module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.Algebra.Ring.Ext | {
"line": 254,
"column": 4
} | {
"line": 254,
"column": 26
} | {
"line": 256,
"column": 2
} | [
{
"pp": "R : Type u\ninst₁ inst₂ : NonAssocRing R\nh_add : HAdd.hAdd = HAdd.hAdd\nh_mul : HMul.hMul = HMul.hMul\nh₁ : inst₁.toNonUnitalNonAssocRing = inst₂.toNonUnitalNonAssocRing\n⊢ inst₁.toNonAssocSemiring = inst₂.toNonAssocSemiring",
"ppTerm": "?m.43",
"assigned": true,
"usedConstants": [
"... | [] | ext : 1 <;> assumption | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Algebra.Ring.Ext | {
"line": 284,
"column": 4
} | {
"line": 284,
"column": 26
} | {
"line": 285,
"column": 2
} | [
{
"pp": "R : Type u\ninst₁ inst₂ : Semiring R\nh_add : HAdd.hAdd = HAdd.hAdd\nh_mul : HMul.hMul = HMul.hMul\n⊢ inst₁.toAddCommMonoid = inst₂.toAddCommMonoid",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"Semiring.toAddCommMonoid",
"AddCommMonoid.ext"
],
"usedFVars": [... | [] | ext : 1 <;> assumption | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Algebra.Ring.Ext | {
"line": 286,
"column": 4
} | {
"line": 286,
"column": 26
} | {
"line": 287,
"column": 2
} | [
{
"pp": "R : Type u\ninst₁ inst₂ : Semiring R\nh_add : HAdd.hAdd = HAdd.hAdd\nh_mul : HMul.hMul = HMul.hMul\nh₀ : inst₁.toAddCommMonoid = inst₂.toAddCommMonoid\n⊢ inst₁.toNonUnitalSemiring = inst₂.toNonUnitalSemiring",
"ppTerm": "?m.42",
"assigned": true,
"usedConstants": [
"NonUnitalSemiring.... | [] | ext : 1 <;> assumption | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Algebra.Ring.Ext | {
"line": 288,
"column": 4
} | {
"line": 288,
"column": 26
} | {
"line": 289,
"column": 2
} | [
{
"pp": "R : Type u\ninst₁ inst₂ : Semiring R\nh_add : HAdd.hAdd = HAdd.hAdd\nh_mul : HMul.hMul = HMul.hMul\nh₀ : inst₁.toAddCommMonoid = inst₂.toAddCommMonoid\nh₁ : inst₁.toNonUnitalSemiring = inst₂.toNonUnitalSemiring\n⊢ inst₁.toNonAssocSemiring = inst₂.toNonAssocSemiring",
"ppTerm": "?m.53",
"assigne... | [] | ext : 1 <;> assumption | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Algebra.Ring.Ext | {
"line": 320,
"column": 4
} | {
"line": 320,
"column": 26
} | {
"line": 321,
"column": 2
} | [
{
"pp": "R : Type u\ninst₁ inst₂ : Ring R\nh_add : HAdd.hAdd = HAdd.hAdd\nh_mul : HMul.hMul = HMul.hMul\n⊢ inst₁.toSemiring = inst₂.toSemiring",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"Semiring.ext",
"Ring.toSemiring"
],
"usedFVars": [
"R",
"inst₁",
... | [] | ext : 1 <;> assumption | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Algebra.Ring.Ext | {
"line": 322,
"column": 4
} | {
"line": 322,
"column": 26
} | {
"line": 325,
"column": 2
} | [
{
"pp": "R : Type u\ninst₁ inst₂ : Ring R\nh_add : HAdd.hAdd = HAdd.hAdd\nh_mul : HMul.hMul = HMul.hMul\nh₁ : inst₁.toSemiring = inst₂.toSemiring\n⊢ toNonAssocRing = toNonAssocRing",
"ppTerm": "?m.43",
"assigned": true,
"usedConstants": [
"NonAssocRing.ext",
"Ring.toNonAssocRing"
],
... | [] | ext : 1 <;> assumption | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Algebra.Ring.Semireal.Defs | {
"line": 43,
"column": 30
} | {
"line": 43,
"column": 41
} | {
"line": 43,
"column": 42
} | [
{
"pp": "R : Type u_1\ninst✝³ : AddGroup R\ninst✝² : One R\ninst✝¹ : Mul R\ninst✝ : IsSemireal R\nx✝ : IsSumSq (-1)\n⊢ False",
"ppTerm": "?m.14",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u_1\ninst✝³ : AddGroup R\ninst✝² : One R\ninst✝¹ : Mul R\ninst✝ : IsSemireal R\nx✝ : IsSumSq (-1)\n⊢ False"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Ring.Semireal.Defs | {
"line": 68,
"column": 6
} | {
"line": 68,
"column": 17
} | {
"line": 68,
"column": 18
} | [
{
"pp": "case succ\nR : Type u_1\ninst✝¹ : NonAssocRing R\ninst✝ : IsSemireal R\nn : ℕ\nhn : 1 + ↑n = 0\n⊢ n + 1 = 0",
"ppTerm": "?succ",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"False",
"Nat.instMulZeroClass",
"Nat.instOne",
"congrArg",
"Nat.add_eq_zero_if... | [
"case succ\nR : Type u_1\ninst✝¹ : NonAssocRing R\ninst✝ : IsSemireal R\nn : ℕ\nhn : 1 + ↑n = 0\n⊢ False"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Ring.IsFormallyReal | {
"line": 63,
"column": 8
} | {
"line": 63,
"column": 19
} | {
"line": 63,
"column": 20
} | [
{
"pp": "case sq_add.inr.inl\nR : Type u_1\ninst✝ : NonUnitalNonAssocSemiring R\ns a : R\nne_a : a ≠ 0\nhs✝ : IsSumSq 0\nih : 0 ≠ 0 → IsSumNonzeroSq 0\nhs : a * a + 0 ≠ 0\n⊢ IsSumNonzeroSq (a * a + 0)",
"ppTerm": "?sq_add.inr.inl",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"IsSumNon... | [
"case sq_add.inr.inl\nR : Type u_1\ninst✝ : NonUnitalNonAssocSemiring R\ns a : R\nne_a : a ≠ 0\nhs✝ : IsSumSq 0\nih : 0 ≠ 0 → IsSumNonzeroSq 0\nhs : a * a + 0 ≠ 0\n⊢ IsSumNonzeroSq (a * a)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Ring.IsFormallyReal | {
"line": 125,
"column": 53
} | {
"line": 125,
"column": 64
} | {
"line": 125,
"column": 65
} | [
{
"pp": "R : Type u_1\ninst✝ : NonUnitalNonAssocSemiring R\nh : ∀ {s a : R}, IsSumSq s → a * a + s = 0 → a = 0\nx a✝ : R\nha : a✝ ≠ 0\nhc : a✝ * a✝ = 0\n⊢ a✝ * a✝ + 0 = 0",
"ppTerm": "?m.65",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"congrArg",
"AddMonoid.t... | [
"R : Type u_1\ninst✝ : NonUnitalNonAssocSemiring R\nh : ∀ {s a : R}, IsSumSq s → a * a + s = 0 → a = 0\nx a✝ : R\nha : a✝ ≠ 0\nhc : a✝ * a✝ = 0\n⊢ a✝ * a✝ = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Ring.IsFormallyReal | {
"line": 136,
"column": 38
} | {
"line": 136,
"column": 65
} | {
"line": 136,
"column": 66
} | [
{
"pp": "R : Type u_1\ninst✝¹ : Ring R\ninst✝ : IsFormallyReal R\nx : R\nhx : x ^ 2 = 0\nhc : x ≠ 0\n⊢ IsSumNonzeroSq 0",
"ppTerm": "?m.29",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u_1\ninst✝¹ : Ring R\ninst✝ : IsFormallyReal R\nx : R\nhx : x ^ 2 = 0\nhc : x ≠ 0\n⊢ IsSumNonzeroSq 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Ring.CentroidHom | {
"line": 496,
"column": 23
} | {
"line": 496,
"column": 39
} | {
"line": 496,
"column": 40
} | [
{
"pp": "α : Type u_5\ninst✝ : NonUnitalNonAssocSemiring α\na : α\nhc : R a = L a\nT : CentroidHom α\nhT : (toEndRingHom α) T = L a\ne1 : ∀ (d : α), T d = a * d\ne2 : ∀ (d : α), T d = d * a\n⊢ ∀ (b c : α), a * (b * c) = a * b * c",
"ppTerm": "?m.141",
"assigned": false,
"usedConstants": [],
"use... | [
"α : Type u_5\ninst✝ : NonUnitalNonAssocSemiring α\na : α\nhc : R a = L a\nT : CentroidHom α\nhT : (toEndRingHom α) T = L a\ne1 : ∀ (d : α), T d = a * d\ne2 : ∀ (d : α), T d = d * a\n⊢ ∀ (b c : α), a * (b * c) = a * b * c"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Ring.CentroidHom | {
"line": 497,
"column": 24
} | {
"line": 497,
"column": 40
} | {
"line": 497,
"column": 41
} | [
{
"pp": "α : Type u_5\ninst✝ : NonUnitalNonAssocSemiring α\na : α\nhc : R a = L a\nT : CentroidHom α\nhT : (toEndRingHom α) T = L a\ne1 : ∀ (d : α), T d = a * d\ne2 : ∀ (d : α), T d = d * a\n⊢ ∀ (a_1 b : α), a_1 * b * a = a_1 * (b * a)",
"ppTerm": "?m.142",
"assigned": false,
"usedConstants": [],
... | [
"α : Type u_5\ninst✝ : NonUnitalNonAssocSemiring α\na : α\nhc : R a = L a\nT : CentroidHom α\nhT : (toEndRingHom α) T = L a\ne1 : ∀ (d : α), T d = a * d\ne2 : ∀ (d : α), T d = d * a\n⊢ ∀ (a_1 b : α), a_1 * b * a = a_1 * (b * a)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.SkewMonoidAlgebra.Support | {
"line": 97,
"column": 4
} | {
"line": 97,
"column": 52
} | {
"line": 97,
"column": 53
} | [
{
"pp": "k : Type u_1\nG : Type u_2\ninst✝³ : Monoid G\ninst✝² : Semiring k\ninst✝¹ : MulSemiringAction G k\nf : SkewMonoidAlgebra k G\ninst✝ : DecidableEq G\nr : k\nx : G\nlx : IsLeftRegular x\nhrx : ∀ (y : k), r * x • y = 0 ↔ y = 0\ny : G\nhy : y ∈ image (fun x_1 ↦ x * x_1) f.support\n⊢ ∃ a ∈ f.support, x * a... | [
"k : Type u_1\nG : Type u_2\ninst✝³ : Monoid G\ninst✝² : Semiring k\ninst✝¹ : MulSemiringAction G k\nf : SkewMonoidAlgebra k G\ninst✝ : DecidableEq G\nr : k\nx : G\nlx : IsLeftRegular x\nhrx : ∀ (y : k), r * x • y = 0 ↔ y = 0\ny : G\nhy : y ∈ image (fun x_1 ↦ x * x_1) f.support\n⊢ ∃ a ∈ f.support, x * a = y"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.SkewMonoidAlgebra.Support | {
"line": 106,
"column": 4
} | {
"line": 106,
"column": 52
} | {
"line": 106,
"column": 53
} | [
{
"pp": "k : Type u_1\nG : Type u_2\ninst✝³ : Monoid G\ninst✝² : Semiring k\ninst✝¹ : MulSemiringAction G k\nf : SkewMonoidAlgebra k G\ninst✝ : DecidableEq G\nr : k\nx : G\nrx : IsRightRegular x\nhrx : ∀ (g : G) (y : k), y * g • r = 0 ↔ y = 0\ny : G\nhy : y ∈ image (fun x_1 ↦ x_1 * x) f.support\n⊢ ∃ a ∈ f.suppo... | [
"k : Type u_1\nG : Type u_2\ninst✝³ : Monoid G\ninst✝² : Semiring k\ninst✝¹ : MulSemiringAction G k\nf : SkewMonoidAlgebra k G\ninst✝ : DecidableEq G\nr : k\nx : G\nrx : IsRightRegular x\nhrx : ∀ (g : G) (y : k), y * g • r = 0 ↔ y = 0\ny : G\nhy : y ∈ image (fun x_1 ↦ x_1 * x) f.support\n⊢ ∃ a ∈ f.support, a * x = ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.RingQuot | {
"line": 160,
"column": 8
} | {
"line": 160,
"column": 83
} | {
"line": 161,
"column": 10
} | [
{
"pp": "case succ\nR : Type uR\ninst✝³ : Semiring R\nS : Type uS\ninst✝² : CommSemiring S\nT : Type uT\nA : Type uA\ninst✝¹ : Semiring A\ninst✝ : Algebra S A\nr : R → R → Prop\nx✝ : RingQuot r\na✝ : Quot (Rel r)\na b : R\nh : Rel r a b\nn : ℕ\nih : Quot.mk (Rel r) (a ^ n) = Quot.mk (Rel r) (b ^ n)\n⊢ Quot.mk (... | [
"case succ\nR : Type uR\ninst✝³ : Semiring R\nS : Type uS\ninst✝² : CommSemiring S\nT : Type uT\nA : Type uA\ninst✝¹ : Semiring A\ninst✝ : Algebra S A\nr : R → R → Prop\nx✝ : RingQuot r\na✝ : Quot (Rel r)\na b : R\nh : Rel r a b\nn : ℕ\nih : Quot.mk (Rel r) (a ^ n) = Quot.mk (Rel r) (b ^ n)\n⊢ Quot.mk (Rel r) (Mul.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.RingQuot | {
"line": 428,
"column": 6
} | {
"line": 431,
"column": 53
} | {
"line": 432,
"column": 6
} | [
{
"pp": "case refine_1\nR : Type uR\ninst✝⁵ : Semiring R\nS : Type uS\ninst✝⁴ : CommSemiring S\nT : Type uT\nA : Type uA\ninst✝³ : Semiring A\ninst✝² : Algebra S A\nr✝ : R → R → Prop\ninst✝¹ : Semiring T\nB : Type uR\ninst✝ : CommRing B\nr : B → B → Prop\nx : B\nh : x ∈ Ideal.ofRel r\n⊢ ∀ x ∈ {x | ∃ a b, r a b ... | [
"case refine_2\nR : Type uR\ninst✝⁵ : Semiring R\nS : Type uS\ninst✝⁴ : CommSemiring S\nT : Type uT\nA : Type uA\ninst✝³ : Semiring A\ninst✝² : Algebra S A\nr✝ : R → R → Prop\ninst✝¹ : Semiring T\nB : Type uR\ninst✝ : CommRing B\nr : B → B → Prop\nx : B\nh : x ∈ Ideal.ofRel r\n⊢ (mkRingHom r) 0 = 0",
"case refine... | · rintro y ⟨a, b, h, su⟩
symm at su
rw [← sub_eq_iff_eq_add] at su
rw [← su, map_sub, mkRingHom_rel h, sub_self] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Star.Subsemiring | {
"line": 122,
"column": 50
} | {
"line": 122,
"column": 66
} | {
"line": 122,
"column": 67
} | [
{
"pp": "R : Type v\ninst✝¹ : NonAssocSemiring R\ninst✝ : StarRing R\nS : StarSubsemiring R\ns : Set R\nhs : s = ↑S\na : R\nha : a ∈ (S.copy s hs).carrier\n⊢ a ∈ S.carrier",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"SetLike.mem_coe._simp_1",
"Submonoid.toSu... | [
"R : Type v\ninst✝¹ : NonAssocSemiring R\ninst✝ : StarRing R\nS : StarSubsemiring R\ns : Set R\nhs : s = ↑S\na : R\nha : a ∈ (S.copy s hs).carrier\n⊢ a ∈ S"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Star.Subsemiring | {
"line": 122,
"column": 50
} | {
"line": 122,
"column": 66
} | {
"line": 122,
"column": 67
} | [
{
"pp": "R : Type v\ninst✝¹ : NonAssocSemiring R\ninst✝ : StarRing R\nS : StarSubsemiring R\na : R\nhs : ↑S = ↑S\n__Subsemiring✝ : Subsemiring R := S.copy (↑S) hs\nha : a ∈ (S.copy (↑S) hs).carrier\n⊢ a ∈ S.carrier",
"ppTerm": "?m.62",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"SetL... | [
"R : Type v\ninst✝¹ : NonAssocSemiring R\ninst✝ : StarRing R\nS : StarSubsemiring R\na : R\nhs : ↑S = ↑S\n__Subsemiring✝ : Subsemiring R := S.copy (↑S) hs\nha : a ∈ (S.copy (↑S) hs).carrier\n⊢ a ∈ S"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Star.CHSH | {
"line": 129,
"column": 4
} | {
"line": 130,
"column": 11
} | {
"line": 130,
"column": 12
} | [
{
"pp": "R : Type u\ninst✝⁵ : CommRing R\ninst✝⁴ : PartialOrder R\ninst✝³ : StarRing R\ninst✝² : StarOrderedRing R\ninst✝¹ : Algebra ℝ R\ninst✝ : IsOrderedModule ℝ R\nA₀ A₁ B₀ B₁ : R\nT : IsCHSHTuple A₀ A₁ B₀ B₁\nP : R := 2 - A₀ * B₀ - A₀ * B₁ - A₁ * B₀ + A₁ * B₁\nidem : P * P = 4 * P\nidem' : P = (1 / 4) • (P ... | [
"R : Type u\ninst✝⁵ : CommRing R\ninst✝⁴ : PartialOrder R\ninst✝³ : StarRing R\ninst✝² : StarOrderedRing R\ninst✝¹ : Algebra ℝ R\ninst✝ : IsOrderedModule ℝ R\nA₀ A₁ B₀ B₁ : R\nT : IsCHSHTuple A₀ A₁ B₀ B₁\nP : R := 2 - A₀ * B₀ - A₀ * B₁ - A₁ * B₀ + A₁ * B₁\nidem : P * P = 4 * P\nidem' : P = (1 / 4) • (P * P)\nsa : s... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Star.CHSH | {
"line": 132,
"column": 2
} | {
"line": 132,
"column": 50
} | {
"line": 132,
"column": 51
} | [
{
"pp": "R : Type u\ninst✝⁵ : CommRing R\ninst✝⁴ : PartialOrder R\ninst✝³ : StarRing R\ninst✝² : StarOrderedRing R\ninst✝¹ : Algebra ℝ R\ninst✝ : IsOrderedModule ℝ R\nA₀ A₁ B₀ B₁ : R\nT : IsCHSHTuple A₀ A₁ B₀ B₁\nP : R := ⋯\ni₁ : 0 ≤ P\n⊢ 0 ≤ 2 - (A₀ * B₀ + A₀ * B₁ + A₁ * B₀ - A₁ * B₁)",
"ppTerm": "?m.235",... | [
"R : Type u\ninst✝⁵ : CommRing R\ninst✝⁴ : PartialOrder R\ninst✝³ : StarRing R\ninst✝² : StarOrderedRing R\ninst✝¹ : Algebra ℝ R\ninst✝ : IsOrderedModule ℝ R\nA₀ A₁ B₀ B₁ : R\nT : IsCHSHTuple A₀ A₁ B₀ B₁\nP : R := 2 - A₀ * B₀ - A₀ * B₁ - A₁ * B₀ + A₁ * B₁\ni₁ : 0 ≤ P\n⊢ 0 ≤ 2 - A₀ * B₀ - A₀ * B₁ - A₁ * B₀ + A₁ * B₁... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.SkewPolynomial.Basic | {
"line": 128,
"column": 2
} | {
"line": 128,
"column": 66
} | {
"line": 128,
"column": 67
} | [
{
"pp": "R : Type u_1\ninst✝ : Semiring R\np q : SkewPolynomial R\n⊢ (p + q).support ⊆ p.support ∪ q.support",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Equiv.instEquivLike",
"Finset.instUnion",
"congrA... | [
"R : Type u_1\ninst✝ : Semiring R\np q : SkewPolynomial R\n⊢ SkewMonoidAlgebra.support (p + q) ⊆ SkewMonoidAlgebra.support p ∪ SkewMonoidAlgebra.support q"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Star.CHSH | {
"line": 195,
"column": 37
} | {
"line": 195,
"column": 64
} | {
"line": 195,
"column": 65
} | [
{
"pp": "R : Type u\ninst✝⁶ : Ring R\ninst✝⁵ : PartialOrder R\ninst✝⁴ : StarRing R\ninst✝³ : StarOrderedRing R\ninst✝² : Algebra ℝ R\ninst✝¹ : IsOrderedModule ℝ R\ninst✝ : StarModule ℝ R\nA₀ A₁ B₀ B₁ : R\nT : IsCHSHTuple A₀ A₁ B₀ B₁\nM : ∀ (m : ℤ) (a : ℝ) (x : R), m • a • x = (↑m * a) • x\nP : R := (√2)⁻¹ • (A₁... | [
"R : Type u\ninst✝⁶ : Ring R\ninst✝⁵ : PartialOrder R\ninst✝⁴ : StarRing R\ninst✝³ : StarOrderedRing R\ninst✝² : Algebra ℝ R\ninst✝¹ : IsOrderedModule ℝ R\ninst✝ : StarModule ℝ R\nA₀ A₁ B₀ B₁ : R\nT : IsCHSHTuple A₀ A₁ B₀ B₁\nM : ∀ (m : ℤ) (a : ℝ) (x : R), m • a • x = (↑m * a) • x\nP : R := (√2)⁻¹ • (A₁ + A₀) - B₀\... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Star.CHSH | {
"line": 196,
"column": 37
} | {
"line": 196,
"column": 64
} | {
"line": 196,
"column": 65
} | [
{
"pp": "R : Type u\ninst✝⁶ : Ring R\ninst✝⁵ : PartialOrder R\ninst✝⁴ : StarRing R\ninst✝³ : StarOrderedRing R\ninst✝² : Algebra ℝ R\ninst✝¹ : IsOrderedModule ℝ R\ninst✝ : StarModule ℝ R\nA₀ A₁ B₀ B₁ : R\nT : IsCHSHTuple A₀ A₁ B₀ B₁\nM : ∀ (m : ℤ) (a : ℝ) (x : R), m • a • x = (↑m * a) • x\nP : R := (√2)⁻¹ • (A₁... | [
"R : Type u\ninst✝⁶ : Ring R\ninst✝⁵ : PartialOrder R\ninst✝⁴ : StarRing R\ninst✝³ : StarOrderedRing R\ninst✝² : Algebra ℝ R\ninst✝¹ : IsOrderedModule ℝ R\ninst✝ : StarModule ℝ R\nA₀ A₁ B₀ B₁ : R\nT : IsCHSHTuple A₀ A₁ B₀ B₁\nM : ∀ (m : ℤ) (a : ℝ) (x : R), m • a • x = (↑m * a) • x\nP : R := (√2)⁻¹ • (A₁ + A₀) - B₀\... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Star.CHSH | {
"line": 199,
"column": 2
} | {
"line": 199,
"column": 68
} | {
"line": 199,
"column": 69
} | [
{
"pp": "R : Type u\ninst✝⁶ : Ring R\ninst✝⁵ : PartialOrder R\ninst✝⁴ : StarRing R\ninst✝³ : StarOrderedRing R\ninst✝² : Algebra ℝ R\ninst✝¹ : IsOrderedModule ℝ R\ninst✝ : StarModule ℝ R\nA₀ A₁ B₀ B₁ : R\nT : IsCHSHTuple A₀ A₁ B₀ B₁\nM : ∀ (m : ℤ) (a : ℝ) (x : R), m • a • x = (↑m * a) • x\nP : R := ⋯\nQ : R := ... | [
"R : Type u\ninst✝⁶ : Ring R\ninst✝⁵ : PartialOrder R\ninst✝⁴ : StarRing R\ninst✝³ : StarOrderedRing R\ninst✝² : Algebra ℝ R\ninst✝¹ : IsOrderedModule ℝ R\ninst✝ : StarModule ℝ R\nA₀ A₁ B₀ B₁ : R\nT : IsCHSHTuple A₀ A₁ B₀ B₁\nM : ∀ (m : ℤ) (a : ℝ) (x : R), m • a • x = (↑m * a) • x\nP : R := (√2)⁻¹ • (A₁ + A₀) - B₀\... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.SkewPolynomial.Basic | {
"line": 462,
"column": 2
} | {
"line": 462,
"column": 35
} | {
"line": 462,
"column": 36
} | [
{
"pp": "R : Type u_1\ninst✝¹ : Semiring R\ninst✝ : MulSemiringAction (Multiplicative ℕ) R\nc : R\n⊢ (C c * X).support ⊆ {1}",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Semiring.toModule",
"Multiplicative.mon... | [
"R : Type u_1\ninst✝¹ : Semiring R\ninst✝ : MulSemiringAction (Multiplicative ℕ) R\nc : R\n⊢ c = 0 ∨ ((monomial 1) c).support = {1}"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.SkewPolynomial.Basic | {
"line": 471,
"column": 2
} | {
"line": 471,
"column": 39
} | {
"line": 471,
"column": 40
} | [
{
"pp": "R : Type u_1\ninst✝¹ : Semiring R\ninst✝ : MulSemiringAction (Multiplicative ℕ) R\nn : ℕ\nc : R\n⊢ (C c * X ^ n).support ⊆ {n}",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Semiring.toModule",
"Multipl... | [
"R : Type u_1\ninst✝¹ : Semiring R\ninst✝ : MulSemiringAction (Multiplicative ℕ) R\nn : ℕ\nc : R\n⊢ c = 0 ∨ ((monomial n) c).support = {n}"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.SkewPolynomial.Basic | {
"line": 621,
"column": 2
} | {
"line": 621,
"column": 52
} | {
"line": 621,
"column": 53
} | [
{
"pp": "R : Type u_1\ninst✝ : Ring R\np : SkewPolynomial R\n⊢ (-p).support = p.support",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"AddGroupWithOne.toAddGroup",
"congrArg",
"Finset",
"Finset.map",
... | [
"R : Type u_1\ninst✝ : Ring R\np : SkewPolynomial R\n⊢ SkewMonoidAlgebra.support (-p) = SkewMonoidAlgebra.support p"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.SkewMonoidAlgebra.Basic | {
"line": 203,
"column": 2
} | {
"line": 203,
"column": 23
} | {
"line": 203,
"column": 24
} | [
{
"pp": "k : Type u_1\nG : Type u_2\ninst✝¹ : AddMonoid k\ninst✝ : DecidableEq G\np q : SkewMonoidAlgebra k G\n⊢ (p + q).support ⊆ p.support ∪ q.support",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Finset.instUnion",
"congrArg",
"Finset",
"AddMon... | [
"k : Type u_1\nG : Type u_2\ninst✝¹ : AddMonoid k\ninst✝ : DecidableEq G\np q : SkewMonoidAlgebra k G\n⊢ (p.coeff + q.coeff).support ⊆ p.coeff.support ∪ q.coeff.support"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.SkewMonoidAlgebra.Basic | {
"line": 226,
"column": 2
} | {
"line": 226,
"column": 21
} | {
"line": 226,
"column": 22
} | [
{
"pp": "k : Type u_1\nG : Type u_2\ninst✝ : AddMonoid k\nf g : G →₀ k\n⊢ { coeff := f } = { coeff := g } ↔ ∀ (n : G), { coeff := f }.coeff n = { coeff := g }.coeff n",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Finsupp.instFunLike",
"Eq.mpr",
"SkewMonoidAlgebra.ofCoe... | [
"k : Type u_1\nG : Type u_2\ninst✝ : AddMonoid k\nf g : G →₀ k\n⊢ f = g ↔ ∀ (n : G), f n = g n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.SkewMonoidAlgebra.Basic | {
"line": 487,
"column": 21
} | {
"line": 487,
"column": 32
} | {
"line": 487,
"column": 33
} | [
{
"pp": "k : Type u_1\nG : Type u_2\ninst✝ : AddCommMonoid k\ninstNonempty : Nonempty G\np : SkewMonoidAlgebra k G → Prop\nf : SkewMonoidAlgebra k G\nsingle : ∀ (g : G) (a : k), p (SkewMonoidAlgebra.single g a)\nadd : ∀ (f g : SkewMonoidAlgebra k G), p f → p g → p (f + g)\n⊢ p 0",
"ppTerm": "?m.21",
"as... | [
"k : Type u_1\nG : Type u_2\ninst✝ : AddCommMonoid k\ninstNonempty : Nonempty G\np : SkewMonoidAlgebra k G → Prop\nf : SkewMonoidAlgebra k G\nsingle : ∀ (g : G) (a : k), p (SkewMonoidAlgebra.single g a)\nadd : ∀ (f g : SkewMonoidAlgebra k G), p f → p g → p (f + g)\n⊢ p 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Star.UnitaryStarAlgAut | {
"line": 96,
"column": 41
} | {
"line": 96,
"column": 66
} | {
"line": 96,
"column": 67
} | [
{
"pp": "R : Type u_3\nS : Type u_4\ninst✝⁸ : Ring R\ninst✝⁷ : StarMul R\ninst✝⁶ : CommRing S\ninst✝⁵ : StarMul S\ninst✝⁴ : Algebra S R\ninst✝³ : StarModule S R\ninst✝² : Algebra.IsCentral S R\ninst✝¹ : IsCancelMulZero S\ninst✝ : Module.IsTorsionFree S R\nu v : ↥(unitary R)\nx✝ : ∃ y, y • 1 = star ↑v * ↑u\ny : ... | [
"R : Type u_3\nS : Type u_4\ninst✝⁸ : Ring R\ninst✝⁷ : StarMul R\ninst✝⁶ : CommRing S\ninst✝⁵ : StarMul S\ninst✝⁴ : Algebra S R\ninst✝³ : StarModule S R\ninst✝² : Algebra.IsCentral S R\ninst✝¹ : IsCancelMulZero S\ninst✝ : Module.IsTorsionFree S R\nu v : ↥(unitary R)\nx✝ : ∃ y, y • 1 = star ↑v * ↑u\ny : S\nh : y • 1... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Star.UnitaryStarAlgAut | {
"line": 97,
"column": 46
} | {
"line": 97,
"column": 71
} | {
"line": 97,
"column": 72
} | [
{
"pp": "R : Type u_3\nS : Type u_4\ninst✝⁸ : Ring R\ninst✝⁷ : StarMul R\ninst✝⁶ : CommRing S\ninst✝⁵ : StarMul S\ninst✝⁴ : Algebra S R\ninst✝³ : StarModule S R\ninst✝² : Algebra.IsCentral S R\ninst✝¹ : IsCancelMulZero S\ninst✝ : Module.IsTorsionFree S R\nu v : ↥(unitary R)\nx✝ : ∃ y, y • 1 = star ↑v * ↑u\ny : ... | [
"R : Type u_3\nS : Type u_4\ninst✝⁸ : Ring R\ninst✝⁷ : StarMul R\ninst✝⁶ : CommRing S\ninst✝⁵ : StarMul S\ninst✝⁴ : Algebra S R\ninst✝³ : StarModule S R\ninst✝² : Algebra.IsCentral S R\ninst✝¹ : IsCancelMulZero S\ninst✝ : Module.IsTorsionFree S R\nu v : ↥(unitary R)\nx✝ : ∃ y, y • 1 = star ↑v * ↑u\ny : S\nh : y • 1... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.SkewMonoidAlgebra.Basic | {
"line": 730,
"column": 42
} | {
"line": 730,
"column": 58
} | {
"line": 730,
"column": 58
} | [
{
"pp": "case single\nk : Type u_1\nG : Type u_2\ninst✝² : Semiring k\ninst✝¹ : Monoid G\ninst✝ : MulSemiringAction G k\ng : G\na : k\n⊢ ((single 1 1).sum fun a₁ b₁ ↦ (single g a).sum fun a₂ b₂ ↦ single (a₁ * a₂) (b₁ * a₁ • b₂)) = single g a",
"ppTerm": "?single",
"assigned": true,
"usedConstants": ... | [
"case single\nk : Type u_1\nG : Type u_2\ninst✝² : Semiring k\ninst✝¹ : Monoid G\ninst✝ : MulSemiringAction G k\ng : G\na : k\n⊢ ((single g a).sum fun a₂ b₂ ↦ single (1 * a₂) (1 * 1 • b₂)) = single g a",
"case single\nk : Type u_1\nG : Type u_2\ninst✝² : Semiring k\ninst✝¹ : Monoid G\ninst✝ : MulSemiringAction G ... | sum_single_index | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.SkewMonoidAlgebra.Basic | {
"line": 734,
"column": 60
} | {
"line": 734,
"column": 76
} | {
"line": 734,
"column": 76
} | [
{
"pp": "case single\nk : Type u_1\nG : Type u_2\ninst✝² : Semiring k\ninst✝¹ : Monoid G\ninst✝ : MulSemiringAction G k\ng : G\na : k\n⊢ ((single 1 1).sum fun a₂ b₂ ↦ single (g * a₂) (a * g • b₂)) = single g a",
"ppTerm": "?single",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAsso... | [
"case single\nk : Type u_1\nG : Type u_2\ninst✝² : Semiring k\ninst✝¹ : Monoid G\ninst✝ : MulSemiringAction G k\ng : G\na : k\n⊢ single (g * 1) (a * g • 1) = single g a",
"case single\nk : Type u_1\nG : Type u_2\ninst✝² : Semiring k\ninst✝¹ : Monoid G\ninst✝ : MulSemiringAction G k\ng : G\na : k\n⊢ single (g * 1)... | sum_single_index | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.SkewMonoidAlgebra.Basic | {
"line": 909,
"column": 8
} | {
"line": 909,
"column": 90
} | {
"line": 910,
"column": 8
} | [
{
"pp": "k : Type u_1\nG : Type u_2\ninst✝² : Semiring k\ninst✝¹ : Mul G\ninst✝ : SMulZeroClass G k\nf g : SkewMonoidAlgebra k G\nx : G\ns : Finset (G × G)\nhs : ∀ {p : G × G}, p ∈ s ↔ p.1 * p.2 = x\nF : G × G → k := fun p ↦ if p.1 * p.2 = x then f.coeff p.1 * p.1 • g.coeff p.2 else 0\np : G × G\nhps : p ∈ s\nh... | [
"k : Type u_1\nG : Type u_2\ninst✝² : Semiring k\ninst✝¹ : Mul G\ninst✝ : SMulZeroClass G k\nf g : SkewMonoidAlgebra k G\nx : G\ns : Finset (G × G)\nhs : ∀ {p : G × G}, p ∈ s ↔ p.1 * p.2 = x\nF : G × G → k := fun p ↦ if p.1 * p.2 = x then f.coeff p.1 * p.1 • g.coeff p.2 else 0\np : G × G\nhps : p ∈ s\nhp : p ∈ s → ... | simp only [Finset.mem_filter, mem_support_iff, not_and, Classical.not_not] at hp ⊢ | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.SkewMonoidAlgebra.Basic | {
"line": 934,
"column": 8
} | {
"line": 934,
"column": 90
} | {
"line": 935,
"column": 8
} | [
{
"pp": "k : Type u_1\nG : Type u_2\ninst✝² : Semiring k\ninst✝¹ : Mul G\ninst✝ : SMulZeroClass G k\nf g : SkewMonoidAlgebra k G\nx : G\nthis : ({p | p.1 * p.2 = x} ∩ Function.support fun p ↦ f.coeff p.1 * p.1 • g.coeff p.2).Finite\ns : Finset (G × G) := this.toFinset\nF : G × G → k := fun p ↦ if p.1 * p.2 = x ... | [
"k : Type u_1\nG : Type u_2\ninst✝² : Semiring k\ninst✝¹ : Mul G\ninst✝ : SMulZeroClass G k\nf g : SkewMonoidAlgebra k G\nx : G\nthis : ({p | p.1 * p.2 = x} ∩ Function.support fun p ↦ f.coeff p.1 * p.1 • g.coeff p.2).Finite\ns : Finset (G × G) := this.toFinset\nF : G × G → k := fun p ↦ if p.1 * p.2 = x then f.coeff... | simp only [Finset.mem_filter, mem_support_iff, not_and, Classical.not_not] at hp ⊢ | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Tropical.Basic | {
"line": 286,
"column": 23
} | {
"line": 286,
"column": 34
} | {
"line": 286,
"column": 35
} | [
{
"pp": "R : Type u\ninst✝ : LinearOrder R\nx y : Tropical R\nh : x ≤ y\n⊢ untrop (x + y) = untrop x",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"PartialOrder.toPreorder",
"Preorder.toLE",
"SemilatticeInf.toPartialOrder",
"DistribLattice.toLattic... | [
"R : Type u\ninst✝ : LinearOrder R\nx y : Tropical R\nh : x ≤ y\n⊢ x ≤ y"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Tropical.Basic | {
"line": 290,
"column": 23
} | {
"line": 290,
"column": 34
} | {
"line": 290,
"column": 35
} | [
{
"pp": "R : Type u\ninst✝ : LinearOrder R\nx y : Tropical R\nh : y ≤ x\n⊢ untrop (x + y) = untrop y",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"PartialOrder.toPreorder",
"Preorder.toLE",
"SemilatticeInf.toPartialOrder",
"DistribLattice.toLattic... | [
"R : Type u\ninst✝ : LinearOrder R\nx y : Tropical R\nh : y ≤ x\n⊢ y ≤ x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Tropical.BigOperators | {
"line": 49,
"column": 29
} | {
"line": 49,
"column": 40
} | {
"line": 49,
"column": 41
} | [
{
"pp": "R : Type u_1\ninst✝ : AddCommMonoid R\ns : Multiset R\n⊢ ∀ (a : List R), trop (sum ⟦a⟧) = (map trop ⟦a⟧).prod",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Multiset.sum",
"Tropical.instCommMonoidTropical",
"Multiset.map",
"Multiset.prod",
"id",
... | [
"R : Type u_1\ninst✝ : AddCommMonoid R\ns : Multiset R\n⊢ ∀ (a : List R), trop a.sum = (List.map trop a).prod"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Tropical.BigOperators | {
"line": 65,
"column": 29
} | {
"line": 65,
"column": 40
} | {
"line": 65,
"column": 41
} | [
{
"pp": "R : Type u_1\ninst✝ : AddCommMonoid R\ns : Multiset (Tropical R)\n⊢ ∀ (a : List (Tropical R)), untrop (prod ⟦a⟧) = (map untrop ⟦a⟧).sum",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Multiset.sum",
"Tropical.instCommMonoidTropical",
"Multiset.map",
"Multi... | [
"R : Type u_1\ninst✝ : AddCommMonoid R\ns : Multiset (Tropical R)\n⊢ ∀ (a : List (Tropical R)), untrop a.prod = (List.map untrop a).sum"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.SkewMonoidAlgebra.Basic | {
"line": 974,
"column": 17
} | {
"line": 974,
"column": 33
} | {
"line": 974,
"column": 33
} | [
{
"pp": "k : Type u_1\nG : Type u_2\ninst✝² : Semiring k\ninst✝¹ : Mul G\ninst✝ : SMulZeroClass G k\nr : k\ng g' : G\nx : SkewMonoidAlgebra k G\nh : ¬∃ d, g' = g * d\n⊢ ((single g r).sum fun a₁ b₁ ↦ x.sum fun a₂ b₂ ↦ if a₁ * a₂ = g' then b₁ * a₁ • b₂ else 0) = 0",
"ppTerm": "?m.35",
"assigned": true,
... | [
"k : Type u_1\nG : Type u_2\ninst✝² : Semiring k\ninst✝¹ : Mul G\ninst✝ : SMulZeroClass G k\nr : k\ng g' : G\nx : SkewMonoidAlgebra k G\nh : ¬∃ d, g' = g * d\n⊢ (x.sum fun a₂ b₂ ↦ if g * a₂ = g' then r * g • b₂ else 0) = 0",
"k : Type u_1\nG : Type u_2\ninst✝² : Semiring k\ninst✝¹ : Mul G\ninst✝ : SMulZeroClass G... | sum_single_index | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Tropical.BigOperators | {
"line": 126,
"column": 2
} | {
"line": 126,
"column": 35
} | {
"line": 126,
"column": 36
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝ : ConditionallyCompleteLinearOrder R\ns : Finset S\nf : S → Tropical (WithTop R)\n⊢ untrop (∑ i ∈ s, f i) = ⨅ i, untrop (f ↑i)",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"WithTop.instInfSet",
"Eq.mpr",
"Lattice.toSemilattice... | [
"R : Type u_1\nS : Type u_2\ninst✝ : ConditionallyCompleteLinearOrder R\ns : Finset S\nf : S → Tropical (WithTop R)\n⊢ ∑ i ∈ s, f i = ∑ i ∈ s.attach, f ↑i"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.SkewMonoidAlgebra.Basic | {
"line": 1147,
"column": 2
} | {
"line": 1147,
"column": 13
} | {
"line": 1147,
"column": 14
} | [
{
"pp": "k : Type u_1\nG : Type u_2\ninst✝³ : Semiring k\ninst✝² : Monoid G\ninst✝¹ : MulSemiringAction G k\ninst✝ : Nontrivial k\na b : G\nh : (single a 1).coeff = (single b 1).coeff\n⊢ a = b",
"ppTerm": "?m.21",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"k : Type u_1\nG : Type u_2\ninst✝³ : Semiring k\ninst✝² : Monoid G\ninst✝¹ : MulSemiringAction G k\ninst✝ : Nontrivial k\na b : G\nh : (single a 1).coeff = (single b 1).coeff\n⊢ a = b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.SkewMonoidAlgebra.Basic | {
"line": 1171,
"column": 57
} | {
"line": 1171,
"column": 68
} | {
"line": 1171,
"column": 69
} | [
{
"pp": "k : Type u_1\nG : Type u_2\ninst✝² : Semiring k\ninst✝¹ : MulOneClass G\nR : Type u_3\ninst✝ : Semiring R\nf : k →+* R\ng : G →* R\nc : k\nφ : SkewMonoidAlgebra k G\nthis : (liftNC ↑f ⇑g).comp ((smulAddHom k (SkewMonoidAlgebra k G)) c) = (AddMonoidHom.mulLeft (f c)).comp (liftNC ↑f ⇑g)\n⊢ (liftNC ↑f ⇑g... | [
"k : Type u_1\nG : Type u_2\ninst✝² : Semiring k\ninst✝¹ : MulOneClass G\nR : Type u_3\ninst✝ : Semiring R\nf : k →+* R\ng : G →* R\nc : k\nφ : SkewMonoidAlgebra k G\nthis : (liftNC ↑f ⇑g).comp ((smulAddHom k (SkewMonoidAlgebra k G)) c) = (AddMonoidHom.mulLeft (f c)).comp (liftNC ↑f ⇑g)\n⊢ (liftNC ↑f ⇑g) (c • φ) = ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Star.LinearMap | {
"line": 89,
"column": 2
} | {
"line": 89,
"column": 13
} | {
"line": 89,
"column": 14
} | [
{
"pp": "R : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝¹³ : Semiring R\ninst✝¹² : InvolutiveStar R\ninst✝¹¹ : AddCommMonoid E\ninst✝¹⁰ : Module R E\ninst✝⁹ : StarAddMonoid E\ninst✝⁸ : StarModule R E\ninst✝⁷ : AddCommMonoid F\ninst✝⁶ : Module R F\ninst✝⁵ : StarAddMonoid F\ninst✝⁴ : StarModule R F\nG : Type u_4\... | [
"R : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝¹³ : Semiring R\ninst✝¹² : InvolutiveStar R\ninst✝¹¹ : AddCommMonoid E\ninst✝¹⁰ : Module R E\ninst✝⁹ : StarAddMonoid E\ninst✝⁸ : StarModule R E\ninst✝⁷ : AddCommMonoid F\ninst✝⁶ : Module R F\ninst✝⁵ : StarAddMonoid F\ninst✝⁴ : StarModule R F\nG : Type u_4\ninst✝³ : Ad... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Star.LinearMap | {
"line": 134,
"column": 2
} | {
"line": 134,
"column": 13
} | {
"line": 134,
"column": 14
} | [
{
"pp": "R : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\nH : Type u_9\ninst✝¹⁷ : CommSemiring R\ninst✝¹⁶ : StarRing R\ninst✝¹⁵ : AddCommMonoid E\ninst✝¹⁴ : StarAddMonoid E\ninst✝¹³ : Module R E\ninst✝¹² : StarModule R E\ninst✝¹¹ : AddCommMonoid F\ninst✝¹⁰ : StarAddMonoid F\ninst✝⁹ : Module R F\ninst✝⁸ :... | [
"R : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\nH : Type u_9\ninst✝¹⁷ : CommSemiring R\ninst✝¹⁶ : StarRing R\ninst✝¹⁵ : AddCommMonoid E\ninst✝¹⁴ : StarAddMonoid E\ninst✝¹³ : Module R E\ninst✝¹² : StarModule R E\ninst✝¹¹ : AddCommMonoid F\ninst✝¹⁰ : StarAddMonoid F\ninst✝⁹ : Module R F\ninst✝⁸ : StarModule ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Star.LinearMap | {
"line": 210,
"column": 13
} | {
"line": 210,
"column": 24
} | {
"line": 210,
"column": 25
} | [
{
"pp": "R : Type u_5\ninst✝⁸ : CommSemiring R\ninst✝⁷ : StarRing R\nn : Type u_8\ninst✝⁶ : DecidableEq n\nB : n → Type u_9\ninst✝⁵ : (i : n) → AddCommMonoid (B i)\ninst✝⁴ : (i : n) → Module R (B i)\ninst✝³ : (i : n) → StarAddMonoid (B i)\ninst✝² : ∀ (i : n), StarModule R (B i)\ninst✝¹ : Fintype n\ninst✝ : (i :... | [
"R : Type u_5\ninst✝⁸ : CommSemiring R\ninst✝⁷ : StarRing R\nn : Type u_8\ninst✝⁶ : DecidableEq n\nB : n → Type u_9\ninst✝⁵ : (i : n) → AddCommMonoid (B i)\ninst✝⁴ : (i : n) → Module R (B i)\ninst✝³ : (i : n) → StarAddMonoid (B i)\ninst✝² : ∀ (i : n), StarModule R (B i)\ninst✝¹ : Fintype n\ninst✝ : (i : n) → Coalge... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Star.LinearMap | {
"line": 278,
"column": 2
} | {
"line": 278,
"column": 18
} | {
"line": 278,
"column": 19
} | [
{
"pp": "R : Type u_1\nE : Type u_2\ninst✝⁵ : Semiring R\ninst✝⁴ : InvolutiveStar R\ninst✝³ : AddCommMonoid E\ninst✝² : Module R E\ninst✝¹ : StarAddMonoid E\ninst✝ : StarModule R E\nf : WithConv (End R E)\nhf : IsUnit f.ofConv\nu : (End R E)ˣ\nhu : ↑u = f.ofConv\nthis : IsUnit (star (toConv ↑u)).ofConv\n⊢ IsUni... | [
"R : Type u_1\nE : Type u_2\ninst✝⁵ : Semiring R\ninst✝⁴ : InvolutiveStar R\ninst✝³ : AddCommMonoid E\ninst✝² : Module R E\ninst✝¹ : StarAddMonoid E\ninst✝ : StarModule R E\nf : WithConv (End R E)\nhf : IsUnit f.ofConv\nu : (End R E)ˣ\nhu : ↑u = f.ofConv\nthis : IsUnit (star (toConv ↑u)).ofConv\n⊢ IsUnit (star f).o... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.RingedSpace.Stalks | {
"line": 84,
"column": 2
} | {
"line": 84,
"column": 87
} | {
"line": 86,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : HasColimits C\nU : TopCat\nX : PresheafedSpace C\nf : U ⟶ ↑X\nh : IsOpenEmbedding ⇑(ConcreteCategory.hom f)\nV : Opens ↑U\nx : ↑U\nhx : x ∈ V\n⊢ X.presheaf.germ (h.functor.obj V) ((ConcreteCategory.hom f) x) ⋯ ≫ (X.restrictStalkIso h x).inv =\n (X.rest... | [] | rw [← restrictStalkIso_hom_eq_germ, Category.assoc, Iso.hom_inv_id, Category.comp_id] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Geometry.RingedSpace.Stalks | {
"line": 84,
"column": 2
} | {
"line": 84,
"column": 87
} | {
"line": 86,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : HasColimits C\nU : TopCat\nX : PresheafedSpace C\nf : U ⟶ ↑X\nh : IsOpenEmbedding ⇑(ConcreteCategory.hom f)\nV : Opens ↑U\nx : ↑U\nhx : x ∈ V\n⊢ X.presheaf.germ (h.functor.obj V) ((ConcreteCategory.hom f) x) ⋯ ≫ (X.restrictStalkIso h x).inv =\n (X.rest... | [] | rw [← restrictStalkIso_hom_eq_germ, Category.assoc, Iso.hom_inv_id, Category.comp_id] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.RingedSpace.Stalks | {
"line": 84,
"column": 2
} | {
"line": 84,
"column": 87
} | {
"line": 86,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : HasColimits C\nU : TopCat\nX : PresheafedSpace C\nf : U ⟶ ↑X\nh : IsOpenEmbedding ⇑(ConcreteCategory.hom f)\nV : Opens ↑U\nx : ↑U\nhx : x ∈ V\n⊢ X.presheaf.germ (h.functor.obj V) ((ConcreteCategory.hom f) x) ⋯ ≫ (X.restrictStalkIso h x).inv =\n (X.rest... | [] | rw [← restrictStalkIso_hom_eq_germ, Category.assoc, Iso.hom_inv_id, Category.comp_id] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.RingedSpace.PresheafedSpace | {
"line": 222,
"column": 4
} | {
"line": 222,
"column": 84
} | {
"line": 223,
"column": 4
} | [
{
"pp": "case h\nC : Type u_1\ninst✝ : Category.{v_1, u_1} C\nX Y : PresheafedSpace C\nH : ↑X ≅ ↑Y\nα : (Presheaf.pushforward C H.hom).obj X.presheaf ≅ Y.presheaf\nU✝ : Opens ↑↑Y\n⊢ α.inv.app (op U✝) ≫\n X.presheaf.map (eqToHom ⋯) ≫\n ((Presheaf.pushforward C H.hom).obj X.presheaf).map ((eqToHom ⋯).... | [
"case h\nC : Type u_1\ninst✝ : Category.{v_1, u_1} C\nX Y : PresheafedSpace C\nH : ↑X ≅ ↑Y\nα : (Presheaf.pushforward C H.hom).obj X.presheaf ≅ Y.presheaf\nU✝ : Opens ↑↑Y\n⊢ α.inv.app (op U✝) ≫ α.hom.app (op ((Opens.map (𝟙 ↑Y)).obj U✝)) = (𝟙 Y).c.app (op U✝)"
] | simp only [eqToHom_map, eqToHom_app, eqToHom_trans_assoc, eqToHom_refl, id_comp] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Geometry.RingedSpace.SheafedSpace | {
"line": 287,
"column": 4
} | {
"line": 287,
"column": 15
} | {
"line": 287,
"column": 16
} | [
{
"pp": "C : Type u\ninst✝⁶ : Category.{v, u} C\nFC : C → C → Type u_1\nCC : C → Type v\ninst✝⁵ : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)\ninstCC : ConcreteCategory C FC\ninst✝⁴ : HasColimits C\ninst✝³ : HasLimits C\ninst✝² : PreservesLimits (CategoryTheory.forget C)\ninst✝¹ : PreservesFilteredColimits (Cate... | [
"C : Type u\ninst✝⁶ : Category.{v, u} C\nFC : C → C → Type u_1\nCC : C → Type v\ninst✝⁵ : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)\ninstCC : ConcreteCategory C FC\ninst✝⁴ : HasColimits C\ninst✝³ : HasLimits C\ninst✝² : PreservesLimits (CategoryTheory.forget C)\ninst✝¹ : PreservesFilteredColimits (CategoryTheory.f... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.RingedSpace.Basic | {
"line": 70,
"column": 2
} | {
"line": 70,
"column": 13
} | {
"line": 70,
"column": 14
} | [
{
"pp": "case h\nX : RingedSpace\nU : Opens ↑↑X.toPresheafedSpace\nf : ↑(X.presheaf.obj (op U))\nx : ↥U\nh : (ConcreteCategory.hom (X.presheaf.germ U ↑x ⋯)) f = 0\nh1 : (ConcreteCategory.hom (X.presheaf.germ U ↑x ⋯)) f = (ConcreteCategory.hom (X.presheaf.germ U ↑x ⋯)) 0\nV : Opens ↑↑X.toPresheafedSpace\nhv : ↑x... | [
"case h\nX : RingedSpace\nU : Opens ↑↑X.toPresheafedSpace\nf : ↑(X.presheaf.obj (op U))\nx : ↥U\nh : (ConcreteCategory.hom (X.presheaf.germ U ↑x ⋯)) f = 0\nh1 : (ConcreteCategory.hom (X.presheaf.germ U ↑x ⋯)) f = (ConcreteCategory.hom (X.presheaf.germ U ↑x ⋯)) 0\nV : Opens ↑↑X.toPresheafedSpace\nhv : ↑x ∈ V\ni w✝ :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.RingedSpace.Basic | {
"line": 90,
"column": 2
} | {
"line": 90,
"column": 13
} | {
"line": 90,
"column": 14
} | [
{
"pp": "case h\nX : RingedSpace\nU : Opens ↑↑X.toPresheafedSpace\nf : ↑(X.presheaf.obj (op U))\nx : ↑↑X.toPresheafedSpace\nhx : x ∈ U\nh : IsUnit ((ConcreteCategory.hom (X.presheaf.germ U x hx)) f)\nV : Opens ↑↑X.toPresheafedSpace\nhxV : x ∈ V\ng : ToType (X.presheaf.obj (op V))\nW : Opens ↑↑X.toPresheafedSpac... | [
"case h\nX : RingedSpace\nU : Opens ↑↑X.toPresheafedSpace\nf : ↑(X.presheaf.obj (op U))\nx : ↑↑X.toPresheafedSpace\nhx : x ∈ U\nh : IsUnit ((ConcreteCategory.hom (X.presheaf.germ U x hx)) f)\nV : Opens ↑↑X.toPresheafedSpace\nhxV : x ∈ V\ng : ToType (X.presheaf.obj (op V))\nW : Opens ↑↑X.toPresheafedSpace := U ⊓ V\n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.RingedSpace.PresheafedSpace | {
"line": 313,
"column": 4
} | {
"line": 313,
"column": 15
} | {
"line": 313,
"column": 16
} | [
{
"pp": "case h\nC : Type u_1\ninst✝ : Category.{v_1, u_1} C\nU : TopCat\nX : PresheafedSpace C\nf : U ⟶ ↑X\nhf : IsOpenEmbedding ⇑(ConcreteCategory.hom f)\nthis✝¹ : Mono f\nZ : PresheafedSpace C\ng₁ g₂ : Z ⟶ X.restrict hf\neq : g₁ ≫ X.ofRestrict hf = g₂ ≫ X.ofRestrict hf\nV : Opens ↑↑(X.restrict hf)\nhV : (Ope... | [
"case h\nC : Type u_1\ninst✝ : Category.{v_1, u_1} C\nU : TopCat\nX : PresheafedSpace C\nf : U ⟶ ↑X\nhf : IsOpenEmbedding ⇑(ConcreteCategory.hom f)\nthis✝¹ : Mono f\nZ : PresheafedSpace C\ng₁ g₂ : Z ⟶ X.restrict hf\neq : g₁ ≫ X.ofRestrict hf = g₂ ≫ X.ofRestrict hf\nV : Opens ↑↑(X.restrict hf)\nhV : (Opens.map (X.of... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Sheaves.LocalPredicate | {
"line": 116,
"column": 6
} | {
"line": 116,
"column": 17
} | {
"line": 116,
"column": 18
} | [
{
"pp": "X : TopCat\nT✝ : ↑X → Type u_1\nT : Type ?u.10\ninst✝ : TopologicalSpace T\nU : Opens ↑X\nf : ↥U → T\nx : ↥U\nV : Opens ↑X\nm : ↑x ∈ V\ni : V ⟶ U\nw : ContinuousAt (fun x ↦ f (i x)) ⟨↑x, m⟩\n⊢ ContinuousAt f x",
"ppTerm": "?m.67",
"assigned": false,
"usedConstants": [],
"usedFVars": [],... | [
"X : TopCat\nT✝ : ↑X → Type u_1\nT : Type ?u.10\ninst✝ : TopologicalSpace T\nU : Opens ↑X\nf : ↥U → T\nx : ↥U\nV : Opens ↑X\nm : ↑x ∈ V\ni : V ⟶ U\nw : ContinuousAt (fun x ↦ f (i x)) ⟨↑x, m⟩\n⊢ ContinuousAt f x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Spec | {
"line": 172,
"column": 2
} | {
"line": 178,
"column": 20
} | {
"line": 182,
"column": 0
} | [
{
"pp": "X : RingedSpace\nR : CommRingCat\nα β : X ⟶ sheafedSpaceObj R\nw : α.hom.base = β.hom.base\nh :\n ∀ (r : ↑R),\n let U := PrimeSpectrum.basicOpen r;\n (CommRingCat.ofHom (algebraMap (↑R) ((structureSheafInType ↑R ↑R).obj.obj (op U))) ≫ α.hom.c.app (op U)) ≫\n X.presheaf.map (eqToHom ⋯) =\n... | [] | ext : 1
· exact w
· apply ((TopCat.Sheaf.pushforward _ β.hom.base).obj X.sheaf).hom_ext _
PrimeSpectrum.isBasis_basic_opens
intro r
apply (StructureSheaf.to_basicOpen_epi R r).1
simpa using! h r | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.Spec | {
"line": 172,
"column": 2
} | {
"line": 178,
"column": 20
} | {
"line": 182,
"column": 0
} | [
{
"pp": "X : RingedSpace\nR : CommRingCat\nα β : X ⟶ sheafedSpaceObj R\nw : α.hom.base = β.hom.base\nh :\n ∀ (r : ↑R),\n let U := PrimeSpectrum.basicOpen r;\n (CommRingCat.ofHom (algebraMap (↑R) ((structureSheafInType ↑R ↑R).obj.obj (op U))) ≫ α.hom.c.app (op U)) ≫\n X.presheaf.map (eqToHom ⋯) =\n... | [] | ext : 1
· exact w
· apply ((TopCat.Sheaf.pushforward _ β.hom.base).obj X.sheaf).hom_ext _
PrimeSpectrum.isBasis_basic_opens
intro r
apply (StructureSheaf.to_basicOpen_epi R r).1
simpa using! h r | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.RingedSpace.OpenImmersion | {
"line": 338,
"column": 8
} | {
"line": 342,
"column": 45
} | {
"line": 343,
"column": 8
} | [
{
"pp": "case op.op\nC : Type u\ninst✝ : Category.{v, u} C\nX Y Z : PresheafedSpace C\nf : X ⟶ Z\nhf : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)\ng : Y ⟶ Z\nunop✝¹ unop✝ : Opens ↑↑X\ni : op unop✝¹ ⟶ op unop✝\n⊢ X.presheaf.map i ≫\n invApp f (unop (op uno... | [
"case op.op\nC : Type u\ninst✝ : Category.{v, u} C\nX Y Z : PresheafedSpace C\nf : X ⟶ Z\nhf : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)\ng : Y ⟶ Z\nunop✝¹ unop✝ : Opens ↑↑X\ni : op unop✝¹ ⟶ op unop✝\n⊢ invApp f unop✝¹ ≫\n g.c.app (op ((opensFunctor f).obj u... | simp only [(inv_naturality_assoc), restrict_carrier, restrict_presheaf,
TopCat.Presheaf.pushforward_obj_obj, Functor.comp_obj, Functor.op_obj,
TopCat.Presheaf.pushforward_obj_map, Functor.comp_map, Functor.op_map, Quiver.Hom.unop_op,
NatTrans.naturality_assoc, TopCat.Presheaf.pushforward_o... | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.AlgebraicGeometry.Scheme | {
"line": 838,
"column": 2
} | {
"line": 838,
"column": 66
} | {
"line": 838,
"column": 67
} | [
{
"pp": "X : Scheme\nU : X.Opens\nι : Type u_1\nf : ι → Set ↑Γ(X, U)\n⊢ X.zeroLocus (⋃ i, f i) = ⋂ i, X.zeroLocus (f i)",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier",
"Opposite",
"CommRi... | [
"X : Scheme\nU : X.Opens\nι : Type u_1\nf : ι → Set ↑Γ(X, U)\n⊢ ⋂ f_1, ⋂ i, ⋂ (_ : f_1 ∈ f i), (↑(X.toRingedSpace.basicOpen f_1))ᶜ =\n ⋂ i, ⋂ f_1 ∈ f i, (↑(X.toRingedSpace.basicOpen f_1))ᶜ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Scheme | {
"line": 846,
"column": 18
} | {
"line": 846,
"column": 29
} | {
"line": 846,
"column": 30
} | [
{
"pp": "X : Scheme\nU : X.Opens\nI : Ideal ↑Γ(X, U)\nx : ↥X\nH : ∀ f ∈ I, x ∉ X.basicOpen f\nf : ↑Γ(X, U)\nhx : x ∈ X.basicOpen f\nhn : f ^ 0 ∈ I\n⊢ f ∈ I",
"ppTerm": "?m.65",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"X : Scheme\nU : X.Opens\nI : Ideal ↑Γ(X, U)\nx : ↥X\nH : ∀ f ∈ I, x ∉ X.basicOpen f\nf : ↑Γ(X, U)\nhx : x ∈ X.basicOpen f\nhn : f ^ 0 ∈ I\n⊢ f ∈ I"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.OpenImmersion | {
"line": 157,
"column": 2
} | {
"line": 157,
"column": 13
} | {
"line": 157,
"column": 14
} | [
{
"pp": "X Y : Scheme\nf : X ⟶ Y\nH : IsOpenImmersion f\nU : X.Opens\n⊢ f ''ᵁ U ≤ opensRange f",
"ppTerm": "?m.17",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"X Y : Scheme\nf : X ⟶ Y\nH : IsOpenImmersion f\nU : X.Opens\n⊢ f ''ᵁ U ≤ opensRange f"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.OpenImmersion | {
"line": 179,
"column": 2
} | {
"line": 179,
"column": 13
} | {
"line": 179,
"column": 14
} | [
{
"pp": "X Y : Scheme\nf : X ⟶ Y\nH : IsOpenImmersion f\nU V : X.Opens\nhUV : (fun x ↦ f ''ᵁ x) U = (fun x ↦ f ''ᵁ x) V\n⊢ U = V",
"ppTerm": "?m.20",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"X Y : Scheme\nf : X ⟶ Y\nH : IsOpenImmersion f\nU V : X.Opens\nhUV : (fun x ↦ f ''ᵁ x) U = (fun x ↦ f ''ᵁ x) V\n⊢ U = V"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Cover.MorphismProperty | {
"line": 217,
"column": 4
} | {
"line": 217,
"column": 18
} | {
"line": 219,
"column": 0
} | [
{
"pp": "K : Precoverage Scheme\nX✝ Y Z : Scheme\n𝒰✝ : Cover K X✝\nf : X✝ ⟶ Z\ng : Y ⟶ Z\ninst✝ : ∀ (x : 𝒰✝.I₀), HasPullback (𝒰✝.f x ≫ f) g\nP Q : MorphismProperty Scheme\nX : Scheme\n𝒰 : AffineCover P X\nx : ↥X\ny : ↥(Spec (𝒰.X (𝒰.idx x)))\nhy : (𝒰.f (𝒰.idx x)) y = x\n⊢ ∃ i, x ∈ Set.range ⇑({ I₀ := 𝒰.... | [] | use 𝒰.idx x, y | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.AlgebraicGeometry.OpenImmersion | {
"line": 667,
"column": 4
} | {
"line": 667,
"column": 19
} | {
"line": 667,
"column": 20
} | [
{
"pp": "X Y Z : Scheme\nf : X ⟶ Z\ng : Y ⟶ Z\nH : IsOpenImmersion f\n⊢ (Scheme.Hom.opensRange f).carrier ∩ Set.range ⇑g = Set.range ⇑g ∩ Set.range ⇑f",
"ppTerm": "?m.71",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"AlgebraicGeometry.PresheafedSpace.carrier",
"congrArg",
... | [
"X Y Z : Scheme\nf : X ⟶ Z\ng : Y ⟶ Z\nH : IsOpenImmersion f\n⊢ Set.range ⇑g ∩ (Scheme.Hom.opensRange f).carrier = Set.range ⇑g ∩ Set.range ⇑f"
] | Set.inter_comm, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.OpenImmersion | {
"line": 759,
"column": 8
} | {
"line": 759,
"column": 39
} | {
"line": 759,
"column": 40
} | [
{
"pp": "U V X Y : Scheme\ng : U ⟶ V\niU : U ⟶ X\niV : V ⟶ Y\nf : X ⟶ Y\ninst✝¹ : IsOpenImmersion iU\ninst✝ : IsOpenImmersion iV\nH : iU ≫ f = g ≫ iV\nH' : f ⁻¹ᵁ Scheme.Hom.opensRange iV = Scheme.Hom.opensRange iU\n⊢ Set.range ⇑(pullback.snd iV f) = Set.range ⇑iU",
"ppTerm": "?m.68",
"assigned": true,
... | [
"U V X Y : Scheme\ng : U ⟶ V\niU : U ⟶ X\niV : V ⟶ Y\nf : X ⟶ Y\ninst✝¹ : IsOpenImmersion iU\ninst✝ : IsOpenImmersion iV\nH : iU ≫ f = g ≫ iV\nH' : f ⁻¹ᵁ Scheme.Hom.opensRange iV = Scheme.Hom.opensRange iU\n⊢ ⇑f ⁻¹' Set.range ⇑iV = Set.range ⇑iU"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Cover.Open | {
"line": 237,
"column": 4
} | {
"line": 237,
"column": 15
} | {
"line": 237,
"column": 16
} | [
{
"pp": "case hcover\nX : Scheme\nU : X.Opens\nf g : ↑Γ(X, U)\n𝒰 : X.OpenCover\nh : ∀ (i : 𝒰.I₀), (ConcreteCategory.hom (Hom.app (𝒰.f i) U)) f = (ConcreteCategory.hom (Hom.app (𝒰.f i) U)) g\nx : ↥X\nhx : x ∈ U\n⊢ ∃ x_1 y, (𝒰.f (Cover.idx 𝒰 x_1)) y = x",
"ppTerm": "?hcover",
"assigned": false,
... | [
"case hcover\nX : Scheme\nU : X.Opens\nf g : ↑Γ(X, U)\n𝒰 : X.OpenCover\nh : ∀ (i : 𝒰.I₀), (ConcreteCategory.hom (Hom.app (𝒰.f i) U)) f = (ConcreteCategory.hom (Hom.app (𝒰.f i) U)) g\nx : ↥X\nhx : x ∈ U\n⊢ ∃ x_1 y, (𝒰.f (Cover.idx 𝒰 x_1)) y = x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Cover.Open | {
"line": 272,
"column": 10
} | {
"line": 277,
"column": 20
} | {
"line": 279,
"column": 0
} | [
{
"pp": "X Y Z : Scheme\n𝒰 : X.OpenCover\nf : X ⟶ Z\ng : Y ⟶ Z\ninst✝ : ∀ (x : 𝒰.I₀), HasPullback (𝒰.f x ≫ f) g\nR : CommRingCat\n⊢ { I₀ := ↑R, X := fun r ↦ Spec (CommRingCat.of (Localization.Away r)),\n f := fun r ↦ Spec.map (CommRingCat.ofHom (algebraMap (↑R) (Localization.Away r))) }.presieve₀ ∈\n ... | [] | by
rw [presieve₀_mem_precoverage_iff]
refine ⟨fun x ↦ ⟨1, ?_⟩, AlgebraicGeometry.Scheme.isOpenImmersion_SpecMap_localizationAway⟩
rw [Set.range_eq_univ.mpr ((TopCat.epi_iff_surjective _).mp _)]
· exact trivial
· infer_instance | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicGeometry.StructureSheaf | {
"line": 160,
"column": 4
} | {
"line": 160,
"column": 41
} | {
"line": 161,
"column": 4
} | [
{
"pp": "R M A : Type u\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : CommRing A\ninst✝ : Algebra R A\nP : ↑(PrimeSpectrum.Top R)\nU : Opens ↑(PrimeSpectrum.Top R)\nr : ↥(sectionsSubalgebra R U)\na : (x : ↥U) → Localizations M ↑x\nha✝ : a ∈ (sectionsSubmodule M U).carrier\nx : ↥U\... | [
"R M A : Type u\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : CommRing A\ninst✝ : Algebra R A\nP : ↑(PrimeSpectrum.Top R)\nU : Opens ↑(PrimeSpectrum.Top R)\nr : ↥(sectionsSubalgebra R U)\na : (x : ↥U) → Localizations M ↑x\nha✝ : a ∈ (sectionsSubmodule M U).carrier\nx : ↥U\nV : Opens ↑... | obtain ⟨hrsy, hry⟩ := hr ⟨y.1, y.2.1⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.AlgebraicGeometry.StructureSheaf | {
"line": 291,
"column": 2
} | {
"line": 292,
"column": 43
} | {
"line": 293,
"column": 2
} | [
{
"pp": "R M : Type u\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nU : Opens ↑(PrimeSpectrum.Top R)\ns : (structureSheafInType R M).obj.obj (op U)\nx : ↑(PrimeSpectrum.Top R)\nhx : x ∈ U\nV : Opens ↑(PrimeSpectrum.Top R)\nhxV : ↑⟨x, hx⟩ ∈ V\niVU : V ⟶ unop (op U)\nf : M\ng : R\nhfg : ∀ (x ... | [
"case refine_1\nR M : Type u\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nU : Opens ↑(PrimeSpectrum.Top R)\ns : (structureSheafInType R M).obj.obj (op U)\nx : ↑(PrimeSpectrum.Top R)\nhx : x ∈ U\nV : Opens ↑(PrimeSpectrum.Top R)\nhxV : ↑⟨x, hx⟩ ∈ V\niVU : V ⟶ unop (op U)\nf : M\ng : R\nhfg : ∀ ... | refine ⟨g' * g, ?_, ?_, g' • f, Subtype.ext <| funext fun ⟨y, hy⟩ ↦ ?_⟩ <;>
simp only [PrimeSpectrum.basicOpen_mul] | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.AlgebraicGeometry.StructureSheaf | {
"line": 322,
"column": 2
} | {
"line": 322,
"column": 13
} | {
"line": 322,
"column": 14
} | [
{
"pp": "R A : Type u\ninst✝² : CommRing R\ninst✝¹ : CommRing A\ninst✝ : Algebra R A\nU : Opens ↑(PrimeSpectrum.Top R)\n⊢ const 1 1 U ⋯ = 1",
"ppTerm": "?m.21",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R A : Type u\ninst✝² : CommRing R\ninst✝¹ : CommRing A\ninst✝ : Algebra R A\nU : Opens ↑(PrimeSpectrum.Top R)\n⊢ const 1 1 U ⋯ = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.RingedSpace.OpenImmersion | {
"line": 1294,
"column": 6
} | {
"line": 1294,
"column": 75
} | {
"line": 1294,
"column": 75
} | [
{
"pp": "X Y : LocallyRingedSpace\nf : X ⟶ Y\nH : IsOpenImmersion f\ninst✝ : Epi f.base\n⊢ IsIso f",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.IsIso",
"AlgebraicGeometry.SheafedSpace",
"congrArg",
"CommRingCat",
"CommRingCat... | [
"X Y : LocallyRingedSpace\nf : X ⟶ Y\nH : IsOpenImmersion f\ninst✝ : Epi f.base\n⊢ IsIso (forgetToSheafedSpace.map f)"
] | ← isIso_iff_of_reflects_iso _ LocallyRingedSpace.forgetToSheafedSpace | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.Restrict | {
"line": 238,
"column": 35
} | {
"line": 238,
"column": 46
} | {
"line": 238,
"column": 47
} | [
{
"pp": "C : Type u₁\ninst✝ : Category.{v, u₁} C\nX✝ : Scheme\nU✝ : X✝.Opens\nX : Scheme\nU V : X.Opens\ne : U ≤ V\n⊢ Set.range ⇑U.ι ⊆ Set.range ⇑V.ι",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier",
... | [
"C : Type u₁\ninst✝ : Category.{v, u₁} C\nX✝ : Scheme\nU✝ : X✝.Opens\nX : Scheme\nU V : X.Opens\ne : U ≤ V\n⊢ U ≤ V"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Restrict | {
"line": 276,
"column": 27
} | {
"line": 276,
"column": 55
} | {
"line": 276,
"column": 56
} | [
{
"pp": "X : Scheme\nU V : X.Opens\ne : U ≤ V\nW : (↑V).Opens\ny : ↥X\nhyU : y ∈ U\nhyW : ⟨y, hyU⟩ ∈ ↑(X.homOfLE e ⁻¹ᵁ W)\n⊢ ⟨y, ⋯⟩ ∈ ↑W",
"ppTerm": "?m.92",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"SetLike.mem_coe._simp_1",
"AlgebraicGeometry.PresheafedSpace.carrier",
... | [
"X : Scheme\nU V : X.Opens\ne : U ≤ V\nW : (↑V).Opens\ny : ↥X\nhyU : y ∈ U\nhyW : ⟨y, hyU⟩ ∈ ↑(X.homOfLE e ⁻¹ᵁ W)\n⊢ ⟨y, ⋯⟩ ∈ W"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Restrict | {
"line": 278,
"column": 26
} | {
"line": 278,
"column": 54
} | {
"line": 278,
"column": 55
} | [
{
"pp": "X : Scheme\nU V : X.Opens\ne : U ≤ V\nW : (↑V).Opens\ny : ↥↑V\nhyW : y ∈ ↑W\nhyU : (ConcreteCategory.hom (LocallyRingedSpace.Hom.toShHom (Hom.toLRSHom V.ι)).hom.base) y ∈ ↑U\n⊢ ⟨↑y, hyU⟩ ∈ ↑(X.homOfLE e ⁻¹ᵁ W)",
"ppTerm": "?m.144",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [
"X : Scheme\nU V : X.Opens\ne : U ≤ V\nW : (↑V).Opens\ny : ↥↑V\nhyW : y ∈ ↑W\nhyU : (ConcreteCategory.hom (LocallyRingedSpace.Hom.toShHom (Hom.toLRSHom V.ι)).hom.base) y ∈ ↑U\n⊢ y ∈ W"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.StructureSheaf | {
"line": 476,
"column": 15
} | {
"line": 476,
"column": 26
} | {
"line": 476,
"column": 27
} | [
{
"pp": "R M : Type u\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nU : Opens ↑(PrimeSpectrum.Top R)\nhU : IsCompact ↑U\ns : (structureSheafInType R M).obj.obj (op U)\ng : ↥U → R\nhxg : ∀ (x : ↥U), ↑x ∈ basicOpen (g x)\nigU : ∀ (x : ↥U), basicOpen (g x) ≤ U\nf : ↥U → M\nH :\n ∀ (x : ↥U),\n... | [
"R M : Type u\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nU : Opens ↑(PrimeSpectrum.Top R)\nhU : IsCompact ↑U\ns : (structureSheafInType R M).obj.obj (op U)\ng : ↥U → R\nhxg : ∀ (x : ↥U), ↑x ∈ basicOpen (g x)\nigU : ∀ (x : ↥U), basicOpen (g x) ≤ U\nf : ↥U → M\nH :\n ∀ (x : ↥U),\n const (f... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.StructureSheaf | {
"line": 476,
"column": 50
} | {
"line": 476,
"column": 61
} | {
"line": 476,
"column": 62
} | [
{
"pp": "R M : Type u\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nU : Opens ↑(PrimeSpectrum.Top R)\nhU : IsCompact ↑U\ns : (structureSheafInType R M).obj.obj (op U)\ng : ↥U → R\nhxg : ∀ (x : ↥U), ↑x ∈ basicOpen (g x)\nigU : ∀ (x : ↥U), basicOpen (g x) ≤ U\nf : ↥U → M\nH :\n ∀ (x : ↥U),\n... | [
"R M : Type u\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nU : Opens ↑(PrimeSpectrum.Top R)\nhU : IsCompact ↑U\ns : (structureSheafInType R M).obj.obj (op U)\ng : ↥U → R\nhxg : ∀ (x : ↥U), ↑x ∈ basicOpen (g x)\nigU : ∀ (x : ↥U), basicOpen (g x) ≤ U\nf : ↥U → M\nH :\n ∀ (x : ↥U),\n const (f... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.StructureSheaf | {
"line": 497,
"column": 6
} | {
"line": 498,
"column": 66
} | {
"line": 498,
"column": 67
} | [
{
"pp": "R M : Type u\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf : R\ns : (structureSheafInType R M).obj.obj (op (basicOpen f))\nι : Type u\nw✝ : Fintype ι\na : ι → M\nb : ι → R\nibU : ∀ (i : ι), basicOpen (b i) ≤ basicOpen f\niU : basicOpen f ≤ ⨆ i, basicOpen (b i)\nhab : ∀ (i j : ι),... | [
"R M : Type u\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf : R\ns : (structureSheafInType R M).obj.obj (op (basicOpen f))\nι : Type u\nw✝ : Fintype ι\na : ι → M\nb : ι → R\nibU : ∀ (i : ι), basicOpen (b i) ≤ basicOpen f\niU : basicOpen f ≤ ⨆ i, basicOpen (b i)\nhab : ∀ (i j : ι), b j • a i =... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.StructureSheaf | {
"line": 501,
"column": 24
} | {
"line": 501,
"column": 35
} | {
"line": 501,
"column": 36
} | [
{
"pp": "R M : Type u\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf : R\ns : (structureSheafInType R M).obj.obj (op (basicOpen f))\nι : Type u\nw✝ : Fintype ι\na : ι → M\nb : ι → R\nibU : ∀ (i : ι), basicOpen (b i) ≤ basicOpen f\niU : basicOpen f ≤ ⨆ i, basicOpen (b i)\nhab : ∀ (i j : ι),... | [
"R M : Type u\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf : R\ns : (structureSheafInType R M).obj.obj (op (basicOpen f))\nι : Type u\nw✝ : Fintype ι\na : ι → M\nb : ι → R\nibU : ∀ (i : ι), basicOpen (b i) ≤ basicOpen f\niU : basicOpen f ≤ ⨆ i, basicOpen (b i)\nhab : ∀ (i j : ι), b j • a i =... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.StructureSheaf | {
"line": 530,
"column": 25
} | {
"line": 530,
"column": 36
} | {
"line": 530,
"column": 37
} | [
{
"pp": "R M : Type u\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nthis : IsLocalizedModule ⊥ (toOpenₗ R M ⊤)\nx y : M\ne : (toOpenₗ R M ⊤) x = (toOpenₗ R M ⊤) y\n⊢ x = y",
"ppTerm": "?m.70",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R M : Type u\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nthis : IsLocalizedModule ⊥ (toOpenₗ R M ⊤)\nx y : M\ne : (toOpenₗ R M ⊤) x = (toOpenₗ R M ⊤) y\n⊢ x = y"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Restrict | {
"line": 872,
"column": 6
} | {
"line": 872,
"column": 17
} | {
"line": 872,
"column": 18
} | [
{
"pp": "case refine_2.refine_1\nX Y S T : Scheme\nf : T ⟶ S\ng : Y ⟶ X\niX : X ⟶ S\niY : Y ⟶ T\nH : IsPullback g iY iX f\nUS : S.Opens\nUT : T.Opens\nUX : X.Opens\nhUST : UT ≤ f ⁻¹ᵁ US\nhUSX : UX ≤ iX ⁻¹ᵁ US\nUY : Y.Opens\nhUY : UY = g ⁻¹ᵁ UX ⊓ iY ⁻¹ᵁ UT\n⊢ IsPullback (g ∣_ UX) (resLE iY (f ⁻¹ᵁ US) (g ⁻¹ᵁ UX) ... | [
"case refine_2.refine_1\nX Y S T : Scheme\nf : T ⟶ S\ng : Y ⟶ X\niX : X ⟶ S\niY : Y ⟶ T\nH : IsPullback g iY iX f\nUS : S.Opens\nUT : T.Opens\nUX : X.Opens\nhUST : UT ≤ f ⁻¹ᵁ US\nhUSX : UX ≤ iX ⁻¹ᵁ US\nUY : Y.Opens\nhUY : UY = g ⁻¹ᵁ UX ⊓ iY ⁻¹ᵁ UT\n⊢ IsPullback (g ∣_ UX) ((g ⁻¹ᵁ UX).ι ≫ iY) (UX.ι ≫ iX) f"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.LocallyDirected | {
"line": 56,
"column": 2
} | {
"line": 56,
"column": 13
} | {
"line": 56,
"column": 14
} | [
{
"pp": "J : Type u_1\ninst✝ : Category.{v_1, u_1} J\nF : Discrete J ⥤ Type u_2\ni : J\n⊢ ∀ (xi xj : F.obj { as := i }),\n (ConcreteCategory.hom (F.map { down := { down := ⋯ } })) xi =\n (ConcreteCategory.hom (F.map { down := { down := ⋯ } })) xj →\n ∃ l fli flj x, (ConcreteCategory.hom (F.map fl... | [
"J : Type u_1\ninst✝ : Category.{v_1, u_1} J\nF : Discrete J ⥤ Type u_2\ni : J\n⊢ ∀ (xi : F.obj { as := i }),\n ∃ a' fli flj x, (ConcreteCategory.hom (F.map fli)) x = xi ∧ (ConcreteCategory.hom (F.map flj)) x = xi"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.LocallyDirected | {
"line": 62,
"column": 4
} | {
"line": 62,
"column": 15
} | {
"line": 62,
"column": 16
} | [
{
"pp": "case id.id\nJ : Type u_1\ninst✝¹ : Category.{v_1, u_1} J\nF : WidePushoutShape J ⥤ Type u_2\ninst✝ : ∀ (i : J), Mono (F.map (WidePushoutShape.Hom.init i))\ni : WidePushoutShape J\n⊢ ∀ (xi xj : F.obj i),\n (ConcreteCategory.hom (F.map (WidePushoutShape.Hom.id i))) xi =\n (ConcreteCategory.hom ... | [
"case id.id\nJ : Type u_1\ninst✝¹ : Category.{v_1, u_1} J\nF : WidePushoutShape J ⥤ Type u_2\ninst✝ : ∀ (i : J), Mono (F.map (WidePushoutShape.Hom.init i))\ni : WidePushoutShape J\n⊢ ∀ (xi : F.obj i), ∃ l fli flj x, (ConcreteCategory.hom (F.map fli)) x = xi ∧ (ConcreteCategory.hom (F.map flj)) x = xi"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Limits.Shapes.Pullback.Equalizer | {
"line": 32,
"column": 38
} | {
"line": 32,
"column": 49
} | {
"line": 32,
"column": 50
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\nX Y : C\nf g : X ⟶ Y\ninst✝¹ : HasEqualizer f g\ninst✝ : HasBinaryProduct Y Y\ns : PullbackCone (prod.lift f g) (prod.lift (𝟙 Y) (𝟙 Y))\n⊢ s.fst ≫ f = s.snd",
"ppTerm": "?m.136",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"used... | [
"C : Type u\ninst✝² : Category.{v, u} C\nX Y : C\nf g : X ⟶ Y\ninst✝¹ : HasEqualizer f g\ninst✝ : HasBinaryProduct Y Y\ns : PullbackCone (prod.lift f g) (prod.lift (𝟙 Y) (𝟙 Y))\n⊢ s.fst ≫ f = s.snd"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Limits.Shapes.Pullback.Equalizer | {
"line": 33,
"column": 38
} | {
"line": 33,
"column": 49
} | {
"line": 33,
"column": 50
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\nX Y : C\nf g : X ⟶ Y\ninst✝¹ : HasEqualizer f g\ninst✝ : HasBinaryProduct Y Y\ns : PullbackCone (prod.lift f g) (prod.lift (𝟙 Y) (𝟙 Y))\nH₁ : s.fst ≫ f = s.snd\n⊢ s.fst ≫ g = s.snd",
"ppTerm": "?m.204",
"assigned": false,
"usedConstants": [],
"u... | [
"C : Type u\ninst✝² : Category.{v, u} C\nX Y : C\nf g : X ⟶ Y\ninst✝¹ : HasEqualizer f g\ninst✝ : HasBinaryProduct Y Y\ns : PullbackCone (prod.lift f g) (prod.lift (𝟙 Y) (𝟙 Y))\nH₁ : s.fst ≫ f = s.snd\n⊢ s.fst ≫ g = s.snd"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Limits.Shapes.Pullback.Equalizer | {
"line": 36,
"column": 21
} | {
"line": 36,
"column": 32
} | {
"line": 36,
"column": 33
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\nX Y : C\nf g : X ⟶ Y\ninst✝¹ : HasEqualizer f g\ninst✝ : HasBinaryProduct Y Y\ns : PullbackCone (prod.lift f g) (prod.lift (𝟙 Y) (𝟙 Y))\n⊢ equalizer.lift s.fst ⋯ ≫ equalizer.ι f g ≫ f = s.snd",
"ppTerm": "?m.263",
"assigned": true,
"usedConstants": ... | [
"C : Type u\ninst✝² : Category.{v, u} C\nX Y : C\nf g : X ⟶ Y\ninst✝¹ : HasEqualizer f g\ninst✝ : HasBinaryProduct Y Y\ns : PullbackCone (prod.lift f g) (prod.lift (𝟙 Y) (𝟙 Y))\n⊢ s.fst ≫ f = s.snd"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Limits.Shapes.Pullback.Equalizer | {
"line": 47,
"column": 38
} | {
"line": 47,
"column": 49
} | {
"line": 47,
"column": 50
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\nX Y : C\nf g : X ⟶ Y\ninst✝¹ : HasCoequalizer f g\ninst✝ : HasBinaryCoproduct X X\ns : PushoutCocone (coprod.desc f g) (coprod.desc (𝟙 X) (𝟙 X))\n⊢ f ≫ s.inl = s.inr",
"ppTerm": "?m.136",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
... | [
"C : Type u\ninst✝² : Category.{v, u} C\nX Y : C\nf g : X ⟶ Y\ninst✝¹ : HasCoequalizer f g\ninst✝ : HasBinaryCoproduct X X\ns : PushoutCocone (coprod.desc f g) (coprod.desc (𝟙 X) (𝟙 X))\n⊢ f ≫ s.inl = s.inr"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Limits.Shapes.Pullback.Equalizer | {
"line": 48,
"column": 38
} | {
"line": 48,
"column": 49
} | {
"line": 48,
"column": 50
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\nX Y : C\nf g : X ⟶ Y\ninst✝¹ : HasCoequalizer f g\ninst✝ : HasBinaryCoproduct X X\ns : PushoutCocone (coprod.desc f g) (coprod.desc (𝟙 X) (𝟙 X))\nH₁ : f ≫ s.inl = s.inr\n⊢ g ≫ s.inl = s.inr",
"ppTerm": "?m.204",
"assigned": false,
"usedConstants": [... | [
"C : Type u\ninst✝² : Category.{v, u} C\nX Y : C\nf g : X ⟶ Y\ninst✝¹ : HasCoequalizer f g\ninst✝ : HasBinaryCoproduct X X\ns : PushoutCocone (coprod.desc f g) (coprod.desc (𝟙 X) (𝟙 X))\nH₁ : f ≫ s.inl = s.inr\n⊢ g ≫ s.inl = s.inr"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Limits.Shapes.Pullback.Equalizer | {
"line": 51,
"column": 21
} | {
"line": 51,
"column": 32
} | {
"line": 51,
"column": 33
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\nX Y : C\nf g : X ⟶ Y\ninst✝¹ : HasCoequalizer f g\ninst✝ : HasBinaryCoproduct X X\ns : PushoutCocone (coprod.desc f g) (coprod.desc (𝟙 X) (𝟙 X))\n⊢ (f ≫ coequalizer.π f g) ≫ coequalizer.desc s.inl ⋯ = s.inr",
"ppTerm": "?m.263",
"assigned": true,
"u... | [
"C : Type u\ninst✝² : Category.{v, u} C\nX Y : C\nf g : X ⟶ Y\ninst✝¹ : HasCoequalizer f g\ninst✝ : HasBinaryCoproduct X X\ns : PushoutCocone (coprod.desc f g) (coprod.desc (𝟙 X) (𝟙 X))\n⊢ f ≫ s.inl = s.inr"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.MorphismProperty.Local | {
"line": 83,
"column": 2
} | {
"line": 83,
"column": 38
} | {
"line": 84,
"column": 2
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\nP : MorphismProperty C\nK : Precoverage C\ninst✝¹ : K.HasPullbacks\ninst✝ : P.RespectsIso\nH : ∀ {X Y : C} (f : X ⟶ Y) (𝒰 : K.ZeroHypercover Y), P f ↔ ∀ (i : 𝒰.I₀), P (pullback.snd f (𝒰.f i))\n⊢ P.IsLocalAtTarget K",
"ppTerm": "?m.29",
"assigned": true... | [
"C : Type u\ninst✝² : Category.{v, u} C\nP : MorphismProperty C\nK : Precoverage C\ninst✝¹ : K.HasPullbacks\ninst✝ : P.RespectsIso\nH : ∀ {X Y : C} (f : X ⟶ Y) (𝒰 : K.ZeroHypercover Y), P f ↔ ∀ (i : 𝒰.I₀), P (pullback.snd f (𝒰.f i))\nX Y : C\nf : X ⟶ Y\nR : Presieve Y\nhR : R ∈ K.coverings Y\n⊢ P f ↔ ∀ {U : C} (... | refine mk_of_iff fun X Y f R hR ↦ ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.CategoryTheory.MorphismProperty.Local | {
"line": 253,
"column": 2
} | {
"line": 253,
"column": 20
} | {
"line": 253,
"column": 21
} | [
{
"pp": "C : Type u\ninst✝⁴ : Category.{v, u} C\ninst✝³ : HasEqualizers C\ninst✝² : HasPullbacks C\nX Y S : C\nf g : X ⟶ Y\ns : X ⟶ S\nt : Y ⟶ S\nhf : f ≫ t = s\nhg : g ≫ t = s\nJ : Precoverage C\n𝒰 : J.ZeroHypercover S\ninst✝¹ : J.IsStableUnderBaseChange\ninst✝ : (MorphismProperty.isomorphisms C).IsLocalAtTar... | [
"C : Type u\ninst✝⁴ : Category.{v, u} C\ninst✝³ : HasEqualizers C\ninst✝² : HasPullbacks C\nX Y S : C\nf g : X ⟶ Y\ns : X ⟶ S\nt : Y ⟶ S\nhf : f ≫ t = s\nhg : g ≫ t = s\nJ : Precoverage C\n𝒰 : J.ZeroHypercover S\ninst✝¹ : J.IsStableUnderBaseChange\ninst✝ : (MorphismProperty.isomorphisms C).IsLocalAtTarget J\nH :\n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.GlueData | {
"line": 110,
"column": 2
} | {
"line": 110,
"column": 13
} | {
"line": 110,
"column": 14
} | [
{
"pp": "C : Type u₁\ninst✝ : Category.{v, u₁} C\nD : GlueData C\ni j : D.J\neq :\n (pullbackSymmetry (D.f i i) (D.f i j)).hom = pullback.snd (D.f i i) (D.f i j) ≫ inv (pullback.fst (D.f i j) (D.f i i))\nthis :\n D.t i j ≫ D.t j i =\n (inv (pullback.fst (D.f i j) (D.f i i)) ≫ 𝟙 (pullback (D.f i j) (D.f i ... | [
"C : Type u₁\ninst✝ : Category.{v, u₁} C\nD : GlueData C\ni j : D.J\neq :\n (pullbackSymmetry (D.f i i) (D.f i j)).hom = pullback.snd (D.f i i) (D.f i j) ≫ inv (pullback.fst (D.f i j) (D.f i i))\nthis :\n D.t i j ≫ D.t j i =\n (inv (pullback.fst (D.f i j) (D.f i i)) ≫ 𝟙 (pullback (D.f i j) (D.f i i))) ≫ pullb... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.GlueData | {
"line": 121,
"column": 49
} | {
"line": 121,
"column": 60
} | {
"line": 121,
"column": 61
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v, u₁} C\nC' : Type u₂\ninst✝ : Category.{v, u₂} C'\nD : GlueData C\ni j k : D.J\n⊢ (D.t' j k i ≫ D.t' k i j) ≫ D.t' i j k = 𝟙 (pullback (D.f j k) (D.f j i))",
"ppTerm": "?m.37",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Cate... | [
"C : Type u₁\ninst✝¹ : Category.{v, u₁} C\nC' : Type u₂\ninst✝ : Category.{v, u₂} C'\nD : GlueData C\ni j k : D.J\n⊢ D.t' j k i ≫ D.t' k i j ≫ D.t' i j k = 𝟙 (pullback (D.f j k) (D.f j i))"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.GlueData | {
"line": 228,
"column": 20
} | {
"line": 228,
"column": 49
} | {
"line": 228,
"column": 50
} | [
{
"pp": "C : Type u₁\ninst✝² : Category.{v, u₁} C\nC' : Type u₂\ninst✝¹ : Category.{v, u₂} C'\nD : GlueData C\nF : C ⥤ C'\ninst✝ : ∀ (i j k : D.J), PreservesLimit (cospan (D.f i j) (D.f i k)) F\ni j k : D.J\n⊢ ((PreservesPullback.iso F (D.f i j) (D.f i k)).inv ≫\n F.map (D.t' i j k) ≫ (PreservesPullback.... | [
"C : Type u₁\ninst✝² : Category.{v, u₁} C\nC' : Type u₂\ninst✝¹ : Category.{v, u₂} C'\nD : GlueData C\nF : C ⥤ C'\ninst✝ : ∀ (i j k : D.J), PreservesLimit (cospan (D.f i j) (D.f i k)) F\ni j k : D.J\n⊢ F.map (D.t' i j k) ≫ F.map (pullback.snd (D.f j k) (D.f j i)) =\n F.map (pullback.fst (D.f i j) (D.f i k)) ≫ F.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.AffineScheme | {
"line": 783,
"column": 2
} | {
"line": 789,
"column": 6
} | {
"line": 791,
"column": 0
} | [
{
"pp": "X Y : Scheme\nf : X ⟶ Y\nx : ↥X\nU : Y.Opens\nhU : IsAffineOpen U\nV : X.Opens\nhV : IsAffineOpen V\nhVU : V ≤ f ⁻¹ᵁ U\nhx : x ∈ V\n⊢ PrimeSpectrum.comap (CommRingCat.Hom.hom (Scheme.Hom.appLE f U V hVU)) (hV.primeIdealOf ⟨x, hx⟩) =\n hU.primeIdealOf ⟨f x, ⋯⟩",
"ppTerm": "?m.53",
"assigned":... | [] | change Spec.map (f.appLE U V hVU) (hV.primeIdealOf ⟨x, hx⟩) = (hU.primeIdealOf ⟨f x, hVU hx⟩)
simp only [IsAffineOpen.primeIdealOf, ← Scheme.Hom.comp_apply, IsAffineOpen.isoSpec_hom,
Scheme.Opens.toSpecΓ_SpecMap_appLE]
simp only [Scheme.Hom.comp_apply]
congr 1
apply Subtype.ext
simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.AffineScheme | {
"line": 783,
"column": 2
} | {
"line": 789,
"column": 6
} | {
"line": 791,
"column": 0
} | [
{
"pp": "X Y : Scheme\nf : X ⟶ Y\nx : ↥X\nU : Y.Opens\nhU : IsAffineOpen U\nV : X.Opens\nhV : IsAffineOpen V\nhVU : V ≤ f ⁻¹ᵁ U\nhx : x ∈ V\n⊢ PrimeSpectrum.comap (CommRingCat.Hom.hom (Scheme.Hom.appLE f U V hVU)) (hV.primeIdealOf ⟨x, hx⟩) =\n hU.primeIdealOf ⟨f x, ⋯⟩",
"ppTerm": "?m.53",
"assigned":... | [] | change Spec.map (f.appLE U V hVU) (hV.primeIdealOf ⟨x, hx⟩) = (hU.primeIdealOf ⟨f x, hVU hx⟩)
simp only [IsAffineOpen.primeIdealOf, ← Scheme.Hom.comp_apply, IsAffineOpen.isoSpec_hom,
Scheme.Opens.toSpecΓ_SpecMap_appLE]
simp only [Scheme.Hom.comp_apply]
congr 1
apply Subtype.ext
simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.AffineScheme | {
"line": 795,
"column": 4
} | {
"line": 796,
"column": 11
} | {
"line": 796,
"column": 12
} | [
{
"pp": "X : Scheme\nU : X.Opens\nhU : IsAffineOpen U\nx : ↥U\nhx : IsClosed {↑x}\n⊢ IsClosed {x}",
"ppTerm": "?m.31",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"X : Scheme\nU : X.Opens\nhU : IsAffineOpen U\nx : ↥U\nhx : IsClosed {↑x}\n⊢ IsClosed {x}"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.AffineScheme | {
"line": 849,
"column": 2
} | {
"line": 849,
"column": 12
} | {
"line": 850,
"column": 2
} | [
{
"pp": "X : Scheme\nU : X.Opens\nhU : IsAffineOpen U\ns : ↑Γ(X, U)\nI : Ideal ↑Γ(X, U)\nH :\n ∀ (x : ↥X) (h : x ∈ U),\n (ConcreteCategory.hom (X.presheaf.germ U x h)) s ∈ Ideal.map (CommRingCat.Hom.hom (X.presheaf.germ U x h)) I\nthis✝ : (x : ↥(Spec Γ(X, U))) → Algebra ↑Γ(X, U) ↑(X.presheaf.stalk (hU.fromS... | [
"X : Scheme\nU : X.Opens\nhU : IsAffineOpen U\ns : ↑Γ(X, U)\nI : Ideal ↑Γ(X, U)\nH :\n ∀ (x : ↥X) (h : x ∈ U),\n (ConcreteCategory.hom (X.presheaf.germ U x h)) s ∈ Ideal.map (CommRingCat.Hom.hom (X.presheaf.germ U x h)) I\nthis✝ : (x : ↥(Spec Γ(X, U))) → Algebra ↑Γ(X, U) ↑(X.presheaf.stalk (hU.fromSpec x)) :=\n... | intro P hP | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
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