Commit 701eaf85 authored by Asma Tafat-Bouzid's avatar Asma Tafat-Bouzid

blocking semantics

parent 39e865f1
......@@ -30,52 +30,35 @@ axiom mident_decide :
type ident = {| ident_index : int |}
(** Terms *)
type term_node =
type term =
| Tvalue value
| Tvar ident
| Tderef mident
| Tbin term operator term
with term = {| term_node : term_node;
term_maxvar : int;
|}
predicate var_occurs_in_term (x:ident) (t:term) =
match t with
| {| term_node = Tvalue _ |} -> false
| {| term_node = Tvar i |} -> x=i
| {| term_node = Tderef _ |} -> false
| {| term_node = Tbin t1 _ t2 |} -> var_occurs_in_term x t1 \/ var_occurs_in_term x t2
| Tvalue _ -> false
| Tvar i -> x=i
| Tderef _ -> false
| Tbin t1 _ t2 -> var_occurs_in_term x t1 \/ var_occurs_in_term x t2
end
predicate term_inv (t:term) =
forall x:ident. var_occurs_in_term x t -> x.ident_index <= t.term_maxvar
(* predicate term_inv (t:term) = *)
(* forall x:ident. var_occurs_in_term x t -> x.ident_index <= t.term_maxvar *)
function mk_tvalue (v:value) : term =
{| term_node = Tvalue v; term_maxvar = -1 |}
lemma mk_tvalue_inv :
forall v:value. term_inv (mk_tvalue v)
Tvalue v
function mk_tvar (i:ident) : term =
{| term_node = Tvar i; term_maxvar = i.ident_index |}
lemma mk_tvar_inv :
forall i:ident. term_inv (mk_tvar i)
Tvar i
function mk_tderef (r:mident) : term =
{| term_node = Tderef r; term_maxvar = -1 |}
lemma mk_tderef_inv :
forall r:mident. term_inv (mk_tderef r)
Tderef r
function mk_tbin (t1:term) (o:operator) (t2:term) : term =
{| term_node = Tbin t1 o t2;
term_maxvar = max t1.term_maxvar t2.term_maxvar |}
lemma mk_tbin_inv :
forall t1 t2:term, o:operator. term_inv t1 /\ term_inv t2 ->
term_inv (mk_tbin t1 o t2)
Tbin t1 o t2
(** Formulas *)
......@@ -126,23 +109,23 @@ function get_reftype (i:mident) (e:type_env) : datatype = IdMap.get e i
inductive type_term type_env type_stack term datatype =
| Type_value :
forall sigma: type_env, pi:type_stack, v:value, m:int.
type_term sigma pi {| term_node = Tvalue v; term_maxvar = m |} (type_value v)
forall sigma: type_env, pi:type_stack, v:value.
type_term sigma pi (Tvalue v) (type_value v)
| Type_var :
forall sigma: type_env, pi:type_stack, v: ident, m:int, ty:datatype.
forall sigma: type_env, pi:type_stack, v: ident, ty:datatype.
(get_vartype v pi = ty) ->
type_term sigma pi {| term_node = Tvar v ; term_maxvar = m |} ty
type_term sigma pi (Tvar v) ty
| Type_deref :
forall sigma: type_env, pi:type_stack, v: mident, m:int, ty:datatype.
forall sigma: type_env, pi:type_stack, v: mident, ty:datatype.
(get_reftype v sigma = ty) ->
type_term sigma pi {| term_node = Tderef v; term_maxvar = m |} ty
type_term sigma pi (Tderef v) ty
| Type_bin :
forall sigma: type_env, pi:type_stack, t1 t2 : term, op:operator,
m:int, ty1 ty2 ty:datatype.
ty1 ty2 ty:datatype.
type_term sigma pi t1 ty1 ->
type_term sigma pi t2 ty2 ->
type_operator op ty1 ty2 ty ->
type_term sigma pi {| term_node = Tbin t1 op t2; term_maxvar = m |} ty
type_term sigma pi (Tbin t1 op t2) ty
inductive type_fmla type_env type_stack fmla =
| Type_term :
......@@ -247,10 +230,10 @@ function eval_bin (x:value) (op:operator) (y:value) : value =
function eval_term (sigma:env) (pi:stack) (t:term) : value =
match t with
| {| term_node = Tvalue v |} -> v
| {| term_node = Tvar id |} -> get_stack id pi
| {| term_node = Tderef id |} -> get_env id sigma
| {| term_node = Tbin t1 op t2 |} ->
| Tvalue v -> v
| Tvar id -> get_stack id pi
| Tderef id -> get_env id sigma
| Tbin t1 op t2 ->
eval_bin (eval_term sigma pi t1) op (eval_term sigma pi t2)
end
......@@ -284,24 +267,24 @@ predicate eval_fmla (sigma:env) (pi:stack) (f:fmla) =
function msubst_term (t:term) (r:mident) (v:ident) : term =
match t with
| {| term_node = Tvalue _ | Tvar _ |} -> t
| {| term_node = Tderef x |} -> if r = x then mk_tvar v else t
| {| term_node = Tbin t1 op t2 |} ->
| Tvalue _ | Tvar _ -> t
| Tderef x -> if r = x then mk_tvar v else t
| Tbin t1 op t2 ->
mk_tbin (msubst_term t1 r v) op (msubst_term t2 r v)
end
function subst_term (t:term) (r:ident) (v:ident) : term =
match t with
| {| term_node = Tvalue _ | Tderef _ |} -> t
| {| term_node = Tvar x |} ->
| Tvalue _ | Tderef _ -> t
| Tvar x ->
if r = x then mk_tvar v else t
| {| term_node = Tbin t1 op t2 |} ->
| Tbin t1 op t2 ->
mk_tbin (subst_term t1 r v) op (subst_term t2 r v)
end
(** [fresh_in_term id t] is true when [id] does not occur in [t] *)
predicate fresh_in_term (id:ident) (t:term) =
id.ident_index > t.term_maxvar
not (var_occurs_in_term id t)
lemma fresh_in_binop:
forall t t':term, op:operator, v:ident.
......@@ -377,6 +360,16 @@ lemma eval_msubst:
(* (eval_fmla sigma pi (subst f x v) <-> *)
(* eval_fmla sigma (Cons(x, (get_stack v pi)) pi) f) *)
lemma eval_same_var_term:
forall t:term, sigma:env, pi:stack, id:ident, v1 v2:value.
eval_term sigma (Cons (id,v1) (Cons (id,v2) pi)) t =
eval_term sigma (Cons (id,v1) pi) t
lemma eval_same_var:
forall f:fmla, sigma:env, pi:stack, id:ident, v1 v2:value.
eval_fmla sigma (Cons (id,v1) (Cons (id,v2) pi)) f <->
eval_fmla sigma (Cons (id,v1) pi) f
lemma eval_swap_term:
forall t:term, sigma:env, pi:stack, id1 id2:ident, v1 v2:value.
id1 <> id2 ->
......@@ -389,11 +382,6 @@ lemma eval_swap:
(eval_fmla sigma (Cons (id1,v1) (Cons (id2,v2) pi)) f <->
eval_fmla sigma (Cons (id2,v2) (Cons (id1,v1) pi)) f)
(* lemma eval_same_var: *)
(* forall f:fmla, sigma:env, pi:stack, id:ident, v1 v2:value. *)
(* eval_fmla sigma (Cons (id,v1) (Cons (id,v2) pi)) f <-> *)
(* eval_fmla sigma (Cons (id,v1) pi) f *)
(* Need it for monotonicity*)
lemma eval_change_free :
forall f:fmla, sigma:env, pi:stack, id:ident, v:value.
......
(* This file is generated by Why3's Coq driver *)
(* Beware! Only edit allowed sections below *)
Require Import BuiltIn.
Require BuiltIn.
Require int.Int.
Require int.MinMax.
(* Why3 assumption *)
Inductive list (a:Type) {a_WT:WhyType a} :=
| Nil : list a
| Cons : a -> (list a) -> list a.
Axiom list_WhyType : forall (a:Type) {a_WT:WhyType a}, WhyType (list a).
Existing Instance list_WhyType.
Implicit Arguments Nil [[a] [a_WT]].
Implicit Arguments Cons [[a] [a_WT]].
Axiom map : forall (a:Type) {a_WT:WhyType a} (b:Type) {b_WT:WhyType b}, Type.
Parameter map_WhyType : forall (a:Type) {a_WT:WhyType a}
(b:Type) {b_WT:WhyType b}, WhyType (map a b).
Existing Instance map_WhyType.
Parameter get: forall {a:Type} {a_WT:WhyType a} {b:Type} {b_WT:WhyType b},
(map a b) -> a -> b.
Parameter set: forall {a:Type} {a_WT:WhyType a} {b:Type} {b_WT:WhyType b},
(map a b) -> a -> b -> (map a b).
Axiom Select_eq : forall {a:Type} {a_WT:WhyType a} {b:Type} {b_WT:WhyType b},
forall (m:(map a b)), forall (a1:a) (a2:a), forall (b1:b), (a1 = a2) ->
((get (set m a1 b1) a2) = b1).
Axiom Select_neq : forall {a:Type} {a_WT:WhyType a}
{b:Type} {b_WT:WhyType b}, forall (m:(map a b)), forall (a1:a) (a2:a),
forall (b1:b), (~ (a1 = a2)) -> ((get (set m a1 b1) a2) = (get m a2)).
Parameter const: forall {a:Type} {a_WT:WhyType a} {b:Type} {b_WT:WhyType b},
b -> (map a b).
Axiom Const : forall {a:Type} {a_WT:WhyType a} {b:Type} {b_WT:WhyType b},
forall (b1:b) (a1:a), ((get (const b1:(map a b)) a1) = b1).
(* Why3 assumption *)
Inductive datatype :=
| TYunit : datatype
| TYint : datatype
| TYbool : datatype .
Axiom datatype_WhyType : WhyType datatype.
Existing Instance datatype_WhyType.
(* Why3 assumption *)
Inductive value :=
| Vvoid : value
| Vint : Z -> value
| Vbool : bool -> value .
Axiom value_WhyType : WhyType value.
Existing Instance value_WhyType.
(* Why3 assumption *)
Inductive operator :=
| Oplus : operator
| Ominus : operator
| Omult : operator
| Ole : operator .
Axiom operator_WhyType : WhyType operator.
Existing Instance operator_WhyType.
Axiom mident : Type.
Parameter mident_WhyType : WhyType mident.
Existing Instance mident_WhyType.
Axiom mident_decide : forall (m1:mident) (m2:mident), (m1 = m2) \/
~ (m1 = m2).
(* Why3 assumption *)
Inductive ident :=
| mk_ident : Z -> ident .
Axiom ident_WhyType : WhyType ident.
Existing Instance ident_WhyType.
(* Why3 assumption *)
Definition ident_index(v:ident): Z := match v with
| (mk_ident x) => x
end.
(* Why3 assumption *)
Inductive term :=
| Tvalue : value -> term
| Tvar : ident -> term
| Tderef : mident -> term
| Tbin : term -> operator -> term -> term .
Axiom term_WhyType : WhyType term.
Existing Instance term_WhyType.
(* Why3 assumption *)
Fixpoint var_occurs_in_term(x:ident) (t:term) {struct t}: Prop :=
match t with
| (Tvalue _) => False
| (Tvar i) => (x = i)
| (Tderef _) => False
| (Tbin t1 _ t2) => (var_occurs_in_term x t1) \/ (var_occurs_in_term x t2)
end.
(* Why3 assumption *)
Inductive fmla :=
| Fterm : term -> fmla
| Fand : fmla -> fmla -> fmla
| Fnot : fmla -> fmla
| Fimplies : fmla -> fmla -> fmla
| Flet : ident -> term -> fmla -> fmla
| Fforall : ident -> datatype -> fmla -> fmla .
Axiom fmla_WhyType : WhyType fmla.
Existing Instance fmla_WhyType.
(* Why3 assumption *)
Inductive stmt :=
| Sskip : stmt
| Sassign : mident -> term -> stmt
| Sseq : stmt -> stmt -> stmt
| Sif : term -> stmt -> stmt -> stmt
| Sassert : fmla -> stmt
| Swhile : term -> fmla -> stmt -> stmt .
Axiom stmt_WhyType : WhyType stmt.
Existing Instance stmt_WhyType.
Axiom decide_is_skip : forall (s:stmt), (s = Sskip) \/ ~ (s = Sskip).
(* Why3 assumption *)
Definition type_value(v:value): datatype :=
match v with
| Vvoid => TYunit
| (Vint int) => TYint
| (Vbool bool1) => TYbool
end.
(* Why3 assumption *)
Inductive type_operator : operator -> datatype -> datatype
-> datatype -> Prop :=
| Type_plus : (type_operator Oplus TYint TYint TYint)
| Type_minus : (type_operator Ominus TYint TYint TYint)
| Type_mult : (type_operator Omult TYint TYint TYint)
| Type_le : (type_operator Ole TYint TYint TYbool).
(* Why3 assumption *)
Definition type_stack := (list (ident* datatype)%type).
Parameter get_vartype: ident -> (list (ident* datatype)%type) -> datatype.
Axiom get_vartype_def : forall (i:ident) (pi:(list (ident* datatype)%type)),
match pi with
| Nil => ((get_vartype i pi) = TYunit)
| (Cons (x, ty) r) => ((x = i) -> ((get_vartype i pi) = ty)) /\
((~ (x = i)) -> ((get_vartype i pi) = (get_vartype i r)))
end.
(* Why3 assumption *)
Definition type_env := (map mident datatype).
(* Why3 assumption *)
Inductive type_term : (map mident datatype) -> (list (ident* datatype)%type)
-> term -> datatype -> Prop :=
| Type_value : forall (sigma:(map mident datatype)) (pi:(list (ident*
datatype)%type)) (v:value), (type_term sigma pi (Tvalue v)
(type_value v))
| Type_var : forall (sigma:(map mident datatype)) (pi:(list (ident*
datatype)%type)) (v:ident) (ty:datatype), ((get_vartype v pi) = ty) ->
(type_term sigma pi (Tvar v) ty)
| Type_deref : forall (sigma:(map mident datatype)) (pi:(list (ident*
datatype)%type)) (v:mident) (ty:datatype), ((get sigma v) = ty) ->
(type_term sigma pi (Tderef v) ty)
| Type_bin : forall (sigma:(map mident datatype)) (pi:(list (ident*
datatype)%type)) (t1:term) (t2:term) (op:operator) (ty1:datatype)
(ty2:datatype) (ty:datatype), (type_term sigma pi t1 ty1) ->
((type_term sigma pi t2 ty2) -> ((type_operator op ty1 ty2 ty) ->
(type_term sigma pi (Tbin t1 op t2) ty))).
(* Why3 assumption *)
Inductive type_fmla : (map mident datatype) -> (list (ident* datatype)%type)
-> fmla -> Prop :=
| Type_term : forall (sigma:(map mident datatype)) (pi:(list (ident*
datatype)%type)) (t:term), (type_term sigma pi t TYbool) ->
(type_fmla sigma pi (Fterm t))
| Type_conj : forall (sigma:(map mident datatype)) (pi:(list (ident*
datatype)%type)) (f1:fmla) (f2:fmla), (type_fmla sigma pi f1) ->
((type_fmla sigma pi f2) -> (type_fmla sigma pi (Fand f1 f2)))
| Type_neg : forall (sigma:(map mident datatype)) (pi:(list (ident*
datatype)%type)) (f:fmla), (type_fmla sigma pi f) -> (type_fmla sigma
pi (Fnot f))
| Type_implies : forall (sigma:(map mident datatype)) (pi:(list (ident*
datatype)%type)) (f1:fmla) (f2:fmla), (type_fmla sigma pi f1) ->
((type_fmla sigma pi f2) -> (type_fmla sigma pi (Fimplies f1 f2)))
| Type_let : forall (sigma:(map mident datatype)) (pi:(list (ident*
datatype)%type)) (x:ident) (t:term) (f:fmla) (ty:datatype),
(type_term sigma pi t ty) -> ((type_fmla sigma (Cons (x, ty) pi) f) ->
(type_fmla sigma pi (Flet x t f)))
| Type_forall1 : forall (sigma:(map mident datatype)) (pi:(list (ident*
datatype)%type)) (x:ident) (f:fmla), (type_fmla sigma (Cons (x, TYint)
pi) f) -> (type_fmla sigma pi (Fforall x TYint f))
| Type_forall2 : forall (sigma:(map mident datatype)) (pi:(list (ident*
datatype)%type)) (x:ident) (f:fmla), (type_fmla sigma (Cons (x, TYbool)
pi) f) -> (type_fmla sigma pi (Fforall x TYbool f))
| Type_forall3 : forall (sigma:(map mident datatype)) (pi:(list (ident*
datatype)%type)) (x:ident) (f:fmla), (type_fmla sigma (Cons (x, TYunit)
pi) f) -> (type_fmla sigma pi (Fforall x TYunit f)).
(* Why3 assumption *)
Inductive type_stmt : (map mident datatype) -> (list (ident* datatype)%type)
-> stmt -> Prop :=
| Type_skip : forall (sigma:(map mident datatype)) (pi:(list (ident*
datatype)%type)), (type_stmt sigma pi Sskip)
| Type_seq : forall (sigma:(map mident datatype)) (pi:(list (ident*
datatype)%type)) (s1:stmt) (s2:stmt), (type_stmt sigma pi s1) ->
((type_stmt sigma pi s2) -> (type_stmt sigma pi (Sseq s1 s2)))
| Type_assigns : forall (sigma:(map mident datatype)) (pi:(list (ident*
datatype)%type)) (x:mident) (t:term) (ty:datatype), ((get sigma
x) = ty) -> ((type_term sigma pi t ty) -> (type_stmt sigma pi
(Sassign x t)))
| Type_if : forall (sigma:(map mident datatype)) (pi:(list (ident*
datatype)%type)) (t:term) (s1:stmt) (s2:stmt), (type_term sigma pi t
TYbool) -> ((type_stmt sigma pi s1) -> ((type_stmt sigma pi s2) ->
(type_stmt sigma pi (Sif t s1 s2))))
| Type_assert : forall (sigma:(map mident datatype)) (pi:(list (ident*
datatype)%type)) (p:fmla), (type_fmla sigma pi p) -> (type_stmt sigma
pi (Sassert p))
| Type_while : forall (sigma:(map mident datatype)) (pi:(list (ident*
datatype)%type)) (guard:term) (body:stmt) (inv:fmla), (type_fmla sigma
pi inv) -> ((type_term sigma pi guard TYbool) -> ((type_stmt sigma pi
body) -> (type_stmt sigma pi (Swhile guard inv body)))).
(* Why3 assumption *)
Definition env := (map mident value).
(* Why3 assumption *)
Definition stack := (list (ident* value)%type).
Parameter get_stack: ident -> (list (ident* value)%type) -> value.
Axiom get_stack_def : forall (i:ident) (pi:(list (ident* value)%type)),
match pi with
| Nil => ((get_stack i pi) = Vvoid)
| (Cons (x, v) r) => ((x = i) -> ((get_stack i pi) = v)) /\ ((~ (x = i)) ->
((get_stack i pi) = (get_stack i r)))
end.
Axiom get_stack_eq : forall (x:ident) (v:value) (r:(list (ident*
value)%type)), ((get_stack x (Cons (x, v) r)) = v).
Axiom get_stack_neq : forall (x:ident) (i:ident) (v:value) (r:(list (ident*
value)%type)), (~ (x = i)) -> ((get_stack i (Cons (x, v) r)) = (get_stack i
r)).
Parameter eval_bin: value -> operator -> value -> value.
Axiom eval_bin_def : forall (x:value) (op:operator) (y:value), match (x,
y) with
| ((Vint x1), (Vint y1)) =>
match op with
| Oplus => ((eval_bin x op y) = (Vint (x1 + y1)%Z))
| Ominus => ((eval_bin x op y) = (Vint (x1 - y1)%Z))
| Omult => ((eval_bin x op y) = (Vint (x1 * y1)%Z))
| Ole => ((x1 <= y1)%Z -> ((eval_bin x op y) = (Vbool true))) /\
((~ (x1 <= y1)%Z) -> ((eval_bin x op y) = (Vbool false)))
end
| (_, _) => ((eval_bin x op y) = Vvoid)
end.
(* Why3 assumption *)
Fixpoint eval_term(sigma:(map mident value)) (pi:(list (ident* value)%type))
(t:term) {struct t}: value :=
match t with
| (Tvalue v) => v
| (Tvar id) => (get_stack id pi)
| (Tderef id) => (get sigma id)
| (Tbin t1 op t2) => (eval_bin (eval_term sigma pi t1) op (eval_term sigma
pi t2))
end.
Axiom eval_bool_term : forall (sigma:(map mident value)) (pi:(list (ident*
value)%type)) (sigmat:(map mident datatype)) (pit:(list (ident*
datatype)%type)) (t:term), (type_term sigmat pit t TYbool) ->
exists b:bool, ((eval_term sigma pi t) = (Vbool b)).
(* Why3 assumption *)
Fixpoint eval_fmla(sigma:(map mident value)) (pi:(list (ident* value)%type))
(f:fmla) {struct f}: Prop :=
match f with
| (Fterm t) => ((eval_term sigma pi t) = (Vbool true))
| (Fand f1 f2) => (eval_fmla sigma pi f1) /\ (eval_fmla sigma pi f2)
| (Fnot f1) => ~ (eval_fmla sigma pi f1)
| (Fimplies f1 f2) => (eval_fmla sigma pi f1) -> (eval_fmla sigma pi f2)
| (Flet x t f1) => (eval_fmla sigma (Cons (x, (eval_term sigma pi t)) pi)
f1)
| (Fforall x TYint f1) => forall (n:Z), (eval_fmla sigma (Cons (x,
(Vint n)) pi) f1)
| (Fforall x TYbool f1) => forall (b:bool), (eval_fmla sigma (Cons (x,
(Vbool b)) pi) f1)
| (Fforall x TYunit f1) => (eval_fmla sigma (Cons (x, Vvoid) pi) f1)
end.
Parameter msubst_term: term -> mident -> ident -> term.
Axiom msubst_term_def : forall (t:term) (r:mident) (v:ident),
match t with
| ((Tvalue _)|(Tvar _)) => ((msubst_term t r v) = t)
| (Tderef x) => ((r = x) -> ((msubst_term t r v) = (Tvar v))) /\
((~ (r = x)) -> ((msubst_term t r v) = t))
| (Tbin t1 op t2) => ((msubst_term t r v) = (Tbin (msubst_term t1 r v) op
(msubst_term t2 r v)))
end.
Parameter subst_term: term -> ident -> ident -> term.
Axiom subst_term_def : forall (t:term) (r:ident) (v:ident),
match t with
| ((Tvalue _)|(Tderef _)) => ((subst_term t r v) = t)
| (Tvar x) => ((r = x) -> ((subst_term t r v) = (Tvar v))) /\
((~ (r = x)) -> ((subst_term t r v) = t))
| (Tbin t1 op t2) => ((subst_term t r v) = (Tbin (subst_term t1 r v) op
(subst_term t2 r v)))
end.
Axiom fresh_in_binop : forall (t:term) (t':term) (op:operator) (v:ident),
(var_occurs_in_term v (Tbin t op t')) -> ((var_occurs_in_term v t) /\
(var_occurs_in_term v t')).
Axiom eval_msubst_term : forall (e:term) (sigma:(map mident value)) (pi:(list
(ident* value)%type)) (x:mident) (v:ident), (var_occurs_in_term v e) ->
((eval_term sigma pi (msubst_term e x v)) = (eval_term (set sigma x
(get_stack v pi)) pi e)).
(* Why3 goal *)
Theorem eval_term_change_free : forall (t:term),
match t with
| (Tvalue v) => True
| (Tvar i) => forall (sigma:(map mident value)) (pi:(list (ident*
value)%type)) (id:ident) (v:value), (var_occurs_in_term id t) ->
((eval_term sigma (Cons (id, v) pi) t) = (eval_term sigma pi t))
| (Tderef m) => True
| (Tbin t1 o t2) => True
end.
destruct t; auto.
intros.
simpl.
Qed.
(* This file is generated by Why3's Coq driver *)
(* Beware! Only edit allowed sections below *)
Require Import BuiltIn.
Require BuiltIn.
Require int.Int.
Require int.MinMax.
(* Why3 assumption *)
Inductive list (a:Type) {a_WT:WhyType a} :=
| Nil : list a
| Cons : a -> (list a) -> list a.
Axiom list_WhyType : forall (a:Type) {a_WT:WhyType a}, WhyType (list a).
Existing Instance list_WhyType.
Implicit Arguments Nil [[a] [a_WT]].
Implicit Arguments Cons [[a] [a_WT]].
Axiom map : forall (a:Type) {a_WT:WhyType a} (b:Type) {b_WT:WhyType b}, Type.
Parameter map_WhyType : forall (a:Type) {a_WT:WhyType a}
(b:Type) {b_WT:WhyType b}, WhyType (map a b).
Existing Instance map_WhyType.
Parameter get: forall {a:Type} {a_WT:WhyType a} {b:Type} {b_WT:WhyType b},
(map a b) -> a -> b.
Parameter set: forall {a:Type} {a_WT:WhyType a} {b:Type} {b_WT:WhyType b},
(map a b) -> a -> b -> (map a b).
Axiom Select_eq : forall {a:Type} {a_WT:WhyType a} {b:Type} {b_WT:WhyType b},
forall (m:(map a b)), forall (a1:a) (a2:a), forall (b1:b), (a1 = a2) ->
((get (set m a1 b1) a2) = b1).
Axiom Select_neq : forall {a:Type} {a_WT:WhyType a}
{b:Type} {b_WT:WhyType b}, forall (m:(map a b)), forall (a1:a) (a2:a),
forall (b1:b), (~ (a1 = a2)) -> ((get (set m a1 b1) a2) = (get m a2)).