module stringlengths 16 90 | startPos dict | endPos dict | nextStartPos dict | goals listlengths 0 96 | goalsAfter listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 375
values | kind stringclasses 379
values |
|---|---|---|---|---|---|---|---|---|
Mathlib.GroupTheory.OreLocalization.Basic | {
"line": 512,
"column": 17
} | {
"line": 512,
"column": 26
} | {
"line": 512,
"column": 26
} | [
{
"pp": "case c.c\nR : Type u_1\ninst✝³ : Monoid R\nS : Submonoid R\ninst✝² : OreSet S\nX : Type ?u.10\ninst✝¹ : MulAction R X\nT : Type u_2\ninst✝ : Monoid T\nf : R →* T\nfS : ↥S →* Tˣ\nhf : ∀ (s : ↥S), f ↑s = ↑(fS s)\nr₁ : R\ns₁ : ↥S\nr₂ : R\ns₂ : ↥S\n⊢ liftExpand (fun r s ↦ ↑(fS s)⁻¹ * f r) ⋯ (r₁ /ₒ s₁ * (r₂... | [
"case c.c\nR : Type u_1\ninst✝³ : Monoid R\nS : Submonoid R\ninst✝² : OreSet S\nX : Type ?u.10\ninst✝¹ : MulAction R X\nT : Type u_2\ninst✝ : Monoid T\nf : R →* T\nfS : ↥S →* Tˣ\nhf : ∀ (s : ↥S), f ↑s = ↑(fS s)\nr₁ : R\ns₁ : ↥S\nr₂ : R\ns₂ : ↥S\n⊢ liftExpand (fun r s ↦ ↑(fS s)⁻¹ * f r) ⋯ (r₁ /ₒ s₁ * (r₂ /ₒ s₂)) =\n... | | _ r₂ s₂
=> | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalCases | null |
Mathlib.GroupTheory.OreLocalization.Basic | {
"line": 607,
"column": 17
} | {
"line": 607,
"column": 26
} | {
"line": 607,
"column": 26
} | [
{
"pp": "case c.c\nR : Type u_1\nR' : Type u_2\nM : Type u_3\nX : Type u_4\ninst✝¹³ : Monoid M\nS : Submonoid M\ninst✝¹² : OreSet S\ninst✝¹¹ : MulAction M X\ninst✝¹⁰ : SMul R X\ninst✝⁹ : SMul R M\ninst✝⁸ : IsScalarTower R M M\ninst✝⁷ : IsScalarTower R M X\ninst✝⁶ : SMul R' X\ninst✝⁵ : SMul R' M\ninst✝⁴ : IsScal... | [
"case c.c\nR : Type u_1\nR' : Type u_2\nM : Type u_3\nX : Type u_4\ninst✝¹³ : Monoid M\nS : Submonoid M\ninst✝¹² : OreSet S\ninst✝¹¹ : MulAction M X\ninst✝¹⁰ : SMul R X\ninst✝⁹ : SMul R M\ninst✝⁸ : IsScalarTower R M M\ninst✝⁷ : IsScalarTower R M X\ninst✝⁶ : SMul R' X\ninst✝⁵ : SMul R' M\ninst✝⁴ : IsScalarTower R' M... | | _ r₂ s₂
=> | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalCases | null |
Mathlib.GroupTheory.OreLocalization.Basic | {
"line": 644,
"column": 17
} | {
"line": 644,
"column": 26
} | {
"line": 644,
"column": 26
} | [
{
"pp": "case c.c\nR : Type u_1\ninst✝¹ : CommMonoid R\nS : Submonoid R\ninst✝ : OreSet S\nr₁ : R\ns₁ : ↥S\nr₂ : R\ns₂ : ↥S\n⊢ r₁ /ₒ s₁ * (r₂ /ₒ s₂) = r₂ /ₒ s₂ * (r₁ /ₒ s₁)",
"ppTerm": "?c.c",
"assigned": true,
"usedConstants": [],
"usedFVars": [],
"usedGoals": [
{
"new": true,... | [
"case c.c\nR : Type u_1\ninst✝¹ : CommMonoid R\nS : Submonoid R\ninst✝ : OreSet S\nr₁ : R\ns₁ : ↥S\nr₂ : R\ns₂ : ↥S\n⊢ r₁ /ₒ s₁ * (r₂ /ₒ s₂) = r₂ /ₒ s₂ * (r₁ /ₒ s₁)"
] | | _ r₂ s₂
=> | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalCases | null |
Mathlib.Order.SuccPred.Basic | {
"line": 386,
"column": 6
} | {
"line": 386,
"column": 24
} | {
"line": 386,
"column": 25
} | [
{
"pp": "α : Type u_1\ninst✝² : PartialOrder α\ninst✝¹ : SuccOrder α\ninst✝ : OrderTop α\n⊢ succ ⊤ = ⊤",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Order.succ",
"congrArg",
"PartialOrder.toPreorder",
"Preorder.toLE",
"id",
"Order.succ... | [
"α : Type u_1\ninst✝² : PartialOrder α\ninst✝¹ : SuccOrder α\ninst✝ : OrderTop α\n⊢ IsMax ⊤"
] | succ_eq_iff_isMax, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Order.SuccPred.Basic | {
"line": 423,
"column": 2
} | {
"line": 423,
"column": 13
} | {
"line": 424,
"column": 2
} | [
{
"pp": "α : Type u_1\ninst✝¹ : LinearOrder α\ninst✝ : SuccOrder α\na b : α\n⊢ a < succ b → a ≤ b",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"Order.succ",
"PartialOrder.toPreorder",
"Preorder.toLE",
"SemilatticeInf.toPartia... | [
"α : Type u_1\ninst✝¹ : LinearOrder α\ninst✝ : SuccOrder α\na b : α\n⊢ b < a → succ b ≤ a"
] | contrapose! | Mathlib.Tactic.Contrapose._aux_Mathlib_Tactic_Contrapose___macroRules_Mathlib_Tactic_Contrapose_contrapose!_1 | Mathlib.Tactic.Contrapose.contrapose! |
Mathlib.Order.SuccPred.Basic | {
"line": 428,
"column": 2
} | {
"line": 428,
"column": 13
} | {
"line": 429,
"column": 2
} | [
{
"pp": "α : Type u_1\ninst✝¹ : LinearOrder α\ninst✝ : SuccOrder α\na b : α\nha : ¬IsMax a\n⊢ b < succ a ↔ b ≤ a",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"Mathlib.Tactic.Contrapose.contrapose_iff₁",
"Order.succ",
"congrArg",
... | [
"α : Type u_1\ninst✝¹ : LinearOrder α\ninst✝ : SuccOrder α\na b : α\nha : ¬IsMax a\n⊢ succ a ≤ b ↔ a < b"
] | contrapose! | Mathlib.Tactic.Contrapose._aux_Mathlib_Tactic_Contrapose___macroRules_Mathlib_Tactic_Contrapose_contrapose!_1 | Mathlib.Tactic.Contrapose.contrapose! |
Mathlib.Order.SuccPred.Basic | {
"line": 433,
"column": 2
} | {
"line": 433,
"column": 13
} | {
"line": 434,
"column": 2
} | [
{
"pp": "α : Type u_1\ninst✝¹ : LinearOrder α\ninst✝ : SuccOrder α\na b : α\nhb : ¬IsMax b\n⊢ b < succ a ↔ b ≤ a",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"Mathlib.Tactic.Contrapose.contrapose_iff₁",
"Order.succ",
"congrArg",
... | [
"α : Type u_1\ninst✝¹ : LinearOrder α\ninst✝ : SuccOrder α\na b : α\nhb : ¬IsMax b\n⊢ succ a ≤ b ↔ a < b"
] | contrapose! | Mathlib.Tactic.Contrapose._aux_Mathlib_Tactic_Contrapose___macroRules_Mathlib_Tactic_Contrapose_contrapose!_1 | Mathlib.Tactic.Contrapose.contrapose! |
Mathlib.Order.SuccPred.Basic | {
"line": 461,
"column": 2
} | {
"line": 462,
"column": 40
} | {
"line": 464,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝¹ : LinearOrder α\ninst✝ : SuccOrder α\na b : α\nha : ¬IsMax a\nhb : ¬IsMax b\n⊢ succ a = succ b ↔ a = b",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"Order.succ",
"congrArg",
"Iff.rfl",
"PartialO... | [] | rw [eq_iff_le_not_lt, eq_iff_le_not_lt, succ_le_succ_iff_of_not_isMax ha hb,
succ_lt_succ_iff_of_not_isMax ha hb] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Order.SuccPred.Basic | {
"line": 461,
"column": 2
} | {
"line": 462,
"column": 40
} | {
"line": 464,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝¹ : LinearOrder α\ninst✝ : SuccOrder α\na b : α\nha : ¬IsMax a\nhb : ¬IsMax b\n⊢ succ a = succ b ↔ a = b",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"Order.succ",
"congrArg",
"Iff.rfl",
"PartialO... | [] | rw [eq_iff_le_not_lt, eq_iff_le_not_lt, succ_le_succ_iff_of_not_isMax ha hb,
succ_lt_succ_iff_of_not_isMax ha hb] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.SuccPred.Basic | {
"line": 461,
"column": 2
} | {
"line": 462,
"column": 40
} | {
"line": 464,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝¹ : LinearOrder α\ninst✝ : SuccOrder α\na b : α\nha : ¬IsMax a\nhb : ¬IsMax b\n⊢ succ a = succ b ↔ a = b",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"Order.succ",
"congrArg",
"Iff.rfl",
"PartialO... | [] | rw [eq_iff_le_not_lt, eq_iff_le_not_lt, succ_le_succ_iff_of_not_isMax ha hb,
succ_lt_succ_iff_of_not_isMax ha hb] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Nat.SuccPred | {
"line": 46,
"column": 4
} | {
"line": 46,
"column": 34
} | {
"line": 47,
"column": 4
} | [
{
"pp": "case zero\nm n a : ℕ\nh : a < 0\n⊢ a ≤ pred 0",
"ppTerm": "?zero",
"assigned": true,
"usedConstants": [
"False.elim",
"Preorder.toLE",
"instOfNatNat",
"LE.le",
"Nat.not_lt_zero",
"Nat.instPreorder",
"Nat",
"OfNat.ofNat",
"Nat.pred"
]... | [
"case succ\nm n a n✝ : ℕ\nh : a < n✝ + 1\n⊢ a ≤ (n✝ + 1).pred"
] | · exact (a.not_lt_zero h).elim | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Order.SuccPred.Archimedean | {
"line": 328,
"column": 6
} | {
"line": 328,
"column": 64
} | {
"line": 329,
"column": 6
} | [
{
"pp": "case h.succ\nα : Type u_1\nβ : Type u_2\ninst✝³ : PartialOrder α\ninst✝² : PredOrder α\ninst✝¹ : IsPredArchimedean α\ns : Set α\ninst✝ : s.OrdConnected\nx✝¹ x✝ : ↑s\nb : α\nhb : b ∈ s\nn : ℕ\nhi : ∀ (c : α) (hc : c ∈ s), b ≤ c → pred^[n] c = b → pred^[n] ⟨c, hc⟩ = ⟨b, hb⟩\nc : α\nhc : c ∈ s\nhbc : b ≤ ... | [
"case h.succ\nα : Type u_1\nβ : Type u_2\ninst✝³ : PartialOrder α\ninst✝² : PredOrder α\ninst✝¹ : IsPredArchimedean α\ns : Set α\ninst✝ : s.OrdConnected\nx✝¹ x✝ : ↑s\nb : α\nhb : b ∈ s\nn : ℕ\nhi : ∀ (c : α) (hc : c ∈ s), b ≤ c → pred^[n] c = b → pred^[n] ⟨c, hc⟩ = ⟨b, hb⟩\nc : α\nhc : c ∈ s\nhbc : b ≤ c\nhn : pred... | simp_all only [Function.iterate_succ, Function.comp_apply] | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Order.SuccPred.Basic | {
"line": 680,
"column": 8
} | {
"line": 680,
"column": 33
} | {
"line": 680,
"column": 34
} | [
{
"pp": "case succ\nα : Type u_1\ninst✝² : PartialOrder α\ninst✝¹ : SuccOrder α\ninst✝ : PredOrder α\ni : α\nn : ℕ\nhn : ¬IsMax (succ^[n - 1] i) → pred^[n] (succ^[n] i) = i\nhin : ¬IsMax (succ^[n + 1 - 1] i)\n⊢ pred^[n + 1] (succ^[n + 1] i) = i",
"ppTerm": "?succ",
"assigned": true,
"usedConstants":... | [
"case succ\nα : Type u_1\ninst✝² : PartialOrder α\ninst✝¹ : SuccOrder α\ninst✝ : PredOrder α\ni : α\nn : ℕ\nhn : ¬IsMax (succ^[n - 1] i) → pred^[n] (succ^[n] i) = i\nhin : ¬IsMax (succ^[n - 0] i)\n⊢ pred^[n + 1] (succ^[n + 1] i) = i"
] | Nat.succ_sub_succ_eq_sub, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Order.SuccPred.Basic | {
"line": 684,
"column": 10
} | {
"line": 684,
"column": 35
} | {
"line": 684,
"column": 36
} | [
{
"pp": "case succ\nα : Type u_1\ninst✝² : PartialOrder α\ninst✝¹ : SuccOrder α\ninst✝ : PredOrder α\ni : α\nn : ℕ\nhn : ¬IsMax (succ^[n + 1 - 1] i) → pred^[n + 1] (succ^[n + 1] i) = i\nhin : ¬IsMax (succ^[n + 1] i)\n⊢ ¬IsMax (succ^[n + 1 - 1] i)",
"ppTerm": "?succ",
"assigned": true,
"usedConstants... | [
"case succ\nα : Type u_1\ninst✝² : PartialOrder α\ninst✝¹ : SuccOrder α\ninst✝ : PredOrder α\ni : α\nn : ℕ\nhn : ¬IsMax (succ^[n - 0] i) → pred^[n + 1] (succ^[n + 1] i) = i\nhin : ¬IsMax (succ^[n + 1] i)\n⊢ ¬IsMax (succ^[n - 0] i)"
] | Nat.succ_sub_succ_eq_sub, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Order.SuccPred | {
"line": 262,
"column": 23
} | {
"line": 262,
"column": 33
} | {
"line": 262,
"column": 33
} | [
{
"pp": "α : Type u_1\nx : α\ninst✝⁴ : LinearOrder α\ninst✝³ : AddMonoidWithOne α\ninst✝² : SuccAddOrder α\ninst✝¹ : IsBotZeroClass α\ninst✝ : NeZero 1\n⊢ x < 1 ∨ x = 1 ↔ x = 0 ∨ x = 1",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"congrArg",
... | [
"α : Type u_1\nx : α\ninst✝⁴ : LinearOrder α\ninst✝³ : AddMonoidWithOne α\ninst✝² : SuccAddOrder α\ninst✝¹ : IsBotZeroClass α\ninst✝ : NeZero 1\n⊢ x = 0 ∨ x = 1 ↔ x = 0 ∨ x = 1"
] | lt_one_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Order.SuccPred.Basic | {
"line": 814,
"column": 4
} | {
"line": 819,
"column": 52
} | {
"line": 819,
"column": 52
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : Preorder α\ninst✝ : NoMaxOrder α\nhα : Nonempty α\na✝ : PredOrder (WithTop α)\n⊢ False",
"ppTerm": "?m.5",
"assigned": true,
"usedConstants": [
"False",
"Eq.ge",
"Preorder.toLT",
"WithTop.coe_lt_top",
"Order.le_pred_of_lt",
... | [] | cases h : pred (⊤ : WithTop α) with
| top => exact hα.elim fun a => (min_of_le_pred h.ge).not_lt <| coe_lt_top a
| coe a =>
obtain ⟨c, hc⟩ := exists_gt a
rw [← coe_lt_coe, ← h] at hc
exact (le_pred_of_lt (coe_lt_top c)).not_gt hc | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalCases | Lean.Parser.Tactic.cases |
Mathlib.Data.ENat.Basic | {
"line": 192,
"column": 63
} | {
"line": 194,
"column": 18
} | {
"line": 196,
"column": 0
} | [
{
"pp": "n : ℕ∞\nhn0 : n ≠ 0\nhxt : n ≠ ⊤\n⊢ 0 < n.toNat",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Nat.instMulZeroClass",
"Preorder.toLT",
"LinearOrderedCommMonoidWithZero.toIsBotZeroClass",
"instTopENat",
"congrArg",
"CommSemiring... | [] | by
rw [pos_iff_ne_zero, ne_eq, ENat.toNat_eq_zero, not_or]
exact ⟨hn0, hxt⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.Finset.Piecewise | {
"line": 151,
"column": 2
} | {
"line": 151,
"column": 22
} | {
"line": 151,
"column": 22
} | [
{
"pp": "ι : Type u_1\ns : Finset ι\ninst✝ : (j : ι) → Decidable (j ∈ s)\nπ : ι → Type u_3\nt : Set ι\nt' : (i : ι) → Set (π i)\nf g : (i : ι) → π i\nhf : f ∈ t.pi t'\nhg : g ∈ t.pi t'\n⊢ s.piecewise f g ∈ t.pi t'",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congr... | [
"ι : Type u_1\ns : Finset ι\ninst✝ : (j : ι) → Decidable (j ∈ s)\nπ : ι → Type u_3\nt : Set ι\nt' : (i : ι) → Set (π i)\nf g : (i : ι) → π i\nhf : f ∈ t.pi t'\nhg : g ∈ t.pi t'\n⊢ (↑s).piecewise f g ∈ t.pi t'"
] | rw [← piecewise_coe] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Data.Vector.Basic | {
"line": 85,
"column": 8
} | {
"line": 85,
"column": 18
} | {
"line": 85,
"column": 18
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nσ : Type u_4\nφ : Type u_5\nm n : ℕ\np : α → Prop\nf : (a : α) → p a → β\na : α\nv : Vector α n\nhp : p a ∧ ∀ (a : α), a ∈ v.toList → p a\n⊢ ∀ (x : α), x ∈ v.toList → p x",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"Membership.me... | [] | exact hp.2 | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Data.Countable.Basic | {
"line": 92,
"column": 2
} | {
"line": 92,
"column": 65
} | {
"line": 94,
"column": 0
} | [
{
"pp": "α : Type u\nβ : Type v\nπ : α → Type w\ninst✝¹ : Countable α\ninst✝ : Countable β\nf : α → ℕ\nhf : Injective f\ng : β → ℕ\nhg : Injective g\n⊢ Countable (α × β)",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Function.Injective.prodMap",
"Function.Injective.countable"... | [] | exact (Nat.pairEquiv.injective.comp <| hf.prodMap hg).countable | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Data.Sym.Basic | {
"line": 626,
"column": 6
} | {
"line": 626,
"column": 36
} | {
"line": 627,
"column": 4
} | [
{
"pp": "case pos\nα : Type u_1\nn : ℕ\ninst✝ : DecidableEq α\ns : Sym α n.succ\nh : none ∈ decode (Sum.inr s)\na : α\nleft✝ : a ∈ ↑s\nha : Embedding.some a = none\n⊢ False",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Option.some_ne_none"
],
"usedFVars": [
"α",
... | [] | exact Option.some_ne_none _ ha | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Order.Hom.Order | {
"line": 110,
"column": 2
} | {
"line": 110,
"column": 29
} | {
"line": 112,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : Preorder α\nι : Sort u_3\ninst✝ : CompleteLattice β\nf : ι → α →o β\n⊢ ⇑(⨆ i, f i) = ⨆ i, ⇑(f i)",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"congrArg",
"iSup",
"OrderHom.instSupSet",
"PartialOrder.toPreorder",
... | [] | funext x; simp [iSup_apply] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.Hom.Order | {
"line": 110,
"column": 2
} | {
"line": 110,
"column": 29
} | {
"line": 112,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : Preorder α\nι : Sort u_3\ninst✝ : CompleteLattice β\nf : ι → α →o β\n⊢ ⇑(⨆ i, f i) = ⨆ i, ⇑(f i)",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"congrArg",
"iSup",
"OrderHom.instSupSet",
"PartialOrder.toPreorder",
... | [] | funext x; simp [iSup_apply] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.OmegaCompletePartialOrder | {
"line": 208,
"column": 33
} | {
"line": 208,
"column": 65
} | {
"line": 208,
"column": 65
} | [
{
"pp": "α : Type u_2\ninst✝ : OmegaCompletePartialOrder α\nc : Chain α\nx : α\nh : ∀ (i : ℕ), c i ≤ x ∨ x ≤ c i\nthis : ¬∀ (i : ℕ), c i ≤ x\n⊢ ∃ i, ¬c i ≤ x",
"ppTerm": "?m.38",
"assigned": true,
"usedConstants": [
"_private.Mathlib.Order.OmegaCompletePartialOrder.0.OmegaCompletePartialOrder.... | [
"α : Type u_2\ninst✝ : OmegaCompletePartialOrder α\nc : Chain α\nx : α\nh : ∀ (i : ℕ), c i ≤ x ∨ x ≤ c i\nthis : ∃ x_1, ¬c x_1 ≤ x\n⊢ ∃ i, ¬c i ≤ x"
] | simp only [not_forall] at this ⊢ | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Order.FixedPoints | {
"line": 151,
"column": 4
} | {
"line": 153,
"column": 15
} | {
"line": 155,
"column": 0
} | [] | [] | h a a = h a (h a).lfp := congr_arg (h a) ha.symm
_ = (h a).lfp := (h a).map_lfp
_ = a := ha | Lean.Elab.Tactic._aux_Mathlib_Tactic_Widget_Calc___elabRules_Lean_calcTactic_1 | Lean.calcSteps |
Mathlib.Order.BourbakiWitt | {
"line": 182,
"column": 6
} | {
"line": 182,
"column": 19
} | {
"line": 183,
"column": 6
} | [
{
"pp": "α : Type u_1\ninst✝ : ChainCompletePartialOrder α\nx : α\nf : α → α\nle_map : ∀ (x : α), x ≤ f x\nc : NonemptyChain α\nhc : ↑c ⊆ {y | IsExtremePt x f y}\ny : α\nhy : y ∈ bot x f\nhy' : y < cSup c\nh : ∀ z ∈ c, f z ≤ y\n⊢ cSup c ≤ y",
"ppTerm": "?m.282",
"assigned": true,
"usedConstants": [
... | [
"α : Type u_1\ninst✝ : ChainCompletePartialOrder α\nx : α\nf : α → α\nle_map : ∀ (x : α), x ≤ f x\nc : NonemptyChain α\nhc : ↑c ⊆ {y | IsExtremePt x f y}\ny : α\nhy : y ∈ bot x f\nhy' : y < cSup c\nh : ∀ z ∈ c, f z ≤ y\n⊢ ∀ y_1 ∈ c.carrier, y_1 ≤ y"
] | apply cSup_le | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.SetTheory.Cardinal.SchroederBernstein | {
"line": 54,
"column": 4
} | {
"line": 54,
"column": 42
} | {
"line": 55,
"column": 4
} | [
{
"pp": "case inl\nα : Type u\nβ : Type v\nf : α → β\ng : β → α\nhf : Injective f\nhg : Injective g\nR : α → β → Prop\nhp₁ : ∀ (a : α), R a (f a)\nhp₂ : ∀ (b : β), R (g b) b\nhβ : IsEmpty β\n⊢ ∃ h, Bijective h ∧ ∀ (a : α), R a (h a)",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Fun... | [
"case inl\nα : Type u\nβ : Type v\nf : α → β\ng : β → α\nhf : Injective f\nhg : Injective g\nR : α → β → Prop\nhp₁ : ∀ (a : α), R a (f a)\nhp₂ : ∀ (b : β), R (g b) b\nhβ : IsEmpty β\nthis : IsEmpty α\n⊢ ∃ h, Bijective h ∧ ∀ (a : α), R a (h a)"
] | have : IsEmpty α := Function.isEmpty f | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.SetTheory.Cardinal.Order | {
"line": 587,
"column": 4
} | {
"line": 601,
"column": 32
} | {
"line": 603,
"column": 0
} | [
{
"pp": "ι : Type u_1\nf g : ι → Cardinal.{u_2}\nH : ∀ (i : ι), f i < g i\nx✝ : prod g ≤ sum f\nF : ((i : ι) → Quotient.out (g i)) ↪ (i : ι) × Quotient.out (f i)\n⊢ False",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Function.invFun",
"Cardinal.mk_ne_zero_iff",
"Eq.mpr... | [] | have : Inhabited (∀ i : ι, (g i).out) := by
refine ⟨fun i => Classical.choice <| mk_ne_zero_iff.1 ?_⟩
rw [mk_out]
exact (H i).ne_bot
let G := invFun F
have sG : Surjective G := invFun_surjective F.2
choose C hc using
show ∀ i, ∃ b, ∀ a, G ⟨i, a⟩ i ≠ b by
intro i
simp ... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.SetTheory.Cardinal.Order | {
"line": 587,
"column": 4
} | {
"line": 601,
"column": 32
} | {
"line": 603,
"column": 0
} | [
{
"pp": "ι : Type u_1\nf g : ι → Cardinal.{u_2}\nH : ∀ (i : ι), f i < g i\nx✝ : prod g ≤ sum f\nF : ((i : ι) → Quotient.out (g i)) ↪ (i : ι) × Quotient.out (f i)\n⊢ False",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Function.invFun",
"Cardinal.mk_ne_zero_iff",
"Eq.mpr... | [] | have : Inhabited (∀ i : ι, (g i).out) := by
refine ⟨fun i => Classical.choice <| mk_ne_zero_iff.1 ?_⟩
rw [mk_out]
exact (H i).ne_bot
let G := invFun F
have sG : Surjective G := invFun_surjective F.2
choose C hc using
show ∀ i, ∃ b, ∀ a, G ⟨i, a⟩ i ≠ b by
intro i
simp ... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Ring.CompTypeclasses | {
"line": 108,
"column": 6
} | {
"line": 108,
"column": 25
} | {
"line": 108,
"column": 25
} | [
{
"pp": "R₁ : Type u_1\nR₂ : Type u_2\nR₃ : Type u_3\ninst✝⁴ : Semiring R₁\ninst✝³ : Semiring R₂\ninst✝² : Semiring R₃\nσ₁₂ : R₁ →+* R₂\nσ₂₃ : R₂ →+* R₃\nσ₁₃ : R₁ →+* R₃\nσ : R₁ →+* R₂\nσ' : R₂ →+* R₁\ninst✝¹ : RingHomInvPair σ σ'\nσ₂₁ : R₂ →+* R₁\ninst✝ : RingHomInvPair σ₁₂ σ₂₁\n⊢ σ₂₁.comp σ₁₂ = RingHom.id R₁"... | [] | simp only [comp_eq] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Ring.CompTypeclasses | {
"line": 108,
"column": 6
} | {
"line": 108,
"column": 25
} | {
"line": 108,
"column": 25
} | [
{
"pp": "R₁ : Type u_1\nR₂ : Type u_2\nR₃ : Type u_3\ninst✝⁴ : Semiring R₁\ninst✝³ : Semiring R₂\ninst✝² : Semiring R₃\nσ₁₂ : R₁ →+* R₂\nσ₂₃ : R₂ →+* R₃\nσ₁₃ : R₁ →+* R₃\nσ : R₁ →+* R₂\nσ' : R₂ →+* R₁\ninst✝¹ : RingHomInvPair σ σ'\nσ₂₁ : R₂ →+* R₁\ninst✝ : RingHomInvPair σ₁₂ σ₂₁\n⊢ σ₂₁.comp σ₁₂ = RingHom.id R₁"... | [] | simp only [comp_eq] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Ring.CompTypeclasses | {
"line": 108,
"column": 6
} | {
"line": 108,
"column": 25
} | {
"line": 108,
"column": 25
} | [
{
"pp": "R₁ : Type u_1\nR₂ : Type u_2\nR₃ : Type u_3\ninst✝⁴ : Semiring R₁\ninst✝³ : Semiring R₂\ninst✝² : Semiring R₃\nσ₁₂ : R₁ →+* R₂\nσ₂₃ : R₂ →+* R₃\nσ₁₃ : R₁ →+* R₃\nσ : R₁ →+* R₂\nσ' : R₂ →+* R₁\ninst✝¹ : RingHomInvPair σ σ'\nσ₂₁ : R₂ →+* R₁\ninst✝ : RingHomInvPair σ₁₂ σ₂₁\n⊢ σ₂₁.comp σ₁₂ = RingHom.id R₁"... | [] | simp only [comp_eq] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.SetTheory.Cardinal.Basic | {
"line": 848,
"column": 66
} | {
"line": 850,
"column": 39
} | {
"line": 852,
"column": 0
} | [
{
"pp": "α : Type u_1\ns : Set α\n⊢ #↑s + #↑sᶜ = #α",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Compl.compl",
"Classical.propDecidable",
"Membership.mem",
"Equiv.Set.sumCompl",
"Set.Elem",
"Sum",
"Set.instCompl",
"Set.instMembership",
... | [] | by
classical
exact mk_congr (Equiv.Set.sumCompl s) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Module.Equiv.Defs | {
"line": 418,
"column": 2
} | {
"line": 420,
"column": 14
} | {
"line": 422,
"column": 0
} | [
{
"pp": "R₁ : Type u_2\nR₂ : Type u_3\nR₃ : Type u_4\nM₁ : Type u_8\nM₂ : Type u_9\nM₃ : Type u_10\ninst✝⁷ : Semiring R₁\ninst✝⁶ : Semiring R₂\ninst✝⁵ : Semiring R₃\ninst✝⁴ : AddCommMonoid M₁\ninst✝³ : AddCommMonoid M₂\ninst✝² : AddCommMonoid M₃\nmodule_M₁ : Module R₁ M₁\nmodule_M₂ : Module R₂ M₂\nmodule_M₃ : M... | [] | constructor <;> intro H <;> ext
· simp [H]
· simp [← H] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Module.Equiv.Defs | {
"line": 418,
"column": 2
} | {
"line": 420,
"column": 14
} | {
"line": 422,
"column": 0
} | [
{
"pp": "R₁ : Type u_2\nR₂ : Type u_3\nR₃ : Type u_4\nM₁ : Type u_8\nM₂ : Type u_9\nM₃ : Type u_10\ninst✝⁷ : Semiring R₁\ninst✝⁶ : Semiring R₂\ninst✝⁵ : Semiring R₃\ninst✝⁴ : AddCommMonoid M₁\ninst✝³ : AddCommMonoid M₂\ninst✝² : AddCommMonoid M₃\nmodule_M₁ : Module R₁ M₁\nmodule_M₂ : Module R₂ M₂\nmodule_M₃ : M... | [] | constructor <;> intro H <;> ext
· simp [H]
· simp [← H] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Module.Equiv.Defs | {
"line": 430,
"column": 2
} | {
"line": 432,
"column": 14
} | {
"line": 434,
"column": 0
} | [
{
"pp": "R₁ : Type u_2\nR₂ : Type u_3\nR₃ : Type u_4\nM₁ : Type u_8\nM₂ : Type u_9\nM₃ : Type u_10\ninst✝⁷ : Semiring R₁\ninst✝⁶ : Semiring R₂\ninst✝⁵ : Semiring R₃\ninst✝⁴ : AddCommMonoid M₁\ninst✝³ : AddCommMonoid M₂\ninst✝² : AddCommMonoid M₃\nmodule_M₁ : Module R₁ M₁\nmodule_M₂ : Module R₂ M₂\nmodule_M₃ : M... | [] | constructor <;> intro H <;> ext
· simp [H]
· simp [← H] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Module.Equiv.Defs | {
"line": 430,
"column": 2
} | {
"line": 432,
"column": 14
} | {
"line": 434,
"column": 0
} | [
{
"pp": "R₁ : Type u_2\nR₂ : Type u_3\nR₃ : Type u_4\nM₁ : Type u_8\nM₂ : Type u_9\nM₃ : Type u_10\ninst✝⁷ : Semiring R₁\ninst✝⁶ : Semiring R₂\ninst✝⁵ : Semiring R₃\ninst✝⁴ : AddCommMonoid M₁\ninst✝³ : AddCommMonoid M₂\ninst✝² : AddCommMonoid M₃\nmodule_M₁ : Module R₁ M₁\nmodule_M₂ : Module R₂ M₂\nmodule_M₃ : M... | [] | constructor <;> intro H <;> ext
· simp [H]
· simp [← H] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Module.Torsion.Free | {
"line": 174,
"column": 7
} | {
"line": 174,
"column": 21
} | {
"line": 174,
"column": 22
} | [
{
"pp": "R : Type u_1\nM : Type u_3\ninst✝⁴ : Ring R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nm : M\ninst✝¹ : IsCancelMulZero R\ninst✝ : IsTorsionFree R M\nhm : m ≠ 0\nr₁ r₂ : R\nhr : r₁ • m = r₂ • m\n⊢ r₁ = r₂",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtrac... | [
"R : Type u_1\nM : Type u_3\ninst✝⁴ : Ring R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nm : M\ninst✝¹ : IsCancelMulZero R\ninst✝ : IsTorsionFree R M\nhm : m ≠ 0\nr₁ r₂ : R\nhr : r₁ • m - r₂ • m = 0\n⊢ r₁ = r₂"
] | ← sub_eq_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Module.Submodule.Lattice | {
"line": 115,
"column": 2
} | {
"line": 117,
"column": 53
} | {
"line": 119,
"column": 0
} | [
{
"pp": "R : Type u_1\nM : Type u_3\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\np : Submodule R M\n⊢ Subsingleton ↥p ↔ p = ⊥",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Submodule",
"Subtype.mk.congr_simp",
"subsingleton_iff",
... | [] | rw [subsingleton_iff, Submodule.eq_bot_iff]
refine ⟨fun h x hx ↦ by simpa using h ⟨x, hx⟩ ⟨0, p.zero_mem⟩,
fun h ⟨x, hx⟩ ⟨y, hy⟩ ↦ by simp [h x hx, h y hy]⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Module.Submodule.Lattice | {
"line": 115,
"column": 2
} | {
"line": 117,
"column": 53
} | {
"line": 119,
"column": 0
} | [
{
"pp": "R : Type u_1\nM : Type u_3\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\np : Submodule R M\n⊢ Subsingleton ↥p ↔ p = ⊥",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Submodule",
"Subtype.mk.congr_simp",
"subsingleton_iff",
... | [] | rw [subsingleton_iff, Submodule.eq_bot_iff]
refine ⟨fun h x hx ↦ by simpa using h ⟨x, hx⟩ ⟨0, p.zero_mem⟩,
fun h ⟨x, hx⟩ ⟨y, hy⟩ ↦ by simp [h x hx, h y hy]⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Module.Submodule.Lattice | {
"line": 226,
"column": 2
} | {
"line": 226,
"column": 44
} | {
"line": 227,
"column": 2
} | [
{
"pp": "R : Type u_1\nM : Type u_3\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nι : Type u_4\ns : Finset ι\np : ι → Submodule R M\nthis : DecidableEq ι := Classical.decEq ι\n⊢ ↑(s.inf p) = ⋂ i ∈ s, ↑(p i)",
"ppTerm": "?m.38",
"assigned": true,
"usedConstants": [
"Submod... | [
"case refine_1\nR : Type u_1\nM : Type u_3\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nι : Type u_4\ns : Finset ι\np : ι → Submodule R M\nthis : DecidableEq ι := Classical.decEq ι\n⊢ ↑(∅.inf p) = ⋂ i ∈ ∅, ↑(p i)",
"case refine_2\nR : Type u_1\nM : Type u_3\ninst✝² : Semiring R\ninst✝¹ : Ad... | refine s.induction_on ?_ fun i s _ ih ↦ ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Algebra.Module.Submodule.Map | {
"line": 119,
"column": 2
} | {
"line": 119,
"column": 59
} | {
"line": 121,
"column": 0
} | [
{
"pp": "R : Type u_1\nR₂ : Type u_3\nM : Type u_5\nM₂ : Type u_7\ninst✝⁶ : Semiring R\ninst✝⁵ : Semiring R₂\ninst✝⁴ : AddCommMonoid M\ninst✝³ : AddCommMonoid M₂\ninst✝² : Module R M\ninst✝¹ : Module R₂ M₂\nσ₁₂ : R →+* R₂\np : Submodule R M\ninst✝ : RingHomSurjective σ₁₂\nf g : M →ₛₗ[σ₁₂] M₂\nm : M\nhm : m ∈ ↑p... | [] | exact add_mem_sup (mem_map_of_mem hm) (mem_map_of_mem hm) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.Module.Submodule.Map | {
"line": 351,
"column": 50
} | {
"line": 355,
"column": 15
} | {
"line": 357,
"column": 0
} | [
{
"pp": "R : Type u_1\nR₁ : Type u_2\nR₂ : Type u_3\nR₃ : Type u_4\nM : Type u_5\nM₁ : Type u_6\nM₂ : Type u_7\nM₃ : Type u_8\ninst✝¹² : Semiring R\ninst✝¹¹ : Semiring R₂\ninst✝¹⁰ : Semiring R₃\ninst✝⁹ : AddCommMonoid M\ninst✝⁸ : AddCommMonoid M₂\ninst✝⁷ : AddCommMonoid M₃\ninst✝⁶ : Module R M\ninst✝⁵ : Module ... | [] | by
simp only [mem_comap, mem_map, forall_exists_index, and_imp]
intro y hy hxy
rw [hf.eq_iff] at hxy
rwa [← hxy] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Ring.CharZero | {
"line": 100,
"column": 6
} | {
"line": 100,
"column": 20
} | {
"line": 100,
"column": 21
} | [
{
"pp": "R : Type u_2\ninst✝² : NonAssocRing R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\nn : ℕ\na b : R\nh : ↑n * a = ↑n * b\n⊢ n = 0 ∨ a = b",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"HMul.hMul",
"AddGroupWithOne.toAddGroup",
... | [
"R : Type u_2\ninst✝² : NonAssocRing R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\nn : ℕ\na b : R\nh : ↑n * a - ↑n * b = 0\n⊢ n = 0 ∨ a = b"
] | ← sub_eq_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Ring.Prod | {
"line": 36,
"column": 35
} | {
"line": 36,
"column": 56
} | {
"line": 37,
"column": 2
} | [
{
"pp": "case fst\nR : Type u_1\nR' : Type u_2\nS : Type u_3\nS' : Type u_4\nT : Type u_5\ninst✝¹ : Distrib R\ninst✝ : Distrib S\nx✝² x✝¹ x✝ : R × S\n⊢ (x✝² * (x✝¹ + x✝)).1 = (x✝² * x✝¹ + x✝² * x✝).1",
"ppTerm": "?fst",
"assigned": true,
"usedConstants": [
"Distrib.leftDistribClass",
"Di... | [] | exact left_distrib .. | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.Ring.Prod | {
"line": 36,
"column": 35
} | {
"line": 36,
"column": 56
} | {
"line": 37,
"column": 2
} | [
{
"pp": "case snd\nR : Type u_1\nR' : Type u_2\nS : Type u_3\nS' : Type u_4\nT : Type u_5\ninst✝¹ : Distrib R\ninst✝ : Distrib S\nx✝² x✝¹ x✝ : R × S\n⊢ (x✝² * (x✝¹ + x✝)).2 = (x✝² * x✝¹ + x✝² * x✝).2",
"ppTerm": "?snd",
"assigned": true,
"usedConstants": [
"Distrib.leftDistribClass",
"Di... | [] | exact left_distrib .. | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.NonUnitalSubsemiring.Basic | {
"line": 461,
"column": 2
} | {
"line": 461,
"column": 19
} | {
"line": 462,
"column": 2
} | [
{
"pp": "R : Type u\ninst✝ : NonUnitalNonAssocSemiring R\ns : Set R\nx : R\nhx : x ∈ closure ↑(AddSubmonoid.closure s)\nH : NonUnitalSubsemiring R\n⊢ x ∈ (fun t ↦ ⋂ (_ : t ∈ {S | s ⊆ ↑S}), ↑t) H",
"ppTerm": "?m.48",
"assigned": true,
"usedConstants": [
"Set.ofPred",
"Membership.mem",
... | [
"R : Type u\ninst✝ : NonUnitalNonAssocSemiring R\ns : Set R\nx : R\nhx : x ∈ closure ↑(AddSubmonoid.closure s)\nH : NonUnitalSubsemiring R\nJ : H ∈ {S | s ⊆ ↑S}\n⊢ x ∈ (fun h ↦ ↑H) J"
] | rintro - ⟨J, rfl⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.Algebra.Algebra.Basic | {
"line": 363,
"column": 8
} | {
"line": 363,
"column": 25
} | {
"line": 363,
"column": 26
} | [
{
"pp": "R : Type u_1\nA : Type u_2\ninst✝⁷ : CommSemiring R\ninst✝⁶ : Semiring A\ninst✝⁵ : Algebra R A\ninst✝⁴ : FaithfulSMul R A\nG : Type u_3\ninst✝³ : Monoid G\ninst✝² : MulSemiringAction G A\ninst✝¹ : SMul G R\ninst✝ : SMulDistribClass G R A\nx✝² x✝¹ : G\nx✝ : R\n⊢ (algebraMap R A) ((x✝² * x✝¹) • x✝) = (al... | [
"R : Type u_1\nA : Type u_2\ninst✝⁷ : CommSemiring R\ninst✝⁶ : Semiring A\ninst✝⁵ : Algebra R A\ninst✝⁴ : FaithfulSMul R A\nG : Type u_3\ninst✝³ : Monoid G\ninst✝² : MulSemiringAction G A\ninst✝¹ : SMul G R\ninst✝ : SMulDistribClass G R A\nx✝² x✝¹ : G\nx✝ : R\n⊢ (x✝² * x✝¹) • (algebraMap R A) x✝ = (algebraMap R A) ... | algebraMap.smul', | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Algebra.Basic | {
"line": 363,
"column": 26
} | {
"line": 363,
"column": 43
} | {
"line": 363,
"column": 44
} | [
{
"pp": "R : Type u_1\nA : Type u_2\ninst✝⁷ : CommSemiring R\ninst✝⁶ : Semiring A\ninst✝⁵ : Algebra R A\ninst✝⁴ : FaithfulSMul R A\nG : Type u_3\ninst✝³ : Monoid G\ninst✝² : MulSemiringAction G A\ninst✝¹ : SMul G R\ninst✝ : SMulDistribClass G R A\nx✝² x✝¹ : G\nx✝ : R\n⊢ (x✝² * x✝¹) • (algebraMap R A) x✝ = (alge... | [
"R : Type u_1\nA : Type u_2\ninst✝⁷ : CommSemiring R\ninst✝⁶ : Semiring A\ninst✝⁵ : Algebra R A\ninst✝⁴ : FaithfulSMul R A\nG : Type u_3\ninst✝³ : Monoid G\ninst✝² : MulSemiringAction G A\ninst✝¹ : SMul G R\ninst✝ : SMulDistribClass G R A\nx✝² x✝¹ : G\nx✝ : R\n⊢ (x✝² * x✝¹) • (algebraMap R A) x✝ = x✝² • (algebraMap... | algebraMap.smul', | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Algebra.Basic | {
"line": 363,
"column": 44
} | {
"line": 363,
"column": 61
} | {
"line": 363,
"column": 62
} | [
{
"pp": "R : Type u_1\nA : Type u_2\ninst✝⁷ : CommSemiring R\ninst✝⁶ : Semiring A\ninst✝⁵ : Algebra R A\ninst✝⁴ : FaithfulSMul R A\nG : Type u_3\ninst✝³ : Monoid G\ninst✝² : MulSemiringAction G A\ninst✝¹ : SMul G R\ninst✝ : SMulDistribClass G R A\nx✝² x✝¹ : G\nx✝ : R\n⊢ (x✝² * x✝¹) • (algebraMap R A) x✝ = x✝² •... | [
"R : Type u_1\nA : Type u_2\ninst✝⁷ : CommSemiring R\ninst✝⁶ : Semiring A\ninst✝⁵ : Algebra R A\ninst✝⁴ : FaithfulSMul R A\nG : Type u_3\ninst✝³ : Monoid G\ninst✝² : MulSemiringAction G A\ninst✝¹ : SMul G R\ninst✝ : SMulDistribClass G R A\nx✝² x✝¹ : G\nx✝ : R\n⊢ (x✝² * x✝¹) • (algebraMap R A) x✝ = x✝² • x✝¹ • (alge... | algebraMap.smul', | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Algebra.Basic | {
"line": 357,
"column": 8
} | {
"line": 357,
"column": 25
} | {
"line": 357,
"column": 26
} | [
{
"pp": "R : Type u_1\nA : Type u_2\ninst✝⁷ : CommSemiring R\ninst✝⁶ : Semiring A\ninst✝⁵ : Algebra R A\ninst✝⁴ : FaithfulSMul R A\nG : Type u_3\ninst✝³ : Monoid G\ninst✝² : MulSemiringAction G A\ninst✝¹ : SMul G R\ninst✝ : SMulDistribClass G R A\nx✝ : R\n⊢ (algebraMap R A) (1 • x✝) = (algebraMap R A) x✝",
... | [
"R : Type u_1\nA : Type u_2\ninst✝⁷ : CommSemiring R\ninst✝⁶ : Semiring A\ninst✝⁵ : Algebra R A\ninst✝⁴ : FaithfulSMul R A\nG : Type u_3\ninst✝³ : Monoid G\ninst✝² : MulSemiringAction G A\ninst✝¹ : SMul G R\ninst✝ : SMulDistribClass G R A\nx✝ : R\n⊢ 1 • (algebraMap R A) x✝ = (algebraMap R A) x✝"
] | algebraMap.smul', | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Algebra.Basic | {
"line": 360,
"column": 8
} | {
"line": 360,
"column": 25
} | {
"line": 360,
"column": 26
} | [
{
"pp": "R : Type u_1\nA : Type u_2\ninst✝⁷ : CommSemiring R\ninst✝⁶ : Semiring A\ninst✝⁵ : Algebra R A\ninst✝⁴ : FaithfulSMul R A\nG : Type u_3\ninst✝³ : Monoid G\ninst✝² : MulSemiringAction G A\ninst✝¹ : SMul G R\ninst✝ : SMulDistribClass G R A\nx✝ : G\n⊢ (algebraMap R A) (x✝ • 0) = (algebraMap R A) 0",
"... | [
"R : Type u_1\nA : Type u_2\ninst✝⁷ : CommSemiring R\ninst✝⁶ : Semiring A\ninst✝⁵ : Algebra R A\ninst✝⁴ : FaithfulSMul R A\nG : Type u_3\ninst✝³ : Monoid G\ninst✝² : MulSemiringAction G A\ninst✝¹ : SMul G R\ninst✝ : SMulDistribClass G R A\nx✝ : G\n⊢ x✝ • (algebraMap R A) 0 = (algebraMap R A) 0"
] | algebraMap.smul', | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Algebra.Basic | {
"line": 366,
"column": 8
} | {
"line": 366,
"column": 25
} | {
"line": 366,
"column": 26
} | [
{
"pp": "R : Type u_1\nA : Type u_2\ninst✝⁷ : CommSemiring R\ninst✝⁶ : Semiring A\ninst✝⁵ : Algebra R A\ninst✝⁴ : FaithfulSMul R A\nG : Type u_3\ninst✝³ : Monoid G\ninst✝² : MulSemiringAction G A\ninst✝¹ : SMul G R\ninst✝ : SMulDistribClass G R A\nx✝² : G\nx✝¹ x✝ : R\n⊢ (algebraMap R A) (x✝² • (x✝¹ + x✝)) = (al... | [
"R : Type u_1\nA : Type u_2\ninst✝⁷ : CommSemiring R\ninst✝⁶ : Semiring A\ninst✝⁵ : Algebra R A\ninst✝⁴ : FaithfulSMul R A\nG : Type u_3\ninst✝³ : Monoid G\ninst✝² : MulSemiringAction G A\ninst✝¹ : SMul G R\ninst✝ : SMulDistribClass G R A\nx✝² : G\nx✝¹ x✝ : R\n⊢ x✝² • (algebraMap R A) (x✝¹ + x✝) = (algebraMap R A) ... | algebraMap.smul', | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Algebra.Basic | {
"line": 369,
"column": 8
} | {
"line": 369,
"column": 25
} | {
"line": 369,
"column": 26
} | [
{
"pp": "R : Type u_1\nA : Type u_2\ninst✝⁷ : CommSemiring R\ninst✝⁶ : Semiring A\ninst✝⁵ : Algebra R A\ninst✝⁴ : FaithfulSMul R A\nG : Type u_3\ninst✝³ : Monoid G\ninst✝² : MulSemiringAction G A\ninst✝¹ : SMul G R\ninst✝ : SMulDistribClass G R A\nx✝ : G\n⊢ (algebraMap R A) (x✝ • 1) = (algebraMap R A) 1",
"... | [
"R : Type u_1\nA : Type u_2\ninst✝⁷ : CommSemiring R\ninst✝⁶ : Semiring A\ninst✝⁵ : Algebra R A\ninst✝⁴ : FaithfulSMul R A\nG : Type u_3\ninst✝³ : Monoid G\ninst✝² : MulSemiringAction G A\ninst✝¹ : SMul G R\ninst✝ : SMulDistribClass G R A\nx✝ : G\n⊢ x✝ • (algebraMap R A) 1 = (algebraMap R A) 1"
] | algebraMap.smul', | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Algebra.Basic | {
"line": 372,
"column": 8
} | {
"line": 372,
"column": 25
} | {
"line": 372,
"column": 26
} | [
{
"pp": "R : Type u_1\nA : Type u_2\ninst✝⁷ : CommSemiring R\ninst✝⁶ : Semiring A\ninst✝⁵ : Algebra R A\ninst✝⁴ : FaithfulSMul R A\nG : Type u_3\ninst✝³ : Monoid G\ninst✝² : MulSemiringAction G A\ninst✝¹ : SMul G R\ninst✝ : SMulDistribClass G R A\nx✝² : G\nx✝¹ x✝ : R\n⊢ (algebraMap R A) (x✝² • (x✝¹ * x✝)) = (al... | [
"R : Type u_1\nA : Type u_2\ninst✝⁷ : CommSemiring R\ninst✝⁶ : Semiring A\ninst✝⁵ : Algebra R A\ninst✝⁴ : FaithfulSMul R A\nG : Type u_3\ninst✝³ : Monoid G\ninst✝² : MulSemiringAction G A\ninst✝¹ : SMul G R\ninst✝ : SMulDistribClass G R A\nx✝² : G\nx✝¹ x✝ : R\n⊢ x✝² • (algebraMap R A) (x✝¹ * x✝) = (algebraMap R A) ... | algebraMap.smul', | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Algebra.Basic | {
"line": 372,
"column": 44
} | {
"line": 372,
"column": 61
} | {
"line": 372,
"column": 62
} | [
{
"pp": "R : Type u_1\nA : Type u_2\ninst✝⁷ : CommSemiring R\ninst✝⁶ : Semiring A\ninst✝⁵ : Algebra R A\ninst✝⁴ : FaithfulSMul R A\nG : Type u_3\ninst✝³ : Monoid G\ninst✝² : MulSemiringAction G A\ninst✝¹ : SMul G R\ninst✝ : SMulDistribClass G R A\nx✝² : G\nx✝¹ x✝ : R\n⊢ x✝² • ((algebraMap R A) x✝¹ * (algebraMap... | [
"R : Type u_1\nA : Type u_2\ninst✝⁷ : CommSemiring R\ninst✝⁶ : Semiring A\ninst✝⁵ : Algebra R A\ninst✝⁴ : FaithfulSMul R A\nG : Type u_3\ninst✝³ : Monoid G\ninst✝² : MulSemiringAction G A\ninst✝¹ : SMul G R\ninst✝ : SMulDistribClass G R A\nx✝² : G\nx✝¹ x✝ : R\n⊢ x✝² • ((algebraMap R A) x✝¹ * (algebraMap R A) x✝) = ... | algebraMap.smul', | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Algebra.Basic | {
"line": 372,
"column": 62
} | {
"line": 372,
"column": 79
} | {
"line": 373,
"column": 6
} | [
{
"pp": "R : Type u_1\nA : Type u_2\ninst✝⁷ : CommSemiring R\ninst✝⁶ : Semiring A\ninst✝⁵ : Algebra R A\ninst✝⁴ : FaithfulSMul R A\nG : Type u_3\ninst✝³ : Monoid G\ninst✝² : MulSemiringAction G A\ninst✝¹ : SMul G R\ninst✝ : SMulDistribClass G R A\nx✝² : G\nx✝¹ x✝ : R\n⊢ x✝² • ((algebraMap R A) x✝¹ * (algebraMap... | [
"R : Type u_1\nA : Type u_2\ninst✝⁷ : CommSemiring R\ninst✝⁶ : Semiring A\ninst✝⁵ : Algebra R A\ninst✝⁴ : FaithfulSMul R A\nG : Type u_3\ninst✝³ : Monoid G\ninst✝² : MulSemiringAction G A\ninst✝¹ : SMul G R\ninst✝ : SMulDistribClass G R A\nx✝² : G\nx✝¹ x✝ : R\n⊢ x✝² • ((algebraMap R A) x✝¹ * (algebraMap R A) x✝) = ... | algebraMap.smul', | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Ring.Subsemiring.Basic | {
"line": 471,
"column": 2
} | {
"line": 471,
"column": 19
} | {
"line": 472,
"column": 2
} | [
{
"pp": "R : Type u\ninst✝ : NonAssocSemiring R\ns : Set R\nx : R\nhx : x ∈ closure ↑(AddSubmonoid.closure s)\nH : Subsemiring R\n⊢ x ∈ (fun t ↦ ⋂ (_ : t ∈ {S | s ⊆ ↑S}), ↑t) H",
"ppTerm": "?m.48",
"assigned": true,
"usedConstants": [
"Subsemiring.instSetLike",
"Set.ofPred",
"Membe... | [
"R : Type u\ninst✝ : NonAssocSemiring R\ns : Set R\nx : R\nhx : x ∈ closure ↑(AddSubmonoid.closure s)\nH : Subsemiring R\nJ : H ∈ {S | s ⊆ ↑S}\n⊢ x ∈ (fun h ↦ ↑H) J"
] | rintro - ⟨J, rfl⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.Algebra.GroupWithZero.Associated | {
"line": 76,
"column": 46
} | {
"line": 76,
"column": 60
} | {
"line": 76,
"column": 61
} | [
{
"pp": "M : Type u_2\nN : Type u_3\ninst✝³ : Monoid M\ninst✝² : Monoid N\nF : Type u_4\ninst✝¹ : FunLike F M N\ninst✝ : MonoidHomClass F M N\nf : F\nx y : M\nu : Mˣ\nha : x * ↑u = y\n⊢ f x * ↑((Units.map ↑f) u) = f x * f ↑u",
"ppTerm": "?m.39",
"assigned": true,
"usedConstants": [
"Units.val"... | [
"M : Type u_2\nN : Type u_3\ninst✝³ : Monoid M\ninst✝² : Monoid N\nF : Type u_4\ninst✝¹ : FunLike F M N\ninst✝ : MonoidHomClass F M N\nf : F\nx y : M\nu : Mˣ\nha : x * ↑u = y\n⊢ f x * ↑f ↑u = f x * f ↑u"
] | Units.coe_map, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.GroupWithZero.Associated | {
"line": 136,
"column": 45
} | {
"line": 138,
"column": 41
} | {
"line": 140,
"column": 0
} | [
{
"pp": "N : Type u_2\ninst✝ : CommMonoid N\nu a b : N\nhu : IsUnit u\n⊢ u * a ~ᵤ b ↔ a ~ᵤ b",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"associated_mul_isUnit_left_iff",
"HMul.hMul",
"CommMonoid.toCommSemigroup",
"Monoid.toMulOneClass",
"c... | [] | by
rw [mul_comm]
exact associated_mul_isUnit_left_iff hu | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.GroupWithZero.Associated | {
"line": 184,
"column": 17
} | {
"line": 184,
"column": 57
} | {
"line": 186,
"column": 0
} | [
{
"pp": "case succ\nM : Type u_1\ninst✝ : CommMonoid M\na b : M\nh : a ~ᵤ b\nn : ℕ\nih : a ^ n ~ᵤ b ^ n\n⊢ a ^ (n + 1) ~ᵤ b ^ (n + 1)",
"ppTerm": "?succ",
"assigned": true,
"usedConstants": [
"Monoid",
"Eq.mpr",
"HMul.hMul",
"Monoid.toMulOneClass",
"congrArg",
"HE... | [] | convert! h.mul_mul ih <;> rw [pow_succ'] | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Algebra.GroupWithZero.Associated | {
"line": 184,
"column": 17
} | {
"line": 184,
"column": 57
} | {
"line": 186,
"column": 0
} | [
{
"pp": "case succ\nM : Type u_1\ninst✝ : CommMonoid M\na b : M\nh : a ~ᵤ b\nn : ℕ\nih : a ^ n ~ᵤ b ^ n\n⊢ a ^ (n + 1) ~ᵤ b ^ (n + 1)",
"ppTerm": "?succ",
"assigned": true,
"usedConstants": [
"Monoid",
"Eq.mpr",
"HMul.hMul",
"Monoid.toMulOneClass",
"congrArg",
"HE... | [] | convert! h.mul_mul ih <;> rw [pow_succ'] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.GroupWithZero.Associated | {
"line": 184,
"column": 17
} | {
"line": 184,
"column": 57
} | {
"line": 186,
"column": 0
} | [
{
"pp": "case succ\nM : Type u_1\ninst✝ : CommMonoid M\na b : M\nh : a ~ᵤ b\nn : ℕ\nih : a ^ n ~ᵤ b ^ n\n⊢ a ^ (n + 1) ~ᵤ b ^ (n + 1)",
"ppTerm": "?succ",
"assigned": true,
"usedConstants": [
"Monoid",
"Eq.mpr",
"HMul.hMul",
"Monoid.toMulOneClass",
"congrArg",
"HE... | [] | convert! h.mul_mul ih <;> rw [pow_succ'] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.GroupWithZero.Associated | {
"line": 525,
"column": 10
} | {
"line": 525,
"column": 28
} | {
"line": 525,
"column": 29
} | [
{
"pp": "M : Type u_1\ninst✝ : CommMonoid M\na : M\n⊢ IsUnit (Associates.mk a) ↔ a ~ᵤ 1",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Associates.mk",
"MulOne.toOne",
"Associates.instCommMonoid",
"Monoid.toMulOneClass",
"congrArg",
"IsU... | [
"M : Type u_1\ninst✝ : CommMonoid M\na : M\n⊢ Associates.mk a = 1 ↔ a ~ᵤ 1"
] | isUnit_iff_eq_one, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.GroupWithZero.NonZeroDivisors | {
"line": 402,
"column": 2
} | {
"line": 402,
"column": 13
} | {
"line": 403,
"column": 2
} | [
{
"pp": "M₀ : Type u_1\ninst✝ : CommMonoidWithZero M₀\na : M₀\n⊢ (∀ (x : Associates M₀), x * Associates.mk a = 0 → x = 0) ↔ ∀ (x : M₀), x * a = 0 → x = 0",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Push.not_forall_eq",
"Eq.mpr",
"Associates.mk",
... | [
"M₀ : Type u_1\ninst✝ : CommMonoidWithZero M₀\na : M₀\n⊢ (∃ x, x * Associates.mk a = 0 ∧ x ≠ 0) ↔ ∃ x, x * a = 0 ∧ x ≠ 0"
] | contrapose! | Mathlib.Tactic.Contrapose._aux_Mathlib_Tactic_Contrapose___macroRules_Mathlib_Tactic_Contrapose_contrapose!_1 | Mathlib.Tactic.Contrapose.contrapose! |
Mathlib.Algebra.GroupWithZero.Associated | {
"line": 649,
"column": 2
} | {
"line": 649,
"column": 76
} | {
"line": 651,
"column": 0
} | [
{
"pp": "M : Type u_1\ninst✝ : CommMonoidWithZero M\np : M\n⊢ (Associates.mk p ≠ 0 ∧\n ¬IsUnit (Associates.mk p) ∧\n ∀ (a : M) (b : Associates M),\n Associates.mk p ∣ Associates.mk a * b → Associates.mk p ∣ Associates.mk a ∨ Associates.mk p ∣ b) ↔\n p ≠ 0 ∧ ¬IsUnit p ∧ ∀ (a b : M), p ∣ a... | [] | simp only [forall_associated, mk_ne_zero, isUnit_mk, mk_mul_mk, mk_dvd_mk] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Module.Submodule.Pointwise | {
"line": 434,
"column": 2
} | {
"line": 434,
"column": 9
} | {
"line": 436,
"column": 0
} | [
{
"pp": "case mpr\nR : Type u_2\nM : Type u_3\ninst✝⁵ : Semiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nS : Type u_4\ninst✝² : Monoid S\ninst✝¹ : DistribMulAction S M\nN : Submodule R M\ninst✝ : SMulCommClass R S M\nr : S\nx : M\n⊢ (∃ m ∈ N, x = r • m) → x ∈ {r} • N",
"ppTerm": "?mpr",
"assig... | [] | · aesop | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Order.ModularLattice | {
"line": 388,
"column": 2
} | {
"line": 388,
"column": 97
} | {
"line": 389,
"column": 2
} | [
{
"pp": "case disjoint\nα : Type u_1\ninst✝² : Lattice α\ninst✝¹ : BoundedOrder α\ninst✝ : IsModularLattice α\na b c : α\nh₁ : IsCompl b c\nh₂ : b ≤ a\n⊢ Disjoint ⟨a ⊓ b, ⋯⟩ ⟨a ⊓ c, ⋯⟩",
"ppTerm": "?disjoint",
"assigned": true,
"usedConstants": [
"Set.Iic.semilatticeInf",
"Subtype.mk.con... | [
"case codisjoint\nα : Type u_1\ninst✝² : Lattice α\ninst✝¹ : BoundedOrder α\ninst✝ : IsModularLattice α\na b c : α\nh₁ : IsCompl b c\nh₂ : b ≤ a\n⊢ Codisjoint ⟨a ⊓ b, ⋯⟩ ⟨a ⊓ c, ⋯⟩"
] | · simp [disjoint_iff, Subtype.ext_iff, inf_comm a c, inf_assoc a, ← inf_assoc b, h₁.inf_eq_bot] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.LinearAlgebra.Span.Defs | {
"line": 233,
"column": 2
} | {
"line": 233,
"column": 52
} | {
"line": 235,
"column": 0
} | [
{
"pp": "M : Type u_4\ninst✝ : AddCommMonoid M\ns : AddSubmonoid M\n⊢ (span ℕ ↑s).toAddSubmonoid = s",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Submodule.toAddSubmonoid",
"congrArg",
"AddMonoid.toAddZeroClass",
"id",
"AddSubmonoid",
... | [] | rw [span_nat_eq_addSubmonoidClosure, s.closure_eq] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.Span.Defs | {
"line": 233,
"column": 2
} | {
"line": 233,
"column": 52
} | {
"line": 235,
"column": 0
} | [
{
"pp": "M : Type u_4\ninst✝ : AddCommMonoid M\ns : AddSubmonoid M\n⊢ (span ℕ ↑s).toAddSubmonoid = s",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Submodule.toAddSubmonoid",
"congrArg",
"AddMonoid.toAddZeroClass",
"id",
"AddSubmonoid",
... | [] | rw [span_nat_eq_addSubmonoidClosure, s.closure_eq] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Span.Defs | {
"line": 233,
"column": 2
} | {
"line": 233,
"column": 52
} | {
"line": 235,
"column": 0
} | [
{
"pp": "M : Type u_4\ninst✝ : AddCommMonoid M\ns : AddSubmonoid M\n⊢ (span ℕ ↑s).toAddSubmonoid = s",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Submodule.toAddSubmonoid",
"congrArg",
"AddMonoid.toAddZeroClass",
"id",
"AddSubmonoid",
... | [] | rw [span_nat_eq_addSubmonoidClosure, s.closure_eq] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Span.Defs | {
"line": 296,
"column": 2
} | {
"line": 298,
"column": 87
} | {
"line": 300,
"column": 0
} | [
{
"pp": "R : Type u_1\nM : Type u_4\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\ns : Set (Set M)\n⊢ span R (⋃₀ s) = sSup (span R '' s)",
"ppTerm": "?m.36",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Submodule",
"congrArg",
"Submodule.completeLattic... | [] | refine le_antisymm ?_ (sSup_le fun P ⟨t, ht, h⟩ ↦ h ▸ span_mono (subset_sUnion_of_mem ht))
rw [span_le]
exact fun x ⟨t, hts, hxt⟩ ↦ le_sSup (mem_image_of_mem (span R) hts) (subset_span hxt) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Span.Defs | {
"line": 296,
"column": 2
} | {
"line": 298,
"column": 87
} | {
"line": 300,
"column": 0
} | [
{
"pp": "R : Type u_1\nM : Type u_4\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\ns : Set (Set M)\n⊢ span R (⋃₀ s) = sSup (span R '' s)",
"ppTerm": "?m.36",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Submodule",
"congrArg",
"Submodule.completeLattic... | [] | refine le_antisymm ?_ (sSup_le fun P ⟨t, ht, h⟩ ↦ h ▸ span_mono (subset_sUnion_of_mem ht))
rw [span_le]
exact fun x ⟨t, hts, hxt⟩ ↦ le_sSup (mem_image_of_mem (span R) hts) (subset_span hxt) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.Atoms | {
"line": 241,
"column": 2
} | {
"line": 241,
"column": 13
} | {
"line": 242,
"column": 2
} | [
{
"pp": "A : Type u_4\nB : Type u_5\ninst✝² : PartialOrder A\ninst✝¹ : SetLike A B\ninst✝ : IsConcreteLE A B\nK L : A\nx✝ : K < L\nH : A\n⊢ K < H → ¬H < L ↔ ∀ (g : B), K ≤ H → H ≤ L → g ∉ K → g ∈ H → H = L",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Push.not_foral... | [
"A : Type u_4\nB : Type u_5\ninst✝² : PartialOrder A\ninst✝¹ : SetLike A B\ninst✝ : IsConcreteLE A B\nK L : A\nx✝ : K < L\nH : A\n⊢ K < H ∧ H < L ↔ ∃ g, K ≤ H ∧ H ≤ L ∧ g ∉ K ∧ g ∈ H ∧ H ≠ L"
] | contrapose! | Mathlib.Tactic.Contrapose._aux_Mathlib_Tactic_Contrapose___macroRules_Mathlib_Tactic_Contrapose_contrapose!_1 | Mathlib.Tactic.Contrapose.contrapose! |
Mathlib.Order.Atoms | {
"line": 1242,
"column": 2
} | {
"line": 1242,
"column": 51
} | {
"line": 1244,
"column": 0
} | [
{
"pp": "ι : Type u_4\nπ : ι → Type u\ninst✝² : DecidableEq ι\ninst✝¹ : (i : ι) → PartialOrder (π i)\ninst✝ : (i : ι) → OrderBot (π i)\nf : (i : ι) → π i\n⊢ IsAtom f ↔ ∃ i a, IsAtom a ∧ f = Function.update ⊥ i a",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Pi.preorder",
"Pr... | [] | simp [← bot_covBy_iff, covBy_iff_exists_right_eq] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Order.Atoms | {
"line": 1242,
"column": 2
} | {
"line": 1242,
"column": 51
} | {
"line": 1244,
"column": 0
} | [
{
"pp": "ι : Type u_4\nπ : ι → Type u\ninst✝² : DecidableEq ι\ninst✝¹ : (i : ι) → PartialOrder (π i)\ninst✝ : (i : ι) → OrderBot (π i)\nf : (i : ι) → π i\n⊢ IsAtom f ↔ ∃ i a, IsAtom a ∧ f = Function.update ⊥ i a",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Pi.preorder",
"Pr... | [] | simp [← bot_covBy_iff, covBy_iff_exists_right_eq] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.Atoms | {
"line": 1242,
"column": 2
} | {
"line": 1242,
"column": 51
} | {
"line": 1244,
"column": 0
} | [
{
"pp": "ι : Type u_4\nπ : ι → Type u\ninst✝² : DecidableEq ι\ninst✝¹ : (i : ι) → PartialOrder (π i)\ninst✝ : (i : ι) → OrderBot (π i)\nf : (i : ι) → π i\n⊢ IsAtom f ↔ ∃ i a, IsAtom a ∧ f = Function.update ⊥ i a",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Pi.preorder",
"Pr... | [] | simp [← bot_covBy_iff, covBy_iff_exists_right_eq] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.SupIndep | {
"line": 182,
"column": 2
} | {
"line": 183,
"column": 57
} | {
"line": 185,
"column": 0
} | [
{
"pp": "case hsup\nα : Type u_1\nι : Type u_3\nι' : Type u_4\ninst✝³ : Lattice α\ninst✝² : IsModularLattice α\ninst✝¹ : OrderBot α\ninst✝ : DecidableEq ι\ns : Finset ι'\ng : ι' → Finset ι\nf : ι → α\nhs : s.SupIndep fun i ↦ (g i).sup f\nhg : ∀ i' ∈ s, (g i').SupIndep f\na : Finset ι\nha : a ⊆ s.biUnion g\nb : ... | [] | · rw [← sup_singleton (f := f) (b := b), ← sup_union, show u ∪ {b} = g i' by grind]
exact (supIndep_iff_disjoint_erase.mp hs i' hi').symm | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Order.SupIndep | {
"line": 228,
"column": 4
} | {
"line": 228,
"column": 63
} | {
"line": 229,
"column": 2
} | [
{
"pp": "case refine_1\nα : Type u_1\nι : Type u_3\ninst✝² : Lattice α\ninst✝¹ : IsModularLattice α\ninst✝ : OrderBot α\nβ : ι → Type u_5\ns : Finset ι\ng : (i : ι) → Finset (β i)\nf : Sigma β → α\nh : (s.sigma g).SupIndep f\nt : Finset ι\nx✝² : t ⊆ s\ni : ι\nx✝¹ : i ∈ s\nx✝ : i ∉ t\nu : Finset ((x : ι) × β x) ... | [] | apply SupIndep.disjoint_sup_sup h <;> grind [disjoint_left] | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Order.SupIndep | {
"line": 394,
"column": 2
} | {
"line": 394,
"column": 86
} | {
"line": 395,
"column": 2
} | [
{
"pp": "α : Type u_1\nι : Type u_3\ninst✝ : CompleteLattice α\nt : ι → α\nh : ∀ (i : { i // t i ≠ ⊥ }), Disjoint (t ↑i) (⨆ j, ⨆ (_ : j ≠ i), t ↑j)\ni : ι\nhi : t i ≠ ⊥\n⊢ ⨆ j, ⨆ (_ : j ≠ i), t j = ⨆ j, ⨆ (_ : j ≠ ⟨i, hi⟩), t ↑j",
"ppTerm": "?m.135",
"assigned": true,
"usedConstants": [
"_priv... | [
"α : Type u_1\nι : Type u_3\ninst✝ : CompleteLattice α\nt : ι → α\nh : ∀ (i : { i // t i ≠ ⊥ }), Disjoint (t ↑i) (⨆ j, ⨆ (_ : j ≠ i), t ↑j)\ni : ι\nhi : t i ≠ ⊥\nthis : ∀ (j : ι), ⨆ (_ : t j = ⊥), t j = ⊥\n⊢ ⨆ j, ⨆ (_ : j ≠ i), t j = ⨆ j, ⨆ (_ : j ≠ ⟨i, hi⟩), t ↑j"
] | have : ∀ j, ⨆ (_ : t j = ⊥), t j = ⊥ := fun j ↦ by simp only [iSup_eq_bot, imp_self] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Order.CompactlyGenerated.Basic | {
"line": 134,
"column": 21
} | {
"line": 134,
"column": 25
} | {
"line": 134,
"column": 26
} | [
{
"pp": "case mpr\nα : Type u_2\ninst✝ : CompleteLattice α\nk : α\nhk : ∀ (s : Set α), k ≤ sSup s → ∃ t, ↑t ⊆ s ∧ k ≤ t.sup id\ns : Set α\nhne : s.Nonempty\n⊢ DirectedOn (fun x1 x2 ↦ x1 ≤ x2) s → k ≤ sSup s → ∃ x ∈ s, k ≤ x",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"PartialOrder... | [
"case mpr\nα : Type u_2\ninst✝ : CompleteLattice α\nk : α\nhk : ∀ (s : Set α), k ≤ sSup s → ∃ t, ↑t ⊆ s ∧ k ≤ t.sup id\ns : Set α\nhne : s.Nonempty\nhdir : DirectedOn (fun x1 x2 ↦ x1 ≤ x2) s\n⊢ k ≤ sSup s → ∃ x ∈ s, k ≤ x"
] | hdir | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Order.CompactlyGenerated.Basic | {
"line": 201,
"column": 17
} | {
"line": 201,
"column": 21
} | {
"line": 201,
"column": 22
} | [
{
"pp": "α : Type u_3\nβ : Type u_4\ninst✝ : CompleteLattice α\nf : β → α\ns : Finset β\nh : ∀ x ∈ s, ∀ (s : Set α), s.Nonempty → DirectedOn (fun x1 x2 ↦ x1 ≤ x2) s → f x ≤ sSup s → ∃ x_1 ∈ s, f x ≤ x_1\nd : Set α\nhemp : d.Nonempty\n⊢ DirectedOn (fun x1 x2 ↦ x1 ≤ x2) d → s.sup f ≤ sSup d → ∃ x ∈ d, s.sup f ≤ x... | [
"α : Type u_3\nβ : Type u_4\ninst✝ : CompleteLattice α\nf : β → α\ns : Finset β\nh : ∀ x ∈ s, ∀ (s : Set α), s.Nonempty → DirectedOn (fun x1 x2 ↦ x1 ≤ x2) s → f x ≤ sSup s → ∃ x_1 ∈ s, f x ≤ x_1\nd : Set α\nhemp : d.Nonempty\nhdir : DirectedOn (fun x1 x2 ↦ x1 ≤ x2) d\n⊢ s.sup f ≤ sSup d → ∃ x ∈ d, s.sup f ≤ x"
] | hdir | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Algebra.Notation.Indicator | {
"line": 194,
"column": 18
} | {
"line": 196,
"column": 38
} | {
"line": 198,
"column": 0
} | [
{
"pp": "α : Type u_1\nM : Type u_3\ninst✝ : One M\ns t : Set α\nf : α → M\nx : α\n⊢ s.mulIndicator (t.mulIndicator f) x = (s ∩ t).mulIndicator f x",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"False",
"Set.decidableInter",
"eq_false",
"congrArg",... | [] | by
simp only [mulIndicator]
split_ifs <;> simp_all +contextual | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.Span.Basic | {
"line": 189,
"column": 2
} | {
"line": 189,
"column": 73
} | {
"line": 190,
"column": 2
} | [
{
"pp": "R : Type u_1\nM : Type u_4\nS : Type u_7\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\ninst✝³ : Semiring S\ninst✝² : SMul R S\ninst✝¹ : Module S M\ninst✝ : IsScalarTower R S M\np : Submodule R M\nthis : ⇑(span S ↑p).subtype '' range ⇑(inclusionSpan S p) = ↑p\n⊢ span S (range ⇑(in... | [
"R : Type u_1\nM : Type u_4\nS : Type u_7\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\ninst✝³ : Semiring S\ninst✝² : SMul R S\ninst✝¹ : Module S M\ninst✝ : IsScalarTower R S M\np : Submodule R M\nthis : ⇑(span S ↑p).subtype '' range ⇑(inclusionSpan S p) = ↑p\n⊢ map (span S ↑p).subtype (span ... | apply map_injective_of_injective (span S (p : Set M)).injective_subtype | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Algebra.BigOperators.Pi | {
"line": 86,
"column": 4
} | {
"line": 86,
"column": 22
} | {
"line": 87,
"column": 4
} | [
{
"pp": "case pos\nι : Type u_1\nκ : Type u_2\nR : Type u_5\ninst✝ : CommSemiring R\ns : Finset ι\nf : ι → Set κ\ng : ι → κ → R\nj : κ\nhj : j ∈ ⋂ x ∈ s, f x\n⊢ ∏ i ∈ s, (f i).indicator (g i) j = ∏ c ∈ s, g c j",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"CommSemiring.toSemiring"... | [
"case pos.a\nι : Type u_1\nκ : Type u_2\nR : Type u_5\ninst✝ : CommSemiring R\ns : Finset ι\nf : ι → Set κ\ng : ι → κ → R\nj : κ\nhj : j ∈ ⋂ x ∈ s, f x\ni : ι\nhi : i ∈ s\n⊢ (f i).indicator (g i) j = g i j"
] | congr! 1 with i hi | Congr!._aux_Mathlib_Tactic_CongrExclamation___elabRules_Congr!_congr!_1 | Congr!.congr! |
Mathlib.LinearAlgebra.Span.Basic | {
"line": 351,
"column": 15
} | {
"line": 351,
"column": 19
} | {
"line": 351,
"column": 20
} | [
{
"pp": "R : Type u_1\nM : Type u_4\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nx : M\nd : Set (Submodule R M)\nhemp : d.Nonempty\n⊢ DirectedOn (fun x1 x2 ↦ x1 ≤ x2) d → R ∙ x ≤ sSup d → ∃ x_1 ∈ d, R ∙ x ≤ x_1",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"S... | [
"R : Type u_1\nM : Type u_4\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nx : M\nd : Set (Submodule R M)\nhemp : d.Nonempty\nhdir : DirectedOn (fun x1 x2 ↦ x1 ≤ x2) d\n⊢ R ∙ x ≤ sSup d → ∃ x_1 ∈ d, R ∙ x ≤ x_1"
] | hdir | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.LinearAlgebra.Span.Basic | {
"line": 455,
"column": 7
} | {
"line": 455,
"column": 30
} | {
"line": 455,
"column": 30
} | [
{
"pp": "R : Type u_1\nR₂ : Type u_2\nM : Type u_4\nM₂ : Type u_5\ninst✝⁵ : Semiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\ninst✝² : Semiring R₂\nσ₁₂ : R →+* R₂\ninst✝¹ : AddCommMonoid M₂\ninst✝ : Module R₂ M₂\nN : Submodule R M\nf : ↥N →ₛₗ[σ₁₂] M₂\nh : f ≠ 0\ns : Set (Submodule R M)\nhs : sSup s = N... | [
"R : Type u_1\nR₂ : Type u_2\nM : Type u_4\nM₂ : Type u_5\ninst✝⁵ : Semiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\ninst✝² : Semiring R₂\nσ₁₂ : R →+* R₂\ninst✝¹ : AddCommMonoid M₂\ninst✝ : Module R₂ M₂\nN : Submodule R M\nf : ↥N →ₛₗ[σ₁₂] M₂\nh : f ≠ 0\ns : Set (Submodule R M)\nhs : ⨆ a ∈ s, a = N\n⊢ sSup... | rw [sSup_eq_iSup] at hs | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.Span.Basic | {
"line": 518,
"column": 60
} | {
"line": 518,
"column": 82
} | {
"line": 520,
"column": 0
} | [
{
"pp": "R : Type u_10\nM : Type u_11\ninst✝² : Ring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\ns : Set M\n⊢ span R (-s) = span R s",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"NegZeroClass.toNeg",
"Submodule",
"congrArg",
"AddCommGroup.toAddCommMonoid",
... | [] | simp [span_neg_eq_neg] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.LinearAlgebra.Span.Basic | {
"line": 518,
"column": 60
} | {
"line": 518,
"column": 82
} | {
"line": 520,
"column": 0
} | [
{
"pp": "R : Type u_10\nM : Type u_11\ninst✝² : Ring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\ns : Set M\n⊢ span R (-s) = span R s",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"NegZeroClass.toNeg",
"Submodule",
"congrArg",
"AddCommGroup.toAddCommMonoid",
... | [] | simp [span_neg_eq_neg] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Span.Basic | {
"line": 518,
"column": 60
} | {
"line": 518,
"column": 82
} | {
"line": 520,
"column": 0
} | [
{
"pp": "R : Type u_10\nM : Type u_11\ninst✝² : Ring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\ns : Set M\n⊢ span R (-s) = span R s",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"NegZeroClass.toNeg",
"Submodule",
"congrArg",
"AddCommGroup.toAddCommMonoid",
... | [] | simp [span_neg_eq_neg] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Finsupp.Defs | {
"line": 185,
"column": 2
} | {
"line": 185,
"column": 13
} | {
"line": 185,
"column": 13
} | [
{
"pp": "α : Type u_1\nM : Type u_4\ninst✝ : Zero M\nf : α →₀ M\n⊢ f.support.Nonempty ↔ f ≠ 0",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Finset",
"Finsupp.support",
"id",
"Finset.instEmptyCollection",
"Iff",
"Finset... | [
"α : Type u_1\nM : Type u_4\ninst✝ : Zero M\nf : α →₀ M\n⊢ f.support = ∅ ↔ f = 0"
] | contrapose! | Mathlib.Tactic.Contrapose._aux_Mathlib_Tactic_Contrapose___macroRules_Mathlib_Tactic_Contrapose_contrapose!_1 | Mathlib.Tactic.Contrapose.contrapose! |
Mathlib.Data.Finsupp.Single | {
"line": 62,
"column": 43
} | {
"line": 62,
"column": 91
} | {
"line": 64,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nM : Type u_5\ninst✝ : Zero M\nf : α → β\nhf : Injective f\nx z : α\ny : M\n⊢ (single (f x) y) (f z) = (single x y) z",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Finsupp.instFunLike",
"congrArg",
"Function.Injective.eq_iff",
... | [] | by classical simp only [single_apply, hf.eq_iff] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Module.Basic | {
"line": 47,
"column": 2
} | {
"line": 51,
"column": 25
} | {
"line": 52,
"column": 2
} | [
{
"pp": "case neg\nM : Type u_3\nM₂ : Type u_4\ninst✝⁷ : AddCommMonoid M\ninst✝⁶ : AddCommMonoid M₂\nF : Type u_5\ninst✝⁵ : FunLike F M M₂\ninst✝⁴ : AddMonoidHomClass F M M₂\nf : F\nR : Type u_6\nS : Type u_7\ninst✝³ : DivisionSemiring R\ninst✝² : DivisionSemiring S\ninst✝¹ : Module R M\ninst✝ : Module S M₂\nn ... | [
"case pos\nM : Type u_3\nM₂ : Type u_4\ninst✝⁷ : AddCommMonoid M\ninst✝⁶ : AddCommMonoid M₂\nF : Type u_5\ninst✝⁵ : FunLike F M M₂\ninst✝⁴ : AddMonoidHomClass F M M₂\nf : F\nR : Type u_6\nS : Type u_7\ninst✝³ : DivisionSemiring R\ninst✝² : DivisionSemiring S\ninst✝¹ : Module R M\ninst✝ : Module S M₂\nn : ℕ\nx : M\n... | · suffices ∀ y, f y = 0 by rw [this, this, smul_zero]
clear x
intro x
rw [← inv_smul_smul₀ hS (f x), ← map_natCast_smul f R S]
simp [hR, map_zero f] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Module.Basic | {
"line": 56,
"column": 8
} | {
"line": 56,
"column": 34
} | {
"line": 56,
"column": 35
} | [
{
"pp": "case neg\nM : Type u_3\nM₂ : Type u_4\ninst✝⁷ : AddCommMonoid M\ninst✝⁶ : AddCommMonoid M₂\nF : Type u_5\ninst✝⁵ : FunLike F M M₂\ninst✝⁴ : AddMonoidHomClass F M M₂\nf : F\nR : Type u_6\nS : Type u_7\ninst✝³ : DivisionSemiring R\ninst✝² : DivisionSemiring S\ninst✝¹ : Module R M\ninst✝ : Module S M₂\nn ... | [
"case neg\nM : Type u_3\nM₂ : Type u_4\ninst✝⁷ : AddCommMonoid M\ninst✝⁶ : AddCommMonoid M₂\nF : Type u_5\ninst✝⁵ : FunLike F M M₂\ninst✝⁴ : AddMonoidHomClass F M M₂\nf : F\nR : Type u_6\nS : Type u_7\ninst✝³ : DivisionSemiring R\ninst✝² : DivisionSemiring S\ninst✝¹ : Module R M\ninst✝ : Module S M₂\nn : ℕ\nx : M\n... | ← inv_smul_smul₀ hS (f _), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.BigOperators.Finsupp.Basic | {
"line": 429,
"column": 2
} | {
"line": 430,
"column": 6
} | {
"line": 432,
"column": 0
} | [
{
"pp": "α : Type u_1\nM : Type u_8\ninst✝¹ : AddCommMonoid M\ninst✝ : Fintype α\ni : α\nm : M\n⊢ ∑ j, (single i m) j = m",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Finsupp.instFunLike",
"Eq.mpr",
"Finset.univ",
"congrArg",
"Finset",
"AddMonoid.toA... | [] | classical rw [single, coe_mk, Finset.sum_pi_single']
simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.BigOperators.Finsupp.Basic | {
"line": 429,
"column": 2
} | {
"line": 430,
"column": 6
} | {
"line": 432,
"column": 0
} | [
{
"pp": "α : Type u_1\nM : Type u_8\ninst✝¹ : AddCommMonoid M\ninst✝ : Fintype α\ni : α\nm : M\n⊢ ∑ j, (single i m) j = m",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Finsupp.instFunLike",
"Eq.mpr",
"Finset.univ",
"congrArg",
"Finset",
"AddMonoid.toA... | [] | classical rw [single, coe_mk, Finset.sum_pi_single']
simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Order.BigOperators.GroupWithZero.Finset | {
"line": 103,
"column": 2
} | {
"line": 107,
"column": 57
} | {
"line": 109,
"column": 0
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\ninst✝³ : CommMonoidWithZero R\ninst✝² : Preorder R\ninst✝¹ : ZeroLEOneClass R\ninst✝ : PosMulMono R\nf : ι → R\ns t : Finset ι\nh : s ⊆ t\nhf0 : ∀ i ∈ s, 0 ≤ f i\nhf : ∀ i ∈ t, i ∉ s → 1 ≤ f i\nthis : MulPosMono R\n⊢ ∏ i ∈ s, f i ≤ ∏ i ∈ t, f i",
"ppTerm": "?m.53",
"... | [] | calc
∏ i ∈ s, f i ≤ (∏ i ∈ t \ s, f i) * ∏ i ∈ s, f i :=
le_mul_of_one_le_left (prod_nonneg hf0) <| one_le_prod <| by simpa only [mem_sdiff, and_imp]
_ = ∏ i ∈ t \ s ∪ s, f i := (prod_union sdiff_disjoint).symm
_ = ∏ i ∈ t, f i := by rw [sdiff_union_of_subset h] | Lean.Elab.Tactic._aux_Mathlib_Tactic_Widget_Calc___elabRules_Lean_calcTactic_1 | Lean.calcTactic |
Mathlib.Algebra.Order.BigOperators.GroupWithZero.Finset | {
"line": 103,
"column": 2
} | {
"line": 107,
"column": 57
} | {
"line": 109,
"column": 0
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\ninst✝³ : CommMonoidWithZero R\ninst✝² : Preorder R\ninst✝¹ : ZeroLEOneClass R\ninst✝ : PosMulMono R\nf : ι → R\ns t : Finset ι\nh : s ⊆ t\nhf0 : ∀ i ∈ s, 0 ≤ f i\nhf : ∀ i ∈ t, i ∉ s → 1 ≤ f i\nthis : MulPosMono R\n⊢ ∏ i ∈ s, f i ≤ ∏ i ∈ t, f i",
"ppTerm": "?m.53",
"... | [] | calc
∏ i ∈ s, f i ≤ (∏ i ∈ t \ s, f i) * ∏ i ∈ s, f i :=
le_mul_of_one_le_left (prod_nonneg hf0) <| one_le_prod <| by simpa only [mem_sdiff, and_imp]
_ = ∏ i ∈ t \ s ∪ s, f i := (prod_union sdiff_disjoint).symm
_ = ∏ i ∈ t, f i := by rw [sdiff_union_of_subset h] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Order.BigOperators.GroupWithZero.Finset | {
"line": 103,
"column": 2
} | {
"line": 107,
"column": 57
} | {
"line": 109,
"column": 0
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\ninst✝³ : CommMonoidWithZero R\ninst✝² : Preorder R\ninst✝¹ : ZeroLEOneClass R\ninst✝ : PosMulMono R\nf : ι → R\ns t : Finset ι\nh : s ⊆ t\nhf0 : ∀ i ∈ s, 0 ≤ f i\nhf : ∀ i ∈ t, i ∉ s → 1 ≤ f i\nthis : MulPosMono R\n⊢ ∏ i ∈ s, f i ≤ ∏ i ∈ t, f i",
"ppTerm": "?m.53",
"... | [] | calc
∏ i ∈ s, f i ≤ (∏ i ∈ t \ s, f i) * ∏ i ∈ s, f i :=
le_mul_of_one_le_left (prod_nonneg hf0) <| one_le_prod <| by simpa only [mem_sdiff, and_imp]
_ = ∏ i ∈ t \ s ∪ s, f i := (prod_union sdiff_disjoint).symm
_ = ∏ i ∈ t, f i := by rw [sdiff_union_of_subset h] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Finsupp.Basic | {
"line": 330,
"column": 18
} | {
"line": 330,
"column": 40
} | {
"line": 330,
"column": 40
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nM : Type u_5\ninst✝ : AddCommMonoid M\nf : α ≃ β\nx : α →₀ M\na : β\n| (mapDomain (⇑f) x) a",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Finsupp.instFunLike",
"Equiv.apply_symm_apply",
"Equiv.instEquivLike",
"congrArg",
... | [
"α : Type u_1\nβ : Type u_2\nM : Type u_5\ninst✝ : AddCommMonoid M\nf : α ≃ β\nx : α →₀ M\na : β\n| (mapDomain (⇑f) x) (f (f.symm a))"
] | ← f.apply_symm_apply a | Lean.Elab.Tactic.Conv.evalRewrite | null |
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