why3doc index index
module Types use option.Option use int.Int use Nat.Nat use Functions.Func use OptionFuncs.Funcs use Sum.Sum use Firstorder_symbol_spec.Spec use Firstorder_symbol_impl.Types use Firstorder_symbol_impl.Logic use Firstorder_symbol_impl.Impl use Firstorder_term_spec.Spec use Firstorder_term_impl.Types use Firstorder_term_impl.Logic use Firstorder_term_impl.Impl use Firstorder_formula_spec.Spec use Firstorder_formula_impl.Types use Firstorder_formula_impl.Logic use Firstorder_formula_impl.Impl use Firstorder_formula_list_spec.Spec type nl_fo_formula_list 'b0 'b1 = | NL_FOFNil | NL_FOFCons (nl_fo_formula 'b0 'b1) (nl_fo_formula_list 'b0 'b1) type nlimpl_fo_formula_list = { nlrepr_fo_formula_list_field : nl_fo_formula_list int int ; nlfree_var_symbol_set_abstraction_fo_formula_list_field : int ; nlfree_var_fo_term_set_abstraction_fo_formula_list_field : int ; ghost model_fo_formula_list_field : fo_formula_list int int ; } type cons_fo_formula_list = | NLC_FOFNil | NLC_FOFCons (nlimpl_fo_formula) (nlimpl_fo_formula_list) end module Logic use option.Option use int.Int use Nat.Nat use Functions.Func use OptionFuncs.Funcs use Sum.Sum use Firstorder_symbol_spec.Spec use Firstorder_symbol_impl.Types use Firstorder_symbol_impl.Logic use Firstorder_symbol_impl.Impl use Firstorder_term_spec.Spec use Firstorder_term_impl.Types use Firstorder_term_impl.Logic use Firstorder_term_impl.Impl use Firstorder_formula_spec.Spec use Firstorder_formula_impl.Types use Firstorder_formula_impl.Logic use Firstorder_formula_impl.Impl use Firstorder_formula_list_spec.Spec use Types function nat_nlsize_fo_formula_list (t:nl_fo_formula_list 'b0 'b1) : nat = match t with | NL_FOFNil -> let s = one_nat in s | NL_FOFCons v0 v1 -> let s = one_nat in let s = add_nat (nat_nlsize_fo_formula_list v1) s in let s = add_nat (nat_nlsize_fo_formula v0) s in s end with nlsize_fo_formula_list (t:nl_fo_formula_list 'b0 'b1) : int = match t with | NL_FOFNil -> let s = 1 in s | NL_FOFCons v0 v1 -> let s = 1 in let s = nlsize_fo_formula_list v1 + s in let s = nlsize_fo_formula v0 + s in s end let rec lemma nlsize_positive_lemma_fo_formula_list (t:nl_fo_formula_list 'b0 'b1) : unit ensures { nlsize_fo_formula_list t > 0 } variant { nat_to_int (nat_nlsize_fo_formula_list t) } = match t with | NL_FOFNil -> () | NL_FOFCons v0 v1 -> nlsize_positive_lemma_fo_formula v0 ; nlsize_positive_lemma_fo_formula_list v1 ; () end function nlmodel_fo_formula_list (t:nl_fo_formula_list 'b0 'b1) (fr0:'b0 -> (symbol 'c0)) (bnd0: int -> (symbol 'c0)) (fr1:'b1 -> (fo_term 'c0 'c1)) (bnd1: int -> (fo_term 'c0 'c1)) : fo_formula_list 'c0 'c1 = match t with | NL_FOFNil -> FOFNil | NL_FOFCons v0 v1 -> FOFCons (nlmodel_fo_formula v0 ((rename_subst_symbol fr0 identity)) ((rename_subst_symbol bnd0 identity)) ((rename_subst_fo_term fr1 identity identity)) ((rename_subst_fo_term bnd1 identity identity))) (nlmodel_fo_formula_list v1 ((rename_subst_symbol fr0 identity)) ((rename_subst_symbol bnd0 identity)) ((rename_subst_fo_term fr1 identity identity)) ((rename_subst_fo_term bnd1 identity identity))) end let rec lemma nlmodel_subst_commutation_lemma_fo_formula_list (t:nl_fo_formula_list 'b0 'b1) (fr0:'b0 -> (symbol 'c0)) (bnd0: int -> (symbol 'c0)) (s0:'c0 -> (symbol 'd0)) (fr1:'b1 -> (fo_term 'c0 'c1)) (bnd1: int -> (fo_term 'c0 'c1)) (s1:'c1 -> (fo_term 'd0 'd1)) : unit ensures { nlmodel_fo_formula_list t (subst_compose_symbol fr0 s0) (subst_compose_symbol bnd0 s0) (subst_compose_fo_term fr1 s0 s1) (subst_compose_fo_term bnd1 s0 s1) = subst_fo_formula_list (nlmodel_fo_formula_list t fr0 bnd0 fr1 bnd1) s0 s1 } variant { nlsize_fo_formula_list t } = match t with | NL_FOFNil -> () | NL_FOFCons v0 v1 -> nlmodel_subst_commutation_lemma_fo_formula v0 ((rename_subst_symbol fr0 identity)) ((rename_subst_symbol bnd0 identity)) ((rename_subst_symbol s0 identity)) ((rename_subst_fo_term fr1 identity identity)) ((rename_subst_fo_term bnd1 identity identity)) ((rename_subst_fo_term s1 identity identity)) ; assert { subst_compose_symbol (rename_subst_symbol fr0 identity) ((rename_subst_symbol s0 identity)) = (rename_subst_symbol (subst_compose_symbol fr0 s0) identity) } ; assert { subst_compose_symbol (rename_subst_symbol bnd0 identity) ((rename_subst_symbol s0 identity)) = (rename_subst_symbol (subst_compose_symbol bnd0 s0) identity) } ; assert { subst_compose_fo_term (rename_subst_fo_term fr1 identity identity) ((rename_subst_symbol s0 identity)) ((rename_subst_fo_term s1 identity identity)) = (rename_subst_fo_term (subst_compose_fo_term fr1 s0 s1) identity identity) } ; assert { subst_compose_fo_term (rename_subst_fo_term bnd1 identity identity) ((rename_subst_symbol s0 identity)) ((rename_subst_fo_term s1 identity identity)) = (rename_subst_fo_term (subst_compose_fo_term bnd1 s0 s1) identity identity) } ; nlmodel_subst_commutation_lemma_fo_formula_list v1 ((rename_subst_symbol fr0 identity)) ((rename_subst_symbol bnd0 identity)) ((rename_subst_symbol s0 identity)) ((rename_subst_fo_term fr1 identity identity)) ((rename_subst_fo_term bnd1 identity identity)) ((rename_subst_fo_term s1 identity identity)) ; assert { subst_compose_symbol (rename_subst_symbol fr0 identity) ((rename_subst_symbol s0 identity)) = (rename_subst_symbol (subst_compose_symbol fr0 s0) identity) } ; assert { subst_compose_symbol (rename_subst_symbol bnd0 identity) ((rename_subst_symbol s0 identity)) = (rename_subst_symbol (subst_compose_symbol bnd0 s0) identity) } ; assert { subst_compose_fo_term (rename_subst_fo_term fr1 identity identity) ((rename_subst_symbol s0 identity)) ((rename_subst_fo_term s1 identity identity)) = (rename_subst_fo_term (subst_compose_fo_term fr1 s0 s1) identity identity) } ; assert { subst_compose_fo_term (rename_subst_fo_term bnd1 identity identity) ((rename_subst_symbol s0 identity)) ((rename_subst_fo_term s1 identity identity)) = (rename_subst_fo_term (subst_compose_fo_term bnd1 s0 s1) identity identity) } ; () end let lemma nlmodel_rename_commutation_lemma_fo_formula_list (t:nl_fo_formula_list 'b0 'b1) (fr0:'b0 -> (symbol 'c0)) (bnd0: int -> (symbol 'c0)) (s0:'c0 -> 'd0) (fr1:'b1 -> (fo_term 'c0 'c1)) (bnd1: int -> (fo_term 'c0 'c1)) (s1:'c1 -> 'd1) : unit ensures { nlmodel_fo_formula_list t (rename_subst_symbol fr0 s0) (rename_subst_symbol bnd0 s0) (rename_subst_fo_term fr1 s0 s1) (rename_subst_fo_term bnd1 s0 s1) = rename_fo_formula_list (nlmodel_fo_formula_list t fr0 bnd0 fr1 bnd1) s0 s1 } = nlmodel_subst_commutation_lemma_fo_formula_list t fr0 bnd0 (subst_of_rename_symbol s0) fr1 bnd1 (subst_of_rename_fo_term s1) predicate correct_indexes_fo_formula_list (t:nl_fo_formula_list 'b0 'b1) = match t with | NL_FOFNil -> true | NL_FOFCons v0 v1 -> correct_indexes_fo_formula v0 /\ correct_indexes_fo_formula_list v1 end function bound_depth_of_symbol_in_fo_formula_list (t:nl_fo_formula_list 'b0 'b1) : int = match t with | NL_FOFNil -> 0 | NL_FOFCons v0 v1 -> let b = bound_depth_of_symbol_in_fo_formula v0 in let a = b in let b = bound_depth_of_symbol_in_fo_formula_list v1 in let a = if a > b then a else b in a end with bound_depth_of_fo_term_in_fo_formula_list (t:nl_fo_formula_list 'b0 'b1) : int = match t with | NL_FOFNil -> 0 | NL_FOFCons v0 v1 -> let b = bound_depth_of_fo_term_in_fo_formula v0 in let a = b in let b = bound_depth_of_fo_term_in_fo_formula_list v1 in let a = if a > b then a else b in a end let rec lemma bound_depth_of_symbol_in_fo_formula_list_nonnegative (t:nl_fo_formula_list 'b0 'b1) : unit requires { correct_indexes_fo_formula_list t } ensures { bound_depth_of_symbol_in_fo_formula_list t >= 0 } variant { nlsize_fo_formula_list t } = match t with | NL_FOFNil -> () | NL_FOFCons v0 v1 -> bound_depth_of_symbol_in_fo_formula_nonnegative v0 ; bound_depth_of_symbol_in_fo_formula_list_nonnegative v1 ; () end with lemma bound_depth_of_fo_term_in_fo_formula_list_nonnegative (t:nl_fo_formula_list 'b0 'b1) : unit requires { correct_indexes_fo_formula_list t } ensures { bound_depth_of_fo_term_in_fo_formula_list t >= 0 } variant { nlsize_fo_formula_list t } = match t with | NL_FOFNil -> () | NL_FOFCons v0 v1 -> bound_depth_of_fo_term_in_fo_formula_nonnegative v0 ; bound_depth_of_fo_term_in_fo_formula_list_nonnegative v1 ; () end let rec lemma model_equal_fo_formula_list (t:nl_fo_formula_list 'b0 'b1) (fr10: 'b0 -> (symbol 'c0)) (fr20: 'b0 -> (symbol 'c0)) (bnd10: int -> (symbol 'c0)) (bnd20: int -> (symbol 'c0)) (fr11: 'b1 -> (fo_term 'c0 'c1)) (fr21: 'b1 -> (fo_term 'c0 'c1)) (bnd11: int -> (fo_term 'c0 'c1)) (bnd21: int -> (fo_term 'c0 'c1)) : unit requires { forall i:int. 0 <= i < bound_depth_of_symbol_in_fo_formula_list t -> bnd10 i = bnd20 i } requires { fr10 = fr20 } requires { forall i:int. 0 <= i < bound_depth_of_fo_term_in_fo_formula_list t -> bnd11 i = bnd21 i } requires { fr11 = fr21 } requires { correct_indexes_fo_formula_list t } ensures { nlmodel_fo_formula_list t fr10 bnd10 fr11 bnd11 = nlmodel_fo_formula_list t fr20 bnd20 fr21 bnd21 } variant { nlsize_fo_formula_list t } = match t with | NL_FOFNil -> () | NL_FOFCons v0 v1 -> model_equal_fo_formula v0 ((rename_subst_symbol fr10 identity)) ((rename_subst_symbol fr20 identity)) ((rename_subst_symbol bnd10 identity)) ((rename_subst_symbol bnd20 identity)) ((rename_subst_fo_term fr11 identity identity)) ((rename_subst_fo_term fr21 identity identity)) ((rename_subst_fo_term bnd11 identity identity)) ((rename_subst_fo_term bnd21 identity identity)) ; model_equal_fo_formula_list v1 ((rename_subst_symbol fr10 identity)) ((rename_subst_symbol fr20 identity)) ((rename_subst_symbol bnd10 identity)) ((rename_subst_symbol bnd20 identity)) ((rename_subst_fo_term fr11 identity identity)) ((rename_subst_fo_term fr21 identity identity)) ((rename_subst_fo_term bnd11 identity identity)) ((rename_subst_fo_term bnd21 identity identity)) ; () end predicate nlimpl_fo_formula_list_ok (t:nlimpl_fo_formula_list) = nlmodel_fo_formula_list t.nlrepr_fo_formula_list_field subst_id_symbol (const (Var_symbol ((-1)))) subst_id_fo_term (const (Var_fo_term ((-1)))) = t.model_fo_formula_list_field /\ correct_indexes_fo_formula_list t.nlrepr_fo_formula_list_field /\ bound_depth_of_symbol_in_fo_formula_list t.nlrepr_fo_formula_list_field = 0 /\ bound_depth_of_fo_term_in_fo_formula_list t.nlrepr_fo_formula_list_field = 0 /\ (forall x:int. is_symbol_free_var_in_fo_formula_list x t.model_fo_formula_list_field -> (x) < (t.nlfree_var_symbol_set_abstraction_fo_formula_list_field)) /\ (forall x:int. is_fo_term_free_var_in_fo_formula_list x t.model_fo_formula_list_field -> (x) < (t.nlfree_var_fo_term_set_abstraction_fo_formula_list_field)) predicate cons_ok_fo_formula_list (c:cons_fo_formula_list) = match c with | NLC_FOFNil -> true | NLC_FOFCons v0 v1 -> nlimpl_fo_formula_ok v0 /\ nlimpl_fo_formula_list_ok v1 end predicate cons_rel_fo_formula_list (c:cons_fo_formula_list) (t:nlimpl_fo_formula_list) = match c with | NLC_FOFNil -> t.model_fo_formula_list_field = FOFNil | NLC_FOFCons v0 v1 -> t.model_fo_formula_list_field = FOFCons (rename_fo_formula v0.model_fo_formula_field identity identity) (rename_fo_formula_list v1.model_fo_formula_list_field identity identity) end predicate cons_open_rel_fo_formula_list (c:cons_fo_formula_list) (t:nlimpl_fo_formula_list) = match c with | NLC_FOFNil -> match t.model_fo_formula_list_field with | FOFNil -> true | FOFCons w0 w1 -> false end | NLC_FOFCons v0 v1 -> match t.model_fo_formula_list_field with | FOFNil -> false | FOFCons w0 w1 -> v0.model_fo_formula_field = (rename_fo_formula w0 identity identity) /\ v1.model_fo_formula_list_field = (rename_fo_formula_list w1 identity identity) end end end module Impl use option.Option use int.Int use Nat.Nat use Functions.Func use OptionFuncs.Funcs use Sum.Sum use Firstorder_symbol_spec.Spec use Firstorder_symbol_impl.Types use Firstorder_symbol_impl.Logic use Firstorder_symbol_impl.Impl use Firstorder_term_spec.Spec use Firstorder_term_impl.Types use Firstorder_term_impl.Logic use Firstorder_term_impl.Impl use Firstorder_formula_spec.Spec use Firstorder_formula_impl.Types use Firstorder_formula_impl.Logic use Firstorder_formula_impl.Impl use Firstorder_formula_list_spec.Spec use Types use Logic let rec bind_var_symbol_in_fo_formula_list (t:nl_fo_formula_list int int) (x:int) (i:int) (ghost fr0: int -> (symbol 'b0)) (ghost bnd0: int -> (symbol 'b0)) (ghost fr1: int -> (fo_term 'b0 'b1)) (ghost bnd1: int -> (fo_term 'b0 'b1)) : nl_fo_formula_list int int requires { correct_indexes_fo_formula_list t } requires { bound_depth_of_symbol_in_fo_formula_list t <= i } variant { nlsize_fo_formula_list t } ensures { bound_depth_of_symbol_in_fo_formula_list result <= i + 1 } ensures { correct_indexes_fo_formula_list result } ensures { bound_depth_of_fo_term_in_fo_formula_list t = bound_depth_of_fo_term_in_fo_formula_list result } ensures { nlmodel_fo_formula_list result fr0 bnd0 fr1 bnd1 = nlmodel_fo_formula_list t (update fr0 x (bnd0 i)) bnd0 fr1 bnd1 } = match t with | NL_FOFNil -> NL_FOFNil | NL_FOFCons v0 v1 -> assert { (rename_symbol (bnd0 i) identity) = (eval ((rename_subst_symbol bnd0 identity)) (i+0)) }; assert { extensionalEqual ((rename_subst_symbol (update fr0 x (bnd0 i)) identity)) ((update ((rename_subst_symbol fr0 identity)) x (rename_symbol (bnd0 i) identity))) }; assert { (rename_subst_symbol (update fr0 x (bnd0 i)) identity) = (update ((rename_subst_symbol fr0 identity)) x (eval ((rename_subst_symbol bnd0 identity)) (i+0))) }; assert { (rename_symbol (bnd0 i) identity) = (eval ((rename_subst_symbol bnd0 identity)) (i+0)) }; assert { extensionalEqual ((rename_subst_symbol (update fr0 x (bnd0 i)) identity)) ((update ((rename_subst_symbol fr0 identity)) x (rename_symbol (bnd0 i) identity))) }; assert { (rename_subst_symbol (update fr0 x (bnd0 i)) identity) = (update ((rename_subst_symbol fr0 identity)) x (eval ((rename_subst_symbol bnd0 identity)) (i+0))) }; NL_FOFCons (bind_var_symbol_in_fo_formula v0 x (i+0) ((rename_subst_symbol fr0 identity)) ((rename_subst_symbol bnd0 identity)) ((rename_subst_fo_term fr1 identity identity)) ((rename_subst_fo_term bnd1 identity identity))) (bind_var_symbol_in_fo_formula_list v1 x (i+0) ((rename_subst_symbol fr0 identity)) ((rename_subst_symbol bnd0 identity)) ((rename_subst_fo_term fr1 identity identity)) ((rename_subst_fo_term bnd1 identity identity))) end with bind_var_fo_term_in_fo_formula_list (t:nl_fo_formula_list int int) (x:int) (i:int) (ghost fr0: int -> (symbol 'b0)) (ghost bnd0: int -> (symbol 'b0)) (ghost fr1: int -> (fo_term 'b0 'b1)) (ghost bnd1: int -> (fo_term 'b0 'b1)) : nl_fo_formula_list int int requires { correct_indexes_fo_formula_list t } requires { bound_depth_of_fo_term_in_fo_formula_list t <= i } variant { nlsize_fo_formula_list t } ensures { bound_depth_of_fo_term_in_fo_formula_list result <= i + 1 } ensures { correct_indexes_fo_formula_list result } ensures { bound_depth_of_symbol_in_fo_formula_list t = bound_depth_of_symbol_in_fo_formula_list result } ensures { nlmodel_fo_formula_list result fr0 bnd0 fr1 bnd1 = nlmodel_fo_formula_list t fr0 bnd0 (update fr1 x (bnd1 i)) bnd1 } = match t with | NL_FOFNil -> NL_FOFNil | NL_FOFCons v0 v1 -> assert { (rename_fo_term (bnd1 i) identity identity) = (eval ((rename_subst_fo_term bnd1 identity identity)) (i+0)) }; assert { extensionalEqual ((rename_subst_fo_term (update fr1 x (bnd1 i)) identity identity)) ((update ((rename_subst_fo_term fr1 identity identity)) x (rename_fo_term (bnd1 i) identity identity))) }; assert { (rename_subst_fo_term (update fr1 x (bnd1 i)) identity identity) = (update ((rename_subst_fo_term fr1 identity identity)) x (eval ((rename_subst_fo_term bnd1 identity identity)) (i+0))) }; assert { (rename_fo_term (bnd1 i) identity identity) = (eval ((rename_subst_fo_term bnd1 identity identity)) (i+0)) }; assert { extensionalEqual ((rename_subst_fo_term (update fr1 x (bnd1 i)) identity identity)) ((update ((rename_subst_fo_term fr1 identity identity)) x (rename_fo_term (bnd1 i) identity identity))) }; assert { (rename_subst_fo_term (update fr1 x (bnd1 i)) identity identity) = (update ((rename_subst_fo_term fr1 identity identity)) x (eval ((rename_subst_fo_term bnd1 identity identity)) (i+0))) }; NL_FOFCons (bind_var_fo_term_in_fo_formula v0 x (i+0) ((rename_subst_symbol fr0 identity)) ((rename_subst_symbol bnd0 identity)) ((rename_subst_fo_term fr1 identity identity)) ((rename_subst_fo_term bnd1 identity identity))) (bind_var_fo_term_in_fo_formula_list v1 x (i+0) ((rename_subst_symbol fr0 identity)) ((rename_subst_symbol bnd0 identity)) ((rename_subst_fo_term fr1 identity identity)) ((rename_subst_fo_term bnd1 identity identity))) end let rec unbind_var_symbol_in_fo_formula_list (t:nl_fo_formula_list int int) (i:int) (x:nl_symbol int) (ghost fr0: int -> (symbol 'b0)) (ghost bnd10: int -> (symbol 'b0)) (ghost fr1: int -> (fo_term 'b0 'b1)) (ghost bnd11: int -> (fo_term 'b0 'b1)) (ghost bnd20: int -> (symbol 'b0)) : nl_fo_formula_list int int requires { i >= 0 } requires { correct_indexes_fo_formula_list t } requires { bound_depth_of_symbol_in_fo_formula_list t <= i + 1 } requires { correct_indexes_symbol x } requires { bound_depth_of_symbol_in_symbol x = 0 } variant { nlsize_fo_formula_list t } ensures { correct_indexes_fo_formula_list result } ensures { bound_depth_of_symbol_in_fo_formula_list result <= i } ensures { bound_depth_of_fo_term_in_fo_formula_list result = bound_depth_of_fo_term_in_fo_formula_list t } ensures { nlmodel_fo_formula_list result fr0 bnd10 fr1 bnd11 = nlmodel_fo_formula_list t fr0 (update bnd10 i (nlmodel_symbol x fr0 bnd20)) fr1 bnd11 } = match t with | NL_FOFNil -> NL_FOFNil | NL_FOFCons v0 v1 -> assert { rename_symbol (nlmodel_symbol x fr0 bnd20) identity = nlmodel_symbol x ((rename_subst_symbol fr0 identity)) ((rename_subst_symbol bnd20 identity)) } ; assert { extensionalEqual ((rename_subst_symbol (update bnd10 i (nlmodel_symbol x fr0 bnd20)) identity)) (update ((rename_subst_symbol bnd10 identity)) (i+0) (rename_symbol (nlmodel_symbol x fr0 bnd20) identity)) } ; assert { (rename_subst_symbol (update bnd10 i (nlmodel_symbol x fr0 bnd20)) identity) = update ((rename_subst_symbol bnd10 identity)) (i+0) (nlmodel_symbol x ((rename_subst_symbol fr0 identity)) ((rename_subst_symbol bnd20 identity))) } ; assert { rename_symbol (nlmodel_symbol x fr0 bnd20) identity = nlmodel_symbol x ((rename_subst_symbol fr0 identity)) ((rename_subst_symbol bnd20 identity)) } ; assert { extensionalEqual ((rename_subst_symbol (update bnd10 i (nlmodel_symbol x fr0 bnd20)) identity)) (update ((rename_subst_symbol bnd10 identity)) (i+0) (rename_symbol (nlmodel_symbol x fr0 bnd20) identity)) } ; assert { (rename_subst_symbol (update bnd10 i (nlmodel_symbol x fr0 bnd20)) identity) = update ((rename_subst_symbol bnd10 identity)) (i+0) (nlmodel_symbol x ((rename_subst_symbol fr0 identity)) ((rename_subst_symbol bnd20 identity))) } ; NL_FOFCons (unbind_var_symbol_in_fo_formula v0 (i+0) x ((rename_subst_symbol fr0 identity)) ((rename_subst_symbol bnd10 identity)) ((rename_subst_fo_term fr1 identity identity)) ((rename_subst_fo_term bnd11 identity identity)) ((rename_subst_symbol bnd20 identity))) (unbind_var_symbol_in_fo_formula_list v1 (i+0) x ((rename_subst_symbol fr0 identity)) ((rename_subst_symbol bnd10 identity)) ((rename_subst_fo_term fr1 identity identity)) ((rename_subst_fo_term bnd11 identity identity)) ((rename_subst_symbol bnd20 identity))) end with unbind_var_fo_term_in_fo_formula_list (t:nl_fo_formula_list int int) (i:int) (x:nl_fo_term int int) (ghost fr0: int -> (symbol 'b0)) (ghost bnd10: int -> (symbol 'b0)) (ghost fr1: int -> (fo_term 'b0 'b1)) (ghost bnd11: int -> (fo_term 'b0 'b1)) (ghost bnd20: int -> (symbol 'b0)) (ghost bnd21: int -> (fo_term 'b0 'b1)) : nl_fo_formula_list int int requires { i >= 0 } requires { correct_indexes_fo_formula_list t } requires { bound_depth_of_fo_term_in_fo_formula_list t <= i + 1 } requires { correct_indexes_fo_term x } requires { bound_depth_of_symbol_in_fo_term x = 0 } requires { bound_depth_of_fo_term_in_fo_term x = 0 } variant { nlsize_fo_formula_list t } ensures { correct_indexes_fo_formula_list result } ensures { bound_depth_of_fo_term_in_fo_formula_list result <= i } ensures { bound_depth_of_symbol_in_fo_formula_list result = bound_depth_of_symbol_in_fo_formula_list t } ensures { nlmodel_fo_formula_list result fr0 bnd10 fr1 bnd11 = nlmodel_fo_formula_list t fr0 bnd10 fr1 (update bnd11 i (nlmodel_fo_term x fr0 bnd20 fr1 bnd21)) } = match t with | NL_FOFNil -> NL_FOFNil | NL_FOFCons v0 v1 -> assert { rename_fo_term (nlmodel_fo_term x fr0 bnd20 fr1 bnd21) identity identity = nlmodel_fo_term x ((rename_subst_symbol fr0 identity)) ((rename_subst_symbol bnd20 identity)) ((rename_subst_fo_term fr1 identity identity)) ((rename_subst_fo_term bnd21 identity identity)) } ; assert { extensionalEqual ((rename_subst_fo_term (update bnd11 i (nlmodel_fo_term x fr0 bnd20 fr1 bnd21)) identity identity)) (update ((rename_subst_fo_term bnd11 identity identity)) (i+0) (rename_fo_term (nlmodel_fo_term x fr0 bnd20 fr1 bnd21) identity identity)) } ; assert { (rename_subst_fo_term (update bnd11 i (nlmodel_fo_term x fr0 bnd20 fr1 bnd21)) identity identity) = update ((rename_subst_fo_term bnd11 identity identity)) (i+0) (nlmodel_fo_term x ((rename_subst_symbol fr0 identity)) ((rename_subst_symbol bnd20 identity)) ((rename_subst_fo_term fr1 identity identity)) ((rename_subst_fo_term bnd21 identity identity))) } ; assert { rename_fo_term (nlmodel_fo_term x fr0 bnd20 fr1 bnd21) identity identity = nlmodel_fo_term x ((rename_subst_symbol fr0 identity)) ((rename_subst_symbol bnd20 identity)) ((rename_subst_fo_term fr1 identity identity)) ((rename_subst_fo_term bnd21 identity identity)) } ; assert { extensionalEqual ((rename_subst_fo_term (update bnd11 i (nlmodel_fo_term x fr0 bnd20 fr1 bnd21)) identity identity)) (update ((rename_subst_fo_term bnd11 identity identity)) (i+0) (rename_fo_term (nlmodel_fo_term x fr0 bnd20 fr1 bnd21) identity identity)) } ; assert { (rename_subst_fo_term (update bnd11 i (nlmodel_fo_term x fr0 bnd20 fr1 bnd21)) identity identity) = update ((rename_subst_fo_term bnd11 identity identity)) (i+0) (nlmodel_fo_term x ((rename_subst_symbol fr0 identity)) ((rename_subst_symbol bnd20 identity)) ((rename_subst_fo_term fr1 identity identity)) ((rename_subst_fo_term bnd21 identity identity))) } ; NL_FOFCons (unbind_var_fo_term_in_fo_formula v0 (i+0) x ((rename_subst_symbol fr0 identity)) ((rename_subst_symbol bnd10 identity)) ((rename_subst_fo_term fr1 identity identity)) ((rename_subst_fo_term bnd11 identity identity)) ((rename_subst_symbol bnd20 identity)) ((rename_subst_fo_term bnd21 identity identity))) (unbind_var_fo_term_in_fo_formula_list v1 (i+0) x ((rename_subst_symbol fr0 identity)) ((rename_subst_symbol bnd10 identity)) ((rename_subst_fo_term fr1 identity identity)) ((rename_subst_fo_term bnd11 identity identity)) ((rename_subst_symbol bnd20 identity)) ((rename_subst_fo_term bnd21 identity identity))) end let rec subst_base_symbol_in_fo_formula_list (t:nl_fo_formula_list int int) (x:int) (u:nl_symbol int) (ghost fr0: int -> (symbol 'b0)) (ghost bnd10: int -> (symbol 'b0)) (ghost fr1: int -> (fo_term 'b0 'b1)) (ghost bnd11: int -> (fo_term 'b0 'b1)) (ghost bnd20: int -> (symbol 'b0)) : nl_fo_formula_list int int requires { correct_indexes_fo_formula_list t } requires { correct_indexes_symbol u } requires { bound_depth_of_symbol_in_symbol u = 0 } variant { nlsize_fo_formula_list t } ensures { correct_indexes_fo_formula_list result } ensures { bound_depth_of_symbol_in_fo_formula_list result = bound_depth_of_symbol_in_fo_formula_list t } ensures { bound_depth_of_fo_term_in_fo_formula_list result = bound_depth_of_fo_term_in_fo_formula_list t } ensures { nlmodel_fo_formula_list result fr0 bnd10 fr1 bnd11 = nlmodel_fo_formula_list t (update fr0 x (nlmodel_symbol u fr0 bnd20)) bnd10 fr1 bnd11 } = match t with | NL_FOFNil -> NL_FOFNil | NL_FOFCons v0 v1 -> assert { rename_symbol (nlmodel_symbol u fr0 bnd20) identity = nlmodel_symbol u ((rename_subst_symbol fr0 identity)) ((rename_subst_symbol bnd20 identity)) } ; assert { extensionalEqual ((rename_subst_symbol (update fr0 x (nlmodel_symbol u fr0 bnd20)) identity)) (update ((rename_subst_symbol fr0 identity)) x (rename_symbol (nlmodel_symbol u fr0 bnd20) identity)) } ; assert { (rename_subst_symbol (update fr0 x (nlmodel_symbol u fr0 bnd20)) identity) = update ((rename_subst_symbol fr0 identity)) x (nlmodel_symbol u ((rename_subst_symbol fr0 identity)) ((rename_subst_symbol bnd20 identity))) } ; assert { rename_symbol (nlmodel_symbol u fr0 bnd20) identity = nlmodel_symbol u ((rename_subst_symbol fr0 identity)) ((rename_subst_symbol bnd20 identity)) } ; assert { extensionalEqual ((rename_subst_symbol (update fr0 x (nlmodel_symbol u fr0 bnd20)) identity)) (update ((rename_subst_symbol fr0 identity)) x (rename_symbol (nlmodel_symbol u fr0 bnd20) identity)) } ; assert { (rename_subst_symbol (update fr0 x (nlmodel_symbol u fr0 bnd20)) identity) = update ((rename_subst_symbol fr0 identity)) x (nlmodel_symbol u ((rename_subst_symbol fr0 identity)) ((rename_subst_symbol bnd20 identity))) } ; NL_FOFCons (subst_base_symbol_in_fo_formula v0 x u ((rename_subst_symbol fr0 identity)) ((rename_subst_symbol bnd10 identity)) ((rename_subst_fo_term fr1 identity identity)) ((rename_subst_fo_term bnd11 identity identity)) ((rename_subst_symbol bnd20 identity))) (subst_base_symbol_in_fo_formula_list v1 x u ((rename_subst_symbol fr0 identity)) ((rename_subst_symbol bnd10 identity)) ((rename_subst_fo_term fr1 identity identity)) ((rename_subst_fo_term bnd11 identity identity)) ((rename_subst_symbol bnd20 identity))) end with subst_base_fo_term_in_fo_formula_list (t:nl_fo_formula_list int int) (x:int) (u:nl_fo_term int int) (ghost fr0: int -> (symbol 'b0)) (ghost bnd10: int -> (symbol 'b0)) (ghost fr1: int -> (fo_term 'b0 'b1)) (ghost bnd11: int -> (fo_term 'b0 'b1)) (ghost bnd20: int -> (symbol 'b0)) (ghost bnd21: int -> (fo_term 'b0 'b1)) : nl_fo_formula_list int int requires { correct_indexes_fo_formula_list t } requires { correct_indexes_fo_term u } requires { bound_depth_of_symbol_in_fo_term u = 0 } requires { bound_depth_of_fo_term_in_fo_term u = 0 } variant { nlsize_fo_formula_list t } ensures { correct_indexes_fo_formula_list result } ensures { bound_depth_of_symbol_in_fo_formula_list result = bound_depth_of_symbol_in_fo_formula_list t } ensures { bound_depth_of_fo_term_in_fo_formula_list result = bound_depth_of_fo_term_in_fo_formula_list t } ensures { nlmodel_fo_formula_list result fr0 bnd10 fr1 bnd11 = nlmodel_fo_formula_list t fr0 bnd10 (update fr1 x (nlmodel_fo_term u fr0 bnd20 fr1 bnd21)) bnd11 } = match t with | NL_FOFNil -> NL_FOFNil | NL_FOFCons v0 v1 -> assert { rename_fo_term (nlmodel_fo_term u fr0 bnd20 fr1 bnd21) identity identity = nlmodel_fo_term u ((rename_subst_symbol fr0 identity)) ((rename_subst_symbol bnd20 identity)) ((rename_subst_fo_term fr1 identity identity)) ((rename_subst_fo_term bnd21 identity identity)) } ; assert { extensionalEqual ((rename_subst_fo_term (update fr1 x (nlmodel_fo_term u fr0 bnd20 fr1 bnd21)) identity identity)) (update ((rename_subst_fo_term fr1 identity identity)) x (rename_fo_term (nlmodel_fo_term u fr0 bnd20 fr1 bnd21) identity identity)) } ; assert { (rename_subst_fo_term (update fr1 x (nlmodel_fo_term u fr0 bnd20 fr1 bnd21)) identity identity) = update ((rename_subst_fo_term fr1 identity identity)) x (nlmodel_fo_term u ((rename_subst_symbol fr0 identity)) ((rename_subst_symbol bnd20 identity)) ((rename_subst_fo_term fr1 identity identity)) ((rename_subst_fo_term bnd21 identity identity))) } ; assert { rename_fo_term (nlmodel_fo_term u fr0 bnd20 fr1 bnd21) identity identity = nlmodel_fo_term u ((rename_subst_symbol fr0 identity)) ((rename_subst_symbol bnd20 identity)) ((rename_subst_fo_term fr1 identity identity)) ((rename_subst_fo_term bnd21 identity identity)) } ; assert { extensionalEqual ((rename_subst_fo_term (update fr1 x (nlmodel_fo_term u fr0 bnd20 fr1 bnd21)) identity identity)) (update ((rename_subst_fo_term fr1 identity identity)) x (rename_fo_term (nlmodel_fo_term u fr0 bnd20 fr1 bnd21) identity identity)) } ; assert { (rename_subst_fo_term (update fr1 x (nlmodel_fo_term u fr0 bnd20 fr1 bnd21)) identity identity) = update ((rename_subst_fo_term fr1 identity identity)) x (nlmodel_fo_term u ((rename_subst_symbol fr0 identity)) ((rename_subst_symbol bnd20 identity)) ((rename_subst_fo_term fr1 identity identity)) ((rename_subst_fo_term bnd21 identity identity))) } ; NL_FOFCons (subst_base_fo_term_in_fo_formula v0 x u ((rename_subst_symbol fr0 identity)) ((rename_subst_symbol bnd10 identity)) ((rename_subst_fo_term fr1 identity identity)) ((rename_subst_fo_term bnd11 identity identity)) ((rename_subst_symbol bnd20 identity)) ((rename_subst_fo_term bnd21 identity identity))) (subst_base_fo_term_in_fo_formula_list v1 x u ((rename_subst_symbol fr0 identity)) ((rename_subst_symbol bnd10 identity)) ((rename_subst_fo_term fr1 identity identity)) ((rename_subst_fo_term bnd11 identity identity)) ((rename_subst_symbol bnd20 identity)) ((rename_subst_fo_term bnd21 identity identity))) end let construct_fo_formula_list (c:cons_fo_formula_list) : nlimpl_fo_formula_list requires { cons_ok_fo_formula_list c } ensures { nlimpl_fo_formula_list_ok result } ensures { cons_rel_fo_formula_list c result } (*ensures { cons_open_rel_fo_formula_list c result }*) = match c with | NLC_FOFNil -> let res = { nlrepr_fo_formula_list_field = (NL_FOFNil) ; nlfree_var_symbol_set_abstraction_fo_formula_list_field = 0 ; nlfree_var_fo_term_set_abstraction_fo_formula_list_field = 0 ; model_fo_formula_list_field = ghost (FOFNil) ; } in assert { forall x:int. is_symbol_free_var_in_fo_formula_list x res.model_fo_formula_list_field -> (false) && (x) < (res.nlfree_var_symbol_set_abstraction_fo_formula_list_field) } ; assert { forall x:int. is_fo_term_free_var_in_fo_formula_list x res.model_fo_formula_list_field -> (false) && (x) < (res.nlfree_var_fo_term_set_abstraction_fo_formula_list_field) } ; res | NLC_FOFCons v0 v1 -> assert { nlimpl_fo_formula_ok v0 } ; assert { nlimpl_fo_formula_list_ok v1 } ; let res = { nlrepr_fo_formula_list_field = (NL_FOFCons (let v0 = v0.nlrepr_fo_formula_field in v0) (let v1 = v1.nlrepr_fo_formula_list_field in v1)) ; nlfree_var_symbol_set_abstraction_fo_formula_list_field = (let aux (a:int) (b:int) : int ensures { result >= a /\ result >= b } = if a < b then b else a in aux (v0.nlfree_var_symbol_set_abstraction_fo_formula_field) (v1.nlfree_var_symbol_set_abstraction_fo_formula_list_field)) ; nlfree_var_fo_term_set_abstraction_fo_formula_list_field = (let aux (a:int) (b:int) : int ensures { result >= a /\ result >= b } = if a < b then b else a in aux (v0.nlfree_var_fo_term_set_abstraction_fo_formula_field) (v1.nlfree_var_fo_term_set_abstraction_fo_formula_list_field)) ; model_fo_formula_list_field = ghost (FOFCons (rename_fo_formula v0.model_fo_formula_field identity identity) (rename_fo_formula_list v1.model_fo_formula_list_field identity identity)) ; } in assert { forall x:int. is_symbol_free_var_in_fo_formula x (rename_fo_formula v0.model_fo_formula_field identity identity) -> (forall y:int. (is_symbol_free_var_in_fo_formula y v0.model_fo_formula_field /\ eval (identity) y = x) -> x = eval (identity) y && x = y && is_symbol_free_var_in_fo_formula x v0.model_fo_formula_field) && is_symbol_free_var_in_fo_formula x v0.model_fo_formula_field && (x) < (v0.nlfree_var_symbol_set_abstraction_fo_formula_field) && (x) < (res.nlfree_var_symbol_set_abstraction_fo_formula_list_field) } ; assert { forall x:int. is_fo_term_free_var_in_fo_formula x (rename_fo_formula v0.model_fo_formula_field identity identity) -> (forall y:int. (is_fo_term_free_var_in_fo_formula y v0.model_fo_formula_field /\ eval (identity) y = x) -> x = eval (identity) y && x = y && is_fo_term_free_var_in_fo_formula x v0.model_fo_formula_field) && is_fo_term_free_var_in_fo_formula x v0.model_fo_formula_field && (x) < (v0.nlfree_var_fo_term_set_abstraction_fo_formula_field) && (x) < (res.nlfree_var_fo_term_set_abstraction_fo_formula_list_field) } ; assert { forall x:int. is_symbol_free_var_in_fo_formula_list x (rename_fo_formula_list v1.model_fo_formula_list_field identity identity) -> (forall y:int. (is_symbol_free_var_in_fo_formula_list y v1.model_fo_formula_list_field /\ eval (identity) y = x) -> x = eval (identity) y && x = y && is_symbol_free_var_in_fo_formula_list x v1.model_fo_formula_list_field) && is_symbol_free_var_in_fo_formula_list x v1.model_fo_formula_list_field && (x) < (v1.nlfree_var_symbol_set_abstraction_fo_formula_list_field) && (x) < (res.nlfree_var_symbol_set_abstraction_fo_formula_list_field) } ; assert { forall x:int. is_fo_term_free_var_in_fo_formula_list x (rename_fo_formula_list v1.model_fo_formula_list_field identity identity) -> (forall y:int. (is_fo_term_free_var_in_fo_formula_list y v1.model_fo_formula_list_field /\ eval (identity) y = x) -> x = eval (identity) y && x = y && is_fo_term_free_var_in_fo_formula_list x v1.model_fo_formula_list_field) && is_fo_term_free_var_in_fo_formula_list x v1.model_fo_formula_list_field && (x) < (v1.nlfree_var_fo_term_set_abstraction_fo_formula_list_field) && (x) < (res.nlfree_var_fo_term_set_abstraction_fo_formula_list_field) } ; assert { forall x:int. is_symbol_free_var_in_fo_formula_list x res.model_fo_formula_list_field -> (is_symbol_free_var_in_fo_formula x (rename_fo_formula v0.model_fo_formula_field identity identity) \/ is_symbol_free_var_in_fo_formula_list x (rename_fo_formula_list v1.model_fo_formula_list_field identity identity)) && (x) < (res.nlfree_var_symbol_set_abstraction_fo_formula_list_field) } ; assert { forall x:int. is_fo_term_free_var_in_fo_formula_list x res.model_fo_formula_list_field -> (is_fo_term_free_var_in_fo_formula x (rename_fo_formula v0.model_fo_formula_field identity identity) \/ is_fo_term_free_var_in_fo_formula_list x (rename_fo_formula_list v1.model_fo_formula_list_field identity identity)) && (x) < (res.nlfree_var_fo_term_set_abstraction_fo_formula_list_field) } ; assert { extensionalEqual (rename_subst_symbol subst_id_symbol identity) ((rename_subst_symbol (subst_id_symbol:(int)->(symbol (int))) identity)) } ; assert { rename_subst_symbol subst_id_symbol identity = (rename_subst_symbol (subst_id_symbol:(int)->(symbol (int))) identity) } ; assert { extensionalEqual (rename_subst_fo_term subst_id_fo_term identity identity) ((rename_subst_fo_term (subst_id_fo_term:(int)->(fo_term (int) (int))) identity identity)) } ; assert { rename_subst_fo_term subst_id_fo_term identity identity = (rename_subst_fo_term (subst_id_fo_term:(int)->(fo_term (int) (int))) identity identity) } ; model_equal_fo_formula v0.nlrepr_fo_formula_field (rename_subst_symbol subst_id_symbol identity) (rename_subst_symbol subst_id_symbol identity) ((rename_subst_symbol (const (Var_symbol (-1))) identity)) (rename_subst_symbol (const (Var_symbol (-1))) identity) (rename_subst_fo_term subst_id_fo_term identity identity) (rename_subst_fo_term subst_id_fo_term identity identity) ((rename_subst_fo_term (const (Var_fo_term (-1))) identity identity)) (rename_subst_fo_term (const (Var_fo_term (-1))) identity identity) ; (*assert { extensionalEqual (rcompose (identity) (identity)) ((identity : (int) -> (int))) } ; assert { rcompose (identity) (identity) = (identity : (int) -> (int)) } ;*) (*assert { extensionalEqual (rcompose (identity) (identity)) ((identity : (int) -> (int))) } ; assert { rcompose (identity) (identity) = (identity : (int) -> (int)) } ;*) assert { extensionalEqual (rename_subst_symbol subst_id_symbol identity) ((rename_subst_symbol (subst_id_symbol:(int)->(symbol (int))) identity)) } ; assert { rename_subst_symbol subst_id_symbol identity = (rename_subst_symbol (subst_id_symbol:(int)->(symbol (int))) identity) } ; assert { extensionalEqual (rename_subst_fo_term subst_id_fo_term identity identity) ((rename_subst_fo_term (subst_id_fo_term:(int)->(fo_term (int) (int))) identity identity)) } ; assert { rename_subst_fo_term subst_id_fo_term identity identity = (rename_subst_fo_term (subst_id_fo_term:(int)->(fo_term (int) (int))) identity identity) } ; model_equal_fo_formula_list v1.nlrepr_fo_formula_list_field (rename_subst_symbol subst_id_symbol identity) (rename_subst_symbol subst_id_symbol identity) ((rename_subst_symbol (const (Var_symbol (-1))) identity)) (rename_subst_symbol (const (Var_symbol (-1))) identity) (rename_subst_fo_term subst_id_fo_term identity identity) (rename_subst_fo_term subst_id_fo_term identity identity) ((rename_subst_fo_term (const (Var_fo_term (-1))) identity identity)) (rename_subst_fo_term (const (Var_fo_term (-1))) identity identity) ; (*assert { extensionalEqual (rcompose (identity) (identity)) ((identity : (int) -> (int))) } ; assert { rcompose (identity) (identity) = (identity : (int) -> (int)) } ;*) (*assert { extensionalEqual (rcompose (identity) (identity)) ((identity : (int) -> (int))) } ; assert { rcompose (identity) (identity) = (identity : (int) -> (int)) } ;*) res end let destruct_fo_formula_list (t:nlimpl_fo_formula_list) : cons_fo_formula_list requires { nlimpl_fo_formula_list_ok t } ensures { cons_ok_fo_formula_list result } ensures { cons_rel_fo_formula_list result t } ensures { cons_open_rel_fo_formula_list result t } = let fv0 = t.nlfree_var_symbol_set_abstraction_fo_formula_list_field in let fv1 = t.nlfree_var_fo_term_set_abstraction_fo_formula_list_field in match t.nlrepr_fo_formula_list_field with | NL_FOFNil -> assert { t.model_fo_formula_list_field = FOFNil } ; let () = match t.model_fo_formula_list_field with | FOFNil -> () | FOFCons x0 x1 -> absurd end in let res = NLC_FOFNil in res | NL_FOFCons v0 v1 -> assert { t.model_fo_formula_list_field = FOFCons (nlmodel_fo_formula v0 (rename_subst_symbol subst_id_symbol identity) (rename_subst_symbol (const (Var_symbol (-1)) : int -> (symbol int)) identity) (rename_subst_fo_term subst_id_fo_term identity identity) (rename_subst_fo_term (const (Var_fo_term (-1)) : int -> (fo_term int int)) identity identity)) (nlmodel_fo_formula_list v1 (rename_subst_symbol subst_id_symbol identity) (rename_subst_symbol (const (Var_symbol (-1)) : int -> (symbol int)) identity) (rename_subst_fo_term subst_id_fo_term identity identity) (rename_subst_fo_term (const (Var_fo_term (-1)) : int -> (fo_term int int)) identity identity)) } ; let (mv0 , mv1) = match t.model_fo_formula_list_field with | FOFNil -> absurd | FOFCons x0 x1 -> (x0 , x1) end in assert { mv0 = nlmodel_fo_formula v0 ((rename_subst_symbol subst_id_symbol identity)) ((rename_subst_symbol (const (Var_symbol (-1)) : int -> (symbol int)) identity)) ((rename_subst_fo_term subst_id_fo_term identity identity)) ((rename_subst_fo_term (const (Var_fo_term (-1)) : int -> (fo_term int int)) identity identity)) } ; assert { mv1 = nlmodel_fo_formula_list v1 ((rename_subst_symbol subst_id_symbol identity)) ((rename_subst_symbol (const (Var_symbol (-1)) : int -> (symbol int)) identity)) ((rename_subst_fo_term subst_id_fo_term identity identity)) ((rename_subst_fo_term (const (Var_fo_term (-1)) : int -> (fo_term int int)) identity identity)) } ; assert { bound_depth_of_symbol_in_fo_formula v0 <= 0 } ; assert { forall x:int. is_symbol_free_var_in_fo_formula x mv0 -> is_symbol_free_var_in_fo_formula_list x t.model_fo_formula_list_field && (x) < (t.nlfree_var_symbol_set_abstraction_fo_formula_list_field) } ; assert { bound_depth_of_fo_term_in_fo_formula v0 <= 0 } ; assert { forall x:int. is_fo_term_free_var_in_fo_formula x mv0 -> is_fo_term_free_var_in_fo_formula_list x t.model_fo_formula_list_field && (x) < (t.nlfree_var_fo_term_set_abstraction_fo_formula_list_field) } ; assert { bound_depth_of_symbol_in_fo_formula_list v1 <= 0 } ; assert { forall x:int. is_symbol_free_var_in_fo_formula_list x mv1 -> is_symbol_free_var_in_fo_formula_list x t.model_fo_formula_list_field && (x) < (t.nlfree_var_symbol_set_abstraction_fo_formula_list_field) } ; assert { bound_depth_of_fo_term_in_fo_formula_list v1 <= 0 } ; assert { forall x:int. is_fo_term_free_var_in_fo_formula_list x mv1 -> is_fo_term_free_var_in_fo_formula_list x t.model_fo_formula_list_field && (x) < (t.nlfree_var_fo_term_set_abstraction_fo_formula_list_field) } ; model_equal_fo_formula v0 subst_id_symbol (rename_subst_symbol ((rename_subst_symbol subst_id_symbol identity)) (identity)) (( const (Var_symbol (-1)) : int -> (symbol int))) (rename_subst_symbol ((rename_subst_symbol (const (Var_symbol (-1)) : int -> (symbol int)) identity)) (identity)) subst_id_fo_term (rename_subst_fo_term ((rename_subst_fo_term subst_id_fo_term identity identity)) (identity) (identity)) ((const (Var_fo_term (-1)) : int -> (fo_term int int))) (rename_subst_fo_term ((rename_subst_fo_term (const (Var_fo_term (-1)) : int -> (fo_term int int)) identity identity)) (identity) (identity)) ; model_equal_fo_formula_list v1 subst_id_symbol (rename_subst_symbol ((rename_subst_symbol subst_id_symbol identity)) (identity)) (( const (Var_symbol (-1)) : int -> (symbol int))) (rename_subst_symbol ((rename_subst_symbol (const (Var_symbol (-1)) : int -> (symbol int)) identity)) (identity)) subst_id_fo_term (rename_subst_fo_term ((rename_subst_fo_term subst_id_fo_term identity identity)) (identity) (identity)) ((const (Var_fo_term (-1)) : int -> (fo_term int int))) (rename_subst_fo_term ((rename_subst_fo_term (const (Var_fo_term (-1)) : int -> (fo_term int int)) identity identity)) (identity) (identity)) ; let ghost mrv0 = rename_fo_formula mv0 identity identity in let ghost mrv1 = rename_fo_formula_list mv1 identity identity in let resv0 = { nlrepr_fo_formula_field = v0 ; nlfree_var_symbol_set_abstraction_fo_formula_field = fv0 ; nlfree_var_fo_term_set_abstraction_fo_formula_field = fv1 ; model_fo_formula_field = ghost mrv0 ; } in let resv1 = { nlrepr_fo_formula_list_field = v1 ; nlfree_var_symbol_set_abstraction_fo_formula_list_field = fv0 ; nlfree_var_fo_term_set_abstraction_fo_formula_list_field = fv1 ; model_fo_formula_list_field = ghost mrv1 ; } in let res = NLC_FOFCons resv0 resv1 in free_var_equivalence_of_rename_fo_formula mv0 (identity) (rcompose (identity) (identity)) (identity) (rcompose (identity) (identity)) ; free_var_equivalence_of_rename_fo_formula_list mv1 (identity) (rcompose (identity) (identity)) (identity) (rcompose (identity) (identity)) ; assert { forall x:int. is_symbol_free_var_in_fo_formula x mrv0 -> (forall y:int. (is_symbol_free_var_in_fo_formula y mv0 /\ eval (identity) y = x) -> x = y && is_symbol_free_var_in_fo_formula_list x t.model_fo_formula_list_field && (x) < (fv0)) && (x) < (fv0) } ; assert { forall x:int. is_fo_term_free_var_in_fo_formula x mrv0 -> (forall y:int. (is_fo_term_free_var_in_fo_formula y mv0 /\ eval (identity) y = x) -> x = y && is_fo_term_free_var_in_fo_formula_list x t.model_fo_formula_list_field && (x) < (fv1)) && (x) < (fv1) } ; assert { forall x:int. is_symbol_free_var_in_fo_formula_list x mrv1 -> (forall y:int. (is_symbol_free_var_in_fo_formula_list y mv1 /\ eval (identity) y = x) -> x = y && is_symbol_free_var_in_fo_formula_list x t.model_fo_formula_list_field && (x) < (fv0)) && (x) < (fv0) } ; assert { forall x:int. is_fo_term_free_var_in_fo_formula_list x mrv1 -> (forall y:int. (is_fo_term_free_var_in_fo_formula_list y mv1 /\ eval (identity) y = x) -> x = y && is_fo_term_free_var_in_fo_formula_list x t.model_fo_formula_list_field && (x) < (fv1)) && (x) < (fv1) } ; res end let nlsubst_symbol_in_fo_formula_list (t:nlimpl_fo_formula_list) (x:int) (u:nlimpl_symbol) : nlimpl_fo_formula_list requires { nlimpl_fo_formula_list_ok t } requires { nlimpl_symbol_ok u } ensures { nlimpl_fo_formula_list_ok result } ensures { result.model_fo_formula_list_field = subst_fo_formula_list t.model_fo_formula_list_field (update (subst_id_symbol: (int) -> (symbol (int))) x u.model_symbol_field) (subst_id_fo_term: (int) -> (fo_term (int) (int))) } = model_equal_fo_formula_list t.nlrepr_fo_formula_list_field (subst_compose_symbol subst_id_symbol ((update (subst_id_symbol: (int) -> (symbol (int))) x u.model_symbol_field))) ((update (subst_id_symbol: (int) -> (symbol (int))) x u.model_symbol_field)) (subst_compose_symbol (const (Var_symbol (-1))) ((update (subst_id_symbol: (int) -> (symbol (int))) x u.model_symbol_field))) ((const (Var_symbol (-1)))) (subst_compose_fo_term subst_id_fo_term ((update (subst_id_symbol: (int) -> (symbol (int))) x u.model_symbol_field)) ((subst_id_fo_term: (int) -> (fo_term (int) (int))))) ((subst_id_fo_term: (int) -> (fo_term (int) (int)))) (subst_compose_fo_term (const (Var_fo_term (-1))) ((update (subst_id_symbol: (int) -> (symbol (int))) x u.model_symbol_field)) ((subst_id_fo_term: (int) -> (fo_term (int) (int))))) ((const (Var_fo_term (-1)))); let res = { nlrepr_fo_formula_list_field = subst_base_symbol_in_fo_formula_list t.nlrepr_fo_formula_list_field x u.nlrepr_symbol_field (subst_id_symbol) ((const (Var_symbol (-1)))) (subst_id_fo_term) ((const (Var_fo_term (-1)))) ((const (Var_symbol (-1)))) ; nlfree_var_symbol_set_abstraction_fo_formula_list_field = (let aux (a:int) (b:int) : int ensures { result >= a /\ result >= b } = if a < b then b else a in aux (t.nlfree_var_symbol_set_abstraction_fo_formula_list_field) (u.nlfree_var_symbol_set_abstraction_symbol_field)) ; nlfree_var_fo_term_set_abstraction_fo_formula_list_field = t.nlfree_var_fo_term_set_abstraction_fo_formula_list_field ; model_fo_formula_list_field = ghost subst_fo_formula_list t.model_fo_formula_list_field (update (subst_id_symbol: (int) -> (symbol (int))) x u.model_symbol_field) (subst_id_fo_term: (int) -> (fo_term (int) (int))) ; } in assert { forall x2:int. is_symbol_free_var_in_fo_formula_list x2 res.model_fo_formula_list_field -> (true /\ (forall y:int. (is_symbol_free_var_in_fo_formula_list y t.model_fo_formula_list_field /\ is_symbol_free_var_in_symbol x2 (eval ((update (subst_id_symbol: (int) -> (symbol (int))) x u.model_symbol_field)) y)) -> ((x = y -> (x2) < (res.nlfree_var_symbol_set_abstraction_fo_formula_list_field)) /\ (x <> y -> x2 = y && (x2) < (res.nlfree_var_symbol_set_abstraction_fo_formula_list_field))) && (x2) < (res.nlfree_var_symbol_set_abstraction_fo_formula_list_field)) /\ (forall y:int. (is_fo_term_free_var_in_fo_formula_list y t.model_fo_formula_list_field /\ is_symbol_free_var_in_fo_term x2 (eval ((subst_id_fo_term: (int) -> (fo_term (int) (int)))) y)) -> false)) && (x2) < (res.nlfree_var_symbol_set_abstraction_fo_formula_list_field) } ; assert { forall x2:int. is_fo_term_free_var_in_fo_formula_list x2 res.model_fo_formula_list_field -> (true /\ (forall y:int. (is_fo_term_free_var_in_fo_formula_list y t.model_fo_formula_list_field /\ is_fo_term_free_var_in_fo_term x2 (eval ((subst_id_fo_term: (int) -> (fo_term (int) (int)))) y)) -> x2 = y && (x2) < (res.nlfree_var_fo_term_set_abstraction_fo_formula_list_field))) && (x2) < (res.nlfree_var_fo_term_set_abstraction_fo_formula_list_field) } ; res let nlsubst_fo_term_in_fo_formula_list (t:nlimpl_fo_formula_list) (x:int) (u:nlimpl_fo_term) : nlimpl_fo_formula_list requires { nlimpl_fo_formula_list_ok t } requires { nlimpl_fo_term_ok u } ensures { nlimpl_fo_formula_list_ok result } ensures { result.model_fo_formula_list_field = subst_fo_formula_list t.model_fo_formula_list_field (subst_id_symbol: (int) -> (symbol (int))) (update (subst_id_fo_term: (int) -> (fo_term (int) (int))) x u.model_fo_term_field) } = model_equal_fo_formula_list t.nlrepr_fo_formula_list_field (subst_compose_symbol subst_id_symbol ((subst_id_symbol: (int) -> (symbol (int))))) ((subst_id_symbol: (int) -> (symbol (int)))) (subst_compose_symbol (const (Var_symbol (-1))) ((subst_id_symbol: (int) -> (symbol (int))))) ((const (Var_symbol (-1)))) (subst_compose_fo_term subst_id_fo_term ((subst_id_symbol: (int) -> (symbol (int)))) ((update (subst_id_fo_term: (int) -> (fo_term (int) (int))) x u.model_fo_term_field))) ((update (subst_id_fo_term: (int) -> (fo_term (int) (int))) x u.model_fo_term_field)) (subst_compose_fo_term (const (Var_fo_term (-1))) ((subst_id_symbol: (int) -> (symbol (int)))) ((update (subst_id_fo_term: (int) -> (fo_term (int) (int))) x u.model_fo_term_field))) ((const (Var_fo_term (-1)))); let res = { nlrepr_fo_formula_list_field = subst_base_fo_term_in_fo_formula_list t.nlrepr_fo_formula_list_field x u.nlrepr_fo_term_field (subst_id_symbol) ((const (Var_symbol (-1)))) (subst_id_fo_term) ((const (Var_fo_term (-1)))) ((const (Var_symbol (-1)))) ((const (Var_fo_term (-1)))) ; nlfree_var_symbol_set_abstraction_fo_formula_list_field = (let aux (a:int) (b:int) : int ensures { result >= a /\ result >= b } = if a < b then b else a in aux (t.nlfree_var_symbol_set_abstraction_fo_formula_list_field) (u.nlfree_var_symbol_set_abstraction_fo_term_field)) ; nlfree_var_fo_term_set_abstraction_fo_formula_list_field = (let aux (a:int) (b:int) : int ensures { result >= a /\ result >= b } = if a < b then b else a in aux (t.nlfree_var_fo_term_set_abstraction_fo_formula_list_field) (u.nlfree_var_fo_term_set_abstraction_fo_term_field)) ; model_fo_formula_list_field = ghost subst_fo_formula_list t.model_fo_formula_list_field (subst_id_symbol: (int) -> (symbol (int))) (update (subst_id_fo_term: (int) -> (fo_term (int) (int))) x u.model_fo_term_field) ; } in assert { forall x2:int. is_symbol_free_var_in_fo_formula_list x2 res.model_fo_formula_list_field -> (true /\ (forall y:int. (is_symbol_free_var_in_fo_formula_list y t.model_fo_formula_list_field /\ is_symbol_free_var_in_symbol x2 (eval ((subst_id_symbol: (int) -> (symbol (int)))) y)) -> x2 = y && (x2) < (res.nlfree_var_symbol_set_abstraction_fo_formula_list_field)) /\ (forall y:int. (is_fo_term_free_var_in_fo_formula_list y t.model_fo_formula_list_field /\ is_symbol_free_var_in_fo_term x2 (eval ((update (subst_id_fo_term: (int) -> (fo_term (int) (int))) x u.model_fo_term_field)) y)) -> ((x = y -> (x2) < (res.nlfree_var_symbol_set_abstraction_fo_formula_list_field)) /\ (x <> y -> false)) && (x2) < (res.nlfree_var_symbol_set_abstraction_fo_formula_list_field))) && (x2) < (res.nlfree_var_symbol_set_abstraction_fo_formula_list_field) } ; assert { forall x2:int. is_fo_term_free_var_in_fo_formula_list x2 res.model_fo_formula_list_field -> (true /\ (forall y:int. (is_fo_term_free_var_in_fo_formula_list y t.model_fo_formula_list_field /\ is_fo_term_free_var_in_fo_term x2 (eval ((update (subst_id_fo_term: (int) -> (fo_term (int) (int))) x u.model_fo_term_field)) y)) -> ((x = y -> (x2) < (res.nlfree_var_fo_term_set_abstraction_fo_formula_list_field)) /\ (x <> y -> x2 = y && (x2) < (res.nlfree_var_fo_term_set_abstraction_fo_formula_list_field))) && (x2) < (res.nlfree_var_fo_term_set_abstraction_fo_formula_list_field))) && (x2) < (res.nlfree_var_fo_term_set_abstraction_fo_formula_list_field) } ; res end
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