module stringlengths 16 90 | startPos dict | endPos dict | goals listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 366
values | kind stringclasses 370
values |
|---|---|---|---|---|---|---|
Mathlib.Order.Filter.Partial | {
"line": 181,
"column": 4
} | {
"line": 181,
"column": 42
} | [
{
"pp": "case mp\nα : Type u\nβ : Type v\nr : SetRel α β\nl₁ : Filter α\nl₂ : Filter β\n⊢ l₁ ≤\n { sets := SetRel.image {(s, t) | r.preimage s ⊆ t} l₂.sets, univ_sets := ⋯, sets_of_superset := ⋯,\n inter_sets := ⋯ } →\n ∀ s ∈ l₂, r.preimage s ∈ l₁",
"usedConstants": [
"Filter.instMember... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Filter.Partial | {
"line": 182,
"column": 4
} | {
"line": 182,
"column": 42
} | [
{
"pp": "case mpr\nα : Type u\nβ : Type v\nr : SetRel α β\nl₁ : Filter α\nl₂ : Filter β\n⊢ (∀ s ∈ l₂, r.preimage s ∈ l₁) →\n l₁ ≤\n { sets := SetRel.image {(s, t) | r.preimage s ⊆ t} l₂.sets, univ_sets := ⋯, sets_of_superset := ⋯,\n inter_sets := ⋯ }",
"usedConstants": [
"Filter.instMem... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Filter.CardinalInter | {
"line": 236,
"column": 2
} | {
"line": 236,
"column": 31
} | [
{
"pp": "ι α β : Type u\nc : Cardinal.{u}\nl✝ : Filter α\ninst✝¹ : CardinalInterFilter l✝ c\nl : Filter β\ninst✝ : CardinalInterFilter l c\nf : α → β\nS : Set (Set α)\nhSc : #↑S < c\nt : Set α → Set β\nhtl : ∀ s ∈ S, t s ∈ l\nht : ∀ s ∈ S, f ⁻¹' t s ⊆ s\n⊢ f ⁻¹' ⋂ i ∈ S, t i ⊆ ⋂₀ S",
"usedConstants": [
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Height | {
"line": 69,
"column": 46
} | {
"line": 69,
"column": 95
} | [
{
"pp": "α : Type u_1\ns : Set α\nr : α → α → Prop\nh : s.chainHeight r ≠ ⊤\nthis : Nonempty { t // t ⊆ s ∧ IsChain r t }\n⊢ ⨆ i, ?m.28 i < ⊤",
"usedConstants": [
"Set.chainHeight",
"Eq.mpr",
"Set.encard",
"Preorder.toLT",
"instCompleteLinearOrderENat",
"ChainCompletePart... | by rwa [← chainHeight_eq_iSup, lt_top_iff_ne_top] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Order.Height | {
"line": 110,
"column": 4
} | {
"line": 110,
"column": 15
} | [
{
"pp": "case refine_1.h\nα : Type u_1\ns : Set α\nr : α → α → Prop\nh : ∀ a ⊆ s, IsChain r a → a = ∅\nx : α\n⊢ x ∈ s ↔ x ∈ ∅",
"usedConstants": [
"Eq.mpr",
"False",
"iff_false",
"Set.mem_empty_iff_false._simp_1",
"congrArg",
"Membership.mem",
"id",
"Iff",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Height | {
"line": 141,
"column": 29
} | {
"line": 141,
"column": 56
} | [
{
"pp": "α : Type u_1\ns : Set α\nr : α → α → Prop\nn : ℕ\nhn : ↑n ≤ s.chainHeight (flip r)\na : Set α\nha₁ : a ⊆ s\nha₂ : a.encard = ↑n\nha₃ : IsChain (flip r) a\nx✝¹ : α\nhx : x✝¹ ∈ a\nx✝ : α\nhy : x✝ ∈ a\nhne : x✝¹ ≠ x✝\n⊢ (fun x y ↦ r x y ∨ r y x) x✝¹ x✝",
"usedConstants": [
"id",
"Or"
]... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Height | {
"line": 141,
"column": 29
} | {
"line": 141,
"column": 56
} | [
{
"pp": "α : Type u_1\ns : Set α\nr : α → α → Prop\nn : ℕ\nhn : ↑n ≤ s.chainHeight r\na : Set α\nha₁ : a ⊆ s\nha₂ : a.encard = ↑n\nha₃ : IsChain r a\nx✝¹ : α\nhx : x✝¹ ∈ a\nx✝ : α\nhy : x✝ ∈ a\nhne : x✝¹ ≠ x✝\n⊢ (fun x y ↦ flip r x y ∨ flip r y x) x✝¹ x✝",
"usedConstants": [
"Eq.mpr",
"_private.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Height | {
"line": 167,
"column": 2
} | {
"line": 167,
"column": 18
} | [
{
"pp": "α : Type u_1\ns : Set α\nr : α → α → Prop\nhc :\n ((⇑(Subtype.relEmbedding (fun x1 x2 ↦ r x1 x2) fun x ↦ x ∈ s) '' univ).chainHeight fun x1 x2 ↦ r x1 x2) =\n univ.chainHeight (Subtype.val ⁻¹'o fun x1 x2 ↦ r x1 x2)\nhs : (Subtype.val ⁻¹'o fun x1 x2 ↦ r x1 x2) = fun x y ↦ r ↑x ↑y\n⊢ (univ.chainHeight... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Height | {
"line": 172,
"column": 2
} | {
"line": 172,
"column": 13
} | [
{
"pp": "α : Type u_1\ns : Set α\ninst✝ : LE α\n⊢ (univ.chainHeight fun x1 x2 ↦ x1 ≤ x2) = s.chainHeight fun x1 x2 ↦ x1 ≤ x2",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Height | {
"line": 177,
"column": 2
} | {
"line": 177,
"column": 13
} | [
{
"pp": "α : Type u_1\ns : Set α\ninst✝ : LT α\n⊢ (univ.chainHeight fun x1 x2 ↦ x1 < x2) = s.chainHeight fun x1 x2 ↦ x1 < x2",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Filter.CardinalInter | {
"line": 313,
"column": 6
} | {
"line": 313,
"column": 31
} | [
{
"pp": "case mp.basic\nα : Type u\nc : Cardinal.{u}\ng : Set (Set α)\ns✝ : Set α\nhreg : c.IsRegular\ns : Set α\nhs : s ∈ g\n⊢ #↑{s} < c ∧ ⋂₀ {s} ⊆ s",
"usedConstants": [
"subset_refl._simp_1",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Set.fintypeSingleton",
"Preor... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Filter.TendstoCofinite | {
"line": 52,
"column": 2
} | {
"line": 52,
"column": 34
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nf : α → β\ninst✝ : TendstoCofinite f\ns : Set β\nhs : s.Finite\n⊢ (f ⁻¹' s).Finite",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Filter.TendstoCofinite | {
"line": 56,
"column": 33
} | {
"line": 56,
"column": 44
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nf : α → β\ninst✝ : TendstoCofinite f\nb : β\n⊢ (f ⁻¹' {b}).Finite",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Filter.TendstoCofinite | {
"line": 74,
"column": 4
} | {
"line": 74,
"column": 15
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nι : Type u_3\nf : α → β\ng : β → ι\ninst✝¹ : TendstoCofinite g\ninst✝ : TendstoCofinite f\nr : ι\n⊢ (g ∘ f ⁻¹' {r}).Finite",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Filter.CardinalInter | {
"line": 316,
"column": 25
} | {
"line": 316,
"column": 55
} | [
{
"pp": "case mp.superset\nα : Type u\nc : Cardinal.{u}\ng : Set (Set α)\ns : Set α\nhreg : c.IsRegular\ns✝ t✝ : Set α\na✝¹ : CardinalGenerateSets g s✝\na✝ : s✝ ⊆ t✝\nih : ∃ S ⊆ g, #↑S < c ∧ ⋂₀ S ⊆ s✝\n⊢ ∃ S ⊆ g, #↑S < c ∧ ⋂₀ S ⊆ t✝",
"usedConstants": [
"Preorder.toLT",
"Cardinal",
"Partia... | exact Exists.imp (by tauto) ih | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Order.Filter.CardinalInter | {
"line": 316,
"column": 25
} | {
"line": 316,
"column": 55
} | [
{
"pp": "case mp.superset\nα : Type u\nc : Cardinal.{u}\ng : Set (Set α)\ns : Set α\nhreg : c.IsRegular\ns✝ t✝ : Set α\na✝¹ : CardinalGenerateSets g s✝\na✝ : s✝ ⊆ t✝\nih : ∃ S ⊆ g, #↑S < c ∧ ⋂₀ S ⊆ s✝\n⊢ ∃ S ⊆ g, #↑S < c ∧ ⋂₀ S ⊆ t✝",
"usedConstants": [
"Preorder.toLT",
"Cardinal",
"Partia... | exact Exists.imp (by tauto) ih | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.Filter.CardinalInter | {
"line": 316,
"column": 25
} | {
"line": 316,
"column": 55
} | [
{
"pp": "case mp.superset\nα : Type u\nc : Cardinal.{u}\ng : Set (Set α)\ns : Set α\nhreg : c.IsRegular\ns✝ t✝ : Set α\na✝¹ : CardinalGenerateSets g s✝\na✝ : s✝ ⊆ t✝\nih : ∃ S ⊆ g, #↑S < c ∧ ⋂₀ S ⊆ s✝\n⊢ ∃ S ⊆ g, #↑S < c ∧ ⋂₀ S ⊆ t✝",
"usedConstants": [
"Preorder.toLT",
"Cardinal",
"Partia... | exact Exists.imp (by tauto) ih | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.Filter.TendstoCofinite | {
"line": 123,
"column": 4
} | {
"line": 123,
"column": 74
} | [
{
"pp": "case refine_2\nα : Type u_1\nβ : Type u_2\nf : α → β\ninst✝ : TendstoCofinite f\nx : β →₀ ℕ\ns : Finset α := x.support.sup fun t ↦ ⋯.toFinset\ne : ↥s ↪ α := Function.Embedding.subtype fun u ↦ u ∈ s\ny : α →₀ ℕ\nhy : mapDomain f y = x\n⊢ y.support ⊆ s",
"usedConstants": [
"Finsupp.instFunLike"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Interval.Set.Nat | {
"line": 23,
"column": 2
} | {
"line": 23,
"column": 38
} | [
{
"pp": "a b : ℕ\n⊢ (Icc a b).ncard = b + 1 - a",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Interval.Set.Nat | {
"line": 26,
"column": 2
} | {
"line": 26,
"column": 38
} | [
{
"pp": "a b : ℕ\n⊢ (Ico a b).ncard = b - a",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Interval.Set.Nat | {
"line": 29,
"column": 2
} | {
"line": 29,
"column": 38
} | [
{
"pp": "a b : ℕ\n⊢ (Ioc a b).ncard = b - a",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Interval.Set.Nat | {
"line": 32,
"column": 2
} | {
"line": 32,
"column": 38
} | [
{
"pp": "a b : ℕ\n⊢ (Ioo a b).ncard = b - a - 1",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Interval.Set.Nat | {
"line": 35,
"column": 2
} | {
"line": 35,
"column": 38
} | [
{
"pp": "a b : ℕ\n⊢ (uIcc a b).ncard = (↑b - ↑a).natAbs + 1",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Interval.Set.Nat | {
"line": 38,
"column": 2
} | {
"line": 38,
"column": 38
} | [
{
"pp": "b : ℕ\n⊢ (Iic b).ncard = b + 1",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Interval.Set.Nat | {
"line": 41,
"column": 2
} | {
"line": 41,
"column": 38
} | [
{
"pp": "b : ℕ\n⊢ (Iio b).ncard = b",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Interval.Set.SurjOn | {
"line": 49,
"column": 2
} | {
"line": 49,
"column": 13
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrder α\ninst✝ : PartialOrder β\nf : α → β\nh_mono : Monotone f\nh_surj : Surjective f\na b : α\n⊢ SurjOn f (Ioc a b) (Ioc (f a) (f b))",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Filter.Cocardinal | {
"line": 100,
"column": 2
} | {
"line": 100,
"column": 38
} | [
{
"pp": "α : Type u\nc : Cardinal.{u}\nhreg : c.IsRegular\nx : α\n⊢ ∀ᶠ (a : α) in cocardinal α hreg, a ≠ x",
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Set.fintypeSingleton",
"Preorder.toLT",
"Classical.not_not._simp_1",
"Cardinal",
"co... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Lattice.Congruence | {
"line": 49,
"column": 22
} | {
"line": 49,
"column": 33
} | [
{
"pp": "α : Type u_2\ninst✝ : Lattice α\nc : LatticeCon α\nx y : α\nh : c.toSetoid (x ⊓ y) (x ⊔ y)\n⊢ x ≈ ?m.23 h",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Lattice.Congruence | {
"line": 49,
"column": 67
} | {
"line": 49,
"column": 78
} | [
{
"pp": "α : Type u_2\ninst✝ : Lattice α\nc : LatticeCon α\nx y : α\nh : c.toSetoid (x ⊓ y) (x ⊔ y)\n⊢ x ⊓ y ≈ y",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Lattice.Congruence | {
"line": 50,
"column": 23
} | {
"line": 50,
"column": 34
} | [
{
"pp": "α : Type u_2\ninst✝ : Lattice α\nc : LatticeCon α\nx y : α\nh : c.toSetoid x y\n⊢ x ⊓ y ≈ ?m.35 h",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Lattice.Congruence | {
"line": 50,
"column": 59
} | {
"line": 50,
"column": 70
} | [
{
"pp": "α : Type u_2\ninst✝ : Lattice α\nc : LatticeCon α\nx y : α\nh : c.toSetoid x y\n⊢ y ≈ x ⊔ y",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Lattice.Congruence | {
"line": 58,
"column": 4
} | {
"line": 58,
"column": 55
} | [
{
"pp": "α : Type u_2\ninst✝ : Lattice α\nr : α → α → Prop\nh₂ : ∀ ⦃x y : α⦄, r x y ↔ r (x ⊓ y) (x ⊔ y)\nh₄ : ∀ ⦃x y t : α⦄, x ≤ y → r x y → r (x ⊓ t) (y ⊓ t) ∧ r (x ⊔ t) (y ⊔ t)\na b c d : α\nhab : a ≤ b\nhbd : b ≤ d\nhac : a ≤ c\nhcd : c ≤ d\nhad : r a d\nthis : r (b ⊓ c ⊓ (b ⊔ c)) (d ⊓ (b ⊔ c))\n⊢ r b c",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Lattice.Congruence | {
"line": 60,
"column": 2
} | {
"line": 60,
"column": 90
} | [
{
"pp": "α : Type u_2\ninst✝ : Lattice α\nr : α → α → Prop\nh₂ : ∀ ⦃x y : α⦄, r x y ↔ r (x ⊓ y) (x ⊔ y)\nh₄ : ∀ ⦃x y t : α⦄, x ≤ y → r x y → r (x ⊓ t) (y ⊓ t) ∧ r (x ⊔ t) (y ⊔ t)\na b c d : α\nhab : a ≤ b\nhbd : b ≤ d\nhac : a ≤ c\nhcd : c ≤ d\nhad : r a d\n⊢ r (b ⊓ c) d",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Lattice.Congruence | {
"line": 72,
"column": 12
} | {
"line": 72,
"column": 35
} | [
{
"pp": "α : Type u_2\ninst✝ : Lattice α\nr : α → α → Prop\nh₂ : ∀ ⦃x y : α⦄, r x y ↔ r (x ⊓ y) (x ⊔ y)\nh₃ : ∀ ⦃x y z : α⦄, x ≤ y → y ≤ z → r x y → r y z → r x z\nh₄ : ∀ ⦃x y t : α⦄, x ≤ y → r x y → r (x ⊓ t) (y ⊓ t) ∧ r (x ⊔ t) (y ⊔ t)\nx y z : α\nhxy : r x y\nhyz : r y z\n⊢ x ⊓ y ⊓ z ≤ y ⊔ z",
"usedConst... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.KonigLemma | {
"line": 69,
"column": 46
} | {
"line": 69,
"column": 57
} | [
{
"pp": "α : Type u_1\ninst✝¹ : PartialOrder α\ninst✝ : IsStronglyAtomic α\nb : α\nhfin : ∀ (a : α), {x | a ⋖ x}.Finite\nhb : (Ici b).Infinite\nh : ∀ (a : { a // (Ici a).Infinite }), ∃ b, ↑a ⋖ b ∧ (Ici b).Infinite :=\n fun a ↦ exists_covby_infinite_Ici_of_infinite_Ici a.property (hfin ↑a)\nks : ℕ → { a // (Ici... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Lattice.Congruence | {
"line": 74,
"column": 8
} | {
"line": 74,
"column": 31
} | [
{
"pp": "α : Type u_2\ninst✝ : Lattice α\nr : α → α → Prop\nh₂ : ∀ ⦃x y : α⦄, r x y ↔ r (x ⊓ y) (x ⊔ y)\nh₃ : ∀ ⦃x y z : α⦄, x ≤ y → y ≤ z → r x y → r y z → r x z\nh₄ : ∀ ⦃x y t : α⦄, x ≤ y → r x y → r (x ⊓ t) (y ⊓ t) ∧ r (x ⊔ t) (y ⊔ t)\nx y z : α\nhxy : r x y\nhyz : r y z\n⊢ x ⊓ y ⊓ z ≤ y ⊓ z",
"usedConst... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.KonigLemma | {
"line": 80,
"column": 62
} | {
"line": 80,
"column": 73
} | [
{
"pp": "α : Type u_1\ninst✝³ : PartialOrder α\ninst✝² : IsStronglyAtomic α\ninst✝¹ : OrderBot α\ninst✝ : Infinite α\nhfin : ∀ (a : α), {x | a ⋖ x}.Finite\n⊢ (Ici ⊥).Infinite",
"usedConstants": [
"Eq.mpr",
"Set.Ici",
"congrArg",
"Set.univ",
"OrderBot.toBot",
"PartialOrder... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.KonigLemma | {
"line": 89,
"column": 17
} | {
"line": 89,
"column": 70
} | [
{
"pp": "case succ\nα : Type u_1\ninst✝⁴ : PartialOrder α\ninst✝³ : IsStronglyAtomic α\ninst✝² : GradeMinOrder ℕ α\ninst✝¹ : OrderBot α\ninst✝ : Infinite α\nhfin : ∀ (a : α), {x | a ⋖ x}.Finite\nf : ℕ ↪o α\nh0 : f 0 = ⊥\nhf : ∀ (i : ℕ), f i ⋖ f (i + 1)\ni : ℕ\nih : grade ℕ (f i) = i\n⊢ grade ℕ (f (i + 1)) = i +... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Lattice.Congruence | {
"line": 77,
"column": 8
} | {
"line": 77,
"column": 56
} | [
{
"pp": "α : Type u_2\ninst✝ : Lattice α\nr : α → α → Prop\nh₂ : ∀ ⦃x y : α⦄, r x y ↔ r (x ⊓ y) (x ⊔ y)\nh₃ : ∀ ⦃x y z : α⦄, x ≤ y → y ≤ z → r x y → r y z → r x z\nh₄ : ∀ ⦃x y t : α⦄, x ≤ y → r x y → r (x ⊓ t) (y ⊓ t) ∧ r (x ⊔ t) (y ⊔ t)\nx y z : α\nhxy : r x y\nhyz : r y z\nthis : r (x ⊓ y ⊓ (y ⊓ z)) ((x ⊔ y) ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Lattice.Congruence | {
"line": 79,
"column": 8
} | {
"line": 79,
"column": 73
} | [
{
"pp": "α : Type u_2\ninst✝ : Lattice α\nr : α → α → Prop\nh₂ : ∀ ⦃x y : α⦄, r x y ↔ r (x ⊓ y) (x ⊔ y)\nh₃ : ∀ ⦃x y z : α⦄, x ≤ y → y ≤ z → r x y → r y z → r x z\nh₄ : ∀ ⦃x y t : α⦄, x ≤ y → r x y → r (x ⊓ t) (y ⊓ t) ∧ r (x ⊔ t) (y ⊔ t)\nx y z : α\nhxy : r x y\nhyz : r y z\n⊢ r (y ⊔ z) (x ⊔ y ⊔ z)",
"usedC... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Lattice.Congruence | {
"line": 91,
"column": 21
} | {
"line": 91,
"column": 63
} | [
{
"pp": "F : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝¹ : Lattice α\ninst✝ : Lattice β\nr : α → α → Prop\nh₁ : Std.Refl r\nh₂ : ∀ ⦃x y : α⦄, r x y ↔ r (x ⊓ y) (x ⊔ y)\nh₃ : ∀ ⦃x y z : α⦄, x ≤ y → y ≤ z → r x y → r y z → r x z\nh₄ : ∀ ⦃x y t : α⦄, x ≤ y → r x y → r (x ⊓ t) (y ⊓ t) ∧ r (x ⊔ t) (y ⊔ t)\nx✝ y✝ : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Lattice.Congruence | {
"line": 101,
"column": 10
} | {
"line": 101,
"column": 34
} | [
{
"pp": "F : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝¹ : Lattice α\ninst✝ : Lattice β\nr : α → α → Prop\nh₁ : Std.Refl r\nh₂ : ∀ ⦃x y : α⦄, r x y ↔ r (x ⊓ y) (x ⊔ y)\nh₃ : ∀ ⦃x y z : α⦄, x ≤ y → y ≤ z → r x y → r y z → r x z\nh₄ : ∀ ⦃x y t : α⦄, x ≤ y → r x y → r (x ⊓ t) (y ⊓ t) ∧ r (x ⊔ t) (y ⊔ t)\nw x✝ y✝ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Lattice.Congruence | {
"line": 110,
"column": 6
} | {
"line": 110,
"column": 30
} | [
{
"pp": "F : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝¹ : Lattice α\ninst✝ : Lattice β\nr : α → α → Prop\nh₁ : Std.Refl r\nh₂ : ∀ ⦃x y : α⦄, r x y ↔ r (x ⊓ y) (x ⊔ y)\nh₃ : ∀ ⦃x y z : α⦄, x ≤ y → y ≤ z → r x y → r y z → r x z\nh₄ : ∀ ⦃x y t : α⦄, x ≤ y → r x y → r (x ⊓ t) (y ⊓ t) ∧ r (x ⊔ t) (y ⊔ t)\nw x✝ y✝ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.KonigLemma | {
"line": 138,
"column": 36
} | {
"line": 138,
"column": 57
} | [
{
"pp": "α : ℕ → Type u_1\ninst✝¹ : Finite (α 0)\ninst✝ : ∀ (i : ℕ), Nonempty (α i)\nπ : {i j : ℕ} → i ≤ j → α j → α i\nπ_refl : ∀ ⦃i : ℕ⦄ (a : α i), π ⋯ a = a\nπ_trans : ∀ ⦃i j k : ℕ⦄ (hij : i ≤ j) (hjk : j ≤ k) (a : α k), π hij (π hjk a) = π ⋯ a\nhfin : ∀ (i : ℕ) (a : α i), {b | π ⋯ b = a}.Finite\nαs : Type u... | by rw [π_trans, ← h2] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Order.Nucleus | {
"line": 216,
"column": 49
} | {
"line": 216,
"column": 65
} | [
{
"pp": "X : Type u_1\ninst✝ : Frame X\nn✝ m✝ : Nucleus X\nx✝ y✝ : X\nm n : Nucleus X\nx y : X\nthis : Nonempty X\nk : X\nhxyk : k ≥ x ⊓ y\nl : X\nhlx : ∀ x_1 ≥ x, l ⊓ m x_1 ≤ n x_1\nhly : ∀ i ≥ y, l ⊓ m i ≤ n i\nhlk : l ≤ m k\n⊢ l = l ⊓ m ((x ⊔ k) ⊓ (y ⊔ k))",
"usedConstants": [
"Eq.mpr",
"Latt... | ← sup_inf_right, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Order.Nucleus | {
"line": 220,
"column": 37
} | {
"line": 220,
"column": 53
} | [
{
"pp": "X : Type u_1\ninst✝ : Frame X\nn✝ m✝ : Nucleus X\nx✝ y✝ : X\nm n : Nucleus X\nx y : X\nthis : Nonempty X\nk : X\nhxyk : k ≥ x ⊓ y\nl : X\nhlx : ∀ x_1 ≥ x, l ⊓ m x_1 ≤ n x_1\nhly : ∀ i ≥ y, l ⊓ m i ≤ n i\nhlk : l ≤ m k\n⊢ n ((x ⊔ k) ⊓ (y ⊔ k)) = n k",
"usedConstants": [
"Eq.mpr",
"Lattic... | ← sup_inf_right, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Order.Nucleus | {
"line": 222,
"column": 6
} | {
"line": 222,
"column": 17
} | [
{
"pp": "X : Type u_1\ninst✝ : Frame X\nn✝ m✝ : Nucleus X\nx y : X\nm n : Nucleus X\n⊢ ∀ (x : X), x ≤ ⨅ y, ⨅ (_ : y ≥ x), m y ⇨ n y",
"usedConstants": [
"Eq.mpr",
"iInf",
"CompleteLattice.toLattice",
"Iff.of_eq",
"congrArg",
"le_himp_iff._simp_1",
"Nucleus",
"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Nucleus | {
"line": 229,
"column": 4
} | {
"line": 230,
"column": 11
} | [
{
"pp": "X : Type u_1\ninst✝ : Frame X\nn✝ m : Nucleus X\nx y : X\nx✝¹ n x✝ : Nucleus X\n⊢ x✝¹ ≤ n ⇨ x✝ ↔ x✝¹ ⊓ n ≤ x✝",
"usedConstants": [
"Eq.mpr",
"Lattice.toSemilatticeSup",
"iInf",
"CompleteLattice.toLattice",
"Iff.of_eq",
"congrArg",
"le_himp_iff._simp_1",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.SaddlePoint | {
"line": 95,
"column": 2
} | {
"line": 97,
"column": 38
} | [
{
"pp": "case refine_3\nE : Type u_1\nF : Type u_2\nβ : Type u_3\nX : Set E\nY : Set F\nf : E → F → β\ninst✝ : CompleteLinearOrder β\na : E\nha : a ∈ X\nb : F\nhb : b ∈ Y\nh : ⨆ y ∈ Y, f a y ≤ ⨅ x ∈ X, f x b\n⊢ ⨅ x ∈ X, f x b = f a b",
"usedConstants": [
"iInf",
"iSup",
"CompletelyDistribL... | · apply le_antisymm
· apply iInf₂_le a ha
· apply le_trans (le_iSup₂ b hb) h | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Order.Partition.Basic | {
"line": 88,
"column": 49
} | {
"line": 88,
"column": 60
} | [
{
"pp": "case mk.mk\nα : Type u_1\ns✝ t x y z : α\nS : Set α\ninst✝ : CompleteLattice α\nP Q : Partition s✝\ns : α\nparts✝¹ : Set α\nsSupIndep'✝¹ : sSupIndep parts✝¹\nbot_notMem'✝¹ : ⊥ ∉ parts✝¹\nsSup_eq'✝¹ : sSup parts✝¹ = s\nparts✝ : Set α\nsSupIndep'✝ : sSupIndep parts✝\nbot_notMem'✝ : ⊥ ∉ parts✝\nsSup_eq'✝ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Partition.Basic | {
"line": 410,
"column": 2
} | {
"line": 410,
"column": 37
} | [
{
"pp": "α : Type u_1\nx : α\nu : Set α\nP : Partition u\nh : x ∈ u\n⊢ (P.partOf x).Nonempty",
"usedConstants": [
"Eq.mpr",
"id",
"_private.Mathlib.Order.Partition.Basic.0.Partition.partOf_nonempty_iff._simp_1_3",
"Partition.partOf",
"Set.Nonempty",
"Set.instEmptyCollecti... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Partition.Basic | {
"line": 549,
"column": 2
} | {
"line": 549,
"column": 13
} | [
{
"pp": "α : Type u_1\nt u : Set α\nP : Partition u\nh : ∀ ⦃x y : α⦄, x ∈ t → y ∈ t → P.Rel x y → x = y\n⊢ ∃ f, P.IsRepFun f ∧ EqOn f id t",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.SuccPred.Tree | {
"line": 70,
"column": 4
} | {
"line": 70,
"column": 22
} | [
{
"pp": "α : Type u_1\ninst✝⁴ : PartialOrder α\ninst✝³ : PredOrder α\ninst✝² : IsPredArchimedean α\ninst✝¹ : OrderBot α\ninst✝ : DecidableEq α\nr : α\nh : Order.pred^[Nat.find ⋯ - 1] r = ⊥\nthis : Nat.find ⋯ = 0\n⊢ r = ⊥",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.SuccPred.Tree | {
"line": 82,
"column": 4
} | {
"line": 82,
"column": 15
} | [
{
"pp": "case right\nα : Type u_1\ninst✝⁴ : PartialOrder α\ninst✝³ : PredOrder α\ninst✝² : IsPredArchimedean α\ninst✝¹ : OrderBot α\ninst✝ : DecidableEq α\nr : α\nhr : r ≠ ⊥\nb : α\nhb : b ≤ Order.pred (findAtom r)\n⊢ b = ⊥",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.SuccPred.Tree | {
"line": 151,
"column": 4
} | {
"line": 151,
"column": 59
} | [
{
"pp": "α : Type u_1\ninst✝² : PartialOrder α\ninst✝¹ : PredOrder α\ninst✝ : IsPredArchimedean α\nt : RootedTree\na₁ a₂ : SubRootedTree t\nh : (fun v ↦ Set.Ici v.root) a₁ = (fun v ↦ Set.Ici v.root) a₂\n⊢ a₁ = a₂",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.SuccPred.Tree | {
"line": 205,
"column": 4
} | {
"line": 205,
"column": 81
} | [
{
"pp": "case inr.h\nt : RootedTree\nt₁ t₂ : SubRootedTree t\nht₁ : t₁ ∈ t.subtrees\nht₂ : t₂ ∈ t.subtrees\nv₁ v₂ : ↑t\nh₁ : t₁.root ≤ v₁\nh₂ : t₂.root ≤ v₂\nh : ⊥ < v₁ ⊓ v₂\noh : t₁.root ≤ v₁ ∧ t₁.root ≤ v₂\n⊢ t₁.root = t₂.root",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Sublocale | {
"line": 83,
"column": 28
} | {
"line": 83,
"column": 39
} | [
{
"pp": "X : Type u_1\ninst✝ : Order.Frame X\nS : Sublocale X\n⊢ ⊤ ∈ S",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Sublocale | {
"line": 212,
"column": 2
} | {
"line": 212,
"column": 77
} | [
{
"pp": "case a\nX : Type u_1\ninst✝ : Order.Frame X\nn : Nucleus X\nx : X\n⊢ ↑(n.toSublocale.restrict x) = ↑⟨n x, ⋯⟩",
"usedConstants": [
"Eq.mpr",
"FrameHom",
"iInf",
"congrArg",
"Nucleus",
"le_iInf_iff._simp_1",
"PartialOrder.toPreorder",
"setOf",
"Su... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Sublocale | {
"line": 213,
"column": 68
} | {
"line": 213,
"column": 79
} | [
{
"pp": "X : Type u_1\ninst✝ : Order.Frame X\nn : Nucleus X\nx y : X\nhxy : x ≤ n y\n⊢ n x ≤ n y",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Sublocale | {
"line": 223,
"column": 27
} | {
"line": 223,
"column": 38
} | [
{
"pp": "case h\nX : Type u_1\ninst✝ : Order.Frame X\nS : Sublocale X\nx : X\n⊢ x ∈ (fun n ↦ (OrderDual.ofDual n).toSublocale) ((fun s ↦ OrderDual.toDual s.toNucleus) S) ↔ x ∈ S",
"usedConstants": [
"OrderDual.toDual",
"Eq.mpr",
"FrameHom",
"Equiv.instEquivLike",
"Sublocale.toN... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Types.Arithmetic | {
"line": 122,
"column": 45
} | {
"line": 122,
"column": 56
} | [
{
"pp": "⊢ card 0 = 0",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Types.Arithmetic | {
"line": 124,
"column": 44
} | {
"line": 124,
"column": 55
} | [
{
"pp": "⊢ card 1 = 1",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Martingale.Centering | {
"line": 100,
"column": 2
} | {
"line": 100,
"column": 36
} | [
{
"pp": "case h\nΩ : Type u_1\nE : Type u_2\nm0 : MeasurableSpace Ω\nμ : Measure Ω\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\nf : ℕ → Ω → E\nℱ : Filtration ℕ m0\ninst✝² : CompleteSpace E\ninst✝¹ : PartialOrder E\ninst✝ : IsOrderedAddMonoid E\nhf : Submartingale f ℱ μ\nω : Ω\nhω : Monotone fun x ↦... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Martingale.Basic | {
"line": 347,
"column": 2
} | {
"line": 347,
"column": 38
} | [
{
"pp": "case h\nΩ : Type u_1\nι : Type u_3\ninst✝⁵ : Preorder ι\nm0 : MeasurableSpace Ω\nμ : Measure Ω\nℱ : Filtration ι m0\nF : Type u_4\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : PartialOrder F\ninst✝² : NormedSpace ℝ F\ninst✝¹ : CompleteSpace F\ninst✝ : IsOrderedModule ℝ F\nf : ι → Ω → F\nc : ℝ\nhc : 0 ≤ c\nh... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Martingale.Basic | {
"line": 393,
"column": 2
} | {
"line": 395,
"column": 62
} | [
{
"pp": "Ω : Type u_1\nm0 : MeasurableSpace Ω\nμ : Measure Ω\n𝒢 : Filtration ℕ m0\ninst✝ : IsFiniteMeasure μ\nf : ℕ → Ω → ℝ\nhadp : StronglyAdapted 𝒢 f\nhint : ∀ (i : ℕ), Integrable (f i) μ\nhf : ∀ (i : ℕ) (s : Set Ω), MeasurableSet s → ∫ (ω : Ω) in s, f i ω ∂μ ≤ ∫ (ω : Ω) in s, f (i + 1) ω ∂μ\ni j : ℕ\nhij :... | induction hij with
| refl => rfl
| step hk₁ hk₂ => exact hk₂.trans (hf _ s (𝒢.mono hk₁ _ hs)) | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.Probability.Process.HittingTime | {
"line": 163,
"column": 2
} | {
"line": 163,
"column": 13
} | [
{
"pp": "case neg\nΩ : Type u_1\nβ : Type u_2\nι : Type u_3\ninst✝ : ConditionallyCompleteLinearOrder ι\nu : ι → Ω → β\ns : Set β\nn m : ι\nhnm : n ≤ m\nω : Ω\nh : ¬∃ j ∈ Set.Icc n m, u j ω ∈ s\n⊢ n ≤ m",
"usedConstants": []
}
] | · exact hnm | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Probability.Martingale.Convergence | {
"line": 150,
"column": 2
} | {
"line": 152,
"column": 99
} | [
{
"pp": "case neg\nΩ : Type u_1\nf : ℕ → Ω → ℝ\nω : Ω\nhf₁ : liminf (fun n ↦ ↑‖f n ω‖₊) atTop < ∞\nhf₂ : ∀ (a b : ℚ), a < b → upcrossings (↑a) (↑b) f ω < ∞\nh : ¬IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) atTop fun n ↦ |f n ω|\n⊢ ∃ c, Tendsto (fun n ↦ f n ω) atTop (𝓝 c)",
"usedConstants": [
"Iff.mpr",
... | · obtain ⟨a, b, hab, h₁, h₂⟩ := ENNReal.exists_upcrossings_of_not_bounded_under hf₁.ne h
exact
False.elim ((hf₂ a b hab).ne (upcrossings_eq_top_of_frequently_lt (Rat.cast_lt.2 hab) h₁ h₂)) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Probability.Martingale.Upcrossing | {
"line": 495,
"column": 2
} | {
"line": 503,
"column": 16
} | [
{
"pp": "Ω : Type u_1\na b : ℝ\nf : ℕ → Ω → ℝ\nN n : ℕ\nω : Ω\nM : ℕ\nhNM : N ≤ M\nh : upperCrossingTime a b f N (n + 1) ω < N\n⊢ upperCrossingTime a b f M (n + 1) ω = upperCrossingTime a b f N (n + 1) ω ∧\n lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω",
"usedConstants": [
"Eq.mpr"... | have := (crossing_eq_crossing_of_lowerCrossingTime_lt hNM
(lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ h)).2
refine ⟨?_, this⟩
rw [upperCrossingTime_succ_eq, upperCrossingTime_succ_eq, eq_comm, this]
refine hittingBtwn_eq_hittingBtwn_of_exists hNM ?_
rw [upperCrossingTime_succ_eq, hittingBtwn... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Martingale.Upcrossing | {
"line": 495,
"column": 2
} | {
"line": 503,
"column": 16
} | [
{
"pp": "Ω : Type u_1\na b : ℝ\nf : ℕ → Ω → ℝ\nN n : ℕ\nω : Ω\nM : ℕ\nhNM : N ≤ M\nh : upperCrossingTime a b f N (n + 1) ω < N\n⊢ upperCrossingTime a b f M (n + 1) ω = upperCrossingTime a b f N (n + 1) ω ∧\n lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω",
"usedConstants": [
"Eq.mpr"... | have := (crossing_eq_crossing_of_lowerCrossingTime_lt hNM
(lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ h)).2
refine ⟨?_, this⟩
rw [upperCrossingTime_succ_eq, upperCrossingTime_succ_eq, eq_comm, this]
refine hittingBtwn_eq_hittingBtwn_of_exists hNM ?_
rw [upperCrossingTime_succ_eq, hittingBtwn... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Process.HittingTime | {
"line": 411,
"column": 4
} | {
"line": 411,
"column": 32
} | [
{
"pp": "case inr\nΩ : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace Ω\ninst✝² : ConditionallyCompleteLinearOrder ι\ninst✝¹ : WellFoundedLT ι\ninst✝ : Countable ι\nx✝ : MeasurableSpace β\nf : Filtration ι m\nu : ι → Ω → β\ns : Set β\nn n' : ι\nhu : Adapted f u\nhs : MeasurableSet s\ni : ι\nhi : i < ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Process.HittingTime | {
"line": 424,
"column": 2
} | {
"line": 424,
"column": 30
} | [
{
"pp": "Ω : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace Ω\ninst✝² : ConditionallyCompleteLinearOrder ι\ninst✝¹ : WellFoundedLT ι\ninst✝ : Countable ι\nx✝ : MeasurableSpace β\nf : Filtration ι m\nu : ι → Ω → β\ns : Set β\nn : ι\nhu : Adapted f u\nhs : MeasurableSet s\ni : ι\nh_set_eq_Union : {ω | ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Process.HittingTime | {
"line": 421,
"column": 2
} | {
"line": 425,
"column": 60
} | [
{
"pp": "Ω : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace Ω\ninst✝² : ConditionallyCompleteLinearOrder ι\ninst✝¹ : WellFoundedLT ι\ninst✝ : Countable ι\nx✝ : MeasurableSpace β\nf : Filtration ι m\nu : ι → Ω → β\ns : Set β\nn : ι\nhu : Adapted f u\nhs : MeasurableSet s\n⊢ IsStoppingTime f (hittingAf... | intro i
have h_set_eq_Union : {ω | hittingAfter u s n ω ≤ i} = ⋃ j ∈ Set.Icc n i, u j ⁻¹' s := by
ext; simp [hittingAfter_le_iff]
simpa [h_set_eq_Union] using MeasurableSet.iUnion fun j =>
MeasurableSet.iUnion fun hj => f.mono hj.2 _ ((hu j) hs) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Process.HittingTime | {
"line": 421,
"column": 2
} | {
"line": 425,
"column": 60
} | [
{
"pp": "Ω : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace Ω\ninst✝² : ConditionallyCompleteLinearOrder ι\ninst✝¹ : WellFoundedLT ι\ninst✝ : Countable ι\nx✝ : MeasurableSpace β\nf : Filtration ι m\nu : ι → Ω → β\ns : Set β\nn : ι\nhu : Adapted f u\nhs : MeasurableSet s\n⊢ IsStoppingTime f (hittingAf... | intro i
have h_set_eq_Union : {ω | hittingAfter u s n ω ≤ i} = ⋃ j ∈ Set.Icc n i, u j ⁻¹' s := by
ext; simp [hittingAfter_le_iff]
simpa [h_set_eq_Union] using MeasurableSet.iUnion fun j =>
MeasurableSet.iUnion fun hj => f.mono hj.2 _ ((hu j) hs) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Martingale.Upcrossing | {
"line": 591,
"column": 6
} | {
"line": 613,
"column": 47
} | [] | ∑ _k ∈ Finset.range (upcrossingsBefore a b f N ω), (b - a) ≤
∑ k ∈ Finset.range (upcrossingsBefore a b f N ω),
(stoppedValue f (fun ω ↦ (upperCrossingTime a b f N (k + 1) ω : ℕ)) ω -
stoppedValue f (fun ω ↦ (lowerCrossingTime a b f N k ω : ℕ)) ω) := by
gcongr ∑ k ∈ _, ?_ with... | Lean.Elab.Tactic._aux_Mathlib_Tactic_Widget_Calc___elabRules_Lean_calcTactic_1 | Lean.calcSteps |
Mathlib.Probability.Martingale.Upcrossing | {
"line": 627,
"column": 8
} | {
"line": 627,
"column": 19
} | [
{
"pp": "case refine_2.h\nΩ : Type u_1\nm0 : MeasurableSpace Ω\nμ : Measure Ω\na b : ℝ\nf : ℕ → Ω → ℝ\nN : ℕ\nℱ : Filtration ℕ m0\ninst✝ : IsFiniteMeasure μ\nhf : Submartingale f ℱ μ\nhfN : ∀ (ω : Ω), a ≤ f N ω\nhfzero : 0 ≤ f 0\nhab : a < b\nω : Ω\n⊢ (b - a) * ↑(upcrossingsBefore a b f N ω) ≤ (∑ k ∈ Finset.ran... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Process.HittingTime | {
"line": 471,
"column": 2
} | {
"line": 471,
"column": 13
} | [
{
"pp": "Ω : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace Ω\ninst✝⁶ : ConditionallyCompleteLinearOrder ι\ninst✝⁵ : WellFoundedLT ι\ninst✝⁴ : Countable ι\ninst✝³ : TopologicalSpace ι\ninst✝² : OrderTopology ι\ninst✝¹ : FirstCountableTopology ι\ninst✝ : MeasurableSpace β\nf : Filtration ι m\nu : ι → ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Process.Stopping | {
"line": 435,
"column": 28
} | {
"line": 435,
"column": 39
} | [
{
"pp": "case h.coe.coe\nΩ : Type u_1\nι : Type u_3\nm : MeasurableSpace Ω\ninst✝⁵ : Add ι\ninst✝⁴ : LinearOrder ι\ninst✝³ : CanonicallyOrderedAdd ι\ninst✝² : Countable ι\ninst✝¹ : TopologicalSpace ι\ninst✝ : OrderTopology ι\nf : Filtration ι m\nτ π : Ω → WithTop ι\nhτ : IsStoppingTime f τ\nhπ : IsStoppingTime ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Process.Stopping | {
"line": 488,
"column": 4
} | {
"line": 488,
"column": 62
} | [
{
"pp": "case h.mp\nΩ : Type u_1\nι : Type u_3\nm : MeasurableSpace Ω\ninst✝ : Preorder ι\nf : Filtration ι m\ni : ι\ns : Set Ω\nh : MeasurableSet s ∧ ∀ (i_1 : ι), MeasurableSet (s ∩ {ω | ↑i ≤ ↑i_1})\nh' : MeasurableSet (s ∩ {ω | ↑i ≤ ↑i})\n⊢ MeasurableSet s",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Process.Stopping | {
"line": 507,
"column": 4
} | {
"line": 507,
"column": 39
} | [
{
"pp": "case mp\nΩ : Type u_1\nι : Type u_3\nm : MeasurableSpace Ω\ninst✝ : Preorder ι\nf : Filtration ι m\nτ : Ω → WithTop ι\nhτ : IsStoppingTime f τ\ns : Set Ω\ni : ι\nthis : ∀ (j : WithTop ι), {ω | τ ω = ↑i} ∩ {ω | τ ω ≤ j} = {ω | τ ω = ↑i} ∩ {_ω | ↑i ≤ j}\nh : MeasurableSet (s ∩ {ω | τ ω = ↑i})\n⊢ Measurab... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Martingale.Upcrossing | {
"line": 759,
"column": 2
} | {
"line": 759,
"column": 50
} | [
{
"pp": "Ω : Type u_1\nm0 : MeasurableSpace Ω\na b : ℝ\nf : ℕ → Ω → ℝ\nN : ℕ\nℱ : Filtration ℕ m0\nhf : StronglyAdapted ℱ f\nhab : a < b\nthis :\n upcrossingsBefore a b f N = fun ω ↦ ∑ i ∈ Finset.Ico 1 (N + 1), {n | upperCrossingTime a b f N n ω < N}.indicator 1 i\ni : ℕ\nx✝ : i ∈ Finset.Ico 1 (N + 1)\n⊢ Measu... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Martingale.Upcrossing | {
"line": 826,
"column": 4
} | {
"line": 828,
"column": 17
} | [
{
"pp": "case neg\nΩ : Type u_1\nm0 : MeasurableSpace Ω\nμ : Measure Ω\nf : ℕ → Ω → ℝ\nℱ : Filtration ℕ m0\ninst✝ : IsFiniteMeasure μ\na b : ℝ\nhf : Submartingale f ℱ μ\nhab : b ≤ a\n⊢ ENNReal.ofReal (b - a) * ∫⁻ (ω : Ω), upcrossings a b f ω ∂μ ≤ ⨆ N, ∫⁻ (ω : Ω), ENNReal.ofReal (f N ω - a)⁺ ∂μ",
"usedConsta... | rw [← sub_nonpos] at hab
rw [ENNReal.ofReal_of_nonpos hab, zero_mul]
exact zero_le | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Martingale.Upcrossing | {
"line": 826,
"column": 4
} | {
"line": 828,
"column": 17
} | [
{
"pp": "case neg\nΩ : Type u_1\nm0 : MeasurableSpace Ω\nμ : Measure Ω\nf : ℕ → Ω → ℝ\nℱ : Filtration ℕ m0\ninst✝ : IsFiniteMeasure μ\na b : ℝ\nhf : Submartingale f ℱ μ\nhab : b ≤ a\n⊢ ENNReal.ofReal (b - a) * ∫⁻ (ω : Ω), upcrossings a b f ω ∂μ ≤ ⨆ N, ∫⁻ (ω : Ω), ENNReal.ofReal (f N ω - a)⁺ ∂μ",
"usedConsta... | rw [← sub_nonpos] at hab
rw [ENNReal.ofReal_of_nonpos hab, zero_mul]
exact zero_le | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.ConditionalExpectation | {
"line": 58,
"column": 6
} | {
"line": 58,
"column": 21
} | [
{
"pp": "case pos.refine_2\nΩ : Type u_1\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\nm₁ m₂ m : MeasurableSpace Ω\nμ : Measure Ω\nf : Ω → E\nhle₁ : m₁ ≤ m\nhle₂ : m₂ ≤ m\ninst✝ : SigmaFinite (μ.trim hle₂)\nhf : StronglyMeasurable f\nhindp : ∀ (t1 t2 : Set Ω),... | Set.inter_comm, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Martingale.BorelCantelli | {
"line": 207,
"column": 4
} | {
"line": 209,
"column": 39
} | [
{
"pp": "case mpr\nΩ : Type u_2\nm0 : MeasurableSpace Ω\nμ : Measure Ω\nℱ : Filtration ℕ m0\nf : ℕ → Ω → ℝ\nR : ℝ≥0\ninst✝ : IsFiniteMeasure μ\nhf : Martingale f ℱ μ\nhbdd : ∀ᵐ (ω : Ω) ∂μ, ∀ (i : ℕ), |f (i + 1) ω - f i ω| ≤ ↑R\nhbdd' : ∀ᵐ (ω : Ω) ∂μ, ∀ (i : ℕ), |(-f) (i + 1) ω - (-f) i ω| ≤ ↑R\nhup : ∀ᵐ (ω : Ω)... | · refine ⟨-c, ?_⟩
convert! hc.neg
simp only [neg_neg, Pi.neg_apply] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Probability.Process.Stopping | {
"line": 737,
"column": 4
} | {
"line": 737,
"column": 15
} | [
{
"pp": "case h\nΩ : Type u_1\nι : Type u_3\nm : MeasurableSpace Ω\ninst✝³ : LinearOrder ι\nf : Filtration ι m\nτ π : Ω → WithTop ι\ninst✝² : TopologicalSpace ι\ninst✝¹ : SecondCountableTopology ι\ninst✝ : OrderTopology ι\nhτ : IsStoppingTime f τ\nhπ : IsStoppingTime f π\nj : ι\nx✝ : Ω\n⊢ x✝ ∈ {ω | τ ω ≤ π ω} ∩... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.ConditionalExpectation | {
"line": 64,
"column": 6
} | {
"line": 64,
"column": 17
} | [
{
"pp": "Ω : Type u_1\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\nm₁ m₂ m : MeasurableSpace Ω\nμ : Measure Ω\nf : Ω → E\nhle₁ : m₁ ≤ m\nhle₂ : m₂ ≤ m\ninst✝ : SigmaFinite (μ.trim hle₂)\nhf : StronglyMeasurable f\nhindp : Indep m₁ m₂ μ\nhfint : MemLp f 1 μ\ns... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.ConditionalExpectation | {
"line": 66,
"column": 6
} | {
"line": 66,
"column": 17
} | [
{
"pp": "Ω : Type u_1\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\nm₁ m₂ m : MeasurableSpace Ω\nμ : Measure Ω\nf : Ω → E\nhle₁ : m₁ ≤ m\nhle₂ : m₂ ≤ m\ninst✝ : SigmaFinite (μ.trim hle₂)\nhf : StronglyMeasurable f\nhindp : Indep m₁ m₂ μ\nhfint : MemLp f 1 μ\ns... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Martingale.OptionalStopping | {
"line": 192,
"column": 8
} | {
"line": 192,
"column": 41
} | [
{
"pp": "case h₂.h.hg\nΩ : Type u_1\nm0 : MeasurableSpace Ω\nμ : Measure Ω\n𝒢 : Filtration ℕ m0\nf : ℕ → Ω → ℝ\ninst✝ : IsFiniteMeasure μ\nhsub : Submartingale f 𝒢 μ\nhnonneg : 0 ≤ f\nε : ℝ≥0\nn : ℕ\n⊢ IntegrableOn (stoppedValue f fun ω ↦ ↑(hittingBtwn f {y | ↑ε ≤ y} 0 n ω))\n {ω | ((range (n + 1)).sup' ⋯ ... | refine Integrable.integrableOn ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Probability.Martingale.BorelCantelli | {
"line": 325,
"column": 2
} | {
"line": 325,
"column": 13
} | [
{
"pp": "case refine_2\nΩ : Type u_2\nm0 : MeasurableSpace Ω\nμ : Measure Ω\nℱ : Filtration ℕ m0\ninst✝ : IsFiniteMeasure μ\ns : ℕ → Set Ω\nhs : ∀ (n : ℕ), MeasurableSet (s n)\nthis :\n ∀ᵐ (ω : Ω) ∂μ,\n Tendsto (fun n ↦ (∑ k ∈ Finset.range n, (s (k + 1)).indicator 1) ω) atTop atTop ↔\n Tendsto (fun n ↦... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Kernel.Disintegration.CDFToKernel | {
"line": 133,
"column": 6
} | {
"line": 133,
"column": 34
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ : Kernel α (β × ℝ)\nν : Kernel α β\nf : α × β → ℚ → ℝ\ninst✝ : IsFiniteKernel κ\nhf : IsRatCondKernelCDF f κ ν\na : α\nx : ℝ\ns : Set β\nhs : MeasurableSet s\nhρ_zero : (ν a).restrict s = 0\nq : ℚ\nhq : x < ↑q\nthis : (κ a) (... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Kernel.Disintegration.CondCDF | {
"line": 96,
"column": 4
} | {
"line": 103,
"column": 45
} | [
{
"pp": "α : Type u_1\nmα : MeasurableSpace α\nρ : Measure (α × ℝ)\ninst✝ : IsFiniteMeasure ρ\ns : Set α\nhs : MeasurableSet s\nh_empty : ρ (s ×ˢ ∅) = 0\nh_neg : Tendsto (fun r ↦ ρ (s ×ˢ Iic ↑(-r))) atTop (𝓝 (ρ (⋂ r, s ×ˢ Iic ↑(-r))))\n⊢ Tendsto (fun r ↦ ρ (s ×ˢ Iic ↑r)) atBot (𝓝 (ρ (⋂ i, s ×ˢ Iic ↑i)))",
... | have h_inter_eq : ⋂ r : ℚ, s ×ˢ Iic ↑(-r) = ⋂ r : ℚ, s ×ˢ Iic (r : ℝ) := by
ext1 x
push _ ∈ _
refine ⟨fun h i ↦ ⟨(h i).1, ?_⟩, fun h i ↦ ⟨(h i).1, ?_⟩⟩ <;> have h' := h (-i)
· rw [neg_neg] at h'; exact h'.2
· exact h'.2
rw [h_inter_eq] at h_neg
exact tendsto_comp_neg_atTop_iff.mp h... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Kernel.Disintegration.CondCDF | {
"line": 96,
"column": 4
} | {
"line": 103,
"column": 45
} | [
{
"pp": "α : Type u_1\nmα : MeasurableSpace α\nρ : Measure (α × ℝ)\ninst✝ : IsFiniteMeasure ρ\ns : Set α\nhs : MeasurableSet s\nh_empty : ρ (s ×ˢ ∅) = 0\nh_neg : Tendsto (fun r ↦ ρ (s ×ˢ Iic ↑(-r))) atTop (𝓝 (ρ (⋂ r, s ×ˢ Iic ↑(-r))))\n⊢ Tendsto (fun r ↦ ρ (s ×ˢ Iic ↑r)) atBot (𝓝 (ρ (⋂ i, s ×ˢ Iic ↑i)))",
... | have h_inter_eq : ⋂ r : ℚ, s ×ˢ Iic ↑(-r) = ⋂ r : ℚ, s ×ˢ Iic (r : ℝ) := by
ext1 x
push _ ∈ _
refine ⟨fun h i ↦ ⟨(h i).1, ?_⟩, fun h i ↦ ⟨(h i).1, ?_⟩⟩ <;> have h' := h (-i)
· rw [neg_neg] at h'; exact h'.2
· exact h'.2
rw [h_inter_eq] at h_neg
exact tendsto_comp_neg_atTop_iff.mp h... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.CDF | {
"line": 78,
"column": 2
} | {
"line": 78,
"column": 80
} | [
{
"pp": "μ : Measure ℝ\ninst✝ : IsProbabilityMeasure μ\nx : ℝ\nh :\n ∫⁻ (a : Unit), ENNReal.ofReal (↑(condCDF ((dirac ()).prod μ) a) x) ∂((dirac ()).prod μ).fst =\n ((dirac ()).prod μ) (univ ×ˢ Iic x)\n⊢ ENNReal.ofReal (↑(cdf μ) x) = μ (Iic x)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Kernel.Disintegration.CondCDF | {
"line": 280,
"column": 2
} | {
"line": 280,
"column": 54
} | [
{
"pp": "case h\nα : Type u_1\nmα : MeasurableSpace α\nρ : Measure (α × ℝ)\ninst✝ : IsFiniteMeasure ρ\nr : ℚ\na : α\nha : ↑(condCDF ρ a) ↑r = (preCDF ρ r a).toReal\nha_le_one : ∀ (r : ℚ), preCDF ρ r a ≤ 1\n⊢ preCDF ρ r a ≠ ∞",
"usedConstants": [
"ProbabilityTheory.preCDF",
"PartialOrder.toPreord... | exact ((ha_le_one r).trans_lt ENNReal.one_lt_top).ne | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Probability.Process.Stopping | {
"line": 1046,
"column": 4
} | {
"line": 1046,
"column": 15
} | [
{
"pp": "case h.h.mpr\nΩ : Type u_1\nι : Type u_3\ninst✝¹ : Nonempty ι\nτ : Ω → WithTop ι\nE : Type u_4\nu : ι → Ω → E\ninst✝ : AddCommMonoid E\ns : Finset ι\ny : Ω\nω i : ι\nhi : ↑i = τ y\nh : ω = (↑i).untopA\nhbdd : ↑i ∈ WithTop.some '' ↑s\n⊢ (↑i).untopA ∈ s ∧ ↑i = ↑(↑i).untopA",
"usedConstants": [
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Kernel.Disintegration.CDFToKernel | {
"line": 326,
"column": 2
} | {
"line": 326,
"column": 37
} | [
{
"pp": "case h\nα : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ : Kernel α (β × ℝ)\nν : Kernel α β\nf : α × β → ℚ → ℝ\nhf : IsRatCondKernelCDFAux f κ ν\ninst✝ : IsFiniteKernel ν\na : α\nq : ℚ\nt : β\nhbdd_below : ∀ (q : ℚ), BddBelow (range fun r ↦ f (a, t) ↑r)\nh_nonneg : ∀ (q : ℚ... | · exact le_ciInf fun r ↦ h_nonneg _ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Probability.Process.Stopping | {
"line": 1364,
"column": 26
} | {
"line": 1364,
"column": 67
} | [
{
"pp": "Ω : Type u_1\nι : Type u_3\nm : MeasurableSpace Ω\ninst✝⁵ : LinearOrder ι\nμ : Measure Ω\nℱ : Filtration ι m\nτ : Ω → WithTop ι\nE : Type u_4\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : CompleteSpace E\nf : Ω → E\ninst✝¹ : SigmaFiniteFiltration μ ℱ\nhτ : IsStoppingTime ℱ τ\nh_cou... | IsStoppingTime.measurableSet_inter_eq_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Process.Stopping | {
"line": 1393,
"column": 26
} | {
"line": 1393,
"column": 67
} | [
{
"pp": "Ω : Type u_1\nι : Type u_3\nm : MeasurableSpace Ω\ninst✝⁸ : LinearOrder ι\nμ : Measure Ω\nℱ : Filtration ι m\nτ : Ω → WithTop ι\nE : Type u_4\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : CompleteSpace E\nf : Ω → E\ninst✝⁴ : TopologicalSpace ι\ninst✝³ : OrderTopology ι\ninst✝² : Fi... | IsStoppingTime.measurableSet_inter_eq_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Moments.ComplexMGF | {
"line": 78,
"column": 2
} | {
"line": 78,
"column": 32
} | [
{
"pp": "Ω : Type u_1\nm : MeasurableSpace Ω\nX : Ω → ℝ\nμ : Measure Ω\nz : ℂ\nhX : AEMeasurable X μ\nh : ¬Integrable (fun ω ↦ rexp (z.re * X ω)) μ\n⊢ ¬Integrable (fun a ↦ ‖cexp (z * ↑(X a))‖) μ",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Real"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Moments.MGFAnalytic | {
"line": 50,
"column": 6
} | {
"line": 50,
"column": 17
} | [
{
"pp": "Ω : Type u_1\nm : MeasurableSpace Ω\nX : Ω → ℝ\nμ : Measure Ω\nt✝ : ℝ\nht : t✝ ∈ interior (integrableExpSet X μ)\nn✝ n : ℕ\nt : ℝ\nht' : t ∈ interior (integrableExpSet X μ)\n⊢ (↑t).re ∈ interior (integrableExpSet X μ)",
"usedConstants": [
"Real",
"PseudoMetricSpace.toUniformSpace",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
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