module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.RingTheory.Flat.EquationalCriterion | {
"line": 240,
"column": 6
} | {
"line": 240,
"column": 17
} | {
"line": 240,
"column": 18
} | [
{
"pp": "case singleton\nR : Type u_1\nM : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\ninst✝³ : Flat R M\nK : Type u_3\ninst✝² : AddCommGroup K\ninst✝¹ : Module R K\ninst✝ : Module.Finite R K\nK' : Submodule R K\nk : K\nn : ℕ\nf : K →ₗ[R] Fin n →₀ R\nx : (Fin n →₀ R) →ₗ[R] M\nh ... | [
"case singleton\nR : Type u_1\nM : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\ninst✝³ : Flat R M\nK : Type u_3\ninst✝² : AddCommGroup K\ninst✝¹ : Module R K\ninst✝ : Module.Finite R K\nK' : Submodule R K\nk : K\nn : ℕ\nf : K →ₗ[R] Fin n →₀ R\nx : (Fin n →₀ R) →ₗ[R] M\nh : x ∘ₗ f = 0... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Module.SpanRankOperations | {
"line": 63,
"column": 58
} | {
"line": 63,
"column": 75
} | {
"line": 63,
"column": 76
} | [
{
"pp": "R : Type u_1\ninst✝³ : CommRing R\nM : Type u_3\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\nN : Submodule R M\ninst✝ : IsLocalRing R\nfg : N.FG\nthis : Module.Finite R ↥N := Module.Finite.iff_fg.mpr fg\ns : Set (𝓀 ⊗[R] ↥N)\nhs₁ : Cardinal.mk ↑s = ⊤.spanRank\nhs₂ : span 𝓀 s = ⊤\n⊢ Cardinal.mk ↑s < ... | [
"R : Type u_1\ninst✝³ : CommRing R\nM : Type u_3\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\nN : Submodule R M\ninst✝ : IsLocalRing R\nfg : N.FG\nthis : Module.Finite R ↥N := Module.Finite.iff_fg.mpr fg\ns : Set (𝓀 ⊗[R] ↥N)\nhs₁ : Cardinal.mk ↑s = ⊤.spanRank\nhs₂ : span 𝓀 s = ⊤\n⊢ ⊤.FG"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Support | {
"line": 254,
"column": 4
} | {
"line": 257,
"column": 87
} | {
"line": 258,
"column": 2
} | [
{
"pp": "case a.refine_2\nR : Type u_1\nM : Type u_2\ninst✝³ : CommRing R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : Module.Finite R M\nI : Ideal R\n⊢ support R (M ⧸ I • ⊤) ⊆ zeroLocus ↑I",
"ppTerm": "?a.refine_2✝",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
... | [] | · rw [support_eq_zeroLocus]
apply PrimeSpectrum.zeroLocus_anti_mono_ideal
rw [Submodule.annihilator_quotient]
exact fun x hx ↦ Submodule.mem_colon.mpr fun p hp ↦ Submodule.smul_mem_smul hx hp | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Data.Fin.Parity | {
"line": 60,
"column": 4
} | {
"line": 60,
"column": 15
} | {
"line": 60,
"column": 16
} | [
{
"pp": "case inr\nn : ℕ\nhn : Odd n\nk : Fin n\nthis : NeZero n\nhk : Odd ↑k\n⊢ Even k",
"ppTerm": "?inr",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"case inr\nn : ℕ\nhn : Odd n\nk : Fin n\nthis : NeZero n\nhk : Odd ↑k\n⊢ Even k"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Fin.Parity | {
"line": 64,
"column": 4
} | {
"line": 64,
"column": 15
} | {
"line": 64,
"column": 16
} | [
{
"pp": "case inl\nn : ℕ\ninst✝ : NeZero n\nhn : Odd n\nk : Fin n\nhk : Even ↑k\n⊢ Odd k",
"ppTerm": "?inl",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"case inl\nn : ℕ\ninst✝ : NeZero n\nhn : Odd n\nk : Fin n\nhk : Even ↑k\n⊢ Odd k"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Flat.EquationalCriterion | {
"line": 264,
"column": 58
} | {
"line": 264,
"column": 82
} | {
"line": 264,
"column": 83
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝¹⁰ : CommRing R\ninst✝⁹ : AddCommGroup M\ninst✝⁸ : Module R M\ninst✝⁷ : Flat R M\nK : Type u_3\nN : Type u_4\ninst✝⁶ : AddCommGroup K\ninst✝⁵ : Module R K\ninst✝⁴ : Module.Finite R K\ninst✝³ : AddCommGroup N\ninst✝² : Module R N\ninst✝¹ : Free R N\ninst✝ : Module.Finite... | [
"R : Type u_1\nM : Type u_2\ninst✝¹⁰ : CommRing R\ninst✝⁹ : AddCommGroup M\ninst✝⁸ : Module R M\ninst✝⁷ : Flat R M\nK : Type u_3\nN : Type u_4\ninst✝⁶ : AddCommGroup K\ninst✝⁵ : Module R K\ninst✝⁴ : Module.Finite R K\ninst✝³ : AddCommGroup N\ninst✝² : Module R N\ninst✝¹ : Free R N\ninst✝ : Module.Finite R N\nf : K ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Alternating.Uncurry.Fin | {
"line": 112,
"column": 8
} | {
"line": 112,
"column": 57
} | {
"line": 113,
"column": 8
} | [
{
"pp": "R : Type u_1\nM : Type u_2\nM₂ : Type u_3\nN : Type u_4\nN₂ : Type u_5\ninst✝⁸ : CommRing R\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : AddCommGroup M₂\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : AddCommGroup N₂\ninst✝³ : Module R M\ninst✝² : Module R M₂\ninst✝¹ : Module R N\ninst✝ : Module R N₂\nn : ℕ\nf : M →ₗ[R] M [⋀... | [
"R : Type u_1\nM : Type u_2\nM₂ : Type u_3\nN : Type u_4\nN₂ : Type u_5\ninst✝⁸ : CommRing R\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : AddCommGroup M₂\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : AddCommGroup N₂\ninst✝³ : Module R M\ninst✝² : Module R M₂\ninst✝¹ : Module R N\ninst✝ : Module R N₂\nn : ℕ\nf : M →ₗ[R] M [⋀^Fin n]→ₗ[R]... | rcases exists_succAbove_eq hkj.symm with ⟨j, rfl⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.LinearAlgebra.ExteriorPower.Pairing | {
"line": 51,
"column": 6
} | {
"line": 51,
"column": 36
} | {
"line": 51,
"column": 37
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nn : ℕ\nf : Fin n → Module.Dual R M\ni j : Fin n\nhf : f i = f j\nhij : i ≠ j\nv : Fin n → M\nthis : (Matrix.of fun i j ↦ (f j) (v i)).det = 0\n⊢ (LinearMap.compAlternatingMap\n (((toTensorPower R M n).d... | [
"R : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nn : ℕ\nf : Fin n → Module.Dual R M\ni j : Fin n\nhf : f i = f j\nhij : i ≠ j\nv : Fin n → M\nthis : (Matrix.of fun i j ↦ (f j) (v i)).det = 0\n⊢ ∑ x, Equiv.Perm.sign x • ∏ i, (f i) (v (x i)) = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.ExteriorPower.Pairing | {
"line": 102,
"column": 15
} | {
"line": 102,
"column": 26
} | {
"line": 102,
"column": 27
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝³ : CommRing R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\nι : Type u_3\ninst✝ : LinearOrder ι\nx : ι → M\nf : ι → Module.Dual R M\nh₁ : ∀ (i : ι), (f i) (x i) = 1\nh₀ : ∀ ⦃i j : ι⦄, i ≠ j → (f i) (x j) = 0\nn : ℕ\na : Fin n ↪o ι\ni j : Fin n\nhij : ¬i = j\n⊢ a j ≠ a... | [
"R : Type u_1\nM : Type u_2\ninst✝³ : CommRing R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\nι : Type u_3\ninst✝ : LinearOrder ι\nx : ι → M\nf : ι → Module.Dual R M\nh₁ : ∀ (i : ι), (f i) (x i) = 1\nh₀ : ∀ ⦃i j : ι⦄, i ≠ j → (f i) (x j) = 0\nn : ℕ\na : Fin n ↪o ι\ni j : Fin n\nhij : ¬i = j\n⊢ ¬j = i"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.ExteriorPower.Basis | {
"line": 59,
"column": 21
} | {
"line": 59,
"column": 44
} | {
"line": 59,
"column": 44
} | [
{
"pp": "R : Type u_1\nM : Type u_3\nn : ℕ\ninst✝³ : CommRing R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\nI : Type u_5\ninst✝ : LinearOrder I\nb : Basis I R M\ns : ↑(powersetCard I n)\n⊢ (ιMultiDual R n b s) ((ιMulti R n) (⇑b ∘ ⇑(ofFinEmbEquiv.symm s))) = 1",
"ppTerm": "?m.37",
"assigned": true,
... | [
"R : Type u_1\nM : Type u_3\nn : ℕ\ninst✝³ : CommRing R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\nI : Type u_5\ninst✝ : LinearOrder I\nb : Basis I R M\ns : ↑(powersetCard I n)\n⊢ (Matrix.of fun i j ↦ (b.coord ((ofFinEmbEquiv.symm s) j)) ((⇑b ∘ ⇑(ofFinEmbEquiv.symm s)) i)).det = 1"
] | ιMultiDual_apply_ιMulti | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.ExteriorPower.Basis | {
"line": 73,
"column": 21
} | {
"line": 73,
"column": 44
} | {
"line": 73,
"column": 44
} | [
{
"pp": "R : Type u_1\nM : Type u_3\nn : ℕ\ninst✝³ : CommRing R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\nI : Type u_5\ninst✝ : LinearOrder I\nb : Basis I R M\ns t : ↑(powersetCard I n)\nhst : s ≠ t\n⊢ (ιMultiDual R n b s) ((ιMulti R n) (⇑b ∘ ⇑(ofFinEmbEquiv.symm t))) = 0",
"ppTerm": "?m.39",
"assi... | [
"R : Type u_1\nM : Type u_3\nn : ℕ\ninst✝³ : CommRing R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\nI : Type u_5\ninst✝ : LinearOrder I\nb : Basis I R M\ns t : ↑(powersetCard I n)\nhst : s ≠ t\n⊢ (Matrix.of fun i j ↦ (b.coord ((ofFinEmbEquiv.symm s) j)) ((⇑b ∘ ⇑(ofFinEmbEquiv.symm t)) i)).det = 0"
] | ιMultiDual_apply_ιMulti | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.LocalRing.Module | {
"line": 66,
"column": 6
} | {
"line": 66,
"column": 17
} | {
"line": 66,
"column": 18
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝³ : CommRing R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : IsLocalRing R\nN₁ N₂ : Submodule R M\nh : N₁ ≤ N₂\nh' : N₂.FG\nhN : Submodule.map (𝔪 • N₂).mkQ N₁ = Submodule.map (𝔪 • N₂).mkQ N₂\n⊢ N₂ ≤ 𝔪 • N₂ ⊔ N₁",
"ppTerm": "?m.104",
"assigned": false... | [
"R : Type u_1\nM : Type u_2\ninst✝³ : CommRing R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : IsLocalRing R\nN₁ N₂ : Submodule R M\nh : N₁ ≤ N₂\nh' : N₂.FG\nhN : Submodule.map (𝔪 • N₂).mkQ N₁ = Submodule.map (𝔪 • N₂).mkQ N₂\n⊢ N₂ ≤ 𝔪 • N₂ ⊔ N₁"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.ExteriorPower.Basis | {
"line": 128,
"column": 2
} | {
"line": 128,
"column": 35
} | {
"line": 128,
"column": 36
} | [
{
"pp": "R : Type u_1\nM : Type u_3\nn : ℕ\ninst✝³ : CommRing R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\nI : Type u_5\ninst✝ : LinearOrder I\nb : Basis I R M\nx : ↥(⋀[R]^n M)\ns : ↑(powersetCard I n)\n⊢ ((Basis.exteriorPower n b).repr x) s = (ιMultiDual R n b s) x",
"ppTerm": "?m.36",
"assigned": ... | [
"R : Type u_1\nM : Type u_3\nn : ℕ\ninst✝³ : CommRing R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\nI : Type u_5\ninst✝ : LinearOrder I\nb : Basis I R M\nx : ↥(⋀[R]^n M)\ns : ↑(powersetCard I n)\n⊢ ((Basis.exteriorPower n b).coord s) x = (ιMultiDual R n b s) x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.ExteriorPower.Basis | {
"line": 133,
"column": 2
} | {
"line": 133,
"column": 32
} | {
"line": 133,
"column": 33
} | [
{
"pp": "R : Type u_1\nM : Type u_3\nn : ℕ\ninst✝³ : CommRing R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\nI : Type u_5\ninst✝ : LinearOrder I\nb : Basis I R M\ns : ↑(powersetCard I n)\n⊢ ((Basis.exteriorPower n b).repr (ιMulti_family R n (⇑b) s)) s = 1",
"ppTerm": "?m.34",
"assigned": true,
"us... | [
"R : Type u_1\nM : Type u_3\nn : ℕ\ninst✝³ : CommRing R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\nI : Type u_5\ninst✝ : LinearOrder I\nb : Basis I R M\ns : ↑(powersetCard I n)\n⊢ (ιMultiDual R n b s) (ιMulti_family R n (⇑b) s) = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.ExteriorPower.Basis | {
"line": 139,
"column": 2
} | {
"line": 139,
"column": 32
} | {
"line": 139,
"column": 33
} | [
{
"pp": "R : Type u_1\nM : Type u_3\nn : ℕ\ninst✝³ : CommRing R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\nI : Type u_5\ninst✝ : LinearOrder I\nb : Basis I R M\ns t : ↑(powersetCard I n)\nhst : s ≠ t\n⊢ ((Basis.exteriorPower n b).repr (ιMulti_family R n (⇑b) s)) t = 0",
"ppTerm": "?m.36",
"assigned"... | [
"R : Type u_1\nM : Type u_3\nn : ℕ\ninst✝³ : CommRing R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\nI : Type u_5\ninst✝ : LinearOrder I\nb : Basis I R M\ns t : ↑(powersetCard I n)\nhst : s ≠ t\n⊢ (ιMultiDual R n b t) (ιMulti_family R n (⇑b) s) = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.TensorProduct.Submodule | {
"line": 159,
"column": 2
} | {
"line": 159,
"column": 13
} | {
"line": 159,
"column": 14
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝² : CommSemiring R\ninst✝¹ : Semiring S\ninst✝ : Algebra R S\nN : Submodule R S\nn : ↥N\n⊢ N.lTensorOne' (1 ⊗ₜ[R] n) = n",
"ppTerm": "?m.26",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u\nS : Type v\ninst✝² : CommSemiring R\ninst✝¹ : Semiring S\ninst✝ : Algebra R S\nN : Submodule R S\nn : ↥N\n⊢ N.lTensorOne' (1 ⊗ₜ[R] n) = n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.TensorProduct.Submodule | {
"line": 211,
"column": 2
} | {
"line": 211,
"column": 13
} | {
"line": 211,
"column": 14
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝² : CommSemiring R\ninst✝¹ : Semiring S\ninst✝ : Algebra R S\nM : Submodule R S\nm : ↥M\n⊢ M.rTensorOne' (m ⊗ₜ[R] 1) = m",
"ppTerm": "?m.26",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u\nS : Type v\ninst✝² : CommSemiring R\ninst✝¹ : Semiring S\ninst✝ : Algebra R S\nM : Submodule R S\nm : ↥M\n⊢ M.rTensorOne' (m ⊗ₜ[R] 1) = m"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.TensorProduct.Submodule | {
"line": 220,
"column": 2
} | {
"line": 223,
"column": 80
} | {
"line": 225,
"column": 0
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝² : CommSemiring R\ninst✝¹ : Semiring S\ninst✝ : Algebra R S\nM N : Submodule R S\nn : ↥M\nr : ↥⊥\n⊢ ((↑(TensorProduct.comm R ↥⊥ ↥M) ∘ₗ (TensorProduct.mk R ↥⊥ ↥M) 1) ∘ₗ M.rTensorOne') (n ⊗ₜ[R] r) =\n LinearMap.id (n ⊗ₜ[R] r)",
"ppTerm": "?m.149",
"assigned": true... | [] | change rTensorOne' M _ ⊗ₜ[R] 1 = n ⊗ₜ[R] r
obtain ⟨x, h⟩ := Algebra.mem_bot.1 r.2
replace h : algebraMap R _ x = r := Subtype.val_injective h
rw [← h, rTensorOne'_tmul, TensorProduct.smul_tmul, Algebra.smul_def, mul_one] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.TensorProduct.Submodule | {
"line": 220,
"column": 2
} | {
"line": 223,
"column": 80
} | {
"line": 225,
"column": 0
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝² : CommSemiring R\ninst✝¹ : Semiring S\ninst✝ : Algebra R S\nM N : Submodule R S\nn : ↥M\nr : ↥⊥\n⊢ ((↑(TensorProduct.comm R ↥⊥ ↥M) ∘ₗ (TensorProduct.mk R ↥⊥ ↥M) 1) ∘ₗ M.rTensorOne') (n ⊗ₜ[R] r) =\n LinearMap.id (n ⊗ₜ[R] r)",
"ppTerm": "?m.149",
"assigned": true... | [] | change rTensorOne' M _ ⊗ₜ[R] 1 = n ⊗ₜ[R] r
obtain ⟨x, h⟩ := Algebra.mem_bot.1 r.2
replace h : algebraMap R _ x = r := Subtype.val_injective h
rw [← h, rTensorOne'_tmul, TensorProduct.smul_tmul, Algebra.smul_def, mul_one] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Ideal.MinimalPrime.Colon | {
"line": 62,
"column": 4
} | {
"line": 62,
"column": 53
} | {
"line": 62,
"column": 54
} | [
{
"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 := ⋯\nhI : I ∈ ann.minimalPrimes\nkey : ∃ n, n ≠ 0 ∧ ∃ J, I ^ n * J ≤ ann ∧ ¬J ≤ I\nJ : Ideal R\nhJI : ¬J ≤ I\... | [
"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 := ⋯\nhI : I ∈ ann.minimalPrimes\nkey : ∃ n, n ≠ 0 ∧ ∃ J, I ^ n * J ≤ ann ∧ ¬J ≤ I\nJ : Ideal R\nhJI : ¬J ≤ I\nn : ℕ := ⋯\... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.LinearDisjoint | {
"line": 536,
"column": 34
} | {
"line": 536,
"column": 68
} | {
"line": 536,
"column": 68
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝² : CommRing R\ninst✝¹ : Ring S\ninst✝ : Algebra R S\nM N : Submodule R S\nH : M.LinearDisjoint N\nhf : Flat R ↥M ∨ Flat R ↥N\nhc : ∀ (m n : ↥(M ⊓ N)), Commute ↑m ↑n\na✝ : Nontrivial R\ns : Finset ↥(M ⊓ N)\nh : LinearIndependent R fun i ↦ ↑i\nhs : 1 < Fintype.card ↥s\n⊢ Fal... | [
"R : Type u\nS : Type v\ninst✝² : CommRing R\ninst✝¹ : Ring S\ninst✝ : Algebra R S\nM N : Submodule R S\nH : M.LinearDisjoint N\nhf : Flat R ↥M ∨ Flat R ↥N\nhc : ∀ (m n : ↥(M ⊓ N)), Commute ↑m ↑n\na✝ : Nontrivial R\ns : Finset ↥(M ⊓ N)\nh : LinearIndependent R fun i ↦ ↑i\nhs : Nontrivial ↥s\n⊢ False"
] | Fintype.one_lt_card_iff_nontrivial | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.PiTensorProduct.Basic | {
"line": 291,
"column": 94
} | {
"line": 294,
"column": 5
} | {
"line": 296,
"column": 0
} | [
{
"pp": "ι : Type u_1\nR : Type u_4\ninst✝² : CommSemiring R\ns : ι → Type u_7\ninst✝¹ : (i : ι) → AddCommMonoid (s i)\ninst✝ : (i : ι) → Module R (s i)\nz : R\nf : (i : ι) → s i\n⊢ tprodCoeff R z f = z • (tprod R) f",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"PiTensorProduct.in... | [] | by
have : z = z • (1 : R) := by simp only [mul_one, smul_eq_mul]
conv_lhs => rw [this]
rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Ideal.MinimalPrime.Colon | {
"line": 90,
"column": 4
} | {
"line": 90,
"column": 67
} | {
"line": 90,
"column": 68
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Ideal.AssociatedPrime.Finiteness | {
"line": 61,
"column": 43
} | {
"line": 61,
"column": 82
} | {
"line": 62,
"column": 2
} | [
{
"pp": "A : Type u\ninst✝² : CommRing A\nM : Type v\ninst✝¹ : AddCommGroup M\ninst✝ : Module A M\nN₁ N₂ : Submodule A M\nf : ↥N₂ ⧸ N₁.submoduleOf N₂ →ₗ[A] M ⧸ N₁ := (N₁.submoduleOf N₂).mapQ N₁ N₂.subtype ⋯\nhf₁ : f.ker = ⊥\n⊢ f.range = map N₁.mkQ N₂",
"ppTerm": "?m.138",
"assigned": true,
"usedCons... | [] | simp [f, mapQ, range_liftQ, range_comp] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.Ideal.AssociatedPrime.Finiteness | {
"line": 62,
"column": 2
} | {
"line": 62,
"column": 96
} | {
"line": 63,
"column": 2
} | [
{
"pp": "A : Type u\ninst✝² : CommRing A\nM : Type v\ninst✝¹ : AddCommGroup M\ninst✝ : Module A M\nN₁ N₂ : Submodule A M\nf : ↥N₂ ⧸ N₁.submoduleOf N₂ →ₗ[A] M ⧸ N₁ := (N₁.submoduleOf N₂).mapQ N₁ N₂.subtype ⋯\nhf₁ : f.ker = ⊥\nhf₂ : f.range = map N₁.mkQ N₂\n⊢ N₁.IsQuotientEquivQuotientPrime N₂ ↔ ∃ x, (⊥.colon {N₁... | [
"case refine_1\nA : Type u\ninst✝² : CommRing A\nM : Type v\ninst✝¹ : AddCommGroup M\ninst✝ : Module A M\nN₁ N₂ : Submodule A M\nf : ↥N₂ ⧸ N₁.submoduleOf N₂ →ₗ[A] M ⧸ N₁ := (N₁.submoduleOf N₂).mapQ N₁ N₂.subtype ⋯\nhf₁ : f.ker = ⊥\nhf₂ : f.range = map N₁.mkQ N₂\nx✝ : N₁.IsQuotientEquivQuotientPrime N₂\nh : N₁ ≤ N₂\... | refine ⟨fun ⟨h, p, ⟨e⟩⟩ ↦ ?_, fun ⟨x, hx, hx'⟩ ↦ ⟨le_sup_left.trans_eq hx'.symm, ⟨_, hx⟩, ?_⟩⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.RingTheory.LocalRing.Module | {
"line": 283,
"column": 47
} | {
"line": 283,
"column": 52
} | {
"line": 283,
"column": 52
} | [
{
"pp": "case insert.specialize_2\nR : Type u_1\nM : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : IsLocalRing R\ninst✝ : Flat R M\nι : Type u\nf : ι → R\nn✝ : ι\ns : Finset ι\nhn : n✝ ∉ s\nih : ∀ (v : ι → M), LinearIndependent k (⇑((TensorProduct.mk R k M) 1) ∘ v) → ∑ i ... | [
"case insert.specialize_2\nR : Type u_1\nM : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : IsLocalRing R\ninst✝ : Flat R M\nι : Type u\nf : ι → R\nn✝ : ι\ns : Finset ι\nhn : n✝ ∉ s\nih : ∀ (v : ι → M), LinearIndependent k (⇑((TensorProduct.mk R k M) 1) ∘ v) → ∑ i ∈ s, f i • v... | ← hfv | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.RingTheory.Ideal.AssociatedPrime.Finiteness | {
"line": 84,
"column": 6
} | {
"line": 84,
"column": 78
} | {
"line": 84,
"column": 79
} | [
{
"pp": "case refine_2.refine_3\nA : Type u\ninst✝² : CommRing A\nM : Type v\ninst✝¹ : AddCommGroup M\ninst✝ : Module A M\nN₁ N₂ : Submodule A M\nf : ↥N₂ ⧸ N₁.submoduleOf N₂ →ₗ[A] M ⧸ N₁ := (N₁.submoduleOf N₂).mapQ N₁ N₂.subtype ⋯\nhf₁ : f.ker = ⊥\nhf₂ : f.range = map N₁.mkQ N₂\nx✝ : ∃ x, (⊥.colon {N₁.mkQ x}).I... | [
"case refine_2.refine_3\nA : Type u\ninst✝² : CommRing A\nM : Type v\ninst✝¹ : AddCommGroup M\ninst✝ : Module A M\nN₁ N₂ : Submodule A M\nf : ↥N₂ ⧸ N₁.submoduleOf N₂ →ₗ[A] M ⧸ N₁ := (N₁.submoduleOf N₂).mapQ N₁ N₂.subtype ⋯\nhf₁ : f.ker = ⊥\nhf₂ : f.range = map N₁.mkQ N₂\nx✝ : ∃ x, (⊥.colon {N₁.mkQ x}).IsPrime ∧ N₂ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Ideal.AssociatedPrime.Basic | {
"line": 234,
"column": 17
} | {
"line": 234,
"column": 28
} | {
"line": 234,
"column": 29
} | [
{
"pp": "R : Type u_1\ninst✝³ : CommSemiring R\nM : Type u_2\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ninst✝ : IsNoetherianRing R\nx : M\nh : (⊥.colon {x}).IsPrime\nr : R\nh' : r ∈ ↑(⊥.colon {x})\n⊢ x ≠ 0",
"ppTerm": "?m.120",
"assigned": true,
"usedConstants": [
"AddMonoid.toAddZeroClas... | [
"R : Type u_1\ninst✝³ : CommSemiring R\nM : Type u_2\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ninst✝ : IsNoetherianRing R\nx : M\nh : (⊥.colon {x}).IsPrime\nr : R\nh' : r ∈ ↑(⊥.colon {x})\n⊢ ¬x = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Ideal.AssociatedPrime.Basic | {
"line": 234,
"column": 42
} | {
"line": 234,
"column": 53
} | {
"line": 234,
"column": 54
} | [
{
"pp": "R : Type u_1\ninst✝³ : CommSemiring R\nM : Type u_2\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ninst✝ : IsNoetherianRing R\nx : M\nh : (⊥.colon {x}).IsPrime\nr : R\nh' : r ∈ ↑(⊥.colon {x})\n⊢ r • x = 0",
"ppTerm": "?m.121",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
... | [
"R : Type u_1\ninst✝³ : CommSemiring R\nM : Type u_2\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ninst✝ : IsNoetherianRing R\nx : M\nh : (⊥.colon {x}).IsPrime\nr : R\nh' : r ∈ ↑(⊥.colon {x})\n⊢ r • x = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.LocalRing.Module | {
"line": 298,
"column": 4
} | {
"line": 298,
"column": 84
} | {
"line": 298,
"column": 85
} | [
{
"pp": "case right.refine_1\nR : Type u_1\nM : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : IsLocalRing R\ninst✝¹ : Module.Finite R M\ninst✝ : Flat R M\nι : Type u\nv : ι → M\nh : Function.Bijective ⇑(linearCombination k (⇑((TensorProduct.mk R k M) 1) ∘ v))\n⊢ ⊤ ≤ Submo... | [
"case right.refine_1\nR : Type u_1\nM : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : IsLocalRing R\ninst✝¹ : Module.Finite R M\ninst✝ : Flat R M\nι : Type u\nv : ι → M\nh : Function.Bijective ⇑(linearCombination k (⇑((TensorProduct.mk R k M) 1) ∘ v))\n⊢ Surjective ⇑(linearCo... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.ClassGroup.Basic | {
"line": 67,
"column": 2
} | {
"line": 67,
"column": 51
} | {
"line": 69,
"column": 0
} | [
{
"pp": "R : Type u_1\nK : Type u_2\ninst✝³ : CommRing R\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\nI : (FractionalIdeal R⁰ K)ˣ\nx : Kˣ\n⊢ (toPrincipalIdeal R K) x = I ↔ spanSingleton R⁰ ↑x = ↑I",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"definition._pr... | [] | simp only [toPrincipalIdeal]; exact Units.ext_iff | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.ClassGroup.Basic | {
"line": 67,
"column": 2
} | {
"line": 67,
"column": 51
} | {
"line": 69,
"column": 0
} | [
{
"pp": "R : Type u_1\nK : Type u_2\ninst✝³ : CommRing R\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\nI : (FractionalIdeal R⁰ K)ˣ\nx : Kˣ\n⊢ (toPrincipalIdeal R K) x = I ↔ spanSingleton R⁰ ↑x = ↑I",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"definition._pr... | [] | simp only [toPrincipalIdeal]; exact Units.ext_iff | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.ClassGroup.Basic | {
"line": 142,
"column": 6
} | {
"line": 142,
"column": 75
} | {
"line": 142,
"column": 76
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nI J : (FractionalIdeal R⁰ (FractionRing R))ˣ\nI' J' : Ideal R\nhI : ↑I = ↑I'\nhJ : ↑J = ↑J'\nx y : R\nhx : x ≠ 0\nhy : y ≠ 0\nh : Ideal.span {x} * I' = Ideal.span {y} * J'\n⊢ IsUnit (mk' (FractionRing R) x ⟨y, ⋯⟩)",
"ppTerm": "?m.230",
"ass... | [
"R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nI J : (FractionalIdeal R⁰ (FractionRing R))ˣ\nI' J' : Ideal R\nhI : ↑I = ↑I'\nhJ : ↑J = ↑J'\nx y : R\nhx : x ≠ 0\nhy : y ≠ 0\nh : Ideal.span {x} * I' = Ideal.span {y} * J'\n⊢ ¬x = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.LocalRing.Module | {
"line": 409,
"column": 47
} | {
"line": 409,
"column": 76
} | {
"line": 409,
"column": 76
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : Finite (MaximalSpectrum R)\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : Module.Finite R M\ninst✝ : Flat R M\nn : ℕ\nrk : ∀ (P : MaximalSpectrum R), finrank (R ⧸ P.asIdeal) ((R ⧸ P.asIdeal) ⊗[R] M) = n\nthis : {R : Type u_1} → [inst : ... | [
"R : Type u_1\nM : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : Finite (MaximalSpectrum R)\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : Module.Finite R M\ninst✝ : Flat R M\nn : ℕ\nrk : ∀ (P : MaximalSpectrum R), finrank (R ⧸ P.asIdeal) ((R ⧸ P.asIdeal) ⊗[R] M) = n\nthis : {R : Type u_1} → [inst : CommRing R] ... | LinearMap.coe_restrictScalars | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.UniqueFactorizationDomain.ClassGroup | {
"line": 104,
"column": 4
} | {
"line": 104,
"column": 21
} | {
"line": 104,
"column": 22
} | [
{
"pp": "case refine_2\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : IsGCDMonoid R\nI : Ideal R\nhI : IsUnit ↑I\na : R\nK : Ideal R\nha0 : a ≠ 0\nh : (↑I)⁻¹ = spanSingleton R⁰ ((algebraMap R (FractionRing R)) a)⁻¹ * ↑K\nhIK : I * K = span {a}\n⊢ Submodule.IsPrincipal (I * K)",
"ppTerm": "... | [
"case refine_2\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : IsGCDMonoid R\nI : Ideal R\nhI : IsUnit ↑I\na : R\nK : Ideal R\nha0 : a ≠ 0\nh : (↑I)⁻¹ = spanSingleton R⁰ ((algebraMap R (FractionRing R)) a)⁻¹ * ↑K\nhIK : I * K = span {a}\n⊢ Submodule.IsPrincipal (span {a})"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Spectrum.Prime.TensorProduct | {
"line": 53,
"column": 6
} | {
"line": 53,
"column": 18
} | {
"line": 53,
"column": 19
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : CommRing T\ninst✝ : Algebra R T\nhRT : (algebraMap R T).SurjectiveOnStalks\np₁ p₂ : PrimeSpectrum (S ⊗[R] T)\nh : tensorProductTo R S T p₁ = tensorProductTo R S T p₂\ng : T →+* S ⊗[R] T :=... | [
"R : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : CommRing T\ninst✝ : Algebra R T\nhRT : (algebraMap R T).SurjectiveOnStalks\np₁ p₂ : PrimeSpectrum (S ⊗[R] T)\nh : tensorProductTo R S T p₁ = tensorProductTo R S T p₂\ng : T →+* S ⊗[R] T := Algebra.Ten... | ← mul_one a, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.ClassGroup.Basic | {
"line": 384,
"column": 8
} | {
"line": 384,
"column": 19
} | {
"line": 384,
"column": 20
} | [
{
"pp": "R : Type u_1\nK : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : Field K\ninst✝³ : Algebra R K\ninst✝² : IsFractionRing R K\ninst✝¹ : IsDomain R\ninst✝ : Subsingleton (ClassGroup R)\nI : Ideal R\nhI : IsUnit ↑I\nhsub : (↑↑I).IsPrincipal\n⊢ (coeSubmodule K I).IsPrincipal",
"ppTerm": "?m.57",
"assigned"... | [
"R : Type u_1\nK : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : Field K\ninst✝³ : Algebra R K\ninst✝² : IsFractionRing R K\ninst✝¹ : IsDomain R\ninst✝ : Subsingleton (ClassGroup R)\nI : Ideal R\nhI : IsUnit ↑I\nhsub : (↑↑I).IsPrincipal\n⊢ Submodule.IsPrincipal I"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.LocalProperties.Projective | {
"line": 171,
"column": 2
} | {
"line": 171,
"column": 12
} | {
"line": 172,
"column": 2
} | [
{
"pp": "R : Type u_1\nM : Type uM\ninst✝¹⁰ : CommRing R\ninst✝⁹ : AddCommGroup M\ninst✝⁸ : Module R M\nRₚ : (P : Ideal R) → [P.IsMaximal] → Type u_4\ninst✝⁷ : (P : Ideal R) → [inst : P.IsMaximal] → CommRing (Rₚ P)\ninst✝⁶ : (P : Ideal R) → [inst : P.IsMaximal] → Algebra R (Rₚ P)\ninst✝⁵ : ∀ (P : Ideal R) [inst... | [
"R : Type u_1\nM : Type uM\ninst✝¹⁰ : CommRing R\ninst✝⁹ : AddCommGroup M\ninst✝⁸ : Module R M\nRₚ : (P : Ideal R) → [P.IsMaximal] → Type u_4\ninst✝⁷ : (P : Ideal R) → [inst : P.IsMaximal] → CommRing (Rₚ P)\ninst✝⁶ : (P : Ideal R) → [inst : P.IsMaximal] → Algebra R (Rₚ P)\ninst✝⁵ : ∀ (P : Ideal R) [inst : P.IsMaxim... | intro P hP | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Topology.LocallyConstant.Basic | {
"line": 416,
"column": 2
} | {
"line": 416,
"column": 20
} | {
"line": 416,
"column": 21
} | [
{
"pp": "X : Type u_1\nY : Type u_2\nZ : Type u_3\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nf : C(X, Y)\nhfs : Function.Surjective f.toFun\na b : LocallyConstant Y Z\nh : comap f a = comap f b\ny : Y\nx : X\nhx : f.toFun x = y\n⊢ a y = b y",
"ppTerm": "?m.38",
"assigned": true,
"used... | [
"X : Type u_1\nY : Type u_2\nZ : Type u_3\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nf : C(X, Y)\nhfs : Function.Surjective f.toFun\na b : LocallyConstant Y Z\nh : comap f a = comap f b\ny : Y\nx : X\nhx : f.toFun x = y\n⊢ a (f x) = b (f x)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.LocallyConstant.Basic | {
"line": 579,
"column": 6
} | {
"line": 579,
"column": 17
} | {
"line": 579,
"column": 18
} | [
{
"pp": "X : Type u_1\nY : Type u_2\nZ : Type u_3\nα : Type u_4\ninst✝¹ : TopologicalSpace X\nC₀ C₁ C₂ : Set X\nh₀ : C₀ ⊆ C₁ ∪ C₂\nh₁ : IsClosed[inst✝¹] C₁\nh₂ : IsClosed[inst✝¹] C₂\nf₁ : LocallyConstant (↑C₁) Z\nf₂ : LocallyConstant (↑C₂) Z\ninst✝ : DecidablePred fun x ↦ x ∈ C₁\nhf : ∀ (x : X) (hx : x ∈ C₁ ∩ C... | [
"X : Type u_1\nY : Type u_2\nZ : Type u_3\nα : Type u_4\ninst✝¹ : TopologicalSpace X\nC₀ C₁ C₂ : Set X\nh₀ : C₀ ⊆ C₁ ∪ C₂\nh₁ : IsClosed[inst✝¹] C₁\nh₂ : IsClosed[inst✝¹] C₂\nf₁ : LocallyConstant (↑C₁) Z\nf₂ : LocallyConstant (↑C₂) Z\ninst✝ : DecidablePred fun x ↦ x ∈ C₁\nhf : ∀ (x : X) (hx : x ∈ C₁ ∩ C₂), f₁ ⟨x, ⋯... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.Minpoly.Finite | {
"line": 35,
"column": 46
} | {
"line": 35,
"column": 57
} | {
"line": 35,
"column": 58
} | [
{
"pp": "A : Type u_1\nB : Type u_2\ninst✝³ : CommRing A\ninst✝² : Ring B\ninst✝¹ : Algebra A B\ninst✝ : Module.Finite A B\nx : B\nf : A[X]\nf_monic : f.Monic\nf_deg : f.natDegree = ⊤.spanFinrank\nf_aeval : (Algebra.lmul A B) ((Polynomial.aeval x) f) = 0\n⊢ (Algebra.lmul A B) ((Polynomial.aeval x) f) = (Algebra... | [
"A : Type u_1\nB : Type u_2\ninst✝³ : CommRing A\ninst✝² : Ring B\ninst✝¹ : Algebra A B\ninst✝ : Module.Finite A B\nx : B\nf : A[X]\nf_monic : f.Monic\nf_deg : f.natDegree = ⊤.spanFinrank\nf_aeval : (Algebra.lmul A B) ((Polynomial.aeval x) f) = 0\n⊢ (LinearMap.mul A B) ((Polynomial.aeval x) f) = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.Minpoly.Finite | {
"line": 39,
"column": 2
} | {
"line": 39,
"column": 53
} | {
"line": 39,
"column": 54
} | [
{
"pp": "A : Type u_1\nB : Type u_2\ninst✝⁴ : CommRing A\ninst✝³ : Ring B\ninst✝² : Algebra A B\ninst✝¹ : Module.Finite A B\nx : B\ninst✝ : Module.Free A B\na✝ : Nontrivial A\n⊢ (minpoly A x).natDegree ≤ Module.finrank A B",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [
"A : Type u_1\nB : Type u_2\ninst✝⁴ : CommRing A\ninst✝³ : Ring B\ninst✝² : Algebra A B\ninst✝¹ : Module.Finite A B\nx : B\ninst✝ : Module.Free A B\na✝ : Nontrivial A\n⊢ (minpoly A x).natDegree ≤ ⊤.spanFinrank"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.Solvable | {
"line": 212,
"column": 40
} | {
"line": 221,
"column": 38
} | {
"line": 221,
"column": 38
} | [
{
"pp": "G : Type u_1\ninst✝¹ : Group G\ninst✝ : IsSimpleGroup G\nx✝ : IsSolvable G\nn : ℕ\nhn : derivedSeries G n = ⊥\n⊢ ∀ (a b : G), a * b = b * a",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"commutatorSet",
"Eq.mpr",
"Semigroup.toMul",
"DivInvMonoid.toInv",
... | [] | by
cases n
· intro a b
refine (mem_bot.1 ?_).trans (mem_bot.1 ?_).symm <;>
· rw [← hn]
exact mem_top _
· rw [IsSimpleGroup.derivedSeries_succ] at hn
intro a b
rw [← mul_inv_eq_one, mul_inv_rev, ← mul_assoc, ← mem_bot, ← hn, commutator_eq_closure]
exact subset_closur... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Spectrum.Prime.FreeLocus | {
"line": 266,
"column": 2
} | {
"line": 276,
"column": 90
} | {
"line": 278,
"column": 0
} | [
{
"pp": "R : Type uR\ninst✝⁵ : CommRing R\nι : Type u_1\ninst✝⁴ : Finite ι\nM : ι → Type u_2\ninst✝³ : (i : ι) → AddCommGroup (M i)\ninst✝² : (i : ι) → Module R (M i)\ninst✝¹ : ∀ (i : ι), Flat R (M i)\ninst✝ : ∀ (i : ι), Module.Finite R (M i)\np : PrimeSpectrum R\n⊢ rankAtStalk ((i : ι) → M i) p = ∑ᶠ (i : ι), r... | [] | cases nonempty_fintype ι
let f : (Π i, M i) →ₗ[R] Π i, LocalizedModule p.asIdeal.primeCompl (M i) :=
.pi (fun i ↦ mkLinearMap p.asIdeal.primeCompl (M i) ∘ₗ LinearMap.proj i)
let e : LocalizedModule p.asIdeal.primeCompl (Π i, M i) ≃ₗ[Localization.AtPrime p.asIdeal]
Π i, LocalizedModule p.asIdeal.primeCompl... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Spectrum.Prime.FreeLocus | {
"line": 266,
"column": 2
} | {
"line": 276,
"column": 90
} | {
"line": 278,
"column": 0
} | [
{
"pp": "R : Type uR\ninst✝⁵ : CommRing R\nι : Type u_1\ninst✝⁴ : Finite ι\nM : ι → Type u_2\ninst✝³ : (i : ι) → AddCommGroup (M i)\ninst✝² : (i : ι) → Module R (M i)\ninst✝¹ : ∀ (i : ι), Flat R (M i)\ninst✝ : ∀ (i : ι), Module.Finite R (M i)\np : PrimeSpectrum R\n⊢ rankAtStalk ((i : ι) → M i) p = ∑ᶠ (i : ι), r... | [] | cases nonempty_fintype ι
let f : (Π i, M i) →ₗ[R] Π i, LocalizedModule p.asIdeal.primeCompl (M i) :=
.pi (fun i ↦ mkLinearMap p.asIdeal.primeCompl (M i) ∘ₗ LinearMap.proj i)
let e : LocalizedModule p.asIdeal.primeCompl (Π i, M i) ≃ₗ[Localization.AtPrime p.asIdeal]
Π i, LocalizedModule p.asIdeal.primeCompl... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.FieldTheory.Normal.Closure | {
"line": 96,
"column": 4
} | {
"line": 96,
"column": 68
} | {
"line": 96,
"column": 69
} | [
{
"pp": "case refine_1\nF : Type u_1\nK : Type u_2\nL : Type u_3\ninst✝⁵ : Field F\ninst✝⁴ : Field K\ninst✝³ : Field L\ninst✝² : Algebra F K\ninst✝¹ : Algebra F L\ninst✝ : Algebra.IsAlgebraic F K\nx✝ : IsNormalClosure F K L\nsplits : ∀ (x : K), (Polynomial.map (algebraMap F L) (minpoly F x)).Splits\nh : ⨆ x, In... | [
"case refine_1\nF : Type u_1\nK : Type u_2\nL : Type u_3\ninst✝⁵ : Field F\ninst✝⁴ : Field K\ninst✝³ : Field L\ninst✝² : Algebra F K\ninst✝¹ : Algebra F L\ninst✝ : Algebra.IsAlgebraic F K\nx✝ : IsNormalClosure F K L\nsplits : ∀ (x : K), (Polynomial.map (algebraMap F L) (minpoly F x)).Splits\nh : ⨆ x, IntermediateFi... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.Normal.Closure | {
"line": 96,
"column": 4
} | {
"line": 96,
"column": 68
} | {
"line": 96,
"column": 69
} | [
{
"pp": "case refine_2\nF : Type u_1\nK : Type u_2\nL : Type u_3\ninst✝⁵ : Field F\ninst✝⁴ : Field K\ninst✝³ : Field L\ninst✝² : Algebra F K\ninst✝¹ : Algebra F L\ninst✝ : Algebra.IsAlgebraic F K\nx✝ : (∀ (x : K), (Polynomial.map (algebraMap F L) (minpoly F x)).Splits) ∧ normalClosure F K L = ⊤\nsplits : ∀ (x :... | [
"case refine_2\nF : Type u_1\nK : Type u_2\nL : Type u_3\ninst✝⁵ : Field F\ninst✝⁴ : Field K\ninst✝³ : Field L\ninst✝² : Algebra F K\ninst✝¹ : Algebra F L\ninst✝ : Algebra.IsAlgebraic F K\nx✝ : (∀ (x : K), (Polynomial.map (algebraMap F L) (minpoly F x)).Splits) ∧ normalClosure F K L = ⊤\nsplits : ∀ (x : K), (Polyno... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.Normal.Basic | {
"line": 64,
"column": 2
} | {
"line": 87,
"column": 35
} | {
"line": 89,
"column": 0
} | [
{
"pp": "F : Type u_1\ninst✝² : Field F\nE : Type u_3\ninst✝¹ : Field E\ninst✝ : Algebra F E\np : F[X]\nhFEp : IsSplittingField F E p\n⊢ Normal F E",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Subalgebra.instSetLike",
"Iff.mpr",
"Eq.mpr",
"Polynomial.SplittingFi... | [] | rcases eq_or_ne p 0 with (rfl | hp)
· have := hFEp.adjoin_rootSet
rw [rootSet_zero, Algebra.adjoin_empty] at this
exact Normal.of_algEquiv
(AlgEquiv.ofBijective (Algebra.ofId F E) (Algebra.bijective_algebraMap_iff.2 this.symm))
refine normal_iff.mpr fun x ↦ ?_
haveI : FiniteDimensional F E := IsSpli... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.FieldTheory.Normal.Basic | {
"line": 64,
"column": 2
} | {
"line": 87,
"column": 35
} | {
"line": 89,
"column": 0
} | [
{
"pp": "F : Type u_1\ninst✝² : Field F\nE : Type u_3\ninst✝¹ : Field E\ninst✝ : Algebra F E\np : F[X]\nhFEp : IsSplittingField F E p\n⊢ Normal F E",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Subalgebra.instSetLike",
"Iff.mpr",
"Eq.mpr",
"Polynomial.SplittingFi... | [] | rcases eq_or_ne p 0 with (rfl | hp)
· have := hFEp.adjoin_rootSet
rw [rootSet_zero, Algebra.adjoin_empty] at this
exact Normal.of_algEquiv
(AlgEquiv.ofBijective (Algebra.ofId F E) (Algebra.bijective_algebraMap_iff.2 this.symm))
refine normal_iff.mpr fun x ↦ ?_
haveI : FiniteDimensional F E := IsSpli... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.PicardGroup | {
"line": 124,
"column": 2
} | {
"line": 124,
"column": 26
} | {
"line": 124,
"column": 27
} | [
{
"pp": "R : Type u\nM : Type v\nN : Type u_1\nP : Type u_2\nQ : Type u_3\ninst✝⁸ : CommSemiring R\ninst✝⁷ : AddCommMonoid M\ninst✝⁶ : AddCommMonoid N\ninst✝⁵ : AddCommMonoid P\ninst✝⁴ : AddCommMonoid Q\ninst✝³ : Module R M\ninst✝² : Module R N\ninst✝¹ : Module R P\ninst✝ : Module R Q\ne : M ⊗[R] N ≃ₗ[R] R\n⊢ F... | [
"R : Type u\nM : Type v\nN : Type u_1\nP : Type u_2\nQ : Type u_3\ninst✝⁸ : CommSemiring R\ninst✝⁷ : AddCommMonoid M\ninst✝⁶ : AddCommMonoid N\ninst✝⁵ : AddCommMonoid P\ninst✝⁴ : AddCommMonoid Q\ninst✝³ : Module R M\ninst✝² : Module R N\ninst✝¹ : Module R P\ninst✝ : Module R Q\ne : M ⊗[R] N ≃ₗ[R] R\n⊢ Function.Inje... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.Normal.Closure | {
"line": 310,
"column": 4
} | {
"line": 310,
"column": 37
} | {
"line": 310,
"column": 38
} | [
{
"pp": "case refine_1\nF : Type u_1\nL : Type u_3\ninst✝⁴ : Field F\ninst✝³ : Field L\ninst✝² : Algebra F L\ninst✝¹ : Normal F L\nK₁ K₂ : IntermediateField F L\ninst✝ : Normal F ↥K₂\nh : ∀ (f : ↥K₁ →ₐ[F] L), f.fieldRange ≤ K₂\n⊢ K₁ ≤ K₂",
"ppTerm": "?refine_1",
"assigned": false,
"usedConstants": [... | [
"case refine_1\nF : Type u_1\nL : Type u_3\ninst✝⁴ : Field F\ninst✝³ : Field L\ninst✝² : Algebra F L\ninst✝¹ : Normal F L\nK₁ K₂ : IntermediateField F L\ninst✝ : Normal F ↥K₂\nh : ∀ (f : ↥K₁ →ₐ[F] L), f.fieldRange ≤ K₂\n⊢ K₁ ≤ K₂"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.Normal.Basic | {
"line": 98,
"column": 51
} | {
"line": 115,
"column": 34
} | {
"line": 117,
"column": 0
} | [
{
"pp": "F : Type u_1\nK : Type u_2\ninst✝² : Field F\ninst✝¹ : Field K\ninst✝ : Algebra F K\nι : Type u_3\nt : ι → IntermediateField F K\nh : ∀ (i : ι), Normal F ↥(t i)\n⊢ Normal F ↥(⨆ i, t i)",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Subtype.coe_mk",
"CommMonoidWithZer... | [] | by
refine { toIsAlgebraic := isAlgebraic_iSup fun i => (h i).1, splits' := fun x => ?_ }
obtain ⟨s, hx⟩ := exists_finset_of_mem_supr'' (fun i => (h i).1) x.2
let E : IntermediateField F K := ⨆ i ∈ s, adjoin F ((minpoly F (i.2 :)).rootSet K)
have hF : Normal F E := by
haveI : IsSplittingField F E (∏ i ∈ s, m... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.FieldTheory.PrimitiveElement | {
"line": 113,
"column": 2
} | {
"line": 113,
"column": 92
} | {
"line": 114,
"column": 2
} | [
{
"pp": "F : Type u_1\ninst✝⁴ : Field F\ninst✝³ : Infinite F\nE : Type u_2\ninst✝² : Field E\nα β : E\ninst✝¹ : Algebra F E\ninst✝ : Algebra.IsSeparable F E\nhα : IsIntegral F α\nhβ : IsIntegral F β\nf : F[X] := minpoly F α\ng : F[X] := minpoly F β\nιFE : F →+* E := algebraMap F E\nιEE' : E →+* (Polynomial.map ... | [
"F : Type u_1\ninst✝⁴ : Field F\ninst✝³ : Infinite F\nE : Type u_2\ninst✝² : Field E\nα β : E\ninst✝¹ : Algebra F E\ninst✝ : Algebra.IsSeparable F E\nhα : IsIntegral F α\nhβ : IsIntegral F β\nf : F[X] := minpoly F α\ng : F[X] := minpoly F β\nιFE : F →+* E := algebraMap F E\nιEE' : E →+* (Polynomial.map ιFE g).Split... | obtain ⟨c, hc⟩ := primitive_element_inf_aux_exists_c (ιEE'.comp ιFE) (ιEE' α) (ιEE' β) f g | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.RingTheory.AlgebraicIndependent.Defs | {
"line": 89,
"column": 2
} | {
"line": 89,
"column": 60
} | {
"line": 89,
"column": 61
} | [
{
"pp": "ι : Type u_1\nι' : Type u_2\nR : Type u_3\nA : Type u_5\nx : ι → A\ninst✝² : CommRing R\ninst✝¹ : CommRing A\ninst✝ : Algebra R A\nhx : AlgebraicIndependent R x\nf : ι' → ι\nhf : Injective f\np q : MvPolynomial ι' R\n⊢ (aeval (x ∘ f)) p = (aeval (x ∘ f)) q → p = q",
"ppTerm": "?m.23",
"assigned... | [
"ι : Type u_1\nι' : Type u_2\nR : Type u_3\nA : Type u_5\nx : ι → A\ninst✝² : CommRing R\ninst✝¹ : CommRing A\ninst✝ : Algebra R A\nhx : AlgebraicIndependent R x\nf : ι' → ι\nhf : Injective f\np q : MvPolynomial ι' R\n⊢ (aeval (x ∘ f)) p = (aeval (x ∘ f)) q → p = q"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.AlgebraicIndependent.Defs | {
"line": 92,
"column": 2
} | {
"line": 92,
"column": 13
} | {
"line": 92,
"column": 14
} | [
{
"pp": "ι : Type u_1\nR : Type u_3\nA : Type u_5\nx : ι → A\ninst✝² : CommRing R\ninst✝¹ : CommRing A\ninst✝ : Algebra R A\nhx : AlgebraicIndependent R x\n⊢ AlgebraicIndependent R Subtype.val",
"ppTerm": "?m.18",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"ι : Type u_1\nR : Type u_3\nA : Type u_5\nx : ι → A\ninst✝² : CommRing R\ninst✝¹ : CommRing A\ninst✝ : Algebra R A\nhx : AlgebraicIndependent R x\n⊢ AlgebraicIndependent R Subtype.val"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.PicardGroup | {
"line": 190,
"column": 38
} | {
"line": 190,
"column": 54
} | {
"line": 190,
"column": 55
} | [
{
"pp": "R : Type u\nM : Type v\ninst✝³ : CommSemiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ninst✝ : Module.Invertible R M\nN : Type (max v u) := Dual R M\ne : M ⊗[R] N ≃ₗ[R] R := TensorProduct.comm R M N ≪≫ₗ linearEquiv R M\nS : Finset (M × N)\nhS : e.symm 1 = ∑ i ∈ S, i.1 ⊗ₜ[R] i.2\nf : (↥S →₀ N) ... | [
"R : Type u\nM : Type v\ninst✝³ : CommSemiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ninst✝ : Module.Invertible R M\nN : Type (max v u) := Dual R M\ne : M ⊗[R] N ≃ₗ[R] R := TensorProduct.comm R M N ≪≫ₗ linearEquiv R M\nS : Finset (M × N)\nhS : e.symm 1 = ∑ i ∈ S, i.1 ⊗ₜ[R] i.2\nf : (↥S →₀ N) →ₗ[R] R := (... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.PicardGroup | {
"line": 206,
"column": 2
} | {
"line": 206,
"column": 13
} | {
"line": 206,
"column": 14
} | [
{
"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\nh : Function.Injective ⇑(LinearMap.lTen... | [
"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\nh : Function.Injective ⇑(LinearMap.lTensor M f)\n⊢ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.PicardGroup | {
"line": 216,
"column": 2
} | {
"line": 216,
"column": 13
} | {
"line": 216,
"column": 14
} | [
{
"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\nh : Function.Surjective ⇑(LinearMap.lTe... | [
"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\nh : Function.Surjective ⇑(LinearMap.lTensor M f)\n⊢... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.PicardGroup | {
"line": 245,
"column": 52
} | {
"line": 245,
"column": 63
} | {
"line": 245,
"column": 64
} | [
{
"pp": "R : Type u\nM : Type v\ninst✝³ : CommSemiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ninst✝ : Module.Invertible R M\nx✝ : Free R M\na✝ : Nontrivial R\ne : M ≃ₗ[R] Free.ChooseBasisIndex R M →₀ R\nthis : Fintype.card (Free.ChooseBasisIndex R M × Free.ChooseBasisIndex R M) = Fintype.card Unit\n⊢... | [
"R : Type u\nM : Type v\ninst✝³ : CommSemiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ninst✝ : Module.Invertible R M\nx✝ : Free R M\na✝ : Nontrivial R\ne : M ≃ₗ[R] Free.ChooseBasisIndex R M →₀ R\nthis : Fintype.card (Free.ChooseBasisIndex R M × Free.ChooseBasisIndex R M) = Fintype.card Unit\n⊢ Fintype.car... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.LocalRing.ResidueField.Fiber | {
"line": 58,
"column": 2
} | {
"line": 60,
"column": 43
} | {
"line": 60,
"column": 44
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\np : Ideal R\ninst✝ : p.IsPrime\nx : S ⊗[R] p.ResidueField\nr : R\na : S\ny : R\nt : ↥p.primeCompl\ne :\n 1 ⊗ₜ[R]\n (r •\n (IsLocalRing.residue (Localization.AtPrime p))\n ((fun x ↦\n ... | [
"R : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\np : Ideal R\ninst✝ : p.IsPrime\nx : S ⊗[R] p.ResidueField\nr : R\na : S\ny : R\nt : ↥p.primeCompl\ne :\n 1 ⊗ₜ[R]\n (r •\n (IsLocalRing.residue (Localization.AtPrime p))\n ((fun x ↦\n ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.PicardGroup | {
"line": 259,
"column": 2
} | {
"line": 259,
"column": 20
} | {
"line": 259,
"column": 21
} | [
{
"pp": "R : Type u\nM : Type v\ninst✝³ : CommSemiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ninst✝ : Module.Invertible R M\nthis :\n ⇑(toModuleEnd R M) =\n ⇑(lid R M).conj ∘\n ⇑(rTensorEquiv R R (TensorProduct.comm R M (Dual R M) ≪≫ₗ linearEquiv R M)) ∘\n ⇑(RingEquiv.moduleEndSelf R)... | [
"R : Type u\nM : Type v\ninst✝³ : CommSemiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ninst✝ : Module.Invertible R M\nthis :\n ⇑(toModuleEnd R M) =\n ⇑(lid R M).conj ∘\n ⇑(rTensorEquiv R R (TensorProduct.comm R M (Dual R M) ≪≫ₗ linearEquiv R M)) ∘\n ⇑(RingEquiv.moduleEndSelf R) ∘ ⇑MulOppos... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.LocalRing.ResidueField.Fiber | {
"line": 81,
"column": 23
} | {
"line": 81,
"column": 34
} | {
"line": 81,
"column": 35
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\np : Ideal R\ninst✝ : p.IsPrime\nx : p.Fiber S\nr : R\nhr : r ∉ p\ns : S\ne : r • (Algebra.TensorProduct.comm R p.ResidueField S) x = s ⊗ₜ[R] 1\n⊢ r • x = 1 ⊗ₜ[R] s",
"ppTerm": "?m.79",
"assigned": false,... | [
"R : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\np : Ideal R\ninst✝ : p.IsPrime\nx : p.Fiber S\nr : R\nhr : r ∉ p\ns : S\ne : r • (Algebra.TensorProduct.comm R p.ResidueField S) x = s ⊗ₜ[R] 1\n⊢ r • x = 1 ⊗ₜ[R] s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.PicardGroup | {
"line": 278,
"column": 2
} | {
"line": 278,
"column": 37
} | {
"line": 278,
"column": 38
} | [
{
"pp": "R : Type u\nM : Type v\nN : Type u_1\ninst✝⁶ : CommSemiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : AddCommMonoid N\ninst✝³ : Module R M\ninst✝² : Module R N\ninst✝¹ : Module.Invertible R M\ninst✝ : Module.Invertible R N\nf : M →ₗ[R] N\nhf : Function.Surjective ⇑f\n⊢ Function.Bijective ⇑f",
"ppTerm":... | [
"R : Type u\nM : Type v\nN : Type u_1\ninst✝⁶ : CommSemiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : AddCommMonoid N\ninst✝³ : Module R M\ninst✝² : Module R N\ninst✝¹ : Module.Invertible R M\ninst✝ : Module.Invertible R N\nf : M →ₗ[R] N\nhf : Function.Surjective ⇑f\n⊢ Function.Bijective ⇑f"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.PicardGroup | {
"line": 279,
"column": 60
} | {
"line": 279,
"column": 96
} | {
"line": 279,
"column": 97
} | [
{
"pp": "R : Type u\nM : Type v\nN : Type u_1\ninst✝⁶ : CommSemiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : AddCommMonoid N\ninst✝³ : Module R M\ninst✝² : Module R N\ninst✝¹ : Module.Invertible R M\ninst✝ : Module.Invertible R N\nf : M →ₗ[R] N\nhf : Function.Surjective ⇑f\n⊢ Function.Surjective ⇑(LinearMap.lTens... | [
"R : Type u\nM : Type v\nN : Type u_1\ninst✝⁶ : CommSemiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : AddCommMonoid N\ninst✝³ : Module R M\ninst✝² : Module R N\ninst✝¹ : Module.Invertible R M\ninst✝ : Module.Invertible R N\nf : M →ₗ[R] N\nhf : Function.Surjective ⇑f\n⊢ Function.Surjective ⇑f"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.PicardGroup | {
"line": 386,
"column": 4
} | {
"line": 387,
"column": 11
} | {
"line": 387,
"column": 12
} | [
{
"pp": "case refine_1\nR : Type u\nM : Type v\ninst✝³ : CommSemiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ninst✝ : Module.Invertible R M\nS : Finset (Dual R M × M)\nhS : (linearEquiv R M).symm 1 = ∑ i ∈ S, i.1 ⊗ₜ[R] i.2\n⊢ Ideal.span ↑(Finset.image (fun i ↦ i.1 i.2) S) = ⊤",
"ppTerm": "?refine_... | [
"case refine_1\nR : Type u\nM : Type v\ninst✝³ : CommSemiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ninst✝ : Module.Invertible R M\nS : Finset (Dual R M × M)\nhS : (linearEquiv R M).symm 1 = ∑ i ∈ S, i.1 ⊗ₜ[R] i.2\n⊢ ∑ i ∈ S, i.1 i.2 ∈ Ideal.span ((fun i ↦ i.1 i.2) '' ↑S)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Matroid.Basic | {
"line": 402,
"column": 2
} | {
"line": 402,
"column": 13
} | {
"line": 402,
"column": 14
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nB₁ B₂ : Set α\ne : α\nhB₁ : M.IsBase B₁\nhB₂ : M.IsBase B₂\nhxB₁ : e ∈ B₁\nhxB₂ : e ∉ B₂\n⊢ ∃ y, (y ∈ B₂ ∧ y ∉ B₁) ∧ M.IsBase (insert y (B₁ \\ {e}))",
"ppTerm": "?m.26",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\nM : Matroid α\nB₁ B₂ : Set α\ne : α\nhB₁ : M.IsBase B₁\nhB₂ : M.IsBase B₂\nhxB₁ : e ∈ B₁\nhxB₂ : e ∉ B₂\n⊢ ∃ y, (y ∈ B₂ ∧ y ∉ B₁) ∧ M.IsBase (insert y (B₁ \\ {e}))"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.PrimitiveElement | {
"line": 152,
"column": 8
} | {
"line": 152,
"column": 31
} | {
"line": 152,
"column": 31
} | [
{
"pp": "F : Type u_1\ninst✝⁴ : Field F\ninst✝³ : Infinite F\nE : Type u_2\ninst✝² : Field E\nα β : E\ninst✝¹ : Algebra F E\ninst✝ : Algebra.IsSeparable F E\nhα : IsIntegral F α\nhβ : IsIntegral F β\nf : F[X] := minpoly F α\ng : F[X] := minpoly F β\nιFE : F →+* E := algebraMap F E\nιEE' : E →+* (Polynomial.map ... | [
"F : Type u_1\ninst✝⁴ : Field F\ninst✝³ : Infinite F\nE : Type u_2\ninst✝² : Field E\nα β : E\ninst✝¹ : Algebra F E\ninst✝ : Algebra.IsSeparable F E\nhα : IsIntegral F α\nhβ : IsIntegral F β\nf : F[X] := minpoly F α\ng : F[X] := minpoly F β\nιFE : F →+* E := algebraMap F E\nιEE' : E →+* (Polynomial.map ιFE g).Split... | mem_roots_map h_ne_zero | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.PicardGroup | {
"line": 538,
"column": 4
} | {
"line": 538,
"column": 67
} | {
"line": 538,
"column": 68
} | [
{
"pp": "R : Type u\nM✝ : Type v\nN : Type u_1\nP : Type u_2\nQ : Type u_3\nA : Type u_4\ninst✝¹¹ : CommSemiring R\ninst✝¹⁰ : AddCommMonoid M✝\ninst✝⁹ : AddCommMonoid N\ninst✝⁸ : AddCommMonoid P\ninst✝⁷ : AddCommMonoid Q\ninst✝⁶ : Module R M✝\ninst✝⁵ : Module R N\ninst✝⁴ : Module R P\ninst✝³ : Module R Q\ninst✝... | [
"R : Type u\nM✝ : Type v\nN : Type u_1\nP : Type u_2\nQ : Type u_3\nA : Type u_4\ninst✝¹¹ : CommSemiring R\ninst✝¹⁰ : AddCommMonoid M✝\ninst✝⁹ : AddCommMonoid N\ninst✝⁸ : AddCommMonoid P\ninst✝⁷ : AddCommMonoid Q\ninst✝⁶ : Module R M✝\ninst✝⁵ : Module R N\ninst✝⁴ : Module R P\ninst✝³ : Module R Q\ninst✝² : Module.I... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Matroid.Basic | {
"line": 666,
"column": 72
} | {
"line": 666,
"column": 85
} | {
"line": 666,
"column": 86
} | [
{
"pp": "case inr\nα : Type u_1\nM : Matroid α\nB : Set α\ne f : α\nhB : M.IsBase B\nhf : f ∉ B\nhI : M.Indep (insert f (B \\ {e}))\nB' : Set α\nhB' : M.IsBase B'\nhfB : f ∈ B'\nh : B \\ B' = {e}\nhx : B' \\ B = {f}\n⊢ M.IsBase (B' \\ B ∪ B ⊓ B')",
"ppTerm": "?inr",
"assigned": true,
"usedConstants"... | [
"case inr\nα : Type u_1\nM : Matroid α\nB : Set α\ne f : α\nhB : M.IsBase B\nhf : f ∉ B\nhI : M.Indep (insert f (B \\ {e}))\nB' : Set α\nhB' : M.IsBase B'\nhfB : f ∈ B'\nh : B \\ B' = {e}\nhx : B' \\ B = {f}\n⊢ M.IsBase (B' \\ B ∪ B ∩ B')"
] | inf_eq_inter, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Combinatorics.Matroid.Basic | {
"line": 681,
"column": 2
} | {
"line": 682,
"column": 9
} | {
"line": 682,
"column": 10
} | [
{
"pp": "case inr\nα : Type u_1\nM : Matroid α\nI : Set α\ne f : α\nhe : e ∉ I\nhf : f ∉ I\nheI : M.IsBase (insert e I)\nhfI : M.Indep (insert f I)\nhef : e ≠ f\n⊢ M.IsBase (insert f I)",
"ppTerm": "?inr",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"case inr\nα : Type u_1\nM : Matroid α\nI : Set α\ne f : α\nhe : e ∉ I\nhf : f ∉ I\nheI : M.IsBase (insert e I)\nhfI : M.Indep (insert f I)\nhef : e ≠ f\n⊢ M.IsBase (insert f I)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.PrimitiveElement | {
"line": 295,
"column": 17
} | {
"line": 295,
"column": 64
} | {
"line": 295,
"column": 65
} | [
{
"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)\nK : IntermediateField F E\nthis : FiniteDimensional F E\nh✝ : Finite F\nα : ↥K\nh : F⟮α⟯ = ⊤\n⊢ F⟮↑α⟯ = K",
"ppTerm": "?m.57",
"assigned": false,
"usedConstants": []... | [
"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)\nK : IntermediateField F E\nthis : FiniteDimensional F E\nh✝ : Finite F\nα : ↥K\nh : F⟮α⟯ = ⊤\n⊢ F⟮↑α⟯ = K"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.PrimitiveElement | {
"line": 303,
"column": 2
} | {
"line": 303,
"column": 24
} | {
"line": 304,
"column": 2
} | [
{
"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\nh : ∃ α, F⟮α⟯ = ⊤\n⊢ FiniteDimensional F E",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Lattice.toSemilatticeSup",
"CompleteLattice.toLattice",
... | [
"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⟯ = ⊤\n⊢ FiniteDimensional F E"
] | obtain ⟨α, hprim⟩ := h | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.FieldTheory.PrimitiveElement | {
"line": 312,
"column": 2
} | {
"line": 312,
"column": 24
} | {
"line": 314,
"column": 2
} | [
{
"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\nh : ∃ α, F⟮α⟯ = ⊤\nthis : FiniteDimensional F E\n⊢ Finite (IntermediateField F E)",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Lattice.toSemilatticeSup"... | [
"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\nthis : FiniteDimensional F E\nα : E\nhprim : F⟮α⟯ = ⊤\n⊢ Finite (IntermediateField F E)"
] | obtain ⟨α, hprim⟩ := h | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.FieldTheory.PrimitiveElement | {
"line": 329,
"column": 4
} | {
"line": 329,
"column": 79
} | {
"line": 329,
"column": 80
} | [
{
"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\nthis : FiniteDimensional F E\nα : E\nhprim : F⟮α⟯ = ⊤\nf : F[X] := minpoly F α\nG : Type (max 0 u_2) := { g // g.Monic ∧ g ∣ Polynomial.map (algebraMap F E) f }\nhfin : Finite G\ng : I... | [
"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\nthis : FiniteDimensional F E\nα : E\nhprim : F⟮α⟯ = ⊤\nf : F[X] := minpoly F α\nG : Type (max 0 u_2) := { g // g.Monic ∧ g ∣ Polynomial.map (algebraMap F E) f }\nhfin : Finite G\ng : IntermediateF... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Matroid.Basic | {
"line": 1044,
"column": 2
} | {
"line": 1044,
"column": 13
} | {
"line": 1044,
"column": 14
} | [
{
"pp": "case refine_2\nα : Type u_1\nM : Matroid α\nI : Set α\nhI : M.Indep I\ne : α\nhe : e ∈ {x | M.IsBasis I (insert x I)} \\ I\nhu : M.Indep (insert e I)\n⊢ e ∈ I",
"ppTerm": "?refine_2",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"case refine_2\nα : Type u_1\nM : Matroid α\nI : Set α\nhI : M.Indep I\ne : α\nhe : e ∈ {x | M.IsBasis I (insert x I)} \\ I\nhu : M.Indep (insert e I)\n⊢ e ∈ I"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Matroid.IndepAxioms | {
"line": 158,
"column": 37
} | {
"line": 158,
"column": 92
} | {
"line": 158,
"column": 93
} | [
{
"pp": "α : Type u_1\nM : IndepMatroid α\nB B' : Set α\nhB : Maximal M.Indep B\nhB' : Maximal M.Indep B'\ne : α\nhe : e ∈ B \\ B'\nhnotmax : ¬Maximal M.Indep (B \\ {e})\nf : α\nhf : f ∈ B' \\ (B \\ {e})\nhfB : M.Indep (insert f (B \\ {e}))\n⊢ f ∈ B' \\ B",
"ppTerm": "?m.222",
"assigned": true,
"use... | [
"α : Type u_1\nM : IndepMatroid α\nB B' : Set α\nhB : Maximal M.Indep B\nhB' : Maximal M.Indep B'\ne : α\nhe : e ∈ B \\ B'\nhnotmax : ¬Maximal M.Indep (B \\ {e})\nf : α\nhf : f ∈ B' \\ (B \\ {e})\nhfB : M.Indep (insert f (B \\ {e}))\n⊢ f ∈ B' ∧ f ∉ B"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Matroid.IndepAxioms | {
"line": 161,
"column": 38
} | {
"line": 161,
"column": 57
} | {
"line": 161,
"column": 58
} | [
{
"pp": "α : Type u_1\nM : IndepMatroid α\nB B' : Set α\nhB : Maximal M.Indep B\nhB' : Maximal M.Indep B'\ne : α\nhe : e ∈ B \\ B'\nhnotmax : ¬Maximal M.Indep (B \\ {e})\nf : α\nhfB : M.Indep (insert f (B \\ {e}))\nhf : f ∈ B' \\ B\nhnot : ¬Maximal M.Indep (insert f (B \\ {e}))\nx : α\nhxB : x ∈ B \\ insert f (... | [
"α : Type u_1\nM : IndepMatroid α\nB B' : Set α\nhB : Maximal M.Indep B\nhB' : Maximal M.Indep B'\ne : α\nhe : e ∈ B \\ B'\nhnotmax : ¬Maximal M.Indep (B \\ {e})\nf : α\nhfB : M.Indep (insert f (B \\ {e}))\nhf : f ∈ B' \\ B\nhnot : ¬Maximal M.Indep (insert f (B \\ {e}))\nx : α\nhxB : x ∈ B \\ insert f (B \\ {e})\nh... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Matroid.Minor.Restrict | {
"line": 278,
"column": 2
} | {
"line": 278,
"column": 13
} | {
"line": 278,
"column": 14
} | [
{
"pp": "α : Type u_1\nM M' : Matroid α\nh : M ≤r M'\nh' : M' ≤r M\n⊢ M = M'",
"ppTerm": "?m.4",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\nM M' : Matroid α\nh : M ≤r M'\nh' : M' ≤r M\n⊢ M = M'"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Matroid.Minor.Restrict | {
"line": 327,
"column": 2
} | {
"line": 327,
"column": 13
} | {
"line": 327,
"column": 14
} | [
{
"pp": "α : Type u_1\nM N : Matroid α\nh : N ≤r M\n⊢ N = M ∨ N <r M",
"ppTerm": "?m.4",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\nM N : Matroid α\nh : N ≤r M\n⊢ N = M ∨ N <r M"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.PicardGroup | {
"line": 747,
"column": 47
} | {
"line": 747,
"column": 74
} | {
"line": 747,
"column": 75
} | [
{
"pp": "R : Type u\nA : Type u_4\ninst✝³ : CommSemiring R\ninst✝² : Semiring A\ninst✝¹ : Algebra R A\ninst✝ : FaithfulSMul R A\nI : (Submodule R A)ˣ\nh✝ : I ∈ (unitsToPic R A).ker\ne : R ≃ₗ[R] ↥↑I\ne' : R ≃ₗ[R] ↥↑I⁻¹\nh : ↑I = R ∙ ↑(↑e 1)\nh' : ↑I⁻¹ = R ∙ ↑(↑e' 1)\n⊢ R ∙ ↑(e 1) * ↑(e' 1) = 1",
"ppTerm": "?... | [
"R : Type u\nA : Type u_4\ninst✝³ : CommSemiring R\ninst✝² : Semiring A\ninst✝¹ : Algebra R A\ninst✝ : FaithfulSMul R A\nI : (Submodule R A)ˣ\nh✝ : I ∈ (unitsToPic R A).ker\ne : R ≃ₗ[R] ↥↑I\ne' : R ≃ₗ[R] ↥↑I⁻¹\nh : ↑I = R ∙ ↑(↑e 1)\nh' : ↑I⁻¹ = R ∙ ↑(↑e' 1)\n⊢ R ∙ ↑(e 1) * ↑(e' 1) = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Matroid.Minor.Restrict | {
"line": 379,
"column": 15
} | {
"line": 379,
"column": 44
} | {
"line": 379,
"column": 45
} | [
{
"pp": "α : Type u_1\nM N : Matroid α\nhMN : N ≤r M\nB : Set α\nh : M.IsBasis B N.E\n⊢ N.IsBase B",
"ppTerm": "?m.13",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\nM N : Matroid α\nhMN : N ≤r M\nB : Set α\nh : M.IsBasis B N.E\n⊢ N.IsBase B"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Matroid.IndepAxioms | {
"line": 197,
"column": 4
} | {
"line": 197,
"column": 64
} | {
"line": 198,
"column": 4
} | [
{
"pp": "α : Type u_1\nE : Set α\nIndep : Set α → Prop\nindep_empty : Indep ∅\nindep_subset : ∀ ⦃I J : Set α⦄, Indep J → I ⊆ J → Indep I\nindep_aug : ∀ ⦃I B : Set α⦄, Indep I → ¬Maximal Indep I → Maximal Indep B → ∃ x ∈ B \\ I, Indep (insert x I)\nindep_compact : ∀ (I : Set α), (∀ J ⊆ I, J.Finite → Indep J) → I... | [
"case inl\nα : Type u_1\nE : Set α\nIndep : Set α → Prop\nindep_empty : Indep ∅\nindep_subset : ∀ ⦃I J : Set α⦄, Indep J → I ⊆ J → Indep I\nindep_aug : ∀ ⦃I B : Set α⦄, Indep I → ¬Maximal Indep I → Maximal Indep B → ∃ x ∈ B \\ I, Indep (insert x I)\nindep_compact : ∀ (I : Set α), (∀ J ⊆ I, J.Finite → Indep J) → Ind... | obtain (hle | hle) := hchain.total (hf _ hxJ).1 (hf _ hyJ).1 | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.RingTheory.PicardGroup | {
"line": 750,
"column": 47
} | {
"line": 750,
"column": 74
} | {
"line": 750,
"column": 75
} | [
{
"pp": "R : Type u\nA : Type u_4\ninst✝³ : CommSemiring R\ninst✝² : Semiring A\ninst✝¹ : Algebra R A\ninst✝ : FaithfulSMul R A\nI : (Submodule R A)ˣ\nh✝ : I ∈ (unitsToPic R A).ker\ne : R ≃ₗ[R] ↥↑I\ne' : R ≃ₗ[R] ↥↑I⁻¹\nh : ↑I = R ∙ ↑(↑e 1)\nh' : ↑I⁻¹ = R ∙ ↑(↑e' 1)\n⊢ R ∙ ↑(e' 1) * ↑(e 1) = 1",
"ppTerm": "?... | [
"R : Type u\nA : Type u_4\ninst✝³ : CommSemiring R\ninst✝² : Semiring A\ninst✝¹ : Algebra R A\ninst✝ : FaithfulSMul R A\nI : (Submodule R A)ˣ\nh✝ : I ∈ (unitsToPic R A).ker\ne : R ≃ₗ[R] ↥↑I\ne' : R ≃ₗ[R] ↥↑I⁻¹\nh : ↑I = R ∙ ↑(↑e 1)\nh' : ↑I⁻¹ = R ∙ ↑(↑e' 1)\n⊢ R ∙ ↑(e' 1) * ↑(e 1) = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.PicardGroup | {
"line": 755,
"column": 61
} | {
"line": 755,
"column": 72
} | {
"line": 755,
"column": 73
} | [
{
"pp": "R : Type u\nA : Type u_4\ninst✝³ : CommSemiring R\ninst✝² : Semiring A\ninst✝¹ : Algebra R A\ninst✝ : FaithfulSMul R A\nx : Aˣ\nx✝¹ x✝ : R\neq : (LinearMap.toSpanSingleton R A ↑x) x✝¹ = (LinearMap.toSpanSingleton R A ↑x) x✝\n⊢ (fun r ↦ r • 1) x✝¹ = (fun r ↦ r • 1) x✝",
"ppTerm": "?m.484",
"assi... | [
"R : Type u\nA : Type u_4\ninst✝³ : CommSemiring R\ninst✝² : Semiring A\ninst✝¹ : Algebra R A\ninst✝ : FaithfulSMul R A\nx : Aˣ\nx✝¹ x✝ : R\neq : (LinearMap.toSpanSingleton R A ↑x) x✝¹ = (LinearMap.toSpanSingleton R A ↑x) x✝\n⊢ x✝¹ • 1 = x✝ • 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Matroid.Constructions | {
"line": 193,
"column": 27
} | {
"line": 193,
"column": 38
} | {
"line": 193,
"column": 39
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nE B I✝ X R J I : Set α\nh : ∀ J ⊆ I, J.Finite → J ⊆ E\ne : α\nheI : e ∈ I\n⊢ e ∈ E",
"ppTerm": "?m.9",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\nM : Matroid α\nE B I✝ X R J I : Set α\nh : ∀ J ⊆ I, J.Finite → J ⊆ E\ne : α\nheI : e ∈ I\n⊢ e ∈ E"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Matroid.Constructions | {
"line": 214,
"column": 2
} | {
"line": 214,
"column": 88
} | {
"line": 215,
"column": 2
} | [
{
"pp": "α : Type u_1\nI E : Set α\n⊢ uniqueBaseOn (I ∩ E) E = uniqueBaseOn I E",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Matroid.Indep",
"id",
"LE.le",
"Set.instInter",
"_private.Mathlib.Combinatorics.Matroid.Constructi... | [
"α : Type u_1\nI E : Set α\n⊢ ∀ I_1 ⊆ E, I_1 ⊆ I ∧ I_1 ⊆ E ↔ I_1 ⊆ I"
] | simp only [uniqueBaseOn, restrict_eq_restrict_iff, freeOn_indep_iff, subset_inter_iff] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.PicardGroup | {
"line": 780,
"column": 2
} | {
"line": 780,
"column": 25
} | {
"line": 781,
"column": 4
} | [
{
"pp": "R : Type u\nM : Type v\nA : Type u_4\ninst✝⁶ : CommSemiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\ninst✝³ : Semiring A\ninst✝² : Algebra R A\ne : A ⊗[R] M ≃ₗ[A] A\ninst✝¹ : Flat R M\ninst✝ : FaithfulSMul R A\n⊢ Function.Injective ⇑(toAlgebra e)",
"ppTerm": "?m.43",
"assigned": true,
... | [
"R : Type u\nM : Type v\nA : Type u_4\ninst✝⁶ : CommSemiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\ninst✝³ : Semiring A\ninst✝² : Algebra R A\ne : A ⊗[R] M ≃ₗ[A] A\ninst✝¹ : Flat R M\ninst✝ : FaithfulSMul R A\n⊢ Function.Injective ⇑(LinearMap.rTensor M (Algebra.ofId R A).toLinearMap)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Matroid.Constructions | {
"line": 245,
"column": 6
} | {
"line": 245,
"column": 17
} | {
"line": 245,
"column": 18
} | [
{
"pp": "α : Type u_1\nI : Set α\n⊢ uniqueBaseOn ∅ I = loopyOn I",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Matroid.loopyOn",
"congrArg",
"Matroid.dual",
"id",
"Matroid.uniqueBaseOn",
"propext",
"Set.instEmptyCollection",
... | [
"α : Type u_1\nI : Set α\n⊢ (uniqueBaseOn ∅ I)✶ = (loopyOn I)✶"
] | ← dual_inj, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.Matroid.Constructions | {
"line": 269,
"column": 25
} | {
"line": 269,
"column": 66
} | {
"line": 269,
"column": 67
} | [
{
"pp": "α : Type u_1\nE I : Set α\nhIE : I ⊆ E\nhI : I.Nonempty\n⊢ ¬(uniqueBaseOn I E).IsBase ∅",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Matroid.IsBase",
"id",
"Matroid.uniqueBaseOn",
"propext",
"Set.instEmptyCollectio... | [
"α : Type u_1\nE I : Set α\nhIE : I ⊆ E\nhI : I.Nonempty\n⊢ ¬∅ = I"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Matroid.Map | {
"line": 134,
"column": 49
} | {
"line": 134,
"column": 60
} | {
"line": 134,
"column": 61
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nf✝ : α → β\nE I✝ : Set α\nM : Matroid α\nN✝ N : Matroid β\nf : α → β\nI B : Set α\nhI : N.Indep (f '' I)\nhIinj : InjOn f I\nhImax : ¬Maximal (fun I ↦ N.Indep (f '' I) ∧ InjOn f I) I\nhBmax : Maximal (fun I ↦ N.Indep (f '' I) ∧ InjOn f I) B\nI' : Set α\nhII' : I ⊂ I'\nhI' : ... | [
"α : Type u_1\nβ : Type u_2\nf✝ : α → β\nE I✝ : Set α\nM : Matroid α\nN✝ N : Matroid β\nf : α → β\nI B : Set α\nhI : N.Indep (f '' I)\nhIinj : InjOn f I\nhImax : ¬Maximal (fun I ↦ N.Indep (f '' I) ∧ InjOn f I) I\nhBmax : Maximal (fun I ↦ N.Indep (f '' I) ∧ InjOn f I) B\nI' : Set α\nhII' : I ⊂ I'\nhI' : N.Indep (f '... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Matroid.Dual | {
"line": 165,
"column": 6
} | {
"line": 165,
"column": 17
} | {
"line": 165,
"column": 18
} | [
{
"pp": "α : Type u_1\nM₁ M₂ : Matroid α\n⊢ M₁ = M₂✶ ↔ M₂ = M₁✶",
"ppTerm": "?m.5",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Matroid.dual",
"id",
"Iff",
"propext",
"Eq.symm",
"Eq",
"Matroid",
"Matroid.dual_inj"
],
... | [
"α : Type u_1\nM₁ M₂ : Matroid α\n⊢ M₁✶ = M₂✶✶ ↔ M₂ = M₁✶"
] | ← dual_inj, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.Matroid.Map | {
"line": 173,
"column": 2
} | {
"line": 173,
"column": 13
} | {
"line": 173,
"column": 14
} | [
{
"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": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Matroid.Dual | {
"line": 200,
"column": 2
} | {
"line": 200,
"column": 96
} | {
"line": 201,
"column": 4
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nB X : Set α\nhB : M.IsBase B\nhX : X ⊆ M.E\nh : M✶.IsBasis (M.E \\ B ∩ (M.E \\ X)) (M.E \\ X)\n⊢ M.IsBasis (B ∩ X) X",
"ppTerm": "?m.29",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\nM : Matroid α\nB X : Set α\nhB : M.IsBase B\nhX : X ⊆ M.E\nh : M✶.IsBasis (M.E \\ B ∩ (M.E \\ X)) (M.E \\ X)\n⊢ M.IsBasis (B ∩ X) X"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Matroid.IndepAxioms | {
"line": 341,
"column": 44
} | {
"line": 341,
"column": 55
} | {
"line": 341,
"column": 56
} | [
{
"pp": "α : Type u_1\nE : Set α\nIndep : Set α → Prop\nindep_empty : Indep ∅\nindep_subset : ∀ ⦃I J : Set α⦄, Indep J → I ⊆ J → Indep I\nindep_aug : ∀ ⦃I B : Set α⦄, Indep I → ¬Maximal Indep I → Maximal Indep B → ∃ x ∈ B \\ I, Indep (insert x I)\nsubset_ground : ∀ (I : Set α), Indep I → I ⊆ E\nB : Set α\nn : ℕ... | [
"α : Type u_1\nE : Set α\nIndep : Set α → Prop\nindep_empty : Indep ∅\nindep_subset : ∀ ⦃I J : Set α⦄, Indep J → I ⊆ J → Indep I\nindep_aug : ∀ ⦃I B : Set α⦄, Indep I → ¬Maximal Indep I → Maximal Indep B → ∃ x ∈ B \\ I, Indep (insert x I)\nsubset_ground : ∀ (I : Set α), Indep I → I ⊆ E\nB : Set α\nn : ℕ\nhn : ∀ (I ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Matroid.Rank.Finite | {
"line": 142,
"column": 73
} | {
"line": 142,
"column": 84
} | {
"line": 142,
"column": 85
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nX : Set α\nh : M.IsRkFinite (X ∩ M.E)\n⊢ M.IsRkFinite (M.closure X)",
"ppTerm": "?m.17",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\nM : Matroid α\nX : Set α\nh : M.IsRkFinite (X ∩ M.E)\n⊢ M.IsRkFinite (M.closure X)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Matroid.IndepAxioms | {
"line": 438,
"column": 42
} | {
"line": 438,
"column": 53
} | {
"line": 438,
"column": 54
} | [
{
"pp": "α : Type u_1\ninst✝ : DecidableEq α\nE : Set α\nIndep : Finset α → Prop\nindep_empty : Indep ∅\nindep_subset : ∀ ⦃I J : Finset α⦄, Indep J → I ⊆ J → Indep I\nindep_aug : ∀ ⦃I J : Finset α⦄, Indep I → Indep J → I.card < J.card → ∃ e ∈ J, e ∉ I ∧ Indep (insert e I)\nsubset_ground : ∀ ⦃I : Finset α⦄, Inde... | [
"α : Type u_1\ninst✝ : DecidableEq α\nE : Set α\nIndep : Finset α → Prop\nindep_empty : Indep ∅\nindep_subset : ∀ ⦃I J : Finset α⦄, Indep J → I ⊆ J → Indep I\nindep_aug : ∀ ⦃I J : Finset α⦄, Indep I → Indep J → I.card < J.card → ∃ e ∈ J, e ∉ I ∧ Indep (insert e I)\nsubset_ground : ∀ ⦃I : Finset α⦄, Indep I → ↑I ⊆ E... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Matroid.Map | {
"line": 411,
"column": 26
} | {
"line": 411,
"column": 57
} | {
"line": 411,
"column": 58
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nI : Set α\nM : Matroid α\nX : Set α\nhIX : M.IsBasis I X\nf : α → β\nhf : InjOn f M.E\ne : α\nhe : e ∈ X\nhe' : f e ∉ f '' I\nhss : insert e I ⊆ M.E\n⊢ insert (f e) (f '' I) ⊆ (M.map f hf).E",
"ppTerm": "?m.88",
"assigned": true,
"usedConstants": [
"Eq.mpr"... | [
"α : Type u_1\nβ : Type u_2\nI : Set α\nM : Matroid α\nX : Set α\nhIX : M.IsBasis I X\nf : α → β\nhf : InjOn f M.E\ne : α\nhe : e ∈ X\nhe' : f e ∉ f '' I\nhss : insert e I ⊆ M.E\n⊢ insert e I ⊆ f ⁻¹' f '' M.E"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Matroid.Circuit | {
"line": 82,
"column": 13
} | {
"line": 82,
"column": 24
} | {
"line": 82,
"column": 25
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nh : M.IsCircuit ∅\n⊢ False",
"ppTerm": "?m.5",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\nM : Matroid α\nh : M.IsCircuit ∅\n⊢ False"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Matroid.Loop | {
"line": 98,
"column": 23
} | {
"line": 98,
"column": 34
} | {
"line": 98,
"column": 35
} | [
{
"pp": "α : Type u_1\nM : Matroid α\ne : α\ntfae_1_iff_2 : M.IsLoop e ↔ e ∈ M.closure ∅\ntfae_2_iff_3 : e ∈ M.closure ∅ ↔ M.IsCircuit {e}\ntfae_2_iff_4 : e ∈ M.closure ∅ ↔ M.Dep {e}\nh : (∀ (x : Set α), M.IsBase x → e ∈ M.E) ∧ ∀ (x : Set α), M.IsBase x → e ∉ x\nhi : M.Indep {e}\nB : Set α\nhB : M.IsBase B\nheB... | [
"α : Type u_1\nM : Matroid α\ne : α\ntfae_1_iff_2 : M.IsLoop e ↔ e ∈ M.closure ∅\ntfae_2_iff_3 : e ∈ M.closure ∅ ↔ M.IsCircuit {e}\ntfae_2_iff_4 : e ∈ M.closure ∅ ↔ M.Dep {e}\nh : (∀ (x : Set α), M.IsBase x → e ∈ M.E) ∧ ∀ (x : Set α), M.IsBase x → e ∉ x\nhi : M.Indep {e}\nB : Set α\nhB : M.IsBase B\nheB : {e} ⊆ B\n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Matroid.Loop | {
"line": 163,
"column": 2
} | {
"line": 163,
"column": 13
} | {
"line": 163,
"column": 14
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nX I : Set α\nhX : X ⊆ M.loops\nh : M.IsBasis I X\nthis : M.IsBasis I M.loops\n⊢ I = ∅",
"ppTerm": "?m.27",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\nM : Matroid α\nX I : Set α\nhX : X ⊆ M.loops\nh : M.IsBasis I X\nthis : M.IsBasis I M.loops\n⊢ I = ∅"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
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