Fixpoint ptree_cmp {T : Type} (p1 p2: Maps.PTree.tree T)
(e : T -> T -> bool) :=
match p1, p2 with
| Maps.PTree.Leaf, Maps.PTree.Leaf => true
| Maps.PTree.Node l1 i1 r1, Maps.PTree.Node l2 i2 r2 =>
match i1, i2 with
| Some v1, Some v2 => andb (e v1 v2)
(andb (ptree_cmp l1 l2 e)
(ptree_cmp r1 r2 e))
| None, None => (andb (ptree_cmp l1 l2 e)
(ptree_cmp r1 r2 e))
| _, _ => false
end
| _, _ => false
end.
Definition const_beq a b :=
match a, b with
| fN c1, fN c2 => beq_nat c1 c2
| fZ c1, fZ c2 => Zbool.Zeq_bool c1 c2
| fPos c1, fPos c2 =>  BinPos.Pos.eqb c1 c2
| fCtype c1, fCtype c2 => expr.eqb_type c1 c2
| fCexpr c1, fCexpr c2 => expr_beq c1 c2
| fenv c1, fenv c2 => ptree_cmp c1 c2 expr.eqb_type
| fllrr c1, fllrr c2 => match c1, c2 with | efield_lemmas.LLLL, efield_lemmas.LLLL | efield_lemmas.RRRR, efield_lemmas.RRRR => true | _, _ => false end
| _, _ => false
end.
Instance const_rel_dec : RelDec (@eq const) := { rel_dec := const_beq }.
Definition beq_z_true : forall a b, Zbool.Zeq_bool a b = true -> a = b.
Hint Resolve beq_nat_true : expr_beq.
Hint Resolve beq_z_true : expr_beq.
Hint Resolve const_beq_sound : expr_beq.
Definition z_op_beq a b : bool :=
match a, b with
| fZ_lt, fZ_lt
| fZ_le, fZ_le
| fZ_gt, fZ_gt
| fZ_ge, fZ_ge
| fZ_add, fZ_add
| fZ_sub, fZ_sub
| fZ_mul, fZ_mul
| fZ_div, fZ_div
| fZ_mod, fZ_mod
| fZ_land, fZ_land
| fZ_max, fZ_max
| fZ_opp, fZ_opp  => true
| _, _ => false
end.
Hint Resolve beq_nat_true : expr_beq.
Hint Resolve z_op_beq_sound : expr_beq.
Definition int_op_beq a b : bool :=
match a, b with
| fint_add, fint_add
| fint_and, fint_and
| fint_lt, fint_lt
| fint_ltu, fint_ltu
| fint_mul, fint_mul
| fint_neg, fint_neg
| fint_sub, fint_sub
| fint_cmp, fint_cmp
| fint_cmpu, fint_cmpu
| fint_repr, fint_repr
| fint_signed, fint_signed
| fint_unsigned, fint_unsigned
| fint_max_unsigned, fint_max_unsigned
| fint64_repr, fint64_repr => true
| _, _ => false
end.
Hint Resolve int_op_beq_sound : expr_beq.
Definition values_beq a b:=
match a, b with
| fVint, fVint
| fVfloat, fVfloat
| fVlong, fVlong
| fVptr, fVptr
| fVundef, fVundef => true
| _, _ => false
end.
Hint Resolve values_beq_sound : expr_beq.
Definition eval_beq a b :=
match a,b with
| feval_cast t1_1 t2_1 , feval_cast t1_2 t2_2 => andb (expr.eqb_type t1_1 t1_2)
(expr.eqb_type t2_1 t2_2)
| fderef_noload t1, fderef_noload t2 => expr.eqb_type t1 t2
| feval_field t1 id1 , feval_field t2 id2 =>  andb (expr.eqb_type t1 t2)
(BinPos.Pos.eqb id1 id2)
| feval_binop op1 t1_1 t2_1, feval_binop op2 t1_2 t2_2 =>
andb (andb (binary_op_beq op1 op2)
(expr.eqb_type t1_1 t1_2))
(expr.eqb_type t2_1 t2_2)
| feval_unop op1 t1 , feval_unop op2 t2 =>
andb (unary_op_beq op1 op2)
(expr.eqb_type t1 t2)
| feval_id id1, feval_id id2 => BinPos.Pos.eqb id1 id2
| _, _ => false
end.
Hint Resolve eval_beq_sound : expr_beq.
Definition other_beq a b :=
match a, b with
| ftwo_power_nat, ftwo_power_nat
| fforce_ptr, fforce_ptr
| fand, fand
| falign, falign
| ftyped_true, ftyped_true => true
| fnone t1, fnone t2
| fsome t1, fsome t2
| feq t1, feq t2 => typ_beq t1 t2
| _, _ => false
end
.
Hint Resolve typ_beq_sound : expr_beq.
Hint Resolve other_beq_sound : expr_beq.
Definition data_beq a b :=
match a, b with
| fnil t1, fnil t2
| fcons t1, fcons t2
| fappend t1, fappend t2 => typ_beq t1 t2
| fnth_error t1 n1, fnth_error t2 n2
| freplace_nth t1 n1, freplace_nth t2 n2 => andb (beq_nat n1 n2) (typ_beq t1 t2)
| fmap ta1 tb1, fmap ta2 tb2
| ffold_right ta1 tb1, ffold_right ta2 tb2
| ffold_left ta1 tb1, ffold_left ta2 tb2 => andb (typ_beq ta1 ta2) (typ_beq tb1 tb2)
| fleaf t, fleaf t'
| fnode t, fnode t'
| fempty t, fempty t' => typ_beq t t'
| _ , _ => false
end.
Hint Resolve data_beq_sound : expr_beq.
Definition sep_beq a b :=
match a, b with
| flocal, flocal
| fprop, fprop => true
| fdata_at ty1, fdata_at ty2 => expr.eqb_type ty1 ty2
| fproj_val ty1, fproj_val ty2 => expr.eqb_type ty1 ty2
| fupd_val ty1, fupd_val ty2 => expr.eqb_type ty1 ty2
(*| ffield_at ty1 li1, ffield_at ty2 li2 => andb (expr.eqb_type ty1 ty2) (li1 ?[eq] li2) | flseg t1 i1 _, flseg t2 i2 _ => andb (expr.eqb_type t1 t2) (BinPos.Pos.eqb i1 i2)*)
| _, _ => false
end
.

Lemma sep_beq_sound : forall a b, sep_beq a b = true -> a = b.