Theorem: HMem.Trace.CostedProgram.splitTimeCost
Proof state before:
case h₀.h
α : Type u_1
inst✝⁵ : Complexity.Coding α Memory
β : Type u_2
inst✝⁴ : Complexity.Encoding β Memory
f : α → β
tp : TracedProgram
γ δ : Type
inst✝³ : Complexity.Coding γ Memory
inst✝² : Complexity.Encoding δ Memory
fs : γ → δ
cs : Complexity.CostFunction γ ℕ
inst✝¹ : Complexity.Computable Encoding.Model fs
inst✝ : Complexity Encoding.Model fs cs
size : α → ℕ
fm : α → Option Memory
fcp : CostedProgram
hf : Program.sound (TracedProgram.toProgram (toTracedProgram fcp)) f size
dst✝ src✝ : Source
next✝ : CostedProgram
h : TracedProgram.sound' f size (toTracedProgram (recurse dst✝ src✝ next✝)) fm
x✝¹ : α
x✝ : Memory
hm : x✝ ∈ fm x✝¹
⊢ Stack.halts
    (Stack.execution
      {
        state :=
          Complexity.encode
            (Option.get (Complexity.decode α (Memory.getms x✝ src✝))
              (_ : Option.isSome (Complexity.decode α (Memory.getms x✝ src✝)) = true)),
        current := toProgram fcp, function := toProgram fcp }
      [])

case h₁
α : Type u_1
inst✝⁵ : Complexity.Coding α Memory
β : Type u_2
inst✝⁴ : Complexity.Encoding β Memory
f : α → β
tp : TracedProgram
γ δ : Type
inst✝³ : Complexity.Coding γ Memory
inst✝² : Complexity.Encoding δ Memory
fs : γ → δ
cs : Complexity.CostFunction γ ℕ
inst✝¹ : Complexity.Computable Encoding.Model fs
inst✝ : Complexity Encoding.Model fs cs
size : α → ℕ
fm : α → Option Memory
fcp : CostedProgram
hf : Program.sound (TracedProgram.toProgram (toTracedProgram fcp)) f size
dst✝ src✝ : Source
next✝ : CostedProgram
h : TracedProgram.sound' f size (toTracedProgram (recurse dst✝ src✝ next✝)) fm
x✝¹ : α
x✝ : Memory
hm : x✝ ∈ fm x✝¹
⊢ Stack.hasTimeCost
    (Stack.execution
      {
        state :=
          Memory.setms x✝ dst✝
            (Complexity.encode
              (f
                (Option.get (Complexity.decode α (Memory.getms x✝ src✝))
                  (_ : Option.isSome (Complexity.decode α (Memory.getms x✝ src✝)) = true)))),
        current := TracedProgram.toProgram (toTracedProgram next✝), function := toProgram fcp }
      [])
    ?m.179013

α : Type u_1
inst✝⁵ : Complexity.Coding α Memory
β : Type u_2
inst✝⁴ : Complexity.Encoding β Memory
f : α → β
tp : TracedProgram
γ δ : Type
inst✝³ : Complexity.Coding γ Memory
inst✝² : Complexity.Encoding δ Memory
fs : γ → δ
cs : Complexity.CostFunction γ ℕ
inst✝¹ : Complexity.Computable Encoding.Model fs
inst✝ : Complexity Encoding.Model fs cs
size : α → ℕ
fm : α → Option Memory
fcp : CostedProgram
hf : Program.sound (TracedProgram.toProgram (toTracedProgram fcp)) f size
dst✝ src✝ : Source
next✝ : CostedProgram
h : TracedProgram.sound' f size (toTracedProgram (recurse dst✝ src✝ next✝)) fm
x✝¹ : α
x✝ : Memory
hm : x✝ ∈ fm x✝¹
⊢ ℕ

α : Type u_1
inst✝⁵ : Complexity.Coding α Memory
β : Type u_2
inst✝⁴ : Complexity.Encoding β Memory
f : α → β
tp : TracedProgram
γ δ : Type
inst✝³ : Complexity.Coding γ Memory
inst✝² : Complexity.Encoding δ Memory
fs : γ → δ
cs : Complexity.CostFunction γ ℕ
inst✝¹ : Complexity.Computable Encoding.Model fs
inst✝ : Complexity Encoding.Model fs cs
size : α → ℕ
fm : α → Option Memory
fcp : CostedProgram
hf : Program.sound (TracedProgram.toProgram (toTracedProgram fcp)) f size
dst✝ src✝ : Source
next✝ : CostedProgram
h : TracedProgram.sound' f size (toTracedProgram (recurse dst✝ src✝ next✝)) fm
x✝¹ : α
x✝ : Memory
hm : x✝ ∈ fm x✝¹
⊢ ?m.179013 +
        Stack.timeCost
          (Stack.execution
            {
              state :=
                Complexity.encode
                  (Option.get (Complexity.decode α (Memory.getms x✝ src✝))
                    (_ : Option.isSome (Complexity.decode α (Memory.getms x✝ src✝)) = true)),
              current := toProgram fcp, function := toProgram fcp }
            [])
          ?h₀.h +
      1 ≤
    (subroutineTimeCost h + recurseTimeCost hf h + localTimeCost h) x✝¹
Tactic:
apply Program.halts_of_sound hf
Proof state after:
case h₁
α : Type u_1
inst✝⁵ : Complexity.Coding α Memory
β : Type u_2
inst✝⁴ : Complexity.Encoding β Memory
f : α → β
tp : TracedProgram
γ δ : Type
inst✝³ : Complexity.Coding γ Memory
inst✝² : Complexity.Encoding δ Memory
fs : γ → δ
cs : Complexity.CostFunction γ ℕ
inst✝¹ : Complexity.Computable Encoding.Model fs
inst✝ : Complexity Encoding.Model fs cs
size : α → ℕ
fm : α → Option Memory
fcp : CostedProgram
hf : Program.sound (TracedProgram.toProgram (toTracedProgram fcp)) f size
dst✝ src✝ : Source
next✝ : CostedProgram
h : TracedProgram.sound' f size (toTracedProgram (recurse dst✝ src✝ next✝)) fm
x✝¹ : α
x✝ : Memory
hm : x✝ ∈ fm x✝¹
⊢ Stack.hasTimeCost
    (Stack.execution
      {
        state :=
          Memory.setms x✝ dst✝
            (Complexity.encode
              (f
                (Option.get (Complexity.decode α (Memory.getms x✝ src✝))
                  (_ : Option.isSome (Complexity.decode α (Memory.getms x✝ src✝)) = true)))),
        current := TracedProgram.toProgram (toTracedProgram next✝), function := toProgram fcp }
      [])
    ?m.179013

α : Type u_1
inst✝⁵ : Complexity.Coding α Memory
β : Type u_2
inst✝⁴ : Complexity.Encoding β Memory
f : α → β
tp : TracedProgram
γ δ : Type
inst✝³ : Complexity.Coding γ Memory
inst✝² : Complexity.Encoding δ Memory
fs : γ → δ
cs : Complexity.CostFunction γ ℕ
inst✝¹ : Complexity.Computable Encoding.Model fs
inst✝ : Complexity Encoding.Model fs cs
size : α → ℕ
fm : α → Option Memory
fcp : CostedProgram
hf : Program.sound (TracedProgram.toProgram (toTracedProgram fcp)) f size
dst✝ src✝ : Source
next✝ : CostedProgram
h : TracedProgram.sound' f size (toTracedProgram (recurse dst✝ src✝ next✝)) fm
x✝¹ : α
x✝ : Memory
hm : x✝ ∈ fm x✝¹
⊢ ℕ

α : Type u_1
inst✝⁵ : Complexity.Coding α Memory
β : Type u_2
inst✝⁴ : Complexity.Encoding β Memory
f : α → β
tp : TracedProgram
γ δ : Type
inst✝³ : Complexity.Coding γ Memory
inst✝² : Complexity.Encoding δ Memory
fs : γ → δ
cs : Complexity.CostFunction γ ℕ
inst✝¹ : Complexity.Computable Encoding.Model fs
inst✝ : Complexity Encoding.Model fs cs
size : α → ℕ
fm : α → Option Memory
fcp : CostedProgram
hf : Program.sound (TracedProgram.toProgram (toTracedProgram fcp)) f size
dst✝ src✝ : Source
next✝ : CostedProgram
h : TracedProgram.sound' f size (toTracedProgram (recurse dst✝ src✝ next✝)) fm
x✝¹ : α
x✝ : Memory
hm : x✝ ∈ fm x✝¹
⊢ ?m.179013 +
        Stack.timeCost
          (Stack.execution
            {
              state :=
                Complexity.encode
                  (Option.get (Complexity.decode α (Memory.getms x✝ src✝))
                    (_ : Option.isSome (Complexity.decode α (Memory.getms x✝ src✝)) = true)),
              current := toProgram fcp, function := toProgram fcp }
            [])
          (_ :
            Program.haltsOn (TracedProgram.toProgram (toTracedProgram fcp))
              (Complexity.encode
                (Option.get (Complexity.decode α (Memory.getms x✝ src✝))
                  (_ : Option.isSome (Complexity.decode α (Memory.getms x✝ src✝)) = true)))) +
      1 ≤
    (subroutineTimeCost h + recurseTimeCost hf h + localTimeCost h) x✝¹