module stringlengths 16 90 | startPos dict | endPos dict | nextStartPos dict | goals listlengths 0 96 | goalsAfter listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 371
values | kind stringclasses 375
values |
|---|---|---|---|---|---|---|---|---|
Mathlib.Data.DFinsupp.Defs | {
"line": 730,
"column": 9
} | {
"line": 731,
"column": 95
} | {
"line": 733,
"column": 0
} | [
{
"pp": "ι : Type u\nβ : ι → Type v\ninst✝¹ : DecidableEq ι\ninst✝ : (i : ι) → AddZeroClass (β i)\ni : ι\nf : Π₀ (i : ι), β i\ni' : ι\nh : ¬i = i'\n⊢ (erase i f + single i (f i)) i' = f i'",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"Eq.recOn",
"dite_congr",
"congrArg... | [] | by
simp only [add_apply, single_apply, erase_apply, dif_neg h, if_neg (Ne.symm h), add_zero] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.SetTheory.Cardinal.Arithmetic | {
"line": 870,
"column": 2
} | {
"line": 870,
"column": 11
} | {
"line": 870,
"column": 11
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ns : Set α\nf : ↑s ↪ β\nh : Nonempty (↑sᶜ ≃ ↑(range ⇑f)ᶜ)\n⊢ ∃ g, ∀ (x : ↑s), g ↑x = f x",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Compl.compl",
"Set.Elem",
"Equiv",
"Set.instCompl",
"Function.Embedding",
"Nonem... | [
"α : Type u_1\nβ : Type u_2\ns : Set α\nf : ↑s ↪ β\nh this : Nonempty (↑sᶜ ≃ ↑(range ⇑f)ᶜ)\n⊢ ∃ g, ∀ (x : ↑s), g ↑x = f x"
] | have := h | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Data.DFinsupp.Defs | {
"line": 790,
"column": 18
} | {
"line": 790,
"column": 49
} | {
"line": 790,
"column": 49
} | [
{
"pp": "ι : Type u\nβ : ι → Type v\ninst✝¹ : DecidableEq ι\ninst✝ : (i : ι) → AddGroup (β i)\ns : Finset ι\nx y : (i : ↑↑s) → β ↑i\ni : ι\n⊢ (mk s (x - y)) i = (mk s x - mk s y) i",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"DFinsupp.mk",
... | [
"ι : Type u\nβ : ι → Type v\ninst✝¹ : DecidableEq ι\ninst✝ : (i : ι) → AddGroup (β i)\ns : Finset ι\nx y : (i : ↑↑s) → β ↑i\ni : ι\n⊢ (if H : i ∈ s then (x - y) ⟨i, H⟩ else 0) = (if H : i ∈ s then x ⟨i, H⟩ else 0) - if H : i ∈ s then y ⟨i, H⟩ else 0"
] | simp only [sub_apply, mk_apply] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Data.DFinsupp.Defs | {
"line": 1156,
"column": 4
} | {
"line": 1156,
"column": 24
} | {
"line": 1157,
"column": 4
} | [
{
"pp": "case some\nι : Type u\nα : Option ι → Type v\ninst✝¹ : DecidableEq ι\ninst✝ : (i : Option ι) → Zero (α i)\ni : ι\nx : α (some i)\nj : ι\n⊢ (extendWith 0 (single i x)) (some j) = (single (some i) x) (some j)",
"ppTerm": "?some",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Opt... | [
"case some\nι : Type u\nα : Option ι → Type v\ninst✝¹ : DecidableEq ι\ninst✝ : (i : Option ι) → Zero (α i)\ni : ι\nx : α (some i)\nj : ι\n⊢ (single i x) j = (single (some i) x) (some j)"
] | rw [extendWith_some] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.Dual.Defs | {
"line": 309,
"column": 53
} | {
"line": 309,
"column": 91
} | {
"line": 311,
"column": 0
} | [
{
"pp": "K : Type u_1\nV : Type u_2\ninst✝⁸ : CommSemiring K\ninst✝⁷ : AddCommMonoid V\ninst✝⁶ : Module K V\nR : Type u_3\nM : Type u_4\nN : Type u_5\ninst✝⁵ : CommSemiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : AddCommMonoid N\ninst✝² : Module R M\ninst✝¹ : Module R N\ninst✝ : IsReflexive R M\nr : R\nhr : IsReg... | [] | simpa [hr.1.eq_iff] using congr(n $hm) | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.LinearAlgebra.DFinsupp | {
"line": 556,
"column": 2
} | {
"line": 556,
"column": 41
} | {
"line": 557,
"column": 2
} | [
{
"pp": "ι : Type u_1\nR : Type u_3\nN : Type u_6\ninst✝³ : DecidableEq ι\ninst✝² : Ring R\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\np : ι → Submodule R N\nh : ∀ (i : ι) (x : ↥(p i)) (v : Π₀ (i : ι), ↥(p i)), ((lsum ℕ) fun i ↦ (p i).subtype) (erase i v) = ↑x → x = 0\nm : Π₀ (i : ι), ↥(p i)\nhm : ((lsum ℕ) f... | [
"ι : Type u_1\nR : Type u_3\nN : Type u_6\ninst✝³ : DecidableEq ι\ninst✝² : Ring R\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\np : ι → Submodule R N\nh : ∀ (i : ι) (x : ↥(p i)) (v : Π₀ (i : ι), ↥(p i)), ((lsum ℕ) fun i ↦ (p i).subtype) (erase i v) = ↑x → x = 0\nm : Π₀ (i : ι), ↥(p i)\nhm : ((lsum ℕ) fun i ↦ (p i)... | rw [DFinsupp.zero_apply, ← neg_eq_zero] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.LinearIndependent.Lemmas | {
"line": 334,
"column": 2
} | {
"line": 345,
"column": 10
} | {
"line": 346,
"column": 2
} | [
{
"pp": "case inl\nR : Type u_2\nM : Type u_4\ninst✝¹⁰ : Ring R\ninst✝⁹ : AddCommGroup M\ninst✝⁸ : Module R M\nx y : M\nS : Type u_6\ninst✝⁷ : CommRing S\ninst✝⁶ : IsDomain S\ninst✝⁵ : Module S R\ninst✝⁴ : Module S M\ninst✝³ : SMulCommClass S R M\ninst✝² : IsScalarTower S R M\ninst✝¹ : IsTorsionFree S R\na b c ... | [
"case inr\nR : Type u_2\nM : Type u_4\ninst✝¹⁰ : Ring R\ninst✝⁹ : AddCommGroup M\ninst✝⁸ : Module R M\nx y : M\nS : Type u_6\ninst✝⁷ : CommRing S\ninst✝⁶ : IsDomain S\ninst✝⁵ : Module S R\ninst✝⁴ : Module S M\ninst✝³ : SMulCommClass S R M\ninst✝² : IsScalarTower S R M\ninst✝¹ : IsTorsionFree S R\na b c d : S\ninst✝... | · suffices ¬ LinearIndependent R ![a • x + b • y, c • x + d • y] by simpa [h]
rw [pair_iff]
push Not
by_cases hbd : b = 0 ∧ d = 0
· simp only [hbd.1, hbd.2, zero_smul, add_zero]
by_cases hac : a = 0 ∧ c = 0; · exact ⟨1, 0, by simp [hac.1, hac.2], by simp⟩
refine ⟨c • 1, -a • 1, ?_, by aesop⟩... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.MonoidAlgebra.Module | {
"line": 226,
"column": 19
} | {
"line": 226,
"column": 87
} | {
"line": 228,
"column": 0
} | [
{
"pp": "k : Type u₁\nG : Type u₂\nH : Type u_1\nR : Type u_2\nS : Type u_3\nM : Type u_4\ninst✝³ : Semiring k\ninst✝² : DistribSMul R k\ninst✝¹ : Mul G\ninst✝ : IsScalarTower R k k\nt : R\na b : k[G]\nm✝ : G\n⊢ ((t • a) • b) m✝ = (t • a • b) m✝",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants"... | [] | simp [mul_apply, sum_smul_index' (b := t), smul_sum, smul_mul_assoc] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.LinearAlgebra.LinearIndependent.Lemmas | {
"line": 695,
"column": 2
} | {
"line": 695,
"column": 10
} | {
"line": 696,
"column": 2
} | [
{
"pp": "R : Type u_6\nK : Type u_7\nM : Type u_8\ninst✝⁷ : CommRing R\ninst✝⁶ : DivisionRing K\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Algebra R K\ninst✝³ : Module K M\ninst✝² : Module R M\ninst✝¹ : IsScalarTower R K M\ninst✝ : FaithfulSMul R K\nn : ℕ\nv : Fin n → M\nhv : LinearIndependent R v\nx : M\nhx : x ∉ span... | [
"R : Type u_6\nK : Type u_7\nM : Type u_8\ninst✝⁷ : CommRing R\ninst✝⁶ : DivisionRing K\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Algebra R K\ninst✝³ : Module K M\ninst✝² : Module R M\ninst✝¹ : IsScalarTower R K M\ninst✝ : FaithfulSMul R K\nn : ℕ\nv : Fin n → M\nhv : LinearIndependent R v\nx : M\nhx : x ∉ span K (range v)... | apply hx | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Data.Finset.NAry | {
"line": 263,
"column": 4
} | {
"line": 263,
"column": 31
} | {
"line": 265,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_3\nγ : Type u_5\nδ : Type u_7\ninst✝¹ : DecidableEq γ\ns : Finset α\nt : Finset β\ninst✝ : DecidableEq δ\nf : γ → β → δ\ng : α → γ\n⊢ image2 f (g '' ↑s) ↑t = image2 (fun a b ↦ f (g a) b) ↑s ↑t",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Finset",... | [] | exact image2_image_left _ _ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Data.Finset.NAry | {
"line": 442,
"column": 2
} | {
"line": 443,
"column": 70
} | {
"line": 444,
"column": 2
} | [
{
"pp": "case insert\nα : Type u_1\nβ : Type u_3\nγ : Type u_5\ninst✝ : DecidableEq γ\nf : α → β → γ\ns✝ : Finset α\nt : Finset β\na : α\ns : Finset α\na✝ : a ∉ s\nih : (∀ a ∈ s, Injective (f a)) → ((fun a ↦ image (f a) t) '' ↑s).PairwiseDisjoint id → #t ∣ #(image₂ f s t)\nhf : ∀ a_1 ∈ insert a s, Injective (f ... | [
"case insert\nα : Type u_1\nβ : Type u_3\nγ : Type u_5\ninst✝ : DecidableEq γ\nf : α → β → γ\ns✝ : Finset α\nt : Finset β\na : α\ns : Finset α\na✝ : a ∉ s\nih : #t ∣ #(image₂ f s t)\nhf : ∀ a_1 ∈ insert a s, Injective (f a_1)\nhs : ((fun a ↦ image (f a) t) '' ↑(insert a s)).PairwiseDisjoint id\n⊢ #t ∣ #(image₂ f (i... | specialize ih (forall_of_forall_insert hf)
(hs.subset <| Set.image_mono <| coe_subset.2 <| subset_insert _ _) | Lean.Elab.Tactic.evalSpecialize | Lean.Parser.Tactic.specialize |
Mathlib.Algebra.Group.UniqueProds.Basic | {
"line": 84,
"column": 43
} | {
"line": 87,
"column": 68
} | {
"line": 89,
"column": 0
} | [
{
"pp": "G : Type u_1\ninst✝ : Mul G\nA B : Finset G\nhA : A.Nonempty\nhB : B.Nonempty\nhA1 : #A ≤ 1\nhB1 : #B ≤ 1\n⊢ ∃ a ∈ A, ∃ b ∈ B, UniqueMul A B a b",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"UniqueMul",
"HMul.hMul",
"congrArg",
"Finset",
"Membershi... | [] | by
rw [Finset.card_le_one_iff] at hA1 hB1
obtain ⟨a, ha⟩ := hA; obtain ⟨b, hb⟩ := hB
exact ⟨a, ha, b, hb, fun _ _ ha' hb' _ ↦ ⟨hA1 ha' ha, hB1 hb' hb⟩⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Group.UniqueProds.Basic | {
"line": 215,
"column": 2
} | {
"line": 215,
"column": 47
} | {
"line": 217,
"column": 0
} | [
{
"pp": "case refine_1\nG : Type u_1\nH : Type u_2\ninst✝² : Mul G\ninst✝¹ : Mul H\ninst✝ : DecidableEq H\nf : G →ₙ* H\nA B : Finset G\naG bG : G\naH bH : H\nhae : f aG = aH\nhbe : f bG = bH\nhuG : UniqueMul ({a ∈ A | f a = aH}) ({b ∈ B | f b = bH}) aG bG\na b : G\nhuH : aH * bH = aH * bH → f a = aH ∧ f b = bH\... | [] | exacts [⟨ha, (huH rfl).1⟩, ⟨hb, (huH rfl).2⟩] | Batteries.Tactic._aux_Batteries_Tactic_Init___elabRules_Batteries_Tactic_exacts_1 | Batteries.Tactic.exacts |
Mathlib.Algebra.Group.Pointwise.Finset.Basic | {
"line": 1184,
"column": 2
} | {
"line": 1184,
"column": 70
} | {
"line": 1186,
"column": 0
} | [
{
"pp": "α : Type u_2\ninst✝² : Mul α\ninst✝¹ : IsRightCancelMul α\ninst✝ : DecidableEq α\ns t : Finset α\na : α\nha : a ∈ ↑s\nb : α\nhb : b ∈ ↑s\nhab : a ≠ b\nc : α\nhc : c ∈ t\n⊢ (s * t).Nontrivial",
"ppTerm": "?m.47",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"... | [] | exact ⟨a * c, mul_mem_mul ha hc, b * c, mul_mem_mul hb hc, by simpa⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.Group.UniqueProds.Basic | {
"line": 443,
"column": 4
} | {
"line": 443,
"column": 32
} | {
"line": 443,
"column": 32
} | [
{
"pp": "case neg\nG✝ : Type u\nH : Type v\ninst✝³ : Mul G✝\ninst✝² : Mul H\nι : Type u_2\nG : ι → Type u_1\ninst✝¹ : (i : ι) → Mul (G i)\ninst✝ : ∀ (i : ι), UniqueProds (G i)\nA✝ : Finset ((i : ι) → G i)\nx✝ : IsWellFounded (Finset ((i : ι) → G i)) fun x1 x2 ↦ x1 ⊂ x2 := isWellFounded_ssubset\nA : Finset ((i :... | [
"case neg\nG✝ : Type u\nH : Type v\ninst✝³ : Mul G✝\ninst✝² : Mul H\nι : Type u_2\nG : ι → Type u_1\ninst✝¹ : (i : ι) → Mul (G i)\ninst✝ : ∀ (i : ι), UniqueProds (G i)\nA✝ : Finset ((i : ι) → G i)\nx✝ : IsWellFounded (Finset ((i : ι) → G i)) fun x1 x2 ↦ x1 ⊂ x2 := isWellFounded_ssubset\nA : Finset ((i : ι) → G i)\n... | let A' := {a ∈ A | a i = ai} | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.Data.Finset.Sort | {
"line": 166,
"column": 2
} | {
"line": 168,
"column": 26
} | {
"line": 169,
"column": 2
} | [
{
"pp": "case a\nα : Type u_1\ninst✝ : LinearOrder α\ns : Finset α\nh : (s.sort fun a b ↦ a ≤ b).length - 1 < (s.sort fun a b ↦ a ≤ b).length\nH : s.Nonempty\nl : List α := ⋯\n⊢ (s.sort fun a b ↦ a ≤ b)[(s.sort fun a b ↦ a ≤ b).length - 1] ≤ s.max' H",
"ppTerm": "?a✝",
"assigned": true,
"usedConstan... | [
"case a\nα : Type u_1\ninst✝ : LinearOrder α\ns : Finset α\nh : (s.sort fun a b ↦ a ≤ b).length - 1 < (s.sort fun a b ↦ a ≤ b).length\nH : s.Nonempty\nl : List α := ⋯\n⊢ s.max' H ≤ (s.sort fun a b ↦ a ≤ b)[(s.sort fun a b ↦ a ≤ b).length - 1]"
] | · have : l.get ⟨s.sort.length - 1, h⟩ ∈ s :=
(s.mem_sort (· ≤ ·)).1 (List.get_mem l _)
exact s.le_max' _ this | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Group.UniqueProds.Basic | {
"line": 487,
"column": 6
} | {
"line": 487,
"column": 49
} | {
"line": 488,
"column": 6
} | [
{
"pp": "case inl\nG : Type u\nH : Type v\ninst✝² : Mul G\ninst✝¹ : Mul H\nf : H →ₙ* G\nhf : ∀ ⦃a b c d : H⦄, a * b = c * d → f a = f c ∧ f b = f d → a = c ∧ b = d\ninst✝ : TwoUniqueProds G\nA B : Finset H\nhc : 1 < #A * #B\nhc' : 1 < #(image (⇑f) A) * #(image (⇑f) B)\na2 b2 : G\nhu2 : UniqueMul (image (⇑f) A) ... | [
"case inl\nG : Type u\nH : Type v\ninst✝² : Mul G\ninst✝¹ : Mul H\nf : H →ₙ* G\nhf : ∀ ⦃a b c d : H⦄, a * b = c * d → f a = f c ∧ f b = f d → a = c ∧ b = d\ninst✝ : TwoUniqueProds G\nA B : Finset H\nhc : 1 < #A * #B\nhc' : 1 < #(image (⇑f) A) * #(image (⇑f) B)\na1 : H\nha1 : a1 ∈ A\nb1 : H\nhb1 : b1 ∈ B\nhu1 : Uniq... | obtain ⟨⟨a2, ha2, rfl⟩, b2, hb2, rfl⟩ := h2 | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Algebra.Polynomial.Basic | {
"line": 429,
"column": 6
} | {
"line": 429,
"column": 22
} | {
"line": 429,
"column": 23
} | [
{
"pp": "R : Type u\ninst✝ : Semiring R\nm n : ℕ\na b : R\n⊢ (monomial m) a = (monomial n) b ↔ m = n ∧ a = b ∨ a = 0 ∧ b = 0",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Semiring.toModule",
"Polynomial.toFinsupp",
"congrArg",
"Polynomial.toFinsup... | [
"R : Type u\ninst✝ : Semiring R\nm n : ℕ\na b : R\n⊢ ((monomial m) a).toFinsupp = ((monomial n) b).toFinsupp ↔ m = n ∧ a = b ∨ a = 0 ∧ b = 0"
] | ← toFinsupp_inj, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.Basic | {
"line": 532,
"column": 2
} | {
"line": 532,
"column": 40
} | {
"line": 534,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝ : Semiring R\np q : R[X]\nn : ℕ\n⊢ p * X ^ n * q = p * q * X ^ n",
"ppTerm": "?m.36",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Semigroup.toMul",
"HMul.hMul",
"congrArg",
"mul_assoc",
"SemigroupWithZero.toSemigroup",
"id",
... | [] | rw [mul_assoc, X_pow_mul, ← mul_assoc] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Polynomial.Basic | {
"line": 532,
"column": 2
} | {
"line": 532,
"column": 40
} | {
"line": 534,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝ : Semiring R\np q : R[X]\nn : ℕ\n⊢ p * X ^ n * q = p * q * X ^ n",
"ppTerm": "?m.36",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Semigroup.toMul",
"HMul.hMul",
"congrArg",
"mul_assoc",
"SemigroupWithZero.toSemigroup",
"id",
... | [] | rw [mul_assoc, X_pow_mul, ← mul_assoc] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Polynomial.Basic | {
"line": 532,
"column": 2
} | {
"line": 532,
"column": 40
} | {
"line": 534,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝ : Semiring R\np q : R[X]\nn : ℕ\n⊢ p * X ^ n * q = p * q * X ^ n",
"ppTerm": "?m.36",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Semigroup.toMul",
"HMul.hMul",
"congrArg",
"mul_assoc",
"SemigroupWithZero.toSemigroup",
"id",
... | [] | rw [mul_assoc, X_pow_mul, ← mul_assoc] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Polynomial.Basic | {
"line": 846,
"column": 36
} | {
"line": 846,
"column": 52
} | {
"line": 846,
"column": 53
} | [
{
"pp": "R : Type u\ninst✝ : Semiring R\nk l m n : ℕ\nu v : R\nhu : u ≠ 0\nhv : v ≠ 0\n⊢ (monomial k) u + (monomial l) v = (monomial m) u + (monomial n) v ↔\n k = m ∧ l = n ∨ u = v ∧ k = n ∧ l = m ∨ u + v = 0 ∧ k = l ∧ m = n",
"ppTerm": "?m.93",
"assigned": true,
"usedConstants": [
"Eq.mpr"... | [
"R : Type u\ninst✝ : Semiring R\nk l m n : ℕ\nu v : R\nhu : u ≠ 0\nhv : v ≠ 0\n⊢ ((monomial k) u + (monomial l) v).toFinsupp = ((monomial m) u + (monomial n) v).toFinsupp ↔\n k = m ∧ l = n ∨ u = v ∧ k = n ∧ l = m ∨ u + v = 0 ∧ k = l ∧ m = n"
] | ← toFinsupp_inj, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Algebra.Polynomial.Basic | {
"line": 1024,
"column": 85
} | {
"line": 1025,
"column": 39
} | {
"line": 1027,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝ : Semiring R\np : R[X]\nn : ℕ\na : R\n⊢ (p.update n a).coeff n = a",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Polynomial.update",
"Polynomial.coeff_update_apply",
"id",
"if_pos",
"Polynomial.coe... | [] | by
rw [p.coeff_update_apply, if_pos rfl] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Polynomial.Basic | {
"line": 1042,
"column": 95
} | {
"line": 1043,
"column": 42
} | {
"line": 1045,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝ : Semiring R\np : R[X]\nn : ℕ\n⊢ (p.update n 0).support = p.support.erase n",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Polynomial.update_zero_eq_erase",
"congrArg",
"Finset",
"Polynomial.update",
"id",
"Po... | [] | by
rw [update_zero_eq_erase, support_erase] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Polynomial.Basic | {
"line": 1046,
"column": 64
} | {
"line": 1046,
"column": 94
} | {
"line": 1048,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝ : Semiring R\np : R[X]\nn : ℕ\na : R\nha : a ≠ 0\n⊢ (p.update n a).support = insert n p.support",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Finset",
"Polynomial.update",
"Classical.propDecidable",
"id... | [] | rw [support_update, if_neg ha] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Polynomial.Basic | {
"line": 1046,
"column": 64
} | {
"line": 1046,
"column": 94
} | {
"line": 1048,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝ : Semiring R\np : R[X]\nn : ℕ\na : R\nha : a ≠ 0\n⊢ (p.update n a).support = insert n p.support",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Finset",
"Polynomial.update",
"Classical.propDecidable",
"id... | [] | rw [support_update, if_neg ha] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Polynomial.Basic | {
"line": 1046,
"column": 64
} | {
"line": 1046,
"column": 94
} | {
"line": 1048,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝ : Semiring R\np : R[X]\nn : ℕ\na : R\nha : a ≠ 0\n⊢ (p.update n a).support = insert n p.support",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Finset",
"Polynomial.update",
"Classical.propDecidable",
"id... | [] | rw [support_update, if_neg ha] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Exact.Basic | {
"line": 162,
"column": 12
} | {
"line": 162,
"column": 14
} | {
"line": 163,
"column": 4
} | [
{
"pp": "case mp\nM₁ : Type u_8\nM₂ : Type u_9\nM₃ : Type u_10\nN₁ : Type u_11\nN₂ : Type u_12\nN₃ : Type u_13\ninst✝⁵ : CommMonoid M₁\ninst✝⁴ : CommMonoid M₂\ninst✝³ : CommMonoid M₃\ninst✝² : CommMonoid N₁\ninst✝¹ : CommMonoid N₂\ninst✝ : CommMonoid N₃\nf : M₁ →* M₂\ng : M₂ →* M₃\nf' : N₁ →* N₂\ng' : N₂ →* N₃\... | [
"case mp\nM₁ : Type u_8\nM₂ : Type u_9\nM₃ : Type u_10\nN₁ : Type u_11\nN₂ : Type u_12\nN₃ : Type u_13\ninst✝⁵ : CommMonoid M₁\ninst✝⁴ : CommMonoid M₂\ninst✝³ : CommMonoid M₃\ninst✝² : CommMonoid N₁\ninst✝¹ : CommMonoid N₂\ninst✝ : CommMonoid N₃\nf : M₁ →* M₂\ng : M₂ →* M₃\nf' : N₁ →* N₂\ng' : N₂ →* N₃\nτ₁ : M₁ →* ... | y₂ | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Algebra.Module.Submodule.Bilinear | {
"line": 64,
"column": 2
} | {
"line": 74,
"column": 82
} | {
"line": 75,
"column": 2
} | [
{
"pp": "case a\nR : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\ninst✝⁶ : CommSemiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : AddCommMonoid N\ninst✝³ : AddCommMonoid P\ninst✝² : Module R M\ninst✝¹ : Module R N\ninst✝ : Module R P\nf : M →ₗ[R] N →ₗ[R] P\ns : Set M\nt : Set N\n⊢ map₂ f (span R s) (span R t... | [
"case a\nR : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\ninst✝⁶ : CommSemiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : AddCommMonoid N\ninst✝³ : AddCommMonoid P\ninst✝² : Module R M\ninst✝¹ : Module R N\ninst✝ : Module R P\nf : M →ₗ[R] N →ₗ[R] P\ns : Set M\nt : Set N\n⊢ span R (image2 (fun m n ↦ (f m) n) s t)... | · rw [map₂_le]
apply @span_induction R M _ _ _ s
on_goal 1 =>
intro a ha
apply @span_induction R N _ _ _ t
· intro b hb
exact subset_span ⟨_, ‹_›, _, ‹_›, rfl⟩
all_goals
intros
simp only [*, add_mem, smul_mem, zero_mem, map_zero, map_add,
LinearMap.zero_apply, L... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Exact.Basic | {
"line": 569,
"column": 6
} | {
"line": 569,
"column": 29
} | {
"line": 569,
"column": 30
} | [
{
"pp": "R : Type u_1\nM : Type u_2\nN : Type u_4\nP : Type u_6\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup N\ninst✝³ : AddCommGroup P\ninst✝² : Module R M\ninst✝¹ : Module R N\ninst✝ : Module R P\nf : M →ₗ[R] N\ng : N →ₗ[R] P\nh : Function.Exact ⇑f ⇑g\n⊢ Function.Surjective ⇑f ↔ g = 0",
... | [
"R : Type u_1\nM : Type u_2\nN : Type u_4\nP : Type u_6\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup N\ninst✝³ : AddCommGroup P\ninst✝² : Module R M\ninst✝¹ : Module R N\ninst✝ : Module R P\nf : M →ₗ[R] N\ng : N →ₗ[R] P\nh : Function.Exact ⇑f ⇑g\n⊢ Function.Surjective ⇑f ↔ g.ker = ⊤"
] | ← LinearMap.ker_eq_top, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.TensorProduct.Basic | {
"line": 153,
"column": 6
} | {
"line": 153,
"column": 22
} | {
"line": 153,
"column": 23
} | [
{
"pp": "R : Type u_1\nR₂ : Type u_2\ninst✝⁷ : CommSemiring R\ninst✝⁶ : CommSemiring R₂\nσ₁₂ : R →+* R₂\nM : Type u_7\nN : Type u_8\nP₂ : Type u_17\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : AddCommMonoid N\ninst✝³ : AddCommMonoid P₂\ninst✝² : Module R M\ninst✝¹ : Module R N\ninst✝ : Module R₂ P₂\ng : M ⊗[R] N →ₛₗ[σ₁₂... | [
"R : Type u_1\nR₂ : Type u_2\ninst✝⁷ : CommSemiring R\ninst✝⁶ : CommSemiring R₂\nσ₁₂ : R →+* R₂\nM : Type u_7\nN : Type u_8\nP₂ : Type u_17\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : AddCommMonoid N\ninst✝³ : AddCommMonoid P₂\ninst✝² : Module R M\ninst✝¹ : Module R N\ninst✝ : Module R₂ P₂\ng : M ⊗[R] N →ₛₗ[σ₁₂] P₂\n⊢ g ∘ₛ... | lift_compr₂ₛₗ g, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.TensorProduct.Defs | {
"line": 445,
"column": 2
} | {
"line": 445,
"column": 15
} | {
"line": 445,
"column": 16
} | [
{
"pp": "case tmul\nR : Type u_1\ninst✝⁴ : CommSemiring R\nM : Type u_7\nN : Type u_8\ninst✝³ : AddCommMonoid M\ninst✝² : AddCommMonoid N\ninst✝¹ : Module R M\ninst✝ : Module R N\nh : ∀ (m₁ m₂ : M) (n₁ n₂ : N), ∃ m n, m₁ ⊗ₜ[R] n₁ + m₂ ⊗ₜ[R] n₂ = m ⊗ₜ[R] n\nm : M\nn : N\n⊢ ∃ m_1 n_1, m ⊗ₜ[R] n = m_1 ⊗ₜ[R] n_1",
... | [] | | tmul m n => | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.LinearAlgebra.TensorProduct.Map | {
"line": 401,
"column": 4
} | {
"line": 401,
"column": 75
} | {
"line": 402,
"column": 2
} | [
{
"pp": "R : Type u_1\nR₂ : Type u_2\nR₃ : Type u_3\nR' : Type u_4\nR'' : Type u_5\ninst✝³⁶ : CommSemiring R\ninst✝³⁵ : CommSemiring R₂\ninst✝³⁴ : CommSemiring R₃\ninst✝³³ : Monoid R'\ninst✝³² : Semiring R''\nσ₁₂ : R →+* R₂\nσ₂₃ : R₂ →+* R₃\nσ₁₃ : R →+* R₃\nA : Type u_6\nM : Type u_7\nN : Type u_8\nP : Type u_9... | [] | simp only [compr₂ₛₗ_apply, mk_apply, add_apply, lTensor_tmul, tmul_add] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.Coprime.Basic | {
"line": 141,
"column": 2
} | {
"line": 141,
"column": 27
} | {
"line": 143,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝ : CommSemiring R\nx y z : R\nH : IsCoprime x (z * y)\n⊢ IsCoprime x z",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"IsCoprime.of_mul_right_left"
],
"usedFVars": [
"R",
"inst✝",
"x",
"z",
"y",
"H"
],
"us... | [] | exact H.of_mul_right_left | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.Coprime.Basic | {
"line": 199,
"column": 2
} | {
"line": 199,
"column": 20
} | {
"line": 200,
"column": 2
} | [
{
"pp": "R : Type u\ninst✝ : CommSemiring R\nx y z : R\nh : IsCoprime (y * z + x) y\n⊢ IsCoprime x y",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"HMul.hMul",
"congrArg",
"CommSemiring.toSemiring",
"Eq.mp",
"add_comm",
"instDistribOfSemiring",
"... | [
"R : Type u\ninst✝ : CommSemiring R\nx y z : R\nh : IsCoprime (x + y * z) y\n⊢ IsCoprime x y"
] | rw [add_comm] at h | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Coprime.Basic | {
"line": 203,
"column": 2
} | {
"line": 203,
"column": 20
} | {
"line": 204,
"column": 2
} | [
{
"pp": "R : Type u\ninst✝ : CommSemiring R\nx y z : R\nh : IsCoprime (z * y + x) y\n⊢ IsCoprime x y",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"HMul.hMul",
"congrArg",
"CommSemiring.toSemiring",
"Eq.mp",
"add_comm",
"instDistribOfSemiring",
"... | [
"R : Type u\ninst✝ : CommSemiring R\nx y z : R\nh : IsCoprime (x + z * y) y\n⊢ IsCoprime x y"
] | rw [add_comm] at h | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Coprime.Basic | {
"line": 207,
"column": 2
} | {
"line": 207,
"column": 20
} | {
"line": 208,
"column": 2
} | [
{
"pp": "R : Type u\ninst✝ : CommSemiring R\nx y z : R\nh : IsCoprime x (x * z + y)\n⊢ IsCoprime x y",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"HMul.hMul",
"congrArg",
"CommSemiring.toSemiring",
"Eq.mp",
"add_comm",
"instDistribOfSemiring",
"... | [
"R : Type u\ninst✝ : CommSemiring R\nx y z : R\nh : IsCoprime x (y + x * z)\n⊢ IsCoprime x y"
] | rw [add_comm] at h | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Coprime.Basic | {
"line": 211,
"column": 2
} | {
"line": 211,
"column": 20
} | {
"line": 212,
"column": 2
} | [
{
"pp": "R : Type u\ninst✝ : CommSemiring R\nx y z : R\nh : IsCoprime x (z * x + y)\n⊢ IsCoprime x y",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"HMul.hMul",
"congrArg",
"CommSemiring.toSemiring",
"Eq.mp",
"add_comm",
"instDistribOfSemiring",
"... | [
"R : Type u\ninst✝ : CommSemiring R\nx y z : R\nh : IsCoprime x (y + z * x)\n⊢ IsCoprime x y"
] | rw [add_comm] at h | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Ideal.Prod | {
"line": 189,
"column": 4
} | {
"line": 189,
"column": 12
} | {
"line": 190,
"column": 4
} | [
{
"pp": "case mp\nR : Type u\nS : Type v\ninst✝¹ : Semiring R\ninst✝ : Semiring S\nI : Ideal (R × S)\n⊢ ((map (RingHom.fst R S) I).prod (map (RingHom.snd R S) I)).IsPrime →\n (∃ p, p.IsPrime ∧ (map (RingHom.fst R S) I).prod (map (RingHom.snd R S) I) = p.prod ⊤) ∨\n ∃ p, p.IsPrime ∧ (map (RingHom.fst R S... | [
"case mp\nR : Type u\nS : Type v\ninst✝¹ : Semiring R\ninst✝ : Semiring S\nI : Ideal (R × S)\nhI : ((map (RingHom.fst R S) I).prod (map (RingHom.snd R S) I)).IsPrime\n⊢ (∃ p, p.IsPrime ∧ (map (RingHom.fst R S) I).prod (map (RingHom.snd R S) I) = p.prod ⊤) ∨\n ∃ p, p.IsPrime ∧ (map (RingHom.fst R S) I).prod (map ... | intro hI | Lean.Elab.Tactic.evalIntro | null |
Mathlib.RingTheory.Ideal.Prod | {
"line": 189,
"column": 4
} | {
"line": 189,
"column": 12
} | {
"line": 190,
"column": 4
} | [
{
"pp": "case mp\nR : Type u\nS : Type v\ninst✝¹ : Semiring R\ninst✝ : Semiring S\nI : Ideal (R × S)\n⊢ ((map (RingHom.fst R S) I).prod (map (RingHom.snd R S) I)).IsPrime →\n (∃ p, p.IsPrime ∧ (map (RingHom.fst R S) I).prod (map (RingHom.snd R S) I) = p.prod ⊤) ∨\n ∃ p, p.IsPrime ∧ (map (RingHom.fst R S... | [
"case mp\nR : Type u\nS : Type v\ninst✝¹ : Semiring R\ninst✝ : Semiring S\nI : Ideal (R × S)\nhI : ((map (RingHom.fst R S) I).prod (map (RingHom.snd R S) I)).IsPrime\n⊢ (∃ p, p.IsPrime ∧ (map (RingHom.fst R S) I).prod (map (RingHom.snd R S) I) = p.prod ⊤) ∨\n ∃ p, p.IsPrime ∧ (map (RingHom.fst R S) I).prod (map ... | intro hI | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.RingTheory.Ideal.Operations | {
"line": 114,
"column": 2
} | {
"line": 116,
"column": 28
} | {
"line": 118,
"column": 0
} | [
{
"pp": "R : Type u\nM : Type v\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nI : Ideal R\nN : Submodule R M\nx : ↥N\n⊢ x ∈ I • ⊤ ↔ ↑x ∈ I • N",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Submo... | [] | have : Submodule.map N.subtype (I • ⊤) = I • N := by
rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype]
simp [← this, -map_smul''] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Ideal.Operations | {
"line": 114,
"column": 2
} | {
"line": 116,
"column": 28
} | {
"line": 118,
"column": 0
} | [
{
"pp": "R : Type u\nM : Type v\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nI : Ideal R\nN : Submodule R M\nx : ↥N\n⊢ x ∈ I • ⊤ ↔ ↑x ∈ I • N",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Submo... | [] | have : Submodule.map N.subtype (I • ⊤) = I • N := by
rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype]
simp [← this, -map_smul''] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Ideal.Maps | {
"line": 638,
"column": 10
} | {
"line": 638,
"column": 22
} | {
"line": 638,
"column": 22
} | [
{
"pp": "S : Type v\nF : Type u_1\ninst✝³ : CommSemiring S\nR : Type u_2\ninst✝² : Semiring R\ninst✝¹ : FunLike F R S\ninst✝ : RingHomClass F R S\nf : F\nI J : Ideal R\nr : R\nhri : r ∈ I\ns : R\nhsj : s ∈ J\n⊢ f (r * s) ∈ map f I * map f J",
"ppTerm": "?m.93",
"assigned": true,
"usedConstants": [
... | [
"S : Type v\nF : Type u_1\ninst✝³ : CommSemiring S\nR : Type u_2\ninst✝² : Semiring R\ninst✝¹ : FunLike F R S\ninst✝ : RingHomClass F R S\nf : F\nI J : Ideal R\nr : R\nhri : r ∈ I\ns : R\nhsj : s ∈ J\n⊢ f r * f s ∈ map f I * map f J"
] | rw [map_mul] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Ideal.Maps | {
"line": 1071,
"column": 4
} | {
"line": 1071,
"column": 73
} | {
"line": 1072,
"column": 2
} | [
{
"pp": "case refine_1\nR : Type u_1\nS : Type u_2\nF : Type u_3\ninst✝² : Ring R\ninst✝¹ : Ring S\ninst✝ : FunLike F R S\nrc : RingHomClass F R S\nA : Set (Ideal R)\nf : F\nhf : Function.Surjective ⇑f\nh : ∀ J ∈ A, RingHom.ker f ≤ J\nj : Ideal S\nhj : j ∈ map f '' A\ny : S\nhy : y ∈ map f (sInf A)\nx : R\nhx :... | [] | exact mem_map_of_mem f (sInf_le_of_le hJ.left (le_of_eq rfl) hx.left) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.Ideal.Operations | {
"line": 541,
"column": 2
} | {
"line": 541,
"column": 60
} | {
"line": 543,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝ : Semiring R\nI : Ideal R\nn m : ℕ\nhnI : I ^ n = 0\nhmn : n ≤ m\nx : R\nhx : x ∈ I\n⊢ x ^ m = 0",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Submodule",
"Semiring.toModule",
"congrArg",
"AddMonoid.toAddZeroClass",
"Submodule.mem... | [] | simpa [hnI] using pow_le_pow_right hmn <| pow_mem_pow hx m | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.RingTheory.Ideal.Operations | {
"line": 541,
"column": 2
} | {
"line": 541,
"column": 60
} | {
"line": 543,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝ : Semiring R\nI : Ideal R\nn m : ℕ\nhnI : I ^ n = 0\nhmn : n ≤ m\nx : R\nhx : x ∈ I\n⊢ x ^ m = 0",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Submodule",
"Semiring.toModule",
"congrArg",
"AddMonoid.toAddZeroClass",
"Submodule.mem... | [] | simpa [hnI] using pow_le_pow_right hmn <| pow_mem_pow hx m | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Ideal.Operations | {
"line": 541,
"column": 2
} | {
"line": 541,
"column": 60
} | {
"line": 543,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝ : Semiring R\nI : Ideal R\nn m : ℕ\nhnI : I ^ n = 0\nhmn : n ≤ m\nx : R\nhx : x ∈ I\n⊢ x ^ m = 0",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Submodule",
"Semiring.toModule",
"congrArg",
"AddMonoid.toAddZeroClass",
"Submodule.mem... | [] | simpa [hnI] using pow_le_pow_right hmn <| pow_mem_pow hx m | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.GCDMonoid.Basic | {
"line": 603,
"column": 2
} | {
"line": 603,
"column": 34
} | {
"line": 605,
"column": 0
} | [
{
"pp": "case h\nα : Type u_1\ninst✝¹ : CommMonoidWithZero α\ninst✝ : GCDMonoid α\na b c : α\nhab : IsUnit (gcd b a)\nk : ℕ\nh✝ : Nontrivial α\nha : ¬a = 0\nhb : ¬b = 0\nhk : k > 0\nhc✝ : c ∣ a * b\nd₁ d₂ : α\nhd₁ : d₁ ∣ a\nhd₂ : d₂ ∣ b\nhc : c = d₂ * d₁\nh0₁ : d₁ ^ k ≠ 0\na' : α\nha' : a = d₁ ^ k * a'\nh0₂ : d... | [] | rw [Units.val_mkOfMulEqOne, ha'] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.GCDMonoid.Basic | {
"line": 951,
"column": 82
} | {
"line": 952,
"column": 60
} | {
"line": 954,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝¹ : CommRing α\ninst✝ : NormalizedGCDMonoid α\na b c : α\nh : a ∣ b - c\n⊢ gcd b a = gcd c a",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"CommSemiring.toCommMonoidWithZero",
"id",
"gcd_comm",
"GCDMono... | [] | by
rw [gcd_comm _ a, gcd_comm _ a, gcd_eq_of_dvd_sub_right h] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.GCDMonoid.Basic | {
"line": 979,
"column": 13
} | {
"line": 979,
"column": 40
} | {
"line": 979,
"column": 41
} | [
{
"pp": "α : Type u_1\ninst✝² : CommMonoidWithZero α\ninst✝¹ : IsCancelMulZero α\ninst✝ : DecidableEq α\nf : Associates α →* α\nhinv : Function.RightInverse (⇑f) Associates.mk\na b : α\nha : a ≠ 0\nhb : b ≠ 0\n⊢ (if a * b = 0 then 1 else Classical.choose ⋯) =\n (if a = 0 then 1 else Classical.choose ⋯) * if ... | [
"α : Type u_1\ninst✝² : CommMonoidWithZero α\ninst✝¹ : IsCancelMulZero α\ninst✝ : DecidableEq α\nf : Associates α →* α\nhinv : Function.RightInverse (⇑f) Associates.mk\na b : α\nha : a ≠ 0\nhb : b ≠ 0\n⊢ Classical.choose ⋯ = (if a = 0 then 1 else Classical.choose ⋯) * if b = 0 then 1 else Classical.choose ⋯"
] | if_neg (mul_ne_zero ha hb), | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.RingTheory.PrincipalIdealDomain | {
"line": 458,
"column": 2
} | {
"line": 458,
"column": 45
} | {
"line": 459,
"column": 2
} | [
{
"pp": "R : Type u\ninst✝³ : CommRing R\ninst✝² : IsBezout R\ninst✝¹ : IsDomain R\ninst✝ : GCDMonoid R\nx y : R\n⊢ Ideal.span {GCDMonoid.gcd x y} = Ideal.span {gcd x y}",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Ideal.span_singleton_eq_span_singleton",
"c... | [
"R : Type u\ninst✝³ : CommRing R\ninst✝² : IsBezout R\ninst✝¹ : IsDomain R\ninst✝ : GCDMonoid R\nx y : R\n⊢ Associated (GCDMonoid.gcd x y) (gcd x y)"
] | rw [Ideal.span_singleton_eq_span_singleton] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.PrincipalIdealDomain | {
"line": 457,
"column": 58
} | {
"line": 461,
"column": 81
} | {
"line": 463,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝³ : CommRing R\ninst✝² : IsBezout R\ninst✝¹ : IsDomain R\ninst✝ : GCDMonoid R\nx y : R\n⊢ Ideal.span {GCDMonoid.gcd x y} = Ideal.span {gcd x y}",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Ideal.span_singleton_eq_span_singleton",
"M... | [] | by
rw [Ideal.span_singleton_eq_span_singleton]
exact associated_of_dvd_dvd
(IsBezout.dvd_gcd (GCDMonoid.gcd_dvd_left _ _) <| GCDMonoid.gcd_dvd_right _ _)
(GCDMonoid.dvd_gcd (IsBezout.gcd_dvd_left _ _) <| IsBezout.gcd_dvd_right _ _) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.Dimension.StrongRankCondition | {
"line": 89,
"column": 4
} | {
"line": 89,
"column": 95
} | {
"line": 90,
"column": 4
} | [
{
"pp": "case inr\nR : Type u\nM : Type v\ninst✝³ : Semiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\nι : Type w\nι' : Type w'\ninst✝ : InvariantBasisNumber R\nv : Basis ι R M\nv' : Basis ι' R M\nthis : Nontrivial R\nval✝ : Infinite ι\n⊢ lift.{w', w} #ι = lift.{w, w'} #ι'",
"ppTerm": "?inr",
"a... | [
"case inr\nR : Type u\nM : Type v\ninst✝³ : Semiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\nι : Type w\nι' : Type w'\ninst✝ : InvariantBasisNumber R\nv : Basis ι R M\nv' : Basis ι' R M\nthis : Nontrivial R\nval✝ : Infinite ι\nw₁ : lift.{w', w} #ι ≤ lift.{w, w'} #ι'\n⊢ lift.{w', w} #ι = lift.{w, w'} #ι'"
... | have w₁ := infinite_basis_le_maximal_linearIndependent' v _ v'.linearIndependent v'.maximal | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Data.Matrix.Mul | {
"line": 88,
"column": 95
} | {
"line": 89,
"column": 19
} | {
"line": 91,
"column": 0
} | [
{
"pp": "α : Type v\ninst✝¹ : AddCommMonoid α\ninst✝ : Mul α\nv w : PUnit.{u_10 + 1} → α\n⊢ v ⬝ᵥ w = v PUnit.unit * w PUnit.unit",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"HMul.hMul",
"Finset.univ",
"dotProduct",
"congrArg",
"Finset",
"PUnit.instUn... | [] | by
simp [dotProduct] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.Matrix.Mul | {
"line": 128,
"column": 87
} | {
"line": 129,
"column": 19
} | {
"line": 131,
"column": 0
} | [
{
"pp": "m : Type u_2\nn : Type u_3\nα : Type v\ninst✝² : Fintype m\ninst✝¹ : Fintype n\ninst✝ : NonUnitalNonAssocSemiring α\nu v : m → α\nx y : n → α\n⊢ Sum.elim u x ⬝ᵥ Sum.elim v y = u ⬝ᵥ v + x ⬝ᵥ y",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"instFintypeSum",
"HMul.hMul"... | [] | by
simp [dotProduct] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.Matrix.Mul | {
"line": 203,
"column": 66
} | {
"line": 204,
"column": 19
} | {
"line": 206,
"column": 0
} | [
{
"pp": "n : Type u_3\nα : Type v\ninst✝¹ : Fintype n\ninst✝ : NonAssocSemiring α\n⊢ 1 ⬝ᵥ 1 = ↑(Fintype.card n)",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"instHSMul",
"HMul.hMul",
"Finset.univ",
"dotProduct",
... | [] | by
simp [dotProduct] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.Matrix.Basis | {
"line": 235,
"column": 2
} | {
"line": 235,
"column": 29
} | {
"line": 236,
"column": 2
} | [
{
"pp": "case intro.intro\nm : Type u_2\nn : Type u_3\nα : Type u_7\nβ : Type u_8\ninst✝⁵ : DecidableEq m\ninst✝⁴ : DecidableEq n\ninst✝³ : Finite m\ninst✝² : Finite n\ninst✝¹ : AddCommMonoid α\ninst✝ : AddCommMonoid β\nf g : Matrix m n α →+ β\nh : ∀ (i : m) (j : n), f.comp (singleAddMonoidHom i j) = g.comp (si... | [
"case intro.intro\nm : Type u_2\nn : Type u_3\nα : Type u_7\nβ : Type u_8\ninst✝⁵ : DecidableEq m\ninst✝⁴ : DecidableEq n\ninst✝³ : Finite m\ninst✝² : Finite n\ninst✝¹ : AddCommMonoid α\ninst✝ : AddCommMonoid β\nf g : Matrix m n α →+ β\nh : ∀ (i : m) (j : n), f.comp (singleAddMonoidHom i j) = g.comp (singleAddMonoi... | rw [matrix_eq_sum_single x] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Data.Matrix.Basis | {
"line": 384,
"column": 2
} | {
"line": 387,
"column": 46
} | {
"line": 388,
"column": 2
} | [
{
"pp": "case inr.inl\nn : Type u_3\nα : Type u_7\ninst✝² : DecidableEq n\ninst✝¹ : Fintype n\ninst✝ : Semiring α\nM : Matrix n n α\nhM : Pairwise fun i j ↦ Commute (single i j 1) M\nh✝ : Nonempty n\ni j : n\n⊢ diagonal (fun x ↦ M i i) j j = M j j",
"ppTerm": "?inr.inl",
"assigned": true,
"usedConst... | [
"case inr.inr\nn : Type u_3\nα : Type u_7\ninst✝² : DecidableEq n\ninst✝¹ : Fintype n\ninst✝ : Semiring α\nM : Matrix n n α\nhM : Pairwise fun i j ↦ Commute (single i j 1) M\nh✝ : Nonempty n\ni j k : n\nhkl : j ≠ k\n⊢ diagonal (fun x ↦ M i i) j k = M j k"
] | · rw [diagonal_apply_eq]
obtain rfl | hij := Decidable.eq_or_ne i j
· rfl
· exact diag_eq_of_commute_single (hM hij) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Data.Set.UnionLift | {
"line": 94,
"column": 66
} | {
"line": 97,
"column": 40
} | {
"line": 99,
"column": 0
} | [
{
"pp": "α : Type u_1\nι : Sort u_3\nβ : Sort u_2\nS : ι → Set α\nf : (i : ι) → ↑(S i) → β\nhf : ∀ (i j : ι) (x : α) (hxi : x ∈ S i) (hxj : x ∈ S j), f i ⟨x, hxi⟩ = f j ⟨x, hxj⟩\nT : Set α\nhT : T ⊆ iUnion S\nc : ↑T\nci : (i : ι) → ↑(S i)\nhci : ∀ (i : ι), ↑(ci i) = ↑c\ncβ : β\nh : ∀ (i : ι), f i (ci i) = cβ\n⊢... | [] | by
let ⟨i, hi⟩ := Set.mem_iUnion.1 (hT c.prop)
have : ci i = ⟨c, hi⟩ := Subtype.ext (hci i)
rw [iUnionLift_of_mem _ hi, ← this, h] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Algebra.Subalgebra.Lattice | {
"line": 128,
"column": 2
} | {
"line": 133,
"column": 49
} | {
"line": 135,
"column": 0
} | [
{
"pp": "R : Type u\nA : Type v\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS T : Subalgebra R A\n⊢ (S ⊔ T).toSubsemiring = S.toSubsemiring ⊔ T.toSubsemiring",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"Subalgebra.instSetLike",
"Eq.mpr",
"NonAs... | [] | rw [← S.toSubsemiring.closure_eq, ← T.toSubsemiring.closure_eq, ← Subsemiring.closure_union]
simp_rw [sup_def, adjoin_toSubsemiring, Subalgebra.coe_toSubsemiring]
congr 1
rw [Set.union_eq_right]
rintro _ ⟨x, rfl⟩
exact Set.mem_union_left _ (algebraMap_mem S x) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Algebra.Subalgebra.Lattice | {
"line": 128,
"column": 2
} | {
"line": 133,
"column": 49
} | {
"line": 135,
"column": 0
} | [
{
"pp": "R : Type u\nA : Type v\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS T : Subalgebra R A\n⊢ (S ⊔ T).toSubsemiring = S.toSubsemiring ⊔ T.toSubsemiring",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"Subalgebra.instSetLike",
"Eq.mpr",
"NonAs... | [] | rw [← S.toSubsemiring.closure_eq, ← T.toSubsemiring.closure_eq, ← Subsemiring.closure_union]
simp_rw [sup_def, adjoin_toSubsemiring, Subalgebra.coe_toSubsemiring]
congr 1
rw [Set.union_eq_right]
rintro _ ⟨x, rfl⟩
exact Set.mem_union_left _ (algebraMap_mem S x) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.DirectSum.Module | {
"line": 222,
"column": 2
} | {
"line": 222,
"column": 27
} | {
"line": 224,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝³ : Semiring R\nι : Type v\nM : ι → Type w\ninst✝² : (i : ι) → AddCommMonoid (M i)\ninst✝¹ : (i : ι) → Module R (M i)\ninst✝ : DecidableEq ι\ni : ι\n⊢ component R ι M i ∘ₗ lof R ι M i = LinearMap.id",
"ppTerm": "?m.42",
"assigned": true,
"usedConstants": [
"dite_cond_... | [] | simp [component_comp_lof] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.DirectSum.Module | {
"line": 222,
"column": 2
} | {
"line": 222,
"column": 27
} | {
"line": 224,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝³ : Semiring R\nι : Type v\nM : ι → Type w\ninst✝² : (i : ι) → AddCommMonoid (M i)\ninst✝¹ : (i : ι) → Module R (M i)\ninst✝ : DecidableEq ι\ni : ι\n⊢ component R ι M i ∘ₗ lof R ι M i = LinearMap.id",
"ppTerm": "?m.42",
"assigned": true,
"usedConstants": [
"dite_cond_... | [] | simp [component_comp_lof] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.DirectSum.Module | {
"line": 222,
"column": 2
} | {
"line": 222,
"column": 27
} | {
"line": 224,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝³ : Semiring R\nι : Type v\nM : ι → Type w\ninst✝² : (i : ι) → AddCommMonoid (M i)\ninst✝¹ : (i : ι) → Module R (M i)\ninst✝ : DecidableEq ι\ni : ι\n⊢ component R ι M i ∘ₗ lof R ι M i = LinearMap.id",
"ppTerm": "?m.42",
"assigned": true,
"usedConstants": [
"dite_cond_... | [] | simp [component_comp_lof] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.DirectSum.Finsupp | {
"line": 113,
"column": 16
} | {
"line": 113,
"column": 52
} | {
"line": 114,
"column": 2
} | [
{
"pp": "case tmul\nR : Type u_1\nS : Type u_2\ninst✝⁹ : CommSemiring R\ninst✝⁸ : Semiring S\ninst✝⁷ : Algebra R S\nM : Type u_3\ninst✝⁶ : AddCommMonoid M\ninst✝⁵ : Module R M\ninst✝⁴ : Module S M\ninst✝³ : IsScalarTower R S M\nN : Type u_4\ninst✝² : AddCommMonoid N\ninst✝¹ : Module R N\nι : Type u_5\ninst✝ : D... | [] | simp [finsuppRight_apply_tmul_apply] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.LinearAlgebra.DirectSum.Finsupp | {
"line": 113,
"column": 16
} | {
"line": 113,
"column": 52
} | {
"line": 114,
"column": 2
} | [
{
"pp": "case tmul\nR : Type u_1\nS : Type u_2\ninst✝⁹ : CommSemiring R\ninst✝⁸ : Semiring S\ninst✝⁷ : Algebra R S\nM : Type u_3\ninst✝⁶ : AddCommMonoid M\ninst✝⁵ : Module R M\ninst✝⁴ : Module S M\ninst✝³ : IsScalarTower R S M\nN : Type u_4\ninst✝² : AddCommMonoid N\ninst✝¹ : Module R N\nι : Type u_5\ninst✝ : D... | [] | simp [finsuppRight_apply_tmul_apply] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.DirectSum.Finsupp | {
"line": 113,
"column": 16
} | {
"line": 113,
"column": 52
} | {
"line": 114,
"column": 2
} | [
{
"pp": "case tmul\nR : Type u_1\nS : Type u_2\ninst✝⁹ : CommSemiring R\ninst✝⁸ : Semiring S\ninst✝⁷ : Algebra R S\nM : Type u_3\ninst✝⁶ : AddCommMonoid M\ninst✝⁵ : Module R M\ninst✝⁴ : Module S M\ninst✝³ : IsScalarTower R S M\nN : Type u_4\ninst✝² : AddCommMonoid N\ninst✝¹ : Module R N\nι : Type u_5\ninst✝ : D... | [] | simp [finsuppRight_apply_tmul_apply] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Module.Rat | {
"line": 27,
"column": 4
} | {
"line": 27,
"column": 27
} | {
"line": 27,
"column": 28
} | [
{
"pp": "M : Type u_1\nM₂ : Type u_2\ninst✝⁷ : AddCommMonoid M\ninst✝⁶ : AddCommMonoid M₂\nF : Type u_3\ninst✝⁵ : FunLike F M M₂\ninst✝⁴ : AddMonoidHomClass F M M₂\nf : F\nR : Type u_4\nS : Type u_5\ninst✝³ : DivisionSemiring R\ninst✝² : DivisionSemiring S\ninst✝¹ : Module R M\ninst✝ : Module S M₂\nc : ℚ≥0\nx :... | [
"M : Type u_1\nM₂ : Type u_2\ninst✝⁷ : AddCommMonoid M\ninst✝⁶ : AddCommMonoid M₂\nF : Type u_3\ninst✝⁵ : FunLike F M M₂\ninst✝⁴ : AddMonoidHomClass F M M₂\nf : F\nR : Type u_4\nS : Type u_5\ninst✝³ : DivisionSemiring R\ninst✝² : DivisionSemiring S\ninst✝¹ : Module R M\ninst✝ : Module S M₂\nc : ℚ≥0\nx : M\n⊢ ↑c.num... | map_natCast_smul f R S, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.Matrix.Block | {
"line": 200,
"column": 2
} | {
"line": 201,
"column": 43
} | {
"line": 203,
"column": 0
} | [
{
"pp": "l : Type u_1\nm : Type u_2\nn : Type u_3\no : Type u_4\nR : Type u_10\ninst✝ : Neg R\nA : Matrix n l R\nB : Matrix n m R\nC : Matrix o l R\nD : Matrix o m R\n⊢ -fromBlocks A B C D = fromBlocks (-A) (-B) (-C) (-D)",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Matrix.fromBl... | [] | ext i j
cases i <;> cases j <;> simp [fromBlocks] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Matrix.Block | {
"line": 200,
"column": 2
} | {
"line": 201,
"column": 43
} | {
"line": 203,
"column": 0
} | [
{
"pp": "l : Type u_1\nm : Type u_2\nn : Type u_3\no : Type u_4\nR : Type u_10\ninst✝ : Neg R\nA : Matrix n l R\nB : Matrix n m R\nC : Matrix o l R\nD : Matrix o m R\n⊢ -fromBlocks A B C D = fromBlocks (-A) (-B) (-C) (-D)",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Matrix.fromBl... | [] | ext i j
cases i <;> cases j <;> simp [fromBlocks] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Matrix.RowCol | {
"line": 344,
"column": 2
} | {
"line": 345,
"column": 93
} | {
"line": 347,
"column": 0
} | [
{
"pp": "l : Type u_1\nm : Type u_2\nn : Type u_3\no : Type u_4\nα : Type v\ninst✝¹ : DecidableEq l\ninst✝ : DecidableEq m\nA : Matrix m n α\ni : l\nr : o → α\ne : l ≃ m\nf : o ≃ n\n⊢ (A.submatrix ⇑e ⇑f).updateRow i r = (A.updateRow (e i) fun j ↦ r (f.symm j)).submatrix ⇑e ⇑f",
"ppTerm": "?m.29",
"assig... | [] | ext i' j
simp only [submatrix_apply, updateRow_apply, Equiv.apply_eq_iff_eq, Equiv.symm_apply_apply] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Matrix.RowCol | {
"line": 344,
"column": 2
} | {
"line": 345,
"column": 93
} | {
"line": 347,
"column": 0
} | [
{
"pp": "l : Type u_1\nm : Type u_2\nn : Type u_3\no : Type u_4\nα : Type v\ninst✝¹ : DecidableEq l\ninst✝ : DecidableEq m\nA : Matrix m n α\ni : l\nr : o → α\ne : l ≃ m\nf : o ≃ n\n⊢ (A.submatrix ⇑e ⇑f).updateRow i r = (A.updateRow (e i) fun j ↦ r (f.symm j)).submatrix ⇑e ⇑f",
"ppTerm": "?m.29",
"assig... | [] | ext i' j
simp only [submatrix_apply, updateRow_apply, Equiv.apply_eq_iff_eq, Equiv.symm_apply_apply] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Matrix.RowCol | {
"line": 462,
"column": 30
} | {
"line": 462,
"column": 50
} | {
"line": 462,
"column": 51
} | [
{
"pp": "l : Type u_1\nm : Type u_2\nn : Type u_3\nα : Type v\ninst✝³ : DecidableEq l\ninst✝² : DecidableEq n\ninst✝¹ : Fintype m\ninst✝ : NonUnitalNonAssocSemiring α\ni : l\nr : m → α\nj : n\nc : m → α\n⊢ (updateCol 0 j 0).updateRow i (Function.update 0 j (r ⬝ᵥ c)) = of (Pi.single i (Pi.single j (r ⬝ᵥ c)))",
... | [
"l : Type u_1\nm : Type u_2\nn : Type u_3\nα : Type v\ninst✝³ : DecidableEq l\ninst✝² : DecidableEq n\ninst✝¹ : Fintype m\ninst✝ : NonUnitalNonAssocSemiring α\ni : l\nr : m → α\nj : n\nc : m → α\n⊢ updateRow 0 i (Function.update 0 j (r ⬝ᵥ c)) = of (Pi.single i (Pi.single j (r ⬝ᵥ c)))"
] | updateCol_zero_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.FreeModule.PID | {
"line": 137,
"column": 4
} | {
"line": 137,
"column": 38
} | {
"line": 139,
"column": 0
} | [
{
"pp": "R : Type u_2\ninst✝⁴ : CommRing R\nM : Type u_3\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : IsPrincipalIdealRing R\nN O : Submodule R M\nhNO : N ≤ O\nϕ : ↥O →ₗ[R] R\nhϕ : ∀ (ψ : ↥O →ₗ[R] R), ¬ϕ.submoduleImage N < ψ.submoduleImage N\ninst✝ : (ϕ.submoduleImage N).IsPrincipal\ny : M\nyN : y ∈ ... | [] | exact subset_span (mem_insert _ _) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.LinearAlgebra.FreeModule.PID | {
"line": 137,
"column": 4
} | {
"line": 137,
"column": 38
} | {
"line": 139,
"column": 0
} | [
{
"pp": "R : Type u_2\ninst✝⁴ : CommRing R\nM : Type u_3\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : IsPrincipalIdealRing R\nN O : Submodule R M\nhNO : N ≤ O\nϕ : ↥O →ₗ[R] R\nhϕ : ∀ (ψ : ↥O →ₗ[R] R), ¬ϕ.submoduleImage N < ψ.submoduleImage N\ninst✝ : (ϕ.submoduleImage N).IsPrincipal\ny : M\nyN : y ∈ ... | [] | exact subset_span (mem_insert _ _) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.FreeModule.PID | {
"line": 137,
"column": 4
} | {
"line": 137,
"column": 38
} | {
"line": 139,
"column": 0
} | [
{
"pp": "R : Type u_2\ninst✝⁴ : CommRing R\nM : Type u_3\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : IsPrincipalIdealRing R\nN O : Submodule R M\nhNO : N ≤ O\nϕ : ↥O →ₗ[R] R\nhϕ : ∀ (ψ : ↥O →ₗ[R] R), ¬ϕ.submoduleImage N < ψ.submoduleImage N\ninst✝ : (ϕ.submoduleImage N).IsPrincipal\ny : M\nyN : y ∈ ... | [] | exact subset_span (mem_insert _ _) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.AffineMonoid.Irreducible | {
"line": 89,
"column": 68
} | {
"line": 89,
"column": 95
} | {
"line": 89,
"column": 95
} | [
{
"pp": "M : Type u_1\ninst✝² : CancelCommMonoid M\ninst✝¹ : Subsingleton Mˣ\ninst✝ : Monoid.FG M\nS : Finset M\nhSgen : closure ↑S = ⊤\nhSmax : ∀ ⦃y : Finset M⦄, (fun S ↦ closure ↑S = ⊤) y → y ≤ S → S ≤ y\nr : M\nhrS : r ∈ ↑S\nhr₀ : r ≠ 1\nhrirred : ¬IsUnit r → ∃ x x_1, ∃ (_ : r = x * x_1), ¬IsUnit x ∧ ¬IsUnit... | [] | rw [hSgen]; exact mem_top _ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.AffineMonoid.Irreducible | {
"line": 89,
"column": 68
} | {
"line": 89,
"column": 95
} | {
"line": 89,
"column": 95
} | [
{
"pp": "M : Type u_1\ninst✝² : CancelCommMonoid M\ninst✝¹ : Subsingleton Mˣ\ninst✝ : Monoid.FG M\nS : Finset M\nhSgen : closure ↑S = ⊤\nhSmax : ∀ ⦃y : Finset M⦄, (fun S ↦ closure ↑S = ⊤) y → y ≤ S → S ≤ y\nr : M\nhrS : r ∈ ↑S\nhr₀ : r ≠ 1\nhrirred : ¬IsUnit r → ∃ x x_1, ∃ (_ : r = x * x_1), ¬IsUnit x ∧ ¬IsUnit... | [] | rw [hSgen]; exact mem_top _ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.AffineMonoid.Irreducible | {
"line": 90,
"column": 68
} | {
"line": 90,
"column": 95
} | {
"line": 90,
"column": 95
} | [
{
"pp": "M : Type u_1\ninst✝² : CancelCommMonoid M\ninst✝¹ : Subsingleton Mˣ\ninst✝ : Monoid.FG M\nS : Finset M\nhSgen : closure ↑S = ⊤\nhSmax : ∀ ⦃y : Finset M⦄, (fun S ↦ closure ↑S = ⊤) y → y ≤ S → S ≤ y\nr : M\nhrS : r ∈ ↑S\nhr₀ : r ≠ 1\nhrirred : ¬IsUnit r → ∃ x x_1, ∃ (_ : r = x * x_1), ¬IsUnit x ∧ ¬IsUnit... | [] | rw [hSgen]; exact mem_top _ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.AffineMonoid.Irreducible | {
"line": 90,
"column": 68
} | {
"line": 90,
"column": 95
} | {
"line": 90,
"column": 95
} | [
{
"pp": "M : Type u_1\ninst✝² : CancelCommMonoid M\ninst✝¹ : Subsingleton Mˣ\ninst✝ : Monoid.FG M\nS : Finset M\nhSgen : closure ↑S = ⊤\nhSmax : ∀ ⦃y : Finset M⦄, (fun S ↦ closure ↑S = ⊤) y → y ≤ S → S ≤ y\nr : M\nhrS : r ∈ ↑S\nhr₀ : r ≠ 1\nhrirred : ¬IsUnit r → ∃ x x_1, ∃ (_ : r = x * x_1), ¬IsUnit x ∧ ¬IsUnit... | [] | rw [hSgen]; exact mem_top _ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.TensorProduct.Tower | {
"line": 477,
"column": 2
} | {
"line": 486,
"column": 34
} | {
"line": 488,
"column": 0
} | [
{
"pp": "R : Type uR\nA : Type uA\nN : Type uN\ninst✝⁴ : CommSemiring R\ninst✝³ : CommSemiring A\ninst✝² : Algebra R A\ninst✝¹ : AddCommMonoid N\ninst✝ : Module R N\nx : A ⊗[A] (A ⊗[R] N)\n⊢ (cancelBaseChange R A A A N) x = (TensorProduct.lid A (A ⊗[R] N)) x",
"ppTerm": "?m.75",
"assigned": true,
"u... | [] | induction x using TensorProduct.induction_on with
| zero => simp only [map_zero]
| tmul b y =>
induction y using TensorProduct.induction_on with
| zero => simp
| tmul a m =>
simp only [cancelBaseChange_tmul, lid_tmul, smul_tmul', smul_eq_mul, mul_comm]
| add x y hx hy =>
simp only [tmul_... | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.LinearAlgebra.FreeModule.PID | {
"line": 252,
"column": 2
} | {
"line": 252,
"column": 10
} | {
"line": 252,
"column": 11
} | [
{
"pp": "case neg.intro.refine_2\nι : Type u_1\nR : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : IsPrincipalIdealRing R\ninst✝³ : IsDomain R\ninst✝² : Finite ι\nO : Type u_4\ninst✝¹ : AddCommGroup O\ninst✝ : Module R O\nM N : Submodule R O\nb'M : Basis ι R ↥M\nN_bot : N ≠ ⊥\nN_le_M : N ≤ M\nthis : ∃ ϕ, ∀ (ψ : ↥M →ₗ[... | [
"case neg.intro.refine_2\nι : Type u_1\nR : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : IsPrincipalIdealRing R\ninst✝³ : IsDomain R\ninst✝² : Finite ι\nO : Type u_4\ninst✝¹ : AddCommGroup O\ninst✝ : Module R O\nM N : Submodule R O\nb'M : Basis ι R ↥M\nN_bot : N ≠ ⊥\nN_le_M : N ≤ M\nthis : ∃ ϕ, ∀ (ψ : ↥M →ₗ[R] R), ¬ϕ.su... | intro m' | Lean.Elab.Tactic.evalIntro | null |
Mathlib.Algebra.Field.Subfield.Basic | {
"line": 343,
"column": 17
} | {
"line": 343,
"column": 40
} | {
"line": 343,
"column": 40
} | [
{
"pp": "K : Type u\ninst✝ : DivisionRing K\ns : Set K\np : (x : K) → x ∈ closure s → Prop\nmem : ∀ (x : K) (hx : x ∈ s), p x ⋯\none : p 1 ⋯\nadd : ∀ (x y : K) (hx : x ∈ closure s) (hy : y ∈ closure s), p x hx → p y hy → p (x + y) ⋯\nneg : ∀ (x : K) (hx : x ∈ closure s), p x hx → p (-x) ⋯\ninv : ∀ (x : K) (hx :... | [
"K : Type u\ninst✝ : DivisionRing K\ns : Set K\np : (x : K) → x ∈ closure s → Prop\nmem : ∀ (x : K) (hx : x ∈ s), p x ⋯\none : p 1 ⋯\nadd : ∀ (x y : K) (hx : x ∈ closure s) (hy : y ∈ closure s), p x hx → p y hy → p (x + y) ⋯\nneg : ∀ (x : K) (hx : x ∈ closure s), p x hx → p (-x) ⋯\ninv : ∀ (x : K) (hx : x ∈ closure... | ← @add_neg_cancel K _ 1 | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.GroupTheory.MonoidLocalization.MonoidWithZero | {
"line": 117,
"column": 20
} | {
"line": 117,
"column": 52
} | {
"line": 117,
"column": 52
} | [
{
"pp": "M : Type u_1\ninst✝² : CommMonoidWithZero M\nS : Submonoid M\nN : Type u_2\ninst✝¹ : CommMonoidWithZero N\nf : S.LocalizationMap N\ninst✝ : IsCancelMulZero M\nn : N\nhn : n ≠ 0\nms : M × ↥S\neq : n * f ↑ms.2 = 0\nh : ms.1 = 0\n⊢ n = 0",
"ppTerm": "?m.85",
"assigned": true,
"usedConstants": ... | [
"M : Type u_1\ninst✝² : CommMonoidWithZero M\nS : Submonoid M\nN : Type u_2\ninst✝¹ : CommMonoidWithZero N\nf : S.LocalizationMap N\ninst✝ : IsCancelMulZero M\nn : N\nhn : n ≠ 0\nms : M × ↥S\neq : n = 0\nh : ms.1 = 0\n⊢ n = 0"
] | (f.map_units _).mul_left_eq_zero | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.MonoidLocalization.MonoidWithZero | {
"line": 144,
"column": 59
} | {
"line": 144,
"column": 91
} | {
"line": 144,
"column": 91
} | [
{
"pp": "M : Type u_1\ninst✝² : CommMonoidWithZero M\nS : Submonoid M\nN : Type u_2\ninst✝¹ : CommMonoidWithZero N\nf : S.LocalizationMap N\ninst✝ : NoZeroDivisors M\nn : N\nhn : n ≠ 0\nms : M × ↥S\neq : n * f ↑ms.2 = 0\nh : ms.1 = 0\n⊢ n = 0",
"ppTerm": "?m.67",
"assigned": true,
"usedConstants": [... | [
"M : Type u_1\ninst✝² : CommMonoidWithZero M\nS : Submonoid M\nN : Type u_2\ninst✝¹ : CommMonoidWithZero N\nf : S.LocalizationMap N\ninst✝ : NoZeroDivisors M\nn : N\nhn : n ≠ 0\nms : M × ↥S\neq : n = 0\nh : ms.1 = 0\n⊢ n = 0"
] | (f.map_units _).mul_left_eq_zero | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.OreLocalization.Ring | {
"line": 141,
"column": 6
} | {
"line": 141,
"column": 32
} | {
"line": 141,
"column": 32
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : Semiring R\nS : Submonoid R\ninst✝³ : OreSet S\nX : Type u_2\ninst✝² : AddCommMonoid X\ninst✝¹ : Module R X\nT : Type u_3\ninst✝ : Semiring T\nf : R →+* T\nfS : ↥S →* Tˣ\nhf : ∀ (s : ↥S), f ↑s = ↑(fS s)\nx y : OreLocalization S R\n⊢ (universalMulHom (↑f) fS ⋯) (x + y) = (universa... | [
"case c\nR : Type u_1\ninst✝⁴ : Semiring R\nS : Submonoid R\ninst✝³ : OreSet S\nX : Type u_2\ninst✝² : AddCommMonoid X\ninst✝¹ : Module R X\nT : Type u_3\ninst✝ : Semiring T\nf : R →+* T\nfS : ↥S →* Tˣ\nhf : ∀ (s : ↥S), f ↑s = ↑(fS s)\ny : OreLocalization S R\nr₁ : R\ns₁ : ↥S\n⊢ (universalMulHom (↑f) fS ⋯) (r₁ /ₒ s... | induction x with | _ r₁ s₁
=> _ | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.RingTheory.Localization.Defs | {
"line": 804,
"column": 4
} | {
"line": 804,
"column": 40
} | {
"line": 805,
"column": 4
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝³ : CommSemiring S\ninst✝² : Algebra R S\nP : Type u_3\ninst✝¹ : CommSemiring P\ninst✝ : Subsingleton R\na : Localization M\n⊢ a = default",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Inhabited.de... | [
"R : Type u_1\ninst✝⁴ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝³ : CommSemiring S\ninst✝² : Algebra R S\nP : Type u_3\ninst✝¹ : CommSemiring P\ninst✝ : Subsingleton R\na : Localization M\n⊢ a = mk 1 1"
] | with_unfolding_all change a = mk 1 1 | Lean.Elab.Tactic.evalWithUnfoldingAll | Lean.Parser.Tactic.withUnfoldingAll |
Mathlib.RingTheory.Localization.Basic | {
"line": 339,
"column": 4
} | {
"line": 340,
"column": 22
} | {
"line": 341,
"column": 4
} | [
{
"pp": "case exists_of_eq\nR : Type u_1\ninst✝⁵ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝⁴ : CommSemiring S\ninst✝³ : Algebra R S\nP : Type u_3\ninst✝² : CommSemiring P\ninst✝¹ : Algebra R P\ninst✝ : IsLocalization M S\nh : S ≃ₐ[R] P\nx y : R\n⊢ (algebraMap R P) x = (algebraMap R P) y → ∃ c, ↑c * ... | [
"case exists_of_eq\nR : Type u_1\ninst✝⁵ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝⁴ : CommSemiring S\ninst✝³ : Algebra R S\nP : Type u_3\ninst✝² : CommSemiring P\ninst✝¹ : Algebra R P\ninst✝ : IsLocalization M S\nh : S ≃ₐ[R] P\nx y : R\n⊢ h.symm.toEquiv ((algebraMap R P) x) = h.symm.toEquiv ((algebraMa... | rw [← h.symm.toEquiv.injective.eq_iff, ← IsLocalization.eq_iff_exists M S, ← h.symm.commutes, ←
h.symm.commutes] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Localization.Basic | {
"line": 398,
"column": 6
} | {
"line": 398,
"column": 18
} | {
"line": 399,
"column": 6
} | [
{
"pp": "case e'_3\nR : Type u_1\ninst✝¹³ : CommSemiring R\nS₁ : Type u_4\nS₂ : Type u_5\nT : Type u_6\ninst✝¹² : CommSemiring S₁\ninst✝¹¹ : CommSemiring S₂\ninst✝¹⁰ : CommSemiring T\ninst✝⁹ : Algebra R S₁\ninst✝⁸ : Algebra R S₂\ninst✝⁷ : Algebra R T\ninst✝⁶ : Algebra S₁ T\ninst✝⁵ : Algebra S₂ T\ninst✝⁴ : IsSca... | [
"case e'_3\nR : Type u_1\ninst✝¹³ : CommSemiring R\nS₁ : Type u_4\nS₂ : Type u_5\nT : Type u_6\ninst✝¹² : CommSemiring S₁\ninst✝¹¹ : CommSemiring S₂\ninst✝¹⁰ : CommSemiring T\ninst✝⁹ : Algebra R S₁\ninst✝⁸ : Algebra R S₂\ninst✝⁷ : Algebra R T\ninst✝⁶ : Algebra S₁ T\ninst✝⁵ : Algebra S₂ T\ninst✝⁴ : IsScalarTower R S... | rw [map_mul] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Localization.FractionRing | {
"line": 278,
"column": 37
} | {
"line": 278,
"column": 46
} | {
"line": 278,
"column": 46
} | [
{
"pp": "A : Type u_4\ninst✝³ : CommRing A\nK : Type u_5\ninst✝² : Field K\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nx : A\ny : ↥A⁰\nthis : Nontrivial A\nhxy : x = ↑y\n⊢ mk' K (↑y) y = 1",
"ppTerm": "?m.42",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddC... | [
"A : Type u_4\ninst✝³ : CommRing A\nK : Type u_5\ninst✝² : Field K\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nx : A\ny : ↥A⁰\nthis : Nontrivial A\nhxy : x = ↑y\n⊢ 1 = 1"
] | mk'_self' | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Star.Pointwise | {
"line": 95,
"column": 2
} | {
"line": 95,
"column": 36
} | {
"line": 97,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝ : InvolutiveStar α\ns t : Set α\n⊢ s⋆ ⊆ t ↔ s ⊆ t⋆",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Set.star",
"star_star",
"congrArg",
"Iff.rfl",
"id",
"HasSubset.Subset",
"Iff",
"Set.instInvolutiv... | [] | rw [← star_subset_star, star_star] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Star.Pointwise | {
"line": 95,
"column": 2
} | {
"line": 95,
"column": 36
} | {
"line": 97,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝ : InvolutiveStar α\ns t : Set α\n⊢ s⋆ ⊆ t ↔ s ⊆ t⋆",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Set.star",
"star_star",
"congrArg",
"Iff.rfl",
"id",
"HasSubset.Subset",
"Iff",
"Set.instInvolutiv... | [] | rw [← star_subset_star, star_star] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Star.Pointwise | {
"line": 95,
"column": 2
} | {
"line": 95,
"column": 36
} | {
"line": 97,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝ : InvolutiveStar α\ns t : Set α\n⊢ s⋆ ⊆ t ↔ s ⊆ t⋆",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Set.star",
"star_star",
"congrArg",
"Iff.rfl",
"id",
"HasSubset.Subset",
"Iff",
"Set.instInvolutiv... | [] | rw [← star_subset_star, star_star] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Algebra.Spectrum.Basic | {
"line": 179,
"column": 4
} | {
"line": 180,
"column": 78
} | {
"line": 181,
"column": 4
} | [
{
"pp": "case neg\nR : Type u\nA : Type v\ninst✝² : CommSemiring R\ninst✝¹ : Ring A\ninst✝ : Algebra R A\nr : Rˣ\ns : R\na : A\nh : s ∉ σ a\n⊢ r • (↑ₐ s - a)⁻¹ʳ = (↑ₐ (r⁻¹ • s) - r⁻¹ • a)⁻¹ʳ",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMo... | [
"case neg\nR : Type u\nA : Type v\ninst✝² : CommSemiring R\ninst✝¹ : Ring A\ninst✝ : Algebra R A\nr : Rˣ\ns : R\na : A\nh : s ∉ σ a\nh' : IsUnit (r • ↑ₐ (r⁻¹ • s) - a)\n⊢ r • (↑ₐ s - a)⁻¹ʳ = (↑ₐ (r⁻¹ • s) - r⁻¹ • a)⁻¹ʳ"
] | have h' : IsUnit (r • algebraMap R A (r⁻¹ • s) - a) := by
simpa [Algebra.algebraMap_eq_smul_one, smul_assoc] using notMem_iff.mp h | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.TensorProduct.Basic | {
"line": 553,
"column": 67
} | {
"line": 553,
"column": 94
} | {
"line": 554,
"column": 4
} | [
{
"pp": "R : Type uR\nA : Type uA\nB : Type uB\ninst✝⁴ : CommSemiring R\ninst✝³ : CommSemiring A\ninst✝² : Semiring B\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\ns : Set B\nhs : adjoin R s = ⊤\n⊢ ⊤ ≤ Subalgebra.toSubmodule (adjoin A ↑includeRight.range)",
"ppTerm": "?m.94",
"assigned": true,
"usedCo... | [
"R : Type uR\nA : Type uA\nB : Type uB\ninst✝⁴ : CommSemiring R\ninst✝³ : CommSemiring A\ninst✝² : Semiring B\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\ns : Set B\nhs : adjoin R s = ⊤\n⊢ Submodule.baseChange A ⊤ ≤ Subalgebra.toSubmodule (adjoin A ↑includeRight.range)"
] | ← Submodule.baseChange_top, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Dimension.Finite | {
"line": 74,
"column": 28
} | {
"line": 74,
"column": 52
} | {
"line": 74,
"column": 53
} | [
{
"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\n⊢ #↑↑⟨s, hs⟩ = 0",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [
"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\n⊢ IsEmpty ↑↑⟨s, hs⟩"
] | Cardinal.mk_eq_zero_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Dimension.Constructions | {
"line": 65,
"column": 2
} | {
"line": 65,
"column": 49
} | {
"line": 66,
"column": 2
} | [
{
"pp": "R : Type u\nM : Type v\nι : Type w\ninst✝² : Ring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\ns t : Set ι\nf : ι → M\nhs : LinearIndepOn R f s\nht : LinearIndepOn R (⇑(span R (f '' s)).mkQ ∘ f) t\n⊢ Disjoint (span R (f '' s)) (span R (f '' t))",
"ppTerm": "?m.57",
"assigned": true,
"use... | [
"case e'_4\nR : Type u\nM : Type v\nι : Type w\ninst✝² : Ring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\ns t : Set ι\nf : ι → M\nhs : LinearIndepOn R f s\nht : LinearIndepOn R (⇑(span R (f '' s)).mkQ ∘ f) t\n⊢ span R (f '' s) = (span R (f '' s)).mkQ.ker",
"case e'_5\nR : Type u\nM : Type v\nι : Type w\ninst✝... | convert! (Submodule.range_ker_disjoint ht).symm | Mathlib.Tactic._aux_Mathlib_Tactic_Convert___macroRules_Mathlib_Tactic_convert!_1 | Mathlib.Tactic.convert! |
Mathlib.LinearAlgebra.Dimension.Finite | {
"line": 123,
"column": 2
} | {
"line": 124,
"column": 84
} | {
"line": 125,
"column": 2
} | [
{
"pp": "R : Type u\nM : Type v\ninst✝³ : Semiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ninst✝ : Free R M\nn : ℕ\nh : Module.rank R M = ↑n\na✝ : Nontrivial R\nι : Type v\nb : Basis ι R M\n⊢ Module.Finite R M",
"ppTerm": "?m.39",
"assigned": true,
"usedConstants": [
"Preorder.toLT",... | [
"R : Type u\nM : Type v\ninst✝³ : Semiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ninst✝ : Free R M\nn : ℕ\nh : Module.rank R M = ↑n\na✝ : Nontrivial R\nι : Type v\nb : Basis ι R M\nthis : Finite ι\n⊢ Module.Finite R M"
] | have := mk_lt_aleph0_iff.mp <|
b.linearIndependent.cardinal_le_rank |>.trans_eq h |>.trans_lt natCast_lt_aleph0 | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.LinearAlgebra.Basis.VectorSpace | {
"line": 67,
"column": 18
} | {
"line": 67,
"column": 44
} | {
"line": 67,
"column": 45
} | [
{
"pp": "K : Type u_3\nV : Type u_4\ninst✝² : DivisionRing K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\ns : Set V\nhs : LinearIndepOn K id s\n⊢ range Subtype.val = hs.extend ⋯",
"ppTerm": "?m.40",
"assigned": true,
"usedConstants": [
"LinearIndepOn.extend",
"Eq.mpr",
"congrArg",... | [
"K : Type u_3\nV : Type u_4\ninst✝² : DivisionRing K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\ns : Set V\nhs : LinearIndepOn K id s\n⊢ {x | x ∈ hs.extend ⋯} = hs.extend ⋯"
] | Subtype.range_coe_subtype, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Basis.VectorSpace | {
"line": 110,
"column": 20
} | {
"line": 110,
"column": 46
} | {
"line": 110,
"column": 47
} | [
{
"pp": "K : Type u_3\nV : Type u_4\ninst✝² : DivisionRing K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\ns t : Set V\nhs : LinearIndepOn K id s\nhst : s ⊆ t\nht : ⊤ ≤ span K t\n⊢ range Subtype.val = hs.extend hst",
"ppTerm": "?m.52",
"assigned": true,
"usedConstants": [
"LinearIndepOn.extend... | [
"K : Type u_3\nV : Type u_4\ninst✝² : DivisionRing K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\ns t : Set V\nhs : LinearIndepOn K id s\nhst : s ⊆ t\nht : ⊤ ≤ span K t\n⊢ {x | x ∈ hs.extend hst} = hs.extend hst"
] | Subtype.range_coe_subtype, | Lean.Elab.Tactic.evalRewriteSeq | null |
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