"congrArg",
"Finset",
"Nat.primeFactorsList_prime",
"Insert.insert",
"List.toFinset",
"Finset.instEmptyCollection",
"Finset.instInsert",
"List... | [] | simp [Nat.primeFactors, primeFactorsList_prime hp] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Nat.PrimeFin | {
"line": 93,
"column": 2
} | {
"line": 93,
"column": 52
} | {
"line": 95,
"column": 0
} | [
{
"pp": "p : ℕ\nhp : Prime p\n⊢ p.primeFactors = {p}",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"congrArg",
"Finset",
"Nat.primeFactorsList_prime",
"Insert.insert",
"List.toFinset",
"Finset.instEmptyCollection",
"Finset.instInsert",
"List... | [] | simp [Nat.primeFactors, primeFactorsList_prime hp] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.Index | {
"line": 283,
"column": 6
} | {
"line": 283,
"column": 28
} | {
"line": 283,
"column": 29
} | [
{
"pp": "G : Type u_1\nG' : Type u_2\ninst✝¹ : Group G\ninst✝ : Group G'\nf : G →* G'\n⊢ f.ker.index = Nat.card ↥f.range",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"MonoidHom.comap_bot",
"Eq.mpr",
"MonoidHom.range",
"Monoid.toMulOneClass",
"congrArg",
... | [
"G : Type u_1\nG' : Type u_2\ninst✝¹ : Group G\ninst✝ : Group G'\nf : G →* G'\n⊢ (comap f ⊥).index = Nat.card ↥f.range"
] | ← MonoidHom.comap_bot, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.Index | {
"line": 287,
"column": 6
} | {
"line": 287,
"column": 28
} | {
"line": 287,
"column": 29
} | [
{
"pp": "G : Type u_1\nG' : Type u_2\ninst✝¹ : Group G\ninst✝ : Group G'\nK : Subgroup G\nf : G →* G'\n⊢ f.ker.relIndex K = Nat.card ↥(map f K)",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"MonoidHom.comap_bot",
"Eq.mpr",
"Subgroup.map",
"Monoid.toMulOneClass",
... | [
"G : Type u_1\nG' : Type u_2\ninst✝¹ : Group G\ninst✝ : Group G'\nK : Subgroup G\nf : G →* G'\n⊢ (comap f ⊥).relIndex K = Nat.card ↥(map f K)"
] | ← MonoidHom.comap_bot, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Algebra.ZMod | {
"line": 66,
"column": 2
} | {
"line": 66,
"column": 33
} | {
"line": 67,
"column": 2
} | [
{
"pp": "R✝ : Type u_1\ninst✝⁴ : Ring R✝\nn : ℕ\nR : Type u_2\nM : Type u_3\ninst✝³ : Ring R\ninst✝² : AddCommGroup M\ninst✝¹ : Module (ZMod n) R\nm₁ : Module (ZMod n) M\ninst✝ : Module R M\n⊢ IsScalarTower (ZMod n) R M",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"ZMod.commRing",... | [
"R✝ : Type u_1\ninst✝⁴ : Ring R✝\nn : ℕ\nR : Type u_2\nM : Type u_3\ninst✝³ : Ring R\ninst✝² : AddCommGroup M\ninst✝¹ : Module (ZMod n) R\nm₁ : Module (ZMod n) M\ninst✝ : Module R M\nthis : Algebra (ZMod n) R := algebraOfModule n R\n⊢ IsScalarTower (ZMod n) R M"
] | let := ZMod.algebraOfModule n R | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.Data.ZMod.Basic | {
"line": 576,
"column": 40
} | {
"line": 576,
"column": 53
} | {
"line": 576,
"column": 54
} | [
{
"pp": "case mpr\np : ℕ\nz : ZMod p\ninst✝ : NeZero p\nk : ℕ\n⊢ z + ↑(p * k) = z",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"HMul.hMul",
"ZMod.commRing",
"AddMonoid.toAddSemigroup",
"congrArg",
... | [
"case mpr\np : ℕ\nz : ZMod p\ninst✝ : NeZero p\nk : ℕ\n⊢ z + ↑p * ↑k = z"
] | Nat.cast_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Order.Ring.GeomSum | {
"line": 134,
"column": 6
} | {
"line": 134,
"column": 19
} | {
"line": 135,
"column": 4
} | [
{
"pp": "case pos\nR : Type u_1\ninst✝² : Ring R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\nx : R\nhx : x + 1 ≠ 0\nn : ℕ\nhn : n + 1 + 1 ≠ 0\nh : x + 1 < 0\nh✝ : Even n.succ.succ\nthis : ∑ i ∈ range n.succ.succ, x ^ i < 0\n⊢ ∑ i ∈ range (n + 1 + 1), x ^ i ≠ 0",
"ppTerm": "?pos✝",
"assigned"... | [] | exact this.ne | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.Order.Ring.GeomSum | {
"line": 134,
"column": 6
} | {
"line": 134,
"column": 19
} | {
"line": 135,
"column": 4
} | [
{
"pp": "case pos\nR : Type u_1\ninst✝² : Ring R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\nx : R\nhx : x + 1 ≠ 0\nn : ℕ\nhn : n + 1 + 1 ≠ 0\nh : x + 1 < 0\nh✝ : Even n.succ.succ\nthis : ∑ i ∈ range n.succ.succ, x ^ i < 0\n⊢ ∑ i ∈ range (n + 1 + 1), x ^ i ≠ 0",
"ppTerm": "?pos✝",
"assigned"... | [] | exact this.ne | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Order.Ring.GeomSum | {
"line": 134,
"column": 6
} | {
"line": 134,
"column": 19
} | {
"line": 135,
"column": 4
} | [
{
"pp": "case pos\nR : Type u_1\ninst✝² : Ring R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\nx : R\nhx : x + 1 ≠ 0\nn : ℕ\nhn : n + 1 + 1 ≠ 0\nh : x + 1 < 0\nh✝ : Even n.succ.succ\nthis : ∑ i ∈ range n.succ.succ, x ^ i < 0\n⊢ ∑ i ∈ range (n + 1 + 1), x ^ i ≠ 0",
"ppTerm": "?pos✝",
"assigned"... | [] | exact this.ne | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.ZMod.Basic | {
"line": 733,
"column": 8
} | {
"line": 733,
"column": 45
} | {
"line": 734,
"column": 6
} | [
{
"pp": "n : ℕ\na : ZMod (n + 1)\n⊢ a * a⁻¹ = a * a⁻¹ + ↑n.succ * ↑(a.val.gcdB n.succ)",
"ppTerm": "?m.73",
"assigned": true,
"usedConstants": [
"Int.cast",
"Eq.mpr",
"HMul.hMul",
"ZMod.instInv",
"ZMod.commRing",
"MulZeroClass.toMul",
"ZMod.natCast_self",
... | [] | rw [natCast_self, zero_mul, add_zero] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Data.ZMod.Basic | {
"line": 733,
"column": 8
} | {
"line": 733,
"column": 45
} | {
"line": 734,
"column": 6
} | [
{
"pp": "n : ℕ\na : ZMod (n + 1)\n⊢ a * a⁻¹ = a * a⁻¹ + ↑n.succ * ↑(a.val.gcdB n.succ)",
"ppTerm": "?m.73",
"assigned": true,
"usedConstants": [
"Int.cast",
"Eq.mpr",
"HMul.hMul",
"ZMod.instInv",
"ZMod.commRing",
"MulZeroClass.toMul",
"ZMod.natCast_self",
... | [] | rw [natCast_self, zero_mul, add_zero] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.ZMod.Basic | {
"line": 733,
"column": 8
} | {
"line": 733,
"column": 45
} | {
"line": 734,
"column": 6
} | [
{
"pp": "n : ℕ\na : ZMod (n + 1)\n⊢ a * a⁻¹ = a * a⁻¹ + ↑n.succ * ↑(a.val.gcdB n.succ)",
"ppTerm": "?m.73",
"assigned": true,
"usedConstants": [
"Int.cast",
"Eq.mpr",
"HMul.hMul",
"ZMod.instInv",
"ZMod.commRing",
"MulZeroClass.toMul",
"ZMod.natCast_self",
... | [] | rw [natCast_self, zero_mul, add_zero] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.ZMod.Basic | {
"line": 892,
"column": 6
} | {
"line": 892,
"column": 68
} | {
"line": 893,
"column": 6
} | [
{
"pp": "m✝ n✝ m n : ℕ\nh : m.Coprime n\nto_fun : ZMod (m * n) → ZMod m × ZMod n := ⇑(castHom ⋯ (ZMod m × ZMod n))\ninv_fun : ZMod m × ZMod n → ZMod (m * n) :=\n fun x ↦\n if m * n = 0 then\n if m = 1 then ((RingHom.snd (ZMod m) (ZMod n)) x).cast else ((RingHom.fst (ZMod m) (ZMod n)) x).cast\n else ... | [
"case inl\nm n : ℕ\nh : Nat.Coprime 0 1\nto_fun : ZMod (0 * 1) → ZMod 0 × ZMod 1 := ⇑(castHom ⋯ (ZMod 0 × ZMod 1))\ninv_fun : ZMod 0 × ZMod 1 → ZMod (0 * 1) :=\n fun x ↦\n if 0 * 1 = 0 then\n if 0 = 1 then ((RingHom.snd (ZMod 0) (ZMod 1)) x).cast else ((RingHom.fst (ZMod 0) (ZMod 1)) x).cast\n else ↑↑(N... | rcases h.eq_of_mul_eq_zero hmn0 with (⟨rfl, rfl⟩ | ⟨rfl, rfl⟩) | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
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