module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.Data.Finsupp.Order | {
"line": 106,
"column": 2
} | {
"line": 106,
"column": 74
} | {
"line": 108,
"column": 0
} | [
{
"pp": "case mpr\nι : Type u_1\nκ : Type u_2\nα : Type u_3\ninst✝² : Zero α\ninst✝¹ : LE α\ninst✝ : Std.Refl fun x1 x2 ↦ x1 ≤ x2\nf : ι ↪ κ\ng₁ g₂ : ι →₀ α\nh : g₁ ≤ g₂\n⊢ embDomain f g₁ ≤ embDomain f g₂",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"Finsupp.embDomain_apply",
... | [] | simp [Finsupp.le_def, embDomain_apply, apply_dite₂, Finsupp.le_def.mp h] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Data.Finsupp.Order | {
"line": 126,
"column": 2
} | {
"line": 126,
"column": 46
} | {
"line": 126,
"column": 47
} | [
{
"pp": "ι : Type u_1\nκ : Type u_2\nα : Type u_3\ninst✝² : AddCommMonoid α\ninst✝¹ : LE α\ninst✝ : Std.Refl fun x1 x2 ↦ x1 ≤ x2\nf : ι → κ\nh : Function.Injective f\ng₁ g₂ : ι →₀ α\n⊢ mapDomain f g₁ ≤ mapDomain f g₂ ↔ g₁ ≤ g₂",
"ppTerm": "?m.21",
"assigned": false,
"usedConstants": [],
"usedFVa... | [
"ι : Type u_1\nκ : Type u_2\nα : Type u_3\ninst✝² : AddCommMonoid α\ninst✝¹ : LE α\ninst✝ : Std.Refl fun x1 x2 ↦ x1 ≤ x2\nf : ι → κ\nh : Function.Injective f\ng₁ g₂ : ι →₀ α\n⊢ mapDomain f g₁ ≤ mapDomain f g₂ ↔ g₁ ≤ g₂"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Finsupp.Order | {
"line": 130,
"column": 2
} | {
"line": 130,
"column": 46
} | {
"line": 130,
"column": 47
} | [
{
"pp": "ι : Type u_1\nκ : Type u_2\nα : Type u_3\ninst✝¹ : AddCommMonoid α\ninst✝ : Preorder α\nf : ι → κ\nh : Function.Injective f\ng₁ g₂ : ι →₀ α\n⊢ mapDomain f g₁ < mapDomain f g₂ ↔ g₁ < g₂",
"ppTerm": "?m.17",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
... | [
"ι : Type u_1\nκ : Type u_2\nα : Type u_3\ninst✝¹ : AddCommMonoid α\ninst✝ : Preorder α\nf : ι → κ\nh : Function.Injective f\ng₁ g₂ : ι →₀ α\n⊢ mapDomain f g₁ < mapDomain f g₂ ↔ g₁ < g₂"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Polynomial.AlgebraMap | {
"line": 350,
"column": 14
} | {
"line": 350,
"column": 25
} | {
"line": 350,
"column": 26
} | [
{
"pp": "R : Type u\ninst✝ : CommSemiring R\np q p' q' : R[X]\nhpq : p.comp q = X\nhqp : q.comp p = X\nhpq' : p'.comp q' = X\nhqp' : q'.comp p' = X\nh : p.algEquivOfCompEqX q hpq hqp = p'.algEquivOfCompEqX q' hpq' hqp'\n⊢ p = p'",
"ppTerm": "?m.40",
"assigned": false,
"usedConstants": [],
"usedF... | [
"R : Type u\ninst✝ : CommSemiring R\np q p' q' : R[X]\nhpq : p.comp q = X\nhqp : q.comp p = X\nhpq' : p'.comp q' = X\nhqp' : q'.comp p' = X\nh : p.algEquivOfCompEqX q hpq hqp = p'.algEquivOfCompEqX q' hpq' hqp'\n⊢ p = p'"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Polynomial.AlgebraMap | {
"line": 350,
"column": 47
} | {
"line": 350,
"column": 64
} | {
"line": 350,
"column": 64
} | [
{
"pp": "R : Type u\ninst✝ : CommSemiring R\np q p' q' : R[X]\nhpq : p.comp q = X\nhqp : q.comp p = X\nhpq' : p'.comp q' = X\nhqp' : q'.comp p' = X\nh : p = p'\n⊢ p.algEquivOfCompEqX q hpq hqp = p'.algEquivOfCompEqX q' hpq' hqp'",
"ppTerm": "?m.43",
"assigned": true,
"usedConstants": [
"congrA... | [] | by ext1; simp [h] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.Finsupp.Order | {
"line": 147,
"column": 62
} | {
"line": 147,
"column": 73
} | {
"line": 147,
"column": 74
} | [
{
"pp": "ι : Type u_1\nκ : Type u_2\nα : Type u_3\ninst✝² : AddCommMonoid α\ninst✝¹ : Preorder α\ninst✝ : IsOrderedAddMonoid α\nf : ι → κ\ng : ι →₀ α\nhg : 0 ≤ g\n⊢ 0 ≤ mapDomain f g",
"ppTerm": "?m.18",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"ι : Type u_1\nκ : Type u_2\nα : Type u_3\ninst✝² : AddCommMonoid α\ninst✝¹ : Preorder α\ninst✝ : IsOrderedAddMonoid α\nf : ι → κ\ng : ι →₀ α\nhg : 0 ≤ g\n⊢ 0 ≤ mapDomain f g"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Finsupp.Order | {
"line": 148,
"column": 62
} | {
"line": 148,
"column": 73
} | {
"line": 148,
"column": 74
} | [
{
"pp": "ι : Type u_1\nκ : Type u_2\nα : Type u_3\ninst✝² : AddCommMonoid α\ninst✝¹ : Preorder α\ninst✝ : IsOrderedAddMonoid α\nf : ι → κ\ng : ι →₀ α\nhg : g ≤ 0\n⊢ mapDomain f g ≤ 0",
"ppTerm": "?m.18",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"ι : Type u_1\nκ : Type u_2\nα : Type u_3\ninst✝² : AddCommMonoid α\ninst✝¹ : Preorder α\ninst✝ : IsOrderedAddMonoid α\nf : ι → κ\ng : ι →₀ α\nhg : g ≤ 0\n⊢ mapDomain f g ≤ 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Finsupp.Order | {
"line": 159,
"column": 4
} | {
"line": 159,
"column": 40
} | {
"line": 159,
"column": 41
} | [
{
"pp": "case inr\nα : Type u_5\nM : Type u_6\nN : Type u_7\ninst✝³ : Zero M\ninst✝² : AddCommMonoid N\ninst✝¹ : PartialOrder N\ninst✝ : IsOrderedAddMonoid N\nf : α →₀ M\ng : α → M → N\nh : 0 ≤ fun x1 x2 ↦ g x1 x2\na : α\nH : f a ≠ 0\n⊢ a ∈ f.support",
"ppTerm": "?inr",
"assigned": true,
"usedConsta... | [
"case inr\nα : Type u_5\nM : Type u_6\nN : Type u_7\ninst✝³ : Zero M\ninst✝² : AddCommMonoid N\ninst✝¹ : PartialOrder N\ninst✝ : IsOrderedAddMonoid N\nf : α →₀ M\ng : α → M → N\nh : 0 ≤ fun x1 x2 ↦ g x1 x2\na : α\nH : f a ≠ 0\n⊢ ¬f a = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Polynomial.AlgebraMap | {
"line": 689,
"column": 2
} | {
"line": 689,
"column": 13
} | {
"line": 689,
"column": 14
} | [
{
"pp": "R : Type u\ninst✝ : CommRing R\np q : R[X]\nx✝¹ : Invertible 1 := invertibleOne\nx✝ : Invertible (-1) := invertibleNeg 1\n⊢ p ∣ q.comp (-X) ↔ p.comp (-X) ∣ q",
"ppTerm": "?m.32",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u\ninst✝ : CommRing R\np q : R[X]\nx✝¹ : Invertible 1 := invertibleOne\nx✝ : Invertible (-1) := invertibleNeg 1\n⊢ p ∣ q.comp (-X) ↔ p.comp (-X) ∣ q"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Polynomial.AlgebraMap | {
"line": 708,
"column": 11
} | {
"line": 708,
"column": 22
} | {
"line": 708,
"column": 23
} | [
{
"pp": "case C\nR : Type u\nA : Type z\nM : Type u_3\ninst✝⁶ : CommSemiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\nq : Submodule R M\nm : M\ninst✝³ : Semiring A\ninst✝² : Algebra R A\ninst✝¹ : Module A M\ninst✝ : IsScalarTower R A M\nhm : m ∈ q\na✝ : A\nhq : q ≤ Submodule.comap ((Algebra.lsmul R R M... | [
"case C\nR : Type u\nA : Type z\nM : Type u_3\ninst✝⁶ : CommSemiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\nq : Submodule R M\nm : M\ninst✝³ : Semiring A\ninst✝² : Algebra R A\ninst✝¹ : Module A M\ninst✝ : IsScalarTower R A M\nhm : m ∈ q\na✝ : A\nhq : q ≤ Submodule.comap ((Algebra.lsmul R R M) a✝) q\na :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.PiLex | {
"line": 254,
"column": 2
} | {
"line": 257,
"column": 7
} | {
"line": 259,
"column": 0
} | [
{
"pp": "ι : Type u_1\nβ : ι → Type u_2\ninst✝¹ : LinearOrder ι\nx y : (i : ι) → β i\ni : ι\ninst✝ : (i : ι) → LinearOrder (β i)\nhxy : toLex x ≤ toLex y\nh : ∀ (j : ι), j < i → x j = y j\n⊢ x i ≤ y i",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
... | [] | contrapose! hxy
apply not_le_of_gt
use i
aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.PiLex | {
"line": 254,
"column": 2
} | {
"line": 257,
"column": 7
} | {
"line": 259,
"column": 0
} | [
{
"pp": "ι : Type u_1\nβ : ι → Type u_2\ninst✝¹ : LinearOrder ι\nx y : (i : ι) → β i\ni : ι\ninst✝ : (i : ι) → LinearOrder (β i)\nhxy : toLex x ≤ toLex y\nh : ∀ (j : ι), j < i → x j = y j\n⊢ x i ≤ y i",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
... | [] | contrapose! hxy
apply not_le_of_gt
use i
aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.PiLex | {
"line": 264,
"column": 2
} | {
"line": 267,
"column": 7
} | {
"line": 269,
"column": 0
} | [
{
"pp": "ι : Type u_1\nβ : ι → Type u_2\ninst✝¹ : LinearOrder ι\nx y : (i : ι) → β i\ni : ι\ninst✝ : (i : ι) → LinearOrder (β i)\nhxy : toColex x ≤ toColex y\nh : ∀ (j : ι), j > i → x j = y j\n⊢ x i ≤ y i",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Preorder.toLT"... | [] | contrapose! hxy
apply not_le_of_gt
use i
aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.PiLex | {
"line": 264,
"column": 2
} | {
"line": 267,
"column": 7
} | {
"line": 269,
"column": 0
} | [
{
"pp": "ι : Type u_1\nβ : ι → Type u_2\ninst✝¹ : LinearOrder ι\nx y : (i : ι) → β i\ni : ι\ninst✝ : (i : ι) → LinearOrder (β i)\nhxy : toColex x ≤ toColex y\nh : ∀ (j : ι), j > i → x j = y j\n⊢ x i ≤ y i",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Preorder.toLT"... | [] | contrapose! hxy
apply not_le_of_gt
use i
aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Polynomial.AlgebraMap | {
"line": 732,
"column": 4
} | {
"line": 732,
"column": 50
} | {
"line": 732,
"column": 51
} | [
{
"pp": "R : Type u\ninst✝ : CommSemiring R\nP : R[X]\nh : ∀ (r : R), r • P = 0 → r = 0\nQ : R[X]\nhQ : P * Q = 0\nthis : ∀ (i : ℕ), P.coeff i • Q = 0\n⊢ Q.leadingCoeff • P = 0",
"ppTerm": "?m.44",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
... | [
"R : Type u\ninst✝ : CommSemiring R\nP : R[X]\nh : ∀ (r : R), r • P = 0 → r = 0\nQ : R[X]\nhQ : P * Q = 0\nthis : ∀ (i : ℕ), P.coeff i • Q = 0\n⊢ ∀ (n : ℕ), P.coeff n * Q.leadingCoeff = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.PiLex | {
"line": 354,
"column": 4
} | {
"line": 354,
"column": 81
} | {
"line": 354,
"column": 82
} | [
{
"pp": "ι : Type u_1\nα : Type u_3\ninst✝² : Preorder ι\ninst✝¹ : DecidableEq ι\ninst✝ : LT α\nf : ι → α\ni j : ι\nh₁ : i ≤ j\nh₂ : f j < f i\n⊢ (fun {i} x1 x2 ↦ x1 < x2) (toLex (f ∘ ⇑(Equiv.swap i j)) i) (toLex f i)",
"ppTerm": "?m.47",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"E... | [
"ι : Type u_1\nα : Type u_3\ninst✝² : Preorder ι\ninst✝¹ : DecidableEq ι\ninst✝ : LT α\nf : ι → α\ni j : ι\nh₁ : i ≤ j\nh₂ : f j < f i\n⊢ f j < f i"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Polynomial.AlgebraMap | {
"line": 769,
"column": 2
} | {
"line": 769,
"column": 27
} | {
"line": 769,
"column": 28
} | [
{
"pp": "R : Type u\ninst✝ : CommSemiring R\nP : R[X]\n⊢ P ∈ R[X]⁰ ↔ ∀ (a : R), a • P = 0 → a = 0",
"ppTerm": "?m.19",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u\ninst✝ : CommSemiring R\nP : R[X]\n⊢ P ∈ R[X]⁰ ↔ ∀ (a : R), a • P = 0 → a = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Polynomial.AlgebraMap | {
"line": 773,
"column": 63
} | {
"line": 773,
"column": 74
} | {
"line": 773,
"column": 75
} | [
{
"pp": "R : Type u\ninst✝ : CommSemiring R\np : R[X]\nn : ℕ\nhp : p.coeff n ∈ R⁰\nr : R\nhr : r • p = 0\n⊢ r * p.coeff n = 0",
"ppTerm": "?m.25",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u\ninst✝ : CommSemiring R\np : R[X]\nn : ℕ\nhp : p.coeff n ∈ R⁰\nr : R\nhr : r • p = 0\n⊢ r * p.coeff n = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.DFinsupp.NeLocus | {
"line": 43,
"column": 2
} | {
"line": 44,
"column": 32
} | {
"line": 44,
"column": 33
} | [
{
"pp": "α : Type u_1\nN : α → Type u_2\ninst✝² : DecidableEq α\ninst✝¹ : (a : α) → DecidableEq (N a)\ninst✝ : (a : α) → Zero (N a)\nf g : Π₀ (a : α), N a\na : α\n⊢ a ∈ f.neLocus g ↔ f a ≠ g a",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instDecidableNot",
"... | [
"α : Type u_1\nN : α → Type u_2\ninst✝² : DecidableEq α\ninst✝¹ : (a : α) → DecidableEq (N a)\ninst✝ : (a : α) → Zero (N a)\nf g : Π₀ (a : α), N a\na : α\n⊢ f a ≠ g a → f a ≠ 0 ∨ g a ≠ 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.DFinsupp.NeLocus | {
"line": 84,
"column": 2
} | {
"line": 84,
"column": 61
} | {
"line": 84,
"column": 62
} | [
{
"pp": "α : Type u_1\nN : α → Type u_2\ninst✝⁴ : DecidableEq α\nM : α → Type u_3\ninst✝³ : (a : α) → Zero (N a)\ninst✝² : (a : α) → Zero (M a)\ninst✝¹ : (a : α) → DecidableEq (N a)\ninst✝ : (a : α) → DecidableEq (M a)\nf g : Π₀ (a : α), N a\nF : (a : α) → N a → M a\nF0 : ∀ (a : α), F a 0 = 0\na : α\n⊢ a ∈ (map... | [
"α : Type u_1\nN : α → Type u_2\ninst✝⁴ : DecidableEq α\nM : α → Type u_3\ninst✝³ : (a : α) → Zero (N a)\ninst✝² : (a : α) → Zero (M a)\ninst✝¹ : (a : α) → DecidableEq (N a)\ninst✝ : (a : α) → DecidableEq (M a)\nf g : Π₀ (a : α), N a\nF : (a : α) → N a → M a\nF0 : ∀ (a : α), F a 0 = 0\na : α\n⊢ f a = g a → F a (f a... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.DFinsupp.NeLocus | {
"line": 139,
"column": 2
} | {
"line": 139,
"column": 35
} | {
"line": 139,
"column": 36
} | [
{
"pp": "α : Type u_1\nN : α → Type u_2\ninst✝² : DecidableEq α\ninst✝¹ : (a : α) → DecidableEq (N a)\ninst✝ : (a : α) → AddGroup (N a)\nf₁ f₂ g : Π₀ (a : α), N a\n⊢ (f₁ - g).neLocus (f₂ - g) = f₁.neLocus f₂",
"ppTerm": "?m.40",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoi... | [
"α : Type u_1\nN : α → Type u_2\ninst✝² : DecidableEq α\ninst✝¹ : (a : α) → DecidableEq (N a)\ninst✝ : (a : α) → AddGroup (N a)\nf₁ f₂ g : Π₀ (a : α), N a\n⊢ (f₁ + -g).neLocus (f₂ + -g) = f₁.neLocus f₂"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.MvPolynomial.Basic | {
"line": 398,
"column": 4
} | {
"line": 398,
"column": 14
} | {
"line": 399,
"column": 2
} | [
{
"pp": "case refine_1\nR : Type u\nσ : Type u_1\ninst✝¹ : CommSemiring R\nA : Type u_2\ninst✝ : Semiring A\nf g : MvPolynomial σ R →+* A\nhC : ∀ (r : R), f (C r) = g (C r)\nhX : ∀ (i : σ), f (X i) = g (X i)\nx : R\n⊢ (f.comp singleZeroRingHom) x = (g.comp singleZeroRingHom) x",
"ppTerm": "?refine_1",
"... | [] | exact hC _ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Data.Finsupp.Lex | {
"line": 142,
"column": 2
} | {
"line": 143,
"column": 86
} | {
"line": 144,
"column": 2
} | [
{
"pp": "case h.left\nα : Type u_1\ninst✝ : LinearOrder α\na b : α\nh : a < b\n⊢ ∀ d < a, (single b 1) d = (single a 1) d",
"ppTerm": "?h.left",
"assigned": true,
"usedConstants": [
"Finsupp.instFunLike",
"Nat.instMulZeroClass",
"Preorder.toLT",
"congrArg",
"PartialOrde... | [
"case h.right\nα : Type u_1\ninst✝ : LinearOrder α\na b : α\nh : a < b\n⊢ (single b 1) a < (single a 1) a"
] | · intro d hd
simp only [Finsupp.single_eq_of_ne hd.ne, Finsupp.single_eq_of_ne (hd.trans h).ne] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.MvPolynomial.Basic | {
"line": 506,
"column": 2
} | {
"line": 506,
"column": 44
} | {
"line": 506,
"column": 45
} | [
{
"pp": "R : Type u\nσ : Type u_1\ninst✝¹ : CommSemiring R\nα : Type u_2\ninst✝ : DecidableEq σ\ns : Finset α\nf : α → MvPolynomial σ R\n⊢ (∑ x ∈ s, f x).support ⊆ s.biUnion fun x ↦ (f x).support",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Nat.instMulZeroClass",
... | [
"R : Type u\nσ : Type u_1\ninst✝¹ : CommSemiring R\nα : Type u_2\ninst✝ : DecidableEq σ\ns : Finset α\nf : α → MvPolynomial σ R\n⊢ (∑ i ∈ s, (f i).coeff).support ⊆ s.biUnion fun x ↦ (f x).coeff.support"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.MvPolynomial.Basic | {
"line": 534,
"column": 2
} | {
"line": 534,
"column": 36
} | {
"line": 534,
"column": 37
} | [
{
"pp": "R : Type u\nσ : Type u_1\ninst✝ : CommSemiring R\na : σ →₀ ℕ\np : MvPolynomial σ R\ns : R\nha : a ∉ p.support\nhs : s ≠ 0\n⊢ Disjoint ((monomial a) s).support p.support",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Nat.instMulZeroClass",
"MvPolynomia... | [
"R : Type u\nσ : Type u_1\ninst✝ : CommSemiring R\na : σ →₀ ℕ\np : MvPolynomial σ R\ns : R\nha : a ∉ p.support\nhs : s ≠ 0\n⊢ coeff a p = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.MvPolynomial.Basic | {
"line": 604,
"column": 33
} | {
"line": 604,
"column": 56
} | {
"line": 604,
"column": 57
} | [
{
"pp": "R : Type u\nσ : Type u_1\ninst✝¹ : CommSemiring R\nn : ℕ\ninst✝ : NeZero n\nm : σ →₀ ℕ\na : R\ni : σ\nH : m + Finsupp.single i n = 0\n⊢ False",
"ppTerm": "?m.29",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u\nσ : Type u_1\ninst✝¹ : CommSemiring R\nn : ℕ\ninst✝ : NeZero n\nm : σ →₀ ℕ\na : R\ni : σ\nH : m + Finsupp.single i n = 0\n⊢ False"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.MvPolynomial.Basic | {
"line": 678,
"column": 2
} | {
"line": 678,
"column": 33
} | {
"line": 678,
"column": 34
} | [
{
"pp": "R : Type u\nσ : Type u_1\ninst✝¹ : CommSemiring R\ninst✝ : DecidableEq σ\ns s' : σ\nn : ℕ\n⊢ coeff (Finsupp.single s' n) (X s) = if n = 1 ∧ s = s' then 1 else 0",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"... | [
"R : Type u\nσ : Type u_1\ninst✝¹ : CommSemiring R\ninst✝ : DecidableEq σ\ns s' : σ\nn : ℕ\n⊢ coeff (Finsupp.single s' n) (X s) = if s = s' ∧ n = 1 then 1 else 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.MvPolynomial.Basic | {
"line": 725,
"column": 6
} | {
"line": 726,
"column": 13
} | {
"line": 726,
"column": 14
} | [
{
"pp": "R : Type u\nσ : Type u_1\ninst✝ : CommSemiring R\nm s : σ →₀ ℕ\nr : R\np : MvPolynomial σ R\nh : m ∈ (p * (monomial s) r).support\n⊢ ∃ j ∈ p.support, j + s = m",
"ppTerm": "?m.102",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Nat.instMulZeroClass",
"congrArg",
"C... | [
"R : Type u\nσ : Type u_1\ninst✝ : CommSemiring R\nm s : σ →₀ ℕ\nr : R\np : MvPolynomial σ R\nh : m ∈ (p * (monomial s) r).support\n⊢ ∃ j, ¬coeff j p = 0 ∧ j + s = m"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.MvPolynomial.Eval | {
"line": 404,
"column": 22
} | {
"line": 404,
"column": 33
} | {
"line": 404,
"column": 34
} | [
{
"pp": "R : Type u\nS₁ : Type v\nσ : Type u_1\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\nf : R →+* S₁\nh : Injective ⇑(map f)\nr r' : R\neq : f r = f r'\n⊢ r = r'",
"ppTerm": "?m.22",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u\nS₁ : Type v\nσ : Type u_1\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\nf : R →+* S₁\nh : Injective ⇑(map f)\nr r' : R\neq : f r = f r'\n⊢ r = r'"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.MvPolynomial.Eval | {
"line": 418,
"column": 51
} | {
"line": 418,
"column": 74
} | {
"line": 418,
"column": 75
} | [
{
"pp": "R : Type u\nS₁ : Type v\nσ : Type u_1\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\nf : R →+* S₁\nh✝ : Surjective ⇑(map f)\ns : S₁\np : MvPolynomial σ R\nh : (map f) p = C s\n⊢ f (coeff 0 p) = s",
"ppTerm": "?m.36",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"used... | [
"R : Type u\nS₁ : Type v\nσ : Type u_1\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\nf : R →+* S₁\nh✝ : Surjective ⇑(map f)\ns : S₁\np : MvPolynomial σ R\nh : (map f) p = C s\n⊢ f (coeff 0 p) = s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.MvPolynomial.Basic | {
"line": 811,
"column": 4
} | {
"line": 821,
"column": 22
} | {
"line": 823,
"column": 0
} | [
{
"pp": "case mpr\nR : Type u\nσ : Type u_1\ninst✝ : CommSemiring R\nr : R\nφ : MvPolynomial σ R\nC : (σ →₀ ℕ) → R\nhc : ∀ (i : σ →₀ ℕ), coeff i φ = r * C i\n⊢ MvPolynomial.C r ∣ φ",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"Finsupp.instAddZeroClass",
"Eq.mpr",
"Nat.i... | [] | classical
let c' : (σ →₀ ℕ) → R := fun i => if i ∈ φ.support then C i else 0
let ψ : MvPolynomial σ R := ∑ i ∈ φ.support, monomial i (c' i)
use ψ
apply MvPolynomial.ext
intro i
simp only [ψ, c', coeff_C_mul, coeff_sum, coeff_monomial, Finset.sum_ite_eq']
split_ifs with hi
... | Lean.Elab.Tactic.evalClassical | Lean.Parser.Tactic.classical |
Mathlib.Algebra.MvPolynomial.Basic | {
"line": 809,
"column": 4
} | {
"line": 821,
"column": 22
} | {
"line": 823,
"column": 0
} | [
{
"pp": "case mpr\nR : Type u\nσ : Type u_1\ninst✝ : CommSemiring R\nr : R\nφ : MvPolynomial σ R\n⊢ (∀ (i : σ →₀ ℕ), r ∣ coeff i φ) → C r ∣ φ",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"Finsupp.instAddZeroClass",
"Eq.mpr",
"Nat.instMulZeroClass",
"Semigroup.toMu... | [] | intro h
choose C hc using h
classical
let c' : (σ →₀ ℕ) → R := fun i => if i ∈ φ.support then C i else 0
let ψ : MvPolynomial σ R := ∑ i ∈ φ.support, monomial i (c' i)
use ψ
apply MvPolynomial.ext
intro i
simp only [ψ, c', coeff_C_mul, coeff_sum, coeff_monomial, Finset.sum_it... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.MvPolynomial.Basic | {
"line": 809,
"column": 4
} | {
"line": 821,
"column": 22
} | {
"line": 823,
"column": 0
} | [
{
"pp": "case mpr\nR : Type u\nσ : Type u_1\ninst✝ : CommSemiring R\nr : R\nφ : MvPolynomial σ R\n⊢ (∀ (i : σ →₀ ℕ), r ∣ coeff i φ) → C r ∣ φ",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"Finsupp.instAddZeroClass",
"Eq.mpr",
"Nat.instMulZeroClass",
"Semigroup.toMu... | [] | intro h
choose C hc using h
classical
let c' : (σ →₀ ℕ) → R := fun i => if i ∈ φ.support then C i else 0
let ψ : MvPolynomial σ R := ∑ i ∈ φ.support, monomial i (c' i)
use ψ
apply MvPolynomial.ext
intro i
simp only [ψ, c', coeff_C_mul, coeff_sum, coeff_monomial, Finset.sum_it... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.MvPolynomial.Degrees | {
"line": 136,
"column": 2
} | {
"line": 136,
"column": 13
} | {
"line": 136,
"column": 14
} | [
{
"pp": "R : Type u\nσ : Type u_1\ninst✝ : CommSemiring R\np : MvPolynomial σ R\nn : ℕ\n⊢ (p ^ n).degrees ≤ n • p.degrees",
"ppTerm": "?m.21",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u\nσ : Type u_1\ninst✝ : CommSemiring R\np : MvPolynomial σ R\nn : ℕ\n⊢ (p ^ n).degrees ≤ n • p.degrees"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.MvPolynomial.Basic | {
"line": 852,
"column": 2
} | {
"line": 852,
"column": 22
} | {
"line": 852,
"column": 23
} | [
{
"pp": "R : Type u\nσ : Type u_1\ninst✝¹ : CommSemiring R\ninst✝ : Subsingleton R\np : MvPolynomial σ R\n⊢ p.coeffs = ∅",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Nat.instMulZeroClass",
"AddMonoidAlgebra.semiring",
"CommSemiring.toSemiring",
"... | [
"R : Type u\nσ : Type u_1\ninst✝¹ : CommSemiring R\ninst✝ : Subsingleton R\np : MvPolynomial σ R\n⊢ p = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.MvPolynomial.Degrees | {
"line": 171,
"column": 2
} | {
"line": 171,
"column": 24
} | {
"line": 171,
"column": 25
} | [
{
"pp": "R : Type u\nσ : Type u_1\ninst✝ : CommSemiring R\np q : MvPolynomial σ R\nh : Disjoint p.degrees q.degrees\n⊢ q.degrees ≤ (p + q).degrees",
"ppTerm": "?m.25",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u\nσ : Type u_1\ninst✝ : CommSemiring R\np q : MvPolynomial σ R\nh : Disjoint p.degrees q.degrees\n⊢ q.degrees ≤ (p + q).degrees"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.MvPolynomial.Variables | {
"line": 122,
"column": 2
} | {
"line": 122,
"column": 13
} | {
"line": 122,
"column": 14
} | [
{
"pp": "R : Type u\nσ : Type u_1\ninst✝¹ : CommSemiring R\ninst✝ : DecidableEq σ\np q : MvPolynomial σ R\nx : σ\nhx : x ∈ (p + q).degrees\n⊢ x ∈ p.degrees ∨ x ∈ q.degrees",
"ppTerm": "?m.32",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u\nσ : Type u_1\ninst✝¹ : CommSemiring R\ninst✝ : DecidableEq σ\np q : MvPolynomial σ R\nx : σ\nhx : x ∈ (p + q).degrees\n⊢ x ∈ p.degrees ∨ x ∈ q.degrees"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.MvPolynomial.Basic | {
"line": 995,
"column": 20
} | {
"line": 995,
"column": 31
} | {
"line": 995,
"column": 32
} | [
{
"pp": "R : Type u_2\nS : Type u_3\nσ : Type u_4\ninst✝² : CommSemiring R\ninst✝¹ : CommSemiring S\ninst✝ : Module R S\nM : Submodule R S\ni : σ →₀ ℕ\nx : S\nh : ∀ (i_1 : σ →₀ ℕ), (if i = i_1 then x else 0) ∈ M\n⊢ x ∈ M",
"ppTerm": "?m.35",
"assigned": false,
"usedConstants": [],
"usedFVars": [... | [
"R : Type u_2\nS : Type u_3\nσ : Type u_4\ninst✝² : CommSemiring R\ninst✝¹ : CommSemiring S\ninst✝ : Module R S\nM : Submodule R S\ni : σ →₀ ℕ\nx : S\nh : ∀ (i_1 : σ →₀ ℕ), (if i = i_1 then x else 0) ∈ M\n⊢ x ∈ M"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.MvPolynomial.Basic | {
"line": 998,
"column": 56
} | {
"line": 998,
"column": 67
} | {
"line": 998,
"column": 68
} | [
{
"pp": "R : Type u_2\nS : Type u_3\nσ : Type u_4\ninst✝² : CommSemiring R\ninst✝¹ : CommSemiring S\ninst✝ : Module R S\nM : Submodule R S\nx : S\n⊢ C x ∈ coeffsIn σ M ↔ x ∈ M",
"ppTerm": "?m.29",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u_2\nS : Type u_3\nσ : Type u_4\ninst✝² : CommSemiring R\ninst✝¹ : CommSemiring S\ninst✝ : Module R S\nM : Submodule R S\nx : S\n⊢ C x ∈ coeffsIn σ M ↔ x ∈ M"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.MvPolynomial.Basic | {
"line": 1001,
"column": 52
} | {
"line": 1001,
"column": 63
} | {
"line": 1001,
"column": 64
} | [
{
"pp": "R : Type u_2\nS : Type u_3\nσ : Type u_4\ninst✝² : CommSemiring R\ninst✝¹ : CommSemiring S\ninst✝ : Module R S\nM : Submodule R S\n⊢ 1 ∈ coeffsIn σ M ↔ 1 ∈ M",
"ppTerm": "?m.29",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u_2\nS : Type u_3\nσ : Type u_4\ninst✝² : CommSemiring R\ninst✝¹ : CommSemiring S\ninst✝ : Module R S\nM : Submodule R S\n⊢ 1 ∈ coeffsIn σ M ↔ 1 ∈ M"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.MvPolynomial.Basic | {
"line": 1009,
"column": 4
} | {
"line": 1009,
"column": 15
} | {
"line": 1009,
"column": 16
} | [
{
"pp": "case mp\nR : Type u_2\nS : Type u_3\nσ : Type u_4\ninst✝² : CommSemiring R\ninst✝¹ : CommSemiring S\ninst✝ : Module R S\nM : Submodule R S\np : MvPolynomial σ S\ni : σ →₀ ℕ\nhp : ∀ (i_1 : σ →₀ ℕ), (if i ≤ i_1 then coeff (i_1 - i) p * 1 else 0) ∈ M\nj : σ →₀ ℕ\n⊢ coeff j p ∈ M",
"ppTerm": "?mp",
... | [
"case mp\nR : Type u_2\nS : Type u_3\nσ : Type u_4\ninst✝² : CommSemiring R\ninst✝¹ : CommSemiring S\ninst✝ : Module R S\nM : Submodule R S\np : MvPolynomial σ S\ni : σ →₀ ℕ\nhp : ∀ (i_1 : σ →₀ ℕ), (if i ≤ i_1 then coeff (i_1 - i) p * 1 else 0) ∈ M\nj : σ →₀ ℕ\n⊢ coeff j p ∈ M"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.MvPolynomial.Degrees | {
"line": 322,
"column": 2
} | {
"line": 322,
"column": 13
} | {
"line": 322,
"column": 14
} | [
{
"pp": "R : Type u\nσ : Type u_1\ninst✝ : CommSemiring R\ni : σ\np : MvPolynomial σ R\nn : ℕ\n⊢ degreeOf i (p ^ n) ≤ n * degreeOf i p",
"ppTerm": "?m.21",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u\nσ : Type u_1\ninst✝ : CommSemiring R\ni : σ\np : MvPolynomial σ R\nn : ℕ\n⊢ degreeOf i (p ^ n) ≤ n * degreeOf i p"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.MvPolynomial.Basic | {
"line": 1008,
"column": 2
} | {
"line": 1009,
"column": 26
} | {
"line": 1010,
"column": 2
} | [
{
"pp": "case mp\nR : Type u_2\nS : Type u_3\nσ : Type u_4\ninst✝² : CommSemiring R\ninst✝¹ : CommSemiring S\ninst✝ : Module R S\nM : Submodule R S\np : MvPolynomial σ S\ni : σ →₀ ℕ\n⊢ (∀ (i_1 : σ →₀ ℕ), (if i ≤ i_1 then coeff (i_1 - i) p * 1 else 0) ∈ M) → ∀ (i : σ →₀ ℕ), coeff i p ∈ M",
"ppTerm": "?mp",
... | [
"case mpr\nR : Type u_2\nS : Type u_3\nσ : Type u_4\ninst✝² : CommSemiring R\ninst✝¹ : CommSemiring S\ninst✝ : Module R S\nM : Submodule R S\np : MvPolynomial σ S\ni : σ →₀ ℕ\n⊢ (∀ (i : σ →₀ ℕ), coeff i p ∈ M) → ∀ (i_1 : σ →₀ ℕ), (if i ≤ i_1 then coeff (i_1 - i) p * 1 else 0) ∈ M"
] | · rintro hp j
simpa using hp (j + i) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.MvPolynomial.Degrees | {
"line": 362,
"column": 2
} | {
"line": 362,
"column": 13
} | {
"line": 362,
"column": 14
} | [
{
"pp": "case h\nR : Type u\nσ : Type u_1\ninst✝ : CommSemiring R\nj : σ\nf : MvPolynomial σ R\nh : f ≠ 0\nthis : (f.support.sup fun m ↦ m j) + 1 = f.support.sup fun m ↦ m j + 1\nx : σ →₀ ℕ\nhx : x ∈ f.support\n⊢ x ∈ f.support ∧ x j + 1 ≤ ((fun m ↦ m j) ∘ ⇑(addRightEmbedding (Finsupp.single j 1))) x",
"ppTe... | [
"case h\nR : Type u\nσ : Type u_1\ninst✝ : CommSemiring R\nj : σ\nf : MvPolynomial σ R\nh : f ≠ 0\nthis : (f.support.sup fun m ↦ m j) + 1 = f.support.sup fun m ↦ m j + 1\nx : σ →₀ ℕ\nhx : x ∈ f.support\n⊢ ¬coeff x f = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.MvPolynomial.Basic | {
"line": 1046,
"column": 4
} | {
"line": 1046,
"column": 15
} | {
"line": 1046,
"column": 16
} | [
{
"pp": "R : Type u_2\nS : Type u_3\nσ : Type u_4\ninst✝² : CommSemiring R\ninst✝¹ : CommSemiring S\ninst✝ : Module R S\nM : Submodule R S\np : MvPolynomial σ S\nh : ↑p.support ⊆ (fun m ↦ coeff m p) ⁻¹' ↑M\ni : σ →₀ ℕ\nhp : i ∉ p.support\n⊢ coeff i p = 0",
"ppTerm": "?m.115",
"assigned": false,
"use... | [
"R : Type u_2\nS : Type u_3\nσ : Type u_4\ninst✝² : CommSemiring R\ninst✝¹ : CommSemiring S\ninst✝ : Module R S\nM : Submodule R S\np : MvPolynomial σ S\nh : ↑p.support ⊆ (fun m ↦ coeff m p) ⁻¹' ↑M\ni : σ →₀ ℕ\nhp : i ∉ p.support\n⊢ coeff i p = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.MvPolynomial.Basic | {
"line": 1063,
"column": 6
} | {
"line": 1063,
"column": 27
} | {
"line": 1063,
"column": 28
} | [
{
"pp": "case refine_1.add\nR : Type u_2\nS : Type u_3\nσ : Type u_4\ninst✝² : CommSemiring R\ninst✝¹ : CommSemiring S\ninst✝ : Algebra R S\nM N : Submodule R S\nr : S\ns : σ →₀ ℕ\nx : S\nhx✝ : x ∈ M * N\ny : S\nhy✝ : y ∈ M * N\nhx : (monomial s) x ∈ coeffsIn σ M * coeffsIn σ N\nhy : (monomial s) y ∈ coeffsIn σ... | [
"case refine_1.add\nR : Type u_2\nS : Type u_3\nσ : Type u_4\ninst✝² : CommSemiring R\ninst✝¹ : CommSemiring S\ninst✝ : Algebra R S\nM N : Submodule R S\nr : S\ns : σ →₀ ℕ\nx : S\nhx✝ : x ∈ M * N\ny : S\nhy✝ : y ∈ M * N\nhx : (monomial s) x ∈ coeffsIn σ M * coeffsIn σ N\nhy : (monomial s) y ∈ coeffsIn σ M * coeffsI... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.MvPolynomial.Basic | {
"line": 1074,
"column": 12
} | {
"line": 1074,
"column": 23
} | {
"line": 1074,
"column": 24
} | [
{
"pp": "R : Type u_2\nS : Type u_3\nσ : Type u_4\ninst✝² : CommSemiring R\ninst✝¹ : CommSemiring S\ninst✝ : Algebra R S\nM : Submodule R S\n⊢ coeffsIn σ M ^ 0 ≤ coeffsIn σ (M ^ 0)",
"ppTerm": "?m.53",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOn... | [
"R : Type u_2\nS : Type u_3\nσ : Type u_4\ninst✝² : CommSemiring R\ninst✝¹ : CommSemiring S\ninst✝ : Algebra R S\nM : Submodule R S\n⊢ ∃ y, (algebraMap R S) y = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.MvPolynomial.Eval | {
"line": 818,
"column": 4
} | {
"line": 818,
"column": 15
} | {
"line": 818,
"column": 16
} | [
{
"pp": "case C\nR : Type u\nσ : Type u_1\ninst✝³ : CommSemiring R\nS : Type u_2\nsubS : Type u_3\ninst✝² : CommSemiring S\ninst✝¹ : SetLike subS S\ninst✝ : SubsemiringClass subS S\nf : R →+* S\ns : subS\nv : σ → S\nhv : ∀ (i : σ), v i ∈ s\na : R\nhs : ∀ (i : σ →₀ ℕ), f (coeff i (C a)) ∈ s\n⊢ eval₂ f v (C a) ∈ ... | [
"case C\nR : Type u\nσ : Type u_1\ninst✝³ : CommSemiring R\nS : Type u_2\nsubS : Type u_3\ninst✝² : CommSemiring S\ninst✝¹ : SetLike subS S\ninst✝ : SubsemiringClass subS S\nf : R →+* S\ns : subS\nv : σ → S\nhv : ∀ (i : σ), v i ∈ s\na : R\nhs : ∀ (i : σ →₀ ℕ), f (coeff i (C a)) ∈ s\n⊢ f a ∈ s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.MvPolynomial.CommRing | {
"line": 103,
"column": 2
} | {
"line": 103,
"column": 49
} | {
"line": 103,
"column": 50
} | [
{
"pp": "R : Type u\nσ : Type u_1\ninst✝ : CommRing R\ni : σ\np q : MvPolynomial σ R\n⊢ degreeOf i (p - q) ≤ max (degreeOf i p) (degreeOf i q)",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Nat.instMulZeroClass",
"AddGroupWithOne.toAddGroup",
"congrArg",... | [
"R : Type u\nσ : Type u_1\ninst✝ : CommRing R\ni : σ\np q : MvPolynomial σ R\n⊢ degreeOf i (p + -q) ≤ max (degreeOf i p) (degreeOf i q)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.MvPolynomial.Degrees | {
"line": 398,
"column": 2
} | {
"line": 398,
"column": 22
} | {
"line": 400,
"column": 0
} | [
{
"pp": "case a\nR : Type u\nσ : Type u_1\ninst✝ : CommSemiring R\np q : MvPolynomial σ R\ni : σ\nh : degreeOf i q < degreeOf i p\nthis✝ : p.support.Nonempty\ns : σ →₀ ℕ\nhs2 : degreeOf i p = s i\nhs1 : ¬coeff s p = 0\nthis : coeff s q = 0\n⊢ ¬coeff s p + coeff s q = 0",
"ppTerm": "?a✝",
"assigned": tru... | [] | rwa [this, add_zero] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.Algebra.MvPolynomial.Eval | {
"line": 822,
"column": 6
} | {
"line": 822,
"column": 56
} | {
"line": 822,
"column": 57
} | [
{
"pp": "case monomial_add.refine_1\nR : Type u\nσ : Type u_1\ninst✝³ : CommSemiring R\nS : Type u_2\nsubS : Type u_3\ninst✝² : CommSemiring S\ninst✝¹ : SetLike subS S\ninst✝ : SubsemiringClass subS S\nf✝ : R →+* S\ns : subS\nv : σ → S\nhv : ∀ (i : σ), v i ∈ s\na : σ →₀ ℕ\nb : R\nf : MvPolynomial σ R\nha : a ∉ ... | [
"case monomial_add.refine_1\nR : Type u\nσ : Type u_1\ninst✝³ : CommSemiring R\nS : Type u_2\nsubS : Type u_3\ninst✝² : CommSemiring S\ninst✝¹ : SetLike subS S\ninst✝ : SubsemiringClass subS S\nf✝ : R →+* S\ns : subS\nv : σ → S\nhv : ∀ (i : σ), v i ∈ s\na : σ →₀ ℕ\nb : R\nf : MvPolynomial σ R\nha : a ∉ f.coeff.supp... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Finsupp.Fin | {
"line": 100,
"column": 2
} | {
"line": 100,
"column": 39
} | {
"line": 100,
"column": 40
} | [
{
"pp": "n : ℕ\nM : Type u_1\ninst✝ : Zero M\ny : M\ns : Fin n →₀ M\ni : Fin n\nhi : i.succ ∈ (cons y s).support\n⊢ ¬s i = 0 ∧ i.succ = i.succ",
"ppTerm": "?m.67",
"assigned": true,
"usedConstants": [
"Finsupp.instFunLike",
"Eq.mpr",
"and_true",
"Fin.succ",
"congrArg",
... | [
"n : ℕ\nM : Type u_1\ninst✝ : Zero M\ny : M\ns : Fin n →₀ M\ni : Fin n\nhi : i.succ ∈ (cons y s).support\n⊢ ¬s i = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.MvPolynomial.Degrees | {
"line": 627,
"column": 4
} | {
"line": 627,
"column": 27
} | {
"line": 627,
"column": 28
} | [
{
"pp": "R : Type u\nS : Type v\nσ : Type u_1\nτ : Type u_2\nr : R\ne : ℕ\nn m : σ\ns✝ : σ →₀ ℕ\ninst✝ : CommSemiring R\np q : MvPolynomial σ R\ns t : Multiset σ\nc : R\nx : MvPolynomial σ R\nhx : x ∈ {p | p.degrees ≤ s}\n⊢ ((algebraMap R (MvPolynomial σ R)) c).degrees + x.degrees ≤ s",
"ppTerm": "?m.71",
... | [
"R : Type u\nS : Type v\nσ : Type u_1\nτ : Type u_2\nr : R\ne : ℕ\nn m : σ\ns✝ : σ →₀ ℕ\ninst✝ : CommSemiring R\np q : MvPolynomial σ R\ns t : Multiset σ\nc : R\nx : MvPolynomial σ R\nhx : x ∈ {p | p.degrees ≤ s}\n⊢ x.degrees ≤ s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.MvPolynomial.Degrees | {
"line": 642,
"column": 35
} | {
"line": 642,
"column": 69
} | {
"line": 642,
"column": 70
} | [
{
"pp": "R : Type u\nσ : Type u_1\ninst✝ : CommSemiring R\ns t : Multiset σ\nx : MvPolynomial σ R\nhx : x ∈ degreesLE R σ (s + t)\ni : σ →₀ ℕ\nhi : toMultiset i ≤ s + t\na : σ →₀ ℕ := Multiset.toFinsupp (toMultiset i - t)\nb : σ →₀ ℕ := Multiset.toFinsupp (toMultiset i ⊓ t)\nthis : a + b = i\n⊢ toMultiset a ≤ s... | [
"R : Type u\nσ : Type u_1\ninst✝ : CommSemiring R\ns t : Multiset σ\nx : MvPolynomial σ R\nhx : x ∈ degreesLE R σ (s + t)\ni : σ →₀ ℕ\nhi : toMultiset i ≤ s + t\na : σ →₀ ℕ := Multiset.toFinsupp (toMultiset i - t)\nb : σ →₀ ℕ := Multiset.toFinsupp (toMultiset i ⊓ t)\nthis : a + b = i\n⊢ toMultiset i ≤ s + t"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.MvPolynomial.Variables | {
"line": 318,
"column": 2
} | {
"line": 318,
"column": 37
} | {
"line": 318,
"column": 38
} | [
{
"pp": "R : Type u\nσ : Type u_1\nτ : Type u_2\ninst✝¹ : CommSemiring R\ninst✝ : DecidableEq τ\nf : σ → τ\nφ : MvPolynomial σ R\ni : τ\nhi : i ∈ ((rename f) φ).degrees\n⊢ ∃ a ∈ φ.degrees, f a = i",
"ppTerm": "?m.31",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
... | [
"R : Type u\nσ : Type u_1\nτ : Type u_2\ninst✝¹ : CommSemiring R\ninst✝ : DecidableEq τ\nf : σ → τ\nφ : MvPolynomial σ R\ni : τ\nhi : i ∈ ((rename f) φ).degrees\n⊢ ∃ a ∈ φ.degrees, f a = i"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.MvPolynomial.Variables | {
"line": 323,
"column": 2
} | {
"line": 323,
"column": 50
} | {
"line": 323,
"column": 51
} | [
{
"pp": "R : Type u\nσ : Type u_1\nτ : Type u_2\ninst✝ : CommSemiring R\nf : σ → τ\nφ : MvPolynomial σ R\nj : τ\nh : j ∈ ((rename f) φ).vars\n⊢ ∃ i ∈ φ.vars, f i = j",
"ppTerm": "?m.24",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u\nσ : Type u_1\nτ : Type u_2\ninst✝ : CommSemiring R\nf : σ → τ\nφ : MvPolynomial σ R\nj : τ\nh : j ∈ ((rename f) φ).vars\n⊢ ∃ i ∈ φ.vars, f i = j"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.MvPolynomial.Variables | {
"line": 332,
"column": 39
} | {
"line": 332,
"column": 98
} | {
"line": 332,
"column": 98
} | [
{
"pp": "R : Type u\nσ : Type u_1\ninst✝¹ : CommSemiring R\nq : MvPolynomial σ R\ns : Set σ\nhs : ↑q.vars ⊆ s\ninst✝ : (i : σ) → Decidable (i ∈ s)\nu : σ →₀ ℕ\nhu : u ∈ q.support\ni : σ\nhi : i ∈ u.support\n⊢ (if i ∈ s then X i else 0) ^ u i = X i ^ u i",
"ppTerm": "?m.110",
"assigned": true,
"usedC... | [] | by simp [hs ((mem_vars_iff_mem_support _).mpr ⟨u, hu, hi⟩)] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Polynomial.Degree.TrailingDegree | {
"line": 128,
"column": 30
} | {
"line": 128,
"column": 41
} | {
"line": 128,
"column": 42
} | [
{
"pp": "R : Type u\ninst✝ : Semiring R\np : R[X]\nh : p.coeff p.natTrailingDegree = 0\nhp : ¬p = 0\n⊢ ?m.25",
"ppTerm": "?m.30",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u\ninst✝ : Semiring R\np : R[X]\nh : p.coeff p.natTrailingDegree = 0\nhp : ¬p = 0\n⊢ ?m.25"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Polynomial.Degree.Lemmas | {
"line": 96,
"column": 2
} | {
"line": 96,
"column": 13
} | {
"line": 96,
"column": 14
} | [
{
"pp": "R : Type u\ninst✝ : Semiring R\na : R\nf : R[X]\n⊢ (C a * f).natDegree ≤ f.natDegree",
"ppTerm": "?m.18",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u\ninst✝ : Semiring R\na : R\nf : R[X]\n⊢ (C a * f).natDegree ≤ f.natDegree"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Polynomial.Degree.Lemmas | {
"line": 99,
"column": 2
} | {
"line": 99,
"column": 13
} | {
"line": 99,
"column": 14
} | [
{
"pp": "R : Type u\ninst✝ : Semiring R\nf : R[X]\na : R\n⊢ (f * C a).natDegree ≤ f.natDegree",
"ppTerm": "?m.18",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u\ninst✝ : Semiring R\nf : R[X]\na : R\n⊢ (f * C a).natDegree ≤ f.natDegree"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Polynomial.Degree.TrailingDegree | {
"line": 237,
"column": 2
} | {
"line": 237,
"column": 33
} | {
"line": 237,
"column": 34
} | [
{
"pp": "R : Type u\ninst✝ : Semiring R\nn : ℕ\n⊢ ↑n ≤ (X ^ n).trailingDegree",
"ppTerm": "?m.15",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u\ninst✝ : Semiring R\nn : ℕ\n⊢ ↑n ≤ (X ^ n).trailingDegree"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Polynomial.Degree.TrailingDegree | {
"line": 281,
"column": 24
} | {
"line": 281,
"column": 41
} | {
"line": 281,
"column": 41
} | [
{
"pp": "case a\nR : Type u\ninst✝ : Semiring R\np : R[X]\nhp : p ≠ 0\nn : ℕ\nh : (p * X ^ n).coeff (p.natTrailingDegree + n) = 0\n⊢ p.coeff p.natTrailingDegree = 0",
"ppTerm": "?a✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"congrArg",
"Polynomial.coeff_mu... | [
"case a\nR : Type u\ninst✝ : Semiring R\np : R[X]\nhp : p ≠ 0\nn : ℕ\nh : (p * X ^ n).coeff (p.natTrailingDegree + n) = 0\n⊢ (p * X ^ ?a.n✝).coeff (p.natTrailingDegree + ?a.n✝) = 0",
"case a.n\nR : Type u\ninst✝ : Semiring R\np : R[X]\nhp : p ≠ 0\nn : ℕ\nh : (p * X ^ n).coeff (p.natTrailingDegree + n) = 0\n⊢ ℕ"
] | ← coeff_mul_X_pow | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.Degree.Lemmas | {
"line": 163,
"column": 4
} | {
"line": 163,
"column": 31
} | {
"line": 163,
"column": 32
} | [
{
"pp": "case inl\nR : Type u\nm n : ℕ\ninst✝ : Semiring R\np : R[X]\npn : p.natDegree ≤ n\nmno : m * n ≤ m * n\n⊢ (p ^ m).coeff (m * n) = if m * n = m * n then p.coeff n ^ m else 0",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"instDecidableTrue",... | [
"case inl\nR : Type u\nm n : ℕ\ninst✝ : Semiring R\np : R[X]\npn : p.natDegree ≤ n\nmno : m * n ≤ m * n\n⊢ (p ^ m).coeff (m * n) = p.coeff n ^ m"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Polynomial.Degree.Lemmas | {
"line": 164,
"column": 4
} | {
"line": 164,
"column": 35
} | {
"line": 164,
"column": 36
} | [
{
"pp": "case inr\nR : Type u\nm n : ℕ\ninst✝ : Semiring R\np : R[X]\no : ℕ\npn : p.natDegree ≤ n\nmno : m * n ≤ o\nh : o ≠ m * n\n⊢ (p ^ m).coeff o = if o = m * n then p.coeff n ^ m else 0",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"False",
"HMul.hMul",
... | [
"case inr\nR : Type u\nm n : ℕ\ninst✝ : Semiring R\np : R[X]\no : ℕ\npn : p.natDegree ≤ n\nmno : m * n ≤ o\nh : o ≠ m * n\n⊢ (p ^ m).coeff o = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Polynomial.Degree.Lemmas | {
"line": 196,
"column": 8
} | {
"line": 196,
"column": 38
} | {
"line": 196,
"column": 39
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝ : Semiring R\nf : S → R[X]\nx : S\ns : Finset S\nhx : x ∉ s\nIH : (s.sum f).degree = s.sup fun i ↦ (f i).degree\nh : {i | i ∈ insert x s ∧ f i ≠ 0}.Pairwise (Ne on degree ∘ f)\nhs : s.Nonempty\ny : S\nhy : y ∈ s\nhy' : (s.sup fun i ↦ (f i).degree) = (f y).degree\nhx0 : ¬f ... | [
"R : Type u\nS : Type v\ninst✝ : Semiring R\nf : S → R[X]\nx : S\ns : Finset S\nhx : x ∉ s\nIH : (s.sum f).degree = s.sup fun i ↦ (f i).degree\nh : {i | i ∈ insert x s ∧ f i ≠ 0}.Pairwise (Ne on degree ∘ f)\nhs : s.Nonempty\ny : S\nhy : y ∈ s\nhy' : (s.sup fun i ↦ (f i).degree) = (f y).degree\nhx0 : ¬f x = 0\nH : f... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Polynomial.Reverse | {
"line": 77,
"column": 14
} | {
"line": 77,
"column": 34
} | {
"line": 77,
"column": 34
} | [
{
"pp": "n o n' : ℕ\nhn : n ≤ n + n'\no' : ℕ\nho : o ≤ o + o'\n⊢ n + o + (n' + o') - (n + o) = n + n' - n + (o + o' - o)",
"ppTerm": "?m.126",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Nat.instOrderedSub",
"Nat.instIsOrderedAddMonoid",
"AddLeftCancelSemigroup.toIsLeftCa... | [
"n o n' : ℕ\nhn : n ≤ n + n'\no' : ℕ\nho : o ≤ o + o'\n⊢ n' + o' = n + n' - n + (o + o' - o)"
] | add_tsub_cancel_left | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.Reverse | {
"line": 77,
"column": 14
} | {
"line": 77,
"column": 34
} | {
"line": 77,
"column": 34
} | [
{
"pp": "n o n' : ℕ\nhn : n ≤ n + n'\no' : ℕ\nho : o ≤ o + o'\n⊢ n' + o' = n + n' - n + (o + o' - o)",
"ppTerm": "?m.135",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Nat.instOrderedSub",
"Nat.instIsOrderedAddMonoid",
"AddLeftCancelSemigroup.toIsLeftCancelAdd",
"con... | [
"n o n' : ℕ\nhn : n ≤ n + n'\no' : ℕ\nho : o ≤ o + o'\n⊢ n' + o' = n' + (o + o' - o)"
] | add_tsub_cancel_left | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.Reverse | {
"line": 77,
"column": 14
} | {
"line": 77,
"column": 34
} | {
"line": 77,
"column": 34
} | [
{
"pp": "n o n' : ℕ\nhn : n ≤ n + n'\no' : ℕ\nho : o ≤ o + o'\n⊢ n' + o' = n' + (o + o' - o)",
"ppTerm": "?m.144",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Nat.instOrderedSub",
"Nat.instIsOrderedAddMonoid",
"AddLeftCancelSemigroup.toIsLeftCancelAdd",
"congrArg",
... | [
"n o n' : ℕ\nhn : n ≤ n + n'\no' : ℕ\nho : o ≤ o + o'\n⊢ n' + o' = n' + o'"
] | add_tsub_cancel_left | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.Reverse | {
"line": 134,
"column": 2
} | {
"line": 134,
"column": 13
} | {
"line": 134,
"column": 14
} | [
{
"pp": "R : Type u_1\ninst✝ : Semiring R\n⊢ reflect 1 X = 1",
"ppTerm": "?m.12",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u_1\ninst✝ : Semiring R\n⊢ reflect 1 X = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Polynomial.EraseLead | {
"line": 155,
"column": 2
} | {
"line": 155,
"column": 13
} | {
"line": 155,
"column": 14
} | [
{
"pp": "R : Type u_1\ninst✝ : Semiring R\nr : R\n⊢ (C r * X).eraseLead = 0",
"ppTerm": "?m.18",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u_1\ninst✝ : Semiring R\nr : R\n⊢ (C r * X).eraseLead = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Polynomial.EraseLead | {
"line": 162,
"column": 4
} | {
"line": 162,
"column": 15
} | {
"line": 162,
"column": 16
} | [
{
"pp": "case pos\nR : Type u_1\ninst✝ : Semiring R\np q : R[X]\npq : q.degree < p.degree\nn : ℕ\nnd : n = p.natDegree\n⊢ 0 = (p.eraseLead + q).coeff p.natDegree",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"congrArg... | [
"case pos\nR : Type u_1\ninst✝ : Semiring R\np q : R[X]\npq : q.degree < p.degree\nn : ℕ\nnd : n = p.natDegree\n⊢ 0 = q.coeff p.natDegree"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Polynomial.EraseLead | {
"line": 176,
"column": 4
} | {
"line": 176,
"column": 15
} | {
"line": 176,
"column": 16
} | [
{
"pp": "case pos\nR : Type u_1\ninst✝ : Semiring R\np q : R[X]\npq : p.degree < q.degree\nn : ℕ\nnd : n = q.natDegree\n⊢ 0 = (p + q.eraseLead).coeff q.natDegree",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"congrArg... | [
"case pos\nR : Type u_1\ninst✝ : Semiring R\np q : R[X]\npq : p.degree < q.degree\nn : ℕ\nnd : n = q.natDegree\n⊢ 0 = p.coeff q.natDegree"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Polynomial.Degree.Lemmas | {
"line": 230,
"column": 2
} | {
"line": 232,
"column": 17
} | {
"line": 234,
"column": 0
} | [
{
"pp": "case neg\nR : Type u\nS : Type v\ninst✝ : Semiring R\nf : S → R[X]\ns : Finset S\nh : {i | i ∈ s ∧ f i ≠ 0}.Pairwise (Ne on natDegree ∘ f)\nH : ∀ x ∈ s, f x = 0\n⊢ (s.sum f).natDegree = s.sup fun i ↦ (f i).natDegree",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [] | · rw [Finset.sum_eq_zero H, natDegree_zero, eq_comm, show 0 = ⊥ from rfl, Finset.sup_eq_bot_iff]
intro x hx
simp [H x hx] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.MvPolynomial.Equiv | {
"line": 112,
"column": 6
} | {
"line": 112,
"column": 17
} | {
"line": 112,
"column": 18
} | [
{
"pp": "case add\nR : Type u\nσ : Type u_1\ninst✝¹ : CommSemiring R\ninst✝ : Unique σ\nn : ℕ\nP Q : MvPolynomial σ R\nhP : ((uniqueAlgEquiv R σ) P).coeff n = coeff (Finsupp.single default n) P\nhQ : ((uniqueAlgEquiv R σ) Q).coeff n = coeff (Finsupp.single default n) Q\n⊢ ((uniqueAlgEquiv R σ) (P + Q)).coeff n ... | [
"case add\nR : Type u\nσ : Type u_1\ninst✝¹ : CommSemiring R\ninst✝ : Unique σ\nn : ℕ\nP Q : MvPolynomial σ R\nhP : ((uniqueAlgEquiv R σ) P).coeff n = coeff (Finsupp.single default n) P\nhQ : ((uniqueAlgEquiv R σ) Q).coeff n = coeff (Finsupp.single default n) Q\n⊢ (eval₂ Polynomial.C (fun x ↦ Polynomial.X) P).coeff... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Polynomial.Monic | {
"line": 146,
"column": 2
} | {
"line": 146,
"column": 22
} | {
"line": 147,
"column": 2
} | [
{
"pp": "case neg\nR : Type u\ninst✝ : Semiring R\np : R[X]\nhp : p.Monic\nq : R[X]\nh : ¬q = 0\n⊢ p.degree + q.degree = q.degree + p.degree",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"WithBot",
"Polynomial.degree",
"WithBot.addCommSemigroup",
"add_comm",
... | [
"case neg\nR : Type u\ninst✝ : Semiring R\np : R[X]\nhp : p.Monic\nq : R[X]\nh : ¬q = 0\n⊢ p.leadingCoeff * q.leadingCoeff ≠ 0"
] | · exact add_comm _ _ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Polynomial.EraseLead | {
"line": 326,
"column": 4
} | {
"line": 326,
"column": 48
} | {
"line": 326,
"column": 49
} | [
{
"pp": "R : Type u_1\ninst✝ : Semiring R\nmotive : R[X] → Prop\nN : ℕ\nzero : motive 0\nC_mul_pow : ∀ (n : ℕ) (r : R), r ≠ 0 → n ≤ N → motive (C r * X ^ n)\nadd : ∀ (f g : R[X]), f.natDegree < g.natDegree → g.natDegree ≤ N → motive f → motive g → motive (f + g)\nf : R[X]\ndf : f.natDegree ≤ N\nhf : #f.support ... | [
"R : Type u_1\ninst✝ : Semiring R\nmotive : R[X] → Prop\nN : ℕ\nzero : motive 0\nC_mul_pow : ∀ (n : ℕ) (r : R), r ≠ 0 → n ≤ N → motive (C r * X ^ n)\nadd : ∀ (f g : R[X]), f.natDegree < g.natDegree → g.natDegree ≤ N → motive f → motive g → motive (f + g)\nf : R[X]\ndf : f.natDegree ≤ N\nhf : #f.support = 0\n⊢ f = 0... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Polynomial.Reverse | {
"line": 258,
"column": 19
} | {
"line": 258,
"column": 31
} | {
"line": 258,
"column": 32
} | [
{
"pp": "R : Type u_1\ninst✝ : Semiring R\nf : R[X]\n⊢ f.coeff ((revAt f.natDegree) ((revAt f.natDegree) f.natTrailingDegree)) = f.trailingCoeff",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Polynomial.revAt",
"congrArg",
"id",
"Polynomial.coeff",... | [
"R : Type u_1\ninst✝ : Semiring R\nf : R[X]\n⊢ f.coeff f.natTrailingDegree = f.trailingCoeff"
] | revAt_invol, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.BigOperators | {
"line": 59,
"column": 29
} | {
"line": 59,
"column": 40
} | {
"line": 59,
"column": 41
} | [
{
"pp": "S : Type u_1\ninst✝ : Semiring S\nl : Multiset S[X]\n⊢ ∀ (a : List S[X]), (Multiset.sum ⟦a⟧).natDegree ≤ foldr max 0 (Multiset.map natDegree ⟦a⟧)",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Multiset.sum",
"Multiset.foldr",
"Multiset.map",
"id",
"... | [
"S : Type u_1\ninst✝ : Semiring S\nl : Multiset S[X]\n⊢ ∀ (a : List S[X]), a.sum.natDegree ≤ List.foldr max 0 (List.map natDegree a)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Polynomial.BigOperators | {
"line": 124,
"column": 6
} | {
"line": 124,
"column": 17
} | {
"line": 124,
"column": 18
} | [
{
"pp": "S : Type u_1\ninst✝ : Semiring S\nn : ℕ\nhd : S[X]\ntl : List S[X]\nIH : (∀ p ∈ tl, p.natDegree ≤ n) → tl.prod.coeff (tl.length * n) = (List.map (fun p ↦ p.coeff n) tl).prod\nhl : ∀ p ∈ hd :: tl, p.natDegree ≤ n\nhl' : ∀ p ∈ tl, p.natDegree ≤ n\n⊢ ∀ x ∈ List.map natDegree tl, x ≤ n",
"ppTerm": "?m.... | [
"S : Type u_1\ninst✝ : Semiring S\nn : ℕ\nhd : S[X]\ntl : List S[X]\nIH : (∀ p ∈ tl, p.natDegree ≤ n) → tl.prod.coeff (tl.length * n) = (List.map (fun p ↦ p.coeff n) tl).prod\nhl : ∀ p ∈ hd :: tl, p.natDegree ≤ n\nhl' : ∀ p ∈ tl, p.natDegree ≤ n\n⊢ ∀ a ∈ tl, a.natDegree ≤ n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Polynomial.BigOperators | {
"line": 134,
"column": 29
} | {
"line": 134,
"column": 40
} | {
"line": 134,
"column": 41
} | [
{
"pp": "R : Type u\ninst✝ : CommSemiring R\nt : Multiset R[X]\n⊢ ∀ (a : List R[X]), (prod ⟦a⟧).natDegree ≤ (Multiset.map natDegree ⟦a⟧).sum",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Multiset.sum",
"Multiset.map",
"CommSemiring.toSemiring",
"Multiset.prod",
... | [
"R : Type u\ninst✝ : CommSemiring R\nt : Multiset R[X]\n⊢ ∀ (a : List R[X]), a.prod.natDegree ≤ (List.map natDegree a).sum"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Polynomial.BigOperators | {
"line": 137,
"column": 2
} | {
"line": 137,
"column": 13
} | {
"line": 137,
"column": 14
} | [
{
"pp": "R : Type u\nι : Type w\ns : Finset ι\ninst✝ : CommSemiring R\nf : ι → R[X]\n⊢ (∏ i ∈ s, f i).natDegree ≤ ∑ i ∈ s, (f i).natDegree",
"ppTerm": "?m.19",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u\nι : Type w\ns : Finset ι\ninst✝ : CommSemiring R\nf : ι → R[X]\n⊢ (∏ i ∈ s, f i).natDegree ≤ ∑ i ∈ s, (f i).natDegree"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Polynomial.BigOperators | {
"line": 143,
"column": 29
} | {
"line": 143,
"column": 40
} | {
"line": 143,
"column": 41
} | [
{
"pp": "R : Type u\ninst✝ : CommSemiring R\nt : Multiset R[X]\n⊢ ∀ (a : List R[X]), (prod ⟦a⟧).degree ≤ (Multiset.map degree ⟦a⟧).sum",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Multiset.sum",
"WithBot.instPreorder",
"WithBot",
"Multiset.map",
"CommSemir... | [
"R : Type u\ninst✝ : CommSemiring R\nt : Multiset R[X]\n⊢ ∀ (a : List R[X]), a.prod.degree ≤ (List.map degree a).sum"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.MvPolynomial.Equiv | {
"line": 594,
"column": 4
} | {
"line": 594,
"column": 79
} | {
"line": 594,
"column": 80
} | [
{
"pp": "case neg.refine_2\nR : Type u\nS₁ : Type v\ninst✝ : CommSemiring R\np : MvPolynomial (Option S₁) R\ni : ℕ\nhi : i ≤ p.totalDegree\nhpi : ¬((optionEquivLeft R S₁) p).coeff i = 0\nσ : S₁ →₀ ℕ\nhσ : σ ∈ (((optionEquivLeft R S₁) p).coeff i).support\n⊢ embDomain Embedding.some σ + Finsupp.single none i ∈ p.... | [
"case neg.refine_2\nR : Type u\nS₁ : Type v\ninst✝ : CommSemiring R\np : MvPolynomial (Option S₁) R\ni : ℕ\nhi : i ≤ p.totalDegree\nhpi : ¬((optionEquivLeft R S₁) p).coeff i = 0\nσ : S₁ →₀ ℕ\nhσ : σ ∈ (((optionEquivLeft R S₁) p).coeff i).support\n⊢ ¬coeff σ (((optionEquivLeft R S₁) p).coeff i) = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.MvPolynomial.Equiv | {
"line": 602,
"column": 2
} | {
"line": 602,
"column": 37
} | {
"line": 603,
"column": 2
} | [
{
"pp": "case neg\nR : Type u\nS₁ : Type v\ninst✝ : CommSemiring R\np : MvPolynomial (Option S₁) R\ni : ℕ\nhpi : ¬((optionEquivLeft R S₁) p).coeff i = 0\n⊢ (((optionEquivLeft R S₁) p).coeff i).totalDegree ≤ p.totalDegree",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [
"case neg\nR : Type u\nS₁ : Type v\ninst✝ : CommSemiring R\np : MvPolynomial (Option S₁) R\ni : ℕ\nhpi : ¬((optionEquivLeft R S₁) p).coeff i = 0\n⊢ ∀ b ∈ (((optionEquivLeft R S₁) p).coeff i).support, (b.sum fun x e ↦ e) ≤ p.totalDegree"
] | rw [totalDegree, Finset.sup_le_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.MvPolynomial.Equiv | {
"line": 606,
"column": 4
} | {
"line": 606,
"column": 79
} | {
"line": 606,
"column": 80
} | [
{
"pp": "case neg.refine_2\nR : Type u\nS₁ : Type v\ninst✝ : CommSemiring R\np : MvPolynomial (Option S₁) R\ni : ℕ\nhpi : ¬((optionEquivLeft R S₁) p).coeff i = 0\nσ : S₁ →₀ ℕ\nhσ : σ ∈ (((optionEquivLeft R S₁) p).coeff i).support\n⊢ embDomain Embedding.some σ + Finsupp.single none i ∈ p.support",
"ppTerm": ... | [
"case neg.refine_2\nR : Type u\nS₁ : Type v\ninst✝ : CommSemiring R\np : MvPolynomial (Option S₁) R\ni : ℕ\nhpi : ¬((optionEquivLeft R S₁) p).coeff i = 0\nσ : S₁ →₀ ℕ\nhσ : σ ∈ (((optionEquivLeft R S₁) p).coeff i).support\n⊢ ¬coeff σ (((optionEquivLeft R S₁) p).coeff i) = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Polynomial.BigOperators | {
"line": 175,
"column": 2
} | {
"line": 175,
"column": 13
} | {
"line": 175,
"column": 14
} | [
{
"pp": "R : Type u\nι : Type w\ns : Finset ι\ninst✝ : CommSemiring R\nf : ι → R[X]\nh : ∏ i ∈ s, (f i).leadingCoeff ≠ 0\n⊢ (∏ i ∈ s, f i).leadingCoeff = ∏ i ∈ s, (f i).leadingCoeff",
"ppTerm": "?m.28",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u\nι : Type w\ns : Finset ι\ninst✝ : CommSemiring R\nf : ι → R[X]\nh : ∏ i ∈ s, (f i).leadingCoeff ≠ 0\n⊢ (∏ i ∈ s, f i).leadingCoeff = ∏ i ∈ s, (f i).leadingCoeff"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Polynomial.BigOperators | {
"line": 175,
"column": 58
} | {
"line": 175,
"column": 69
} | {
"line": 175,
"column": 70
} | [
{
"pp": "R : Type u\nι : Type w\ns : Finset ι\ninst✝ : CommSemiring R\nf : ι → R[X]\nh : ∏ i ∈ s, (f i).leadingCoeff ≠ 0\n⊢ (Multiset.map leadingCoeff (Multiset.map f s.val)).prod ≠ 0",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Multiset.map",
"congrArg",
... | [
"R : Type u\nι : Type w\ns : Finset ι\ninst✝ : CommSemiring R\nf : ι → R[X]\nh : ∏ i ∈ s, (f i).leadingCoeff ≠ 0\n⊢ ¬∏ x ∈ s, (f x).leadingCoeff = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Polynomial.BigOperators | {
"line": 201,
"column": 2
} | {
"line": 201,
"column": 13
} | {
"line": 201,
"column": 14
} | [
{
"pp": "R : Type u\nι : Type w\ns : Finset ι\ninst✝ : CommSemiring R\nf : ι → R[X]\nh : ∏ i ∈ s, (f i).leadingCoeff ≠ 0\n⊢ (∏ i ∈ s, f i).natDegree = ∑ i ∈ s, (f i).natDegree",
"ppTerm": "?m.28",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u\nι : Type w\ns : Finset ι\ninst✝ : CommSemiring R\nf : ι → R[X]\nh : ∏ i ∈ s, (f i).leadingCoeff ≠ 0\n⊢ (∏ i ∈ s, f i).natDegree = ∑ i ∈ s, (f i).natDegree"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Polynomial.BigOperators | {
"line": 201,
"column": 55
} | {
"line": 201,
"column": 66
} | {
"line": 201,
"column": 67
} | [
{
"pp": "R : Type u\nι : Type w\ns : Finset ι\ninst✝ : CommSemiring R\nf : ι → R[X]\nh : ∏ i ∈ s, (f i).leadingCoeff ≠ 0\n⊢ (Multiset.map (fun f ↦ f.leadingCoeff) (Multiset.map f s.val)).prod ≠ 0",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Multiset.map",
"c... | [
"R : Type u\nι : Type w\ns : Finset ι\ninst✝ : CommSemiring R\nf : ι → R[X]\nh : ∏ i ∈ s, (f i).leadingCoeff ≠ 0\n⊢ ¬∏ x ∈ s, (f x).leadingCoeff = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Polynomial.BigOperators | {
"line": 211,
"column": 43
} | {
"line": 211,
"column": 54
} | {
"line": 211,
"column": 55
} | [
{
"pp": "R : Type u\ninst✝¹ : CommSemiring R\nt : Multiset R[X]\ninst✝ : Nontrivial R\nh : ∀ f ∈ t, f.Monic\n⊢ t.prod.Monic",
"ppTerm": "?m.30",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u\ninst✝¹ : CommSemiring R\nt : Multiset R[X]\ninst✝ : Nontrivial R\nh : ∀ f ∈ t, f.Monic\n⊢ t.prod.Monic"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Polynomial.Monic | {
"line": 425,
"column": 2
} | {
"line": 425,
"column": 42
} | {
"line": 425,
"column": 43
} | [
{
"pp": "R : Type u\ninst✝ : Ring R\nx : R\n⊢ (X - C x).Monic",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Polynomial.C",
"AddGroupWithOne.toAddGroup",
"congrArg",
"AddMonoid.toAddZeroClass",
"sub_eq_add_neg",
"HSub.hSub",
"Ring... | [
"R : Type u\ninst✝ : Ring R\nx : R\n⊢ (X + -C x).Monic"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Polynomial.Monic | {
"line": 428,
"column": 2
} | {
"line": 428,
"column": 30
} | {
"line": 428,
"column": 31
} | [
{
"pp": "R : Type u\ninst✝ : Ring R\np : R[X]\nn : ℕ\nH : p.degree < ↑n\n⊢ (X ^ n - p).Monic",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"AddGroupWithOne.toAddGroup",
"congrArg",
"AddMonoid.toAddZeroClass",
"sub_eq_add_neg",
"HSub.hSub",
... | [
"R : Type u\ninst✝ : Ring R\np : R[X]\nn : ℕ\nH : p.degree < ↑n\n⊢ (X ^ n + -p).Monic"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Polynomial.Monic | {
"line": 433,
"column": 2
} | {
"line": 433,
"column": 46
} | {
"line": 433,
"column": 47
} | [
{
"pp": "R : Type u\ninst✝ : Ring R\na : R\nn : ℕ\nh : n ≠ 0\n⊢ (X ^ n - C a).Monic",
"ppTerm": "?m.23",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u\ninst✝ : Ring R\na : R\nn : ℕ\nh : n ≠ 0\n⊢ (X ^ n - C a).Monic"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Polynomial.Monic | {
"line": 443,
"column": 89
} | {
"line": 444,
"column": 35
} | {
"line": 446,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝ : Ring R\np : R[X]\nhp : p.Monic\nr : R\n⊢ (p.comp (X - C r)).Monic",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Polynomial.C",
"NegZeroClass.toNeg",
"RingHom.instRingHomClass",
"RingHomClass.toAddMonoidHomClass",
"AddGroupWithOn... | [] | by
simpa using! hp.comp_X_add_C (-r) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Polynomial.BigOperators | {
"line": 218,
"column": 2
} | {
"line": 218,
"column": 13
} | {
"line": 218,
"column": 14
} | [
{
"pp": "R : Type u\nι : Type w\ns : Finset ι\ninst✝ : CommSemiring R\nf : ι → R[X]\nh : ∀ i ∈ s, (f i).Monic\n⊢ (∏ i ∈ s, f i).natDegree = ∑ i ∈ s, (f i).natDegree",
"ppTerm": "?m.24",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u\nι : Type w\ns : Finset ι\ninst✝ : CommSemiring R\nf : ι → R[X]\nh : ∀ i ∈ s, (f i).Monic\n⊢ (∏ i ∈ s, f i).natDegree = ∑ i ∈ s, (f i).natDegree"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Polynomial.BigOperators | {
"line": 218,
"column": 63
} | {
"line": 218,
"column": 74
} | {
"line": 218,
"column": 75
} | [
{
"pp": "R : Type u\nι : Type w\ns : Finset ι\ninst✝ : CommSemiring R\nf : ι → R[X]\nh : ∀ i ∈ s, (f i).Monic\n⊢ ∀ f_1 ∈ Multiset.map f s.val, f_1.Monic",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Multiset.map",
"CommSemiring.toSemiring",
"Finset",
... | [
"R : Type u\nι : Type w\ns : Finset ι\ninst✝ : CommSemiring R\nf : ι → R[X]\nh : ∀ i ∈ s, (f i).Monic\n⊢ ∀ a ∈ s, (f a).Monic"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Polynomial.BigOperators | {
"line": 222,
"column": 2
} | {
"line": 222,
"column": 13
} | {
"line": 222,
"column": 14
} | [
{
"pp": "R : Type u\nι : Type w\ns : Finset ι\ninst✝¹ : CommSemiring R\nf : ι → R[X]\ninst✝ : Nontrivial R\nh : ∀ i ∈ s, (f i).Monic\n⊢ (∏ i ∈ s, f i).degree = ∑ i ∈ s, (f i).degree",
"ppTerm": "?m.24",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u\nι : Type w\ns : Finset ι\ninst✝¹ : CommSemiring R\nf : ι → R[X]\ninst✝ : Nontrivial R\nh : ∀ i ∈ s, (f i).Monic\n⊢ (∏ i ∈ s, f i).degree = ∑ i ∈ s, (f i).degree"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Polynomial.BigOperators | {
"line": 222,
"column": 60
} | {
"line": 222,
"column": 71
} | {
"line": 222,
"column": 72
} | [
{
"pp": "R : Type u\nι : Type w\ns : Finset ι\ninst✝¹ : CommSemiring R\nf : ι → R[X]\ninst✝ : Nontrivial R\nh : ∀ i ∈ s, (f i).Monic\n⊢ ∀ f_1 ∈ Multiset.map f s.val, f_1.Monic",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Multiset.map",
"CommSemiring.toSemiri... | [
"R : Type u\nι : Type w\ns : Finset ι\ninst✝¹ : CommSemiring R\nf : ι → R[X]\ninst✝ : Nontrivial R\nh : ∀ i ∈ s, (f i).Monic\n⊢ ∀ a ∈ s, (f a).Monic"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Polynomial.BigOperators | {
"line": 227,
"column": 2
} | {
"line": 227,
"column": 13
} | {
"line": 227,
"column": 14
} | [
{
"pp": "case h\nR : Type u\ninst✝ : CommSemiring R\nt : Multiset R[X]\nn : ℕ\na✝ : List R[X]\nhl : ∀ p ∈ ⟦a✝⟧, p.natDegree ≤ n\n⊢ (prod ⟦a✝⟧).coeff (Multiset.card ⟦a✝⟧ * n) = (Multiset.map (fun p ↦ p.coeff n) ⟦a✝⟧).prod",
"ppTerm": "?h",
"assigned": true,
"usedConstants": [
"HMul.hMul",
... | [
"case h\nR : Type u\ninst✝ : CommSemiring R\nt : Multiset R[X]\nn : ℕ\na✝ : List R[X]\nhl : ∀ p ∈ ⟦a✝⟧, p.natDegree ≤ n\n⊢ a✝.prod.coeff (a✝.length * n) = (List.map (fun p ↦ p.coeff n) a✝).prod"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.MvPolynomial.Equiv | {
"line": 738,
"column": 4
} | {
"line": 738,
"column": 44
} | {
"line": 738,
"column": 45
} | [
{
"pp": "case mp\nR : Type u\ninst✝ : CommSemiring R\nn : ℕ\nf : MvPolynomial (Fin (n + 1)) R\ni : ℕ\nm : Fin n →₀ ℕ\nh : m ∈ (((finSuccEquiv R n) f).coeff i).support\n⊢ cons i m ∈ f.support",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Nat.instMulZeroClass",
"... | [
"case mp\nR : Type u\ninst✝ : CommSemiring R\nn : ℕ\nf : MvPolynomial (Fin (n + 1)) R\ni : ℕ\nm : Fin n →₀ ℕ\nh : m ∈ (((finSuccEquiv R n) f).coeff i).support\n⊢ ¬coeff m (((finSuccEquiv R n) f).coeff i) = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
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