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+Require Import ZArith.
+
+Definition zeq := Z_eq_dec.
+
+Definition update (A: Set) (x: Z) (v: A) (s: Z -> A) : Z -> A :=
+ fun y => if zeq x y then v else s y.
+
+Implicit Arguments update [A].
+
+Definition ident := Z.
+Parameter operator: Set.
+Parameter value: Set.
+Parameter is_true: value -> Prop.
+Definition label := Z.
+
+Inductive expr : Set :=
+ | Evar: ident -> expr
+ | Econst: value -> expr
+ | Eop: operator -> expr -> expr -> expr.
+
+Inductive stmt : Set :=
+ | Sskip: stmt
+ | Sassign: ident -> expr -> stmt
+ | Scall: ident -> ident -> expr -> stmt (* x := f(e) *)
+ | Sreturn: expr -> stmt
+ | Sseq: stmt -> stmt -> stmt
+ | Sifthenelse: expr -> stmt -> stmt -> stmt
+ | Sloop: stmt -> stmt
+ | Sblock: stmt -> stmt
+ | Sexit: nat -> stmt
+ | Slabel: label -> stmt -> stmt
+ | Sgoto: label -> stmt.
+
+Record function : Set := mkfunction {
+ fn_param: ident;
+ fn_body: stmt
+}.
+
+Parameter program: ident -> option function.
+
+Parameter main_function: ident.
+
+Definition store := ident -> value.
+
+Parameter empty_store : store.
+
+Parameter eval_op: operator -> value -> value -> option value.
+
+Fixpoint eval_expr (st: store) (e: expr) {struct e} : option value :=
+ match e with
+ | Evar v => Some (st v)
+ | Econst v => Some v
+ | Eop op e1 e2 =>
+ match eval_expr st e1, eval_expr st e2 with
+ | Some v1, Some v2 => eval_op op v1 v2
+ | _, _ => None
+ end
+ end.
+
+Inductive outcome: Set :=
+ | Onormal: outcome
+ | Oexit: nat -> outcome
+ | Ogoto: label -> outcome
+ | Oreturn: value -> outcome.
+
+Definition outcome_block (out: outcome) : outcome :=
+ match out with
+ | Onormal => Onormal
+ | Oexit O => Onormal
+ | Oexit (S m) => Oexit m
+ | Ogoto lbl => Ogoto lbl
+ | Oreturn v => Oreturn v
+ end.
+
+Fixpoint label_defined (lbl: label) (s: stmt) {struct s}: Prop :=
+ match s with
+ | Sskip => False
+ | Sassign id e => False
+ | Scall id fn e => False
+ | Sreturn e => False
+ | Sseq s1 s2 => label_defined lbl s1 \/ label_defined lbl s2
+ | Sifthenelse e s1 s2 => label_defined lbl s1 \/ label_defined lbl s2
+ | Sloop s1 => label_defined lbl s1
+ | Sblock s1 => label_defined lbl s1
+ | Sexit n => False
+ | Slabel lbl1 s1 => lbl1 = lbl \/ label_defined lbl s1
+ | Sgoto lbl => False
+ end.
+
+Inductive exec : stmt -> store -> outcome -> store -> Prop :=
+ | exec_skip: forall st,
+ exec Sskip st Onormal st
+ | exec_assign: forall id e st v,
+ eval_expr st e = Some v ->
+ exec (Sassign id e) st Onormal (update id v st)
+ | exec_call: forall id fn e st v1 f v2 st',
+ eval_expr st e = Some v1 ->
+ program fn = Some f ->
+ exec_function f (update f.(fn_param) v1 empty_store) v2 st' ->
+ exec (Scall id fn e) st Onormal (update id v2 st)
+ | exec_return: forall e st v,
+ eval_expr st e = Some v ->
+ exec (Sreturn e) st (Oreturn v) st
+ | exec_seq_2: forall s1 s2 st st1 out' st',
+ exec s1 st Onormal st1 -> exec s2 st1 out' st' ->
+ exec (Sseq s1 s2) st out' st'
+ | exec_seq_1: forall s1 s2 st out st',
+ exec s1 st out st' -> out <> Onormal ->
+ exec (Sseq s1 s2) st out st'
+ | exec_ifthenelse_true: forall e s1 s2 st out st' v,
+ eval_expr st e = Some v -> is_true v -> exec s1 st out st' ->
+ exec (Sifthenelse e s1 s2) st out st'
+ | exec_ifthenelse_false: forall e s1 s2 st out st' v,
+ eval_expr st e = Some v -> ~is_true v -> exec s2 st out st' ->
+ exec (Sifthenelse e s1 s2) st out st'
+ | exec_loop_loop: forall s st st1 out' st',
+ exec s st Onormal st1 ->
+ exec (Sloop s) st1 out' st' ->
+ exec (Sloop s) st out' st'
+ | exec_loop_stop: forall s st st' out,
+ exec s st out st' -> out <> Onormal ->
+ exec (Sloop s) st out st'
+ | exec_block: forall s st out st',
+ exec s st out st' ->
+ exec (Sblock s) st (outcome_block out) st'
+ | exec_exit: forall n st,
+ exec (Sexit n) st (Oexit n) st
+ | exec_label: forall s lbl st st' out,
+ exec s st out st' ->
+ exec (Slabel lbl s) st out st'
+ | exec_goto: forall st lbl,
+ exec (Sgoto lbl) st (Ogoto lbl) st
+
+(** [execg lbl stmt st out st'] starts executing at label [lbl] within [s],
+ in initial store [st]. The result of the execution is the outcome
+ [out] with final store [st']. *)
+
+with execg: label -> stmt -> store -> outcome -> store -> Prop :=
+ | execg_left_seq_2: forall lbl s1 s2 st st1 out' st',
+ execg lbl s1 st Onormal st1 -> exec s2 st1 out' st' ->
+ execg lbl (Sseq s1 s2) st out' st'
+ | execg_left_seq_1: forall lbl s1 s2 st out st',
+ execg lbl s1 st out st' -> out <> Onormal ->
+ execg lbl (Sseq s1 s2) st out st'
+ | execg_right_seq: forall lbl s1 s2 st out st',
+ ~(label_defined lbl s1) ->
+ execg lbl s2 st out st' ->
+ execg lbl (Sseq s1 s2) st out st'
+ | execg_ifthenelse_left: forall lbl e s1 s2 st out st',
+ execg lbl s1 st out st' ->
+ execg lbl (Sifthenelse e s1 s2) st out st'
+ | execg_ifthenelse_right: forall lbl e s1 s2 st out st',
+ ~(label_defined lbl s1) ->
+ execg lbl s2 st out st' ->
+ execg lbl (Sifthenelse e s1 s2) st out st'
+ | execg_loop_loop: forall lbl s st st1 out' st',
+ execg lbl s st Onormal st1 ->
+ exec (Sloop s) st1 out' st' ->
+ execg lbl (Sloop s) st out' st'
+ | execg_loop_stop: forall lbl s st st' out,
+ execg lbl s st out st' -> out <> Onormal ->
+ execg lbl (Sloop s) st out st'
+ | execg_block: forall lbl s st out st',
+ execg lbl s st out st' ->
+ execg lbl (Sblock s) st (outcome_block out) st'
+ | execg_label_found: forall lbl s st st' out,
+ exec s st out st' ->
+ execg lbl (Slabel lbl s) st out st'
+ | execg_label_notfound: forall lbl s lbl' st st' out,
+ lbl' <> lbl ->
+ execg lbl s st out st' ->
+ execg lbl (Slabel lbl' s) st out st'
+
+(** [exec_finish out st st'] takes the outcome [out] and the store [st]
+ at the end of the evaluation of the program. If [out] is a [goto],
+ execute again the program starting at the corresponding label.
+ Iterate this way until [out] is [Onormal]. *)
+
+with exec_finish: function -> outcome -> store -> value -> store -> Prop :=
+ | exec_finish_normal: forall f st v,
+ exec_finish f (Oreturn v) st v st
+ | exec_finish_goto: forall f lbl st out v st1 st',
+ execg lbl f.(fn_body) st out st1 ->
+ exec_finish f out st1 v st' ->
+ exec_finish f (Ogoto lbl) st v st'
+
+(** Execution of a function *)
+
+with exec_function: function -> store -> value -> store -> Prop :=
+ | exec_function_intro: forall f st out st1 v st',
+ exec f.(fn_body) st out st1 ->
+ exec_finish f out st1 v st' ->
+ exec_function f st v st'.
+
+Scheme exec_ind4:= Minimality for exec Sort Prop
+ with execg_ind4:= Minimality for execg Sort Prop
+ with exec_finish_ind4 := Minimality for exec_finish Sort Prop
+ with exec_function_ind4 := Minimality for exec_function Sort Prop.
+
+Scheme exec_dind4:= Induction for exec Sort Prop
+ with execg_dind4:= Minimality for execg Sort Prop
+ with exec_finish_dind4 := Induction for exec_finish Sort Prop
+ with exec_function_dind4 := Induction for exec_function Sort Prop.
+
+Combined Scheme exec_inductiond from exec_dind4, execg_dind4, exec_finish_dind4,
+ exec_function_dind4.
+
+Scheme exec_dind4' := Induction for exec Sort Prop
+ with execg_dind4' := Induction for execg Sort Prop
+ with exec_finish_dind4' := Induction for exec_finish Sort Prop
+ with exec_function_dind4' := Induction for exec_function Sort Prop.
+
+Combined Scheme exec_induction from exec_ind4, execg_ind4, exec_finish_ind4,
+ exec_function_ind4.
+
+Combined Scheme exec_inductiond' from exec_dind4', execg_dind4', exec_finish_dind4',
+ exec_function_dind4'.