module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.Algebra.EuclideanDomain.Basic | {
"line": 164,
"column": 34
} | {
"line": 164,
"column": 81
} | {
"line": 164,
"column": 81
} | [
{
"pp": "R : Type u\ninst✝¹ : EuclideanDomain R\ninst✝ : DecidableEq R\na b : R\nh : a ∣ b\n⊢ gcd a b = a",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"Dvd.dvd",
"CommRing.toNonUnitalCommRing",
"congrArg",
"CommSemiring.toSemiring... | [] | by rw [gcd_val, mod_eq_zero.2 h, gcd_zero_left] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.EuclideanDomain.Basic | {
"line": 209,
"column": 21
} | {
"line": 209,
"column": 53
} | {
"line": 209,
"column": 53
} | [
{
"pp": "R : Type u\ninst✝¹ : EuclideanDomain R\ninst✝ : DecidableEq R\na b : R\n⊢ b = a * 0 + b * 1",
"ppTerm": "?m.53",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"HMul.hMul",
"MulZeroClass.toMul",
"congrArg",
"CommSemiring.toSemiring",
... | [] | rw [mul_one, mul_zero, zero_add] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Ideal.Maps | {
"line": 891,
"column": 4
} | {
"line": 892,
"column": 60
} | {
"line": 893,
"column": 2
} | [
{
"pp": "case refine_1\nR : Type u_4\nM : Type u_5\ninst✝² : Ring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nH : annihilator R M ≤ ⊥\nr s : R\nH' : ∀ (a : M), r • a = s • a\n⊢ r - s = 0",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Module.mem_annihilator"... | [] | exact H (Module.mem_annihilator (r := r - s).mpr
(by simp only [sub_smul, H', sub_self, implies_true])) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.Ideal.Operations | {
"line": 193,
"column": 4
} | {
"line": 194,
"column": 12
} | {
"line": 194,
"column": 13
} | [
{
"pp": "case mp.refine_3\nR : Type u\nM : Type v\ninst✝³ : Semiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\nI : Ideal R\ninst✝ : I.IsTwoSided\nι : Type u_4\nf : ι → M\nx : M\nhx : x ∈ I • span R (Set.range f)\nax : ι →₀ R\nhax : ∀ (i : ι), ax i ∈ I\nay : ι →₀ R\nhay : ∀ (i : ι), ay i ∈ I\n⊢ ∃ a, ∃ (_... | [
"case mp.refine_3.refine_1\nR : Type u\nM : Type v\ninst✝³ : Semiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\nI : Ideal R\ninst✝ : I.IsTwoSided\nι : Type u_4\nf : ι → M\nx : M\nhx : x ∈ I • span R (Set.range f)\nax : ι →₀ R\nhax : ∀ (i : ι), ax i ∈ I\nay : ι →₀ R\nhay : ∀ (i : ι), ay i ∈ I\na✝ : ι\n⊢ 0 • ... | refine ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' ?_ ?_⟩ <;>
intros | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.RingTheory.Ideal.Operations | {
"line": 669,
"column": 8
} | {
"line": 669,
"column": 25
} | {
"line": 669,
"column": 26
} | [
{
"pp": "case mp\nR : Type u_2\ninst✝ : CommSemiring R\nx y u : R\nhu : ∃ a, a * x = u\nv : R\nhv : ∃ a, a * y = v\nh1 : u + v = 1\n⊢ IsCoprime x y",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"Exists.choose_spec",
"HMul.hMul",
... | [
"case mp\nR : Type u_2\ninst✝ : CommSemiring R\nx y u : R\nhu : ∃ a, a * x = u\nv : R\nhv : ∃ a, a * y = v\nh1 : hu.choose * x + v = 1\n⊢ IsCoprime x y"
] | ← hu.choose_spec, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.GCDMonoid.Basic | {
"line": 581,
"column": 6
} | {
"line": 581,
"column": 40
} | {
"line": 582,
"column": 6
} | [
{
"pp": "case neg\nα : Type u_1\ninst✝ : CommMonoidWithZero α\nk m n : α\nH : k ∣ m * n\na✝ : GCDMonoid α\nh0 : ¬gcd k m = 0\n⊢ ∃ a₁ a₂, a₁ ∣ m ∧ a₂ ∣ n ∧ k = a₁ * a₂",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"Semigroup.toMul",
"Dvd.dvd",
"HMul.hMul",
"semigro... | [
"case neg\nα : Type u_1\ninst✝ : CommMonoidWithZero α\nk m n : α\nH : k ∣ m * n\na✝ : GCDMonoid α\nh0 : ¬gcd k m = 0\na : α\nha : k = gcd k m * a\n⊢ ∃ a₁ a₂, a₁ ∣ m ∧ a₂ ∣ n ∧ k = a₁ * a₂"
] | obtain ⟨a, ha⟩ := gcd_dvd_left k m | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Algebra.GCDMonoid.Basic | {
"line": 606,
"column": 14
} | {
"line": 606,
"column": 67
} | {
"line": 607,
"column": 4
} | [
{
"pp": "case neg.zero\nα : Type u_1\ninst✝¹ : CommMonoidWithZero α\ninst✝ : GCDMonoid α\na b : α\nhg : ¬gcd a b = 0\n⊢ gcd a (b ^ 0) ∣ gcd a b ^ 0",
"ppTerm": "?neg.zero✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"Dvd.dvd",
"Associated.dvd",
"Mon... | [] | rw [pow_zero, pow_zero]; exact (gcd_one_right' a).dvd | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.GCDMonoid.Basic | {
"line": 606,
"column": 14
} | {
"line": 606,
"column": 67
} | {
"line": 607,
"column": 4
} | [
{
"pp": "case neg.zero\nα : Type u_1\ninst✝¹ : CommMonoidWithZero α\ninst✝ : GCDMonoid α\na b : α\nhg : ¬gcd a b = 0\n⊢ gcd a (b ^ 0) ∣ gcd a b ^ 0",
"ppTerm": "?neg.zero✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"Dvd.dvd",
"Associated.dvd",
"Mon... | [] | rw [pow_zero, pow_zero]; exact (gcd_one_right' a).dvd | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Ideal.Operations | {
"line": 792,
"column": 2
} | {
"line": 792,
"column": 23
} | {
"line": 793,
"column": 4
} | [
{
"pp": "case insert\nR : Type u\nι : Type u_1\ninst✝ : CommSemiring R\nJ : ι → Ideal R\na : ι\ns : Finset ι\nhs : a ∉ s\nih : (↑s).Pairwise (IsCoprime on J) → ∏ i ∈ s, J i = ⨅ i ∈ s, J i\nhp : (↑(insert a s)).Pairwise (IsCoprime on J)\n⊢ ∏ i ∈ insert a s, J i = ⨅ i ∈ insert a s, J i",
"ppTerm": "?insert",
... | [] | | insert a s hs ih => | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.Algebra.GCDMonoid.Basic | {
"line": 986,
"column": 4
} | {
"line": 986,
"column": 12
} | {
"line": 987,
"column": 4
} | [
{
"pp": "case mk\nα : Type u_1\ninst✝¹ : CommMonoidWithZero α\ninst✝ : Subsingleton αˣ\ng₂ : GCDMonoid α\ntoIsCancelMulZero✝ : IsCancelMulZero α\ngcd✝ lcm✝ : α → α → α\ngcd_dvd_left✝ : ∀ (a b : α), gcd✝ a b ∣ a\ngcd_dvd_right✝ : ∀ (a b : α), gcd✝ a b ∣ b\ndvd_gcd✝ : ∀ {a b c : α}, a ∣ c → a ∣ b → a ∣ gcd✝ c b\n... | [
"case mk.mk\nα : Type u_1\ninst✝¹ : CommMonoidWithZero α\ninst✝ : Subsingleton αˣ\ntoIsCancelMulZero✝¹ : IsCancelMulZero α\ngcd✝¹ lcm✝¹ : α → α → α\ngcd_dvd_left✝¹ : ∀ (a b : α), gcd✝¹ a b ∣ a\ngcd_dvd_right✝¹ : ∀ (a b : α), gcd✝¹ a b ∣ b\ndvd_gcd✝¹ : ∀ {a b c : α}, a ∣ c → a ∣ b → a ∣ gcd✝¹ c b\ngcd_mul_lcm✝¹ : ∀ ... | cases g₂ | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalCases | Lean.Parser.Tactic.cases |
Mathlib.Algebra.GCDMonoid.Basic | {
"line": 1109,
"column": 6
} | {
"line": 1109,
"column": 89
} | {
"line": 1110,
"column": 6
} | [
{
"pp": "case neg\nα : Type u_1\ninst✝² : CommMonoidWithZero α\ninst✝¹ : IsCancelMulZero α\ninst✝ : DecidableEq α\ngcd : α → α → α\ngcd_dvd_left : ∀ (a b : α), gcd a b ∣ a\ngcd_dvd_right : ∀ (a b : α), gcd a b ∣ b\ndvd_gcd : ∀ {a b c : α}, a ∣ c → a ∣ b → a ∣ gcd c b\na : α\na0 : ¬a = 0\n⊢ Classical.choose ⋯ = ... | [
"case neg\nα : Type u_1\ninst✝² : CommMonoidWithZero α\ninst✝¹ : IsCancelMulZero α\ninst✝ : DecidableEq α\ngcd : α → α → α\ngcd_dvd_left : ∀ (a b : α), gcd a b ∣ a\ngcd_dvd_right : ∀ (a b : α), gcd a b ∣ b\ndvd_gcd : ∀ {a b c : α}, a ∣ c → a ∣ b → a ∣ gcd c b\na : α\na0 : ¬a = 0\nh : gcd a 0 * Classical.choose ⋯ = ... | have h := (Classical.choose_spec ((gcd_dvd_left a 0).trans (Dvd.intro 0 rfl))).symm | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.PrincipalIdealDomain | {
"line": 209,
"column": 72
} | {
"line": 209,
"column": 91
} | {
"line": 210,
"column": 4
} | [
{
"pp": "R : Type u\ninst✝¹ : CommRing R\nx y : R\ninst✝ : IsPrincipal (Ideal.span {x, y})\nh : IsRelPrime x y\n⊢ Ideal.span {x} ⊔ Ideal.span {y} = ⊤",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Lattice.toSemilatticeSup",
"Semiring.toModule",
"congrArg... | [
"R : Type u\ninst✝¹ : CommRing R\nx y : R\ninst✝ : IsPrincipal (Ideal.span {x, y})\nh : IsRelPrime x y\n⊢ Ideal.span ({x} ∪ {y}) = ⊤"
] | ← Ideal.span_union, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Dimension.StrongRankCondition | {
"line": 283,
"column": 2
} | {
"line": 283,
"column": 75
} | {
"line": 284,
"column": 2
} | [
{
"pp": "R : Type u\ninst✝¹ : Semiring R\ninst✝ : StrongRankCondition R\nα β : Type v\nf : (α →₀ R) →ₗ[R] β →₀ R\ni : Injective ⇑f\nb : Basis β R (β →₀ R) := { repr := 1 }\n⊢ #α ≤ #β",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"Sem... | [
"R : Type u\ninst✝¹ : Semiring R\ninst✝ : StrongRankCondition R\nα β : Type v\nf : (α →₀ R) →ₗ[R] β →₀ R\ni : Injective ⇑f\nb : Basis β R (β →₀ R) := ⋯\n⊢ LinearIndependent R fun i ↦ f (single i 1)"
] | apply linearIndependent_le_basis b (fun (i : α) ↦ f (Finsupp.single i 1)) | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.LinearAlgebra.Dimension.StrongRankCondition | {
"line": 443,
"column": 2
} | {
"line": 444,
"column": 60
} | {
"line": 446,
"column": 0
} | [
{
"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✝ : Free R M\n⊢ Module.rank R M = 0 ↔ Subsingleton M",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Nontrivial",
"Eq.mpr",
"NonAssocS... | [] | rw [← not_nontrivial_iff_subsingleton, iff_not_comm,
← Module.rank_pos_iff_of_free (R := R), pos_iff_ne_zero] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.Dimension.StrongRankCondition | {
"line": 443,
"column": 2
} | {
"line": 444,
"column": 60
} | {
"line": 446,
"column": 0
} | [
{
"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✝ : Free R M\n⊢ Module.rank R M = 0 ↔ Subsingleton M",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Nontrivial",
"Eq.mpr",
"NonAssocS... | [] | rw [← not_nontrivial_iff_subsingleton, iff_not_comm,
← Module.rank_pos_iff_of_free (R := R), pos_iff_ne_zero] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Dimension.StrongRankCondition | {
"line": 443,
"column": 2
} | {
"line": 444,
"column": 60
} | {
"line": 446,
"column": 0
} | [
{
"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✝ : Free R M\n⊢ Module.rank R M = 0 ↔ Subsingleton M",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Nontrivial",
"Eq.mpr",
"NonAssocS... | [] | rw [← not_nontrivial_iff_subsingleton, iff_not_comm,
← Module.rank_pos_iff_of_free (R := R), pos_iff_ne_zero] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Ideal.Operations | {
"line": 1127,
"column": 6
} | {
"line": 1127,
"column": 67
} | {
"line": 1128,
"column": 4
} | [
{
"pp": "case pos.inr.inr\nι : Type u_1\nR : Type u\ninst✝ : CommRing R\nf : ι → Ideal R\nI : Ideal R\nn : ℕ\na b i : ι\nt : Finset ι\nhit : i ∉ t\nhn : t.card = n\nh : ↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ ⋃ i_1 ∈ ↑(insert i t), ↑(f i_1)\nhp : (f i).IsPrime ∧ ∀ x ∈ t, (f x).IsPrime\nHt : ¬∃ j ∈ t, f j ≤ f i\nHa : ¬f a ≤ f i\... | [] | · exact Or.inr (Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Data.Matrix.Basis | {
"line": 136,
"column": 73
} | {
"line": 141,
"column": 18
} | {
"line": 143,
"column": 0
} | [
{
"pp": "m : Type u_2\nn : Type u_3\nα : Type u_7\ninst✝³ : DecidableEq m\ninst✝² : DecidableEq n\ninst✝¹ : NonUnitalNonAssocSemiring α\ninst✝ : Fintype m\ni : n\nj : m\nc : α\nx : m → α\n⊢ single i j c *ᵥ x = Function.update 0 i (c * x j)",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
... | [] | by
ext i'
simp only [mulVec, dotProduct, single, of_apply, ite_mul, zero_mul]
rcases eq_or_ne i i' with rfl | h
· simp
simp [h, h.symm] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.Matrix.Basis | {
"line": 410,
"column": 11
} | {
"line": 410,
"column": 25
} | {
"line": 410,
"column": 26
} | [
{
"pp": "n : Type u_3\nα : Type u_7\ninst✝² : DecidableEq n\ninst✝¹ : Fintype n\ninst✝ : Semiring α\nx : Matrix n n α\n⊢ x ∈ Set.center (Matrix n n α) ↔ x ∈ ⇑(scalar n) '' Set.center α",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Matrix.scalar",
"congrArg",
... | [
"n : Type u_3\nα : Type u_7\ninst✝² : DecidableEq n\ninst✝¹ : Fintype n\ninst✝ : Semiring α\nx : Matrix n n α\n⊢ x ∈ Set.center (Matrix n n α) ↔ ∃ x_1 ∈ Set.center α, (scalar n) x_1 = x"
] | Set.mem_image, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.RingTheory.AlgebraTower | {
"line": 104,
"column": 6
} | {
"line": 104,
"column": 90
} | {
"line": 105,
"column": 4
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nA : Type u_3\ninst✝⁶ : Semiring R\ninst✝⁵ : Semiring S\ninst✝⁴ : AddCommMonoid A\ninst✝³ : Module R S\ninst✝² : Module S A\ninst✝¹ : Module R A\ninst✝ : IsScalarTower R S A\nι : Type u_5\nb : ι → S\nι' : Type u_6\nc : ι' → A\nhb : LinearIndependent R b\nhc : LinearIndependen... | [
"R : Type u_1\nS : Type u_2\nA : Type u_3\ninst✝⁶ : Semiring R\ninst✝⁵ : Semiring S\ninst✝⁴ : AddCommMonoid A\ninst✝³ : Module R S\ninst✝² : Module S A\ninst✝¹ : Module R A\ninst✝ : IsScalarTower R S A\nι : Type u_5\nb : ι → S\nι' : Type u_6\nc : ι' → A\nhb : LinearIndependent R b\nhc : LinearIndependent S c\n⊢ Lin... | ← linearIndependent_equiv' (.prodComm ..) (g := fun p : ι' × ι ↦ b p.2 • c p.1) rfl, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.Matrix.Basic | {
"line": 663,
"column": 89
} | {
"line": 663,
"column": 94
} | {
"line": 663,
"column": 94
} | [
{
"pp": "α : Type u_14\ninst✝¹ : MulOne α\ninst✝ : AddCommMonoid α\nx✝ : IsStablyFiniteRing αᵐᵒᵖ\nn : ℕ\n⊢ map 1 (op ∘ op) = 1",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"MulOne.toOne",
"Matrix.instMulOneOfFintypeOfDecidableEqOfAddCommMonoid",
"congrArg",
"Matr... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Data.Matrix.Basic | {
"line": 663,
"column": 89
} | {
"line": 663,
"column": 94
} | {
"line": 663,
"column": 94
} | [
{
"pp": "α : Type u_14\ninst✝¹ : MulOne α\ninst✝ : AddCommMonoid α\nx✝ : IsStablyFiniteRing αᵐᵒᵖ\nn : ℕ\n⊢ map 1 (op ∘ op) = 1",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"MulOne.toOne",
"Matrix.instMulOneOfFintypeOfDecidableEqOfAddCommMonoid",
"congrArg",
"Matr... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Matrix.Basic | {
"line": 663,
"column": 89
} | {
"line": 663,
"column": 94
} | {
"line": 663,
"column": 94
} | [
{
"pp": "α : Type u_14\ninst✝¹ : MulOne α\ninst✝ : AddCommMonoid α\nx✝ : IsStablyFiniteRing αᵐᵒᵖ\nn : ℕ\n⊢ map 1 (op ∘ op) = 1",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"MulOne.toOne",
"Matrix.instMulOneOfFintypeOfDecidableEqOfAddCommMonoid",
"congrArg",
"Matr... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Set.UnionLift | {
"line": 143,
"column": 2
} | {
"line": 144,
"column": 40
} | {
"line": 145,
"column": 2
} | [
{
"pp": "α : Type u_1\nι : Sort u_2\nβ : Sort u_3\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⟩\ndir : Directed (fun x1 x2 ↦ x1 ⊆ x2) S\nopi : (i : ι) → ↑(S i) → ↑(S i) → ↑(S i)\nopβ : β → β → β\nh : ∀ (i : ι) (x y : ↑(S i)), f i ... | [
"α : Type u_1\nι : Sort u_2\nβ : Sort u_3\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⟩\ndir : Directed (fun x1 x2 ↦ x1 ⊆ x2) S\nopi : (i : ι) → ↑(S i) → ↑(S i) → ↑(S i)\nopβ : β → β → β\nh : ∀ (i : ι) (x y : ↑(S i)), f i (opi i x y) ... | have hxy : (Set.inclusion (Set.subset_iUnion S k) (opi k ⟨x, hik hi⟩ ⟨y, hjk hj⟩) : α) ∈ S k :=
(opi k ⟨x, hik hi⟩ ⟨y, hjk hj⟩).prop | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.TwoSidedIdeal.Basic | {
"line": 54,
"column": 36
} | {
"line": 54,
"column": 41
} | {
"line": 54,
"column": 41
} | [
{
"pp": "R : Type u_1\ninst✝¹ : NonUnitalNonAssocRing R\nI✝ : TwoSidedIdeal R\ninst✝ : Nontrivial R\nI J : RingCon R\nh : { ringCon := I } = { ringCon := J }\n⊢ I = J",
"ppTerm": "?m.45",
"assigned": true,
"usedConstants": [
"congrArg",
"TwoSidedIdeal",
"RingCon",
"Eq.mp",
... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.RingTheory.TwoSidedIdeal.Kernel | {
"line": 37,
"column": 32
} | {
"line": 37,
"column": 37
} | {
"line": 38,
"column": 4
} | [
{
"pp": "case refl\nR : Type u_1\nS : Type u_2\ninst✝³ : NonUnitalNonAssocRing R\ninst✝² : NonUnitalNonAssocSemiring S\nF : Type u_3\ninst✝¹ : FunLike F R S\ninst✝ : NonUnitalRingHomClass F R S\nf : F\n⊢ ∀ (x : R), f x = f x",
"ppTerm": "?refl",
"assigned": true,
"usedConstants": [
"eq_self",
... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.RingTheory.TwoSidedIdeal.Kernel | {
"line": 37,
"column": 32
} | {
"line": 37,
"column": 37
} | {
"line": 38,
"column": 4
} | [
{
"pp": "case symm\nR : Type u_1\nS : Type u_2\ninst✝³ : NonUnitalNonAssocRing R\ninst✝² : NonUnitalNonAssocSemiring S\nF : Type u_3\ninst✝¹ : FunLike F R S\ninst✝ : NonUnitalRingHomClass F R S\nf : F\n⊢ ∀ {x y : R}, f x = f y → f y = f x",
"ppTerm": "?symm",
"assigned": true,
"usedConstants": [
... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.RingTheory.TwoSidedIdeal.Kernel | {
"line": 37,
"column": 32
} | {
"line": 37,
"column": 37
} | {
"line": 38,
"column": 4
} | [
{
"pp": "case trans\nR : Type u_1\nS : Type u_2\ninst✝³ : NonUnitalNonAssocRing R\ninst✝² : NonUnitalNonAssocSemiring S\nF : Type u_3\ninst✝¹ : FunLike F R S\ninst✝ : NonUnitalRingHomClass F R S\nf : F\n⊢ ∀ {x y z : R}, f x = f y → f y = f z → f x = f z",
"ppTerm": "?trans",
"assigned": true,
"usedC... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.DirectSum.Module | {
"line": 577,
"column": 19
} | {
"line": 577,
"column": 24
} | {
"line": 578,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : Semiring R\nι : Type u_2\nN : ι → Type u_3\ninst✝³ : (i : ι) → AddCommMonoid (N i)\ninst✝² : (i : ι) → Module R (N i)\nP : ι → Type u_4\ninst✝¹ : (i : ι) → AddCommMonoid (P i)\ninst✝ : (i : ι) → Module R (P i)\nu : (i : ι) → N i ≃+ P i\nx : ⨁ (i : ι), N i\n⊢ (map fun i ↦ (u i).sy... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.DirectSum.Module | {
"line": 577,
"column": 19
} | {
"line": 577,
"column": 24
} | {
"line": 578,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : Semiring R\nι : Type u_2\nN : ι → Type u_3\ninst✝³ : (i : ι) → AddCommMonoid (N i)\ninst✝² : (i : ι) → Module R (N i)\nP : ι → Type u_4\ninst✝¹ : (i : ι) → AddCommMonoid (P i)\ninst✝ : (i : ι) → Module R (P i)\nu : (i : ι) → N i ≃+ P i\nx : ⨁ (i : ι), N i\n⊢ (map fun i ↦ (u i).sy... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.DirectSum.Module | {
"line": 577,
"column": 19
} | {
"line": 577,
"column": 24
} | {
"line": 578,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : Semiring R\nι : Type u_2\nN : ι → Type u_3\ninst✝³ : (i : ι) → AddCommMonoid (N i)\ninst✝² : (i : ι) → Module R (N i)\nP : ι → Type u_4\ninst✝¹ : (i : ι) → AddCommMonoid (P i)\ninst✝ : (i : ι) → Module R (P i)\nu : (i : ι) → N i ≃+ P i\nx : ⨁ (i : ι), N i\n⊢ (map fun i ↦ (u i).sy... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.DirectSum.Module | {
"line": 578,
"column": 20
} | {
"line": 578,
"column": 25
} | {
"line": 580,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : Semiring R\nι : Type u_2\nN : ι → Type u_3\ninst✝³ : (i : ι) → AddCommMonoid (N i)\ninst✝² : (i : ι) → Module R (N i)\nP : ι → Type u_4\ninst✝¹ : (i : ι) → AddCommMonoid (P i)\ninst✝ : (i : ι) → Module R (P i)\nu : (i : ι) → N i ≃+ P i\ny : ⨁ (i : ι), P i\n⊢ (↑(map fun i ↦ (u i).... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.DirectSum.Module | {
"line": 578,
"column": 20
} | {
"line": 578,
"column": 25
} | {
"line": 580,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : Semiring R\nι : Type u_2\nN : ι → Type u_3\ninst✝³ : (i : ι) → AddCommMonoid (N i)\ninst✝² : (i : ι) → Module R (N i)\nP : ι → Type u_4\ninst✝¹ : (i : ι) → AddCommMonoid (P i)\ninst✝ : (i : ι) → Module R (P i)\nu : (i : ι) → N i ≃+ P i\ny : ⨁ (i : ι), P i\n⊢ (↑(map fun i ↦ (u i).... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.DirectSum.Module | {
"line": 578,
"column": 20
} | {
"line": 578,
"column": 25
} | {
"line": 580,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : Semiring R\nι : Type u_2\nN : ι → Type u_3\ninst✝³ : (i : ι) → AddCommMonoid (N i)\ninst✝² : (i : ι) → Module R (N i)\nP : ι → Type u_4\ninst✝¹ : (i : ι) → AddCommMonoid (P i)\ninst✝ : (i : ι) → Module R (P i)\nu : (i : ι) → N i ≃+ P i\ny : ⨁ (i : ι), P i\n⊢ (↑(map fun i ↦ (u i).... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Algebra.Subalgebra.Lattice | {
"line": 689,
"column": 4
} | {
"line": 689,
"column": 35
} | {
"line": 690,
"column": 4
} | [
{
"pp": "case a.cons\nR : Type uR\nA : Type uA\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\ns : Set A\nhd : List A\ntl : List (List A)\nih :\n (∀ t ∈ tl, ∀ y ∈ t, y ∈ Set.range ⇑(algebraMap R A) ∪ s) → (List.map List.prod tl).sum ∈ span R ↑(Submonoid.closure s)\nHL : ∀ t ∈ hd :: tl, ∀ y ... | [
"case a.cons\nR : Type uR\nA : Type uA\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\ns : Set A\nhd : List A\ntl : List (List A)\nih :\n (∀ t ∈ tl, ∀ y ∈ t, y ∈ Set.range ⇑(algebraMap R A) ∪ s) → (List.map List.prod tl).sum ∈ span R ↑(Submonoid.closure s)\nHL : (∀ y ∈ hd, y ∈ Set.range ⇑(algeb... | rw [List.forall_mem_cons] at HL | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Algebra.Subalgebra.Lattice | {
"line": 701,
"column": 4
} | {
"line": 701,
"column": 35
} | {
"line": 702,
"column": 4
} | [
{
"pp": "case a.cons.cons\nR : Type uR\nA : Type uA\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\ns : Set A\nhd : A\ntl : List A\nih : (∀ y ∈ tl, y ∈ Set.range ⇑(algebraMap R A) ∪ s) → ∃ z r, ∃ (_ : r ∈ Submonoid.closure s), z • r = tl.prod\nHL : ∀ y ∈ hd :: tl, y ∈ Set.range ⇑(algebraMap ... | [
"case a.cons.cons\nR : Type uR\nA : Type uA\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\ns : Set A\nhd : A\ntl : List A\nih : (∀ y ∈ tl, y ∈ Set.range ⇑(algebraMap R A) ∪ s) → ∃ z r, ∃ (_ : r ∈ Submonoid.closure s), z • r = tl.prod\nHL : hd ∈ Set.range ⇑(algebraMap R A) ∪ s ∧ ∀ x ∈ tl, x ∈ Se... | rw [List.forall_mem_cons] at HL | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.DirectSum.Finsupp | {
"line": 67,
"column": 21
} | {
"line": 67,
"column": 76
} | {
"line": 68,
"column": 2
} | [
{
"pp": "case add\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✝ : De... | [] | simp [add_tmul, map_add, hf, hg, Finsupp.sum_add_index] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.LinearAlgebra.DirectSum.Finsupp | {
"line": 67,
"column": 21
} | {
"line": 67,
"column": 76
} | {
"line": 68,
"column": 2
} | [
{
"pp": "case add\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✝ : De... | [] | simp [add_tmul, map_add, hf, hg, Finsupp.sum_add_index] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.DirectSum.Finsupp | {
"line": 67,
"column": 21
} | {
"line": 67,
"column": 76
} | {
"line": 68,
"column": 2
} | [
{
"pp": "case add\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✝ : De... | [] | simp [add_tmul, map_add, hf, hg, Finsupp.sum_add_index] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Module.Projective | {
"line": 255,
"column": 4
} | {
"line": 255,
"column": 21
} | {
"line": 257,
"column": 0
} | [
{
"pp": "case H\nR : Type u\ninst✝¹⁰ : Semiring R\nP : Type v\ninst✝⁹ : AddCommMonoid P\ninst✝⁸ : Module R P\nR₀ : Type u_2\nM : Type u_1\nN : Type u_3\ninst✝⁷ : CommSemiring R₀\ninst✝⁶ : Algebra R₀ R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R₀ M\ninst✝³ : Module R M\ninst✝² : IsScalarTower R₀ R M\ninst✝¹ : A... | [] | ext; simp [hsN _] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Module.Projective | {
"line": 255,
"column": 4
} | {
"line": 255,
"column": 21
} | {
"line": 257,
"column": 0
} | [
{
"pp": "case H\nR : Type u\ninst✝¹⁰ : Semiring R\nP : Type v\ninst✝⁹ : AddCommMonoid P\ninst✝⁸ : Module R P\nR₀ : Type u_2\nM : Type u_1\nN : Type u_3\ninst✝⁷ : CommSemiring R₀\ninst✝⁶ : Algebra R₀ R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R₀ M\ninst✝³ : Module R M\ninst✝² : IsScalarTower R₀ R M\ninst✝¹ : A... | [] | ext; simp [hsN _] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Matrix.Composition | {
"line": 45,
"column": 64
} | {
"line": 45,
"column": 69
} | {
"line": 47,
"column": 0
} | [
{
"pp": "I : Type u_1\nJ : Type u_2\nR : Type u_5\ninst✝³ : DecidableEq I\ninst✝² : DecidableEq J\ninst✝¹ : Zero R\ninst✝ : One R\ni✝ j✝ : I × J\n⊢ if i✝ = j✝ then (if i✝.1 = j✝.1 then 1 else 0) i✝.2 j✝.2 = 1 else (if i✝.1 = j✝.1 then 1 else 0) i✝.2 j✝.2 = 0",
"ppTerm": "?m.16",
"assigned": true,
"u... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Star.SelfAdjoint | {
"line": 655,
"column": 38
} | {
"line": 655,
"column": 45
} | {
"line": 655,
"column": 46
} | [
{
"pp": "R : Type u_1\ninst✝¹ : NonUnitalNonAssocSemiring R\ninst✝ : StarRing R\na b : R\nhab : Commute a (star b)\nha : Commute (star a) a\nhb : Commute (star b) b\nthis : Commute (star a) b\n⊢ a * star a + star a * b + (star b * a + b * star b) = a * star a + b * star a + (a * star b + b * star b)",
"ppTe... | [
"R : Type u_1\ninst✝¹ : NonUnitalNonAssocSemiring R\ninst✝ : StarRing R\na b : R\nhab : Commute a (star b)\nha : Commute (star a) a\nhb : Commute (star b) b\nthis : Commute (star a) b\n⊢ a * star a + star a * b + (star b * a + b * star b) = a * star a + b * star a + (star b * a + b * star b)"
] | hab.eq, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Basis.Fin | {
"line": 51,
"column": 7
} | {
"line": 54,
"column": 27
} | {
"line": 54,
"column": 27
} | [
{
"pp": "ι : Type u_1\nι' : Type u_2\nR : Type u_3\nR₂ : Type u_4\nM : Type u_5\nM' : Type u_6\nv : ι → M\ninst✝⁴ : Ring R\ninst✝³ : CommRing R₂\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : Module R₂ M\nx y✝ : M\nb✝ : Basis ι R M\nn : ℕ\nN : Submodule R M\ny : M\nb : Basis (Fin n) R ↥N\nhli : ∀ (c : R... | [] | by
intro c x hx hc
rw [← span_b] at hx
exact hli c x hx hc | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.Basis.Fin | {
"line": 92,
"column": 7
} | {
"line": 95,
"column": 27
} | {
"line": 95,
"column": 27
} | [
{
"pp": "ι : Type u_1\nι' : Type u_2\nR : Type u_3\nR₂ : Type u_4\nM : Type u_5\nM' : Type u_6\nv : ι → M\ninst✝⁴ : Ring R\ninst✝³ : CommRing R₂\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : Module R₂ M\nx y✝ : M\nb✝ : Basis ι R M\nn : ℕ\nN : Submodule R M\nb : Basis (Fin n) R ↥N\ny : M\nhli : ∀ (c : R... | [] | by
intro c x hx hc
rw [← span_b] at hx
exact hli c x hx hc | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.Matrix.Block | {
"line": 640,
"column": 4
} | {
"line": 640,
"column": 32
} | {
"line": 642,
"column": 0
} | [
{
"pp": "o : Type u_4\nm' : o → Type u_7\nα : Type u_12\ninst✝³ : DecidableEq o\ninst✝² : Zero α\ninst✝¹ : (i : o) → DecidableEq (m' i)\ninst✝ : One α\n⊢ (blockDiagonal' fun i ↦ diagonal fun x ↦ 1) = diagonal fun x ↦ 1",
"ppTerm": "?m.38",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"... | [] | rw [blockDiagonal'_diagonal] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Data.Matrix.Block | {
"line": 640,
"column": 4
} | {
"line": 640,
"column": 32
} | {
"line": 642,
"column": 0
} | [
{
"pp": "o : Type u_4\nm' : o → Type u_7\nα : Type u_12\ninst✝³ : DecidableEq o\ninst✝² : Zero α\ninst✝¹ : (i : o) → DecidableEq (m' i)\ninst✝ : One α\n⊢ (blockDiagonal' fun i ↦ diagonal fun x ↦ 1) = diagonal fun x ↦ 1",
"ppTerm": "?m.38",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"... | [] | rw [blockDiagonal'_diagonal] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Matrix.Block | {
"line": 640,
"column": 4
} | {
"line": 640,
"column": 32
} | {
"line": 642,
"column": 0
} | [
{
"pp": "o : Type u_4\nm' : o → Type u_7\nα : Type u_12\ninst✝³ : DecidableEq o\ninst✝² : Zero α\ninst✝¹ : (i : o) → DecidableEq (m' i)\ninst✝ : One α\n⊢ (blockDiagonal' fun i ↦ diagonal fun x ↦ 1) = diagonal fun x ↦ 1",
"ppTerm": "?m.38",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"... | [] | rw [blockDiagonal'_diagonal] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Matrix.RowCol | {
"line": 398,
"column": 66
} | {
"line": 402,
"column": 21
} | {
"line": 404,
"column": 0
} | [
{
"pp": "l : Type u_1\nm : Type u_2\nα : Type v\ninst✝² : DecidableEq l\ninst✝¹ : Fintype m\ninst✝ : NonUnitalNonAssocSemiring α\nA : Matrix l m α\ni : l\nc v : m → α\n⊢ A.updateRow i c *ᵥ v = Function.update (A *ᵥ v) i (c ⬝ᵥ v)",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"False"... | [] | by
ext i'
obtain rfl | hi := eq_or_ne i' i
· simp [mulVec]
· simp [mulVec, hi] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.Matrix.Notation | {
"line": 547,
"column": 36
} | {
"line": 547,
"column": 41
} | {
"line": 548,
"column": 0
} | [
{
"pp": "case refine_2.«0».«0»\nα : Type u_1\nx y : α\nh : x ≠ y\nh' : ![x, y] ((fun i ↦ i) ⟨0, ⋯⟩) = ![x, y] ((fun i ↦ i) ⟨0, ⋯⟩)\n⊢ (fun i ↦ i) ⟨0, ⋯⟩ = (fun i ↦ i) ⟨0, ⋯⟩",
"ppTerm": "?refine_2.«0».«0»",
"assigned": true,
"usedConstants": [
"instNeZeroNatHAdd_1",
"Nat.le_refl",
... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.LinearAlgebra.Matrix.Notation | {
"line": 547,
"column": 36
} | {
"line": 547,
"column": 41
} | {
"line": 548,
"column": 0
} | [
{
"pp": "case refine_2.«0».«1»\nα : Type u_1\nx y : α\nh : x ≠ y\nh' : ![x, y] ((fun i ↦ i) ⟨0, ⋯⟩) = ![x, y] ((fun i ↦ i) ⟨1, ⋯⟩)\n⊢ (fun i ↦ i) ⟨0, ⋯⟩ = (fun i ↦ i) ⟨1, ⋯⟩",
"ppTerm": "?refine_2.«0».«1»",
"assigned": true,
"usedConstants": [
"False",
"eq_false",
"Nat.le_refl",
... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.LinearAlgebra.Matrix.Notation | {
"line": 547,
"column": 36
} | {
"line": 547,
"column": 41
} | {
"line": 548,
"column": 0
} | [
{
"pp": "case refine_2.«1».«0»\nα : Type u_1\nx y : α\nh : x ≠ y\nh' : ![x, y] ((fun i ↦ i) ⟨1, ⋯⟩) = ![x, y] ((fun i ↦ i) ⟨0, ⋯⟩)\n⊢ (fun i ↦ i) ⟨1, ⋯⟩ = (fun i ↦ i) ⟨0, ⋯⟩",
"ppTerm": "?refine_2.«1».«0»",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instNeZeroNatHAdd_1",
"Fals... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.LinearAlgebra.Matrix.Notation | {
"line": 547,
"column": 36
} | {
"line": 547,
"column": 41
} | {
"line": 548,
"column": 0
} | [
{
"pp": "case refine_2.«1».«1»\nα : Type u_1\nx y : α\nh : x ≠ y\nh' : ![x, y] ((fun i ↦ i) ⟨1, ⋯⟩) = ![x, y] ((fun i ↦ i) ⟨1, ⋯⟩)\n⊢ (fun i ↦ i) ⟨1, ⋯⟩ = (fun i ↦ i) ⟨1, ⋯⟩",
"ppTerm": "?refine_2.«1».«1»",
"assigned": true,
"usedConstants": [
"instNeZeroNatHAdd_1",
"Nat.le_refl",
... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.LinearAlgebra.Matrix.ToLin | {
"line": 186,
"column": 2
} | {
"line": 188,
"column": 48
} | {
"line": 190,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝¹ : Semiring R\nm : Type u_3\nn : Type u_4\ninst✝ : Fintype m\nM : Matrix m n R\n⊢ (Function.Injective fun v ↦ v ᵥ* M) ↔ LinearIndependent R M.row",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Pi.Function.module",
"Semiring.toModul... | [] | rw [← coe_vecMulLinear, linearIndependent_iff_injective_fintypeLinearCombination]
congr! 1
exact funext fun _ => Matrix.vecMul_eq_sum _ _ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Matrix.ToLin | {
"line": 186,
"column": 2
} | {
"line": 188,
"column": 48
} | {
"line": 190,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝¹ : Semiring R\nm : Type u_3\nn : Type u_4\ninst✝ : Fintype m\nM : Matrix m n R\n⊢ (Function.Injective fun v ↦ v ᵥ* M) ↔ LinearIndependent R M.row",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Pi.Function.module",
"Semiring.toModul... | [] | rw [← coe_vecMulLinear, linearIndependent_iff_injective_fintypeLinearCombination]
congr! 1
exact funext fun _ => Matrix.vecMul_eq_sum _ _ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Matrix.ToLin | {
"line": 611,
"column": 27
} | {
"line": 611,
"column": 55
} | {
"line": 611,
"column": 55
} | [
{
"pp": "R : Type u_1\ninst✝⁷ : CommSemiring R\nm : Type u_3\nn : Type u_4\ninst✝⁶ : Fintype n\ninst✝⁵ : Finite m\ninst✝⁴ : DecidableEq n\nM₁ : Type u_5\nM₂ : Type u_6\ninst✝³ : AddCommMonoid M₁\ninst✝² : AddCommMonoid M₂\ninst✝¹ : Module R M₁\ninst✝ : Module R M₂\nv₁ : Basis n R M₁\nv₂ : Basis m R M₂\nf : M₁ →... | [
"R : Type u_1\ninst✝⁷ : CommSemiring R\nm : Type u_3\nn : Type u_4\ninst✝⁶ : Fintype n\ninst✝⁵ : Finite m\ninst✝⁴ : DecidableEq n\nM₁ : Type u_5\nM₂ : Type u_6\ninst✝³ : AddCommMonoid M₁\ninst✝² : AddCommMonoid M₂\ninst✝¹ : Module R M₁\ninst✝ : Module R M₂\nv₁ : Basis n R M₁\nv₂ : Basis m R M₂\nf : M₁ →ₗ[R] M₂\n⊢ f... | LinearEquiv.apply_symm_apply | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Matrix.ToLin | {
"line": 958,
"column": 4
} | {
"line": 960,
"column": 36
} | {
"line": 962,
"column": 0
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁴ : CommSemiring R\ninst✝³ : Semiring S\ninst✝² : Algebra R S\nm : Type u_3\ninst✝¹ : Fintype m\ninst✝ : DecidableEq m\nb : Basis m R S\nr : R\n⊢ (toMatrix b b) ((lmul R S) ((algebraMap R S) r)) = (algebraMap R (Matrix m m R)) r",
"ppTerm": "?m.137",
"assigned":... | [] | ext
rw [lmul_algebraMap, toMatrix_lsmul, algebraMap_eq_diagonal, Pi.algebraMap_def,
Algebra.algebraMap_self_apply] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Matrix.ToLin | {
"line": 958,
"column": 4
} | {
"line": 960,
"column": 36
} | {
"line": 962,
"column": 0
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁴ : CommSemiring R\ninst✝³ : Semiring S\ninst✝² : Algebra R S\nm : Type u_3\ninst✝¹ : Fintype m\ninst✝ : DecidableEq m\nb : Basis m R S\nr : R\n⊢ (toMatrix b b) ((lmul R S) ((algebraMap R S) r)) = (algebraMap R (Matrix m m R)) r",
"ppTerm": "?m.137",
"assigned":... | [] | ext
rw [lmul_algebraMap, toMatrix_lsmul, algebraMap_eq_diagonal, Pi.algebraMap_def,
Algebra.algebraMap_self_apply] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Matrix.ToLin | {
"line": 1155,
"column": 27
} | {
"line": 1155,
"column": 32
} | {
"line": 1155,
"column": 32
} | [
{
"pp": "ι : Type u_1\ninst✝⁸ : Fintype ι\ninst✝⁷ : DecidableEq ι\nR : Type u_2\ninst✝⁶ : CommSemiring R\nA : Type u_3\ninst✝⁵ : Semiring A\ninst✝⁴ : Algebra R A\nM : Type u_4\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\ninst✝¹ : Module A M\ninst✝ : IsScalarTower R A M\nf : End A (ι → M)\ni : ι\nx : M\nj : ι... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.LinearAlgebra.Matrix.ToLin | {
"line": 1155,
"column": 27
} | {
"line": 1155,
"column": 32
} | {
"line": 1155,
"column": 32
} | [
{
"pp": "ι : Type u_1\ninst✝⁸ : Fintype ι\ninst✝⁷ : DecidableEq ι\nR : Type u_2\ninst✝⁶ : CommSemiring R\nA : Type u_3\ninst✝⁵ : Semiring A\ninst✝⁴ : Algebra R A\nM : Type u_4\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\ninst✝¹ : Module A M\ninst✝ : IsScalarTower R A M\nf : End A (ι → M)\ni : ι\nx : M\nj : ι... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Matrix.ToLin | {
"line": 1155,
"column": 27
} | {
"line": 1155,
"column": 32
} | {
"line": 1155,
"column": 32
} | [
{
"pp": "ι : Type u_1\ninst✝⁸ : Fintype ι\ninst✝⁷ : DecidableEq ι\nR : Type u_2\ninst✝⁶ : CommSemiring R\nA : Type u_3\ninst✝⁵ : Semiring A\ninst✝⁴ : Algebra R A\nM : Type u_4\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\ninst✝¹ : Module A M\ninst✝ : IsScalarTower R A M\nf : End A (ι → M)\ni : ι\nx : M\nj : ι... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Matrix.ToLin | {
"line": 1162,
"column": 27
} | {
"line": 1162,
"column": 32
} | {
"line": 1162,
"column": 32
} | [
{
"pp": "ι : Type u_1\ninst✝⁸ : Fintype ι\ninst✝⁷ : DecidableEq ι\nR : Type u_2\ninst✝⁶ : CommSemiring R\nA : Type u_3\ninst✝⁵ : Semiring A\ninst✝⁴ : Algebra R A\nM : Type u_4\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\ninst✝¹ : Module A M\ninst✝ : IsScalarTower R A M\nf g : End A (ι → M)\ni✝ j✝ : ι\nx✝ : M... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.LinearAlgebra.Matrix.ToLin | {
"line": 1162,
"column": 27
} | {
"line": 1162,
"column": 32
} | {
"line": 1162,
"column": 32
} | [
{
"pp": "ι : Type u_1\ninst✝⁸ : Fintype ι\ninst✝⁷ : DecidableEq ι\nR : Type u_2\ninst✝⁶ : CommSemiring R\nA : Type u_3\ninst✝⁵ : Semiring A\ninst✝⁴ : Algebra R A\nM : Type u_4\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\ninst✝¹ : Module A M\ninst✝ : IsScalarTower R A M\nf g : End A (ι → M)\ni✝ j✝ : ι\nx✝ : M... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Matrix.ToLin | {
"line": 1162,
"column": 27
} | {
"line": 1162,
"column": 32
} | {
"line": 1162,
"column": 32
} | [
{
"pp": "ι : Type u_1\ninst✝⁸ : Fintype ι\ninst✝⁷ : DecidableEq ι\nR : Type u_2\ninst✝⁶ : CommSemiring R\nA : Type u_3\ninst✝⁵ : Semiring A\ninst✝⁴ : Algebra R A\nM : Type u_4\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\ninst✝¹ : Module A M\ninst✝ : IsScalarTower R A M\nf g : End A (ι → M)\ni✝ j✝ : ι\nx✝ : M... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Localization.FractionRing | {
"line": 788,
"column": 4
} | {
"line": 789,
"column": 70
} | {
"line": 791,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝¹³ : CommRing R\nM : Submonoid R\nS : Type u_2\ninst✝¹² : CommRing S\ninst✝¹¹ : Algebra R S\nP : Type u_3\ninst✝¹⁰ : CommRing P\nA : Type u_4\ninst✝⁹ : CommRing A\nK✝ : Type u_5\ninst✝⁸ : IsDomain A\nk : Type u_6\nK : Type u_7\ninst✝⁷ : Field k\ninst✝⁶ : Field K\ninst✝⁵ : Algebra A k... | [] | simp_rw [← smul_assoc, Localization.smul_mk, smul_eq_mul, Localization.mk_eq_mk',
IsLocalization.mk'_mul_cancel_left, algebraMap_smul, smul_assoc] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.RingTheory.Adjoin.Basic | {
"line": 172,
"column": 6
} | {
"line": 172,
"column": 32
} | {
"line": 172,
"column": 32
} | [
{
"pp": "R : Type uR\nA : Type uA\ninst✝² : CommSemiring R\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\ns : NonUnitalSubalgebra R A\n⊢ R ∙ 1 ⊔ (NonUnitalAlgebra.adjoin R ↑s).toSubmodule = R ∙ 1 ⊔ s.toSubmodule",
"ppTerm": "?m.61",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAsso... | [
"R : Type uR\nA : Type uA\ninst✝² : CommSemiring R\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\ns : NonUnitalSubalgebra R A\n⊢ R ∙ 1 ⊔ s.toSubmodule = R ∙ 1 ⊔ s.toSubmodule"
] | NonUnitalAlgebra.adjoin_eq | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Algebra.Epi | {
"line": 50,
"column": 2
} | {
"line": 50,
"column": 7
} | {
"line": 52,
"column": 0
} | [
{
"pp": "R : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nh : ∀ (a : A), 1 ⊗ₜ[R] a = a ⊗ₜ[R] 1\nh' : ∀ (x : A ⊗[R] A), ∃ a, x = a ⊗ₜ[R] 1\na b : A\nhxy : (lift (LinearMap.mul R A)) (a ⊗ₜ[R] 1) = (lift (LinearMap.mul R A)) (b ⊗ₜ[R] 1)\n⊢ a ⊗ₜ[R] 1 = b ⊗ₜ[R] 1",
"... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Algebra.Epi | {
"line": 55,
"column": 2
} | {
"line": 57,
"column": 40
} | {
"line": 59,
"column": 0
} | [
{
"pp": "R : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nh : Surjective ⇑(algebraMap R A)\n⊢ Algebra.IsEpi R A",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
... | [] | refine (isEpi_iff_forall_one_tmul_eq R A).mpr fun a ↦ ?_
obtain ⟨r, rfl⟩ := h a
rw [algebraMap_eq_smul_one, smul_tmul] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Algebra.Epi | {
"line": 55,
"column": 2
} | {
"line": 57,
"column": 40
} | {
"line": 59,
"column": 0
} | [
{
"pp": "R : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nh : Surjective ⇑(algebraMap R A)\n⊢ Algebra.IsEpi R A",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
... | [] | refine (isEpi_iff_forall_one_tmul_eq R A).mpr fun a ↦ ?_
obtain ⟨r, rfl⟩ := h a
rw [algebraMap_eq_smul_one, smul_tmul] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Algebra.Epi | {
"line": 67,
"column": 29
} | {
"line": 67,
"column": 34
} | {
"line": 68,
"column": 2
} | [
{
"pp": "R : Type u_1\nA : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : IsDomain R\ninst✝² : Field A\ninst✝¹ : Algebra R A\ninst✝ : IsFractionRing R A\na b : R\nhb : b ∈ nonZeroDivisors R\nf : R →+* A := algebraMap R A\nhf : f = algebraMap R A\n⊢ f b ≠ 0",
"ppTerm": "?m.77",
"assigned": true,
"usedConsta... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Algebra.Epi | {
"line": 116,
"column": 30
} | {
"line": 116,
"column": 56
} | {
"line": 117,
"column": 4
} | [
{
"pp": "R : Type u_1\nA : Type u_2\ninst✝³ : CommSemiring R\ninst✝² : CommSemiring A\ninst✝¹ : Algebra R A\ninst✝ : Algebra.IsEpi R A\na✝ b✝ a b : A\n⊢ a ⊗ₜ[R] b = a • 1 ⊗ₜ[R] b",
"ppTerm": "?m.44",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne"... | [] | rw [tmul_eq_smul_one_tmul] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Algebra.Epi | {
"line": 116,
"column": 30
} | {
"line": 116,
"column": 56
} | {
"line": 117,
"column": 4
} | [
{
"pp": "R : Type u_1\nA : Type u_2\ninst✝³ : CommSemiring R\ninst✝² : CommSemiring A\ninst✝¹ : Algebra R A\ninst✝ : Algebra.IsEpi R A\na✝ b✝ a b : A\n⊢ a ⊗ₜ[R] b = a • 1 ⊗ₜ[R] b",
"ppTerm": "?m.44",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne"... | [] | rw [tmul_eq_smul_one_tmul] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Algebra.Epi | {
"line": 116,
"column": 30
} | {
"line": 116,
"column": 56
} | {
"line": 117,
"column": 4
} | [
{
"pp": "R : Type u_1\nA : Type u_2\ninst✝³ : CommSemiring R\ninst✝² : CommSemiring A\ninst✝¹ : Algebra R A\ninst✝ : Algebra.IsEpi R A\na✝ b✝ a b : A\n⊢ a ⊗ₜ[R] b = a • 1 ⊗ₜ[R] b",
"ppTerm": "?m.44",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne"... | [] | rw [tmul_eq_smul_one_tmul] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Algebra.Epi | {
"line": 118,
"column": 36
} | {
"line": 118,
"column": 62
} | {
"line": 119,
"column": 2
} | [
{
"pp": "R : Type u_1\nA : Type u_2\ninst✝³ : CommSemiring R\ninst✝² : CommSemiring A\ninst✝¹ : Algebra R A\ninst✝ : Algebra.IsEpi R A\na✝ b✝ a b : A\n⊢ a • b ⊗ₜ[R] 1 = a • b • 1 ⊗ₜ[R] 1",
"ppTerm": "?m.88",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoid... | [] | rw [tmul_eq_smul_one_tmul] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Algebra.Epi | {
"line": 118,
"column": 36
} | {
"line": 118,
"column": 62
} | {
"line": 119,
"column": 2
} | [
{
"pp": "R : Type u_1\nA : Type u_2\ninst✝³ : CommSemiring R\ninst✝² : CommSemiring A\ninst✝¹ : Algebra R A\ninst✝ : Algebra.IsEpi R A\na✝ b✝ a b : A\n⊢ a • b ⊗ₜ[R] 1 = a • b • 1 ⊗ₜ[R] 1",
"ppTerm": "?m.88",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoid... | [] | rw [tmul_eq_smul_one_tmul] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Algebra.Epi | {
"line": 118,
"column": 36
} | {
"line": 118,
"column": 62
} | {
"line": 119,
"column": 2
} | [
{
"pp": "R : Type u_1\nA : Type u_2\ninst✝³ : CommSemiring R\ninst✝² : CommSemiring A\ninst✝¹ : Algebra R A\ninst✝ : Algebra.IsEpi R A\na✝ b✝ a b : A\n⊢ a • b ⊗ₜ[R] 1 = a • b • 1 ⊗ₜ[R] 1",
"ppTerm": "?m.88",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoid... | [] | rw [tmul_eq_smul_one_tmul] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Ring.TransferInstance | {
"line": 136,
"column": 2
} | {
"line": 138,
"column": 76
} | {
"line": 140,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ne : α ≃ β\ninst✝ : NonAssocRing β\n⊢ NonAssocRing α",
"ppTerm": "?m.1",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Int.cast",
"NegZeroClass.toNeg",
"Equiv.apply_symm_apply",
"instHSMul",
"Equiv.instEq... | [] | let add_group_with_one := e.addGroupWithOne
let mul := e.mul
apply e.injective.nonAssocRing _ <;> intros <;> exact e.apply_symm_apply _ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Ring.TransferInstance | {
"line": 136,
"column": 2
} | {
"line": 138,
"column": 76
} | {
"line": 140,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ne : α ≃ β\ninst✝ : NonAssocRing β\n⊢ NonAssocRing α",
"ppTerm": "?m.1",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Int.cast",
"NegZeroClass.toNeg",
"Equiv.apply_symm_apply",
"instHSMul",
"Equiv.instEq... | [] | let add_group_with_one := e.addGroupWithOne
let mul := e.mul
apply e.injective.nonAssocRing _ <;> intros <;> exact e.apply_symm_apply _ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Ring.TransferInstance | {
"line": 156,
"column": 2
} | {
"line": 156,
"column": 50
} | {
"line": 156,
"column": 51
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ne : α ≃ β\ninst✝ : NonUnitalCommRing β\nzero : Zero α := e.zero\nadd : Add α := e.add\nmul : Mul α := e.mul\nneg : Neg α := e.Neg\nsub : Sub α := e.sub\nnsmul : SMul ℕ α := Equiv.smul ℕ e\nzsmul : SMul ℤ α := Equiv.smul ℤ e\n⊢ NonUnitalCommRing α",
"ppTerm": "?m.42",
... | [
"case zero\nα : Type u_1\nβ : Type u_2\ne : α ≃ β\ninst✝ : NonUnitalCommRing β\nzero : Zero α := ⋯\nadd : Add α := ⋯\nmul : Mul α := ⋯\nneg : Neg α := ⋯\nsub : Sub α := ⋯\nnsmul : SMul ℕ α := ⋯\nzsmul : SMul ℤ α := ⋯\n⊢ e 0 = 0",
"case add\nα : Type u_1\nβ : Type u_2\ne : α ≃ β\ninst✝ : NonUnitalCommRing β\nzero ... | apply e.injective.nonUnitalCommRing _ <;> intros | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.RingTheory.TensorProduct.Basic | {
"line": 267,
"column": 2
} | {
"line": 269,
"column": 36
} | {
"line": 271,
"column": 0
} | [
{
"pp": "R : Type uR\nA : Type uA\nB : Type uB\ninst✝⁴ : CommSemiring R\ninst✝³ : Semiring A\ninst✝² : Algebra R A\ninst✝¹ : Semiring B\ninst✝ : Algebra R B\na : A\nb : B\nk : ℕ\n⊢ a ⊗ₜ[R] b ^ k = (a ^ k) ⊗ₜ[R] (b ^ k)",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"NonAssocSemiring... | [] | induction k with
| zero => simp [one_def]
| succ k ih => simp [pow_succ, ih] | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.RingTheory.TensorProduct.Basic | {
"line": 267,
"column": 2
} | {
"line": 269,
"column": 36
} | {
"line": 271,
"column": 0
} | [
{
"pp": "R : Type uR\nA : Type uA\nB : Type uB\ninst✝⁴ : CommSemiring R\ninst✝³ : Semiring A\ninst✝² : Algebra R A\ninst✝¹ : Semiring B\ninst✝ : Algebra R B\na : A\nb : B\nk : ℕ\n⊢ a ⊗ₜ[R] b ^ k = (a ^ k) ⊗ₜ[R] (b ^ k)",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"NonAssocSemiring... | [] | induction k with
| zero => simp [one_def]
| succ k ih => simp [pow_succ, ih] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.TensorProduct.Basic | {
"line": 267,
"column": 2
} | {
"line": 269,
"column": 36
} | {
"line": 271,
"column": 0
} | [
{
"pp": "R : Type uR\nA : Type uA\nB : Type uB\ninst✝⁴ : CommSemiring R\ninst✝³ : Semiring A\ninst✝² : Algebra R A\ninst✝¹ : Semiring B\ninst✝ : Algebra R B\na : A\nb : B\nk : ℕ\n⊢ a ⊗ₜ[R] b ^ k = (a ^ k) ⊗ₜ[R] (b ^ k)",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"NonAssocSemiring... | [] | induction k with
| zero => simp [one_def]
| succ k ih => simp [pow_succ, ih] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Algebra.Subalgebra.Centralizer | {
"line": 44,
"column": 57
} | {
"line": 44,
"column": 77
} | {
"line": 44,
"column": 77
} | [
{
"pp": "R : Type u_1\ninst✝² : CommSemiring R\nA : Type u_2\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS T K : Subalgebra R A\n⊢ S ≤ centralizer R ↑K ∧ T ≤ centralizer R ↑K ↔ K ≤ centralizer R ↑S ∧ K ≤ centralizer R ↑T",
"ppTerm": "?m.40",
"assigned": true,
"usedConstants": [
"Subalgebra.inst... | [] | K.le_centralizer_iff | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Algebra.Algebra.Subalgebra.Centralizer | {
"line": 49,
"column": 59
} | {
"line": 49,
"column": 79
} | {
"line": 49,
"column": 79
} | [
{
"pp": "R : Type u_1\ninst✝² : CommSemiring R\nA : Type u_2\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nι : Sort u_3\nS : ι → Subalgebra R A\nK : Subalgebra R A\n⊢ (∀ (i : ι), S i ≤ centralizer R ↑K) ↔ ∀ (i : ι), K ≤ centralizer R ↑(S i)",
"ppTerm": "?m.41",
"assigned": true,
"usedConstants": [
... | [] | K.le_centralizer_iff | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Algebra.Order.Star.Basic | {
"line": 62,
"column": 29
} | {
"line": 62,
"column": 67
} | {
"line": 63,
"column": 2
} | [
{
"pp": "case add\nR : Type u_1\nA : Type u_2\ninst✝⁷ : Semiring 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\nr : R\na : A\nha : a ∈ AddSubmonoid.closure (range fun s ↦ star s *... | [] | simpa [add_smul] using add_mem hr₁ hr₂ | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Algebra.Order.Star.Basic | {
"line": 62,
"column": 29
} | {
"line": 62,
"column": 67
} | {
"line": 63,
"column": 2
} | [
{
"pp": "case add\nR : Type u_1\nA : Type u_2\ninst✝⁷ : Semiring 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\nr : R\na : A\nha : a ∈ AddSubmonoid.closure (range fun s ↦ star s *... | [] | simpa [add_smul] using add_mem hr₁ hr₂ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Order.Star.Basic | {
"line": 62,
"column": 29
} | {
"line": 62,
"column": 67
} | {
"line": 63,
"column": 2
} | [
{
"pp": "case add\nR : Type u_1\nA : Type u_2\ninst✝⁷ : Semiring 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\nr : R\na : A\nha : a ∈ AddSubmonoid.closure (range fun s ↦ star s *... | [] | simpa [add_smul] using add_mem hr₁ hr₂ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Algebra.Subalgebra.Centralizer | {
"line": 117,
"column": 76
} | {
"line": 117,
"column": 81
} | {
"line": 117,
"column": 81
} | [
{
"pp": "R : Type u_1\ninst✝⁵ : CommSemiring R\nA : Type u_2\ninst✝⁴ : Semiring A\ninst✝³ : Algebra R A\nB : Type u_3\ninst✝² : Semiring B\ninst✝¹ : Algebra R B\nS : Set B\ninst✝ : Module.Free R A\neq1 :\n comap (↑(Algebra.TensorProduct.comm R A B)) (centralizer R (⇑includeLeft '' S)) =\n comap (↑(Algebra.T... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Algebra.Subalgebra.Centralizer | {
"line": 117,
"column": 76
} | {
"line": 117,
"column": 81
} | {
"line": 117,
"column": 81
} | [
{
"pp": "R : Type u_1\ninst✝⁵ : CommSemiring R\nA : Type u_2\ninst✝⁴ : Semiring A\ninst✝³ : Algebra R A\nB : Type u_3\ninst✝² : Semiring B\ninst✝¹ : Algebra R B\nS : Set B\ninst✝ : Module.Free R A\neq1 :\n comap (↑(Algebra.TensorProduct.comm R A B)) (centralizer R (⇑includeLeft '' S)) =\n comap (↑(Algebra.T... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Algebra.Subalgebra.Centralizer | {
"line": 117,
"column": 76
} | {
"line": 117,
"column": 81
} | {
"line": 117,
"column": 81
} | [
{
"pp": "R : Type u_1\ninst✝⁵ : CommSemiring R\nA : Type u_2\ninst✝⁴ : Semiring A\ninst✝³ : Algebra R A\nB : Type u_3\ninst✝² : Semiring B\ninst✝¹ : Algebra R B\nS : Set B\ninst✝ : Module.Free R A\neq1 :\n comap (↑(Algebra.TensorProduct.comm R A B)) (centralizer R (⇑includeLeft '' S)) =\n comap (↑(Algebra.T... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Algebra.Subalgebra.Centralizer | {
"line": 118,
"column": 59
} | {
"line": 118,
"column": 64
} | {
"line": 118,
"column": 64
} | [
{
"pp": "R : Type u_1\ninst✝⁵ : CommSemiring R\nA : Type u_2\ninst✝⁴ : Semiring A\ninst✝³ : Algebra R A\nB : Type u_3\ninst✝² : Semiring B\ninst✝¹ : Algebra R B\nS : Set B\ninst✝ : Module.Free R A\neq1 :\n comap (↑(Algebra.TensorProduct.comm R A B)) (centralizer R (⇑includeLeft '' S)) =\n comap (↑(Algebra.T... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Algebra.Subalgebra.Centralizer | {
"line": 118,
"column": 59
} | {
"line": 118,
"column": 64
} | {
"line": 118,
"column": 64
} | [
{
"pp": "R : Type u_1\ninst✝⁵ : CommSemiring R\nA : Type u_2\ninst✝⁴ : Semiring A\ninst✝³ : Algebra R A\nB : Type u_3\ninst✝² : Semiring B\ninst✝¹ : Algebra R B\nS : Set B\ninst✝ : Module.Free R A\neq1 :\n comap (↑(Algebra.TensorProduct.comm R A B)) (centralizer R (⇑includeLeft '' S)) =\n comap (↑(Algebra.T... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Algebra.Subalgebra.Centralizer | {
"line": 118,
"column": 59
} | {
"line": 118,
"column": 64
} | {
"line": 118,
"column": 64
} | [
{
"pp": "R : Type u_1\ninst✝⁵ : CommSemiring R\nA : Type u_2\ninst✝⁴ : Semiring A\ninst✝³ : Algebra R A\nB : Type u_3\ninst✝² : Semiring B\ninst✝¹ : Algebra R B\nS : Set B\ninst✝ : Module.Free R A\neq1 :\n comap (↑(Algebra.TensorProduct.comm R A B)) (centralizer R (⇑includeLeft '' S)) =\n comap (↑(Algebra.T... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Algebra.Subalgebra.Directed | {
"line": 33,
"column": 2
} | {
"line": 36,
"column": 64
} | {
"line": 37,
"column": 2
} | [
{
"pp": "R : Type u_1\nA : Type u_2\ninst✝³ : CommSemiring R\ninst✝² : Semiring A\ninst✝¹ : Algebra R A\nι : Type u_4\ninst✝ : Nonempty ι\nK : ι → Subalgebra R A\ndir : Directed (fun x1 x2 ↦ x1 ≤ x2) K\n⊢ ↑(iSup K) = ⋃ i, ↑(K i)",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"Iff.mp... | [
"R : Type u_1\nA : Type u_2\ninst✝³ : CommSemiring R\ninst✝² : Semiring A\ninst✝¹ : Algebra R A\nι : Type u_4\ninst✝ : Nonempty ι\nK : ι → Subalgebra R A\ndir : Directed (fun x1 x2 ↦ x1 ≤ x2) K\ns : Subalgebra R A :=\n let __spread.0 := (⨆ i, (K i).toSubsemiring).copy (⋃ i, ↑(K i).toSubsemiring) ⋯;\n { toSubsemir... | let s : Subalgebra R A :=
{ __ := Subsemiring.copy _ _ (Subsemiring.coe_iSup_of_directed dir).symm
algebraMap_mem' := fun _ ↦ Set.mem_iUnion.2
⟨Classical.arbitrary ι, Subalgebra.algebraMap_mem _ _⟩ } | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.Algebra.Order.Star.Basic | {
"line": 279,
"column": 8
} | {
"line": 279,
"column": 59
} | {
"line": 279,
"column": 59
} | [
{
"pp": "R : Type u_1\ninst✝³ : NonUnitalSemiring R\ninst✝² : PartialOrder R\ninst✝¹ : StarRing R\ninst✝ : StarOrderedRing R\nx y : R\nthis : ∀ (x y : R), x ≤ y → star x ≤ star y\n⊢ star x ≤ star y → x ≤ y",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"star_star",
"congrArg",... | [] | simpa only [star_star] using this (star x) (star y) | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Algebra.Order.Star.Basic | {
"line": 279,
"column": 8
} | {
"line": 279,
"column": 59
} | {
"line": 279,
"column": 59
} | [
{
"pp": "R : Type u_1\ninst✝³ : NonUnitalSemiring R\ninst✝² : PartialOrder R\ninst✝¹ : StarRing R\ninst✝ : StarOrderedRing R\nx y : R\nthis : ∀ (x y : R), x ≤ y → star x ≤ star y\n⊢ star x ≤ star y → x ≤ y",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"star_star",
"congrArg",... | [] | simpa only [star_star] using this (star x) (star y) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Order.Star.Basic | {
"line": 279,
"column": 8
} | {
"line": 279,
"column": 59
} | {
"line": 279,
"column": 59
} | [
{
"pp": "R : Type u_1\ninst✝³ : NonUnitalSemiring R\ninst✝² : PartialOrder R\ninst✝¹ : StarRing R\ninst✝ : StarOrderedRing R\nx y : R\nthis : ∀ (x y : R), x ≤ y → star x ≤ star y\n⊢ star x ≤ star y → x ≤ y",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"star_star",
"congrArg",... | [] | simpa only [star_star] using this (star x) (star y) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Order.Star.Basic | {
"line": 307,
"column": 51
} | {
"line": 308,
"column": 48
} | {
"line": 310,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝³ : NonUnitalSemiring R\ninst✝² : PartialOrder R\ninst✝¹ : StarRing R\ninst✝ : StarOrderedRing R\nx : R\n⊢ 0 < star x ↔ 0 < x",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Preorder.toLT",
"congrArg",
"AddMonoid.toAddZeroClass",
"Partia... | [] | by
simpa using star_lt_star_iff (x := 0) (y := x) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.Dimension.Constructions | {
"line": 67,
"column": 2
} | {
"line": 67,
"column": 7
} | {
"line": 69,
"column": 0
} | [
{
"pp": "case e'_5\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⊢ f '' t = range fun x ↦ f ↑x",
"ppTerm": "?e'_5",
"assigned": true,
"usedConst... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Order.Star.Basic | {
"line": 458,
"column": 12
} | {
"line": 458,
"column": 17
} | {
"line": 460,
"column": 0
} | [
{
"pp": "case mem\nR : Type u_4\nS : Type u_5\ninst✝⁷ : NonUnitalSemiring R\ninst✝⁶ : PartialOrder R\ninst✝⁵ : StarRing R\ninst✝⁴ : StarOrderedRing R\ninst✝³ : NonUnitalSemiring S\ninst✝² : PartialOrder S\ninst✝¹ : StarRing S\ninst✝ : StarOrderedRing S\nf : R →⋆ₙ+* S\nx p : R\nhf : ∀ (r : R), f (star r) = star ... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
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