module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.LinearAlgebra.Dimension.Constructions | {
"line": 138,
"column": 2
} | {
"line": 138,
"column": 33
} | {
"line": 139,
"column": 2
} | [
{
"pp": "R : Type u\nM : Type v\nM' : Type v'\ninst✝⁵ : Semiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : AddCommMonoid M'\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Nontrivial R\n⊢ lift.{v', v} (Module.rank R M) + lift.{v, v'} (Module.rank R M') ≤ Module.rank R (M × M')",
"ppTerm": "?m.22",
"assi... | [
"R : Type u\nM : Type v\nM' : Type v'\ninst✝⁵ : Semiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : AddCommMonoid M'\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Nontrivial R\n⊢ Module.rank R (ULift.{v', v} M) + Module.rank R (ULift.{v, v'} M') ≤ Module.rank R (M × M')"
] | rw [← rank_ulift, ← rank_ulift] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.Dimension.Constructions | {
"line": 152,
"column": 2
} | {
"line": 153,
"column": 9
} | {
"line": 153,
"column": 10
} | [
{
"pp": "R : Type u\nM : Type v\nM' : Type v'\ninst✝⁷ : Semiring R\ninst✝⁶ : AddCommMonoid M\ninst✝⁵ : AddCommMonoid M'\ninst✝⁴ : Module R M\ninst✝³ : Module R M'\ninst✝² : StrongRankCondition R\ninst✝¹ : Free R M\ninst✝ : Free R M'\n⊢ Module.rank R (M × M') = lift.{v', v} (Module.rank R M) + lift.{v, v'} (Modu... | [
"R : Type u\nM : Type v\nM' : Type v'\ninst✝⁷ : Semiring R\ninst✝⁶ : AddCommMonoid M\ninst✝⁵ : AddCommMonoid M'\ninst✝⁴ : Module R M\ninst✝³ : Module R M'\ninst✝² : StrongRankCondition R\ninst✝¹ : Free R M\ninst✝ : Free R M'\n⊢ Module.rank R (M × M') = lift.{v', v} #(ChooseBasisIndex R M) + lift.{v, v'} #(ChooseBas... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.TensorProduct.Maps | {
"line": 659,
"column": 59
} | {
"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... | [] | by
have : (includeLeft : A →ₐ[S] A ⊗[R] B).comp (TensorProduct.rid R S A).toAlgHom =
map (.id S A) (Algebra.ofId R B) := by ext; simp
rw [← Function.Bijective.of_comp_iff _ (TensorProduct.rid R S A).bijective]
convert_to Function.Bijective (map (.id R A) (Algebra.ofId R B))
· exact DFunLike.coe_fn_eq.mpr ... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.Dimension.Finite | {
"line": 78,
"column": 4
} | {
"line": 78,
"column": 15
} | {
"line": 78,
"column": 16
} | [
{
"pp": "case mpr\nR : Type u_1\nM : Type u_2\ninst✝² : Ring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\na✝ : Nontrivial R\nh : ∀ (x : M), ∃ a, a ≠ 0 ∧ a • x = 0\ns : Set M\nhs : LinearIndepOn R id s\ni : ↑s\na : R\nha : a ≠ 0\nha' : a • ↑i = 0\n⊢ a = 0",
"ppTerm": "?mpr",
"assigned": false,
"us... | [
"case mpr\nR : Type u_1\nM : Type u_2\ninst✝² : Ring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\na✝ : Nontrivial R\nh : ∀ (x : M), ∃ a, a ≠ 0 ∧ a • x = 0\ns : Set M\nhs : LinearIndepOn R id s\ni : ↑s\na : R\nha : a ≠ 0\nha' : a • ↑i = 0\n⊢ a = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Dimension.Finite | {
"line": 159,
"column": 2
} | {
"line": 159,
"column": 69
} | {
"line": 159,
"column": 70
} | [
{
"pp": "R : Type u\nM : Type v\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\ninst✝¹ : StrongRankCondition R\ninst✝ : Module.Finite R M\nι : Type w\nb : ι → M\nh : LinearIndependent R b\n⊢ lift.{max v w, w} #ι ≤ lift.{max v w, w} ↑(finrank R M)",
"ppTerm": "?m.20",
"assigned": tru... | [
"R : Type u\nM : Type v\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\ninst✝¹ : StrongRankCondition R\ninst✝ : Module.Finite R M\nι : Type w\nb : ι → M\nh : LinearIndependent R b\n⊢ #ι ≤ ↑(finrank R M)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Dimension.Finite | {
"line": 164,
"column": 2
} | {
"line": 164,
"column": 13
} | {
"line": 164,
"column": 14
} | [
{
"pp": "R : Type u\nM : Type v\ninst✝⁵ : Semiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\ninst✝² : StrongRankCondition R\ninst✝¹ : Module.Finite R M\nι : Type u_1\ninst✝ : Fintype ι\nb : ι → M\nh : LinearIndependent R b\n⊢ Fintype.card ι ≤ finrank R M",
"ppTerm": "?m.18",
"assigned": false,
... | [
"R : Type u\nM : Type v\ninst✝⁵ : Semiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\ninst✝² : StrongRankCondition R\ninst✝¹ : Module.Finite R M\nι : Type u_1\ninst✝ : Fintype ι\nb : ι → M\nh : LinearIndependent R b\n⊢ Fintype.card ι ≤ finrank R M"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Dimension.Finite | {
"line": 198,
"column": 27
} | {
"line": 198,
"column": 38
} | {
"line": 198,
"column": 39
} | [
{
"pp": "R : Type u\nM : Type v\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nn : ℕ\nhn : ↑n ≤ Module.rank R M\nh : ↑n = Module.rank R M\ns : Set M\nhs : LinearIndepOn R id s\nhs' : #↑↑⟨s, hs⟩ = ↑n\nthis : Finite ↑s\nval✝ : Fintype ↑s\n⊢ s.toFinset.card = n",
"ppTerm": "?m.117",
"a... | [
"R : Type u\nM : Type v\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nn : ℕ\nhn : ↑n ≤ Module.rank R M\nh : ↑n = Module.rank R M\ns : Set M\nhs : LinearIndepOn R id s\nhs' : #↑↑⟨s, hs⟩ = ↑n\nthis : Finite ↑s\nval✝ : Fintype ↑s\n⊢ s.ncard = n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Dimension.Constructions | {
"line": 282,
"column": 49
} | {
"line": 286,
"column": 54
} | {
"line": 288,
"column": 0
} | [
{
"pp": "R : Type u\nη : Type u₁'\nφ : η → Type u_1\ninst✝⁵ : Semiring R\ninst✝⁴ : StrongRankCondition R\ninst✝³ : (i : η) → AddCommMonoid (φ i)\ninst✝² : (i : η) → Module R (φ i)\ninst✝¹ : ∀ (i : η), Free R (φ i)\ninst✝ : Finite η\n⊢ Module.rank R ((i : η) → φ i) = sum fun i ↦ Module.rank R (φ i)",
"ppTerm... | [] | by
cases nonempty_fintype η
let B i := chooseBasis R (φ i)
let b : Basis _ R (∀ i, φ i) := Pi.basis fun i => B i
simp [← b.mk_eq_rank'', fun i => (B i).mk_eq_rank''] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.Dimension.Finite | {
"line": 202,
"column": 26
} | {
"line": 202,
"column": 37
} | {
"line": 202,
"column": 38
} | [
{
"pp": "R : Type u\nM : Type v\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nn : ℕ\nhn : ↑n ≤ Module.rank R M\nh : ↑n < Module.rank R M\ns : Set M\nhs : #↑s = ↑n\nhs' : LinearIndepOn R id s\nthis : Finite ↑s\nval✝ : Fintype ↑s\n⊢ s.toFinset.card = n",
"ppTerm": "?m.187",
"assigned... | [
"R : Type u\nM : Type v\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nn : ℕ\nhn : ↑n ≤ Module.rank R M\nh : ↑n < Module.rank R M\ns : Set M\nhs : #↑s = ↑n\nhs' : LinearIndepOn R id s\nthis : Finite ↑s\nval✝ : Fintype ↑s\n⊢ s.ncard = n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Dimension.Finite | {
"line": 215,
"column": 15
} | {
"line": 215,
"column": 26
} | {
"line": 215,
"column": 27
} | [
{
"pp": "R : Type u\nM : Type v\ninst✝³ : Semiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ninst✝ : Nontrivial R\nn : ℕ\nH : ∃ f, LinearIndependent R f\n⊢ ↑n ≤ Module.rank R M",
"ppTerm": "?m.23",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u\nM : Type v\ninst✝³ : Semiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ninst✝ : Nontrivial R\nn : ℕ\nH : ∃ f, LinearIndependent R f\n⊢ ↑n ≤ Module.rank R M"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Dimension.Finite | {
"line": 220,
"column": 25
} | {
"line": 220,
"column": 41
} | {
"line": 220,
"column": 42
} | [
{
"pp": "R : Type u\nM : Type v\ninst✝³ : Semiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ninst✝ : Nontrivial R\nn : ℕ\nx✝ : ∃ s, s.card = n ∧ LinearIndependent R Subtype.val\ns : Finset M\nh₁ : s.card = n\nh₂ : LinearIndependent R Subtype.val\n⊢ ↑n ≤ Module.rank R M",
"ppTerm": "?m.33",
"assi... | [
"R : Type u\nM : Type v\ninst✝³ : Semiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ninst✝ : Nontrivial R\nn : ℕ\nx✝ : ∃ s, s.card = n ∧ LinearIndependent R Subtype.val\ns : Finset M\nh₁ : s.card = n\nh₂ : LinearIndependent R Subtype.val\n⊢ ↑n ≤ Module.rank R M"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Dimension.Constructions | {
"line": 433,
"column": 2
} | {
"line": 433,
"column": 13
} | {
"line": 433,
"column": 14
} | [
{
"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": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Dimension.Constructions | {
"line": 446,
"column": 28
} | {
"line": 446,
"column": 39
} | {
"line": 446,
"column": 40
} | [
{
"pp": "R : Type u\nM : Type v\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\ninst✝¹ : StrongRankCondition R\ns : Set M\ninst✝ : Fintype ↑s\n⊢ Module.rank R ↥(span R s) ≤ ↑s.toFinset.card",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Submodul... | [
"R : Type u\nM : Type v\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\ninst✝¹ : StrongRankCondition R\ns : Set M\ninst✝ : Fintype ↑s\n⊢ Module.rank R ↥(span R s) ≤ ↑s.ncard"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Dimension.Constructions | {
"line": 472,
"column": 5
} | {
"line": 474,
"column": 60
} | {
"line": 474,
"column": 60
} | [
{
"pp": "R : Type u\nM : Type v\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\ninst✝¹ : StrongRankCondition R\ns : Set M\ninst✝ : Fintype ↑s\nhs : LinearIndepOn R id s\n⊢ Module.rank R ↥(span R s) = ↑s.toFinset.card",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
... | [] | by
have : Module.rank R (span R s) = #s := rank_span_set hs
rwa [Cardinal.mk_fintype, ← Set.toFinset_card] at this | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.Dimension.Constructions | {
"line": 598,
"column": 2
} | {
"line": 599,
"column": 9
} | {
"line": 599,
"column": 10
} | [
{
"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)\nj : n\n⊢ Submodule.Quotient.mk ((bW.sumQuot bQ) (Sum.inr j)) = bQ j",
"ppTerm": "?m.35",
"assigned": true,
... | [
"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)\nj : n\n⊢ W.mkQ (surjInv ⋯ (bQ j)) = bQ j"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Dimension.Finite | {
"line": 286,
"column": 54
} | {
"line": 286,
"column": 65
} | {
"line": 286,
"column": 66
} | [
{
"pp": "ι : Type w\nR : Type u\nM : Type v\ninst✝⁷ : Ring R\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : Module R M\ninst✝⁴ : IsDomain R\ninst✝³ : IsTorsionFree R M\ninst✝² : Module.Finite R M\ninst✝¹ : StrongRankCondition R\np : ι → Submodule R M\nhp : iSupIndep p\ninst✝ : Fintype { i // p i ≠ ⊥ }\n⊢ Fintype.card { i /... | [
"ι : Type w\nR : Type u\nM : Type v\ninst✝⁷ : Ring R\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : Module R M\ninst✝⁴ : IsDomain R\ninst✝³ : IsTorsionFree R M\ninst✝² : Module.Finite R M\ninst✝¹ : StrongRankCondition R\np : ι → Submodule R M\nhp : iSupIndep p\ninst✝ : Fintype { i // p i ≠ ⊥ }\n⊢ Fintype.card { i // ¬p i = ⊥ }... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Basis.VectorSpace | {
"line": 56,
"column": 37
} | {
"line": 56,
"column": 48
} | {
"line": 56,
"column": 49
} | [
{
"pp": "ι : Type u_1\nι' : Type u_2\nK : 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'\nv : ι → V\ns t : Set V\nx y z : V\nhs : LinearIndepOn K id s\n⊢ ↑⊤ ⊆ ↑(span K (range ((hs.extend ⋯).restrict id))... | [
"ι : Type u_1\nι' : Type u_2\nK : 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'\nv : ι → V\ns t : Set V\nx y z : V\nhs : LinearIndepOn K id s\n⊢ span K (hs.extend ⋯) = ⊤"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Basis.VectorSpace | {
"line": 97,
"column": 46
} | {
"line": 97,
"column": 57
} | {
"line": 97,
"column": 58
} | [
{
"pp": "ι : Type u_1\nι' : Type u_2\nK : 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'\nv : ι → V\ns t : Set V\nx y z : V\nhs : LinearIndepOn K id s\nhst : s ⊆ t\nht : ⊤ ≤ span K t\n⊢ t ⊆ ↑(span K (ran... | [
"ι : Type u_1\nι' : Type u_2\nK : 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'\nv : ι → V\ns t : Set V\nx y z : V\nhs : LinearIndepOn K id s\nhst : s ⊆ t\nht : ⊤ ≤ span K t\n⊢ t ⊆ ↑(span K (hs.extend hst))... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Dimension.Finite | {
"line": 505,
"column": 2
} | {
"line": 507,
"column": 61
} | {
"line": 509,
"column": 0
} | [
{
"pp": "R : Type u\nM : Type v\ninst✝⁵ : Ring R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : IsDomain R\ninst✝¹ : IsTorsionFree R M\ninst✝ : StrongRankCondition R\nv : M\nn : v ≠ 0\nh : ∀ (w : M), ∃ c, c • v = w\n⊢ Module.rank R M = 1",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants... | [] | haveI := nontrivial_of_invariantBasisNumber R
obtain ⟨b⟩ := (Basis.basis_singleton_iff.{_, _, u} PUnit).mpr ⟨v, n, h⟩
rw [rank_eq_card_basis b, Fintype.card_punit, Nat.cast_one] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Dimension.Finite | {
"line": 505,
"column": 2
} | {
"line": 507,
"column": 61
} | {
"line": 509,
"column": 0
} | [
{
"pp": "R : Type u\nM : Type v\ninst✝⁵ : Ring R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : IsDomain R\ninst✝¹ : IsTorsionFree R M\ninst✝ : StrongRankCondition R\nv : M\nn : v ≠ 0\nh : ∀ (w : M), ∃ c, c • v = w\n⊢ Module.rank R M = 1",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants... | [] | haveI := nontrivial_of_invariantBasisNumber R
obtain ⟨b⟩ := (Basis.basis_singleton_iff.{_, _, u} PUnit).mpr ⟨v, n, h⟩
rw [rank_eq_card_basis b, Fintype.card_punit, Nat.cast_one] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Dimension.Finite | {
"line": 521,
"column": 2
} | {
"line": 521,
"column": 13
} | {
"line": 521,
"column": 14
} | [
{
"pp": "R : Type u\nM : Type v\ninst✝³ : Semiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ninst✝ : StrongRankCondition R\nv : M\nh : ∀ (w : M), ∃ c, c • v = w\n⊢ Module.rank R M ≤ 1",
"ppTerm": "?m.17",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u\nM : Type v\ninst✝³ : Semiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ninst✝ : StrongRankCondition R\nv : M\nh : ∀ (w : M), ∃ c, c • v = w\n⊢ Module.rank R M ≤ 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Basis.VectorSpace | {
"line": 209,
"column": 4
} | {
"line": 209,
"column": 50
} | {
"line": 210,
"column": 4
} | [
{
"pp": "case right.inl\nK : Type u_3\nV : Type u_4\ninst✝² : DivisionRing K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nv : V\nhv : v ≠ 0\nT : Submodule K V\nhT : T < K ∙ v\nhs : 0 • v ∈ T\nhz : 0 • v ≠ 0\n⊢ v ∈ T",
"ppTerm": "?right.inl",
"assigned": true,
"usedConstants": [
"Submodule",
... | [
"case right.inr\nK : Type u_3\nV : Type u_4\ninst✝² : DivisionRing K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nv : V\nhv : v ≠ 0\nT : Submodule K V\nhT : T < K ∙ v\na : K\nhs : a • v ∈ T\nhz : a • v ≠ 0\nh : a ≠ 0\n⊢ v ∈ T"
] | · simp only [zero_smul, ne_eq, not_true] at hz | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Star.NonUnitalSubalgebra | {
"line": 573,
"column": 39
} | {
"line": 574,
"column": 13
} | {
"line": 574,
"column": 14
} | [
{
"pp": "F : Type v'\nR' : Type u'\nR : Type u\nA : Type v\nB : Type w\nC : Type w'\ninst✝⁵ : CommSemiring R\ninst✝⁴ : StarRing R\ninst✝³ : NonUnitalSemiring A\ninst✝² : StarRing A\ninst✝¹ : Module R A\ninst✝ : StarModule R A\nS : NonUnitalSubalgebra R A\nx y : A\nhx : x ∈ star S.carrier\nhy : y ∈ star S.carrie... | [
"F : Type v'\nR' : Type u'\nR : Type u\nA : Type v\nB : Type w\nC : Type w'\ninst✝⁵ : CommSemiring R\ninst✝⁴ : StarRing R\ninst✝³ : NonUnitalSemiring A\ninst✝² : StarRing A\ninst✝¹ : Module R A\ninst✝ : StarModule R A\nS : NonUnitalSubalgebra R A\nx y : A\nhx : x ∈ star S.carrier\nhy : y ∈ star S.carrier\n⊢ star (x... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Star.NonUnitalSubalgebra | {
"line": 571,
"column": 39
} | {
"line": 572,
"column": 13
} | {
"line": 572,
"column": 14
} | [
{
"pp": "F : Type v'\nR' : Type u'\nR : Type u\nA : Type v\nB : Type w\nC : Type w'\ninst✝⁵ : CommSemiring R\ninst✝⁴ : StarRing R\ninst✝³ : NonUnitalSemiring A\ninst✝² : StarRing A\ninst✝¹ : Module R A\ninst✝ : StarModule R A\nS : NonUnitalSubalgebra R A\nx y : A\nhx : x ∈ star S.carrier\nhy : y ∈ star S.carrie... | [
"F : Type v'\nR' : Type u'\nR : Type u\nA : Type v\nB : Type w\nC : Type w'\ninst✝⁵ : CommSemiring R\ninst✝⁴ : StarRing R\ninst✝³ : NonUnitalSemiring A\ninst✝² : StarRing A\ninst✝¹ : Module R A\ninst✝ : StarModule R A\nS : NonUnitalSubalgebra R A\nx y : A\nhx : x ∈ star S.carrier\nhy : y ∈ star S.carrier\n⊢ star (x... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Star.NonUnitalSubalgebra | {
"line": 576,
"column": 36
} | {
"line": 577,
"column": 13
} | {
"line": 577,
"column": 14
} | [
{
"pp": "F : Type v'\nR' : Type u'\nR : Type u\nA : Type v\nB : Type w\nC : Type w'\ninst✝⁵ : CommSemiring R\ninst✝⁴ : StarRing R\ninst✝³ : NonUnitalSemiring A\ninst✝² : StarRing A\ninst✝¹ : Module R A\ninst✝ : StarModule R A\nS : NonUnitalSubalgebra R A\nr : R\nx : A\nhx : x ∈ star S.carrier\n⊢ r • x ∈ star S.... | [
"F : Type v'\nR' : Type u'\nR : Type u\nA : Type v\nB : Type w\nC : Type w'\ninst✝⁵ : CommSemiring R\ninst✝⁴ : StarRing R\ninst✝³ : NonUnitalSemiring A\ninst✝² : StarRing A\ninst✝¹ : Module R A\ninst✝ : StarModule R A\nS : NonUnitalSubalgebra R A\nr : R\nx : A\nhx : x ∈ star S.carrier\n⊢ star (r • x) ∈ S"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Star.NonUnitalSubalgebra | {
"line": 604,
"column": 18
} | {
"line": 604,
"column": 46
} | {
"line": 604,
"column": 47
} | [
{
"pp": "R : Type u\nA : Type v\ninst✝⁷ : CommSemiring R\ninst✝⁶ : StarRing R\ninst✝⁵ : NonUnitalSemiring A\ninst✝⁴ : StarRing A\ninst✝³ : Module R A\ninst✝² : StarModule R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\ns : Set A\nthis : ∀ (t : Set A), NonUnitalAlgebra.adjoin R (star t) ≤ star (N... | [
"R : Type u\nA : Type v\ninst✝⁷ : CommSemiring R\ninst✝⁶ : StarRing R\ninst✝⁵ : NonUnitalSemiring A\ninst✝⁴ : StarRing A\ninst✝³ : Module R A\ninst✝² : StarModule R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\ns : Set A\nthis : ∀ (t : Set A), NonUnitalAlgebra.adjoin R (star t) ≤ star (NonUnitalAlge... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Star.NonUnitalSubalgebra | {
"line": 614,
"column": 4
} | {
"line": 615,
"column": 27
} | {
"line": 615,
"column": 28
} | [
{
"pp": "F : Type v'\nR' : Type u'\nR : Type u\nA : Type v\nB : Type w\nC : Type w'\ninst✝⁷ : CommSemiring R\ninst✝⁶ : StarRing R\ninst✝⁵ : NonUnitalSemiring A\ninst✝⁴ : StarRing A\ninst✝³ : Module R A\ninst✝² : StarModule R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\nS : NonUnitalSubalgebra R... | [
"F : Type v'\nR' : Type u'\nR : Type u\nA : Type v\nB : Type w\nC : Type w'\ninst✝⁷ : CommSemiring R\ninst✝⁶ : StarRing R\ninst✝⁵ : NonUnitalSemiring A\ninst✝⁴ : StarRing A\ninst✝³ : Module R A\ninst✝² : StarModule R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\nS : NonUnitalSubalgebra R A\na : A\nh... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Star.NonUnitalSubalgebra | {
"line": 710,
"column": 4
} | {
"line": 710,
"column": 15
} | {
"line": 710,
"column": 16
} | [
{
"pp": "case inr\nR : Type u\nA : Type v\ninst✝⁷ : CommSemiring R\ninst✝⁶ : StarRing R\ninst✝⁵ : NonUnitalSemiring A\ninst✝⁴ : StarRing A\ninst✝³ : Module R A\ninst✝² : IsScalarTower R A A\ninst✝¹ : SMulCommClass R A A\ninst✝ : StarModule R A\ns : Set A\np : (x : A) → x ∈ adjoin R s → Prop\nmem : ∀ (x : A) (hx... | [
"case inr\nR : Type u\nA : Type v\ninst✝⁷ : CommSemiring R\ninst✝⁶ : StarRing R\ninst✝⁵ : NonUnitalSemiring A\ninst✝⁴ : StarRing A\ninst✝³ : Module R A\ninst✝² : IsScalarTower R A A\ninst✝¹ : SMulCommClass R A A\ninst✝ : StarModule R A\ns : Set A\np : (x : A) → x ∈ adjoin R s → Prop\nmem : ∀ (x : A) (hx : x ∈ s), p... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Star.NonUnitalSubalgebra | {
"line": 710,
"column": 61
} | {
"line": 710,
"column": 72
} | {
"line": 710,
"column": 73
} | [
{
"pp": "R : Type u\nA : Type v\ninst✝⁷ : CommSemiring R\ninst✝⁶ : StarRing R\ninst✝⁵ : NonUnitalSemiring A\ninst✝⁴ : StarRing A\ninst✝³ : Module R A\ninst✝² : IsScalarTower R A A\ninst✝¹ : SMulCommClass R A A\ninst✝ : StarModule R A\ns : Set A\np : (x : A) → x ∈ adjoin R s → Prop\nmem : ∀ (x : A) (hx : x ∈ s),... | [
"R : Type u\nA : Type v\ninst✝⁷ : CommSemiring R\ninst✝⁶ : StarRing R\ninst✝⁵ : NonUnitalSemiring A\ninst✝⁴ : StarRing A\ninst✝³ : Module R A\ninst✝² : IsScalarTower R A A\ninst✝¹ : SMulCommClass R A A\ninst✝ : StarModule R A\ns : Set A\np : (x : A) → x ∈ adjoin R s → Prop\nmem : ∀ (x : A) (hx : x ∈ s), p x ⋯\nadd ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Basis.VectorSpace | {
"line": 271,
"column": 2
} | {
"line": 271,
"column": 34
} | {
"line": 271,
"column": 35
} | [
{
"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'\nf : V →ₗ[K] V'\nh_inj : f.ker = ⊥\n⊢ f.leftInverse ∘ₗ f = id",
"ppTerm": "?m.64",
"assigned": true,
"usedConstants": [
"dite... | [
"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'\nf : V →ₗ[K] V'\nh_inj : f.ker = ⊥\n⊢ ⋯.choose ∘ₗ f = id"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Basis.VectorSpace | {
"line": 308,
"column": 2
} | {
"line": 308,
"column": 13
} | {
"line": 308,
"column": 14
} | [
{
"pp": "K : Type u_3\nV : Type u_4\ninst✝² : DivisionRing K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\np : Submodule K V\nv : V\nhv : v ∉ p\nf : V →ₗ[K] K\nhpf : f ∘ₗ p.subtype = 0\nhfv : f v = 1\nx : V\nhx : x ∈ p\n⊢ x ∈ f.ker",
"ppTerm": "?m.86",
"assigned": true,
"usedConstants": [
"Eq.... | [
"K : Type u_3\nV : Type u_4\ninst✝² : DivisionRing K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\np : Submodule K V\nv : V\nhv : v ∉ p\nf : V →ₗ[K] K\nhpf : f ∘ₗ p.subtype = 0\nhfv : f v = 1\nx : V\nhx : x ∈ p\n⊢ f x = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Star.NonUnitalSubalgebra | {
"line": 839,
"column": 2
} | {
"line": 839,
"column": 13
} | {
"line": 839,
"column": 14
} | [
{
"pp": "F : Type v'\nR : Type u\nA : Type v\nB : Type w\ninst✝¹⁷ : CommSemiring R\ninst✝¹⁶ : StarRing R\ninst✝¹⁵ : NonUnitalSemiring A\ninst✝¹⁴ : StarRing A\ninst✝¹³ : Module R A\ninst✝¹² : NonUnitalSemiring B\ninst✝¹¹ : StarRing B\ninst✝¹⁰ : Module R B\ninst✝⁹ : FunLike F A B\ninst✝⁸ : NonUnitalAlgHomClass F ... | [
"F : Type v'\nR : Type u\nA : Type v\nB : Type w\ninst✝¹⁷ : CommSemiring R\ninst✝¹⁶ : StarRing R\ninst✝¹⁵ : NonUnitalSemiring A\ninst✝¹⁴ : StarRing A\ninst✝¹³ : Module R A\ninst✝¹² : NonUnitalSemiring B\ninst✝¹¹ : StarRing B\ninst✝¹⁰ : Module R B\ninst✝⁹ : FunLike F A B\ninst✝⁸ : NonUnitalAlgHomClass F R A B\ninst✝... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Star.NonUnitalSubalgebra | {
"line": 1044,
"column": 2
} | {
"line": 1045,
"column": 35
} | {
"line": 1045,
"column": 36
} | [
{
"pp": "R : Type u\nA : Type v\ninst✝⁸ : CommSemiring R\ninst✝⁷ : NonUnitalSemiring A\ninst✝⁶ : StarRing A\ninst✝⁵ : Module R A\nι : Type u_1\ninst✝⁴ : StarRing R\ninst✝³ : IsScalarTower R A A\ninst✝² : SMulCommClass R A A\ninst✝¹ : StarModule R A\ninst✝ : Nonempty ι\nS : ι → NonUnitalStarSubalgebra R A\nhS : ... | [
"R : Type u\nA : Type v\ninst✝⁸ : CommSemiring R\ninst✝⁷ : NonUnitalSemiring A\ninst✝⁶ : StarRing A\ninst✝⁵ : Module R A\nι : Type u_1\ninst✝⁴ : StarRing R\ninst✝³ : IsScalarTower R A A\ninst✝² : SMulCommClass R A A\ninst✝¹ : StarModule R A\ninst✝ : Nonempty ι\nS : ι → NonUnitalStarSubalgebra R A\nhS : ∀ (i : ι), I... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Basis.VectorSpace | {
"line": 369,
"column": 27
} | {
"line": 369,
"column": 38
} | {
"line": 369,
"column": 39
} | [
{
"pp": "K : Type u_6\nV : Type u_7\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : V →ₗ[K] K\nv : V\nhfv : f v ≠ 0\nb₁ : Basis (↑(Basis.ofVectorSpaceIndex K ↥f.ker)) K ↥f.ker := Basis.ofVectorSpace K ↥f.ker\ns : Set V := ⇑f.ker.subtype '' Set.range ⇑b₁\nhs : span K s = f.ker\nn : Set V := i... | [
"K : Type u_6\nV : Type u_7\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : V →ₗ[K] K\nv : V\nhfv : f v ≠ 0\nb₁ : Basis (↑(Basis.ofVectorSpaceIndex K ↥f.ker)) K ↥f.ker := Basis.ofVectorSpace K ↥f.ker\ns : Set V := ⇑f.ker.subtype '' Set.range ⇑b₁\nhs : span K s = f.ker\nn : Set V := insert v s\nH... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Basis.VectorSpace | {
"line": 394,
"column": 45
} | {
"line": 394,
"column": 56
} | {
"line": 394,
"column": 57
} | [
{
"pp": "K : Type u_6\nV : Type u_7\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : V →ₗ[K] K\nhf : f ≠ 0\nv : ↥f.ker\nhv : ↑v ≠ 0\n⊢ LinearIndepOn K _root_.id {v}",
"ppTerm": "?m.86",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Submodule",
"Semiring.toModul... | [
"K : Type u_6\nV : Type u_7\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : V →ₗ[K] K\nhf : f ≠ 0\nv : ↥f.ker\nhv : ↑v ≠ 0\n⊢ ¬v = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.FiniteDimensional.Basic | {
"line": 74,
"column": 4
} | {
"line": 74,
"column": 20
} | {
"line": 74,
"column": 21
} | [
{
"pp": "K : Type u\nV : Type v\ninst✝³ : DivisionRing K\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\ninst✝ : FiniteDimensional K V\nS : Submodule K V\nh : finrank K ↥S = finrank K V\nbS : Basis (↑(Basis.ofVectorSpaceIndex K ↥S)) K ↥S := Basis.ofVectorSpace K ↥S\nbS_eq : bS = Basis.ofVectorSpace K ↥S\n⊢ Linea... | [
"K : Type u\nV : Type v\ninst✝³ : DivisionRing K\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\ninst✝ : FiniteDimensional K V\nS : Submodule K V\nh : finrank K ↥S = finrank K V\nbS : Basis (↑(Basis.ofVectorSpaceIndex K ↥S)) K ↥S := Basis.ofVectorSpace K ↥S\nbS_eq : bS = Basis.ofVectorSpace K ↥S\n⊢ LinearIndepOn K i... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Star.NonUnitalSubalgebra | {
"line": 1239,
"column": 22
} | {
"line": 1239,
"column": 33
} | {
"line": 1239,
"column": 34
} | [
{
"pp": "R : Type u\nA : Type v\ninst✝⁷ : CommSemiring R\ninst✝⁶ : StarRing R\ninst✝⁵ : NonUnitalSemiring A\ninst✝⁴ : StarRing A\ninst✝³ : Module R A\ninst✝² : IsScalarTower R A A\ninst✝¹ : SMulCommClass R A A\ninst✝ : StarModule R A\na b✝ : A\ns : Set A\nhb✝ : b✝ ∈ adjoin R s\nh : ∀ b ∈ s, Commute a b\nh_star ... | [
"R : Type u\nA : Type v\ninst✝⁷ : CommSemiring R\ninst✝⁶ : StarRing R\ninst✝⁵ : NonUnitalSemiring A\ninst✝⁴ : StarRing A\ninst✝³ : Module R A\ninst✝² : IsScalarTower R A A\ninst✝¹ : SMulCommClass R A A\ninst✝ : StarModule R A\na b✝ : A\ns : Set A\nhb✝ : b✝ ∈ adjoin R s\nh : ∀ b ∈ s, Commute a b\nh_star : ∀ b ∈ s, C... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Basis.VectorSpace | {
"line": 420,
"column": 27
} | {
"line": 420,
"column": 38
} | {
"line": 420,
"column": 39
} | [
{
"pp": "K : Type u_6\nV : Type u_7\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : V →ₗ[K] K\nhf : f ≠ 0\nv : ↥f.ker\nhv : ↑v ≠ 0\nthis : LinearIndepOn K _root_.id {v}\nb₁ : Basis (↑(this.extend ⋯)) K ↥f.ker := Basis.extend this\nw : V\nhw : f w = 1\ns : Set V := ⇑f.ker.subtype '' Set.range... | [
"K : Type u_6\nV : Type u_7\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : V →ₗ[K] K\nhf : f ≠ 0\nv : ↥f.ker\nhv : ↑v ≠ 0\nthis : LinearIndepOn K _root_.id {v}\nb₁ : Basis (↑(this.extend ⋯)) K ↥f.ker := Basis.extend this\nw : V\nhw : f w = 1\ns : Set V := ⇑f.ker.subtype '' Set.range ⇑b₁\nhs : s... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.FiniteDimensional.Basic | {
"line": 88,
"column": 2
} | {
"line": 88,
"column": 67
} | {
"line": 90,
"column": 0
} | [
{
"pp": "K : Type u\nV : Type v\ninst✝³ : DivisionRing K\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\ninst✝ : FiniteDimensional K V\nS : Submodule K V\nh : finrank K ↥S = finrank K V\nbS : Basis (↑(Basis.ofVectorSpaceIndex K ↥S)) K ↥S := Basis.ofVectorSpace K ↥S\nbS_eq : bS = Basis.ofVectorSpace K ↥S\nthis✝¹ ... | [] | rw [this, Submodule.map_top (Submodule.subtype S), range_subtype] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.Dimension.FreeAndStrongRankCondition | {
"line": 86,
"column": 8
} | {
"line": 86,
"column": 58
} | {
"line": 86,
"column": 59
} | [
{
"pp": "K : 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\nκ : Type v\nb : Basis κ K V\nhd : Subsingleton κ\nhb : IsEmpty κ\n⊢ ∀ (v : V), v = 0",
"ppTerm": "?m.98",
"assigned": false,
"usedConstants": [],
"use... | [
"K : 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\nκ : Type v\nb : Basis κ K V\nhd : Subsingleton κ\nhb : IsEmpty κ\n⊢ ∀ (v : V), v = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Dimension.FreeAndStrongRankCondition | {
"line": 135,
"column": 53
} | {
"line": 136,
"column": 43
} | {
"line": 136,
"column": 43
} | [
{
"pp": "K : Type u\nV : Type v\ninst✝⁴ : Ring K\ninst✝³ : StrongRankCondition K\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\ns : Submodule K V\ninst✝ : Free K ↥s\nx✝ : ∃ v₀, v₀ ≠ 0 ∧ ∀ (v : ↥s), ∃ r, r • v₀ = v\nv₀ : V\nhv₀ : v₀ ∈ s\nH : ⟨v₀, hv₀⟩ ≠ 0\nh : ∀ (v : ↥s), ∃ r, r • ⟨v₀, hv₀⟩ = v\nh' : v₀ = 0\n⊢ F... | [] | by
simp only [h', ne_eq] at H; exact H rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.FiniteDimensional.Basic | {
"line": 387,
"column": 32
} | {
"line": 387,
"column": 61
} | {
"line": 387,
"column": 62
} | [
{
"pp": "K : Type u\nV : Type v\ninst✝³ : DivisionRing K\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\np : Submodule K V\ninst✝ : FiniteDimensional K ↥p\nf : V →ₗ[K] V\nh : ∀ x ∈ p, f x ∈ p\nh' : Disjoint p f.ker\nx : V\nhx : x ∈ comap f p\ny : V\nhy : y ∈ p\nhxy : (f.restrict h) ⟨y, hy⟩ = ⟨f x, hx⟩\n⊢ f y = f... | [
"K : Type u\nV : Type v\ninst✝³ : DivisionRing K\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\np : Submodule K V\ninst✝ : FiniteDimensional K ↥p\nf : V →ₗ[K] V\nh : ∀ x ∈ p, f x ∈ p\nh' : Disjoint p f.ker\nx : V\nhx : x ∈ comap f p\ny : V\nhy : y ∈ p\nhxy : (f.restrict h) ⟨y, hy⟩ = ⟨f x, hx⟩\n⊢ f y = f x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Dimension.FreeAndStrongRankCondition | {
"line": 156,
"column": 4
} | {
"line": 156,
"column": 32
} | {
"line": 156,
"column": 33
} | [
{
"pp": "case mpr\nK : Type u\nV : Type v\ninst✝⁴ : Ring K\ninst✝³ : StrongRankCondition K\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\ns : Submodule K V\ninst✝ : Free K ↥s\nthis : Nontrivial K\nv₀ : V\nh : s ≤ K ∙ v₀\nκ : Type v\nb : Basis κ K ↥s\n⊢ Module.rank K ↥s ≤ 1",
"ppTerm": "?mpr",
"assigned"... | [
"case mpr\nK : Type u\nV : Type v\ninst✝⁴ : Ring K\ninst✝³ : StrongRankCondition K\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\ns : Submodule K V\ninst✝ : Free K ↥s\nthis : Nontrivial K\nv₀ : V\nh : s ≤ K ∙ v₀\nκ : Type v\nb : Basis κ K ↥s\n⊢ Module.rank K ↥s ≤ 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.LinearPMap | {
"line": 238,
"column": 6
} | {
"line": 238,
"column": 61
} | {
"line": 238,
"column": 61
} | [
{
"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 :... | [] | rw [this hfx, this hgx, f.map_smulₛₗ, g.map_smulₛₗ, hx] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.Dimension.FreeAndStrongRankCondition | {
"line": 167,
"column": 2
} | {
"line": 169,
"column": 67
} | {
"line": 171,
"column": 0
} | [
{
"pp": "case mpr\nK : Type u\nV : Type v\ninst✝⁴ : Ring K\ninst✝³ : StrongRankCondition K\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\nW : Submodule K V\ninst✝ : Free K ↥W\n⊢ (∃ a, (∀ v ∈ W, ∃ r, r • a = v) ∧ a ∈ W) → ∃ v₀, ∀ (v : ↥W), ∃ r, r • v₀ = v",
"ppTerm": "?mpr",
"assigned": true,
"usedCo... | [] | · rintro ⟨a, ⟨h, ha⟩⟩
choose f hf using h
exact ⟨⟨a, ha⟩, fun v => ⟨f v.1 v.2, Subtype.ext (hf v.1 v.2)⟩⟩ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.LinearAlgebra.Dimension.FreeAndStrongRankCondition | {
"line": 185,
"column": 4
} | {
"line": 185,
"column": 15
} | {
"line": 185,
"column": 16
} | [
{
"pp": "case mpr\nK : Type u\nV : Type v\ninst✝⁵ : Ring K\ninst✝⁴ : StrongRankCondition K\ninst✝³ : AddCommGroup V\ninst✝² : Module K V\ninst✝¹ : Free K V\nι : Type u_1\ninst✝ : Unique ι\nb : Basis ι K V\n⊢ finrank K V = 1",
"ppTerm": "?mpr",
"assigned": false,
"usedConstants": [],
"usedFVars":... | [
"case mpr\nK : Type u\nV : Type v\ninst✝⁵ : Ring K\ninst✝⁴ : StrongRankCondition K\ninst✝³ : AddCommGroup V\ninst✝² : Module K V\ninst✝¹ : Free K V\nι : Type u_1\ninst✝ : Unique ι\nb : Basis ι K V\n⊢ finrank K V = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.FiniteDimensional.Basic | {
"line": 406,
"column": 56
} | {
"line": 406,
"column": 79
} | {
"line": 407,
"column": 6
} | [
{
"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\ncomm : (↑(insert i s)).Pairwise (Commute on f)\nh : (insert i s).SupIndep fun i ↦ (f i).ker\nih : ... | [
"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... | Module.End.mul_eq_comp, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Dimension.FreeAndStrongRankCondition | {
"line": 226,
"column": 2
} | {
"line": 226,
"column": 13
} | {
"line": 226,
"column": 14
} | [
{
"pp": "K V : Type u\ninst✝⁵ : Ring K\ninst✝⁴ : StrongRankCondition K\ninst✝³ : AddCommGroup V\ninst✝² : Module K V\ninst✝¹ : Free K V\ninst✝ : Module.Finite K V\n⊢ #V = #K ^ Module.rank K V",
"ppTerm": "?m.21",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"K V : Type u\ninst✝⁵ : Ring K\ninst✝⁴ : StrongRankCondition K\ninst✝³ : AddCommGroup V\ninst✝² : Module K V\ninst✝¹ : Free K V\ninst✝ : Module.Finite K V\n⊢ #V = #K ^ Module.rank K V"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.FiniteDimensional.Basic | {
"line": 412,
"column": 6
} | {
"line": 412,
"column": 46
} | {
"line": 412,
"column": 47
} | [
{
"pp": "case insert.h'\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)\nih : (s.noncommProd f ⋯).ker = ⨆ x ∈ s, (f x).k... | [
"case insert.h'\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)\nih : (s.noncommProd f ⋯).ker = ⨆ x ∈ s, (f x).ker\nh : Disj... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.FreeAlgebra | {
"line": 237,
"column": 4
} | {
"line": 238,
"column": 33
} | {
"line": 239,
"column": 2
} | [
{
"pp": "R : Type u_1\nX : Type u_2\ninst✝ : CommSemiring R\n⊢ ∀ (a b : FreeAlgebra R X), a + b = b + a",
"ppTerm": "?m.212",
"assigned": true,
"usedConstants": [
"Quot.sound",
"Quot.ind",
"FreeAlgebra.Pre.hasAdd",
"FreeAlgebra.Pre",
"FreeAlgebra",
"instHAdd",
... | [] | rintro ⟨⟩ ⟨⟩
exact Quot.sound Rel.add_comm | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.FreeAlgebra | {
"line": 237,
"column": 4
} | {
"line": 238,
"column": 33
} | {
"line": 239,
"column": 2
} | [
{
"pp": "R : Type u_1\nX : Type u_2\ninst✝ : CommSemiring R\n⊢ ∀ (a b : FreeAlgebra R X), a + b = b + a",
"ppTerm": "?m.212",
"assigned": true,
"usedConstants": [
"Quot.sound",
"Quot.ind",
"FreeAlgebra.Pre.hasAdd",
"FreeAlgebra.Pre",
"FreeAlgebra",
"instHAdd",
... | [] | rintro ⟨⟩ ⟨⟩
exact Quot.sound Rel.add_comm | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.FiniteDimensional.Basic | {
"line": 610,
"column": 13
} | {
"line": 610,
"column": 24
} | {
"line": 610,
"column": 25
} | [
{
"pp": "K : Type u\nV : Type v\ninst✝² : DivisionRing K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nv : V\nnz : v ≠ 0\nh : finrank K V = 1\n⊢ K ∙ v = ⊤",
"ppTerm": "?m.25",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"K : Type u\nV : Type v\ninst✝² : DivisionRing K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nv : V\nnz : v ≠ 0\nh : finrank K V = 1\n⊢ K ∙ v = ⊤"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.FiniteDimensional.Basic | {
"line": 627,
"column": 2
} | {
"line": 627,
"column": 39
} | {
"line": 628,
"column": 2
} | [
{
"pp": "K : Type u\nV : Type v\ninst✝⁸ : DivisionRing K\ninst✝⁷ : AddCommGroup V\ninst✝⁶ : Module K V\nW : Type u_1\nA : Type u_2\ninst✝⁵ : Semiring A\ninst✝⁴ : Module A V\ninst✝³ : AddCommGroup W\ninst✝² : Module K W\ninst✝¹ : Module A W\ninst✝ : LinearMap.CompatibleSMul V W K A\nh : finrank K W = 1\nf : V →ₗ... | [
"K : Type u\nV : Type v\ninst✝⁸ : DivisionRing K\ninst✝⁷ : AddCommGroup V\ninst✝⁶ : Module K V\nW : Type u_1\nA : Type u_2\ninst✝⁵ : Semiring A\ninst✝⁴ : Module A V\ninst✝³ : AddCommGroup W\ninst✝² : Module K W\ninst✝¹ : Module A W\ninst✝ : LinearMap.CompatibleSMul V W K A\nh : finrank K W = 1\nf : V →ₗ[A] W\nw : f... | obtain ⟨v, n⟩ := DFunLike.ne_iff.mp w | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.LinearAlgebra.FiniteDimensional.Basic | {
"line": 671,
"column": 8
} | {
"line": 671,
"column": 31
} | {
"line": 671,
"column": 32
} | [
{
"pp": "case a\nK : Type u\nV : Type v\ninst✝² : DivisionRing K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : End K V\nk : ℕ\nh : LinearMap.ker (f ^ k) = LinearMap.ker (f ^ k.succ)\nm : ℕ\n⊢ LinearMap.ker (f ^ (k + 1) * f ^ m) ≤ LinearMap.ker (f ^ k * f ^ m)",
"ppTerm": "?a✝",
"assigned": true,
... | [
"case a\nK : Type u\nV : Type v\ninst✝² : DivisionRing K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : End K V\nk : ℕ\nh : LinearMap.ker (f ^ k) = LinearMap.ker (f ^ k.succ)\nm : ℕ\n⊢ ((f ^ (k + 1)) ∘ₗ f ^ m).ker ≤ LinearMap.ker (f ^ k * f ^ m)"
] | Module.End.mul_eq_comp, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.FiniteDimensional.Basic | {
"line": 671,
"column": 32
} | {
"line": 671,
"column": 55
} | {
"line": 671,
"column": 56
} | [
{
"pp": "case a\nK : Type u\nV : Type v\ninst✝² : DivisionRing K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : End K V\nk : ℕ\nh : LinearMap.ker (f ^ k) = LinearMap.ker (f ^ k.succ)\nm : ℕ\n⊢ ((f ^ (k + 1)) ∘ₗ f ^ m).ker ≤ LinearMap.ker (f ^ k * f ^ m)",
"ppTerm": "?a✝",
"assigned": true,
"usedC... | [
"case a\nK : Type u\nV : Type v\ninst✝² : DivisionRing K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : End K V\nk : ℕ\nh : LinearMap.ker (f ^ k) = LinearMap.ker (f ^ k.succ)\nm : ℕ\n⊢ ((f ^ (k + 1)) ∘ₗ f ^ m).ker ≤ ((f ^ k) ∘ₗ f ^ m).ker"
] | Module.End.mul_eq_comp, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.Eval.Defs | {
"line": 372,
"column": 68
} | {
"line": 372,
"column": 79
} | {
"line": 372,
"column": 80
} | [
{
"pp": "R : Type u\ninst✝ : Semiring R\nr a : R\nhr : r ≠ 0\n⊢ ¬(C r).IsRoot a",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Polynomial.C",
"Polynomial.eval",
"Polynomial.eval_C",
"congrArg",
"Polynomial.IsRoot",
"RingHom",
"id"... | [
"R : Type u\ninst✝ : Semiring R\nr a : R\nhr : r ≠ 0\n⊢ ¬r = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Nat.Prime.Defs | {
"line": 249,
"column": 6
} | {
"line": 249,
"column": 29
} | {
"line": 250,
"column": 8
} | [
{
"pp": "case pos\nn : ℕ\nn2 : 2 ≤ n\nx✝ : ℕ\nk : ℕ := x✝\ni : ℕ\ne : x✝ = 2 * i + 3\na : ∀ (m : ℕ), 2 ≤ m → m ∣ n → x✝ ≤ m\nh : n < k * k\npp : Prime n\n⊢ n.minFacProp (if n < x✝ * x✝ then n else if x✝ ∣ n then x✝ else n.minFacAux (x✝ + 2))",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
... | [
"case pos\nn : ℕ\nn2 : 2 ≤ n\nx✝ : ℕ\nk : ℕ := x✝\ni : ℕ\ne : x✝ = 2 * i + 3\na : ∀ (m : ℕ), 2 ≤ m → m ∣ n → x✝ ≤ m\nh : n < k * k\npp : Prime n\n⊢ n.minFacProp (if True then n else if x✝ ∣ n then x✝ else n.minFacAux (x✝ + 2))"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.MonoidAlgebra.Basic | {
"line": 68,
"column": 6
} | {
"line": 69,
"column": 33
} | {
"line": 69,
"column": 34
} | [
{
"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✝⁵ : Semiring R\ninst✝⁴ : Mul M\ninst✝³ : NonUnitalNonAssocSemiring A\ninst✝² : Module R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\nf : M →ₙ* A\na₁ a₂ : R... | [
"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✝⁵ : Semiring R\ninst✝⁴ : Mul M\ninst✝³ : NonUnitalNonAssocSemiring A\ninst✝² : Module R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\nf : M →ₙ* A\na₁ a₂ : R[M]\n⊢ (a₁.c... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.LinearPMap | {
"line": 538,
"column": 32
} | {
"line": 538,
"column": 69
} | {
"line": 538,
"column": 70
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝¹² : Ring R\ninst✝¹¹ : Ring S\ninst✝¹⁰ : Ring T\nσ✝ : R →+* S\nτ : S →+* T\nE : Type u_4\ninst✝⁹ : AddCommGroup E\ninst✝⁸ : Module R E\nF : Type u_5\ninst✝⁷ : AddCommGroup F\ninst✝⁶ : Module S F\nG : Type u_6\ninst✝⁵ : AddCommGroup G\ninst✝⁴ : Module T G\n... | [
"R : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝¹² : Ring R\ninst✝¹¹ : Ring S\ninst✝¹⁰ : Ring T\nσ✝ : R →+* S\nτ : S →+* T\nE : Type u_4\ninst✝⁹ : AddCommGroup E\ninst✝⁸ : Module R E\nF : Type u_5\ninst✝⁷ : AddCommGroup F\ninst✝⁶ : Module S F\nG : Type u_6\ninst✝⁵ : AddCommGroup G\ninst✝⁴ : Module T G\nK : Type u_7... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.LinearPMap | {
"line": 552,
"column": 2
} | {
"line": 552,
"column": 63
} | {
"line": 553,
"column": 2
} | [
{
"pp": "E : Type u_4\ninst✝⁵ : AddCommGroup E\nF : Type u_5\ninst✝⁴ : AddCommGroup F\nK : Type u_7\nL : Type u_8\ninst✝³ : DivisionRing K\ninst✝² : DivisionRing L\nσ : K →+* L\ninst✝¹ : Module K E\ninst✝ : Module L F\nf : E →ₛₗ.[σ] F\nx : E\ny : F\nhx : x ∉ f.domain\nx' : E\nhx' : x' ∈ f.domain\nc : K\n⊢ ↑(f.s... | [
"case hz\nE : Type u_4\ninst✝⁵ : AddCommGroup E\nF : Type u_5\ninst✝⁴ : AddCommGroup F\nK : Type u_7\nL : Type u_8\ninst✝³ : DivisionRing K\ninst✝² : DivisionRing L\nσ : K →+* L\ninst✝¹ : Module K E\ninst✝ : Module L F\nf : E →ₛₗ.[σ] F\nx : E\ny : F\nhx : x ∉ f.domain\nx' : E\nhx' : x' ∈ f.domain\nc : K\n⊢ ↑⟨x', hx... | rw [sup_apply _ ⟨x', hx'⟩ ⟨c • x, _⟩, mkSpanSingleton'_apply] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.CharP.Defs | {
"line": 71,
"column": 4
} | {
"line": 71,
"column": 30
} | {
"line": 71,
"column": 31
} | [
{
"pp": "case inr\nR : Type u_1\ninst✝² : AddMonoidWithOne R\np : ℕ\ninst✝¹ : CharP R p\na b : ℕ\ninst✝ : IsLeftCancelAdd R\nthis :\n ∀ (R : Type u_1) [inst : AddMonoidWithOne R] (p : ℕ) [CharP R p] {a b : ℕ} [IsLeftCancelAdd R],\n a ≤ b → (↑a = ↑b ↔ a % p = b % p)\nhle : b < a\n⊢ ↑a = ↑b ↔ a % p = b % p",
... | [
"case inr\nR : Type u_1\ninst✝² : AddMonoidWithOne R\np : ℕ\ninst✝¹ : CharP R p\na b : ℕ\ninst✝ : IsLeftCancelAdd R\nthis :\n ∀ (R : Type u_1) [inst : AddMonoidWithOne R] (p : ℕ) [CharP R p] {a b : ℕ} [IsLeftCancelAdd R],\n a ≤ b → (↑a = ↑b ↔ a % p = b % p)\nhle : b < a\n⊢ ↑a = ↑b ↔ a % p = b % p"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.LinearPMap | {
"line": 561,
"column": 2
} | {
"line": 561,
"column": 40
} | {
"line": 561,
"column": 41
} | [
{
"pp": "E : Type u_4\ninst✝⁵ : AddCommGroup E\nF : Type u_5\ninst✝⁴ : AddCommGroup F\nK : Type u_7\nL : Type u_8\ninst✝³ : DivisionRing K\ninst✝² : DivisionRing L\nσ : K →+* L\ninst✝¹ : Module K E\ninst✝ : Module L F\nf : E →ₛₗ.[σ] F\nx : E\ny : F\nhx : x ∉ f.domain\nc : K\n⊢ ↑(f.supSpanSingleton x y hx) ⟨c • ... | [
"E : Type u_4\ninst✝⁵ : AddCommGroup E\nF : Type u_5\ninst✝⁴ : AddCommGroup F\nK : Type u_7\nL : Type u_8\ninst✝³ : DivisionRing K\ninst✝² : DivisionRing L\nσ : K →+* L\ninst✝¹ : Module K E\ninst✝ : Module L F\nf : E →ₛₗ.[σ] F\nx : E\ny : F\nhx : x ∉ f.domain\nc : K\n⊢ ↑(f.supSpanSingleton x y hx) ⟨c • x, ⋯⟩ = σ c ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.CharP.Defs | {
"line": 78,
"column": 2
} | {
"line": 78,
"column": 39
} | {
"line": 78,
"column": 40
} | [
{
"pp": "case mpr\nR✝ : Type u_1\ninst✝³ : AddMonoidWithOne R✝\np✝ a✝ b : ℕ\nR : Type u_1\ninst✝² : AddMonoidWithOne R\np : ℕ\ninst✝¹ : CharP R p\na : ℕ\ninst✝ : IsLeftCancelAdd R\nc : ℕ\nhle : a ≤ a + c\nh : a % p = (a + c) % p\nthis : (a + c - a) % p = 0\n⊢ p ∣ c",
"ppTerm": "?mpr",
"assigned": true,
... | [
"case mpr\nR✝ : Type u_1\ninst✝³ : AddMonoidWithOne R✝\np✝ a✝ b : ℕ\nR : Type u_1\ninst✝² : AddMonoidWithOne R\np : ℕ\ninst✝¹ : CharP R p\na : ℕ\ninst✝ : IsLeftCancelAdd R\nc : ℕ\nhle : a ≤ a + c\nh : a % p = (a + c) % p\nthis : (a + c - a) % p = 0\n⊢ c % p = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.LinearPMap | {
"line": 566,
"column": 2
} | {
"line": 566,
"column": 13
} | {
"line": 566,
"column": 14
} | [
{
"pp": "E : Type u_4\ninst✝⁵ : AddCommGroup E\nF : Type u_5\ninst✝⁴ : AddCommGroup F\nK : Type u_7\nL : Type u_8\ninst✝³ : DivisionRing K\ninst✝² : DivisionRing L\nσ : K →+* L\ninst✝¹ : Module K E\ninst✝ : Module L F\nf : E →ₛₗ.[σ] F\nx : E\ny : F\nhx : x ∉ f.domain\n⊢ ↑(f.supSpanSingleton x y hx) ⟨x, ⋯⟩ = y",... | [
"E : Type u_4\ninst✝⁵ : AddCommGroup E\nF : Type u_5\ninst✝⁴ : AddCommGroup F\nK : Type u_7\nL : Type u_8\ninst✝³ : DivisionRing K\ninst✝² : DivisionRing L\nσ : K →+* L\ninst✝¹ : Module K E\ninst✝ : Module L F\nf : E →ₛₗ.[σ] F\nx : E\ny : F\nhx : x ∉ f.domain\n⊢ ↑(f.supSpanSingleton x y hx) ⟨x, ⋯⟩ = y"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.LinearPMap | {
"line": 571,
"column": 2
} | {
"line": 571,
"column": 13
} | {
"line": 571,
"column": 14
} | [
{
"pp": "E : Type u_4\ninst✝⁵ : AddCommGroup E\nF : Type u_5\ninst✝⁴ : AddCommGroup F\nK : Type u_7\nL : Type u_8\ninst✝³ : DivisionRing K\ninst✝² : DivisionRing L\nσ : K →+* L\ninst✝¹ : Module K E\ninst✝ : Module L F\nf : E →ₛₗ.[σ] F\nx : E\ny : F\nhx : x ∉ f.domain\nx' : ↥(f.supSpanSingleton x y hx).domain\nh... | [
"E : Type u_4\ninst✝⁵ : AddCommGroup E\nF : Type u_5\ninst✝⁴ : AddCommGroup F\nK : Type u_7\nL : Type u_8\ninst✝³ : DivisionRing K\ninst✝² : DivisionRing L\nσ : K →+* L\ninst✝¹ : Module K E\ninst✝ : Module L F\nf : E →ₛₗ.[σ] F\nx : E\ny : F\nhx : x ∉ f.domain\nx' : ↥(f.supSpanSingleton x y hx).domain\nhx' : ↑x' ∈ f... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.CharP.Defs | {
"line": 216,
"column": 27
} | {
"line": 216,
"column": 38
} | {
"line": 216,
"column": 39
} | [
{
"pp": "R : Type u_1\ninst✝¹ : NonAssocSemiring R\ninst✝ : Nontrivial R\np : ℕ\nhc : CharP R p\nhp : p = 1\n⊢ 1 = 0",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"False",
"NeZero.one",
"NonUnitalNonAssocS... | [
"R : Type u_1\ninst✝¹ : NonAssocSemiring R\ninst✝ : Nontrivial R\np : ℕ\nhc : CharP R p\nhp : p = 1\n⊢ False"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.MonoidAlgebra.Basic | {
"line": 225,
"column": 18
} | {
"line": 225,
"column": 41
} | {
"line": 225,
"column": 42
} | [
{
"pp": "case single\nR : Type u_1\nA : Type u_4\nB : Type u_5\nM : Type u_7\ninst✝⁵ : CommSemiring R\ninst✝⁴ : Semiring A\ninst✝³ : Semiring B\ninst✝² : Algebra R A\ninst✝¹ : Algebra R B\ninst✝ : Monoid M\nφ₁ φ₂ : A[M] →ₐ[R] B\nsingle_one_right : ∀ (m : M), φ₁ (single m 1) = φ₂ (single m 1)\nsingle_one_left : ... | [
"case single\nR : Type u_1\nA : Type u_4\nB : Type u_5\nM : Type u_7\ninst✝⁵ : CommSemiring R\ninst✝⁴ : Semiring A\ninst✝³ : Semiring B\ninst✝² : Algebra R A\ninst✝¹ : Algebra R B\ninst✝ : Monoid M\nφ₁ φ₂ : A[M] →ₐ[R] B\nsingle_one_right : ∀ (m : M), φ₁ (single m 1) = φ₂ (single m 1)\nsingle_one_left : φ₁.comp sing... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.CharP.Defs | {
"line": 291,
"column": 2
} | {
"line": 291,
"column": 13
} | {
"line": 291,
"column": 14
} | [
{
"pp": "R : Type u_1\ninst✝¹ : NonAssocSemiring R\ninst✝ : Nontrivial R\n⊢ ringChar R ≠ 1",
"ppTerm": "?m.5",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"id",
"Ne",
"instOfNatNat",
"ringChar",
"Nat",
"ringChar.ringChar_eq_one._simp_1",... | [
"R : Type u_1\ninst✝¹ : NonAssocSemiring R\ninst✝ : Nontrivial R\n⊢ ¬Subsingleton R"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.CharP.Defs | {
"line": 375,
"column": 2
} | {
"line": 375,
"column": 39
} | {
"line": 377,
"column": 0
} | [
{
"pp": "case inl\nR : Type u_1\ninst✝¹ : AddMonoidWithOne R\nq : ℕ\ninst✝ : ExpChar R q\nh : Nat.Prime q\n⊢ 0 < q",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Nat.Prime.pos"
],
"usedFVars": [
"q",
"h"
],
"usedGoals": []
},
{
"pp": "case inr\nR ... | [] | exacts [Nat.Prime.pos h, Nat.one_pos] | Batteries.Tactic._aux_Batteries_Tactic_Init___elabRules_Batteries_Tactic_exacts_1 | Batteries.Tactic.exacts |
Mathlib.Algebra.CharP.Defs | {
"line": 420,
"column": 4
} | {
"line": 421,
"column": 38
} | {
"line": 423,
"column": 0
} | [
{
"pp": "case mpr\nR : Type u_1\ninst✝³ : NonAssocSemiring R\ninst✝² : Nontrivial R\np q : ℕ\ninst✝¹ : CharP R p\ninst✝ : ExpChar R q\n⊢ p = 0 → q = 1",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"expChar_one_of_char_zero",
"in... | [] | rintro rfl
exact expChar_one_of_char_zero R q | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.CharP.Defs | {
"line": 420,
"column": 4
} | {
"line": 421,
"column": 38
} | {
"line": 423,
"column": 0
} | [
{
"pp": "case mpr\nR : Type u_1\ninst✝³ : NonAssocSemiring R\ninst✝² : Nontrivial R\np q : ℕ\ninst✝¹ : CharP R p\ninst✝ : ExpChar R q\n⊢ p = 0 → q = 1",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"expChar_one_of_char_zero",
"in... | [] | rintro rfl
exact expChar_one_of_char_zero R q | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.Filter.Extr | {
"line": 446,
"column": 47
} | {
"line": 446,
"column": 80
} | {
"line": 446,
"column": 81
} | [
{
"pp": "α : Type u\nβ : Type v\ninst✝² : AddCommGroup β\ninst✝¹ : PartialOrder β\ninst✝ : IsOrderedAddMonoid β\nf g : α → β\na : α\nl : Filter α\nhf : IsMinFilter f l a\nhg : IsMaxFilter g l a\n⊢ IsMinFilter (fun x ↦ f x - g x) l a",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Eq... | [
"α : Type u\nβ : Type v\ninst✝² : AddCommGroup β\ninst✝¹ : PartialOrder β\ninst✝ : IsOrderedAddMonoid β\nf g : α → β\na : α\nl : Filter α\nhf : IsMinFilter f l a\nhg : IsMaxFilter g l a\n⊢ IsMinFilter (fun x ↦ f x + -g x) l a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Filter.Extr | {
"line": 449,
"column": 47
} | {
"line": 449,
"column": 80
} | {
"line": 449,
"column": 81
} | [
{
"pp": "α : Type u\nβ : Type v\ninst✝² : AddCommGroup β\ninst✝¹ : PartialOrder β\ninst✝ : IsOrderedAddMonoid β\nf g : α → β\na : α\nl : Filter α\nhf : IsMaxFilter f l a\nhg : IsMinFilter g l a\n⊢ IsMaxFilter (fun x ↦ f x - g x) l a",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Eq... | [
"α : Type u\nβ : Type v\ninst✝² : AddCommGroup β\ninst✝¹ : PartialOrder β\ninst✝ : IsOrderedAddMonoid β\nf g : α → β\na : α\nl : Filter α\nhf : IsMaxFilter f l a\nhg : IsMinFilter g l a\n⊢ IsMaxFilter (fun x ↦ f x + -g x) l a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Filter.Extr | {
"line": 453,
"column": 2
} | {
"line": 453,
"column": 35
} | {
"line": 453,
"column": 36
} | [
{
"pp": "α : Type u\nβ : Type v\ninst✝² : AddCommGroup β\ninst✝¹ : PartialOrder β\ninst✝ : IsOrderedAddMonoid β\nf g : α → β\na : α\ns : Set α\nhf : IsMinOn f s a\nhg : IsMaxOn g s a\n⊢ IsMinOn (fun x ↦ f x - g x) s a",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"c... | [
"α : Type u\nβ : Type v\ninst✝² : AddCommGroup β\ninst✝¹ : PartialOrder β\ninst✝ : IsOrderedAddMonoid β\nf g : α → β\na : α\ns : Set α\nhf : IsMinOn f s a\nhg : IsMaxOn g s a\n⊢ IsMinOn (fun x ↦ f x + -g x) s a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Filter.Extr | {
"line": 457,
"column": 2
} | {
"line": 457,
"column": 35
} | {
"line": 457,
"column": 36
} | [
{
"pp": "α : Type u\nβ : Type v\ninst✝² : AddCommGroup β\ninst✝¹ : PartialOrder β\ninst✝ : IsOrderedAddMonoid β\nf g : α → β\na : α\ns : Set α\nhf : IsMaxOn f s a\nhg : IsMinOn g s a\n⊢ IsMaxOn (fun x ↦ f x - g x) s a",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"c... | [
"α : Type u\nβ : Type v\ninst✝² : AddCommGroup β\ninst✝¹ : PartialOrder β\ninst✝ : IsOrderedAddMonoid β\nf g : α → β\na : α\ns : Set α\nhf : IsMaxOn f s a\nhg : IsMinOn g s a\n⊢ IsMaxOn (fun x ↦ f x + -g x) s a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.MonoidAlgebra.Basic | {
"line": 566,
"column": 6
} | {
"line": 567,
"column": 33
} | {
"line": 567,
"column": 34
} | [
{
"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✝⁵ : Semiring R\ninst✝⁴ : Add M\ninst✝³ : NonUnitalNonAssocSemiring A\ninst✝² : Module R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\nf : Multiplicative M →... | [
"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✝⁵ : Semiring R\ninst✝⁴ : Add M\ninst✝³ : NonUnitalNonAssocSemiring A\ninst✝² : Module R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\nf : Multiplicative M →ₙ* A\na₁ a₂ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Polynomial.Coeff | {
"line": 254,
"column": 2
} | {
"line": 254,
"column": 28
} | {
"line": 254,
"column": 29
} | [
{
"pp": "R : Type u\ninst✝ : Semiring R\np : R[X]\nn : ℕ\n⊢ (p * X).coeff (n + 1) = p.coeff n",
"ppTerm": "?m.24",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u\ninst✝ : Semiring R\np : R[X]\nn : ℕ\n⊢ (p * X).coeff (n + 1) = p.coeff n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Polynomial.Coeff | {
"line": 353,
"column": 4
} | {
"line": 353,
"column": 15
} | {
"line": 353,
"column": 16
} | [
{
"pp": "case mp\nm n : ℕ\nR : Type u_1\ninst✝¹ : Semiring R\ninst✝ : CharZero R\nh : (↑m).coeff 0 = (↑n).coeff 0\n⊢ m = n",
"ppTerm": "?mp",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"case mp\nm n : ℕ\nR : Type u_1\ninst✝¹ : Semiring R\ninst✝ : CharZero R\nh : (↑m).coeff 0 = (↑n).coeff 0\n⊢ m = n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Polynomial.Coeff | {
"line": 366,
"column": 4
} | {
"line": 366,
"column": 15
} | {
"line": 366,
"column": 16
} | [
{
"pp": "case mp\nm n : ℤ\nR : Type u_1\ninst✝¹ : Ring R\ninst✝ : CharZero R\nh : (↑m).coeff 0 = (↑n).coeff 0\n⊢ m = n",
"ppTerm": "?mp",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"case mp\nm n : ℤ\nR : Type u_1\ninst✝¹ : Ring R\ninst✝ : CharZero R\nh : (↑m).coeff 0 = (↑n).coeff 0\n⊢ m = n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Polynomial.Degree.Defs | {
"line": 215,
"column": 2
} | {
"line": 215,
"column": 28
} | {
"line": 215,
"column": 29
} | [
{
"pp": "R : Type u\ninst✝ : Semiring R\na : R\nha : a ≠ 0\n⊢ (C a * X).natDegree = 1",
"ppTerm": "?m.23",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u\ninst✝ : Semiring R\na : R\nha : a ≠ 0\n⊢ (C a * X).natDegree = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Polynomial.Degree.Defs | {
"line": 238,
"column": 2
} | {
"line": 238,
"column": 33
} | {
"line": 238,
"column": 34
} | [
{
"pp": "R : Type u\ninst✝ : Semiring R\nn : ℕ\n⊢ (X ^ n).degree ≤ ↑n",
"ppTerm": "?m.15",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u\ninst✝ : Semiring R\nn : ℕ\n⊢ (X ^ n).degree ≤ ↑n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Polynomial.Degree.Defs | {
"line": 370,
"column": 2
} | {
"line": 370,
"column": 13
} | {
"line": 370,
"column": 14
} | [
{
"pp": "R : Type u\ninst✝ : Semiring R\np : R[X]\nn : ℕ\n⊢ (erase n p).support ⊆ p.support",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Finset",
"PartialOrder.toPreorder",
"Preorder.toLE",
"id",
"LE.le",
"Polynomial... | [
"R : Type u\ninst✝ : Semiring R\np : R[X]\nn : ℕ\n⊢ p.support.erase n ⊆ p.support"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Polynomial.Degree.Defs | {
"line": 424,
"column": 2
} | {
"line": 424,
"column": 28
} | {
"line": 424,
"column": 29
} | [
{
"pp": "R : Type u\ninst✝ : Semiring R\na : R\n⊢ (C a * X).leadingCoeff = a",
"ppTerm": "?m.18",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u\ninst✝ : Semiring R\na : R\n⊢ (C a * X).leadingCoeff = a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Polynomial.Degree.Defs | {
"line": 431,
"column": 2
} | {
"line": 431,
"column": 33
} | {
"line": 431,
"column": 34
} | [
{
"pp": "R : Type u\ninst✝ : Semiring R\nn : ℕ\n⊢ (X ^ n).leadingCoeff = 1",
"ppTerm": "?m.16",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u\ninst✝ : Semiring R\nn : ℕ\n⊢ (X ^ n).leadingCoeff = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Polynomial.Degree.Defs | {
"line": 434,
"column": 2
} | {
"line": 434,
"column": 28
} | {
"line": 434,
"column": 29
} | [
{
"pp": "R : Type u\ninst✝ : Semiring R\n⊢ X.leadingCoeff = 1",
"ppTerm": "?m.12",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u\ninst✝ : Semiring R\n⊢ X.leadingCoeff = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Polynomial.Degree.Defs | {
"line": 499,
"column": 2
} | {
"line": 500,
"column": 87
} | {
"line": 502,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝ : Semiring R\nf : R[X]\nn : ℕ\n⊢ f.degree < ↑n ↔ ∀ (m : ℕ), n ≤ m → f.coeff m = 0",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"not_le",
"WithBot.instPreorder",
"WithBot.some",
"WithBot",
"Preorder.toLT",
"Lattice.toSemilatt... | [] | simp only [degree, Finset.sup_lt_iff (WithBot.bot_lt_coe n), mem_support_iff,
WithBot.coe_lt_coe, ← @not_le ℕ, max_eq_sup_coe, Nat.cast_withBot, Ne, not_imp_not] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Polynomial.Degree.Defs | {
"line": 499,
"column": 2
} | {
"line": 500,
"column": 87
} | {
"line": 502,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝ : Semiring R\nf : R[X]\nn : ℕ\n⊢ f.degree < ↑n ↔ ∀ (m : ℕ), n ≤ m → f.coeff m = 0",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"not_le",
"WithBot.instPreorder",
"WithBot.some",
"WithBot",
"Preorder.toLT",
"Lattice.toSemilatt... | [] | simp only [degree, Finset.sup_lt_iff (WithBot.bot_lt_coe n), mem_support_iff,
WithBot.coe_lt_coe, ← @not_le ℕ, max_eq_sup_coe, Nat.cast_withBot, Ne, not_imp_not] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Polynomial.Degree.Defs | {
"line": 499,
"column": 2
} | {
"line": 500,
"column": 87
} | {
"line": 502,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝ : Semiring R\nf : R[X]\nn : ℕ\n⊢ f.degree < ↑n ↔ ∀ (m : ℕ), n ≤ m → f.coeff m = 0",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"not_le",
"WithBot.instPreorder",
"WithBot.some",
"WithBot",
"Preorder.toLT",
"Lattice.toSemilatt... | [] | simp only [degree, Finset.sup_lt_iff (WithBot.bot_lt_coe n), mem_support_iff,
WithBot.coe_lt_coe, ← @not_le ℕ, max_eq_sup_coe, Nat.cast_withBot, Ne, not_imp_not] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Nat.WithBot | {
"line": 73,
"column": 21
} | {
"line": 73,
"column": 32
} | {
"line": 73,
"column": 33
} | [
{
"pp": "x : WithBot ℕ\n⊢ ¬x < 1 ↔ ¬x ≤ 0",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"WithBot.instPreorder",
"Eq.mpr",
"Nat.instMulZeroClass",
"WithBot",
"Preorder.toLT",
"Nat.instOne",
"congrArg",
"PartialOrder.toPreorder",
"WithB... | [
"x : WithBot ℕ\n⊢ 1 ≤ x ↔ 0 < x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Polynomial.Degree.Units | {
"line": 92,
"column": 2
} | {
"line": 92,
"column": 45
} | {
"line": 92,
"column": 46
} | [
{
"pp": "R : Type u\ninst✝ : CommSemiring R\na p : R[X]\nhp : p.Monic\nhap : C (a.coeff 0) ∣ p\nh : a.degree ≤ 0\n⊢ IsUnit (C (a.coeff 0))",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Polynomial.C",
"CommSemiring.toSemiring",
"IsUnit",
"RingHom",... | [
"R : Type u\ninst✝ : CommSemiring R\na p : R[X]\nhp : p.Monic\nhap : C (a.coeff 0) ∣ p\nh : a.degree ≤ 0\n⊢ IsUnit (a.coeff 0)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.MonoidAlgebra.Degree | {
"line": 422,
"column": 2
} | {
"line": 422,
"column": 58
} | {
"line": 422,
"column": 59
} | [
{
"pp": "R : Type u_1\nA : Type u_3\nB : Type u_5\ninst✝³ : Semiring R\ninst✝² : LinearOrder B\ninst✝¹ : OrderBot B\np q : R[A]\nD : A → B\ninst✝ : AddZeroClass A\na : A\nh : supDegree D q < D a\nha : a ∈ p.coeff.support\nhe : supDegree D p = D a\n⊢ a ∈ (p + q).coeff.support",
"ppTerm": "?m.123",
"assig... | [
"R : Type u_1\nA : Type u_3\nB : Type u_5\ninst✝³ : Semiring R\ninst✝² : LinearOrder B\ninst✝¹ : OrderBot B\np q : R[A]\nD : A → B\ninst✝ : AddZeroClass A\na : A\nh : supDegree D q < D a\nha : a ∈ p.coeff.support\nhe : supDegree D p = D a\n⊢ ¬p.coeff a = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Polynomial.Eval.Coeff | {
"line": 48,
"column": 47
} | {
"line": 48,
"column": 63
} | {
"line": 48,
"column": 63
} | [
{
"pp": "R : Type u\ninst✝ : Semiring R\np✝ p q : R[X]\nhp : eval₂ C X p = p\nhq : eval₂ C X q = q\n⊢ eval₂ C X (p + q) = p + q",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Polynomial.C",
"congrArg",
"Distrib.toAdd",
"Polynomial.eval₂",
"Polynomial.instAdd... | [] | by simp [hp, hq] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Polynomial.Eval.Coeff | {
"line": 97,
"column": 29
} | {
"line": 97,
"column": 40
} | {
"line": 97,
"column": 41
} | [
{
"pp": "R✝ : Type u\nS : Type v\nT : Type w\nι✝ : Type y\na b : R✝\nm n✝ : ℕ\ninst✝³ : Semiring R✝\np✝ q✝ r : R✝[X]\ninst✝² : Semiring S\nf : R✝ →+* S\nι : Type ?u.38\ninst✝¹ : Finite ι\nR : ι → Type u_1\ninst✝ : (i : ι) → Semiring (R i)\np q : ((i : ι) → R i)[X]\nh : (RingHom.pi fun i ↦ mapRingHom (Pi.evalRin... | [
"R✝ : Type u\nS : Type v\nT : Type w\nι✝ : Type y\na b : R✝\nm n✝ : ℕ\ninst✝³ : Semiring R✝\np✝ q✝ r : R✝[X]\ninst✝² : Semiring S\nf : R✝ →+* S\nι : Type ?u.38\ninst✝¹ : Finite ι\nR : ι → Type u_1\ninst✝ : (i : ι) → Semiring (R i)\np q : ((i : ι) → R i)[X]\nh : (RingHom.pi fun i ↦ mapRingHom (Pi.evalRingHom R i)) p... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Polynomial.Eval.Coeff | {
"line": 107,
"column": 22
} | {
"line": 107,
"column": 33
} | {
"line": 107,
"column": 34
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝¹ : Semiring R\ninst✝ : Semiring S\nf : R →+* S\nh : Function.Injective (map f)\nr r' : R\neq : f r = f r'\n⊢ r = r'",
"ppTerm": "?m.21",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u\nS : Type v\ninst✝¹ : Semiring R\ninst✝ : Semiring S\nf : R →+* S\nh : Function.Injective (map f)\nr r' : R\neq : f r = f r'\n⊢ r = r'"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Polynomial.Eval.Coeff | {
"line": 117,
"column": 51
} | {
"line": 117,
"column": 62
} | {
"line": 117,
"column": 63
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝¹ : Semiring R\ninst✝ : Semiring S\nf : R →+* S\nh✝ : Function.Surjective (map f)\ns : S\np : R[X]\nh : map f p = C s\n⊢ f (p.coeff 0) = s",
"ppTerm": "?m.33",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u\nS : Type v\ninst✝¹ : Semiring R\ninst✝ : Semiring S\nf : R →+* S\nh✝ : Function.Surjective (map f)\ns : S\np : R[X]\nh : map f p = C s\n⊢ f (p.coeff 0) = s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.MonoidAlgebra.Degree | {
"line": 472,
"column": 2
} | {
"line": 472,
"column": 45
} | {
"line": 473,
"column": 2
} | [
{
"pp": "case neg\nR : Type u_1\nA : Type u_3\nB : Type u_5\ninst✝³ : Semiring R\ninst✝² : LinearOrder B\ninst✝¹ : OrderBot B\nD : A → B\nι : Type u_7\ns : Finset ι\ni : ι\nf : ι → R[A]\ninst✝ : AddZeroClass A\nhi : i ∈ s\nhmax : ∀ j ∈ s, j ≠ i → supDegree D (f j) < supDegree D (f i)\nhs : (s.erase i).Nonempty\... | [
"case neg.refine_1\nR : Type u_1\nA : Type u_3\nB : Type u_5\ninst✝³ : Semiring R\ninst✝² : LinearOrder B\ninst✝¹ : OrderBot B\nD : A → B\nι : Type u_7\ns : Finset ι\ni : ι\nf : ι → R[A]\ninst✝ : AddZeroClass A\nhi : i ∈ s\nhmax : ∀ j ∈ s, j ≠ i → supDegree D (f j) < supDegree D (f i)\nhs : (s.erase i).Nonempty\n⊢ ... | refine supDegree_sum_lt ?_ (fun j hj => ?_) | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Algebra.Polynomial.Monomial | {
"line": 33,
"column": 64
} | {
"line": 33,
"column": 97
} | {
"line": 33,
"column": 98
} | [
{
"pp": "R : Type u\na b : R\nm✝ n✝ : ℕ\ninst✝¹ : Semiring R\np q r : R[X]\ninst✝ : Nontrivial R\nm n : ℕ\nh : (fun i ↦ (monomial i) 1) m = (fun i ↦ (monomial i) 1) n\n⊢ m = n",
"ppTerm": "?m.17",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u\na b : R\nm✝ n✝ : ℕ\ninst✝¹ : Semiring R\np q r : R[X]\ninst✝ : Nontrivial R\nm n : ℕ\nh : (fun i ↦ (monomial i) 1) m = (fun i ↦ (monomial i) 1) n\n⊢ m = n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Polynomial.Degree.Operations | {
"line": 506,
"column": 2
} | {
"line": 506,
"column": 38
} | {
"line": 506,
"column": 39
} | [
{
"pp": "R : Type u\na b : R\ninst✝ : Semiring R\nha : a ≠ 0\n⊢ (C b).degree < (C a * X).degree",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"WithBot.instPreorder",
"Eq.mpr",
"Polynomial.C",
"WithBot",
"Preorder.toLT",
"HMul.hMul",
"Nat.instOne"... | [
"R : Type u\na b : R\ninst✝ : Semiring R\nha : a ≠ 0\n⊢ (C b).degree < 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Polynomial.Degree.Operations | {
"line": 638,
"column": 15
} | {
"line": 638,
"column": 70
} | {
"line": 638,
"column": 71
} | [
{
"pp": "R : Type u\ninst✝¹ : Nontrivial R\ninst✝ : Semiring R\nn : ℕ\nhn : 0 < n\na : R\nh : X ^ n + C a = 1\n⊢ n = 0",
"ppTerm": "?m.27",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u\ninst✝¹ : Nontrivial R\ninst✝ : Semiring R\nn : ℕ\nhn : 0 < n\na : R\nh : X ^ n + C a = 1\n⊢ n = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Finsupp.Order | {
"line": 104,
"column": 4
} | {
"line": 104,
"column": 41
} | {
"line": 104,
"column": 42
} | [
{
"pp": "case mp\nι : Type u_1\nκ : Type u_2\nα : Type u_3\ninst✝² : Zero α\ninst✝¹ : LE α\ninst✝ : Std.Refl fun x1 x2 ↦ x1 ≤ x2\nf : ι ↪ κ\ng₁ g₂ : ι →₀ α\nh' : ∀ (i : κ), (embDomain f g₁) i ≤ (embDomain f g₂) i\nx : ι\n⊢ g₁ x ≤ g₂ x",
"ppTerm": "?mp",
"assigned": false,
"usedConstants": [],
"u... | [
"case mp\nι : Type u_1\nκ : Type u_2\nα : Type u_3\ninst✝² : Zero α\ninst✝¹ : LE α\ninst✝ : Std.Refl fun x1 x2 ↦ x1 ≤ x2\nf : ι ↪ κ\ng₁ g₂ : ι →₀ α\nh' : ∀ (i : κ), (embDomain f g₁) i ≤ (embDomain f g₂) i\nx : ι\n⊢ g₁ x ≤ g₂ x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
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