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reflectionScript.sml
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reflectionScript.sml
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open preamble countableLib countableTheory
open basicReflectionLib
open setSpecTheory holSyntaxLibTheory holSyntaxTheory holSyntaxExtraTheory holSemanticsTheory holSemanticsExtraTheory
open holBoolSyntaxTheory holBoolTheory holExtensionTheory holConsistencyTheory holAxiomsSyntaxTheory holAxiomsTheory
open holConstrainedExtensionTheory
local open holSyntaxLib in end
val _ = temp_tight_equality()
val _ = new_theory"reflection"
val _ = Parse.remove_type_abbrev"reln"
(* TODO: this is a hack... *)
val tyvar_inst_exists2 = store_thm("tyvar_inst_exists2",
``∃i. tyvar = REV_ASSOCD b1 i b1 ∧
tyvar = REV_ASSOCD b2 i b2``,
qexists_tac`[(tyvar,b1);(tyvar,b2)]` >>
EVAL_TAC)
val tyvar_inst_exists2_diff = store_thm("tyvar_inst_exists2_diff",
``b1 ≠ b2 ⇒
∃i. ty1 = REV_ASSOCD b1 i b1 ∧
ty2 = REV_ASSOCD b2 i b2``,
strip_tac >>
qexists_tac`[(ty1,b1);(ty2,b2)]` >>
EVAL_TAC >> rw[])
(* -- *)
val provable_imp_eq_true = store_thm("provable_imp_eq_true",
``∀thy i v.
is_set_theory ^mem ⇒
i models thy ⇒
is_valuation (tysof (sigof thy)) (tyaof i) v
⇒
∀p. (thy,[]) |- p ⇒ termsem (tmsof (sigof thy)) i v p = True``,
rw[] >>
imp_res_tac holSoundnessTheory.proves_sound >>
fs[entails_def] >> res_tac >>
fs[satisfies_def])
val mp_lemma = store_thm("mp_lemma",
``(a ==> b) /\ (c ==> a) ==> (c ==> b)``,
metis_tac[])
val good_context_def = Define`
good_context ^mem (^tysig,^tmsig) (^tyass,^tmass) ⇔
is_set_theory ^mem ∧
is_std_sig ^signatur ∧
is_interpretation ^signatur ^interpretation ∧
is_std_interpretation ^interpretation`
val good_context = good_context_def |> concl |> strip_forall |> snd |> lhs
val is_valuation = ``is_valuation ^tysig ^tyass ^valuation``
val good_context_unpaired = store_thm("good_context_unpaired",
``good_context mem sig i ⇔
is_set_theory mem ∧
is_std_sig sig ∧
is_interpretation sig i ∧
is_std_interpretation i``,
map_every PairCases_on[`sig`,`i`]>>rw[good_context_def])
val finv_def = Define`
finv f x = @y. f y = x`
val wf_to_inner_def = xDefine"wf_to_inner"`
wf_to_inner0 ^mem f = ∃x. BIJ f UNIV {a | mem a x}`
val _ = Parse.overload_on("wf_to_inner",``wf_to_inner0 ^mem``)
val wf_to_inner_finv_left = store_thm("wf_to_inner_finv_left",
``∀ina.
wf_to_inner ina ⇒ ∀x. finv ina (ina x) = x``,
rw[finv_def] >>
SELECT_ELIM_TAC >>
conj_tac >-metis_tac[] >>
fs[wf_to_inner_def,BIJ_DEF,INJ_DEF])
val ext_def = xDefine"ext"`
ext0 ^mem s = { a | mem a s }`
val _ = Parse.overload_on("ext",``ext0 ^mem``)
val range_def = xDefine"range"`
range0 ^mem (f : α -> 'U) = @x. BIJ f UNIV {a | mem a x}`
val _ = Parse.overload_on("range",``range0 ^mem``)
val wf_to_inner_bij_thm = store_thm("wf_to_inner_bij_thm",
``∀f. wf_to_inner f ⇒ BIJ f UNIV (ext (range f))``,
rw[wf_to_inner_def,range_def] >>
SELECT_ELIM_TAC >> conj_tac >- metis_tac[] >>
rw[ext_def])
val wf_to_inner_range_thm = store_thm("wf_to_inner_range_thm",
``∀f x. wf_to_inner f ⇒ f x <: range f``,
rw[] >>
imp_res_tac wf_to_inner_bij_thm >>
fs[BIJ_DEF,ext_def,INJ_DEF])
val wf_to_inner_finv_right = store_thm("wf_to_inner_finv_right",
``∀ina.
wf_to_inner ina ⇒ ∀x. x <: range ina ⇒
(ina (finv ina x)) = x``,
rw[finv_def] >>
SELECT_ELIM_TAC >>
conj_tac >-(
imp_res_tac wf_to_inner_bij_thm >>
fs[ext_def,BIJ_DEF,SURJ_DEF] ) >>
rw[])
val bool_to_inner_def = xDefine"bool_to_inner"`
bool_to_inner0 ^mem = Boolean`
val _ = Parse.overload_on("bool_to_inner",``bool_to_inner0 ^mem``)
val wf_to_inner_bool_to_inner = store_thm("wf_to_inner_bool_to_inner",
``is_set_theory ^mem ⇒
wf_to_inner bool_to_inner``,
rw[wf_to_inner_def,BIJ_IFF_INV] >>
qexists_tac`boolset` >>
rw[bool_to_inner_def,boolean_in_boolset] >>
qexists_tac`λx. x = True` >>
rw[bool_to_inner_def,boolean_eq_true] >>
rfs[mem_boolset,boolean_eq_true,true_neq_false,boolean_def])
val fun_to_inner_def = xDefine"fun_to_inner"`
fun_to_inner0 ^mem ina inb f =
Abstract (range ina) (range inb) (λx. inb (f (finv ina x)))`
val _ = Parse.overload_on("fun_to_inner",``fun_to_inner0 ^mem``)
val out_fun_def = xDefine"out_fun"`
out_fun0 ^mem ina inb mf x = finv inb (mf ' (ina x))`
val _ = Parse.overload_on("out_fun",``out_fun0 ^mem``)
val wf_to_inner_fun_to_inner = store_thm("wf_to_inner_fun_to_inner",
``is_set_theory ^mem ⇒
∀ina inb. wf_to_inner ina ∧ wf_to_inner inb ⇒ wf_to_inner (fun_to_inner ina inb)``,
rw[] >>
rw[wf_to_inner_def,BIJ_IFF_INV] >>
qexists_tac`Funspace (range ina) (range inb)` >>
conj_tac >- (
rw[fun_to_inner_def] >>
match_mp_tac (UNDISCH abstract_in_funspace) >>
simp[range_def] >>
SELECT_ELIM_TAC >>
conj_tac >- metis_tac[wf_to_inner_def] >>
rw[] >>
SELECT_ELIM_TAC >>
conj_tac >- metis_tac[wf_to_inner_def] >>
rw[] >>
fs[BIJ_IFF_INV] ) >>
qexists_tac`out_fun ina inb` >>
conj_tac >- (
rw[out_fun_def,fun_to_inner_def,FUN_EQ_THM] >>
qmatch_abbrev_tac`finv invb (Abstract s t f ' a) = Z` >>
rfs[] >>
`Abstract s t f ' a = f a` by (
match_mp_tac (UNDISCH apply_abstract) >>
imp_res_tac wf_to_inner_bij_thm >>
fs[ext_def,BIJ_IFF_INV] >>
unabbrev_all_tac >> fs[] ) >>
rw[Abbr`Z`,Abbr`f`,Abbr`a`,Abbr`invb`] >>
imp_res_tac wf_to_inner_finv_left >>
simp[] ) >>
rw[fun_to_inner_def,out_fun_def] >>
first_x_assum(mp_tac o
MATCH_MP(REWRITE_RULE[GSYM AND_IMP_INTRO](UNDISCH in_funspace_abstract))) >>
rw[] >>
match_mp_tac (UNDISCH abstract_eq) >>
gen_tac >>
qspecl_then[`f`,`ina (finv ina x)`,`range ina`,`range inb`]mp_tac
(UNDISCH apply_abstract) >>
impl_tac >- (
imp_res_tac wf_to_inner_bij_thm >>
fs[ext_def,BIJ_DEF,INJ_DEF] ) >>
rw[] >>
imp_res_tac wf_to_inner_finv_right >>
rw[])
val range_bool_to_inner = store_thm("range_bool_to_inner",
``is_set_theory ^mem ⇒
range bool_to_inner = boolset``,
strip_tac >>
imp_res_tac wf_to_inner_bool_to_inner >>
imp_res_tac wf_to_inner_bij_thm >>
imp_res_tac is_extensional >>
pop_assum mp_tac >>
simp[extensional_def] >>
disch_then kall_tac >>
fs[ext_def,BIJ_IFF_INV,mem_boolset] >>
fs[bool_to_inner_def,boolean_def] >>
metis_tac[] )
val range_fun_to_inner = store_thm("range_fun_to_inner",
``is_set_theory ^mem ∧ wf_to_inner ina ∧ wf_to_inner inb ⇒
range (fun_to_inner ina inb) = Funspace (range ina) (range inb)``,
rw[] >>
strip_assume_tac(SPEC_ALL (UNDISCH wf_to_inner_fun_to_inner)) >> rfs[] >>
imp_res_tac wf_to_inner_bij_thm >>
imp_res_tac is_extensional >>
pop_assum mp_tac >>
simp[extensional_def] >>
disch_then kall_tac >>
fs[ext_def,BIJ_IFF_INV] >>
rw[EQ_IMP_THM] >- (
fs[fun_to_inner_def] >>
res_tac >>
pop_assum(SUBST1_TAC o SYM) >>
match_mp_tac (UNDISCH abstract_in_funspace) >>
rw[] ) >>
qspecl_then[`a`,`range ina`,`range inb`]mp_tac (UNDISCH in_funspace_abstract) >>
simp[] >> strip_tac >>
qpat_x_assum`a = X`(SUBST1_TAC) >>
qsuff_tac`∃x. Abstract (range ina) (range inb) f = fun_to_inner ina inb x` >- metis_tac[] >>
rw[fun_to_inner_def] >>
qexists_tac`finv inb o f o ina` >>
match_mp_tac (UNDISCH abstract_eq) >> simp[] >>
metis_tac[wf_to_inner_finv_right,wf_to_inner_finv_left])
val finv_bool_to_inner_eq_true = store_thm("finv_bool_to_inner_eq_true",
``is_set_theory ^mem ⇒
∀x.
((x = True) ⇒ finv bool_to_inner x) ∧
(x <: boolset ∧ finv bool_to_inner x ⇒ (x = True))``,
metis_tac[wf_to_inner_finv_right,bool_to_inner_def,boolean_def,range_bool_to_inner,wf_to_inner_bool_to_inner,wf_to_inner_finv_left])
val finv_bool_to_inner_True = prove(
``^good_context ⇒
(finv bool_to_inner x ⇒ y) ⇒ ((x = True) ⇒ y)``,
metis_tac[finv_bool_to_inner_eq_true,good_context_def]) |> UNDISCH
val _ = save_thm("finv_bool_to_inner_True",finv_bool_to_inner_True)
val [count_mlstring_aux_inj_rwt] = mk_count_aux_inj_rwt[``:mlstring``]
val [count_type_aux_inj_rwt] = mk_count_aux_inj_rwt_ttac[``:type``]
(SOME(WF_REL_TAC`measure type_size`>>gen_tac>>Induct>>
rw[type_size_def]>>res_tac>>simp[]))
val countable_type = inj_rwt_to_countable count_type_aux_inj_rwt
val type_rep_def = new_specification("type_rep_def",["type_rep"],
countable_type |> REWRITE_RULE[countable_def])
val num_def = xDefine"num"`
(num0 ^mem (0:num) = empty mem) ∧
(num0 ^mem (SUC n) = num0 mem n + one mem)`
val _ = Parse.overload_on("num",``num0 ^mem``)
val num_not_one = store_thm("num_not_one",
``is_set_theory ^mem ⇒ ∀n. num n ≠ One``,
strip_tac >> Induct >> rw[num_def] >>
imp_res_tac is_extensional >> fs[extensional_def] >> pop_assum kall_tac >>
rfs[mem_empty,mem_one,mem_upair] >>
`One ≠ ∅` by (
imp_res_tac is_extensional >> fs[extensional_def] >> pop_assum kall_tac >>
simp[mem_one,mem_empty] ) >>
fs[EQ_IMP_THM] >> metis_tac[])
val num_suc_not_empty = store_thm("num_suc_not_empty",
``is_set_theory ^mem ⇒ ∀n. num (SUC n) ≠ ∅``,
rw[num_def] >>
imp_res_tac is_extensional >> fs[extensional_def] >> pop_assum kall_tac >>
simp[mem_empty,mem_upair] >> metis_tac[])
val num_inj = store_thm("num_inj",
``is_set_theory ^mem ⇒
∀m n. (num m = num n) ⇔ (m = n)``,
strip_tac >>
Induct >> simp[num_def] >- (
Cases >> simp[num_def] >> rw[] >>
imp_res_tac is_extensional >>
fs[extensional_def] >> pop_assum kall_tac >>
simp[mem_empty,mem_upair] >>
metis_tac[] ) >>
Cases >> rw[num_def] >- (
imp_res_tac is_extensional >>
fs[extensional_def] >> pop_assum kall_tac >>
simp[mem_empty,mem_upair] >>
metis_tac[] ) >>
fs[EQ_IMP_THM] >> rw[] >>
first_x_assum match_mp_tac >>
imp_res_tac is_extensional >>
fs[extensional_def] >> pop_assum kall_tac >>
rfs[mem_upair] >> metis_tac[num_not_one])
val pair_not_empty = store_thm("pair_not_empty",
``is_set_theory ^mem ⇒ ∀x y. (x,y) ≠ ∅``,
rw[] >>
imp_res_tac is_extensional >> fs[extensional_def] >> pop_assum kall_tac >>
simp[mem_empty,pair_def,mem_upair] >> metis_tac[])
val pair_not_one = store_thm("pair_not_one",
``is_set_theory ^mem ⇒ ∀x y. (x,y) ≠ One``,
rw[] >>
imp_res_tac is_extensional >> fs[extensional_def] >> pop_assum kall_tac >>
rfs[mem_one,mem_empty,mem_upair,pair_def] >> rw[] >>
spose_not_then strip_assume_tac >>
first_x_assum(qspec_then`∅`mp_tac) >> simp[] >>
imp_res_tac is_extensional >> fs[extensional_def] >> pop_assum kall_tac >>
simp[mem_empty,mem_unit,mem_upair] >>
metis_tac[])
val tag_exists = prove(
``∃tag:('U->'U->bool)->type->'U->'U.
∀mem. is_set_theory mem ⇒
(∀ty1 v1 ty2 v2.
(((ty1,v1) ≠ (ty2,v2)) ⇒ tag mem ty1 v1 ≠ tag mem ty2 v2)) ∧
(∀ty v. ∃u. IMAGE (tag mem ty) (ext v) = ext u) ∧
(∀ty v. ¬ (tag mem ty v <: boolset) ∧
(∀x y. ¬ (tag mem ty v <: Funspace x y)))``,
qexists_tac`λmem ty x. Two ∪ (Unit(num (type_rep ty), x))` >>
gen_tac >> strip_tac >> simp[] >>
conj_tac >- (
rpt gen_tac >>
qmatch_abbrev_tac`p ⇒ q` >> strip_tac >>
qunabbrev_tac`q` >>
imp_res_tac is_extensional >>
pop_assum mp_tac >>
simp[extensional_def] >>
disch_then kall_tac >>
simp[mem_binary_union,mem_boolset,mem_unit] >>
qexists_tac`num(type_rep ty1),v1` >>
simp[pair_inj,num_inj] >>
conj_tac >- (
simp[true_def,false_def,pair_not_empty,pair_not_one] ) >>
fs[Abbr`p`] >>
metis_tac[type_rep_def,INJ_DEF,IN_UNIV]) >>
conj_tac >- (
rw[EXTENSION,PULL_EXISTS,ext_def] >>
qexists_tac`Pow (boolset ∪ Unit(num (type_rep ty)) × v) suchthat
λx. ∃a. a <: v ∧ x = boolset ∪ Unit (num (type_rep ty),a)` >>
simp[mem_sub,mem_power] >> gen_tac >>
reverse EQ_TAC >- metis_tac[] >> strip_tac >>
reverse conj_tac >- metis_tac[] >>
rfs[mem_binary_union,mem_unit,mem_product] >> rw[] >>
rw[pair_inj]) >>
rw[] >- (
simp[mem_boolset,true_def,false_def] >>
imp_res_tac is_extensional >> fs[extensional_def] >> pop_assum kall_tac >>
simp[mem_binary_union,mem_unit,mem_boolset,true_def,false_def,mem_empty,mem_one] >>
simp[EQ_IMP_THM] >>
metis_tac[pair_not_empty] ) >>
strip_tac >>
imp_res_tac (UNDISCH in_funspace_abstract) >>
qpat_x_assum`X = Y`mp_tac >>
imp_res_tac is_extensional >> fs[extensional_def] >> pop_assum kall_tac >>
simp[EQ_IMP_THM,EXISTS_OR_THM] >> disj1_tac >>
srw_tac[boolSimps.DNF_ss][mem_binary_union,mem_boolset,true_def] >> disj1_tac >>
simp[abstract_def,mem_sub,mem_product,pair_not_empty])
val tag_def =
new_specification("tag_def",["tag0"],tag_exists)
|> SPEC mem
val _ = overload_on("tag",``tag0 ^mem``)
val _ = save_thm("tag_def",tag_def)
val to_inner_def = xDefine"to_inner"`
to_inner0 ^mem (ty:type) = (tag ty) o (@f. wf_to_inner f)`
val _ = overload_on("to_inner",``to_inner0 ^mem``)
val Var_thm = prove(
``^tmval (x,ty) = inty v ⇒
∀mem. inty v = termsem0 mem ^tmsig ^interpretation ^valuation (Var x ty)``,
rw[termsem_def])
val Const_thm = prove(
``instance ^tmsig ^interpretation name ty ^tyval = inty c ⇒
∀mem. inty c = termsem0 mem ^tmsig ^interpretation ^valuation (Const name ty)``,
rw[termsem_def])
val Comb_thm = prove(
``^good_context ⇒
termsem ^tmsig ^interpretation ^valuation ftm =
fun_to_inner ina inb f ∧
termsem ^tmsig ^interpretation ^valuation xtm = ina x ⇒
wf_to_inner ina ⇒ wf_to_inner inb
⇒
termsem ^tmsig ^interpretation ^valuation (Comb ftm xtm) =
inb (f x)``,
rw[good_context_def,termsem_def] >>
first_assum(SUBST1_TAC o SYM) >>
rw[fun_to_inner_def] >>
match_mp_tac apply_abstract_matchable >>
simp[] >>
rw[wf_to_inner_range_thm] >>
AP_TERM_TAC >>
AP_TERM_TAC >>
match_mp_tac wf_to_inner_finv_left >>
simp[]) |> UNDISCH
val Abs_thm = prove(
``^good_context ⇒ ^is_valuation ⇒
∀ina inb f x xty b bty.
typesem tyass tyval xty = range ina ∧
typesem tyass tyval bty = range inb ⇒
(*
wf_to_inner ina ⇒ (* these are unnecessary for this theorem *)
wf_to_inner inb ⇒ (* but useful for the automation *)
*)
term_ok (tysig,tmsig) b ∧
typeof b = bty ∧
(∀m. m <: range ina ⇒
termsem tmsig (tyass,tmass) (tyval,((x,xty) =+ m) tmval) b =
inb (f (finv ina m)))
⇒
termsem tmsig (tyass,tmass) (tyval,tmval) (Abs (Var x xty) b) =
fun_to_inner ina inb f
``,
rw[termsem_def,fun_to_inner_def,good_context_def] >>
match_mp_tac (UNDISCH abstract_eq) >> simp[] >>
rw[] >>
res_tac >> first_x_assum(SUBST1_TAC o SYM) >>
first_assum(SUBST1_TAC o SYM) >>
match_mp_tac (UNDISCH termsem_typesem) >>
simp[] >>
qexists_tac`(tysig,tmsig)` >> simp[] >>
fs[is_std_interpretation_def] >>
fs[is_valuation_def,is_term_valuation_def] >>
simp[combinTheory.APPLY_UPDATE_THM] >>
rw[] >> metis_tac[]) |> funpow 2 UNDISCH
val save_thms = map2 (curry save_thm)
val _ = save_thms ["Var_thm","Const_thm","Comb_thm","Abs_thm"]
[ Var_thm , Const_thm , Comb_thm , Abs_thm ]
val tyass_bool_thm = prove(
``is_set_theory ^mem ⇒ is_std_type_assignment tyass ==> (tyass (strlit"bool") [] = range bool_to_inner)``,
rw[is_std_type_assignment_def,range_bool_to_inner]) |> funpow 2 UNDISCH
val tyass_fun_thm = prove(
``is_set_theory ^mem ⇒ is_std_type_assignment tyass ==> !tya tyb ina inb.
wf_to_inner ina /\ wf_to_inner inb /\ tya = range ina /\ tyb = range inb ==>
tyass (strlit"fun") [tya; tyb] = range (fun_to_inner ina inb)``,
rw[is_std_type_assignment_def,range_fun_to_inner]
) |> funpow 2 UNDISCH
val good_context_lookup_bool = prove(
``^good_context ⇒ FLOOKUP ^tysig (strlit "bool") = SOME 0``,
rw[good_context_def,is_std_sig_def]) |> UNDISCH
val good_context_lookup_fun = prove(
``^good_context ⇒ FLOOKUP ^tysig (strlit "fun") = SOME 2``,
rw[good_context_def,is_std_sig_def]) |> UNDISCH
val is_valuation_extend = prove(
``^is_valuation ∧ m <: typesem ^tyass ^tyval ty ⇒
is_valuation ^tysig ^tyass (^tyval,(((x,ty) =+ m) ^tmval))``,
rw[is_valuation_def,is_term_valuation_def,combinTheory.APPLY_UPDATE_THM] >>
rw[] >> rw[])
val good_context_instance_equality = prove(
``∀ty ina.
^good_context ∧ ^is_valuation ∧
type_ok ^tysig ty ∧
typesem ^tyass ^tyval ty = range ina ∧
wf_to_inner ina ⇒
instance ^tmsig ^interpretation (strlit"=") (Fun ty (Fun ty Bool)) ^tyval =
fun_to_inner ina (fun_to_inner ina bool_to_inner) $=``,
rw[good_context_def] >>
fs[is_std_sig_def] >>
imp_res_tac instance_def >>
first_x_assum(qspec_then`[ty,Tyvar (strlit"A")]`mp_tac) >>
simp[holSyntaxLibTheory.REV_ASSOCD] >>
disch_then(mp_tac o SPEC interpretation) >>
simp[] >> disch_then kall_tac >>
EVAL_STRING_SORT >> simp[holSyntaxLibTheory.REV_ASSOCD] >>
fs[is_std_interpretation_def,interprets_def] >>
first_x_assum(qspec_then`(strlit"A"=+ typesem ^tyass ^tyval ty)(K boolset)`mp_tac) >>
impl_tac >- (
simp[is_type_valuation_def,combinTheory.APPLY_UPDATE_THM] >>
reverse(rw[mem_boolset]) >- metis_tac[] >>
qpat_x_assum`X = Y` (SUBST1_TAC o SYM) >>
match_mp_tac (UNDISCH typesem_inhabited) >>
fs[is_valuation_def,is_interpretation_def] >>
metis_tac[] ) >>
simp[combinTheory.APPLY_UPDATE_THM,mlstringTheory.implode_def] >>
disch_then kall_tac >>
simp[fun_to_inner_def] >>
match_mp_tac (UNDISCH abstract_eq) >>
simp[] >> gen_tac >> strip_tac >>
conj_tac >- (
match_mp_tac (UNDISCH abstract_in_funspace) >>
simp[boolean_in_boolset] ) >>
Q.ISPECL_THEN[`mem`,`bool_to_inner`,`ina`]mp_tac (GEN_ALL range_fun_to_inner) >>
impl_tac >- ( simp[wf_to_inner_bool_to_inner] ) >>
strip_tac >> simp[range_bool_to_inner] >>
Q.ISPECL_THEN[`mem`,`bool_to_inner`,`ina`]mp_tac (GEN_ALL range_fun_to_inner) >>
impl_tac >- ( simp[wf_to_inner_bool_to_inner] ) >>
strip_tac >> simp[range_bool_to_inner] >>
conj_tac >- (
match_mp_tac (UNDISCH abstract_in_funspace) >>
simp[bool_to_inner_def,boolean_in_boolset] ) >>
match_mp_tac (UNDISCH abstract_eq) >>
simp[bool_to_inner_def,boolean_in_boolset] >>
simp[boolean_def] >> rw[true_neq_false] >>
spose_not_then strip_assume_tac >>
metis_tac[wf_to_inner_finv_right])
val _ = save_thms
["tyass_bool_thm", "tyass_fun_thm",
"good_context_lookup_bool","good_context_lookup_fun",
"is_valuation_extend", "good_context_instance_equality"]
[ tyass_bool_thm , tyass_fun_thm ,
good_context_lookup_bool , good_context_lookup_fun ,
is_valuation_extend, good_context_instance_equality ]
val base_tyval_exists = prove(
``∃τ. ∀mem. is_set_theory mem ⇒ is_type_valuation0 mem (τ mem)``,
rw[GSYM SKOLEM_THM,is_type_valuation_def] >>
qexists_tac`K (one mem)` >> rw[] >>
qexists_tac`empty mem` >>
metis_tac[mem_one])
val base_tyval_prim_def = new_specification("base_tyval_prim_def",["base_tyval0"],base_tyval_exists)
val _ = overload_on("base_tyval",``base_tyval0 ^mem``)
val base_tyval_def = save_thm("base_tyval_def",base_tyval_prim_def |> ISPEC mem |> UNDISCH)
val is_type_valuation_update_list = store_thm("is_type_valuation_update_list",
``∀ls t. is_type_valuation t ⇒ EVERY (inhabited o SND) ls ⇒ is_type_valuation (t =++ ls)``,
simp[AND_IMP_INTRO] >>
Induct >> simp[UPDATE_LIST_THM] >> rw[] >>
first_x_assum match_mp_tac >> rw[] >>
fs[is_type_valuation_def,combinTheory.APPLY_UPDATE_THM] >>
rw[] >> metis_tac[])
val inhabited_range = store_thm("inhabited_range",
``∀inx. wf_to_inner inx ⇒ inhabited (range inx)``,
rw[] >> imp_res_tac wf_to_inner_range_thm >>
metis_tac[] )
val init_model_def = new_specification("init_model_def",["init_model0"],
SIMP_RULE std_ss [GSYM RIGHT_EXISTS_IMP_THM,SKOLEM_THM] (GEN_ALL init_ctxt_has_model))
val _ = overload_on("init_model",``init_model0 ^mem``)
val bool_ctxt_no_new_axioms =
``(∀p. MEM (NewAxiom p) (mk_bool_ctxt init_ctxt) ⇒
MEM (NewAxiom p) init_ctxt)``
|> (EVAL THENC (SIMP_CONV std_ss []))
|> EQT_ELIM
val bool_model_exists =
extends_consistent
|> UNDISCH
|> Q.SPECL[`init_ctxt`,`mk_bool_ctxt init_ctxt`]
|> C MATCH_MP bool_extends_init
|> SPEC ``init_model``
|> REWRITE_RULE[GSYM AND_IMP_INTRO]
|> C MATCH_MP init_theory_ok
|> C MATCH_MP (UNDISCH (SPEC mem init_model_def))
|> C MATCH_MP bool_ctxt_no_new_axioms
|> DISCH_ALL |> GEN_ALL
|> SIMP_RULE std_ss [GSYM RIGHT_EXISTS_IMP_THM,SKOLEM_THM]
val bool_model_def = new_specification("bool_model_def",["bool_model0"],
bool_model_exists)
val _ = overload_on("bool_model",``bool_model0 ^mem``)
val bool_model_models = UNDISCH (SPEC mem bool_model_def)
val bool_theory_ok =
extends_theory_ok
|> Q.SPECL[`init_ctxt`,`mk_bool_ctxt init_ctxt`]
|> SIMP_RULE std_ss [bool_extends_init,init_theory_ok]
val bool_model_interpretation =
bool_has_bool_interpretation
|> UNDISCH
|> Q.SPEC`init_ctxt`
|> Q.SPEC`bool_model`
|> SIMP_RULE std_ss [bool_model_models,bool_theory_ok]
val _ = map2 (curry save_thm)
["bool_theory_ok","bool_model_interpretation","bool_model_models"]
[ bool_theory_ok , bool_model_interpretation , bool_model_models]
val not_thm = prove(
``is_set_theory ^mem ⇒
(Abstract boolset boolset (λx. Boolean (¬finv bool_to_inner x)) =
Abstract boolset boolset (λp. Boolean (p ≠ True)))``,
rw[] >>
match_mp_tac (UNDISCH abstract_eq) >>
simp[boolean_in_boolset] >>
rw[boolean_def] >>
metis_tac[finv_bool_to_inner_eq_true])
val fun_to_inner_not =
``fun_to_inner bool_to_inner bool_to_inner $~``
|> SIMP_CONV std_ss [Once bool_to_inner_def,fun_to_inner_def,UNDISCH range_bool_to_inner,UNDISCH not_thm]
val bool_to_inner_false =
``bool_to_inner F``
|> SIMP_CONV std_ss [bool_to_inner_def,boolean_def]
val bool_to_inner_true =
``bool_to_inner T``
|> SIMP_CONV std_ss [bool_to_inner_def,boolean_def]
val range_fun_to_inner_ina_bool_to_inner =
range_fun_to_inner |> GEN_ALL |> SPEC mem
|> Q.ISPECL[`bool_to_inner`,`ina:'a -> 'U`]
|> SIMP_RULE std_ss [UNDISCH wf_to_inner_bool_to_inner,GSYM AND_IMP_INTRO]
|> UNDISCH |> UNDISCH
val forall_thm = prove(
``is_set_theory ^mem ⇒ wf_to_inner ina ⇒
(Abstract (Funspace (range ina) boolset) boolset
(λP. Boolean (∀x. x <: range ina ⇒ Holds P x)) =
Abstract (Funspace (range ina) boolset) boolset
(λx. bool_to_inner ($! (finv (fun_to_inner ina bool_to_inner) x))))``,
rw[] >>
match_mp_tac (UNDISCH abstract_eq) >>
rw[boolean_in_boolset,Once bool_to_inner_def] >>
rw[Once bool_to_inner_def] >> AP_TERM_TAC >>
`∃f. (x = fun_to_inner ina Boolean (λa. (f (ina a)) = True)) ∧
(∀a. f (ina a) <: boolset)` by (
simp[UNDISCH range_bool_to_inner,fun_to_inner_def,GSYM bool_to_inner_def] >>
qspecl_then[`x`,`range ina`,`boolset`]mp_tac (UNDISCH in_funspace_abstract) >>
impl_tac >- metis_tac[inhabited_range,mem_boolset] >> rw[] >>
qexists_tac`f` >> simp[bool_to_inner_def] >>
reverse conj_tac >- metis_tac[wf_to_inner_range_thm] >>
match_mp_tac (UNDISCH abstract_eq) >>
simp[boolean_in_boolset] >> rw[] >>
simp[boolean_def] >> rw[] >>
imp_res_tac wf_to_inner_finv_right >>
metis_tac[mem_boolset] ) >>
Q.ISPEC_THEN`fun_to_inner ina Boolean`mp_tac wf_to_inner_finv_left >>
impl_tac >- metis_tac[wf_to_inner_fun_to_inner,wf_to_inner_bool_to_inner,bool_to_inner_def] >>
simp[holds_def,GSYM bool_to_inner_def] >>
disch_then kall_tac >>
rw[EQ_IMP_THM] >- (
first_x_assum(qspec_then`ina a`mp_tac) >>
impl_tac >- metis_tac[wf_to_inner_range_thm] >>
simp[fun_to_inner_def] >>
disch_then (SUBST1_TAC o SYM) >>
match_mp_tac EQ_SYM >>
match_mp_tac apply_abstract_matchable >>
simp[wf_to_inner_range_thm,GSYM bool_to_inner_def,range_bool_to_inner] >>
simp[bool_to_inner_def,boolean_in_boolset] >>
Q.ISPEC_THEN`ina`mp_tac wf_to_inner_finv_left >> rw[] >>
rw[boolean_def] >>
metis_tac[mem_boolset] ) >>
rw[fun_to_inner_def] >>
match_mp_tac apply_abstract_matchable >>
simp[wf_to_inner_range_thm,GSYM bool_to_inner_def,range_bool_to_inner] >>
simp[bool_to_inner_def,boolean_in_boolset] >>
rw[boolean_def]) |> UNDISCH |> UNDISCH
val fun_to_inner_forall =
``fun_to_inner (fun_to_inner ina bool_to_inner) bool_to_inner $!``
|> SIMP_CONV std_ss [fun_to_inner_def,UNDISCH range_bool_to_inner,range_fun_to_inner_ina_bool_to_inner,GSYM forall_thm]
val exists_thm = prove(
``is_set_theory ^mem ⇒ wf_to_inner ina ⇒
(Abstract (Funspace (range ina) boolset) boolset
(λP. Boolean (?x. x <: range ina ∧ Holds P x)) =
Abstract (Funspace (range ina) boolset) boolset
(λx. bool_to_inner ($? (finv (fun_to_inner ina bool_to_inner) x))))``,
rw[] >>
match_mp_tac (UNDISCH abstract_eq) >>
rw[boolean_in_boolset,Once bool_to_inner_def] >>
rw[Once bool_to_inner_def] >> AP_TERM_TAC >>
`∃f. (x = fun_to_inner ina Boolean (λa. (f (ina a)) = True)) ∧
(∀a. f (ina a) <: boolset)` by (
simp[UNDISCH range_bool_to_inner,fun_to_inner_def,GSYM bool_to_inner_def] >>
qspecl_then[`x`,`range ina`,`boolset`]mp_tac (UNDISCH in_funspace_abstract) >>
impl_tac >- metis_tac[inhabited_range,mem_boolset] >> rw[] >>
qexists_tac`f` >> simp[bool_to_inner_def] >>
reverse conj_tac >- metis_tac[wf_to_inner_range_thm] >>
match_mp_tac (UNDISCH abstract_eq) >>
simp[boolean_in_boolset] >> rw[] >>
simp[boolean_def] >> rw[] >>
imp_res_tac wf_to_inner_finv_right >>
metis_tac[mem_boolset] ) >>
Q.ISPEC_THEN`fun_to_inner ina Boolean`mp_tac wf_to_inner_finv_left >>
impl_tac >- metis_tac[wf_to_inner_fun_to_inner,wf_to_inner_bool_to_inner,bool_to_inner_def] >>
simp[holds_def,GSYM bool_to_inner_def] >>
disch_then kall_tac >>
rw[EQ_IMP_THM] >- (
qmatch_assum_rename_tac`z <: range ina` >>
qexists_tac`finv ina z` >>
pop_assum mp_tac >>
simp[fun_to_inner_def] >>
disch_then (SUBST1_TAC o SYM) >>
match_mp_tac EQ_SYM >>
match_mp_tac apply_abstract_matchable >>
simp[wf_to_inner_range_thm,GSYM bool_to_inner_def,range_bool_to_inner] >>
simp[bool_to_inner_def,boolean_in_boolset] >>
rw[boolean_def] >>
metis_tac[mem_boolset] ) >>
rw[fun_to_inner_def] >>
qexists_tac`ina a` >>
conj_tac >- metis_tac[wf_to_inner_range_thm] >>
match_mp_tac apply_abstract_matchable >>
simp[wf_to_inner_range_thm,GSYM bool_to_inner_def,range_bool_to_inner] >>
simp[bool_to_inner_def,boolean_in_boolset] >>
rw[boolean_def] >>
metis_tac[wf_to_inner_finv_left]) |> UNDISCH |> UNDISCH
val fun_to_inner_exists =
``fun_to_inner (fun_to_inner ina bool_to_inner) bool_to_inner $?``
|> SIMP_CONV std_ss [fun_to_inner_def,UNDISCH range_bool_to_inner,range_fun_to_inner_ina_bool_to_inner,GSYM exists_thm]
val range_fun_to_inner_bool_to_inner_bool_to_inner =
range_fun_to_inner |> GEN_ALL |> SPEC mem
|> Q.ISPECL[`bool_to_inner`,`bool_to_inner`]
|> SIMP_RULE std_ss [UNDISCH wf_to_inner_bool_to_inner]
|> UNDISCH
val binop_thm1 = prove(
``is_set_theory ^mem ∧ p <: boolset ⇒
(Abstract boolset boolset (λx. bool_to_inner (op (finv bool_to_inner p) (finv bool_to_inner x))) =
Abstract boolset boolset (λq. Boolean (op (p = True) (q = True))))``,
rw[] >>
match_mp_tac (UNDISCH abstract_eq) >>
rw[boolean_in_boolset] >>
rw[Once bool_to_inner_def,boolean_in_boolset] >>
`EVERY (λz. z = True ⇔ finv bool_to_inner z) [p;x]` by (
simp[] >> metis_tac[finv_bool_to_inner_eq_true]) >>
fs[boolean_def])
val binop_thm = prove(
``is_set_theory ^mem ⇒
(Abstract boolset (Funspace boolset boolset)
(λy. Abstract boolset boolset (λx. bool_to_inner (op (finv bool_to_inner y) (finv bool_to_inner x)))) =
Abstract boolset (Funspace boolset boolset)
(λp. Abstract boolset boolset (λq. Boolean (op (p = True) (q = True)))))``,
rw[] >>
match_mp_tac (UNDISCH abstract_eq) >>
rw[binop_thm1] >>
match_mp_tac (UNDISCH abstract_in_funspace) >>
rw[boolean_in_boolset])
val fun_to_inner_binop =
``fun_to_inner bool_to_inner (fun_to_inner bool_to_inner bool_to_inner) op``
|> SIMP_CONV std_ss [fun_to_inner_def,UNDISCH range_bool_to_inner,range_fun_to_inner_bool_to_inner_bool_to_inner,UNDISCH binop_thm]
val fun_to_inner_select = prove(
``is_set_theory ^mem ⇒ wf_to_inner ina ⇒
(fun_to_inner (fun_to_inner ina bool_to_inner) ina $@ =
Abstract (range (fun_to_inner ina bool_to_inner)) (range ina)
(λp. ina (@x. Holds p (ina x))))``,
rw[fun_to_inner_def] >>
match_mp_tac (UNDISCH abstract_eq) >>
simp[bool_to_inner_def,boolean_in_boolset] >>
simp[wf_to_inner_range_thm] >>
simp[GSYM bool_to_inner_def] >>
Q.ISPEC_THEN`bool_to_inner`mp_tac(Q.GEN`inb`range_fun_to_inner) >>
impl_tac >- metis_tac[wf_to_inner_bool_to_inner] >> rw[] >>
AP_TERM_TAC >> AP_TERM_TAC >>
qmatch_abbrev_tac`l = r` >>
qsuff_tac`fun_to_inner ina bool_to_inner l = fun_to_inner ina bool_to_inner r` >- (
`wf_to_inner (fun_to_inner ina bool_to_inner)` by metis_tac[wf_to_inner_fun_to_inner,wf_to_inner_bool_to_inner] >>
fs[wf_to_inner_def,BIJ_DEF,INJ_DEF] ) >>
Q.ISPEC_THEN`fun_to_inner ina bool_to_inner`mp_tac wf_to_inner_finv_right >>
impl_tac >- metis_tac[wf_to_inner_fun_to_inner,wf_to_inner_bool_to_inner] >>
simp[range_fun_to_inner] >> disch_then(qspec_then`x`mp_tac) >>
impl_tac >- simp[] >>
simp[Abbr`l`] >> disch_then kall_tac >>
simp[Abbr`r`] >>
Q.ISPECL_THEN[`x`,`range ina`,`range bool_to_inner`]mp_tac(UNDISCH in_funspace_abstract) >>
impl_tac >- ( metis_tac[inhabited_range,wf_to_inner_bool_to_inner] ) >>
rw[] >>
simp[fun_to_inner_def] >>
match_mp_tac (UNDISCH abstract_eq) >>
simp[range_bool_to_inner] >>
simp[bool_to_inner_def,boolean_in_boolset] >>
rw[holds_def] >>
qmatch_abbrev_tac`f x = Boolean (b:'U = True)` >>
`b = f x` by (
simp[Abbr`b`] >>
match_mp_tac apply_abstract_matchable >>
metis_tac[wf_to_inner_finv_right,range_bool_to_inner] ) >>
rw[boolean_def] >>
metis_tac[range_bool_to_inner,mem_boolset]) |> UNDISCH
local
val dest_fun_to_inner = dest_triop ``fun_to_inner0`` (mk_HOL_ERR"""dest_fun_to_inner""")
val range_fun_to_inner0 =
range_fun_to_inner
|> Q.GENL[`inb`,`ina`,`mem`]
|> SIMP_RULE std_ss [GSYM AND_IMP_INTRO]
in
fun range_fun_to_inner_conv tm =
let
val fun_to_inner_ina_inb = rand tm
val (mem,ina,inb) = dest_fun_to_inner fun_to_inner_ina_inb
val th = ISPECL[mem,ina,inb] range_fun_to_inner0 |> funpow 3 UNDISCH
in
REWR_CONV th tm
end
end
val fun_to_inner_equals = prove(
``is_set_theory ^mem ⇒ wf_to_inner ina ⇒
(fun_to_inner ina (fun_to_inner ina bool_to_inner) $= =
Abstract (range ina) (Funspace (range ina) boolset)
(λx. Abstract (range ina) boolset (λy. Boolean (x = y))))``,
rw[] >>
rw[fun_to_inner_def] >>
assume_tac (UNDISCH wf_to_inner_bool_to_inner) >>
CONV_TAC(DEPTH_CONV range_fun_to_inner_conv) >>
simp[range_bool_to_inner] >>
match_mp_tac (UNDISCH abstract_eq) >>
simp[] >> rw[] >>
TRY (
match_mp_tac (UNDISCH abstract_in_funspace) >>
simp[bool_to_inner_def,boolean_in_boolset] ) >>
match_mp_tac (UNDISCH abstract_eq) >>
simp[bool_to_inner_def,boolean_in_boolset] >>
rw[boolean_def] >>
metis_tac[wf_to_inner_finv_right]) |> funpow 2 UNDISCH
val _ = map2 (curry save_thm)
["fun_to_inner_not","fun_to_inner_forall","fun_to_inner_exists","fun_to_inner_binop","bool_to_inner_false","bool_to_inner_true","fun_to_inner_select","fun_to_inner_equals"]
[ fun_to_inner_not , fun_to_inner_forall , fun_to_inner_exists , fun_to_inner_binop , bool_to_inner_false , bool_to_inner_true , fun_to_inner_select , fun_to_inner_equals ]
val std_sig_instances = store_thm("std_sig_instances",
``is_std_sig sig ⇒
(instance (tmsof sig) i (strlit"=") (Fun ty (Fun ty Bool)) =
(λτ. tmaof i (strlit"=") [typesem (tyaof i) τ ty]))``,
rw[is_std_sig_def] >>
Q.ISPECL_THEN[`tmsof sig`,`i`,`strlit"="`]mp_tac instance_def >> simp[] >>
disch_then(qspec_then`[ty,Tyvar (strlit"A")]`mp_tac) >>
EVAL_TAC >> simp[])
val is_select_sig_def = Define`
is_select_sig sig ⇔
is_bool_sig sig ∧
(FLOOKUP (tmsof sig) (strlit"@") = SOME (Fun (Fun (Tyvar (strlit"A")) Bool) (Tyvar (strlit"A"))))`
val select_sig_instances = store_thm("select_sig_instances",
``is_select_sig sig ⇒
(instance (tmsof sig) i (strlit"@") (Fun (Fun ty Bool) ty) =
(λτ. tmaof i (strlit"@") [typesem (tyaof i) τ ty]))``,
rw[is_select_sig_def] >>
Q.ISPECL_THEN[`tmsof sig`,`i`,`strlit"@"`]mp_tac instance_def >> simp[] >>
disch_then(qspec_then`[ty,Tyvar (strlit"A")]`mp_tac) >>
EVAL_TAC >> rw[])
val bool_sig_defs = [is_true_sig_def,is_false_sig_def,is_implies_sig_def,
is_and_sig_def,is_or_sig_def,is_not_sig_def,is_forall_sig_def,is_exists_sig_def]
val select_has_select_sig = store_thm("select_has_select_sig",
``is_bool_sig (sigof ctxt) ⇒ is_select_sig (sigof (mk_select_ctxt ctxt))``,
rw[is_select_sig_def] >- (
fs([is_bool_sig_def,mk_select_ctxt_def,FLOOKUP_UPDATE]@bool_sig_defs) >>
fs[is_std_sig_def,FLOOKUP_UPDATE] ) >>
EVAL_TAC)
val eta_theory_ok = prove(
``theory_ok (thyof (mk_eta_ctxt (mk_bool_ctxt init_ctxt)))``,
match_mp_tac (MP_CANON extends_theory_ok) >>
REWRITE_TAC[Once CONJ_COMM] >>
match_exists_tac (concl bool_theory_ok) >>
conj_tac >- ACCEPT_TAC bool_theory_ok >>
match_mp_tac eta_extends >>
match_mp_tac is_bool_sig_std >>
match_mp_tac bool_has_bool_sig >>
ACCEPT_TAC (MATCH_MP theory_ok_sig init_theory_ok |> SIMP_RULE std_ss[]) )
val select_model_exists = prove(
``∃f. ∀mem select. is_set_theory mem ⇒ good_select select ⇒
equal_on (sigof (mk_eta_ctxt (mk_bool_ctxt init_ctxt))) (bool_model0 mem) (f mem select) ∧
f mem select models thyof (mk_select_ctxt (mk_eta_ctxt (mk_bool_ctxt init_ctxt))) ∧
(tmaof (f mem select) (strlit"@") = λls.
Abstract (Funspace (HD ls) boolset) (HD ls)
(λp. select (HD ls) (Holds p)))``,
rw[GSYM SKOLEM_THM,RIGHT_EXISTS_IMP_THM] >>
qspec_then`mk_eta_ctxt (mk_bool_ctxt init_ctxt)`mp_tac(UNDISCH select_has_model_gen) >>
impl_keep_tac >- (
simp[eta_theory_ok] >>
EVAL_TAC ) >>
disch_then match_mp_tac >>
conj_asm1_tac >- (
match_mp_tac (MP_CANON (UNDISCH eta_has_model)) >>
conj_tac >- (
match_mp_tac is_bool_sig_std >>
match_mp_tac bool_has_bool_sig >>
ACCEPT_TAC (MATCH_MP theory_ok_sig init_theory_ok |> SIMP_RULE std_ss[]) ) >>
simp[bool_model_def] ) >>
assume_tac bool_model_interpretation >>
fs[is_bool_interpretation_def])
val select_model_def = new_specification("select_model_def",["select_model0"],
select_model_exists)
val _ = overload_on("select_model",``select_model0 ^mem``)
val select_model_models = GEN_ALL (UNDISCH (SPEC_ALL (SPEC mem select_model_def)))
val select_extends_bool = prove(
``mk_select_ctxt (mk_bool_ctxt init_ctxt) extends mk_bool_ctxt init_ctxt``,
match_mp_tac select_extends >>
conj_tac >- (
match_mp_tac is_bool_sig_std >>
match_mp_tac bool_has_bool_sig >>
ACCEPT_TAC (MATCH_MP theory_ok_sig init_theory_ok |> SIMP_RULE std_ss[]) ) >>
EVAL_TAC )
val select_extends_eta = prove(
``mk_select_ctxt (mk_eta_ctxt (mk_bool_ctxt init_ctxt)) extends mk_eta_ctxt (mk_bool_ctxt init_ctxt)``,
match_mp_tac select_extends >>
conj_tac >- (
ACCEPT_TAC (MATCH_MP theory_ok_sig eta_theory_ok |> SIMP_RULE std_ss[])) >>
EVAL_TAC )
val select_theory_ok =
extends_theory_ok
|> Q.SPECL[`mk_eta_ctxt (mk_bool_ctxt init_ctxt)`,`mk_select_ctxt (mk_eta_ctxt (mk_bool_ctxt init_ctxt))`]
|> SIMP_RULE std_ss [eta_theory_ok,select_extends_eta]
val bool_interpretation_defs =
[is_true_interpretation_def,
is_and_interpretation_def,
is_implies_interpretation_def,
is_forall_interpretation_def,
is_exists_interpretation_def,
is_or_interpretation_def,
is_false_interpretation_def,
is_not_interpretation_def]
val extends_bool_interpretation = prove(
``is_set_theory ^mem ⇒
∀model.
is_std_interpretation model ∧
equal_on (sigof (mk_bool_ctxt init_ctxt)) bool_model model ⇒
is_bool_interpretation model``,
rw[] >>
assume_tac bool_model_interpretation >>
fs([equal_on_def,is_bool_interpretation_def]@bool_interpretation_defs) >>
fs[term_ok_def] >>
rpt conj_tac >>
qmatch_abbrev_tac`tmaof model interprets name on args as val` >>
first_x_assum(qspec_then`name`mp_tac) >>
qunabbrev_tac`name` >>
CONV_TAC(LAND_CONV(LAND_CONV EVAL)) >>
simp[Abbr`args`,Abbr`val`,type_ok_def,FLOOKUP_UPDATE] >>
fs[interprets_def] >> rw[] >>
TRY( first_x_assum match_mp_tac >>
metis_tac[base_tyval_def] ) >>
fs[PULL_EXISTS,type_ok_def,FLOOKUP_UPDATE] >>
first_x_assum(qspec_then`[]`mp_tac)>>
(impl_tac >- EVAL_TAC) >> rw[]) |> UNDISCH
val select_bool_interpretation = prove(
``is_set_theory ^mem ⇒
good_select select ⇒
is_bool_interpretation (select_model select)``,
rw[] >>
match_mp_tac (MP_CANON extends_bool_interpretation) >>
first_assum(strip_assume_tac o MATCH_MP select_model_models) >>
conj_tac >- fs[models_def] >>
match_mp_tac equal_on_reduce >>
fs[mk_eta_ctxt_def] >>
qexists_tac`[]`>>simp[]) |> UNDISCH |> UNDISCH
val infinity_extends_select = prove(
``mk_infinity_ctxt (mk_select_ctxt (mk_eta_ctxt (mk_bool_ctxt init_ctxt))) extends
(mk_select_ctxt (mk_eta_ctxt (mk_bool_ctxt init_ctxt)))``,
match_mp_tac infinity_extends >>
conj_tac >- (
ACCEPT_TAC select_theory_ok ) >>
EVAL_TAC)
val hol_theory_ok = save_thm("hol_theory_ok",
extends_theory_ok
|> Q.SPECL[`mk_select_ctxt (mk_eta_ctxt (mk_bool_ctxt init_ctxt))`,`mk_infinity_ctxt (mk_select_ctxt (mk_eta_ctxt (mk_bool_ctxt init_ctxt)))`]
|> SIMP_RULE std_ss [select_theory_ok,infinity_extends_select]
|> SIMP_RULE std_ss [GSYM hol_ctxt_def])
(* probably not true
val is_bool_interpretation_subinterpretation = store_thm("is_bool_interpretation_subinterpretation",
``is_set_theory ^mem ⇒
(is_bool_interpretation model ⇔
subinterpretation (mk_bool_ctxt init_ctxt) bool_model model)``,
strip_tac >> EQ_TAC >> strip_tac >- (
simp[subinterpretation_def] >>
assume_tac bool_model_interpretation >>
simp[term_ok_def,type_ok_def] >>
conj_tac >> rpt gen_tac >>
CONV_TAC(LAND_CONV EVAL) >>
rw[] >>
fs[is_bool_interpretation_def,is_std_interpretation_def,is_std_type_assignment_def] >>
fs[interprets_nil,type_ok_def,FLOOKUP_UPDATE] >>
fs[]
)
*)
val good_select_extend_base_select = store_thm("good_select_extend_base_select",
``∀ina. wf_to_inner ina ⇒
∀s. good_select s ⇒
good_select ((range ina =+ (λp. ina (@x. p (ina x)))) s)``,
rw[good_select_def,APPLY_UPDATE_THM] >> rw[] >>
TRY (
SELECT_ELIM_TAC >> simp[] >>
qexists_tac`finv ina x` >>
metis_tac[wf_to_inner_finv_right] ) >>
metis_tac[wf_to_inner_range_thm])
val select_instance_thm = prove(
``is_set_theory ^mem ⇒
is_select_sig ^signatur ⇒
good_select select_fun ⇒
(select_fun (range inty) = λp. inty (@x. p (inty x))) ⇒
(typesem (tyaof (select_model select_fun)) ^tyval ty = range inty) ⇒
wf_to_inner inty
⇒
(instance ^tmsig (select_model select_fun) (strlit "@") (Fun (Fun ty Bool) ty) ^tyval =
fun_to_inner (fun_to_inner inty bool_to_inner) inty $@)``,
rw[is_select_sig_def] >>
qspecl_then[`tmsig`,`select_model select_fun`,`strlit"@"`]mp_tac instance_def >>
simp[] >>