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 375
values | kind stringclasses 379
values |
|---|---|---|---|---|---|---|---|---|
Mathlib.Algebra.Lie.Cochain | {
"line": 108,
"column": 33
} | {
"line": 108,
"column": 48
} | {
"line": 108,
"column": 49
} | [
{
"pp": "R : Type u_1\ninst✝⁶ : CommRing R\nL : Type u_2\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra R L\nM : Type u_3\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\ninst✝ : LieModule R L M\nf : oneCochain R L M\nx✝² : R\nx✝¹ x✝ : L\n⊢ { toFun := fun y ↦ ⁅x✝² • x✝¹, f y⁆ - ⁅y, f (x✝² • ... | [] | simp [smul_sub] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Lie.Cochain | {
"line": 111,
"column": 27
} | {
"line": 111,
"column": 42
} | {
"line": 113,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝⁶ : CommRing R\nL : Type u_2\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra R L\nM : Type u_3\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\ninst✝ : LieModule R L M\nx✝³ : R\nx✝² : oneCochain R L M\nx✝¹ x✝ : L\n⊢ (↑⟨{\n toFun := fun x ↦\n ... | [] | simp [smul_sub] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.LinearAlgebra.Matrix.Rank | {
"line": 472,
"column": 56
} | {
"line": 472,
"column": 68
} | {
"line": 473,
"column": 4
} | [
{
"pp": "m : Type um\nR : Type uR\ninst✝¹ : Field R\ninst✝ : DecidableEq m\nw : m → R\nw' : { i // w i ≠ 0 } → m → R := fun i ↦ diagonal w ↑i\nh : LinearIndependent R w'\nhrw : insert 0 (range (diagonal w).col) = insert 0 (range w')\n⊢ Module.rank R ↥(span R (range w')) = lift.{uR, um} #{ i // w i ≠ 0 }",
"... | [
"m : Type um\nR : Type uR\ninst✝¹ : Field R\ninst✝ : DecidableEq m\nw : m → R\nw' : { i // w i ≠ 0 } → m → R := fun i ↦ diagonal w ↑i\nh : LinearIndependent R w'\nhrw : insert 0 (range (diagonal w).col) = insert 0 (range w')\n⊢ #↑(range w') = lift.{uR, um} #{ i // w i ≠ 0 }"
] | rank_span h, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Lie.DirectSum | {
"line": 52,
"column": 19
} | {
"line": 54,
"column": 50
} | {
"line": 55,
"column": 2
} | [
{
"pp": "R : Type u\nι : Type v\ninst✝⁶ : CommRing R\nL : Type w₁\nM : ι → Type w\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra R L\ninst✝³ : (i : ι) → AddCommGroup (M i)\ninst✝² : (i : ι) → Module R (M i)\ninst✝¹ : (i : ι) → LieRingModule L (M i)\ninst✝ : ∀ (i : ι), LieModule R L (M i)\nx : L\nm n : ⨁ (i : ι), M i\... | [] | by
ext
simp only [mapRange_apply, add_apply, lie_add] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Lie.DirectSum | {
"line": 212,
"column": 6
} | {
"line": 213,
"column": 50
} | {
"line": 213,
"column": 51
} | [
{
"pp": "case inr\nR : Type u\nι : Type v\ninst✝⁵ : CommRing R\nL : ι → Type w\ninst✝⁴ : (i : ι) → LieRing (L i)\ninst✝³ : (i : ι) → LieAlgebra R (L i)\ninst✝² : DecidableEq ι\nL' : Type w₁\ninst✝¹ : LieRing L'\ninst✝ : LieAlgebra R L'\nf : (i : ι) → L i →ₗ⁅R⁆ L'\nhf : Pairwise fun i j ↦ ∀ (x : L i) (y : L j), ... | [] | · simp_rw [lie_of_of_ne _ hij.symm, map_zero, LinearMap.toAddMonoidHom_coe,
LieHom.coe_toLinearMap, hf hij.symm x y] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Lie.UniversalEnveloping | {
"line": 85,
"column": 8
} | {
"line": 85,
"column": 80
} | {
"line": 86,
"column": 6
} | [
{
"pp": "R : Type u₁\nL : Type u₂\ninst✝² : CommRing R\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nx y : L\nthis : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)\n⊢ failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)",
... | [] | rw [map_mul] at this; simp [LieRing.of_associative_ring_bracket, ← this] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Lie.UniversalEnveloping | {
"line": 85,
"column": 8
} | {
"line": 85,
"column": 80
} | {
"line": 86,
"column": 6
} | [
{
"pp": "R : Type u₁\nL : Type u₂\ninst✝² : CommRing R\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nx y : L\nthis : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)\n⊢ failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)",
... | [] | rw [map_mul] at this; simp [LieRing.of_associative_ring_bracket, ← this] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Lie.Free | {
"line": 255,
"column": 6
} | {
"line": 255,
"column": 23
} | {
"line": 255,
"column": 23
} | [
{
"pp": "R : Type u\nX : Type v\ninst✝² : CommRing R\nL : Type w\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nF : FreeLieAlgebra R X →ₗ⁅R⁆ L\n⊢ (lift R) (⇑F ∘ of R) = F",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"LieHom",
"Eq.mpr",
"Equiv.instEquivLike",
"FreeL... | [
"R : Type u\nX : Type v\ninst✝² : CommRing R\nL : Type w\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nF : FreeLieAlgebra R X →ₗ⁅R⁆ L\n⊢ (lift R) ((lift R).symm F) = F"
] | ← lift_symm_apply | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Lie.LieTheorem | {
"line": 62,
"column": 51
} | {
"line": 62,
"column": 54
} | {
"line": 62,
"column": 55
} | [
{
"pp": "R : Type u_1\nL : Type u_2\nA : Type u_3\nV : Type u_4\ninst✝¹⁹ : CommRing R\ninst✝¹⁸ : IsPrincipalIdealRing R\ninst✝¹⁷ : IsDomain R\ninst✝¹⁶ : CharZero R\ninst✝¹⁵ : LieRing L\ninst✝¹⁴ : LieAlgebra R L\ninst✝¹³ : LieRing A\ninst✝¹² : LieAlgebra R A\ninst✝¹¹ : Bracket L A\ninst✝¹⁰ : Bracket A L\ninst✝⁹ ... | [
"R : Type u_1\nL : Type u_2\nA : Type u_3\nV : Type u_4\ninst✝¹⁹ : CommRing R\ninst✝¹⁸ : IsPrincipalIdealRing R\ninst✝¹⁷ : IsDomain R\ninst✝¹⁶ : CharZero R\ninst✝¹⁵ : LieRing L\ninst✝¹⁴ : LieAlgebra R L\ninst✝¹³ : LieRing A\ninst✝¹² : LieAlgebra R A\ninst✝¹¹ : Bracket L A\ninst✝¹⁰ : Bracket A L\ninst✝⁹ : AddCommGro... | hv, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.TensorProduct.Decomposition | {
"line": 115,
"column": 4
} | {
"line": 118,
"column": 64
} | {
"line": 119,
"column": 2
} | [
{
"pp": "case of\nι : Type u_1\nR : Type u_2\nM : Type u_3\ninst✝⁶ : CommSemiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\nℳ : ι → Submodule R M\nN : Type u_5\ninst✝³ : AddCommMonoid N\ninst✝² : Module R N\ninst✝¹ : DecidableEq ι\ninst✝ : Decomposition ℳ\ni : ι\ny : ↥(ℳ i) ⊗[R] N\nthis :\n (rTensor N ... | [] | rw [coeAddMonoidHom_of, LinearEquiv.eq_symm_apply, LinearEquiv.eq_symm_apply,
← (LinearEquiv.rTensor N _).coe_coe, LinearEquiv.coe_rTensor, ← rTensor_comp_apply,
decomposeLinearEquiv_comp_subtype, this, LinearEquiv.apply_symm_apply,
decomposeTensorEquiv_of_apply, decomposeTensorEquiv_apply] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.GroupTheory.Perm.Cycle.Concrete | {
"line": 81,
"column": 6
} | {
"line": 81,
"column": 16
} | {
"line": 82,
"column": 6
} | [
{
"pp": "case h.right\nα : Type u_1\ninst✝ : DecidableEq α\ny : α\nl : List α\ntail_ih✝ : ∀ (x : α), (x :: l).Nodup → 2 ≤ (x :: l).length → (x :: l).formPerm.IsCycle\nx : α\nhl : (x :: y :: l).Nodup\nhn : 2 ≤ (x :: y :: l).length\n⊢ ∀ ⦃y_1 : α⦄, (x :: y :: l).formPerm y_1 ≠ y_1 → (x :: y :: l).formPerm.SameCycl... | [
"case h.right\nα : Type u_1\ninst✝ : DecidableEq α\ny : α\nl : List α\ntail_ih✝ : ∀ (x : α), (x :: l).Nodup → 2 ≤ (x :: l).length → (x :: l).formPerm.IsCycle\nx : α\nhl : (x :: y :: l).Nodup\nhn : 2 ≤ (x :: y :: l).length\nw : α\nhw : (x :: y :: l).formPerm w ≠ w\n⊢ (x :: y :: l).formPerm.SameCycle x w"
] | intro w hw | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Algebra.Lie.Extension | {
"line": 378,
"column": 55
} | {
"line": 378,
"column": 67
} | {
"line": 378,
"column": 67
} | [
{
"pp": "R : Type u_1\nL : Type u_3\nM : Type u_4\ninst✝⁵ : CommRing R\ninst✝⁴ : LieRing L\ninst✝³ : LieAlgebra R L\ninst✝² : LieRing M\ninst✝¹ : LieAlgebra R M\ninst✝ : IsLieAbelian M\nE : Extension R M L\ns : L →ₗ[R] E.L\nhs : LeftInverse ⇑E.proj ⇑s\nx : E.L\ny : ↥E.proj.ker\n⊢ s (E.proj x) - x ∈ E.proj.ker",... | [] | simp [hs.eq] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Lie.Extension | {
"line": 378,
"column": 55
} | {
"line": 378,
"column": 67
} | {
"line": 378,
"column": 67
} | [
{
"pp": "R : Type u_1\nL : Type u_3\nM : Type u_4\ninst✝⁵ : CommRing R\ninst✝⁴ : LieRing L\ninst✝³ : LieAlgebra R L\ninst✝² : LieRing M\ninst✝¹ : LieAlgebra R M\ninst✝ : IsLieAbelian M\nE : Extension R M L\ns : L →ₗ[R] E.L\nhs : LeftInverse ⇑E.proj ⇑s\nx : E.L\ny : ↥E.proj.ker\n⊢ s (E.proj x) - x ∈ E.proj.ker",... | [] | simp [hs.eq] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Lie.Extension | {
"line": 378,
"column": 55
} | {
"line": 378,
"column": 67
} | {
"line": 378,
"column": 67
} | [
{
"pp": "R : Type u_1\nL : Type u_3\nM : Type u_4\ninst✝⁵ : CommRing R\ninst✝⁴ : LieRing L\ninst✝³ : LieAlgebra R L\ninst✝² : LieRing M\ninst✝¹ : LieAlgebra R M\ninst✝ : IsLieAbelian M\nE : Extension R M L\ns : L →ₗ[R] E.L\nhs : LeftInverse ⇑E.proj ⇑s\nx : E.L\ny : ↥E.proj.ker\n⊢ s (E.proj x) - x ∈ E.proj.ker",... | [] | simp [hs.eq] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Lie.Extension | {
"line": 386,
"column": 44
} | {
"line": 386,
"column": 56
} | {
"line": 386,
"column": 56
} | [
{
"pp": "R : Type u_1\nN : Type u_2\nL : Type u_3\nM : Type u_4\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieAlgebra R L\ninst✝¹ : LieRing M\ninst✝ : LieAlgebra R M\nE : Extension R M L\ns : L →ₗ[R] E.L\nhs : LeftInverse ⇑E.proj ⇑s\nx y : L\n⊢ ⁅s x, s y⁆ - s ⁅x, y⁆ ∈ E.proj.ker",
"ppTerm": "?m.112"... | [] | simp [hs.eq] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Lie.Extension | {
"line": 386,
"column": 44
} | {
"line": 386,
"column": 56
} | {
"line": 386,
"column": 56
} | [
{
"pp": "R : Type u_1\nN : Type u_2\nL : Type u_3\nM : Type u_4\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieAlgebra R L\ninst✝¹ : LieRing M\ninst✝ : LieAlgebra R M\nE : Extension R M L\ns : L →ₗ[R] E.L\nhs : LeftInverse ⇑E.proj ⇑s\nx y : L\n⊢ ⁅s x, s y⁆ - s ⁅x, y⁆ ∈ E.proj.ker",
"ppTerm": "?m.112"... | [] | simp [hs.eq] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Lie.Extension | {
"line": 386,
"column": 44
} | {
"line": 386,
"column": 56
} | {
"line": 386,
"column": 56
} | [
{
"pp": "R : Type u_1\nN : Type u_2\nL : Type u_3\nM : Type u_4\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieAlgebra R L\ninst✝¹ : LieRing M\ninst✝ : LieAlgebra R M\nE : Extension R M L\ns : L →ₗ[R] E.L\nhs : LeftInverse ⇑E.proj ⇑s\nx y : L\n⊢ ⁅s x, s y⁆ - s ⁅x, y⁆ ∈ E.proj.ker",
"ppTerm": "?m.112"... | [] | simp [hs.eq] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Lie.Extension | {
"line": 388,
"column": 26
} | {
"line": 388,
"column": 41
} | {
"line": 388,
"column": 42
} | [
{
"pp": "R : Type u_1\nN : Type u_2\nL : Type u_3\nM : Type u_4\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieAlgebra R L\ninst✝¹ : LieRing M\ninst✝ : LieAlgebra R M\nE : Extension R M L\ns : L →ₗ[R] E.L\nhs : LeftInverse ⇑E.proj ⇑s\nx : L\nx✝¹ : R\nx✝ : L\n⊢ ⟨⁅s x, s (x✝¹ • x✝)⁆ - s ⁅x, x✝¹ • x✝⁆, ⋯⟩ =... | [] | simp [smul_sub] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Lie.Extension | {
"line": 388,
"column": 26
} | {
"line": 388,
"column": 41
} | {
"line": 388,
"column": 42
} | [
{
"pp": "R : Type u_1\nN : Type u_2\nL : Type u_3\nM : Type u_4\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieAlgebra R L\ninst✝¹ : LieRing M\ninst✝ : LieAlgebra R M\nE : Extension R M L\ns : L →ₗ[R] E.L\nhs : LeftInverse ⇑E.proj ⇑s\nx : L\nx✝¹ : R\nx✝ : L\n⊢ ⟨⁅s x, s (x✝¹ • x✝)⁆ - s ⁅x, x✝¹ • x✝⁆, ⋯⟩ =... | [] | simp [smul_sub] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Lie.Extension | {
"line": 388,
"column": 26
} | {
"line": 388,
"column": 41
} | {
"line": 388,
"column": 42
} | [
{
"pp": "R : Type u_1\nN : Type u_2\nL : Type u_3\nM : Type u_4\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieAlgebra R L\ninst✝¹ : LieRing M\ninst✝ : LieAlgebra R M\nE : Extension R M L\ns : L →ₗ[R] E.L\nhs : LeftInverse ⇑E.proj ⇑s\nx : L\nx✝¹ : R\nx✝ : L\n⊢ ⟨⁅s x, s (x✝¹ • x✝)⁆ - s ⁅x, x✝¹ • x✝⁆, ⋯⟩ =... | [] | simp [smul_sub] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Lie.Extension | {
"line": 390,
"column": 27
} | {
"line": 390,
"column": 42
} | {
"line": 392,
"column": 0
} | [
{
"pp": "R : Type u_1\nN : Type u_2\nL : Type u_3\nM : Type u_4\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieAlgebra R L\ninst✝¹ : LieRing M\ninst✝ : LieAlgebra R M\nE : Extension R M L\ns : L →ₗ[R] E.L\nhs : LeftInverse ⇑E.proj ⇑s\nx✝² : R\nx✝¹ x✝ : L\n⊢ ↑({ toFun := fun y ↦ ⟨⁅s (x✝² • x✝¹), s y⁆ - s ... | [] | simp [smul_sub] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Lie.Extension | {
"line": 446,
"column": 2
} | {
"line": 446,
"column": 37
} | {
"line": 447,
"column": 2
} | [
{
"pp": "R : Type u_1\nL : Type u_3\nM : Type u_4\ninst✝⁵ : CommRing R\ninst✝⁴ : LieRing L\ninst✝³ : LieAlgebra R L\ninst✝² : LieRing M\ninst✝¹ : LieAlgebra R M\ninst✝ : IsLieAbelian M\nE : Extension R M L\ns₁ s₂ : L →ₗ[R] E.L\nhs₁ : LeftInverse ⇑E.proj ⇑s₁\nhs₂ : LeftInverse ⇑E.proj ⇑s₂\nx y : L\n⊢ (↑((d₁₂ R L... | [
"R : Type u_1\nL : Type u_3\nM : Type u_4\ninst✝⁵ : CommRing R\ninst✝⁴ : LieRing L\ninst✝³ : LieAlgebra R L\ninst✝² : LieRing M\ninst✝¹ : LieAlgebra R M\ninst✝ : IsLieAbelian M\nE : Extension R M L\ns₁ s₂ : L →ₗ[R] E.L\nhs₁ : LeftInverse ⇑E.proj ⇑s₁\nhs₂ : LeftInverse ⇑E.proj ⇑s₂\nx y : L\ns : L → E.L\nhs : ∀ (b : ... | choose s hs using E.proj_surjective | Mathlib.Tactic.Choose._aux_Mathlib_Tactic_Choose___elabRules_Mathlib_Tactic_Choose_choose_1 | Mathlib.Tactic.Choose.choose |
Mathlib.LinearAlgebra.Matrix.Determinant.Bird.Correctness | {
"line": 274,
"column": 6
} | {
"line": 274,
"column": 60
} | {
"line": 275,
"column": 6
} | [
{
"pp": "case neg\nR : Type u_1\ninst✝ : CommRing R\nn : ℕ\nA : Matrix (Fin n) (Fin n) R\np : ℕ\ni j : Fin n\nα : Fin p → Fin n\nk : Fin n\nhα : α ∈ S p i\nhk : i < k\nhoccurs : k ∉ Set.range α\nhnotmem : ¬bminor A i (α, k).2 (α, k).1 * A (α, k).2 j = 0\n⊢ (α, k) ∈\n (fun x ↦\n match x with\n |... | [
"case neg\nR : Type u_1\ninst✝ : CommRing R\nn : ℕ\nA : Matrix (Fin n) (Fin n) R\np : ℕ\ni j : Fin n\nα : Fin p → Fin n\nk : Fin n\nhα : α ∈ S p i\nhk : i < k\nhoccurs : k ∉ Set.range α\nhnotmem : ¬bminor A i (α, k).2 (α, k).1 * A (α, k).2 j = 0\nt : Fin (p + 1)\nht : t.insertNth k α ∈ S (p + 1) i\n⊢ (α, k) ∈\n ... | obtain ⟨t, ht⟩ := exists_insertNth_mem_S hα hk hoccurs | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.RingTheory.ChainOfDivisors | {
"line": 52,
"column": 16
} | {
"line": 52,
"column": 18
} | {
"line": 52,
"column": 18
} | [
{
"pp": "M : Type u_1\ninst✝¹ : CommMonoidWithZero M\ninst✝ : IsCancelMulZero M\np : Associates M\nh₁ : p ≠ 0\nhp : IsAtom p\na b : Associates M\nh : p = a * b\nha : a = p\n⊢ IsUnit b",
"ppTerm": "?m.80",
"assigned": true,
"usedConstants": [
"HMul.hMul",
"Monoid.toMulOneClass",
"co... | [
"M : Type u_1\ninst✝¹ : CommMonoidWithZero M\ninst✝ : IsCancelMulZero M\np : Associates M\nh₁ : p ≠ 0\nhp : IsAtom p\na b : Associates M\nh : p = p * b\nha : a = p\n⊢ IsUnit b"
] | ha | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.ChainOfDivisors | {
"line": 192,
"column": 6
} | {
"line": 197,
"column": 36
} | {
"line": 198,
"column": 4
} | [] | [
"M : Type u_1\ninst✝¹ : CommMonoidWithZero M\ninst✝ : UniqueFactorizationMonoid M\nq : Associates M\nn : ℕ\nhn : n ≠ 0\nc : Fin (n + 1) → Associates M\nh₁ : StrictMono c\nh₂ : ∀ {r : Associates M}, r ≤ q ↔ ∃ i, r = c i\nhq : q ≠ 0\ni : Fin (n + 1)\nhi' : q = c 1 ^ ↑i\n⊢ Finset.image c Finset.univ ⊆ Finset.image (fu... | n + 1 = (Finset.univ : Finset (Fin (n + 1))).card := (Finset.card_fin _).symm
_ = (Finset.univ.image c).card := (Finset.card_image_iff.mpr h₁.injective.injOn).symm
_ ≤ (Finset.univ.image fun m : Fin (i + 1) => c 1 ^ (m : ℕ)).card :=
(Finset.card_le_card ?_)
_ ≤ (Finset.univ : Finset (Fin (i + ... | Lean.Elab.Tactic._aux_Mathlib_Tactic_Widget_Calc___elabRules_Lean_calcTactic_1 | Lean.calcSteps |
Mathlib.RingTheory.FractionalIdeal.Basic | {
"line": 649,
"column": 16
} | {
"line": 649,
"column": 18
} | {
"line": 649,
"column": 19
} | [
{
"pp": "case mp.refine_1\nR : Type u_1\ninst✝² : CommRing R\nS : Submonoid R\nP : Type u_2\ninst✝¹ : CommRing P\ninst✝ : Algebra R P\nJ : FractionalIdeal S P\nhJ : J ≤ 1\na b : R\n⊢ a ∈ {x | (algebraMap R P) x ∈ J} → b ∈ {x | (algebraMap R P) x ∈ J} → a + b ∈ {x | (algebraMap R P) x ∈ J}",
"ppTerm": "?mp.r... | [
"case mp.refine_1\nR : Type u_1\ninst✝² : CommRing R\nS : Submonoid R\nP : Type u_2\ninst✝¹ : CommRing P\ninst✝ : Algebra R P\nJ : FractionalIdeal S P\nhJ : J ≤ 1\na b : R\nha : a ∈ {x | (algebraMap R P) x ∈ J}\n⊢ b ∈ {x | (algebraMap R P) x ∈ J} → a + b ∈ {x | (algebraMap R P) x ∈ J}"
] | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.RingTheory.ChainOfDivisors | {
"line": 277,
"column": 2
} | {
"line": 279,
"column": 51
} | {
"line": 280,
"column": 2
} | [
{
"pp": "case neg.refine_3\nM : Type u_1\ninst✝⁴ : CommMonoidWithZero M\ninst✝³ : IsCancelMulZero M\nN : Type u_2\ninst✝² : CommMonoidWithZero N\ninst✝¹ : UniqueFactorizationMonoid N\ninst✝ : UniqueFactorizationMonoid M\nm : Associates M\nn : Associates N\nhn : n ≠ 0\nd : ↑(Set.Iic m) ≃o ↑(Set.Iic n)\ns : ℕ\nhs... | [
"case neg.refine_4\nM : Type u_1\ninst✝⁴ : CommMonoidWithZero M\ninst✝³ : IsCancelMulZero M\nN : Type u_2\ninst✝² : CommMonoidWithZero N\ninst✝¹ : UniqueFactorizationMonoid N\ninst✝ : UniqueFactorizationMonoid M\nm : Associates M\nn : Associates N\nhn : n ≠ 0\nd : ↑(Set.Iic m) ≃o ↑(Set.Iic n)\ns : ℕ\nhs : ¬s = 0\nc... | · rintro ⟨i, hr⟩
rw [hr, c₂_def, Subtype.coe_le_coe, d.le_iff_le]
simpa [Subtype.mk_le_mk] using hc₁''.2 ⟨i, rfl⟩ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.ChainOfDivisors | {
"line": 298,
"column": 19
} | {
"line": 301,
"column": 10
} | {
"line": 302,
"column": 4
} | [
{
"pp": "M : Type u_1\ninst✝⁴ : CommMonoidWithZero M\ninst✝³ : IsCancelMulZero M\nN : Type u_2\ninst✝² : CommMonoidWithZero N\ninst✝¹ : UniqueFactorizationMonoid N\ninst✝ : UniqueFactorizationMonoid M\nm p : Associates M\nn : Associates N\nhn : n ≠ 0\nhp : p ∈ normalizedFactors m\nd : ↑(Set.Iic m) ≃o ↑(Set.Iic ... | [] | by
rw [this, OrderIso.map_bot d] at hx
refine (Subtype.mk_eq_bot_iff ?_ _).mp hx.symm
simp | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.Matrix.Cartan | {
"line": 309,
"column": 6
} | {
"line": 309,
"column": 58
} | {
"line": 309,
"column": 58
} | [
{
"pp": "⊢ E₆.IsSimplyLaced",
"ppTerm": "?m.2",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"congrArg",
"PartialOrder.toPreorder",
"Matrix.IsSimplyLaced",
"SemilatticeInf.toPartialOrder",
"DistribLattice.toLattice",
"id",
"Int... | [
"⊢ ∀ ⦃i j : Fin 6⦄, j < i → E₆ i j = 0 ∨ E₆ i j = -1"
] | Matrix.isSimplyLaced_iff_of_linearOrder E₆ E₆_isSymm | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.DedekindDomain.Ideal.Basic | {
"line": 162,
"column": 4
} | {
"line": 162,
"column": 72
} | {
"line": 163,
"column": 4
} | [
{
"pp": "case refine_3\nA : Type u_2\nK : Type u_3\ninst✝⁴ : CommRing A\ninst✝³ : Field K\ninst✝² : Algebra A K\ninst✝¹ : IsFractionRing A K\ninst✝ : IsDedekindDomain A\nI✝ : Ideal A\nhNF : ¬IsField A\nI : Ideal A\nhI0✝ : I ≠ ⊥\nhI1 : I ≠ ⊤\nhM : I.IsMaximal\nhI0 : ⊥ < I\na : A\nhaI : a ∈ I\nha0 : a ≠ 0\nJ : Id... | [
"case refine_3\nA : Type u_2\nK : Type u_3\ninst✝⁴ : CommRing A\ninst✝³ : Field K\ninst✝² : Algebra A K\ninst✝¹ : IsFractionRing A K\ninst✝ : IsDedekindDomain A\nI✝ : Ideal A\nhNF : ¬IsField A\nI : Ideal A\nhI0✝ : I ≠ ⊥\nhI1 : I ≠ ⊤\nhM : I.IsMaximal\nhI0 : ⊥ < I\na : A\nhaI : a ∈ I\nha0 : a ≠ 0\nJ : Ideal A := spa... | rw [← div_eq_mul_inv, eq_div_iff_mul_eq hnz_fa, ← map_mul] at h₂_abs | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.FractionalIdeal.Operations | {
"line": 826,
"column": 6
} | {
"line": 826,
"column": 31
} | {
"line": 826,
"column": 32
} | [
{
"pp": "R : Type u_1\ninst✝³ : CommRing R\nS : Submonoid R\nP : Type u_2\ninst✝² : CommRing P\ninst✝¹ : Algebra R P\ninst✝ : IsLocalization S P\nI : FractionalIdeal S P\n⊢ ↑I.num ≤ I",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"FractionalIdeal.num",
"Fracti... | [
"R : Type u_1\ninst✝³ : CommRing R\nS : Submonoid R\nP : Type u_2\ninst✝² : CommRing P\ninst✝¹ : Algebra R P\ninst✝ : IsLocalization S P\nI : FractionalIdeal S P\n⊢ spanSingleton S ((algebraMap R P) ↑I.den) * I ≤ I"
] | ← I.den_mul_self_eq_num', | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Module.Lattice | {
"line": 174,
"column": 2
} | {
"line": 174,
"column": 55
} | {
"line": 176,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝¹⁰ : CommRing R\nK : Type u_2\ninst✝⁹ : Field K\ninst✝⁸ : Algebra R K\nV : Type u_3\ninst✝⁷ : AddCommGroup V\ninst✝⁶ : Module K V\ninst✝⁵ : Module R V\ninst✝⁴ : IsScalarTower R K V\ninst✝³ : IsDomain R\ninst✝² : IsPrincipalIdealRing R\ninst✝¹ : IsTorsionFree R K\nM : Submodule R V\ni... | [
"R : Type u_1\ninst✝¹⁰ : CommRing R\nK : Type u_2\ninst✝⁹ : Field K\ninst✝⁸ : Algebra R K\nV : Type u_3\ninst✝⁷ : AddCommGroup V\ninst✝⁶ : Module K V\ninst✝⁵ : Module R V\ninst✝⁴ : IsScalarTower R K V\ninst✝³ : IsDomain R\ninst✝² : IsPrincipalIdealRing R\ninst✝¹ : IsTorsionFree R K\nM : Submodule R V\ninst✝ : IsLat... | have := Module.IsTorsionFree.trans_faithfulSMul R K V | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.Finiteness.Cofinite | {
"line": 109,
"column": 4
} | {
"line": 109,
"column": 29
} | {
"line": 111,
"column": 0
} | [
{
"pp": "case insert\nR : Type u_1\ninst✝³ : Ring R\nM : Type u_2\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : IsNoetherianRing R\nw : Submodule R M\ns : Finset (Submodule R M)\nhws : w ∉ s\nhs' : (∀ S ∈ s, S.CoFG) → (sInf ↑s).CoFG\nhs : w.CoFG ∧ ∀ a ∈ s, a.CoFG\n⊢ (w ⊓ sInf ↑s).CoFG",
"ppTerm": "... | [] | exact hs.1.inf (hs' hs.2) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.Module.LinearMap.FiniteRange | {
"line": 238,
"column": 91
} | {
"line": 239,
"column": 37
} | {
"line": 241,
"column": 0
} | [
{
"pp": "K : Type u_1\nV : Type u_2\nV₂ : Type u_4\ninst✝⁴ : CommRing K\ninst✝³ : AddCommGroup V\ninst✝² : Module K V\ninst✝¹ : AddCommGroup V₂\ninst✝ : Module K V₂\nu : V →ₗ[K] V₂\n⊢ u ≈ 0 ↔ u.HasNoetherianRange",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"congrArg",
"Comm... | [] | by
simp [equiv_iff_hasNoetherianRange] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Module.LinearMap.Index | {
"line": 114,
"column": 2
} | {
"line": 114,
"column": 49
} | {
"line": 115,
"column": 2
} | [
{
"pp": "M : Type u_1\nN : Type u_2\ninst✝¹⁰ : AddCommGroup M\ninst✝⁹ : AddCommGroup N\nk : Type u_3\ninst✝⁸ : DivisionRing k\ninst✝⁷ : Module k M\ninst✝⁶ : Module k N\nf : M →ₗ[k] N\nP : Type u_4\ninst✝⁵ : AddCommGroup P\ninst✝⁴ : Module k P\ng : N →ₗ[k] P\ninst✝³ : FiniteDimensional k ↥f.ker\ninst✝² : FiniteD... | [
"M : Type u_1\nN : Type u_2\ninst✝¹⁰ : AddCommGroup M\ninst✝⁹ : AddCommGroup N\nk : Type u_3\ninst✝⁸ : DivisionRing k\ninst✝⁷ : Module k M\ninst✝⁶ : Module k N\nf : M →ₗ[k] N\nP : Type u_4\ninst✝⁵ : AddCommGroup P\ninst✝⁴ : Module k P\ng : N →ₗ[k] P\ninst✝³ : FiniteDimensional k ↥f.ker\ninst✝² : FiniteDimensional k... | have h₀ : Injective f₀ := inclusion_injective _ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Algebra.Module.FinitePresentation | {
"line": 178,
"column": 2
} | {
"line": 178,
"column": 23
} | {
"line": 179,
"column": 2
} | [
{
"pp": "R : Type u_1\nM : Type u_2\nN : Type u_3\ninst✝⁴ : Ring R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\nl : M →ₗ[R] N\nhl : Function.Surjective ⇑l\nhl' : l.ker.FG\ns : Finset M\nhs : Submodule.span R ↑s = ⊤\nhs' : (linearCombination R Subtype.val).ker.FG\n⊢... | [
"R : Type u_1\nM : Type u_2\nN : Type u_3\ninst✝⁴ : Ring R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\nl : M →ₗ[R] N\nhl : Function.Surjective ⇑l\ns : Finset M\nhs : Submodule.span R ↑s = ⊤\nhs' : (linearCombination R Subtype.val).ker.FG\nt : Finset M\nht : Submodule.... | obtain ⟨t, ht⟩ := hl' | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Algebra.Module.LinearMap.Prod | {
"line": 40,
"column": 4
} | {
"line": 40,
"column": 19
} | {
"line": 42,
"column": 0
} | [
{
"pp": "case map_smul\nR : Type u_1\nM : Type u_2\ninst✝² : Semiring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\n⊢ ∀ (c : R) (x : M × M), (c • x).1 - (c • x).2 = c • (x.1 - x.2)",
"ppTerm": "?map_smul",
"assigned": true,
"usedConstants": [
"instHSMul",
"congrArg",
"DistribMulA... | [] | simp [smul_sub] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Module.LinearMap.Prod | {
"line": 40,
"column": 4
} | {
"line": 40,
"column": 19
} | {
"line": 42,
"column": 0
} | [
{
"pp": "case map_smul\nR : Type u_1\nM : Type u_2\ninst✝² : Semiring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\n⊢ ∀ (c : R) (x : M × M), (c • x).1 - (c • x).2 = c • (x.1 - x.2)",
"ppTerm": "?map_smul",
"assigned": true,
"usedConstants": [
"instHSMul",
"congrArg",
"DistribMulA... | [] | simp [smul_sub] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Module.LinearMap.Prod | {
"line": 40,
"column": 4
} | {
"line": 40,
"column": 19
} | {
"line": 42,
"column": 0
} | [
{
"pp": "case map_smul\nR : Type u_1\nM : Type u_2\ninst✝² : Semiring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\n⊢ ∀ (c : R) (x : M × M), (c • x).1 - (c • x).2 = c • (x.1 - x.2)",
"ppTerm": "?map_smul",
"assigned": true,
"usedConstants": [
"instHSMul",
"congrArg",
"DistribMulA... | [] | simp [smul_sub] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Module.PID | {
"line": 162,
"column": 59
} | {
"line": 162,
"column": 61
} | {
"line": 162,
"column": 61
} | [
{
"pp": "R : Type u\ninst✝⁴ : CommRing R\ninst✝³ : IsPrincipalIdealRing R\nM : Type v\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : IsDomain R\np : R\nhp : Irreducible p\nhM : IsTorsion' M ↥(Submonoid.powers p)\ndec : (x : M) → Decidable (x = 0)\nz : M\nhz : IsTorsionBy R M (p ^ pOrder hM z)\nk : ℕ\nf ... | [
"R : Type u\ninst✝⁴ : CommRing R\ninst✝³ : IsPrincipalIdealRing R\nM : Type v\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : IsDomain R\np : R\nhp : Irreducible p\nhM : IsTorsion' M ↥(Submonoid.powers p)\ndec : (x : M) → Decidable (x = 0)\nz : M\nhz : IsTorsionBy R M (p ^ pOrder hM z)\nk : ℕ\nf : R ⧸ R ∙ p ... | ha | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Module.FinitePresentation | {
"line": 704,
"column": 2
} | {
"line": 705,
"column": 63
} | {
"line": 706,
"column": 2
} | [
{
"pp": "case refine_1\nR : Type u_4\nM : Type u_5\nN : Type u_6\ninst✝⁸ : CommRing R\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\nS : Type u_3\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : Flat R S\ninst✝ : FinitePresentation R M\nn m : ℕ\nf : (Fin n → R) ... | [
"case refine_2\nR : Type u_4\nM : Type u_5\nN : Type u_6\ninst✝⁸ : CommRing R\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\nS : Type u_3\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : Flat R S\ninst✝ : FinitePresentation R M\nn m : ℕ\nf : (Fin n → R) →ₗ[R] M\ng :... | · exact LinearMap.ext fun φ ↦ TensorProduct.AlgebraTensorModule.curry_injective
(LinearMap.ext fun s ↦ (LinearMap.ext fun m ↦ (by simp))) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Module.FinitePresentation | {
"line": 706,
"column": 2
} | {
"line": 707,
"column": 63
} | {
"line": 708,
"column": 2
} | [
{
"pp": "case refine_2\nR : Type u_4\nM : Type u_5\nN : Type u_6\ninst✝⁸ : CommRing R\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\nS : Type u_3\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : Flat R S\ninst✝ : FinitePresentation R M\nn m : ℕ\nf : (Fin n → R) ... | [
"case refine_3\nR : Type u_4\nM : Type u_5\nN : Type u_6\ninst✝⁸ : CommRing R\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\nS : Type u_3\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : Flat R S\ninst✝ : FinitePresentation R M\nn m : ℕ\nf : (Fin n → R) →ₗ[R] M\ng :... | · exact LinearMap.ext fun φ ↦ TensorProduct.AlgebraTensorModule.curry_injective
(LinearMap.ext fun s ↦ (LinearMap.ext fun m ↦ (by simp))) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.Extension.Basic | {
"line": 531,
"column": 4
} | {
"line": 532,
"column": 34
} | {
"line": 533,
"column": 4
} | [
{
"pp": "case refine_1\nR : Type u\nS : Type v\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nP : Extension R S\nP' : Extension R S\nf : P.Hom P'\nh : Function.Surjective ⇑f\neq : Ideal.comap f.toRingHom P'.ker = RingHom.ker f.toRingHom ⊔ P.ker\neq_map : P'.ker = Ideal.map f.toRingHom P.ker\nx ... | [
"case refine_1\nR : Type u\nS : Type v\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nP : Extension R S\nP' : Extension R S\nf : P.Hom P'\nh : Function.Surjective ⇑f\neq : Ideal.comap f.toRingHom P'.ker = RingHom.ker f.toRingHom ⊔ P.ker\neq_map : P'.ker = Ideal.map f.toRingHom P.ker\nx : ↥P.ker\ny ... | suffices ∃ a, a ∈ RingHom.ker f.toRingHom ∧ a ∈ P.ker ∧ a - x ∈ P.ker ^ 2 by
simpa [mk_eq_mk_iff_sub_mem] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticSuffices__1 | Lean.Parser.Tactic.tacticSuffices_ |
Mathlib.Algebra.Module.Presentation.Differentials | {
"line": 146,
"column": 4
} | {
"line": 147,
"column": 58
} | {
"line": 149,
"column": 0
} | [
{
"pp": "R : Type u\nS : Type v\nι : Type w\nσ : Type t\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\npres : Presentation R S ι σ\nr : pres.differentialsRelations.R\n⊢ pres.toExtension.toKaehler (pres.toExtension.cotangentComplex ((hom₁ pres) (Finsupp.single r 1))) = 0",
"ppTerm": "?m.43",... | [] | apply DFunLike.congr_fun (Function.Exact.linearMap_comp_eq_zero
(pres.toExtension.exact_cotangentComplex_toKaehler)) | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Algebra.Module.Presentation.Differentials | {
"line": 162,
"column": 4
} | {
"line": 162,
"column": 37
} | {
"line": 163,
"column": 4
} | [
{
"pp": "case right\nR : Type u\nS : Type v\nι : Type w\nσ : Type t\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\npres : Presentation R S ι σ\n⊢ pres.differentialsSolution.π.ker = Submodule.span S (Set.range pres.differentialsRelations.relation)",
"ppTerm": "?right",
"assigned": true,
... | [
"case right\nR : Type u\nS : Type v\nι : Type w\nσ : Type t\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\npres : Presentation R S ι σ\n⊢ pres.differentialsSolution.π.ker = pres.differentialsRelations.map.range"
] | rw [← Module.Relations.range_map] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Module.Presentation.Free | {
"line": 75,
"column": 14
} | {
"line": 75,
"column": 26
} | {
"line": 76,
"column": 2
} | [
{
"pp": "A : Type u\ninst✝² : Ring A\nrelations : Relations A\nM : Type v\ninst✝¹ : AddCommGroup M\ninst✝ : Module A M\nG : Type w₀\n⊢ PEmpty.{w₁ + 1} → G →₀ A",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"PEmpty",
"PEmpty.casesOn",
"Ring.toSemiring",
"Finsupp",
... | [] | by rintro ⟨⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Extension.Basic | {
"line": 558,
"column": 14
} | {
"line": 558,
"column": 16
} | {
"line": 558,
"column": 17
} | [
{
"pp": "case refine_1.refine_2\nR : Type u\nS : Type v\ninst✝²² : CommRing R\ninst✝²¹ : CommRing S\ninst✝²⁰ : Algebra R S\nP : Extension R S\nR' : Type ?u.16\nS' : Type ?u.18\ninst✝¹⁹ : CommRing R'\ninst✝¹⁸ : CommRing S'\ninst✝¹⁷ : Algebra R' S'\nP' : Extension R' S'\nR'' : Type ?u.29\nS'' : Type ?u.31\ninst✝¹... | [
"case refine_1.refine_2\nR : Type u\nS : Type v\ninst✝²² : CommRing R\ninst✝²¹ : CommRing S\ninst✝²⁰ : Algebra R S\nP : Extension R S\nR' : Type ?u.16\nS' : Type ?u.18\ninst✝¹⁹ : CommRing R'\ninst✝¹⁸ : CommRing S'\ninst✝¹⁷ : Algebra R' S'\nP' : Extension R' S'\nR'' : Type ?u.29\nS'' : Type ?u.31\ninst✝¹⁶ : CommRing... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.RingTheory.Extension.Generators | {
"line": 660,
"column": 4
} | {
"line": 660,
"column": 26
} | {
"line": 660,
"column": 26
} | [
{
"pp": "R : Type u\nS : Type v\nι : Type w\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\nP : Generators R S ι\nT : Type u_8\ninst✝¹ : CommRing T\ninst✝ : Algebra R T\ne : S ≃ₐ[R] T\n⊢ RingHom.ker (aeval P.val) = P.ker",
"ppTerm": "?m.173",
"assigned": true,
"usedConstants": [
... | [
"R : Type u\nS : Type v\nι : Type w\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\nP : Generators R S ι\nT : Type u_8\ninst✝¹ : CommRing T\ninst✝ : Algebra R T\ne : S ≃ₐ[R] T\n⊢ P.ker = P.ker"
] | ← ker_eq_ker_aeval_val | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Module.Presentation.Tensor | {
"line": 88,
"column": 8
} | {
"line": 88,
"column": 14
} | {
"line": 89,
"column": 8
} | [
{
"pp": "case e'_7\nA : Type u\ninst✝⁶ : CommRing A\nM₁ : Type v₁\nM₂ : Type v₂\ninst✝⁵ : AddCommGroup M₁\ninst✝⁴ : AddCommGroup M₂\ninst✝³ : Module A M₁\ninst✝² : Module A M₂\nrelations₁ : Relations A\nrelations₂ : Relations A\nsolution₁ : relations₁.Solution M₁\nsolution₂ : relations₂.Solution M₂\nh₁ : soluti... | [
"case e'_7\nA : Type u\ninst✝⁶ : CommRing A\nM₁ : Type v₁\nM₂ : Type v₂\ninst✝⁵ : AddCommGroup M₁\ninst✝⁴ : AddCommGroup M₂\ninst✝³ : Module A M₁\ninst✝² : Module A M₂\nrelations₁ : Relations A\nrelations₂ : Relations A\nsolution₁ : relations₁.Solution M₁\nsolution₂ : relations₂.Solution M₂\nh₁ : solution₁.IsPresen... | ext g₁ | _private.Lean.Elab.Tactic.Ext.0.Lean.Elab.Tactic.Ext.evalExt | Lean.Elab.Tactic.Ext.ext |
Mathlib.Data.Fin.Parity | {
"line": 63,
"column": 2
} | {
"line": 65,
"column": 23
} | {
"line": 67,
"column": 0
} | [
{
"pp": "n : ℕ\ninst✝ : NeZero n\nhn : Odd n\nk : Fin n\n⊢ Odd k",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"AddMonoid.toAddSemigroup",
"Fin.odd_of_val",
"congrArg",
"CommSemiring.toSemiring",
"Fin.instCommR... | [] | rcases k.val.even_or_odd with hk | hk
· simpa using (Even.add_odd hk hn).natCast (R := Fin n)
· exact odd_of_val hk | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Fin.Parity | {
"line": 63,
"column": 2
} | {
"line": 65,
"column": 23
} | {
"line": 67,
"column": 0
} | [
{
"pp": "n : ℕ\ninst✝ : NeZero n\nhn : Odd n\nk : Fin n\n⊢ Odd k",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"AddMonoid.toAddSemigroup",
"Fin.odd_of_val",
"congrArg",
"CommSemiring.toSemiring",
"Fin.instCommR... | [] | rcases k.val.even_or_odd with hk | hk
· simpa using (Even.add_odd hk hn).natCast (R := Fin n)
· exact odd_of_val hk | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.TensorPower.Basic | {
"line": 150,
"column": 2
} | {
"line": 150,
"column": 25
} | {
"line": 151,
"column": 2
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nn : ℕ\na : ⨂[R]^n M\n⊢ (cast R M ⋯) (GradedMonoid.GMul.mul GradedMonoid.GOne.one a) = a",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"PiTensorProduct.instModule",
"Eq.... | [
"R : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nn : ℕ\na : ⨂[R]^n M\n⊢ (cast R M ⋯) (mulEquiv ((tprod R) Fin.elim0 ⊗ₜ[R] a)) = a"
] | rw [gMul_def, gOne_def] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.TensorPower.Basic | {
"line": 156,
"column": 4
} | {
"line": 156,
"column": 35
} | {
"line": 157,
"column": 4
} | [
{
"pp": "case smul_tprod.e_a.e_6\nR : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nn : ℕ\nr : R\na : Fin n → M\ni : Fin n\n⊢ ((a ∘ Fin.cast ⋯) ∘ Fin.cast ⋯) i = a i",
"ppTerm": "?smul_tprod.e_a.e_6",
"assigned": true,
"usedConstants": [
"AddMon... | [
"case smul_tprod.e_a.e_6\nR : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nn : ℕ\nr : R\na : Fin n → M\ni : Fin n\n⊢ ↑(Fin.cast ⋯ (Fin.cast ⋯ i)) = ↑i"
] | refine congr_arg a (Fin.ext ?_) | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.LinearAlgebra.TensorPower.Basic | {
"line": 162,
"column": 2
} | {
"line": 162,
"column": 25
} | {
"line": 163,
"column": 2
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nn : ℕ\na : ⨂[R]^n M\n⊢ (cast R M ⋯) (GradedMonoid.GMul.mul a GradedMonoid.GOne.one) = a",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"PiTensorProduct.instModule",
"Eq.... | [
"R : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nn : ℕ\na : ⨂[R]^n M\n⊢ (cast R M ⋯) (mulEquiv (a ⊗ₜ[R] (tprod R) Fin.elim0)) = a"
] | rw [gMul_def, gOne_def] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Ideal.MinimalPrime.Noetherian | {
"line": 40,
"column": 2
} | {
"line": 41,
"column": 55
} | {
"line": 43,
"column": 0
} | [
{
"pp": "case inr\nR : Type u_1\ninst✝ : CommSemiring R\nhR : IsNoetherianRing R\nI✝ : Ideal R\nhI✝¹ : ¬I✝.minimalPrimes.Finite\nI : Ideal R\nhI✝ : ¬I.minimalPrimes.Finite\nhmax : ∀ (I_1 : Submodule R R), I < I_1 → (Ideal.minimalPrimes I_1).Finite\nh1 : ¬I.IsPrime\nh2 : I ≠ ⊤\nx : R\nhx : I < I ⊔ span {x}\ny : ... | [] | · exact Or.inr ⟨⟨hp, sup_le hI (p.span_singleton_le_iff_mem.mpr hyp)⟩,
fun q hq hqp ↦ hmin ⟨hq.1, hy.le.trans hq.2⟩ hqp⟩ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.Ideal.AssociatedPrime.Basic | {
"line": 80,
"column": 8
} | {
"line": 80,
"column": 38
} | {
"line": 80,
"column": 39
} | [
{
"pp": "case mp\nR : Type u_1\nM : Type u_2\ninst✝³ : CommSemiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\nN : Submodule R M\ninst✝ : IsNoetherianRing R\nx : M\nhx : (N.colon {x}).radical.IsPrime\n⊢ (N.colon {x}).radical ∈ (N.colon {x}).minimalPrimes",
"ppTerm": "?mp",
"assigned": true,
"... | [
"case mp\nR : Type u_1\nM : Type u_2\ninst✝³ : CommSemiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\nN : Submodule R M\ninst✝ : IsNoetherianRing R\nx : M\nhx : (N.colon {x}).radical.IsPrime\n⊢ (N.colon {x}).radical ∈ (N.colon {x}).radical.minimalPrimes"
] | ← Ideal.radical_minimalPrimes, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Ideal.MinimalPrime.Colon | {
"line": 93,
"column": 4
} | {
"line": 93,
"column": 80
} | {
"line": 94,
"column": 4
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝³ : CommSemiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\nN : Submodule R M\nI : Ideal R\nx : M\ninst✝ : IsNoetherianRing R\nhx : x ∉ N\nann : Ideal R := N.colon {x}\nhI : I ∈ ann.minimalPrimes\nkey : ∃ n, n ≠ 0 ∧ ∃ J, I ^ n * J ≤ ann ∧ ¬J ≤ I\nJ : Ideal R\nhJI... | [
"R : Type u_1\nM : Type u_2\ninst✝³ : CommSemiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\nN : Submodule R M\nI : Ideal R\nx : M\ninst✝ : IsNoetherianRing R\nhx : x ∉ N\nann : Ideal R := N.colon {x}\nhI : I ∈ ann.minimalPrimes\nkey : ∃ n, n ≠ 0 ∧ ∃ J, I ^ n * J ≤ ann ∧ ¬J ≤ I\nJ : Ideal R\nhJI : ¬J ≤ I\nn... | rw [← mul_assoc, hI.isPrime.mul_le, not_or, Ideal.span_singleton_le_iff_mem] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.LocalRing.Module | {
"line": 340,
"column": 6
} | {
"line": 340,
"column": 61
} | {
"line": 341,
"column": 4
} | [
{
"pp": "R : Type u_1\nM : Type u_2\nN : Type u_3\ninst✝⁸ : CommRing R\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : AddCommGroup N\ninst✝⁵ : Module R M\ninst✝⁴ : Module R N\ninst✝³ : IsLocalRing R\ninst✝² : Module.Finite R M\ninst✝¹ : Module.Finite R N\ninst✝ : Free R N\nl : M →ₗ[R] N\nl' : N →ₗ[R] M\nhl : l' ∘ₗ l = Line... | [] | rw [← LinearMap.lTensor_comp, hl, LinearMap.lTensor_id] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.UniqueFactorizationDomain.ClassGroup | {
"line": 41,
"column": 33
} | {
"line": 41,
"column": 43
} | {
"line": 43,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : IsGCDMonoid R\nJ : Ideal R\nthis : NormalizedGCDMonoid R := Classical.arbitrary (NormalizedGCDMonoid R)\nK : Ideal R\nhJK0 : J * K ≠ 0\nhK : Submodule.IsPrincipal (J * K)\nx : R\nhJK : J * K = R ∙ x\n⊢ x ∈ J * K",
"ppTerm": "?m.71",
... | [] | simp [hJK] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.Ideal.AssociatedPrime.Finiteness | {
"line": 163,
"column": 4
} | {
"line": 163,
"column": 68
} | {
"line": 164,
"column": 4
} | [
{
"pp": "case quotient\nA : Type u\ninst✝⁶ : CommRing A\nM : Type v\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module A M\ninst✝³ : IsNoetherianRing A\nN : Type v\ninst✝² : AddCommGroup N\ninst✝¹ : Module A N\ninst✝ : Module.Finite A N\np : PrimeSpectrum A\nf : N ≃ₗ[A] A ⧸ p.asIdeal\n⊢ (associatedPrimes A N).Finite",
... | [
"case quotient\nA : Type u\ninst✝⁶ : CommRing A\nM : Type v\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module A M\ninst✝³ : IsNoetherianRing A\nN : Type v\ninst✝² : AddCommGroup N\ninst✝¹ : Module A N\ninst✝ : Module.Finite A N\np : PrimeSpectrum A\nf : N ≃ₗ[A] A ⧸ p.asIdeal\nthis : associatedPrimes A (A ⧸ p.asIdeal) = {p.... | have := associatedPrimes.eq_singleton_of_isPrimary p.2.isPrimary | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.ClassGroup.Basic | {
"line": 161,
"column": 51
} | {
"line": 161,
"column": 88
} | {
"line": 161,
"column": 88
} | [
{
"pp": "case mp\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nI : (FractionalIdeal R⁰ (FractionRing R))ˣ\nI' : Ideal R\nhI : ↑I = ↑I'\nx : R\nhx : x ≠ 0\ni : R\n_hi : i ∈ I'\nhy : x * i ≠ 0\nh : Ideal.span {x} * I' = Ideal.span {x} * Ideal.span {i}\n⊢ ∃ x, x ≠ 0 ∧ I' = Ideal.span {x}",
"ppTerm": ... | [
"case mp\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nI : (FractionalIdeal R⁰ (FractionRing R))ˣ\nI' : Ideal R\nhI : ↑I = ↑I'\nx : R\nhx : x ≠ 0\ni : R\n_hi : i ∈ I'\nhy : x * i ≠ 0\nh : I' = Ideal.span {i}\n⊢ ∃ x, x ≠ 0 ∧ I' = Ideal.span {x}"
] | Ideal.span_singleton_mul_right_inj hx | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.Perm.ViaEmbedding | {
"line": 38,
"column": 2
} | {
"line": 38,
"column": 65
} | {
"line": 40,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ne : Perm α\nι : α ↪ β\nx : β\nhx : ¬x ∈ Set.range ⇑ι\n⊢ (e.viaEmbedding ι) x = x",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Classical.propDecidable",
"Membership.mem",
"Function.Embedding.inj'",
"Equiv.Perm.extendDomain_app... | [] | exact extendDomain_apply_not_subtype e (ofInjective ι.1 ι.2) hx | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.GroupTheory.Perm.ViaEmbedding | {
"line": 38,
"column": 2
} | {
"line": 38,
"column": 65
} | {
"line": 40,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ne : Perm α\nι : α ↪ β\nx : β\nhx : ¬x ∈ Set.range ⇑ι\n⊢ (e.viaEmbedding ι) x = x",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Classical.propDecidable",
"Membership.mem",
"Function.Embedding.inj'",
"Equiv.Perm.extendDomain_app... | [] | exact extendDomain_apply_not_subtype e (ofInjective ι.1 ι.2) hx | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.Perm.ViaEmbedding | {
"line": 38,
"column": 2
} | {
"line": 38,
"column": 65
} | {
"line": 40,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ne : Perm α\nι : α ↪ β\nx : β\nhx : ¬x ∈ Set.range ⇑ι\n⊢ (e.viaEmbedding ι) x = x",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Classical.propDecidable",
"Membership.mem",
"Function.Embedding.inj'",
"Equiv.Perm.extendDomain_app... | [] | exact extendDomain_apply_not_subtype e (ofInjective ι.1 ι.2) hx | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Spectrum.Prime.FreeLocus | {
"line": 325,
"column": 2
} | {
"line": 325,
"column": 67
} | {
"line": 326,
"column": 2
} | [
{
"pp": "R : Type uR\nM : Type uM\ninst✝⁶ : CommRing R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\ninst✝³ : Flat R M\ninst✝² : Module.Finite R M\nS : Type u_1\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\np : PrimeSpectrum S\nq : PrimeSpectrum R := comap (algebraMap R S) p\n⊢ rankAtStalk (S ⊗[R] M) p = rankAtSt... | [
"R : Type uR\nM : Type uM\ninst✝⁶ : CommRing R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\ninst✝³ : Flat R M\ninst✝² : Module.Finite R M\nS : Type u_1\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\np : PrimeSpectrum S\nq : PrimeSpectrum R := comap (algebraMap R S) p\nthis : Algebra (Localization.AtPrime q.asIdeal) (... | let := Localization.AtPrime.algebraOfLiesOver q.asIdeal p.asIdeal | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.RingTheory.PicardGroup | {
"line": 211,
"column": 63
} | {
"line": 212,
"column": 69
} | {
"line": 214,
"column": 0
} | [
{
"pp": "R : Type u\nM : Type v\nN : Type u_1\nP : Type u_2\ninst✝⁷ : CommSemiring R\ninst✝⁶ : AddCommMonoid M\ninst✝⁵ : AddCommMonoid N\ninst✝⁴ : AddCommMonoid P\ninst✝³ : Module R M\ninst✝² : Module R N\ninst✝¹ : Module R P\ninst✝ : Module.Invertible R M\nf : N →ₗ[R] P\n⊢ Function.Injective ⇑(LinearMap.rTenso... | [] | by
rw [← LinearMap.lTensor_inj_iff_rTensor_inj, lTensor_injective_iff] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.FieldTheory.PrimitiveElement | {
"line": 284,
"column": 2
} | {
"line": 284,
"column": 48
} | {
"line": 286,
"column": 0
} | [
{
"pp": "F : Type u_1\nE : Type u_2\ninst✝³ : Field F\ninst✝² : Field E\ninst✝¹ : Algebra F E\ninst✝ : Finite (IntermediateField F E)\nIF : Type (max 0 u_2) := { K // ∃ x, K = F⟮x⟯ }\nthis✝ : Algebra.IsAlgebraic F E\nthis : ∀ (K : IF), FiniteDimensional F ↥↑K\nhfin : FiniteDimensional F ↥⊤\nhtop : ⨆ K, ↑K = ⊤\n... | [] | exact topEquiv.toLinearEquiv.finiteDimensional | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.FieldTheory.PrimitiveElement | {
"line": 302,
"column": 2
} | {
"line": 302,
"column": 48
} | {
"line": 305,
"column": 0
} | [
{
"pp": "F : Type u_1\nE : Type u_2\ninst✝³ : Field F\ninst✝² : Field E\ninst✝¹ : Algebra F E\ninst✝ : Algebra.IsAlgebraic F E\nα : E\nhprim : F⟮α⟯ = ⊤\nhfin : FiniteDimensional F ↥⊤\n⊢ FiniteDimensional F E",
"ppTerm": "?m.46",
"assigned": true,
"usedConstants": [
"LinearEquiv.finiteDimension... | [] | exact topEquiv.toLinearEquiv.finiteDimensional | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Combinatorics.Matroid.Basic | {
"line": 1101,
"column": 19
} | {
"line": 1101,
"column": 42
} | {
"line": 1101,
"column": 43
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nB X : Set α\nhX : X ⊆ M.E\nhB : M.IsBase B\nhBX : B ⊆ X\n⊢ M.Indep B ∧ B ⊆ X ∧ ∀ (J : Set α), M.Indep J → B ⊆ J → J ⊆ X → B = J",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Matroid.Indep",
"id",
"... | [
"α : Type u_1\nM : Matroid α\nB X : Set α\nhX : X ⊆ M.E\nhB : M.IsBase B\nhBX : B ⊆ X\n⊢ B ⊆ X ∧ ∀ (J : Set α), M.Indep J → B ⊆ J → J ⊆ X → B = J"
] | and_iff_right hB.indep, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.Matroid.Basic | {
"line": 1125,
"column": 2
} | {
"line": 1125,
"column": 76
} | {
"line": 1126,
"column": 2
} | [
{
"pp": "α : Type u_1\nE : Set α\nhE : E.Finite\nf : Matroid α → Set α × Set (Set α) := fun M ↦ (M.E, {B | M.IsBase B})\nhf : Function.Injective f\n⊢ (f '' {M | M.E ⊆ E}).Finite",
"ppTerm": "?m.75",
"assigned": true,
"usedConstants": [
"Set.instSProd",
"SProd.sprod",
"Set.ofPred",
... | [
"α : Type u_1\nE : Set α\nhE : E.Finite\nf : Matroid α → Set α × Set (Set α) := fun M ↦ (M.E, {B | M.IsBase B})\nhf : Function.Injective f\n⊢ f '' {M | M.E ⊆ E} ⊆ {b | b ⊆ E} ×ˢ {b | b ⊆ {b | b ⊆ E}}"
] | refine (hE.finite_subsets.prod hE.finite_subsets.finite_subsets).subset ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.RingTheory.PicardGroup | {
"line": 929,
"column": 4
} | {
"line": 931,
"column": 99
} | {
"line": 932,
"column": 2
} | [
{
"pp": "R : Type u_5\ninst✝³ : CommRing R\ninst✝² : IsFractionRing R R\nI J : Ideal R\ninst✝¹ : Module.Invertible R ↥I\ninst✝ : Module.Invertible R ↥J\nh : Pic.mk R (↥I ⊗[R] ↥J) = 1\ne✝ : ↥I ⊗[R] ↥J ≃ₗ[R] R\ne : R ≃ₗ[R] ↥(I * J)\n⊢ IsRightRegular ↑(e 1)",
"ppTerm": "?m.185",
"assigned": true,
"used... | [] | rw [IsRightRegular]
convert! Subtype.val_injective.comp e.injective using 2
rw [← smul_eq_mul, ← Submodule.coe_smul, ← map_smul, smul_eq_mul, mul_one, Function.comp_apply] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.PicardGroup | {
"line": 929,
"column": 4
} | {
"line": 931,
"column": 99
} | {
"line": 932,
"column": 2
} | [
{
"pp": "R : Type u_5\ninst✝³ : CommRing R\ninst✝² : IsFractionRing R R\nI J : Ideal R\ninst✝¹ : Module.Invertible R ↥I\ninst✝ : Module.Invertible R ↥J\nh : Pic.mk R (↥I ⊗[R] ↥J) = 1\ne✝ : ↥I ⊗[R] ↥J ≃ₗ[R] R\ne : R ≃ₗ[R] ↥(I * J)\n⊢ IsRightRegular ↑(e 1)",
"ppTerm": "?m.185",
"assigned": true,
"used... | [] | rw [IsRightRegular]
convert! Subtype.val_injective.comp e.injective using 2
rw [← smul_eq_mul, ← Submodule.coe_smul, ← map_smul, smul_eq_mul, mul_one, Function.comp_apply] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.Matroid.Map | {
"line": 173,
"column": 2
} | {
"line": 173,
"column": 32
} | {
"line": 175,
"column": 0
} | [
{
"pp": "case inr\nα : Type u_1\nβ : Type u_2\nf : α → β\nI : Set α\nN : Matroid β\nhI : N.Indep (f '' I) ∧ ¬InjOn f I\n⊢ I ⊆ f ⁻¹' N.E",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Matroid.E",
"Eq.mp",
"Matroid.Indep",
"LE.le",
"Set.image_subset_iff._simp_1... | [] | simpa using hI.1.subset_ground | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Combinatorics.Matroid.Map | {
"line": 230,
"column": 2
} | {
"line": 230,
"column": 66
} | {
"line": 232,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nf✝ : α → β\nE I✝ : Set α\nM : Matroid α\nN✝ N : Matroid β\ninst✝ : N.Finitary\nf : α → β\nI : Set α\nhI : ∀ J ⊆ I, J.Finite → (N.comap f).Indep J\nJ : Set α\nhJ : J ⊆ I\nJ' : Set α\nhfin : (f '' J').Finite\nhJ'J : J' ⊆ J\nhJ' : BijOn f J' (f '' J)\n⊢ N.Indep (f '' J')",
... | [] | exact (hI J' (hJ'J.trans hJ) (hfin.of_finite_image hJ'.injOn)).1 | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Combinatorics.Matroid.Map | {
"line": 329,
"column": 4
} | {
"line": 329,
"column": 23
} | {
"line": 330,
"column": 4
} | [
{
"pp": "case mp\nα : Type u_1\nβ : Type u_2\nM : Matroid α\nf : ↑M.E ↪ β\nI : Set β\n⊢ (M.Indep (Subtype.val '' ⇑f ⁻¹' I) ∧ ∃ t, ⇑f '' t = I) → ∃ I₀, M.Indep (Subtype.val '' I₀) ∧ I = ⇑f '' I₀",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
"Matroid.E",
"Membership.mem",
"... | [
"case mp\nα : Type u_1\nβ : Type u_2\nM : Matroid α\nf : ↑M.E ↪ β\nI : Set ↑M.E\nhI : M.Indep (Subtype.val '' ⇑f ⁻¹' ⇑f '' I)\n⊢ ∃ I₀, M.Indep (Subtype.val '' I₀) ∧ ⇑f '' I = ⇑f '' I₀"
] | rintro ⟨hI, I, rfl⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.Combinatorics.Matroid.Map | {
"line": 480,
"column": 2
} | {
"line": 480,
"column": 27
} | {
"line": 481,
"column": 2
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nf✝ : α → β\nE I✝ : Set α\nM : Matroid α\nN : Matroid β\ninst✝ : M.Finitary\nf : α → β\nhf : InjOn f M.E\nI : Set β\nhI : ∀ J ⊆ I, J.Finite → (M.map f hf).Indep J\n⊢ (M.map f hf).Indep I",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [
"α : Type u_1\nβ : Type u_2\nf✝ : α → β\nE I✝ : Set α\nM : Matroid α\nN : Matroid β\ninst✝ : M.Finitary\nf : α → β\nhf : InjOn f M.E\nI : Set β\nhI : ∀ J ⊆ I, J.Finite → (M.map f hf).Indep J\n⊢ ∃ I₀, M.Indep I₀ ∧ I = f '' I₀"
] | simp only [map_indep_iff] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Combinatorics.Matroid.Map | {
"line": 681,
"column": 23
} | {
"line": 681,
"column": 45
} | {
"line": 681,
"column": 46
} | [
{
"pp": "α : Type u_1\nM : Matroid α\n⊢ ((M ↾ M.E).comap Subtype.val)✶ = M✶.restrictSubtype M.E",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Matroid.comapOn",
"congrArg",
"Matroid.E",
"Matroid.dual",
"Membership.mem",
"Set.Elem",
... | [
"α : Type u_1\nM : Matroid α\n⊢ ((M ↾ M.E).comapOn (M.E ↓∩ (M ↾ M.E).E) Subtype.val)✶ = M✶.restrictSubtype M.E"
] | ← comapOn_preimage_eq, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.Matroid.Map | {
"line": 681,
"column": 2
} | {
"line": 683,
"column": 73
} | {
"line": 685,
"column": 0
} | [
{
"pp": "α : Type u_1\nM : Matroid α\n⊢ (M.restrictSubtype M.E)✶ = M✶.restrictSubtype M.E",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Set.image_univ",
"ChainCompletePartialOrder.instOfCompleteLattice",
"Matroid.comapOn",
"instReflLe",
"cong... | [] | rw [restrictSubtype, ← comapOn_preimage_eq, comapOn_dual_eq_of_bijOn, restrict_ground_eq_self,
← dual_ground, comapOn_preimage_eq, restrictSubtype, restrict_ground_eq_self]
exact ⟨by simp [MapsTo], Subtype.val_injective.injOn, by simp [SurjOn]⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.Matroid.Map | {
"line": 681,
"column": 2
} | {
"line": 683,
"column": 73
} | {
"line": 685,
"column": 0
} | [
{
"pp": "α : Type u_1\nM : Matroid α\n⊢ (M.restrictSubtype M.E)✶ = M✶.restrictSubtype M.E",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Set.image_univ",
"ChainCompletePartialOrder.instOfCompleteLattice",
"Matroid.comapOn",
"instReflLe",
"cong... | [] | rw [restrictSubtype, ← comapOn_preimage_eq, comapOn_dual_eq_of_bijOn, restrict_ground_eq_self,
← dual_ground, comapOn_preimage_eq, restrictSubtype, restrict_ground_eq_self]
exact ⟨by simp [MapsTo], Subtype.val_injective.injOn, by simp [SurjOn]⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.Matroid.Loop | {
"line": 343,
"column": 14
} | {
"line": 343,
"column": 59
} | {
"line": 343,
"column": 59
} | [
{
"pp": "α : Type u_1\nM : Matroid α\ne : α\nC : Set α\nhC : M.IsCircuit C\nhC' : C.Nontrivial\nhe : e ∈ C\nhL : M.IsLoop e\n⊢ False",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"False",
"Matroid.IsLoop.eq_of_isCircuit_mem",
"congrArg",
"False.elim",
"Eq.mp... | [] | by simp [hL.eq_of_isCircuit_mem hC he] at hC' | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.Matroid.Rank.ENat | {
"line": 214,
"column": 2
} | {
"line": 214,
"column": 29
} | {
"line": 216,
"column": 0
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nX I : Set α\nhI : M.IsBasis' I X\n⊢ I.encard ≤ X.encard",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Set.encard_mono",
"Matroid.IsBasis'.subset"
],
"usedFVars": [
"α",
"I",
"X",
"M",
"hI"
],
... | [] | exact encard_mono hI.subset | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Combinatorics.Matroid.Loop | {
"line": 527,
"column": 4
} | {
"line": 527,
"column": 31
} | {
"line": 528,
"column": 4
} | [
{
"pp": "α : Type u_1\nM : Matroid α\ne : α\ntfae_1_iff_2 : M.IsColoop e ↔ e ∈ M.coloops\ntfae_1_iff_3 : M.IsColoop e ↔ M.IsCocircuit {e}\ntfae_1_iff_4 : M.IsColoop e ↔ ∀ ⦃B : Set α⦄, M.IsBase B → e ∈ B\ntfae_3_to_5 : M.IsCocircuit {e} → (∀ ⦃C : Set α⦄, M.IsCircuit C → e ∉ C) ∧ e ∈ M.E\nx✝ : (∀ ⦃C : Set α⦄, M.I... | [
"α : Type u_1\nM : Matroid α\ne : α\ntfae_1_iff_2 : M.IsColoop e ↔ e ∈ M.coloops\ntfae_1_iff_3 : M.IsColoop e ↔ M.IsCocircuit {e}\ntfae_1_iff_4 : M.IsColoop e ↔ ∀ ⦃B : Set α⦄, M.IsBase B → e ∈ B\ntfae_3_to_5 : M.IsCocircuit {e} → (∀ ⦃C : Set α⦄, M.IsCircuit C → e ∉ C) ∧ e ∈ M.E\nx✝ : (∀ ⦃C : Set α⦄, M.IsCircuit C →... | rw [← hB.closure_eq] at heE | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Combinatorics.Matroid.Closure | {
"line": 291,
"column": 44
} | {
"line": 291,
"column": 75
} | {
"line": 291,
"column": 75
} | [
{
"pp": "α : Type u_2\nM : Matroid α\nX Y X' : Set α\nh : M.closure X = M.closure X'\n⊢ M.closure (M.closure X' ∪ Y) = M.closure (X' ∪ Y)",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Set.instUnion",
"id",
"Matroid.closure_union_closur... | [
"α : Type u_2\nM : Matroid α\nX Y X' : Set α\nh : M.closure X = M.closure X'\n⊢ M.closure (X' ∪ Y) = M.closure (X' ∪ Y)"
] | M.closure_union_closure_left_eq | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Algebraic.MvPolynomial | {
"line": 32,
"column": 84
} | {
"line": 64,
"column": 67
} | {
"line": 66,
"column": 0
} | [
{
"pp": "σ : Type u_1\nR : Type u_2\ninst✝ : CommRing R\ni : σ\ns : Set σ\nh : i ∉ s\nf : R[X]\nhf : Transcendental R f\n⊢ Transcendental (↥(supported R s)) ((Polynomial.aeval (X i)) f)",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Subalgebra.instSetLike",
"Finsupp.instAddZe... | [] | by
classical
rw [transcendental_iff_injective] at hf ⊢
let g := MvPolynomial.mapAlgHom (R := R) (σ := s) (Polynomial.aeval (R := R) f)
replace hf : Function.Injective g := MvPolynomial.map_injective _ hf
let u := (Subalgebra.val _).comp
((optionEquivRight R s).symm |>.trans
(renameEquiv R (Set.subty... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.Matroid.Rank.Cardinal | {
"line": 352,
"column": 7
} | {
"line": 352,
"column": 25
} | {
"line": 353,
"column": 4
} | [
{
"pp": "α : Type u\nβ : Type v\nf : α → β\nM✝ : Matroid α\nI✝ J✝ B B' X✝ Y : Set α\nM : Matroid α\ninst✝ : M.InvariantCardinalRank\nhf : InjOn f M.E\nI X : Set α\nhIX : M.IsBasis I X\nhI : (M.map f hf).IsBasis (f '' I) (f '' X)\nJ : Set α\nhJ : (M.map f hf).IsBasis (f '' J) (f '' X)\nhJX : M.IsBasis J X\nh' : ... | [
"α : Type u\nβ : Type v\nf : α → β\nM✝ : Matroid α\nI✝ J✝ B B' X✝ Y : Set α\nM : Matroid α\ninst✝ : M.InvariantCardinalRank\nhf : InjOn f M.E\nI X : Set α\nhIX : M.IsBasis I X\nhI : (M.map f hf).IsBasis (f '' I) (f '' X)\nJ : Set α\nhJ : (M.map f hf).IsBasis (f '' J) (f '' X)\nhJX : M.IsBasis J X\nh' : f '' X = f '... | ← lift_inj.{u, v}, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.Matroid.Rank.Cardinal | {
"line": 365,
"column": 6
} | {
"line": 365,
"column": 24
} | {
"line": 365,
"column": 25
} | [
{
"pp": "α : Type u\nβ : Type v\nf✝ : α → β\nM✝ : Matroid α\nI✝ J✝ B B' X✝ Y : Set α\nM : Matroid β\ninst✝ : M.InvariantCardinalRank\nf : α → β\nI J X : Set α\nhI✝ : (M.comap f).IsBasis I X\nhJ✝ : (M.comap f).IsBasis J X\nhI : M.IsBasis (f '' I) (f '' X)\nhfI : InjOn f I\nhIX : I ⊆ X\nhJ : M.IsBasis (f '' J) (f... | [
"α : Type u\nβ : Type v\nf✝ : α → β\nM✝ : Matroid α\nI✝ J✝ B B' X✝ Y : Set α\nM : Matroid β\ninst✝ : M.InvariantCardinalRank\nf : α → β\nI J X : Set α\nhI✝ : (M.comap f).IsBasis I X\nhJ✝ : (M.comap f).IsBasis J X\nhI : M.IsBasis (f '' I) (f '' X)\nhfI : InjOn f I\nhIX : I ⊆ X\nhJ : M.IsBasis (f '' J) (f '' X)\nhfJ ... | ← lift_inj.{u, v}, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.Matroid.Circuit | {
"line": 762,
"column": 6
} | {
"line": 762,
"column": 63
} | {
"line": 762,
"column": 63
} | [
{
"pp": "case inr.inr.inr\nα : Type u_1\nM : Matroid α\nB : Set α\nhB : M.IsBase B\ne f : α\nhe : M✶.Indep (insert f (M✶.E \\ B) \\ {e})\nhne : e ≠ f\nhB' : M✶.IsBase (M✶.E \\ B)\nhfE : f ∈ M.E\nhfB : f ∈ B\nheE : e ∈ M.E\nheB : e ∉ B\n⊢ f ∈ M.fundCircuit e B",
"ppTerm": "?inr.inr.inr",
"assigned": true... | [
"case inr.inr.inr\nα : Type u_1\nM : Matroid α\nB : Set α\nhB : M.IsBase B\ne f : α\nhe : M✶.Indep (insert f (M✶.E \\ B) \\ {e})\nhne : e ≠ f\nhB' : M✶.IsBase (M✶.E \\ B)\nhfE : f ∈ M.E\nhfB : f ∈ B\nheE : e ∈ M.E\nheB : e ∉ B\n⊢ M.Indep (insert e B \\ {f})"
] | hB.indep.mem_fundCircuit_iff (by rwa [hB.closure_eq]) heB | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.AlgebraicIndependent.Transcendental | {
"line": 67,
"column": 59
} | {
"line": 70,
"column": 49
} | {
"line": 72,
"column": 0
} | [
{
"pp": "ι : Type u_1\nR : Type u_3\nA : Type v\nx : ι → A\ninst✝² : CommRing R\ninst✝¹ : CommRing A\ninst✝ : Algebra R A\nhx : AlgebraicIndependent R x\ni : ι\n⊢ Transcendental R (x i)",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"FinVec.map",
"AlgebraicInde... | [] | by
have := hx.comp ![i] (Function.injective_of_subsingleton _)
have : AlgebraicIndependent R ![x i] := by rwa [← FinVec.map_eq] at this
rwa [← algebraicIndependent_iff_transcendental] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.AlgebraicIndependent.Transcendental | {
"line": 97,
"column": 2
} | {
"line": 97,
"column": 72
} | {
"line": 99,
"column": 0
} | [
{
"pp": "R : Type u_3\nA : Type v\ninst✝² : CommRing R\ninst✝¹ : CommRing A\ninst✝ : Algebra R A\n⊢ trdeg R A ≠ 0 ↔ Algebra.Transcendental R A",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Cardinal",
"congrArg",
"Iff.rfl",
"Algebra.Transcendental"... | [] | rw [Algebra.transcendental_iff_not_isAlgebraic, Ne, trdeg_eq_zero_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.AlgebraicIndependent.Transcendental | {
"line": 97,
"column": 2
} | {
"line": 97,
"column": 72
} | {
"line": 99,
"column": 0
} | [
{
"pp": "R : Type u_3\nA : Type v\ninst✝² : CommRing R\ninst✝¹ : CommRing A\ninst✝ : Algebra R A\n⊢ trdeg R A ≠ 0 ↔ Algebra.Transcendental R A",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Cardinal",
"congrArg",
"Iff.rfl",
"Algebra.Transcendental"... | [] | rw [Algebra.transcendental_iff_not_isAlgebraic, Ne, trdeg_eq_zero_iff] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.AlgebraicIndependent.Transcendental | {
"line": 97,
"column": 2
} | {
"line": 97,
"column": 72
} | {
"line": 99,
"column": 0
} | [
{
"pp": "R : Type u_3\nA : Type v\ninst✝² : CommRing R\ninst✝¹ : CommRing A\ninst✝ : Algebra R A\n⊢ trdeg R A ≠ 0 ↔ Algebra.Transcendental R A",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Cardinal",
"congrArg",
"Iff.rfl",
"Algebra.Transcendental"... | [] | rw [Algebra.transcendental_iff_not_isAlgebraic, Ne, trdeg_eq_zero_iff] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.Matroid.Closure | {
"line": 787,
"column": 6
} | {
"line": 787,
"column": 30
} | {
"line": 787,
"column": 31
} | [
{
"pp": "α : Type u_2\nM : Matroid α\nX Y : Set α\nh : Y ⊆ M.closure (X \\ Y)\n⊢ M.closure (X \\ Y) = M.closure X",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Set.instUnion",
"Set.sdiff_union_inter",
"id",
"Set.instInter",
... | [
"α : Type u_2\nM : Matroid α\nX Y : Set α\nh : Y ⊆ M.closure (X \\ Y)\n⊢ M.closure ((X \\ Y ∪ X ∩ Y) \\ Y) = M.closure (X \\ Y ∪ X ∩ Y)"
] | ← sdiff_union_inter X Y, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.Matroid.Closure | {
"line": 1024,
"column": 30
} | {
"line": 1024,
"column": 40
} | {
"line": 1024,
"column": 41
} | [
{
"pp": "α : Type u_2\nβ : Type u_3\nM : Matroid β\nf : α → β\nX I : Set α\nhI : (M.comap f).IsBasis' I X\nhI' : M.IsBasis' (f '' I) (f '' X)\nhIinj : InjOn f I\n⊢ ∀ (x : α),\n f x ∈ M.E ∧ (M.Indep (f '' insert x I) → InjOn f (insert x I) → x ∈ I) ↔\n f x ∈ M.E ∧ (M.Indep (insert (f x) (f '' I)) → f x ∈... | [
"α : Type u_2\nβ : Type u_3\nM : Matroid β\nf : α → β\nX I : Set α\nhI : (M.comap f).IsBasis' I X\nhI' : M.IsBasis' (f '' I) (f '' X)\nhIinj : InjOn f I\n⊢ ∀ (x : α),\n f x ∈ M.E ∧ (M.Indep (f '' insert x I) → InjOn f (insert x I) → x ∈ I) ↔\n f x ∈ M.E ∧ (M.Indep (insert (f x) (f '' I)) → ∃ x_1 ∈ I, f x_1 ... | mem_image, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Combinatorics.Matroid.Closure | {
"line": 1035,
"column": 4
} | {
"line": 1035,
"column": 54
} | {
"line": 1036,
"column": 4
} | [
{
"pp": "α : Type u_2\nβ : Type u_3\nM : Matroid α\nf : α → β\nhf : InjOn f M.E\nX : Set β\naux : ∀ ⦃I : Set α⦄, M.Indep I → (M.map f hf).closure (f '' I) = f '' M.closure I\n⊢ (M.map f hf).closure X = f '' M.closure (f ⁻¹' X)",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Set.inte... | [
"α : Type u_2\nβ : Type u_3\nM : Matroid α\nf : α → β\nhf : InjOn f M.E\nX : Set β\naux : ∀ ⦃I : Set α⦄, M.Indep I → (M.map f hf).closure (f '' I) = f '' M.closure I\nI : Set α\nhI : M.IsBasis I (f ⁻¹' X ∩ M.E)\n⊢ (M.map f hf).closure X = f '' M.closure (f ⁻¹' X)"
] | obtain ⟨I, hI⟩ := M.exists_isBasis (f ⁻¹' X ∩ M.E) | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Combinatorics.Matroid.Closure | {
"line": 1035,
"column": 4
} | {
"line": 1037,
"column": 79
} | {
"line": 1039,
"column": 2
} | [
{
"pp": "α : Type u_2\nβ : Type u_3\nM : Matroid α\nf : α → β\nhf : InjOn f M.E\nX : Set β\naux : ∀ ⦃I : Set α⦄, M.Indep I → (M.map f hf).closure (f '' I) = f '' M.closure I\n⊢ (M.map f hf).closure X = f '' M.closure (f ⁻¹' X)",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Eq.mpr",... | [] | obtain ⟨I, hI⟩ := M.exists_isBasis (f ⁻¹' X ∩ M.E)
rw [← closure_inter_ground, map_ground, ← M.closure_inter_ground, ← hI.closure_eq_closure,
← aux hI.indep, ← image_preimage_inter, ← (hI.map hf).closure_eq_closure] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.Matroid.Closure | {
"line": 1035,
"column": 4
} | {
"line": 1037,
"column": 79
} | {
"line": 1039,
"column": 2
} | [
{
"pp": "α : Type u_2\nβ : Type u_3\nM : Matroid α\nf : α → β\nhf : InjOn f M.E\nX : Set β\naux : ∀ ⦃I : Set α⦄, M.Indep I → (M.map f hf).closure (f '' I) = f '' M.closure I\n⊢ (M.map f hf).closure X = f '' M.closure (f ⁻¹' X)",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Eq.mpr",... | [] | obtain ⟨I, hI⟩ := M.exists_isBasis (f ⁻¹' X ∩ M.E)
rw [← closure_inter_ground, map_ground, ← M.closure_inter_ground, ← hI.closure_eq_closure,
← aux hI.indep, ← image_preimage_inter, ← (hI.map hf).closure_eq_closure] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.Matroid.Closure | {
"line": 1040,
"column": 2
} | {
"line": 1041,
"column": 54
} | {
"line": 1043,
"column": 2
} | [
{
"pp": "α : Type u_2\nβ : Type u_3\nM : Matroid α\nf : α → β\nhf : InjOn f M.E\nX : Set β\nI : Set α\nhI : M.Indep I\ne : β\n⊢ e ∈ (M.map f hf).closure (f '' I) ↔ e ∈ f '' M.closure I",
"ppTerm": "?m.90",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Matroid.Indep.mem_closure_iff'",
... | [
"α : Type u_2\nβ : Type u_3\nM : Matroid α\nf : α → β\nhf : InjOn f M.E\nX : Set β\nI : Set α\nhI : M.Indep I\ne : β\n⊢ ((∃ x ∈ M.E, f x = e) ∧ ∀ (x : Set α), M.Indep x → insert e (f '' I) = f '' x → ∃ x ∈ I, f x = e) ↔\n ∃ x, (x ∈ M.E ∧ (M.Indep (insert x I) → x ∈ I)) ∧ f x = e"
] | simp only [(hI.map f hf).mem_closure_iff', map_ground, mem_image, map_indep_iff,
forall_exists_index, and_imp, hI.mem_closure_iff'] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.AlgebraicIndependent.TranscendenceBasis | {
"line": 122,
"column": 2
} | {
"line": 128,
"column": 17
} | {
"line": 130,
"column": 0
} | [
{
"pp": "ι : Type u\nR : Type u_1\nA : Type w\nx : ι → A\ninst✝³ : CommRing R\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\ninst✝ : Nontrivial R\nind : AlgebraicIndependent R x\n⊢ IsTranscendenceBasis R x ↔ Algebra.IsAlgebraic (↥(adjoin R (range x))) A",
"ppTerm": "?m.29",
"assigned": true,
"usedConst... | [] | refine ⟨(·.isAlgebraic), fun alg ↦ ⟨ind, fun s ind_s hxs ↦ of_not_not fun hxs' ↦ ?_⟩⟩
have : ¬ s ⊆ range x := (hxs' <| hxs.antisymm ·)
have ⟨a, has, hax⟩ := not_subset.mp this
rw [show range x = Subtype.val '' range (Set.inclusion hxs) by
rw [← range_comp, val_comp_inclusion, Subtype.range_val]] at alg
refi... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.AlgebraicIndependent.TranscendenceBasis | {
"line": 122,
"column": 2
} | {
"line": 128,
"column": 17
} | {
"line": 130,
"column": 0
} | [
{
"pp": "ι : Type u\nR : Type u_1\nA : Type w\nx : ι → A\ninst✝³ : CommRing R\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\ninst✝ : Nontrivial R\nind : AlgebraicIndependent R x\n⊢ IsTranscendenceBasis R x ↔ Algebra.IsAlgebraic (↥(adjoin R (range x))) A",
"ppTerm": "?m.29",
"assigned": true,
"usedConst... | [] | refine ⟨(·.isAlgebraic), fun alg ↦ ⟨ind, fun s ind_s hxs ↦ of_not_not fun hxs' ↦ ?_⟩⟩
have : ¬ s ⊆ range x := (hxs' <| hxs.antisymm ·)
have ⟨a, has, hax⟩ := not_subset.mp this
rw [show range x = Subtype.val '' range (Set.inclusion hxs) by
rw [← range_comp, val_comp_inclusion, Subtype.range_val]] at alg
refi... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.AlgebraicIndependent.TranscendenceBasis | {
"line": 577,
"column": 4
} | {
"line": 584,
"column": 75
} | {
"line": 585,
"column": 2
} | [
{
"pp": "case inr.inl\nι : Type u\nR : Type u_1\nS : Type v\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : FaithfulSMul R S\ninst✝ : NoZeroDivisors S\ns : Set ι\ni j : ι\nv : ι → S\nhj✝ : j ∈ insert i s\nH₁ : IsTranscendenceBasis R fun x ↦ v ↑x\nthis✝² : Nontrivial ↥(adjoin R (v '' (i... | [] | convert!
H₁.comp_equiv <|
.symm <|
((Equiv.swap j i).image s).trans <|
.setCongr <| Equiv.image_swap_of_mem_of_notMem hj hi with
⟨x, rfl | hxi, hxj⟩
· simp [eq]
· simp [Equiv.swap_apply_of_ne_of_ne hxj (ne_of_mem_of_not_mem hxi hi)] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
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