module stringlengths 16 90 | startPos dict | endPos dict | goals listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 366
values | kind stringclasses 370
values |
|---|---|---|---|---|---|---|
Mathlib.LinearAlgebra.Matrix.Ideal | {
"line": 181,
"column": 29
} | {
"line": 181,
"column": 53
} | [
{
"pp": "R : Type u_1\nn : Type u_2\ninst✝² : NonUnitalNonAssocSemiring R\ninst✝¹ : Fintype n\ninst✝ : DecidableEq n\nc : RingCon (Matrix n n R)\nw✝ x✝² y✝ z✝ : R\nh₁ : ∀ (i j : n), c (single i j w✝) (single i j x✝²)\nh₂ : ∀ (i j : n), c (single i j y✝) (single i j z✝)\nx✝¹ x✝ : n\n⊢ c (single x✝¹ x✝ (w✝ + y✝))... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.Ideal | {
"line": 196,
"column": 4
} | {
"line": 196,
"column": 15
} | [
{
"pp": "case H.mp\nR : Type u_1\nn : Type u_2\ninst✝³ : NonUnitalNonAssocSemiring R\ninst✝² : Fintype n\ninst✝¹ : DecidableEq n\ninst✝ : Nonempty n\nc : RingCon R\nx y : R\nh : (matrix n c).ofMatrix x y\ninhabited_h : Inhabited n\n⊢ c x y",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.Ideal | {
"line": 219,
"column": 4
} | {
"line": 219,
"column": 15
} | [
{
"pp": "case H.mpr\nR : Type u_1\nn : Type u_2\ninst✝² : NonAssocSemiring R\ninst✝¹ : Fintype n\ninst✝ : DecidableEq n\nc : RingCon (Matrix n n R)\nx y : Matrix n n R\nh : c x y\ni' j' i j : n\n⊢ c (single i j (x i' j')) (single i j (y i' j'))",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.Ideal | {
"line": 225,
"column": 2
} | {
"line": 225,
"column": 13
} | [
{
"pp": "R : Type u_1\nn : Type u_2\ninst✝² : NonAssocSemiring R\ninst✝¹ : Fintype n\ninst✝ : DecidableEq n\nc : RingCon (Matrix n n R)\nx y : R\ni j : n\nh : c (single i j x) (single i j y)\ni' j' : n\n⊢ c (single i' j' x) (single i' j' y)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.FixedDetMatrices | {
"line": 252,
"column": 2
} | {
"line": 252,
"column": 68
} | [
{
"pp": "case step\nm : ℤ\nC : FixedDetMatrix (Fin 2) ℤ m → Prop\nA✝ : FixedDetMatrix (Fin 2) ℤ m\nhm : m ≠ 0\nh0 : ∀ (A : FixedDetMatrix (Fin 2) ℤ m), ↑A 1 0 = 0 → 0 < ↑A 0 0 → 0 ≤ ↑A 0 1 → |↑A 0 1| < |↑A 1 1| → C A\nhS : ∀ (B : FixedDetMatrix (Fin 2) ℤ m), C B → C (S • B)\nhT : ∀ (B : FixedDetMatrix (Fin 2) ℤ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.Ideal | {
"line": 235,
"column": 4
} | {
"line": 235,
"column": 15
} | [
{
"pp": "case h.h.a.mpr\nR : Type u_1\nn : Type u_2\ninst✝² : NonAssocSemiring R\ninst✝¹ : Fintype n\ninst✝ : DecidableEq n\nc : RingCon (Matrix n n R)\ni j : n\nX Y : Matrix n n R\nh : c X Y\ni' j' : n\n⊢ c (single i' j' ((fun x ↦ x i j) X)) (single i' j' ((fun x ↦ x i j) Y))",
"usedConstants": [
"Ma... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.Ideal | {
"line": 298,
"column": 28
} | {
"line": 298,
"column": 39
} | [
{
"pp": "R : Type u_1\nn : Type u_2\ninst✝³ : NonAssocRing R\ninst✝² : Fintype n\ninst✝¹ : Nonempty n\ninst✝ : DecidableEq n\nI : TwoSidedIdeal (Matrix n n R)\ni j : n\nr : R\nh : r ∈ ↑(equivMatrix.symm I)\n⊢ single i j r ∈ I",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.Ideal | {
"line": 316,
"column": 6
} | {
"line": 316,
"column": 17
} | [
{
"pp": "case mp\nR : Type u_1\nn : Type u_2\ninst✝³ : NonAssocRing R\ninst✝² : Fintype n\ninst✝¹ : Nonempty n\ninst✝ : DecidableEq n\nI J : TwoSidedIdeal R\nx : R\nxI : x ∈ I\nle : (of fun x_1 x_2 ↦ x) ∈ matrix n J\n⊢ x ∈ J",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.Ideal | {
"line": 314,
"column": 6
} | {
"line": 316,
"column": 20
} | [
{
"pp": "case mp\nR : Type u_1\nn : Type u_2\ninst✝³ : NonAssocRing R\ninst✝² : Fintype n\ninst✝¹ : Nonempty n\ninst✝ : DecidableEq n\nI J : TwoSidedIdeal R\n⊢ matrix n I ≤ matrix n J → I ≤ J",
"usedConstants": [
"Equiv.instEquivLike",
"TwoSidedIdeal",
"Matrix",
"Matrix.of",
"P... | intro le x xI
specialize @le (of fun _ _ => x) (by simp [xI])
simpa using le | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Matrix.Ideal | {
"line": 314,
"column": 6
} | {
"line": 316,
"column": 20
} | [
{
"pp": "case mp\nR : Type u_1\nn : Type u_2\ninst✝³ : NonAssocRing R\ninst✝² : Fintype n\ninst✝¹ : Nonempty n\ninst✝ : DecidableEq n\nI J : TwoSidedIdeal R\n⊢ matrix n I ≤ matrix n J → I ≤ J",
"usedConstants": [
"Equiv.instEquivLike",
"TwoSidedIdeal",
"Matrix",
"Matrix.of",
"P... | intro le x xI
specialize @le (of fun _ _ => x) (by simp [xI])
simpa using le | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Matrix.Ideal | {
"line": 353,
"column": 2
} | {
"line": 353,
"column": 42
} | [
{
"pp": "case h\nR : Type u_1\ninst✝² : Ring R\nn : Type u_2\ninst✝¹ : Fintype n\ninst✝ : DecidableEq n\nI : TwoSidedIdeal R\nM : Matrix n n R\np q : n\ny : R\nMmem : ∃ z, z * y • single p p 1 * (M * single q p 1) + z - 1 ∈ matrix n I\nN : Matrix n n R\nNxMI : N * y • single p p 1 * (M * single q p 1) + N - 1 ∈... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.Permanent | {
"line": 37,
"column": 2
} | {
"line": 38,
"column": 67
} | [
{
"pp": "case refine_1\nn : Type u_1\ninst✝² : DecidableEq n\ninst✝¹ : Fintype n\nR : Type u_2\ninst✝ : CommSemiring R\nd : n → R\nσ : Perm n\nx✝ : σ ∈ univ\nhσ : σ ≠ 1\n⊢ ∏ i, diagonal d (σ i) i = 0",
"usedConstants": [
"Finset.mem_univ",
"Equiv.instEquivLike",
"Equiv.Perm.instOne",
... | · match not_forall.mp (mt Equiv.ext hσ) with
| ⟨x, hx⟩ => exact Finset.prod_eq_zero (mem_univ x) (if_neg hx) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.LinearAlgebra.Matrix.Swap | {
"line": 43,
"column": 2
} | {
"line": 43,
"column": 35
} | [
{
"pp": "R : Type u_1\nn : Type u_2\ninst✝² : Zero R\ninst✝¹ : One R\ninst✝ : DecidableEq n\ni j : n\n⊢ swap R i j = swap R j i",
"usedConstants": [
"One",
"congrArg",
"Equiv.swap_comm",
"Matrix",
"Equiv.swap",
"Equiv.Perm.permMatrix",
"congrFun",
"Equiv.Perm"... | simp only [swap, Equiv.swap_comm] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.LinearAlgebra.Matrix.Swap | {
"line": 43,
"column": 2
} | {
"line": 43,
"column": 35
} | [
{
"pp": "R : Type u_1\nn : Type u_2\ninst✝² : Zero R\ninst✝¹ : One R\ninst✝ : DecidableEq n\ni j : n\n⊢ swap R i j = swap R j i",
"usedConstants": [
"One",
"congrArg",
"Equiv.swap_comm",
"Matrix",
"Equiv.swap",
"Equiv.Perm.permMatrix",
"congrFun",
"Equiv.Perm"... | simp only [swap, Equiv.swap_comm] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Matrix.Swap | {
"line": 43,
"column": 2
} | {
"line": 43,
"column": 35
} | [
{
"pp": "R : Type u_1\nn : Type u_2\ninst✝² : Zero R\ninst✝¹ : One R\ninst✝ : DecidableEq n\ni j : n\n⊢ swap R i j = swap R j i",
"usedConstants": [
"One",
"congrArg",
"Equiv.swap_comm",
"Matrix",
"Equiv.swap",
"Equiv.Perm.permMatrix",
"congrFun",
"Equiv.Perm"... | simp only [swap, Equiv.swap_comm] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.PiTensorProduct.DFinsupp | {
"line": 60,
"column": 2
} | {
"line": 60,
"column": 71
} | [
{
"pp": "R : Type u_1\nι : Type u_2\nκ : ι → Type u_3\nM : (i : ι) → κ i → Type u_4\ninst✝⁵ : CommSemiring R\ninst✝⁴ : (i : ι) → (j : κ i) → AddCommMonoid (M i j)\ninst✝³ : (i : ι) → (j : κ i) → Module R (M i j)\ninst✝² : Fintype ι\ninst✝¹ : DecidableEq ι\ninst✝ : (i : ι) → DecidableEq (κ i)\nx : (i : ι) → Π₀ (... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.PiTensorProduct | {
"line": 98,
"column": 69
} | {
"line": 101,
"column": 43
} | [
{
"pp": "ι : Type u_1\nR : Type u_3\nA : ι → Type u_4\ninst✝⁴ : CommSemiring R\ninst✝³ : (i : ι) → NonAssocSemiring (A i)\ninst✝² : (i : ι) → Module R (A i)\ninst✝¹ : ∀ (i : ι), SMulCommClass R (A i) (A i)\ninst✝ : ∀ (i : ι), IsScalarTower R (A i) (A i)\nx : ⨂[R] (i : ι), A i\n⊢ (mul ((tprod R) 1)) x = x",
... | by
induction x using PiTensorProduct.induction_on with
| smul_tprod => simp
| add _ _ h1 h2 => simp [map_add, h1, h2] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.PiTensorProduct | {
"line": 136,
"column": 2
} | {
"line": 136,
"column": 36
} | [
{
"pp": "case H.H.H.H.H.H\nι : Type u_1\nR : Type u_3\nA : ι → Type u_4\ninst✝⁴ : CommSemiring R\ninst✝³ : (i : ι) → NonUnitalSemiring (A i)\ninst✝² : (i : ι) → Module R (A i)\ninst✝¹ : ∀ (i : ι), SMulCommClass R (A i) (A i)\ninst✝ : ∀ (i : ι), IsScalarTower R (A i) (A i)\nx✝ y✝ z✝ : ⨂[R] (i : ι), A i\nx y z : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Projectivization.Cardinality | {
"line": 45,
"column": 2
} | {
"line": 45,
"column": 13
} | [
{
"pp": "k : Type u_1\nV : Type u_2\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : Subsingleton V\nthis : IsEmpty { v // v ≠ 0 }\n⊢ IsEmpty (ℙ k V)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Projectivization.Cardinality | {
"line": 72,
"column": 2
} | {
"line": 72,
"column": 33
} | [
{
"pp": "case inl\nk : Type u_1\nV : Type u_2\ninst✝² : DivisionRing k\ninst✝¹ : AddCommGroup V\ninst✝ : Module k V\na✝ : Nontrivial V\nh : Finite k\nx✝ : Finite V ∨ Infinite V\n⊢ Nat.card V - 1 = Nat.card (ℙ k V) * (Nat.card k - 1)",
"usedConstants": [
"Fintype.card_congr",
"not_iff_not",
... | cases finite_or_infinite V with | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalCases | null |
Mathlib.LinearAlgebra.Projectivization.Cardinality | {
"line": 115,
"column": 6
} | {
"line": 115,
"column": 17
} | [
{
"pp": "case h\nk : Type u_1\nV : Type u_2\ninst✝³ : Field k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : Finite k\nn : ℕ\nh : Module.finrank k V = n\nthis✝ :\n ∀ (k : Type u_1) (V : Type u_2) [inst : Field k] [inst_1 : AddCommGroup V] [inst_2 : Module k V] [Finite k] {n : ℕ},\n Module.finrank k ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Projectivization.Independence | {
"line": 70,
"column": 6
} | {
"line": 70,
"column": 58
} | [
{
"pp": "case refine_2.refine_1\nι : Type u_1\nK : Type u_2\nV : Type u_3\ninst✝² : DivisionRing K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : ι → ℙ K V\nh : iSupIndep fun i ↦ (f i).submodule\ni : ι\n⊢ (Projectivization.rep ∘ f) i ∈ (Projectivization.submodule ∘ f) i",
"usedConstants": [
"Eq.mpr... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Projectivization.Subspace | {
"line": 143,
"column": 10
} | {
"line": 143,
"column": 13
} | [
{
"pp": "K : Type u_1\nV : Type u_2\ninst✝² : DivisionRing K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nx : ℙ K V\n⊢ x ∈ ⊤ → x ∈ span Set.univ",
"usedConstants": [
"PartialOrder.toPreorder",
"Preorder.toLE",
"Membership.mem",
"CompleteLattice.toBoundedOrder",
"Projectivizati... | _hx | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.LinearAlgebra.Projectivization.Action | {
"line": 131,
"column": 4
} | {
"line": 131,
"column": 65
} | [
{
"pp": "case pos\nK : Type u_1\nV : Type u_2\ninst✝² : AddCommGroup V\ninst✝¹ : Field K\ninst✝ : Module K V\nthis✝ : ∀ {a b c d : ℙ K V}, a ≠ b → c ≠ d → ∃ g, g • a = c ∧ g • b = d\nD D' E E' : ℙ K V\nhD : LinearIndependent K ![D.rep, D'.rep]\nhE : E ≠ E'\ng : V ≃ₗ[K] V\ngD : g • D = E\ngE : g • D' = E'\nhV : ... | let s := (linearIndepOn_pair D D').extend (Set.subset_univ _) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.LinearAlgebra.QuadraticForm.AlgClosed | {
"line": 38,
"column": 6
} | {
"line": 38,
"column": 17
} | [
{
"pp": "ι : Type u_1\ninst✝³ : Fintype ι\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : IsAlgClosed K\ninst✝ : DecidableEq K\nw : ι → K\ni : ι\nh : ¬w i = 0\n⊢ w i ≠ 0",
"usedConstants": [
"id",
"Ne",
"Field.toSemifield",
"Semifield.toDivisionSemiring",
"DivisionSemiring.toSemiring... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.SpecialLinearGroup | {
"line": 91,
"column": 4
} | {
"line": 92,
"column": 54
} | [
{
"pp": "case neg.h\nR : Type u_1\nV : Type u_2\ninst✝³ : CommRing R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\ninst✝ : Module.Free R V\nd1 : Module.finrank R V = 1\nu✝ v : SpecialLinearGroup R V\na✝ : Nontrivial R\nx✝ : V\nhx : ¬x✝ = 0\nu : SpecialLinearGroup R V\nx : V\nc : R := (LinearEquiv.smul_id_of_fi... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.SpecialLinearGroup | {
"line": 227,
"column": 4
} | {
"line": 227,
"column": 12
} | [
{
"pp": "case mpr\nR : Type u_1\nV : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup V\ninst✝ : Module R V\nu : LinearMap.GeneralLinearGroup R V\n⊢ LinearEquiv.det u.toLinearEquiv = 1 → u ∈ Set.range ⇑toGeneralLinearGroup",
"usedConstants": [
"LinearEquiv.det",
"MonoidHom.instFunLike",
... | intro hu | Lean.Elab.Tactic.evalIntro | null |
Mathlib.LinearAlgebra.SpecialLinearGroup | {
"line": 227,
"column": 4
} | {
"line": 227,
"column": 12
} | [
{
"pp": "case mpr\nR : Type u_1\nV : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup V\ninst✝ : Module R V\nu : LinearMap.GeneralLinearGroup R V\n⊢ LinearEquiv.det u.toLinearEquiv = 1 → u ∈ Set.range ⇑toGeneralLinearGroup",
"usedConstants": [
"LinearEquiv.det",
"MonoidHom.instFunLike",
... | intro hu | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.LinearAlgebra.QuadraticForm.Basis | {
"line": 109,
"column": 4
} | {
"line": 109,
"column": 15
} | [
{
"pp": "case H.h\nι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁵ : LinearOrder ι\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : AddCommGroup N\ninst✝¹ : Module R M\ninst✝ : Module R N\nQ : QuadraticMap R M N\nbm : Basis ι R M\nx✝ : M\nx : ι × ι\nhx : x.1 ∈ (bm.repr x✝).support ∧ x.2 ∈ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.SpecialLinearGroup | {
"line": 460,
"column": 6
} | {
"line": 460,
"column": 22
} | [
{
"pp": "case pos\nR : Type u_1\nV : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup V\ninst✝² : Module R V\ninst✝¹ : Module.Free R V\ninst✝ : Module.Finite R V\nr : ↥(rootsOfUnity (max (Module.finrank R V) 1) R)\nhV : Module.finrank R V = 0\nhR : Nontrivial R\n⊢ r = 1",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.SpecialLinearGroup | {
"line": 517,
"column": 2
} | {
"line": 517,
"column": 50
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nn : ℕ\nr : ↥(rootsOfUnity n R)\nhn : n = 1\nh : ↑r ∈ rootsOfUnity n R\n⊢ ↑r = 1",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.QuadraticForm.Radical | {
"line": 191,
"column": 4
} | {
"line": 191,
"column": 69
} | [
{
"pp": "case h.mp\n𝕜 : Type u_1\nι : Type u_2\ninst✝² : Field 𝕜\ninst✝¹ : NeZero 2\ninst✝ : Fintype ι\nw v : ι → 𝕜\nhv : ∑ x, w x * v x ^ 2 = 0\nhvv' : ∀ (n : ι → 𝕜), ∑ x, w x * v x ^ 2 + ∑ x, w x * (2 * v x * n x) = 0\ni : ι\n⊢ i ∉ {i | w i = 0} → v i = 0",
"usedConstants": [
"setOf",
"Mem... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.QuadraticForm.Radical | {
"line": 192,
"column": 4
} | {
"line": 193,
"column": 11
} | [
{
"pp": "case h.mpr\n𝕜 : Type u_1\nι : Type u_2\ninst✝² : Field 𝕜\ninst✝¹ : NeZero 2\ninst✝ : Fintype ι\nw v : ι → 𝕜\n⊢ (∀ i ∉ {i | w i = 0}, v i = 0) →\n ∑ x, w x * v x ^ 2 = 0 ∧ ∀ (n : ι → 𝕜), ∑ x, w x * v x ^ 2 + ∑ x, w x * (2 * v x * n x) = 0",
"usedConstants": [
"Eq.mpr",
"NonAssocSe... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.QuadraticForm.Signature | {
"line": 155,
"column": 30
} | {
"line": 155,
"column": 41
} | [
{
"pp": "M : Type u_2\ninst✝⁴ : AddCommGroup M\n𝕜 : Type u_4\ninst✝³ : Field 𝕜\ninst✝² : LinearOrder 𝕜\ninst✝¹ : Module 𝕜 M\nQ : QuadraticForm 𝕜 M\ninst✝ : FiniteDimensional 𝕜 M\nV : Subspace 𝕜 M\nhV : ∀ x ∈ V, Q x ≤ 0\nVp : Submodule 𝕜 M\nhr : Module.finrank 𝕜 ↥Vp = sigPos Q\nhVp : (QuadraticMap.restr... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.QuadraticForm.Signature | {
"line": 203,
"column": 2
} | {
"line": 203,
"column": 13
} | [
{
"pp": "case convert_8\n𝕜 : Type u_4\ninst✝³ : Field 𝕜\ninst✝² : LinearOrder 𝕜\nι : Type u_5\ninst✝¹ : Fintype ι\nw : ι → 𝕜\ninst✝ : IsStrictOrderedRing 𝕜\np : Set ι := {i | 0 < w i}\nm : Set ι := {i | w i ≤ 0}\nthis✝ : p.ncard + m.ncard = Nat.card ι\nthis : p.ncard ≤ sigPos (weightedSumSquares 𝕜 w)\n⊢ s... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Lemmas | {
"line": 51,
"column": 53
} | {
"line": 51,
"column": 93
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁸ : CommRing R\ninst✝⁷ : CharZero R\ninst✝⁶ : IsDomain R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\ninst✝³ : AddCommGroup N\ninst✝² : Module R N\nP : RootPairing ι R M N\ninst✝¹ : Finite ι\ninst✝ : P.IsCrystallographic\nb : P.Base\ni j : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Basic | {
"line": 113,
"column": 81
} | {
"line": 113,
"column": 96
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁶ : CommRing R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\ninst✝³ : AddCommGroup N\ninst✝² : Module R N\nP : RootPairing ι R M N\ninst✝¹ : P.IsCrystallographic\nb : P.Base\ninst✝ : DecidableEq ι\nd : ι → R\ng : Matrix ι ι R →ₗ[R] Matrix (↥... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Lemmas | {
"line": 61,
"column": 4
} | {
"line": 61,
"column": 15
} | [
{
"pp": "case a\nι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁸ : CommRing R\ninst✝⁷ : CharZero R\ninst✝⁶ : IsDomain R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\ninst✝³ : AddCommGroup N\ninst✝² : Module R N\nP : RootPairing ι R M N\ninst✝¹ : Finite ι\ninst✝ : P.IsCrystallographic\nb : P.Base... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Basic | {
"line": 134,
"column": 2
} | {
"line": 134,
"column": 56
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁶ : CommRing R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\ninst✝³ : AddCommGroup N\ninst✝² : Module R N\nP : RootPairing ι R M N\ninst✝¹ : P.IsCrystallographic\nb : P.Base\ninst✝ : Fintype ι\ni j : ↥b.support\n⊢ ⁅h i, h j⁆ = 0",
"usedC... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Lemmas | {
"line": 152,
"column": 4
} | {
"line": 154,
"column": 43
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CharZero R\ninst✝⁸ : IsDomain R\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\nP : RootPairing ι R M N\ninst✝³ : Finite ι\ninst✝² : P.IsCrystallographic\nb : P.Base\ni j ... | replace h₁ : P.root k = 2 • P.root l := by
rwa [← neg_eq_iff_eq_neg.mpr contra, ← sub_eq_add_neg, sub_eq_iff_eq_add, ← two_nsmul] at h₁
exact P.nsmul_notMem_range_root ⟨_, h₁⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Lemmas | {
"line": 152,
"column": 4
} | {
"line": 154,
"column": 43
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CharZero R\ninst✝⁸ : IsDomain R\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\nP : RootPairing ι R M N\ninst✝³ : Finite ι\ninst✝² : P.IsCrystallographic\nb : P.Base\ni j ... | replace h₁ : P.root k = 2 • P.root l := by
rwa [← neg_eq_iff_eq_neg.mpr contra, ← sub_eq_add_neg, sub_eq_iff_eq_add, ← two_nsmul] at h₁
exact P.nsmul_notMem_range_root ⟨_, h₁⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Basic | {
"line": 280,
"column": 2
} | {
"line": 280,
"column": 34
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁹ : CommRing R\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : Module R M\ninst✝⁶ : AddCommGroup N\ninst✝⁵ : Module R N\nP : RootPairing ι R M N\ninst✝⁴ : P.IsCrystallographic\nb : P.Base\ninst✝³ : Finite ι\ninst✝² : IsDomain R\ninst✝¹ : CharZero R\ninst✝... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Basic | {
"line": 331,
"column": 27
} | {
"line": 331,
"column": 64
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹⁰ : CommRing R\ninst✝⁹ : AddCommGroup M\ninst✝⁸ : Module R M\ninst✝⁷ : AddCommGroup N\ninst✝⁶ : Module R N\nP : RootPairing ι R M N\ninst✝⁵ : P.IsCrystallographic\nb : P.Base\ninst✝⁴ : Finite ι\ninst✝³ : IsDomain R\ninst✝² : CharZero R\ninst... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Basic | {
"line": 337,
"column": 41
} | {
"line": 337,
"column": 52
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹⁰ : CommRing R\ninst✝⁹ : AddCommGroup M\ninst✝⁸ : Module R M\ninst✝⁷ : AddCommGroup N\ninst✝⁶ : Module R N\nP : RootPairing ι R M N\ninst✝⁵ : P.IsCrystallographic\nb : P.Base\ninst✝⁴ : Finite ι\ninst✝³ : IsDomain R\ninst✝² : CharZero R\ninst... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Basic | {
"line": 348,
"column": 2
} | {
"line": 348,
"column": 43
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹⁰ : CommRing R\ninst✝⁹ : AddCommGroup M\ninst✝⁸ : Module R M\ninst✝⁷ : AddCommGroup N\ninst✝⁶ : Module R N\nP : RootPairing ι R M N\ninst✝⁵ : P.IsCrystallographic\nb : P.Base\ninst✝⁴ : Finite ι\ninst✝³ : IsDomain R\ninst✝² : CharZero R\ninst... | ext (k | k) <;> simp [e, Pi.single_apply] | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Basic | {
"line": 348,
"column": 2
} | {
"line": 348,
"column": 43
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹⁰ : CommRing R\ninst✝⁹ : AddCommGroup M\ninst✝⁸ : Module R M\ninst✝⁷ : AddCommGroup N\ninst✝⁶ : Module R N\nP : RootPairing ι R M N\ninst✝⁵ : P.IsCrystallographic\nb : P.Base\ninst✝⁴ : Finite ι\ninst✝³ : IsDomain R\ninst✝² : CharZero R\ninst... | ext (k | k) <;> simp [e, Pi.single_apply] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Basic | {
"line": 348,
"column": 2
} | {
"line": 348,
"column": 43
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹⁰ : CommRing R\ninst✝⁹ : AddCommGroup M\ninst✝⁸ : Module R M\ninst✝⁷ : AddCommGroup N\ninst✝⁶ : Module R N\nP : RootPairing ι R M N\ninst✝⁵ : P.IsCrystallographic\nb : P.Base\ninst✝⁴ : Finite ι\ninst✝³ : IsDomain R\ninst✝² : CharZero R\ninst... | ext (k | k) <;> simp [e, Pi.single_apply] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Basic | {
"line": 354,
"column": 65
} | {
"line": 354,
"column": 76
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹⁰ : CommRing R\ninst✝⁹ : AddCommGroup M\ninst✝⁸ : Module R M\ninst✝⁷ : AddCommGroup N\ninst✝⁶ : Module R N\nP : RootPairing ι R M N\ninst✝⁵ : P.IsCrystallographic\nb : P.Base\ninst✝⁴ : Finite ι\ninst✝³ : IsDomain R\ninst✝² : CharZero R\ninst... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Basic | {
"line": 367,
"column": 64
} | {
"line": 367,
"column": 75
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹⁰ : CommRing R\ninst✝⁹ : AddCommGroup M\ninst✝⁸ : Module R M\ninst✝⁷ : AddCommGroup N\ninst✝⁶ : Module R N\nP : RootPairing ι R M N\ninst✝⁵ : P.IsCrystallographic\nb : P.Base\ninst✝⁴ : Finite ι\ninst✝³ : IsDomain R\ninst✝² : CharZero R\ninst... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Basic | {
"line": 400,
"column": 6
} | {
"line": 400,
"column": 55
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹⁰ : CommRing R\ninst✝⁹ : AddCommGroup M\ninst✝⁸ : Module R M\ninst✝⁷ : AddCommGroup N\ninst✝⁶ : Module R N\nP : RootPairing ι R M N\ninst✝⁵ : P.IsCrystallographic\nb : P.Base\ninst✝⁴ : Finite ι\ninst✝³ : IsDomain R\ninst✝² : CharZero R\ninst... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Lemmas | {
"line": 177,
"column": 24
} | {
"line": 177,
"column": 40
} | [
{
"pp": "case h.e'_5\nι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CharZero R\ninst✝⁸ : IsDomain R\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\nP : RootPairing ι R M N\ninst✝³ : Finite ι\ninst✝² : P.IsCrystallographic\nb :... | Int.zsmul_eq_mul | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Basic | {
"line": 406,
"column": 25
} | {
"line": 406,
"column": 55
} | [
{
"pp": "case add\nι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹⁰ : CommRing R\ninst✝⁹ : AddCommGroup M\ninst✝⁸ : Module R M\ninst✝⁷ : AddCommGroup N\ninst✝⁶ : Module R N\nP : RootPairing ι R M N\ninst✝⁵ : P.IsCrystallographic\nb : P.Base\ninst✝⁴ : Finite ι\ninst✝³ : IsDomain R\ninst✝² : CharZe... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Basic | {
"line": 407,
"column": 21
} | {
"line": 407,
"column": 32
} | [
{
"pp": "case smul\nι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹⁰ : CommRing R\ninst✝⁹ : AddCommGroup M\ninst✝⁸ : Module R M\ninst✝⁷ : AddCommGroup N\ninst✝⁶ : Module R N\nP : RootPairing ι R M N\ninst✝⁵ : P.IsCrystallographic\nb : P.Base\ninst✝⁴ : Finite ι\ninst✝³ : IsDomain R\ninst✝² : CharZ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Basic | {
"line": 420,
"column": 35
} | {
"line": 420,
"column": 62
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN✝ : Type u_4\ninst✝¹⁰ : CommRing R\ninst✝⁹ : AddCommGroup M\ninst✝⁸ : Module R M\ninst✝⁷ : AddCommGroup N✝\ninst✝⁶ : Module R N✝\nP : RootPairing ι R M N✝\ninst✝⁵ : P.IsCrystallographic\nb : P.Base\ninst✝⁴ : Finite ι\ninst✝³ : IsDomain R\ninst✝² : CharZero R\n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Lemmas | {
"line": 183,
"column": 24
} | {
"line": 183,
"column": 40
} | [
{
"pp": "case h.e'_5\nι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CharZero R\ninst✝⁸ : IsDomain R\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\nP : RootPairing ι R M N\ninst✝³ : Finite ι\ninst✝² : P.IsCrystallographic\nb :... | Int.zsmul_eq_mul | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Relations | {
"line": 62,
"column": 40
} | {
"line": 62,
"column": 60
} | [
{
"pp": "case a.inr.inr.inl\nι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁹ : Finite ι\ninst✝⁸ : CommRing R\ninst✝⁷ : IsDomain R\ninst✝⁶ : CharZero R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\ninst✝³ : AddCommGroup N\ninst✝² : Module R N\nP : RootPairing ι R M N\ninst✝¹ : P.IsCrystallographi... | · simp [P.ne_zero i] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Lemmas | {
"line": 189,
"column": 24
} | {
"line": 189,
"column": 40
} | [
{
"pp": "case h.e'_5\nι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CharZero R\ninst✝⁸ : IsDomain R\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\nP : RootPairing ι R M N\ninst✝³ : Finite ι\ninst✝² : P.IsCrystallographic\nb :... | Int.zsmul_eq_mul | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Lemmas | {
"line": 194,
"column": 24
} | {
"line": 194,
"column": 40
} | [
{
"pp": "case h.e'_5\nι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CharZero R\ninst✝⁸ : IsDomain R\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\nP : RootPairing ι R M N\ninst✝³ : Finite ι\ninst✝² : P.IsCrystallographic\nb :... | Int.zsmul_eq_mul | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Relations | {
"line": 77,
"column": 4
} | {
"line": 77,
"column": 38
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁹ : Finite ι\ninst✝⁸ : CommRing R\ninst✝⁷ : IsDomain R\ninst✝⁶ : CharZero R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\ninst✝³ : AddCommGroup N\ninst✝² : Module R N\nP : RootPairing ι R M N\ninst✝¹ : P.IsCrystallographic\nb : P.Base\ninst✝... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Relations | {
"line": 111,
"column": 71
} | {
"line": 111,
"column": 87
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹⁰ : Finite ι\ninst✝⁹ : CommRing R\ninst✝⁸ : IsDomain R\ninst✝⁷ : CharZero R\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : Module R M\ninst✝⁴ : AddCommGroup N\ninst✝³ : Module R N\nP : RootPairing ι R M N\ninst✝² : P.IsCrystallographic\nb : P.Base\ninst... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Relations | {
"line": 124,
"column": 71
} | {
"line": 124,
"column": 87
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹⁰ : Finite ι\ninst✝⁹ : CommRing R\ninst✝⁸ : IsDomain R\ninst✝⁷ : CharZero R\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : Module R M\ninst✝⁴ : AddCommGroup N\ninst✝³ : Module R N\nP : RootPairing ι R M N\ninst✝² : P.IsCrystallographic\nb : P.Base\ninst... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Lemmas | {
"line": 238,
"column": 72
} | {
"line": 238,
"column": 90
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CharZero R\ninst✝⁸ : IsDomain R\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\nP : RootPairing ι R M N\ninst✝³ : Finite ι\ninst✝² : P.IsCrystallographic\nb : P.Base\ni j ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Relations | {
"line": 173,
"column": 10
} | {
"line": 173,
"column": 84
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹⁰ : Finite ι\ninst✝⁹ : CommRing R\ninst✝⁸ : IsDomain R\ninst✝⁷ : CharZero R\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : Module R M\ninst✝⁴ : AddCommGroup N\ninst✝³ : Module R N\nP : RootPairing ι R M N\ninst✝² : P.IsCrystallographic\nb : P.Base\ninst... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Relations | {
"line": 189,
"column": 31
} | {
"line": 189,
"column": 46
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹¹ : Finite ι\ninst✝¹⁰ : CommRing R\ninst✝⁹ : IsDomain R\ninst✝⁸ : CharZero R\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\nP : RootPairing ι R M N\ninst✝³ : P.IsCrystallographic\nb : P.Base\nins... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Relations | {
"line": 212,
"column": 54
} | {
"line": 212,
"column": 85
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁹ : Finite ι\ninst✝⁸ : CommRing R\ninst✝⁷ : IsDomain R\ninst✝⁶ : CharZero R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\ninst✝³ : AddCommGroup N\ninst✝² : Module R N\nP : RootPairing ι R M N\ninst✝¹ : P.IsCrystallographic\nb : P.Base\ninst✝... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Lemmas | {
"line": 286,
"column": 6
} | {
"line": 286,
"column": 65
} | [
{
"pp": "case inl\nι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CharZero R\ninst✝⁸ : IsDomain R\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\nP : RootPairing ι R M N\ninst✝³ : Finite ι\ninst✝² : P.IsCrystallographic\nb : P.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Relations | {
"line": 229,
"column": 36
} | {
"line": 229,
"column": 47
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁹ : Finite ι\ninst✝⁸ : CommRing R\ninst✝⁷ : IsDomain R\ninst✝⁶ : CharZero R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\ninst✝³ : AddCommGroup N\ninst✝² : Module R N\nP : RootPairing ι R M N\ninst✝¹ : P.IsCrystallographic\nb : P.Base\ninst✝... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.RootSystem.Finite.G2 | {
"line": 172,
"column": 2
} | {
"line": 172,
"column": 13
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁸ : CommRing R\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\nP : RootPairing ι R M N\ninst✝³ : Finite ι\ninst✝² : CharZero R\ninst✝¹ : IsDomain R\ni j : ι\ninst✝ : P.IsNotG2\nh : P.root i + P.roo... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.RootSystem.Finite.G2 | {
"line": 194,
"column": 58
} | {
"line": 194,
"column": 88
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁸ : CommRing R\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\nP : RootPairing ι R M N\ninst✝³ : Finite ι\ninst✝² : CharZero R\ninst✝¹ : IsDomain R\ni j : ι\ninst✝ : P.IsNotG2\nh : P.root i - P.roo... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.RootSystem.Finite.G2 | {
"line": 195,
"column": 2
} | {
"line": 195,
"column": 13
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁸ : CommRing R\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\nP : RootPairing ι R M N\ninst✝³ : Finite ι\ninst✝² : CharZero R\ninst✝¹ : IsDomain R\ni j : ι\ninst✝ : P.IsNotG2\nthis : InvolutiveNeg... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.RootSystem.Finite.G2 | {
"line": 338,
"column": 2
} | {
"line": 338,
"column": 13
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁸ : CommRing R\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\nP : RootPairing ι R M N\ninst✝³ : P.EmbeddedG2\ninst✝² : Finite ι\ninst✝¹ : CharZero R\ninst✝ : IsDomain R\nB : P.InvariantForm\n⊢ (B.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Lemmas | {
"line": 318,
"column": 4
} | {
"line": 318,
"column": 37
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CharZero R\ninst✝⁸ : IsDomain R\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\nP : RootPairing ι R M N\ninst✝³ : Finite ι\ninst✝² : P.IsCrystallographic\nb : P.Base\ni j ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Lemmas | {
"line": 319,
"column": 10
} | {
"line": 319,
"column": 43
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CharZero R\ninst✝⁸ : IsDomain R\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\nP : RootPairing ι R M N\ninst✝³ : Finite ι\ninst✝² : P.IsCrystallographic\nb : P.Base\ni j ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Lemmas | {
"line": 320,
"column": 10
} | {
"line": 320,
"column": 43
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CharZero R\ninst✝⁸ : IsDomain R\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\nP : RootPairing ι R M N\ninst✝³ : Finite ι\ninst✝² : P.IsCrystallographic\nb : P.Base\ni j ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Lemmas | {
"line": 325,
"column": 59
} | {
"line": 325,
"column": 92
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CharZero R\ninst✝⁸ : IsDomain R\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\nP : RootPairing ι R M N\ninst✝³ : Finite ι\ninst✝² : P.IsCrystallographic\nb : P.Base\ni j ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Lemmas | {
"line": 347,
"column": 42
} | {
"line": 347,
"column": 70
} | [
{
"pp": "case h.e'_5\nι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CharZero R\ninst✝⁸ : IsDomain R\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\nP : RootPairing ι R M N\ninst✝³ : Finite ι\ninst✝² : P.IsCrystallographic\nb :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Lemmas | {
"line": 362,
"column": 4
} | {
"line": 362,
"column": 30
} | [
{
"pp": "case inr.inl\nι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CharZero R\ninst✝⁸ : IsDomain R\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\nP : RootPairing ι R M N\ninst✝³ : Finite ι\ninst✝² : P.IsCrystallographic\nb ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Lemmas | {
"line": 367,
"column": 37
} | {
"line": 367,
"column": 48
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CharZero R\ninst✝⁸ : IsDomain R\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\nP : RootPairing ι R M N\ninst✝³ : Finite ι\ninst✝² : P.IsCrystallographic\nb : P.Base\ni j ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Lemmas | {
"line": 369,
"column": 4
} | {
"line": 369,
"column": 15
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CharZero R\ninst✝⁸ : IsDomain R\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\nP : RootPairing ι R M N\ninst✝³ : Finite ι\ninst✝² : P.IsCrystallographic\nb : P.Base\ni j ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.RootSystem.Finite.G2 | {
"line": 566,
"column": 2
} | {
"line": 566,
"column": 13
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁹ : CommRing R\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : Module R M\ninst✝⁶ : AddCommGroup N\ninst✝⁵ : Module R N\nP : RootPairing ι R M N\ninst✝⁴ : P.EmbeddedG2\ninst✝³ : Finite ι\ninst✝² : CharZero R\ninst✝¹ : IsDomain R\ninst✝ : P.IsIrreducible\n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.RootSystem.Finite.G2 | {
"line": 597,
"column": 36
} | {
"line": 597,
"column": 47
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁹ : CommRing R\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : Module R M\ninst✝⁶ : AddCommGroup N\ninst✝⁵ : Module R N\nP : RootPairing ι R M N\ninst✝⁴ : P.EmbeddedG2\ninst✝³ : Finite ι\ninst✝² : CharZero R\ninst✝¹ : IsDomain R\ninst✝ : P.IsIrreducible\n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.RootSystem.Finite.G2 | {
"line": 597,
"column": 33
} | {
"line": 597,
"column": 64
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁹ : CommRing R\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : Module R M\ninst✝⁶ : AddCommGroup N\ninst✝⁵ : Module R N\nP : RootPairing ι R M N\ninst✝⁴ : P.EmbeddedG2\ninst✝³ : Finite ι\ninst✝² : CharZero R\ninst✝¹ : IsDomain R\ninst✝ : P.IsIrreducible\n... | by simpa using mem_allRoots P i | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.RootSystem.Finite.G2 | {
"line": 613,
"column": 50
} | {
"line": 613,
"column": 66
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁸ : CommRing R\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\nP : RootPairing ι R M N\ninst✝³ : P.IsG2\nb : P.Base\ninst✝² : Finite ι\ninst✝¹ : CharZero R\ninst✝ : IsDomain R\n_i✝ : P.EmbeddedG2\n... | Fintype.card_coe | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.SModEq.Pow | {
"line": 27,
"column": 34
} | {
"line": 27,
"column": 45
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nI J : Ideal R\np : ℕ\nhpI : ↑p ∈ I\nx y : R\nh : x - y ∈ J\nhJI : J ≤ I\nh₁ : (Ideal.Quotient.mk I) x = (Ideal.Quotient.mk I) y\n⊢ ↑p = 0",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.SesquilinearForm.Star | {
"line": 61,
"column": 2
} | {
"line": 68,
"column": 13
} | [
{
"pp": "M : Type u_2\nn : Type u_3\ninst✝⁶ : AddCommMonoid M\ninst✝⁵ : Fintype n\ninst✝⁴ : DecidableEq n\nR : Type u_4\ninst✝³ : CommRing R\ninst✝² : StarRing R\ninst✝¹ : PartialOrder R\ninst✝ : Module R M\nB : M →ₗ⋆[R] M →ₗ[R] R\nb : Basis n R M\n⊢ B.IsPosSemidef ↔ ((toMatrix₂ b b) B).PosSemidef",
"usedCo... | rw [isPosSemidef_def, Matrix.posSemidef_iff_dotProduct_mulVec]
apply and_congr (B.isSymm_iff_isHermitian_toMatrix b)
rw [isNonneg_def]
refine ⟨fun h x ↦ ?_, fun h x ↦ ?_⟩
· rw [star_dotProduct_toMatrix₂_mulVec]
exact h _
· rw [apply_eq_star_dotProduct_toMatrix₂_mulVec b]
exact h _ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.SesquilinearForm.Star | {
"line": 61,
"column": 2
} | {
"line": 68,
"column": 13
} | [
{
"pp": "M : Type u_2\nn : Type u_3\ninst✝⁶ : AddCommMonoid M\ninst✝⁵ : Fintype n\ninst✝⁴ : DecidableEq n\nR : Type u_4\ninst✝³ : CommRing R\ninst✝² : StarRing R\ninst✝¹ : PartialOrder R\ninst✝ : Module R M\nB : M →ₗ⋆[R] M →ₗ[R] R\nb : Basis n R M\n⊢ B.IsPosSemidef ↔ ((toMatrix₂ b b) B).PosSemidef",
"usedCo... | rw [isPosSemidef_def, Matrix.posSemidef_iff_dotProduct_mulVec]
apply and_congr (B.isSymm_iff_isHermitian_toMatrix b)
rw [isNonneg_def]
refine ⟨fun h x ↦ ?_, fun h x ↦ ?_⟩
· rw [star_dotProduct_toMatrix₂_mulVec]
exact h _
· rw [apply_eq_star_dotProduct_toMatrix₂_mulVec b]
exact h _ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.TensorAlgebra.Grading | {
"line": 32,
"column": 36
} | {
"line": 32,
"column": 62
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nm : M\n⊢ (TensorAlgebra.ι R) m ∈ (TensorAlgebra.ι R).range ^ 1",
"usedConstants": [
"Eq.mpr",
"Submodule",
"RingHomSurjective.ids",
"IsScalarTower.right",
"congrArg",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.TensorAlgebra.Grading | {
"line": 37,
"column": 33
} | {
"line": 37,
"column": 59
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nm : M\n⊢ (TensorAlgebra.ι R) m ∈ (TensorAlgebra.ι R).range ^ 1",
"usedConstants": [
"Eq.mpr",
"Submodule",
"RingHomSurjective.ids",
"IsScalarTower.right",
"congrArg",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Semisimple | {
"line": 68,
"column": 6
} | {
"line": 72,
"column": 14
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹⁰ : CommRing R\ninst✝⁹ : IsDomain R\ninst✝⁸ : CharZero R\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\nP : RootPairing ι R M N\ninst✝³ : P.IsCrystallographic\ninst✝² : P.IsReduced\nb : P.Base\ni... | replace hk₁ : P.root (-j) = (n + 1) • P.root i := by
simp only [indexNeg_neg, root_reflectionPerm, reflection_apply_self, neg_eq_iff_add_eq_zero,
add_smul, one_smul] at hk₁ ⊢
rw [← hk₁]
module | Lean.Elab.Tactic.evalReplace | Lean.Parser.Tactic.replace |
Mathlib.LinearAlgebra.SymmetricAlgebra.Basic | {
"line": 69,
"column": 11
} | {
"line": 69,
"column": 20
} | [
{
"pp": "case ι\nR : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nmotive : SymmetricAlgebra R M → Prop\nalgebraMap : ∀ (r : R), motive ((Algebra.algebraMap R (SymmetricAlgebra R M)) r)\nι : ∀ (x : M), motive ((SymmetricAlgebra.ι R M) x)\nmul : ∀ (a b : Symmetric... | exact ι x | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.LinearAlgebra.SymmetricAlgebra.Basic | {
"line": 69,
"column": 11
} | {
"line": 69,
"column": 20
} | [
{
"pp": "case ι\nR : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nmotive : SymmetricAlgebra R M → Prop\nalgebraMap : ∀ (r : R), motive ((Algebra.algebraMap R (SymmetricAlgebra R M)) r)\nι : ∀ (x : M), motive ((SymmetricAlgebra.ι R M) x)\nmul : ∀ (a b : Symmetric... | exact ι x | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.SymmetricAlgebra.Basic | {
"line": 69,
"column": 11
} | {
"line": 69,
"column": 20
} | [
{
"pp": "case ι\nR : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nmotive : SymmetricAlgebra R M → Prop\nalgebraMap : ∀ (r : R), motive ((Algebra.algebraMap R (SymmetricAlgebra R M)) r)\nι : ∀ (x : M), motive ((SymmetricAlgebra.ι R M) x)\nmul : ∀ (a b : Symmetric... | exact ι x | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.SymmetricAlgebra.Basic | {
"line": 134,
"column": 2
} | {
"line": 134,
"column": 22
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nx : R\n⊢ algebraMapInv ((algebraMap R (SymmetricAlgebra R M)) x) = x",
"usedConstants": [
"TensorAlgebra.SymRel",
"Semiring.toModule",
"Equiv.instEquivLike",
"Algebra.algebraMa... | simp [algebraMapInv] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.LinearAlgebra.SymmetricAlgebra.Basic | {
"line": 134,
"column": 2
} | {
"line": 134,
"column": 22
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nx : R\n⊢ algebraMapInv ((algebraMap R (SymmetricAlgebra R M)) x) = x",
"usedConstants": [
"TensorAlgebra.SymRel",
"Semiring.toModule",
"Equiv.instEquivLike",
"Algebra.algebraMa... | simp [algebraMapInv] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.SymmetricAlgebra.Basic | {
"line": 134,
"column": 2
} | {
"line": 134,
"column": 22
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nx : R\n⊢ algebraMapInv ((algebraMap R (SymmetricAlgebra R M)) x) = x",
"usedConstants": [
"TensorAlgebra.SymRel",
"Semiring.toModule",
"Equiv.instEquivLike",
"Algebra.algebraMa... | simp [algebraMapInv] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.SymmetricAlgebra.Basic | {
"line": 199,
"column": 2
} | {
"line": 199,
"column": 13
} | [
{
"pp": "case h.h\nR : Type u_1\nM : Type u_2\ninst✝⁴ : CommSemiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\nA : Type u_3\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nf : M →ₗ[R] A\ne : SymmetricAlgebra R M ≃ₐ[R] A\nhe : ↑↑e ∘ₗ SymmetricAlgebra.ι R M = f\nx : M\n⊢ (↑↑e ∘ₗ SymmetricAlgebra.ι R M) x =... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.SymmetricAlgebra.Basic | {
"line": 226,
"column": 2
} | {
"line": 226,
"column": 13
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝⁶ : CommSemiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\nA : Type u_3\ninst✝³ : CommSemiring A\ninst✝² : Algebra R A\nf : M →ₗ[R] A\nA' : Type u_4\ninst✝¹ : CommSemiring A'\ninst✝ : Algebra R A'\nh : IsSymmetricAlgebra f\nF G : A →ₐ[R] A'\nhFG : ↑F ∘ₗ f = ↑G ∘... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.SymmetricAlgebra.Basic | {
"line": 245,
"column": 20
} | {
"line": 245,
"column": 31
} | [
{
"pp": "case algebraMap\nR : Type u_1\nM : Type u_2\ninst✝⁴ : CommSemiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\nA : Type u_3\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nf : M →ₗ[R] A\nh : IsSymmetricAlgebra f\nmotive : A → Prop\nalgebraMap : ∀ (r : R), motive ((Algebra.algebraMap R A) r)\nι : ∀... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.SymmetricAlgebra.Basic | {
"line": 246,
"column": 11
} | {
"line": 246,
"column": 22
} | [
{
"pp": "case ι\nR : Type u_1\nM : Type u_2\ninst✝⁴ : CommSemiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\nA : Type u_3\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nf : M →ₗ[R] A\nh : IsSymmetricAlgebra f\nmotive : A → Prop\nalgebraMap : ∀ (r : R), motive ((Algebra.algebraMap R A) r)\nι : ∀ (x : M),... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.SymmetricAlgebra.Basic | {
"line": 247,
"column": 21
} | {
"line": 247,
"column": 32
} | [
{
"pp": "case mul\nR : Type u_1\nM : Type u_2\ninst✝⁴ : CommSemiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\nA : Type u_3\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nf : M →ₗ[R] A\nh : IsSymmetricAlgebra f\nmotive : A → Prop\nalgebraMap : ∀ (r : R), motive ((Algebra.algebraMap R A) r)\nι : ∀ (x : M... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
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