statement stringlengths 4 1.37k | proof stringclasses 1
value | type stringclasses 6
values | symbolic_name stringlengths 1 27 | library stringclasses 4
values | filename stringclasses 35
values | imports listlengths 0 0 | deps listlengths 0 25 | docstring stringclasses 1
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|---|---|---|---|---|---|---|---|---|---|---|
wf_type : TY_ENV → TYPE SET
----
tyenv •
wf_type tyenv = ¥{tyset|
(s • mk_var_type s tyset)
∧
s tyl • s í Length tyl tyenv ∧ ( t • t Elems tyl ⇒ t tyset)
⇒ mk_type(s, tyl) tyset} | HOLCONST | wf_type | hol | src/hol/spc001.doc | [] | [
"Elems",
"mk_type",
"mk_var_type"
] | https://github.com/RobArthan/pp | dabebf8fdbdede1a3ad6333500c6663ecd15c98f | ||
wf_term: TY_ENV → CON_ENV → TERM SET
----
tyenv conenv •
wf_term tyenv conenv = ¥{tmset|
(s ty • ty wf_type tyenv ⇒ mk_var(s, ty) tmset)
∧
(s ty • (ty wf_type tyenv ∧ ty' tysubs • s í ty' conenv ∧ inst_type tysubs ty' = ty)
⇒ mk_const(s, ty) tmset)
∧
(f a tm • (has_mk_comb(f, a) tm ... | HOLCONST | wf_term | hol | src/hol/spc001.doc | [] | [
"has_mk_abs",
"has_mk_comb",
"inst_type",
"mk_const",
"mk_var",
"wf_type"
] | https://github.com/RobArthan/pp | dabebf8fdbdede1a3ad6333500c6663ecd15c98f | ||
wf_seq: TY_ENV → CON_ENV → SEQ SET
----
seq tyenv conenv •
seq wf_seq tyenv conenv ¤
let ok = {tm | tm wf_term tyenv conenv ∧ type_of_term tm = Bool}
in concl seq ok ∧ tm • tm hyp seq ⇒ tm ok | HOLCONST | wf_seq | hol | src/hol/spc001.doc | [] | [
"Bool",
"concl",
"hyp",
"seq",
"type_of_term",
"wf_term"
] | https://github.com/RobArthan/pp | dabebf8fdbdede1a3ad6333500c6663ecd15c98f | ||
wf_seqs: TY_ENV → CON_ENV → SEQS SET
----
tyenv conenv •
wf_seqs tyenv conenv = ℙ (wf_seq tyenv conenv) | HOLCONST | wf_seqs | hol | src/hol/spc001.doc | [] | [
"wf_seq"
] | https://github.com/RobArthan/pp | dabebf8fdbdede1a3ad6333500c6663ecd15c98f | ||
wf_con_env: TY_ENV → CON_ENV SET
----
conenv tyenv •
conenv wf_con_env tyenv
¤ conenv Functional
∧ con ty • con í ty conenv ⇒ ty wf_type tyenv | HOLCONST | wf_con_env | hol | src/hol/spc001.doc | [] | [
"Functional",
"wf_type"
] | https://github.com/RobArthan/pp | dabebf8fdbdede1a3ad6333500c6663ecd15c98f | ||
wf_ty_env: TY_ENV SET
----
wf_ty_env = Functional | HOLCONST | wf_ty_env | hol | src/hol/spc001.doc | [] | [
"Functional"
] | https://github.com/RobArthan/pp | dabebf8fdbdede1a3ad6333500c6663ecd15c98f | ||
is_theory: (TY_ENV × CON_ENV × SEQS) SET
----
ty_env con_env axioms •
(ty_env, con_env, axioms) is_theory ¤
ty_env wf_ty_env ∧
con_env wf_con_env ty_env ∧
axioms wf_seqs ty_env con_env | HOLCONST | is_theory | hol | src/hol/spc001.doc | [] | [
"axioms",
"wf_con_env",
"wf_seqs",
"wf_ty_env"
] | https://github.com/RobArthan/pp | dabebf8fdbdede1a3ad6333500c6663ecd15c98f | ||
axioms : THEORY → SEQS
----
thy • axioms thy = Snd(Snd(rep_theory thy)) | HOLCONST | axioms | hol | src/hol/spc001.doc | [] | [] | https://github.com/RobArthan/pp | dabebf8fdbdede1a3ad6333500c6663ecd15c98f | ||
types : THEORY → TY_ENV
----
thy • types thy = Fst(rep_theory thy) | HOLCONST | types | hol | src/hol/spc001.doc | [] | [] | https://github.com/RobArthan/pp | dabebf8fdbdede1a3ad6333500c6663ecd15c98f | ||
constants : THEORY → CON_ENV
----
thy • constants thy = Fst(Snd(rep_theory thy)) | HOLCONST | constants | hol | src/hol/spc001.doc | [] | [] | https://github.com/RobArthan/pp | dabebf8fdbdede1a3ad6333500c6663ecd15c98f | ||
sequents : THEORY → SEQS
----
seq thy •
seq sequents thy ¤
seq wf_seq (types thy) (constants thy) | HOLCONST | sequents | hol | src/hol/spc001.doc | [] | [
"constants",
"seq",
"types",
"wf_seq"
] | https://github.com/RobArthan/pp | dabebf8fdbdede1a3ad6333500c6663ecd15c98f | ||
extensional : ('U → 'U → BOOL) → BOOL
----
mem • extensional mem ¤
x y:'U • x = y ¤ a:'U • mem a x ¤ mem a y | HOLCONST | extensional | hol | src/hol/spc002.doc | [] | [
"mem"
] | https://github.com/RobArthan/pp | dabebf8fdbdede1a3ad6333500c6663ecd15c98f | ||
is_separation : ('U → 'U → BOOL) → ('U → ('U → BOOL) → 'U) → BOOL
----
mem sub • is_separation mem sub ¤
x P • a • mem a (sub x P) ¤ mem a x ∧ P a | HOLCONST | is_separation | hol | src/hol/spc002.doc | [] | [
"mem",
"sub"
] | https://github.com/RobArthan/pp | dabebf8fdbdede1a3ad6333500c6663ecd15c98f | ||
is_power : ('U → 'U → BOOL) → ('U → 'U) → BOOL
----
mem power • is_power mem power ¤
x • a • mem a (power x) ¤ b • mem b a ⇒ mem b x | HOLCONST | is_power | hol | src/hol/spc002.doc | [] | [
"mem"
] | https://github.com/RobArthan/pp | dabebf8fdbdede1a3ad6333500c6663ecd15c98f | ||
is_union : ('U → 'U → BOOL) → ('U → 'U) → BOOL
----
mem union • is_union mem union ¤
x • a • mem a (union x) ¤ b • mem a b ∧ mem b x | HOLCONST | is_union | hol | src/hol/spc002.doc | [] | [
"mem"
] | https://github.com/RobArthan/pp | dabebf8fdbdede1a3ad6333500c6663ecd15c98f | ||
is_pair : ('U → 'U → BOOL) → ('U → 'U → 'U) → BOOL
----
mem pair • is_pair mem pair ¤
x y • a • mem a (pair x y) ¤ a = x ² a = y | HOLCONST | is_pair | hol | src/hol/spc002.doc | [] | [
"mem"
] | https://github.com/RobArthan/pp | dabebf8fdbdede1a3ad6333500c6663ecd15c98f | ||
is_set_theory : ('U → 'U → BOOL) → BOOL
----
mem • is_set_theory mem ¤
extensional mem
∧ ( sub • is_separation mem sub)
∧ ( power • is_power mem power)
∧ ( union • is_union mem union)
∧ ( pair • is_pair mem pair) | HOLCONST | is_set_theory | hol | src/hol/spc002.doc | [] | [
"extensional",
"is_pair",
"is_power",
"is_separation",
"is_union",
"mem",
"sub"
] | https://github.com/RobArthan/pp | dabebf8fdbdede1a3ad6333500c6663ecd15c98f | ||
Subs : 'U → ('U → BOOL) → 'U
----
( mem:'U → 'U → BOOL • is_set_theory mem) ⇒
is_separation ($s) (Subs:'U → ('U → BOOL) → 'U) | HOLCONST | Subs | hol | src/hol/spc002.doc | [] | [
"is_separation",
"is_set_theory",
"mem"
] | https://github.com/RobArthan/pp | dabebf8fdbdede1a3ad6333500c6663ecd15c98f | ||
Units : 'U → 'U
----
x:'U • Units x = x «s x | HOLCONST | Units | hol | src/hol/spc002.doc | [] | [] | https://github.com/RobArthan/pp | dabebf8fdbdede1a3ad6333500c6663ecd15c98f | ||
Bools : 'U LIST → 'U
----
(Bools [] = 2s)
∧ (x t • Bools (Cons x t) = s) | HOLCONST | Bools | hol | src/hol/spc002.doc | [] | [
"Bool"
] | https://github.com/RobArthan/pp | dabebf8fdbdede1a3ad6333500c6663ecd15c98f | ||
Funs : 'U LIST → 'U
----
args • Funs args =
let dom = Hd args
in let rng = Hd (Tl args)
in if args = [dom; rng] ∧ ³dom = s ∧ ³rng = s
then dom → s rng
else s | HOLCONST | Funs | hol | src/hol/spc002.doc | [] | [
"Fun"
] | https://github.com/RobArthan/pp | dabebf8fdbdede1a3ad6333500c6663ecd15c98f | ||
Abss : ('U → 'U) → 'U → 'U → 'U
----
f dom rng • Abss f dom rng = Subs (dom × s rng) ( ⇔ x • a • x = a ís f a) | HOLCONST | Abss | hol | src/hol/spc002.doc | [] | [] | https://github.com/RobArthan/pp | dabebf8fdbdede1a3ad6333500c6663ecd15c98f | ||
EqRels : 'U → 'U
----
x:'U • EqRels x = Subs (x × s x × s 2s)
( ⇔ a • b c • a = (b ís c ís if b = c then 1s else s)) | HOLCONST | EqRels | hol | src/hol/spc002.doc | [] | [] | https://github.com/RobArthan/pp | dabebf8fdbdede1a3ad6333500c6663ecd15c98f | ||
ImpRels : 'U
----
ImpRels =
Units(1s ís 1s ís 1s)
Às Units(1s ís s ís s)
Às Units(s ís 1s ís 1s)
Às Units(s ís s ís 1s) | HOLCONST | ImpRels | hol | src/hol/spc002.doc | [] | [] | https://github.com/RobArthan/pp | dabebf8fdbdede1a3ad6333500c6663ecd15c98f | ||
is_wf_ty_con_assignment: TY_ENV → 'U TY_CON_ASSIGNMENT → BOOL
----
tyenv tyconass • is_wf_ty_con_assignment tyenv tyconass ¤
(tycon args •
(³tyconass tycon args = s)
¤ (tycon í Length args tyenv ∧ a • a Elems args ⇒ ³a = s))
∧ tyconass "bool" = Bools
∧ tyconass " → " = Funs | HOLCONST | is_wf_ty_con_assignment | hol | src/hol/spc002.doc | [] | [
"Bool",
"Elems",
"Fun"
] | https://github.com/RobArthan/pp | dabebf8fdbdede1a3ad6333500c6663ecd15c98f | ||
$supports: STRING SET → 'U POLY_VALUE → BOOL
----
X v • X supports v ¤ (a1 a2 • (x • x X ⇒ a1 x = a2 x) ⇒ v a1 = v a2) | HOLCONST | supports | hol | src/hol/spc002.doc | [] | [
"STRING"
] | https://github.com/RobArthan/pp | dabebf8fdbdede1a3ad6333500c6663ecd15c98f | ||
type_value : 'U TY_CON_ASSIGNMENT → TYPE → 'U POLY_VALUE
----
tyconass s tyl •
type_value tyconass (mk_var_type s) = ( ⇔ tyvarass • tyvarass s)
∧ type_value tyconass (mk_type(s, tyl)) =
( ⇔ tyvarass • tyconass s (Map( ⇔ ty • type_value tyconass ty tyvarass) tyl)) | HOLCONST | type_value | hol | src/hol/spc002.doc | [] | [
"mk_type",
"mk_var_type"
] | https://github.com/RobArthan/pp | dabebf8fdbdede1a3ad6333500c6663ecd15c98f | ||
is_wf_con_assignment: CON_ENV → 'U CON_ASSIGNMENT → BOOL
----
conenv conass • is_wf_con_assignment conenv conass ¤
(tyvarass •
conass "=" tyvarass = EqRels (tyvarass "*")
∧ conass " ⇒ " tyvarass = ImpRels)
∧ (con ty • con í ty conenv ⇒ type_ty_vars ty supports (conass con)) | HOLCONST | is_wf_con_assignment | hol | src/hol/spc002.doc | [] | [
"supports",
"type_ty_vars"
] | https://github.com/RobArthan/pp | dabebf8fdbdede1a3ad6333500c6663ecd15c98f | ||
is_wf_interpretation : THEORY → 'U INTERPRETATION → BOOL
----
thy tyconass conass •
is_wf_interpretation thy (tyconass, conass)
¤ ( is_wf_ty_con_assignment (types thy) tyconass
∧ is_wf_con_assignment (constants thy) conass
∧ s ty • s í ty constants thy ⇒
tyvarass • is_wf_ty_var_assignment tyva... | HOLCONST | is_wf_interpretation | hol | src/hol/spc002.doc | [] | [
"constants",
"is_wf_con_assignment",
"is_wf_ty_con_assignment",
"type_value",
"types"
] | https://github.com/RobArthan/pp | dabebf8fdbdede1a3ad6333500c6663ecd15c98f | ||
is_wf_var_assignment : 'U TY_CON_ASSIGNMENT →
'U VAR_ASSIGNMENT → BOOL
----
tyconass varass •
is_wf_var_assignment tyconass varass
¤ s ty tyvarass • is_wf_ty_var_assignment tyvarass ⇒
varass (s, ty) tyvarass s type_value tyconass ty tyvarass | HOLCONST | is_wf_var_assignment | hol | src/hol/spc002.doc | [] | [
"type_value"
] | https://github.com/RobArthan/pp | dabebf8fdbdede1a3ad6333500c6663ecd15c98f | ||
instance : THEORY → 'U INTERPRETATION →
STRING × TYPE → 'U POLY_VALUE
----
thy tyconass conass s instty f declty •
is_wf_con_assignment (constants thy) conass ⇒
s í declty constants thy ∧ inst_type f declty = instty ⇒
instance thy (tyconass, conass) (s, instty)
= ( ⇔ tyvarass • conass s ( ⇔ ... | HOLCONST | instance | hol | src/hol/spc002.doc | [] | [
"STRING",
"constants",
"inst_type",
"is_wf_con_assignment",
"type_value"
] | https://github.com/RobArthan/pp | dabebf8fdbdede1a3ad6333500c6663ecd15c98f | ||
term_value : THEORY → 'U INTERPRETATION →
TERM → 'U VAR_ASSIGNMENT → 'U POLY_VALUE
----
thy tyconass conass tm varass sty f a v b •
(term_value thy (tyconass, conass) (mk_var sty) varass
= ( ⇔ tyvarass • varass sty tyvarass))
∧ (term_value thy (tyconass, conass)(mk_const sty) varass
= instance th... | HOLCONST | term_value | hol | src/hol/spc002.doc | [] | [
"has_mk_abs",
"has_mk_comb",
"instance",
"mk_const",
"mk_var",
"type_of_term",
"type_value"
] | https://github.com/RobArthan/pp | dabebf8fdbdede1a3ad6333500c6663ecd15c98f | ||
$satisfies : THEORY × 'U INTERPRETATION → SEQ → BOOL
----
thy int seq • (thy, int) satisfies seq ¤
varass tyvarass •
let val_of = ⇔ tm • term_value thy int tm varass tyvarass
and (tyconass, _) = int
in is_wf_ty_var_assignment tyvarass
∧ is_wf_var_assignment tyconass varass
∧ (asm • asm hyp s... | HOLCONST | satisfies | hol | src/hol/spc002.doc | [] | [
"and",
"concl",
"hyp",
"is_wf_var_assignment",
"seq",
"term_value"
] | https://github.com/RobArthan/pp | dabebf8fdbdede1a3ad6333500c6663ecd15c98f | ||
is_model : THEORY → 'U INTERPRETATION SET
----
thy int •
int is_model thy ¤
(is_wf_interpretation thy int ∧
seq • seq axioms thy ⇒ (thy, int) satisfies seq) | HOLCONST | is_model | hol | src/hol/spc002.doc | [] | [
"axioms",
"is_wf_interpretation",
"satisfies",
"seq"
] | https://github.com/RobArthan/pp | dabebf8fdbdede1a3ad6333500c6663ecd15c98f | ||
valid : 'U → THEORY → SEQ SET
----
v thy seq •
seq valid v thy ¤
int:'U INTERPRETATION • int is_model thy ⇒ (thy, int) satisfies seq | HOLCONST | valid | hol | src/hol/spc002.doc | [] | [
"is_model",
"satisfies",
"seq"
] | https://github.com/RobArthan/pp | dabebf8fdbdede1a3ad6333500c6663ecd15c98f | ||
freevars_list: TERM → ((STRING × TYPE)LIST)
----
s : STRING; ty : TYPE; tm f a vty b : TERM •
freevars_list (mk_var(s, ty)) = [(s, ty)]
∧
freevars_list (mk_const(s, ty)) = []
∧
(has_mk_comb(f, a) tm ⇒ freevars_list tm = freevars_list f ë freevars_list a)
∧
((has_mk_abs(vty, b) tm ∧ mk_var(s, ty) =... | HOLCONST | freevars_list | hol | src/hol/spc003.doc | [] | [
"STRING",
"has_mk_abs",
"has_mk_comb",
"mk_const",
"mk_var"
] | https://github.com/RobArthan/pp | dabebf8fdbdede1a3ad6333500c6663ecd15c98f | ||
freevars_set: TERM → (STRING × TYPE) SET
----
tm : TERM • freevars_set tm = Elems(freevars_list tm) | HOLCONST | freevars_set | hol | src/hol/spc003.doc | [] | [
"Elems",
"STRING",
"freevars_list"
] | https://github.com/RobArthan/pp | dabebf8fdbdede1a3ad6333500c6663ecd15c98f | ||
Equality : TYPE → TERM
----
ty • Equality ty = mk_const("=", Fun ty (Fun ty Bool)) | HOLCONST | Equality | hol | src/hol/spc003.doc | [] | [
"Bool",
"Fun",
"mk_const"
] | https://github.com/RobArthan/pp | dabebf8fdbdede1a3ad6333500c6663ecd15c98f | ||
has_mk_eq : (TERM × TERM) → TERM → BOOL
----
lhs rhs tm • has_mk_eq(lhs, rhs) tm ¤
tm2 •
has_mk_comb(Equality(type_of_term lhs), lhs) tm2
∧ has_mk_comb(tm2, rhs) tm | HOLCONST | has_mk_eq | hol | src/hol/spc003.doc | [] | [
"Equality",
"has_mk_comb",
"type_of_term"
] | https://github.com/RobArthan/pp | dabebf8fdbdede1a3ad6333500c6663ecd15c98f | ||
Implication : TERM
----
Implication = mk_const(" ⇒ ", Fun Bool (Fun Bool Bool)) | HOLCONST | Implication | hol | src/hol/spc003.doc | [] | [
"Bool",
"Fun",
"mk_const"
] | https://github.com/RobArthan/pp | dabebf8fdbdede1a3ad6333500c6663ecd15c98f | ||
has_mk_imp : (TERM × TERM) → TERM → BOOL
----
lhs rhs tm • has_mk_imp(lhs, rhs) tm ¤
tm2 •
has_mk_comb(Implication, lhs) tm2
∧ has_mk_comb(tm2, rhs) tm | HOLCONST | has_mk_imp | hol | src/hol/spc003.doc | [] | [
"Implication",
"has_mk_comb"
] | https://github.com/RobArthan/pp | dabebf8fdbdede1a3ad6333500c6663ecd15c98f | ||
variant : ((STRING × TYPE) SET) → (STRING × TYPE) → STRING
----
vs v ty •
if ³(v, ty) vs
then variant vs (v, ty) = v
else ³(variant vs (v, ty), ty) vs | HOLCONST | variant | hol | src/hol/spc003.doc | [] | [
"STRING"
] | https://github.com/RobArthan/pp | dabebf8fdbdede1a3ad6333500c6663ecd15c98f | ||
subst : ((STRING × TYPE) → TERM) → TERM → TERM
----
R :(STRING × TYPE) → TERM; tm : TERM;
s : STRING; ty : TYPE; vty : TERM;
f : TERM; a : TERM; b : TERM
•
subst R (mk_var(s, ty)) = R(s,ty)
∧
subst R (mk_const(s, ty)) = mk_const(s, ty)
∧
(has_mk_comb(f, a) tm ⇒
(subst R tm = Åt • has_mk_com... | HOLCONST | subst | hol | src/hol/spc003.doc | [] | [
"Graph",
"Image",
"STRING",
"freevars_set",
"has_mk_abs",
"has_mk_comb",
"mk_const",
"mk_var",
"variant"
] | https://github.com/RobArthan/pp | dabebf8fdbdede1a3ad6333500c6663ecd15c98f | ||
rename : (STRING × TYPE) → STRING → TERM → TERM
----
v : STRING; ty : TYPE; w: STRING
•
rename (v, ty) w =
subst ( ⇔ x • if x = (v, ty) then mk_var(w, ty) else mk_var x) | HOLCONST | rename | hol | src/hol/spc003.doc | [] | [
"STRING",
"mk_var",
"subst"
] | https://github.com/RobArthan/pp | dabebf8fdbdede1a3ad6333500c6663ecd15c98f | ||
aconv : TERM → TERM → BOOL
----
t1 t2 : TERM •
aconv t1 t2 ¤
(t1 = t2)
² ( t1f t1a t2f t2a •
has_mk_comb(t1f, t1a)t1
∧ has_mk_comb(t2f, t2a)t2
∧ aconv t1f t2f ∧ aconv t1a t2a)
² ( v1 v2 ty v1ty v2ty b1 b2 •
has_mk_abs(v1ty, b1)t1 ∧ has_mk_abs(v2ty, b2)t2
∧ mk_var(v1, ty) = v1ty ∧ mk_... | HOLCONST | aconv | hol | src/hol/spc003.doc | [] | [
"freevars_set",
"has_mk_abs",
"has_mk_comb",
"mk_var",
"rename"
] | https://github.com/RobArthan/pp | dabebf8fdbdede1a3ad6333500c6663ecd15c98f | ||
SUBST_rule : ((STRING × TYPE) → SEQ) →
TERM → SEQ → SEQ → BOOL
----
eqs tm old_asms old_conc new_asms new_conc •
SUBST_rule eqs tm (old_asms, old_conc) (new_asms, new_conc) ¤
(v ty •
lhs rhs •
has_mk_eq(lhs, rhs)(concl(eqs(v, ty))) ∧
(type_of_term lhs = ty))
∧
(aconv old_conc (subst( ⇔ ... | HOLCONST | SUBST_rule | hol | src/hol/spc003.doc | [] | [
"STRING",
"aconv",
"concl",
"has_mk_eq",
"hyp",
"subst",
"type_of_term"
] | https://github.com/RobArthan/pp | dabebf8fdbdede1a3ad6333500c6663ecd15c98f | ||
ABS_rule : (STRING × TYPE) → SEQ → SEQ → BOOL
----
vty old_asms old_conc new_asms new_conc •
ABS_rule vty (old_asms, old_conc) (new_asms, new_conc) ¤
( old_lhs old_rhs new_lhs new_rhs v •
has_mk_eq(old_lhs, old_rhs)old_conc ∧
has_mk_eq(new_lhs, new_rhs)new_conc ∧
mk_var vty = v ∧
has_mk_abs(v... | HOLCONST | ABS_rule | hol | src/hol/spc003.doc | [] | [
"Graph",
"Image",
"STRING",
"freevars_set",
"has_mk_abs",
"has_mk_eq",
"mk_var"
] | https://github.com/RobArthan/pp | dabebf8fdbdede1a3ad6333500c6663ecd15c98f | ||
inst_loc1 : (STRING → TYPE) → TERM → TERM → BOOL
----
tysubs : STRING → TYPE;
tm1 tm2 : TERM •
inst_loc1 tysubs tm1 tm2 ¤
( s ty1 ty2 mk_X •
((mk_X = mk_var) ² (mk_X = mk_const))
∧ mk_X(s, ty1) = tm1 ∧ mk_X(s, ty2) = tm2
∧ (ty2 = inst_type tysubs ty1))
² ( tm1f tm1a tm2f tm2a •
has_mk_... | HOLCONST | inst_loc1 | hol | src/hol/spc003.doc | [] | [
"STRING",
"freevars_set",
"has_mk_abs",
"has_mk_comb",
"inst_type",
"mk_const",
"mk_var",
"rename"
] | https://github.com/RobArthan/pp | dabebf8fdbdede1a3ad6333500c6663ecd15c98f | ||
inst_loc2 : ((STRING × TYPE) SET) →
(STRING → TYPE) →
(((STRING × TYPE) × (STRING × TYPE)) LIST) →
TERM → TERM → BOOL
----
avoid : (STRING × TYPE) SET;
tysubs :STRING → TYPE;
v1 : STRING; ty1 : TYPE;
v2 : STRING; ty2 : TYPE;
rest : ((STRING × TYPE) × (STRING × TYPE)) LIST;
tm1 t... | HOLCONST | inst_loc2 | hol | src/hol/spc003.doc | [] | [
"STRING",
"inst_loc1",
"inst_type",
"rename"
] | https://github.com/RobArthan/pp | dabebf8fdbdede1a3ad6333500c6663ecd15c98f | ||
inst : ((STRING × TYPE) SET) →
(STRING → TYPE) → TERM → TERM
----
avoid : (STRING × TYPE) SET;
tysubs :STRING → TYPE; tm1 : TERM •
let tm2 = inst avoid tysubs tm1
in let fl1 = freevars_list tm1
in let fl2 = freevars_list tm2
in
((Length fl1 = Length fl2)
∧ inst_loc2 avoid tysubs (Combine fl1... | HOLCONST | inst | hol | src/hol/spc003.doc | [] | [
"STRING",
"freevars_list",
"inst_loc2"
] | https://github.com/RobArthan/pp | dabebf8fdbdede1a3ad6333500c6663ecd15c98f | ||
type_tyvars : TYPE → (STRING SET)
----
(s • type_tyvars (mk_var_type s) = {s})
∧ (s tl • type_tyvars (mk_type(s, tl)) =
Þ (Elems (Map type_tyvars tl))) | HOLCONST | type_tyvars | hol | src/hol/spc003.doc | [] | [
"Elems",
"STRING",
"mk_type",
"mk_var_type"
] | https://github.com/RobArthan/pp | dabebf8fdbdede1a3ad6333500c6663ecd15c98f | ||
term_types : TERM → (TYPE SET)
----
tm : TERM; s: STRING; ty : TYPE;
f : TERM; a : TERM; v: TERM; b: TERM •
term_types (mk_var(s, ty)) = {ty}
∧
term_types (mk_const(s, ty)) = {ty}
∧
(has_mk_comb(f, a) tm ⇒ (term_types tm = term_types f À term_types a))
∧
(has_mk_abs(v, b) tm ⇒ (term_types tm = ter... | HOLCONST | term_types | hol | src/hol/spc003.doc | [] | [
"STRING",
"has_mk_abs",
"has_mk_comb",
"mk_const",
"mk_var"
] | https://github.com/RobArthan/pp | dabebf8fdbdede1a3ad6333500c6663ecd15c98f | ||
term_tyvars : TERM → (STRING SET)
----
tm • term_tyvars tm = Þ(Graph type_tyvars Image (term_types tm)) | HOLCONST | term_tyvars | hol | src/hol/spc003.doc | [] | [
"Graph",
"Image",
"STRING",
"term_types",
"type_tyvars"
] | https://github.com/RobArthan/pp | dabebf8fdbdede1a3ad6333500c6663ecd15c98f | ||
INST_TYPE_rule : (STRING → TYPE) → SEQ → SEQ → BOOL
----
tysubs old_asms old_conc new_seq •
INST_TYPE_rule tysubs (old_asms, old_conc) new_seq ¤
( tyv •
(tyv Þ (Graph term_tyvars Image old_asms)) ⇒
(tysubs tyv = mk_var_type tyv))
∧
let asms_frees = Þ (Graph freevars_set Image old_asms)
in
new... | HOLCONST | INST_TYPE_rule | hol | src/hol/spc003.doc | [] | [
"Graph",
"Image",
"STRING",
"freevars_set",
"inst",
"mk_var_type",
"term_tyvars"
] | https://github.com/RobArthan/pp | dabebf8fdbdede1a3ad6333500c6663ecd15c98f | ||
DISCH_rule : TERM → SEQ → SEQ → BOOL
----
tm old_asms old_conc new_seq •
DISCH_rule tm (old_asms, old_conc) new_seq ¤
(type_of_term tm = Bool) ∧
(new_seq = ((old_asms \ {tm}), Åt • has_mk_imp(tm, old_conc)t)) | HOLCONST | DISCH_rule | hol | src/hol/spc003.doc | [] | [
"Bool",
"has_mk_imp",
"type_of_term"
] | https://github.com/RobArthan/pp | dabebf8fdbdede1a3ad6333500c6663ecd15c98f | ||
MP_rule : SEQ → SEQ → SEQ → BOOL
----
imp_asms imp_conc ant_asms ant_conc new_asms new_conc •
MP_rule (imp_asms, imp_conc) (ant_asms, ant_conc) (new_asms, new_conc) ¤
(has_mk_imp(ant_conc, new_conc)imp_conc) ∧
(new_asms = imp_asms À ant_asms) | HOLCONST | MP_rule | hol | src/hol/spc003.doc | [] | [
"has_mk_imp"
] | https://github.com/RobArthan/pp | dabebf8fdbdede1a3ad6333500c6663ecd15c98f | ||
ASSUME_axiom : TERM → SEQ → BOOL
----
tm seq • ASSUME_axiom tm seq ¤
(type_of_term tm = Bool) ∧
(seq = ({tm}, tm)) | HOLCONST | ASSUME_axiom | hol | src/hol/spc003.doc | [] | [
"Bool",
"seq",
"type_of_term"
] | https://github.com/RobArthan/pp | dabebf8fdbdede1a3ad6333500c6663ecd15c98f | ||
REFL_axiom : TERM → SEQ
----
tm • REFL_axiom tm = ({}, Åt • has_mk_eq(tm, tm)t) | HOLCONST | REFL_axiom | hol | src/hol/spc003.doc | [] | [
"has_mk_eq"
] | https://github.com/RobArthan/pp | dabebf8fdbdede1a3ad6333500c6663ecd15c98f | ||
BETA_CONV_axiom : TERM → SEQ → BOOL
----
tm new_seq •
BETA_CONV_axiom tm new_seq ¤
v ty vty b abs a •
mk_var(v, ty) = vty ∧
has_mk_abs(vty, b)abs ∧
has_mk_comb(abs, a)tm ∧
(new_seq =
let subs: ((STRING × TYPE) → TERM) =
( ⇔ (vx, tyx) • if vx = v ∧ tyx = ty then a else mk_var(vx, tyx))
in
... | HOLCONST | BETA_CONV_axiom | hol | src/hol/spc003.doc | [] | [
"STRING",
"has_mk_abs",
"has_mk_comb",
"has_mk_eq",
"mk_var",
"subst"
] | https://github.com/RobArthan/pp | dabebf8fdbdede1a3ad6333500c6663ecd15c98f | ||
directly_derivable_from : SEQ → (SEQ SET) → BOOL
----
seq seqs •
directly_derivable_from seq seqs ¤
( eqs tm old_seq •
Ran (Graph eqs) seqs ∧ old_seq seqs ∧ SUBST_rule eqs tm old_seq seq)
²
( vty old_seq • old_seq seqs ∧ ABS_rule vty old_seq seq)
²
( tysubs old_seq • old_seq seqs ∧ ... | HOLCONST | directly_derivable_from | hol | src/hol/spc003.doc | [] | [
"ABS_rule",
"ASSUME_axiom",
"BETA_CONV_axiom",
"DISCH_rule",
"Graph",
"INST_TYPE_rule",
"MP_rule",
"REFL_axiom",
"Ran",
"SUBST_rule",
"seq"
] | https://github.com/RobArthan/pp | dabebf8fdbdede1a3ad6333500c6663ecd15c98f | ||
premisses : (SEQ LIST) → (SEQ SET)
----
seq rest •
premisses [] = {}
∧
premisses (Cons seq rest) =
if directly_derivable_from seq (Elems rest)
then premisses rest
else {seq} À premisses rest | HOLCONST | premisses | hol | src/hol/spc003.doc | [] | [
"Elems",
"directly_derivable_from",
"seq"
] | https://github.com/RobArthan/pp | dabebf8fdbdede1a3ad6333500c6663ecd15c98f | ||
derivable_from : SEQ → (SEQ SET) → BOOL
----
seq seqs •
derivable_from seq seqs =
seql • premisses (Cons seq seql) seqs | HOLCONST | derivable_from | hol | src/hol/spc003.doc | [] | [
"premisses",
"seq"
] | https://github.com/RobArthan/pp | dabebf8fdbdede1a3ad6333500c6663ecd15c98f | ||
Star : TYPE
----
Star = mk_var_type "*" | HOLCONST | Star | hol | src/hol/spc003.doc | [] | [
"mk_var_type"
] | https://github.com/RobArthan/pp | dabebf8fdbdede1a3ad6333500c6663ecd15c98f | ||
Choice : TERM
----
Choice = mk_const(("Å", Fun (Fun Star Bool) Star)) | HOLCONST | Choice | hol | src/hol/spc003.doc | [] | [
"Bool",
"Fun",
"Star",
"mk_const"
] | https://github.com/RobArthan/pp | dabebf8fdbdede1a3ad6333500c6663ecd15c98f | ||
Ind : TYPE
----
Ind = mk_type("ind", []) | HOLCONST | Ind | hol | src/hol/spc003.doc | [] | [
"mk_type"
] | https://github.com/RobArthan/pp | dabebf8fdbdede1a3ad6333500c6663ecd15c98f | ||
MIN_REP : TY_ENV × CON_ENV × SEQS
----
MIN_REP = (
{("bool", 0); (" → ", 2 ); ("ind", 0 )},
{("=", Fun Star (Fun Star Bool));
(" ⇒ ", Fun Bool (Fun Bool Bool));
("Å", Fun (Fun Star Bool) Star)},
{}
) | HOLCONST | MIN_REP | hol | src/hol/spc003.doc | [] | [
"Bool",
"Fun",
"Star"
] | https://github.com/RobArthan/pp | dabebf8fdbdede1a3ad6333500c6663ecd15c98f | ||
MIN : THEORY
----
MIN = abs_theory MIN_REP | HOLCONST | MIN | hol | src/hol/spc003.doc | [] | [
"MIN_REP"
] | https://github.com/RobArthan/pp | dabebf8fdbdede1a3ad6333500c6663ecd15c98f | ||
$extends : THEORY → THEORY → BOOL
----
thy1 thy2 •
thy1 extends thy2 ¤
(types thy2 types thy1) ∧
(constants thy2 constants thy1) ∧
(axioms thy2 axioms thy1) | HOLCONST | extends | hol | src/hol/spc003.doc | [] | [
"axioms",
"constants",
"types"
] | https://github.com/RobArthan/pp | dabebf8fdbdede1a3ad6333500c6663ecd15c98f | ||
is_normal_theory : THEORY SET
----
thy • thy is_normal_theory = thy extends MIN | HOLCONST | is_normal_theory | hol | src/hol/spc003.doc | [] | [
"MIN",
"extends"
] | https://github.com/RobArthan/pp | dabebf8fdbdede1a3ad6333500c6663ecd15c98f | ||
is_thm : (SEQ × THEORY) SET
----
seq thy •
(seq, thy) is_thm ¤
thy is_normal_theory
∧
seq sequents thy
∧
derivable_from seq (axioms thy) | HOLCONST | is_thm | hol | src/hol/spc003.doc | [] | [
"axioms",
"derivable_from",
"is_normal_theory",
"seq",
"sequents"
] | https://github.com/RobArthan/pp | dabebf8fdbdede1a3ad6333500c6663ecd15c98f | ||
thm_seq : THM → SEQ
----
thm •
thm_seq thm = Fst(rep_thm thm) | HOLCONST | thm_seq | hol | src/hol/spc003.doc | [] | [] | https://github.com/RobArthan/pp | dabebf8fdbdede1a3ad6333500c6663ecd15c98f | ||
thm_thy : THM → THEORY
----
thm •
thm_thy thm = Snd(rep_thm thm) | HOLCONST | thm_thy | hol | src/hol/spc003.doc | [] | [] | https://github.com/RobArthan/pp | dabebf8fdbdede1a3ad6333500c6663ecd15c98f | ||
consistent_theory : THEORY SET
----
thy •
thy consistent_theory ¤
seq •
(seq sequents thy)
∧
³(derivable_from seq (axioms thy)) | HOLCONST | consistent_theory | hol | src/hol/spc003.doc | [] | [
"axioms",
"derivable_from",
"seq",
"sequents"
] | https://github.com/RobArthan/pp | dabebf8fdbdede1a3ad6333500c6663ecd15c98f | ||
$conservatively_extends : THEORY → THEORY → BOOL
----
thy1 thy2 •
thy1 conservatively_extends thy2 ¤
(thy1 extends thy2) ∧
( seq •
(seq sequents thy2) ⇒
(derivable_from seq (axioms thy1)) ⇒
(derivable_from seq (axioms thy2))) | HOLCONST | conservatively_extends | hol | src/hol/spc003.doc | [] | [
"axioms",
"derivable_from",
"extends",
"seq",
"sequents"
] | https://github.com/RobArthan/pp | dabebf8fdbdede1a3ad6333500c6663ecd15c98f | ||
mk_comb : (TERM × TERM) → TERM
----
mk_comb = $Å o has_mk_comb | HOLCONST | mk_comb | hol | src/hol/spc003.doc | [] | [
"has_mk_comb"
] | https://github.com/RobArthan/pp | dabebf8fdbdede1a3ad6333500c6663ecd15c98f | ||
mk_abs : (TERM × TERM) → TERM
----
mk_abs = $Å o has_mk_abs | HOLCONST | mk_abs | hol | src/hol/spc003.doc | [] | [
"has_mk_abs"
] | https://github.com/RobArthan/pp | dabebf8fdbdede1a3ad6333500c6663ecd15c98f | ||
mk_eq : (TERM × TERM) → TERM
----
mk_eq = $Å o has_mk_eq | HOLCONST | mk_eq | hol | src/hol/spc003.doc | [] | [
"has_mk_eq"
] | https://github.com/RobArthan/pp | dabebf8fdbdede1a3ad6333500c6663ecd15c98f | ||
mk_imp : (TERM × TERM) → TERM
----
mk_imp = $Å o has_mk_imp | HOLCONST | mk_imp | hol | src/hol/spc003.doc | [] | [
"has_mk_imp"
] | https://github.com/RobArthan/pp | dabebf8fdbdede1a3ad6333500c6663ecd15c98f | ||
Truth : TERM
----
Truth = mk_const("T", Bool) | HOLCONST | Truth | hol | src/hol/spc003.doc | [] | [
"Bool",
"mk_const"
] | https://github.com/RobArthan/pp | dabebf8fdbdede1a3ad6333500c6663ecd15c98f | ||
Truth_def : TERM
----
Truth_def =
let x = mk_var("x", Bool)
in
mk_eq(mk_abs(x, x), mk_abs(x, x)) | HOLCONST | Truth_def | hol | src/hol/spc003.doc | [] | [
"Bool",
"mk_abs",
"mk_eq",
"mk_var"
] | https://github.com/RobArthan/pp | dabebf8fdbdede1a3ad6333500c6663ecd15c98f | ||
Forall : TYPE → TERM
----
ty • Forall ty = mk_const("", Fun (Fun ty Bool) Bool) | HOLCONST | Forall | hol | src/hol/spc003.doc | [] | [
"Bool",
"Fun",
"mk_const"
] | https://github.com/RobArthan/pp | dabebf8fdbdede1a3ad6333500c6663ecd15c98f | ||
Forall_def : TERM
----
Forall_def =
let P = mk_var("P", Fun Star Bool)
in let x = mk_var("x", Star)
in
mk_abs(P, mk_eq(P, mk_abs(x, Truth))) | HOLCONST | Forall_def | hol | src/hol/spc003.doc | [] | [
"Bool",
"Fun",
"Star",
"Truth",
"mk_abs",
"mk_eq",
"mk_var"
] | https://github.com/RobArthan/pp | dabebf8fdbdede1a3ad6333500c6663ecd15c98f | ||
mk_forall : (TERM × TERM) → TERM
----
tm1 tm2 • mk_forall(tm1, tm2) =
mk_comb(Forall (type_of_term tm1), mk_abs(tm1, tm2)) | HOLCONST | mk_forall | hol | src/hol/spc003.doc | [] | [
"Forall",
"mk_abs",
"mk_comb",
"type_of_term"
] | https://github.com/RobArthan/pp | dabebf8fdbdede1a3ad6333500c6663ecd15c98f | ||
Exists : TYPE → TERM
----
ty • Exists ty = mk_const(" ", Fun (Fun ty Bool) Bool) | HOLCONST | Exists | hol | src/hol/spc003.doc | [] | [
"Bool",
"Fun",
"mk_const"
] | https://github.com/RobArthan/pp | dabebf8fdbdede1a3ad6333500c6663ecd15c98f | ||
Exists_def : TERM
----
Exists_def =
let P = mk_var("P", Fun Star Bool)
in let PchoiceP = mk_comb(P,mk_comb(Choice, P))
in
mk_abs(P, PchoiceP) | HOLCONST | Exists_def | hol | src/hol/spc003.doc | [] | [
"Bool",
"Choice",
"Fun",
"Star",
"mk_abs",
"mk_comb",
"mk_var"
] | https://github.com/RobArthan/pp | dabebf8fdbdede1a3ad6333500c6663ecd15c98f | ||
has_mk_exists : (TERM × TERM) → TERM → BOOL
----
tm1 tm2 tm3 •
has_mk_exists(tm1, tm2) tm3 =
has_mk_comb(Exists (type_of_term tm1), mk_abs(tm1, tm2))tm3 | HOLCONST | has_mk_exists | hol | src/hol/spc003.doc | [] | [
"Exists",
"has_mk_comb",
"mk_abs",
"type_of_term"
] | https://github.com/RobArthan/pp | dabebf8fdbdede1a3ad6333500c6663ecd15c98f | ||
mk_exists : (TERM × TERM) → TERM
----
tm1 tm2 • mk_exists(tm1, tm2) =
mk_comb(Exists (type_of_term tm1), mk_abs(tm1, tm2)) | HOLCONST | mk_exists | hol | src/hol/spc003.doc | [] | [
"Exists",
"mk_abs",
"mk_comb",
"type_of_term"
] | https://github.com/RobArthan/pp | dabebf8fdbdede1a3ad6333500c6663ecd15c98f | ||
Falsity : TERM
----
Falsity = mk_const("F", Bool) | HOLCONST | Falsity | hol | src/hol/spc003.doc | [] | [
"Bool",
"mk_const"
] | https://github.com/RobArthan/pp | dabebf8fdbdede1a3ad6333500c6663ecd15c98f | ||
Falsity_def : TERM
----
Falsity_def =
let x = mk_var("x", Bool)
in
mk_forall(x, x) | HOLCONST | Falsity_def | hol | src/hol/spc003.doc | [] | [
"Bool",
"mk_forall",
"mk_var"
] | https://github.com/RobArthan/pp | dabebf8fdbdede1a3ad6333500c6663ecd15c98f | ||
Negation : TERM
----
Negation = mk_const("³", Fun Bool Bool) | HOLCONST | Negation | hol | src/hol/spc003.doc | [] | [
"Bool",
"Fun",
"mk_const"
] | https://github.com/RobArthan/pp | dabebf8fdbdede1a3ad6333500c6663ecd15c98f | ||
Negation_def : TERM
----
Negation_def =
let b = mk_var("b", Bool)
in
mk_abs(b, mk_imp(b, Falsity)) | HOLCONST | Negation_def | hol | src/hol/spc003.doc | [] | [
"Bool",
"Falsity",
"mk_abs",
"mk_imp",
"mk_var"
] | https://github.com/RobArthan/pp | dabebf8fdbdede1a3ad6333500c6663ecd15c98f | ||
Conjunction : TERM
----
Conjunction = mk_const("/\\", Fun Bool (Fun Bool Bool)) | HOLCONST | Conjunction | hol | src/hol/spc003.doc | [] | [
"Bool",
"Fun",
"mk_const"
] | https://github.com/RobArthan/pp | dabebf8fdbdede1a3ad6333500c6663ecd15c98f | ||
Conjunction_def : TERM
----
Conjunction_def =
let b = mk_var("b", Bool)
in let b1 = mk_var("b1", Bool)
in let b2 = mk_var("b2", Bool)
in
mk_abs(b1, mk_abs(b2, mk_forall(b, mk_imp(mk_imp(b1, mk_imp(b2, b)), b)))) | HOLCONST | Conjunction_def | hol | src/hol/spc003.doc | [] | [
"Bool",
"mk_abs",
"mk_forall",
"mk_imp",
"mk_var"
] | https://github.com/RobArthan/pp | dabebf8fdbdede1a3ad6333500c6663ecd15c98f | ||
mk_conj : (TERM × TERM) → TERM
----
tm1 tm2 •
mk_conj(tm1, tm2) = mk_comb(mk_comb(Conjunction, tm1),tm2) | HOLCONST | mk_conj | hol | src/hol/spc003.doc | [] | [
"Conjunction",
"mk_comb"
] | https://github.com/RobArthan/pp | dabebf8fdbdede1a3ad6333500c6663ecd15c98f | ||
Disjunction : TERM
----
Disjunction = mk_const("\\/", Fun Bool (Fun Bool Bool)) | HOLCONST | Disjunction | hol | src/hol/spc003.doc | [] | [
"Bool",
"Fun",
"mk_const"
] | https://github.com/RobArthan/pp | dabebf8fdbdede1a3ad6333500c6663ecd15c98f | ||
Disjunction_def : TERM
----
Disjunction_def =
let b = mk_var("b", Bool)
in let b1 = mk_var("b1", Bool)
in let b2 = mk_var("b2", Bool)
in
mk_abs(b1, mk_abs(b2, mk_forall(b, mk_imp(mk_imp(b1, b),
mk_imp(mk_imp(b2, b), b))))) | HOLCONST | Disjunction_def | hol | src/hol/spc003.doc | [] | [
"Bool",
"mk_abs",
"mk_forall",
"mk_imp",
"mk_var"
] | https://github.com/RobArthan/pp | dabebf8fdbdede1a3ad6333500c6663ecd15c98f | ||
mk_disj : (TERM × TERM) → TERM
----
tm1 tm2 •
mk_disj(tm1, tm2) = mk_comb(mk_comb(Disjunction, tm1),tm2) | HOLCONST | mk_disj | hol | src/hol/spc003.doc | [] | [
"Disjunction",
"mk_comb"
] | https://github.com/RobArthan/pp | dabebf8fdbdede1a3ad6333500c6663ecd15c98f | ||
StarStar : TYPE
----
StarStar = mk_var_type "**" | HOLCONST | StarStar | hol | src/hol/spc003.doc | [] | [
"mk_var_type"
] | https://github.com/RobArthan/pp | dabebf8fdbdede1a3ad6333500c6663ecd15c98f | ||
One_One : TERM
----
One_One = mk_const("ONE_ONE", Fun(Fun Star StarStar)Bool) | HOLCONST | One_One | hol | src/hol/spc003.doc | [] | [
"Bool",
"Fun",
"Star",
"StarStar",
"mk_const"
] | https://github.com/RobArthan/pp | dabebf8fdbdede1a3ad6333500c6663ecd15c98f | ||
One_One_def : TERM
----
One_One_def =
let f = mk_var("f",Fun Star StarStar)
in let x1 = mk_var("x1",Star)
in let x2 = mk_var("x2",Star) in
mk_abs(f, mk_forall(x1, mk_forall(x2,
mk_imp(mk_eq(mk_comb(f, x1), mk_comb(f, x2)),
mk_eq(x1, x2))))) | HOLCONST | One_One_def | hol | src/hol/spc003.doc | [] | [
"Fun",
"Star",
"StarStar",
"mk_abs",
"mk_comb",
"mk_eq",
"mk_forall",
"mk_imp",
"mk_var"
] | https://github.com/RobArthan/pp | dabebf8fdbdede1a3ad6333500c6663ecd15c98f |
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