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https://github.com/calcu16/lean_complexity.git
d267eb4f3952c654f130a119311692c1c1a7cd71
HMem/Trace/Cost.lean
HMem.Trace.CostedProgram.splitTimeCost
[244, 1]
[349, 16]
apply hcost.cost_ale.le_bound
case h₀ α : Type u_1 inst✝⁷ : Complexity.Coding α Memory β : Type u_2 inst✝⁶ : Complexity.Encoding β Memory f : α → β tp : TracedProgram γ δ : Type inst✝⁵ : Complexity.Coding γ Memory inst✝⁴ : Complexity.Encoding δ Memory fs : γ → δ cs : Complexity.CostFunction γ ℕ inst✝³ : Complexity.Computable Encoding.Model fs inst✝² : 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 γ✝ δ✝ : Type inst✝¹ : Complexity.Coding γ✝ Memory inst✝ : Complexity.Encoding δ✝ Memory fs✝ : γ✝ → δ✝ cs✝ : Complexity.CostFunction γ✝ ℕ hcost : Complexity Encoding.Model fs✝ cs✝ next : CostedProgram h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm a : α m : Memory hm : m ∈ fm a ⊢ Stack.timeCost (Stack.execution { state := Complexity.encode (Option.get (Complexity.decode γ✝ (Memory.getms m src✝)) (_ : Option.isSome (Complexity.decode γ✝ (Memory.getms m src✝)) = true)), current := Encoding.getProgram fs✝, function := Encoding.getProgram fs✝ } []) ?m.173938 ≤ ?m.173011 α : Type u_1 inst✝⁷ : Complexity.Coding α Memory β : Type u_2 inst✝⁶ : Complexity.Encoding β Memory f : α → β tp : TracedProgram γ δ : Type inst✝⁵ : Complexity.Coding γ Memory inst✝⁴ : Complexity.Encoding δ Memory fs : γ → δ cs : Complexity.CostFunction γ ℕ inst✝³ : Complexity.Computable Encoding.Model fs inst✝² : 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 γ✝ δ✝ : Type inst✝¹ : Complexity.Coding γ✝ Memory inst✝ : Complexity.Encoding δ✝ Memory fs✝ : γ✝ → δ✝ cs✝ : Complexity.CostFunction γ✝ ℕ hcost : Complexity Encoding.Model fs✝ cs✝ next : CostedProgram h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm a : α m : Memory hm : m ∈ fm a ⊢ ℕ α : Type u_1 inst✝⁷ : Complexity.Coding α Memory β : Type u_2 inst✝⁶ : Complexity.Encoding β Memory f : α → β tp : TracedProgram γ δ : Type inst✝⁵ : Complexity.Coding γ Memory inst✝⁴ : Complexity.Encoding δ Memory fs : γ → δ cs : Complexity.CostFunction γ ℕ inst✝³ : Complexity.Computable Encoding.Model fs inst✝² : 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 γ✝ δ✝ : Type inst✝¹ : Complexity.Coding γ✝ Memory inst✝ : Complexity.Encoding δ✝ Memory fs✝ : γ✝ → δ✝ cs✝ : Complexity.CostFunction γ✝ ℕ hcost : Complexity Encoding.Model fs✝ cs✝ next : CostedProgram h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm a : α m : Memory hm : m ∈ fm a ⊢ Stack.halts (Stack.execution { state := Complexity.encode (Option.get (Complexity.decode γ✝ (Memory.getms m src✝)) (_ : Option.isSome (Complexity.decode γ✝ (Memory.getms m src✝)) = true)), current := Encoding.getProgram fs✝, function := Encoding.getProgram fs✝ } []) α : Type u_1 inst✝⁷ : Complexity.Coding α Memory β : Type u_2 inst✝⁶ : Complexity.Encoding β Memory f : α → β tp : TracedProgram γ δ : Type inst✝⁵ : Complexity.Coding γ Memory inst✝⁴ : Complexity.Encoding δ Memory fs : γ → δ cs : Complexity.CostFunction γ ℕ inst✝³ : Complexity.Computable Encoding.Model fs inst✝² : 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 γ✝ δ✝ : Type inst✝¹ : Complexity.Coding γ✝ Memory inst✝ : Complexity.Encoding δ✝ Memory fs✝ : γ✝ → δ✝ cs✝ : Complexity.CostFunction γ✝ ℕ hcost : Complexity Encoding.Model fs✝ cs✝ next : CostedProgram h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm a : α m : Memory hm : m ∈ fm a ⊢ ℕ case h₁ α : Type u_1 inst✝⁷ : Complexity.Coding α Memory β : Type u_2 inst✝⁶ : Complexity.Encoding β Memory f : α → β tp : TracedProgram γ δ : Type inst✝⁵ : Complexity.Coding γ Memory inst✝⁴ : Complexity.Encoding δ Memory fs : γ → δ cs : Complexity.CostFunction γ ℕ inst✝³ : Complexity.Computable Encoding.Model fs inst✝² : 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 γ✝ δ✝ : Type inst✝¹ : Complexity.Coding γ✝ Memory inst✝ : Complexity.Encoding δ✝ Memory fs✝ : γ✝ → δ✝ cs✝ : Complexity.CostFunction γ✝ ℕ hcost : Complexity Encoding.Model fs✝ cs✝ next : CostedProgram h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm a : α m : Memory hm : m ∈ fm a ⊢ Stack.hasTimeCost (Stack.execution { state := Memory.setms m dst✝ (Complexity.encode (fs✝ (Option.get (Complexity.decode γ✝ (Memory.getms m src✝)) (_ : Option.isSome (Complexity.decode γ✝ (Memory.getms m src✝)) = true)))), current := TracedProgram.toProgram (toTracedProgram next), function := toProgram fcp } []) ?m.173010 α : Type u_1 inst✝⁷ : Complexity.Coding α Memory β : Type u_2 inst✝⁶ : Complexity.Encoding β Memory f : α → β tp : TracedProgram γ δ : Type inst✝⁵ : Complexity.Coding γ Memory inst✝⁴ : Complexity.Encoding δ Memory fs : γ → δ cs : Complexity.CostFunction γ ℕ inst✝³ : Complexity.Computable Encoding.Model fs inst✝² : 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 γ✝ δ✝ : Type inst✝¹ : Complexity.Coding γ✝ Memory inst✝ : Complexity.Encoding δ✝ Memory fs✝ : γ✝ → δ✝ cs✝ : Complexity.CostFunction γ✝ ℕ hcost : Complexity Encoding.Model fs✝ cs✝ next : CostedProgram h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm a : α m : Memory hm : m ∈ fm a ⊢ ℕ α : Type u_1 inst✝⁷ : Complexity.Coding α Memory β : Type u_2 inst✝⁶ : Complexity.Encoding β Memory f : α → β tp : TracedProgram γ δ : Type inst✝⁵ : Complexity.Coding γ Memory inst✝⁴ : Complexity.Encoding δ Memory fs : γ → δ cs : Complexity.CostFunction γ ℕ inst✝³ : Complexity.Computable Encoding.Model fs inst✝² : 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 γ✝ δ✝ : Type inst✝¹ : Complexity.Coding γ✝ Memory inst✝ : Complexity.Encoding δ✝ Memory fs✝ : γ✝ → δ✝ cs✝ : Complexity.CostFunction γ✝ ℕ hcost : Complexity Encoding.Model fs✝ cs✝ next : CostedProgram h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm a : α m : Memory hm : m ∈ fm a ⊢ ℕ α : Type u_1 inst✝⁷ : Complexity.Coding α Memory β : Type u_2 inst✝⁶ : Complexity.Encoding β Memory f : α → β tp : TracedProgram γ δ : Type inst✝⁵ : Complexity.Coding γ Memory inst✝⁴ : Complexity.Encoding δ Memory fs : γ → δ cs : Complexity.CostFunction γ ℕ inst✝³ : Complexity.Computable Encoding.Model fs inst✝² : 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 γ✝ δ✝ : Type inst✝¹ : Complexity.Coding γ✝ Memory inst✝ : Complexity.Encoding δ✝ Memory fs✝ : γ✝ → δ✝ cs✝ : Complexity.CostFunction γ✝ ℕ hcost : Complexity Encoding.Model fs✝ cs✝ next : CostedProgram h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm a : α m : Memory hm : m ∈ fm a ⊢ ?m.173010 + ?m.173011 + 1 ≤ (subroutineTimeCost h + recurseTimeCost hf h + localTimeCost h) a
α : Type u_1 inst✝⁷ : Complexity.Coding α Memory β : Type u_2 inst✝⁶ : Complexity.Encoding β Memory f : α → β tp : TracedProgram γ δ : Type inst✝⁵ : Complexity.Coding γ Memory inst✝⁴ : Complexity.Encoding δ Memory fs : γ → δ cs : Complexity.CostFunction γ ℕ inst✝³ : Complexity.Computable Encoding.Model fs inst✝² : 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 γ✝ δ✝ : Type inst✝¹ : Complexity.Coding γ✝ Memory inst✝ : Complexity.Encoding δ✝ Memory fs✝ : γ✝ → δ✝ cs✝ : Complexity.CostFunction γ✝ ℕ hcost : Complexity Encoding.Model fs✝ cs✝ next : CostedProgram h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm a : α m : Memory hm : m ∈ fm a ⊢ Stack.halts (Stack.execution { state := Complexity.encode (Option.get (Complexity.decode γ✝ (Memory.getms m src✝)) (_ : Option.isSome (Complexity.decode γ✝ (Memory.getms m src✝)) = true)), current := Encoding.getProgram fs✝, function := Encoding.getProgram fs✝ } []) case h₁ α : Type u_1 inst✝⁷ : Complexity.Coding α Memory β : Type u_2 inst✝⁶ : Complexity.Encoding β Memory f : α → β tp : TracedProgram γ δ : Type inst✝⁵ : Complexity.Coding γ Memory inst✝⁴ : Complexity.Encoding δ Memory fs : γ → δ cs : Complexity.CostFunction γ ℕ inst✝³ : Complexity.Computable Encoding.Model fs inst✝² : 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 γ✝ δ✝ : Type inst✝¹ : Complexity.Coding γ✝ Memory inst✝ : Complexity.Encoding δ✝ Memory fs✝ : γ✝ → δ✝ cs✝ : Complexity.CostFunction γ✝ ℕ hcost : Complexity Encoding.Model fs✝ cs✝ next : CostedProgram h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm a : α m : Memory hm : m ∈ fm a ⊢ Stack.hasTimeCost (Stack.execution { state := Memory.setms m dst✝ (Complexity.encode (fs✝ (Option.get (Complexity.decode γ✝ (Memory.getms m src✝)) (_ : Option.isSome (Complexity.decode γ✝ (Memory.getms m src✝)) = true)))), current := TracedProgram.toProgram (toTracedProgram next), function := toProgram fcp } []) ?m.173010 α : Type u_1 inst✝⁷ : Complexity.Coding α Memory β : Type u_2 inst✝⁶ : Complexity.Encoding β Memory f : α → β tp : TracedProgram γ δ : Type inst✝⁵ : Complexity.Coding γ Memory inst✝⁴ : Complexity.Encoding δ Memory fs : γ → δ cs : Complexity.CostFunction γ ℕ inst✝³ : Complexity.Computable Encoding.Model fs inst✝² : 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 γ✝ δ✝ : Type inst✝¹ : Complexity.Coding γ✝ Memory inst✝ : Complexity.Encoding δ✝ Memory fs✝ : γ✝ → δ✝ cs✝ : Complexity.CostFunction γ✝ ℕ hcost : Complexity Encoding.Model fs✝ cs✝ next : CostedProgram h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm a : α m : Memory hm : m ∈ fm a ⊢ ℕ α : Type u_1 inst✝⁷ : Complexity.Coding α Memory β : Type u_2 inst✝⁶ : Complexity.Encoding β Memory f : α → β tp : TracedProgram γ δ : Type inst✝⁵ : Complexity.Coding γ Memory inst✝⁴ : Complexity.Encoding δ Memory fs : γ → δ cs : Complexity.CostFunction γ ℕ inst✝³ : Complexity.Computable Encoding.Model fs inst✝² : 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 γ✝ δ✝ : Type inst✝¹ : Complexity.Coding γ✝ Memory inst✝ : Complexity.Encoding δ✝ Memory fs✝ : γ✝ → δ✝ cs✝ : Complexity.CostFunction γ✝ ℕ hcost : Complexity Encoding.Model fs✝ cs✝ next : CostedProgram h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm a : α m : Memory hm : m ∈ fm a ⊢ ?m.173010 + Complexity.ALE.bound Complexity.cost_ale (Option.get (Complexity.decode γ✝ (Memory.getms m src✝)) (_ : Option.isSome (Complexity.decode γ✝ (Memory.getms m src✝)) = true)) + 1 ≤ (subroutineTimeCost h + recurseTimeCost hf h + localTimeCost h) a
https://github.com/calcu16/lean_complexity.git
d267eb4f3952c654f130a119311692c1c1a7cd71
HMem/Trace/Cost.lean
HMem.Trace.CostedProgram.splitTimeCost
[244, 1]
[349, 16]
exact (hcost.toComputable.has_result _).elim λ _ h ↦ ⟨(_, _), h⟩
α : Type u_1 inst✝⁷ : Complexity.Coding α Memory β : Type u_2 inst✝⁶ : Complexity.Encoding β Memory f : α → β tp : TracedProgram γ δ : Type inst✝⁵ : Complexity.Coding γ Memory inst✝⁴ : Complexity.Encoding δ Memory fs : γ → δ cs : Complexity.CostFunction γ ℕ inst✝³ : Complexity.Computable Encoding.Model fs inst✝² : 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 γ✝ δ✝ : Type inst✝¹ : Complexity.Coding γ✝ Memory inst✝ : Complexity.Encoding δ✝ Memory fs✝ : γ✝ → δ✝ cs✝ : Complexity.CostFunction γ✝ ℕ hcost : Complexity Encoding.Model fs✝ cs✝ next : CostedProgram h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm a : α m : Memory hm : m ∈ fm a ⊢ Stack.halts (Stack.execution { state := Complexity.encode (Option.get (Complexity.decode γ✝ (Memory.getms m src✝)) (_ : Option.isSome (Complexity.decode γ✝ (Memory.getms m src✝)) = true)), current := Encoding.getProgram fs✝, function := Encoding.getProgram fs✝ } []) case h₁ α : Type u_1 inst✝⁷ : Complexity.Coding α Memory β : Type u_2 inst✝⁶ : Complexity.Encoding β Memory f : α → β tp : TracedProgram γ δ : Type inst✝⁵ : Complexity.Coding γ Memory inst✝⁴ : Complexity.Encoding δ Memory fs : γ → δ cs : Complexity.CostFunction γ ℕ inst✝³ : Complexity.Computable Encoding.Model fs inst✝² : 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 γ✝ δ✝ : Type inst✝¹ : Complexity.Coding γ✝ Memory inst✝ : Complexity.Encoding δ✝ Memory fs✝ : γ✝ → δ✝ cs✝ : Complexity.CostFunction γ✝ ℕ hcost : Complexity Encoding.Model fs✝ cs✝ next : CostedProgram h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm a : α m : Memory hm : m ∈ fm a ⊢ Stack.hasTimeCost (Stack.execution { state := Memory.setms m dst✝ (Complexity.encode (fs✝ (Option.get (Complexity.decode γ✝ (Memory.getms m src✝)) (_ : Option.isSome (Complexity.decode γ✝ (Memory.getms m src✝)) = true)))), current := TracedProgram.toProgram (toTracedProgram next), function := toProgram fcp } []) ?m.173010 α : Type u_1 inst✝⁷ : Complexity.Coding α Memory β : Type u_2 inst✝⁶ : Complexity.Encoding β Memory f : α → β tp : TracedProgram γ δ : Type inst✝⁵ : Complexity.Coding γ Memory inst✝⁴ : Complexity.Encoding δ Memory fs : γ → δ cs : Complexity.CostFunction γ ℕ inst✝³ : Complexity.Computable Encoding.Model fs inst✝² : 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 γ✝ δ✝ : Type inst✝¹ : Complexity.Coding γ✝ Memory inst✝ : Complexity.Encoding δ✝ Memory fs✝ : γ✝ → δ✝ cs✝ : Complexity.CostFunction γ✝ ℕ hcost : Complexity Encoding.Model fs✝ cs✝ next : CostedProgram h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm a : α m : Memory hm : m ∈ fm a ⊢ ℕ α : Type u_1 inst✝⁷ : Complexity.Coding α Memory β : Type u_2 inst✝⁶ : Complexity.Encoding β Memory f : α → β tp : TracedProgram γ δ : Type inst✝⁵ : Complexity.Coding γ Memory inst✝⁴ : Complexity.Encoding δ Memory fs : γ → δ cs : Complexity.CostFunction γ ℕ inst✝³ : Complexity.Computable Encoding.Model fs inst✝² : 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 γ✝ δ✝ : Type inst✝¹ : Complexity.Coding γ✝ Memory inst✝ : Complexity.Encoding δ✝ Memory fs✝ : γ✝ → δ✝ cs✝ : Complexity.CostFunction γ✝ ℕ hcost : Complexity Encoding.Model fs✝ cs✝ next : CostedProgram h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm a : α m : Memory hm : m ∈ fm a ⊢ ?m.173010 + Complexity.ALE.bound Complexity.cost_ale (Option.get (Complexity.decode γ✝ (Memory.getms m src✝)) (_ : Option.isSome (Complexity.decode γ✝ (Memory.getms m src✝)) = true)) + 1 ≤ (subroutineTimeCost h + recurseTimeCost hf h + localTimeCost h) a
case h₁ α : Type u_1 inst✝⁷ : Complexity.Coding α Memory β : Type u_2 inst✝⁶ : Complexity.Encoding β Memory f : α → β tp : TracedProgram γ δ : Type inst✝⁵ : Complexity.Coding γ Memory inst✝⁴ : Complexity.Encoding δ Memory fs : γ → δ cs : Complexity.CostFunction γ ℕ inst✝³ : Complexity.Computable Encoding.Model fs inst✝² : 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 γ✝ δ✝ : Type inst✝¹ : Complexity.Coding γ✝ Memory inst✝ : Complexity.Encoding δ✝ Memory fs✝ : γ✝ → δ✝ cs✝ : Complexity.CostFunction γ✝ ℕ hcost : Complexity Encoding.Model fs✝ cs✝ next : CostedProgram h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm a : α m : Memory hm : m ∈ fm a ⊢ Stack.hasTimeCost (Stack.execution { state := Memory.setms m dst✝ (Complexity.encode (fs✝ (Option.get (Complexity.decode γ✝ (Memory.getms m src✝)) (_ : Option.isSome (Complexity.decode γ✝ (Memory.getms m src✝)) = true)))), current := TracedProgram.toProgram (toTracedProgram next), function := toProgram fcp } []) ?m.173010 α : Type u_1 inst✝⁷ : Complexity.Coding α Memory β : Type u_2 inst✝⁶ : Complexity.Encoding β Memory f : α → β tp : TracedProgram γ δ : Type inst✝⁵ : Complexity.Coding γ Memory inst✝⁴ : Complexity.Encoding δ Memory fs : γ → δ cs : Complexity.CostFunction γ ℕ inst✝³ : Complexity.Computable Encoding.Model fs inst✝² : 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 γ✝ δ✝ : Type inst✝¹ : Complexity.Coding γ✝ Memory inst✝ : Complexity.Encoding δ✝ Memory fs✝ : γ✝ → δ✝ cs✝ : Complexity.CostFunction γ✝ ℕ hcost : Complexity Encoding.Model fs✝ cs✝ next : CostedProgram h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm a : α m : Memory hm : m ∈ fm a ⊢ ℕ α : Type u_1 inst✝⁷ : Complexity.Coding α Memory β : Type u_2 inst✝⁶ : Complexity.Encoding β Memory f : α → β tp : TracedProgram γ δ : Type inst✝⁵ : Complexity.Coding γ Memory inst✝⁴ : Complexity.Encoding δ Memory fs : γ → δ cs : Complexity.CostFunction γ ℕ inst✝³ : Complexity.Computable Encoding.Model fs inst✝² : 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 γ✝ δ✝ : Type inst✝¹ : Complexity.Coding γ✝ Memory inst✝ : Complexity.Encoding δ✝ Memory fs✝ : γ✝ → δ✝ cs✝ : Complexity.CostFunction γ✝ ℕ hcost : Complexity Encoding.Model fs✝ cs✝ next : CostedProgram h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm a : α m : Memory hm : m ∈ fm a ⊢ ?m.173010 + Complexity.ALE.bound Complexity.cost_ale (Option.get (Complexity.decode γ✝ (Memory.getms m src✝)) (_ : Option.isSome (Complexity.decode γ✝ (Memory.getms m src✝)) = true)) + 1 ≤ (subroutineTimeCost h + recurseTimeCost hf h + localTimeCost h) a
https://github.com/calcu16/lean_complexity.git
d267eb4f3952c654f130a119311692c1c1a7cd71
HMem/Trace/Cost.lean
HMem.Trace.CostedProgram.splitTimeCost
[244, 1]
[349, 16]
apply splitTimeCost hf (TracedProgram.sound'_subroutine_next h)
case h₁ α : Type u_1 inst✝⁷ : Complexity.Coding α Memory β : Type u_2 inst✝⁶ : Complexity.Encoding β Memory f : α → β tp : TracedProgram γ δ : Type inst✝⁵ : Complexity.Coding γ Memory inst✝⁴ : Complexity.Encoding δ Memory fs : γ → δ cs : Complexity.CostFunction γ ℕ inst✝³ : Complexity.Computable Encoding.Model fs inst✝² : 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 γ✝ δ✝ : Type inst✝¹ : Complexity.Coding γ✝ Memory inst✝ : Complexity.Encoding δ✝ Memory fs✝ : γ✝ → δ✝ cs✝ : Complexity.CostFunction γ✝ ℕ hcost : Complexity Encoding.Model fs✝ cs✝ next : CostedProgram h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm a : α m : Memory hm : m ∈ fm a ⊢ Stack.hasTimeCost (Stack.execution { state := Memory.setms m dst✝ (Complexity.encode (fs✝ (Option.get (Complexity.decode γ✝ (Memory.getms m src✝)) (_ : Option.isSome (Complexity.decode γ✝ (Memory.getms m src✝)) = true)))), current := TracedProgram.toProgram (toTracedProgram next), function := toProgram fcp } []) ?m.173010 α : Type u_1 inst✝⁷ : Complexity.Coding α Memory β : Type u_2 inst✝⁶ : Complexity.Encoding β Memory f : α → β tp : TracedProgram γ δ : Type inst✝⁵ : Complexity.Coding γ Memory inst✝⁴ : Complexity.Encoding δ Memory fs : γ → δ cs : Complexity.CostFunction γ ℕ inst✝³ : Complexity.Computable Encoding.Model fs inst✝² : 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 γ✝ δ✝ : Type inst✝¹ : Complexity.Coding γ✝ Memory inst✝ : Complexity.Encoding δ✝ Memory fs✝ : γ✝ → δ✝ cs✝ : Complexity.CostFunction γ✝ ℕ hcost : Complexity Encoding.Model fs✝ cs✝ next : CostedProgram h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm a : α m : Memory hm : m ∈ fm a ⊢ ℕ α : Type u_1 inst✝⁷ : Complexity.Coding α Memory β : Type u_2 inst✝⁶ : Complexity.Encoding β Memory f : α → β tp : TracedProgram γ δ : Type inst✝⁵ : Complexity.Coding γ Memory inst✝⁴ : Complexity.Encoding δ Memory fs : γ → δ cs : Complexity.CostFunction γ ℕ inst✝³ : Complexity.Computable Encoding.Model fs inst✝² : 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 γ✝ δ✝ : Type inst✝¹ : Complexity.Coding γ✝ Memory inst✝ : Complexity.Encoding δ✝ Memory fs✝ : γ✝ → δ✝ cs✝ : Complexity.CostFunction γ✝ ℕ hcost : Complexity Encoding.Model fs✝ cs✝ next : CostedProgram h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm a : α m : Memory hm : m ∈ fm a ⊢ ?m.173010 + Complexity.ALE.bound Complexity.cost_ale (Option.get (Complexity.decode γ✝ (Memory.getms m src✝)) (_ : Option.isSome (Complexity.decode γ✝ (Memory.getms m src✝)) = true)) + 1 ≤ (subroutineTimeCost h + recurseTimeCost hf h + localTimeCost h) a
case h₁.a α : Type u_1 inst✝⁷ : Complexity.Coding α Memory β : Type u_2 inst✝⁶ : Complexity.Encoding β Memory f : α → β tp : TracedProgram γ δ : Type inst✝⁵ : Complexity.Coding γ Memory inst✝⁴ : Complexity.Encoding δ Memory fs : γ → δ cs : Complexity.CostFunction γ ℕ inst✝³ : Complexity.Computable Encoding.Model fs inst✝² : 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 γ✝ δ✝ : Type inst✝¹ : Complexity.Coding γ✝ Memory inst✝ : Complexity.Encoding δ✝ Memory fs✝ : γ✝ → δ✝ cs✝ : Complexity.CostFunction γ✝ ℕ hcost : Complexity Encoding.Model fs✝ cs✝ next : CostedProgram h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm a : α m : Memory hm : m ∈ fm a ⊢ Memory.setms m dst✝ (Complexity.encode (fs✝ (Option.get (Complexity.decode γ✝ (Memory.getms m src✝)) (_ : Option.isSome (Complexity.decode γ✝ (Memory.getms m src✝)) = true)))) ∈ ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ fs✝) (Complexity.decode γ✝ (Memory.getms m src✝))) ∘ fm) ?h₁.a case h₁.a α : Type u_1 inst✝⁷ : Complexity.Coding α Memory β : Type u_2 inst✝⁶ : Complexity.Encoding β Memory f : α → β tp : TracedProgram γ δ : Type inst✝⁵ : Complexity.Coding γ Memory inst✝⁴ : Complexity.Encoding δ Memory fs : γ → δ cs : Complexity.CostFunction γ ℕ inst✝³ : Complexity.Computable Encoding.Model fs inst✝² : 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 γ✝ δ✝ : Type inst✝¹ : Complexity.Coding γ✝ Memory inst✝ : Complexity.Encoding δ✝ Memory fs✝ : γ✝ → δ✝ cs✝ : Complexity.CostFunction γ✝ ℕ hcost : Complexity Encoding.Model fs✝ cs✝ next : CostedProgram h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm a : α m : Memory hm : m ∈ fm a ⊢ α α : Type u_1 inst✝⁷ : Complexity.Coding α Memory β : Type u_2 inst✝⁶ : Complexity.Encoding β Memory f : α → β tp : TracedProgram γ δ : Type inst✝⁵ : Complexity.Coding γ Memory inst✝⁴ : Complexity.Encoding δ Memory fs : γ → δ cs : Complexity.CostFunction γ ℕ inst✝³ : Complexity.Computable Encoding.Model fs inst✝² : 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 γ✝ δ✝ : Type inst✝¹ : Complexity.Coding γ✝ Memory inst✝ : Complexity.Encoding δ✝ Memory fs✝ : γ✝ → δ✝ cs✝ : Complexity.CostFunction γ✝ ℕ hcost : Complexity Encoding.Model fs✝ cs✝ next : CostedProgram h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm a : α m : Memory hm : m ∈ fm a ⊢ (subroutineTimeCost (_ : TracedProgram.sound' f size (toTracedProgram next) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ fs✝) (Complexity.decode γ✝ (Memory.getms m src✝))) ∘ fm)) + recurseTimeCost hf (_ : TracedProgram.sound' f size (toTracedProgram next) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ fs✝) (Complexity.decode γ✝ (Memory.getms m src✝))) ∘ fm)) + localTimeCost (_ : TracedProgram.sound' f size (toTracedProgram next) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ fs✝) (Complexity.decode γ✝ (Memory.getms m src✝))) ∘ fm))) ?h₁.a + Complexity.ALE.bound Complexity.cost_ale (Option.get (Complexity.decode γ✝ (Memory.getms m src✝)) (_ : Option.isSome (Complexity.decode γ✝ (Memory.getms m src✝)) = true)) + 1 ≤ (subroutineTimeCost h + recurseTimeCost hf h + localTimeCost h) a
https://github.com/calcu16/lean_complexity.git
d267eb4f3952c654f130a119311692c1c1a7cd71
HMem/Trace/Cost.lean
HMem.Trace.CostedProgram.splitTimeCost
[244, 1]
[349, 16]
apply Option.mem_bind_of_mem hm
case h₁.a α : Type u_1 inst✝⁷ : Complexity.Coding α Memory β : Type u_2 inst✝⁶ : Complexity.Encoding β Memory f : α → β tp : TracedProgram γ δ : Type inst✝⁵ : Complexity.Coding γ Memory inst✝⁴ : Complexity.Encoding δ Memory fs : γ → δ cs : Complexity.CostFunction γ ℕ inst✝³ : Complexity.Computable Encoding.Model fs inst✝² : 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 γ✝ δ✝ : Type inst✝¹ : Complexity.Coding γ✝ Memory inst✝ : Complexity.Encoding δ✝ Memory fs✝ : γ✝ → δ✝ cs✝ : Complexity.CostFunction γ✝ ℕ hcost : Complexity Encoding.Model fs✝ cs✝ next : CostedProgram h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm a : α m : Memory hm : m ∈ fm a ⊢ Memory.setms m dst✝ (Complexity.encode (fs✝ (Option.get (Complexity.decode γ✝ (Memory.getms m src✝)) (_ : Option.isSome (Complexity.decode γ✝ (Memory.getms m src✝)) = true)))) ∈ ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ fs✝) (Complexity.decode γ✝ (Memory.getms m src✝))) ∘ fm) ?h₁.a case h₁.a α : Type u_1 inst✝⁷ : Complexity.Coding α Memory β : Type u_2 inst✝⁶ : Complexity.Encoding β Memory f : α → β tp : TracedProgram γ δ : Type inst✝⁵ : Complexity.Coding γ Memory inst✝⁴ : Complexity.Encoding δ Memory fs : γ → δ cs : Complexity.CostFunction γ ℕ inst✝³ : Complexity.Computable Encoding.Model fs inst✝² : 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 γ✝ δ✝ : Type inst✝¹ : Complexity.Coding γ✝ Memory inst✝ : Complexity.Encoding δ✝ Memory fs✝ : γ✝ → δ✝ cs✝ : Complexity.CostFunction γ✝ ℕ hcost : Complexity Encoding.Model fs✝ cs✝ next : CostedProgram h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm a : α m : Memory hm : m ∈ fm a ⊢ α α : Type u_1 inst✝⁷ : Complexity.Coding α Memory β : Type u_2 inst✝⁶ : Complexity.Encoding β Memory f : α → β tp : TracedProgram γ δ : Type inst✝⁵ : Complexity.Coding γ Memory inst✝⁴ : Complexity.Encoding δ Memory fs : γ → δ cs : Complexity.CostFunction γ ℕ inst✝³ : Complexity.Computable Encoding.Model fs inst✝² : 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 γ✝ δ✝ : Type inst✝¹ : Complexity.Coding γ✝ Memory inst✝ : Complexity.Encoding δ✝ Memory fs✝ : γ✝ → δ✝ cs✝ : Complexity.CostFunction γ✝ ℕ hcost : Complexity Encoding.Model fs✝ cs✝ next : CostedProgram h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm a : α m : Memory hm : m ∈ fm a ⊢ (subroutineTimeCost (_ : TracedProgram.sound' f size (toTracedProgram next) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ fs✝) (Complexity.decode γ✝ (Memory.getms m src✝))) ∘ fm)) + recurseTimeCost hf (_ : TracedProgram.sound' f size (toTracedProgram next) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ fs✝) (Complexity.decode γ✝ (Memory.getms m src✝))) ∘ fm)) + localTimeCost (_ : TracedProgram.sound' f size (toTracedProgram next) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ fs✝) (Complexity.decode γ✝ (Memory.getms m src✝))) ∘ fm))) ?h₁.a + Complexity.ALE.bound Complexity.cost_ale (Option.get (Complexity.decode γ✝ (Memory.getms m src✝)) (_ : Option.isSome (Complexity.decode γ✝ (Memory.getms m src✝)) = true)) + 1 ≤ (subroutineTimeCost h + recurseTimeCost hf h + localTimeCost h) a
case h₁.a α : Type u_1 inst✝⁷ : Complexity.Coding α Memory β : Type u_2 inst✝⁶ : Complexity.Encoding β Memory f : α → β tp : TracedProgram γ δ : Type inst✝⁵ : Complexity.Coding γ Memory inst✝⁴ : Complexity.Encoding δ Memory fs : γ → δ cs : Complexity.CostFunction γ ℕ inst✝³ : Complexity.Computable Encoding.Model fs inst✝² : 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 γ✝ δ✝ : Type inst✝¹ : Complexity.Coding γ✝ Memory inst✝ : Complexity.Encoding δ✝ Memory fs✝ : γ✝ → δ✝ cs✝ : Complexity.CostFunction γ✝ ℕ hcost : Complexity Encoding.Model fs✝ cs✝ next : CostedProgram h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm a : α m : Memory hm : m ∈ fm a ⊢ Memory.setms m dst✝ (Complexity.encode (fs✝ (Option.get (Complexity.decode γ✝ (Memory.getms m src✝)) (_ : Option.isSome (Complexity.decode γ✝ (Memory.getms m src✝)) = true)))) ∈ Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ fs✝) (Complexity.decode γ✝ (Memory.getms m src✝)) α : Type u_1 inst✝⁷ : Complexity.Coding α Memory β : Type u_2 inst✝⁶ : Complexity.Encoding β Memory f : α → β tp : TracedProgram γ δ : Type inst✝⁵ : Complexity.Coding γ Memory inst✝⁴ : Complexity.Encoding δ Memory fs : γ → δ cs : Complexity.CostFunction γ ℕ inst✝³ : Complexity.Computable Encoding.Model fs inst✝² : 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 γ✝ δ✝ : Type inst✝¹ : Complexity.Coding γ✝ Memory inst✝ : Complexity.Encoding δ✝ Memory fs✝ : γ✝ → δ✝ cs✝ : Complexity.CostFunction γ✝ ℕ hcost : Complexity Encoding.Model fs✝ cs✝ next : CostedProgram h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm a : α m : Memory hm : m ∈ fm a ⊢ (subroutineTimeCost (_ : TracedProgram.sound' f size (toTracedProgram next) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ fs✝) (Complexity.decode γ✝ (Memory.getms m src✝))) ∘ fm)) + recurseTimeCost hf (_ : TracedProgram.sound' f size (toTracedProgram next) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ fs✝) (Complexity.decode γ✝ (Memory.getms m src✝))) ∘ fm)) + localTimeCost (_ : TracedProgram.sound' f size (toTracedProgram next) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ fs✝) (Complexity.decode γ✝ (Memory.getms m src✝))) ∘ fm))) a + Complexity.ALE.bound Complexity.cost_ale (Option.get (Complexity.decode γ✝ (Memory.getms m src✝)) (_ : Option.isSome (Complexity.decode γ✝ (Memory.getms m src✝)) = true)) + 1 ≤ (subroutineTimeCost h + recurseTimeCost hf h + localTimeCost h) a
https://github.com/calcu16/lean_complexity.git
d267eb4f3952c654f130a119311692c1c1a7cd71
HMem/Trace/Cost.lean
HMem.Trace.CostedProgram.splitTimeCost
[244, 1]
[349, 16]
apply Option.mem_map_of_mem
case h₁.a α : Type u_1 inst✝⁷ : Complexity.Coding α Memory β : Type u_2 inst✝⁶ : Complexity.Encoding β Memory f : α → β tp : TracedProgram γ δ : Type inst✝⁵ : Complexity.Coding γ Memory inst✝⁴ : Complexity.Encoding δ Memory fs : γ → δ cs : Complexity.CostFunction γ ℕ inst✝³ : Complexity.Computable Encoding.Model fs inst✝² : 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 γ✝ δ✝ : Type inst✝¹ : Complexity.Coding γ✝ Memory inst✝ : Complexity.Encoding δ✝ Memory fs✝ : γ✝ → δ✝ cs✝ : Complexity.CostFunction γ✝ ℕ hcost : Complexity Encoding.Model fs✝ cs✝ next : CostedProgram h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm a : α m : Memory hm : m ∈ fm a ⊢ Memory.setms m dst✝ (Complexity.encode (fs✝ (Option.get (Complexity.decode γ✝ (Memory.getms m src✝)) (_ : Option.isSome (Complexity.decode γ✝ (Memory.getms m src✝)) = true)))) ∈ Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ fs✝) (Complexity.decode γ✝ (Memory.getms m src✝)) α : Type u_1 inst✝⁷ : Complexity.Coding α Memory β : Type u_2 inst✝⁶ : Complexity.Encoding β Memory f : α → β tp : TracedProgram γ δ : Type inst✝⁵ : Complexity.Coding γ Memory inst✝⁴ : Complexity.Encoding δ Memory fs : γ → δ cs : Complexity.CostFunction γ ℕ inst✝³ : Complexity.Computable Encoding.Model fs inst✝² : 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 γ✝ δ✝ : Type inst✝¹ : Complexity.Coding γ✝ Memory inst✝ : Complexity.Encoding δ✝ Memory fs✝ : γ✝ → δ✝ cs✝ : Complexity.CostFunction γ✝ ℕ hcost : Complexity Encoding.Model fs✝ cs✝ next : CostedProgram h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm a : α m : Memory hm : m ∈ fm a ⊢ (subroutineTimeCost (_ : TracedProgram.sound' f size (toTracedProgram next) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ fs✝) (Complexity.decode γ✝ (Memory.getms m src✝))) ∘ fm)) + recurseTimeCost hf (_ : TracedProgram.sound' f size (toTracedProgram next) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ fs✝) (Complexity.decode γ✝ (Memory.getms m src✝))) ∘ fm)) + localTimeCost (_ : TracedProgram.sound' f size (toTracedProgram next) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ fs✝) (Complexity.decode γ✝ (Memory.getms m src✝))) ∘ fm))) a + Complexity.ALE.bound Complexity.cost_ale (Option.get (Complexity.decode γ✝ (Memory.getms m src✝)) (_ : Option.isSome (Complexity.decode γ✝ (Memory.getms m src✝)) = true)) + 1 ≤ (subroutineTimeCost h + recurseTimeCost hf h + localTimeCost h) a
case h₁.a.h α : Type u_1 inst✝⁷ : Complexity.Coding α Memory β : Type u_2 inst✝⁶ : Complexity.Encoding β Memory f : α → β tp : TracedProgram γ δ : Type inst✝⁵ : Complexity.Coding γ Memory inst✝⁴ : Complexity.Encoding δ Memory fs : γ → δ cs : Complexity.CostFunction γ ℕ inst✝³ : Complexity.Computable Encoding.Model fs inst✝² : 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 γ✝ δ✝ : Type inst✝¹ : Complexity.Coding γ✝ Memory inst✝ : Complexity.Encoding δ✝ Memory fs✝ : γ✝ → δ✝ cs✝ : Complexity.CostFunction γ✝ ℕ hcost : Complexity Encoding.Model fs✝ cs✝ next : CostedProgram h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm a : α m : Memory hm : m ∈ fm a ⊢ Option.get (Complexity.decode γ✝ (Memory.getms m src✝)) (_ : Option.isSome (Complexity.decode γ✝ (Memory.getms m src✝)) = true) ∈ Complexity.decode γ✝ (Memory.getms m src✝) α : Type u_1 inst✝⁷ : Complexity.Coding α Memory β : Type u_2 inst✝⁶ : Complexity.Encoding β Memory f : α → β tp : TracedProgram γ δ : Type inst✝⁵ : Complexity.Coding γ Memory inst✝⁴ : Complexity.Encoding δ Memory fs : γ → δ cs : Complexity.CostFunction γ ℕ inst✝³ : Complexity.Computable Encoding.Model fs inst✝² : 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 γ✝ δ✝ : Type inst✝¹ : Complexity.Coding γ✝ Memory inst✝ : Complexity.Encoding δ✝ Memory fs✝ : γ✝ → δ✝ cs✝ : Complexity.CostFunction γ✝ ℕ hcost : Complexity Encoding.Model fs✝ cs✝ next : CostedProgram h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm a : α m : Memory hm : m ∈ fm a ⊢ (subroutineTimeCost (_ : TracedProgram.sound' f size (toTracedProgram next) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ fs✝) (Complexity.decode γ✝ (Memory.getms m src✝))) ∘ fm)) + recurseTimeCost hf (_ : TracedProgram.sound' f size (toTracedProgram next) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ fs✝) (Complexity.decode γ✝ (Memory.getms m src✝))) ∘ fm)) + localTimeCost (_ : TracedProgram.sound' f size (toTracedProgram next) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ fs✝) (Complexity.decode γ✝ (Memory.getms m src✝))) ∘ fm))) a + Complexity.ALE.bound Complexity.cost_ale (Option.get (Complexity.decode γ✝ (Memory.getms m src✝)) (_ : Option.isSome (Complexity.decode γ✝ (Memory.getms m src✝)) = true)) + 1 ≤ (subroutineTimeCost h + recurseTimeCost hf h + localTimeCost h) a
https://github.com/calcu16/lean_complexity.git
d267eb4f3952c654f130a119311692c1c1a7cd71
HMem/Trace/Cost.lean
HMem.Trace.CostedProgram.splitTimeCost
[244, 1]
[349, 16]
apply Option.get_mem
case h₁.a.h α : Type u_1 inst✝⁷ : Complexity.Coding α Memory β : Type u_2 inst✝⁶ : Complexity.Encoding β Memory f : α → β tp : TracedProgram γ δ : Type inst✝⁵ : Complexity.Coding γ Memory inst✝⁴ : Complexity.Encoding δ Memory fs : γ → δ cs : Complexity.CostFunction γ ℕ inst✝³ : Complexity.Computable Encoding.Model fs inst✝² : 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 γ✝ δ✝ : Type inst✝¹ : Complexity.Coding γ✝ Memory inst✝ : Complexity.Encoding δ✝ Memory fs✝ : γ✝ → δ✝ cs✝ : Complexity.CostFunction γ✝ ℕ hcost : Complexity Encoding.Model fs✝ cs✝ next : CostedProgram h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm a : α m : Memory hm : m ∈ fm a ⊢ Option.get (Complexity.decode γ✝ (Memory.getms m src✝)) (_ : Option.isSome (Complexity.decode γ✝ (Memory.getms m src✝)) = true) ∈ Complexity.decode γ✝ (Memory.getms m src✝) α : Type u_1 inst✝⁷ : Complexity.Coding α Memory β : Type u_2 inst✝⁶ : Complexity.Encoding β Memory f : α → β tp : TracedProgram γ δ : Type inst✝⁵ : Complexity.Coding γ Memory inst✝⁴ : Complexity.Encoding δ Memory fs : γ → δ cs : Complexity.CostFunction γ ℕ inst✝³ : Complexity.Computable Encoding.Model fs inst✝² : 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 γ✝ δ✝ : Type inst✝¹ : Complexity.Coding γ✝ Memory inst✝ : Complexity.Encoding δ✝ Memory fs✝ : γ✝ → δ✝ cs✝ : Complexity.CostFunction γ✝ ℕ hcost : Complexity Encoding.Model fs✝ cs✝ next : CostedProgram h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm a : α m : Memory hm : m ∈ fm a ⊢ (subroutineTimeCost (_ : TracedProgram.sound' f size (toTracedProgram next) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ fs✝) (Complexity.decode γ✝ (Memory.getms m src✝))) ∘ fm)) + recurseTimeCost hf (_ : TracedProgram.sound' f size (toTracedProgram next) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ fs✝) (Complexity.decode γ✝ (Memory.getms m src✝))) ∘ fm)) + localTimeCost (_ : TracedProgram.sound' f size (toTracedProgram next) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ fs✝) (Complexity.decode γ✝ (Memory.getms m src✝))) ∘ fm))) a + Complexity.ALE.bound Complexity.cost_ale (Option.get (Complexity.decode γ✝ (Memory.getms m src✝)) (_ : Option.isSome (Complexity.decode γ✝ (Memory.getms m src✝)) = true)) + 1 ≤ (subroutineTimeCost h + recurseTimeCost hf h + localTimeCost h) a
α : Type u_1 inst✝⁷ : Complexity.Coding α Memory β : Type u_2 inst✝⁶ : Complexity.Encoding β Memory f : α → β tp : TracedProgram γ δ : Type inst✝⁵ : Complexity.Coding γ Memory inst✝⁴ : Complexity.Encoding δ Memory fs : γ → δ cs : Complexity.CostFunction γ ℕ inst✝³ : Complexity.Computable Encoding.Model fs inst✝² : 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 γ✝ δ✝ : Type inst✝¹ : Complexity.Coding γ✝ Memory inst✝ : Complexity.Encoding δ✝ Memory fs✝ : γ✝ → δ✝ cs✝ : Complexity.CostFunction γ✝ ℕ hcost : Complexity Encoding.Model fs✝ cs✝ next : CostedProgram h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm a : α m : Memory hm : m ∈ fm a ⊢ (subroutineTimeCost (_ : TracedProgram.sound' f size (toTracedProgram next) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ fs✝) (Complexity.decode γ✝ (Memory.getms m src✝))) ∘ fm)) + recurseTimeCost hf (_ : TracedProgram.sound' f size (toTracedProgram next) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ fs✝) (Complexity.decode γ✝ (Memory.getms m src✝))) ∘ fm)) + localTimeCost (_ : TracedProgram.sound' f size (toTracedProgram next) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ fs✝) (Complexity.decode γ✝ (Memory.getms m src✝))) ∘ fm))) a + Complexity.ALE.bound Complexity.cost_ale (Option.get (Complexity.decode γ✝ (Memory.getms m src✝)) (_ : Option.isSome (Complexity.decode γ✝ (Memory.getms m src✝)) = true)) + 1 ≤ (subroutineTimeCost h + recurseTimeCost hf h + localTimeCost h) a
https://github.com/calcu16/lean_complexity.git
d267eb4f3952c654f130a119311692c1c1a7cd71
HMem/Trace/Cost.lean
HMem.Trace.CostedProgram.splitTimeCost
[244, 1]
[349, 16]
apply le_of_le_of_eq _
α : Type u_1 inst✝⁷ : Complexity.Coding α Memory β : Type u_2 inst✝⁶ : Complexity.Encoding β Memory f : α → β tp : TracedProgram γ δ : Type inst✝⁵ : Complexity.Coding γ Memory inst✝⁴ : Complexity.Encoding δ Memory fs : γ → δ cs : Complexity.CostFunction γ ℕ inst✝³ : Complexity.Computable Encoding.Model fs inst✝² : 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 γ✝ δ✝ : Type inst✝¹ : Complexity.Coding γ✝ Memory inst✝ : Complexity.Encoding δ✝ Memory fs✝ : γ✝ → δ✝ cs✝ : Complexity.CostFunction γ✝ ℕ hcost : Complexity Encoding.Model fs✝ cs✝ next : CostedProgram h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm a : α m : Memory hm : m ∈ fm a ⊢ (subroutineTimeCost (_ : TracedProgram.sound' f size (toTracedProgram next) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ fs✝) (Complexity.decode γ✝ (Memory.getms m src✝))) ∘ fm)) + recurseTimeCost hf (_ : TracedProgram.sound' f size (toTracedProgram next) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ fs✝) (Complexity.decode γ✝ (Memory.getms m src✝))) ∘ fm)) + localTimeCost (_ : TracedProgram.sound' f size (toTracedProgram next) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ fs✝) (Complexity.decode γ✝ (Memory.getms m src✝))) ∘ fm))) a + Complexity.ALE.bound Complexity.cost_ale (Option.get (Complexity.decode γ✝ (Memory.getms m src✝)) (_ : Option.isSome (Complexity.decode γ✝ (Memory.getms m src✝)) = true)) + 1 ≤ (subroutineTimeCost h + recurseTimeCost hf h + localTimeCost h) a
α : Type u_1 inst✝⁷ : Complexity.Coding α Memory β : Type u_2 inst✝⁶ : Complexity.Encoding β Memory f : α → β tp : TracedProgram γ δ : Type inst✝⁵ : Complexity.Coding γ Memory inst✝⁴ : Complexity.Encoding δ Memory fs : γ → δ cs : Complexity.CostFunction γ ℕ inst✝³ : Complexity.Computable Encoding.Model fs inst✝² : 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 γ✝ δ✝ : Type inst✝¹ : Complexity.Coding γ✝ Memory inst✝ : Complexity.Encoding δ✝ Memory fs✝ : γ✝ → δ✝ cs✝ : Complexity.CostFunction γ✝ ℕ hcost : Complexity Encoding.Model fs✝ cs✝ next : CostedProgram h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm a : α m : Memory hm : m ∈ fm a ⊢ ?m.174514 = (subroutineTimeCost h + recurseTimeCost hf h + localTimeCost h) a α : Type u_1 inst✝⁷ : Complexity.Coding α Memory β : Type u_2 inst✝⁶ : Complexity.Encoding β Memory f : α → β tp : TracedProgram γ δ : Type inst✝⁵ : Complexity.Coding γ Memory inst✝⁴ : Complexity.Encoding δ Memory fs : γ → δ cs : Complexity.CostFunction γ ℕ inst✝³ : Complexity.Computable Encoding.Model fs inst✝² : 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 γ✝ δ✝ : Type inst✝¹ : Complexity.Coding γ✝ Memory inst✝ : Complexity.Encoding δ✝ Memory fs✝ : γ✝ → δ✝ cs✝ : Complexity.CostFunction γ✝ ℕ hcost : Complexity Encoding.Model fs✝ cs✝ next : CostedProgram h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm a : α m : Memory hm : m ∈ fm a ⊢ ℕ α : Type u_1 inst✝⁷ : Complexity.Coding α Memory β : Type u_2 inst✝⁶ : Complexity.Encoding β Memory f : α → β tp : TracedProgram γ δ : Type inst✝⁵ : Complexity.Coding γ Memory inst✝⁴ : Complexity.Encoding δ Memory fs : γ → δ cs : Complexity.CostFunction γ ℕ inst✝³ : Complexity.Computable Encoding.Model fs inst✝² : 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 γ✝ δ✝ : Type inst✝¹ : Complexity.Coding γ✝ Memory inst✝ : Complexity.Encoding δ✝ Memory fs✝ : γ✝ → δ✝ cs✝ : Complexity.CostFunction γ✝ ℕ hcost : Complexity Encoding.Model fs✝ cs✝ next : CostedProgram h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm a : α m : Memory hm : m ∈ fm a ⊢ (subroutineTimeCost (_ : TracedProgram.sound' f size (toTracedProgram next) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ fs✝) (Complexity.decode γ✝ (Memory.getms m src✝))) ∘ fm)) + recurseTimeCost hf (_ : TracedProgram.sound' f size (toTracedProgram next) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ fs✝) (Complexity.decode γ✝ (Memory.getms m src✝))) ∘ fm)) + localTimeCost (_ : TracedProgram.sound' f size (toTracedProgram next) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ fs✝) (Complexity.decode γ✝ (Memory.getms m src✝))) ∘ fm))) a + Complexity.ALE.bound Complexity.cost_ale (Option.get (Complexity.decode γ✝ (Memory.getms m src✝)) (_ : Option.isSome (Complexity.decode γ✝ (Memory.getms m src✝)) = true)) + 1 ≤ ?m.174514
https://github.com/calcu16/lean_complexity.git
d267eb4f3952c654f130a119311692c1c1a7cd71
HMem/Trace/Cost.lean
HMem.Trace.CostedProgram.splitTimeCost
[244, 1]
[349, 16]
apply Nat.add_assoc
α : Type u_1 inst✝⁷ : Complexity.Coding α Memory β : Type u_2 inst✝⁶ : Complexity.Encoding β Memory f : α → β tp : TracedProgram γ δ : Type inst✝⁵ : Complexity.Coding γ Memory inst✝⁴ : Complexity.Encoding δ Memory fs : γ → δ cs : Complexity.CostFunction γ ℕ inst✝³ : Complexity.Computable Encoding.Model fs inst✝² : 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 γ✝ δ✝ : Type inst✝¹ : Complexity.Coding γ✝ Memory inst✝ : Complexity.Encoding δ✝ Memory fs✝ : γ✝ → δ✝ cs✝ : Complexity.CostFunction γ✝ ℕ hcost : Complexity Encoding.Model fs✝ cs✝ next : CostedProgram h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm a : α m : Memory hm : m ∈ fm a ⊢ ?m.174514 = (subroutineTimeCost h + recurseTimeCost hf h + localTimeCost h) a α : Type u_1 inst✝⁷ : Complexity.Coding α Memory β : Type u_2 inst✝⁶ : Complexity.Encoding β Memory f : α → β tp : TracedProgram γ δ : Type inst✝⁵ : Complexity.Coding γ Memory inst✝⁴ : Complexity.Encoding δ Memory fs : γ → δ cs : Complexity.CostFunction γ ℕ inst✝³ : Complexity.Computable Encoding.Model fs inst✝² : 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 γ✝ δ✝ : Type inst✝¹ : Complexity.Coding γ✝ Memory inst✝ : Complexity.Encoding δ✝ Memory fs✝ : γ✝ → δ✝ cs✝ : Complexity.CostFunction γ✝ ℕ hcost : Complexity Encoding.Model fs✝ cs✝ next : CostedProgram h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm a : α m : Memory hm : m ∈ fm a ⊢ ℕ α : Type u_1 inst✝⁷ : Complexity.Coding α Memory β : Type u_2 inst✝⁶ : Complexity.Encoding β Memory f : α → β tp : TracedProgram γ δ : Type inst✝⁵ : Complexity.Coding γ Memory inst✝⁴ : Complexity.Encoding δ Memory fs : γ → δ cs : Complexity.CostFunction γ ℕ inst✝³ : Complexity.Computable Encoding.Model fs inst✝² : 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 γ✝ δ✝ : Type inst✝¹ : Complexity.Coding γ✝ Memory inst✝ : Complexity.Encoding δ✝ Memory fs✝ : γ✝ → δ✝ cs✝ : Complexity.CostFunction γ✝ ℕ hcost : Complexity Encoding.Model fs✝ cs✝ next : CostedProgram h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm a : α m : Memory hm : m ∈ fm a ⊢ (subroutineTimeCost (_ : TracedProgram.sound' f size (toTracedProgram next) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ fs✝) (Complexity.decode γ✝ (Memory.getms m src✝))) ∘ fm)) + recurseTimeCost hf (_ : TracedProgram.sound' f size (toTracedProgram next) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ fs✝) (Complexity.decode γ✝ (Memory.getms m src✝))) ∘ fm)) + localTimeCost (_ : TracedProgram.sound' f size (toTracedProgram next) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ fs✝) (Complexity.decode γ✝ (Memory.getms m src✝))) ∘ fm))) a + Complexity.ALE.bound Complexity.cost_ale (Option.get (Complexity.decode γ✝ (Memory.getms m src✝)) (_ : Option.isSome (Complexity.decode γ✝ (Memory.getms m src✝)) = true)) + 1 ≤ ?m.174514
α : Type u_1 inst✝⁷ : Complexity.Coding α Memory β : Type u_2 inst✝⁶ : Complexity.Encoding β Memory f : α → β tp : TracedProgram γ δ : Type inst✝⁵ : Complexity.Coding γ Memory inst✝⁴ : Complexity.Encoding δ Memory fs : γ → δ cs : Complexity.CostFunction γ ℕ inst✝³ : Complexity.Computable Encoding.Model fs inst✝² : 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 γ✝ δ✝ : Type inst✝¹ : Complexity.Coding γ✝ Memory inst✝ : Complexity.Encoding δ✝ Memory fs✝ : γ✝ → δ✝ cs✝ : Complexity.CostFunction γ✝ ℕ hcost : Complexity Encoding.Model fs✝ cs✝ next : CostedProgram h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm a : α m : Memory hm : m ∈ fm a ⊢ (subroutineTimeCost (_ : TracedProgram.sound' f size (toTracedProgram next) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ fs✝) (Complexity.decode γ✝ (Memory.getms m src✝))) ∘ fm)) + recurseTimeCost hf (_ : TracedProgram.sound' f size (toTracedProgram next) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ fs✝) (Complexity.decode γ✝ (Memory.getms m src✝))) ∘ fm)) + localTimeCost (_ : TracedProgram.sound' f size (toTracedProgram next) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ fs✝) (Complexity.decode γ✝ (Memory.getms m src✝))) ∘ fm))) a + Complexity.ALE.bound Complexity.cost_ale (Option.get (Complexity.decode γ✝ (Memory.getms m src✝)) (_ : Option.isSome (Complexity.decode γ✝ (Memory.getms m src✝)) = true)) + 1 ≤ (subroutineTimeCost h + recurseTimeCost hf h) a + localTimeCost (_ : TracedProgram.sound' f size (toTracedProgram next) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ fs✝) (Complexity.decode γ✝ (Memory.getms m src✝))) ∘ fm)) a + Complexity.CostFunction.flatMap fm 1 a
https://github.com/calcu16/lean_complexity.git
d267eb4f3952c654f130a119311692c1c1a7cd71
HMem/Trace/Cost.lean
HMem.Trace.CostedProgram.splitTimeCost
[244, 1]
[349, 16]
apply add_le_add _
α : Type u_1 inst✝⁷ : Complexity.Coding α Memory β : Type u_2 inst✝⁶ : Complexity.Encoding β Memory f : α → β tp : TracedProgram γ δ : Type inst✝⁵ : Complexity.Coding γ Memory inst✝⁴ : Complexity.Encoding δ Memory fs : γ → δ cs : Complexity.CostFunction γ ℕ inst✝³ : Complexity.Computable Encoding.Model fs inst✝² : 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 γ✝ δ✝ : Type inst✝¹ : Complexity.Coding γ✝ Memory inst✝ : Complexity.Encoding δ✝ Memory fs✝ : γ✝ → δ✝ cs✝ : Complexity.CostFunction γ✝ ℕ hcost : Complexity Encoding.Model fs✝ cs✝ next : CostedProgram h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm a : α m : Memory hm : m ∈ fm a ⊢ (subroutineTimeCost (_ : TracedProgram.sound' f size (toTracedProgram next) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ fs✝) (Complexity.decode γ✝ (Memory.getms m src✝))) ∘ fm)) + recurseTimeCost hf (_ : TracedProgram.sound' f size (toTracedProgram next) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ fs✝) (Complexity.decode γ✝ (Memory.getms m src✝))) ∘ fm)) + localTimeCost (_ : TracedProgram.sound' f size (toTracedProgram next) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ fs✝) (Complexity.decode γ✝ (Memory.getms m src✝))) ∘ fm))) a + Complexity.ALE.bound Complexity.cost_ale (Option.get (Complexity.decode γ✝ (Memory.getms m src✝)) (_ : Option.isSome (Complexity.decode γ✝ (Memory.getms m src✝)) = true)) + 1 ≤ (subroutineTimeCost h + recurseTimeCost hf h) a + localTimeCost (_ : TracedProgram.sound' f size (toTracedProgram next) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ fs✝) (Complexity.decode γ✝ (Memory.getms m src✝))) ∘ fm)) a + Complexity.CostFunction.flatMap fm 1 a
α : Type u_1 inst✝⁷ : Complexity.Coding α Memory β : Type u_2 inst✝⁶ : Complexity.Encoding β Memory f : α → β tp : TracedProgram γ δ : Type inst✝⁵ : Complexity.Coding γ Memory inst✝⁴ : Complexity.Encoding δ Memory fs : γ → δ cs : Complexity.CostFunction γ ℕ inst✝³ : Complexity.Computable Encoding.Model fs inst✝² : 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 γ✝ δ✝ : Type inst✝¹ : Complexity.Coding γ✝ Memory inst✝ : Complexity.Encoding δ✝ Memory fs✝ : γ✝ → δ✝ cs✝ : Complexity.CostFunction γ✝ ℕ hcost : Complexity Encoding.Model fs✝ cs✝ next : CostedProgram h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm a : α m : Memory hm : m ∈ fm a ⊢ 1 ≤ Complexity.CostFunction.flatMap fm 1 a α : Type u_1 inst✝⁷ : Complexity.Coding α Memory β : Type u_2 inst✝⁶ : Complexity.Encoding β Memory f : α → β tp : TracedProgram γ δ : Type inst✝⁵ : Complexity.Coding γ Memory inst✝⁴ : Complexity.Encoding δ Memory fs : γ → δ cs : Complexity.CostFunction γ ℕ inst✝³ : Complexity.Computable Encoding.Model fs inst✝² : 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 γ✝ δ✝ : Type inst✝¹ : Complexity.Coding γ✝ Memory inst✝ : Complexity.Encoding δ✝ Memory fs✝ : γ✝ → δ✝ cs✝ : Complexity.CostFunction γ✝ ℕ hcost : Complexity Encoding.Model fs✝ cs✝ next : CostedProgram h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm a : α m : Memory hm : m ∈ fm a ⊢ (subroutineTimeCost (_ : TracedProgram.sound' f size (toTracedProgram next) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ fs✝) (Complexity.decode γ✝ (Memory.getms m src✝))) ∘ fm)) + recurseTimeCost hf (_ : TracedProgram.sound' f size (toTracedProgram next) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ fs✝) (Complexity.decode γ✝ (Memory.getms m src✝))) ∘ fm)) + localTimeCost (_ : TracedProgram.sound' f size (toTracedProgram next) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ fs✝) (Complexity.decode γ✝ (Memory.getms m src✝))) ∘ fm))) a + Complexity.ALE.bound Complexity.cost_ale (Option.get (Complexity.decode γ✝ (Memory.getms m src✝)) (_ : Option.isSome (Complexity.decode γ✝ (Memory.getms m src✝)) = true)) ≤ (subroutineTimeCost h + recurseTimeCost hf h) a + localTimeCost (_ : TracedProgram.sound' f size (toTracedProgram next) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ fs✝) (Complexity.decode γ✝ (Memory.getms m src✝))) ∘ fm)) a
https://github.com/calcu16/lean_complexity.git
d267eb4f3952c654f130a119311692c1c1a7cd71
HMem/Trace/Cost.lean
HMem.Trace.CostedProgram.splitTimeCost
[244, 1]
[349, 16]
apply (le_of_eq (Eq.symm (Complexity.CostFunction.flatMap_some hm _)))
α : Type u_1 inst✝⁷ : Complexity.Coding α Memory β : Type u_2 inst✝⁶ : Complexity.Encoding β Memory f : α → β tp : TracedProgram γ δ : Type inst✝⁵ : Complexity.Coding γ Memory inst✝⁴ : Complexity.Encoding δ Memory fs : γ → δ cs : Complexity.CostFunction γ ℕ inst✝³ : Complexity.Computable Encoding.Model fs inst✝² : 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 γ✝ δ✝ : Type inst✝¹ : Complexity.Coding γ✝ Memory inst✝ : Complexity.Encoding δ✝ Memory fs✝ : γ✝ → δ✝ cs✝ : Complexity.CostFunction γ✝ ℕ hcost : Complexity Encoding.Model fs✝ cs✝ next : CostedProgram h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm a : α m : Memory hm : m ∈ fm a ⊢ 1 ≤ Complexity.CostFunction.flatMap fm 1 a α : Type u_1 inst✝⁷ : Complexity.Coding α Memory β : Type u_2 inst✝⁶ : Complexity.Encoding β Memory f : α → β tp : TracedProgram γ δ : Type inst✝⁵ : Complexity.Coding γ Memory inst✝⁴ : Complexity.Encoding δ Memory fs : γ → δ cs : Complexity.CostFunction γ ℕ inst✝³ : Complexity.Computable Encoding.Model fs inst✝² : 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 γ✝ δ✝ : Type inst✝¹ : Complexity.Coding γ✝ Memory inst✝ : Complexity.Encoding δ✝ Memory fs✝ : γ✝ → δ✝ cs✝ : Complexity.CostFunction γ✝ ℕ hcost : Complexity Encoding.Model fs✝ cs✝ next : CostedProgram h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm a : α m : Memory hm : m ∈ fm a ⊢ (subroutineTimeCost (_ : TracedProgram.sound' f size (toTracedProgram next) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ fs✝) (Complexity.decode γ✝ (Memory.getms m src✝))) ∘ fm)) + recurseTimeCost hf (_ : TracedProgram.sound' f size (toTracedProgram next) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ fs✝) (Complexity.decode γ✝ (Memory.getms m src✝))) ∘ fm)) + localTimeCost (_ : TracedProgram.sound' f size (toTracedProgram next) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ fs✝) (Complexity.decode γ✝ (Memory.getms m src✝))) ∘ fm))) a + Complexity.ALE.bound Complexity.cost_ale (Option.get (Complexity.decode γ✝ (Memory.getms m src✝)) (_ : Option.isSome (Complexity.decode γ✝ (Memory.getms m src✝)) = true)) ≤ (subroutineTimeCost h + recurseTimeCost hf h) a + localTimeCost (_ : TracedProgram.sound' f size (toTracedProgram next) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ fs✝) (Complexity.decode γ✝ (Memory.getms m src✝))) ∘ fm)) a
α : Type u_1 inst✝⁷ : Complexity.Coding α Memory β : Type u_2 inst✝⁶ : Complexity.Encoding β Memory f : α → β tp : TracedProgram γ δ : Type inst✝⁵ : Complexity.Coding γ Memory inst✝⁴ : Complexity.Encoding δ Memory fs : γ → δ cs : Complexity.CostFunction γ ℕ inst✝³ : Complexity.Computable Encoding.Model fs inst✝² : 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 γ✝ δ✝ : Type inst✝¹ : Complexity.Coding γ✝ Memory inst✝ : Complexity.Encoding δ✝ Memory fs✝ : γ✝ → δ✝ cs✝ : Complexity.CostFunction γ✝ ℕ hcost : Complexity Encoding.Model fs✝ cs✝ next : CostedProgram h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm a : α m : Memory hm : m ∈ fm a ⊢ (subroutineTimeCost (_ : TracedProgram.sound' f size (toTracedProgram next) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ fs✝) (Complexity.decode γ✝ (Memory.getms m src✝))) ∘ fm)) + recurseTimeCost hf (_ : TracedProgram.sound' f size (toTracedProgram next) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ fs✝) (Complexity.decode γ✝ (Memory.getms m src✝))) ∘ fm)) + localTimeCost (_ : TracedProgram.sound' f size (toTracedProgram next) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ fs✝) (Complexity.decode γ✝ (Memory.getms m src✝))) ∘ fm))) a + Complexity.ALE.bound Complexity.cost_ale (Option.get (Complexity.decode γ✝ (Memory.getms m src✝)) (_ : Option.isSome (Complexity.decode γ✝ (Memory.getms m src✝)) = true)) ≤ (subroutineTimeCost h + recurseTimeCost hf h) a + localTimeCost (_ : TracedProgram.sound' f size (toTracedProgram next) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ fs✝) (Complexity.decode γ✝ (Memory.getms m src✝))) ∘ fm)) a
https://github.com/calcu16/lean_complexity.git
d267eb4f3952c654f130a119311692c1c1a7cd71
HMem/Trace/Cost.lean
HMem.Trace.CostedProgram.splitTimeCost
[244, 1]
[349, 16]
apply le_of_eq_of_le
α : Type u_1 inst✝⁷ : Complexity.Coding α Memory β : Type u_2 inst✝⁶ : Complexity.Encoding β Memory f : α → β tp : TracedProgram γ δ : Type inst✝⁵ : Complexity.Coding γ Memory inst✝⁴ : Complexity.Encoding δ Memory fs : γ → δ cs : Complexity.CostFunction γ ℕ inst✝³ : Complexity.Computable Encoding.Model fs inst✝² : 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 γ✝ δ✝ : Type inst✝¹ : Complexity.Coding γ✝ Memory inst✝ : Complexity.Encoding δ✝ Memory fs✝ : γ✝ → δ✝ cs✝ : Complexity.CostFunction γ✝ ℕ hcost : Complexity Encoding.Model fs✝ cs✝ next : CostedProgram h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm a : α m : Memory hm : m ∈ fm a ⊢ (subroutineTimeCost (_ : TracedProgram.sound' f size (toTracedProgram next) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ fs✝) (Complexity.decode γ✝ (Memory.getms m src✝))) ∘ fm)) + recurseTimeCost hf (_ : TracedProgram.sound' f size (toTracedProgram next) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ fs✝) (Complexity.decode γ✝ (Memory.getms m src✝))) ∘ fm)) + localTimeCost (_ : TracedProgram.sound' f size (toTracedProgram next) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ fs✝) (Complexity.decode γ✝ (Memory.getms m src✝))) ∘ fm))) a + Complexity.ALE.bound Complexity.cost_ale (Option.get (Complexity.decode γ✝ (Memory.getms m src✝)) (_ : Option.isSome (Complexity.decode γ✝ (Memory.getms m src✝)) = true)) ≤ (subroutineTimeCost h + recurseTimeCost hf h) a + localTimeCost (_ : TracedProgram.sound' f size (toTracedProgram next) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ fs✝) (Complexity.decode γ✝ (Memory.getms m src✝))) ∘ fm)) a
case h₁ α : Type u_1 inst✝⁷ : Complexity.Coding α Memory β : Type u_2 inst✝⁶ : Complexity.Encoding β Memory f : α → β tp : TracedProgram γ δ : Type inst✝⁵ : Complexity.Coding γ Memory inst✝⁴ : Complexity.Encoding δ Memory fs : γ → δ cs : Complexity.CostFunction γ ℕ inst✝³ : Complexity.Computable Encoding.Model fs inst✝² : 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 γ✝ δ✝ : Type inst✝¹ : Complexity.Coding γ✝ Memory inst✝ : Complexity.Encoding δ✝ Memory fs✝ : γ✝ → δ✝ cs✝ : Complexity.CostFunction γ✝ ℕ hcost : Complexity Encoding.Model fs✝ cs✝ next : CostedProgram h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm a : α m : Memory hm : m ∈ fm a ⊢ (subroutineTimeCost (_ : TracedProgram.sound' f size (toTracedProgram next) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ fs✝) (Complexity.decode γ✝ (Memory.getms m src✝))) ∘ fm)) + recurseTimeCost hf (_ : TracedProgram.sound' f size (toTracedProgram next) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ fs✝) (Complexity.decode γ✝ (Memory.getms m src✝))) ∘ fm)) + localTimeCost (_ : TracedProgram.sound' f size (toTracedProgram next) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ fs✝) (Complexity.decode γ✝ (Memory.getms m src✝))) ∘ fm))) a + Complexity.ALE.bound Complexity.cost_ale (Option.get (Complexity.decode γ✝ (Memory.getms m src✝)) (_ : Option.isSome (Complexity.decode γ✝ (Memory.getms m src✝)) = true)) = ?b case h₂ α : Type u_1 inst✝⁷ : Complexity.Coding α Memory β : Type u_2 inst✝⁶ : Complexity.Encoding β Memory f : α → β tp : TracedProgram γ δ : Type inst✝⁵ : Complexity.Coding γ Memory inst✝⁴ : Complexity.Encoding δ Memory fs : γ → δ cs : Complexity.CostFunction γ ℕ inst✝³ : Complexity.Computable Encoding.Model fs inst✝² : 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 γ✝ δ✝ : Type inst✝¹ : Complexity.Coding γ✝ Memory inst✝ : Complexity.Encoding δ✝ Memory fs✝ : γ✝ → δ✝ cs✝ : Complexity.CostFunction γ✝ ℕ hcost : Complexity Encoding.Model fs✝ cs✝ next : CostedProgram h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm a : α m : Memory hm : m ∈ fm a ⊢ ?b ≤ (subroutineTimeCost h + recurseTimeCost hf h) a + localTimeCost (_ : TracedProgram.sound' f size (toTracedProgram next) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ fs✝) (Complexity.decode γ✝ (Memory.getms m src✝))) ∘ fm)) a case b α : Type u_1 inst✝⁷ : Complexity.Coding α Memory β : Type u_2 inst✝⁶ : Complexity.Encoding β Memory f : α → β tp : TracedProgram γ δ : Type inst✝⁵ : Complexity.Coding γ Memory inst✝⁴ : Complexity.Encoding δ Memory fs : γ → δ cs : Complexity.CostFunction γ ℕ inst✝³ : Complexity.Computable Encoding.Model fs inst✝² : 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 γ✝ δ✝ : Type inst✝¹ : Complexity.Coding γ✝ Memory inst✝ : Complexity.Encoding δ✝ Memory fs✝ : γ✝ → δ✝ cs✝ : Complexity.CostFunction γ✝ ℕ hcost : Complexity Encoding.Model fs✝ cs✝ next : CostedProgram h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm a : α m : Memory hm : m ∈ fm a ⊢ ℕ
https://github.com/calcu16/lean_complexity.git
d267eb4f3952c654f130a119311692c1c1a7cd71
HMem/Trace/Cost.lean
HMem.Trace.CostedProgram.splitTimeCost
[244, 1]
[349, 16]
apply Eq.trans
case h₁ α : Type u_1 inst✝⁷ : Complexity.Coding α Memory β : Type u_2 inst✝⁶ : Complexity.Encoding β Memory f : α → β tp : TracedProgram γ δ : Type inst✝⁵ : Complexity.Coding γ Memory inst✝⁴ : Complexity.Encoding δ Memory fs : γ → δ cs : Complexity.CostFunction γ ℕ inst✝³ : Complexity.Computable Encoding.Model fs inst✝² : 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 γ✝ δ✝ : Type inst✝¹ : Complexity.Coding γ✝ Memory inst✝ : Complexity.Encoding δ✝ Memory fs✝ : γ✝ → δ✝ cs✝ : Complexity.CostFunction γ✝ ℕ hcost : Complexity Encoding.Model fs✝ cs✝ next : CostedProgram h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm a : α m : Memory hm : m ∈ fm a ⊢ (subroutineTimeCost (_ : TracedProgram.sound' f size (toTracedProgram next) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ fs✝) (Complexity.decode γ✝ (Memory.getms m src✝))) ∘ fm)) + recurseTimeCost hf (_ : TracedProgram.sound' f size (toTracedProgram next) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ fs✝) (Complexity.decode γ✝ (Memory.getms m src✝))) ∘ fm)) + localTimeCost (_ : TracedProgram.sound' f size (toTracedProgram next) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ fs✝) (Complexity.decode γ✝ (Memory.getms m src✝))) ∘ fm))) a + Complexity.ALE.bound Complexity.cost_ale (Option.get (Complexity.decode γ✝ (Memory.getms m src✝)) (_ : Option.isSome (Complexity.decode γ✝ (Memory.getms m src✝)) = true)) = ?b case h₂ α : Type u_1 inst✝⁷ : Complexity.Coding α Memory β : Type u_2 inst✝⁶ : Complexity.Encoding β Memory f : α → β tp : TracedProgram γ δ : Type inst✝⁵ : Complexity.Coding γ Memory inst✝⁴ : Complexity.Encoding δ Memory fs : γ → δ cs : Complexity.CostFunction γ ℕ inst✝³ : Complexity.Computable Encoding.Model fs inst✝² : 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 γ✝ δ✝ : Type inst✝¹ : Complexity.Coding γ✝ Memory inst✝ : Complexity.Encoding δ✝ Memory fs✝ : γ✝ → δ✝ cs✝ : Complexity.CostFunction γ✝ ℕ hcost : Complexity Encoding.Model fs✝ cs✝ next : CostedProgram h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm a : α m : Memory hm : m ∈ fm a ⊢ ?b ≤ (subroutineTimeCost h + recurseTimeCost hf h) a + localTimeCost (_ : TracedProgram.sound' f size (toTracedProgram next) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ fs✝) (Complexity.decode γ✝ (Memory.getms m src✝))) ∘ fm)) a case b α : Type u_1 inst✝⁷ : Complexity.Coding α Memory β : Type u_2 inst✝⁶ : Complexity.Encoding β Memory f : α → β tp : TracedProgram γ δ : Type inst✝⁵ : Complexity.Coding γ Memory inst✝⁴ : Complexity.Encoding δ Memory fs : γ → δ cs : Complexity.CostFunction γ ℕ inst✝³ : Complexity.Computable Encoding.Model fs inst✝² : 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 γ✝ δ✝ : Type inst✝¹ : Complexity.Coding γ✝ Memory inst✝ : Complexity.Encoding δ✝ Memory fs✝ : γ✝ → δ✝ cs✝ : Complexity.CostFunction γ✝ ℕ hcost : Complexity Encoding.Model fs✝ cs✝ next : CostedProgram h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm a : α m : Memory hm : m ∈ fm a ⊢ ℕ
case h₁.h₁ α : Type u_1 inst✝⁷ : Complexity.Coding α Memory β : Type u_2 inst✝⁶ : Complexity.Encoding β Memory f : α → β tp : TracedProgram γ δ : Type inst✝⁵ : Complexity.Coding γ Memory inst✝⁴ : Complexity.Encoding δ Memory fs : γ → δ cs : Complexity.CostFunction γ ℕ inst✝³ : Complexity.Computable Encoding.Model fs inst✝² : 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 γ✝ δ✝ : Type inst✝¹ : Complexity.Coding γ✝ Memory inst✝ : Complexity.Encoding δ✝ Memory fs✝ : γ✝ → δ✝ cs✝ : Complexity.CostFunction γ✝ ℕ hcost : Complexity Encoding.Model fs✝ cs✝ next : CostedProgram h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm a : α m : Memory hm : m ∈ fm a ⊢ (subroutineTimeCost (_ : TracedProgram.sound' f size (toTracedProgram next) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ fs✝) (Complexity.decode γ✝ (Memory.getms m src✝))) ∘ fm)) + recurseTimeCost hf (_ : TracedProgram.sound' f size (toTracedProgram next) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ fs✝) (Complexity.decode γ✝ (Memory.getms m src✝))) ∘ fm)) + localTimeCost (_ : TracedProgram.sound' f size (toTracedProgram next) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ fs✝) (Complexity.decode γ✝ (Memory.getms m src✝))) ∘ fm))) a + Complexity.ALE.bound Complexity.cost_ale (Option.get (Complexity.decode γ✝ (Memory.getms m src✝)) (_ : Option.isSome (Complexity.decode γ✝ (Memory.getms m src✝)) = true)) = ?h₁.b case h₁.h₂ α : Type u_1 inst✝⁷ : Complexity.Coding α Memory β : Type u_2 inst✝⁶ : Complexity.Encoding β Memory f : α → β tp : TracedProgram γ δ : Type inst✝⁵ : Complexity.Coding γ Memory inst✝⁴ : Complexity.Encoding δ Memory fs : γ → δ cs : Complexity.CostFunction γ ℕ inst✝³ : Complexity.Computable Encoding.Model fs inst✝² : 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 γ✝ δ✝ : Type inst✝¹ : Complexity.Coding γ✝ Memory inst✝ : Complexity.Encoding δ✝ Memory fs✝ : γ✝ → δ✝ cs✝ : Complexity.CostFunction γ✝ ℕ hcost : Complexity Encoding.Model fs✝ cs✝ next : CostedProgram h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm a : α m : Memory hm : m ∈ fm a ⊢ ?h₁.b = ?b case h₁.b α : Type u_1 inst✝⁷ : Complexity.Coding α Memory β : Type u_2 inst✝⁶ : Complexity.Encoding β Memory f : α → β tp : TracedProgram γ δ : Type inst✝⁵ : Complexity.Coding γ Memory inst✝⁴ : Complexity.Encoding δ Memory fs : γ → δ cs : Complexity.CostFunction γ ℕ inst✝³ : Complexity.Computable Encoding.Model fs inst✝² : 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 γ✝ δ✝ : Type inst✝¹ : Complexity.Coding γ✝ Memory inst✝ : Complexity.Encoding δ✝ Memory fs✝ : γ✝ → δ✝ cs✝ : Complexity.CostFunction γ✝ ℕ hcost : Complexity Encoding.Model fs✝ cs✝ next : CostedProgram h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm a : α m : Memory hm : m ∈ fm a ⊢ ℕ case h₂ α : Type u_1 inst✝⁷ : Complexity.Coding α Memory β : Type u_2 inst✝⁶ : Complexity.Encoding β Memory f : α → β tp : TracedProgram γ δ : Type inst✝⁵ : Complexity.Coding γ Memory inst✝⁴ : Complexity.Encoding δ Memory fs : γ → δ cs : Complexity.CostFunction γ ℕ inst✝³ : Complexity.Computable Encoding.Model fs inst✝² : 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 γ✝ δ✝ : Type inst✝¹ : Complexity.Coding γ✝ Memory inst✝ : Complexity.Encoding δ✝ Memory fs✝ : γ✝ → δ✝ cs✝ : Complexity.CostFunction γ✝ ℕ hcost : Complexity Encoding.Model fs✝ cs✝ next : CostedProgram h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm a : α m : Memory hm : m ∈ fm a ⊢ ?b ≤ (subroutineTimeCost h + recurseTimeCost hf h) a + localTimeCost (_ : TracedProgram.sound' f size (toTracedProgram next) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ fs✝) (Complexity.decode γ✝ (Memory.getms m src✝))) ∘ fm)) a case b α : Type u_1 inst✝⁷ : Complexity.Coding α Memory β : Type u_2 inst✝⁶ : Complexity.Encoding β Memory f : α → β tp : TracedProgram γ δ : Type inst✝⁵ : Complexity.Coding γ Memory inst✝⁴ : Complexity.Encoding δ Memory fs : γ → δ cs : Complexity.CostFunction γ ℕ inst✝³ : Complexity.Computable Encoding.Model fs inst✝² : 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 γ✝ δ✝ : Type inst✝¹ : Complexity.Coding γ✝ Memory inst✝ : Complexity.Encoding δ✝ Memory fs✝ : γ✝ → δ✝ cs✝ : Complexity.CostFunction γ✝ ℕ hcost : Complexity Encoding.Model fs✝ cs✝ next : CostedProgram h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm a : α m : Memory hm : m ∈ fm a ⊢ ℕ
https://github.com/calcu16/lean_complexity.git
d267eb4f3952c654f130a119311692c1c1a7cd71
HMem/Trace/Cost.lean
HMem.Trace.CostedProgram.splitTimeCost
[244, 1]
[349, 16]
apply add_comm
case h₁.h₁ α : Type u_1 inst✝⁷ : Complexity.Coding α Memory β : Type u_2 inst✝⁶ : Complexity.Encoding β Memory f : α → β tp : TracedProgram γ δ : Type inst✝⁵ : Complexity.Coding γ Memory inst✝⁴ : Complexity.Encoding δ Memory fs : γ → δ cs : Complexity.CostFunction γ ℕ inst✝³ : Complexity.Computable Encoding.Model fs inst✝² : 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 γ✝ δ✝ : Type inst✝¹ : Complexity.Coding γ✝ Memory inst✝ : Complexity.Encoding δ✝ Memory fs✝ : γ✝ → δ✝ cs✝ : Complexity.CostFunction γ✝ ℕ hcost : Complexity Encoding.Model fs✝ cs✝ next : CostedProgram h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm a : α m : Memory hm : m ∈ fm a ⊢ (subroutineTimeCost (_ : TracedProgram.sound' f size (toTracedProgram next) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ fs✝) (Complexity.decode γ✝ (Memory.getms m src✝))) ∘ fm)) + recurseTimeCost hf (_ : TracedProgram.sound' f size (toTracedProgram next) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ fs✝) (Complexity.decode γ✝ (Memory.getms m src✝))) ∘ fm)) + localTimeCost (_ : TracedProgram.sound' f size (toTracedProgram next) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ fs✝) (Complexity.decode γ✝ (Memory.getms m src✝))) ∘ fm))) a + Complexity.ALE.bound Complexity.cost_ale (Option.get (Complexity.decode γ✝ (Memory.getms m src✝)) (_ : Option.isSome (Complexity.decode γ✝ (Memory.getms m src✝)) = true)) = ?h₁.b case h₁.h₂ α : Type u_1 inst✝⁷ : Complexity.Coding α Memory β : Type u_2 inst✝⁶ : Complexity.Encoding β Memory f : α → β tp : TracedProgram γ δ : Type inst✝⁵ : Complexity.Coding γ Memory inst✝⁴ : Complexity.Encoding δ Memory fs : γ → δ cs : Complexity.CostFunction γ ℕ inst✝³ : Complexity.Computable Encoding.Model fs inst✝² : 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 γ✝ δ✝ : Type inst✝¹ : Complexity.Coding γ✝ Memory inst✝ : Complexity.Encoding δ✝ Memory fs✝ : γ✝ → δ✝ cs✝ : Complexity.CostFunction γ✝ ℕ hcost : Complexity Encoding.Model fs✝ cs✝ next : CostedProgram h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm a : α m : Memory hm : m ∈ fm a ⊢ ?h₁.b = ?b case h₁.b α : Type u_1 inst✝⁷ : Complexity.Coding α Memory β : Type u_2 inst✝⁶ : Complexity.Encoding β Memory f : α → β tp : TracedProgram γ δ : Type inst✝⁵ : Complexity.Coding γ Memory inst✝⁴ : Complexity.Encoding δ Memory fs : γ → δ cs : Complexity.CostFunction γ ℕ inst✝³ : Complexity.Computable Encoding.Model fs inst✝² : 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 γ✝ δ✝ : Type inst✝¹ : Complexity.Coding γ✝ Memory inst✝ : Complexity.Encoding δ✝ Memory fs✝ : γ✝ → δ✝ cs✝ : Complexity.CostFunction γ✝ ℕ hcost : Complexity Encoding.Model fs✝ cs✝ next : CostedProgram h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm a : α m : Memory hm : m ∈ fm a ⊢ ℕ case h₂ α : Type u_1 inst✝⁷ : Complexity.Coding α Memory β : Type u_2 inst✝⁶ : Complexity.Encoding β Memory f : α → β tp : TracedProgram γ δ : Type inst✝⁵ : Complexity.Coding γ Memory inst✝⁴ : Complexity.Encoding δ Memory fs : γ → δ cs : Complexity.CostFunction γ ℕ inst✝³ : Complexity.Computable Encoding.Model fs inst✝² : 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 γ✝ δ✝ : Type inst✝¹ : Complexity.Coding γ✝ Memory inst✝ : Complexity.Encoding δ✝ Memory fs✝ : γ✝ → δ✝ cs✝ : Complexity.CostFunction γ✝ ℕ hcost : Complexity Encoding.Model fs✝ cs✝ next : CostedProgram h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm a : α m : Memory hm : m ∈ fm a ⊢ ?b ≤ (subroutineTimeCost h + recurseTimeCost hf h) a + localTimeCost (_ : TracedProgram.sound' f size (toTracedProgram next) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ fs✝) (Complexity.decode γ✝ (Memory.getms m src✝))) ∘ fm)) a case b α : Type u_1 inst✝⁷ : Complexity.Coding α Memory β : Type u_2 inst✝⁶ : Complexity.Encoding β Memory f : α → β tp : TracedProgram γ δ : Type inst✝⁵ : Complexity.Coding γ Memory inst✝⁴ : Complexity.Encoding δ Memory fs : γ → δ cs : Complexity.CostFunction γ ℕ inst✝³ : Complexity.Computable Encoding.Model fs inst✝² : 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 γ✝ δ✝ : Type inst✝¹ : Complexity.Coding γ✝ Memory inst✝ : Complexity.Encoding δ✝ Memory fs✝ : γ✝ → δ✝ cs✝ : Complexity.CostFunction γ✝ ℕ hcost : Complexity Encoding.Model fs✝ cs✝ next : CostedProgram h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm a : α m : Memory hm : m ∈ fm a ⊢ ℕ
case h₁.h₂ α : Type u_1 inst✝⁷ : Complexity.Coding α Memory β : Type u_2 inst✝⁶ : Complexity.Encoding β Memory f : α → β tp : TracedProgram γ δ : Type inst✝⁵ : Complexity.Coding γ Memory inst✝⁴ : Complexity.Encoding δ Memory fs : γ → δ cs : Complexity.CostFunction γ ℕ inst✝³ : Complexity.Computable Encoding.Model fs inst✝² : 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 γ✝ δ✝ : Type inst✝¹ : Complexity.Coding γ✝ Memory inst✝ : Complexity.Encoding δ✝ Memory fs✝ : γ✝ → δ✝ cs✝ : Complexity.CostFunction γ✝ ℕ hcost : Complexity Encoding.Model fs✝ cs✝ next : CostedProgram h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm a : α m : Memory hm : m ∈ fm a ⊢ Complexity.ALE.bound Complexity.cost_ale (Option.get (Complexity.decode γ✝ (Memory.getms m src✝)) (_ : Option.isSome (Complexity.decode γ✝ (Memory.getms m src✝)) = true)) + (subroutineTimeCost (_ : TracedProgram.sound' f size (toTracedProgram next) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ fs✝) (Complexity.decode γ✝ (Memory.getms m src✝))) ∘ fm)) + recurseTimeCost hf (_ : TracedProgram.sound' f size (toTracedProgram next) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ fs✝) (Complexity.decode γ✝ (Memory.getms m src✝))) ∘ fm)) + localTimeCost (_ : TracedProgram.sound' f size (toTracedProgram next) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ fs✝) (Complexity.decode γ✝ (Memory.getms m src✝))) ∘ fm))) a = ?b case h₂ α : Type u_1 inst✝⁷ : Complexity.Coding α Memory β : Type u_2 inst✝⁶ : Complexity.Encoding β Memory f : α → β tp : TracedProgram γ δ : Type inst✝⁵ : Complexity.Coding γ Memory inst✝⁴ : Complexity.Encoding δ Memory fs : γ → δ cs : Complexity.CostFunction γ ℕ inst✝³ : Complexity.Computable Encoding.Model fs inst✝² : 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 γ✝ δ✝ : Type inst✝¹ : Complexity.Coding γ✝ Memory inst✝ : Complexity.Encoding δ✝ Memory fs✝ : γ✝ → δ✝ cs✝ : Complexity.CostFunction γ✝ ℕ hcost : Complexity Encoding.Model fs✝ cs✝ next : CostedProgram h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm a : α m : Memory hm : m ∈ fm a ⊢ ?b ≤ (subroutineTimeCost h + recurseTimeCost hf h) a + localTimeCost (_ : TracedProgram.sound' f size (toTracedProgram next) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ fs✝) (Complexity.decode γ✝ (Memory.getms m src✝))) ∘ fm)) a case b α : Type u_1 inst✝⁷ : Complexity.Coding α Memory β : Type u_2 inst✝⁶ : Complexity.Encoding β Memory f : α → β tp : TracedProgram γ δ : Type inst✝⁵ : Complexity.Coding γ Memory inst✝⁴ : Complexity.Encoding δ Memory fs : γ → δ cs : Complexity.CostFunction γ ℕ inst✝³ : Complexity.Computable Encoding.Model fs inst✝² : 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 γ✝ δ✝ : Type inst✝¹ : Complexity.Coding γ✝ Memory inst✝ : Complexity.Encoding δ✝ Memory fs✝ : γ✝ → δ✝ cs✝ : Complexity.CostFunction γ✝ ℕ hcost : Complexity Encoding.Model fs✝ cs✝ next : CostedProgram h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm a : α m : Memory hm : m ∈ fm a ⊢ ℕ
https://github.com/calcu16/lean_complexity.git
d267eb4f3952c654f130a119311692c1c1a7cd71
HMem/Trace/Cost.lean
HMem.Trace.CostedProgram.splitTimeCost
[244, 1]
[349, 16]
apply (Nat.add_assoc _ _ _).symm
case h₁.h₂ α : Type u_1 inst✝⁷ : Complexity.Coding α Memory β : Type u_2 inst✝⁶ : Complexity.Encoding β Memory f : α → β tp : TracedProgram γ δ : Type inst✝⁵ : Complexity.Coding γ Memory inst✝⁴ : Complexity.Encoding δ Memory fs : γ → δ cs : Complexity.CostFunction γ ℕ inst✝³ : Complexity.Computable Encoding.Model fs inst✝² : 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 γ✝ δ✝ : Type inst✝¹ : Complexity.Coding γ✝ Memory inst✝ : Complexity.Encoding δ✝ Memory fs✝ : γ✝ → δ✝ cs✝ : Complexity.CostFunction γ✝ ℕ hcost : Complexity Encoding.Model fs✝ cs✝ next : CostedProgram h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm a : α m : Memory hm : m ∈ fm a ⊢ Complexity.ALE.bound Complexity.cost_ale (Option.get (Complexity.decode γ✝ (Memory.getms m src✝)) (_ : Option.isSome (Complexity.decode γ✝ (Memory.getms m src✝)) = true)) + (subroutineTimeCost (_ : TracedProgram.sound' f size (toTracedProgram next) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ fs✝) (Complexity.decode γ✝ (Memory.getms m src✝))) ∘ fm)) + recurseTimeCost hf (_ : TracedProgram.sound' f size (toTracedProgram next) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ fs✝) (Complexity.decode γ✝ (Memory.getms m src✝))) ∘ fm)) + localTimeCost (_ : TracedProgram.sound' f size (toTracedProgram next) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ fs✝) (Complexity.decode γ✝ (Memory.getms m src✝))) ∘ fm))) a = ?b case h₂ α : Type u_1 inst✝⁷ : Complexity.Coding α Memory β : Type u_2 inst✝⁶ : Complexity.Encoding β Memory f : α → β tp : TracedProgram γ δ : Type inst✝⁵ : Complexity.Coding γ Memory inst✝⁴ : Complexity.Encoding δ Memory fs : γ → δ cs : Complexity.CostFunction γ ℕ inst✝³ : Complexity.Computable Encoding.Model fs inst✝² : 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 γ✝ δ✝ : Type inst✝¹ : Complexity.Coding γ✝ Memory inst✝ : Complexity.Encoding δ✝ Memory fs✝ : γ✝ → δ✝ cs✝ : Complexity.CostFunction γ✝ ℕ hcost : Complexity Encoding.Model fs✝ cs✝ next : CostedProgram h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm a : α m : Memory hm : m ∈ fm a ⊢ ?b ≤ (subroutineTimeCost h + recurseTimeCost hf h) a + localTimeCost (_ : TracedProgram.sound' f size (toTracedProgram next) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ fs✝) (Complexity.decode γ✝ (Memory.getms m src✝))) ∘ fm)) a case b α : Type u_1 inst✝⁷ : Complexity.Coding α Memory β : Type u_2 inst✝⁶ : Complexity.Encoding β Memory f : α → β tp : TracedProgram γ δ : Type inst✝⁵ : Complexity.Coding γ Memory inst✝⁴ : Complexity.Encoding δ Memory fs : γ → δ cs : Complexity.CostFunction γ ℕ inst✝³ : Complexity.Computable Encoding.Model fs inst✝² : 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 γ✝ δ✝ : Type inst✝¹ : Complexity.Coding γ✝ Memory inst✝ : Complexity.Encoding δ✝ Memory fs✝ : γ✝ → δ✝ cs✝ : Complexity.CostFunction γ✝ ℕ hcost : Complexity Encoding.Model fs✝ cs✝ next : CostedProgram h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm a : α m : Memory hm : m ∈ fm a ⊢ ℕ
case h₂ α : Type u_1 inst✝⁷ : Complexity.Coding α Memory β : Type u_2 inst✝⁶ : Complexity.Encoding β Memory f : α → β tp : TracedProgram γ δ : Type inst✝⁵ : Complexity.Coding γ Memory inst✝⁴ : Complexity.Encoding δ Memory fs : γ → δ cs : Complexity.CostFunction γ ℕ inst✝³ : Complexity.Computable Encoding.Model fs inst✝² : 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 γ✝ δ✝ : Type inst✝¹ : Complexity.Coding γ✝ Memory inst✝ : Complexity.Encoding δ✝ Memory fs✝ : γ✝ → δ✝ cs✝ : Complexity.CostFunction γ✝ ℕ hcost : Complexity Encoding.Model fs✝ cs✝ next : CostedProgram h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm a : α m : Memory hm : m ∈ fm a ⊢ Complexity.ALE.bound Complexity.cost_ale (Option.get (Complexity.decode γ✝ (Memory.getms m src✝)) (_ : Option.isSome (Complexity.decode γ✝ (Memory.getms m src✝)) = true)) + (subroutineTimeCost (_ : TracedProgram.sound' f size (toTracedProgram next) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ fs✝) (Complexity.decode γ✝ (Memory.getms m src✝))) ∘ fm)) + recurseTimeCost hf (_ : TracedProgram.sound' f size (toTracedProgram next) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ fs✝) (Complexity.decode γ✝ (Memory.getms m src✝))) ∘ fm))) a + localTimeCost (_ : TracedProgram.sound' f size (toTracedProgram next) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ fs✝) (Complexity.decode γ✝ (Memory.getms m src✝))) ∘ fm)) a ≤ (subroutineTimeCost h + recurseTimeCost hf h) a + localTimeCost (_ : TracedProgram.sound' f size (toTracedProgram next) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ fs✝) (Complexity.decode γ✝ (Memory.getms m src✝))) ∘ fm)) a
https://github.com/calcu16/lean_complexity.git
d267eb4f3952c654f130a119311692c1c1a7cd71
HMem/Trace/Cost.lean
HMem.Trace.CostedProgram.splitTimeCost
[244, 1]
[349, 16]
apply add_le_add _ (le_refl _)
case h₂ α : Type u_1 inst✝⁷ : Complexity.Coding α Memory β : Type u_2 inst✝⁶ : Complexity.Encoding β Memory f : α → β tp : TracedProgram γ δ : Type inst✝⁵ : Complexity.Coding γ Memory inst✝⁴ : Complexity.Encoding δ Memory fs : γ → δ cs : Complexity.CostFunction γ ℕ inst✝³ : Complexity.Computable Encoding.Model fs inst✝² : 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 γ✝ δ✝ : Type inst✝¹ : Complexity.Coding γ✝ Memory inst✝ : Complexity.Encoding δ✝ Memory fs✝ : γ✝ → δ✝ cs✝ : Complexity.CostFunction γ✝ ℕ hcost : Complexity Encoding.Model fs✝ cs✝ next : CostedProgram h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm a : α m : Memory hm : m ∈ fm a ⊢ Complexity.ALE.bound Complexity.cost_ale (Option.get (Complexity.decode γ✝ (Memory.getms m src✝)) (_ : Option.isSome (Complexity.decode γ✝ (Memory.getms m src✝)) = true)) + (subroutineTimeCost (_ : TracedProgram.sound' f size (toTracedProgram next) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ fs✝) (Complexity.decode γ✝ (Memory.getms m src✝))) ∘ fm)) + recurseTimeCost hf (_ : TracedProgram.sound' f size (toTracedProgram next) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ fs✝) (Complexity.decode γ✝ (Memory.getms m src✝))) ∘ fm))) a + localTimeCost (_ : TracedProgram.sound' f size (toTracedProgram next) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ fs✝) (Complexity.decode γ✝ (Memory.getms m src✝))) ∘ fm)) a ≤ (subroutineTimeCost h + recurseTimeCost hf h) a + localTimeCost (_ : TracedProgram.sound' f size (toTracedProgram next) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ fs✝) (Complexity.decode γ✝ (Memory.getms m src✝))) ∘ fm)) a
α : Type u_1 inst✝⁷ : Complexity.Coding α Memory β : Type u_2 inst✝⁶ : Complexity.Encoding β Memory f : α → β tp : TracedProgram γ δ : Type inst✝⁵ : Complexity.Coding γ Memory inst✝⁴ : Complexity.Encoding δ Memory fs : γ → δ cs : Complexity.CostFunction γ ℕ inst✝³ : Complexity.Computable Encoding.Model fs inst✝² : 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 γ✝ δ✝ : Type inst✝¹ : Complexity.Coding γ✝ Memory inst✝ : Complexity.Encoding δ✝ Memory fs✝ : γ✝ → δ✝ cs✝ : Complexity.CostFunction γ✝ ℕ hcost : Complexity Encoding.Model fs✝ cs✝ next : CostedProgram h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm a : α m : Memory hm : m ∈ fm a ⊢ Complexity.ALE.bound Complexity.cost_ale (Option.get (Complexity.decode γ✝ (Memory.getms m src✝)) (_ : Option.isSome (Complexity.decode γ✝ (Memory.getms m src✝)) = true)) + (subroutineTimeCost (_ : TracedProgram.sound' f size (toTracedProgram next) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ fs✝) (Complexity.decode γ✝ (Memory.getms m src✝))) ∘ fm)) + recurseTimeCost hf (_ : TracedProgram.sound' f size (toTracedProgram next) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ fs✝) (Complexity.decode γ✝ (Memory.getms m src✝))) ∘ fm))) a ≤ (subroutineTimeCost h + recurseTimeCost hf h) a
https://github.com/calcu16/lean_complexity.git
d267eb4f3952c654f130a119311692c1c1a7cd71
HMem/Trace/Cost.lean
HMem.Trace.CostedProgram.splitTimeCost
[244, 1]
[349, 16]
apply le_of_eq_of_le
α : Type u_1 inst✝⁷ : Complexity.Coding α Memory β : Type u_2 inst✝⁶ : Complexity.Encoding β Memory f : α → β tp : TracedProgram γ δ : Type inst✝⁵ : Complexity.Coding γ Memory inst✝⁴ : Complexity.Encoding δ Memory fs : γ → δ cs : Complexity.CostFunction γ ℕ inst✝³ : Complexity.Computable Encoding.Model fs inst✝² : 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 γ✝ δ✝ : Type inst✝¹ : Complexity.Coding γ✝ Memory inst✝ : Complexity.Encoding δ✝ Memory fs✝ : γ✝ → δ✝ cs✝ : Complexity.CostFunction γ✝ ℕ hcost : Complexity Encoding.Model fs✝ cs✝ next : CostedProgram h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm a : α m : Memory hm : m ∈ fm a ⊢ Complexity.ALE.bound Complexity.cost_ale (Option.get (Complexity.decode γ✝ (Memory.getms m src✝)) (_ : Option.isSome (Complexity.decode γ✝ (Memory.getms m src✝)) = true)) + (subroutineTimeCost (_ : TracedProgram.sound' f size (toTracedProgram next) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ fs✝) (Complexity.decode γ✝ (Memory.getms m src✝))) ∘ fm)) + recurseTimeCost hf (_ : TracedProgram.sound' f size (toTracedProgram next) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ fs✝) (Complexity.decode γ✝ (Memory.getms m src✝))) ∘ fm))) a ≤ (subroutineTimeCost h + recurseTimeCost hf h) a
case h₁ α : Type u_1 inst✝⁷ : Complexity.Coding α Memory β : Type u_2 inst✝⁶ : Complexity.Encoding β Memory f : α → β tp : TracedProgram γ δ : Type inst✝⁵ : Complexity.Coding γ Memory inst✝⁴ : Complexity.Encoding δ Memory fs : γ → δ cs : Complexity.CostFunction γ ℕ inst✝³ : Complexity.Computable Encoding.Model fs inst✝² : 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 γ✝ δ✝ : Type inst✝¹ : Complexity.Coding γ✝ Memory inst✝ : Complexity.Encoding δ✝ Memory fs✝ : γ✝ → δ✝ cs✝ : Complexity.CostFunction γ✝ ℕ hcost : Complexity Encoding.Model fs✝ cs✝ next : CostedProgram h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm a : α m : Memory hm : m ∈ fm a ⊢ Complexity.ALE.bound Complexity.cost_ale (Option.get (Complexity.decode γ✝ (Memory.getms m src✝)) (_ : Option.isSome (Complexity.decode γ✝ (Memory.getms m src✝)) = true)) + (subroutineTimeCost (_ : TracedProgram.sound' f size (toTracedProgram next) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ fs✝) (Complexity.decode γ✝ (Memory.getms m src✝))) ∘ fm)) + recurseTimeCost hf (_ : TracedProgram.sound' f size (toTracedProgram next) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ fs✝) (Complexity.decode γ✝ (Memory.getms m src✝))) ∘ fm))) a = ?b case h₂ α : Type u_1 inst✝⁷ : Complexity.Coding α Memory β : Type u_2 inst✝⁶ : Complexity.Encoding β Memory f : α → β tp : TracedProgram γ δ : Type inst✝⁵ : Complexity.Coding γ Memory inst✝⁴ : Complexity.Encoding δ Memory fs : γ → δ cs : Complexity.CostFunction γ ℕ inst✝³ : Complexity.Computable Encoding.Model fs inst✝² : 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 γ✝ δ✝ : Type inst✝¹ : Complexity.Coding γ✝ Memory inst✝ : Complexity.Encoding δ✝ Memory fs✝ : γ✝ → δ✝ cs✝ : Complexity.CostFunction γ✝ ℕ hcost : Complexity Encoding.Model fs✝ cs✝ next : CostedProgram h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm a : α m : Memory hm : m ∈ fm a ⊢ ?b ≤ (subroutineTimeCost h + recurseTimeCost hf h) a case b α : Type u_1 inst✝⁷ : Complexity.Coding α Memory β : Type u_2 inst✝⁶ : Complexity.Encoding β Memory f : α → β tp : TracedProgram γ δ : Type inst✝⁵ : Complexity.Coding γ Memory inst✝⁴ : Complexity.Encoding δ Memory fs : γ → δ cs : Complexity.CostFunction γ ℕ inst✝³ : Complexity.Computable Encoding.Model fs inst✝² : 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 γ✝ δ✝ : Type inst✝¹ : Complexity.Coding γ✝ Memory inst✝ : Complexity.Encoding δ✝ Memory fs✝ : γ✝ → δ✝ cs✝ : Complexity.CostFunction γ✝ ℕ hcost : Complexity Encoding.Model fs✝ cs✝ next : CostedProgram h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm a : α m : Memory hm : m ∈ fm a ⊢ ℕ
https://github.com/calcu16/lean_complexity.git
d267eb4f3952c654f130a119311692c1c1a7cd71
HMem/Trace/Cost.lean
HMem.Trace.CostedProgram.splitTimeCost
[244, 1]
[349, 16]
apply (Nat.add_assoc _ _ _).symm
case h₁ α : Type u_1 inst✝⁷ : Complexity.Coding α Memory β : Type u_2 inst✝⁶ : Complexity.Encoding β Memory f : α → β tp : TracedProgram γ δ : Type inst✝⁵ : Complexity.Coding γ Memory inst✝⁴ : Complexity.Encoding δ Memory fs : γ → δ cs : Complexity.CostFunction γ ℕ inst✝³ : Complexity.Computable Encoding.Model fs inst✝² : 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 γ✝ δ✝ : Type inst✝¹ : Complexity.Coding γ✝ Memory inst✝ : Complexity.Encoding δ✝ Memory fs✝ : γ✝ → δ✝ cs✝ : Complexity.CostFunction γ✝ ℕ hcost : Complexity Encoding.Model fs✝ cs✝ next : CostedProgram h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm a : α m : Memory hm : m ∈ fm a ⊢ Complexity.ALE.bound Complexity.cost_ale (Option.get (Complexity.decode γ✝ (Memory.getms m src✝)) (_ : Option.isSome (Complexity.decode γ✝ (Memory.getms m src✝)) = true)) + (subroutineTimeCost (_ : TracedProgram.sound' f size (toTracedProgram next) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ fs✝) (Complexity.decode γ✝ (Memory.getms m src✝))) ∘ fm)) + recurseTimeCost hf (_ : TracedProgram.sound' f size (toTracedProgram next) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ fs✝) (Complexity.decode γ✝ (Memory.getms m src✝))) ∘ fm))) a = ?b case h₂ α : Type u_1 inst✝⁷ : Complexity.Coding α Memory β : Type u_2 inst✝⁶ : Complexity.Encoding β Memory f : α → β tp : TracedProgram γ δ : Type inst✝⁵ : Complexity.Coding γ Memory inst✝⁴ : Complexity.Encoding δ Memory fs : γ → δ cs : Complexity.CostFunction γ ℕ inst✝³ : Complexity.Computable Encoding.Model fs inst✝² : 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 γ✝ δ✝ : Type inst✝¹ : Complexity.Coding γ✝ Memory inst✝ : Complexity.Encoding δ✝ Memory fs✝ : γ✝ → δ✝ cs✝ : Complexity.CostFunction γ✝ ℕ hcost : Complexity Encoding.Model fs✝ cs✝ next : CostedProgram h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm a : α m : Memory hm : m ∈ fm a ⊢ ?b ≤ (subroutineTimeCost h + recurseTimeCost hf h) a case b α : Type u_1 inst✝⁷ : Complexity.Coding α Memory β : Type u_2 inst✝⁶ : Complexity.Encoding β Memory f : α → β tp : TracedProgram γ δ : Type inst✝⁵ : Complexity.Coding γ Memory inst✝⁴ : Complexity.Encoding δ Memory fs : γ → δ cs : Complexity.CostFunction γ ℕ inst✝³ : Complexity.Computable Encoding.Model fs inst✝² : 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 γ✝ δ✝ : Type inst✝¹ : Complexity.Coding γ✝ Memory inst✝ : Complexity.Encoding δ✝ Memory fs✝ : γ✝ → δ✝ cs✝ : Complexity.CostFunction γ✝ ℕ hcost : Complexity Encoding.Model fs✝ cs✝ next : CostedProgram h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm a : α m : Memory hm : m ∈ fm a ⊢ ℕ
case h₂ α : Type u_1 inst✝⁷ : Complexity.Coding α Memory β : Type u_2 inst✝⁶ : Complexity.Encoding β Memory f : α → β tp : TracedProgram γ δ : Type inst✝⁵ : Complexity.Coding γ Memory inst✝⁴ : Complexity.Encoding δ Memory fs : γ → δ cs : Complexity.CostFunction γ ℕ inst✝³ : Complexity.Computable Encoding.Model fs inst✝² : 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 γ✝ δ✝ : Type inst✝¹ : Complexity.Coding γ✝ Memory inst✝ : Complexity.Encoding δ✝ Memory fs✝ : γ✝ → δ✝ cs✝ : Complexity.CostFunction γ✝ ℕ hcost : Complexity Encoding.Model fs✝ cs✝ next : CostedProgram h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm a : α m : Memory hm : m ∈ fm a ⊢ Complexity.ALE.bound Complexity.cost_ale (Option.get (Complexity.decode γ✝ (Memory.getms m src✝)) (_ : Option.isSome (Complexity.decode γ✝ (Memory.getms m src✝)) = true)) + subroutineTimeCost (_ : TracedProgram.sound' f size (toTracedProgram next) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ fs✝) (Complexity.decode γ✝ (Memory.getms m src✝))) ∘ fm)) a + recurseTimeCost hf (_ : TracedProgram.sound' f size (toTracedProgram next) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ fs✝) (Complexity.decode γ✝ (Memory.getms m src✝))) ∘ fm)) a ≤ (subroutineTimeCost h + recurseTimeCost hf h) a
https://github.com/calcu16/lean_complexity.git
d267eb4f3952c654f130a119311692c1c1a7cd71
HMem/Trace/Cost.lean
HMem.Trace.CostedProgram.splitTimeCost
[244, 1]
[349, 16]
apply add_le_add _ (le_refl _)
case h₂ α : Type u_1 inst✝⁷ : Complexity.Coding α Memory β : Type u_2 inst✝⁶ : Complexity.Encoding β Memory f : α → β tp : TracedProgram γ δ : Type inst✝⁵ : Complexity.Coding γ Memory inst✝⁴ : Complexity.Encoding δ Memory fs : γ → δ cs : Complexity.CostFunction γ ℕ inst✝³ : Complexity.Computable Encoding.Model fs inst✝² : 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 γ✝ δ✝ : Type inst✝¹ : Complexity.Coding γ✝ Memory inst✝ : Complexity.Encoding δ✝ Memory fs✝ : γ✝ → δ✝ cs✝ : Complexity.CostFunction γ✝ ℕ hcost : Complexity Encoding.Model fs✝ cs✝ next : CostedProgram h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm a : α m : Memory hm : m ∈ fm a ⊢ Complexity.ALE.bound Complexity.cost_ale (Option.get (Complexity.decode γ✝ (Memory.getms m src✝)) (_ : Option.isSome (Complexity.decode γ✝ (Memory.getms m src✝)) = true)) + subroutineTimeCost (_ : TracedProgram.sound' f size (toTracedProgram next) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ fs✝) (Complexity.decode γ✝ (Memory.getms m src✝))) ∘ fm)) a + recurseTimeCost hf (_ : TracedProgram.sound' f size (toTracedProgram next) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ fs✝) (Complexity.decode γ✝ (Memory.getms m src✝))) ∘ fm)) a ≤ (subroutineTimeCost h + recurseTimeCost hf h) a
α : Type u_1 inst✝⁷ : Complexity.Coding α Memory β : Type u_2 inst✝⁶ : Complexity.Encoding β Memory f : α → β tp : TracedProgram γ δ : Type inst✝⁵ : Complexity.Coding γ Memory inst✝⁴ : Complexity.Encoding δ Memory fs : γ → δ cs : Complexity.CostFunction γ ℕ inst✝³ : Complexity.Computable Encoding.Model fs inst✝² : 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 γ✝ δ✝ : Type inst✝¹ : Complexity.Coding γ✝ Memory inst✝ : Complexity.Encoding δ✝ Memory fs✝ : γ✝ → δ✝ cs✝ : Complexity.CostFunction γ✝ ℕ hcost : Complexity Encoding.Model fs✝ cs✝ next : CostedProgram h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm a : α m : Memory hm : m ∈ fm a ⊢ Complexity.ALE.bound Complexity.cost_ale (Option.get (Complexity.decode γ✝ (Memory.getms m src✝)) (_ : Option.isSome (Complexity.decode γ✝ (Memory.getms m src✝)) = true)) + subroutineTimeCost (_ : TracedProgram.sound' f size (toTracedProgram next) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ fs✝) (Complexity.decode γ✝ (Memory.getms m src✝))) ∘ fm)) a ≤ subroutineTimeCost h a
https://github.com/calcu16/lean_complexity.git
d267eb4f3952c654f130a119311692c1c1a7cd71
HMem/Trace/Cost.lean
HMem.Trace.CostedProgram.splitTimeCost
[244, 1]
[349, 16]
apply le_of_eq_of_le
α : Type u_1 inst✝⁷ : Complexity.Coding α Memory β : Type u_2 inst✝⁶ : Complexity.Encoding β Memory f : α → β tp : TracedProgram γ δ : Type inst✝⁵ : Complexity.Coding γ Memory inst✝⁴ : Complexity.Encoding δ Memory fs : γ → δ cs : Complexity.CostFunction γ ℕ inst✝³ : Complexity.Computable Encoding.Model fs inst✝² : 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 γ✝ δ✝ : Type inst✝¹ : Complexity.Coding γ✝ Memory inst✝ : Complexity.Encoding δ✝ Memory fs✝ : γ✝ → δ✝ cs✝ : Complexity.CostFunction γ✝ ℕ hcost : Complexity Encoding.Model fs✝ cs✝ next : CostedProgram h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm a : α m : Memory hm : m ∈ fm a ⊢ Complexity.ALE.bound Complexity.cost_ale (Option.get (Complexity.decode γ✝ (Memory.getms m src✝)) (_ : Option.isSome (Complexity.decode γ✝ (Memory.getms m src✝)) = true)) + subroutineTimeCost (_ : TracedProgram.sound' f size (toTracedProgram next) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ fs✝) (Complexity.decode γ✝ (Memory.getms m src✝))) ∘ fm)) a ≤ subroutineTimeCost h a
case h₁ α : Type u_1 inst✝⁷ : Complexity.Coding α Memory β : Type u_2 inst✝⁶ : Complexity.Encoding β Memory f : α → β tp : TracedProgram γ δ : Type inst✝⁵ : Complexity.Coding γ Memory inst✝⁴ : Complexity.Encoding δ Memory fs : γ → δ cs : Complexity.CostFunction γ ℕ inst✝³ : Complexity.Computable Encoding.Model fs inst✝² : 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 γ✝ δ✝ : Type inst✝¹ : Complexity.Coding γ✝ Memory inst✝ : Complexity.Encoding δ✝ Memory fs✝ : γ✝ → δ✝ cs✝ : Complexity.CostFunction γ✝ ℕ hcost : Complexity Encoding.Model fs✝ cs✝ next : CostedProgram h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm a : α m : Memory hm : m ∈ fm a ⊢ Complexity.ALE.bound Complexity.cost_ale (Option.get (Complexity.decode γ✝ (Memory.getms m src✝)) (_ : Option.isSome (Complexity.decode γ✝ (Memory.getms m src✝)) = true)) + subroutineTimeCost (_ : TracedProgram.sound' f size (toTracedProgram next) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ fs✝) (Complexity.decode γ✝ (Memory.getms m src✝))) ∘ fm)) a = ?b case h₂ α : Type u_1 inst✝⁷ : Complexity.Coding α Memory β : Type u_2 inst✝⁶ : Complexity.Encoding β Memory f : α → β tp : TracedProgram γ δ : Type inst✝⁵ : Complexity.Coding γ Memory inst✝⁴ : Complexity.Encoding δ Memory fs : γ → δ cs : Complexity.CostFunction γ ℕ inst✝³ : Complexity.Computable Encoding.Model fs inst✝² : 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 γ✝ δ✝ : Type inst✝¹ : Complexity.Coding γ✝ Memory inst✝ : Complexity.Encoding δ✝ Memory fs✝ : γ✝ → δ✝ cs✝ : Complexity.CostFunction γ✝ ℕ hcost : Complexity Encoding.Model fs✝ cs✝ next : CostedProgram h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm a : α m : Memory hm : m ∈ fm a ⊢ ?b ≤ subroutineTimeCost h a case b α : Type u_1 inst✝⁷ : Complexity.Coding α Memory β : Type u_2 inst✝⁶ : Complexity.Encoding β Memory f : α → β tp : TracedProgram γ δ : Type inst✝⁵ : Complexity.Coding γ Memory inst✝⁴ : Complexity.Encoding δ Memory fs : γ → δ cs : Complexity.CostFunction γ ℕ inst✝³ : Complexity.Computable Encoding.Model fs inst✝² : 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 γ✝ δ✝ : Type inst✝¹ : Complexity.Coding γ✝ Memory inst✝ : Complexity.Encoding δ✝ Memory fs✝ : γ✝ → δ✝ cs✝ : Complexity.CostFunction γ✝ ℕ hcost : Complexity Encoding.Model fs✝ cs✝ next : CostedProgram h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm a : α m : Memory hm : m ∈ fm a ⊢ ℕ
https://github.com/calcu16/lean_complexity.git
d267eb4f3952c654f130a119311692c1c1a7cd71
HMem/Trace/Cost.lean
HMem.Trace.CostedProgram.splitTimeCost
[244, 1]
[349, 16]
apply add_comm
case h₁ α : Type u_1 inst✝⁷ : Complexity.Coding α Memory β : Type u_2 inst✝⁶ : Complexity.Encoding β Memory f : α → β tp : TracedProgram γ δ : Type inst✝⁵ : Complexity.Coding γ Memory inst✝⁴ : Complexity.Encoding δ Memory fs : γ → δ cs : Complexity.CostFunction γ ℕ inst✝³ : Complexity.Computable Encoding.Model fs inst✝² : 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 γ✝ δ✝ : Type inst✝¹ : Complexity.Coding γ✝ Memory inst✝ : Complexity.Encoding δ✝ Memory fs✝ : γ✝ → δ✝ cs✝ : Complexity.CostFunction γ✝ ℕ hcost : Complexity Encoding.Model fs✝ cs✝ next : CostedProgram h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm a : α m : Memory hm : m ∈ fm a ⊢ Complexity.ALE.bound Complexity.cost_ale (Option.get (Complexity.decode γ✝ (Memory.getms m src✝)) (_ : Option.isSome (Complexity.decode γ✝ (Memory.getms m src✝)) = true)) + subroutineTimeCost (_ : TracedProgram.sound' f size (toTracedProgram next) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ fs✝) (Complexity.decode γ✝ (Memory.getms m src✝))) ∘ fm)) a = ?b case h₂ α : Type u_1 inst✝⁷ : Complexity.Coding α Memory β : Type u_2 inst✝⁶ : Complexity.Encoding β Memory f : α → β tp : TracedProgram γ δ : Type inst✝⁵ : Complexity.Coding γ Memory inst✝⁴ : Complexity.Encoding δ Memory fs : γ → δ cs : Complexity.CostFunction γ ℕ inst✝³ : Complexity.Computable Encoding.Model fs inst✝² : 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 γ✝ δ✝ : Type inst✝¹ : Complexity.Coding γ✝ Memory inst✝ : Complexity.Encoding δ✝ Memory fs✝ : γ✝ → δ✝ cs✝ : Complexity.CostFunction γ✝ ℕ hcost : Complexity Encoding.Model fs✝ cs✝ next : CostedProgram h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm a : α m : Memory hm : m ∈ fm a ⊢ ?b ≤ subroutineTimeCost h a case b α : Type u_1 inst✝⁷ : Complexity.Coding α Memory β : Type u_2 inst✝⁶ : Complexity.Encoding β Memory f : α → β tp : TracedProgram γ δ : Type inst✝⁵ : Complexity.Coding γ Memory inst✝⁴ : Complexity.Encoding δ Memory fs : γ → δ cs : Complexity.CostFunction γ ℕ inst✝³ : Complexity.Computable Encoding.Model fs inst✝² : 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 γ✝ δ✝ : Type inst✝¹ : Complexity.Coding γ✝ Memory inst✝ : Complexity.Encoding δ✝ Memory fs✝ : γ✝ → δ✝ cs✝ : Complexity.CostFunction γ✝ ℕ hcost : Complexity Encoding.Model fs✝ cs✝ next : CostedProgram h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm a : α m : Memory hm : m ∈ fm a ⊢ ℕ
case h₂ α : Type u_1 inst✝⁷ : Complexity.Coding α Memory β : Type u_2 inst✝⁶ : Complexity.Encoding β Memory f : α → β tp : TracedProgram γ δ : Type inst✝⁵ : Complexity.Coding γ Memory inst✝⁴ : Complexity.Encoding δ Memory fs : γ → δ cs : Complexity.CostFunction γ ℕ inst✝³ : Complexity.Computable Encoding.Model fs inst✝² : 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 γ✝ δ✝ : Type inst✝¹ : Complexity.Coding γ✝ Memory inst✝ : Complexity.Encoding δ✝ Memory fs✝ : γ✝ → δ✝ cs✝ : Complexity.CostFunction γ✝ ℕ hcost : Complexity Encoding.Model fs✝ cs✝ next : CostedProgram h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm a : α m : Memory hm : m ∈ fm a ⊢ subroutineTimeCost (_ : TracedProgram.sound' f size (toTracedProgram next) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ fs✝) (Complexity.decode γ✝ (Memory.getms m src✝))) ∘ fm)) a + Complexity.ALE.bound Complexity.cost_ale (Option.get (Complexity.decode γ✝ (Memory.getms m src✝)) (_ : Option.isSome (Complexity.decode γ✝ (Memory.getms m src✝)) = true)) ≤ subroutineTimeCost h a
https://github.com/calcu16/lean_complexity.git
d267eb4f3952c654f130a119311692c1c1a7cd71
HMem/Trace/Cost.lean
HMem.Trace.CostedProgram.splitTimeCost
[244, 1]
[349, 16]
apply add_le_add (le_refl _)
case h₂ α : Type u_1 inst✝⁷ : Complexity.Coding α Memory β : Type u_2 inst✝⁶ : Complexity.Encoding β Memory f : α → β tp : TracedProgram γ δ : Type inst✝⁵ : Complexity.Coding γ Memory inst✝⁴ : Complexity.Encoding δ Memory fs : γ → δ cs : Complexity.CostFunction γ ℕ inst✝³ : Complexity.Computable Encoding.Model fs inst✝² : 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 γ✝ δ✝ : Type inst✝¹ : Complexity.Coding γ✝ Memory inst✝ : Complexity.Encoding δ✝ Memory fs✝ : γ✝ → δ✝ cs✝ : Complexity.CostFunction γ✝ ℕ hcost : Complexity Encoding.Model fs✝ cs✝ next : CostedProgram h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm a : α m : Memory hm : m ∈ fm a ⊢ subroutineTimeCost (_ : TracedProgram.sound' f size (toTracedProgram next) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ fs✝) (Complexity.decode γ✝ (Memory.getms m src✝))) ∘ fm)) a + Complexity.ALE.bound Complexity.cost_ale (Option.get (Complexity.decode γ✝ (Memory.getms m src✝)) (_ : Option.isSome (Complexity.decode γ✝ (Memory.getms m src✝)) = true)) ≤ subroutineTimeCost h a
case h₂ α : Type u_1 inst✝⁷ : Complexity.Coding α Memory β : Type u_2 inst✝⁶ : Complexity.Encoding β Memory f : α → β tp : TracedProgram γ δ : Type inst✝⁵ : Complexity.Coding γ Memory inst✝⁴ : Complexity.Encoding δ Memory fs : γ → δ cs : Complexity.CostFunction γ ℕ inst✝³ : Complexity.Computable Encoding.Model fs inst✝² : 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 γ✝ δ✝ : Type inst✝¹ : Complexity.Coding γ✝ Memory inst✝ : Complexity.Encoding δ✝ Memory fs✝ : γ✝ → δ✝ cs✝ : Complexity.CostFunction γ✝ ℕ hcost : Complexity Encoding.Model fs✝ cs✝ next : CostedProgram h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm a : α m : Memory hm : m ∈ fm a ⊢ Complexity.ALE.bound Complexity.cost_ale (Option.get (Complexity.decode γ✝ (Memory.getms m src✝)) (_ : Option.isSome (Complexity.decode γ✝ (Memory.getms m src✝)) = true)) ≤ Complexity.CostFunction.flatMap (TracedProgram.sound'_subroutine_arg h) (Encoding.getBound hcost) a
https://github.com/calcu16/lean_complexity.git
d267eb4f3952c654f130a119311692c1c1a7cd71
HMem/Trace/Cost.lean
HMem.Trace.CostedProgram.splitTimeCost
[244, 1]
[349, 16]
apply le_of_le_of_eq _
case h₂ α : Type u_1 inst✝⁷ : Complexity.Coding α Memory β : Type u_2 inst✝⁶ : Complexity.Encoding β Memory f : α → β tp : TracedProgram γ δ : Type inst✝⁵ : Complexity.Coding γ Memory inst✝⁴ : Complexity.Encoding δ Memory fs : γ → δ cs : Complexity.CostFunction γ ℕ inst✝³ : Complexity.Computable Encoding.Model fs inst✝² : 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 γ✝ δ✝ : Type inst✝¹ : Complexity.Coding γ✝ Memory inst✝ : Complexity.Encoding δ✝ Memory fs✝ : γ✝ → δ✝ cs✝ : Complexity.CostFunction γ✝ ℕ hcost : Complexity Encoding.Model fs✝ cs✝ next : CostedProgram h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm a : α m : Memory hm : m ∈ fm a ⊢ Complexity.ALE.bound Complexity.cost_ale (Option.get (Complexity.decode γ✝ (Memory.getms m src✝)) (_ : Option.isSome (Complexity.decode γ✝ (Memory.getms m src✝)) = true)) ≤ Complexity.CostFunction.flatMap (TracedProgram.sound'_subroutine_arg h) (Encoding.getBound hcost) a
case h₂ α : Type u_1 inst✝⁷ : Complexity.Coding α Memory β : Type u_2 inst✝⁶ : Complexity.Encoding β Memory f : α → β tp : TracedProgram γ δ : Type inst✝⁵ : Complexity.Coding γ Memory inst✝⁴ : Complexity.Encoding δ Memory fs : γ → δ cs : Complexity.CostFunction γ ℕ inst✝³ : Complexity.Computable Encoding.Model fs inst✝² : 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 γ✝ δ✝ : Type inst✝¹ : Complexity.Coding γ✝ Memory inst✝ : Complexity.Encoding δ✝ Memory fs✝ : γ✝ → δ✝ cs✝ : Complexity.CostFunction γ✝ ℕ hcost : Complexity Encoding.Model fs✝ cs✝ next : CostedProgram h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm a : α m : Memory hm : m ∈ fm a ⊢ ?m.178694 = Complexity.CostFunction.flatMap (TracedProgram.sound'_subroutine_arg h) (Encoding.getBound hcost) a α : Type u_1 inst✝⁷ : Complexity.Coding α Memory β : Type u_2 inst✝⁶ : Complexity.Encoding β Memory f : α → β tp : TracedProgram γ δ : Type inst✝⁵ : Complexity.Coding γ Memory inst✝⁴ : Complexity.Encoding δ Memory fs : γ → δ cs : Complexity.CostFunction γ ℕ inst✝³ : Complexity.Computable Encoding.Model fs inst✝² : 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 γ✝ δ✝ : Type inst✝¹ : Complexity.Coding γ✝ Memory inst✝ : Complexity.Encoding δ✝ Memory fs✝ : γ✝ → δ✝ cs✝ : Complexity.CostFunction γ✝ ℕ hcost : Complexity Encoding.Model fs✝ cs✝ next : CostedProgram h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm a : α m : Memory hm : m ∈ fm a ⊢ ℕ α : Type u_1 inst✝⁷ : Complexity.Coding α Memory β : Type u_2 inst✝⁶ : Complexity.Encoding β Memory f : α → β tp : TracedProgram γ δ : Type inst✝⁵ : Complexity.Coding γ Memory inst✝⁴ : Complexity.Encoding δ Memory fs : γ → δ cs : Complexity.CostFunction γ ℕ inst✝³ : Complexity.Computable Encoding.Model fs inst✝² : 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 γ✝ δ✝ : Type inst✝¹ : Complexity.Coding γ✝ Memory inst✝ : Complexity.Encoding δ✝ Memory fs✝ : γ✝ → δ✝ cs✝ : Complexity.CostFunction γ✝ ℕ hcost : Complexity Encoding.Model fs✝ cs✝ next : CostedProgram h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm a : α m : Memory hm : m ∈ fm a ⊢ Complexity.ALE.bound Complexity.cost_ale (Option.get (Complexity.decode γ✝ (Memory.getms m src✝)) (_ : Option.isSome (Complexity.decode γ✝ (Memory.getms m src✝)) = true)) ≤ ?m.178694
https://github.com/calcu16/lean_complexity.git
d267eb4f3952c654f130a119311692c1c1a7cd71
HMem/Trace/Cost.lean
HMem.Trace.CostedProgram.splitTimeCost
[244, 1]
[349, 16]
apply Eq.symm
case h₂ α : Type u_1 inst✝⁷ : Complexity.Coding α Memory β : Type u_2 inst✝⁶ : Complexity.Encoding β Memory f : α → β tp : TracedProgram γ δ : Type inst✝⁵ : Complexity.Coding γ Memory inst✝⁴ : Complexity.Encoding δ Memory fs : γ → δ cs : Complexity.CostFunction γ ℕ inst✝³ : Complexity.Computable Encoding.Model fs inst✝² : 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 γ✝ δ✝ : Type inst✝¹ : Complexity.Coding γ✝ Memory inst✝ : Complexity.Encoding δ✝ Memory fs✝ : γ✝ → δ✝ cs✝ : Complexity.CostFunction γ✝ ℕ hcost : Complexity Encoding.Model fs✝ cs✝ next : CostedProgram h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm a : α m : Memory hm : m ∈ fm a ⊢ ?m.178694 = Complexity.CostFunction.flatMap (TracedProgram.sound'_subroutine_arg h) (Encoding.getBound hcost) a α : Type u_1 inst✝⁷ : Complexity.Coding α Memory β : Type u_2 inst✝⁶ : Complexity.Encoding β Memory f : α → β tp : TracedProgram γ δ : Type inst✝⁵ : Complexity.Coding γ Memory inst✝⁴ : Complexity.Encoding δ Memory fs : γ → δ cs : Complexity.CostFunction γ ℕ inst✝³ : Complexity.Computable Encoding.Model fs inst✝² : 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 γ✝ δ✝ : Type inst✝¹ : Complexity.Coding γ✝ Memory inst✝ : Complexity.Encoding δ✝ Memory fs✝ : γ✝ → δ✝ cs✝ : Complexity.CostFunction γ✝ ℕ hcost : Complexity Encoding.Model fs✝ cs✝ next : CostedProgram h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm a : α m : Memory hm : m ∈ fm a ⊢ ℕ α : Type u_1 inst✝⁷ : Complexity.Coding α Memory β : Type u_2 inst✝⁶ : Complexity.Encoding β Memory f : α → β tp : TracedProgram γ δ : Type inst✝⁵ : Complexity.Coding γ Memory inst✝⁴ : Complexity.Encoding δ Memory fs : γ → δ cs : Complexity.CostFunction γ ℕ inst✝³ : Complexity.Computable Encoding.Model fs inst✝² : 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 γ✝ δ✝ : Type inst✝¹ : Complexity.Coding γ✝ Memory inst✝ : Complexity.Encoding δ✝ Memory fs✝ : γ✝ → δ✝ cs✝ : Complexity.CostFunction γ✝ ℕ hcost : Complexity Encoding.Model fs✝ cs✝ next : CostedProgram h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm a : α m : Memory hm : m ∈ fm a ⊢ Complexity.ALE.bound Complexity.cost_ale (Option.get (Complexity.decode γ✝ (Memory.getms m src✝)) (_ : Option.isSome (Complexity.decode γ✝ (Memory.getms m src✝)) = true)) ≤ ?m.178694
case h₂.h α : Type u_1 inst✝⁷ : Complexity.Coding α Memory β : Type u_2 inst✝⁶ : Complexity.Encoding β Memory f : α → β tp : TracedProgram γ δ : Type inst✝⁵ : Complexity.Coding γ Memory inst✝⁴ : Complexity.Encoding δ Memory fs : γ → δ cs : Complexity.CostFunction γ ℕ inst✝³ : Complexity.Computable Encoding.Model fs inst✝² : 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 γ✝ δ✝ : Type inst✝¹ : Complexity.Coding γ✝ Memory inst✝ : Complexity.Encoding δ✝ Memory fs✝ : γ✝ → δ✝ cs✝ : Complexity.CostFunction γ✝ ℕ hcost : Complexity Encoding.Model fs✝ cs✝ next : CostedProgram h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm a : α m : Memory hm : m ∈ fm a ⊢ Complexity.CostFunction.flatMap (TracedProgram.sound'_subroutine_arg h) (Encoding.getBound hcost) a = ?m.178694 α : Type u_1 inst✝⁷ : Complexity.Coding α Memory β : Type u_2 inst✝⁶ : Complexity.Encoding β Memory f : α → β tp : TracedProgram γ δ : Type inst✝⁵ : Complexity.Coding γ Memory inst✝⁴ : Complexity.Encoding δ Memory fs : γ → δ cs : Complexity.CostFunction γ ℕ inst✝³ : Complexity.Computable Encoding.Model fs inst✝² : 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 γ✝ δ✝ : Type inst✝¹ : Complexity.Coding γ✝ Memory inst✝ : Complexity.Encoding δ✝ Memory fs✝ : γ✝ → δ✝ cs✝ : Complexity.CostFunction γ✝ ℕ hcost : Complexity Encoding.Model fs✝ cs✝ next : CostedProgram h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm a : α m : Memory hm : m ∈ fm a ⊢ ℕ α : Type u_1 inst✝⁷ : Complexity.Coding α Memory β : Type u_2 inst✝⁶ : Complexity.Encoding β Memory f : α → β tp : TracedProgram γ δ : Type inst✝⁵ : Complexity.Coding γ Memory inst✝⁴ : Complexity.Encoding δ Memory fs : γ → δ cs : Complexity.CostFunction γ ℕ inst✝³ : Complexity.Computable Encoding.Model fs inst✝² : 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 γ✝ δ✝ : Type inst✝¹ : Complexity.Coding γ✝ Memory inst✝ : Complexity.Encoding δ✝ Memory fs✝ : γ✝ → δ✝ cs✝ : Complexity.CostFunction γ✝ ℕ hcost : Complexity Encoding.Model fs✝ cs✝ next : CostedProgram h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm a : α m : Memory hm : m ∈ fm a ⊢ Complexity.ALE.bound Complexity.cost_ale (Option.get (Complexity.decode γ✝ (Memory.getms m src✝)) (_ : Option.isSome (Complexity.decode γ✝ (Memory.getms m src✝)) = true)) ≤ ?m.178694
https://github.com/calcu16/lean_complexity.git
d267eb4f3952c654f130a119311692c1c1a7cd71
HMem/Trace/Cost.lean
HMem.Trace.CostedProgram.splitTimeCost
[244, 1]
[349, 16]
apply Complexity.CostFunction.flatMap_some
case h₂.h α : Type u_1 inst✝⁷ : Complexity.Coding α Memory β : Type u_2 inst✝⁶ : Complexity.Encoding β Memory f : α → β tp : TracedProgram γ δ : Type inst✝⁵ : Complexity.Coding γ Memory inst✝⁴ : Complexity.Encoding δ Memory fs : γ → δ cs : Complexity.CostFunction γ ℕ inst✝³ : Complexity.Computable Encoding.Model fs inst✝² : 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 γ✝ δ✝ : Type inst✝¹ : Complexity.Coding γ✝ Memory inst✝ : Complexity.Encoding δ✝ Memory fs✝ : γ✝ → δ✝ cs✝ : Complexity.CostFunction γ✝ ℕ hcost : Complexity Encoding.Model fs✝ cs✝ next : CostedProgram h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm a : α m : Memory hm : m ∈ fm a ⊢ Complexity.CostFunction.flatMap (TracedProgram.sound'_subroutine_arg h) (Encoding.getBound hcost) a = ?m.178694 α : Type u_1 inst✝⁷ : Complexity.Coding α Memory β : Type u_2 inst✝⁶ : Complexity.Encoding β Memory f : α → β tp : TracedProgram γ δ : Type inst✝⁵ : Complexity.Coding γ Memory inst✝⁴ : Complexity.Encoding δ Memory fs : γ → δ cs : Complexity.CostFunction γ ℕ inst✝³ : Complexity.Computable Encoding.Model fs inst✝² : 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 γ✝ δ✝ : Type inst✝¹ : Complexity.Coding γ✝ Memory inst✝ : Complexity.Encoding δ✝ Memory fs✝ : γ✝ → δ✝ cs✝ : Complexity.CostFunction γ✝ ℕ hcost : Complexity Encoding.Model fs✝ cs✝ next : CostedProgram h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm a : α m : Memory hm : m ∈ fm a ⊢ ℕ α : Type u_1 inst✝⁷ : Complexity.Coding α Memory β : Type u_2 inst✝⁶ : Complexity.Encoding β Memory f : α → β tp : TracedProgram γ δ : Type inst✝⁵ : Complexity.Coding γ Memory inst✝⁴ : Complexity.Encoding δ Memory fs : γ → δ cs : Complexity.CostFunction γ ℕ inst✝³ : Complexity.Computable Encoding.Model fs inst✝² : 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 γ✝ δ✝ : Type inst✝¹ : Complexity.Coding γ✝ Memory inst✝ : Complexity.Encoding δ✝ Memory fs✝ : γ✝ → δ✝ cs✝ : Complexity.CostFunction γ✝ ℕ hcost : Complexity Encoding.Model fs✝ cs✝ next : CostedProgram h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm a : α m : Memory hm : m ∈ fm a ⊢ Complexity.ALE.bound Complexity.cost_ale (Option.get (Complexity.decode γ✝ (Memory.getms m src✝)) (_ : Option.isSome (Complexity.decode γ✝ (Memory.getms m src✝)) = true)) ≤ ?m.178694
case h₂.h.h α : Type u_1 inst✝⁷ : Complexity.Coding α Memory β : Type u_2 inst✝⁶ : Complexity.Encoding β Memory f : α → β tp : TracedProgram γ δ : Type inst✝⁵ : Complexity.Coding γ Memory inst✝⁴ : Complexity.Encoding δ Memory fs : γ → δ cs : Complexity.CostFunction γ ℕ inst✝³ : Complexity.Computable Encoding.Model fs inst✝² : 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 γ✝ δ✝ : Type inst✝¹ : Complexity.Coding γ✝ Memory inst✝ : Complexity.Encoding δ✝ Memory fs✝ : γ✝ → δ✝ cs✝ : Complexity.CostFunction γ✝ ℕ hcost : Complexity Encoding.Model fs✝ cs✝ next : CostedProgram h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm a : α m : Memory hm : m ∈ fm a ⊢ TracedProgram.sound'_subroutine_arg h a = some ?h₂.h.b case h₂.h.b α : Type u_1 inst✝⁷ : Complexity.Coding α Memory β : Type u_2 inst✝⁶ : Complexity.Encoding β Memory f : α → β tp : TracedProgram γ δ : Type inst✝⁵ : Complexity.Coding γ Memory inst✝⁴ : Complexity.Encoding δ Memory fs : γ → δ cs : Complexity.CostFunction γ ℕ inst✝³ : Complexity.Computable Encoding.Model fs inst✝² : 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 γ✝ δ✝ : Type inst✝¹ : Complexity.Coding γ✝ Memory inst✝ : Complexity.Encoding δ✝ Memory fs✝ : γ✝ → δ✝ cs✝ : Complexity.CostFunction γ✝ ℕ hcost : Complexity Encoding.Model fs✝ cs✝ next : CostedProgram h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm a : α m : Memory hm : m ∈ fm a ⊢ γ✝ α : Type u_1 inst✝⁷ : Complexity.Coding α Memory β : Type u_2 inst✝⁶ : Complexity.Encoding β Memory f : α → β tp : TracedProgram γ δ : Type inst✝⁵ : Complexity.Coding γ Memory inst✝⁴ : Complexity.Encoding δ Memory fs : γ → δ cs : Complexity.CostFunction γ ℕ inst✝³ : Complexity.Computable Encoding.Model fs inst✝² : 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 γ✝ δ✝ : Type inst✝¹ : Complexity.Coding γ✝ Memory inst✝ : Complexity.Encoding δ✝ Memory fs✝ : γ✝ → δ✝ cs✝ : Complexity.CostFunction γ✝ ℕ hcost : Complexity Encoding.Model fs✝ cs✝ next : CostedProgram h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm a : α m : Memory hm : m ∈ fm a ⊢ Complexity.ALE.bound Complexity.cost_ale (Option.get (Complexity.decode γ✝ (Memory.getms m src✝)) (_ : Option.isSome (Complexity.decode γ✝ (Memory.getms m src✝)) = true)) ≤ Encoding.getBound hcost ?h₂.h.b
https://github.com/calcu16/lean_complexity.git
d267eb4f3952c654f130a119311692c1c1a7cd71
HMem/Trace/Cost.lean
HMem.Trace.CostedProgram.splitTimeCost
[244, 1]
[349, 16]
apply Option.mem_bind_of_mem
case h₂.h.h α : Type u_1 inst✝⁷ : Complexity.Coding α Memory β : Type u_2 inst✝⁶ : Complexity.Encoding β Memory f : α → β tp : TracedProgram γ δ : Type inst✝⁵ : Complexity.Coding γ Memory inst✝⁴ : Complexity.Encoding δ Memory fs : γ → δ cs : Complexity.CostFunction γ ℕ inst✝³ : Complexity.Computable Encoding.Model fs inst✝² : 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 γ✝ δ✝ : Type inst✝¹ : Complexity.Coding γ✝ Memory inst✝ : Complexity.Encoding δ✝ Memory fs✝ : γ✝ → δ✝ cs✝ : Complexity.CostFunction γ✝ ℕ hcost : Complexity Encoding.Model fs✝ cs✝ next : CostedProgram h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm a : α m : Memory hm : m ∈ fm a ⊢ TracedProgram.sound'_subroutine_arg h a = some ?h₂.h.b case h₂.h.b α : Type u_1 inst✝⁷ : Complexity.Coding α Memory β : Type u_2 inst✝⁶ : Complexity.Encoding β Memory f : α → β tp : TracedProgram γ δ : Type inst✝⁵ : Complexity.Coding γ Memory inst✝⁴ : Complexity.Encoding δ Memory fs : γ → δ cs : Complexity.CostFunction γ ℕ inst✝³ : Complexity.Computable Encoding.Model fs inst✝² : 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 γ✝ δ✝ : Type inst✝¹ : Complexity.Coding γ✝ Memory inst✝ : Complexity.Encoding δ✝ Memory fs✝ : γ✝ → δ✝ cs✝ : Complexity.CostFunction γ✝ ℕ hcost : Complexity Encoding.Model fs✝ cs✝ next : CostedProgram h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm a : α m : Memory hm : m ∈ fm a ⊢ γ✝ α : Type u_1 inst✝⁷ : Complexity.Coding α Memory β : Type u_2 inst✝⁶ : Complexity.Encoding β Memory f : α → β tp : TracedProgram γ δ : Type inst✝⁵ : Complexity.Coding γ Memory inst✝⁴ : Complexity.Encoding δ Memory fs : γ → δ cs : Complexity.CostFunction γ ℕ inst✝³ : Complexity.Computable Encoding.Model fs inst✝² : 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 γ✝ δ✝ : Type inst✝¹ : Complexity.Coding γ✝ Memory inst✝ : Complexity.Encoding δ✝ Memory fs✝ : γ✝ → δ✝ cs✝ : Complexity.CostFunction γ✝ ℕ hcost : Complexity Encoding.Model fs✝ cs✝ next : CostedProgram h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm a : α m : Memory hm : m ∈ fm a ⊢ Complexity.ALE.bound Complexity.cost_ale (Option.get (Complexity.decode γ✝ (Memory.getms m src✝)) (_ : Option.isSome (Complexity.decode γ✝ (Memory.getms m src✝)) = true)) ≤ Encoding.getBound hcost ?h₂.h.b
case h₂.h.h.ha α : Type u_1 inst✝⁷ : Complexity.Coding α Memory β : Type u_2 inst✝⁶ : Complexity.Encoding β Memory f : α → β tp : TracedProgram γ δ : Type inst✝⁵ : Complexity.Coding γ Memory inst✝⁴ : Complexity.Encoding δ Memory fs : γ → δ cs : Complexity.CostFunction γ ℕ inst✝³ : Complexity.Computable Encoding.Model fs inst✝² : 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 γ✝ δ✝ : Type inst✝¹ : Complexity.Coding γ✝ Memory inst✝ : Complexity.Encoding δ✝ Memory fs✝ : γ✝ → δ✝ cs✝ : Complexity.CostFunction γ✝ ℕ hcost : Complexity Encoding.Model fs✝ cs✝ next : CostedProgram h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm a : α m : Memory hm : m ∈ fm a ⊢ ?h₂.h.h.a ∈ fm a case h₂.h.h.hf α : Type u_1 inst✝⁷ : Complexity.Coding α Memory β : Type u_2 inst✝⁶ : Complexity.Encoding β Memory f : α → β tp : TracedProgram γ δ : Type inst✝⁵ : Complexity.Coding γ Memory inst✝⁴ : Complexity.Encoding δ Memory fs : γ → δ cs : Complexity.CostFunction γ ℕ inst✝³ : Complexity.Computable Encoding.Model fs inst✝² : 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 γ✝ δ✝ : Type inst✝¹ : Complexity.Coding γ✝ Memory inst✝ : Complexity.Encoding δ✝ Memory fs✝ : γ✝ → δ✝ cs✝ : Complexity.CostFunction γ✝ ℕ hcost : Complexity Encoding.Model fs✝ cs✝ next : CostedProgram h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm a : α m : Memory hm : m ∈ fm a ⊢ ?h₂.h.b ∈ (Complexity.decode γ✝ ∘ flip Memory.getms src✝) ?h₂.h.h.a case h₂.h.h.a α : Type u_1 inst✝⁷ : Complexity.Coding α Memory β : Type u_2 inst✝⁶ : Complexity.Encoding β Memory f : α → β tp : TracedProgram γ δ : Type inst✝⁵ : Complexity.Coding γ Memory inst✝⁴ : Complexity.Encoding δ Memory fs : γ → δ cs : Complexity.CostFunction γ ℕ inst✝³ : Complexity.Computable Encoding.Model fs inst✝² : 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 γ✝ δ✝ : Type inst✝¹ : Complexity.Coding γ✝ Memory inst✝ : Complexity.Encoding δ✝ Memory fs✝ : γ✝ → δ✝ cs✝ : Complexity.CostFunction γ✝ ℕ hcost : Complexity Encoding.Model fs✝ cs✝ next : CostedProgram h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm a : α m : Memory hm : m ∈ fm a ⊢ Memory case h₂.h.b α : Type u_1 inst✝⁷ : Complexity.Coding α Memory β : Type u_2 inst✝⁶ : Complexity.Encoding β Memory f : α → β tp : TracedProgram γ δ : Type inst✝⁵ : Complexity.Coding γ Memory inst✝⁴ : Complexity.Encoding δ Memory fs : γ → δ cs : Complexity.CostFunction γ ℕ inst✝³ : Complexity.Computable Encoding.Model fs inst✝² : 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 γ✝ δ✝ : Type inst✝¹ : Complexity.Coding γ✝ Memory inst✝ : Complexity.Encoding δ✝ Memory fs✝ : γ✝ → δ✝ cs✝ : Complexity.CostFunction γ✝ ℕ hcost : Complexity Encoding.Model fs✝ cs✝ next : CostedProgram h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm a : α m : Memory hm : m ∈ fm a ⊢ γ✝ α : Type u_1 inst✝⁷ : Complexity.Coding α Memory β : Type u_2 inst✝⁶ : Complexity.Encoding β Memory f : α → β tp : TracedProgram γ δ : Type inst✝⁵ : Complexity.Coding γ Memory inst✝⁴ : Complexity.Encoding δ Memory fs : γ → δ cs : Complexity.CostFunction γ ℕ inst✝³ : Complexity.Computable Encoding.Model fs inst✝² : 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 γ✝ δ✝ : Type inst✝¹ : Complexity.Coding γ✝ Memory inst✝ : Complexity.Encoding δ✝ Memory fs✝ : γ✝ → δ✝ cs✝ : Complexity.CostFunction γ✝ ℕ hcost : Complexity Encoding.Model fs✝ cs✝ next : CostedProgram h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm a : α m : Memory hm : m ∈ fm a ⊢ Complexity.ALE.bound Complexity.cost_ale (Option.get (Complexity.decode γ✝ (Memory.getms m src✝)) (_ : Option.isSome (Complexity.decode γ✝ (Memory.getms m src✝)) = true)) ≤ Encoding.getBound hcost ?h₂.h.b
https://github.com/calcu16/lean_complexity.git
d267eb4f3952c654f130a119311692c1c1a7cd71
HMem/Trace/Cost.lean
HMem.Trace.CostedProgram.splitTimeCost
[244, 1]
[349, 16]
apply hm
case h₂.h.h.ha α : Type u_1 inst✝⁷ : Complexity.Coding α Memory β : Type u_2 inst✝⁶ : Complexity.Encoding β Memory f : α → β tp : TracedProgram γ δ : Type inst✝⁵ : Complexity.Coding γ Memory inst✝⁴ : Complexity.Encoding δ Memory fs : γ → δ cs : Complexity.CostFunction γ ℕ inst✝³ : Complexity.Computable Encoding.Model fs inst✝² : 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 γ✝ δ✝ : Type inst✝¹ : Complexity.Coding γ✝ Memory inst✝ : Complexity.Encoding δ✝ Memory fs✝ : γ✝ → δ✝ cs✝ : Complexity.CostFunction γ✝ ℕ hcost : Complexity Encoding.Model fs✝ cs✝ next : CostedProgram h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm a : α m : Memory hm : m ∈ fm a ⊢ ?h₂.h.h.a ∈ fm a case h₂.h.h.hf α : Type u_1 inst✝⁷ : Complexity.Coding α Memory β : Type u_2 inst✝⁶ : Complexity.Encoding β Memory f : α → β tp : TracedProgram γ δ : Type inst✝⁵ : Complexity.Coding γ Memory inst✝⁴ : Complexity.Encoding δ Memory fs : γ → δ cs : Complexity.CostFunction γ ℕ inst✝³ : Complexity.Computable Encoding.Model fs inst✝² : 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 γ✝ δ✝ : Type inst✝¹ : Complexity.Coding γ✝ Memory inst✝ : Complexity.Encoding δ✝ Memory fs✝ : γ✝ → δ✝ cs✝ : Complexity.CostFunction γ✝ ℕ hcost : Complexity Encoding.Model fs✝ cs✝ next : CostedProgram h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm a : α m : Memory hm : m ∈ fm a ⊢ ?h₂.h.b ∈ (Complexity.decode γ✝ ∘ flip Memory.getms src✝) ?h₂.h.h.a case h₂.h.h.a α : Type u_1 inst✝⁷ : Complexity.Coding α Memory β : Type u_2 inst✝⁶ : Complexity.Encoding β Memory f : α → β tp : TracedProgram γ δ : Type inst✝⁵ : Complexity.Coding γ Memory inst✝⁴ : Complexity.Encoding δ Memory fs : γ → δ cs : Complexity.CostFunction γ ℕ inst✝³ : Complexity.Computable Encoding.Model fs inst✝² : 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 γ✝ δ✝ : Type inst✝¹ : Complexity.Coding γ✝ Memory inst✝ : Complexity.Encoding δ✝ Memory fs✝ : γ✝ → δ✝ cs✝ : Complexity.CostFunction γ✝ ℕ hcost : Complexity Encoding.Model fs✝ cs✝ next : CostedProgram h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm a : α m : Memory hm : m ∈ fm a ⊢ Memory case h₂.h.b α : Type u_1 inst✝⁷ : Complexity.Coding α Memory β : Type u_2 inst✝⁶ : Complexity.Encoding β Memory f : α → β tp : TracedProgram γ δ : Type inst✝⁵ : Complexity.Coding γ Memory inst✝⁴ : Complexity.Encoding δ Memory fs : γ → δ cs : Complexity.CostFunction γ ℕ inst✝³ : Complexity.Computable Encoding.Model fs inst✝² : 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 γ✝ δ✝ : Type inst✝¹ : Complexity.Coding γ✝ Memory inst✝ : Complexity.Encoding δ✝ Memory fs✝ : γ✝ → δ✝ cs✝ : Complexity.CostFunction γ✝ ℕ hcost : Complexity Encoding.Model fs✝ cs✝ next : CostedProgram h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm a : α m : Memory hm : m ∈ fm a ⊢ γ✝ α : Type u_1 inst✝⁷ : Complexity.Coding α Memory β : Type u_2 inst✝⁶ : Complexity.Encoding β Memory f : α → β tp : TracedProgram γ δ : Type inst✝⁵ : Complexity.Coding γ Memory inst✝⁴ : Complexity.Encoding δ Memory fs : γ → δ cs : Complexity.CostFunction γ ℕ inst✝³ : Complexity.Computable Encoding.Model fs inst✝² : 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 γ✝ δ✝ : Type inst✝¹ : Complexity.Coding γ✝ Memory inst✝ : Complexity.Encoding δ✝ Memory fs✝ : γ✝ → δ✝ cs✝ : Complexity.CostFunction γ✝ ℕ hcost : Complexity Encoding.Model fs✝ cs✝ next : CostedProgram h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm a : α m : Memory hm : m ∈ fm a ⊢ Complexity.ALE.bound Complexity.cost_ale (Option.get (Complexity.decode γ✝ (Memory.getms m src✝)) (_ : Option.isSome (Complexity.decode γ✝ (Memory.getms m src✝)) = true)) ≤ Encoding.getBound hcost ?h₂.h.b
case h₂.h.h.hf α : Type u_1 inst✝⁷ : Complexity.Coding α Memory β : Type u_2 inst✝⁶ : Complexity.Encoding β Memory f : α → β tp : TracedProgram γ δ : Type inst✝⁵ : Complexity.Coding γ Memory inst✝⁴ : Complexity.Encoding δ Memory fs : γ → δ cs : Complexity.CostFunction γ ℕ inst✝³ : Complexity.Computable Encoding.Model fs inst✝² : 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 γ✝ δ✝ : Type inst✝¹ : Complexity.Coding γ✝ Memory inst✝ : Complexity.Encoding δ✝ Memory fs✝ : γ✝ → δ✝ cs✝ : Complexity.CostFunction γ✝ ℕ hcost : Complexity Encoding.Model fs✝ cs✝ next : CostedProgram h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm a : α m : Memory hm : m ∈ fm a ⊢ ?h₂.h.b ∈ (Complexity.decode γ✝ ∘ flip Memory.getms src✝) m case h₂.h.b α : Type u_1 inst✝⁷ : Complexity.Coding α Memory β : Type u_2 inst✝⁶ : Complexity.Encoding β Memory f : α → β tp : TracedProgram γ δ : Type inst✝⁵ : Complexity.Coding γ Memory inst✝⁴ : Complexity.Encoding δ Memory fs : γ → δ cs : Complexity.CostFunction γ ℕ inst✝³ : Complexity.Computable Encoding.Model fs inst✝² : 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 γ✝ δ✝ : Type inst✝¹ : Complexity.Coding γ✝ Memory inst✝ : Complexity.Encoding δ✝ Memory fs✝ : γ✝ → δ✝ cs✝ : Complexity.CostFunction γ✝ ℕ hcost : Complexity Encoding.Model fs✝ cs✝ next : CostedProgram h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm a : α m : Memory hm : m ∈ fm a ⊢ γ✝ α : Type u_1 inst✝⁷ : Complexity.Coding α Memory β : Type u_2 inst✝⁶ : Complexity.Encoding β Memory f : α → β tp : TracedProgram γ δ : Type inst✝⁵ : Complexity.Coding γ Memory inst✝⁴ : Complexity.Encoding δ Memory fs : γ → δ cs : Complexity.CostFunction γ ℕ inst✝³ : Complexity.Computable Encoding.Model fs inst✝² : 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 γ✝ δ✝ : Type inst✝¹ : Complexity.Coding γ✝ Memory inst✝ : Complexity.Encoding δ✝ Memory fs✝ : γ✝ → δ✝ cs✝ : Complexity.CostFunction γ✝ ℕ hcost : Complexity Encoding.Model fs✝ cs✝ next : CostedProgram h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm a : α m : Memory hm : m ∈ fm a ⊢ Complexity.ALE.bound Complexity.cost_ale (Option.get (Complexity.decode γ✝ (Memory.getms m src✝)) (_ : Option.isSome (Complexity.decode γ✝ (Memory.getms m src✝)) = true)) ≤ Encoding.getBound hcost ?h₂.h.b
https://github.com/calcu16/lean_complexity.git
d267eb4f3952c654f130a119311692c1c1a7cd71
HMem/Trace/Cost.lean
HMem.Trace.CostedProgram.splitTimeCost
[244, 1]
[349, 16]
apply Option.get_mem
case h₂.h.h.hf α : Type u_1 inst✝⁷ : Complexity.Coding α Memory β : Type u_2 inst✝⁶ : Complexity.Encoding β Memory f : α → β tp : TracedProgram γ δ : Type inst✝⁵ : Complexity.Coding γ Memory inst✝⁴ : Complexity.Encoding δ Memory fs : γ → δ cs : Complexity.CostFunction γ ℕ inst✝³ : Complexity.Computable Encoding.Model fs inst✝² : 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 γ✝ δ✝ : Type inst✝¹ : Complexity.Coding γ✝ Memory inst✝ : Complexity.Encoding δ✝ Memory fs✝ : γ✝ → δ✝ cs✝ : Complexity.CostFunction γ✝ ℕ hcost : Complexity Encoding.Model fs✝ cs✝ next : CostedProgram h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm a : α m : Memory hm : m ∈ fm a ⊢ ?h₂.h.b ∈ (Complexity.decode γ✝ ∘ flip Memory.getms src✝) m case h₂.h.b α : Type u_1 inst✝⁷ : Complexity.Coding α Memory β : Type u_2 inst✝⁶ : Complexity.Encoding β Memory f : α → β tp : TracedProgram γ δ : Type inst✝⁵ : Complexity.Coding γ Memory inst✝⁴ : Complexity.Encoding δ Memory fs : γ → δ cs : Complexity.CostFunction γ ℕ inst✝³ : Complexity.Computable Encoding.Model fs inst✝² : 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 γ✝ δ✝ : Type inst✝¹ : Complexity.Coding γ✝ Memory inst✝ : Complexity.Encoding δ✝ Memory fs✝ : γ✝ → δ✝ cs✝ : Complexity.CostFunction γ✝ ℕ hcost : Complexity Encoding.Model fs✝ cs✝ next : CostedProgram h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm a : α m : Memory hm : m ∈ fm a ⊢ γ✝ α : Type u_1 inst✝⁷ : Complexity.Coding α Memory β : Type u_2 inst✝⁶ : Complexity.Encoding β Memory f : α → β tp : TracedProgram γ δ : Type inst✝⁵ : Complexity.Coding γ Memory inst✝⁴ : Complexity.Encoding δ Memory fs : γ → δ cs : Complexity.CostFunction γ ℕ inst✝³ : Complexity.Computable Encoding.Model fs inst✝² : 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 γ✝ δ✝ : Type inst✝¹ : Complexity.Coding γ✝ Memory inst✝ : Complexity.Encoding δ✝ Memory fs✝ : γ✝ → δ✝ cs✝ : Complexity.CostFunction γ✝ ℕ hcost : Complexity Encoding.Model fs✝ cs✝ next : CostedProgram h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm a : α m : Memory hm : m ∈ fm a ⊢ Complexity.ALE.bound Complexity.cost_ale (Option.get (Complexity.decode γ✝ (Memory.getms m src✝)) (_ : Option.isSome (Complexity.decode γ✝ (Memory.getms m src✝)) = true)) ≤ Encoding.getBound hcost ?h₂.h.b
case h₂.h.h.hf.h α : Type u_1 inst✝⁷ : Complexity.Coding α Memory β : Type u_2 inst✝⁶ : Complexity.Encoding β Memory f : α → β tp : TracedProgram γ δ : Type inst✝⁵ : Complexity.Coding γ Memory inst✝⁴ : Complexity.Encoding δ Memory fs : γ → δ cs : Complexity.CostFunction γ ℕ inst✝³ : Complexity.Computable Encoding.Model fs inst✝² : 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 γ✝ δ✝ : Type inst✝¹ : Complexity.Coding γ✝ Memory inst✝ : Complexity.Encoding δ✝ Memory fs✝ : γ✝ → δ✝ cs✝ : Complexity.CostFunction γ✝ ℕ hcost : Complexity Encoding.Model fs✝ cs✝ next : CostedProgram h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm a : α m : Memory hm : m ∈ fm a ⊢ Option.isSome ((Complexity.decode γ✝ ∘ flip Memory.getms src✝) m) = true α : Type u_1 inst✝⁷ : Complexity.Coding α Memory β : Type u_2 inst✝⁶ : Complexity.Encoding β Memory f : α → β tp : TracedProgram γ δ : Type inst✝⁵ : Complexity.Coding γ Memory inst✝⁴ : Complexity.Encoding δ Memory fs : γ → δ cs : Complexity.CostFunction γ ℕ inst✝³ : Complexity.Computable Encoding.Model fs inst✝² : 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 γ✝ δ✝ : Type inst✝¹ : Complexity.Coding γ✝ Memory inst✝ : Complexity.Encoding δ✝ Memory fs✝ : γ✝ → δ✝ cs✝ : Complexity.CostFunction γ✝ ℕ hcost : Complexity Encoding.Model fs✝ cs✝ next : CostedProgram h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm a : α m : Memory hm : m ∈ fm a ⊢ Complexity.ALE.bound Complexity.cost_ale (Option.get (Complexity.decode γ✝ (Memory.getms m src✝)) (_ : Option.isSome (Complexity.decode γ✝ (Memory.getms m src✝)) = true)) ≤ Encoding.getBound hcost (Option.get ((Complexity.decode γ✝ ∘ flip Memory.getms src✝) m) ?h₂.h.h.hf.h)
https://github.com/calcu16/lean_complexity.git
d267eb4f3952c654f130a119311692c1c1a7cd71
HMem/Trace/Cost.lean
HMem.Trace.CostedProgram.splitTimeCost
[244, 1]
[349, 16]
apply (h.left _ _ hm).elim λ hsome _ ↦ hsome
case h₂.h.h.hf.h α : Type u_1 inst✝⁷ : Complexity.Coding α Memory β : Type u_2 inst✝⁶ : Complexity.Encoding β Memory f : α → β tp : TracedProgram γ δ : Type inst✝⁵ : Complexity.Coding γ Memory inst✝⁴ : Complexity.Encoding δ Memory fs : γ → δ cs : Complexity.CostFunction γ ℕ inst✝³ : Complexity.Computable Encoding.Model fs inst✝² : 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 γ✝ δ✝ : Type inst✝¹ : Complexity.Coding γ✝ Memory inst✝ : Complexity.Encoding δ✝ Memory fs✝ : γ✝ → δ✝ cs✝ : Complexity.CostFunction γ✝ ℕ hcost : Complexity Encoding.Model fs✝ cs✝ next : CostedProgram h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm a : α m : Memory hm : m ∈ fm a ⊢ Option.isSome ((Complexity.decode γ✝ ∘ flip Memory.getms src✝) m) = true α : Type u_1 inst✝⁷ : Complexity.Coding α Memory β : Type u_2 inst✝⁶ : Complexity.Encoding β Memory f : α → β tp : TracedProgram γ δ : Type inst✝⁵ : Complexity.Coding γ Memory inst✝⁴ : Complexity.Encoding δ Memory fs : γ → δ cs : Complexity.CostFunction γ ℕ inst✝³ : Complexity.Computable Encoding.Model fs inst✝² : 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 γ✝ δ✝ : Type inst✝¹ : Complexity.Coding γ✝ Memory inst✝ : Complexity.Encoding δ✝ Memory fs✝ : γ✝ → δ✝ cs✝ : Complexity.CostFunction γ✝ ℕ hcost : Complexity Encoding.Model fs✝ cs✝ next : CostedProgram h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm a : α m : Memory hm : m ∈ fm a ⊢ Complexity.ALE.bound Complexity.cost_ale (Option.get (Complexity.decode γ✝ (Memory.getms m src✝)) (_ : Option.isSome (Complexity.decode γ✝ (Memory.getms m src✝)) = true)) ≤ Encoding.getBound hcost (Option.get ((Complexity.decode γ✝ ∘ flip Memory.getms src✝) m) ?h₂.h.h.hf.h)
α : Type u_1 inst✝⁷ : Complexity.Coding α Memory β : Type u_2 inst✝⁶ : Complexity.Encoding β Memory f : α → β tp : TracedProgram γ δ : Type inst✝⁵ : Complexity.Coding γ Memory inst✝⁴ : Complexity.Encoding δ Memory fs : γ → δ cs : Complexity.CostFunction γ ℕ inst✝³ : Complexity.Computable Encoding.Model fs inst✝² : 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 γ✝ δ✝ : Type inst✝¹ : Complexity.Coding γ✝ Memory inst✝ : Complexity.Encoding δ✝ Memory fs✝ : γ✝ → δ✝ cs✝ : Complexity.CostFunction γ✝ ℕ hcost : Complexity Encoding.Model fs✝ cs✝ next : CostedProgram h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm a : α m : Memory hm : m ∈ fm a ⊢ Complexity.ALE.bound Complexity.cost_ale (Option.get (Complexity.decode γ✝ (Memory.getms m src✝)) (_ : Option.isSome (Complexity.decode γ✝ (Memory.getms m src✝)) = true)) ≤ Encoding.getBound hcost (Option.get ((Complexity.decode γ✝ ∘ flip Memory.getms src✝) m) (_ : Option.isSome (Complexity.decode γ✝ (Memory.getms m src✝)) = true))
https://github.com/calcu16/lean_complexity.git
d267eb4f3952c654f130a119311692c1c1a7cd71
HMem/Trace/Cost.lean
HMem.Trace.CostedProgram.splitTimeCost
[244, 1]
[349, 16]
apply le_refl
α : Type u_1 inst✝⁷ : Complexity.Coding α Memory β : Type u_2 inst✝⁶ : Complexity.Encoding β Memory f : α → β tp : TracedProgram γ δ : Type inst✝⁵ : Complexity.Coding γ Memory inst✝⁴ : Complexity.Encoding δ Memory fs : γ → δ cs : Complexity.CostFunction γ ℕ inst✝³ : Complexity.Computable Encoding.Model fs inst✝² : 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 γ✝ δ✝ : Type inst✝¹ : Complexity.Coding γ✝ Memory inst✝ : Complexity.Encoding δ✝ Memory fs✝ : γ✝ → δ✝ cs✝ : Complexity.CostFunction γ✝ ℕ hcost : Complexity Encoding.Model fs✝ cs✝ next : CostedProgram h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm a : α m : Memory hm : m ∈ fm a ⊢ Complexity.ALE.bound Complexity.cost_ale (Option.get (Complexity.decode γ✝ (Memory.getms m src✝)) (_ : Option.isSome (Complexity.decode γ✝ (Memory.getms m src✝)) = true)) ≤ Encoding.getBound hcost (Option.get ((Complexity.decode γ✝ ∘ flip Memory.getms src✝) m) (_ : Option.isSome (Complexity.decode γ✝ (Memory.getms m src✝)) = true))
no goals
https://github.com/calcu16/lean_complexity.git
d267eb4f3952c654f130a119311692c1c1a7cd71
HMem/Trace/Cost.lean
HMem.Trace.CostedProgram.splitTimeCost
[244, 1]
[349, 16]
apply Stack.hasTimeCost_recurse_le _
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 := x✝, current := toProgram (recurse dst✝ src✝ next✝), function := toProgram fcp } []) ((subroutineTimeCost h + recurseTimeCost hf h + localTimeCost h) x✝¹)
case hr 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.hasResult (Stack.execution { state := Memory.getms x✝ src✝, current := toProgram fcp, function := toProgram fcp } []) ?m.179017 case h₀ 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.getms x✝ src✝, current := toProgram fcp, function := toProgram fcp } []) ?m.179014 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✝ ?m.179017, 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 ⊢ 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 ⊢ Memory 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 + ?m.179014 + 1 ≤ (subroutineTimeCost h + recurseTimeCost hf h + localTimeCost h) x✝¹
https://github.com/calcu16/lean_complexity.git
d267eb4f3952c654f130a119311692c1c1a7cd71
HMem/Trace/Cost.lean
HMem.Trace.CostedProgram.splitTimeCost
[244, 1]
[349, 16]
rw [TracedProgram.sound_recurse_arg (TracedProgram.sound_of_sound' h _ _ hm)]
case hr 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.hasResult (Stack.execution { state := Memory.getms x✝ src✝, current := toProgram fcp, function := toProgram fcp } []) ?m.179017 case h₀ 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.getms x✝ src✝, current := toProgram fcp, function := toProgram fcp } []) ?m.179014 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✝ ?m.179017, 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 ⊢ 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 ⊢ Memory 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 + ?m.179014 + 1 ≤ (subroutineTimeCost h + recurseTimeCost hf h + localTimeCost h) x✝¹
case hr 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.hasResult (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 } []) ?m.179017 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 ⊢ Memory case h₀ 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.getms x✝ src✝, current := toProgram fcp, function := toProgram fcp } []) ?m.179014 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✝ ?m.179017, 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 ⊢ 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 ⊢ Memory 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 + ?m.179014 + 1 ≤ (subroutineTimeCost h + recurseTimeCost hf h + localTimeCost h) x✝¹
https://github.com/calcu16/lean_complexity.git
d267eb4f3952c654f130a119311692c1c1a7cd71
HMem/Trace/Cost.lean
HMem.Trace.CostedProgram.splitTimeCost
[244, 1]
[349, 16]
apply Program.hasResult_of_sound hf
case hr 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.hasResult (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 } []) ?m.179017 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 ⊢ Memory case h₀ 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.getms x✝ src✝, current := toProgram fcp, function := toProgram fcp } []) ?m.179014 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✝ ?m.179017, 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 ⊢ 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 ⊢ Memory 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 + ?m.179014 + 1 ≤ (subroutineTimeCost h + recurseTimeCost hf h + localTimeCost h) x✝¹
case h₀ 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.getms x✝ src✝, current := toProgram fcp, function := toProgram fcp } []) ?m.179014 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 ⊢ 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 + ?m.179014 + 1 ≤ (subroutineTimeCost h + recurseTimeCost hf h + localTimeCost h) x✝¹
https://github.com/calcu16/lean_complexity.git
d267eb4f3952c654f130a119311692c1c1a7cd71
HMem/Trace/Cost.lean
HMem.Trace.CostedProgram.splitTimeCost
[244, 1]
[349, 16]
rw [TracedProgram.sound_recurse_arg (TracedProgram.sound_of_sound' h _ _ hm)]
case h₀ 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.getms x✝ src✝, current := toProgram fcp, function := toProgram fcp } []) ?m.179014 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 ⊢ 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 + ?m.179014 + 1 ≤ (subroutineTimeCost h + recurseTimeCost hf h + localTimeCost h) x✝¹
case h₀ 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 := 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 } []) ?m.179014 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 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 ⊢ 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 + ?m.179014 + 1 ≤ (subroutineTimeCost h + recurseTimeCost hf h + localTimeCost h) x✝¹
https://github.com/calcu16/lean_complexity.git
d267eb4f3952c654f130a119311692c1c1a7cd71
HMem/Trace/Cost.lean
HMem.Trace.CostedProgram.splitTimeCost
[244, 1]
[349, 16]
apply Stack.timeCost_sound
case h₀ 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 := 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 } []) ?m.179014 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 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 ⊢ 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 + ?m.179014 + 1 ≤ (subroutineTimeCost h + recurseTimeCost hf h + localTimeCost h) x✝¹
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
https://github.com/calcu16/lean_complexity.git
d267eb4f3952c654f130a119311692c1c1a7cd71
HMem/Trace/Cost.lean
HMem.Trace.CostedProgram.splitTimeCost
[244, 1]
[349, 16]
apply Program.halts_of_sound hf
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
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
https://github.com/calcu16/lean_complexity.git
d267eb4f3952c654f130a119311692c1c1a7cd71
HMem/Trace/Cost.lean
HMem.Trace.CostedProgram.splitTimeCost
[244, 1]
[349, 16]
apply splitTimeCost hf (TracedProgram.sound'_recurse_next h)
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
case h₁.a 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 ⊢ 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)))) ∈ ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ f) (Complexity.decode α (Memory.getms m src✝))) ∘ fm) ?h₁.a case h₁.a 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 ⊢ α 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 ⊢ (subroutineTimeCost (_ : TracedProgram.sound' f size (toTracedProgram next✝) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ f) (Complexity.decode α (Memory.getms m src✝))) ∘ fm)) + recurseTimeCost hf (_ : TracedProgram.sound' f size (toTracedProgram next✝) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ f) (Complexity.decode α (Memory.getms m src✝))) ∘ fm)) + localTimeCost (_ : TracedProgram.sound' f size (toTracedProgram next✝) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ f) (Complexity.decode α (Memory.getms m src✝))) ∘ fm))) ?h₁.a + 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
https://github.com/calcu16/lean_complexity.git
d267eb4f3952c654f130a119311692c1c1a7cd71
HMem/Trace/Cost.lean
HMem.Trace.CostedProgram.splitTimeCost
[244, 1]
[349, 16]
apply Option.mem_bind_of_mem hm
case h₁.a 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 ⊢ 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)))) ∈ ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ f) (Complexity.decode α (Memory.getms m src✝))) ∘ fm) ?h₁.a case h₁.a 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 ⊢ α 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 ⊢ (subroutineTimeCost (_ : TracedProgram.sound' f size (toTracedProgram next✝) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ f) (Complexity.decode α (Memory.getms m src✝))) ∘ fm)) + recurseTimeCost hf (_ : TracedProgram.sound' f size (toTracedProgram next✝) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ f) (Complexity.decode α (Memory.getms m src✝))) ∘ fm)) + localTimeCost (_ : TracedProgram.sound' f size (toTracedProgram next✝) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ f) (Complexity.decode α (Memory.getms m src✝))) ∘ fm))) ?h₁.a + 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
case h₁.a 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 ⊢ 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)))) ∈ Option.map (Memory.setms x✝ dst✝ ∘ Complexity.encode ∘ f) (Complexity.decode α (Memory.getms x✝ src✝)) 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 ⊢ (subroutineTimeCost (_ : TracedProgram.sound' f size (toTracedProgram next✝) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ f) (Complexity.decode α (Memory.getms m src✝))) ∘ fm)) + recurseTimeCost hf (_ : TracedProgram.sound' f size (toTracedProgram next✝) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ f) (Complexity.decode α (Memory.getms m src✝))) ∘ fm)) + localTimeCost (_ : TracedProgram.sound' f size (toTracedProgram next✝) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ f) (Complexity.decode α (Memory.getms m src✝))) ∘ fm))) x✝¹ + 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
https://github.com/calcu16/lean_complexity.git
d267eb4f3952c654f130a119311692c1c1a7cd71
HMem/Trace/Cost.lean
HMem.Trace.CostedProgram.splitTimeCost
[244, 1]
[349, 16]
apply Option.mem_map_of_mem
case h₁.a 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 ⊢ 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)))) ∈ Option.map (Memory.setms x✝ dst✝ ∘ Complexity.encode ∘ f) (Complexity.decode α (Memory.getms x✝ src✝)) 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 ⊢ (subroutineTimeCost (_ : TracedProgram.sound' f size (toTracedProgram next✝) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ f) (Complexity.decode α (Memory.getms m src✝))) ∘ fm)) + recurseTimeCost hf (_ : TracedProgram.sound' f size (toTracedProgram next✝) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ f) (Complexity.decode α (Memory.getms m src✝))) ∘ fm)) + localTimeCost (_ : TracedProgram.sound' f size (toTracedProgram next✝) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ f) (Complexity.decode α (Memory.getms m src✝))) ∘ fm))) x✝¹ + 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
case h₁.a.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 ⊢ Option.get (Complexity.decode α (Memory.getms x✝ src✝)) (_ : Option.isSome (Complexity.decode α (Memory.getms x✝ src✝)) = true) ∈ Complexity.decode α (Memory.getms x✝ src✝) 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 ⊢ (subroutineTimeCost (_ : TracedProgram.sound' f size (toTracedProgram next✝) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ f) (Complexity.decode α (Memory.getms m src✝))) ∘ fm)) + recurseTimeCost hf (_ : TracedProgram.sound' f size (toTracedProgram next✝) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ f) (Complexity.decode α (Memory.getms m src✝))) ∘ fm)) + localTimeCost (_ : TracedProgram.sound' f size (toTracedProgram next✝) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ f) (Complexity.decode α (Memory.getms m src✝))) ∘ fm))) x✝¹ + 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
https://github.com/calcu16/lean_complexity.git
d267eb4f3952c654f130a119311692c1c1a7cd71
HMem/Trace/Cost.lean
HMem.Trace.CostedProgram.splitTimeCost
[244, 1]
[349, 16]
apply Option.get_mem
case h₁.a.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 ⊢ Option.get (Complexity.decode α (Memory.getms x✝ src✝)) (_ : Option.isSome (Complexity.decode α (Memory.getms x✝ src✝)) = true) ∈ Complexity.decode α (Memory.getms x✝ src✝) 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 ⊢ (subroutineTimeCost (_ : TracedProgram.sound' f size (toTracedProgram next✝) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ f) (Complexity.decode α (Memory.getms m src✝))) ∘ fm)) + recurseTimeCost hf (_ : TracedProgram.sound' f size (toTracedProgram next✝) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ f) (Complexity.decode α (Memory.getms m src✝))) ∘ fm)) + localTimeCost (_ : TracedProgram.sound' f size (toTracedProgram next✝) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ f) (Complexity.decode α (Memory.getms m src✝))) ∘ fm))) x✝¹ + 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
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 ⊢ (subroutineTimeCost (_ : TracedProgram.sound' f size (toTracedProgram next✝) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ f) (Complexity.decode α (Memory.getms m src✝))) ∘ fm)) + recurseTimeCost hf (_ : TracedProgram.sound' f size (toTracedProgram next✝) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ f) (Complexity.decode α (Memory.getms m src✝))) ∘ fm)) + localTimeCost (_ : TracedProgram.sound' f size (toTracedProgram next✝) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ f) (Complexity.decode α (Memory.getms m src✝))) ∘ fm))) x✝¹ + 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
https://github.com/calcu16/lean_complexity.git
d267eb4f3952c654f130a119311692c1c1a7cd71
HMem/Trace/Cost.lean
HMem.Trace.CostedProgram.splitTimeCost
[244, 1]
[349, 16]
apply le_of_le_of_eq _
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 ⊢ (subroutineTimeCost (_ : TracedProgram.sound' f size (toTracedProgram next✝) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ f) (Complexity.decode α (Memory.getms m src✝))) ∘ fm)) + recurseTimeCost hf (_ : TracedProgram.sound' f size (toTracedProgram next✝) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ f) (Complexity.decode α (Memory.getms m src✝))) ∘ fm)) + localTimeCost (_ : TracedProgram.sound' f size (toTracedProgram next✝) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ f) (Complexity.decode α (Memory.getms m src✝))) ∘ fm))) x✝¹ + 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
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.179858 = (subroutineTimeCost h + recurseTimeCost hf h + localTimeCost h) x✝¹ 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 ⊢ (subroutineTimeCost (_ : TracedProgram.sound' f size (toTracedProgram next✝) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ f) (Complexity.decode α (Memory.getms m src✝))) ∘ fm)) + recurseTimeCost hf (_ : TracedProgram.sound' f size (toTracedProgram next✝) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ f) (Complexity.decode α (Memory.getms m src✝))) ∘ fm)) + localTimeCost (_ : TracedProgram.sound' f size (toTracedProgram next✝) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ f) (Complexity.decode α (Memory.getms m src✝))) ∘ fm))) x✝¹ + 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 ≤ ?m.179858
https://github.com/calcu16/lean_complexity.git
d267eb4f3952c654f130a119311692c1c1a7cd71
HMem/Trace/Cost.lean
HMem.Trace.CostedProgram.splitTimeCost
[244, 1]
[349, 16]
apply Nat.add_assoc
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.179858 = (subroutineTimeCost h + recurseTimeCost hf h + localTimeCost h) x✝¹ 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 ⊢ (subroutineTimeCost (_ : TracedProgram.sound' f size (toTracedProgram next✝) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ f) (Complexity.decode α (Memory.getms m src✝))) ∘ fm)) + recurseTimeCost hf (_ : TracedProgram.sound' f size (toTracedProgram next✝) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ f) (Complexity.decode α (Memory.getms m src✝))) ∘ fm)) + localTimeCost (_ : TracedProgram.sound' f size (toTracedProgram next✝) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ f) (Complexity.decode α (Memory.getms m src✝))) ∘ fm))) x✝¹ + 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 ≤ ?m.179858
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 ⊢ (subroutineTimeCost (_ : TracedProgram.sound' f size (toTracedProgram next✝) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ f) (Complexity.decode α (Memory.getms m src✝))) ∘ fm)) + recurseTimeCost hf (_ : TracedProgram.sound' f size (toTracedProgram next✝) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ f) (Complexity.decode α (Memory.getms m src✝))) ∘ fm)) + localTimeCost (_ : TracedProgram.sound' f size (toTracedProgram next✝) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ f) (Complexity.decode α (Memory.getms m src✝))) ∘ fm))) x✝¹ + 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) x✝¹ + localTimeCost (_ : TracedProgram.sound' f size (toTracedProgram next✝) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ f) (Complexity.decode α (Memory.getms m src✝))) ∘ fm)) x✝¹ + Complexity.CostFunction.flatMap fm 1 x✝¹
https://github.com/calcu16/lean_complexity.git
d267eb4f3952c654f130a119311692c1c1a7cd71
HMem/Trace/Cost.lean
HMem.Trace.CostedProgram.splitTimeCost
[244, 1]
[349, 16]
apply add_le_add _
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 ⊢ (subroutineTimeCost (_ : TracedProgram.sound' f size (toTracedProgram next✝) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ f) (Complexity.decode α (Memory.getms m src✝))) ∘ fm)) + recurseTimeCost hf (_ : TracedProgram.sound' f size (toTracedProgram next✝) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ f) (Complexity.decode α (Memory.getms m src✝))) ∘ fm)) + localTimeCost (_ : TracedProgram.sound' f size (toTracedProgram next✝) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ f) (Complexity.decode α (Memory.getms m src✝))) ∘ fm))) x✝¹ + 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) x✝¹ + localTimeCost (_ : TracedProgram.sound' f size (toTracedProgram next✝) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ f) (Complexity.decode α (Memory.getms m src✝))) ∘ fm)) x✝¹ + Complexity.CostFunction.flatMap fm 1 x✝¹
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 ⊢ 1 ≤ Complexity.CostFunction.flatMap fm 1 x✝¹ 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 ⊢ (subroutineTimeCost (_ : TracedProgram.sound' f size (toTracedProgram next✝) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ f) (Complexity.decode α (Memory.getms m src✝))) ∘ fm)) + recurseTimeCost hf (_ : TracedProgram.sound' f size (toTracedProgram next✝) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ f) (Complexity.decode α (Memory.getms m src✝))) ∘ fm)) + localTimeCost (_ : TracedProgram.sound' f size (toTracedProgram next✝) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ f) (Complexity.decode α (Memory.getms m src✝))) ∘ fm))) x✝¹ + 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)))) ≤ (subroutineTimeCost h + recurseTimeCost hf h) x✝¹ + localTimeCost (_ : TracedProgram.sound' f size (toTracedProgram next✝) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ f) (Complexity.decode α (Memory.getms m src✝))) ∘ fm)) x✝¹
https://github.com/calcu16/lean_complexity.git
d267eb4f3952c654f130a119311692c1c1a7cd71
HMem/Trace/Cost.lean
HMem.Trace.CostedProgram.splitTimeCost
[244, 1]
[349, 16]
apply (le_of_eq (Eq.symm (Complexity.CostFunction.flatMap_some hm _)))
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 ⊢ 1 ≤ Complexity.CostFunction.flatMap fm 1 x✝¹ 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 ⊢ (subroutineTimeCost (_ : TracedProgram.sound' f size (toTracedProgram next✝) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ f) (Complexity.decode α (Memory.getms m src✝))) ∘ fm)) + recurseTimeCost hf (_ : TracedProgram.sound' f size (toTracedProgram next✝) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ f) (Complexity.decode α (Memory.getms m src✝))) ∘ fm)) + localTimeCost (_ : TracedProgram.sound' f size (toTracedProgram next✝) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ f) (Complexity.decode α (Memory.getms m src✝))) ∘ fm))) x✝¹ + 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)))) ≤ (subroutineTimeCost h + recurseTimeCost hf h) x✝¹ + localTimeCost (_ : TracedProgram.sound' f size (toTracedProgram next✝) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ f) (Complexity.decode α (Memory.getms m src✝))) ∘ fm)) x✝¹
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 ⊢ (subroutineTimeCost (_ : TracedProgram.sound' f size (toTracedProgram next✝) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ f) (Complexity.decode α (Memory.getms m src✝))) ∘ fm)) + recurseTimeCost hf (_ : TracedProgram.sound' f size (toTracedProgram next✝) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ f) (Complexity.decode α (Memory.getms m src✝))) ∘ fm)) + localTimeCost (_ : TracedProgram.sound' f size (toTracedProgram next✝) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ f) (Complexity.decode α (Memory.getms m src✝))) ∘ fm))) x✝¹ + 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)))) ≤ (subroutineTimeCost h + recurseTimeCost hf h) x✝¹ + localTimeCost (_ : TracedProgram.sound' f size (toTracedProgram next✝) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ f) (Complexity.decode α (Memory.getms m src✝))) ∘ fm)) x✝¹
https://github.com/calcu16/lean_complexity.git
d267eb4f3952c654f130a119311692c1c1a7cd71
HMem/Trace/Cost.lean
HMem.Trace.CostedProgram.splitTimeCost
[244, 1]
[349, 16]
apply le_of_eq_of_le
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 ⊢ (subroutineTimeCost (_ : TracedProgram.sound' f size (toTracedProgram next✝) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ f) (Complexity.decode α (Memory.getms m src✝))) ∘ fm)) + recurseTimeCost hf (_ : TracedProgram.sound' f size (toTracedProgram next✝) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ f) (Complexity.decode α (Memory.getms m src✝))) ∘ fm)) + localTimeCost (_ : TracedProgram.sound' f size (toTracedProgram next✝) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ f) (Complexity.decode α (Memory.getms m src✝))) ∘ fm))) x✝¹ + 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)))) ≤ (subroutineTimeCost h + recurseTimeCost hf h) x✝¹ + localTimeCost (_ : TracedProgram.sound' f size (toTracedProgram next✝) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ f) (Complexity.decode α (Memory.getms m src✝))) ∘ fm)) x✝¹
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 ⊢ (subroutineTimeCost (_ : TracedProgram.sound' f size (toTracedProgram next✝) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ f) (Complexity.decode α (Memory.getms m src✝))) ∘ fm)) + recurseTimeCost hf (_ : TracedProgram.sound' f size (toTracedProgram next✝) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ f) (Complexity.decode α (Memory.getms m src✝))) ∘ fm)) + localTimeCost (_ : TracedProgram.sound' f size (toTracedProgram next✝) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ f) (Complexity.decode α (Memory.getms m src✝))) ∘ fm))) x✝¹ + 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)))) = ?b case h₂ 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 ⊢ ?b ≤ (subroutineTimeCost h + recurseTimeCost hf h) x✝¹ + localTimeCost (_ : TracedProgram.sound' f size (toTracedProgram next✝) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ f) (Complexity.decode α (Memory.getms m src✝))) ∘ fm)) x✝¹ case b 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
https://github.com/calcu16/lean_complexity.git
d267eb4f3952c654f130a119311692c1c1a7cd71
HMem/Trace/Cost.lean
HMem.Trace.CostedProgram.splitTimeCost
[244, 1]
[349, 16]
apply Eq.trans
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 ⊢ (subroutineTimeCost (_ : TracedProgram.sound' f size (toTracedProgram next✝) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ f) (Complexity.decode α (Memory.getms m src✝))) ∘ fm)) + recurseTimeCost hf (_ : TracedProgram.sound' f size (toTracedProgram next✝) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ f) (Complexity.decode α (Memory.getms m src✝))) ∘ fm)) + localTimeCost (_ : TracedProgram.sound' f size (toTracedProgram next✝) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ f) (Complexity.decode α (Memory.getms m src✝))) ∘ fm))) x✝¹ + 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)))) = ?b case h₂ 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 ⊢ ?b ≤ (subroutineTimeCost h + recurseTimeCost hf h) x✝¹ + localTimeCost (_ : TracedProgram.sound' f size (toTracedProgram next✝) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ f) (Complexity.decode α (Memory.getms m src✝))) ∘ fm)) x✝¹ case b 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
case h₁.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 ⊢ (subroutineTimeCost (_ : TracedProgram.sound' f size (toTracedProgram next✝) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ f) (Complexity.decode α (Memory.getms m src✝))) ∘ fm)) + recurseTimeCost hf (_ : TracedProgram.sound' f size (toTracedProgram next✝) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ f) (Complexity.decode α (Memory.getms m src✝))) ∘ fm)) + localTimeCost (_ : TracedProgram.sound' f size (toTracedProgram next✝) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ f) (Complexity.decode α (Memory.getms m src✝))) ∘ fm))) x✝¹ + 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)))) = ?h₁.b case h₁.h₂ 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 ⊢ ?h₁.b = ?b case h₁.b 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 case h₂ 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 ⊢ ?b ≤ (subroutineTimeCost h + recurseTimeCost hf h) x✝¹ + localTimeCost (_ : TracedProgram.sound' f size (toTracedProgram next✝) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ f) (Complexity.decode α (Memory.getms m src✝))) ∘ fm)) x✝¹ case b 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
https://github.com/calcu16/lean_complexity.git
d267eb4f3952c654f130a119311692c1c1a7cd71
HMem/Trace/Cost.lean
HMem.Trace.CostedProgram.splitTimeCost
[244, 1]
[349, 16]
apply add_comm
case h₁.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 ⊢ (subroutineTimeCost (_ : TracedProgram.sound' f size (toTracedProgram next✝) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ f) (Complexity.decode α (Memory.getms m src✝))) ∘ fm)) + recurseTimeCost hf (_ : TracedProgram.sound' f size (toTracedProgram next✝) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ f) (Complexity.decode α (Memory.getms m src✝))) ∘ fm)) + localTimeCost (_ : TracedProgram.sound' f size (toTracedProgram next✝) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ f) (Complexity.decode α (Memory.getms m src✝))) ∘ fm))) x✝¹ + 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)))) = ?h₁.b case h₁.h₂ 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 ⊢ ?h₁.b = ?b case h₁.b 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 case h₂ 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 ⊢ ?b ≤ (subroutineTimeCost h + recurseTimeCost hf h) x✝¹ + localTimeCost (_ : TracedProgram.sound' f size (toTracedProgram next✝) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ f) (Complexity.decode α (Memory.getms m src✝))) ∘ fm)) x✝¹ case b 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
case h₁.h₂ 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.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)))) + (subroutineTimeCost (_ : TracedProgram.sound' f size (toTracedProgram next✝) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ f) (Complexity.decode α (Memory.getms m src✝))) ∘ fm)) + recurseTimeCost hf (_ : TracedProgram.sound' f size (toTracedProgram next✝) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ f) (Complexity.decode α (Memory.getms m src✝))) ∘ fm)) + localTimeCost (_ : TracedProgram.sound' f size (toTracedProgram next✝) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ f) (Complexity.decode α (Memory.getms m src✝))) ∘ fm))) x✝¹ = ?b case h₂ 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 ⊢ ?b ≤ (subroutineTimeCost h + recurseTimeCost hf h) x✝¹ + localTimeCost (_ : TracedProgram.sound' f size (toTracedProgram next✝) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ f) (Complexity.decode α (Memory.getms m src✝))) ∘ fm)) x✝¹ case b 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
https://github.com/calcu16/lean_complexity.git
d267eb4f3952c654f130a119311692c1c1a7cd71
HMem/Trace/Cost.lean
HMem.Trace.CostedProgram.splitTimeCost
[244, 1]
[349, 16]
apply (Nat.add_assoc _ _ _).symm
case h₁.h₂ 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.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)))) + (subroutineTimeCost (_ : TracedProgram.sound' f size (toTracedProgram next✝) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ f) (Complexity.decode α (Memory.getms m src✝))) ∘ fm)) + recurseTimeCost hf (_ : TracedProgram.sound' f size (toTracedProgram next✝) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ f) (Complexity.decode α (Memory.getms m src✝))) ∘ fm)) + localTimeCost (_ : TracedProgram.sound' f size (toTracedProgram next✝) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ f) (Complexity.decode α (Memory.getms m src✝))) ∘ fm))) x✝¹ = ?b case h₂ 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 ⊢ ?b ≤ (subroutineTimeCost h + recurseTimeCost hf h) x✝¹ + localTimeCost (_ : TracedProgram.sound' f size (toTracedProgram next✝) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ f) (Complexity.decode α (Memory.getms m src✝))) ∘ fm)) x✝¹ case b 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
case h₂ 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.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)))) + (subroutineTimeCost (_ : TracedProgram.sound' f size (toTracedProgram next✝) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ f) (Complexity.decode α (Memory.getms m src✝))) ∘ fm)) + recurseTimeCost hf (_ : TracedProgram.sound' f size (toTracedProgram next✝) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ f) (Complexity.decode α (Memory.getms m src✝))) ∘ fm))) x✝¹ + localTimeCost (_ : TracedProgram.sound' f size (toTracedProgram next✝) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ f) (Complexity.decode α (Memory.getms m src✝))) ∘ fm)) x✝¹ ≤ (subroutineTimeCost h + recurseTimeCost hf h) x✝¹ + localTimeCost (_ : TracedProgram.sound' f size (toTracedProgram next✝) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ f) (Complexity.decode α (Memory.getms m src✝))) ∘ fm)) x✝¹
https://github.com/calcu16/lean_complexity.git
d267eb4f3952c654f130a119311692c1c1a7cd71
HMem/Trace/Cost.lean
HMem.Trace.CostedProgram.splitTimeCost
[244, 1]
[349, 16]
apply add_le_add _ (le_refl _)
case h₂ 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.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)))) + (subroutineTimeCost (_ : TracedProgram.sound' f size (toTracedProgram next✝) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ f) (Complexity.decode α (Memory.getms m src✝))) ∘ fm)) + recurseTimeCost hf (_ : TracedProgram.sound' f size (toTracedProgram next✝) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ f) (Complexity.decode α (Memory.getms m src✝))) ∘ fm))) x✝¹ + localTimeCost (_ : TracedProgram.sound' f size (toTracedProgram next✝) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ f) (Complexity.decode α (Memory.getms m src✝))) ∘ fm)) x✝¹ ≤ (subroutineTimeCost h + recurseTimeCost hf h) x✝¹ + localTimeCost (_ : TracedProgram.sound' f size (toTracedProgram next✝) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ f) (Complexity.decode α (Memory.getms m src✝))) ∘ fm)) x✝¹
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.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)))) + (subroutineTimeCost (_ : TracedProgram.sound' f size (toTracedProgram next✝) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ f) (Complexity.decode α (Memory.getms m src✝))) ∘ fm)) + recurseTimeCost hf (_ : TracedProgram.sound' f size (toTracedProgram next✝) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ f) (Complexity.decode α (Memory.getms m src✝))) ∘ fm))) x✝¹ ≤ (subroutineTimeCost h + recurseTimeCost hf h) x✝¹
https://github.com/calcu16/lean_complexity.git
d267eb4f3952c654f130a119311692c1c1a7cd71
HMem/Trace/Cost.lean
HMem.Trace.CostedProgram.splitTimeCost
[244, 1]
[349, 16]
apply le_of_eq_of_le
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.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)))) + (subroutineTimeCost (_ : TracedProgram.sound' f size (toTracedProgram next✝) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ f) (Complexity.decode α (Memory.getms m src✝))) ∘ fm)) + recurseTimeCost hf (_ : TracedProgram.sound' f size (toTracedProgram next✝) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ f) (Complexity.decode α (Memory.getms m src✝))) ∘ fm))) x✝¹ ≤ (subroutineTimeCost h + recurseTimeCost hf h) x✝¹
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.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)))) + (subroutineTimeCost (_ : TracedProgram.sound' f size (toTracedProgram next✝) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ f) (Complexity.decode α (Memory.getms m src✝))) ∘ fm)) + recurseTimeCost hf (_ : TracedProgram.sound' f size (toTracedProgram next✝) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ f) (Complexity.decode α (Memory.getms m src✝))) ∘ fm))) x✝¹ = ?b case h₂ 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 ⊢ ?b ≤ (subroutineTimeCost h + recurseTimeCost hf h) x✝¹ case b 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
https://github.com/calcu16/lean_complexity.git
d267eb4f3952c654f130a119311692c1c1a7cd71
HMem/Trace/Cost.lean
HMem.Trace.CostedProgram.splitTimeCost
[244, 1]
[349, 16]
apply Eq.trans
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.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)))) + (subroutineTimeCost (_ : TracedProgram.sound' f size (toTracedProgram next✝) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ f) (Complexity.decode α (Memory.getms m src✝))) ∘ fm)) + recurseTimeCost hf (_ : TracedProgram.sound' f size (toTracedProgram next✝) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ f) (Complexity.decode α (Memory.getms m src✝))) ∘ fm))) x✝¹ = ?b case h₂ 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 ⊢ ?b ≤ (subroutineTimeCost h + recurseTimeCost hf h) x✝¹ case b 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
case h₁.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.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)))) + (subroutineTimeCost (_ : TracedProgram.sound' f size (toTracedProgram next✝) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ f) (Complexity.decode α (Memory.getms m src✝))) ∘ fm)) + recurseTimeCost hf (_ : TracedProgram.sound' f size (toTracedProgram next✝) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ f) (Complexity.decode α (Memory.getms m src✝))) ∘ fm))) x✝¹ = ?h₁.b case h₁.h₂ 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 ⊢ ?h₁.b = ?b case h₁.b 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 case h₂ 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 ⊢ ?b ≤ (subroutineTimeCost h + recurseTimeCost hf h) x✝¹ case b 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
https://github.com/calcu16/lean_complexity.git
d267eb4f3952c654f130a119311692c1c1a7cd71
HMem/Trace/Cost.lean
HMem.Trace.CostedProgram.splitTimeCost
[244, 1]
[349, 16]
apply add_comm
case h₁.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.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)))) + (subroutineTimeCost (_ : TracedProgram.sound' f size (toTracedProgram next✝) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ f) (Complexity.decode α (Memory.getms m src✝))) ∘ fm)) + recurseTimeCost hf (_ : TracedProgram.sound' f size (toTracedProgram next✝) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ f) (Complexity.decode α (Memory.getms m src✝))) ∘ fm))) x✝¹ = ?h₁.b case h₁.h₂ 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 ⊢ ?h₁.b = ?b case h₁.b 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 case h₂ 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 ⊢ ?b ≤ (subroutineTimeCost h + recurseTimeCost hf h) x✝¹ case b 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
case h₁.h₂ 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 ⊢ (subroutineTimeCost (_ : TracedProgram.sound' f size (toTracedProgram next✝) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ f) (Complexity.decode α (Memory.getms m src✝))) ∘ fm)) + recurseTimeCost hf (_ : TracedProgram.sound' f size (toTracedProgram next✝) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ f) (Complexity.decode α (Memory.getms m src✝))) ∘ fm))) x✝¹ + 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)))) = ?b case h₂ 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 ⊢ ?b ≤ (subroutineTimeCost h + recurseTimeCost hf h) x✝¹ case b 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
https://github.com/calcu16/lean_complexity.git
d267eb4f3952c654f130a119311692c1c1a7cd71
HMem/Trace/Cost.lean
HMem.Trace.CostedProgram.splitTimeCost
[244, 1]
[349, 16]
apply Nat.add_assoc _ _ _
case h₁.h₂ 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 ⊢ (subroutineTimeCost (_ : TracedProgram.sound' f size (toTracedProgram next✝) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ f) (Complexity.decode α (Memory.getms m src✝))) ∘ fm)) + recurseTimeCost hf (_ : TracedProgram.sound' f size (toTracedProgram next✝) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ f) (Complexity.decode α (Memory.getms m src✝))) ∘ fm))) x✝¹ + 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)))) = ?b case h₂ 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 ⊢ ?b ≤ (subroutineTimeCost h + recurseTimeCost hf h) x✝¹ case b 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
case h₂ 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 ⊢ subroutineTimeCost (_ : TracedProgram.sound' f size (toTracedProgram next✝) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ f) (Complexity.decode α (Memory.getms m src✝))) ∘ fm)) x✝¹ + (recurseTimeCost hf (_ : TracedProgram.sound' f size (toTracedProgram next✝) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ f) (Complexity.decode α (Memory.getms m src✝))) ∘ fm)) x✝¹ + 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))))) ≤ (subroutineTimeCost h + recurseTimeCost hf h) x✝¹
https://github.com/calcu16/lean_complexity.git
d267eb4f3952c654f130a119311692c1c1a7cd71
HMem/Trace/Cost.lean
HMem.Trace.CostedProgram.splitTimeCost
[244, 1]
[349, 16]
apply add_le_add (le_refl _)
case h₂ 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 ⊢ subroutineTimeCost (_ : TracedProgram.sound' f size (toTracedProgram next✝) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ f) (Complexity.decode α (Memory.getms m src✝))) ∘ fm)) x✝¹ + (recurseTimeCost hf (_ : TracedProgram.sound' f size (toTracedProgram next✝) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ f) (Complexity.decode α (Memory.getms m src✝))) ∘ fm)) x✝¹ + 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))))) ≤ (subroutineTimeCost h + recurseTimeCost hf h) x✝¹
case h₂ 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 ⊢ recurseTimeCost hf (_ : TracedProgram.sound' f size (toTracedProgram next✝) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ f) (Complexity.decode α (Memory.getms m src✝))) ∘ fm)) x✝¹ + 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)))) ≤ recurseTimeCost hf h x✝¹
https://github.com/calcu16/lean_complexity.git
d267eb4f3952c654f130a119311692c1c1a7cd71
HMem/Trace/Cost.lean
HMem.Trace.CostedProgram.splitTimeCost
[244, 1]
[349, 16]
apply add_le_add (le_refl _)
case h₂ 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 ⊢ recurseTimeCost hf (_ : TracedProgram.sound' f size (toTracedProgram next✝) ((flip Option.bind fun m => Option.map (Memory.setms m dst✝ ∘ Complexity.encode ∘ f) (Complexity.decode α (Memory.getms m src✝))) ∘ fm)) x✝¹ + 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)))) ≤ recurseTimeCost hf h x✝¹
case h₂ 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.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)))) ≤ Complexity.CostFunction.flatMap (TracedProgram.sound'_recurse_arg h) (fun arg => Program.timeCost (TracedProgram.toProgram (toTracedProgram fcp)) (_ : Program.haltsOn (TracedProgram.toProgram (toTracedProgram fcp)) (Complexity.encode arg))) x✝¹
https://github.com/calcu16/lean_complexity.git
d267eb4f3952c654f130a119311692c1c1a7cd71
HMem/Trace/Cost.lean
HMem.Trace.CostedProgram.splitTimeCost
[244, 1]
[349, 16]
apply le_of_le_of_eq _
case h₂ 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.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)))) ≤ Complexity.CostFunction.flatMap (TracedProgram.sound'_recurse_arg h) (fun arg => Program.timeCost (TracedProgram.toProgram (toTracedProgram fcp)) (_ : Program.haltsOn (TracedProgram.toProgram (toTracedProgram fcp)) (Complexity.encode arg))) x✝¹
case h₂ 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.183728 = Complexity.CostFunction.flatMap (TracedProgram.sound'_recurse_arg h) (fun arg => Program.timeCost (TracedProgram.toProgram (toTracedProgram fcp)) (_ : Program.haltsOn (TracedProgram.toProgram (toTracedProgram fcp)) (Complexity.encode arg))) x✝¹ 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 ⊢ 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)))) ≤ ?m.183728
https://github.com/calcu16/lean_complexity.git
d267eb4f3952c654f130a119311692c1c1a7cd71
HMem/Trace/Cost.lean
HMem.Trace.CostedProgram.splitTimeCost
[244, 1]
[349, 16]
apply Eq.symm
case h₂ 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.183728 = Complexity.CostFunction.flatMap (TracedProgram.sound'_recurse_arg h) (fun arg => Program.timeCost (TracedProgram.toProgram (toTracedProgram fcp)) (_ : Program.haltsOn (TracedProgram.toProgram (toTracedProgram fcp)) (Complexity.encode arg))) x✝¹ 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 ⊢ 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)))) ≤ ?m.183728
case h₂.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 ⊢ Complexity.CostFunction.flatMap (TracedProgram.sound'_recurse_arg h) (fun arg => Program.timeCost (TracedProgram.toProgram (toTracedProgram fcp)) (_ : Program.haltsOn (TracedProgram.toProgram (toTracedProgram fcp)) (Complexity.encode arg))) x✝¹ = ?m.183728 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 ⊢ 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)))) ≤ ?m.183728
https://github.com/calcu16/lean_complexity.git
d267eb4f3952c654f130a119311692c1c1a7cd71
HMem/Trace/Cost.lean
HMem.Trace.CostedProgram.splitTimeCost
[244, 1]
[349, 16]
apply Complexity.CostFunction.flatMap_some
case h₂.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 ⊢ Complexity.CostFunction.flatMap (TracedProgram.sound'_recurse_arg h) (fun arg => Program.timeCost (TracedProgram.toProgram (toTracedProgram fcp)) (_ : Program.haltsOn (TracedProgram.toProgram (toTracedProgram fcp)) (Complexity.encode arg))) x✝¹ = ?m.183728 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 ⊢ 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)))) ≤ ?m.183728
case h₂.h.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 ⊢ TracedProgram.sound'_recurse_arg h x✝¹ = some ?h₂.h.b case h₂.h.b 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 ⊢ α 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.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)))) ≤ Program.timeCost (TracedProgram.toProgram (toTracedProgram fcp)) (_ : Program.haltsOn (TracedProgram.toProgram (toTracedProgram fcp)) (Complexity.encode ?h₂.h.b))
https://github.com/calcu16/lean_complexity.git
d267eb4f3952c654f130a119311692c1c1a7cd71
HMem/Trace/Cost.lean
HMem.Trace.CostedProgram.splitTimeCost
[244, 1]
[349, 16]
apply Option.mem_bind_of_mem
case h₂.h.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 ⊢ TracedProgram.sound'_recurse_arg h x✝¹ = some ?h₂.h.b case h₂.h.b 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 ⊢ α 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.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)))) ≤ Program.timeCost (TracedProgram.toProgram (toTracedProgram fcp)) (_ : Program.haltsOn (TracedProgram.toProgram (toTracedProgram fcp)) (Complexity.encode ?h₂.h.b))
case h₂.h.h.ha 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 ⊢ ?h₂.h.h.a ∈ fm x✝¹ case h₂.h.h.hf 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 ⊢ ?h₂.h.b ∈ (Complexity.decode α ∘ flip Memory.getms src✝) ?h₂.h.h.a case h₂.h.h.a 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 ⊢ Memory case h₂.h.b 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 ⊢ α 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.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)))) ≤ Program.timeCost (TracedProgram.toProgram (toTracedProgram fcp)) (_ : Program.haltsOn (TracedProgram.toProgram (toTracedProgram fcp)) (Complexity.encode ?h₂.h.b))
https://github.com/calcu16/lean_complexity.git
d267eb4f3952c654f130a119311692c1c1a7cd71
HMem/Trace/Cost.lean
HMem.Trace.CostedProgram.splitTimeCost
[244, 1]
[349, 16]
apply hm
case h₂.h.h.ha 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 ⊢ ?h₂.h.h.a ∈ fm x✝¹ case h₂.h.h.hf 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 ⊢ ?h₂.h.b ∈ (Complexity.decode α ∘ flip Memory.getms src✝) ?h₂.h.h.a case h₂.h.h.a 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 ⊢ Memory case h₂.h.b 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 ⊢ α 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.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)))) ≤ Program.timeCost (TracedProgram.toProgram (toTracedProgram fcp)) (_ : Program.haltsOn (TracedProgram.toProgram (toTracedProgram fcp)) (Complexity.encode ?h₂.h.b))
case h₂.h.h.hf 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 ⊢ ?h₂.h.b ∈ (Complexity.decode α ∘ flip Memory.getms src✝) x✝ case h₂.h.b 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 ⊢ α 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.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)))) ≤ Program.timeCost (TracedProgram.toProgram (toTracedProgram fcp)) (_ : Program.haltsOn (TracedProgram.toProgram (toTracedProgram fcp)) (Complexity.encode ?h₂.h.b))
https://github.com/calcu16/lean_complexity.git
d267eb4f3952c654f130a119311692c1c1a7cd71
HMem/Trace/Cost.lean
HMem.Trace.CostedProgram.splitTimeCost
[244, 1]
[349, 16]
apply Option.get_mem
case h₂.h.h.hf 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 ⊢ ?h₂.h.b ∈ (Complexity.decode α ∘ flip Memory.getms src✝) x✝ case h₂.h.b 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 ⊢ α 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.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)))) ≤ Program.timeCost (TracedProgram.toProgram (toTracedProgram fcp)) (_ : Program.haltsOn (TracedProgram.toProgram (toTracedProgram fcp)) (Complexity.encode ?h₂.h.b))
case h₂.h.h.hf.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 ⊢ Option.isSome ((Complexity.decode α ∘ flip Memory.getms src✝) x✝) = true 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.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)))) ≤ Program.timeCost (TracedProgram.toProgram (toTracedProgram fcp)) (_ : Program.haltsOn (TracedProgram.toProgram (toTracedProgram fcp)) (Complexity.encode (Option.get ((Complexity.decode α ∘ flip Memory.getms src✝) x✝) ?h₂.h.h.hf.h)))
https://github.com/calcu16/lean_complexity.git
d267eb4f3952c654f130a119311692c1c1a7cd71
HMem/Trace/Cost.lean
HMem.Trace.CostedProgram.splitTimeCost
[244, 1]
[349, 16]
apply (h.left _ _ hm).elim λ hsome _ ↦ hsome
case h₂.h.h.hf.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 ⊢ Option.isSome ((Complexity.decode α ∘ flip Memory.getms src✝) x✝) = true 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.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)))) ≤ Program.timeCost (TracedProgram.toProgram (toTracedProgram fcp)) (_ : Program.haltsOn (TracedProgram.toProgram (toTracedProgram fcp)) (Complexity.encode (Option.get ((Complexity.decode α ∘ flip Memory.getms src✝) x✝) ?h₂.h.h.hf.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.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)))) ≤ Program.timeCost (TracedProgram.toProgram (toTracedProgram fcp)) (_ : Program.haltsOn (TracedProgram.toProgram (toTracedProgram fcp)) (Complexity.encode (Option.get ((Complexity.decode α ∘ flip Memory.getms src✝) x✝) (_ : Option.isSome (Complexity.decode α (Memory.getms x✝ src✝)) = true))))
https://github.com/calcu16/lean_complexity.git
d267eb4f3952c654f130a119311692c1c1a7cd71
HMem/Trace/Cost.lean
HMem.Trace.CostedProgram.splitTimeCost
[244, 1]
[349, 16]
apply le_refl
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.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)))) ≤ Program.timeCost (TracedProgram.toProgram (toTracedProgram fcp)) (_ : Program.haltsOn (TracedProgram.toProgram (toTracedProgram fcp)) (Complexity.encode (Option.get ((Complexity.decode α ∘ flip Memory.getms src✝) x✝) (_ : Option.isSome (Complexity.decode α (Memory.getms x✝ src✝)) = true))))
no goals
https://github.com/calcu16/lean_complexity.git
d267eb4f3952c654f130a119311692c1c1a7cd71
HMem/Trace/Cost.lean
HMem.Program.splitTimeCost
[433, 1]
[438, 18]
intro a
TracedProgram : Sort ?u.211566 α : Type u_1 inst✝⁶ : Complexity.Coding α Memory β : Type u_2 inst✝⁵ : Complexity.Encoding β Memory f : α → β tp : TracedProgram γ δ : Type inst✝⁴ : Complexity.Coding γ Memory inst✝³ : Complexity.Encoding δ Memory fs : γ → δ cs : Complexity.CostFunction γ ℕ inst✝² : Complexity.Computable Encoding.Model fs inst✝¹ : Complexity Encoding.Model fs cs sz : α → ℕ p : Program inst✝ : Trace.HasCostedProgram p h : sound p f sz ⊢ timeCost' p h ≤ subroutineTimeCost h + recurseTimeCost h + localTimeCost h
TracedProgram : Sort ?u.211566 α : Type u_1 inst✝⁶ : Complexity.Coding α Memory β : Type u_2 inst✝⁵ : Complexity.Encoding β Memory f : α → β tp : TracedProgram γ δ : Type inst✝⁴ : Complexity.Coding γ Memory inst✝³ : Complexity.Encoding δ Memory fs : γ → δ cs : Complexity.CostFunction γ ℕ inst✝² : Complexity.Computable Encoding.Model fs inst✝¹ : Complexity Encoding.Model fs cs sz : α → ℕ p : Program inst✝ : Trace.HasCostedProgram p h : sound p f sz a : α ⊢ timeCost' p h a ≤ (subroutineTimeCost h + recurseTimeCost h + localTimeCost h) a
https://github.com/calcu16/lean_complexity.git
d267eb4f3952c654f130a119311692c1c1a7cd71
HMem/Trace/Cost.lean
HMem.Program.splitTimeCost
[433, 1]
[438, 18]
apply le_trans
TracedProgram : Sort ?u.211566 α : Type u_1 inst✝⁶ : Complexity.Coding α Memory β : Type u_2 inst✝⁵ : Complexity.Encoding β Memory f : α → β tp : TracedProgram γ δ : Type inst✝⁴ : Complexity.Coding γ Memory inst✝³ : Complexity.Encoding δ Memory fs : γ → δ cs : Complexity.CostFunction γ ℕ inst✝² : Complexity.Computable Encoding.Model fs inst✝¹ : Complexity Encoding.Model fs cs sz : α → ℕ p : Program inst✝ : Trace.HasCostedProgram p h : sound p f sz a : α ⊢ timeCost' p h a ≤ (subroutineTimeCost h + recurseTimeCost h + localTimeCost h) a
case a TracedProgram : Sort ?u.211566 α : Type u_1 inst✝⁶ : Complexity.Coding α Memory β : Type u_2 inst✝⁵ : Complexity.Encoding β Memory f : α → β tp : TracedProgram γ δ : Type inst✝⁴ : Complexity.Coding γ Memory inst✝³ : Complexity.Encoding δ Memory fs : γ → δ cs : Complexity.CostFunction γ ℕ inst✝² : Complexity.Computable Encoding.Model fs inst✝¹ : Complexity Encoding.Model fs cs sz : α → ℕ p : Program inst✝ : Trace.HasCostedProgram p h : sound p f sz a : α ⊢ timeCost' p h a ≤ ?b case a TracedProgram : Sort ?u.211566 α : Type u_1 inst✝⁶ : Complexity.Coding α Memory β : Type u_2 inst✝⁵ : Complexity.Encoding β Memory f : α → β tp : TracedProgram γ δ : Type inst✝⁴ : Complexity.Coding γ Memory inst✝³ : Complexity.Encoding δ Memory fs : γ → δ cs : Complexity.CostFunction γ ℕ inst✝² : Complexity.Computable Encoding.Model fs inst✝¹ : Complexity Encoding.Model fs cs sz : α → ℕ p : Program inst✝ : Trace.HasCostedProgram p h : sound p f sz a : α ⊢ ?b ≤ (subroutineTimeCost h + recurseTimeCost h + localTimeCost h) a case b TracedProgram : Sort ?u.211566 α : Type u_1 inst✝⁶ : Complexity.Coding α Memory β : Type u_2 inst✝⁵ : Complexity.Encoding β Memory f : α → β tp : TracedProgram γ δ : Type inst✝⁴ : Complexity.Coding γ Memory inst✝³ : Complexity.Encoding δ Memory fs : γ → δ cs : Complexity.CostFunction γ ℕ inst✝² : Complexity.Computable Encoding.Model fs inst✝¹ : Complexity Encoding.Model fs cs sz : α → ℕ p : Program inst✝ : Trace.HasCostedProgram p h : sound p f sz a : α ⊢ ℕ
https://github.com/calcu16/lean_complexity.git
d267eb4f3952c654f130a119311692c1c1a7cd71
HMem/Trace/Cost.lean
HMem.Program.splitTimeCost
[433, 1]
[438, 18]
exact (Stack.timeCost_le_iff _ (Program.halts_of_sound h _)).mp (p.costedMatches ▸ p.costed.splitTimeCost (costed_sound h) (costed_sound' h) a _ (Option.mem_some_iff.mpr rfl))
case a TracedProgram : Sort ?u.211566 α : Type u_1 inst✝⁶ : Complexity.Coding α Memory β : Type u_2 inst✝⁵ : Complexity.Encoding β Memory f : α → β tp : TracedProgram γ δ : Type inst✝⁴ : Complexity.Coding γ Memory inst✝³ : Complexity.Encoding δ Memory fs : γ → δ cs : Complexity.CostFunction γ ℕ inst✝² : Complexity.Computable Encoding.Model fs inst✝¹ : Complexity Encoding.Model fs cs sz : α → ℕ p : Program inst✝ : Trace.HasCostedProgram p h : sound p f sz a : α ⊢ timeCost' p h a ≤ ?b case a TracedProgram : Sort ?u.211566 α : Type u_1 inst✝⁶ : Complexity.Coding α Memory β : Type u_2 inst✝⁵ : Complexity.Encoding β Memory f : α → β tp : TracedProgram γ δ : Type inst✝⁴ : Complexity.Coding γ Memory inst✝³ : Complexity.Encoding δ Memory fs : γ → δ cs : Complexity.CostFunction γ ℕ inst✝² : Complexity.Computable Encoding.Model fs inst✝¹ : Complexity Encoding.Model fs cs sz : α → ℕ p : Program inst✝ : Trace.HasCostedProgram p h : sound p f sz a : α ⊢ ?b ≤ (subroutineTimeCost h + recurseTimeCost h + localTimeCost h) a case b TracedProgram : Sort ?u.211566 α : Type u_1 inst✝⁶ : Complexity.Coding α Memory β : Type u_2 inst✝⁵ : Complexity.Encoding β Memory f : α → β tp : TracedProgram γ δ : Type inst✝⁴ : Complexity.Coding γ Memory inst✝³ : Complexity.Encoding δ Memory fs : γ → δ cs : Complexity.CostFunction γ ℕ inst✝² : Complexity.Computable Encoding.Model fs inst✝¹ : Complexity Encoding.Model fs cs sz : α → ℕ p : Program inst✝ : Trace.HasCostedProgram p h : sound p f sz a : α ⊢ ℕ
case a TracedProgram : Sort ?u.211566 α : Type u_1 inst✝⁶ : Complexity.Coding α Memory β : Type u_2 inst✝⁵ : Complexity.Encoding β Memory f : α → β tp : TracedProgram γ δ : Type inst✝⁴ : Complexity.Coding γ Memory inst✝³ : Complexity.Encoding δ Memory fs : γ → δ cs : Complexity.CostFunction γ ℕ inst✝² : Complexity.Computable Encoding.Model fs inst✝¹ : Complexity Encoding.Model fs cs sz : α → ℕ p : Program inst✝ : Trace.HasCostedProgram p h : sound p f sz a : α ⊢ (Trace.CostedProgram.subroutineTimeCost (_ : Trace.TracedProgram.sound' f sz (Trace.CostedProgram.toTracedProgram (costed p)) (some ∘ Complexity.encode)) + Trace.CostedProgram.recurseTimeCost (_ : sound (Trace.TracedProgram.toProgram (Trace.CostedProgram.toTracedProgram (costed p))) f sz) (_ : Trace.TracedProgram.sound' f sz (Trace.CostedProgram.toTracedProgram (costed p)) (some ∘ Complexity.encode)) + Trace.CostedProgram.localTimeCost (_ : Trace.TracedProgram.sound' f sz (Trace.CostedProgram.toTracedProgram (costed p)) (some ∘ Complexity.encode))) a ≤ (subroutineTimeCost h + recurseTimeCost h + localTimeCost h) a
https://github.com/calcu16/lean_complexity.git
d267eb4f3952c654f130a119311692c1c1a7cd71
HMem/Trace/Cost.lean
HMem.Program.splitTimeCost
[433, 1]
[438, 18]
exact le_refl _
case a TracedProgram : Sort ?u.211566 α : Type u_1 inst✝⁶ : Complexity.Coding α Memory β : Type u_2 inst✝⁵ : Complexity.Encoding β Memory f : α → β tp : TracedProgram γ δ : Type inst✝⁴ : Complexity.Coding γ Memory inst✝³ : Complexity.Encoding δ Memory fs : γ → δ cs : Complexity.CostFunction γ ℕ inst✝² : Complexity.Computable Encoding.Model fs inst✝¹ : Complexity Encoding.Model fs cs sz : α → ℕ p : Program inst✝ : Trace.HasCostedProgram p h : sound p f sz a : α ⊢ (Trace.CostedProgram.subroutineTimeCost (_ : Trace.TracedProgram.sound' f sz (Trace.CostedProgram.toTracedProgram (costed p)) (some ∘ Complexity.encode)) + Trace.CostedProgram.recurseTimeCost (_ : sound (Trace.TracedProgram.toProgram (Trace.CostedProgram.toTracedProgram (costed p))) f sz) (_ : Trace.TracedProgram.sound' f sz (Trace.CostedProgram.toTracedProgram (costed p)) (some ∘ Complexity.encode)) + Trace.CostedProgram.localTimeCost (_ : Trace.TracedProgram.sound' f sz (Trace.CostedProgram.toTracedProgram (costed p)) (some ∘ Complexity.encode))) a ≤ (subroutineTimeCost h + recurseTimeCost h + localTimeCost h) a
no goals
https://github.com/calcu16/lean_complexity.git
d267eb4f3952c654f130a119311692c1c1a7cd71
HMem/Encoding/Basic.lean
HMem.Encoding.encode_sum_getm_t
[73, 9]
[74, 59]
cases v <;> simp
α : Type u_1 β : Type u_2 inst✝¹ : Encoding α Memory inst✝ : Encoding β Memory v : α ⊕ β ⊢ Memory.getm (encode v) true = 0
no goals
https://github.com/calcu16/lean_complexity.git
d267eb4f3952c654f130a119311692c1c1a7cd71
HMem/Encoding/Basic.lean
HMem.Encoding.encodeList_inj
[88, 1]
[92, 90]
simp[encodeList]
α : Type u_1 inst✝ : Encoding α Memory head✝ : α tail✝ : List α ⊢ encodeList [] = encodeList (head✝ :: tail✝) → [] = head✝ :: tail✝
no goals
https://github.com/calcu16/lean_complexity.git
d267eb4f3952c654f130a119311692c1c1a7cd71
HMem/Encoding/Basic.lean
HMem.Encoding.encodeList_inj
[88, 1]
[92, 90]
simp[encodeList]
α : Type u_1 inst✝ : Encoding α Memory head✝ : α tail✝ : List α ⊢ encodeList (head✝ :: tail✝) = encodeList [] → head✝ :: tail✝ = []
no goals
https://github.com/calcu16/lean_complexity.git
d267eb4f3952c654f130a119311692c1c1a7cd71
HMem/Encoding/Basic.lean
HMem.Encoding.encodeList_inj
[88, 1]
[92, 90]
simp[encodeList, Iff.intro (encodeList_inj lhs rhs) λ h ↦ h ▸ rfl]
α : Type u_1 inst✝ : Encoding α Memory head✝¹ : α lhs : List α head✝ : α rhs : List α ⊢ encodeList (head✝¹ :: lhs) = encodeList (head✝ :: rhs) → head✝¹ :: lhs = head✝ :: rhs
no goals
https://github.com/calcu16/lean_complexity.git
d267eb4f3952c654f130a119311692c1c1a7cd71
HMem/Encoding/Basic.lean
HMem.Encoding.rawDecodeList_inv
[101, 1]
[103, 87]
simpa [encodeList, decodeList, rawDecodeList] using rawDecodeList_inv tl
α : Type u_1 inst✝ : Coding α Memory head✝ : α tl : List α ⊢ decodeList (encodeList (head✝ :: tl)) = some (head✝ :: tl)
no goals
https://github.com/calcu16/lean_complexity.git
d267eb4f3952c654f130a119311692c1c1a7cd71
HMem/Encoding/Basic.lean
HMem.Encoding.tupleInj_eq
[145, 9]
[147, 44]
simp [encodeFinTuple_def, List.ofFn_inj']
α : Type u_1 inst✝ : Encoding α Memory N₀ N₁ : ℕ f₀ : Fin N₀ → α f₁ : Fin N₁ → α ⊢ (encode f₀ = encode f₁) = (N₀ = N₁ ∧ HEq f₀ f₁)
no goals
https://github.com/calcu16/lean_complexity.git
d267eb4f3952c654f130a119311692c1c1a7cd71
HMem/Encoding/Basic.lean
HMem.Encoding.encodeOp_inj
[168, 1]
[180, 41]
simpa [encodeOp] using λ h ↦ funext λ x ↦ Subsingleton.elim _ x ▸ h
α : Type u_1 inst✝ : Encoding α Memory f₀ f₁ : (Fin 0 → Bool) → α ⊢ encodeOp f₀ = encodeOp f₁ → f₀ = f₁
no goals
https://github.com/calcu16/lean_complexity.git
d267eb4f3952c654f130a119311692c1c1a7cd71
HMem/Encoding/Basic.lean
HMem.Encoding.encodeOp_inj
[168, 1]
[180, 41]
simpa [encodeOp, λ b ↦ Iff.intro (encodeOp_inj (f₀ ∘ (Fin.cons (α := λ _ ↦ Bool) b)) (f₁ ∘ (Fin.cons (α := λ _ ↦ Bool) b))) (λ h ↦ h ▸ rfl)] using λ hf ht ↦ funext λ y ↦ Fin.cons_self_tail y ▸ match y 0 with | false => congrFun hf (Fin.tail y) | true => congrFun ht (Fin.tail y)
α : Type u_1 inst✝ : Encoding α Memory N : ℕ f₀ f₁ : (Fin (N + 1) → Bool) → α ⊢ encodeOp f₀ = encodeOp f₁ → f₀ = f₁
no goals
https://github.com/calcu16/lean_complexity.git
d267eb4f3952c654f130a119311692c1c1a7cd71
HMem/Encoding/Basic.lean
HMem.Encoding.decodeOp_inv
[195, 1]
[209, 35]
simpa [decodeOp, encodeOp] using funext λ _ ↦ congrArg _ (funext finZeroElim)
α : Type u_1 inst✝ : Coding α Memory x✝ : (Fin 0 → Bool) → α ⊢ decodeOp (encodeOp x✝) = some x✝
no goals
https://github.com/calcu16/lean_complexity.git
d267eb4f3952c654f130a119311692c1c1a7cd71
HMem/Encoding/Basic.lean
HMem.Encoding.decodeOp_inv
[195, 1]
[209, 35]
simp [decodeOp, encodeOp, decodeOp_inv (N := N)]
α : Type u_1 inst✝ : Coding α Memory N : ℕ g : (Fin (N + 1) → Bool) → α ⊢ decodeOp (encodeOp g) = some g
α : Type u_1 inst✝ : Coding α Memory N : ℕ g : (Fin (N + 1) → Bool) → α ⊢ (fun g_1 => match g_1 0 with | false => g (Fin.cons false (Fin.tail g_1)) | true => g (Fin.cons true (Fin.tail g_1))) = g
https://github.com/calcu16/lean_complexity.git
d267eb4f3952c654f130a119311692c1c1a7cd71
HMem/Encoding/Basic.lean
HMem.Encoding.decodeOp_inv
[195, 1]
[209, 35]
apply funext
α : Type u_1 inst✝ : Coding α Memory N : ℕ g : (Fin (N + 1) → Bool) → α ⊢ (fun g_1 => match g_1 0 with | false => g (Fin.cons false (Fin.tail g_1)) | true => g (Fin.cons true (Fin.tail g_1))) = g
case h α : Type u_1 inst✝ : Coding α Memory N : ℕ g : (Fin (N + 1) → Bool) → α ⊢ ∀ (x : Fin (N + 1) → Bool), (match x 0 with | false => g (Fin.cons false (Fin.tail x)) | true => g (Fin.cons true (Fin.tail x))) = g x
https://github.com/calcu16/lean_complexity.git
d267eb4f3952c654f130a119311692c1c1a7cd71
HMem/Encoding/Basic.lean
HMem.Encoding.decodeOp_inv
[195, 1]
[209, 35]
intro x
case h α : Type u_1 inst✝ : Coding α Memory N : ℕ g : (Fin (N + 1) → Bool) → α ⊢ ∀ (x : Fin (N + 1) → Bool), (match x 0 with | false => g (Fin.cons false (Fin.tail x)) | true => g (Fin.cons true (Fin.tail x))) = g x
case h α : Type u_1 inst✝ : Coding α Memory N : ℕ g : (Fin (N + 1) → Bool) → α x : Fin (N + 1) → Bool ⊢ (match x 0 with | false => g (Fin.cons false (Fin.tail x)) | true => g (Fin.cons true (Fin.tail x))) = g x
https://github.com/calcu16/lean_complexity.git
d267eb4f3952c654f130a119311692c1c1a7cd71
HMem/Encoding/Basic.lean
HMem.Encoding.decodeOp_inv
[195, 1]
[209, 35]
match h:x 0 with | true => apply congrArg g exact h ▸ Fin.cons_self_tail _ | false => apply congrArg g exact h ▸ Fin.cons_self_tail _
case h α : Type u_1 inst✝ : Coding α Memory N : ℕ g : (Fin (N + 1) → Bool) → α x : Fin (N + 1) → Bool ⊢ (match x 0 with | false => g (Fin.cons false (Fin.tail x)) | true => g (Fin.cons true (Fin.tail x))) = g x
no goals
https://github.com/calcu16/lean_complexity.git
d267eb4f3952c654f130a119311692c1c1a7cd71
HMem/Encoding/Basic.lean
HMem.Encoding.decodeOp_inv
[195, 1]
[209, 35]
apply congrArg g
α : Type u_1 inst✝ : Coding α Memory N : ℕ g : (Fin (N + 1) → Bool) → α x : Fin (N + 1) → Bool h : x 0 = true ⊢ (match true with | false => g (Fin.cons false (Fin.tail x)) | true => g (Fin.cons true (Fin.tail x))) = g x
α : Type u_1 inst✝ : Coding α Memory N : ℕ g : (Fin (N + 1) → Bool) → α x : Fin (N + 1) → Bool h : x 0 = true ⊢ Fin.cons true (Fin.tail x) = x
https://github.com/calcu16/lean_complexity.git
d267eb4f3952c654f130a119311692c1c1a7cd71
HMem/Encoding/Basic.lean
HMem.Encoding.decodeOp_inv
[195, 1]
[209, 35]
exact h ▸ Fin.cons_self_tail _
α : Type u_1 inst✝ : Coding α Memory N : ℕ g : (Fin (N + 1) → Bool) → α x : Fin (N + 1) → Bool h : x 0 = true ⊢ Fin.cons true (Fin.tail x) = x
no goals
https://github.com/calcu16/lean_complexity.git
d267eb4f3952c654f130a119311692c1c1a7cd71
HMem/Encoding/Basic.lean
HMem.Encoding.decodeOp_inv
[195, 1]
[209, 35]
apply congrArg g
α : Type u_1 inst✝ : Coding α Memory N : ℕ g : (Fin (N + 1) → Bool) → α x : Fin (N + 1) → Bool h : x 0 = false ⊢ (match false with | false => g (Fin.cons false (Fin.tail x)) | true => g (Fin.cons true (Fin.tail x))) = g x
α : Type u_1 inst✝ : Coding α Memory N : ℕ g : (Fin (N + 1) → Bool) → α x : Fin (N + 1) → Bool h : x 0 = false ⊢ Fin.cons false (Fin.tail x) = x
https://github.com/calcu16/lean_complexity.git
d267eb4f3952c654f130a119311692c1c1a7cd71
HMem/Encoding/Basic.lean
HMem.Encoding.decodeOp_inv
[195, 1]
[209, 35]
exact h ▸ Fin.cons_self_tail _
α : Type u_1 inst✝ : Coding α Memory N : ℕ g : (Fin (N + 1) → Bool) → α x : Fin (N + 1) → Bool h : x 0 = false ⊢ Fin.cons false (Fin.tail x) = x
no goals
https://github.com/calcu16/lean_complexity.git
d267eb4f3952c654f130a119311692c1c1a7cd71
HMem/Encoding/Basic.lean
HMem.Encoding.encodeOp_injN
[221, 1]
[226, 53]
cases N₁ <;> simp [encodeOp_zeroDef, encodeOp_succDef]
α : Type u_1 inst✝ : Encoding α Memory N₁ : ℕ x✝¹ : (Fin 0 → Bool) → α x✝ : (Fin N₁ → Bool) → α ⊢ encode x✝¹ = encode x✝ → 0 = N₁
no goals
https://github.com/calcu16/lean_complexity.git
d267eb4f3952c654f130a119311692c1c1a7cd71
HMem/Encoding/Basic.lean
HMem.Encoding.encodeOp_injN
[221, 1]
[226, 53]
simp [encodeOp_zeroDef, encodeOp_succDef]
α : Type u_1 inst✝ : Encoding α Memory N₀ : ℕ x✝¹ : (Fin (N₀ + 1) → Bool) → α x✝ : (Fin 0 → Bool) → α ⊢ encode x✝¹ = encode x✝ → N₀ + 1 = 0
no goals
https://github.com/calcu16/lean_complexity.git
d267eb4f3952c654f130a119311692c1c1a7cd71
HMem/Encoding/Basic.lean
HMem.Encoding.encodeOp_injN
[221, 1]
[226, 53]
simpa [encodeOp_succDef] using λ _ ↦ encodeOp_injN
α : Type u_1 inst✝ : Encoding α Memory N₀ N₁ : ℕ f₀ : (Fin (N₀ + 1) → Bool) → α f₁ : (Fin (N₁ + 1) → Bool) → α ⊢ encode f₀ = encode f₁ → N₀ + 1 = N₁ + 1
no goals
https://github.com/calcu16/lean_complexity.git
d267eb4f3952c654f130a119311692c1c1a7cd71
HMem/Encoding/Basic.lean
HMem.Encoding.encodeOp_injEq
[228, 9]
[232, 48]
cases (encodeOp_injN h)
α : Type u_1 inst✝ : Encoding α Memory N₀ N₁ : ℕ f₀ : (Fin N₀ → Bool) → α f₁ : (Fin N₁ → Bool) → α h : encode f₀ = encode f₁ ⊢ N₀ = N₁ ∧ HEq f₀ f₁
case refl α : Type u_1 inst✝ : Encoding α Memory N₀ : ℕ f₀ f₁ : (Fin N₀ → Bool) → α h : encode f₀ = encode f₁ ⊢ N₀ = N₀ ∧ HEq f₀ f₁
https://github.com/calcu16/lean_complexity.git
d267eb4f3952c654f130a119311692c1c1a7cd71
HMem/Encoding/Basic.lean
HMem.Encoding.encodeOp_injEq
[228, 9]
[232, 48]
simp [h, ← encodeInjEq]
case refl α : Type u_1 inst✝ : Encoding α Memory N₀ : ℕ f₀ f₁ : (Fin N₀ → Bool) → α h : encode f₀ = encode f₁ ⊢ N₀ = N₀ ∧ HEq f₀ f₁
no goals
https://github.com/calcu16/lean_complexity.git
d267eb4f3952c654f130a119311692c1c1a7cd71
HMem/Encoding/Basic.lean
HMem.Encoding.encodeOp_injEq
[228, 9]
[232, 48]
cases h.left
α : Type u_1 inst✝ : Encoding α Memory N₀ N₁ : ℕ f₀ : (Fin N₀ → Bool) → α f₁ : (Fin N₁ → Bool) → α h : N₀ = N₁ ∧ HEq f₀ f₁ ⊢ encode f₀ = encode f₁
case refl α : Type u_1 inst✝ : Encoding α Memory N₀ : ℕ f₀ f₁ : (Fin N₀ → Bool) → α h : N₀ = N₀ ∧ HEq f₀ f₁ ⊢ encode f₀ = encode f₁
https://github.com/calcu16/lean_complexity.git
d267eb4f3952c654f130a119311692c1c1a7cd71
HMem/Encoding/Basic.lean
HMem.Encoding.encodeOp_injEq
[228, 9]
[232, 48]
cases h.right
case refl α : Type u_1 inst✝ : Encoding α Memory N₀ : ℕ f₀ f₁ : (Fin N₀ → Bool) → α h : N₀ = N₀ ∧ HEq f₀ f₁ ⊢ encode f₀ = encode f₁
case refl.refl α : Type u_1 inst✝ : Encoding α Memory N₀ : ℕ f₀ : (Fin N₀ → Bool) → α h : N₀ = N₀ ∧ HEq f₀ f₀ ⊢ encode f₀ = encode f₀
https://github.com/calcu16/lean_complexity.git
d267eb4f3952c654f130a119311692c1c1a7cd71
HMem/Encoding/Basic.lean
HMem.Encoding.encodeOp_injEq
[228, 9]
[232, 48]
rfl
case refl.refl α : Type u_1 inst✝ : Encoding α Memory N₀ : ℕ f₀ : (Fin N₀ → Bool) → α h : N₀ = N₀ ∧ HEq f₀ f₀ ⊢ encode f₀ = encode f₀
no goals
https://github.com/calcu16/lean_complexity.git
d267eb4f3952c654f130a119311692c1c1a7cd71
HMem/Encoding/Basic.lean
HMem.Encoding.encode_nat_def
[244, 1]
[255, 6]
refine Nat.binaryRec' (n := n) Memory.zero_def'.symm ?hi
n : ℕ ⊢ encode n = Memory.mk (decide (n ≠ 0)) (encode (decide (n % 2 = 1))) (encode (n / 2))
case hi n : ℕ ⊢ ∀ (b : Bool) (n : ℕ), (n = 0 → b = true) → encode n = Memory.mk (decide (n ≠ 0)) (encode (decide (n % 2 = 1))) (encode (n / 2)) → encode (Nat.bit b n) = Memory.mk (decide (Nat.bit b n ≠ 0)) (encode (decide (Nat.bit b n % 2 = 1))) (encode (Nat.bit b n / 2))
https://github.com/calcu16/lean_complexity.git
d267eb4f3952c654f130a119311692c1c1a7cd71
HMem/Encoding/Basic.lean
HMem.Encoding.encode_nat_def
[244, 1]
[255, 6]
intro b n hn _
case hi n : ℕ ⊢ ∀ (b : Bool) (n : ℕ), (n = 0 → b = true) → encode n = Memory.mk (decide (n ≠ 0)) (encode (decide (n % 2 = 1))) (encode (n / 2)) → encode (Nat.bit b n) = Memory.mk (decide (Nat.bit b n ≠ 0)) (encode (decide (Nat.bit b n % 2 = 1))) (encode (Nat.bit b n / 2))
case hi n✝ : ℕ b : Bool n : ℕ hn : n = 0 → b = true a✝ : encode n = Memory.mk (decide (n ≠ 0)) (encode (decide (n % 2 = 1))) (encode (n / 2)) ⊢ encode (Nat.bit b n) = Memory.mk (decide (Nat.bit b n ≠ 0)) (encode (decide (Nat.bit b n % 2 = 1))) (encode (Nat.bit b n / 2))
https://github.com/calcu16/lean_complexity.git
d267eb4f3952c654f130a119311692c1c1a7cd71
HMem/Encoding/Basic.lean
HMem.Encoding.encode_nat_def
[244, 1]
[255, 6]
apply Eq.trans
case hi n✝ : ℕ b : Bool n : ℕ hn : n = 0 → b = true a✝ : encode n = Memory.mk (decide (n ≠ 0)) (encode (decide (n % 2 = 1))) (encode (n / 2)) ⊢ encode (Nat.bit b n) = Memory.mk (decide (Nat.bit b n ≠ 0)) (encode (decide (Nat.bit b n % 2 = 1))) (encode (Nat.bit b n / 2))
case hi.h₁ n✝ : ℕ b : Bool n : ℕ hn : n = 0 → b = true a✝ : encode n = Memory.mk (decide (n ≠ 0)) (encode (decide (n % 2 = 1))) (encode (n / 2)) ⊢ encode (Nat.bit b n) = ?hi.b case hi.h₂ n✝ : ℕ b : Bool n : ℕ hn : n = 0 → b = true a✝ : encode n = Memory.mk (decide (n ≠ 0)) (encode (decide (n % 2 = 1))) (encode (n / 2)) ⊢ ?hi.b = Memory.mk (decide (Nat.bit b n ≠ 0)) (encode (decide (Nat.bit b n % 2 = 1))) (encode (Nat.bit b n / 2)) case hi.b n✝ : ℕ b : Bool n : ℕ hn : n = 0 → b = true a✝ : encode n = Memory.mk (decide (n ≠ 0)) (encode (decide (n % 2 = 1))) (encode (n / 2)) ⊢ Memory
https://github.com/calcu16/lean_complexity.git
d267eb4f3952c654f130a119311692c1c1a7cd71
HMem/Encoding/Basic.lean
HMem.Encoding.encode_nat_def
[244, 1]
[255, 6]
apply congrArg (encode (α := List Bool))
case hi.h₁ n✝ : ℕ b : Bool n : ℕ hn : n = 0 → b = true a✝ : encode n = Memory.mk (decide (n ≠ 0)) (encode (decide (n % 2 = 1))) (encode (n / 2)) ⊢ encode (Nat.bit b n) = ?hi.b case hi.h₂ n✝ : ℕ b : Bool n : ℕ hn : n = 0 → b = true a✝ : encode n = Memory.mk (decide (n ≠ 0)) (encode (decide (n % 2 = 1))) (encode (n / 2)) ⊢ ?hi.b = Memory.mk (decide (Nat.bit b n ≠ 0)) (encode (decide (Nat.bit b n % 2 = 1))) (encode (Nat.bit b n / 2)) case hi.b n✝ : ℕ b : Bool n : ℕ hn : n = 0 → b = true a✝ : encode n = Memory.mk (decide (n ≠ 0)) (encode (decide (n % 2 = 1))) (encode (n / 2)) ⊢ Memory
case hi.h₁ n✝ : ℕ b : Bool n : ℕ hn : n = 0 → b = true a✝ : encode n = Memory.mk (decide (n ≠ 0)) (encode (decide (n % 2 = 1))) (encode (n / 2)) ⊢ Nat.bits (Nat.bit b n) = ?m.80295 n✝ : ℕ b : Bool n : ℕ hn : n = 0 → b = true a✝ : encode n = Memory.mk (decide (n ≠ 0)) (encode (decide (n % 2 = 1))) (encode (n / 2)) ⊢ List Bool case hi.h₂ n✝ : ℕ b : Bool n : ℕ hn : n = 0 → b = true a✝ : encode n = Memory.mk (decide (n ≠ 0)) (encode (decide (n % 2 = 1))) (encode (n / 2)) ⊢ encode ?m.80295 = Memory.mk (decide (Nat.bit b n ≠ 0)) (encode (decide (Nat.bit b n % 2 = 1))) (encode (Nat.bit b n / 2))
https://github.com/calcu16/lean_complexity.git
d267eb4f3952c654f130a119311692c1c1a7cd71
HMem/Encoding/Basic.lean
HMem.Encoding.encode_nat_def
[244, 1]
[255, 6]
apply Nat.bits_append_bit _ _ hn
case hi.h₁ n✝ : ℕ b : Bool n : ℕ hn : n = 0 → b = true a✝ : encode n = Memory.mk (decide (n ≠ 0)) (encode (decide (n % 2 = 1))) (encode (n / 2)) ⊢ Nat.bits (Nat.bit b n) = ?m.80295 n✝ : ℕ b : Bool n : ℕ hn : n = 0 → b = true a✝ : encode n = Memory.mk (decide (n ≠ 0)) (encode (decide (n % 2 = 1))) (encode (n / 2)) ⊢ List Bool case hi.h₂ n✝ : ℕ b : Bool n : ℕ hn : n = 0 → b = true a✝ : encode n = Memory.mk (decide (n ≠ 0)) (encode (decide (n % 2 = 1))) (encode (n / 2)) ⊢ encode ?m.80295 = Memory.mk (decide (Nat.bit b n ≠ 0)) (encode (decide (Nat.bit b n % 2 = 1))) (encode (Nat.bit b n / 2))
case hi.h₂ n✝ : ℕ b : Bool n : ℕ hn : n = 0 → b = true a✝ : encode n = Memory.mk (decide (n ≠ 0)) (encode (decide (n % 2 = 1))) (encode (n / 2)) ⊢ encode (b :: Nat.bits n) = Memory.mk (decide (Nat.bit b n ≠ 0)) (encode (decide (Nat.bit b n % 2 = 1))) (encode (Nat.bit b n / 2))
https://github.com/calcu16/lean_complexity.git
d267eb4f3952c654f130a119311692c1c1a7cd71
HMem/Encoding/Basic.lean
HMem.Encoding.encode_nat_def
[244, 1]
[255, 6]
apply Memory.mk_inj_iff.mpr
case hi.h₂ n✝ : ℕ b : Bool n : ℕ hn : n = 0 → b = true a✝ : encode n = Memory.mk (decide (n ≠ 0)) (encode (decide (n % 2 = 1))) (encode (n / 2)) ⊢ encode (b :: Nat.bits n) = Memory.mk (decide (Nat.bit b n ≠ 0)) (encode (decide (Nat.bit b n % 2 = 1))) (encode (Nat.bit b n / 2))
case hi.h₂ n✝ : ℕ b : Bool n : ℕ hn : n = 0 → b = true a✝ : encode n = Memory.mk (decide (n ≠ 0)) (encode (decide (n % 2 = 1))) (encode (n / 2)) ⊢ true = decide (Nat.bit b n ≠ 0) ∧ encode b = encode (decide (Nat.bit b n % 2 = 1)) ∧ encodeList (Nat.bits n) = encode (Nat.bit b n / 2)
https://github.com/calcu16/lean_complexity.git
d267eb4f3952c654f130a119311692c1c1a7cd71
HMem/Encoding/Basic.lean
HMem.Encoding.encode_nat_def
[244, 1]
[255, 6]
apply And.intro
case hi.h₂ n✝ : ℕ b : Bool n : ℕ hn : n = 0 → b = true a✝ : encode n = Memory.mk (decide (n ≠ 0)) (encode (decide (n % 2 = 1))) (encode (n / 2)) ⊢ true = decide (Nat.bit b n ≠ 0) ∧ encode b = encode (decide (Nat.bit b n % 2 = 1)) ∧ encodeList (Nat.bits n) = encode (Nat.bit b n / 2)
case hi.h₂.left n✝ : ℕ b : Bool n : ℕ hn : n = 0 → b = true a✝ : encode n = Memory.mk (decide (n ≠ 0)) (encode (decide (n % 2 = 1))) (encode (n / 2)) ⊢ true = decide (Nat.bit b n ≠ 0) case hi.h₂.right n✝ : ℕ b : Bool n : ℕ hn : n = 0 → b = true a✝ : encode n = Memory.mk (decide (n ≠ 0)) (encode (decide (n % 2 = 1))) (encode (n / 2)) ⊢ encode b = encode (decide (Nat.bit b n % 2 = 1)) ∧ encodeList (Nat.bits n) = encode (Nat.bit b n / 2)
https://github.com/calcu16/lean_complexity.git
d267eb4f3952c654f130a119311692c1c1a7cd71
HMem/Encoding/Basic.lean
HMem.Encoding.encode_nat_def
[244, 1]
[255, 6]
exact Eq.symm (decide_eq_true λ h ↦ absurd ((Nat.bit_eq_zero.mp h).right.symm.trans (hn (Nat.bit_eq_zero.mp h).left)) Bool.noConfusion)
case hi.h₂.left n✝ : ℕ b : Bool n : ℕ hn : n = 0 → b = true a✝ : encode n = Memory.mk (decide (n ≠ 0)) (encode (decide (n % 2 = 1))) (encode (n / 2)) ⊢ true = decide (Nat.bit b n ≠ 0) case hi.h₂.right n✝ : ℕ b : Bool n : ℕ hn : n = 0 → b = true a✝ : encode n = Memory.mk (decide (n ≠ 0)) (encode (decide (n % 2 = 1))) (encode (n / 2)) ⊢ encode b = encode (decide (Nat.bit b n % 2 = 1)) ∧ encodeList (Nat.bits n) = encode (Nat.bit b n / 2)
case hi.h₂.right n✝ : ℕ b : Bool n : ℕ hn : n = 0 → b = true a✝ : encode n = Memory.mk (decide (n ≠ 0)) (encode (decide (n % 2 = 1))) (encode (n / 2)) ⊢ encode b = encode (decide (Nat.bit b n % 2 = 1)) ∧ encodeList (Nat.bits n) = encode (Nat.bit b n / 2)
https://github.com/calcu16/lean_complexity.git
d267eb4f3952c654f130a119311692c1c1a7cd71
HMem/Encoding/Basic.lean
HMem.Encoding.encode_nat_def
[244, 1]
[255, 6]
cases b <;> simp [-Nat.bit_false, -Nat.bit_true, Nat.bit_mod_two, Nat.bit_div_2]
case hi.h₂.right n✝ : ℕ b : Bool n : ℕ hn : n = 0 → b = true a✝ : encode n = Memory.mk (decide (n ≠ 0)) (encode (decide (n % 2 = 1))) (encode (n / 2)) ⊢ encode b = encode (decide (Nat.bit b n % 2 = 1)) ∧ encodeList (Nat.bits n) = encode (Nat.bit b n / 2)
case hi.h₂.right.false n✝ n : ℕ a✝ : encode n = Memory.mk (decide (n ≠ 0)) (encode (decide (n % 2 = 1))) (encode (n / 2)) hn : n = 0 → false = true ⊢ encodeList (Nat.bits n) = encode n case hi.h₂.right.true n✝ n : ℕ a✝ : encode n = Memory.mk (decide (n ≠ 0)) (encode (decide (n % 2 = 1))) (encode (n / 2)) hn : n = 0 → true = true ⊢ encodeList (Nat.bits n) = encode n