module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.Order.RelSeries | {
"line": 255,
"column": 33
} | {
"line": 255,
"column": 44
} | {
"line": 255,
"column": 45
} | [
{
"pp": "α : Type u_1\nr : SetRel α α\nβ : Type u_2\ns✝ : SetRel β β\ns : RelSeries r\nx : α\np : RelSeries r\ni : ℕ\nhi : i < p.toList.length\n⊢ i < p.length + 1",
"ppTerm": "?m.21",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\nr : SetRel α α\nβ : Type u_2\ns✝ : SetRel β β\ns : RelSeries r\nx : α\np : RelSeries r\ni : ℕ\nhi : i < p.toList.length\n⊢ i < p.length + 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.RelSeries | {
"line": 264,
"column": 20
} | {
"line": 264,
"column": 31
} | {
"line": 264,
"column": 32
} | [
{
"pp": "α : Type u_1\nr : SetRel α α\nβ : Type u_2\ns✝ : SetRel β β\ns : RelSeries r\nx : α\np : RelSeries r\ni : ℕ\nhi : i < p.length + 1\n⊢ i < p.toList.length",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"id",
"RelSeries.length",
"... | [
"α : Type u_1\nr : SetRel α α\nβ : Type u_2\ns✝ : SetRel β β\ns : RelSeries r\nx : α\np : RelSeries r\ni : ℕ\nhi : i < p.length + 1\n⊢ i < p.length + 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Rel | {
"line": 463,
"column": 60
} | {
"line": 463,
"column": 71
} | {
"line": 463,
"column": 72
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nι : Sort u_5\nR✝ R₁✝ R₂✝ : SetRel α β\nS✝ : SetRel β γ\ns s₁ s₂ : Set α\nt t₁ t₂ : Set β\nu : Set γ\na✝ a₁ a₂ : α\nb✝ : β\nc✝ : γ\nR R₁ R₂ : SetRel α α\nS : SetRel β β\na b c : α\ninst✝ : R.IsSymm\n⊢ (R ○ R).IsSymm",
"ppTerm": "?m.6",
"ass... | [
"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nι : Sort u_5\nR✝ R₁✝ R₂✝ : SetRel α β\nS✝ : SetRel β γ\ns s₁ s₂ : Set α\nt t₁ t₂ : Set β\nu : Set γ\na✝ a₁ a₂ : α\nb✝ : β\nc✝ : γ\nR R₁ R₂ : SetRel α α\nS : SetRel β β\na b c : α\ninst✝ : R.IsSymm\n⊢ (R ○ R).IsSymm"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Exact.Basic | {
"line": 394,
"column": 6
} | {
"line": 394,
"column": 26
} | {
"line": 394,
"column": 27
} | [
{
"pp": "case refine_1.left\nR : Type u_1\nM : Type u_2\nM' : Type u_3\nN : Type u_4\nN' : Type u_5\nP : Type u_6\nP' : Type u_7\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup N\ninst✝³ : AddCommGroup P\ninst✝² : Module R M\ninst✝¹ : Module R N\ninst✝ : Module R P\nf : M →ₗ[R] N\ng : N →ₗ[... | [
"case refine_1.left\nR : Type u_1\nM : Type u_2\nM' : Type u_3\nN : Type u_4\nN' : Type u_5\nP : Type u_6\nP' : Type u_7\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup N\ninst✝³ : AddCommGroup P\ninst✝² : Module R M\ninst✝¹ : Module R N\ninst✝ : Module R P\nf : M →ₗ[R] N\ng : N →ₗ[R] P\nh : Ex... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Exact.Basic | {
"line": 432,
"column": 43
} | {
"line": 432,
"column": 68
} | {
"line": 432,
"column": 69
} | [
{
"pp": "R✝ : Type u_1\nM✝ : Type u_2\nM' : Type u_3\nN✝ : Type u_4\nN' : Type u_5\nP✝ : Type u_6\nP' : Type u_7\ninst✝¹³ : Semiring R✝\ninst✝¹² : AddCommGroup M✝\ninst✝¹¹ : AddCommGroup N✝\ninst✝¹⁰ : AddCommGroup P✝\ninst✝⁹ : Module R✝ M✝\ninst✝⁸ : Module R✝ N✝\ninst✝⁷ : Module R✝ P✝\nf✝ : M✝ →ₗ[R✝] N✝\ng✝ : N... | [
"R✝ : Type u_1\nM✝ : Type u_2\nM' : Type u_3\nN✝ : Type u_4\nN' : Type u_5\nP✝ : Type u_6\nP' : Type u_7\ninst✝¹³ : Semiring R✝\ninst✝¹² : AddCommGroup M✝\ninst✝¹¹ : AddCommGroup N✝\ninst✝¹⁰ : AddCommGroup P✝\ninst✝⁹ : Module R✝ M✝\ninst✝⁸ : Module R✝ N✝\ninst✝⁷ : Module R✝ P✝\nf✝ : M✝ →ₗ[R✝] N✝\ng✝ : N✝ →ₗ[R✝] P✝\... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Exact.Basic | {
"line": 433,
"column": 6
} | {
"line": 433,
"column": 76
} | {
"line": 434,
"column": 4
} | [
{
"pp": "case refine_1.left\nR✝ : Type u_1\nM✝ : Type u_2\nM' : Type u_3\nN✝ : Type u_4\nN' : Type u_5\nP✝ : Type u_6\nP' : Type u_7\ninst✝¹³ : Semiring R✝\ninst✝¹² : AddCommGroup M✝\ninst✝¹¹ : AddCommGroup N✝\ninst✝¹⁰ : AddCommGroup P✝\ninst✝⁹ : Module R✝ M✝\ninst✝⁸ : Module R✝ N✝\ninst✝⁷ : Module R✝ P✝\nf✝ : ... | [] | rw [← sub_eq_zero, ← hz, ← h₁ z, hz, map_sub, e.1, sub_self, map_zero] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Order.RelSeries | {
"line": 583,
"column": 36
} | {
"line": 583,
"column": 69
} | {
"line": 583,
"column": 70
} | [
{
"pp": "α : Type u_1\nr : SetRel α α\np : RelSeries r\nlen_pos : p.length ≠ 0\n⊢ 0 < p.length.sub 0",
"ppTerm": "?m.37",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Nat.instOrderedSub",
"congrArg",
"id",
"RelSeries.length",
"instSubNat",
"Ne",
"in... | [
"α : Type u_1\nr : SetRel α α\np : RelSeries r\nlen_pos : p.length ≠ 0\n⊢ ¬p.length = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.RelSeries | {
"line": 620,
"column": 6
} | {
"line": 623,
"column": 42
} | {
"line": 624,
"column": 4
} | [
{
"pp": "case zero\nα : Type u_1\nr : SetRel α α\nβ : Type u_2\ns : SetRel β β\nmotive : RelSeries r → Sort u_3\nsingleton : (x : α) → motive (RelSeries.singleton r x)\ncons : (p : RelSeries r) → (x : α) → (hx : (x, p.head) ∈ r) → motive p → motive (p.cons x hx)\np : RelSeries r\nheq : p.length = 0\n⊢ motive p"... | [] | convert! singleton p.head
ext n
· exact heq
simp [show n = 0 by lia, apply_zero] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.RelSeries | {
"line": 620,
"column": 6
} | {
"line": 623,
"column": 42
} | {
"line": 624,
"column": 4
} | [
{
"pp": "case zero\nα : Type u_1\nr : SetRel α α\nβ : Type u_2\ns : SetRel β β\nmotive : RelSeries r → Sort u_3\nsingleton : (x : α) → motive (RelSeries.singleton r x)\ncons : (p : RelSeries r) → (x : α) → (hx : (x, p.head) ∈ r) → motive p → motive (p.cons x hx)\np : RelSeries r\nheq : p.length = 0\n⊢ motive p"... | [] | convert! singleton p.head
ext n
· exact heq
simp [show n = 0 by lia, apply_zero] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Exact.Basic | {
"line": 477,
"column": 4
} | {
"line": 477,
"column": 15
} | {
"line": 477,
"column": 16
} | [
{
"pp": "R : Type u_8\nM : Type u_9\nN : Type u_10\nP : Type u_11\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup N\ninst✝³ : AddCommGroup P\ninst✝² : Module R M\ninst✝¹ : Module R N\ninst✝ : Module R P\nf : M →ₗ[R] N\ng : N →ₗ[R] P\nh : Exact ⇑f ⇑g\nhf : Injective ⇑f\nhg : Surjective ⇑g\n⊢... | [
"R : Type u_8\nM : Type u_9\nN : Type u_10\nP : Type u_11\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup N\ninst✝³ : AddCommGroup P\ninst✝² : Module R M\ninst✝¹ : Module R N\ninst✝ : Module R P\nf : M →ₗ[R] N\ng : N →ₗ[R] P\nh : Exact ⇑f ⇑g\nhf : Injective ⇑f\nhg : Surjective ⇑g\n⊢ (∃ l, g ∘ₗ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Exact.Basic | {
"line": 479,
"column": 4
} | {
"line": 479,
"column": 15
} | {
"line": 479,
"column": 16
} | [
{
"pp": "R : Type u_8\nM : Type u_9\nN : Type u_10\nP : Type u_11\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup N\ninst✝³ : AddCommGroup P\ninst✝² : Module R M\ninst✝¹ : Module R N\ninst✝ : Module R P\nf : M →ₗ[R] N\ng : N →ₗ[R] P\nh : Exact ⇑f ⇑g\nhf : Injective ⇑f\nhg : Surjective ⇑g\nt... | [
"R : Type u_8\nM : Type u_9\nN : Type u_10\nP : Type u_11\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup N\ninst✝³ : AddCommGroup P\ninst✝² : Module R M\ninst✝¹ : Module R N\ninst✝ : Module R P\nf : M →ₗ[R] N\ng : N →ₗ[R] P\nh : Exact ⇑f ⇑g\nhf : Injective ⇑f\nhg : Surjective ⇑g\ntfae_1_iff_3 ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.RelSeries | {
"line": 663,
"column": 4
} | {
"line": 663,
"column": 15
} | {
"line": 663,
"column": 16
} | [
{
"pp": "case hl\nα : Type u_1\nr : SetRel α α\np : RelSeries r\nhp : p.length ≠ 0\n⊢ p.eraseLast.toList.length = p.toList.dropLast.length",
"ppTerm": "?hl",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Nat.instOrderedSub",
"Nat.instIsOrderedAddMonoid",
"AddLeftCancelSemig... | [
"case hl\nα : Type u_1\nr : SetRel α α\np : RelSeries r\nhp : p.length ≠ 0\n⊢ p.length - 1 + 1 = p.length"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.RelSeries | {
"line": 711,
"column": 4
} | {
"line": 711,
"column": 79
} | {
"line": 711,
"column": 80
} | [
{
"pp": "case right\nα : Type u_1\nr : SetRel α α\nβ : Type u_2\ns : SetRel β β\np q : RelSeries r\nconnect : p.last = q.head\ni : Fin q.length\n⊢ (Fin.addCases (p.toFun ∘ Fin.castSucc) q.toFun (Fin.natAdd p.length i).castSucc,\n Fin.addCases (p.toFun ∘ Fin.castSucc) q.toFun (Fin.natAdd p.length i).succ) ∈... | [
"case right\nα : Type u_1\nr : SetRel α α\nβ : Type u_2\ns : SetRel β β\np q : RelSeries r\nconnect : p.last = q.head\ni : Fin q.length\n⊢ (q.toFun i.castSucc, q.toFun i.succ) ∈ r"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.RelSeries | {
"line": 797,
"column": 32
} | {
"line": 797,
"column": 48
} | {
"line": 797,
"column": 49
} | [
{
"pp": "α : Type u_1\nr : SetRel α α\ninst✝ : Nonempty α\nH : ∀ (x : RelSeries r), ∃ y, x.length < y.length\nn : ℕ\nl : RelSeries r\nhl : l.length = n\nl' : RelSeries r\nhl' : l.length < l'.length\n⊢ n + 1 < l'.length + 1",
"ppTerm": "?m.72",
"assigned": true,
"usedConstants": [
"IsRightCance... | [
"α : Type u_1\nr : SetRel α α\ninst✝ : Nonempty α\nH : ∀ (x : RelSeries r), ∃ y, x.length < y.length\nn : ℕ\nl : RelSeries r\nhl : l.length = n\nl' : RelSeries r\nhl' : l.length < l'.length\n⊢ n < l'.length"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.RelSeries | {
"line": 957,
"column": 27
} | {
"line": 957,
"column": 72
} | {
"line": 957,
"column": 73
} | [
{
"pp": "α : Type u_1\nr : SetRel α α\nβ : Type u_2\ns : SetRel β β\ninst✝¹ : Preorder α\ninst✝ : Preorder β\np : LTSeries β\nf : α → β\ncomap : ∀ ⦃x y : α⦄, f x < f y → x < y\nsurjective : Function.Surjective f\ni j : Fin (p.length + 1)\nh : i < j\n⊢ f ((fun i ↦ ⋯.choose) i) < f ((fun i ↦ ⋯.choose) j)",
"p... | [
"α : Type u_1\nr : SetRel α α\nβ : Type u_2\ns : SetRel β β\ninst✝¹ : Preorder α\ninst✝ : Preorder β\np : LTSeries β\nf : α → β\ncomap : ∀ ⦃x y : α⦄, f x < f y → x < y\nsurjective : Function.Surjective f\ni j : Fin (p.length + 1)\nh : i < j\n⊢ p.toFun i < p.toFun j"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Exact.Basic | {
"line": 558,
"column": 4
} | {
"line": 558,
"column": 15
} | {
"line": 558,
"column": 16
} | [
{
"pp": "case mp\nR : Type u_1\nM : Type u_2\nN : Type u_4\nP : Type u_6\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup N\ninst✝³ : AddCommGroup P\ninst✝² : Module R M\ninst✝¹ : Module R N\ninst✝ : Module R P\nf : M →ₗ[R] N\ng : N →ₗ[R] P\nh : f.range ≤ g.ker\nhfg : ∀ (x : N ⧸ f.range), (f.ran... | [
"case mp\nR : Type u_1\nM : Type u_2\nN : Type u_4\nP : Type u_6\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup N\ninst✝³ : AddCommGroup P\ninst✝² : Module R M\ninst✝¹ : Module R N\ninst✝ : Module R P\nf : M →ₗ[R] N\ng : N →ₗ[R] P\nh : f.range ≤ g.ker\nhfg : ∀ (x : N ⧸ f.range), (f.range.liftQ g h... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Exact.Basic | {
"line": 561,
"column": 4
} | {
"line": 561,
"column": 15
} | {
"line": 561,
"column": 16
} | [
{
"pp": "case mpr\nR : Type u_1\nM : Type u_2\nN : Type u_4\nP : Type u_6\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup N\ninst✝³ : AddCommGroup P\ninst✝² : Module R M\ninst✝¹ : Module R N\ninst✝ : Module R P\nf : M →ₗ[R] N\ng : N →ₗ[R] P\nh : f.range ≤ g.ker\nhfg : ∀ (x : N), g x = 0 ↔ ∃ y, ... | [
"case mpr\nR : Type u_1\nM : Type u_2\nN : Type u_4\nP : Type u_6\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup N\ninst✝³ : AddCommGroup P\ninst✝² : Module R M\ninst✝¹ : Module R N\ninst✝ : Module R P\nf : M →ₗ[R] N\ng : N →ₗ[R] P\nh : f.range ≤ g.ker\nhfg : ∀ (x : N), g x = 0 ↔ ∃ y, f y = x\nx :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Exact.Basic | {
"line": 565,
"column": 2
} | {
"line": 565,
"column": 82
} | {
"line": 565,
"column": 83
} | [
{
"pp": "R : Type u_1\nM : Type u_2\nN : Type u_4\nP : Type u_6\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup N\ninst✝³ : AddCommGroup P\ninst✝² : Module R M\ninst✝¹ : Module R N\ninst✝ : Module R P\nf : M →ₗ[R] N\ng : N →ₗ[R] P\nh : Function.Exact ⇑f ⇑g\n⊢ Function.Injective ⇑(f.range.liftQ ... | [
"R : Type u_1\nM : Type u_2\nN : Type u_4\nP : Type u_6\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup N\ninst✝³ : AddCommGroup P\ninst✝² : Module R M\ninst✝¹ : Module R N\ninst✝ : Module R P\nf : M →ₗ[R] N\ng : N →ₗ[R] P\nh : Function.Exact ⇑f ⇑g\n⊢ g.ker = f.range"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.KrullDimension | {
"line": 160,
"column": 4
} | {
"line": 160,
"column": 19
} | {
"line": 162,
"column": 0
} | [
{
"pp": "case mpr\nα : Type u_1\ninst✝ : Preorder α\na : α\nn : ℕ∞\n⊢ (∀ ⦃p : LTSeries α⦄, RelSeries.last p = a → ↑p.length ≤ n) → height a ≤ n",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"Order.height_le"
],
"usedFVars": [
"α",
"inst✝",
"a",
"n"
... | [] | exact height_le | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Order.KrullDimension | {
"line": 160,
"column": 4
} | {
"line": 160,
"column": 19
} | {
"line": 162,
"column": 0
} | [
{
"pp": "case mpr\nα : Type u_1\ninst✝ : Preorder α\na : α\nn : ℕ∞\n⊢ (∀ ⦃p : LTSeries α⦄, RelSeries.last p = a → ↑p.length ≤ n) → height a ≤ n",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"Order.height_le"
],
"usedFVars": [
"α",
"inst✝",
"a",
"n"
... | [] | exact height_le | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.KrullDimension | {
"line": 160,
"column": 4
} | {
"line": 160,
"column": 19
} | {
"line": 162,
"column": 0
} | [
{
"pp": "case mpr\nα : Type u_1\ninst✝ : Preorder α\na : α\nn : ℕ∞\n⊢ (∀ ⦃p : LTSeries α⦄, RelSeries.last p = a → ↑p.length ≤ n) → height a ≤ n",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"Order.height_le"
],
"usedFVars": [
"α",
"inst✝",
"a",
"n"
... | [] | exact height_le | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Module.Submodule.Bilinear | {
"line": 140,
"column": 4
} | {
"line": 140,
"column": 30
} | {
"line": 140,
"column": 31
} | [
{
"pp": "ι : Sort uι\nR : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\ninst✝⁶ : CommSemiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : AddCommMonoid N\ninst✝³ : AddCommMonoid P\ninst✝² : Module R M\ninst✝¹ : Module R N\ninst✝ : Module R P\nf : M →ₗ[R] N →ₗ[R] P\ns : ι → Submodule R M\nt : Submodule R N\nthis... | [
"ι : Sort uι\nR : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\ninst✝⁶ : CommSemiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : AddCommMonoid N\ninst✝³ : AddCommMonoid P\ninst✝² : Module R M\ninst✝¹ : Module R N\ninst✝ : Module R P\nf : M →ₗ[R] N →ₗ[R] P\ns : ι → Submodule R M\nt : Submodule R N\nthis : map₂ f (⨆... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Module.Submodule.Bilinear | {
"line": 146,
"column": 4
} | {
"line": 146,
"column": 30
} | {
"line": 146,
"column": 31
} | [
{
"pp": "ι : Sort uι\nR : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\ninst✝⁶ : CommSemiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : AddCommMonoid N\ninst✝³ : AddCommMonoid P\ninst✝² : Module R M\ninst✝¹ : Module R N\ninst✝ : Module R P\nf : M →ₗ[R] N →ₗ[R] P\ns : Submodule R M\nt : ι → Submodule R N\nthis... | [
"ι : Sort uι\nR : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\ninst✝⁶ : CommSemiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : AddCommMonoid N\ninst✝³ : AddCommMonoid P\ninst✝² : Module R M\ninst✝¹ : Module R N\ninst✝ : Module R P\nf : M →ₗ[R] N →ₗ[R] P\ns : Submodule R M\nt : ι → Submodule R N\nthis : map₂ f (s... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.KrullDimension | {
"line": 284,
"column": 2
} | {
"line": 284,
"column": 13
} | {
"line": 284,
"column": 14
} | [
{
"pp": "α : Type u_1\ninst✝ : Preorder α\nx y : α\nhxy : x < y\nhfin : height x < ⊤\np : LTSeries α\nhlast : RelSeries.last p = x\nthis : ↑(RelSeries.snoc p y ⋯).length ≤ height (RelSeries.snoc p y ⋯).last\n⊢ ↑p.length + 1 ≤ height y",
"ppTerm": "?m.45",
"assigned": true,
"usedConstants": [
"... | [
"α : Type u_1\ninst✝ : Preorder α\nx y : α\nhxy : x < y\nhfin : height x < ⊤\np : LTSeries α\nhlast : RelSeries.last p = x\nthis : ↑(RelSeries.snoc p y ⋯).length ≤ height (RelSeries.snoc p y ⋯).last\n⊢ ↑p.length + 1 ≤ height y"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.KrullDimension | {
"line": 293,
"column": 2
} | {
"line": 293,
"column": 12
} | {
"line": 294,
"column": 4
} | [
{
"pp": "case coe\nα : Type u_1\ninst✝ : Preorder α\na b : α\nhab : a < b\nn : ℕ\nhfin : height a = ↑n\n⊢ ↑n + 1 ≤ height b",
"ppTerm": "?coe",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"instAddMonoidWithOneENat",
"ENat.instNatCast",
"instTopENat",... | [] | | coe n => | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalCases | null |
Mathlib.Order.KrullDimension | {
"line": 311,
"column": 2
} | {
"line": 311,
"column": 12
} | {
"line": 312,
"column": 4
} | [
{
"pp": "case coe\nα : Type u_1\ninst✝ : Preorder α\na b : α\nhab : b < a\nn : ℕ\nhfin : coheight a = ↑n\n⊢ ↑n + 1 ≤ coheight b",
"ppTerm": "?coe",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"instAddMonoidWithOneENat",
"ENat.instNatCast",
"instTopEN... | [] | | coe n => | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalCases | null |
Mathlib.Order.KrullDimension | {
"line": 338,
"column": 4
} | {
"line": 338,
"column": 51
} | {
"line": 339,
"column": 4
} | [
{
"pp": "case cons\nα : Type u_1\nβ : Type u_2\ninst✝¹ : Preorder α\ninst✝ : Preorder β\nf : α → β\nhf : StrictMono f\nh : ∀ (a : α) (b : β), f a < b → ∃ a', a < a' ∧ f a' = b\np : RelSeries {(a, b) | a < b}\nx : β\nhx : (x, p.head) ∈ {(a, b) | a < b}\nih : ∀ (a : α), p.head = f a → ↑p.length ≤ coheight a\na : ... | [
"case cons\nα : Type u_1\nβ : Type u_2\ninst✝¹ : Preorder α\ninst✝ : Preorder β\nf : α → β\nhf : StrictMono f\nh : ∀ (a : α) (b : β), f a < b → ∃ a', a < a' ∧ f a' = b\np : RelSeries {(a, b) | a < b}\nx : β\nhx : (x, p.head) ∈ {(a, b) | a < b}\nih : ∀ (a : α), p.head = f a → ↑p.length ≤ coheight a\na : α\nhp : x = ... | obtain ⟨a', haa', ha'⟩ := h a p.head (by grind) | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Order.KrullDimension | {
"line": 348,
"column": 2
} | {
"line": 348,
"column": 37
} | {
"line": 348,
"column": 38
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : Preorder α\ninst✝ : Preorder β\nf : α → β\nhf : StrictMono f\nh : ∀ (a : α), ∀ b < f a, ∃ a' < a, f a' = b\na : α\nthis : coheight (OrderDual.toDual a) = coheight (OrderDual.toDual (f a))\n⊢ height a = height (f a)",
"ppTerm": "?m.49",
"assigned": false,
... | [
"α : Type u_1\nβ : Type u_2\ninst✝¹ : Preorder α\ninst✝ : Preorder β\nf : α → β\nhf : StrictMono f\nh : ∀ (a : α), ∀ b < f a, ∃ a' < a, f a' = b\na : α\nthis : coheight (OrderDual.toDual a) = coheight (OrderDual.toDual (f a))\n⊢ height a = height (f a)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.KrullDimension | {
"line": 353,
"column": 4
} | {
"line": 353,
"column": 15
} | {
"line": 353,
"column": 16
} | [
{
"pp": "case a\nα : Type u_1\nβ : Type u_2\ninst✝¹ : Preorder α\ninst✝ : Preorder β\nf : α ≃o β\nx : α\n⊢ height (f x) ≤ height x",
"ppTerm": "?a✝",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"case a\nα : Type u_1\nβ : Type u_2\ninst✝¹ : Preorder α\ninst✝ : Preorder β\nf : α ≃o β\nx : α\n⊢ height (f x) ≤ height x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.KrullDimension | {
"line": 464,
"column": 2
} | {
"line": 464,
"column": 39
} | {
"line": 464,
"column": 40
} | [
{
"pp": "α : Type u_1\ninst✝ : Preorder α\nx : α\n⊢ height x = 0 ↔ IsMin x",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"_private.Mathlib.Order.KrullDimension.0.Order.height_eq_zero._simp_1_1",
"Eq.mpr",
"Preorder.toLT",
"congrArg",
"CommSemiring.toSemiring"... | [
"α : Type u_1\ninst✝ : Preorder α\nx : α\n⊢ height x = 0 ↔ ∀ (b : α), ¬b < x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.KrullDimension | {
"line": 525,
"column": 6
} | {
"line": 525,
"column": 45
} | {
"line": 525,
"column": 46
} | [
{
"pp": "case e'_2\nα✝ : Type u_1\ninst✝¹ : Preorder α✝\nα : Type u_1\ninst✝ : Preorder α\nx : α\nn : ℕ\nhfin : height x < ⊤\n⊢ height x ≤ ↑n + 1 ↔ ∀ y < x, height y ≤ ↑n",
"ppTerm": "?e'_2",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"case e'_2\nα✝ : Type u_1\ninst✝¹ : Preorder α✝\nα : Type u_1\ninst✝ : Preorder α\nx : α\nn : ℕ\nhfin : height x < ⊤\n⊢ height x ≤ ↑n + 1 ↔ ∀ y < x, height y ≤ ↑n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.KrullDimension | {
"line": 557,
"column": 4
} | {
"line": 557,
"column": 21
} | {
"line": 558,
"column": 4
} | [
{
"pp": "case pos\nα : Type u_1\ninst✝ : Preorder α\na : α\nn : ℕ\nhfin : height a < ⊤\nx✝ : ℕ\nhn : n = x✝\n⊢ height a = ↑x✝ ↔ Minimal (fun y ↦ ↑x✝ ≤ height y) a",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"ENat.coe_ne_top._simp_1",
"False",
"Preorder.toLT",
"i... | [] | cases hn : n with | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalCases | null |
Mathlib.Order.KrullDimension | {
"line": 566,
"column": 4
} | {
"line": 566,
"column": 20
} | {
"line": 566,
"column": 21
} | [
{
"pp": "case right\nα : Type u_1\ninst✝ : Preorder α\nn : ℕ\np : LTSeries α\nhp : p.length = n + 1\nhfin : ∀ (n : ℕ), ∃ p_1, RelSeries.last p_1 = RelSeries.last p ∧ p_1.length = n\n⊢ ↑n ≤ height (RelSeries.eraseLast p).last",
"ppTerm": "?right",
"assigned": true,
"usedConstants": [
"Iff.mpr",... | [
"case right\nα : Type u_1\ninst✝ : Preorder α\nn : ℕ\np : LTSeries α\nhp : p.length = n + 1\nhfin : ∀ (n : ℕ), ∃ p_1, RelSeries.last p_1 = RelSeries.last p ∧ p_1.length = n\n⊢ ↑n ≤ height (p.toFun ⟨n, ⋯⟩)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.KrullDimension | {
"line": 640,
"column": 2
} | {
"line": 641,
"column": 70
} | {
"line": 643,
"column": 0
} | [
{
"pp": "case mpr\nα : Type u_1\ninst✝ : Preorder α\n⊢ (∃ x, (∃ b, b < x) ∧ ∃ b, x < b) → ∃ i, 1 < ↑i.length",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"WithBot.addMonoidWithOne",
"WithBot.zeroLEOneClass",
"WithBot.addLeftMono",
"WithBot",
"Preorder.toLT",... | [] | · rintro ⟨x, ⟨y, hxy⟩, z, hzx⟩
exact ⟨⟨2, ![y, x, z], fun i ↦ by fin_cases i <;> simpa⟩, by simp⟩ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Order.KrullDimension | {
"line": 718,
"column": 4
} | {
"line": 718,
"column": 15
} | {
"line": 718,
"column": 16
} | [
{
"pp": "α : Type u_1\ninst✝¹ : Preorder α\ninst✝ : InfiniteDimensionalOrder α\nm✝ : WithBot ℕ∞\nhm✝ : ∀ (i : LTSeries α), ↑i.length ≤ m✝\nm : WithBot ℕ∞\nhm : ∀ (i : LTSeries α), ↑i.length ≤ m\nn : ℕ\n⊢ ↑n ≤ m",
"ppTerm": "?m.101",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"u... | [
"α : Type u_1\ninst✝¹ : Preorder α\ninst✝ : InfiniteDimensionalOrder α\nm✝ : WithBot ℕ∞\nhm✝ : ∀ (i : LTSeries α), ↑i.length ≤ m✝\nm : WithBot ℕ∞\nhm : ∀ (i : LTSeries α), ↑i.length ≤ m\nn : ℕ\n⊢ ↑n ≤ m"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.KrullDimension | {
"line": 737,
"column": 4
} | {
"line": 737,
"column": 33
} | {
"line": 737,
"column": 34
} | [
{
"pp": "case inr.inr\nα : Type u_1\ninst✝ : Preorder α\nn : ℕ\nh✝¹ : Nonempty α\nh✝ : InfiniteDimensionalOrder α\n⊢ ↑n ≤ krullDim α ↔ ∃ l, l.length = n",
"ppTerm": "?inr.inr",
"assigned": true,
"usedConstants": [
"WithBot.instPreorder",
"Eq.mpr",
"WithBot",
"Preorder.toLT",
... | [
"case inr.inr\nα : Type u_1\ninst✝ : Preorder α\nn : ℕ\nh✝¹ : Nonempty α\nh✝ : InfiniteDimensionalOrder α\n⊢ ∃ l, l.length = n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.TensorProduct.Basic | {
"line": 336,
"column": 41
} | {
"line": 336,
"column": 52
} | {
"line": 336,
"column": 53
} | [
{
"pp": "R : Type u_1\ninst✝¹³ : CommSemiring R\nA : Type u_22\nS : Type u_23\nM : Type u_24\nN : Type u_25\ninst✝¹² : AddCommMonoid M\ninst✝¹¹ : AddCommMonoid N\ninst✝¹⁰ : Module R M\ninst✝⁹ : Module R N\ninst✝⁸ : CommSemiring A\ninst✝⁷ : Module A M\ninst✝⁶ : Module A N\ninst✝⁵ : SMulCommClass R A M\ninst✝⁴ : ... | [
"R : Type u_1\ninst✝¹³ : CommSemiring R\nA : Type u_22\nS : Type u_23\nM : Type u_24\nN : Type u_25\ninst✝¹² : AddCommMonoid M\ninst✝¹¹ : AddCommMonoid N\ninst✝¹⁰ : Module R M\ninst✝⁹ : Module R N\ninst✝⁸ : CommSemiring A\ninst✝⁷ : Module A M\ninst✝⁶ : Module A N\ninst✝⁵ : SMulCommClass R A M\ninst✝⁴ : CommSemiring... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.TensorProduct.Basic | {
"line": 350,
"column": 22
} | {
"line": 350,
"column": 33
} | {
"line": 350,
"column": 34
} | [
{
"pp": "R : Type u_1\nR₂ : Type u_2\nR₃ : Type u_3\nR' : Type u_4\nR'' : Type u_5\ninst✝⁵⁰ : CommSemiring R\ninst✝⁴⁹ : CommSemiring R₂\ninst✝⁴⁸ : CommSemiring R₃\ninst✝⁴⁷ : Monoid R'\ninst✝⁴⁶ : Semiring R''\nσ₁₂ : R →+* R₂\nσ₂₃ : R₂ →+* R₃\nσ₁₃ : R →+* R₃\nA✝ : Type u_6\nM✝ : Type u_7\nN✝ : Type u_8\nP : Type ... | [
"R : Type u_1\nR₂ : Type u_2\nR₃ : Type u_3\nR' : Type u_4\nR'' : Type u_5\ninst✝⁵⁰ : CommSemiring R\ninst✝⁴⁹ : CommSemiring R₂\ninst✝⁴⁸ : CommSemiring R₃\ninst✝⁴⁷ : Monoid R'\ninst✝⁴⁶ : Semiring R''\nσ₁₂ : R →+* R₂\nσ₂₃ : R₂ →+* R₃\nσ₁₃ : R →+* R₃\nA✝ : Type u_6\nM✝ : Type u_7\nN✝ : Type u_8\nP : Type u_9\nQ : Typ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.TensorProduct.Basic | {
"line": 352,
"column": 22
} | {
"line": 352,
"column": 33
} | {
"line": 352,
"column": 34
} | [
{
"pp": "R : Type u_1\nR₂ : Type u_2\nR₃ : Type u_3\nR' : Type u_4\nR'' : Type u_5\ninst✝⁵⁰ : CommSemiring R\ninst✝⁴⁹ : CommSemiring R₂\ninst✝⁴⁸ : CommSemiring R₃\ninst✝⁴⁷ : Monoid R'\ninst✝⁴⁶ : Semiring R''\nσ₁₂ : R →+* R₂\nσ₂₃ : R₂ →+* R₃\nσ₁₃ : R →+* R₃\nA✝ : Type u_6\nM✝ : Type u_7\nN✝ : Type u_8\nP : Type ... | [
"R : Type u_1\nR₂ : Type u_2\nR₃ : Type u_3\nR' : Type u_4\nR'' : Type u_5\ninst✝⁵⁰ : CommSemiring R\ninst✝⁴⁹ : CommSemiring R₂\ninst✝⁴⁸ : CommSemiring R₃\ninst✝⁴⁷ : Monoid R'\ninst✝⁴⁶ : Semiring R''\nσ₁₂ : R →+* R₂\nσ₂₃ : R₂ →+* R₃\nσ₁₃ : R →+* R₃\nA✝ : Type u_6\nM✝ : Type u_7\nN✝ : Type u_8\nP : Type u_9\nQ : Typ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.KrullDimension | {
"line": 833,
"column": 4
} | {
"line": 833,
"column": 14
} | {
"line": 834,
"column": 6
} | [
{
"pp": "case coe\nα✝ : Type u_1\ninst✝² : Preorder α✝\nα : Type u_1\ninst✝¹ : Preorder α\ninst✝ : Nonempty α\nhnottop : krullDim α < ⊤\na : α\nthis✝ : height a < ⊤\nthis : coheight a < ⊤\nn : ℕ\nhh : height a = ↑n\n⊢ ↑n + coheight a ≤ ⨆ p, ↑p.length",
"ppTerm": "?coe",
"assigned": true,
"usedConsta... | [] | | coe n => | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalCases | null |
Mathlib.Order.KrullDimension | {
"line": 820,
"column": 56
} | {
"line": 839,
"column": 71
} | {
"line": 841,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝¹ : Preorder α\ninst✝ : Nonempty α\n⊢ krullDim α = ↑(⨆ a, height a + coheight a)",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"WithBot.instPreorder",
"Eq.mpr",
"lt_of_le_of_lt",
"False",
"WithBot.some",
"WithBot",
"Pr... | [] | by
apply le_antisymm
· rw [krullDim_eq_iSup_height_of_nonempty, WithBot.coe_le_coe]
apply ciSup_mono (by bddDefault) (by simp)
· wlog hnottop : krullDim α < ⊤
· simp_all
rw [krullDim_eq_iSup_length, WithBot.coe_le_coe]
apply iSup_le
intro a
have : height a < ⊤ := WithBot.coe_lt_coe.mp (lt_... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.TensorProduct.Basic | {
"line": 426,
"column": 49
} | {
"line": 426,
"column": 91
} | {
"line": 426,
"column": 92
} | [
{
"pp": "R : Type u_1\ninst✝⁸ : CommSemiring R\nM : Type u_2\nN : Type u_3\nP : Type u_4\nQ : Type u_5\nS : Type u_6\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : AddCommMonoid N\ninst✝⁵ : AddCommGroup P\ninst✝⁴ : AddCommMonoid Q\ninst✝³ : Module R M\ninst✝² : Module R N\ninst✝¹ : Module R P\ninst✝ : Module R Q\nr✝ : ℤ\nm... | [
"R : Type u_1\ninst✝⁸ : CommSemiring R\nM : Type u_2\nN : Type u_3\nP : Type u_4\nQ : Type u_5\nS : Type u_6\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : AddCommMonoid N\ninst✝⁵ : AddCommGroup P\ninst✝⁴ : AddCommMonoid Q\ninst✝³ : Module R M\ninst✝² : Module R N\ninst✝¹ : Module R P\ninst✝ : Module R Q\nr✝ : ℤ\nm : M\np : P\... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.TensorProduct.Basic | {
"line": 427,
"column": 21
} | {
"line": 427,
"column": 63
} | {
"line": 427,
"column": 64
} | [
{
"pp": "R : Type u_1\ninst✝⁸ : CommSemiring R\nM : Type u_2\nN : Type u_3\nP : Type u_4\nQ : Type u_5\nS : Type u_6\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : AddCommMonoid N\ninst✝⁵ : AddCommGroup P\ninst✝⁴ : AddCommMonoid Q\ninst✝³ : Module R M\ninst✝² : Module R N\ninst✝¹ : Module R P\ninst✝ : Module R Q\nr✝ : ℤ\nm... | [
"R : Type u_1\ninst✝⁸ : CommSemiring R\nM : Type u_2\nN : Type u_3\nP : Type u_4\nQ : Type u_5\nS : Type u_6\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : AddCommMonoid N\ninst✝⁵ : AddCommGroup P\ninst✝⁴ : AddCommMonoid Q\ninst✝³ : Module R M\ninst✝² : Module R N\ninst✝¹ : Module R P\ninst✝ : Module R Q\nr✝ : ℤ\nm : M\np : P\... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Algebra.Bilinear | {
"line": 67,
"column": 2
} | {
"line": 67,
"column": 11
} | {
"line": 68,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝² : CommSemiring R\nM : Type u_4\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nr : R\n⊢ ∀ (x : R) (y : M), (lift (lsmul R M ∘ₗ (mul R R) r)) (x ⊗ₜ[R] y) = ((lsmul R M) r ∘ₗ lift (lsmul R M)) (x ⊗ₜ[R] y)",
"ppTerm": "?m.116",
"assigned": true,
"usedConstants": [],
"us... | [
"R : Type u_1\ninst✝² : CommSemiring R\nM : Type u_4\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nr x : R\na : M\n⊢ (lift (lsmul R M ∘ₗ (mul R R) r)) (x ⊗ₜ[R] a) = ((lsmul R M) r ∘ₗ lift (lsmul R M)) (x ⊗ₜ[R] a)"
] | intro x a | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Algebra.Algebra.Bilinear | {
"line": 120,
"column": 55
} | {
"line": 120,
"column": 78
} | {
"line": 120,
"column": 79
} | [
{
"pp": "R : Type u_1\nA : Type u_2\ninst✝³ : Semiring R\ninst✝² : Semiring A\ninst✝¹ : Module R A\ninst✝ : SMulCommClass R A A\na : A\nn✝ n : ℕ\n⊢ mulLeft R a ^ n * mulLeft R a = mulLeft R (a ^ n) ∘ₗ mulLeft R a",
"ppTerm": "?m.86",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Module... | [
"R : Type u_1\nA : Type u_2\ninst✝³ : Semiring R\ninst✝² : Semiring A\ninst✝¹ : Module R A\ninst✝ : SMulCommClass R A A\na : A\nn✝ n : ℕ\n⊢ (mulLeft R a ^ n) ∘ₗ mulLeft R a = mulLeft R (a ^ n) ∘ₗ mulLeft R a"
] | Module.End.mul_eq_comp, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Algebra.Bilinear | {
"line": 131,
"column": 57
} | {
"line": 131,
"column": 80
} | {
"line": 131,
"column": 81
} | [
{
"pp": "R : Type u_1\nA : Type u_2\ninst✝³ : Semiring R\ninst✝² : Semiring A\ninst✝¹ : Module R A\ninst✝ : IsScalarTower R A A\na : A\nn✝ n : ℕ\n⊢ mulRight R a ^ n * mulRight R a = mulRight R (a ^ n) ∘ₗ mulRight R a",
"ppTerm": "?m.87",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Mo... | [
"R : Type u_1\nA : Type u_2\ninst✝³ : Semiring R\ninst✝² : Semiring A\ninst✝¹ : Module R A\ninst✝ : IsScalarTower R A A\na : A\nn✝ n : ℕ\n⊢ (mulRight R a ^ n) ∘ₗ mulRight R a = mulRight R (a ^ n) ∘ₗ mulRight R a"
] | Module.End.mul_eq_comp, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Algebra.Bilinear | {
"line": 155,
"column": 19
} | {
"line": 155,
"column": 30
} | {
"line": 155,
"column": 31
} | [
{
"pp": "R : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\na₁ a₂ : A\nh : (Algebra.lmul R A) a₁ = (Algebra.lmul R A) a₂\n⊢ a₁ = a₂",
"ppTerm": "?m.12",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\na₁ a₂ : A\nh : (Algebra.lmul R A) a₁ = (Algebra.lmul R A) a₂\n⊢ a₁ = a₂"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Group.Pointwise.Set.BigOperators | {
"line": 45,
"column": 4
} | {
"line": 45,
"column": 83
} | {
"line": 46,
"column": 6
} | [
{
"pp": "α : Type u_2\nβ : Type u_3\nF : Type u_4\ninst✝³ : FunLike F α β\ninst✝² : CommMonoid α\ninst✝¹ : CommMonoid β\ninst✝ : MonoidHomClass F α β\nf : F\n⊢ ∀ (a : List (Set α)), ⇑f '' Multiset.prod ⟦a⟧ = (Multiset.map (fun s ↦ ⇑f '' s) ⟦a⟧).prod",
"ppTerm": "?m.22",
"assigned": true,
"usedConsta... | [
"α : Type u_2\nβ : Type u_3\nF : Type u_4\ninst✝³ : FunLike F α β\ninst✝² : CommMonoid α\ninst✝¹ : CommMonoid β\ninst✝ : MonoidHomClass F α β\nf : F\n⊢ ∀ (a : List (Set α)), ⇑f '' a.prod = (List.map (fun s ↦ ⇑f '' s) a).prod"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.KrullDimension | {
"line": 918,
"column": 2
} | {
"line": 918,
"column": 13
} | {
"line": 918,
"column": 14
} | [
{
"pp": "α : Type u_1\ninst✝¹ : Preorder α\ninst✝ : FiniteDimensionalOrder α\nx : α\n⊢ krullDim α < ↑⊤",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"WithBot.instPreorder",
"WithBot.some",
"WithBot",
"Preorder.toLT",
"instTopENat",
"instPreorderENat",
... | [
"α : Type u_1\ninst✝¹ : Preorder α\ninst✝ : FiniteDimensionalOrder α\nx : α\n⊢ krullDim α < ⊤"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Group.Pointwise.Set.BigOperators | {
"line": 93,
"column": 2
} | {
"line": 93,
"column": 13
} | {
"line": 93,
"column": 14
} | [
{
"pp": "α : Type u_2\ninst✝ : CommMonoid α\nn : ℕ\ns : Set α\na : α\n⊢ a ∈ s ^ n ↔ ∃ f, (∀ (i : Fin n), f i ∈ s) ∧ ∏ i, f i = a",
"ppTerm": "?m.23",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_2\ninst✝ : CommMonoid α\nn : ℕ\ns : Set α\na : α\n⊢ a ∈ s ^ n ↔ ∃ f, (∀ (i : Fin n), f i ∈ s) ∧ ∏ i, f i = a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Group.Pointwise.Set.BigOperators | {
"line": 199,
"column": 2
} | {
"line": 199,
"column": 36
} | {
"line": 199,
"column": 37
} | [
{
"pp": "ι : Type u_1\nα : Type u_2\ninst✝¹ : CommMonoid α\ninst✝ : Fintype ι\nS : ι → Set α\n⊢ (fun f ↦ ∏ i, f i) '' univ.pi S = ∏ i, S i",
"ppTerm": "?m.24",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"ι : Type u_1\nα : Type u_2\ninst✝¹ : CommMonoid α\ninst✝ : Fintype ι\nS : ι → Set α\n⊢ (fun f ↦ ∏ i, f i) '' univ.pi S = ∏ i, S i"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.KrullDimension | {
"line": 1051,
"column": 26
} | {
"line": 1051,
"column": 79
} | {
"line": 1051,
"column": 80
} | [
{
"pp": "α : Type u_1\ninst✝ : Preorder α\nx : α\np : LTSeries (WithTop α)\nhlast : RelSeries.last p = ↑x\ni : Fin p.length\n⊢ ((p.toFun i.castSucc).untop ⋯, (p.toFun i.succ).untop ⋯) ∈ {(a, b) | a < b}",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"lt_of_le_of_lt",... | [
"α : Type u_1\ninst✝ : Preorder α\nx : α\np : LTSeries (WithTop α)\nhlast : RelSeries.last p = ↑x\ni : Fin p.length\n⊢ p.toFun i.castSucc < p.toFun i.succ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.KrullDimension | {
"line": 1056,
"column": 6
} | {
"line": 1056,
"column": 22
} | {
"line": 1056,
"column": 23
} | [
{
"pp": "α : Type u_1\ninst✝ : Preorder α\nx : α\np : LTSeries (WithTop α)\nhlast : RelSeries.last p = ↑x\np' : LTSeries α := { length := p.length, toFun := fun i ↦ (p.toFun i).untop ⋯, step := ⋯ }\nhlast' : RelSeries.last p' = x\nthis : ↑p'.length ≤ height x\n⊢ ↑p.length ≤ height x",
"ppTerm": "?m.95",
... | [
"α : Type u_1\ninst✝ : Preorder α\nx : α\np : LTSeries (WithTop α)\nhlast : RelSeries.last p = ↑x\np' : LTSeries α := { length := p.length, toFun := fun i ↦ (p.toFun i).untop ⋯, step := ⋯ }\nhlast' : RelSeries.last p' = x\nthis : ↑p'.length ≤ height x\n⊢ ↑p.length ≤ height x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.KrullDimension | {
"line": 1089,
"column": 2
} | {
"line": 1089,
"column": 12
} | {
"line": 1089,
"column": 13
} | [
{
"pp": "case coe\nn : ℕ\n⊢ height ↑n = ↑n",
"ppTerm": "?coe",
"assigned": true,
"usedConstants": [
"Order.height_nat",
"ENat.instNatCast",
"WithTop.instPreorder",
"Order.height_coe_withTop",
"Nat.cast",
"WithTop.some",
"Nat.instPreorder",
"Nat",
... | [] | | coe n => | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalCases | null |
Mathlib.LinearAlgebra.TensorProduct.Defs | {
"line": 428,
"column": 2
} | {
"line": 429,
"column": 37
} | {
"line": 431,
"column": 0
} | [
{
"pp": "case refine_3\nR : Type u_1\ninst✝⁴ : CommSemiring R\nM : Type u_7\nN : Type u_8\ninst✝³ : AddCommMonoid M\ninst✝² : AddCommMonoid N\ninst✝¹ : Module R M\ninst✝ : Module R N\nt : M ⊗[R] N\n⊢ ∀ (x y : M ⊗[R] N),\n x ∈ Submodule.span R {t | ∃ m n, m ⊗ₜ[R] n = t} →\n y ∈ Submodule.span R {t | ∃ m ... | [] | · intro t₁ t₂ ht₁ ht₂
exact Submodule.add_mem _ ht₁ ht₂ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Order.KrullDimension | {
"line": 1107,
"column": 29
} | {
"line": 1107,
"column": 39
} | {
"line": 1108,
"column": 4
} | [
{
"pp": "case coe\nα : Type u_1\nβ : Type u_2\ninst✝¹ : Preorder α\ninst✝ : PartialOrder β\nm : ℕ\nf : α →o β\nh : ∀ (x : β), krullDim ↑(⇑f ⁻¹' {x}) ≤ ↑m\nx : α\nn : ℕ\nh' : height (f x) = ↑n\n⊢ height x ≤ (↑m + 1) * ↑n + ↑m",
"ppTerm": "?coe",
"assigned": true,
"usedConstants": [
"Order.LTSer... | [] | | coe n => | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalCases | null |
Mathlib.Order.KrullDimension | {
"line": 1158,
"column": 2
} | {
"line": 1158,
"column": 13
} | {
"line": 1158,
"column": 14
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : Preorder α\ninst✝ : PartialOrder β\ne : α ↪o β\nthis : ∀ (b : β), Subsingleton ↑(⇑e ⁻¹' {b})\n⊢ krullDim α ≤ krullDim β",
"ppTerm": "?m.37",
"assigned": true,
"usedConstants": [
"WithBot.instPreorder",
"Eq.mpr",
"WithBot",
"Partia... | [
"α : Type u_1\nβ : Type u_2\ninst✝¹ : Preorder α\ninst✝ : PartialOrder β\ne : α ↪o β\nthis : ∀ (b : β), Subsingleton ↑(⇑e ⁻¹' {b})\n⊢ krullDim α ≤ krullDim β"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.TensorProduct.Map | {
"line": 461,
"column": 2
} | {
"line": 461,
"column": 64
} | {
"line": 463,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝⁸ : CommSemiring R\nM : Type u_7\nN : Type u_8\nP : Type u_9\nQ : Type u_10\ninst✝⁷ : AddCommMonoid M\ninst✝⁶ : AddCommMonoid N\ninst✝⁵ : AddCommMonoid P\ninst✝⁴ : AddCommMonoid Q\ninst✝³ : Module R M\ninst✝² : Module R N\ninst✝¹ : Module R P\ninst✝ : Module R Q\ng : P →ₗ[R] Q\nf : N... | [] | simp only [compr₂ₛₗ_apply, mk_apply, comp_apply, rTensor_tmul] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Order.Kleene | {
"line": 191,
"column": 2
} | {
"line": 191,
"column": 13
} | {
"line": 191,
"column": 14
} | [
{
"pp": "α : Type u_1\ninst✝ : KleeneAlgebra α\na b : α\nhb : 1 ≤ b\n⊢ b * a ≤ b → a∗ ≤ b",
"ppTerm": "?m.15",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\ninst✝ : KleeneAlgebra α\na b : α\nhb : 1 ≤ b\n⊢ b * a ≤ b → a∗ ≤ b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Kleene | {
"line": 191,
"column": 2
} | {
"line": 191,
"column": 29
} | {
"line": 193,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝ : KleeneAlgebra α\na b : α\nhb : 1 ≤ b\n⊢ b * a ≤ b → a∗ ≤ b",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"HMul.hMul",
"congrArg",
"KStar.kstar",
"PartialOrder.toPreorder",
"Kleen... | [] | simpa using mul_kstar_le hb | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Algebra.Order.Kleene | {
"line": 191,
"column": 2
} | {
"line": 191,
"column": 29
} | {
"line": 193,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝ : KleeneAlgebra α\na b : α\nhb : 1 ≤ b\n⊢ b * a ≤ b → a∗ ≤ b",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"HMul.hMul",
"congrArg",
"KStar.kstar",
"PartialOrder.toPreorder",
"Kleen... | [] | simpa using mul_kstar_le hb | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Order.Kleene | {
"line": 191,
"column": 2
} | {
"line": 191,
"column": 29
} | {
"line": 193,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝ : KleeneAlgebra α\na b : α\nhb : 1 ≤ b\n⊢ b * a ≤ b → a∗ ≤ b",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"HMul.hMul",
"congrArg",
"KStar.kstar",
"PartialOrder.toPreorder",
"Kleen... | [] | simpa using mul_kstar_le hb | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Order.Kleene | {
"line": 194,
"column": 2
} | {
"line": 194,
"column": 13
} | {
"line": 194,
"column": 14
} | [
{
"pp": "α : Type u_1\ninst✝ : KleeneAlgebra α\na b : α\nhb : 1 ≤ b\n⊢ a * b ≤ b → a∗ ≤ b",
"ppTerm": "?m.15",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\ninst✝ : KleeneAlgebra α\na b : α\nhb : 1 ≤ b\n⊢ a * b ≤ b → a∗ ≤ b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Ring.Submonoid.Pointwise | {
"line": 157,
"column": 4
} | {
"line": 157,
"column": 45
} | {
"line": 158,
"column": 4
} | [
{
"pp": "case a\nR : Type u_2\ninst✝ : NonUnitalNonAssocSemiring R\nS T : Set R\na : R\nha : a ∈ closure S\nb : R\nhb : b ∈ closure T\na' : R\nha' : a' ∈ S\n⊢ b ∈ comap (AddMonoidHom.mulLeft a') (closure (S * T))",
"ppTerm": "?a✝",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"AddMono... | [
"case a\nR : Type u_2\ninst✝ : NonUnitalNonAssocSemiring R\nS T : Set R\na : R\nha : a ∈ closure S\nb : R\nhb : b ∈ closure T\na' : R\nha' : a' ∈ S\nb' : R\nhb' : b' ∈ T\n⊢ b' ∈ ↑(comap (AddMonoidHom.mulLeft a') (closure (S * T)))"
] | refine (closure_le.2 fun b' hb' => ?_) hb | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Algebra.Ring.Submonoid.Pointwise | {
"line": 189,
"column": 24
} | {
"line": 189,
"column": 82
} | {
"line": 189,
"column": 82
} | [
{
"pp": "R : Type u_2\ninst✝ : NonUnitalNonAssocSemiring R\nM N P : AddSubmonoid R\np : R\nhp : p ∈ P\nm : R\nhm : m ∈ M\nn : R\nhn : n ∈ N\nhmn : m + n ∈ M ⊔ N\n⊢ m * p + n * p ∈ M * P ⊔ N * P",
"ppTerm": "?m.118",
"assigned": true,
"usedConstants": [
"HMul.hMul",
"AddSubmonoid.mul",
... | [] | exact add_mem_sup (mul_mem_mul hm hp) <| mul_mem_mul hn hp | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Data.Fintype.Lattice | {
"line": 59,
"column": 2
} | {
"line": 59,
"column": 13
} | {
"line": 59,
"column": 14
} | [
{
"pp": "case intro\nα : Type u_2\nβ : Type u_3\ninst✝² : Finite α\ninst✝¹ : Nonempty α\ninst✝ : LinearOrder β\nf : α → β\nval✝ : Fintype α\n⊢ ∃ x₀, ∀ (x : α), f x ≤ f x₀",
"ppTerm": "?intro",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"case intro\nα : Type u_2\nβ : Type u_3\ninst✝² : Finite α\ninst✝¹ : Nonempty α\ninst✝ : LinearOrder β\nf : α → β\nval✝ : Fintype α\n⊢ ∃ x₀, ∀ (x : α), f x ≤ f x₀"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Fintype.Lattice | {
"line": 64,
"column": 2
} | {
"line": 64,
"column": 13
} | {
"line": 64,
"column": 14
} | [
{
"pp": "case intro\nα : Type u_2\nβ : Type u_3\ninst✝² : Finite α\ninst✝¹ : Nonempty α\ninst✝ : LinearOrder β\nf : α → β\nval✝ : Fintype α\n⊢ ∃ x₀, ∀ (x : α), f x₀ ≤ f x",
"ppTerm": "?intro",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"case intro\nα : Type u_2\nβ : Type u_3\ninst✝² : Finite α\ninst✝¹ : Nonempty α\ninst✝ : LinearOrder β\nf : α → β\nval✝ : Fintype α\n⊢ ∃ x₀, ∀ (x : α), f x₀ ≤ f x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Coprime.Basic | {
"line": 126,
"column": 4
} | {
"line": 127,
"column": 46
} | {
"line": 129,
"column": 0
} | [
{
"pp": "case h₂\nR : Type u\ninst✝ : CommSemiring R\nx y z : R\nH1 : x ∣ z\nH2 : y ∣ z\na b : R\nh : a * x + b * y = 1\n⊢ x * y ∣ z * (b * y)",
"ppTerm": "?h₂",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Semigroup.toMul",
"Dvd.dvd",
"HMul.hMul",
"Monoid.toMulOneCl... | [] | rw [mul_comm b, ← mul_assoc]
exact (mul_dvd_mul_right H1 _).mul_right _ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Coprime.Basic | {
"line": 126,
"column": 4
} | {
"line": 127,
"column": 46
} | {
"line": 129,
"column": 0
} | [
{
"pp": "case h₂\nR : Type u\ninst✝ : CommSemiring R\nx y z : R\nH1 : x ∣ z\nH2 : y ∣ z\na b : R\nh : a * x + b * y = 1\n⊢ x * y ∣ z * (b * y)",
"ppTerm": "?h₂",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Semigroup.toMul",
"Dvd.dvd",
"HMul.hMul",
"Monoid.toMulOneCl... | [] | rw [mul_comm b, ← mul_assoc]
exact (mul_dvd_mul_right H1 _).mul_right _ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Coprime.Basic | {
"line": 185,
"column": 4
} | {
"line": 186,
"column": 26
} | {
"line": 186,
"column": 27
} | [
{
"pp": "R : Type u\ninst✝ : CommSemiring R\nx y z : R\nh : IsCoprime (x + y * z) y\na b : R\nH : a * (x + y * z) + b * y = 1\n⊢ a * x + (a * z + b) * y = 1",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"Distrib.leftDistribClass",
"Eq.mpr",
"NonAssocSemiring.toAddCommMo... | [
"R : Type u\ninst✝ : CommSemiring R\nx y z : R\nh : IsCoprime (x + y * z) y\na b : R\nH : a * (x + y * z) + b * y = 1\n⊢ x * a + (y * b + y * (z * a)) = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Coprime.Basic | {
"line": 205,
"column": 2
} | {
"line": 206,
"column": 31
} | {
"line": 208,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝ : CommSemiring R\nx y z : R\nh : IsCoprime (z * y + x) y\n⊢ IsCoprime x y",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"HMul.hMul",
"congrArg",
"CommSemiring.toSemiring",
"IsCoprime.of_add_mul_right_left",
"Eq.mp",
"add_comm... | [] | rw [add_comm] at h
exact h.of_add_mul_right_left | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Coprime.Basic | {
"line": 205,
"column": 2
} | {
"line": 206,
"column": 31
} | {
"line": 208,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝ : CommSemiring R\nx y z : R\nh : IsCoprime (z * y + x) y\n⊢ IsCoprime x y",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"HMul.hMul",
"congrArg",
"CommSemiring.toSemiring",
"IsCoprime.of_add_mul_right_left",
"Eq.mp",
"add_comm... | [] | rw [add_comm] at h
exact h.of_add_mul_right_left | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Coprime.Basic | {
"line": 311,
"column": 43
} | {
"line": 311,
"column": 91
} | {
"line": 311,
"column": 92
} | [
{
"pp": "R : Type u\ninst✝ : CommRing R\nx y : R\nh : IsCoprime x y\nz : R\n⊢ IsCoprime (x + y * z + y * -z) y",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NegZeroClass.toNeg",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"HMul.hMul",
"CommRi... | [
"R : Type u\ninst✝ : CommRing R\nx y : R\nh : IsCoprime x y\nz : R\n⊢ IsCoprime x y"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Coprime.Basic | {
"line": 425,
"column": 2
} | {
"line": 425,
"column": 48
} | {
"line": 425,
"column": 49
} | [
{
"pp": "R : Type u\ninst✝ : CommRing R\nx : R\nh : 2 ∣ x\n⊢ IsCoprime (x + 1) (x - 1)",
"ppTerm": "?m.19",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u\ninst✝ : CommRing R\nx : R\nh : 2 ∣ x\n⊢ IsCoprime (x + 1) (x - 1)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Coprime.Basic | {
"line": 491,
"column": 43
} | {
"line": 491,
"column": 91
} | {
"line": 491,
"column": 92
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nx y : R\nh : IsRelPrime x y\nz : R\n⊢ IsRelPrime (x + y * z + y * -z) y",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NegZeroClass.toNeg",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"HMul.hMul",
"Co... | [
"R : Type u_1\ninst✝ : CommRing R\nx y : R\nh : IsRelPrime x y\nz : R\n⊢ IsRelPrime x y"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Coprime.Lemmas | {
"line": 40,
"column": 2
} | {
"line": 42,
"column": 41
} | {
"line": 43,
"column": 2
} | [
{
"pp": "case mp\nm n : ℤ\n⊢ IsCoprime m n → m.gcd n = 1",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
"Int.gcd",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Dvd.dvd",
... | [
"case mpr\nm n : ℤ\n⊢ m.gcd n = 1 → IsCoprime m n"
] | · rintro ⟨a, b, h⟩
refine Nat.dvd_one.mp (Int.gcd_dvd_iff.mpr ⟨a, b, ?_⟩)
rwa [mul_comm m, mul_comm n, eq_comm] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.Coprime.Basic | {
"line": 593,
"column": 2
} | {
"line": 593,
"column": 48
} | {
"line": 593,
"column": 49
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nx : R\nh : 2 ∣ x\n⊢ IsRelPrime (x + 1) (x - 1)",
"ppTerm": "?m.21",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u_1\ninst✝ : CommRing R\nx : R\nh : 2 ∣ x\n⊢ IsRelPrime (x + 1) (x - 1)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Coprime.Lemmas | {
"line": 74,
"column": 4
} | {
"line": 74,
"column": 34
} | {
"line": 74,
"column": 35
} | [
{
"pp": "A : Type u\ninst✝¹ : CommRing A\ninst✝ : Nontrivial A\na b : ℕ\nh : a.Coprime b\n⊢ IsCoprime ↑a ↑b",
"ppTerm": "?m.11",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"A : Type u\ninst✝¹ : CommRing A\ninst✝ : Nontrivial A\na b : ℕ\nh : a.Coprime b\n⊢ IsCoprime ↑a ↑b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Coprime.Lemmas | {
"line": 85,
"column": 2
} | {
"line": 85,
"column": 35
} | {
"line": 85,
"column": 36
} | [
{
"pp": "R : Type u\nI : Type v\ninst✝ : CommSemiring R\nx : R\ns : I → R\nt : Finset I\n⊢ (∀ i ∈ t, IsCoprime x (s i)) → IsCoprime x (∏ i ∈ t, s i)",
"ppTerm": "?m.14",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u\nI : Type v\ninst✝ : CommSemiring R\nx : R\ns : I → R\nt : Finset I\n⊢ (∀ i ∈ t, IsCoprime x (s i)) → IsCoprime x (∏ i ∈ t, s i)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Coprime.Lemmas | {
"line": 93,
"column": 2
} | {
"line": 93,
"column": 35
} | {
"line": 93,
"column": 36
} | [
{
"pp": "R : Type u\nI : Type v\ninst✝ : CommSemiring R\nx : R\ns : I → R\nt : Finset I\n⊢ IsCoprime x (∏ i ∈ t, s i) ↔ ∀ i ∈ t, IsCoprime x (s i)",
"ppTerm": "?m.14",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u\nI : Type v\ninst✝ : CommSemiring R\nx : R\ns : I → R\nt : Finset I\n⊢ IsCoprime x (∏ i ∈ t, s i) ↔ ∀ i ∈ t, IsCoprime x (s i)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Tactic.Order.ToInt | {
"line": 58,
"column": 22
} | {
"line": 58,
"column": 33
} | {
"line": 58,
"column": 34
} | [
{
"pp": "α : Type u_1\ninst✝ : LinearOrder α\nn : ℕ\nval : Fin n → α\nli : List α := List.ofFn val\nsli : List α := li.mergeSort fun a b ↦ decide (a ≤ b)\nthis : ∀ (i : Fin n), ∃ j, sli[j] = val i\ni j : Fin n\npf✝ : ∀ (j : Fin sli.length), ↑j < sli.length\nhi : ∃ j, sli[↑j] = val i\nhj : ∃ j_1, sli[↑j_1] = val... | [
"α : Type u_1\ninst✝ : LinearOrder α\nn : ℕ\nval : Fin n → α\nli : List α := List.ofFn val\nsli : List α := li.mergeSort fun a b ↦ decide (a ≤ b)\nthis : ∀ (i : Fin n), ∃ j, sli[j] = val i\ni j : Fin n\npf✝ : ∀ (j : Fin sli.length), ↑j < sli.length\nhi : ∃ j, sli[↑j] = val i\nhj : ∃ j_1, sli[↑j_1] = val j\nh_eq : ¬... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Tactic.Order.ToInt | {
"line": 58,
"column": 48
} | {
"line": 58,
"column": 59
} | {
"line": 58,
"column": 60
} | [
{
"pp": "α : Type u_1\ninst✝ : LinearOrder α\nn : ℕ\nval : Fin n → α\nli : List α := List.ofFn val\nsli : List α := li.mergeSort fun a b ↦ decide (a ≤ b)\nthis : ∀ (i : Fin n), ∃ j, sli[j] = val i\ni j : Fin n\npf✝ : ∀ (j : Fin sli.length), ↑j < sli.length\nhi : ∃ j, sli[↑j] = val i\nhj : ∃ j_1, sli[↑j_1] = val... | [
"α : Type u_1\ninst✝ : LinearOrder α\nn : ℕ\nval : Fin n → α\nli : List α := List.ofFn val\nsli : List α := li.mergeSort fun a b ↦ decide (a ≤ b)\nthis : ∀ (i : Fin n), ∃ j, sli[j] = val i\ni j : Fin n\npf✝ : ∀ (j : Fin sli.length), ↑j < sli.length\nhi : ∃ j, sli[↑j] = val i\nhj : ∃ j_1, sli[↑j_1] = val j\nh_eq : ¬... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Tactic.Order.ToInt | {
"line": 62,
"column": 29
} | {
"line": 62,
"column": 40
} | {
"line": 62,
"column": 41
} | [
{
"pp": "α : Type u_1\ninst✝ : LinearOrder α\nn : ℕ\nval : Fin n → α\nli : List α := List.ofFn val\nsli : List α := li.mergeSort fun a b ↦ decide (a ≤ b)\nthis✝ : ∀ (i : Fin n), ∃ j, sli[j] = val i\ni j : Fin n\npf✝ : ∀ (j : Fin sli.length), ↑j < sli.length\nhi : ∃ j, sli[↑j] = val i\nhj : ∃ j_1, sli[↑j_1] = va... | [
"α : Type u_1\ninst✝ : LinearOrder α\nn : ℕ\nval : Fin n → α\nli : List α := List.ofFn val\nsli : List α := li.mergeSort fun a b ↦ decide (a ≤ b)\nthis✝ : ∀ (i : Fin n), ∃ j, sli[j] = val i\ni j : Fin n\npf✝ : ∀ (j : Fin sli.length), ↑j < sli.length\nhi : ∃ j, sli[↑j] = val i\nhj : ∃ j_1, sli[↑j_1] = val j\nh_eq : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Tactic.Order.ToInt | {
"line": 64,
"column": 4
} | {
"line": 64,
"column": 15
} | {
"line": 64,
"column": 16
} | [
{
"pp": "case neg.refine_2\nα : Type u_1\ninst✝ : LinearOrder α\nn : ℕ\nval : Fin n → α\nli : List α := List.ofFn val\nsli : List α := li.mergeSort fun a b ↦ decide (a ≤ b)\nthis✝ : ∀ (i : Fin n), ∃ j, sli[j] = val i\ni j : Fin n\npf✝ : ∀ (j : Fin sli.length), ↑j < sli.length\nhi : ∃ j, sli[↑j] = val i\nhj : ∃ ... | [
"case neg.refine_2\nα : Type u_1\ninst✝ : LinearOrder α\nn : ℕ\nval : Fin n → α\nli : List α := List.ofFn val\nsli : List α := li.mergeSort fun a b ↦ decide (a ≤ b)\nthis✝ : ∀ (i : Fin n), ∃ j, sli[j] = val i\ni j : Fin n\npf✝ : ∀ (j : Fin sli.length), ↑j < sli.length\nhi : ∃ j, sli[↑j] = val i\nhj : ∃ j_1, sli[↑j_... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Tactic.Order.ToInt | {
"line": 76,
"column": 2
} | {
"line": 76,
"column": 13
} | {
"line": 76,
"column": 14
} | [
{
"pp": "α : Type u_1\ninst✝ : LinearOrder α\nn : ℕ\nval : Fin n → α\ni j : Fin n\n⊢ toInt val i < toInt val j ↔ val i < val j",
"ppTerm": "?m.11",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\ninst✝ : LinearOrder α\nn : ℕ\nval : Fin n → α\ni j : Fin n\n⊢ toInt val i < toInt val j ↔ val i < val j"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Tactic.Order.ToInt | {
"line": 82,
"column": 2
} | {
"line": 82,
"column": 13
} | {
"line": 82,
"column": 14
} | [
{
"pp": "α : Type u_1\ninst✝ : LinearOrder α\nn : ℕ\nval : Fin n → α\ni j : Fin n\n⊢ toInt val i ≠ toInt val j ↔ val i ≠ val j",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"id",
"Ne",
"Int",
"Iff",
"Mathlib.Tactic.Order.ToInt.toInt"
],
"usedFVars": [... | [
"α : Type u_1\ninst✝ : LinearOrder α\nn : ℕ\nval : Fin n → α\ni j : Fin n\n⊢ ¬toInt val i = toInt val j ↔ ¬val i = val j"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Tactic.Order.ToInt | {
"line": 85,
"column": 2
} | {
"line": 85,
"column": 13
} | {
"line": 85,
"column": 14
} | [
{
"pp": "α : Type u_1\ninst✝ : LinearOrder α\nn : ℕ\nval : Fin n → α\ni j : Fin n\n⊢ ¬toInt val i ≤ toInt val j ↔ ¬val i ≤ val j",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"congrArg",
"Int.instLinearOrder",
"PartialOrder.toPreor... | [
"α : Type u_1\ninst✝ : LinearOrder α\nn : ℕ\nval : Fin n → α\ni j : Fin n\n⊢ toInt val j < toInt val i ↔ val j < val i"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Tactic.Order.ToInt | {
"line": 88,
"column": 2
} | {
"line": 88,
"column": 13
} | {
"line": 88,
"column": 14
} | [
{
"pp": "α : Type u_1\ninst✝ : LinearOrder α\nn : ℕ\nval : Fin n → α\ni j : Fin n\n⊢ ¬toInt val i < toInt val j ↔ ¬val i < val j",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"congrArg",
"Int.instLinearOrder",
"PartialOrder.toPreor... | [
"α : Type u_1\ninst✝ : LinearOrder α\nn : ℕ\nval : Fin n → α\ni j : Fin n\n⊢ toInt val j ≤ toInt val i ↔ val j ≤ val i"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Tactic.Order.ToInt | {
"line": 106,
"column": 43
} | {
"line": 106,
"column": 75
} | {
"line": 106,
"column": 76
} | [
{
"pp": "α✝ : Type u_1\ninst✝ : LinearOrder α✝\nn : ℕ\nval : Fin n → α✝\ni j k : Fin n\nu : Level\nα :\n let u := u;\n Q(Type u)\natoms : Array Q(«$α»)\nh : ¬atoms.isEmpty = true\n⊢ 0 < atoms.size",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"id",
"instOfNa... | [
"α✝ : Type u_1\ninst✝ : LinearOrder α✝\nn : ℕ\nval : Fin n → α✝\ni j k : Fin n\nu : Level\nα :\n let u := u;\n Q(Type u)\natoms : Array Q(«$α»)\nh : ¬atoms.isEmpty = true\n⊢ ¬atoms = #[]"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Algebra.Operations | {
"line": 141,
"column": 83
} | {
"line": 144,
"column": 52
} | {
"line": 146,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝⁶ : Semiring R\nA : Type v\ninst✝⁵ : Semiring A\ninst✝⁴ : Module R A\nM : Type u_1\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\ninst✝¹ : Module A M\ninst✝ : IsScalarTower R A M\nI : Submodule R A\nN : Submodule R M\nx : M\nhx : x ∈ I • N\np : (x : M) → x ∈ I • N → Prop\nsmul : ∀ (r ... | [] | by
refine Exists.elim ?_ fun (h : x ∈ I • N) (H : p x h) ↦ H
exact smul_induction_on hx (fun a ha x hx ↦ ⟨_, smul _ ha _ hx⟩)
fun x y ⟨_, hx⟩ ⟨_, hy⟩ ↦ ⟨_, add _ _ _ _ hx hy⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Coprime.Lemmas | {
"line": 232,
"column": 2
} | {
"line": 232,
"column": 36
} | {
"line": 232,
"column": 37
} | [
{
"pp": "α : Type u_2\nI : Type u_1\ninst✝¹ : CommMonoid α\ninst✝ : DecompositionMonoid α\nx : α\ns : I → α\nt : Finset I\n⊢ (∀ i ∈ t, IsRelPrime x (s i)) → IsRelPrime x (∏ i ∈ t, s i)",
"ppTerm": "?m.17",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_2\nI : Type u_1\ninst✝¹ : CommMonoid α\ninst✝ : DecompositionMonoid α\nx : α\ns : I → α\nt : Finset I\n⊢ (∀ i ∈ t, IsRelPrime x (s i)) → IsRelPrime x (∏ i ∈ t, s i)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Coprime.Lemmas | {
"line": 240,
"column": 2
} | {
"line": 240,
"column": 36
} | {
"line": 240,
"column": 37
} | [
{
"pp": "α : Type u_1\nI : Type u_2\ninst✝¹ : CommMonoid α\ninst✝ : DecompositionMonoid α\nx : α\ns : I → α\nt : Finset I\n⊢ IsRelPrime x (∏ i ∈ t, s i) ↔ ∀ i ∈ t, IsRelPrime x (s i)",
"ppTerm": "?m.17",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\nI : Type u_2\ninst✝¹ : CommMonoid α\ninst✝ : DecompositionMonoid α\nx : α\ns : I → α\nt : Finset I\n⊢ IsRelPrime x (∏ i ∈ t, s i) ↔ ∀ i ∈ t, IsRelPrime x (s i)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Algebra.Operations | {
"line": 572,
"column": 86
} | {
"line": 574,
"column": 30
} | {
"line": 576,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nP : Submodule R A\nx y : A\n⊢ x ∈ P * (R ∙ y) ↔ ∃ z ∈ P, z * y = x",
"ppTerm": "?m.39",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Submodule",
"RingHomSurjective.ids",
"Al... | [] | by
simp_rw [mul_eq_map₂, map₂_span_singleton_eq_map_flip, mem_map, LinearMap.flip_apply,
LinearMap.mul_apply_apply] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Ideal.Prod | {
"line": 73,
"column": 2
} | {
"line": 73,
"column": 13
} | {
"line": 73,
"column": 14
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝¹ : Semiring R\ninst✝ : Semiring S\nI : Ideal (R × S)\nr : R\ns' : S\nh₁ : (r, s') ∈ I\nr' : R\ns : S\nh₂ : (r', s) ∈ I\n⊢ ((RingHom.fst R S) (r, s'), (RingHom.snd R S) (r', s)) ∈ I",
"ppTerm": "?m.175",
"assigned": true,
"usedConstants": [
"Semiring.toMod... | [
"R : Type u\nS : Type v\ninst✝¹ : Semiring R\ninst✝ : Semiring S\nI : Ideal (R × S)\nr : R\ns' : S\nh₁ : (r, s') ∈ I\nr' : R\ns : S\nh₂ : (r', s) ∈ I\n⊢ (r, s) ∈ I"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Algebra.Operations | {
"line": 764,
"column": 7
} | {
"line": 764,
"column": 43
} | {
"line": 764,
"column": 44
} | [
{
"pp": "R : Type u\ninst✝³ : CommSemiring R\nA : Type v\ninst✝² : Semiring A\ninst✝¹ : Algebra R A\ninst✝ : FaithfulSMul R A\nx✝ : Aˣ\n⊢ x✝ ∈ (Units.map ↑(spanSingleton R)).ker ↔ x✝ ∈ (Units.map ↑(algebraMap R A)).range",
"ppTerm": "?m.54",
"assigned": true,
"usedConstants": [
"Units.val",
... | [
"R : Type u\ninst✝³ : CommSemiring R\nA : Type v\ninst✝² : Semiring A\ninst✝¹ : Algebra R A\ninst✝ : FaithfulSMul R A\nx✝ : Aˣ\n⊢ 1 = R ∙ ↑x✝ ↔ ∃ x, ↑x✝ = (algebraMap R A) ↑x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Ideal.Prod | {
"line": 161,
"column": 4
} | {
"line": 161,
"column": 15
} | {
"line": 161,
"column": 16
} | [
{
"pp": "case mem_or_mem'\nR : Type u\nS : Type v\ninst✝¹ : Semiring R\ninst✝ : Semiring S\nI : Ideal R\nh : (I.prod ⊤).IsPrime\nx y : R\nhxy : x * y ∈ I\nthis : (x, 1) * (y, 1) ∈ I.prod ⊤\n⊢ x ∈ I ∨ y ∈ I",
"ppTerm": "?mem_or_mem'",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"... | [
"case mem_or_mem'\nR : Type u\nS : Type v\ninst✝¹ : Semiring R\ninst✝ : Semiring S\nI : Ideal R\nh : (I.prod ⊤).IsPrime\nx y : R\nhxy : x * y ∈ I\nthis : (x, 1) * (y, 1) ∈ I.prod ⊤\n⊢ x ∈ I ∨ y ∈ I"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Ideal.Prod | {
"line": 171,
"column": 16
} | {
"line": 171,
"column": 27
} | {
"line": 171,
"column": 28
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝¹ : Semiring R\ninst✝ : Semiring S\nI : Ideal R\nh : I.IsPrime\n⊢ I.prod ⊤ ≠ ⊤",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Semiring.toModule",
"and_true",
"congrArg",
"id",
"Prod.instSemiring",
"... | [
"R : Type u\nS : Type v\ninst✝¹ : Semiring R\ninst✝ : Semiring S\nI : Ideal R\nh : I.IsPrime\n⊢ ¬I = ⊤"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Ideal.Prod | {
"line": 172,
"column": 26
} | {
"line": 172,
"column": 37
} | {
"line": 172,
"column": 38
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝¹ : Semiring R\ninst✝ : Semiring S\nI : Ideal R\nh : I.IsPrime\nx y : R × S\n⊢ x * y ∈ I.prod ⊤ → x ∈ I.prod ⊤ ∨ y ∈ I.prod ⊤",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Submodule.mem_top._simp_1",
"Semiring.toModule",
... | [
"R : Type u\nS : Type v\ninst✝¹ : Semiring R\ninst✝ : Semiring S\nI : Ideal R\nh : I.IsPrime\nx y : R × S\n⊢ x.1 * y.1 ∈ I → x.1 ∈ I ∨ y.1 ∈ I"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Ideal.Prod | {
"line": 211,
"column": 8
} | {
"line": 211,
"column": 24
} | {
"line": 211,
"column": 25
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝³ : Semiring R\ninst✝² : Semiring S\nI✝ : Ideal R\nJ : Ideal S\ninst✝¹ : IsPrincipalIdealRing R\ninst✝ : IsPrincipalIdealRing S\nI : Ideal (R × S)\n⊢ Submodule.IsPrincipal I",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Semiring... | [
"R : Type u\nS : Type v\ninst✝³ : Semiring R\ninst✝² : Semiring S\nI✝ : Ideal R\nJ : Ideal S\ninst✝¹ : IsPrincipalIdealRing R\ninst✝ : IsPrincipalIdealRing S\nI : Ideal (R × S)\n⊢ Submodule.IsPrincipal ((Ideal.map (RingHom.fst R S) I).prod (Ideal.map (RingHom.snd R S) I))"
] | I.ideal_prod_eq, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Algebra.Operations | {
"line": 895,
"column": 2
} | {
"line": 895,
"column": 13
} | {
"line": 895,
"column": 14
} | [
{
"pp": "R : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nI : Submodule R A\nhI : I ≤ 1\n⊢ I ≤ I * (1 / I)",
"ppTerm": "?m.28",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nI : Submodule R A\nhI : I ≤ 1\n⊢ I ≤ I * (1 / I)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.