module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.Data.Part | {
"line": 740,
"column": 20
} | {
"line": 740,
"column": 25
} | {
"line": 742,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝ : Union α\na b : Part α\nhab : (a ∪ b).Dom\n⊢ (a.bind fun y ↦ map (fun x ↦ y ∪ x) b).get ⋯ = a.get ⋯ ∪ b.get ⋯",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Part",
"Part.bind",
"id",
"Part.get",
"Part.Dom",
"Eq.ndrec",
... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Data.Part | {
"line": 745,
"column": 44
} | {
"line": 745,
"column": 49
} | {
"line": 747,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝ : SDiff α\na b : Part α\nma mb : α\nha : ma ∈ a\nhb : mb ∈ b\n⊢ ∃ a_1, a_1 ∈ a ∧ ∃ a, a ∈ b ∧ a_1 \\ a = ma \\ mb",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Part",
"Membership.mem",
"Exists",
"Part.instMembership",
"SDiff.sdi... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Data.Part | {
"line": 754,
"column": 20
} | {
"line": 754,
"column": 25
} | {
"line": 756,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝ : SDiff α\na b : Part α\nhab : (a \\ b).Dom\n⊢ (a.bind fun y ↦ map (fun x ↦ y \\ x) b).get ⋯ = a.get ⋯ \\ b.get ⋯",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Part",
"Part.bind",
"id",
"Part.get",
"SDiff.sdiff",
"Part.Dom... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Order.FixedPoints | {
"line": 273,
"column": 28
} | {
"line": 280,
"column": 87
} | {
"line": 282,
"column": 0
} | [
{
"pp": "α : Type u\ninst✝ : CompleteLattice α\nf : α →o α\nh : ωScottContinuous ⇑f\n⊢ lfp f = ⨆ n, (⇑f)^[n] ⊥",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"OmegaCompletePartialOrder.fixedPoints.ωSup_iterate_le_prefixedPoint",
"Lattice.toSemilatticeSup",
"ChainComplete... | [] | by
apply le_antisymm
· apply lfp_le_fixed
exact Function.mem_fixedPoints.mp (ωSup_iterate_mem_fixedPoint
⟨f, h.map_ωSup_of_orderHom⟩ ⊥ bot_le)
· apply le_lfp
intro a h_a
exact ωSup_iterate_le_prefixedPoint ⟨f, h.map_ωSup_of_orderHom⟩ ⊥ bot_le h_a bot_le | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Order.OmegaCompletePartialOrder | {
"line": 156,
"column": 24
} | {
"line": 156,
"column": 29
} | {
"line": 158,
"column": 0
} | [
{
"pp": "ι : Sort u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\nδ : Type u_5\ninst✝² : Preorder α\ninst✝¹ : Preorder β\ninst✝ : Preorder γ\nc c' : Chain α\nf : α →o β\ng : β →o γ\na b : α\nhab : a ≤ b\nx✝² x✝¹ : ℕ\nx✝ : x✝² ≤ x✝¹\n⊢ (fun x ↦\n match x with\n | 0 => a\n | x => b)\n x✝²... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Order.OmegaCompletePartialOrder | {
"line": 156,
"column": 24
} | {
"line": 156,
"column": 29
} | {
"line": 158,
"column": 0
} | [
{
"pp": "ι : Sort u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\nδ : Type u_5\ninst✝² : Preorder α\ninst✝¹ : Preorder β\ninst✝ : Preorder γ\nc c' : Chain α\nf : α →o β\ng : β →o γ\na b : α\nhab : a ≤ b\nx✝² x✝¹ : ℕ\nx✝ : x✝² ≤ x✝¹\n⊢ (fun x ↦\n match x with\n | 0 => a\n | x => b)\n x✝²... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.OmegaCompletePartialOrder | {
"line": 156,
"column": 24
} | {
"line": 156,
"column": 29
} | {
"line": 158,
"column": 0
} | [
{
"pp": "ι : Sort u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\nδ : Type u_5\ninst✝² : Preorder α\ninst✝¹ : Preorder β\ninst✝ : Preorder γ\nc c' : Chain α\nf : α →o β\ng : β →o γ\na b : α\nhab : a ≤ b\nx✝² x✝¹ : ℕ\nx✝ : x✝² ≤ x✝¹\n⊢ (fun x ↦\n match x with\n | 0 => a\n | x => b)\n x✝²... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.OmegaCompletePartialOrder | {
"line": 162,
"column": 50
} | {
"line": 162,
"column": 55
} | {
"line": 162,
"column": 55
} | [
{
"pp": "α : Type u_2\ninst✝ : Preorder α\na b : α\nhab : a ≤ b\nx✝ : α\n⊢ ((pair a b hab) 0 = x✝ ∨ ∃ n, (pair a b hab) (n + 1) = x✝) ↔ x✝ ∈ {a, b}",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"true_or",
"exists_const._simp_1",
"OmegaC... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Order.OmegaCompletePartialOrder | {
"line": 162,
"column": 50
} | {
"line": 162,
"column": 55
} | {
"line": 162,
"column": 55
} | [
{
"pp": "α : Type u_2\ninst✝ : Preorder α\na b : α\nhab : a ≤ b\nx✝ : α\n⊢ ((pair a b hab) 0 = x✝ ∨ ∃ n, (pair a b hab) (n + 1) = x✝) ↔ x✝ ∈ {a, b}",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"true_or",
"exists_const._simp_1",
"OmegaC... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.OmegaCompletePartialOrder | {
"line": 162,
"column": 50
} | {
"line": 162,
"column": 55
} | {
"line": 162,
"column": 55
} | [
{
"pp": "α : Type u_2\ninst✝ : Preorder α\na b : α\nhab : a ≤ b\nx✝ : α\n⊢ ((pair a b hab) 0 = x✝ ∨ ∃ n, (pair a b hab) (n + 1) = x✝) ↔ x✝ ∈ {a, b}",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"true_or",
"exists_const._simp_1",
"OmegaC... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.OmegaCompletePartialOrder | {
"line": 663,
"column": 2
} | {
"line": 663,
"column": 39
} | {
"line": 665,
"column": 0
} | [
{
"pp": "α : Type u_2\nβ : Type u_3\ninst✝¹ : OmegaCompletePartialOrder α\ninst✝ : OmegaCompletePartialOrder β\nc₀ : Chain (α →𝒄 β)\nc₁ : Chain α\nz : β\n⊢ (∀ (j i : ℕ), (c₀ i) (c₁ j) ≤ z) ↔ ∀ (i : ℕ), (c₀ i) (c₁ i) ≤ z",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [] | rw [forall_comm, forall_forall_merge] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Order.OmegaCompletePartialOrder | {
"line": 663,
"column": 2
} | {
"line": 663,
"column": 39
} | {
"line": 665,
"column": 0
} | [
{
"pp": "α : Type u_2\nβ : Type u_3\ninst✝¹ : OmegaCompletePartialOrder α\ninst✝ : OmegaCompletePartialOrder β\nc₀ : Chain (α →𝒄 β)\nc₁ : Chain α\nz : β\n⊢ (∀ (j i : ℕ), (c₀ i) (c₁ j) ≤ z) ↔ ∀ (i : ℕ), (c₀ i) (c₁ i) ≤ z",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [] | rw [forall_comm, forall_forall_merge] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.OmegaCompletePartialOrder | {
"line": 663,
"column": 2
} | {
"line": 663,
"column": 39
} | {
"line": 665,
"column": 0
} | [
{
"pp": "α : Type u_2\nβ : Type u_3\ninst✝¹ : OmegaCompletePartialOrder α\ninst✝ : OmegaCompletePartialOrder β\nc₀ : Chain (α →𝒄 β)\nc₁ : Chain α\nz : β\n⊢ (∀ (j i : ℕ), (c₀ i) (c₁ j) ≤ z) ↔ ∀ (i : ℕ), (c₀ i) (c₁ i) ≤ z",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [] | rw [forall_comm, forall_forall_merge] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.SetTheory.Cardinal.ENat | {
"line": 242,
"column": 15
} | {
"line": 242,
"column": 36
} | {
"line": 242,
"column": 36
} | [
{
"pp": "a : Cardinal.{u_1}\n⊢ a = ↑(toENat a) ↔ a ≤ ℵ₀",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"GaloisConnection.exists_eq_l",
"Cardinal",
"congrArg",
"CommSemiring.toSemiring",
"Cardinal.commSemiring",
"PartialOrder.toPreorder",
... | [
"a : Cardinal.{u_1}\n⊢ (∃ b, a = ↑b) ↔ a ≤ ℵ₀"
] | ← enat_gc.exists_eq_l | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Order.OmegaCompletePartialOrder | {
"line": 740,
"column": 6
} | {
"line": 742,
"column": 47
} | {
"line": 744,
"column": 0
} | [
{
"pp": "ι : Sort u_1\nα : Type u_2\nβ✝ : Type u_3\nγ✝ : Type u_4\nδ : Type u_5\ninst✝³ : OmegaCompletePartialOrder α\ninst✝² : OmegaCompletePartialOrder β✝\ninst✝¹ : OmegaCompletePartialOrder γ✝\ninst✝ : OmegaCompletePartialOrder δ\nβ γ : Type v\nf : α →𝒄 Part (β → γ)\ng : α →𝒄 Part β\n⊢ (fun x ↦ f x <*> g x... | [] | ext
simp only [seq_eq_bind_map, Part.bind_eq_bind, Part.mem_bind_iff, flip_apply, _root_.flip,
map_apply, bind_apply, Part.map_eq_map] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.OmegaCompletePartialOrder | {
"line": 740,
"column": 6
} | {
"line": 742,
"column": 47
} | {
"line": 744,
"column": 0
} | [
{
"pp": "ι : Sort u_1\nα : Type u_2\nβ✝ : Type u_3\nγ✝ : Type u_4\nδ : Type u_5\ninst✝³ : OmegaCompletePartialOrder α\ninst✝² : OmegaCompletePartialOrder β✝\ninst✝¹ : OmegaCompletePartialOrder γ✝\ninst✝ : OmegaCompletePartialOrder δ\nβ γ : Type v\nf : α →𝒄 Part (β → γ)\ng : α →𝒄 Part β\n⊢ (fun x ↦ f x <*> g x... | [] | ext
simp only [seq_eq_bind_map, Part.bind_eq_bind, Part.mem_bind_iff, flip_apply, _root_.flip,
map_apply, bind_apply, Part.map_eq_map] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.SetTheory.Cardinal.Basic | {
"line": 83,
"column": 4
} | {
"line": 83,
"column": 36
} | {
"line": 84,
"column": 4
} | [
{
"pp": "case refine_1\nα✝ : Type u\nh✝ : Fintype α✝\nα β : Type u\nhβ : Fintype β\ne : α ≃ β\nh : ∀ (f : α → Cardinal.{v}), prod f = lift.{u, v} (∏ i, f i)\nf : β → Cardinal.{v}\n⊢ prod f = lift.{u, v} (∏ i, f i)",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"Fintype.ofEquiv",... | [
"case refine_1\nα✝ : Type u\nh✝ : Fintype α✝\nα β : Type u\nhβ : Fintype β\ne : α ≃ β\nh : ∀ (f : α → Cardinal.{v}), prod f = lift.{u, v} (∏ i, f i)\nf : β → Cardinal.{v}\nthis : Fintype α := Fintype.ofEquiv β e.symm\n⊢ prod f = lift.{u, v} (∏ i, f i)"
] | letI := Fintype.ofEquiv β e.symm | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLetI___1 | Lean.Parser.Tactic.tacticLetI__ |
Mathlib.SetTheory.Cardinal.Basic | {
"line": 284,
"column": 75
} | {
"line": 285,
"column": 40
} | {
"line": 287,
"column": 0
} | [
{
"pp": "n : ℕ\nc : Cardinal.{u_1}\n⊢ c < ↑n + 1 ↔ c ≤ ↑n",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"Cardinal.instOne",
"Order.succ",
"Cardinal.succ_natCast",
"Cardinal",
"congrArg",
"Iff.rfl",
"PartialO... | [] | by
rw [← Order.lt_succ_iff, succ_natCast] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.SetTheory.Cardinal.Basic | {
"line": 420,
"column": 78
} | {
"line": 421,
"column": 50
} | {
"line": 423,
"column": 0
} | [
{
"pp": "α : Type u\nn : ℕ\n⊢ #α = ↑n ↔ Nonempty (α ≃ Fin n)",
"ppTerm": "?m.2",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Cardinal",
"congrArg",
"Iff.rfl",
"Cardinal.lift",
"Cardinal.mk",
"id",
"Equiv",
"Cardinal.lift_mk_fin",
"Cardi... | [] | by
rw [← lift_mk_fin, ← lift_uzero #α, lift_mk_eq'] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Regular.SMul | {
"line": 73,
"column": 4
} | {
"line": 73,
"column": 41
} | {
"line": 74,
"column": 2
} | [
{
"pp": "case inr.inl\nM : Type u_3\ninst✝ : SubtractionMonoid M\nn : ℤ\nh : n.sign = 0\n⊢ (∀ ⦃a₁ a₂ : M⦄, n.natAbs • a₁ = n.natAbs • a₂ → a₁ = a₂) ↔\n ∀ ⦃a₁ a₂ : M⦄, (n.sign * ↑n.natAbs) • a₁ = (n.sign * ↑n.natAbs) • a₂ → a₁ = a₂",
"ppTerm": "?inr.inl",
"assigned": true,
"usedConstants": [
... | [] | simp [Int.sign_eq_zero_iff_zero.mp h] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Regular.SMul | {
"line": 73,
"column": 4
} | {
"line": 73,
"column": 41
} | {
"line": 74,
"column": 2
} | [
{
"pp": "case inr.inl\nM : Type u_3\ninst✝ : SubtractionMonoid M\nn : ℤ\nh : n.sign = 0\n⊢ (∀ ⦃a₁ a₂ : M⦄, n.natAbs • a₁ = n.natAbs • a₂ → a₁ = a₂) ↔\n ∀ ⦃a₁ a₂ : M⦄, (n.sign * ↑n.natAbs) • a₁ = (n.sign * ↑n.natAbs) • a₂ → a₁ = a₂",
"ppTerm": "?inr.inl",
"assigned": true,
"usedConstants": [
... | [] | simp [Int.sign_eq_zero_iff_zero.mp h] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Regular.SMul | {
"line": 73,
"column": 4
} | {
"line": 73,
"column": 41
} | {
"line": 74,
"column": 2
} | [
{
"pp": "case inr.inl\nM : Type u_3\ninst✝ : SubtractionMonoid M\nn : ℤ\nh : n.sign = 0\n⊢ (∀ ⦃a₁ a₂ : M⦄, n.natAbs • a₁ = n.natAbs • a₂ → a₁ = a₂) ↔\n ∀ ⦃a₁ a₂ : M⦄, (n.sign * ↑n.natAbs) • a₁ = (n.sign * ↑n.natAbs) • a₂ → a₁ = a₂",
"ppTerm": "?inr.inl",
"assigned": true,
"usedConstants": [
... | [] | simp [Int.sign_eq_zero_iff_zero.mp h] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Algebra.Defs | {
"line": 333,
"column": 30
} | {
"line": 333,
"column": 52
} | {
"line": 333,
"column": 53
} | [
{
"pp": "R : Type u\nA : Type w\ninst✝⁵ : CommSemiring R\ninst✝⁴ : Semiring A\ninst✝³ : Algebra R A\nα : Type u_2\ninst✝² : Monoid α\ninst✝¹ : MulDistribMulAction α A\ninst✝ : SMulCommClass α R A\na : α\nr : R\n⊢ a • r • 1 = r • 1",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"Eq.m... | [
"R : Type u\nA : Type w\ninst✝⁵ : CommSemiring R\ninst✝⁴ : Semiring A\ninst✝³ : Algebra R A\nα : Type u_2\ninst✝² : Monoid α\ninst✝¹ : MulDistribMulAction α A\ninst✝ : SMulCommClass α R A\na : α\nr : R\n⊢ r • a • 1 = r • 1"
] | smul_comm a r (1 : A), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.GroupAction.Hom | {
"line": 334,
"column": 2
} | {
"line": 334,
"column": 21
} | {
"line": 336,
"column": 0
} | [
{
"pp": "M : Type u_2\nN : Type u_3\nφ : M → N\nX : Type u_5\ninst✝¹ : SMul M X\nY : Type u_6\ninst✝ : SMul N Y\nφ' : N → M\nf : X →ₑ[φ] Y\ng : Y → X\nk₁ : Function.LeftInverse φ' φ\nk₂ : Function.RightInverse φ' φ\nh₁ : Function.LeftInverse g ⇑f\nh₂ : Function.RightInverse g ⇑f\nx✝ : X\n⊢ ((f.inverse' g k₂ h₁ ... | [] | simpa using h₁.eq _ | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.GroupTheory.GroupAction.Hom | {
"line": 865,
"column": 34
} | {
"line": 865,
"column": 46
} | {
"line": 865,
"column": 47
} | [
{
"pp": "R : Type u_10\ninst✝³ : Semiring R\nS : Type u_11\ninst✝² : Semiring S\nN' : Type u_13\ninst✝¹ : AddMonoid N'\ninst✝ : DistribMulAction S N'\nσ : R →* S\nf g : R →ₑ+[σ] N'\nh : f 1 = g 1\nx : R\n⊢ f (x • 1) = g (x • 1)",
"ppTerm": "?m.80",
"assigned": true,
"usedConstants": [
"Eq.mpr"... | [
"R : Type u_10\ninst✝³ : Semiring R\nS : Type u_11\ninst✝² : Semiring S\nN' : Type u_13\ninst✝¹ : AddMonoid N'\ninst✝ : DistribMulAction S N'\nσ : R →* S\nf g : R →ₑ+[σ] N'\nh : f 1 = g 1\nx : R\n⊢ σ x • f 1 = g (x • 1)"
] | f.map_smulₑ, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Module.LinearMap.Defs | {
"line": 641,
"column": 13
} | {
"line": 643,
"column": 31
} | {
"line": 645,
"column": 0
} | [
{
"pp": "R : Type u_1\nS : Type u_5\nM : Type u_8\nM₂ : Type u_10\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : AddCommMonoid M₂\ninst✝³ : Semiring R\ninst✝² : Module R M\ninst✝¹ : Semiring S\ninst✝ : Module S M₂\nσ : R →+* S\nf g : M →ₑ+[↑σ] M₂\nh : ↑f = ↑g\n⊢ f = g",
"ppTerm": "?m.51",
"assigned": true,
"us... | [] | by
ext m
exact LinearMap.congr_fun h m | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Module.LinearMap.Defs | {
"line": 789,
"column": 2
} | {
"line": 789,
"column": 7
} | {
"line": 791,
"column": 0
} | [
{
"pp": "R₁ : Type u_2\nR₂ : Type u_3\nM : Type u_8\nM₂ : Type u_10\ninst✝⁵ : Semiring R₁\ninst✝⁴ : Semiring R₂\ninst✝³ : AddCommMonoid M\ninst✝² : AddCommMonoid M₂\ninst✝¹ : Module R₁ M\ninst✝ : Module R₂ M₂\nσ₁₂ : R₁ →+* R₂\nf : M →ₛₗ[σ₁₂] M₂\n⊢ ⇑f = 0 ↔ f = 0",
"ppTerm": "?m.38",
"assigned": true,
... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Module.LinearMap.Defs | {
"line": 789,
"column": 2
} | {
"line": 789,
"column": 7
} | {
"line": 791,
"column": 0
} | [
{
"pp": "R₁ : Type u_2\nR₂ : Type u_3\nM : Type u_8\nM₂ : Type u_10\ninst✝⁵ : Semiring R₁\ninst✝⁴ : Semiring R₂\ninst✝³ : AddCommMonoid M\ninst✝² : AddCommMonoid M₂\ninst✝¹ : Module R₁ M\ninst✝ : Module R₂ M₂\nσ₁₂ : R₁ →+* R₂\nf : M →ₛₗ[σ₁₂] M₂\n⊢ ⇑f = 0 ↔ f = 0",
"ppTerm": "?m.38",
"assigned": true,
... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Module.LinearMap.Defs | {
"line": 789,
"column": 2
} | {
"line": 789,
"column": 7
} | {
"line": 791,
"column": 0
} | [
{
"pp": "R₁ : Type u_2\nR₂ : Type u_3\nM : Type u_8\nM₂ : Type u_10\ninst✝⁵ : Semiring R₁\ninst✝⁴ : Semiring R₂\ninst✝³ : AddCommMonoid M\ninst✝² : AddCommMonoid M₂\ninst✝¹ : Module R₁ M\ninst✝ : Module R₂ M₂\nσ₁₂ : R₁ →+* R₂\nf : M →ₛₗ[σ₁₂] M₂\n⊢ ⇑f = 0 ↔ f = 0",
"ppTerm": "?m.38",
"assigned": true,
... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Module.LinearMap.Basic | {
"line": 145,
"column": 15
} | {
"line": 145,
"column": 51
} | {
"line": 145,
"column": 51
} | [
{
"pp": "R : Type u_6\nA : Type u_7\ninst✝³ : Semiring R\ninst✝² : NonAssocSemiring A\ninst✝¹ : Module R A\ninst✝ : SMulCommClass R A A\na b : A\nh : mulLeft R a = mulLeft R b\n⊢ a = b",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"c... | [] | simpa using LinearMap.ext_iff.mp h 1 | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Algebra.Module.LinearMap.Basic | {
"line": 145,
"column": 15
} | {
"line": 145,
"column": 51
} | {
"line": 145,
"column": 51
} | [
{
"pp": "R : Type u_6\nA : Type u_7\ninst✝³ : Semiring R\ninst✝² : NonAssocSemiring A\ninst✝¹ : Module R A\ninst✝ : SMulCommClass R A A\na b : A\nh : mulLeft R a = mulLeft R b\n⊢ a = b",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"c... | [] | simpa using LinearMap.ext_iff.mp h 1 | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Module.LinearMap.Basic | {
"line": 145,
"column": 15
} | {
"line": 145,
"column": 51
} | {
"line": 145,
"column": 51
} | [
{
"pp": "R : Type u_6\nA : Type u_7\ninst✝³ : Semiring R\ninst✝² : NonAssocSemiring A\ninst✝¹ : Module R A\ninst✝ : SMulCommClass R A A\na b : A\nh : mulLeft R a = mulLeft R b\n⊢ a = b",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"c... | [] | simpa using LinearMap.ext_iff.mp h 1 | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Module.LinearMap.Basic | {
"line": 149,
"column": 15
} | {
"line": 149,
"column": 51
} | {
"line": 149,
"column": 51
} | [
{
"pp": "R : Type u_6\nA : Type u_7\ninst✝³ : Semiring R\ninst✝² : NonAssocSemiring A\ninst✝¹ : Module R A\ninst✝ : IsScalarTower R A A\na b : A\nh : mulRight R a = mulRight R b\n⊢ a = b",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
... | [] | simpa using LinearMap.ext_iff.mp h 1 | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Algebra.Module.LinearMap.Basic | {
"line": 149,
"column": 15
} | {
"line": 149,
"column": 51
} | {
"line": 149,
"column": 51
} | [
{
"pp": "R : Type u_6\nA : Type u_7\ninst✝³ : Semiring R\ninst✝² : NonAssocSemiring A\ninst✝¹ : Module R A\ninst✝ : IsScalarTower R A A\na b : A\nh : mulRight R a = mulRight R b\n⊢ a = b",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
... | [] | simpa using LinearMap.ext_iff.mp h 1 | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Module.LinearMap.Basic | {
"line": 149,
"column": 15
} | {
"line": 149,
"column": 51
} | {
"line": 149,
"column": 51
} | [
{
"pp": "R : Type u_6\nA : Type u_7\ninst✝³ : Semiring R\ninst✝² : NonAssocSemiring A\ninst✝¹ : Module R A\ninst✝ : IsScalarTower R A A\na b : A\nh : mulRight R a = mulRight R b\n⊢ a = b",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
... | [] | simpa using LinearMap.ext_iff.mp h 1 | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Module.LinearMap.End | {
"line": 401,
"column": 30
} | {
"line": 403,
"column": 22
} | {
"line": 404,
"column": 6
} | [
{
"pp": "R : Type u_1\nR₂ : Type u_2\nS : Type u_3\nM : Type u_4\nM₁ : Type u_5\nM₂ : Type u_6\nM₃ : Type u_7\nN₁ : Type u_8\ninst✝⁶ : CommSemiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : AddCommMonoid M₂\ninst✝³ : AddCommMonoid M₃\ninst✝² : Module R M\ninst✝¹ : Module R M₂\ninst✝ : Module R M₃\nf✝ : M →ₗ[R] M₂\n... | [] | by
ext
apply smul_add | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Module.PUnit | {
"line": 46,
"column": 15
} | {
"line": 46,
"column": 30
} | {
"line": 47,
"column": 2
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝ : Zero R\n⊢ ∀ (a : R), a • 0 = 0",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"instHSMul",
"instSubsingletonPUnit",
"PUnit",
"PUnit.smul",
"PUnit.instZero_mathlib",
"Zero.toOfNat0",
"HSMul.hSMul",
... | [] | by subsingleton | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Module.PUnit | {
"line": 47,
"column": 15
} | {
"line": 47,
"column": 30
} | {
"line": 49,
"column": 0
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝ : Zero R\n⊢ ∀ (m : PUnit), 0 • m = 0",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"instHSMul",
"instSubsingletonPUnit",
"PUnit",
"PUnit.smul",
"PUnit.instZero_mathlib",
"Zero.toOfNat0",
"HSMul.hSMul",
... | [] | by subsingleton | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Module.PUnit | {
"line": 52,
"column": 14
} | {
"line": 52,
"column": 29
} | {
"line": 54,
"column": 0
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝ : Monoid R\n⊢ ∀ (x y : R) (b : PUnit), (x * y) • b = x • y • b",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Semigroup.toMul",
"instHSMul",
"HMul.hMul",
"instSubsingletonPUnit",
"PUnit",
"PUnit.smul",
"... | [] | by subsingleton | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Module.PUnit | {
"line": 51,
"column": 14
} | {
"line": 51,
"column": 29
} | {
"line": 52,
"column": 2
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝ : Monoid R\n⊢ ∀ (b : PUnit), 1 • b = b",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"MulOne.toOne",
"instHSMul",
"Monoid.toMulOneClass",
"instSubsingletonPUnit",
"MulOneClass.toMulOne",
"PUnit",
"PUnit.... | [] | by subsingleton | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Module.PUnit | {
"line": 56,
"column": 15
} | {
"line": 56,
"column": 30
} | {
"line": 57,
"column": 2
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝ : Monoid R\n⊢ ∀ (a : R), a • 0 = 0",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"instHSMul",
"AddMonoid.toAddZeroClass",
"PUnit.commRing",
"AddGroupWithOne.toAddMonoidWithOne",
"PUnit.mulAction",
"AddZeroClas... | [] | by subsingleton | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Module.PUnit | {
"line": 57,
"column": 14
} | {
"line": 57,
"column": 29
} | {
"line": 59,
"column": 0
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝ : Monoid R\n⊢ ∀ (a : R) (x y : PUnit), a • (x + y) = a • x + a • y",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"instHSMul",
"AddMonoid.toAddSemigroup",
"PUnit.commRing",
"AddGroupWithOne.toAddMonoidWithOne",
"PUni... | [] | by subsingleton | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Module.PUnit | {
"line": 62,
"column": 14
} | {
"line": 62,
"column": 29
} | {
"line": 64,
"column": 0
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝ : Monoid R\n⊢ ∀ (r : R), r • 1 = 1",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"MulOne.toOne",
"instHSMul",
"Monoid.toMulOneClass",
"CommSemiring.toSemiring",
"PUnit.commRing",
"PUnit.mulAction",
"inst... | [] | by subsingleton | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Module.PUnit | {
"line": 61,
"column": 14
} | {
"line": 61,
"column": 29
} | {
"line": 62,
"column": 2
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝ : Monoid R\n⊢ ∀ (r : R) (x y : PUnit), r • (x * y) = r • x * r • y",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"instHSMul",
"HMul.hMul",
"Monoid.toMulOneClass",
"CommSemiring.toSemiring",
"PUnit.commRing",
"... | [] | by subsingleton | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Module.PUnit | {
"line": 72,
"column": 14
} | {
"line": 72,
"column": 29
} | {
"line": 73,
"column": 2
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝ : Semiring R\n⊢ ∀ (r s : R) (x : PUnit), (r + s) • x = r • x + s • x",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"instHSMul",
"CommSemiring.toSemiring",
"PUnit.commRing",
"AddGroupWithOne.toAddMonoidWithOne",
"Dis... | [] | by subsingleton | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Module.PUnit | {
"line": 73,
"column": 15
} | {
"line": 73,
"column": 30
} | {
"line": 75,
"column": 0
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝ : Semiring R\n⊢ ∀ (x : PUnit), 0 • x = 0",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"instHSMul",
"CommSemiring.toSemiring",
"AddMonoid.toAddZeroClass",
"PUnit.commRing",
"AddGroupWithOne.toAddMonoidWithOne",
... | [] | by subsingleton | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.GroupTheory.GroupAction.SubMulAction | {
"line": 258,
"column": 25
} | {
"line": 258,
"column": 30
} | {
"line": 258,
"column": 30
} | [
{
"pp": "S : Type u'\nT : Type u''\nR : Type u\nM : Type v\ninst✝ : SMul R M\ns t : SubMulAction R M\n⊢ ∀ (c : R) {x : M}, x ∈ ↑s ∪ ↑t → c • x ∈ ↑s ∪ ↑t",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"SMulMemClass.smul_mem",
"SubMulAction.instSetLike",
"Eq.mpr",
"S... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.GroupTheory.GroupAction.SubMulAction | {
"line": 258,
"column": 25
} | {
"line": 258,
"column": 30
} | {
"line": 258,
"column": 30
} | [
{
"pp": "S : Type u'\nT : Type u''\nR : Type u\nM : Type v\ninst✝ : SMul R M\ns t : SubMulAction R M\n⊢ ∀ (c : R) {x : M}, x ∈ ↑s ∪ ↑t → c • x ∈ ↑s ∪ ↑t",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"SMulMemClass.smul_mem",
"SubMulAction.instSetLike",
"Eq.mpr",
"S... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.GroupAction.SubMulAction | {
"line": 258,
"column": 25
} | {
"line": 258,
"column": 30
} | {
"line": 258,
"column": 30
} | [
{
"pp": "S : Type u'\nT : Type u''\nR : Type u\nM : Type v\ninst✝ : SMul R M\ns t : SubMulAction R M\n⊢ ∀ (c : R) {x : M}, x ∈ ↑s ∪ ↑t → c • x ∈ ↑s ∪ ↑t",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"SMulMemClass.smul_mem",
"SubMulAction.instSetLike",
"Eq.mpr",
"S... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.GroupAction.SubMulAction | {
"line": 262,
"column": 25
} | {
"line": 262,
"column": 30
} | {
"line": 262,
"column": 30
} | [
{
"pp": "S : Type u'\nT : Type u''\nR : Type u\nM : Type v\ninst✝ : SMul R M\ns t : SubMulAction R M\n⊢ ∀ (c : R) {x : M}, x ∈ ↑s ∩ ↑t → c • x ∈ ↑s ∩ ↑t",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"SMulMemClass.smul_mem",
"SubMulAction.instSetLike",
"Eq.mpr",
"S... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.GroupTheory.GroupAction.SubMulAction | {
"line": 262,
"column": 25
} | {
"line": 262,
"column": 30
} | {
"line": 262,
"column": 30
} | [
{
"pp": "S : Type u'\nT : Type u''\nR : Type u\nM : Type v\ninst✝ : SMul R M\ns t : SubMulAction R M\n⊢ ∀ (c : R) {x : M}, x ∈ ↑s ∩ ↑t → c • x ∈ ↑s ∩ ↑t",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"SMulMemClass.smul_mem",
"SubMulAction.instSetLike",
"Eq.mpr",
"S... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.GroupAction.SubMulAction | {
"line": 262,
"column": 25
} | {
"line": 262,
"column": 30
} | {
"line": 262,
"column": 30
} | [
{
"pp": "S : Type u'\nT : Type u''\nR : Type u\nM : Type v\ninst✝ : SMul R M\ns t : SubMulAction R M\n⊢ ∀ (c : R) {x : M}, x ∈ ↑s ∩ ↑t → c • x ∈ ↑s ∩ ↑t",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"SMulMemClass.smul_mem",
"SubMulAction.instSetLike",
"Eq.mpr",
"S... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.GroupAction.SubMulAction | {
"line": 266,
"column": 28
} | {
"line": 266,
"column": 33
} | {
"line": 266,
"column": 33
} | [
{
"pp": "S✝ : Type u'\nT : Type u''\nR : Type u\nM : Type v\ninst✝ : SMul R M\nS : Set (SubMulAction R M)\n⊢ ∀ (c : R) {x : M}, x ∈ ⋃ s ∈ S, ↑s → c • x ∈ ⋃ s ∈ S, ↑s",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"SMulMemClass.smul_mem",
"SubMulAction.instSetLike",
"Eq.m... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.GroupTheory.GroupAction.SubMulAction | {
"line": 266,
"column": 28
} | {
"line": 266,
"column": 33
} | {
"line": 266,
"column": 33
} | [
{
"pp": "S✝ : Type u'\nT : Type u''\nR : Type u\nM : Type v\ninst✝ : SMul R M\nS : Set (SubMulAction R M)\n⊢ ∀ (c : R) {x : M}, x ∈ ⋃ s ∈ S, ↑s → c • x ∈ ⋃ s ∈ S, ↑s",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"SMulMemClass.smul_mem",
"SubMulAction.instSetLike",
"Eq.m... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.GroupAction.SubMulAction | {
"line": 266,
"column": 28
} | {
"line": 266,
"column": 33
} | {
"line": 266,
"column": 33
} | [
{
"pp": "S✝ : Type u'\nT : Type u''\nR : Type u\nM : Type v\ninst✝ : SMul R M\nS : Set (SubMulAction R M)\n⊢ ∀ (c : R) {x : M}, x ∈ ⋃ s ∈ S, ↑s → c • x ∈ ⋃ s ∈ S, ↑s",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"SMulMemClass.smul_mem",
"SubMulAction.instSetLike",
"Eq.m... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.GroupAction.SubMulAction | {
"line": 270,
"column": 29
} | {
"line": 270,
"column": 34
} | {
"line": 270,
"column": 34
} | [
{
"pp": "S✝ : Type u'\nT : Type u''\nR : Type u\nM : Type v\ninst✝ : SMul R M\nS : Set (SubMulAction R M)\n⊢ ∀ (c : R) {x : M}, x ∈ ⋂ s ∈ S, ↑s → c • x ∈ ⋂ s ∈ S, ↑s",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"SMulMemClass.smul_mem",
"SubMulAction.instSetLike",
"Eq.m... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.GroupTheory.GroupAction.SubMulAction | {
"line": 270,
"column": 29
} | {
"line": 270,
"column": 34
} | {
"line": 270,
"column": 34
} | [
{
"pp": "S✝ : Type u'\nT : Type u''\nR : Type u\nM : Type v\ninst✝ : SMul R M\nS : Set (SubMulAction R M)\n⊢ ∀ (c : R) {x : M}, x ∈ ⋂ s ∈ S, ↑s → c • x ∈ ⋂ s ∈ S, ↑s",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"SMulMemClass.smul_mem",
"SubMulAction.instSetLike",
"Eq.m... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.GroupAction.SubMulAction | {
"line": 270,
"column": 29
} | {
"line": 270,
"column": 34
} | {
"line": 270,
"column": 34
} | [
{
"pp": "S✝ : Type u'\nT : Type u''\nR : Type u\nM : Type v\ninst✝ : SMul R M\nS : Set (SubMulAction R M)\n⊢ ∀ (c : R) {x : M}, x ∈ ⋂ s ∈ S, ↑s → c • x ∈ ⋂ s ∈ S, ↑s",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"SMulMemClass.smul_mem",
"SubMulAction.instSetLike",
"Eq.m... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Module.Submodule.Map | {
"line": 117,
"column": 80
} | {
"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\n⊢ map (f + g) p ≤ ... | [] | by
rintro x ⟨m, hm, rfl⟩
exact add_mem_sup (mem_map_of_mem hm) (mem_map_of_mem hm) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Module.Submodule.Map | {
"line": 168,
"column": 14
} | {
"line": 168,
"column": 42
} | {
"line": 168,
"column": 42
} | [
{
"pp": "R : Type u_1\nR₂ : Type u_3\nM : Type u_5\nM₂ : Type u_7\ninst✝⁷ : Semiring R\ninst✝⁶ : Semiring R₂\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : AddCommMonoid M₂\ninst✝³ : Module R M\ninst✝² : Module R₂ M₂\nσ₁₂ : R →+* R₂\nσ₂₁ : R₂ →+* R\ninst✝¹ : RingHomInvPair σ₁₂ σ₂₁\ninst✝ : RingHomInvPair σ₂₁ σ₁₂\nf : M →ₛ... | [
"R : Type u_1\nR₂ : Type u_3\nM : Type u_5\nM₂ : Type u_7\ninst✝⁷ : Semiring R\ninst✝⁶ : Semiring R₂\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : AddCommMonoid M₂\ninst✝³ : Module R M\ninst✝² : Module R₂ M₂\nσ₁₂ : R →+* R₂\nσ₂₁ : R₂ →+* R\ninst✝¹ : RingHomInvPair σ₁₂ σ₂₁\ninst✝ : RingHomInvPair σ₂₁ σ₁₂\nf : M →ₛₗ[σ₁₂] M₂\ni... | LinearEquiv.apply_symm_apply | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Module.Submodule.Map | {
"line": 235,
"column": 36
} | {
"line": 235,
"column": 56
} | {
"line": 235,
"column": 57
} | [
{
"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\ninst✝ : RingHomSurjective σ₁₂\nf : M →ₛₗ[σ₁₂] M₂\nhf : Injective ⇑f\np q : Submodule R M\... | [
"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\ninst✝ : RingHomSurjective σ₁₂\nf : M →ₛₗ[σ₁₂] M₂\nhf : Injective ⇑f\np q : Submodule R M\nhpq : Disjo... | disjoint_iff.mp hpq, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Ring.CharZero | {
"line": 101,
"column": 2
} | {
"line": 101,
"column": 18
} | {
"line": 103,
"column": 0
} | [
{
"pp": "R : Type u_2\ninst✝² : NonAssocRing R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\nn : ℕ\na b : R\nh : ↑n = 0 ∨ a = b\n⊢ n = 0 ∨ a = b",
"ppTerm": "?m.51",
"assigned": true,
"usedConstants": [
"congrArg",
"AddMonoid.toAddZeroClass",
"AddGroupWithOne.toAddMonoidWithOne",... | [] | exact mod_cast h | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.NonUnitalSubsemiring.Defs | {
"line": 44,
"column": 27
} | {
"line": 44,
"column": 32
} | {
"line": 46,
"column": 0
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝³ : Mul R\ninst✝² : HasDistribNeg R\ninst✝¹ : SetLike S R\ninst✝ : MulMemClass S R\ns : S\nx y z : R\nhx : x ∈ s\nhy : -y ∈ s\nhz : z ∈ s\n⊢ x * -y * z ∈ s",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"co... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.RingTheory.NonUnitalSubsemiring.Defs | {
"line": 44,
"column": 27
} | {
"line": 44,
"column": 32
} | {
"line": 46,
"column": 0
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝³ : Mul R\ninst✝² : HasDistribNeg R\ninst✝¹ : SetLike S R\ninst✝ : MulMemClass S R\ns : S\nx y z : R\nhx : x ∈ s\nhy : -y ∈ s\nhz : z ∈ s\n⊢ x * -y * z ∈ s",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"co... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.NonUnitalSubsemiring.Defs | {
"line": 44,
"column": 27
} | {
"line": 44,
"column": 32
} | {
"line": 46,
"column": 0
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝³ : Mul R\ninst✝² : HasDistribNeg R\ninst✝¹ : SetLike S R\ninst✝ : MulMemClass S R\ns : S\nx y z : R\nhx : x ∈ s\nhy : -y ∈ s\nhz : z ∈ s\n⊢ x * -y * z ∈ s",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"co... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Ring.Center | {
"line": 64,
"column": 15
} | {
"line": 64,
"column": 70
} | {
"line": 65,
"column": 2
} | [
{
"pp": "M : Type u_1\ninst✝ : Distrib M\na b : M\nha : a ∈ center M\nhb : b ∈ center M\nx✝ : M\n⊢ Commute (a + b) x✝",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"add_mul",
"Distrib.leftDistribClass",
"Eq.mpr",
"HMul.hMul",
"congrArg",
"commute_iff_e... | [] | rw [commute_iff_eq, add_mul, mul_add, ha.comm, hb.comm] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Ring.Center | {
"line": 64,
"column": 15
} | {
"line": 64,
"column": 70
} | {
"line": 65,
"column": 2
} | [
{
"pp": "M : Type u_1\ninst✝ : Distrib M\na b : M\nha : a ∈ center M\nhb : b ∈ center M\nx✝ : M\n⊢ Commute (a + b) x✝",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"add_mul",
"Distrib.leftDistribClass",
"Eq.mpr",
"HMul.hMul",
"congrArg",
"commute_iff_e... | [] | rw [commute_iff_eq, add_mul, mul_add, ha.comm, hb.comm] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Ring.Center | {
"line": 64,
"column": 15
} | {
"line": 64,
"column": 70
} | {
"line": 65,
"column": 2
} | [
{
"pp": "M : Type u_1\ninst✝ : Distrib M\na b : M\nha : a ∈ center M\nhb : b ∈ center M\nx✝ : M\n⊢ Commute (a + b) x✝",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"add_mul",
"Distrib.leftDistribClass",
"Eq.mpr",
"HMul.hMul",
"congrArg",
"commute_iff_e... | [] | rw [commute_iff_eq, add_mul, mul_add, ha.comm, hb.comm] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Ring.Subsemiring.Basic | {
"line": 430,
"column": 58
} | {
"line": 430,
"column": 63
} | {
"line": 432,
"column": 0
} | [
{
"pp": "case refine_2.mem\nR : Type u\ninst✝ : NonAssocSemiring R\nM : Submonoid R\nx✝¹ x✝ : R\nh✝ : x✝ ∈ ↑M\n⊢ x✝ ∈ NonUnitalSubsemiring.closure ↑M",
"ppTerm": "?refine_2.mem",
"assigned": true,
"usedConstants": [
"SetLike.mem_coe._simp_1",
"NonUnitalSubsemiring.mem_closure_of_mem",
... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Ring.Subsemiring.Basic | {
"line": 430,
"column": 58
} | {
"line": 430,
"column": 63
} | {
"line": 432,
"column": 0
} | [
{
"pp": "case refine_2.zero\nR : Type u\ninst✝ : NonAssocSemiring R\nM : Submonoid R\nx✝ : R\n⊢ 0 ∈ NonUnitalSubsemiring.closure ↑M",
"ppTerm": "?refine_2.zero",
"assigned": true,
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"AddMonoid.toAddZeroClass",
"Membership.... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Ring.Subsemiring.Basic | {
"line": 430,
"column": 58
} | {
"line": 430,
"column": 63
} | {
"line": 432,
"column": 0
} | [
{
"pp": "case refine_2.add\nR : Type u\ninst✝ : NonAssocSemiring R\nM : Submonoid R\nx✝¹ x✝ y✝ : R\nhx✝ : x✝ ∈ AddSubmonoid.closure ↑M\nhy✝ : y✝ ∈ AddSubmonoid.closure ↑M\na✝¹ : x✝ ∈ NonUnitalSubsemiring.closure ↑M\na✝ : y✝ ∈ NonUnitalSubsemiring.closure ↑M\n⊢ x✝ + y✝ ∈ NonUnitalSubsemiring.closure ↑M",
"pp... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.GroupWithZero.Associated | {
"line": 248,
"column": 4
} | {
"line": 248,
"column": 66
} | {
"line": 249,
"column": 2
} | [
{
"pp": "case refine_1.inr\nM : Type u_1\ninst✝¹ : CommMonoidWithZero M\ninst✝ : IsCancelMulZero M\nx y : M\nh : Prime (x * y)\nhy : IsUnit y\n⊢ Prime x ∧ IsUnit y ∨ IsUnit x ∧ Prime y",
"ppTerm": "?refine_1.inr",
"assigned": true,
"usedConstants": [
"HMul.hMul",
"Monoid.toMulOneClass",
... | [] | · exact Or.inl ⟨(associated_mul_unit_left x y hy).prime h, hy⟩ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Ring.Subring.Basic | {
"line": 1008,
"column": 2
} | {
"line": 1008,
"column": 33
} | {
"line": 1009,
"column": 2
} | [
{
"pp": "case cons\nR : Type u_1\ninst✝ : Ring R\ns : Set R\nC : R → Prop\nh1 : C 1\nhneg1 : C (-1)\nhs : ∀ z ∈ s, ∀ (n : R), C n → C (z * n)\nha : ∀ {x y : R}, C x → C y → C (x + y)\nh0 : C 0\nhd : List R\ntl : List (List R)\nih : (∀ t ∈ tl, ∀ y ∈ t, y ∈ s ∨ y = -1) → C (List.map List.prod tl).sum\nHL : ∀ t ∈ ... | [
"case cons\nR : Type u_1\ninst✝ : Ring R\ns : Set R\nC : R → Prop\nh1 : C 1\nhneg1 : C (-1)\nhs : ∀ z ∈ s, ∀ (n : R), C n → C (z * n)\nha : ∀ {x y : R}, C x → C y → C (x + y)\nh0 : C 0\nhd : List R\ntl : List (List R)\nih : (∀ t ∈ tl, ∀ y ∈ t, y ∈ s ∨ y = -1) → C (List.map List.prod tl).sum\nHL : (∀ y ∈ hd, y ∈ s ∨... | rw [List.forall_mem_cons] at HL | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Ring.Subring.Basic | {
"line": 1035,
"column": 2
} | {
"line": 1035,
"column": 33
} | {
"line": 1036,
"column": 2
} | [
{
"pp": "case cons\nR : Type u_1\ninst✝ : Ring R\ns : Set R\nC : R → Prop\nh1 : C 1\nhneg1 : C (-1)\nhs : ∀ z ∈ s, ∀ (n : R), C n → C (z * n)\nha : ∀ {x y : R}, C x → C y → C (x + y)\nh0 : C 0\nhd : R\ntl : List R\nih : (∀ y ∈ tl, y ∈ s ∨ y = -1) → ∃ L, (∀ x ∈ L, x ∈ s) ∧ (tl.prod = L.prod ∨ tl.prod = -L.prod)\... | [
"case cons\nR : Type u_1\ninst✝ : Ring R\ns : Set R\nC : R → Prop\nh1 : C 1\nhneg1 : C (-1)\nhs : ∀ z ∈ s, ∀ (n : R), C n → C (z * n)\nha : ∀ {x y : R}, C x → C y → C (x + y)\nh0 : C 0\nhd : R\ntl : List R\nih : (∀ y ∈ tl, y ∈ s ∨ y = -1) → ∃ L, (∀ x ∈ L, x ∈ s) ∧ (tl.prod = L.prod ∨ tl.prod = -L.prod)\nHL : (hd ∈ ... | rw [List.forall_mem_cons] at HL | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Module.Submodule.Range | {
"line": 431,
"column": 2
} | {
"line": 431,
"column": 75
} | {
"line": 432,
"column": 2
} | [
{
"pp": "R : Type u_1\nM : Type u_5\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\nM' : Type u_10\ninst✝¹ : AddCommMonoid M'\ninst✝ : Module R M'\nO : Submodule R M\nϕ : ↥O →ₗ[R] M'\nN : Submodule R M\nx : M'\n⊢ x ∈ ϕ.submoduleImage N ↔ ∃ y, ∃ (yO : y ∈ O), y ∈ N ∧ ϕ ⟨y, yO⟩ = x",
"ppT... | [
"case refine_1\nR : Type u_1\nM : Type u_5\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\nM' : Type u_10\ninst✝¹ : AddCommMonoid M'\ninst✝ : Module R M'\nO : Submodule R M\nϕ : ↥O →ₗ[R] M'\nN : Submodule R M\nx : M'\n⊢ (∃ y, O.subtype y ∈ N ∧ ϕ y = x) → ∃ y, ∃ (yO : y ∈ O), y ∈ N ∧ ϕ ⟨y, yO⟩ =... | refine Submodule.mem_map.trans ⟨?_, ?_⟩ <;> simp_rw [Submodule.mem_comap] | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Algebra.Module.Submodule.Pointwise | {
"line": 411,
"column": 78
} | {
"line": 411,
"column": 83
} | {
"line": 413,
"column": 0
} | [
{
"pp": "case smul₀\nR : Type u_2\nM : Type u_3\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\nS : Type u_4\ninst✝¹ : Monoid S\ninst✝ : DistribMulAction S M\ns : Set S\nx : M\nr✝ : S\nn✝ : M\nmem₁✝ : r✝ ∈ s\nmem₂✝ : n✝ ∈ ⊥\n⊢ r✝ • n✝ ∈ ⊥",
"ppTerm": "?smul₀",
"assigned": true,
... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Module.Submodule.Pointwise | {
"line": 411,
"column": 78
} | {
"line": 411,
"column": 83
} | {
"line": 413,
"column": 0
} | [
{
"pp": "case smul₁\nR : Type u_2\nM : Type u_3\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\nS : Type u_4\ninst✝¹ : Monoid S\ninst✝ : DistribMulAction S M\ns : Set S\nx : M\nr✝ : R\nm✝ : M\nmem✝ : m✝ ∈ s • ⊥\na✝ : m✝ ∈ ⊥\n⊢ r✝ • m✝ ∈ ⊥",
"ppTerm": "?smul₁",
"assigned": true,
... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Module.Submodule.Pointwise | {
"line": 411,
"column": 78
} | {
"line": 411,
"column": 83
} | {
"line": 413,
"column": 0
} | [
{
"pp": "case add\nR : Type u_2\nM : Type u_3\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\nS : Type u_4\ninst✝¹ : Monoid S\ninst✝ : DistribMulAction S M\ns : Set S\nx m₁✝ m₂✝ : M\nmem₁✝ : m₁✝ ∈ s • ⊥\nmem₂✝ : m₂✝ ∈ s • ⊥\na✝¹ : m₁✝ ∈ ⊥\na✝ : m₂✝ ∈ ⊥\n⊢ m₁✝ + m₂✝ ∈ ⊥",
"ppTerm": "?add... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Module.Submodule.Pointwise | {
"line": 411,
"column": 78
} | {
"line": 411,
"column": 83
} | {
"line": 413,
"column": 0
} | [
{
"pp": "case zero\nR : Type u_2\nM : Type u_3\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\nS : Type u_4\ninst✝¹ : Monoid S\ninst✝ : DistribMulAction S M\ns : Set S\nx : M\n⊢ 0 ∈ ⊥",
"ppTerm": "?zero",
"assigned": true,
"usedConstants": [
"Submodule",
"Submodule.a... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Module.Submodule.Pointwise | {
"line": 425,
"column": 15
} | {
"line": 425,
"column": 20
} | {
"line": 426,
"column": 4
} | [
{
"pp": "case mp.smul₀\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\nr✝ : S\nn✝ : M\nmem₁✝ : r✝ ∈ {r}\nmem₂✝ : n✝ ∈ N\n⊢ ∃ m ∈ N, r✝ • ... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Module.Submodule.Pointwise | {
"line": 425,
"column": 15
} | {
"line": 425,
"column": 20
} | {
"line": 426,
"column": 4
} | [
{
"pp": "case mp.smul₀\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\nr✝ : S\nn✝ : M\nmem₁✝ : r✝ ∈ {r}\nmem₂✝ : n✝ ∈ N\n⊢ ∃ m ∈ N, r✝ • ... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Module.Submodule.Pointwise | {
"line": 425,
"column": 15
} | {
"line": 425,
"column": 20
} | {
"line": 426,
"column": 4
} | [
{
"pp": "case mp.smul₀\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\nr✝ : S\nn✝ : M\nmem₁✝ : r✝ ∈ {r}\nmem₂✝ : n✝ ∈ N\n⊢ ∃ m ∈ N, r✝ • ... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Module.Submodule.Pointwise | {
"line": 428,
"column": 23
} | {
"line": 428,
"column": 28
} | {
"line": 428,
"column": 28
} | [
{
"pp": "R : 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\nt : R\nn : M\nhn : n ∈ N\nmem : r • n ∈ {r} • N\n⊢ t • n ∈ N",
"ppTerm": "?m.... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Module.Submodule.Pointwise | {
"line": 428,
"column": 23
} | {
"line": 428,
"column": 28
} | {
"line": 428,
"column": 28
} | [
{
"pp": "R : 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\nt : R\nn : M\nhn : n ∈ N\nmem : r • n ∈ {r} • N\n⊢ t • n ∈ N",
"ppTerm": "?m.... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Module.Submodule.Pointwise | {
"line": 428,
"column": 23
} | {
"line": 428,
"column": 28
} | {
"line": 428,
"column": 28
} | [
{
"pp": "R : 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\nt : R\nn : M\nhn : n ∈ N\nmem : r • n ∈ {r} • N\n⊢ t • n ∈ N",
"ppTerm": "?m.... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Module.Submodule.Pointwise | {
"line": 434,
"column": 4
} | {
"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 | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Module.Submodule.Pointwise | {
"line": 434,
"column": 4
} | {
"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.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Module.Submodule.Pointwise | {
"line": 434,
"column": 4
} | {
"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.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.SupClosed | {
"line": 103,
"column": 2
} | {
"line": 103,
"column": 7
} | {
"line": 105,
"column": 0
} | [
{
"pp": "α : Type u_3\ninst✝ : SemilatticeSup α\ns : Set α\na : α\nhs : SupClosed s\nha : a ∈ upperBounds s\n⊢ ∀ ⦃a_1 : α⦄, a_1 ∈ insert a s → ∀ ⦃b : α⦄, b ∈ insert a s → a_1 ⊔ b ∈ insert a s",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instReflLe",
"congrAr... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Order.SupClosed | {
"line": 110,
"column": 2
} | {
"line": 110,
"column": 7
} | {
"line": 112,
"column": 0
} | [
{
"pp": "α : Type u_3\ninst✝ : SemilatticeSup α\ns : Set α\na : α\nh : SupClosed s\nha : a ∈ lowerBounds s\nha' : ∀ b ∈ s, a ≤ b\n⊢ ∀ ⦃a_1 : α⦄, a_1 ∈ insert a s → ∀ ⦃b : α⦄, b ∈ insert a s → a_1 ⊔ b ∈ insert a s",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"lowerB... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.LinearAlgebra.Span.Defs | {
"line": 409,
"column": 2
} | {
"line": 409,
"column": 44
} | {
"line": 410,
"column": 2
} | [
{
"pp": "R : Type u_1\nM : Type u_4\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\np p' : Submodule R M\nx✝ : M\n⊢ x✝ ∈ ↑(p ⊔ p') ↔ x✝ ∈ ↑p + ↑p'",
"ppTerm": "?m.36",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Submodule",
"Lattice.toSemilatticeSup",
... | [
"R : Type u_1\nM : Type u_4\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\np p' : Submodule R M\nx✝ : M\n⊢ (∃ y ∈ p, ∃ z ∈ p', y + z = x✝) ↔ ∃ x ∈ ↑p, ∃ y ∈ ↑p', x + y = x✝"
] | rw [SetLike.mem_coe, mem_sup, Set.mem_add] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.Span.Defs | {
"line": 408,
"column": 2
} | {
"line": 410,
"column": 6
} | {
"line": 412,
"column": 0
} | [
{
"pp": "R : Type u_1\nM : Type u_4\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\np p' : Submodule R M\n⊢ ↑(p ⊔ p') = ↑p + ↑p'",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"Set.ext",
"Eq.mpr",
"Submodule",
"SetLike.mem_coe._simp_1",
"L... | [] | ext
rw [SetLike.mem_coe, mem_sup, Set.mem_add]
simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Span.Defs | {
"line": 408,
"column": 2
} | {
"line": 410,
"column": 6
} | {
"line": 412,
"column": 0
} | [
{
"pp": "R : Type u_1\nM : Type u_4\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\np p' : Submodule R M\n⊢ ↑(p ⊔ p') = ↑p + ↑p'",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"Set.ext",
"Eq.mpr",
"Submodule",
"SetLike.mem_coe._simp_1",
"L... | [] | ext
rw [SetLike.mem_coe, mem_sup, Set.mem_add]
simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Span.Defs | {
"line": 428,
"column": 2
} | {
"line": 428,
"column": 7
} | {
"line": 430,
"column": 0
} | [
{
"pp": "R : Type u_1\nM : Type u_4\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\np p' : Submodule R M\nP : M → Prop\n⊢ (∀ (x x_1 : M), x_1 ∈ p → ∀ x_2 ∈ p', x_1 + x_2 = x → P x) ↔ ∀ x₁ ∈ p, ∀ x₂ ∈ p', P (x₁ + x₂)",
"ppTerm": "?m.41",
"assigned": true,
"usedConstants": [
... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Order.Atoms | {
"line": 242,
"column": 2
} | {
"line": 242,
"column": 59
} | {
"line": 243,
"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, K ≤ H ∧ H ≤ L ∧ g ∉ K ∧ g ∈ H ∧ H ≠ L",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"lt_iff_le_and_ne",
"Eq.mpr",
... | [
"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) ∧ ¬H ≤ K ∧ H ≠ L ↔ ∃ g, K ≤ H ∧ H ≤ L ∧ g ∉ K ∧ g ∈ H ∧ H ≠ L"
] | rw [lt_iff_le_not_ge, lt_iff_le_and_ne, and_and_and_comm] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Order.Atoms | {
"line": 622,
"column": 2
} | {
"line": 622,
"column": 62
} | {
"line": 623,
"column": 2
} | [
{
"pp": "α : Type u_2\ninst✝² : PartialOrder α\ninst✝¹ : OrderBot α\ninst✝ : IsAtomistic α\na b : α\nh : ∀ (c : α), IsAtom c → (c ≤ a ↔ c ≤ b)\n⊢ a = b",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"le_iff_atom_le_imp",
"congrArg",
"PartialOrder.toPreord... | [
"α : Type u_2\ninst✝² : PartialOrder α\ninst✝¹ : OrderBot α\ninst✝ : IsAtomistic α\na b : α\nh : ∀ (c : α), IsAtom c → (c ≤ a ↔ c ≤ b)\n⊢ (∀ (c : α), IsAtom c → c ≤ a → c ≤ b) ∧ ∀ (c : α), IsAtom c → c ≤ b → c ≤ a"
] | rw [le_antisymm_iff, le_iff_atom_le_imp, le_iff_atom_le_imp] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Order.Atoms | {
"line": 638,
"column": 4
} | {
"line": 641,
"column": 41
} | {
"line": 641,
"column": 41
} | [
{
"pp": "ι : Sort u_1\nα : Type u_2\nβ : Type u_3\ninst✝² : PartialOrder α\ninst✝¹ : OrderTop α\ninst✝ : IsCoatomistic α\nb : α\n⊢ b = ⊤ ∨ ∃ a, IsCoatom a ∧ b ≤ a",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"lowerBounds",
"congrArg",
"true_or",
"PartialOrder.toP... | [] | rcases isGLB_coatoms b with ⟨s, hsb, hs⟩
rcases s.eq_empty_or_nonempty with rfl | ⟨a, ha⟩
· simp_all
· exact Or.inr ⟨a, hs _ ha, hsb.1 ha⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.Atoms | {
"line": 638,
"column": 4
} | {
"line": 641,
"column": 41
} | {
"line": 641,
"column": 41
} | [
{
"pp": "ι : Sort u_1\nα : Type u_2\nβ : Type u_3\ninst✝² : PartialOrder α\ninst✝¹ : OrderTop α\ninst✝ : IsCoatomistic α\nb : α\n⊢ b = ⊤ ∨ ∃ a, IsCoatom a ∧ b ≤ a",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"lowerBounds",
"congrArg",
"true_or",
"PartialOrder.toP... | [] | rcases isGLB_coatoms b with ⟨s, hsb, hs⟩
rcases s.eq_empty_or_nonempty with rfl | ⟨a, ha⟩
· simp_all
· exact Or.inr ⟨a, hs _ ha, hsb.1 ha⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Span.Defs | {
"line": 709,
"column": 15
} | {
"line": 709,
"column": 20
} | {
"line": 710,
"column": 2
} | [
{
"pp": "case mem\nR : Type u_8\nA : Type u_9\ninst✝⁴ : Semiring R\ninst✝³ : NonUnitalNonAssocSemiring A\ninst✝² : Module R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\ns : Set A\nx : A\nh : ∀ y ∈ s, Commute y x\ny x✝ : A\nh✝ : x✝ ∈ s\n⊢ Commute x✝ x",
"ppTerm": "?mem",
"assigned": true... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.LinearAlgebra.Span.Defs | {
"line": 709,
"column": 15
} | {
"line": 709,
"column": 20
} | {
"line": 710,
"column": 2
} | [
{
"pp": "case mem\nR : Type u_8\nA : Type u_9\ninst✝⁴ : Semiring R\ninst✝³ : NonUnitalNonAssocSemiring A\ninst✝² : Module R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\ns : Set A\nx : A\nh : ∀ y ∈ s, Commute y x\ny x✝ : A\nh✝ : x✝ ∈ s\n⊢ Commute x✝ x",
"ppTerm": "?mem",
"assigned": true... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
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