Project-declaredLean 4.32.0 · mathlib@81a5d257

Simulate Q option T for In yield pure none

simulateQ_optionT_forIn_yield_pure_none

program verification separation logic cryptography basic

Plain-language statement

Failing companion to simulateQ_optionT_forIn_yield_pure_some: if each loop body, under simulateQ, resolves to pure (some (ForInStep.yield init)) when its per-element condition cond a holds and to pure none otherwise, and some element of the list fails its condition, then the whole OptionT-monadic forIn resolves to pure none (the failure...

Exact Lean statement

lemma simulateQ_optionT_forIn_yield_pure_none (xs : List α) (init : β)
    (body : α → β → OptionT (OracleComp spec) (ForInStep β))
    (cond : α → Prop) [DecidablePred cond]
    (hbody : ∀ a, simulateQ impl ((body a init : OptionT (OracleComp spec) (ForInStep β)) :
        OracleComp spec (Option (ForInStep β)))
      = (pure (if cond a then some (ForInStep.yield init) else none) :
          n (Option (ForInStep β))))
    (hfail : ¬ ∀ a ∈ xs, cond a) :
    simulateQ impl ((forIn xs init body : OptionT (OracleComp spec) β) :
        OracleComp spec (Option β))
      = (pure none : n (Option β))

Formal artifact

Lean source

Canonical source

Full Lean sourceLean 4

lemma simulateQ_optionT_forIn_yield_pure_none (xs : List α) (init : β)    (body : α → β → OptionT (OracleComp spec) (ForInStep β))    (cond : α → Prop) [DecidablePred cond]    (hbody : ∀ a, simulateQ impl ((body a init : OptionT (OracleComp spec) (ForInStep β)) :        OracleComp spec (Option (ForInStep β)))      = (pure (if cond a then some (ForInStep.yield init) else none) :          n (Option (ForInStep β))))    (hfail : ¬ ∀ a ∈ xs, cond a) :    simulateQ impl ((forIn xs init body : OptionT (OracleComp spec) β) :        OracleComp spec (Option β))      = (pure none : n (Option β)) := by  rw [simulateQ_optionT_list_forIn]  induction xs with  | nil => exact absurd (List.forall_mem_nil _) hfail  | cons x rest ih =>      rw [List.forIn_cons]      by_cases hx : cond x      · change ((simulateQ impl ((body x init : OptionT (OracleComp spec) (ForInStep β)) :            OracleComp spec (Option (ForInStep β))) : OptionT n (ForInStep β)) >>= _ :            OptionT n β) = _        rw [show (simulateQ impl ((body x init : OptionT (OracleComp spec) (ForInStep β)) :            OracleComp spec (Option (ForInStep β))) : OptionT n (ForInStep β))            = (pure (ForInStep.yield init) : OptionT n (ForInStep β)) by          rw [hbody x, if_pos hx]          rfl, pure_bind]        exact ih (fun hall ↦ hfail (List.forall_mem_cons.mpr ⟨hx, hall⟩))      · change ((simulateQ impl ((body x init : OptionT (OracleComp spec) (ForInStep β)) :            OracleComp spec (Option (ForInStep β))) : OptionT n (ForInStep β)) >>= _ :            OptionT n β) = _        rw [show (simulateQ impl ((body x init : OptionT (OracleComp spec) (ForInStep β)) :            OracleComp spec (Option (ForInStep β))) : OptionT n (ForInStep β))            = (failure : OptionT n (ForInStep β)) by          rw [hbody x, if_neg hx]          rfl, failure_bind]        rfl

Project: VCVio
License: Apache-2.0
Commit: 2ceb2d825ee3
Source: VCVio/OracleComp/SimSemantics/OptionT/Basic.lean:222-256

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