module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.Order.Filter.Bases.Basic | {
"line": 697,
"column": 2
} | {
"line": 697,
"column": 93
} | {
"line": 699,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nι' : Sort u_5\nlb : Filter β\npb : ι' → Prop\nsb : ι' → Set β\nsa : Set α\nh : lb.HasBasis pb sb\n⊢ (𝓟 sa ×ˢ lb).HasBasis pb fun x ↦ sa ×ˢ sb x",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Set.instSProd",
"Eq.mpr",
"SProd.sprod",
... | [] | simpa only [prod_eq_inf, comap_principal, prod_eq] using (h.comap Prod.snd).principal_inf _ | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Order.Filter.Bases.Basic | {
"line": 697,
"column": 2
} | {
"line": 697,
"column": 93
} | {
"line": 699,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nι' : Sort u_5\nlb : Filter β\npb : ι' → Prop\nsb : ι' → Set β\nsa : Set α\nh : lb.HasBasis pb sb\n⊢ (𝓟 sa ×ˢ lb).HasBasis pb fun x ↦ sa ×ˢ sb x",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Set.instSProd",
"Eq.mpr",
"SProd.sprod",
... | [] | simpa only [prod_eq_inf, comap_principal, prod_eq] using (h.comap Prod.snd).principal_inf _ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.Filter.Bases.Basic | {
"line": 697,
"column": 2
} | {
"line": 697,
"column": 93
} | {
"line": 699,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nι' : Sort u_5\nlb : Filter β\npb : ι' → Prop\nsb : ι' → Set β\nsa : Set α\nh : lb.HasBasis pb sb\n⊢ (𝓟 sa ×ˢ lb).HasBasis pb fun x ↦ sa ×ˢ sb x",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Set.instSProd",
"Eq.mpr",
"SProd.sprod",
... | [] | simpa only [prod_eq_inf, comap_principal, prod_eq] using (h.comap Prod.snd).principal_inf _ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.Filter.Bases.Basic | {
"line": 701,
"column": 2
} | {
"line": 701,
"column": 58
} | {
"line": 701,
"column": 59
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nι : Sort u_4\nla : Filter α\npa : ι → Prop\nsa : ι → Set α\nh : la.HasBasis pa sa\nsb : Set β\n⊢ (la ×ˢ 𝓟 sb).HasBasis pa fun x ↦ sa x ×ˢ sb",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Set.instSProd",
"Eq.mpr",
"SProd.sprod",
... | [
"α : Type u_1\nβ : Type u_2\nι : Sort u_4\nla : Filter α\npa : ι → Prop\nsa : ι → Set α\nh : la.HasBasis pa sa\nsb : Set β\n⊢ (Filter.comap Prod.fst la ⊓ 𝓟 (Prod.snd ⁻¹' sb)).HasBasis pa fun x ↦ Prod.fst ⁻¹' sa x ∩ Prod.snd ⁻¹' sb"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Filter.Bases.Basic | {
"line": 705,
"column": 2
} | {
"line": 705,
"column": 35
} | {
"line": 705,
"column": 36
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nι' : Sort u_5\nlb : Filter β\npb : ι' → Prop\nsb : ι' → Set β\nh : lb.HasBasis pb sb\n⊢ (⊤ ×ˢ lb).HasBasis pb fun x ↦ univ ×ˢ sb x",
"ppTerm": "?m.19",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\nβ : Type u_2\nι' : Sort u_5\nlb : Filter β\npb : ι' → Prop\nsb : ι' → Set β\nh : lb.HasBasis pb sb\n⊢ (⊤ ×ˢ lb).HasBasis pb fun x ↦ univ ×ˢ sb x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Filter.Bases.Basic | {
"line": 709,
"column": 2
} | {
"line": 709,
"column": 35
} | {
"line": 709,
"column": 36
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nι : Sort u_4\nla : Filter α\npa : ι → Prop\nsa : ι → Set α\nh : la.HasBasis pa sa\n⊢ (la ×ˢ ⊤).HasBasis pa fun x ↦ sa x ×ˢ univ",
"ppTerm": "?m.19",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\nβ : Type u_2\nι : Sort u_4\nla : Filter α\npa : ι → Prop\nsa : ι → Set α\nh : la.HasBasis pa sa\n⊢ (la ×ˢ ⊤).HasBasis pa fun x ↦ sa x ×ˢ univ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Filter.Bases.Basic | {
"line": 739,
"column": 4
} | {
"line": 739,
"column": 52
} | {
"line": 740,
"column": 6
} | [
{
"pp": "α : Type u_1\nι : Sort u_4\nla : Filter α\npa : ι → Prop\nsa : ι → Set α\nhl : la.HasBasis pa sa\ni j : ι\nhi : pa i\nhj : pa j\n⊢ ∃ k, pa k ∧ sa k ⊆ sa i ∧ sa k ⊆ sa j",
"ppTerm": "?m.26",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\nι : Sort u_4\nla : Filter α\npa : ι → Prop\nsa : ι → Set α\nhl : la.HasBasis pa sa\ni j : ι\nhi : pa i\nhj : pa j\n⊢ ∃ k, pa k ∧ sa k ⊆ sa i ∧ sa k ⊆ sa j"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Filter.Map | {
"line": 696,
"column": 14
} | {
"line": 696,
"column": 18
} | {
"line": 697,
"column": 4
} | [
{
"pp": "case mp\nα : Type u_1\nβ : Type u_2\nf : α → β\nF : Filter β\nx : α\nh : ∀ (A : Set α), ∀ B ∈ F, f ⁻¹' B ⊆ A → x ∈ A\nU : Set β\n⊢ U ∈ F → f x ∈ U",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
"Filter.instMembership",
"Membership.mem",
"Filter",
"Set"
]... | [
"case mp\nα : Type u_1\nβ : Type u_2\nf : α → β\nF : Filter β\nx : α\nh : ∀ (A : Set α), ∀ B ∈ F, f ⁻¹' B ⊆ A → x ∈ A\nU : Set β\nU_in : U ∈ F\n⊢ f x ∈ U"
] | U_in | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Order.Filter.Map | {
"line": 697,
"column": 4
} | {
"line": 697,
"column": 68
} | {
"line": 697,
"column": 69
} | [
{
"pp": "case mp\nα : Type u_1\nβ : Type u_2\nf : α → β\nF : Filter β\nx : α\nh : ∀ (A : Set α), ∀ B ∈ F, f ⁻¹' B ⊆ A → x ∈ A\nU : Set β\nU_in : U ∈ F\n⊢ f x ∈ U",
"ppTerm": "?mp",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"case mp\nα : Type u_1\nβ : Type u_2\nf : α → β\nF : Filter β\nx : α\nh : ∀ (A : Set α), ∀ B ∈ F, f ⁻¹' B ⊆ A → x ∈ A\nU : Set β\nU_in : U ∈ F\n⊢ f x ∈ U"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Filter.Map | {
"line": 698,
"column": 16
} | {
"line": 698,
"column": 20
} | {
"line": 698,
"column": 21
} | [
{
"pp": "case mpr\nα : Type u_1\nβ : Type u_2\nf : α → β\nF : Filter β\nx : α\nh : ∀ B ∈ F, f x ∈ B\nV : Set α\nU : Set β\n⊢ U ∈ F → f ⁻¹' U ⊆ V → x ∈ V",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"Filter.instMembership",
"Membership.mem",
"Filter",
"Set"
],
... | [
"case mpr\nα : Type u_1\nβ : Type u_2\nf : α → β\nF : Filter β\nx : α\nh : ∀ B ∈ F, f x ∈ B\nV : Set α\nU : Set β\nU_in : U ∈ F\n⊢ f ⁻¹' U ⊆ V → x ∈ V"
] | U_in | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Order.Filter.Basic | {
"line": 1103,
"column": 24
} | {
"line": 1103,
"column": 35
} | {
"line": 1103,
"column": 36
} | [
{
"pp": "α : Type u\nβ : Type v\ninst✝ : AddGroup β\nf g : α → β\nl : Filter α\nh : f =ᶠ[l] g\n⊢ f - g =ᶠ[l] 0",
"ppTerm": "?m.11",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u\nβ : Type v\ninst✝ : AddGroup β\nf g : α → β\nl : Filter α\nh : f =ᶠ[l] g\n⊢ f - g =ᶠ[l] 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Filter.Basic | {
"line": 1107,
"column": 34
} | {
"line": 1107,
"column": 45
} | {
"line": 1107,
"column": 46
} | [
{
"pp": "α : Type u\nβ : Type v\ninst✝ : AddGroup β\nf g : α → β\nl : Filter α\nh : f - g =ᶠ[l] 0\n⊢ f =ᶠ[l] g",
"ppTerm": "?m.21",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u\nβ : Type v\ninst✝ : AddGroup β\nf g : α → β\nl : Filter α\nh : f - g =ᶠ[l] 0\n⊢ f =ᶠ[l] g"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Filter.Tendsto | {
"line": 171,
"column": 24
} | {
"line": 171,
"column": 49
} | {
"line": 171,
"column": 50
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nf : α → β\ng : β → γ\na : Filter α\nb : Filter β\nc : Filter γ\nhfg : a ≤ comap (g ∘ f) c\nhg : comap g c ≤ b\n⊢ comap (g ∘ f) c ≤ comap f b",
"ppTerm": "?m.44",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nf : α → β\ng : β → γ\na : Filter α\nb : Filter β\nc : Filter γ\nhfg : a ≤ comap (g ∘ f) c\nhg : comap g c ≤ b\n⊢ comap (g ∘ f) c ≤ comap f b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Filter.Map | {
"line": 1004,
"column": 31
} | {
"line": 1004,
"column": 61
} | {
"line": 1004,
"column": 62
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nF : Filter α\nG : Filter β\nm : α → β\nhm : G ≤ map m F\nH : Filter β\nhHG : H ≤ G\n⊢ map m ((fun x ↦ F ⊓ comap m x) H) = H",
"ppTerm": "?m.75",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CompleteLattice.toLattice",
"congrArg",
"Filte... | [
"α : Type u_1\nβ : Type u_2\nF : Filter α\nG : Filter β\nm : α → β\nhm : G ≤ map m F\nH : Filter β\nhHG : H ≤ G\n⊢ H ≤ map m F"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Filter.Map | {
"line": 1014,
"column": 46
} | {
"line": 1018,
"column": 87
} | {
"line": 1020,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ns : Set α\nm : α → β\n⊢ InjOn (map m) (Iic (𝓟 s)) ↔ InjOn m s",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Pure.pure",
"Filter.instMembership",
"Eq.mpr",
"Filter.le_principal_iff",
"Set.InjOn.eq_iff",
"congrArg",... | [] | by
refine ⟨fun hm x hx y hy hxy ↦ ?_, fun hm F hF G hG ↦ ?_⟩
· rwa [← pure_injective.eq_iff, ← map_pure, ← map_pure, hm.eq_iff, pure_injective.eq_iff]
at hxy <;> rwa [mem_Iic, pure_le_principal]
· simp [map_eq_map_iff_of_injOn (le_principal_iff.mp hF) (le_principal_iff.mp hG) hm] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Group.Action.TransferInstance | {
"line": 90,
"column": 6
} | {
"line": 90,
"column": 52
} | {
"line": 90,
"column": 53
} | [
{
"pp": "M : Type u_1\nα : Type u_4\nβ : Type u_5\ninst✝¹ : SMul M β\ne : α ≃ β\ninst✝ : FaithfulSMul M β\nthis : SMul M α := Equiv.smul M e\nm₁ m₂ : M\n⊢ (∀ (a : α), m₁ • a = m₂ • a) → m₁ = m₂",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instHSMul",
"Equiv.... | [
"M : Type u_1\nα : Type u_4\nβ : Type u_5\ninst✝¹ : SMul M β\ne : α ≃ β\ninst✝ : FaithfulSMul M β\nthis : SMul M α := Equiv.smul M e\nm₁ m₂ : M\n⊢ (∀ (a : α), m₁ • e a = m₂ • e a) → m₁ = m₂"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.NoZeroSMulDivisors.Defs | {
"line": 61,
"column": 4
} | {
"line": 61,
"column": 41
} | {
"line": 61,
"column": 42
} | [
{
"pp": "R : Type u_1\nM : Type u_2\nG : Type u_3\ninst✝⁴ : Semiring R\ninst✝³ : IsDomain R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : NoZeroSMulDivisors R M\nr : R\nhr : IsRegular r\nm₁ m₂ : M\nhm : r • (m₁ - m₂) = 0\n⊢ m₁ = m₂",
"ppTerm": "?m.41",
"assigned": false,
"usedConstants": []... | [
"R : Type u_1\nM : Type u_2\nG : Type u_3\ninst✝⁴ : Semiring R\ninst✝³ : IsDomain R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : NoZeroSMulDivisors R M\nr : R\nhr : IsRegular r\nm₁ m₂ : M\nhm : r • (m₁ - m₂) = 0\n⊢ m₁ = m₂"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.NoZeroSMulDivisors.Defs | {
"line": 71,
"column": 20
} | {
"line": 71,
"column": 31
} | {
"line": 71,
"column": 32
} | [
{
"pp": "R : Type u_1\nM : Type u_2\nG : Type u_3\ninst✝¹ : AddMonoid M\ninst✝ : IsAddTorsionFree M\nn : ℕ\nx : M\nhx : n ≠ 0 ∧ x ≠ 0\n⊢ n • x ≠ 0",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instHSMul",
"congrArg",
"AddMonoid.toAddZeroClass",
"A... | [
"R : Type u_1\nM : Type u_2\nG : Type u_3\ninst✝¹ : AddMonoid M\ninst✝ : IsAddTorsionFree M\nn : ℕ\nx : M\nhx : n ≠ 0 ∧ x ≠ 0\n⊢ ¬x = 0 ∧ ¬n = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.NoZeroSMulDivisors.Defs | {
"line": 76,
"column": 20
} | {
"line": 76,
"column": 31
} | {
"line": 76,
"column": 32
} | [
{
"pp": "R : Type u_1\nM : Type u_2\nG : Type u_3\ninst✝¹ : AddGroup G\ninst✝ : IsAddTorsionFree G\nn : ℤ\nx : G\nhx : n ≠ 0 ∧ x ≠ 0\n⊢ n • x ≠ 0",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"instHSMul",
"congrArg",
... | [
"R : Type u_1\nM : Type u_2\nG : Type u_3\ninst✝¹ : AddGroup G\ninst✝ : IsAddTorsionFree G\nn : ℤ\nx : G\nhx : n ≠ 0 ∧ x ≠ 0\n⊢ ¬x = 0 ∧ ¬n = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Filter.AtTopBot.Basic | {
"line": 226,
"column": 2
} | {
"line": 226,
"column": 59
} | {
"line": 227,
"column": 2
} | [
{
"pp": "α : Type u_3\nβ : Type u_4\ninst✝³ : Preorder α\ninst✝² : IsDirectedOrder α\ninst✝¹ : Preorder β\ninst✝ : IsDirectedOrder β\nf : α → β\nhf : Monotone f\nb : β\nthis : Nonempty α\ng : β → α\nhfg : ∀ (c : β), b ≤ c → f (g c) = c\nhgle : ∀ (c : β), b ≤ c → ∀ (a : α), f a ≤ c ↔ a ≤ g c\n⊢ map f atTop = atT... | [
"case refine_1\nα : Type u_3\nβ : Type u_4\ninst✝³ : Preorder α\ninst✝² : IsDirectedOrder α\ninst✝¹ : Preorder β\ninst✝ : IsDirectedOrder β\nf : α → β\nhf : Monotone f\nb : β\nthis : Nonempty α\ng : β → α\nhfg : ∀ (c : β), b ≤ c → f (g c) = c\nhgle : ∀ (c : β), b ≤ c → ∀ (a : α), f a ≤ c ↔ a ≤ g c\nc : β\n⊢ ∃ a, c ... | refine le_antisymm (hf.tendsto_atTop_atTop fun c ↦ ?_) ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Order.Filter.AtTopBot.Basic | {
"line": 231,
"column": 4
} | {
"line": 231,
"column": 13
} | {
"line": 232,
"column": 4
} | [
{
"pp": "case refine_2\nα : Type u_3\nβ : Type u_4\ninst✝³ : Preorder α\ninst✝² : IsDirectedOrder α\ninst✝¹ : Preorder β\ninst✝ : IsDirectedOrder β\nf : α → β\nhf : Monotone f\nb : β\nthis✝ : Nonempty α\ng : β → α\nhfg : ∀ (c : β), b ≤ c → f (g c) = c\nhgle : ∀ (c : β), b ≤ c → ∀ (a : α), f a ≤ c ↔ a ≤ g c\nthi... | [
"case refine_2\nα : Type u_3\nβ : Type u_4\ninst✝³ : Preorder α\ninst✝² : IsDirectedOrder α\ninst✝¹ : Preorder β\ninst✝ : IsDirectedOrder β\nf : α → β\nhf : Monotone f\nb : β\nthis✝ : Nonempty α\ng : β → α\nhfg : ∀ (c : β), b ≤ c → f (g c) = c\nhgle : ∀ (c : β), b ≤ c → ∀ (a : α), f a ≤ c ↔ a ≤ g c\nthis : Nonempty... | intro a _ | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Algebra.Module.TransferInstance | {
"line": 37,
"column": 2
} | {
"line": 37,
"column": 55
} | {
"line": 37,
"column": 56
} | [
{
"pp": "α : Type u_2\nβ : Type u_3\ne : α ≃ β\nR : Type u_4\ninst✝³ : Zero R\ninst✝² : Zero β\ninst✝¹ : SMul R β\ninst✝ : NoZeroSMulDivisors R β\nthis✝¹ : Zero α := e.zero\nthis✝ : SMul R α := Equiv.smul R e\nr : R\nm : α\n⊢ r • m = 0 → r = 0 ∨ m = 0",
"ppTerm": "?m.25",
"assigned": true,
"usedCons... | [
"α : Type u_2\nβ : Type u_3\ne : α ≃ β\nR : Type u_4\ninst✝³ : Zero R\ninst✝² : Zero β\ninst✝¹ : SMul R β\ninst✝ : NoZeroSMulDivisors R β\nthis✝¹ : Zero α := e.zero\nthis✝ : SMul R α := Equiv.smul R e\nr : R\nm : α\n⊢ r • e m = 0 → r = 0 ∨ e m = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Set.Finite.Lemmas | {
"line": 78,
"column": 4
} | {
"line": 78,
"column": 55
} | {
"line": 79,
"column": 6
} | [
{
"pp": "α : Type u\nβ : Type v\ninst✝ : LinearOrder β\ns : Set α\nf : α → β\nh1 : s.Finite\nx : α\nhx : x ∈ s\n⊢ ∃ a ∈ s, ∀ b ∈ s, f a ≤ f b",
"ppTerm": "?m.19",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u\nβ : Type v\ninst✝ : LinearOrder β\ns : Set α\nf : α → β\nh1 : s.Finite\nx : α\nhx : x ∈ s\n⊢ ∃ a ∈ s, ∀ b ∈ s, f a ≤ f b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Set.Finite.Lemmas | {
"line": 84,
"column": 4
} | {
"line": 84,
"column": 55
} | {
"line": 85,
"column": 6
} | [
{
"pp": "α : Type u\nβ : Type v\ninst✝ : LinearOrder β\ns : Set α\nf : α → β\nh1 : s.Finite\nx : α\nhx : x ∈ s\n⊢ ∃ a ∈ s, ∀ b ∈ s, f b ≤ f a",
"ppTerm": "?m.19",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u\nβ : Type v\ninst✝ : LinearOrder β\ns : Set α\nf : α → β\nh1 : s.Finite\nx : α\nhx : x ∈ s\n⊢ ∃ a ∈ s, ∀ b ∈ s, f b ≤ f a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Finiteness.Defs | {
"line": 53,
"column": 26
} | {
"line": 53,
"column": 73
} | {
"line": 53,
"column": 74
} | [
{
"pp": "M : Type u_2\ninst✝ : AddCommMonoid M\nP : Submodule ℕ M\nx✝ : P.FG\nS : Finset M\nhS : AddSubmonoid.closure ↑S = P.toAddSubmonoid\n⊢ span ℕ ↑S = P",
"ppTerm": "?m.46",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"M : Type u_2\ninst✝ : AddCommMonoid M\nP : Submodule ℕ M\nx✝ : P.FG\nS : Finset M\nhS : AddSubmonoid.closure ↑S = P.toAddSubmonoid\n⊢ span ℕ ↑S = P"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Finiteness.Defs | {
"line": 58,
"column": 26
} | {
"line": 58,
"column": 72
} | {
"line": 58,
"column": 73
} | [
{
"pp": "G : Type u_3\ninst✝ : AddCommGroup G\nP : Submodule ℤ G\nx✝ : P.toAddSubgroup.FG\nS : Finset G\nhS : AddSubgroup.closure ↑S = P.toAddSubgroup\n⊢ span ℤ ↑S = P",
"ppTerm": "?m.46",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"G : Type u_3\ninst✝ : AddCommGroup G\nP : Submodule ℤ G\nx✝ : P.toAddSubgroup.FG\nS : Finset G\nhS : AddSubgroup.closure ↑S = P.toAddSubgroup\n⊢ span ℤ ↑S = P"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Noetherian.Defs | {
"line": 178,
"column": 4
} | {
"line": 178,
"column": 90
} | {
"line": 178,
"column": 91
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝³ : Semiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ninst✝ : IsNoetherian R M\nf : End R M\nn : ℕ\nhn : ∀ (m : ℕ), n ≤ m → LinearMap.ker (f ^ n) = LinearMap.ker (f ^ m)\nm : ℕ\nhm : n ≤ m\nx : M\nhx : (f ^ m) x ∈ ↑(LinearMap.ker (f ^ n))\n⊢ x ∈ LinearMap.ker (... | [
"R : Type u_1\nM : Type u_2\ninst✝³ : Semiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ninst✝ : IsNoetherian R M\nf : End R M\nn : ℕ\nhn : ∀ (m : ℕ), n ≤ m → LinearMap.ker (f ^ n) = LinearMap.ker (f ^ m)\nm : ℕ\nhm : n ≤ m\nx : M\nhx : (f ^ m) x ∈ ↑(LinearMap.ker (f ^ n))\n⊢ (f ^ (n + m)) x = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Cofinal | {
"line": 41,
"column": 2
} | {
"line": 41,
"column": 13
} | {
"line": 41,
"column": 14
} | [
{
"pp": "α : Type u_1\ninst✝ : LE α\nh : IsCofinal ∅\na : α\n⊢ False",
"ppTerm": "?m.11",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\ninst✝ : LE α\nh : IsCofinal ∅\na : α\n⊢ False"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Cofinal | {
"line": 99,
"column": 15
} | {
"line": 99,
"column": 26
} | {
"line": 99,
"column": 27
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : Preorder α\ninst✝ : Preorder β\ne : α ≃o β\ns : Set α\nhs : IsCofinal (⇑e '' s)\n⊢ IsCofinal s",
"ppTerm": "?m.13",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\nβ : Type u_2\ninst✝¹ : Preorder α\ninst✝ : Preorder β\ne : α ≃o β\ns : Set α\nhs : IsCofinal (⇑e '' s)\n⊢ IsCofinal s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Cofinal | {
"line": 127,
"column": 2
} | {
"line": 128,
"column": 29
} | {
"line": 130,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝ : PartialOrder α\ns : Set α\na : α\nha : IsMax a\nhs : IsCofinal s\n⊢ a ∈ s",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"congrArg",
"PartialOrder.toPreorder",
"Preorder.toLE",
"Membership.mem",
"Exists",
"Eq.mp",
"LE... | [] | obtain ⟨b, hb, hb'⟩ := hs a
rwa [ha.eq_of_ge hb'] at hb | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.Cofinal | {
"line": 127,
"column": 2
} | {
"line": 128,
"column": 29
} | {
"line": 130,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝ : PartialOrder α\ns : Set α\na : α\nha : IsMax a\nhs : IsCofinal s\n⊢ a ∈ s",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"congrArg",
"PartialOrder.toPreorder",
"Preorder.toLE",
"Membership.mem",
"Exists",
"Eq.mp",
"LE... | [] | obtain ⟨b, hb, hb'⟩ := hs a
rwa [ha.eq_of_ge hb'] at hb | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.SetTheory.Cardinal.Cofinality.Basic | {
"line": 47,
"column": 2
} | {
"line": 47,
"column": 13
} | {
"line": 47,
"column": 14
} | [
{
"pp": "α : Type u\ninst✝ : Preorder α\nc : Cardinal.{u}\n⊢ c ≤ cof α ↔ ∀ (s : Set α), IsCofinal s → c ≤ #↑s",
"ppTerm": "?m.8",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u\ninst✝ : Preorder α\nc : Cardinal.{u}\n⊢ c ≤ cof α ↔ ∀ (s : Set α), IsCofinal s → c ≤ #↑s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Cardinal.Cofinality.Basic | {
"line": 73,
"column": 2
} | {
"line": 73,
"column": 13
} | {
"line": 73,
"column": 14
} | [
{
"pp": "α : Type u\ninst✝ : Preorder α\n⊢ cof α ≠ 0 ↔ Nonempty α",
"ppTerm": "?m.5",
"assigned": true,
"usedConstants": [
"Cardinal",
"id",
"Ne",
"Iff",
"Nonempty",
"Zero.toOfNat0",
"Order.cof",
"OfNat.ofNat",
"Cardinal.instZero"
],
"use... | [
"α : Type u\ninst✝ : Preorder α\n⊢ ¬cof α = 0 ↔ Nonempty α"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Finiteness.Basic | {
"line": 53,
"column": 28
} | {
"line": 53,
"column": 65
} | {
"line": 53,
"column": 66
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nι : Type u_3\ns : Finset ι\nN : ι → Submodule R M\nh : ∀ i ∈ s, (N i).FG\n⊢ (⨆ i ∈ s, N i).FG",
"ppTerm": "?m.31",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
... | [
"R : Type u_1\nM : Type u_2\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nι : Type u_3\ns : Finset ι\nN : ι → Submodule R M\nh : ∀ i ∈ s, (N i).FG\n⊢ (⨆ i ∈ s, N i).FG"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Finiteness.Basic | {
"line": 58,
"column": 2
} | {
"line": 58,
"column": 31
} | {
"line": 58,
"column": 32
} | [
{
"pp": "case intro\nR : Type u_1\nM : Type u_2\ninst✝³ : Semiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\nι : Sort u_3\ninst✝ : Finite ι\nN : ι → Submodule R M\nh : ∀ (i : ι), (N i).FG\nval✝ : Fintype (PLift ι)\n⊢ (iSup N).FG",
"ppTerm": "?intro",
"assigned": false,
"usedConstants": [],
... | [
"case intro\nR : Type u_1\nM : Type u_2\ninst✝³ : Semiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\nι : Sort u_3\ninst✝ : Finite ι\nN : ι → Submodule R M\nh : ∀ (i : ι), (N i).FG\nval✝ : Fintype (PLift ι)\n⊢ (iSup N).FG"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Cardinal.Cofinality.Basic | {
"line": 124,
"column": 44
} | {
"line": 124,
"column": 55
} | {
"line": 124,
"column": 56
} | [
{
"pp": "α : Type u\nβ : Type v\ninst✝¹ : LinearOrder α\ninst✝ : LinearOrder β\nf : α → β\nhf : StrictMono f\nhf' : IsCofinal (range f)\ns : Set β\nhs : IsCofinal s\nx : ↑s\n⊢ ∃ y, ↑x ≤ f y",
"ppTerm": "?m.39",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u\nβ : Type v\ninst✝¹ : LinearOrder α\ninst✝ : LinearOrder β\nf : α → β\nhf : StrictMono f\nhf' : IsCofinal (range f)\ns : Set β\nhs : IsCofinal s\nx : ↑s\n⊢ ∃ y, ↑x ≤ f y"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Cardinal.Cofinality.Basic | {
"line": 136,
"column": 2
} | {
"line": 136,
"column": 13
} | {
"line": 136,
"column": 14
} | [
{
"pp": "α γ : Type u\ninst✝¹ : LinearOrder α\ninst✝ : LinearOrder γ\nf : α → γ\nhf : StrictMono f\nhf' : IsCofinal (range f)\n⊢ cof α = cof γ",
"ppTerm": "?m.12",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α γ : Type u\ninst✝¹ : LinearOrder α\ninst✝ : LinearOrder γ\nf : α → γ\nhf : StrictMono f\nhf' : IsCofinal (range f)\n⊢ cof α = cof γ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Finiteness.Basic | {
"line": 138,
"column": 13
} | {
"line": 138,
"column": 24
} | {
"line": 138,
"column": 25
} | [
{
"pp": "case empty\nR : Type u_4\nM : Type u_5\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nmotive : (N : Submodule R M) → N.FG → Prop\nsingleton : ∀ (x : M), motive (R ∙ x) ⋯\nsup : ∀ (N₁ N₂ : Submodule R M) (hN₁ : N₁.FG) (hN₂ : N₂.FG), motive N₁ hN₁ → motive N₂ hN₂ → motive (N₁ ⊔ N₂) ⋯... | [
"case empty\nR : Type u_4\nM : Type u_5\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nmotive : (N : Submodule R M) → N.FG → Prop\nsingleton : ∀ (x : M), motive (R ∙ x) ⋯\nsup : ∀ (N₁ N₂ : Submodule R M) (hN₁ : N₁.FG) (hN₂ : N₂.FG), motive N₁ hN₁ → motive N₂ hN₂ → motive (N₁ ⊔ N₂) ⋯\n⊢ motive ⊥... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Finiteness.Basic | {
"line": 140,
"column": 4
} | {
"line": 140,
"column": 39
} | {
"line": 141,
"column": 6
} | [
{
"pp": "case insert\nR : Type u_4\nM : Type u_5\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nmotive : (N : Submodule R M) → N.FG → Prop\nsingleton : ∀ (x : M), motive (R ∙ x) ⋯\nsup : ∀ (N₁ N₂ : Submodule R M) (hN₁ : N₁.FG) (hN₂ : N₂.FG), motive N₁ hN₁ → motive N₂ hN₂ → motive (N₁ ⊔ N₂) ... | [
"case insert\nR : Type u_4\nM : Type u_5\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nmotive : (N : Submodule R M) → N.FG → Prop\nsingleton : ∀ (x : M), motive (R ∙ x) ⋯\nsup : ∀ (N₁ N₂ : Submodule R M) (hN₁ : N₁.FG) (hN₂ : N₂.FG), motive N₁ hN₁ → motive N₂ hN₂ → motive (N₁ ⊔ N₂) ⋯\nx : M\ns ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Cardinal.Cofinality.Basic | {
"line": 178,
"column": 2
} | {
"line": 178,
"column": 13
} | {
"line": 178,
"column": 14
} | [
{
"pp": "α γ : Type u\ninst✝¹ : Preorder α\ninst✝ : Preorder γ\nf : γ → α\ng : α → γ\nh : GaloisConnection f g\n⊢ cof γ ≤ cof α",
"ppTerm": "?m.9",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α γ : Type u\ninst✝¹ : Preorder α\ninst✝ : Preorder γ\nf : γ → α\ng : α → γ\nh : GaloisConnection f g\n⊢ cof γ ≤ cof α"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Cardinal.Cofinality.Basic | {
"line": 187,
"column": 2
} | {
"line": 187,
"column": 13
} | {
"line": 187,
"column": 14
} | [
{
"pp": "α γ : Type u\ninst✝¹ : Preorder α\ninst✝ : Preorder γ\nf : α ≃o γ\n⊢ cof α = cof γ",
"ppTerm": "?m.6",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α γ : Type u\ninst✝¹ : Preorder α\ninst✝ : Preorder γ\nf : α ≃o γ\n⊢ cof α = cof γ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Finiteness.Basic | {
"line": 152,
"column": 25
} | {
"line": 152,
"column": 60
} | {
"line": 152,
"column": 61
} | [
{
"pp": "case insert\nR : Type u_4\nM : Type u_5\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nmotive : (N : Submodule R M) → N.FG → Prop\nbot : motive ⊥ ⋯\nsup : ∀ (N : Submodule R M) (x : M) (hN : N.FG), motive N hN → motive (N ⊔ R ∙ x) ⋯\nx : M\ns : Finset M\nhxs : x ∉ s\nih : motive (s... | [
"case insert\nR : Type u_4\nM : Type u_5\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nmotive : (N : Submodule R M) → N.FG → Prop\nbot : motive ⊥ ⋯\nsup : ∀ (N : Submodule R M) (x : M) (hN : N.FG), motive N hN → motive (N ⊔ R ∙ x) ⋯\nx : M\ns : Finset M\nhxs : x ∉ s\nih : motive (span R ↑s) ⋯\... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Cardinal.Cofinality.Basic | {
"line": 205,
"column": 2
} | {
"line": 205,
"column": 13
} | {
"line": 205,
"column": 14
} | [
{
"pp": "α : Type u_1\ninst✝ : LinearOrder α\ns : Set (Set α)\nh₁ : IsCofinal (⋃₀ s)\nf : (x : Set α) → x ∈ s → α\nhf : ∀ (x : Set α) (_ : x ∈ s), ∀ y ∈ x, y < f x _\na b : α\nhab : a ≤ b\nt : Set α\nht : t ∈ s\nhb : b ∈ t\n⊢ ∃ y ∈ range fun x ↦ f ↑x ⋯, a ≤ y",
"ppTerm": "?m.83",
"assigned": true,
"... | [
"α : Type u_1\ninst✝ : LinearOrder α\ns : Set (Set α)\nh₁ : IsCofinal (⋃₀ s)\nf : (x : Set α) → x ∈ s → α\nhf : ∀ (x : Set α) (_ : x ∈ s), ∀ y ∈ x, y < f x _\na b : α\nhab : a ≤ b\nt : Set α\nht : t ∈ s\nhb : b ∈ t\n⊢ ∃ a_1, ∃ (h : a_1 ∈ s), a ≤ f a_1 ⋯"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Finiteness.Basic | {
"line": 183,
"column": 35
} | {
"line": 183,
"column": 56
} | {
"line": 183,
"column": 57
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nM' : Submodule R M\nN : ℕ →o Submodule R M\nH : iSup ⇑N = M'\nS : Finset M\nhS : span R ↑S = M'\ns : ↥S\n⊢ ↑s ∈ ⨆ k, N k",
"ppTerm": "?m.59",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [
"R : Type u_1\nM : Type u_2\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nM' : Submodule R M\nN : ℕ →o Submodule R M\nH : iSup ⇑N = M'\nS : Finset M\nhS : span R ↑S = M'\ns : ↥S\n⊢ ↑s ∈ span R ↑S"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Module.Defs | {
"line": 583,
"column": 2
} | {
"line": 583,
"column": 30
} | {
"line": 583,
"column": 31
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\na : α\nb : β\ninst✝⁵ : Zero α\ninst✝⁴ : Zero β\ninst✝³ : SMulZeroClass α β\ninst✝² : Preorder α\ninst✝¹ : Preorder β\ninst✝ : PosSMulStrictMono α β\nha : 0 < a\nhb : 0 < b\n⊢ 0 < a • b",
"ppTerm": "?m.22",
"assigned": false,
"usedConstants": [],
"usedFVars": ... | [
"α : Type u_1\nβ : Type u_2\na : α\nb : β\ninst✝⁵ : Zero α\ninst✝⁴ : Zero β\ninst✝³ : SMulZeroClass α β\ninst✝² : Preorder α\ninst✝¹ : Preorder β\ninst✝ : PosSMulStrictMono α β\nha : 0 < a\nhb : 0 < b\n⊢ 0 < a • b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Module.Defs | {
"line": 586,
"column": 2
} | {
"line": 586,
"column": 30
} | {
"line": 586,
"column": 31
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\na : α\nb : β\ninst✝⁵ : Zero α\ninst✝⁴ : Zero β\ninst✝³ : SMulZeroClass α β\ninst✝² : Preorder α\ninst✝¹ : Preorder β\ninst✝ : PosSMulStrictMono α β\nha : 0 < a\nhb : b < 0\n⊢ a • b < 0",
"ppTerm": "?m.22",
"assigned": false,
"usedConstants": [],
"usedFVars": ... | [
"α : Type u_1\nβ : Type u_2\na : α\nb : β\ninst✝⁵ : Zero α\ninst✝⁴ : Zero β\ninst✝³ : SMulZeroClass α β\ninst✝² : Preorder α\ninst✝¹ : Preorder β\ninst✝ : PosSMulStrictMono α β\nha : 0 < a\nhb : b < 0\n⊢ a • b < 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Module.Defs | {
"line": 591,
"column": 2
} | {
"line": 591,
"column": 30
} | {
"line": 591,
"column": 31
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\na : α\nb : β\ninst✝⁶ : Zero α\ninst✝⁵ : Zero β\ninst✝⁴ : SMulZeroClass α β\ninst✝³ : Preorder α\ninst✝² : Preorder β\ninst✝¹ : PosSMulStrictMono α β\ninst✝ : PosSMulReflectLT α β\nha : 0 < a\n⊢ 0 < a • b ↔ 0 < b",
"ppTerm": "?m.26",
"assigned": false,
"usedConsta... | [
"α : Type u_1\nβ : Type u_2\na : α\nb : β\ninst✝⁶ : Zero α\ninst✝⁵ : Zero β\ninst✝⁴ : SMulZeroClass α β\ninst✝³ : Preorder α\ninst✝² : Preorder β\ninst✝¹ : PosSMulStrictMono α β\ninst✝ : PosSMulReflectLT α β\nha : 0 < a\n⊢ 0 < a • b ↔ 0 < b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Module.Defs | {
"line": 595,
"column": 2
} | {
"line": 595,
"column": 30
} | {
"line": 595,
"column": 31
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\na : α\nb : β\ninst✝⁶ : Zero α\ninst✝⁵ : Zero β\ninst✝⁴ : SMulZeroClass α β\ninst✝³ : Preorder α\ninst✝² : Preorder β\ninst✝¹ : PosSMulStrictMono α β\ninst✝ : PosSMulReflectLT α β\nha : 0 < a\n⊢ a • b < 0 ↔ b < 0",
"ppTerm": "?m.26",
"assigned": false,
"usedConsta... | [
"α : Type u_1\nβ : Type u_2\na : α\nb : β\ninst✝⁶ : Zero α\ninst✝⁵ : Zero β\ninst✝⁴ : SMulZeroClass α β\ninst✝³ : Preorder α\ninst✝² : Preorder β\ninst✝¹ : PosSMulStrictMono α β\ninst✝ : PosSMulReflectLT α β\nha : 0 < a\n⊢ a • b < 0 ↔ b < 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Module.Defs | {
"line": 598,
"column": 2
} | {
"line": 598,
"column": 30
} | {
"line": 598,
"column": 31
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\na : α\nb₁ : β\ninst✝⁵ : Zero α\ninst✝⁴ : Zero β\ninst✝³ : SMulZeroClass α β\ninst✝² : Preorder α\ninst✝¹ : Preorder β\ninst✝ : PosSMulMono α β\nha : 0 ≤ a\nhb : 0 ≤ b₁\n⊢ 0 ≤ a • b₁",
"ppTerm": "?m.22",
"assigned": false,
"usedConstants": [],
"usedFVars": [],... | [
"α : Type u_1\nβ : Type u_2\na : α\nb₁ : β\ninst✝⁵ : Zero α\ninst✝⁴ : Zero β\ninst✝³ : SMulZeroClass α β\ninst✝² : Preorder α\ninst✝¹ : Preorder β\ninst✝ : PosSMulMono α β\nha : 0 ≤ a\nhb : 0 ≤ b₁\n⊢ 0 ≤ a • b₁"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Module.Defs | {
"line": 601,
"column": 2
} | {
"line": 601,
"column": 30
} | {
"line": 601,
"column": 31
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\na : α\nb : β\ninst✝⁵ : Zero α\ninst✝⁴ : Zero β\ninst✝³ : SMulZeroClass α β\ninst✝² : Preorder α\ninst✝¹ : Preorder β\ninst✝ : PosSMulMono α β\nha : 0 ≤ a\nhb : b ≤ 0\n⊢ a • b ≤ 0",
"ppTerm": "?m.22",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
... | [
"α : Type u_1\nβ : Type u_2\na : α\nb : β\ninst✝⁵ : Zero α\ninst✝⁴ : Zero β\ninst✝³ : SMulZeroClass α β\ninst✝² : Preorder α\ninst✝¹ : Preorder β\ninst✝ : PosSMulMono α β\nha : 0 ≤ a\nhb : b ≤ 0\n⊢ a • b ≤ 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Module.Defs | {
"line": 635,
"column": 2
} | {
"line": 635,
"column": 30
} | {
"line": 635,
"column": 31
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\na : α\nb : β\ninst✝⁵ : Zero α\ninst✝⁴ : Zero β\ninst✝³ : SMulWithZero α β\ninst✝² : Preorder α\ninst✝¹ : Preorder β\ninst✝ : SMulPosStrictMono α β\nha : 0 < a\nhb : 0 < b\n⊢ 0 < a • b",
"ppTerm": "?m.23",
"assigned": false,
"usedConstants": [],
"usedFVars": [... | [
"α : Type u_1\nβ : Type u_2\na : α\nb : β\ninst✝⁵ : Zero α\ninst✝⁴ : Zero β\ninst✝³ : SMulWithZero α β\ninst✝² : Preorder α\ninst✝¹ : Preorder β\ninst✝ : SMulPosStrictMono α β\nha : 0 < a\nhb : 0 < b\n⊢ 0 < a • b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.DirSupClosed | {
"line": 121,
"column": 2
} | {
"line": 121,
"column": 13
} | {
"line": 121,
"column": 14
} | [
{
"pp": "α : Type u_1\ninst✝ : Preorder α\ns : Set (Set α)\nhs : ∀ x ∈ s, DirSupClosed x\n⊢ DirSupClosed (⋂₀ s)",
"ppTerm": "?m.10",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\ninst✝ : Preorder α\ns : Set (Set α)\nhs : ∀ x ∈ s, DirSupClosed x\n⊢ DirSupClosed (⋂₀ s)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Module.Defs | {
"line": 638,
"column": 2
} | {
"line": 638,
"column": 30
} | {
"line": 638,
"column": 31
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\na : α\nb : β\ninst✝⁵ : Zero α\ninst✝⁴ : Zero β\ninst✝³ : SMulWithZero α β\ninst✝² : Preorder α\ninst✝¹ : Preorder β\ninst✝ : SMulPosStrictMono α β\nha : a < 0\nhb : 0 < b\n⊢ a • b < 0",
"ppTerm": "?m.23",
"assigned": false,
"usedConstants": [],
"usedFVars": [... | [
"α : Type u_1\nβ : Type u_2\na : α\nb : β\ninst✝⁵ : Zero α\ninst✝⁴ : Zero β\ninst✝³ : SMulWithZero α β\ninst✝² : Preorder α\ninst✝¹ : Preorder β\ninst✝ : SMulPosStrictMono α β\nha : a < 0\nhb : 0 < b\n⊢ a • b < 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Module.Defs | {
"line": 643,
"column": 2
} | {
"line": 643,
"column": 30
} | {
"line": 643,
"column": 31
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\na : α\nb : β\ninst✝⁶ : Zero α\ninst✝⁵ : Zero β\ninst✝⁴ : SMulWithZero α β\ninst✝³ : Preorder α\ninst✝² : Preorder β\ninst✝¹ : SMulPosStrictMono α β\ninst✝ : SMulPosReflectLT α β\nhb : 0 < b\n⊢ 0 < a • b ↔ 0 < a",
"ppTerm": "?m.27",
"assigned": false,
"usedConstan... | [
"α : Type u_1\nβ : Type u_2\na : α\nb : β\ninst✝⁶ : Zero α\ninst✝⁵ : Zero β\ninst✝⁴ : SMulWithZero α β\ninst✝³ : Preorder α\ninst✝² : Preorder β\ninst✝¹ : SMulPosStrictMono α β\ninst✝ : SMulPosReflectLT α β\nhb : 0 < b\n⊢ 0 < a • b ↔ 0 < a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.DirSupClosed | {
"line": 132,
"column": 2
} | {
"line": 132,
"column": 13
} | {
"line": 132,
"column": 14
} | [
{
"pp": "α : Type u_1\ninst✝ : Preorder α\ns : Set (Set α)\nhs : ∀ x ∈ s, DirSupInacc x\n⊢ DirSupInacc (⋃₀ s)",
"ppTerm": "?m.10",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\ninst✝ : Preorder α\ns : Set (Set α)\nhs : ∀ x ∈ s, DirSupInacc x\n⊢ DirSupInacc (⋃₀ s)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Module.Defs | {
"line": 646,
"column": 2
} | {
"line": 646,
"column": 30
} | {
"line": 646,
"column": 31
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\na : α\nb₁ : β\ninst✝⁵ : Zero α\ninst✝⁴ : Zero β\ninst✝³ : SMulWithZero α β\ninst✝² : Preorder α\ninst✝¹ : Preorder β\ninst✝ : SMulPosMono α β\nha : 0 ≤ a\nhb : 0 ≤ b₁\n⊢ 0 ≤ a • b₁",
"ppTerm": "?m.23",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
... | [
"α : Type u_1\nβ : Type u_2\na : α\nb₁ : β\ninst✝⁵ : Zero α\ninst✝⁴ : Zero β\ninst✝³ : SMulWithZero α β\ninst✝² : Preorder α\ninst✝¹ : Preorder β\ninst✝ : SMulPosMono α β\nha : 0 ≤ a\nhb : 0 ≤ b₁\n⊢ 0 ≤ a • b₁"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Module.Defs | {
"line": 649,
"column": 2
} | {
"line": 649,
"column": 30
} | {
"line": 649,
"column": 31
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\na : α\nb : β\ninst✝⁵ : Zero α\ninst✝⁴ : Zero β\ninst✝³ : SMulWithZero α β\ninst✝² : Preorder α\ninst✝¹ : Preorder β\ninst✝ : SMulPosMono α β\nha : a ≤ 0\nhb : 0 ≤ b\n⊢ a • b ≤ 0",
"ppTerm": "?m.23",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
... | [
"α : Type u_1\nβ : Type u_2\na : α\nb : β\ninst✝⁵ : Zero α\ninst✝⁴ : Zero β\ninst✝³ : SMulWithZero α β\ninst✝² : Preorder α\ninst✝¹ : Preorder β\ninst✝ : SMulPosMono α β\nha : a ≤ 0\nhb : 0 ≤ b\n⊢ a • b ≤ 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Module.Defs | {
"line": 681,
"column": 45
} | {
"line": 681,
"column": 56
} | {
"line": 681,
"column": 57
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝⁸ : Zero α\ninst✝⁷ : Zero β\ninst✝⁶ : SMulWithZero α β\ninst✝⁵ : Preorder α\ninst✝⁴ : Preorder β\ninst✝³ : MulOneClass β\ninst✝² : PosMulMono β\ninst✝¹ : MulPosMono β\ninst✝ : IsScalarTower α β β\nh : Monotone fun x ↦ x • 1\nx✝² : α\nha : 0 ≤ x✝²\nx✝¹ x✝ : β\nhb : x✝¹ ≤... | [
"α : Type u_1\nβ : Type u_2\ninst✝⁸ : Zero α\ninst✝⁷ : Zero β\ninst✝⁶ : SMulWithZero α β\ninst✝⁵ : Preorder α\ninst✝⁴ : Preorder β\ninst✝³ : MulOneClass β\ninst✝² : PosMulMono β\ninst✝¹ : MulPosMono β\ninst✝ : IsScalarTower α β β\nh : Monotone fun x ↦ x • 1\nx✝² : α\nha : 0 ≤ x✝²\nx✝¹ x✝ : β\nhb : x✝¹ ≤ x✝\n⊢ 0 ≤ ?... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.DirSupClosed | {
"line": 161,
"column": 2
} | {
"line": 161,
"column": 13
} | {
"line": 161,
"column": 14
} | [
{
"pp": "α : Type u_1\ns t : Set α\ninst✝ : Preorder α\nhs : DirSupClosed s\nht : DirSupClosed t\n⊢ DirSupClosed (s ∩ t)",
"ppTerm": "?m.9",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\ns t : Set α\ninst✝ : Preorder α\nhs : DirSupClosed s\nht : DirSupClosed t\n⊢ DirSupClosed (s ∩ t)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.DirSupClosed | {
"line": 168,
"column": 2
} | {
"line": 168,
"column": 13
} | {
"line": 168,
"column": 14
} | [
{
"pp": "α : Type u_1\ns t : Set α\ninst✝ : Preorder α\nhs : DirSupInacc s\nht : DirSupInacc t\n⊢ DirSupInacc (s ∪ t)",
"ppTerm": "?m.9",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\ns t : Set α\ninst✝ : Preorder α\nhs : DirSupInacc s\nht : DirSupInacc t\n⊢ DirSupInacc (s ∪ t)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Module.Defs | {
"line": 684,
"column": 4
} | {
"line": 684,
"column": 15
} | {
"line": 684,
"column": 16
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝⁸ : Zero α\ninst✝⁷ : Zero β\ninst✝⁶ : SMulWithZero α β\ninst✝⁵ : Preorder α\ninst✝⁴ : Preorder β\ninst✝³ : MulOneClass β\ninst✝² : PosMulMono β\ninst✝¹ : MulPosMono β\ninst✝ : IsScalarTower α β β\nh : Monotone fun x ↦ x • 1\nx✝² : β\nha : 0 ≤ x✝²\nx✝¹ x✝ : α\nhb : x✝¹ ≤... | [
"α : Type u_1\nβ : Type u_2\ninst✝⁸ : Zero α\ninst✝⁷ : Zero β\ninst✝⁶ : SMulWithZero α β\ninst✝⁵ : Preorder α\ninst✝⁴ : Preorder β\ninst✝³ : MulOneClass β\ninst✝² : PosMulMono β\ninst✝¹ : MulPosMono β\ninst✝ : IsScalarTower α β β\nh : Monotone fun x ↦ x • 1\nx✝² : β\nha : 0 ≤ x✝²\nx✝¹ x✝ : α\nhb : x✝¹ ≤ x✝\n⊢ x✝¹ •... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.IsNormal | {
"line": 63,
"column": 2
} | {
"line": 63,
"column": 31
} | {
"line": 63,
"column": 32
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\na : α\nf : α → β\ninst✝¹ : LinearOrder α\ninst✝ : LinearOrder β\nhf : IsNormal f\nha : IsSuccLimit a\nb : β\n⊢ f a ≤ b ↔ ∀ a' < a, f a' ≤ b",
"ppTerm": "?m.14",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\nβ : Type u_2\na : α\nf : α → β\ninst✝¹ : LinearOrder α\ninst✝ : LinearOrder β\nhf : IsNormal f\nha : IsSuccLimit a\nb : β\n⊢ f a ≤ b ↔ ∀ a' < a, f a' ≤ b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.IsNormal | {
"line": 67,
"column": 2
} | {
"line": 67,
"column": 31
} | {
"line": 67,
"column": 32
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\na : α\nf : α → β\ninst✝¹ : LinearOrder α\ninst✝ : LinearOrder β\nhf : IsNormal f\nha : IsSuccLimit a\nb : β\n⊢ b < f a ↔ ∃ a' < a, b < f a'",
"ppTerm": "?m.15",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\nβ : Type u_2\na : α\nf : α → β\ninst✝¹ : LinearOrder α\ninst✝ : LinearOrder β\nhf : IsNormal f\nha : IsSuccLimit a\nb : β\n⊢ b < f a ↔ ∃ a' < a, b < f a'"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.IsNormal | {
"line": 81,
"column": 4
} | {
"line": 81,
"column": 58
} | {
"line": 81,
"column": 59
} | [
{
"pp": "case refine_1\nα : Type u_1\nβ : Type u_2\na : α\nf : α → β\ninst✝¹ : LinearOrder α\ninst✝ : LinearOrder β\nhf : IsNormal f\ns : Set α\nhs : IsLUB s a\nhs' : s.Nonempty\n⊢ f a ∈ upperBounds (f '' s)",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"_privat... | [
"case refine_1\nα : Type u_1\nβ : Type u_2\na : α\nf : α → β\ninst✝¹ : LinearOrder α\ninst✝ : LinearOrder β\nhf : IsNormal f\ns : Set α\nhs : IsLUB s a\nhs' : s.Nonempty\n⊢ ∀ a_1 ∈ s, a_1 ≤ a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.DirSupClosed | {
"line": 188,
"column": 2
} | {
"line": 188,
"column": 13
} | {
"line": 188,
"column": 14
} | [
{
"pp": "α : Type u_1\ns t : Set α\ninst✝ : Preorder α\nhs : DirSupClosed s\nht : DirSupClosed t\n⊢ DirSupClosed (s ∪ t)",
"ppTerm": "?m.9",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\ns t : Set α\ninst✝ : Preorder α\nhs : DirSupClosed s\nht : DirSupClosed t\n⊢ DirSupClosed (s ∪ t)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.DirSupClosed | {
"line": 191,
"column": 2
} | {
"line": 191,
"column": 13
} | {
"line": 191,
"column": 14
} | [
{
"pp": "α : Type u_1\ns t : Set α\ninst✝ : Preorder α\nhs : DirSupInacc s\nht : DirSupInacc t\n⊢ DirSupInacc (s ∩ t)",
"ppTerm": "?m.9",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\ns t : Set α\ninst✝ : Preorder α\nhs : DirSupInacc s\nht : DirSupInacc t\n⊢ DirSupInacc (s ∩ t)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.IsNormal | {
"line": 109,
"column": 2
} | {
"line": 109,
"column": 83
} | {
"line": 110,
"column": 4
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nf : α → β\ng : β → γ\ninst✝² : LinearOrder α\ninst✝¹ : LinearOrder β\ninst✝ : LinearOrder γ\nhg : IsNormal g\nhf : IsNormal f\na✝ : α\nha : IsSuccLimit a✝\nb : γ\nhb : ∀ ⦃x : α⦄, x ∈ Iio a✝ → g (f x) ≤ b\n⊢ (g ∘ f) a✝ ≤ b",
"ppTerm": "?m.57",
"assigned"... | [
"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nf : α → β\ng : β → γ\ninst✝² : LinearOrder α\ninst✝¹ : LinearOrder β\ninst✝ : LinearOrder γ\nhg : IsNormal g\nhf : IsNormal f\na✝ : α\nha : IsSuccLimit a✝\nb : γ\nhb : ∀ ⦃x : α⦄, x ∈ Iio a✝ → g (f x) ≤ b\n⊢ ∀ (a' : β), ∀ x < a✝, a' < f x → g a' ≤ b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.DirSupClosed | {
"line": 224,
"column": 2
} | {
"line": 224,
"column": 13
} | {
"line": 224,
"column": 14
} | [
{
"pp": "α : Type u_1\ns : Set α\ninst✝ : Preorder α\n⊢ DirSupInacc s ↔\n ∀ ⦃d : Set α⦄,\n d.Nonempty → DirectedOn (fun x1 x2 ↦ x1 ≤ x2) d → ∀ ⦃a : α⦄, IsLUB d a → a ∈ s → ∃ b ∈ d, Ici b ∩ d ⊆ s",
"ppTerm": "?m.28",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals"... | [
"α : Type u_1\ns : Set α\ninst✝ : Preorder α\n⊢ DirSupInacc s ↔\n ∀ ⦃d : Set α⦄,\n d.Nonempty → DirectedOn (fun x1 x2 ↦ x1 ≤ x2) d → ∀ ⦃a : α⦄, IsLUB d a → a ∈ s → ∃ b ∈ d, Ici b ∩ d ⊆ s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.DirSupClosed | {
"line": 250,
"column": 2
} | {
"line": 250,
"column": 27
} | {
"line": 250,
"column": 28
} | [
{
"pp": "α : Type u_1\ns : Set α\ninst✝ : Preorder α\nhs : DirSupInacc s\na b : α\nh : AntisymmRel (fun x1 x2 ↦ x1 ≤ x2) a b\n⊢ a ∈ s ↔ b ∈ s",
"ppTerm": "?m.16",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\ns : Set α\ninst✝ : Preorder α\nhs : DirSupInacc s\na b : α\nh : AntisymmRel (fun x1 x2 ↦ x1 ≤ x2) a b\n⊢ a ∈ s ↔ b ∈ s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Module.Defs | {
"line": 784,
"column": 4
} | {
"line": 784,
"column": 91
} | {
"line": 785,
"column": 2
} | [
{
"pp": "case refine_1\nα : Type u_1\nβ : Type u_2\na₁ a₂ : α\nb₁ b₂ : β\ninst✝⁶ : Zero α\ninst✝⁵ : Zero β\ninst✝⁴ : SMulWithZero α β\ninst✝³ : PartialOrder α\ninst✝² : PartialOrder β\ninst✝¹ : PosSMulStrictMono α β\ninst✝ : SMulPosStrictMono α β\nha✝ : a₁ ≤ a₂\nhb : b₁ ≤ b₂\nh₂ : 0 < a₂\nh₁ : 0 < b₁\nh : a₁ • ... | [] | exact (smul_lt_smul_of_pos_right ha h₁).trans_le (smul_le_smul_of_nonneg_left hb h₂.le) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.Order.Module.Defs | {
"line": 784,
"column": 4
} | {
"line": 784,
"column": 91
} | {
"line": 785,
"column": 2
} | [
{
"pp": "case refine_1\nα : Type u_1\nβ : Type u_2\na₁ a₂ : α\nb₁ b₂ : β\ninst✝⁶ : Zero α\ninst✝⁵ : Zero β\ninst✝⁴ : SMulWithZero α β\ninst✝³ : PartialOrder α\ninst✝² : PartialOrder β\ninst✝¹ : PosSMulStrictMono α β\ninst✝ : SMulPosStrictMono α β\nha✝ : a₁ ≤ a₂\nhb : b₁ ≤ b₂\nh₂ : 0 < a₂\nh₁ : 0 < b₁\nh : a₁ • ... | [] | exact (smul_lt_smul_of_pos_right ha h₁).trans_le (smul_le_smul_of_nonneg_left hb h₂.le) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Order.Module.Defs | {
"line": 784,
"column": 4
} | {
"line": 784,
"column": 91
} | {
"line": 785,
"column": 2
} | [
{
"pp": "case refine_1\nα : Type u_1\nβ : Type u_2\na₁ a₂ : α\nb₁ b₂ : β\ninst✝⁶ : Zero α\ninst✝⁵ : Zero β\ninst✝⁴ : SMulWithZero α β\ninst✝³ : PartialOrder α\ninst✝² : PartialOrder β\ninst✝¹ : PosSMulStrictMono α β\ninst✝ : SMulPosStrictMono α β\nha✝ : a₁ ≤ a₂\nhb : b₁ ≤ b₂\nh₂ : 0 < a₂\nh₁ : 0 < b₁\nh : a₁ • ... | [] | exact (smul_lt_smul_of_pos_right ha h₁).trans_le (smul_le_smul_of_nonneg_left hb h₂.le) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Finiteness.Basic | {
"line": 296,
"column": 11
} | {
"line": 296,
"column": 50
} | {
"line": 296,
"column": 51
} | [
{
"pp": "R : Type u_1\ninst✝ : Semiring R\n⊢ Ideal.span ↑{1} = ⊤",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Finset.coe_singleton",
"Semiring.toModule",
"congrArg",
"Finset",
"Set.instSingle... | [
"R : Type u_1\ninst✝ : Semiring R\n⊢ Ideal.span {1} = ⊤"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Module.Defs | {
"line": 844,
"column": 2
} | {
"line": 844,
"column": 29
} | {
"line": 844,
"column": 30
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\na : α\nb : β\ninst✝⁵ : Monoid α\ninst✝⁴ : Zero β\ninst✝³ : MulAction α β\ninst✝² : Preorder α\ninst✝¹ : Preorder β\ninst✝ : SMulPosMono α β\nhb : 0 ≤ b\nh : a ≤ 1\n⊢ a • b ≤ b",
"ppTerm": "?m.20",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"... | [
"α : Type u_1\nβ : Type u_2\na : α\nb : β\ninst✝⁵ : Monoid α\ninst✝⁴ : Zero β\ninst✝³ : MulAction α β\ninst✝² : Preorder α\ninst✝¹ : Preorder β\ninst✝ : SMulPosMono α β\nhb : 0 ≤ b\nh : a ≤ 1\n⊢ a • b ≤ b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Module.Defs | {
"line": 847,
"column": 2
} | {
"line": 847,
"column": 29
} | {
"line": 847,
"column": 30
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\na : α\nb : β\ninst✝⁵ : Monoid α\ninst✝⁴ : Zero β\ninst✝³ : MulAction α β\ninst✝² : Preorder α\ninst✝¹ : Preorder β\ninst✝ : SMulPosMono α β\nhb : 0 ≤ b\nh : 1 ≤ a\n⊢ b ≤ a • b",
"ppTerm": "?m.20",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"... | [
"α : Type u_1\nβ : Type u_2\na : α\nb : β\ninst✝⁵ : Monoid α\ninst✝⁴ : Zero β\ninst✝³ : MulAction α β\ninst✝² : Preorder α\ninst✝¹ : Preorder β\ninst✝ : SMulPosMono α β\nhb : 0 ≤ b\nh : 1 ≤ a\n⊢ b ≤ a • b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Module.Defs | {
"line": 850,
"column": 2
} | {
"line": 850,
"column": 29
} | {
"line": 850,
"column": 30
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\na : α\nb : β\ninst✝⁵ : Monoid α\ninst✝⁴ : Zero β\ninst✝³ : MulAction α β\ninst✝² : Preorder α\ninst✝¹ : Preorder β\ninst✝ : SMulPosStrictMono α β\nhb : 0 < b\nh : a < 1\n⊢ a • b < b",
"ppTerm": "?m.20",
"assigned": false,
"usedConstants": [],
"usedFVars": [],... | [
"α : Type u_1\nβ : Type u_2\na : α\nb : β\ninst✝⁵ : Monoid α\ninst✝⁴ : Zero β\ninst✝³ : MulAction α β\ninst✝² : Preorder α\ninst✝¹ : Preorder β\ninst✝ : SMulPosStrictMono α β\nhb : 0 < b\nh : a < 1\n⊢ a • b < b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Module.Defs | {
"line": 853,
"column": 2
} | {
"line": 853,
"column": 29
} | {
"line": 853,
"column": 30
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\na : α\nb : β\ninst✝⁵ : Monoid α\ninst✝⁴ : Zero β\ninst✝³ : MulAction α β\ninst✝² : Preorder α\ninst✝¹ : Preorder β\ninst✝ : SMulPosStrictMono α β\nhb : 0 < b\nh : 1 < a\n⊢ b < a • b",
"ppTerm": "?m.20",
"assigned": false,
"usedConstants": [],
"usedFVars": [],... | [
"α : Type u_1\nβ : Type u_2\na : α\nb : β\ninst✝⁵ : Monoid α\ninst✝⁴ : Zero β\ninst✝³ : MulAction α β\ninst✝² : Preorder α\ninst✝¹ : Preorder β\ninst✝ : SMulPosStrictMono α β\nhb : 0 < b\nh : 1 < a\n⊢ b < a • b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Module.Defs | {
"line": 864,
"column": 48
} | {
"line": 864,
"column": 82
} | {
"line": 864,
"column": 83
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝⁵ : Semiring α\ninst✝⁴ : AddCommGroup β\ninst✝³ : Module α β\ninst✝² : PartialOrder α\ninst✝¹ : PartialOrder β\ninst✝ : IsOrderedAddMonoid β\nh : ∀ (a : α), 0 ≤ a → ∀ (b : β), 0 ≤ b → 0 ≤ a • b\n_a : α\nha : 0 ≤ _a\nb₁ b₂ : β\n⊢ b₁ ≤ b₂ → _a • b₁ ≤ _a • b₂",
"ppTerm... | [
"α : Type u_1\nβ : Type u_2\ninst✝⁵ : Semiring α\ninst✝⁴ : AddCommGroup β\ninst✝³ : Module α β\ninst✝² : PartialOrder α\ninst✝¹ : PartialOrder β\ninst✝ : IsOrderedAddMonoid β\nh : ∀ (a : α), 0 ≤ a → ∀ (b : β), 0 ≤ b → 0 ≤ a • b\n_a : α\nha : 0 ≤ _a\nb₁ b₂ : β\n⊢ b₁ ≤ b₂ → _a • b₁ ≤ _a • b₂"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Module.Defs | {
"line": 902,
"column": 4
} | {
"line": 902,
"column": 38
} | {
"line": 902,
"column": 39
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝⁶ : Ring α\ninst✝⁵ : AddCommGroup β\ninst✝⁴ : Module α β\ninst✝³ : PartialOrder α\ninst✝² : PartialOrder β\ninst✝¹ : IsOrderedAddMonoid α\ninst✝ : IsOrderedAddMonoid β\nh : ∀ (a : α), 0 ≤ a → ∀ (b : β), 0 ≤ b → 0 ≤ a • b\n_b : β\nhb : 0 ≤ _b\na₁ a₂ : α\n⊢ a₁ ≤ a₂ → a₁ •... | [
"α : Type u_1\nβ : Type u_2\ninst✝⁶ : Ring α\ninst✝⁵ : AddCommGroup β\ninst✝⁴ : Module α β\ninst✝³ : PartialOrder α\ninst✝² : PartialOrder β\ninst✝¹ : IsOrderedAddMonoid α\ninst✝ : IsOrderedAddMonoid β\nh : ∀ (a : α), 0 ≤ a → ∀ (b : β), 0 ≤ b → 0 ≤ a • b\n_b : β\nhb : 0 ≤ _b\na₁ a₂ : α\n⊢ a₁ ≤ a₂ → a₁ • _b ≤ a₂ • _... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Finiteness.Basic | {
"line": 392,
"column": 8
} | {
"line": 392,
"column": 50
} | {
"line": 392,
"column": 51
} | [
{
"pp": "A₁ : Type u_6\nB₁ : Type u_7\nA₂ : Type u_8\nB₂ : Type u_9\ninst✝⁶ : CommSemiring A₁\ninst✝⁵ : CommSemiring B₁\ninst✝⁴ : CommSemiring A₂\ninst✝³ : Semiring B₂\ninst✝² : Algebra A₁ B₁\ninst✝¹ : Algebra A₂ B₂\ne₁ : A₁ ≃+* A₂\ne₂ : B₁ ≃+* B₂\nhe : (algebraMap A₂ B₂).comp ↑e₁ = (↑e₂).comp (algebraMap A₁ B₁... | [
"A₁ : Type u_6\nB₁ : Type u_7\nA₂ : Type u_8\nB₂ : Type u_9\ninst✝⁶ : CommSemiring A₁\ninst✝⁵ : CommSemiring B₁\ninst✝⁴ : CommSemiring A₂\ninst✝³ : Semiring B₂\ninst✝² : Algebra A₁ B₁\ninst✝¹ : Algebra A₂ B₂\ne₁ : A₁ ≃+* A₂\ne₂ : B₁ ≃+* B₂\nhe : (algebraMap A₂ B₂).comp ↑e₁ = (↑e₂).comp (algebraMap A₁ B₁)\ninst✝ : M... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Module.Defs | {
"line": 1101,
"column": 2
} | {
"line": 1101,
"column": 30
} | {
"line": 1101,
"column": 31
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\na : α\nb : β\ninst✝⁸ : Ring α\ninst✝⁷ : PartialOrder α\ninst✝⁶ : IsOrderedRing α\ninst✝⁵ : AddCommGroup β\ninst✝⁴ : PartialOrder β\ninst✝³ : IsOrderedAddMonoid β\ninst✝² : Module α β\ninst✝¹ : PosSMulStrictMono α β\ninst✝ : PosSMulReflectLT α β\nha : a < 0\n⊢ 0 < a • b ↔ b <... | [
"α : Type u_1\nβ : Type u_2\na : α\nb : β\ninst✝⁸ : Ring α\ninst✝⁷ : PartialOrder α\ninst✝⁶ : IsOrderedRing α\ninst✝⁵ : AddCommGroup β\ninst✝⁴ : PartialOrder β\ninst✝³ : IsOrderedAddMonoid β\ninst✝² : Module α β\ninst✝¹ : PosSMulStrictMono α β\ninst✝ : PosSMulReflectLT α β\nha : a < 0\n⊢ 0 < a • b ↔ b < 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Module.Defs | {
"line": 1106,
"column": 2
} | {
"line": 1106,
"column": 30
} | {
"line": 1106,
"column": 31
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\na : α\nb : β\ninst✝⁸ : Ring α\ninst✝⁷ : PartialOrder α\ninst✝⁶ : IsOrderedRing α\ninst✝⁵ : AddCommGroup β\ninst✝⁴ : PartialOrder β\ninst✝³ : IsOrderedAddMonoid β\ninst✝² : Module α β\ninst✝¹ : PosSMulStrictMono α β\ninst✝ : PosSMulReflectLT α β\nha : a < 0\n⊢ a • b < 0 ↔ 0 <... | [
"α : Type u_1\nβ : Type u_2\na : α\nb : β\ninst✝⁸ : Ring α\ninst✝⁷ : PartialOrder α\ninst✝⁶ : IsOrderedRing α\ninst✝⁵ : AddCommGroup β\ninst✝⁴ : PartialOrder β\ninst✝³ : IsOrderedAddMonoid β\ninst✝² : Module α β\ninst✝¹ : PosSMulStrictMono α β\ninst✝ : PosSMulReflectLT α β\nha : a < 0\n⊢ a • b < 0 ↔ 0 < b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Finiteness.Basic | {
"line": 475,
"column": 2
} | {
"line": 476,
"column": 18
} | {
"line": 478,
"column": 0
} | [
{
"pp": "A : Type u_1\nB : Type u_2\nC : Type u_3\ninst✝² : CommRing A\ninst✝¹ : CommRing B\ninst✝ : CommRing C\ng : B →+* C\nf : A →+* B\nhg : g.Finite\nhf : f.Finite\n⊢ (g.comp f).Finite",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Algebra.algebraMap",
"CommSemiring.toSem... | [] | algebraize [f, g, g.comp f]
exact .trans B C | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Finiteness.Basic | {
"line": 475,
"column": 2
} | {
"line": 476,
"column": 18
} | {
"line": 478,
"column": 0
} | [
{
"pp": "A : Type u_1\nB : Type u_2\nC : Type u_3\ninst✝² : CommRing A\ninst✝¹ : CommRing B\ninst✝ : CommRing C\ng : B →+* C\nf : A →+* B\nhg : g.Finite\nhf : f.Finite\n⊢ (g.comp f).Finite",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Algebra.algebraMap",
"CommSemiring.toSem... | [] | algebraize [f, g, g.comp f]
exact .trans B C | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Finiteness.Basic | {
"line": 523,
"column": 4
} | {
"line": 523,
"column": 15
} | {
"line": 523,
"column": 16
} | [
{
"pp": "case inl\nR : Type u_1\nE : Type u_2\ninst✝⁴ : Ring R\ninst✝³ : LinearOrder R\ninst✝² : IsOrderedRing R\ninst✝¹ : AddCommMonoid E\ninst✝ : Module R E\nx : R\nhx : 0 ≤ x\n⊢ x ∈ Submodule.span R≥0 {1, -1}",
"ppTerm": "?inl",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"us... | [
"case inl\nR : Type u_1\nE : Type u_2\ninst✝⁴ : Ring R\ninst✝³ : LinearOrder R\ninst✝² : IsOrderedRing R\ninst✝¹ : AddCommMonoid E\ninst✝ : Module R E\nx : R\nhx : 0 ≤ x\n⊢ x ∈ Submodule.span R≥0 {1, -1}"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Finiteness.Basic | {
"line": 525,
"column": 4
} | {
"line": 525,
"column": 15
} | {
"line": 525,
"column": 16
} | [
{
"pp": "case inr\nR : Type u_1\nE : Type u_2\ninst✝⁴ : Ring R\ninst✝³ : LinearOrder R\ninst✝² : IsOrderedRing R\ninst✝¹ : AddCommMonoid E\ninst✝ : Module R E\nx : R\nhx : x ≤ 0\n⊢ x ∈ Submodule.span R≥0 {1, -1}",
"ppTerm": "?inr",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"us... | [
"case inr\nR : Type u_1\nE : Type u_2\ninst✝⁴ : Ring R\ninst✝³ : LinearOrder R\ninst✝² : IsOrderedRing R\ninst✝¹ : AddCommMonoid E\ninst✝ : Module R E\nx : R\nhx : x ≤ 0\n⊢ x ∈ Submodule.span R≥0 {1, -1}"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Module.Defs | {
"line": 1295,
"column": 40
} | {
"line": 1295,
"column": 63
} | {
"line": 1295,
"column": 64
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝⁶ : Preorder α\ninst✝⁵ : Preorder β\ninst✝⁴ : Preorder γ\ninst✝³ : SMul α β\ninst✝² : SMul α γ\nf : β → γ\ninst✝¹ : Zero α\ninst✝ : PosSMulReflectLE α γ\nhf : ∀ {b₁ b₂ : β}, f b₁ ≤ f b₂ ↔ b₁ ≤ b₂\nsmul : ∀ (a : α) (b : β), f (a • b) = a • f b\na : α\nha : ... | [
"α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝⁶ : Preorder α\ninst✝⁵ : Preorder β\ninst✝⁴ : Preorder γ\ninst✝³ : SMul α β\ninst✝² : SMul α γ\nf : β → γ\ninst✝¹ : Zero α\ninst✝ : PosSMulReflectLE α γ\nhf : ∀ {b₁ b₂ : β}, f b₁ ≤ f b₂ ↔ b₁ ≤ b₂\nsmul : ∀ (a : α) (b : β), f (a • b) = a • f b\na : α\nha : 0 < a\nb₁ b₂... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Ordinal.Univ | {
"line": 145,
"column": 2
} | {
"line": 145,
"column": 13
} | {
"line": 145,
"column": 14
} | [
{
"pp": "⊢ ℵ₀ < univ.{u, v}",
"ppTerm": "?m.3",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"⊢ ℵ₀ < univ.{u, v}"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Ordinal.Univ | {
"line": 161,
"column": 60
} | {
"line": 161,
"column": 71
} | {
"line": 161,
"column": 72
} | [
{
"pp": "o : Ordinal.{max (u + 1) v}\nh : o < Ordinal.univ.{u, v}\n⊢ ?m.25 < Ordinal.liftPrincipalSeg.top",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"Ordinal.partialOrder",
"PartialOrder.toPreorder",
"Ordinal.univ",
"id",
... | [
"o : Ordinal.{max (u + 1) v}\nh : o < Ordinal.univ.{u, v}\n⊢ ?m.25 < Ordinal.univ.{?u.12, ?u.11}"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Ordinal.Arithmetic | {
"line": 79,
"column": 6
} | {
"line": 79,
"column": 17
} | {
"line": 79,
"column": 18
} | [
{
"pp": "α✝ : Type u_1\nβ✝ : Type u_2\nγ✝ : Type u_3\nr✝ : α✝ → α✝ → Prop\ns✝ : β✝ → β✝ → Prop\nt✝ : γ✝ → γ✝ → Prop\nc a✝ b : Ordinal.{u}\nα : Type u\nr : α → α → Prop\nx✝³ : IsWellOrder α r\nβ : Type u\ns : β → β → Prop\nx✝² : IsWellOrder β s\nγ : Type u\nt : γ → γ → Prop\nx✝¹ : IsWellOrder γ t\nx✝ : type t + ... | [
"α✝ : Type u_1\nβ✝ : Type u_2\nγ✝ : Type u_3\nr✝ : α✝ → α✝ → Prop\ns✝ : β✝ → β✝ → Prop\nt✝ : γ✝ → γ✝ → Prop\nc a✝ b : Ordinal.{u}\nα : Type u\nr : α → α → Prop\nx✝³ : IsWellOrder α r\nβ : Type u\ns : β → β → Prop\nx✝² : IsWellOrder β s\nγ : Type u\nt : γ → γ → Prop\nx✝¹ : IsWellOrder γ t\nx✝ : type t + type r ≤ typ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Ordinal.Arithmetic | {
"line": 91,
"column": 32
} | {
"line": 91,
"column": 87
} | {
"line": 91,
"column": 88
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\na b c : Ordinal.{u_4}\nh : (fun x ↦ a + x) b = (fun x ↦ a + x) c\n⊢ b = c",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Ordinal.partialOrder",
"PartialOrde... | [
"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\na b c : Ordinal.{u_4}\nh : (fun x ↦ a + x) b = (fun x ↦ a + x) c\n⊢ b ≤ c ∧ c ≤ b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Ordinal.Arithmetic | {
"line": 107,
"column": 16
} | {
"line": 107,
"column": 41
} | {
"line": 107,
"column": 42
} | [
{
"pp": "a b : Ordinal.{u_4}\nn : ℕ\n⊢ a + ↑(n + 1) ≤ b + ↑(n + 1) ↔ a ≤ b",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"_private.Mathlib.SetTheory.Ordinal.Arithmetic.0.Ordinal.add_le_add_iff_right._simp_1_3",
"Eq.mpr",
"Ordinal.instLinearOrder",
"Preorder.toLT",... | [
"a b : Ordinal.{u_4}\nn : ℕ\n⊢ a + ↑n ≤ b + ↑n ↔ a ≤ b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Ordinal.Arithmetic | {
"line": 144,
"column": 2
} | {
"line": 144,
"column": 13
} | {
"line": 144,
"column": 14
} | [
{
"pp": "o : Ordinal.{u_4}\nh : IsSuccLimit o\nn : ℕ\n⊢ ↑n < o",
"ppTerm": "?m.5",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"o : Ordinal.{u_4}\nh : IsSuccLimit o\nn : ℕ\n⊢ ↑n < o"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Ordinal.Arithmetic | {
"line": 150,
"column": 2
} | {
"line": 150,
"column": 13
} | {
"line": 150,
"column": 14
} | [
{
"pp": "o : Ordinal.{u_4}\n⊢ o = 0 ∨ o ∈ range succ ∨ IsSuccLimit o",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Order.succ",
"Order.succ_eq_add_one",
"Ordinal.partialOrder",
"congrArg",
"PartialOrder.toPreorder",
"Membership.mem",
... | [
"o : Ordinal.{u_4}\n⊢ o = 0 ∨ (∃ y, y + 1 = o) ∨ IsSuccLimit o"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Ordinal.Family | {
"line": 150,
"column": 2
} | {
"line": 150,
"column": 22
} | {
"line": 150,
"column": 23
} | [
{
"pp": "s : Set Ordinal.{u}\ninst✝ : Small.{u, u + 1} ↑s\na : Cardinal.{u}\nha : a ∈ upperBounds (succ ∘ card '' s)\nb : Ordinal.{u}\nhb : b ∈ s\n⊢ b < a.ord",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"Ordinal.partialOrder",
"Cardina... | [
"s : Set Ordinal.{u}\ninst✝ : Small.{u, u + 1} ↑s\na : Cardinal.{u}\nha : a ∈ upperBounds (succ ∘ card '' s)\nb : Ordinal.{u}\nhb : b ∈ s\n⊢ b.card < a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Ordinal.Family | {
"line": 160,
"column": 65
} | {
"line": 162,
"column": 20
} | {
"line": 164,
"column": 0
} | [
{
"pp": "s : Set Ordinal.{u}\nhf : BddAbove s\nf : Ordinal.{u} → Ordinal.{max u v}\n⊢ BddAbove (f '' s)",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Ordinal.partialOrder",
"congrArg",
"PartialOrder.toPreorder",
"Preorder.toLE",
"small_image"... | [] | by
rw [bddAbove_iff_small] at hf ⊢
exact small_lift _ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.SetTheory.Ordinal.Arithmetic | {
"line": 382,
"column": 2
} | {
"line": 382,
"column": 18
} | {
"line": 382,
"column": 19
} | [
{
"pp": "α : Type u_1\nf : Ordinal.{u_4} → Ordinal.{u_5}\no : Ordinal.{u_5}\nH : Ordinal.IsNormal f\np : Set α\np0 : p.Nonempty\ng : α → Ordinal.{u_4}\nb : Ordinal.{u_4}\nH₂ : ∀ (o : Ordinal.{u_4}), b ≤ o ↔ ∀ a ∈ p, g a ≤ o\n⊢ f b ≤ o ↔ ∀ a ∈ p, f (g a) ≤ o",
"ppTerm": "?m.24",
"assigned": false,
"u... | [
"α : Type u_1\nf : Ordinal.{u_4} → Ordinal.{u_5}\no : Ordinal.{u_5}\nH : Ordinal.IsNormal f\np : Set α\np0 : p.Nonempty\ng : α → Ordinal.{u_4}\nb : Ordinal.{u_4}\nH₂ : ∀ (o : Ordinal.{u_4}), b ≤ o ↔ ∀ a ∈ p, g a ≤ o\n⊢ f b ≤ o ↔ ∀ a ∈ p, f (g a) ≤ o"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Ordinal.Arithmetic | {
"line": 414,
"column": 2
} | {
"line": 414,
"column": 13
} | {
"line": 414,
"column": 14
} | [
{
"pp": "a b : Ordinal.{u_4}\n⊢ a + b - a = b",
"ppTerm": "?m.10",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"a b : Ordinal.{u_4}\n⊢ a + b - a = b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Ordinal.Arithmetic | {
"line": 419,
"column": 4
} | {
"line": 419,
"column": 37
} | {
"line": 419,
"column": 38
} | [
{
"pp": "case inr\na b : Ordinal.{u_4}\nh : a < b\n⊢ a ≤ b + (a - b)",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Ordinal.partialOrder",
"congrArg",
"_private.Mathlib.SetTheory.Ordinal.Arithmetic.0.Ordinal.sub_eq_zero_of_lt",
"AddMonoid.toAddZeroC... | [
"case inr\na b : Ordinal.{u_4}\nh : a < b\n⊢ a ≤ b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Ordinal.Arithmetic | {
"line": 442,
"column": 49
} | {
"line": 442,
"column": 76
} | {
"line": 442,
"column": 77
} | [
{
"pp": "a : Ordinal.{u_4}\n⊢ a - 0 = a",
"ppTerm": "?m.8",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"a : Ordinal.{u_4}\n⊢ a - 0 = a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.