mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Data.WSeq.Basic

Goal state before:
pp: α : Type u
s : WSeq α
a : α
h : a ∈ s.tail
⊢ a ∈ s

Tactic:
by

Goal state after:
Module: Mathlib.Data.Seq.Parallel

Goal state before:
pp: α : Type u
S✝ : WSeq (Computation α)
c✝ : Computation α
h✝ : c✝ ∈ S✝
T✝ : c✝.Terminates
n : ℕ
IH :
  ∀ (l : List (Computation α)) (S : Stream'.Seq (Option (Computation α))) (c : Computation α),
    c ∈ l ∨ some (some c) = S.get? n → c.Terminates → (corec parallel.aux1 (l, S)).Terminates
l : List (Computation α)
S : Stream'.Seq (Option (Computation α))
c : Computation α
T : c.Terminates
a : some (some c) = S.get? (n + 1)
l' : List (Computation α)
h : parallel.aux2 l = Sum.inr l'
⊢ (corec parallel.aux1 (l, S)).destruct = Sum.inr ?m.247

Tactic:
by

Goal state after:
Module: Mathlib.Data.WSeq.Basic

Goal state before:
pp: α : Type u
s : WSeq α
a : α
n : ℕ
⊢ some a ∈ s.get? n → a ∈ s

Tactic:
by

Goal state after:
Module: Mathlib.Data.WSeq.Relation

Goal state before:
pp: α : Type u
s t : WSeq α
h : s ~ʷ t
⊢ s.tail ~ʷ t.tail

Tactic:
by

Goal state after:
Module: Mathlib.Data.WSeq.Relation

Goal state before:
pp: α : Type u
s t : WSeq α
h : s ~ʷ t
n : ℕ
⊢ s.drop n ~ʷ t.drop n

Tactic:
by

Goal state after:
Module: Mathlib.Data.WSeq.Basic

Goal state before:
pp: α : Type u
s : WSeq α
a : α
h : a ∈ s
⊢ ∃ n, some a ∈ s.get? n

Tactic:
by

Goal state after:
Module: Mathlib.Data.Seq.Parallel

Goal state before:
pp: α : Type u
S✝ : WSeq (Computation α)
c✝ : Computation α
h✝ : c✝ ∈ S✝
T✝ : c✝.Terminates
n : ℕ
IH :
  ∀ (l : List (Computation α)) (S : Stream'.Seq (Option (Computation α))) (c : Computation α),
    c ∈ l ∨ some (some c) = S.get? n → c.Terminates → (corec parallel.aux1 (l, S)).Terminates
l : List (Computation α)
S : Stream'.Seq (Option (Computation α))
c : Computation α
T : c.Terminates
a : some (some c) = S.get? (n + 1)
l' : List (Computation α)
h : parallel.aux2 l = Sum.inr l'
C :
  corec parallel.aux1 (l, S) =
    (corec parallel.aux1
        (match S.destruct with
        | none => (l', Seq.nil)
        | some (none, S') => (l', S')
        | some (some c, S') => (c :: l', S'))).think
⊢ ∀ (l' : List (Computation α)), (corec parallel.aux1 (l', S.tail)).Terminates

Tactic:
by

Goal state after:
Module: Mathlib.Data.WSeq.Basic

Goal state before:
pp: α : Type u
s : WSeq α
a : α
h✝ : a ∈ s
n : ℕ
h : some a ∈ s.get? n
⊢ ∃ s', some (a, s') ∈ (s.drop n).destruct

Tactic:
by

Goal state after:
Module: Mathlib.Data.WSeq.Basic

Goal state before:
pp: α : Type u
β : Type v
f : α → β
b : β
g : Stream' (Option (Option α))
al : g.IsSeq
h : b ∈ map f ⟨g, al⟩
⊢ ∃ a ∈ ⟨g, al⟩, f a = b

Tactic:
by

Goal state after:
Module: Mathlib.Data.WSeq.Basic

Goal state before:
pp: α : Type u
l : List α
s : WSeq α
a : α
⊢ (Computation.corec
        (fun x ↦
          match x with
          | (l, s) =>
            match Seq.destruct s with
            | none => Sum.inl l.reverse
            | some (none, s') => Sum.inr (l, s')
            | some (some a, s') => Sum.inr (a :: l, s'))
        (l, cons a s)).destruct =
    Sum.inr
      (Computation.corec
        (fun x ↦
          match x with
          | (l, s) =>
            match Seq.destruct s with
            | none => Sum.inl l.reverse
            | some (none, s') => Sum.inr (l, s')
            | some (some a, s') => Sum.inr (a :: l, s'))
        (a :: l, s))

Tactic:
by

Goal state after:
Module: Mathlib.Data.WSeq.Basic

Goal state before:
pp: α : Type u
l : List α
s : WSeq α
⊢ (Computation.corec
        (fun x ↦
          match x with
          | (l, s) =>
            match Seq.destruct s with
            | none => Sum.inl l.reverse
            | some (none, s') => Sum.inr (l, s')
            | some (some a, s') => Sum.inr (a :: l, s'))
        (l, s.think)).destruct =
    Sum.inr
      (Computation.corec
        (fun x ↦
          match x with
          | (l, s) =>
            match Seq.destruct s with
            | none => Sum.inl l.reverse
            | some (none, s') => Sum.inr (l, s')
            | some (some a, s') => Sum.inr (a :: l, s'))
        (l, s))

Tactic:
by

Goal state after:
Module: Mathlib.Data.Seq.Parallel

Goal state before:
pp: α : Type u
S✝ : WSeq (Computation α)
c✝ : Computation α
h✝ : c✝ ∈ S✝
T✝ : c✝.Terminates
n : ℕ
IH :
  ∀ (l : List (Computation α)) (S : Stream'.Seq (Option (Computation α))) (c : Computation α),
    c ∈ l ∨ some (some c) = S.get? n → c.Terminates → (corec parallel.aux1 (l, S)).Terminates
l : List (Computation α)
S : Stream'.Seq (Option (Computation α))
c : Computation α
T : c.Terminates
a : some (some c) = S.get? (n + 1)
l' : List (Computation α)
h : parallel.aux2 l = Sum.inr l'
C :
  corec parallel.aux1 (l, S) =
    (corec parallel.aux1
        (match S.destruct with
        | none => (l', Seq.nil)
        | some (none, S') => (l', S')
        | some (some c, S') => (c :: l', S'))).think
TT : ∀ (l' : List (Computation α)), (corec parallel.aux1 (l', S.tail)).Terminates
o : Option (Computation α)
e : S.get? 0 = some o
⊢ S.destruct = some (o, S.tail)

Tactic:
by

Goal state after:
Module: Mathlib.Data.WSeq.Relation

Goal state before:
pp: α : Type u
s✝ t✝ : WSeq α
h✝ : ∀ (n : ℕ), s✝.get? n ~ t✝.get? n
s t : WSeq α
h : (fun s t ↦ ∀ (n : ℕ), s.get? n ~ t.get? n) s t
⊢ Computation.LiftRel (LiftRelO Eq fun s t ↦ ∀ (n : ℕ), s.get? n ~ t.get? n) s.destruct t.destruct

Tactic:
by

Goal state after:
Module: Mathlib.Data.Setoid.Partition.Card

Goal state before:
pp: α : Type u_1
P : Set (Set α)
hP : IsPartition P
s : Set α
hs : s.Finite
hst : ∀ (t : Set α), (s ∩ t).Finite
hst' : ∀ (t : Set α), Nat.card ↑(s ∩ t) = ⋯.toFinset.card
hs' : Function.HasFiniteSupport fun t ↦ (s ∩ ↑t).ncard
t : ↑P
ht : t ∈ ⋯.toFinset
⊢ Nat.card ↑(s ∩ ↑t) = ?m.54 t

Tactic:
by

Goal state after:
Module: Mathlib.Data.Seq.Parallel

Goal state before:
pp: α : Type u
S : WSeq (Computation α)
c : Computation α
h : c ∈ S
T : c.Terminates
⊢ (parallel S).Terminates

Tactic:
by

Goal state after:
Module: Mathlib.Data.Seq.Parallel

Goal state before:
pp: α : Type u
S : WSeq (Computation α)
a : α
h : a ∈ parallel S
⊢ List (Computation α) → α ⊕ List (Computation α) → Prop

Tactic:
by

Goal state after:
Module: Mathlib.Data.WSeq.Relation

Goal state before:
pp: α : Type u
β : Type v
γ : Type w
δ : Type u_1
R : α → β → Prop
S : γ → δ → Prop
s1✝ : WSeq α
s2✝ : WSeq β
f1 : α → γ
f2 : β → δ
h1 : LiftRel R s1✝ s2✝
h2 : ∀ {a : α} {b : β}, R a b → S (f1 a) (f2 b)
s1 : WSeq γ
s2 : WSeq δ
h✝ : (fun s1 s2 ↦ ∃ s t, s1 = map f1 s ∧ s2 = map f2 t ∧ LiftRel R s t) s1 s2
s : WSeq α
t : WSeq β
h : LiftRel R s t
⊢ Computation.LiftRel (LiftRelO S fun s1 s2 ↦ ∃ s t, s1 = map f1 s ∧ s2 = map f2 t ∧ LiftRel R s t) (map f1 s).destruct
    (map f2 t).destruct

Tactic:
by

Goal state after:
Module: Mathlib.Data.Seq.Parallel

Goal state before:
pp: case inl
α : Type u
S : WSeq (Computation α)
a : α
h : a ∈ parallel S
F : List (Computation α) → α ⊕ List (Computation α) → Prop :=
  fun l a ↦ Sum.casesOn a (fun a ↦ ∃ c ∈ l, a ∈ c) fun l' ↦ ∀ (a' : α), (∃ c ∈ l', a' ∈ c) → ∃ c ∈ l, a' ∈ c
c : Computation α
l : List (Computation α)
val✝ : α
IH : F l (Sum.inl val✝)
⊢ F (c :: l)
    (match Sum.inl val✝ with
    | Sum.inl a => Sum.inl a
    | Sum.inr ls => rmap (fun c' ↦ c' :: ls) c.destruct)

Tactic:
simp only

Goal state after:
case inl
α : Type u
S : WSeq (Computation α)
a : α
h : a ∈ parallel S
F : List (Computation α) → α ⊕ List (Computation α) → Prop :=
  fun l a ↦ Sum.casesOn a (fun a ↦ ∃ c ∈ l, a ∈ c) fun l' ↦ ∀ (a' : α), (∃ c ∈ l', a' ∈ c) → ∃ c ∈ l, a' ∈ c
c : Computation α
l : List (Computation α)
val✝ : α
IH : F l (Sum.inl val✝)
⊢ F (c :: l) (Sum.inl val✝)
Module: Mathlib.Data.Seq.Parallel

Goal state before:
pp: case inr
α : Type u
S : WSeq (Computation α)
a : α
h : a ∈ parallel S
F : List (Computation α) → α ⊕ List (Computation α) → Prop :=
  fun l a ↦ Sum.casesOn a (fun a ↦ ∃ c ∈ l, a ∈ c) fun l' ↦ ∀ (a' : α), (∃ c ∈ l', a' ∈ c) → ∃ c ∈ l, a' ∈ c
c : Computation α
l val✝ : List (Computation α)
IH : F l (Sum.inr val✝)
⊢ F (c :: l)
    (match Sum.inr val✝ with
    | Sum.inl a => Sum.inl a
    | Sum.inr ls => rmap (fun c' ↦ c' :: ls) c.destruct)

Tactic:
simp only

Goal state after:
case inr
α : Type u
S : WSeq (Computation α)
a : α
h : a ∈ parallel S
F : List (Computation α) → α ⊕ List (Computation α) → Prop :=
  fun l a ↦ Sum.casesOn a (fun a ↦ ∃ c ∈ l, a ∈ c) fun l' ↦ ∀ (a' : α), (∃ c ∈ l', a' ∈ c) → ∃ c ∈ l, a' ∈ c
c : Computation α
l val✝ : List (Computation α)
IH : F l (Sum.inr val✝)
⊢ F (c :: l) (rmap (fun c' ↦ c' :: val✝) c.destruct)
Module: Mathlib.Data.WSeq.Relation

Goal state before:
pp: α : Type u
β : Type v
R : α → β → Prop
s1 s2 : WSeq α
t1 t2 : WSeq β
h1 : LiftRel R s1 t1
h2 : LiftRel R s2 t2
s✝ : WSeq α
t✝ : WSeq β
h✝ : (fun s t ↦ LiftRel R s t ∨ ∃ s1 t1, s = s1.append s2 ∧ t = t1.append t2 ∧ LiftRel R s1 t1) s✝ t✝
s : WSeq α
t : WSeq β
h : LiftRel R s t
⊢ Computation.LiftRel
    (LiftRelO R fun s t ↦ LiftRel R s t ∨ ∃ s1 t1, s = s1.append s2 ∧ t = t1.append t2 ∧ LiftRel R s1 t1) s.destruct
    t.destruct

Tactic:
by

Goal state after:
Module: Mathlib.Data.Setoid.Partition.Card

Goal state before:
pp: α : Type u_1
P : Set (Set α)
hP : IsPartition P
s : Set α
hs : s.Finite
hst : ∀ (t : Set α), (s ∩ t).Finite
hst' : ∀ (t : Set α), Nat.card ↑(s ∩ t) = ⋯.toFinset.card
hs' : ?m.31
⊢ s.ncard = ∑ᶠ (t : ↑P), (s ∩ ↑t).ncard

Tactic:
by

Goal state after:
Module: Mathlib.Data.Setoid.Partition.Card

Goal state before:
pp: α : Type u_1
P : Set (Set α)
hP : IsPartition P
s : Set α
hs : s.Finite
⊢ s.ncard = ∑ᶠ (t : ↑P), (s ∩ ↑t).ncard

Tactic:
by

Goal state after:
Module: Mathlib.Data.Sigma.Order

Goal state before:
pp: ι : Type u_1
α : ι → Type u_2
inst✝ : (i : ι) → LE (α i)
a b : (i : ι) × α i
⊢ a ≤ b ↔ ∃ h, h ▸ a.snd ≤ b.snd

Tactic:
by

Goal state after:
Module: Mathlib.Data.Sigma.Order

Goal state before:
pp: ι : Type u_1
α : ι → Type u_2
inst✝ : (i : ι) → LT (α i)
a b : (i : ι) × α i
⊢ a < b ↔ ∃ h, h ▸ a.snd < b.snd

Tactic:
by

Goal state after:
Module: Mathlib.Data.Sigma.Order

Goal state before:
pp: ι : Type u_1
α : ι → Type u_2
inst✝ : (i : ι) → Preorder (α i)
⊢ ∀ (a b c : (i : ι) × α i), a ≤ b → b ≤ c → a ≤ c

Tactic:
by

Goal state after:
Module: Mathlib.Data.Sigma.Order

Goal state before:
pp: ι : Type u_1
α : ι → Type u_2
inst✝ : (i : ι) → Preorder (α i)
x✝¹ x✝ : (i : ι) × α i
⊢ x✝¹ < x✝ ↔ x✝¹ ≤ x✝ ∧ ¬x✝ ≤ x✝¹

Tactic:
by

Goal state after:
Module: Mathlib.Data.Sigma.Order

Goal state before:
pp: ι : Type u_1
α : ι → Type u_2
inst✝ : (i : ι) → PartialOrder (α i)
⊢ ∀ (a b : (i : ι) × α i), a ≤ b → b ≤ a → a = b

Tactic:
by

Goal state after:
Module: Mathlib.Data.Sigma.Order

Goal state before:
pp: ι : Type u_1
α : ι → Type u_2
inst✝¹ : (i : ι) → Preorder (α i)
inst✝ : ∀ (i : ι), DenselyOrdered (α i)
⊢ ∀ (a₁ a₂ : (i : ι) × α i), a₁ < a₂ → ∃ a, a₁ < a ∧ a < a₂

Tactic:
by

Goal state after:
Module: Mathlib.Data.Sigma.Order

Goal state before:
pp: ι : Type u_1
α : ι → Type u_2
inst✝¹ : Preorder ι
inst✝ : (i : ι) → Preorder (α i)
⊢ ∀ (a b : Σₗ (i : ι), α i), a < b ↔ a ≤ b ∧ ¬b ≤ a

Tactic:
by

Goal state after:
Module: Mathlib.Data.WSeq.Basic

Goal state before:
pp: α : Type u
l : List α
s : WSeq α
⊢ Computation.corec
      (fun x ↦
        match x with
        | (l, s) =>
          match Seq.destruct s with
          | none => Sum.inl l.reverse
          | some (none, s') => Sum.inr (l, s')
          | some (some a, s') => Sum.inr (a :: l, s'))
      (l, s) =
    (fun x ↦ l.reverse ++ x) <$> s.toList

Tactic:
by

Goal state after:
Module: Mathlib.Data.Seq.Parallel

Goal state before:
pp: α : Type u
S : WSeq (Computation α)
a : α
h : a ∈ parallel S
F : List (Computation α) → α ⊕ List (Computation α) → Prop :=
  fun l a ↦ Sum.casesOn a (fun a ↦ ∃ c ∈ l, a ∈ c) fun l' ↦ ∀ (a' : α), (∃ c ∈ l', a' ∈ c) → ∃ c ∈ l, a' ∈ c
l : List (Computation α)
⊢ F l (parallel.aux2 l)

Tactic:
by

Goal state after:
Module: Mathlib.Data.Sigma.Order

Goal state before:
pp: ι : Type u_1
α : ι → Type u_2
inst✝³ : PartialOrder ι
inst✝² : OrderBot ι
inst✝¹ : (i : ι) → Preorder (α i)
inst✝ : OrderBot (α ⊥)
x✝ : Σₗ (i : ι), α i
a : ι
b : α a
⊢ ⟨⊥, ⊥⟩ ≤ ⟨a, b⟩

Tactic:
by

Goal state after:
Module: Mathlib.Data.WSeq.Basic

Goal state before:
pp: α : Type u
a : α
s : WSeq α
⊢ (cons a s).toList.destruct = Sum.inr (List.cons a <$> s.toList)

Tactic:
by

Goal state after:
Module: Mathlib.Data.WSeq.Basic

Goal state before:
pp: α : Type u
l : List α
⊢ l ∈ (↑l).toList

Tactic:
by

Goal state after:
Module: Mathlib.Data.Sigma.Order

Goal state before:
pp: ι : Type u_1
α : ι → Type u_2
inst✝³ : PartialOrder ι
inst✝² : OrderTop ι
inst✝¹ : (i : ι) → Preorder (α i)
inst✝ : OrderTop (α ⊤)
x✝ : Σₗ (i : ι), α i
a : ι
b : α a
⊢ ⟨a, b⟩ ≤ ⟨⊤, ⊤⟩

Tactic:
by

Goal state after:
Module: Mathlib.Data.Sigma.Order

Goal state before:
pp: ι : Type u_1
α : ι → Type u_2
inst✝⁴ : Preorder ι
inst✝³ : DenselyOrdered ι
inst✝² : ∀ (i : ι), Nonempty (α i)
inst✝¹ : (i : ι) → Preorder (α i)
inst✝ : ∀ (i : ι), DenselyOrdered (α i)
⊢ ∀ (a₁ a₂ : Σₗ (i : ι), α i), a₁ < a₂ → ∃ a, a₁ < a ∧ a < a₂

Tactic:
by

Goal state after:
Module: Mathlib.Data.Sigma.Order

Goal state before:
pp: ι : Type u_1
α : ι → Type u_2
inst✝³ : Preorder ι
inst✝² : (i : ι) → Preorder (α i)
inst✝¹ : ∀ (i : ι), DenselyOrdered (α i)
inst✝ : ∀ (i : ι), NoMaxOrder (α i)
⊢ ∀ (a₁ a₂ : Σₗ (i : ι), α i), a₁ < a₂ → ∃ a, a₁ < a ∧ a < a₂

Tactic:
by

Goal state after:
Module: Mathlib.Data.WSeq.Basic

Goal state before:
pp: α : Type u
s : Seq α
⊢ (↑s).destruct.destruct = Sum.inl (Option.map (fun a ↦ (a, ↑s.tail)) s.head)

Tactic:
by

Goal state after:
Module: Mathlib.Data.WSeq.Basic

Goal state before:
pp: α : Type u
s : Seq α
⊢ (↑s).head = Computation.pure s.head

Tactic:
by

Goal state after:
Module: Mathlib.Data.Sigma.Order

Goal state before:
pp: ι : Type u_1
α : ι → Type u_2
inst✝³ : Preorder ι
inst✝² : (i : ι) → Preorder (α i)
inst✝¹ : ∀ (i : ι), DenselyOrdered (α i)
inst✝ : ∀ (i : ι), NoMinOrder (α i)
⊢ ∀ (a₁ a₂ : Σₗ (i : ι), α i), a₁ < a₂ → ∃ a, a₁ < a ∧ a < a₂

Tactic:
by

Goal state after:
Module: Mathlib.Data.WSeq.Basic

Goal state before:
pp: α : Type u
s : Seq α
⊢ (↑s).tail = ↑s.tail

Tactic:
by

Goal state after:
Module: Mathlib.Data.WSeq.Basic

Goal state before:
pp: α : Type u
s : Seq α
n : ℕ
⊢ (↑s).drop (n + 1) = ↑(s.drop (n + 1))

Tactic:
by

Goal state after:
Module: Mathlib.Data.Sigma.Order

Goal state before:
pp: ι : Type u_1
α : ι → Type u_2
inst✝³ : Preorder ι
inst✝² : (i : ι) → Preorder (α i)
inst✝¹ : NoMaxOrder ι
inst✝ : ∀ (i : ι), Nonempty (α i)
⊢ ∀ (a : Σₗ (i : ι), α i), ∃ b, a < b

Tactic:
by

Goal state after:
Module: Mathlib.Data.WSeq.Basic

Goal state before:
pp: α : Type u
s : Seq α
n : ℕ
⊢ (↑s).get? n = Computation.pure (s.get? n)

Tactic:
by

Goal state after:
Module: Mathlib.Data.WSeq.Basic

Goal state before:
pp: α : Type u
s : WSeq α
⊢ map id s = s

Tactic:
by

Goal state after:
Module: Mathlib.Data.WSeq.Basic

Goal state before:
pp: α : Type u
β : Type v
f : α → β
a : α
⊢ map f (ret a) = ret (f a)

Tactic:
by

Goal state after:
Module: Mathlib.Data.Sigma.Order

Goal state before:
pp: ι : Type u_1
α : ι → Type u_2
inst✝³ : Preorder ι
inst✝² : (i : ι) → Preorder (α i)
inst✝¹ : NoMinOrder ι
inst✝ : ∀ (i : ι), Nonempty (α i)
⊢ ∀ (a : Σₗ (i : ι), α i), ∃ b, b < a

Tactic:
by

Goal state after:
Module: Mathlib.Data.Sigma.Order

Goal state before:
pp: ι : Type u_1
α : ι → Type u_2
inst✝² : Preorder ι
inst✝¹ : (i : ι) → Preorder (α i)
inst✝ : ∀ (i : ι), NoMaxOrder (α i)
⊢ ∀ (a : Σₗ (i : ι), α i), ∃ b, a < b

Tactic:
by

Goal state after:
Module: Mathlib.Data.WSeq.Basic

Goal state before:
pp: α : Type u
β : Type v
γ : Type w
f : α → β
g : β → γ
s : WSeq α
⊢ map (g ∘ f) s = map g (map f s)

Tactic:
by

Goal state after:
Module: Mathlib.Data.Sigma.Order

Goal state before:
pp: ι : Type u_1
α : ι → Type u_2
inst✝² : Preorder ι
inst✝¹ : (i : ι) → Preorder (α i)
inst✝ : ∀ (i : ι), NoMinOrder (α i)
⊢ ∀ (a : Σₗ (i : ι), α i), ∃ b, b < a

Tactic:
by

Goal state after:
Module: Mathlib.Data.WSeq.Basic

Goal state before:
pp: α : Type u
a : α
this :
  ∀ (ss : WSeq α),
    a ∈ ss → ∀ (s : WSeq α) (S : WSeq (WSeq α)), s.append S.join = ss → a ∈ s.append S.join → a ∈ s ∨ ∃ s ∈ S, a ∈ s
S : WSeq (WSeq α)
h : a ∈ S.join
⊢ nil.append S.join = S.join

Tactic:
by

Goal state after:
Module: Mathlib.Data.WSeq.Basic

Goal state before:
pp: α : Type u
a : α
this :
  ∀ (ss : WSeq α),
    a ∈ ss → ∀ (s : WSeq α) (S : WSeq (WSeq α)), s.append S.join = ss → a ∈ s.append S.join → a ∈ s ∨ ∃ s ∈ S, a ∈ s
S : WSeq (WSeq α)
h : a ∈ S.join
⊢ a ∈ nil.append S.join

Tactic:
by

Goal state after:
Module: Mathlib.Data.Sigma.Interval

Goal state before:
pp: ι : Type u_1
α : ι → Type u_2
inst✝² : DecidableEq ι
inst✝¹ : (i : ι) → Preorder (α i)
inst✝ : (i : ι) → LocallyFiniteOrder (α i)
x✝⁴ x✝³ x✝² : (i : ι) × α i
i : ι
a : α i
j : ι
b : α j
k : ι
c : α k
x✝¹ : i = k
x✝ : ⋯ ▸ a ≤ c
⊢ (∃ (x : j = k), c ≤ ⋯ ▸ b) ↔ ∃ (h : k = j), ⋯ ▸ c ≤ b

Tactic:
by

Goal state after:
Module: Mathlib.Data.Sigma.Interval

Goal state before:
pp: ι : Type u_1
α : ι → Type u_2
inst✝² : DecidableEq ι
inst✝¹ : (i : ι) → Preorder (α i)
inst✝ : (i : ι) → LocallyFiniteOrder (α i)
x✝² x✝¹ x✝ : (i : ι) × α i
i : ι
a : α i
j : ι
b : α j
k : ι
c : α k
⊢ ⟨k, c⟩ ∈ sigmaLift (fun x ↦ Icc) ⟨i, a⟩ ⟨j, b⟩ ↔ ⟨i, a⟩ ≤ ⟨k, c⟩ ∧ ⟨k, c⟩ ≤ ⟨j, b⟩

Tactic:
by

Goal state after:
Module: Mathlib.Data.WSeq.Relation

Goal state before:
pp: α : Type u
β : Type v
R : α → β → Prop
s1✝ s2 : WSeq α
t1✝ t2 : WSeq β
h1 : LiftRel R s1✝ t1✝
h2 : LiftRel R s2 t2
s : WSeq α
t : WSeq β
h✝ : (fun s t ↦ LiftRel R s t ∨ ∃ s1 t1, s = s1.append s2 ∧ t = t1.append t2 ∧ LiftRel R s1 t1) s t
s1 : WSeq α
t1 : WSeq β
h : LiftRel R s1 t1
⊢ Computation.LiftRel
    (LiftRelO R fun s t ↦ LiftRel R s t ∨ ∃ s1 t1, s = s1.append s2 ∧ t = t1.append t2 ∧ LiftRel R s1 t1)
    (s1.append s2).destruct (t1.append t2).destruct

Tactic:
by

Goal state after:
Module: Mathlib.Data.Sigma.Interval

Goal state before:
pp: ι : Type u_1
α : ι → Type u_2
inst✝² : DecidableEq ι
inst✝¹ : (i : ι) → Preorder (α i)
inst✝ : (i : ι) → LocallyFiniteOrder (α i)
x✝⁴ x✝³ x✝² : (i : ι) × α i
i : ι
a : α i
j : ι
b : α j
k : ι
c : α k
x✝¹ : i = k
x✝ : ⋯ ▸ a ≤ c
⊢ (∃ (x : j = k), c < ⋯ ▸ b) ↔ ∃ (h : k = j), ⋯ ▸ c < b

Tactic:
by

Goal state after:
Module: Mathlib.Data.Sigma.Interval

Goal state before:
pp: ι : Type u_1
α : ι → Type u_2
inst✝² : DecidableEq ι
inst✝¹ : (i : ι) → Preorder (α i)
inst✝ : (i : ι) → LocallyFiniteOrder (α i)
x✝² x✝¹ x✝ : (i : ι) × α i
i : ι
a : α i
j : ι
b : α j
k : ι
c : α k
⊢ ⟨k, c⟩ ∈ sigmaLift (fun x ↦ Ico) ⟨i, a⟩ ⟨j, b⟩ ↔ ⟨i, a⟩ ≤ ⟨k, c⟩ ∧ ⟨k, c⟩ < ⟨j, b⟩

Tactic:
by

Goal state after:
Module: Mathlib.Data.Sigma.Interval

Goal state before:
pp: ι : Type u_1
α : ι → Type u_2
inst✝² : DecidableEq ι
inst✝¹ : (i : ι) → Preorder (α i)
inst✝ : (i : ι) → LocallyFiniteOrder (α i)
x✝⁴ x✝³ x✝² : (i : ι) × α i
i : ι
a : α i
j : ι
b : α j
k : ι
c : α k
x✝¹ : i = k
x✝ : ⋯ ▸ a < c
⊢ (∃ (x : j = k), c ≤ ⋯ ▸ b) ↔ ∃ (h : k = j), ⋯ ▸ c ≤ b

Tactic:
by

Goal state after:
Module: Mathlib.Data.Sigma.Interval

Goal state before:
pp: ι : Type u_1
α : ι → Type u_2
inst✝² : DecidableEq ι
inst✝¹ : (i : ι) → Preorder (α i)
inst✝ : (i : ι) → LocallyFiniteOrder (α i)
x✝² x✝¹ x✝ : (i : ι) × α i
i : ι
a : α i
j : ι
b : α j
k : ι
c : α k
⊢ ⟨k, c⟩ ∈ sigmaLift (fun x ↦ Ioc) ⟨i, a⟩ ⟨j, b⟩ ↔ ⟨i, a⟩ < ⟨k, c⟩ ∧ ⟨k, c⟩ ≤ ⟨j, b⟩

Tactic:
by

Goal state after:
Module: Mathlib.Data.Sigma.Interval

Goal state before:
pp: ι : Type u_1
α : ι → Type u_2
inst✝² : DecidableEq ι
inst✝¹ : (i : ι) → Preorder (α i)
inst✝ : (i : ι) → LocallyFiniteOrder (α i)
x✝⁴ x✝³ x✝² : (i : ι) × α i
i : ι
a : α i
j : ι
b : α j
k : ι
c : α k
x✝¹ : i = k
x✝ : ⋯ ▸ a < c
⊢ (∃ (x : j = k), c < ⋯ ▸ b) ↔ ∃ (h : k = j), ⋯ ▸ c < b

Tactic:
by

Goal state after:
Module: Mathlib.Data.Sigma.Interval

Goal state before:
pp: ι : Type u_1
α : ι → Type u_2
inst✝² : DecidableEq ι
inst✝¹ : (i : ι) → Preorder (α i)
inst✝ : (i : ι) → LocallyFiniteOrder (α i)
x✝² x✝¹ x✝ : (i : ι) × α i
i : ι
a : α i
j : ι
b : α j
k : ι
c : α k
⊢ ⟨k, c⟩ ∈ sigmaLift (fun x ↦ Ioo) ⟨i, a⟩ ⟨j, b⟩ ↔ ⟨i, a⟩ < ⟨k, c⟩ ∧ ⟨k, c⟩ < ⟨j, b⟩

Tactic:
by

Goal state after:
Module: Mathlib.Data.WSeq.Relation

Goal state before:
pp: α : Type u
β : Type v
R : α → β → Prop
U : WSeq α → WSeq β → Prop
HU :
  ∀ (s1 : WSeq α) (s2 : WSeq β),
    (∃ s t S T, s1 = s.append S.join ∧ s2 = t.append T.join ∧ LiftRel R s t ∧ LiftRel (LiftRel R) S T) → U s1 s2
n : ℕ
IH :
  ∀ m < n,
    ∀ {S : WSeq (WSeq α)} {T : WSeq (WSeq β)},
      LiftRel (LiftRel R) S T →
        ∀ {a : Option (α × WSeq α)}, S.join.destruct.Results a m → ∃ b ∈ T.join.destruct, LiftRelO R U a b
S : WSeq (WSeq α)
T : WSeq (WSeq β)
ST : LiftRel (LiftRel R) S T
a : Option (α × WSeq α)
ra : (S.destruct.bind destruct_join.aux).Results a n
o : Option (WSeq α × WSeq (WSeq α))
m k : ℕ
rs1 : S.destruct.Results o m
rs2✝ : (destruct_join.aux o).Results a k
en : n = k + m
p : Option (WSeq β × WSeq (WSeq β))
mT✝ : p ∈ T.destruct
rop : LiftRelO (LiftRel R) (LiftRel (LiftRel R)) o p
x✝¹ : LiftRelO (LiftRel R) (LiftRel (LiftRel R)) none none
x✝ : S.destruct.Results none m
rs2 : (destruct_join.aux none).Results a k
mT : none ∈ T.destruct
⊢ LiftRelO R U a none

Tactic:
by

Goal state after:
Module: Mathlib.Data.Sigma.Interval

Goal state before:
pp: ι : Type u_1
α : ι → Type u_2
inst✝¹ : (i : ι) → Preorder (α i)
inst✝ : (i : ι) → LocallyFiniteOrderBot (α i)
x✝¹ x✝ : (i : ι) × α i
i : ι
a : α i
j : ι
b : α j
⊢ (⟨j, b⟩ ∈
      match ⟨i, a⟩ with
      | ⟨i, a⟩ => Finset.map (Embedding.sigmaMk i) (Iic a)) ↔
    ⟨j, b⟩ ≤ ⟨i, a⟩

Tactic:
by

Goal state after:
Module: Mathlib.Data.WSeq.Relation

Goal state before:
pp: α : Type u
β : Type v
R : α → β → Prop
U : WSeq α → WSeq β → Prop
HU :
  ∀ (s1 : WSeq α) (s2 : WSeq β),
    (∃ s t S T, s1 = s.append S.join ∧ s2 = t.append T.join ∧ LiftRel R s t ∧ LiftRel (LiftRel R) S T) → U s1 s2
n : ℕ
IH :
  ∀ m < n,
    ∀ {S : WSeq (WSeq α)} {T : WSeq (WSeq β)},
      LiftRel (LiftRel R) S T →
        ∀ {a : Option (α × WSeq α)}, S.join.destruct.Results a m → ∃ b ∈ T.join.destruct, LiftRelO R U a b
S : WSeq (WSeq α)
T : WSeq (WSeq β)
ST : LiftRel (LiftRel R) S T
a : Option (α × WSeq α)
ra : (S.destruct.bind destruct_join.aux).Results a n
o : Option (WSeq α × WSeq (WSeq α))
m k : ℕ
rs1 : S.destruct.Results o m
rs2✝ : (destruct_join.aux o).Results a k
en : n = k + m
p : Option (WSeq β × WSeq (WSeq β))
mT✝ : p ∈ T.destruct
rop : LiftRelO (LiftRel R) (LiftRel (LiftRel R)) o p
x✝¹ : LiftRelO (LiftRel R) (LiftRel (LiftRel R)) none none
x✝ : S.destruct.Results none m
rs2 : (destruct_join.aux none).Results a k
mT : none ∈ T.destruct
⊢ ∃ b ∈ T.join.destruct, LiftRelO R U a b

Tactic:
by

Goal state after:
Module: Mathlib.Data.Sigma.Interval

Goal state before:
pp: ι : Type u_1
α : ι → Type u_2
inst✝¹ : (i : ι) → Preorder (α i)
inst✝ : (i : ι) → LocallyFiniteOrderBot (α i)
x✝¹ x✝ : (i : ι) × α i
i : ι
a : α i
j : ι
b : α j
⊢ (⟨j, b⟩ ∈
      match ⟨i, a⟩ with
      | ⟨i, a⟩ => Finset.map (Embedding.sigmaMk i) (Iio a)) ↔
    ⟨j, b⟩ < ⟨i, a⟩

Tactic:
by

Goal state after: