(* ** Abstract specification using Hoare triples *)
From Undecidability.TM Require Import TM Util.TM_facts.
(* Abstract Assertions over Tapes *)
Definition Assert (sig : Type) (n : nat) : Type := tapes sig n -> Prop.
(* [{P} pM {fun c => Q c}] means that whenever [pM] starts with tapes that satisfy [P] and terminates in label [c] and tapes [tp'] that satisfy [Q c]. *)
Definition Triple_Rel {sig : finType} {n : nat} {F : Type} (P : Assert sig n) (Q : F -> Assert sig n) : pRel sig F n :=
fun tin '(yout, tout) => P tin -> Q yout tout.
Inductive Triple {sig : finType} {n : nat} {F : Type} P (pM : pTM sig F n) Q :=
TripleI : pM ⊨ Triple_Rel P Q -> Triple P pM Q.
Inductive Entails (sig : Type) (n : nat) (P1 P2 : Assert sig n) :=
EntailsI : (forall tin, P1 tin -> P2 tin) -> Entails P1 P2.

Lemma asPointwise A X (R: A -> A -> Prop) f g: (forall (x:X), R (f x) (g x)) -> pointwise_relation X R f g.