module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.Algebra.Order.Star.Basic | {
"line": 458,
"column": 12
} | {
"line": 458,
"column": 17
} | {
"line": 460,
"column": 0
} | [
{
"pp": "case mem\nR : Type u_4\nS : Type u_5\ninst✝⁷ : NonUnitalSemiring R\ninst✝⁶ : PartialOrder R\ninst✝⁵ : StarRing R\ninst✝⁴ : StarOrderedRing R\ninst✝³ : NonUnitalSemiring S\ninst✝² : PartialOrder S\ninst✝¹ : StarRing S\ninst✝ : StarOrderedRing S\nf : R →⋆ₙ+* S\nx p : R\nhf : ∀ (r : R), f (star r) = star ... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Order.Star.Basic | {
"line": 458,
"column": 12
} | {
"line": 458,
"column": 17
} | {
"line": 460,
"column": 0
} | [
{
"pp": "case mem\nR : Type u_4\nS : Type u_5\ninst✝⁷ : NonUnitalSemiring R\ninst✝⁶ : PartialOrder R\ninst✝⁵ : StarRing R\ninst✝⁴ : StarOrderedRing R\ninst✝³ : NonUnitalSemiring S\ninst✝² : PartialOrder S\ninst✝¹ : StarRing S\ninst✝ : StarOrderedRing S\nf : R →⋆ₙ+* S\nx p : R\nhf : ∀ (r : R), f (star r) = star ... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Order.Star.Basic | {
"line": 458,
"column": 12
} | {
"line": 458,
"column": 17
} | {
"line": 460,
"column": 0
} | [
{
"pp": "case zero\nR : Type u_4\nS : Type u_5\ninst✝⁷ : NonUnitalSemiring R\ninst✝⁶ : PartialOrder R\ninst✝⁵ : StarRing R\ninst✝⁴ : StarOrderedRing R\ninst✝³ : NonUnitalSemiring S\ninst✝² : PartialOrder S\ninst✝¹ : StarRing S\ninst✝ : StarOrderedRing S\nf : R →⋆ₙ+* S\nx p : R\nhf : ∀ (r : R), f (star r) = star... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Order.Star.Basic | {
"line": 458,
"column": 12
} | {
"line": 458,
"column": 17
} | {
"line": 460,
"column": 0
} | [
{
"pp": "case zero\nR : Type u_4\nS : Type u_5\ninst✝⁷ : NonUnitalSemiring R\ninst✝⁶ : PartialOrder R\ninst✝⁵ : StarRing R\ninst✝⁴ : StarOrderedRing R\ninst✝³ : NonUnitalSemiring S\ninst✝² : PartialOrder S\ninst✝¹ : StarRing S\ninst✝ : StarOrderedRing S\nf : R →⋆ₙ+* S\nx p : R\nhf : ∀ (r : R), f (star r) = star... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Order.Star.Basic | {
"line": 458,
"column": 12
} | {
"line": 458,
"column": 17
} | {
"line": 460,
"column": 0
} | [
{
"pp": "case zero\nR : Type u_4\nS : Type u_5\ninst✝⁷ : NonUnitalSemiring R\ninst✝⁶ : PartialOrder R\ninst✝⁵ : StarRing R\ninst✝⁴ : StarOrderedRing R\ninst✝³ : NonUnitalSemiring S\ninst✝² : PartialOrder S\ninst✝¹ : StarRing S\ninst✝ : StarOrderedRing S\nf : R →⋆ₙ+* S\nx p : R\nhf : ∀ (r : R), f (star r) = star... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Order.Star.Basic | {
"line": 458,
"column": 12
} | {
"line": 458,
"column": 17
} | {
"line": 460,
"column": 0
} | [
{
"pp": "case add\nR : Type u_4\nS : Type u_5\ninst✝⁷ : NonUnitalSemiring R\ninst✝⁶ : PartialOrder R\ninst✝⁵ : StarRing R\ninst✝⁴ : StarOrderedRing R\ninst✝³ : NonUnitalSemiring S\ninst✝² : PartialOrder S\ninst✝¹ : StarRing S\ninst✝ : StarOrderedRing S\nf : R →⋆ₙ+* S\nx p : R\nhf : ∀ (r : R), f (star r) = star ... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Order.Star.Basic | {
"line": 458,
"column": 12
} | {
"line": 458,
"column": 17
} | {
"line": 460,
"column": 0
} | [
{
"pp": "case add\nR : Type u_4\nS : Type u_5\ninst✝⁷ : NonUnitalSemiring R\ninst✝⁶ : PartialOrder R\ninst✝⁵ : StarRing R\ninst✝⁴ : StarOrderedRing R\ninst✝³ : NonUnitalSemiring S\ninst✝² : PartialOrder S\ninst✝¹ : StarRing S\ninst✝ : StarOrderedRing S\nf : R →⋆ₙ+* S\nx p : R\nhf : ∀ (r : R), f (star r) = star ... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Order.Star.Basic | {
"line": 458,
"column": 12
} | {
"line": 458,
"column": 17
} | {
"line": 460,
"column": 0
} | [
{
"pp": "case add\nR : Type u_4\nS : Type u_5\ninst✝⁷ : NonUnitalSemiring R\ninst✝⁶ : PartialOrder R\ninst✝⁵ : StarRing R\ninst✝⁴ : StarOrderedRing R\ninst✝³ : NonUnitalSemiring S\ninst✝² : PartialOrder S\ninst✝¹ : StarRing S\ninst✝ : StarOrderedRing S\nf : R →⋆ₙ+* S\nx p : R\nhf : ∀ (r : R), f (star r) = star ... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Dimension.Finite | {
"line": 226,
"column": 71
} | {
"line": 226,
"column": 76
} | {
"line": 226,
"column": 76
} | [
{
"pp": "case e'_1.e'_2\nR : Type u\nM : Type v\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nn : ℕ\nhn : n ≤ finrank R M\nh : finrank R M = 0\n⊢ (fun x ↦ x ∈ ∅) = ∅",
"ppTerm": "?e'_1.e'_2",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"False",
"congrArg",
... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.LinearAlgebra.Dimension.Finite | {
"line": 226,
"column": 71
} | {
"line": 226,
"column": 76
} | {
"line": 226,
"column": 76
} | [
{
"pp": "case e'_4\nR : Type u\nM : Type v\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nn : ℕ\nhn : n ≤ finrank R M\nh : finrank R M = 0\ne_1✝ : ↥∅ = ↑∅\n⊢ (fun x ↦ x ∈ ∅) = fun x ↦ x ∈ ∅",
"ppTerm": "?e'_4",
"assigned": true,
"usedConstants": [
"False",
"Set.mem_e... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.LinearAlgebra.Dimension.Constructions | {
"line": 585,
"column": 87
} | {
"line": 585,
"column": 92
} | {
"line": 585,
"column": 92
} | [
{
"pp": "R✝ : Type u\nS : Type u'\nM : Type v\nM' : Type v'\nM₁ : Type v\nι : Type w\nι' : Type w'\nη : Type u₁'\nφ : η → Type u_1\ninst✝⁸ : Semiring R✝\ninst✝⁷ : CommSemiring S\ninst✝⁶ : AddCommMonoid M\ninst✝⁵ : AddCommMonoid M'\ninst✝⁴ : AddCommMonoid M₁\ninst✝³ : Module R✝ M\nR : Type u_2\nV : Type u_3\nins... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.LinearAlgebra.Dimension.Constructions | {
"line": 585,
"column": 87
} | {
"line": 585,
"column": 92
} | {
"line": 585,
"column": 92
} | [
{
"pp": "R✝ : Type u\nS : Type u'\nM : Type v\nM' : Type v'\nM₁ : Type v\nι : Type w\nι' : Type w'\nη : Type u₁'\nφ : η → Type u_1\ninst✝⁸ : Semiring R✝\ninst✝⁷ : CommSemiring S\ninst✝⁶ : AddCommMonoid M\ninst✝⁵ : AddCommMonoid M'\ninst✝⁴ : AddCommMonoid M₁\ninst✝³ : Module R✝ M\nR : Type u_2\nV : Type u_3\nins... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Dimension.Constructions | {
"line": 585,
"column": 87
} | {
"line": 585,
"column": 92
} | {
"line": 585,
"column": 92
} | [
{
"pp": "R✝ : Type u\nS : Type u'\nM : Type v\nM' : Type v'\nM₁ : Type v\nι : Type w\nι' : Type w'\nη : Type u₁'\nφ : η → Type u_1\ninst✝⁸ : Semiring R✝\ninst✝⁷ : CommSemiring S\ninst✝⁶ : AddCommMonoid M\ninst✝⁵ : AddCommMonoid M'\ninst✝⁴ : AddCommMonoid M₁\ninst✝³ : Module R✝ M\nR : Type u_2\nV : Type u_3\nins... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Dimension.Constructions | {
"line": 607,
"column": 56
} | {
"line": 611,
"column": 39
} | {
"line": 613,
"column": 0
} | [
{
"pp": "R : Type u_2\nV : Type u_3\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup V\ninst✝ : Module R V\nW : Submodule R V\nm : Type u_4\nn : Type u_5\nbW : Basis m R ↥W\nbQ : Basis n R (V ⧸ W)\nw : ↥W\ni : m\n⊢ ((bW.sumQuot bQ).repr ↑w) (Sum.inl i) = (bW.repr w) i",
"ppTerm": "?m.44",
"assigned": true,
... | [] | by
classical
refine Eq.symm <| (bW.repr_apply_eq
(fun w i => (sumQuot bW bQ).repr (W.subtype w) (Sum.inl i)) ?_ ?_ ?_ w i) <;>
aesop (add simp Finsupp.single_apply) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.Dimension.Constructions | {
"line": 638,
"column": 2
} | {
"line": 638,
"column": 7
} | {
"line": 640,
"column": 0
} | [
{
"pp": "R : Type u_2\nV : Type u_3\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup V\ninst✝ : Module R V\nW : Submodule R V\nm : Type u_4\nn : Type u_5\nbW : Basis m R ↥W\nbQ : Basis n R (V ⧸ W)\nv : V\nhv : v ∈ W\nj : n\n⊢ W.mkQ v = 0",
"ppTerm": "?m.50",
"assigned": true,
"usedConstants": [
"Su... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.LinearAlgebra.FiniteDimensional.Basic | {
"line": 98,
"column": 65
} | {
"line": 98,
"column": 70
} | {
"line": 98,
"column": 70
} | [
{
"pp": "K : Type u\nV : Type v\ninst✝³ : DivisionRing K\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\nW W' : Submodule K V\ninst✝ : FiniteDimensional K ↥W\nf : ↥W ≃ₗ[K] ↥W'\nQ : Submodule K V\nhQ : IsCompl W Q\neQ : (↥W × ↥Q) ≃ₗ[K] V := W.prodEquivOfIsCompl Q hQ\nQ' : Submodule K V\nhQ' : IsCompl W' Q'\neQ' :... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.LinearAlgebra.FiniteDimensional.Basic | {
"line": 98,
"column": 65
} | {
"line": 98,
"column": 70
} | {
"line": 98,
"column": 70
} | [
{
"pp": "K : Type u\nV : Type v\ninst✝³ : DivisionRing K\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\nW W' : Submodule K V\ninst✝ : FiniteDimensional K ↥W\nf : ↥W ≃ₗ[K] ↥W'\nQ : Submodule K V\nhQ : IsCompl W Q\neQ : (↥W × ↥Q) ≃ₗ[K] V := W.prodEquivOfIsCompl Q hQ\nQ' : Submodule K V\nhQ' : IsCompl W' Q'\neQ' :... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.FiniteDimensional.Basic | {
"line": 98,
"column": 65
} | {
"line": 98,
"column": 70
} | {
"line": 98,
"column": 70
} | [
{
"pp": "K : Type u\nV : Type v\ninst✝³ : DivisionRing K\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\nW W' : Submodule K V\ninst✝ : FiniteDimensional K ↥W\nf : ↥W ≃ₗ[K] ↥W'\nQ : Submodule K V\nhQ : IsCompl W Q\neQ : (↥W × ↥Q) ≃ₗ[K] V := W.prodEquivOfIsCompl Q hQ\nQ' : Submodule K V\nhQ' : IsCompl W' Q'\neQ' :... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.FiniteDimensional.Basic | {
"line": 101,
"column": 28
} | {
"line": 101,
"column": 76
} | {
"line": 102,
"column": 4
} | [
{
"pp": "K : Type u\nV : Type v\ninst✝³ : DivisionRing K\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\nW W' : Submodule K V\ninst✝ : FiniteDimensional K ↥W\nf : ↥W ≃ₗ[K] ↥W'\nQ : Submodule K V\nhQ : IsCompl W Q\neQ : (↥W × ↥Q) ≃ₗ[K] V := W.prodEquivOfIsCompl Q hQ\nQ' : Submodule K V\nhQ' : IsCompl W' Q'\neQ' :... | [
"K : Type u\nV : Type v\ninst✝³ : DivisionRing K\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\nW W' : Submodule K V\ninst✝ : FiniteDimensional K ↥W\nf : ↥W ≃ₗ[K] ↥W'\nQ : Submodule K V\nhQ : IsCompl W Q\neQ : (↥W × ↥Q) ≃ₗ[K] V := W.prodEquivOfIsCompl Q hQ\nQ' : Submodule K V\nhQ' : IsCompl W' Q'\neQ' : (↥W' × ↥Q')... | Module.nonempty_linearEquiv_iff_rank_eq.mp ⟨eQ⟩, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.LinearPMap | {
"line": 106,
"column": 8
} | {
"line": 106,
"column": 24
} | {
"line": 106,
"column": 24
} | [
{
"pp": "R : 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\nf g : E →ₛₗ.[σ] F\nEQ : f = g\nx y : ↥f.domain\nh : ↑x = ↑y\n⊢ x = y",
"ppTerm": "?m.138",
"assigned"... | [] | exact mod_cast h | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.LinearAlgebra.Dimension.FreeAndStrongRankCondition | {
"line": 109,
"column": 2
} | {
"line": 113,
"column": 50
} | {
"line": 114,
"column": 2
} | [
{
"pp": "case refine_1\nK : Type u\nV : Type v\ninst✝⁴ : Ring K\ninst✝³ : StrongRankCondition K\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\ninst✝ : Free K V\nthis : Nontrivial K\nh : Module.rank K V = 1\n⊢ ∃ v₀, v₀ ≠ 0 ∧ ∀ (v : V), ∃ r, r • v₀ = v",
"ppTerm": "?refine_1",
"assigned": true,
"usedC... | [
"case refine_2\nK : Type u\nV : Type v\ninst✝⁴ : Ring K\ninst✝³ : StrongRankCondition K\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\ninst✝ : Free K V\nthis : Nontrivial K\nx✝ : ∃ v₀, v₀ ≠ 0 ∧ ∀ (v : V), ∃ r, r • v₀ = v\nv₀ : V\nh : v₀ ≠ 0\nhv : ∀ (v : V), ∃ r, r • v₀ = v\n⊢ 1 ≤ Module.rank K V"
] | · obtain ⟨v₀, hv⟩ := rank_le_one_iff.1 h.le
refine ⟨v₀, fun hzero ↦ ?_, hv⟩
simp_rw [hzero, smul_zero, exists_const] at hv
haveI : Subsingleton V := .intro fun _ _ ↦ by simp_rw [← hv]
exact one_ne_zero (h ▸ rank_subsingleton' K V) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.LinearAlgebra.FiniteDimensional.Basic | {
"line": 404,
"column": 4
} | {
"line": 405,
"column": 65
} | {
"line": 406,
"column": 4
} | [
{
"pp": "case insert\nK : Type u\nV : Type v\ninst✝³ : DivisionRing K\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\ninst✝ : FiniteDimensional K V\nι : Type u_1\nf : ι → V →ₗ[K] V\ni : ι\ns : Finset ι\nhi : i ∉ s\nih :\n ∀ (comm : (↑s).Pairwise (Commute on f)),\n (s.SupIndep fun i ↦ (f i).ker) → (s.noncommP... | [
"case insert\nK : Type u\nV : Type v\ninst✝³ : DivisionRing K\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\ninst✝ : FiniteDimensional K V\nι : Type u_1\nf : ι → V →ₗ[K] V\ni : ι\ns : Finset ι\nhi : i ∉ s\ncomm : (↑(insert i s)).Pairwise (Commute on f)\nh : (insert i s).SupIndep fun i ↦ (f i).ker\nih : (s.noncommPr... | replace ih : ker (Finset.noncommProd s f <| Set.Pairwise.mono (s.subset_insert i) comm) =
⨆ x ∈ s, ker (f x) := ih _ (h.subset (s.subset_insert i)) | Lean.Elab.Tactic.evalReplace | Lean.Parser.Tactic.replace |
Mathlib.Algebra.FreeAlgebra | {
"line": 231,
"column": 4
} | {
"line": 231,
"column": 33
} | {
"line": 232,
"column": 2
} | [
{
"pp": "case mk\nR : Type u_1\nX : Type u_2\ninst✝ : CommSemiring R\na✝¹ : FreeAlgebra R X\na✝ : Pre R X\n⊢ 0 + Quot.mk (Rel R X) a✝ = Quot.mk (Rel R X) a✝",
"ppTerm": "?mk",
"assigned": true,
"usedConstants": [
"Quot.sound",
"FreeAlgebra.Pre.hasAdd",
"FreeAlgebra.Pre",
"ins... | [] | exact Quot.sound Rel.zero_add | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.LinearAlgebra.FiniteDimensional.Basic | {
"line": 560,
"column": 36
} | {
"line": 560,
"column": 41
} | {
"line": 560,
"column": 41
} | [
{
"pp": "K : Type u\nV : Type v\ninst✝² : DivisionRing K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nS : Submodule K V\nhS : finrank K ↥S = 1\n⊢ S ≠ ⊥",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Submodule",
"False",
"Nat.instMulZeroClass",
"Nat.instOne",
... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.LinearAlgebra.FiniteDimensional.Basic | {
"line": 560,
"column": 36
} | {
"line": 560,
"column": 41
} | {
"line": 560,
"column": 41
} | [
{
"pp": "K : Type u\nV : Type v\ninst✝² : DivisionRing K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nS : Submodule K V\nhS : finrank K ↥S = 1\n⊢ S ≠ ⊥",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Submodule",
"False",
"Nat.instMulZeroClass",
"Nat.instOne",
... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.FiniteDimensional.Basic | {
"line": 560,
"column": 36
} | {
"line": 560,
"column": 41
} | {
"line": 560,
"column": 41
} | [
{
"pp": "K : Type u\nV : Type v\ninst✝² : DivisionRing K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nS : Submodule K V\nhS : finrank K ↥S = 1\n⊢ S ≠ ⊥",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Submodule",
"False",
"Nat.instMulZeroClass",
"Nat.instOne",
... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.MonoidAlgebra.Basic | {
"line": 438,
"column": 19
} | {
"line": 438,
"column": 24
} | {
"line": 439,
"column": 2
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nT : Type u_3\nA : Type u_4\nB : Type u_5\nC : Type u_6\nM : Type u_7\nN : Type u_8\nO : Type u_9\ninst✝⁹ : CommSemiring R\ninst✝⁸ : CommSemiring S\ninst✝⁷ : Semiring A\ninst✝⁶ : Semiring B\ninst✝⁵ : Semiring C\ninst✝⁴ : Algebra R A\ninst✝³ : Algebra R B\ninst✝² : Algebra R C... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.MonoidAlgebra.Basic | {
"line": 438,
"column": 19
} | {
"line": 438,
"column": 24
} | {
"line": 439,
"column": 2
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nT : Type u_3\nA : Type u_4\nB : Type u_5\nC : Type u_6\nM : Type u_7\nN : Type u_8\nO : Type u_9\ninst✝⁹ : CommSemiring R\ninst✝⁸ : CommSemiring S\ninst✝⁷ : Semiring A\ninst✝⁶ : Semiring B\ninst✝⁵ : Semiring C\ninst✝⁴ : Algebra R A\ninst✝³ : Algebra R B\ninst✝² : Algebra R C... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.MonoidAlgebra.Basic | {
"line": 438,
"column": 19
} | {
"line": 438,
"column": 24
} | {
"line": 439,
"column": 2
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nT : Type u_3\nA : Type u_4\nB : Type u_5\nC : Type u_6\nM : Type u_7\nN : Type u_8\nO : Type u_9\ninst✝⁹ : CommSemiring R\ninst✝⁸ : CommSemiring S\ninst✝⁷ : Semiring A\ninst✝⁶ : Semiring B\ninst✝⁵ : Semiring C\ninst✝⁴ : Algebra R A\ninst✝³ : Algebra R B\ninst✝² : Algebra R C... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.MonoidAlgebra.Basic | {
"line": 439,
"column": 20
} | {
"line": 439,
"column": 25
} | {
"line": 441,
"column": 0
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nT : Type u_3\nA : Type u_4\nB : Type u_5\nC : Type u_6\nM : Type u_7\nN : Type u_8\nO : Type u_9\ninst✝⁹ : CommSemiring R\ninst✝⁸ : CommSemiring S\ninst✝⁷ : Semiring A\ninst✝⁶ : Semiring B\ninst✝⁵ : Semiring C\ninst✝⁴ : Algebra R A\ninst✝³ : Algebra R B\ninst✝² : Algebra R C... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.MonoidAlgebra.Basic | {
"line": 439,
"column": 20
} | {
"line": 439,
"column": 25
} | {
"line": 441,
"column": 0
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nT : Type u_3\nA : Type u_4\nB : Type u_5\nC : Type u_6\nM : Type u_7\nN : Type u_8\nO : Type u_9\ninst✝⁹ : CommSemiring R\ninst✝⁸ : CommSemiring S\ninst✝⁷ : Semiring A\ninst✝⁶ : Semiring B\ninst✝⁵ : Semiring C\ninst✝⁴ : Algebra R A\ninst✝³ : Algebra R B\ninst✝² : Algebra R C... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.MonoidAlgebra.Basic | {
"line": 439,
"column": 20
} | {
"line": 439,
"column": 25
} | {
"line": 441,
"column": 0
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nT : Type u_3\nA : Type u_4\nB : Type u_5\nC : Type u_6\nM : Type u_7\nN : Type u_8\nO : Type u_9\ninst✝⁹ : CommSemiring R\ninst✝⁸ : CommSemiring S\ninst✝⁷ : Semiring A\ninst✝⁶ : Semiring B\ninst✝⁵ : Semiring C\ninst✝⁴ : Algebra R A\ninst✝³ : Algebra R B\ninst✝² : Algebra R C... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.LinearPMap | {
"line": 626,
"column": 19
} | {
"line": 627,
"column": 61
} | {
"line": 627,
"column": 61
} | [
{
"pp": "R : 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)\nhnonempty : c.Nonempty\nhc : DirectedOn (fun x1 x2 ↦ x1 ≤ x2) c\nx : E\n⊢ x ∈ sSup (do... | [
"R : 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)\nhnonempty : c.Nonempty\nhc : DirectedOn (fun x1 x2 ↦ x1 ≤ x2) c\nx : E\n⊢ (∃ y ∈ domain '' c, x ∈ ... | Submodule.mem_sSup_of_directed (hnonempty.image _)
(DirectedOn.mono_comp LinearPMap.domain_mono.monotone hc) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Order.Filter.Extr | {
"line": 679,
"column": 40
} | {
"line": 679,
"column": 45
} | {
"line": 681,
"column": 0
} | [
{
"pp": "case inl\nα : Type u\nβ : Type u_1\ninst✝¹ : LinearOrder α\ninst✝ : Preorder β\nb : α\nf : α → β\nh₀ : MonotoneOn f (Iic b)\nh₁ : AntitoneOn f (Ici b)\nx : α\nx✝ : x ∈ univ\nh✝ : x ≤ b\n⊢ x ∈ {x | (fun x ↦ f x ≤ f b) x}",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"instRef... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Order.Filter.Extr | {
"line": 679,
"column": 40
} | {
"line": 679,
"column": 45
} | {
"line": 681,
"column": 0
} | [
{
"pp": "case inr\nα : Type u\nβ : Type u_1\ninst✝¹ : LinearOrder α\ninst✝ : Preorder β\nb : α\nf : α → β\nh₀ : MonotoneOn f (Iic b)\nh₁ : AntitoneOn f (Ici b)\nx : α\nx✝ : x ∈ univ\nh✝ : b ≤ x\n⊢ x ∈ {x | (fun x ↦ f x ≤ f b) x}",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Set.Ici... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.LinearAlgebra.LinearPMap | {
"line": 833,
"column": 6
} | {
"line": 833,
"column": 20
} | {
"line": 833,
"column": 20
} | [
{
"pp": "R : 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 : E →ₗ.[R] F\nx : E\ny : F\nh : (x, y) ∈ f.graph\n⊢ x ∈ f.domain",
"ppTerm": "?m.53",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [
"R : 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 : E →ₗ.[R] F\nx : E\ny : F\nh : (x, y) ∈ f.graph\n⊢ ∃ y, (x, y) ∈ f.graph"
] | mem_domain_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.Degree.Defs | {
"line": 177,
"column": 2
} | {
"line": 177,
"column": 41
} | {
"line": 179,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝ : Semiring R\nn : ℕ\n⊢ (↑n).natDegree = 0",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Polynomial.C",
"NonAssocSemiring.toAddCommMonoidWithOne",
"congrArg",
"RingHom",
"Polynomial.natDegree_C",
"AddMonoidWithOne.toNatCast",... | [] | simp only [← C_eq_natCast, natDegree_C] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Polynomial.Degree.Defs | {
"line": 177,
"column": 2
} | {
"line": 177,
"column": 41
} | {
"line": 179,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝ : Semiring R\nn : ℕ\n⊢ (↑n).natDegree = 0",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Polynomial.C",
"NonAssocSemiring.toAddCommMonoidWithOne",
"congrArg",
"RingHom",
"Polynomial.natDegree_C",
"AddMonoidWithOne.toNatCast",... | [] | simp only [← C_eq_natCast, natDegree_C] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Polynomial.Degree.Defs | {
"line": 177,
"column": 2
} | {
"line": 177,
"column": 41
} | {
"line": 179,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝ : Semiring R\nn : ℕ\n⊢ (↑n).natDegree = 0",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Polynomial.C",
"NonAssocSemiring.toAddCommMonoidWithOne",
"congrArg",
"RingHom",
"Polynomial.natDegree_C",
"AddMonoidWithOne.toNatCast",... | [] | simp only [← C_eq_natCast, natDegree_C] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.LinearPMap | {
"line": 855,
"column": 8
} | {
"line": 855,
"column": 22
} | {
"line": 855,
"column": 22
} | [
{
"pp": "case left\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\nx : E\nhx : x ∈ f.domain\n⊢ x ∈ g.domain",
"ppTerm": "?left",
"assigned": true,
"usedCons... | [
"case left\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\nx : E\nhx : ∃ y, (x, y) ∈ f.graph\n⊢ ∃ y, (x, y) ∈ g.graph"
] | mem_domain_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.MonoidAlgebra.Degree | {
"line": 115,
"column": 35
} | {
"line": 115,
"column": 54
} | {
"line": 115,
"column": 54
} | [
{
"pp": "R : Type u_1\nA : Type u_3\nB : Type u_5\ninst✝⁶ : SemilatticeSup B\ninst✝⁵ : OrderBot B\ninst✝⁴ : Semiring R\ninst✝³ : Add A\ninst✝² : Add B\ninst✝¹ : AddLeftMono B\ninst✝ : AddRightMono B\ndegb : A → B\ndegbm : ∀ (a b : A), degb (a + b) ≤ degb a + degb b\nf g : R[A]\n_fd : A\nfds : _fd ∈ f.coeff.supp... | [
"R : Type u_1\nA : Type u_3\nB : Type u_5\ninst✝⁶ : SemilatticeSup B\ninst✝⁵ : OrderBot B\ninst✝⁴ : Semiring R\ninst✝³ : Add A\ninst✝² : Add B\ninst✝¹ : AddLeftMono B\ninst✝ : AddRightMono B\ndegb : A → B\ndegbm : ∀ (a b : A), degb (a + b) ≤ degb a + degb b\nf g : R[A]\n_fd : A\nfds : _fd ∈ f.coeff.support\n_gd : A... | ← Finset.le_sup gds | Mathlib.Tactic.GRewrite.evalGRewriteSeq | null |
Mathlib.Algebra.Polynomial.Degree.Units | {
"line": 71,
"column": 2
} | {
"line": 71,
"column": 23
} | {
"line": 73,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝² : Semiring R\ninst✝¹ : NoZeroDivisors R\ninst✝ : Nontrivial R\nu : R[X]ˣ\n⊢ ¬↑u = 0",
"ppTerm": "?m.36",
"assigned": true,
"usedConstants": [
"Units.ne_zero",
"Polynomial.nontrivial",
"Polynomial",
"Polynomial.semiring",
"Semiring.toMonoidWit... | [] | exact Units.ne_zero _ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.Polynomial.Eval.Coeff | {
"line": 63,
"column": 50
} | {
"line": 63,
"column": 71
} | {
"line": 63,
"column": 71
} | [
{
"pp": "R : Type u\ninst✝ : Semiring R\np : R[X]\nb : ℕ\nx✝ : b ∈ p.support\nhb : b ≠ 0\n⊢ (fun e a ↦ a * 0 ^ e) b (p.coeff b) = 0",
"ppTerm": "?m.68",
"assigned": true,
"usedConstants": [
"HMul.hMul",
"congrArg",
"AddMonoid.toAddZeroClass",
"AddZeroClass.toAddZero",
"... | [] | by simp [zero_pow hb] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Polynomial.Eval.Coeff | {
"line": 87,
"column": 7
} | {
"line": 87,
"column": 26
} | {
"line": 87,
"column": 26
} | [
{
"pp": "R : Type u\nS : Type v\nT : Type w\ninst✝² : Semiring R\ninst✝¹ : Semiring S\nf : R →+* S\ninst✝ : Semiring T\ng : S →+* T\np : R[X]\n⊢ ∀ (n : ℕ), (map g (map f p)).coeff n = (map (g.comp f) p).coeff n",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"Polynomial.coeff_map",
... | [] | by simp [coeff_map] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Polynomial.Degree.Operations | {
"line": 468,
"column": 2
} | {
"line": 470,
"column": 42
} | {
"line": 472,
"column": 0
} | [
{
"pp": "R : Type u\na : R\ninst✝ : Semiring R\np : R[X]\nha : a ≠ 0\nhp : IsRightRegular p.leadingCoeff\n⊢ (a • p).degree = p.degree",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Polynomial.degree_l... | [] | refine le_antisymm (degree_smul_le a p) <| degree_le_degree ?_
rw [coeff_smul, coeff_natDegree, smul_eq_mul, ne_eq]
exact hp.mul_right_eq_zero_iff.ne.mpr ha | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Polynomial.Degree.Operations | {
"line": 468,
"column": 2
} | {
"line": 470,
"column": 42
} | {
"line": 472,
"column": 0
} | [
{
"pp": "R : Type u\na : R\ninst✝ : Semiring R\np : R[X]\nha : a ≠ 0\nhp : IsRightRegular p.leadingCoeff\n⊢ (a • p).degree = p.degree",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Polynomial.degree_l... | [] | refine le_antisymm (degree_smul_le a p) <| degree_le_degree ?_
rw [coeff_smul, coeff_natDegree, smul_eq_mul, ne_eq]
exact hp.mul_right_eq_zero_iff.ne.mpr ha | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Polynomial.Degree.Operations | {
"line": 486,
"column": 55
} | {
"line": 486,
"column": 60
} | {
"line": 486,
"column": 60
} | [
{
"pp": "R : Type u\ninst✝ : Semiring R\np : R[X]\n⊢ (∃ x, C x = p) → p.natDegree = 0",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Polynomial.C",
"congrArg",
"RingHom",
"Exists",
"Polynomial.natDegree_C",
"instOfNatNat",
"Polynomial",
"Ri... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Polynomial.Degree.Operations | {
"line": 486,
"column": 55
} | {
"line": 486,
"column": 60
} | {
"line": 486,
"column": 60
} | [
{
"pp": "R : Type u\ninst✝ : Semiring R\np : R[X]\n⊢ (∃ x, C x = p) → p.natDegree = 0",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Polynomial.C",
"congrArg",
"RingHom",
"Exists",
"Polynomial.natDegree_C",
"instOfNatNat",
"Polynomial",
"Ri... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Polynomial.Degree.Operations | {
"line": 486,
"column": 55
} | {
"line": 486,
"column": 60
} | {
"line": 486,
"column": 60
} | [
{
"pp": "R : Type u\ninst✝ : Semiring R\np : R[X]\n⊢ (∃ x, C x = p) → p.natDegree = 0",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Polynomial.C",
"congrArg",
"RingHom",
"Exists",
"Polynomial.natDegree_C",
"instOfNatNat",
"Polynomial",
"Ri... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.MonoidAlgebra.Degree | {
"line": 604,
"column": 89
} | {
"line": 604,
"column": 94
} | {
"line": 604,
"column": 94
} | [
{
"pp": "case zero\nR : Type u_1\ninst✝⁷ : Semiring R\nA : Type u_8\nB : Type u_9\ninst✝⁶ : AddMonoid A\ninst✝⁵ : AddMonoid B\ninst✝⁴ : LinearOrder B\ninst✝³ : OrderBot B\ninst✝² : AddLeftStrictMono B\ninst✝¹ : AddRightStrictMono B\nD : A → B\np : R[A]\nn : ℕ\nhzero : D 0 = 0\nhadd : ∀ (a1 a2 : A), D (a1 + a2) ... | [
"case zero\nR : Type u_1\ninst✝⁷ : Semiring R\nA : Type u_8\nB : Type u_9\ninst✝⁶ : AddMonoid A\ninst✝⁵ : AddMonoid B\ninst✝⁴ : LinearOrder B\ninst✝³ : OrderBot B\ninst✝² : AddLeftStrictMono B\ninst✝¹ : AddRightStrictMono B\nD : A → B\np : R[A]\nn : ℕ\nhzero : D 0 = 0\nhadd : ∀ (a1 a2 : A), D (a1 + a2) = D a1 + D a... | hzero | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.Finsupp.Antidiagonal | {
"line": 72,
"column": 6
} | {
"line": 72,
"column": 46
} | {
"line": 73,
"column": 6
} | [
{
"pp": "case mp.inr\nα : Type u\ninst✝ : DecidableEq α\na : α\nn : ℕ\nx y : α →₀ ℕ\ni : α\nh : x i + y i = if a = i then n else 0\nhai : a ≠ i\n⊢ (if a = i then x a else 0) = x i ∧ (if a = i then y a else 0) = y i",
"ppTerm": "?mp.inr",
"assigned": true,
"usedConstants": [
"Finsupp.instFunLik... | [
"case mp.inr\nα : Type u\ninst✝ : DecidableEq α\na : α\nn : ℕ\nx y : α →₀ ℕ\ni : α\nhai : a ≠ i\nh : x i = 0 ∧ y i = 0\n⊢ 0 = x i ∧ 0 = y i"
] | simp_rw [if_neg hai, add_eq_zero] at h ⊢ | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Algebra.Polynomial.AlgebraMap | {
"line": 365,
"column": 2
} | {
"line": 365,
"column": 11
} | {
"line": 366,
"column": 2
} | [
{
"pp": "R : Type u_3\ninst✝¹ : CommRing R\na b : R\ninst✝ : Invertible a\n⊢ (algEquivCMulXAddC a b).symm = algEquivCMulXAddC (⅟a) (-⅟a * b)",
"ppTerm": "?m.39",
"assigned": true,
"usedConstants": [
"NegZeroClass.toNeg",
"HMul.hMul",
"AlgEquiv.symm",
"CommSemiring.toSemiring"... | [
"R : Type u_3\ninst✝¹ : CommRing R\na b : R\ninst✝ : Invertible a\np : R[X]\n⊢ (algEquivCMulXAddC a b).symm p = (algEquivCMulXAddC (⅟a) (-⅟a * b)) p"
] | ext p : 1 | _private.Lean.Elab.Tactic.Ext.0.Lean.Elab.Tactic.Ext.evalExt | Lean.Elab.Tactic.Ext.ext |
Mathlib.Order.PiLex | {
"line": 257,
"column": 2
} | {
"line": 257,
"column": 7
} | {
"line": 259,
"column": 0
} | [
{
"pp": "case h\nι : Type u_1\nβ : ι → Type u_2\ninst✝¹ : LinearOrder ι\nx y : (i : ι) → β i\ni : ι\ninst✝ : (i : ι) → LinearOrder (β i)\nh : ∀ (j : ι), j < i → x j = y j\nhxy : y i < x i\n⊢ (∀ (j : ι), (fun x1 x2 ↦ x1 < x2) j i → toLex y j = toLex x j) ∧ (fun {i} x1 x2 ↦ x1 < x2) (toLex y i) (toLex x i)",
... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Order.PiLex | {
"line": 267,
"column": 2
} | {
"line": 267,
"column": 7
} | {
"line": 269,
"column": 0
} | [
{
"pp": "case h\nι : Type u_1\nβ : ι → Type u_2\ninst✝¹ : LinearOrder ι\nx y : (i : ι) → β i\ni : ι\ninst✝ : (i : ι) → LinearOrder (β i)\nh : ∀ (j : ι), j > i → x j = y j\nhxy : y i < x i\n⊢ (∀ (j : ι), (fun x1 x2 ↦ x1 > x2) j i → toColex y j = toColex x j) ∧\n (fun {i} x1 x2 ↦ x1 < x2) (toColex y i) (toCole... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.MvPolynomial.Eval | {
"line": 120,
"column": 2
} | {
"line": 120,
"column": 35
} | {
"line": 121,
"column": 2
} | [
{
"pp": "R : Type u\nS₁ : Type v\nσ : Type u_1\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\np : MvPolynomial σ R\nf : R →+* S₁\ng : σ → S₁\n⊢ ∀ {s : σ →₀ ℕ} {a : R}, eval₂ f g (p * (monomial s) a) = eval₂ f g p * f a * s.prod fun n e ↦ g n ^ e",
"ppTerm": "?m.56",
"assigned": true,
"usedConsta... | [
"case C\nR : Type u\nS₁ : Type v\nσ : Type u_1\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\np : MvPolynomial σ R\nf : R →+* S₁\ng : σ → S₁\n⊢ ∀ (a : R) {s : σ →₀ ℕ} {a_1 : R},\n eval₂ f g (C a * (monomial s) a_1) = eval₂ f g (C a) * f a_1 * s.prod fun n e ↦ g n ^ e",
"case add\nR : Type u\nS₁ : Type v\nσ... | apply MvPolynomial.induction_on p | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Algebra.MvPolynomial.Eval | {
"line": 121,
"column": 4
} | {
"line": 122,
"column": 52
} | {
"line": 123,
"column": 2
} | [
{
"pp": "case C\nR : Type u\nS₁ : Type v\nσ : Type u_1\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\np : MvPolynomial σ R\nf : R →+* S₁\ng : σ → S₁\n⊢ ∀ (a : R) {s : σ →₀ ℕ} {a_1 : R},\n eval₂ f g (C a * (monomial s) a_1) = eval₂ f g (C a) * f a_1 * s.prod fun n e ↦ g n ^ e",
"ppTerm": "?C",
"as... | [] | intro a' s a
simp [C_mul_monomial, eval₂_monomial, f.map_mul] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.MvPolynomial.Eval | {
"line": 121,
"column": 4
} | {
"line": 122,
"column": 52
} | {
"line": 123,
"column": 2
} | [
{
"pp": "case C\nR : Type u\nS₁ : Type v\nσ : Type u_1\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\np : MvPolynomial σ R\nf : R →+* S₁\ng : σ → S₁\n⊢ ∀ (a : R) {s : σ →₀ ℕ} {a_1 : R},\n eval₂ f g (C a * (monomial s) a_1) = eval₂ f g (C a) * f a_1 * s.prod fun n e ↦ g n ^ e",
"ppTerm": "?C",
"as... | [] | intro a' s a
simp [C_mul_monomial, eval₂_monomial, f.map_mul] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.MvPolynomial.Eval | {
"line": 216,
"column": 2
} | {
"line": 216,
"column": 35
} | {
"line": 216,
"column": 36
} | [
{
"pp": "R : Type u\nS₁ : Type v\nσ : Type u_1\ninst✝² : CommSemiring R\ninst✝¹ : CommSemiring S₁\nS₂ : Type u_2\ninst✝ : CommSemiring S₂\nk : S₁ →+* S₂\nf : R →+* S₁\ng : σ → S₁\np : MvPolynomial σ R\n⊢ k (eval₂ f g p) = eval₂ (k.comp f) (⇑k ∘ g) p",
"ppTerm": "?m.34",
"assigned": true,
"usedConsta... | [
"case C\nR : Type u\nS₁ : Type v\nσ : Type u_1\ninst✝² : CommSemiring R\ninst✝¹ : CommSemiring S₁\nS₂ : Type u_2\ninst✝ : CommSemiring S₂\nk : S₁ →+* S₂\nf : R →+* S₁\ng : σ → S₁\np : MvPolynomial σ R\n⊢ ∀ (a : R), k (eval₂ f g (C a)) = eval₂ (k.comp f) (⇑k ∘ g) (C a)",
"case add\nR : Type u\nS₁ : Type v\nσ : Typ... | apply MvPolynomial.induction_on p | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Algebra.MvPolynomial.Eval | {
"line": 217,
"column": 4
} | {
"line": 217,
"column": 65
} | {
"line": 219,
"column": 0
} | [
{
"pp": "case C\nR : Type u\nS₁ : Type v\nσ : Type u_1\ninst✝² : CommSemiring R\ninst✝¹ : CommSemiring S₁\nS₂ : Type u_2\ninst✝ : CommSemiring S₂\nk : S₁ →+* S₂\nf : R →+* S₁\ng : σ → S₁\np : MvPolynomial σ R\n⊢ ∀ (a : R), k (eval₂ f g (C a)) = eval₂ (k.comp f) (⇑k ∘ g) (C a)",
"ppTerm": "?C",
"assigned... | [] | simp +contextual [eval₂_add, k.map_add, eval₂_mul, k.map_mul] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.MvPolynomial.Eval | {
"line": 217,
"column": 4
} | {
"line": 217,
"column": 65
} | {
"line": 219,
"column": 0
} | [
{
"pp": "case add\nR : Type u\nS₁ : Type v\nσ : Type u_1\ninst✝² : CommSemiring R\ninst✝¹ : CommSemiring S₁\nS₂ : Type u_2\ninst✝ : CommSemiring S₂\nk : S₁ →+* S₂\nf : R →+* S₁\ng : σ → S₁\np : MvPolynomial σ R\n⊢ ∀ (p q : MvPolynomial σ R),\n k (eval₂ f g p) = eval₂ (k.comp f) (⇑k ∘ g) p →\n k (eval₂ f... | [] | simp +contextual [eval₂_add, k.map_add, eval₂_mul, k.map_mul] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.MvPolynomial.Eval | {
"line": 217,
"column": 4
} | {
"line": 217,
"column": 65
} | {
"line": 219,
"column": 0
} | [
{
"pp": "case mul_X\nR : Type u\nS₁ : Type v\nσ : Type u_1\ninst✝² : CommSemiring R\ninst✝¹ : CommSemiring S₁\nS₂ : Type u_2\ninst✝ : CommSemiring S₂\nk : S₁ →+* S₂\nf : R →+* S₁\ng : σ → S₁\np : MvPolynomial σ R\n⊢ ∀ (p : MvPolynomial σ R) (n : σ),\n k (eval₂ f g p) = eval₂ (k.comp f) (⇑k ∘ g) p → k (eval₂ ... | [] | simp +contextual [eval₂_add, k.map_add, eval₂_mul, k.map_mul] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.MvPolynomial.Eval | {
"line": 223,
"column": 2
} | {
"line": 223,
"column": 35
} | {
"line": 223,
"column": 36
} | [
{
"pp": "R : Type u\nσ : Type u_1\ninst✝ : CommSemiring R\np : MvPolynomial σ R\n⊢ eval₂ C X p = p",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Nat.instMulZeroClass",
"MvPolynomial.X",
"AddMonoidAlgebra.commSemiring",
"MvPolynomial.eval₂",
"MvPolynomial.in... | [
"case C\nR : Type u\nσ : Type u_1\ninst✝ : CommSemiring R\np : MvPolynomial σ R\n⊢ ∀ (a : R), eval₂ C X (C a) = C a",
"case add\nR : Type u\nσ : Type u_1\ninst✝ : CommSemiring R\np : MvPolynomial σ R\n⊢ ∀ (p q : MvPolynomial σ R), eval₂ C X p = p → eval₂ C X q = q → eval₂ C X (p + q) = p + q",
"case mul_X\nR : ... | apply MvPolynomial.induction_on p | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Algebra.MvPolynomial.Basic | {
"line": 327,
"column": 4
} | {
"line": 333,
"column": 46
} | {
"line": 335,
"column": 0
} | [
{
"pp": "case single_add\nR : Type u\nσ : Type u_1\ninst✝ : CommSemiring R\nmotive : MvPolynomial σ R → Prop\nC : ∀ (a : R), motive (MvPolynomial.C a)\nmul_X : ∀ (p : MvPolynomial σ R) (n : σ), motive p → motive (p * X n)\ns : σ →₀ ℕ\na : R\n⊢ ∀ (a_1 : σ) (b : ℕ) (f : σ →₀ ℕ),\n a_1 ∉ f.support → b ≠ 0 → mot... | [] | intro n e p _hpn _he ih
have : ∀ e : ℕ, motive (monomial p a * X n ^ e) := by
intro e
induction e with
| zero => simp [ih]
| succ e e_ih => simp [pow_succ, (mul_assoc _ _ _).symm, mul_X, e_ih]
simp [add_comm, monomial_add_single, this] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.MvPolynomial.Basic | {
"line": 327,
"column": 4
} | {
"line": 333,
"column": 46
} | {
"line": 335,
"column": 0
} | [
{
"pp": "case single_add\nR : Type u\nσ : Type u_1\ninst✝ : CommSemiring R\nmotive : MvPolynomial σ R → Prop\nC : ∀ (a : R), motive (MvPolynomial.C a)\nmul_X : ∀ (p : MvPolynomial σ R) (n : σ), motive p → motive (p * X n)\ns : σ →₀ ℕ\na : R\n⊢ ∀ (a_1 : σ) (b : ℕ) (f : σ →₀ ℕ),\n a_1 ∉ f.support → b ≠ 0 → mot... | [] | intro n e p _hpn _he ih
have : ∀ e : ℕ, motive (monomial p a * X n ^ e) := by
intro e
induction e with
| zero => simp [ih]
| succ e e_ih => simp [pow_succ, (mul_assoc _ _ _).symm, mul_X, e_ih]
simp [add_comm, monomial_add_single, this] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.MvPolynomial.Eval | {
"line": 357,
"column": 2
} | {
"line": 357,
"column": 35
} | {
"line": 358,
"column": 2
} | [
{
"pp": "R : Type u\nS₁ : Type v\nσ : Type u_1\ninst✝² : CommSemiring R\ninst✝¹ : CommSemiring S₁\nS₂ : Type u_2\ninst✝ : CommSemiring S₂\nk : S₁ →+* S₂\nf : R →+* S₁\ng : σ → S₁\np : MvPolynomial σ R\n⊢ k (eval₂ f g p) = eval₂ k (⇑k ∘ g) ((map f) p)",
"ppTerm": "?m.34",
"assigned": true,
"usedConst... | [
"case C\nR : Type u\nS₁ : Type v\nσ : Type u_1\ninst✝² : CommSemiring R\ninst✝¹ : CommSemiring S₁\nS₂ : Type u_2\ninst✝ : CommSemiring S₂\nk : S₁ →+* S₂\nf : R →+* S₁\ng : σ → S₁\np : MvPolynomial σ R\n⊢ ∀ (a : R), k (eval₂ f g (C a)) = eval₂ k (⇑k ∘ g) ((map f) (C a))",
"case add\nR : Type u\nS₁ : Type v\nσ : Ty... | apply MvPolynomial.induction_on p | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Algebra.MvPolynomial.Basic | {
"line": 505,
"column": 65
} | {
"line": 506,
"column": 70
} | {
"line": 508,
"column": 0
} | [
{
"pp": "R : Type u\nσ : Type u_1\ninst✝¹ : CommSemiring R\nα : Type u_2\ninst✝ : DecidableEq σ\ns : Finset α\nf : α → MvPolynomial σ R\n⊢ (∑ x ∈ s, f x).support ⊆ s.biUnion fun x ↦ (f x).support",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Nat.instMulZeroClass",
... | [] | by
simpa [support, coeff, MvPolynomial] using Finsupp.support_finsetSum | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.MvPolynomial.Eval | {
"line": 367,
"column": 2
} | {
"line": 367,
"column": 35
} | {
"line": 368,
"column": 2
} | [
{
"pp": "R : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\nf : R →+* S₁\ng : S₂ → MvPolynomial S₃ R\np : MvPolynomial S₂ R\n⊢ (map f) (eval₂ C g p) = eval₂ C (⇑(map f) ∘ g) ((map f) p)",
"ppTerm": "?m.48",
"assigned": true,
"usedConstants": [
... | [
"case C\nR : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\nf : R →+* S₁\ng : S₂ → MvPolynomial S₃ R\np : MvPolynomial S₂ R\n⊢ ∀ (a : R), (map f) (eval₂ C g (C a)) = eval₂ C (⇑(map f) ∘ g) ((map f) (C a))",
"case add\nR : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Typ... | apply MvPolynomial.induction_on p | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Algebra.MvPolynomial.Rename | {
"line": 188,
"column": 4
} | {
"line": 188,
"column": 27
} | {
"line": 189,
"column": 4
} | [
{
"pp": "case monomial\nσ : Type u_1\nτ : Type u_2\nR : Type u_4\ninst✝ : CommSemiring R\nf : σ → τ\nhf : Injective f\np : MvPolynomial τ R\ns : σ →₀ ℕ\nu : τ →₀ ℕ\nr : R\n⊢ coeff s ((killCompl hf) ((monomial u) r)) = coeff (Finsupp.mapDomain f s) ((monomial u) r)",
"ppTerm": "?monomial",
"assigned": tr... | [
"case monomial\nσ : Type u_1\nτ : Type u_2\nR : Type u_4\ninst✝ : CommSemiring R\nf : σ → τ\nhf : Injective f\np : MvPolynomial τ R\ns : σ →₀ ℕ\nu : τ →₀ ℕ\nr : R\n⊢ coeff s (if ↑u.support ⊆ range f then (monomial (Finsupp.comapDomain f u ⋯)) r else 0) =\n coeff (Finsupp.mapDomain f s) ((monomial u) r)"
] | rw [killCompl_monomial] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.MvPolynomial.Basic | {
"line": 710,
"column": 2
} | {
"line": 710,
"column": 40
} | {
"line": 711,
"column": 2
} | [
{
"pp": "R : Type u\nσ : Type u_1\ninst✝¹ : CommSemiring R\ninst✝ : DecidableEq σ\np q : MvPolynomial σ R\n⊢ p.support ∆ q.support ⊆ (p + q).support",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"symmDiff_def",
"Eq.mpr",
"Nat.instMulZeroClass",
"AddMonoidAlgebra.s... | [
"R : Type u\nσ : Type u_1\ninst✝¹ : CommSemiring R\ninst✝ : DecidableEq σ\np q : MvPolynomial σ R\n⊢ p.support \\ q.support ∪ q.support \\ p.support ⊆ (p + q).support"
] | rw [symmDiff_def, Finset.sup_eq_union] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.MvPolynomial.Rename | {
"line": 217,
"column": 26
} | {
"line": 217,
"column": 82
} | {
"line": 217,
"column": 83
} | [
{
"pp": "σ : Type u_1\nτ : Type u_2\nα : Type u_3\nR : Type u_4\nS : Type u_5\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S\nf : σ ≃ τ\np : MvPolynomial τ R\n⊢ (rename ⇑f) ((rename ⇑f.symm) p) = p",
"ppTerm": "?m.62",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Nat.instMulZeroClas... | [] | by rw [rename_rename, f.self_comp_symm, rename_id_apply] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.MvPolynomial.Rename | {
"line": 239,
"column": 2
} | {
"line": 239,
"column": 35
} | {
"line": 239,
"column": 36
} | [
{
"pp": "σ : Type u_1\nτ : Type u_2\nR : Type u_4\nS : Type u_5\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S\nf : R →+* S\nk : σ → τ\ng : τ → S\np : MvPolynomial σ R\n⊢ eval₂ f g ((rename k) p) = eval₂ f (g ∘ k) p",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Nat.instMulZeroCl... | [
"case C\nσ : Type u_1\nτ : Type u_2\nR : Type u_4\nS : Type u_5\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S\nf : R →+* S\nk : σ → τ\ng : τ → S\np : MvPolynomial σ R\n⊢ ∀ (a : R), eval₂ f g ((rename k) (C a)) = eval₂ f (g ∘ k) (C a)",
"case add\nσ : Type u_1\nτ : Type u_2\nR : Type u_4\nS : Type u_5\ninst✝¹ :... | apply MvPolynomial.induction_on p | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Algebra.MvPolynomial.Eval | {
"line": 386,
"column": 2
} | {
"line": 386,
"column": 35
} | {
"line": 386,
"column": 36
} | [
{
"pp": "R : Type u\nS₁ : Type v\nσ : Type u_1\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\nf : R →+* S₁\np : MvPolynomial σ R\n⊢ ∀ (m : σ →₀ ℕ), coeff m ((map f) p) = f (coeff m p)",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Finsupp.instAddZeroClass",
"Nat.instMulZe... | [
"case C\nR : Type u\nS₁ : Type v\nσ : Type u_1\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\nf : R →+* S₁\np : MvPolynomial σ R\n⊢ ∀ (a : R) (m : σ →₀ ℕ), coeff m ((map f) (C a)) = f (coeff m (C a))",
"case add\nR : Type u\nS₁ : Type v\nσ : Type u_1\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\nf : R →+... | apply MvPolynomial.induction_on p | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Algebra.MvPolynomial.Rename | {
"line": 258,
"column": 2
} | {
"line": 258,
"column": 35
} | {
"line": 258,
"column": 36
} | [
{
"pp": "σ : Type u_1\nτ : Type u_2\nR : Type u_4\ninst✝ : CommSemiring R\nk : σ → τ\np : MvPolynomial σ R\ng : τ → MvPolynomial σ R\n⊢ (rename k) (eval₂ C (g ∘ k) p) = eval₂ C (⇑(rename k) ∘ g) ((rename k) p)",
"ppTerm": "?m.43",
"assigned": true,
"usedConstants": [
"Nat.instMulZeroClass",
... | [
"case C\nσ : Type u_1\nτ : Type u_2\nR : Type u_4\ninst✝ : CommSemiring R\nk : σ → τ\np : MvPolynomial σ R\ng : τ → MvPolynomial σ R\n⊢ ∀ (a : R), (rename k) (eval₂ C (g ∘ k) (C a)) = eval₂ C (⇑(rename k) ∘ g) ((rename k) (C a))",
"case add\nσ : Type u_1\nτ : Type u_2\nR : Type u_4\ninst✝ : CommSemiring R\nk : σ ... | apply MvPolynomial.induction_on p | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Algebra.MvPolynomial.Rename | {
"line": 264,
"column": 2
} | {
"line": 264,
"column": 35
} | {
"line": 264,
"column": 36
} | [
{
"pp": "σ : Type u_1\nτ : Type u_2\nR : Type u_4\ninst✝ : CommSemiring R\np : MvPolynomial σ R\nj : τ\ng : σ → MvPolynomial σ R\n⊢ (rename (Prod.mk j)) (eval₂ C g p) = eval₂ C (fun x ↦ (rename (Prod.mk j)) (g x)) p",
"ppTerm": "?m.40",
"assigned": true,
"usedConstants": [
"Nat.instMulZeroClas... | [
"case C\nσ : Type u_1\nτ : Type u_2\nR : Type u_4\ninst✝ : CommSemiring R\np : MvPolynomial σ R\nj : τ\ng : σ → MvPolynomial σ R\n⊢ ∀ (a : R), (rename (Prod.mk j)) (eval₂ C g (C a)) = eval₂ C (fun x ↦ (rename (Prod.mk j)) (g x)) (C a)",
"case add\nσ : Type u_1\nτ : Type u_2\nR : Type u_4\ninst✝ : CommSemiring R\n... | apply MvPolynomial.induction_on p | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Algebra.MvPolynomial.Basic | {
"line": 765,
"column": 75
} | {
"line": 765,
"column": 92
} | {
"line": 767,
"column": 0
} | [
{
"pp": "R : Type u\nσ : Type u_1\ninst✝ : CommSemiring R\np : MvPolynomial σ R\n⊢ p.support = ∅ ↔ p = 0",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Finsupp.support_eq_empty._simp_1",
"AddMonoidAlgebra.instAddMonoid",
"Nat.instMulZeroClass",
"AddMonoidAlgebra.s... | [] | by simp [support] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.MvPolynomial.Rename | {
"line": 270,
"column": 2
} | {
"line": 270,
"column": 35
} | {
"line": 270,
"column": 36
} | [
{
"pp": "σ : Type u_1\nτ : Type u_2\nR : Type u_4\nS : Type u_5\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S\nf : R →+* S\ng : σ × τ → S\ni : σ\np : MvPolynomial τ R\n⊢ eval₂ f g ((rename (Prod.mk i)) p) = eval₂ f (fun j ↦ g (i, j)) p",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
... | [
"case C\nσ : Type u_1\nτ : Type u_2\nR : Type u_4\nS : Type u_5\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S\nf : R →+* S\ng : σ × τ → S\ni : σ\np : MvPolynomial τ R\n⊢ ∀ (a : R), eval₂ f g ((rename (Prod.mk i)) (C a)) = eval₂ f (fun j ↦ g (i, j)) (C a)",
"case add\nσ : Type u_1\nτ : Type u_2\nR : Type u_4\nS... | apply MvPolynomial.induction_on p | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Algebra.MvPolynomial.Eval | {
"line": 434,
"column": 2
} | {
"line": 434,
"column": 35
} | {
"line": 434,
"column": 36
} | [
{
"pp": "R : Type u\nS₁ : Type v\nσ : Type u_1\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\nf : R →+* S₁\ng : σ → S₁\np : MvPolynomial σ R\n⊢ (eval g) ((map f) p) = eval₂ f g p",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Finsupp.instAddZeroClass",
"Nat.instMulZeroCla... | [
"case C\nR : Type u\nS₁ : Type v\nσ : Type u_1\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\nf : R →+* S₁\ng : σ → S₁\np : MvPolynomial σ R\n⊢ ∀ (a : R), (eval g) ((map f) (C a)) = eval₂ f g (C a)",
"case add\nR : Type u\nS₁ : Type v\nσ : Type u_1\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\nf : R →+* ... | apply MvPolynomial.induction_on p | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Algebra.MvPolynomial.Degrees | {
"line": 263,
"column": 2
} | {
"line": 263,
"column": 7
} | {
"line": 265,
"column": 0
} | [
{
"pp": "R : Type u\nσ : Type u_1\ninst✝ : CommSemiring R\np : MvPolynomial σ R\ni : σ\n⊢ degreeOf i p ≠ 0 → p ≠ 0",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"False",
"Nat.instMulZeroClass",
"AddMonoidAlgebra.semiring",
"congrArg",
"CommSemiring.toSemirin... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.MvPolynomial.Degrees | {
"line": 263,
"column": 2
} | {
"line": 263,
"column": 7
} | {
"line": 265,
"column": 0
} | [
{
"pp": "R : Type u\nσ : Type u_1\ninst✝ : CommSemiring R\np : MvPolynomial σ R\ni : σ\n⊢ degreeOf i p ≠ 0 → p ≠ 0",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"False",
"Nat.instMulZeroClass",
"AddMonoidAlgebra.semiring",
"congrArg",
"CommSemiring.toSemirin... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.MvPolynomial.Degrees | {
"line": 263,
"column": 2
} | {
"line": 263,
"column": 7
} | {
"line": 265,
"column": 0
} | [
{
"pp": "R : Type u\nσ : Type u_1\ninst✝ : CommSemiring R\np : MvPolynomial σ R\ni : σ\n⊢ degreeOf i p ≠ 0 → p ≠ 0",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"False",
"Nat.instMulZeroClass",
"AddMonoidAlgebra.semiring",
"congrArg",
"CommSemiring.toSemirin... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.MvPolynomial.Eval | {
"line": 657,
"column": 16
} | {
"line": 657,
"column": 21
} | {
"line": 659,
"column": 0
} | [
{
"pp": "R : Type u\nS₁ : Type v\nσ : Type u_1\ninst✝² : CommSemiring R\ninst✝¹ : CommSemiring S₁\ninst✝ : Algebra R S₁\nf : σ → S₁\ni : σ\n⊢ (aeval f).toRingHom (X i) = f i",
"ppTerm": "?m.163",
"assigned": true,
"usedConstants": [
"Nat.instMulZeroClass",
"AddMonoidAlgebra.semiring",
... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.MvPolynomial.Eval | {
"line": 657,
"column": 16
} | {
"line": 657,
"column": 21
} | {
"line": 659,
"column": 0
} | [
{
"pp": "R : Type u\nS₁ : Type v\nσ : Type u_1\ninst✝² : CommSemiring R\ninst✝¹ : CommSemiring S₁\ninst✝ : Algebra R S₁\nf : σ → S₁\ni : σ\n⊢ (aeval f).toRingHom (X i) = f i",
"ppTerm": "?m.163",
"assigned": true,
"usedConstants": [
"Nat.instMulZeroClass",
"AddMonoidAlgebra.semiring",
... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.MvPolynomial.Eval | {
"line": 657,
"column": 16
} | {
"line": 657,
"column": 21
} | {
"line": 659,
"column": 0
} | [
{
"pp": "R : Type u\nS₁ : Type v\nσ : Type u_1\ninst✝² : CommSemiring R\ninst✝¹ : CommSemiring S₁\ninst✝ : Algebra R S₁\nf : σ → S₁\ni : σ\n⊢ (aeval f).toRingHom (X i) = f i",
"ppTerm": "?m.163",
"assigned": true,
"usedConstants": [
"Nat.instMulZeroClass",
"AddMonoidAlgebra.semiring",
... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.MvPolynomial.Degrees | {
"line": 389,
"column": 34
} | {
"line": 389,
"column": 39
} | {
"line": 390,
"column": 2
} | [
{
"pp": "R : Type u\nσ : Type u_1\ninst✝ : CommSemiring R\np q : MvPolynomial σ R\ni : σ\nh : degreeOf i q < degreeOf i p\n⊢ p.support.Nonempty",
"ppTerm": "?m.60",
"assigned": true,
"usedConstants": [
"not_lt_zero._simp_1",
"Eq.mpr",
"Nat.instCanonicallyOrderedAdd",
"False",... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Data.Finsupp.Fin | {
"line": 96,
"column": 2
} | {
"line": 100,
"column": 42
} | {
"line": 102,
"column": 0
} | [
{
"pp": "n : ℕ\nM : Type u_1\ninst✝ : Zero M\ny : M\ns : Fin n →₀ M\n⊢ (cons y s).support ⊆ insert 0 (Finset.map (Fin.succEmb n) s.support)",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Finsupp.instFunLike",
"Eq.mpr",
"instNeZeroNatHAdd_1",
"and_true",
"Fin... | [] | intro i hi
suffices i = 0 ∨ ∃ a, ¬s a = 0 ∧ a.succ = i by simpa
apply (Fin.eq_zero_or_eq_succ i).imp id (Exists.imp _)
rintro i rfl
simpa [Finsupp.mem_support_iff] using hi | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Finsupp.Fin | {
"line": 96,
"column": 2
} | {
"line": 100,
"column": 42
} | {
"line": 102,
"column": 0
} | [
{
"pp": "n : ℕ\nM : Type u_1\ninst✝ : Zero M\ny : M\ns : Fin n →₀ M\n⊢ (cons y s).support ⊆ insert 0 (Finset.map (Fin.succEmb n) s.support)",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Finsupp.instFunLike",
"Eq.mpr",
"instNeZeroNatHAdd_1",
"and_true",
"Fin... | [] | intro i hi
suffices i = 0 ∨ ∃ a, ¬s a = 0 ∧ a.succ = i by simpa
apply (Fin.eq_zero_or_eq_succ i).imp id (Exists.imp _)
rintro i rfl
simpa [Finsupp.mem_support_iff] using hi | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Polynomial.Degree.TrailingDegree | {
"line": 174,
"column": 90
} | {
"line": 176,
"column": 5
} | {
"line": 178,
"column": 0
} | [
{
"pp": "R : Type u\na : R\nn : ℕ\ninst✝ : Semiring R\nha : a ≠ 0\n⊢ ((monomial n) a).natTrailingDegree = n",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Semiring.toModule",
"ENat.instNatCast",
"congrArg",
"LinearMap.instFunLike",
"Polynomia... | [] | by
rw [natTrailingDegree, trailingDegree_monomial ha]
rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Polynomial.Degree.TrailingDegree | {
"line": 303,
"column": 2
} | {
"line": 304,
"column": 86
} | {
"line": 305,
"column": 2
} | [
{
"pp": "R : Type u\ninst✝ : Semiring R\np q : R[X]\nh : p * q ≠ 0\nhp : p ≠ 0\nhq : q ≠ 0\n⊢ p.natTrailingDegree + q.natTrailingDegree ≤ (p * q).natTrailingDegree",
"ppTerm": "?m.47",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"ENat.instNatCast",
"congrArg",... | [
"R : Type u\ninst✝ : Semiring R\np q : R[X]\nh : p * q ≠ 0\nhp : p ≠ 0\nhq : q ≠ 0\n⊢ p.trailingDegree + q.trailingDegree ≤ (p * q).trailingDegree"
] | rw [← ENat.coe_le_coe, ENat.coe_add, ← trailingDegree_eq_natTrailingDegree hp,
← trailingDegree_eq_natTrailingDegree hq, ← trailingDegree_eq_natTrailingDegree h] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Polynomial.Degree.TrailingDegree | {
"line": 439,
"column": 4
} | {
"line": 444,
"column": 58
} | {
"line": 446,
"column": 0
} | [
{
"pp": "case refine_2\nR : Type u\ninst✝ : Semiring R\np : R[X]\nh₁ : p.Monic\nh : p.natDegree ≤ p.natTrailingDegree\n⊢ p = X ^ p.natDegree",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Preorder.toLT",
"in... | [] | ext n
rw [coeff_X_pow]
obtain hn | rfl | hn := lt_trichotomy n p.natDegree
· rw [if_neg hn.ne, coeff_eq_zero_of_lt_natTrailingDegree (hn.trans_le h)]
· simpa only [if_pos rfl] using! h₁.leadingCoeff
· rw [if_neg hn.ne', coeff_eq_zero_of_natDegree_lt hn] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Polynomial.Degree.TrailingDegree | {
"line": 439,
"column": 4
} | {
"line": 444,
"column": 58
} | {
"line": 446,
"column": 0
} | [
{
"pp": "case refine_2\nR : Type u\ninst✝ : Semiring R\np : R[X]\nh₁ : p.Monic\nh : p.natDegree ≤ p.natTrailingDegree\n⊢ p = X ^ p.natDegree",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Preorder.toLT",
"in... | [] | ext n
rw [coeff_X_pow]
obtain hn | rfl | hn := lt_trichotomy n p.natDegree
· rw [if_neg hn.ne, coeff_eq_zero_of_lt_natTrailingDegree (hn.trans_le h)]
· simpa only [if_pos rfl] using! h₁.leadingCoeff
· rw [if_neg hn.ne', coeff_eq_zero_of_natDegree_lt hn] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Polynomial.Reverse | {
"line": 99,
"column": 61
} | {
"line": 99,
"column": 80
} | {
"line": 100,
"column": 4
} | [
{
"pp": "R : Type u_1\ninst✝ : Semiring R\nN i : ℕ\nf : AddMonoidAlgebra R ℕ\n⊢ (embDomain (revAt N) f.coeff) i = (embDomain (revAt N) f.coeff) ((revAt N) ((revAt N) i))",
"ppTerm": "?m.40",
"assigned": true,
"usedConstants": [
"Finsupp.instFunLike",
"Eq.mpr",
"Polynomial.revAt",
... | [] | by rw [revAt_invol] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Polynomial.EraseLead | {
"line": 125,
"column": 91
} | {
"line": 128,
"column": 77
} | {
"line": 130,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : Semiring R\nf : R[X]\nh : f.eraseLead = 0\n⊢ #f.support ≤ 1",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Nat.instMulZeroClass",
"LinearOrderedCommMonoidWithZero.toIsBotZeroClass",
"congrArg",
"zero_le._simp_1",
"Polynomial.in... | [] | by
by_cases hpz : f = 0
case pos => simp [hpz]
case neg => exact le_of_eq (card_support_eq_one_of_eraseLead_eq_zero hpz h) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Polynomial.Degree.Lemmas | {
"line": 218,
"column": 8
} | {
"line": 218,
"column": 31
} | {
"line": 219,
"column": 8
} | [
{
"pp": "case pos\nR : Type u\nS : Type v\ninst✝ : Semiring R\nf : S → R[X]\ns : Finset S\nh : {i | i ∈ s ∧ f i ≠ 0}.Pairwise (Ne on natDegree ∘ f)\nx : S\nhx : x ∈ s\nhx' : f x ≠ 0\nhs : s.Nonempty\nb : S\nhb : b ∈ s\nhb' : f b = 0\n⊢ (f b).degree ≤ s.sup' hs (WithBot.some ∘ fun i ↦ (f i).natDegree)",
"ppT... | [
"case neg\nR : Type u\nS : Type v\ninst✝ : Semiring R\nf : S → R[X]\ns : Finset S\nh : {i | i ∈ s ∧ f i ≠ 0}.Pairwise (Ne on natDegree ∘ f)\nx : S\nhx : x ∈ s\nhx' : f x ≠ 0\nhs : s.Nonempty\nb : S\nhb : b ∈ s\nhb' : ¬f b = 0\n⊢ (f b).degree ≤ s.sup' hs (WithBot.some ∘ fun i ↦ (f i).natDegree)"
] | · simpa [hb'] using! hs | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Polynomial.EraseLead | {
"line": 207,
"column": 4
} | {
"line": 209,
"column": 8
} | {
"line": 210,
"column": 2
} | [
{
"pp": "case pos\nR : Type u_1\ninst✝ : Semiring R\nf : R[X]\nh : #f.support ≤ 1\n⊢ f.eraseLead.natDegree < f.natDegree ∨ f.eraseLead = 0",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Polynomial.C",
"HMul.hMul",
"congrArg",
"RingHom",
"id",... | [] | right
rw [← C_mul_X_pow_eq_self h]
simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Polynomial.EraseLead | {
"line": 207,
"column": 4
} | {
"line": 209,
"column": 8
} | {
"line": 210,
"column": 2
} | [
{
"pp": "case pos\nR : Type u_1\ninst✝ : Semiring R\nf : R[X]\nh : #f.support ≤ 1\n⊢ f.eraseLead.natDegree < f.natDegree ∨ f.eraseLead = 0",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Polynomial.C",
"HMul.hMul",
"congrArg",
"RingHom",
"id",... | [] | right
rw [← C_mul_X_pow_eq_self h]
simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Polynomial.Degree.Lemmas | {
"line": 329,
"column": 22
} | {
"line": 329,
"column": 48
} | {
"line": 329,
"column": 48
} | [
{
"pp": "R : Type u\nn : ℕ\ninst✝ : Ring R\np q : R[X]\ndg : (-q).natDegree < n\n⊢ (p + -q).coeff n = p.coeff n",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"Polynomial.coeff_add_eq_left_of_lt",
"Eq.mpr",
"Polynomial.instNeg",
"AddGroupWithOne.toAddGroup",
... | [
"R : Type u\nn : ℕ\ninst✝ : Ring R\np q : R[X]\ndg : (-q).natDegree < n\n⊢ p.coeff n = p.coeff n"
] | coeff_add_eq_left_of_lt dg | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.Reverse | {
"line": 170,
"column": 4
} | {
"line": 170,
"column": 77
} | {
"line": 171,
"column": 4
} | [
{
"pp": "case neg.Nf\nR : Type u_1\ninst✝ : Semiring R\ncg N O : ℕ\ng : R[X]\nCg : #g.support ≤ cg.succ\nOg : g.natDegree ≤ O\ncf : ℕ\nhcf : ∀ (f : R[X]), #f.support ≤ cf.succ → f.natDegree ≤ N → reflect (N + O) (f * g) = reflect N f * reflect O g\nf : R[X]\nCf : #f.support ≤ (cf + 1).succ\nNf : f.natDegree ≤ N... | [
"case neg.Cf\nR : Type u_1\ninst✝ : Semiring R\ncg N O : ℕ\ng : R[X]\nCg : #g.support ≤ cg.succ\nOg : g.natDegree ≤ O\ncf : ℕ\nhcf : ∀ (f : R[X]), #f.support ≤ cf.succ → f.natDegree ≤ N → reflect (N + O) (f * g) = reflect N f * reflect O g\nf : R[X]\nCf : #f.support ≤ (cf + 1).succ\nNf : f.natDegree ≤ N\nf0 : ¬f = ... | · exact le_trans (natDegree_C_mul_X_pow_le f.leadingCoeff f.natDegree) Nf | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Polynomial.EraseLead | {
"line": 308,
"column": 6
} | {
"line": 309,
"column": 31
} | {
"line": 310,
"column": 4
} | [
{
"pp": "R : Type u_2\ninst✝² : Ring R\ninst✝¹ : NoZeroDivisors R\ninst✝ : Nontrivial R\nx : R\nP : R[X]\nhx : x ≠ 0\nh : P.nextCoeff = 0\nhp : ¬P = 0\nhe : ¬P.eraseLead = 0\nh₁ : ((X - C x) * P).natDegree = P.natDegree + 1\ndP : ℕ\nhdP : P.natDegree = dP + 2\nh₂ : ((X - C x) * P).nextCoeff ≠ 0\nn : ℕ\nhn : n ≥... | [] | grw [eraseLead_natDegree_le, eraseLead_natDegree_le]
simpa [h₁, hdP] using! hn | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Polynomial.EraseLead | {
"line": 308,
"column": 6
} | {
"line": 309,
"column": 31
} | {
"line": 310,
"column": 4
} | [
{
"pp": "R : Type u_2\ninst✝² : Ring R\ninst✝¹ : NoZeroDivisors R\ninst✝ : Nontrivial R\nx : R\nP : R[X]\nhx : x ≠ 0\nh : P.nextCoeff = 0\nhp : ¬P = 0\nhe : ¬P.eraseLead = 0\nh₁ : ((X - C x) * P).natDegree = P.natDegree + 1\ndP : ℕ\nhdP : P.natDegree = dP + 2\nh₂ : ((X - C x) * P).nextCoeff ≠ 0\nn : ℕ\nhn : n ≥... | [] | grw [eraseLead_natDegree_le, eraseLead_natDegree_le]
simpa [h₁, hdP] using! hn | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Polynomial.Reverse | {
"line": 223,
"column": 6
} | {
"line": 223,
"column": 20
} | {
"line": 223,
"column": 21
} | [
{
"pp": "R : Type u_1\ninst✝ : Semiring R\nf : R[X]\n⊢ f.reverse.coeff 0 = f.leadingCoeff",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Polynomial.revAt",
"congrArg",
"id",
"Polynomial.leadingCoeff",
"instOfNatNat",
"Polynomial.coeff",... | [
"R : Type u_1\ninst✝ : Semiring R\nf : R[X]\n⊢ f.coeff ((revAt f.natDegree) 0) = f.leadingCoeff"
] | coeff_reverse, | Lean.Elab.Tactic.evalRewriteSeq | null |
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