+ +
+Require Import Arith.
+Require Import Bool.
+Require Import List.
+Require Import Ott.ott_list_core.
+ +
+Require Export Ascii.
+Require Export String.
+Require Export ZArith.
+Require Import Lia.
+Require Coq.Program.Wf.
+ +
+Require Import FMapList OrderedTypeEx.
+Module Map <: FMapInterface.S := FMapList.Make String_as_OT.
+ +
+#[export] Hint Resolve bool_dec : ott_coq_equality.
+#[export] Hint Resolve ascii_dec : ott_coq_equality.
+#[export] Hint Resolve Pos.eq_dec : ott_coq_equality.
+ +
+
+Definition i : Set := nat. (*r index variables (subscripts) *)
+Lemma eq_i: forall (x y : i), {x = y} + {x <> y}.
+Proof.
+ decide equality; auto with ott_coq_equality arith.
+Defined.
+Hint Resolve eq_i : ott_coq_equality.
+Definition fn : Set := string. (*r function name *)
+Lemma eq_fn: forall (x y : fn), {x = y} + {x <> y}.
+Proof.
+ decide equality; auto with ott_coq_equality arith.
+Defined.
+Hint Resolve eq_fn : ott_coq_equality.
+Definition x : Set := string. (*r variable *)
+Lemma eq_x: forall (x' y : x), {x' = y} + {x' <> y}.
+Proof.
+ decide equality; auto with ott_coq_equality arith.
+Defined.
+Hint Resolve eq_x : ott_coq_equality.
+Definition b : Set := bool. (*r boolean *)
+Lemma eq_b: forall (x y : b), {x = y} + {x <> y}.
+Proof.
+ decide equality; auto with ott_coq_equality arith.
+Defined.
+Hint Resolve eq_b : ott_coq_equality.
+Definition z : Set := Z. (*r integer *)
+Lemma eq_z: forall (x y : z), {x = y} + {x <> y}.
+Proof.
+ decide equality; auto with ott_coq_equality arith.
+Defined.
+Hint Resolve eq_z : ott_coq_equality.
+ +
+Inductive T : Set := (*r ground type *)
+ | T_bool : T
+ | T_int : T.
+ +
+Inductive t : Set := (*r ground term *)
+ | t_b (b5:b) (*r boolean *)
+ | t_int (z5:z) (*r integer *).
+ +
+Inductive sig : Set :=
+ | sig_sig (_:list T) (T_5:T).
+ +
+Inductive e : Set := (*r expression *)
+ | e_t (t5:t) (*r term *)
+ | e_var (x5:x) (*r variable *)
+ | e_fn_call (fn5:fn) (_:list e) (*r function call *).
+ +
+Inductive ctxv : Set :=
+ | ctxv_T (T5:T)
+ | ctxv_sig (sig5:sig).
+ +
+Inductive F : Set := (*r function definition *)
+ | F_fn (T_5:T) (fn5:fn) (_:list (T*x)) (e5:e).
+ +
+Definition s : Type := Map.t t.
+ +
+Definition G : Type := Map.t ctxv.
+
+ +
+Variables
+ (P_e : e -> Prop)
+ (P_list_e : list e -> Prop).
+ +
+Hypothesis
+ (H_e_t : forall (t5:t), P_e (e_t t5))
+ (H_e_var : forall (x5:x), P_e (e_var x5))
+ (H_e_fn_call : forall (e_list:list e), P_list_e e_list -> forall (fn5:fn), P_e (e_fn_call fn5 e_list))
+ (H_list_e_nil : P_list_e nil)
+ (H_list_e_cons : forall (e0:e), P_e e0 -> forall (e_l:list e), P_list_e e_l -> P_list_e (cons e0 e_l)).
+ +
+Fixpoint e_ott_ind (n:e) : P_e n :=
+ match n as x return P_e x with
+ | (e_t t5) => H_e_t t5
+ | (e_var x5) => H_e_var x5
+ | (e_fn_call fn5 e_list) => H_e_fn_call e_list (((fix e_list_ott_ind (e_l:list e) : P_list_e e_l := match e_l as x return P_list_e x with nil => H_list_e_nil | cons e1 xl => H_list_e_cons e1(e_ott_ind e1)xl (e_list_ott_ind xl) end)) e_list) fn5
+end.
+ +
+End e_rect.
+Fixpoint get_val (k:x) (l:list(x*x)): option x :=
+ match l with
+ | nil => None
+ | (k', v) :: l' =>
+ match eq_x k k' with
+ | left _ => Some v
+ | _ => get_val k l'
+ end
+ end.
+ +
+Fixpoint e_var_subst (e5:e) (l:list (x*x)) : e :=
+ match e5 with
+ | e_t t => e_t t
+ | e_var x =>
+ match get_val x l with
+ | None => e_var x
+ | Some y => e_var y
+ end
+ | e_fn_call fn e_list =>
+ e_fn_call fn (map (fun e => e_var_subst e l) e_list)
+ end.
+ +
+(* we actually need to do some work to convince Coq fresh_vars is well founded *)
+Fixpoint e_size (e0 : e) : nat :=
+ match e0 with
+ | e_t _ => 1
+ | e_var _ => 1
+ | e_fn_call _ es => 1 + list_sum (map e_size es)
+ end.
+ +
+Lemma size_ge_1: forall e0,
+ 1 <= e_size e0.
+Proof. destruct e0;cbn;lia. Qed.
+ +
+Program Fixpoint fresh_vars (l : list x) (e0 : e) {measure (e_size e0)} : Prop :=
+ match e0 with
+ | e_t _ => True
+ | e_var x => ~ In x l
+ | e_fn_call fn es =>
+ match es with
+ | nil => True
+ | e1::es' => fresh_vars l e1
+ /\ fresh_vars l (e_fn_call fn es')
+ end
+ end.
+Next Obligation. cbn;lia. Qed.
+Next Obligation.
+ induction es';cbn; specialize (size_ge_1 e1);intro; lia.
+Qed.
+ +
+
+(* defns expression_well_typing *)
+Inductive typ_e : G -> e -> T -> Prop := (* defn e *)
+ | typ_bool : forall (G5:G) (b5:b),
+ typ_e G5 (e_t (t_b b5)) T_bool
+ | typ_int : forall (G5:G) (z5:z),
+ typ_e G5 (e_t (t_int z5)) T_int
+ | typ_var : forall (G5:G) (x5:x) (T5:T),
+ (Map.find x5 G5 = Some (ctxv_T T5 )) ->
+ typ_e G5 (e_var x5) T5
+ | typ_func_expr : forall (e_T_list:list (e*T)) (G5:G) (fn5:fn) (T_5:T),
+ (forall e_ T_, In (e_,T_) (map (fun (pat_: (e*T)) => match pat_ with (e_,T_) => (e_,T_) end) e_T_list) -> (typ_e G5 e_ T_)) ->
+ (Map.find fn5 G5 = Some (ctxv_sig (sig_sig (map (fun (pat_:(e*T)) => match pat_ with (e_,T_) => T_ end ) e_T_list) T_5) )) ->
+ typ_e G5 (e_fn_call fn5 (map (fun (pat_:(e*T)) => match pat_ with (e_,T_) => e_ end ) e_T_list)) T_5.
+
+(* defns function_well_typing *)
+Inductive typ_F : G -> F -> Prop := (* defn F *)
+ | typ_func_decl : forall (T_x_list:list (T*x)) (G5:G) (T_5:T) (fn5:fn) (e5:e),
+ (Map.find fn5 G5 = Some (ctxv_sig (sig_sig (map (fun (pat_:(T*x)) => match pat_ with (T_,x_) => T_ end ) T_x_list) T_5) )) ->
+ typ_e (fold_right (fun (ax : x * T) (G5 : G) => Map.add (fst ax) (ctxv_T (snd ax)) G5) G5 (map (fun (pat_:(T*x)) => match pat_ with (T_,x_) => (x_,T_) end ) T_x_list) ) e5 T_5 ->
+ typ_F G5 (F_fn T_5 fn5 T_x_list e5).
+
+(* defns evaluation_reduction *)
+Inductive red_e : list F -> s -> e -> s -> e -> Prop := (* defn e *)
+ | red_var : forall (F_list:list F) (s5:s) (x5:x) (t5:t),
+ (Map.find x5 s5 = Some ( t5 )) ->
+ red_e F_list s5 (e_var x5) s5 (e_t t5)
+ | red_fun_exp : forall (e'_list e_list:list e) (F_list:list F) (s5:s) (fn5:fn) (e_5:e) (s':s) (e':e),
+ red_e F_list s5 e_5 s' e' ->
+ red_e F_list s5 (e_fn_call fn5 ((app e_list (app (cons e_5 nil) (app e'_list nil))))) s' (e_fn_call fn5 ((app e_list (app (cons e' nil) (app e'_list nil)))))
+ | red_fun_ground : forall (T_x_t_y_list:list (T*x*t*x)) (F'_list F_list:list F) (T_5:T) (fn5:fn) (e5:e) (s5:s),
+ (fresh_vars (map (fun (pat_:(T*x*t*x)) => match pat_ with (T_,x_,t_,y_) => y_ end ) T_x_t_y_list) e5 ) ->
+ red_e ((app F_list (app (cons (F_fn T_5 fn5 (map (fun (pat_:(T*x*t*x)) => match pat_ with (T_,x_,t_,y_) => (T_,x_) end ) T_x_t_y_list) e5) nil) (app F'_list nil)))) s5 (e_fn_call fn5 (map (fun (pat_:(T*x*t*x)) => match pat_ with (T_,x_,t_,y_) => (e_t t_) end ) T_x_t_y_list)) (fold_right (fun (xt : x * t) (s5 : s) => Map.add (fst xt) (snd xt) s5) s5 (map (fun (pat_:(T*x*t*x)) => match pat_ with (T_,x_,t_,y_) => (y_,t_) end ) T_x_t_y_list) ) (e_var_subst e5 (map (fun (pat_:(T*x*t*x)) => match pat_ with (T_,x_,t_,y_) => (x_,y_) end ) T_x_t_y_list) ) .
+ +
+