(* ** Some Hoare Rules *)
From Undecidability Require Export TM.Compound.Multi TM.Combinators.Combinators.
From Undecidability.TM.Hoare Require Import HoareLogic HoareRegister.
(* *** Nop *)
(* A classic example for Hoare logic *)
(* *** [Relabel] *)
(* *** [Return] *)
(* *** Sequential Composition *)

Lemma Seq_SpecT (sig : finType) (n : nat) (F1 F2 : finType) (pM1 : pTM sig F1 n) (pM2 : pTM sig F2 n)
(P : Assert sig n) (Q : F1 -> Assert sig n) (R : F2 -> Assert sig n)
(k k1 k2 : nat) :
TripleT P k1 pM1 Q -> (* Correctness of [pM1] *)
(forall (ymid : F1), TripleT (Q ymid) k2 pM2 R) -> (* Correctness of [pM2] (for every output label of [pM1]) *)
1 + k1 + k2 <= k ->
TripleT P k (pM1;; pM2) R.