mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Computability.TuringMachine.Config

Goal state before:
pp: case neg.refl
f : Code
k : Cont
v✝ : List ℕ
fok : f.Ok
x : Cfg
v' : List ℕ
he✝ : v' ∈ f.fix.eval v✝
hr✝ : x ∈ eval step (Cfg.ret k v')
hr : x ∈ eval step (stepRet k v')
v : List ℕ
he : v' ∈ f.fix.eval v
IH :
  ∀ (a'' : List ℕ),
    Sum.inr a'' ∈ Part.map (fun v ↦ if v.headI = 0 then Sum.inl v.tail else Sum.inr v.tail) (f.eval v) →
      x ∈ eval step (stepNormal f (Cont.fix f k) a'')
v₁ : List ℕ
he₁ : v₁ ∈ f.eval v
h : ¬v₁.headI = 0
right✝ :
  v' ∈ PFun.fix (fun v ↦ Part.map (fun v ↦ if v.headI = 0 then Sum.inl v.tail else Sum.inr v.tail) (f.eval v)) v₁.tail
⊢ Reaches step (Cfg.ret (Cont.fix f k) v₁) ?m.887

pp: case neg.refl
f : Code
k : Cont
v✝ : List ℕ
fok : f.Ok
x : Cfg
v' : List ℕ
he✝ : v' ∈ f.fix.eval v✝
hr✝ : x ∈ eval step (Cfg.ret k v')
hr : x ∈ eval step (stepRet k v')
v : List ℕ
he : v' ∈ f.fix.eval v
IH :
  ∀ (a'' : List ℕ),
    Sum.inr a'' ∈ Part.map (fun v ↦ if v.headI = 0 then Sum.inl v.tail else Sum.inr v.tail) (f.eval v) →
      x ∈ eval step (stepNormal f (Cont.fix f k) a'')
v₁ : List ℕ
he₁ : v₁ ∈ f.eval v
h : ¬v₁.headI = 0
right✝ :
  v' ∈ PFun.fix (fun v ↦ Part.map (fun v ↦ if v.headI = 0 then Sum.inl v.tail else Sum.inr v.tail) (f.eval v)) v₁.tail
⊢ x ∈ eval step ?m.887

pp: f : Code
k : Cont
v✝ : List ℕ
fok : f.Ok
x : Cfg
v' : List ℕ
he✝ : v' ∈ f.fix.eval v✝
hr✝ : x ∈ eval step (Cfg.ret k v')
hr : x ∈ eval step (stepRet k v')
v : List ℕ
he : v' ∈ f.fix.eval v
IH :
  ∀ (a'' : List ℕ),
    Sum.inr a'' ∈ Part.map (fun v ↦ if v.headI = 0 then Sum.inl v.tail else Sum.inr v.tail) (f.eval v) →
      x ∈ eval step (stepNormal f (Cont.fix f k) a'')
v₁ : List ℕ
he₁ : v₁ ∈ f.eval v
h : ¬v₁.headI = 0
right✝ :
  v' ∈ PFun.fix (fun v ↦ Part.map (fun v ↦ if v.headI = 0 then Sum.inl v.tail else Sum.inr v.tail) (f.eval v)) v₁.tail
⊢ Cfg

Tactic:
·

Goal state after:
case neg.refl
f : Code
k : Cont
v✝ : List ℕ
fok : f.Ok
x : Cfg
v' : List ℕ
he✝ : v' ∈ f.fix.eval v✝
hr✝ : x ∈ eval step (Cfg.ret k v')
hr : x ∈ eval step (stepRet k v')
v : List ℕ
he : v' ∈ f.fix.eval v
IH :
  ∀ (a'' : List ℕ),
    Sum.inr a'' ∈ Part.map (fun v ↦ if v.headI = 0 then Sum.inl v.tail else Sum.inr v.tail) (f.eval v) →
      x ∈ eval step (stepNormal f (Cont.fix f k) a'')
v₁ : List ℕ
he₁ : v₁ ∈ f.eval v
h : ¬v₁.headI = 0
right✝ :
  v' ∈ PFun.fix (fun v ↦ Part.map (fun v ↦ if v.headI = 0 then Sum.inl v.tail else Sum.inr v.tail) (f.eval v)) v₁.tail
⊢ Reaches step (Cfg.ret (Cont.fix f k) v₁) ?m.887
Module: Mathlib.Computability.TuringMachine.StackTuringMachine

Goal state before:
pp: case refine_1.inl.inl
K : Type u_1
Γ : K → Type u_2
Λ : Type u_3
σ : Type u_4
inst✝¹ : DecidableEq K
M : Λ → TM2.Stmt Γ Λ σ
inst✝ : Inhabited Λ
S : Finset Λ
ss : TM2.Supports M S
k✝ : K
s : StAct K Γ σ k✝
q✝ : TM2.Stmt Γ Λ σ
IH :
  TM2.SupportsStmt S q✝ →
    (∀ x ∈ trStmts₁ q✝, x ∈ trSupp M S) →
      TM1.SupportsStmt (trSupp M S) (trNormal q✝) ∧ ∀ l' ∈ trStmts₁ q✝, TM1.SupportsStmt (trSupp M S) (tr M l')
ss' : TM2.SupportsStmt S q✝
sub : ∀ (x : Λ' K Γ Λ σ), (x = go k✝ s q✝ ∨ x = ret q✝) ∨ x ∈ trStmts₁ q✝ → x ∈ trSupp M S
hgo : go k✝ s q✝ ∈ trSupp M S
hret : ret q✝ ∈ trSupp M S
IH₁ : TM1.SupportsStmt (trSupp M S) (trNormal q✝)
IH₂ : ∀ l' ∈ trStmts₁ q✝, TM1.SupportsStmt (trSupp M S) (tr M l')
⊢ TM1.SupportsStmt (trSupp M S) (tr M (go k✝ s q✝))

pp: case refine_1.inl.inr
K : Type u_1
Γ : K → Type u_2
Λ : Type u_3
σ : Type u_4
inst✝¹ : DecidableEq K
M : Λ → TM2.Stmt Γ Λ σ
inst✝ : Inhabited Λ
S : Finset Λ
ss : TM2.Supports M S
k✝ : K
s : StAct K Γ σ k✝
q✝ : TM2.Stmt Γ Λ σ
IH :
  TM2.SupportsStmt S q✝ →
    (∀ x ∈ trStmts₁ q✝, x ∈ trSupp M S) →
      TM1.SupportsStmt (trSupp M S) (trNormal q✝) ∧ ∀ l' ∈ trStmts₁ q✝, TM1.SupportsStmt (trSupp M S) (tr M l')
ss' : TM2.SupportsStmt S q✝
sub : ∀ (x : Λ' K Γ Λ σ), (x = go k✝ s q✝ ∨ x = ret q✝) ∨ x ∈ trStmts₁ q✝ → x ∈ trSupp M S
hgo : go k✝ s q✝ ∈ trSupp M S
hret : ret q✝ ∈ trSupp M S
IH₁ : TM1.SupportsStmt (trSupp M S) (trNormal q✝)
IH₂ : ∀ l' ∈ trStmts₁ q✝, TM1.SupportsStmt (trSupp M S) (tr M l')
⊢ TM1.SupportsStmt (trSupp M S) (tr M (ret q✝))

pp: case refine_1.inr
K : Type u_1
Γ : K → Type u_2
Λ : Type u_3
σ : Type u_4
inst✝¹ : DecidableEq K
M : Λ → TM2.Stmt Γ Λ σ
inst✝ : Inhabited Λ
S : Finset Λ
ss : TM2.Supports M S
k✝ : K
s : StAct K Γ σ k✝
q✝ : TM2.Stmt Γ Λ σ
IH :
  TM2.SupportsStmt S q✝ →
    (∀ x ∈ trStmts₁ q✝, x ∈ trSupp M S) →
      TM1.SupportsStmt (trSupp M S) (trNormal q✝) ∧ ∀ l' ∈ trStmts₁ q✝, TM1.SupportsStmt (trSupp M S) (tr M l')
ss' : TM2.SupportsStmt S q✝
sub : ∀ (x : Λ' K Γ Λ σ), (x = go k✝ s q✝ ∨ x = ret q✝) ∨ x ∈ trStmts₁ q✝ → x ∈ trSupp M S
hgo : go k✝ s q✝ ∈ trSupp M S
hret : ret q✝ ∈ trSupp M S
IH₁ : TM1.SupportsStmt (trSupp M S) (trNormal q✝)
IH₂ : ∀ l' ∈ trStmts₁ q✝, TM1.SupportsStmt (trSupp M S) (tr M l')
l : Λ' K Γ Λ σ
h : l ∈ trStmts₁ q✝
⊢ TM1.SupportsStmt (trSupp M S) (tr M l)

Tactic:
·

Goal state after:
case refine_1.inl.inl
K : Type u_1
Γ : K → Type u_2
Λ : Type u_3
σ : Type u_4
inst✝¹ : DecidableEq K
M : Λ → TM2.Stmt Γ Λ σ
inst✝ : Inhabited Λ
S : Finset Λ
ss : TM2.Supports M S
k✝ : K
s : StAct K Γ σ k✝
q✝ : TM2.Stmt Γ Λ σ
IH :
  TM2.SupportsStmt S q✝ →
    (∀ x ∈ trStmts₁ q✝, x ∈ trSupp M S) →
      TM1.SupportsStmt (trSupp M S) (trNormal q✝) ∧ ∀ l' ∈ trStmts₁ q✝, TM1.SupportsStmt (trSupp M S) (tr M l')
ss' : TM2.SupportsStmt S q✝
sub : ∀ (x : Λ' K Γ Λ σ), (x = go k✝ s q✝ ∨ x = ret q✝) ∨ x ∈ trStmts₁ q✝ → x ∈ trSupp M S
hgo : go k✝ s q✝ ∈ trSupp M S
hret : ret q✝ ∈ trSupp M S
IH₁ : TM1.SupportsStmt (trSupp M S) (trNormal q✝)
IH₂ : ∀ l' ∈ trStmts₁ q✝, TM1.SupportsStmt (trSupp M S) (tr M l')
⊢ TM1.SupportsStmt (trSupp M S) (tr M (go k✝ s q✝))
Module: Mathlib.Computability.TuringMachine.StackTuringMachine

Goal state before:
pp: case refine_1.inl.inl.peek
K : Type u_1
Γ : K → Type u_2
Λ : Type u_3
σ : Type u_4
inst✝¹ : DecidableEq K
M : Λ → TM2.Stmt Γ Λ σ
inst✝ : Inhabited Λ
S : Finset Λ
ss : TM2.Supports M S
k✝ : K
q✝ : TM2.Stmt Γ Λ σ
IH :
  TM2.SupportsStmt S q✝ →
    (∀ x ∈ trStmts₁ q✝, x ∈ trSupp M S) →
      TM1.SupportsStmt (trSupp M S) (trNormal q✝) ∧ ∀ l' ∈ trStmts₁ q✝, TM1.SupportsStmt (trSupp M S) (tr M l')
ss' : TM2.SupportsStmt S q✝
hret : ret q✝ ∈ trSupp M S
IH₁ : TM1.SupportsStmt (trSupp M S) (trNormal q✝)
IH₂ : ∀ l' ∈ trStmts₁ q✝, TM1.SupportsStmt (trSupp M S) (tr M l')
a✝ : σ → Option (Γ k✝) → σ
sub : ∀ (x : Λ' K Γ Λ σ), (x = go k✝ (StAct.peek a✝) q✝ ∨ x = ret q✝) ∨ x ∈ trStmts₁ q✝ → x ∈ trSupp M S
hgo : go k✝ (StAct.peek a✝) q✝ ∈ trSupp M S
⊢ TM1.SupportsStmt (trSupp M S) (tr M (go k✝ (StAct.peek a✝) q✝))

pp: case refine_1.inl.inl.pop
K : Type u_1
Γ : K → Type u_2
Λ : Type u_3
σ : Type u_4
inst✝¹ : DecidableEq K
M : Λ → TM2.Stmt Γ Λ σ
inst✝ : Inhabited Λ
S : Finset Λ
ss : TM2.Supports M S
k✝ : K
q✝ : TM2.Stmt Γ Λ σ
IH :
  TM2.SupportsStmt S q✝ →
    (∀ x ∈ trStmts₁ q✝, x ∈ trSupp M S) →
      TM1.SupportsStmt (trSupp M S) (trNormal q✝) ∧ ∀ l' ∈ trStmts₁ q✝, TM1.SupportsStmt (trSupp M S) (tr M l')
ss' : TM2.SupportsStmt S q✝
hret : ret q✝ ∈ trSupp M S
IH₁ : TM1.SupportsStmt (trSupp M S) (trNormal q✝)
IH₂ : ∀ l' ∈ trStmts₁ q✝, TM1.SupportsStmt (trSupp M S) (tr M l')
a✝ : σ → Option (Γ k✝) → σ
sub : ∀ (x : Λ' K Γ Λ σ), (x = go k✝ (StAct.pop a✝) q✝ ∨ x = ret q✝) ∨ x ∈ trStmts₁ q✝ → x ∈ trSupp M S
hgo : go k✝ (StAct.pop a✝) q✝ ∈ trSupp M S
⊢ TM1.SupportsStmt (trSupp M S) (tr M (go k✝ (StAct.pop a✝) q✝))

Tactic:
·

Goal state after:
case refine_1.inl.inl.peek
K : Type u_1
Γ : K → Type u_2
Λ : Type u_3
σ : Type u_4
inst✝¹ : DecidableEq K
M : Λ → TM2.Stmt Γ Λ σ
inst✝ : Inhabited Λ
S : Finset Λ
ss : TM2.Supports M S
k✝ : K
q✝ : TM2.Stmt Γ Λ σ
IH :
  TM2.SupportsStmt S q✝ →
    (∀ x ∈ trStmts₁ q✝, x ∈ trSupp M S) →
      TM1.SupportsStmt (trSupp M S) (trNormal q✝) ∧ ∀ l' ∈ trStmts₁ q✝, TM1.SupportsStmt (trSupp M S) (tr M l')
ss' : TM2.SupportsStmt S q✝
hret : ret q✝ ∈ trSupp M S
IH₁ : TM1.SupportsStmt (trSupp M S) (trNormal q✝)
IH₂ : ∀ l' ∈ trStmts₁ q✝, TM1.SupportsStmt (trSupp M S) (tr M l')
a✝ : σ → Option (Γ k✝) → σ
sub : ∀ (x : Λ' K Γ Λ σ), (x = go k✝ (StAct.peek a✝) q✝ ∨ x = ret q✝) ∨ x ∈ trStmts₁ q✝ → x ∈ trSupp M S
hgo : go k✝ (StAct.peek a✝) q✝ ∈ trSupp M S
⊢ TM1.SupportsStmt (trSupp M S) (tr M (go k✝ (StAct.peek a✝) q✝))
Module: Mathlib.Computability.TuringMachine.StackTuringMachine

Goal state before:
pp: case refine_1.inl.inl.pop
K : Type u_1
Γ : K → Type u_2
Λ : Type u_3
σ : Type u_4
inst✝¹ : DecidableEq K
M : Λ → TM2.Stmt Γ Λ σ
inst✝ : Inhabited Λ
S : Finset Λ
ss : TM2.Supports M S
k✝ : K
q✝ : TM2.Stmt Γ Λ σ
IH :
  TM2.SupportsStmt S q✝ →
    (∀ x ∈ trStmts₁ q✝, x ∈ trSupp M S) →
      TM1.SupportsStmt (trSupp M S) (trNormal q✝) ∧ ∀ l' ∈ trStmts₁ q✝, TM1.SupportsStmt (trSupp M S) (tr M l')
ss' : TM2.SupportsStmt S q✝
hret : ret q✝ ∈ trSupp M S
IH₁ : TM1.SupportsStmt (trSupp M S) (trNormal q✝)
IH₂ : ∀ l' ∈ trStmts₁ q✝, TM1.SupportsStmt (trSupp M S) (tr M l')
a✝ : σ → Option (Γ k✝) → σ
sub : ∀ (x : Λ' K Γ Λ σ), (x = go k✝ (StAct.pop a✝) q✝ ∨ x = ret q✝) ∨ x ∈ trStmts₁ q✝ → x ∈ trSupp M S
hgo : go k✝ (StAct.pop a✝) q✝ ∈ trSupp M S
⊢ TM1.SupportsStmt (trSupp M S) (tr M (go k✝ (StAct.pop a✝) q✝))

Tactic:
·

Goal state after:
case refine_1.inl.inl.pop
K : Type u_1
Γ : K → Type u_2
Λ : Type u_3
σ : Type u_4
inst✝¹ : DecidableEq K
M : Λ → TM2.Stmt Γ Λ σ
inst✝ : Inhabited Λ
S : Finset Λ
ss : TM2.Supports M S
k✝ : K
q✝ : TM2.Stmt Γ Λ σ
IH :
  TM2.SupportsStmt S q✝ →
    (∀ x ∈ trStmts₁ q✝, x ∈ trSupp M S) →
      TM1.SupportsStmt (trSupp M S) (trNormal q✝) ∧ ∀ l' ∈ trStmts₁ q✝, TM1.SupportsStmt (trSupp M S) (tr M l')
ss' : TM2.SupportsStmt S q✝
hret : ret q✝ ∈ trSupp M S
IH₁ : TM1.SupportsStmt (trSupp M S) (trNormal q✝)
IH₂ : ∀ l' ∈ trStmts₁ q✝, TM1.SupportsStmt (trSupp M S) (tr M l')
a✝ : σ → Option (Γ k✝) → σ
sub : ∀ (x : Λ' K Γ Λ σ), (x = go k✝ (StAct.pop a✝) q✝ ∨ x = ret q✝) ∨ x ∈ trStmts₁ q✝ → x ∈ trSupp M S
hgo : go k✝ (StAct.pop a✝) q✝ ∈ trSupp M S
⊢ TM1.SupportsStmt (trSupp M S) (tr M (go k✝ (StAct.pop a✝) q✝))
Module: Mathlib.Computability.TuringMachine.StackTuringMachine

Goal state before:
pp: case refine_1.inl.inr
K : Type u_1
Γ : K → Type u_2
Λ : Type u_3
σ : Type u_4
inst✝¹ : DecidableEq K
M : Λ → TM2.Stmt Γ Λ σ
inst✝ : Inhabited Λ
S : Finset Λ
ss : TM2.Supports M S
k✝ : K
s : StAct K Γ σ k✝
q✝ : TM2.Stmt Γ Λ σ
IH :
  TM2.SupportsStmt S q✝ →
    (∀ x ∈ trStmts₁ q✝, x ∈ trSupp M S) →
      TM1.SupportsStmt (trSupp M S) (trNormal q✝) ∧ ∀ l' ∈ trStmts₁ q✝, TM1.SupportsStmt (trSupp M S) (tr M l')
ss' : TM2.SupportsStmt S q✝
sub : ∀ (x : Λ' K Γ Λ σ), (x = go k✝ s q✝ ∨ x = ret q✝) ∨ x ∈ trStmts₁ q✝ → x ∈ trSupp M S
hgo : go k✝ s q✝ ∈ trSupp M S
hret : ret q✝ ∈ trSupp M S
IH₁ : TM1.SupportsStmt (trSupp M S) (trNormal q✝)
IH₂ : ∀ l' ∈ trStmts₁ q✝, TM1.SupportsStmt (trSupp M S) (tr M l')
⊢ TM1.SupportsStmt (trSupp M S) (tr M (ret q✝))

pp: case refine_1.inr
K : Type u_1
Γ : K → Type u_2
Λ : Type u_3
σ : Type u_4
inst✝¹ : DecidableEq K
M : Λ → TM2.Stmt Γ Λ σ
inst✝ : Inhabited Λ
S : Finset Λ
ss : TM2.Supports M S
k✝ : K
s : StAct K Γ σ k✝
q✝ : TM2.Stmt Γ Λ σ
IH :
  TM2.SupportsStmt S q✝ →
    (∀ x ∈ trStmts₁ q✝, x ∈ trSupp M S) →
      TM1.SupportsStmt (trSupp M S) (trNormal q✝) ∧ ∀ l' ∈ trStmts₁ q✝, TM1.SupportsStmt (trSupp M S) (tr M l')
ss' : TM2.SupportsStmt S q✝
sub : ∀ (x : Λ' K Γ Λ σ), (x = go k✝ s q✝ ∨ x = ret q✝) ∨ x ∈ trStmts₁ q✝ → x ∈ trSupp M S
hgo : go k✝ s q✝ ∈ trSupp M S
hret : ret q✝ ∈ trSupp M S
IH₁ : TM1.SupportsStmt (trSupp M S) (trNormal q✝)
IH₂ : ∀ l' ∈ trStmts₁ q✝, TM1.SupportsStmt (trSupp M S) (tr M l')
l : Λ' K Γ Λ σ
h : l ∈ trStmts₁ q✝
⊢ TM1.SupportsStmt (trSupp M S) (tr M l)

Tactic:
·

Goal state after:
case refine_1.inl.inr
K : Type u_1
Γ : K → Type u_2
Λ : Type u_3
σ : Type u_4
inst✝¹ : DecidableEq K
M : Λ → TM2.Stmt Γ Λ σ
inst✝ : Inhabited Λ
S : Finset Λ
ss : TM2.Supports M S
k✝ : K
s : StAct K Γ σ k✝
q✝ : TM2.Stmt Γ Λ σ
IH :
  TM2.SupportsStmt S q✝ →
    (∀ x ∈ trStmts₁ q✝, x ∈ trSupp M S) →
      TM1.SupportsStmt (trSupp M S) (trNormal q✝) ∧ ∀ l' ∈ trStmts₁ q✝, TM1.SupportsStmt (trSupp M S) (tr M l')
ss' : TM2.SupportsStmt S q✝
sub : ∀ (x : Λ' K Γ Λ σ), (x = go k✝ s q✝ ∨ x = ret q✝) ∨ x ∈ trStmts₁ q✝ → x ∈ trSupp M S
hgo : go k✝ s q✝ ∈ trSupp M S
hret : ret q✝ ∈ trSupp M S
IH₁ : TM1.SupportsStmt (trSupp M S) (trNormal q✝)
IH₂ : ∀ l' ∈ trStmts₁ q✝, TM1.SupportsStmt (trSupp M S) (tr M l')
⊢ TM1.SupportsStmt (trSupp M S) (tr M (ret q✝))
Module: Mathlib.Computability.TuringMachine.Config

Goal state before:
pp: case e_a
f fs : Code
IHf : f.Ok
IHfs : fs.Ok
k : Cont
v✝ v : List ℕ
⊢ Reaches step (Cfg.ret (Cont.cons₁ fs v✝ k) v) ?m.61

pp: case e_a
f fs : Code
IHf : f.Ok
IHfs : fs.Ok
k : Cont
v✝ v : List ℕ
⊢ eval step ?m.61 = do
    let x ← fs.eval v✝
    eval step (Cfg.ret k (v.headI :: x))

pp: f fs : Code
IHf : f.Ok
IHfs : fs.Ok
k : Cont
v✝ v : List ℕ
⊢ Cfg

Tactic:
·

Goal state after:
case e_a
f fs : Code
IHf : f.Ok
IHfs : fs.Ok
k : Cont
v✝ v : List ℕ
⊢ Reaches step (Cfg.ret (Cont.cons₁ fs v✝ k) v) ?m.61
Module: Mathlib.Computability.TuringMachine.StackTuringMachine

Goal state before:
pp: case refine_1.inr
K : Type u_1
Γ : K → Type u_2
Λ : Type u_3
σ : Type u_4
inst✝¹ : DecidableEq K
M : Λ → TM2.Stmt Γ Λ σ
inst✝ : Inhabited Λ
S : Finset Λ
ss : TM2.Supports M S
k✝ : K
s : StAct K Γ σ k✝
q✝ : TM2.Stmt Γ Λ σ
IH :
  TM2.SupportsStmt S q✝ →
    (∀ x ∈ trStmts₁ q✝, x ∈ trSupp M S) →
      TM1.SupportsStmt (trSupp M S) (trNormal q✝) ∧ ∀ l' ∈ trStmts₁ q✝, TM1.SupportsStmt (trSupp M S) (tr M l')
ss' : TM2.SupportsStmt S q✝
sub : ∀ (x : Λ' K Γ Λ σ), (x = go k✝ s q✝ ∨ x = ret q✝) ∨ x ∈ trStmts₁ q✝ → x ∈ trSupp M S
hgo : go k✝ s q✝ ∈ trSupp M S
hret : ret q✝ ∈ trSupp M S
IH₁ : TM1.SupportsStmt (trSupp M S) (trNormal q✝)
IH₂ : ∀ l' ∈ trStmts₁ q✝, TM1.SupportsStmt (trSupp M S) (tr M l')
l : Λ' K Γ Λ σ
h : l ∈ trStmts₁ q✝
⊢ TM1.SupportsStmt (trSupp M S) (tr M l)

Tactic:
·

Goal state after:
case refine_1.inr
K : Type u_1
Γ : K → Type u_2
Λ : Type u_3
σ : Type u_4
inst✝¹ : DecidableEq K
M : Λ → TM2.Stmt Γ Λ σ
inst✝ : Inhabited Λ
S : Finset Λ
ss : TM2.Supports M S
k✝ : K
s : StAct K Γ σ k✝
q✝ : TM2.Stmt Γ Λ σ
IH :
  TM2.SupportsStmt S q✝ →
    (∀ x ∈ trStmts₁ q✝, x ∈ trSupp M S) →
      TM1.SupportsStmt (trSupp M S) (trNormal q✝) ∧ ∀ l' ∈ trStmts₁ q✝, TM1.SupportsStmt (trSupp M S) (tr M l')
ss' : TM2.SupportsStmt S q✝
sub : ∀ (x : Λ' K Γ Λ σ), (x = go k✝ s q✝ ∨ x = ret q✝) ∨ x ∈ trStmts₁ q✝ → x ∈ trSupp M S
hgo : go k✝ s q✝ ∈ trSupp M S
hret : ret q✝ ∈ trSupp M S
IH₁ : TM1.SupportsStmt (trSupp M S) (trNormal q✝)
IH₂ : ∀ l' ∈ trStmts₁ q✝, TM1.SupportsStmt (trSupp M S) (tr M l')
l : Λ' K Γ Λ σ
h : l ∈ trStmts₁ q✝
⊢ TM1.SupportsStmt (trSupp M S) (tr M l)
Module: Mathlib.Computability.TuringMachine.ToPartrec

Goal state before:
pp: case cons.nil
p : Γ' → Bool
k₁ k₂ : K'
q : Λ'
o : Option Γ'
L₂ : List Γ'
h₁ : k₁ ≠ k₂
a : Γ'
L₁ : List Γ'
IH :
  ∀ {s : Option Γ'} {S : K' → List Γ'},
    splitAtPred p (S k₁) = (L₁, o, L₂) →
      Reaches₁ (TM2.step tr) { l := some (Λ'.move p k₁ k₂ q), var := s, stk := S }
        { l := some q, var := o, stk := update (update S k₁ L₂) k₂ (L₁.reverseAux (S k₂)) }
s : Option Γ'
S : K' → List Γ'
e : ([], none, []) = (a :: L₁, o, L₂)
e₁ : S k₁ = []
⊢ TransGen (fun a b ↦ b ∈ TM2.step tr a)
    (bif
        match [].head?, true, p with
        | some x, x_1, f => f x
        | none, y, x => y then
      { l := some q, var := [].head?, stk := update S k₁ [].tail }
    else
      { l := some (Λ'.move p k₁ k₂ q), var := [].head?,
        stk := update (update S k₁ [].tail) k₂ ([].head?.getD default :: update S k₁ [].tail k₂) })
    { l := some q, var := o, stk := update (update S k₁ L₂) k₂ ((a :: L₁).reverseAux (S k₂)) }

pp: case cons.cons
p : Γ' → Bool
k₁ k₂ : K'
q : Λ'
o : Option Γ'
L₂ : List Γ'
h₁ : k₁ ≠ k₂
a : Γ'
L₁ : List Γ'
IH :
  ∀ {s : Option Γ'} {S : K' → List Γ'},
    splitAtPred p (S k₁) = (L₁, o, L₂) →
      Reaches₁ (TM2.step tr) { l := some (Λ'.move p k₁ k₂ q), var := s, stk := S }
        { l := some q, var := o, stk := update (update S k₁ L₂) k₂ (L₁.reverseAux (S k₂)) }
s : Option Γ'
S : K' → List Γ'
a' : Γ'
Sk : List Γ'
e :
  (bif p a' then ([], some a', Sk)
    else
      match splitAtPred p Sk with
      | (l₁, o, l₂) => (a' :: l₁, o, l₂)) =
    (a :: L₁, o, L₂)
e₁ : S k₁ = a' :: Sk
⊢ TransGen (fun a b ↦ b ∈ TM2.step tr a)
    (bif
        match (a' :: Sk).head?, true, p with
        | some x, x_1, f => f x
        | none, y, x => y then
      { l := some q, var := (a' :: Sk).head?, stk := update S k₁ (a' :: Sk).tail }
    else
      { l := some (Λ'.move p k₁ k₂ q), var := (a' :: Sk).head?,
        stk :=
          update (update S k₁ (a' :: Sk).tail) k₂ ((a' :: Sk).head?.getD default :: update S k₁ (a' :: Sk).tail k₂) })
    { l := some q, var := o, stk := update (update S k₁ L₂) k₂ ((a :: L₁).reverseAux (S k₂)) }

Tactic:
·

Goal state after:
case cons.nil
p : Γ' → Bool
k₁ k₂ : K'
q : Λ'
o : Option Γ'
L₂ : List Γ'
h₁ : k₁ ≠ k₂
a : Γ'
L₁ : List Γ'
IH :
  ∀ {s : Option Γ'} {S : K' → List Γ'},
    splitAtPred p (S k₁) = (L₁, o, L₂) →
      Reaches₁ (TM2.step tr) { l := some (Λ'.move p k₁ k₂ q), var := s, stk := S }
        { l := some q, var := o, stk := update (update S k₁ L₂) k₂ (L₁.reverseAux (S k₂)) }
s : Option Γ'
S : K' → List Γ'
e : ([], none, []) = (a :: L₁, o, L₂)
e₁ : S k₁ = []
⊢ TransGen (fun a b ↦ b ∈ TM2.step tr a)
    (bif
        match [].head?, true, p with
        | some x, x_1, f => f x
        | none, y, x => y then
      { l := some q, var := [].head?, stk := update S k₁ [].tail }
    else
      { l := some (Λ'.move p k₁ k₂ q), var := [].head?,
        stk := update (update S k₁ [].tail) k₂ ([].head?.getD default :: update S k₁ [].tail k₂) })
    { l := some q, var := o, stk := update (update S k₁ L₂) k₂ ((a :: L₁).reverseAux (S k₂)) }
Module: Mathlib.Computability.TuringMachine.Config

Goal state before:
pp: case e_a
f g : Code
IHf : f.Ok
IHg : g.Ok
k : Cont
v✝ v : List ℕ
⊢ Reaches step (Cfg.ret (Cont.comp f k) v) ?m.171

pp: case e_a
f g : Code
IHf : f.Ok
IHg : g.Ok
k : Cont
v✝ v : List ℕ
⊢ eval step ?m.171 = do
    let v ← f.eval v
    eval step (Cfg.ret k v)

pp: f g : Code
IHf : f.Ok
IHg : g.Ok
k : Cont
v✝ v : List ℕ
⊢ Cfg

Tactic:
·

Goal state after:
case e_a
f g : Code
IHf : f.Ok
IHg : g.Ok
k : Cont
v✝ v : List ℕ
⊢ Reaches step (Cfg.ret (Cont.comp f k) v) ?m.171
Module: Mathlib.Computability.TuringMachine.StackTuringMachine

Goal state before:
pp: case refine_3
K : Type u_1
Γ : K → Type u_2
Λ : Type u_3
σ : Type u_4
inst✝¹ : DecidableEq K
M : Λ → TM2.Stmt Γ Λ σ
inst✝ : Inhabited Λ
S : Finset Λ
ss : TM2.Supports M S
⊢ ∀ (p : σ → Bool) (q₁ q₂ : TM2.Stmt Γ Λ σ),
    (TM2.SupportsStmt S q₁ →
        (∀ x ∈ trStmts₁ q₁, x ∈ trSupp M S) →
          TM1.SupportsStmt (trSupp M S) (trNormal q₁) ∧ ∀ l' ∈ trStmts₁ q₁, TM1.SupportsStmt (trSupp M S) (tr M l')) →
      (TM2.SupportsStmt S q₂ →
          (∀ x ∈ trStmts₁ q₂, x ∈ trSupp M S) →
            TM1.SupportsStmt (trSupp M S) (trNormal q₂) ∧ ∀ l' ∈ trStmts₁ q₂, TM1.SupportsStmt (trSupp M S) (tr M l')) →
        TM2.SupportsStmt S (TM2.Stmt.branch p q₁ q₂) →
          (∀ x ∈ trStmts₁ (TM2.Stmt.branch p q₁ q₂), x ∈ trSupp M S) →
            TM1.SupportsStmt (trSupp M S) (trNormal (TM2.Stmt.branch p q₁ q₂)) ∧
              ∀ l' ∈ trStmts₁ (TM2.Stmt.branch p q₁ q₂), TM1.SupportsStmt (trSupp M S) (tr M l')

pp: case refine_4
K : Type u_1
Γ : K → Type u_2
Λ : Type u_3
σ : Type u_4
inst✝¹ : DecidableEq K
M : Λ → TM2.Stmt Γ Λ σ
inst✝ : Inhabited Λ
S : Finset Λ
ss : TM2.Supports M S
⊢ ∀ (l : σ → Λ),
    TM2.SupportsStmt S (TM2.Stmt.goto l) →
      (∀ x ∈ trStmts₁ (TM2.Stmt.goto l), x ∈ trSupp M S) →
        TM1.SupportsStmt (trSupp M S) (trNormal (TM2.Stmt.goto l)) ∧
          ∀ l' ∈ trStmts₁ (TM2.Stmt.goto l), TM1.SupportsStmt (trSupp M S) (tr M l')

pp: case refine_5
K : Type u_1
Γ : K → Type u_2
Λ : Type u_3
σ : Type u_4
inst✝¹ : DecidableEq K
M : Λ → TM2.Stmt Γ Λ σ
inst✝ : Inhabited Λ
S : Finset Λ
ss : TM2.Supports M S
⊢ TM2.SupportsStmt S TM2.Stmt.halt →
    (∀ x ∈ trStmts₁ TM2.Stmt.halt, x ∈ trSupp M S) →
      TM1.SupportsStmt (trSupp M S) (trNormal TM2.Stmt.halt) ∧
        ∀ l' ∈ trStmts₁ TM2.Stmt.halt, TM1.SupportsStmt (trSupp M S) (tr M l')

Tactic:
·

Goal state after:
case refine_3
K : Type u_1
Γ : K → Type u_2
Λ : Type u_3
σ : Type u_4
inst✝¹ : DecidableEq K
M : Λ → TM2.Stmt Γ Λ σ
inst✝ : Inhabited Λ
S : Finset Λ
ss : TM2.Supports M S
⊢ ∀ (p : σ → Bool) (q₁ q₂ : TM2.Stmt Γ Λ σ),
    (TM2.SupportsStmt S q₁ →
        (∀ x ∈ trStmts₁ q₁, x ∈ trSupp M S) →
          TM1.SupportsStmt (trSupp M S) (trNormal q₁) ∧ ∀ l' ∈ trStmts₁ q₁, TM1.SupportsStmt (trSupp M S) (tr M l')) →
      (TM2.SupportsStmt S q₂ →
          (∀ x ∈ trStmts₁ q₂, x ∈ trSupp M S) →
            TM1.SupportsStmt (trSupp M S) (trNormal q₂) ∧ ∀ l' ∈ trStmts₁ q₂, TM1.SupportsStmt (trSupp M S) (tr M l')) →
        TM2.SupportsStmt S (TM2.Stmt.branch p q₁ q₂) →
          (∀ x ∈ trStmts₁ (TM2.Stmt.branch p q₁ q₂), x ∈ trSupp M S) →
            TM1.SupportsStmt (trSupp M S) (trNormal (TM2.Stmt.branch p q₁ q₂)) ∧
              ∀ l' ∈ trStmts₁ (TM2.Stmt.branch p q₁ q₂), TM1.SupportsStmt (trSupp M S) (tr M l')
Module: Mathlib.Computability.TuringMachine.Config

Goal state before:
pp: case e_a
fs : Code
as : List ℕ
k : Cont
IH : ∀ {v : List ℕ}, eval step (stepRet k v) = Cfg.halt <$> k.eval v
v v' : List ℕ
⊢ Reaches step (Cfg.ret (Cont.cons₂ v k) v') ?m.89

pp: case e_a
fs : Code
as : List ℕ
k : Cont
IH : ∀ {v : List ℕ}, eval step (stepRet k v) = Cfg.halt <$> k.eval v
v v' : List ℕ
⊢ eval step ?m.89 = do
    let a ← k.eval (v.headI :: v')
    pure (Cfg.halt a)

pp: fs : Code
as : List ℕ
k : Cont
IH : ∀ {v : List ℕ}, eval step (stepRet k v) = Cfg.halt <$> k.eval v
v v' : List ℕ
⊢ Cfg

Tactic:
·

Goal state after:
case e_a
fs : Code
as : List ℕ
k : Cont
IH : ∀ {v : List ℕ}, eval step (stepRet k v) = Cfg.halt <$> k.eval v
v v' : List ℕ
⊢ Reaches step (Cfg.ret (Cont.cons₂ v k) v') ?m.89
Module: Mathlib.Computability.TuringMachine.StackTuringMachine

Goal state before:
pp: case refine_4
K : Type u_1
Γ : K → Type u_2
Λ : Type u_3
σ : Type u_4
inst✝¹ : DecidableEq K
M : Λ → TM2.Stmt Γ Λ σ
inst✝ : Inhabited Λ
S : Finset Λ
ss : TM2.Supports M S
⊢ ∀ (l : σ → Λ),
    TM2.SupportsStmt S (TM2.Stmt.goto l) →
      (∀ x ∈ trStmts₁ (TM2.Stmt.goto l), x ∈ trSupp M S) →
        TM1.SupportsStmt (trSupp M S) (trNormal (TM2.Stmt.goto l)) ∧
          ∀ l' ∈ trStmts₁ (TM2.Stmt.goto l), TM1.SupportsStmt (trSupp M S) (tr M l')

pp: case refine_5
K : Type u_1
Γ : K → Type u_2
Λ : Type u_3
σ : Type u_4
inst✝¹ : DecidableEq K
M : Λ → TM2.Stmt Γ Λ σ
inst✝ : Inhabited Λ
S : Finset Λ
ss : TM2.Supports M S
⊢ TM2.SupportsStmt S TM2.Stmt.halt →
    (∀ x ∈ trStmts₁ TM2.Stmt.halt, x ∈ trSupp M S) →
      TM1.SupportsStmt (trSupp M S) (trNormal TM2.Stmt.halt) ∧
        ∀ l' ∈ trStmts₁ TM2.Stmt.halt, TM1.SupportsStmt (trSupp M S) (tr M l')

Tactic:
·

Goal state after:
case refine_4
K : Type u_1
Γ : K → Type u_2
Λ : Type u_3
σ : Type u_4
inst✝¹ : DecidableEq K
M : Λ → TM2.Stmt Γ Λ σ
inst✝ : Inhabited Λ
S : Finset Λ
ss : TM2.Supports M S
⊢ ∀ (l : σ → Λ),
    TM2.SupportsStmt S (TM2.Stmt.goto l) →
      (∀ x ∈ trStmts₁ (TM2.Stmt.goto l), x ∈ trSupp M S) →
        TM1.SupportsStmt (trSupp M S) (trNormal (TM2.Stmt.goto l)) ∧
          ∀ l' ∈ trStmts₁ (TM2.Stmt.goto l), TM1.SupportsStmt (trSupp M S) (tr M l')
Module: Mathlib.Computability.TuringMachine.Config

Goal state before:
pp: case e_a
f : Code
k : Cont
IH : ∀ {v : List ℕ}, eval step (stepRet k v) = Cfg.halt <$> k.eval v
v v' : List ℕ
⊢ Reaches step (Cfg.ret k v') ?m.167

pp: case e_a
f : Code
k : Cont
IH : ∀ {v : List ℕ}, eval step (stepRet k v) = Cfg.halt <$> k.eval v
v v' : List ℕ
⊢ eval step ?m.167 = do
    let a ← k.eval v'
    pure (Cfg.halt a)

pp: f : Code
k : Cont
IH : ∀ {v : List ℕ}, eval step (stepRet k v) = Cfg.halt <$> k.eval v
v v' : List ℕ
⊢ Cfg

Tactic:
·

Goal state after:
case e_a
f : Code
k : Cont
IH : ∀ {v : List ℕ}, eval step (stepRet k v) = Cfg.halt <$> k.eval v
v v' : List ℕ
⊢ Reaches step (Cfg.ret k v') ?m.167