| | Who Cites col? |
|
| col |
Def Collection(T) == T  Prop |
| | | Thm*  T:Type{i'}. Collection{i}(T)  Type{i'} |
|
| col_subst |
Def col_subst(c;r)
== col_map_subst(as.rel_subst(as;r); < zip(rel_vars(r);s) | s col_list_prod(map(c;rel_vars(r))) > ) |
| | | Thm* c:(LabelCollection(Term)), r:rel(). col_subst(c;r) Fmla |
| | |
Thm* c:(LabelCollection(Term)), r:rel(). col_subst(c;r) Collection(rel()) |
|
| col_list_prod |
Def col_list_prod(l)(x) == ||x|| = ||l||  & (i:. i < ||x||   x[i] l[i]) |
| | | Thm* T:Type, l:Collection(T) List. col_list_prod(l) Collection(T List) |
|
| col_map_subst |
Def col_map_subst(x.f(x);c) == < f(x) | x c > |
| | | Thm* f:(((Label Term) List)rel()), c:Collection((LabelTerm) List).
col_map_subst(x.f(x);c) Collection(rel()) |
|
| col_map |
Def < f(x) | x c > (y) ==  x:T. x c & y = f(x) T' |
| | | Thm* T,T':Type, f:(TT'), c:Collection(T). < f(x) | x c > Collection(T') |
|
| col_member |
Def x c == c(x) |
| | | Thm* T:Type, x:T, c:Collection(T). x c Prop |
|
| iff |
Def P  Q == (P Q) & (P Q) |
| | | Thm* A,B:Prop. (A B) Prop |
|
| rel |
Def rel() == relname()(Term List) |
| | | Thm* rel() Type |
|
| term |
Def Term == Tree(ts()) |
| | | Thm* Term Type |
|
| relname |
Def relname() == SimpleType+Label |
| | | Thm* relname() Type |
|
| ts |
Def ts() == Label+Label+Label+Label+Label |
| | | Thm* ts() Type |
|
| st |
Def SimpleType == Tree(Label+Unit) |
| | | Thm* SimpleType Type |
|
| lbl |
Def Label == {p:Pattern|  ground_ptn(p) } |
| | | Thm* Label Type |
|
| length |
Def ||as|| == Case of as; nil  0 ; a.as' ||as'||+1 (recursive) |
| | |
Thm* A:Type, l:A List. ||l||  |
| | |
Thm* ||nil|| |
|
| rel_subst |
Def rel_subst(as;r) == mk_rel(r.name, map( t.term_subst(as;t);r.args)) |
| | | Thm* r:rel(), as:(LabelTerm) List. rel_subst(as;r) rel() |
|
| unzip |
Def unzip(as) == < map(p.1of(p);as),map(p.2of(p);as) > |
| | | Thm* T1,T2:Type, as:(T1T2) List. unzip(as) (T1 List)(T2 List) |
|
| map |
Def map(f;as) == Case of as; nil nil ; a.as' [(f(a)) / map(f;as')] (recursive) |
| | |
Thm* A,B:Type, f:(AB), l:A List. map(f;l) B List |
| | |
Thm* A,B:Type, f:(AB), l:A List . map(f;l) B List |
|
| nat |
Def == {i:| 0 i } |
| | | Thm* Type |
|
| term_subst |
Def term_subst(as;t)
== iterate(statevar v- > apply_alist(as;v;v)
statevar v'- > apply_alist(as;v;v')
funsymbol f- > f
freevar f- > f
trace(P)- > trace(P)
x(y)- > x y
over t) |
| | | Thm* t:Term, as:(LabelTerm) List. term_subst(as;t) Term |
|
| rel_name |
Def t.name == 1of(t) |
| | |
Thm* t:rel(). t.name relname() |
|
| apply_alist |
Def apply_alist(as;l;d) == 2of((first p as s.t. 1of(p) = l else < l,d > )) |
| | | Thm* T:Type, as:(LabelT) List, l:Label, d:T. apply_alist(as;l;d) T |
|
| pi1 |
Def 1of(t) == t.1 |
| | | Thm* A:Type, B:(AType), p:(a:AB(a)). 1of(p) A |
|
| rel_vars |
Def rel_vars(r) == reduce(t,vs. term_vars(t) @ vs;nil;r.args) |
| | | Thm* r:rel(). rel_vars(r) Label List |
|
| rel_args |
Def t.args == 2of(t) |
| | |
Thm* t:rel(). t.args Term List |
|
| pi2 |
Def 2of(t) == t.2 |
| | |
Thm* A:Type, B:(AType), p:(a:AB(a)). 2of(p) B(1of(p)) |
|
| select |
Def l[i] == hd(nth_tl(i;l)) |
| | |
Thm* A:Type, l:A List, n:. 0n n < ||l|| l[n] A |
|
| zip |
Def zip(as;bs)
== Case of as
nil nil
a.as' Case of bs; nil nil ; b.bs' [ < a,b > / zip(as';bs')]
(recursive) |
| | |
Thm* T1,T2:Type, as:T1 List, bs:T2 List. zip(as;bs) (T1T2) List |
|
| rev_implies |
Def P Q == Q P |
| | | Thm* A,B:Prop. (A B) Prop |
|
| ground_ptn |
Def ground_ptn(p)
== Case(p)
Case ptn_var(v) = >
false
Case ptn_pr( < x, y > ) = >
ground_ptn(x) ground_ptn(y)
Default = > true
(recursive) |
| | |
Thm* p:Pattern. ground_ptn(p)  |
|
| assert |
Def b == if b True else False fi |
| | | Thm* b:. b Prop |
|
| ptn |
Def Pattern == rec(T.ptn_con(T)) |
| | |
Thm* Pattern Type |
|
| le |
Def AB ==  B < A |
| | | Thm* i,j:. (ij) Prop |
|
| mk_rel |
Def mk_rel(name, args) == < name,args > |
| | | Thm* name:relname(), args:Term List. mk_rel(name, args) rel() |
|
| term_vars |
Def term_vars(t)
== iterate(statevar v- > [v]
statevar v'- > [v]
funsymbol f- > nil
freevar f- > nil
trace(P)- > nil
x(y)- > x @ y
over t) |
| | | Thm* t:Term. term_vars(t) Label List |
|
| append |
Def as @ bs == Case of as; nil bs ; a.as' [a / (as' @ bs)] (recursive) |
| | |
Thm* T:Type, as,bs:T List. (as @ bs) T List |
|
| find |
Def (first x as s.t. P(x) else d) == Case of filter(x.P(x);as); nil d ; a.b a |
| | | Thm* T:Type, P:(T), as:T List, d:T. (first a as s.t. P(a) else d) T |
|
| filter |
Def filter(P;l) == reduce(a,v. if P(a) [a / v] else v fi;nil;l) |
| | | Thm* T:Type, P:(T), l:T List. filter(P;l) T List |
|
| reduce |
Def reduce(f;k;as) == Case of as; nil k ; a.as' f(a,reduce(f;k;as')) (recursive) |
| | |
Thm* A,B:Type, f:(ABB), k:B, as:A List. reduce(f;k;as) B |
|
| nth_tl |
Def nth_tl(n;as) == if n0 as else nth_tl(n-1;tl(as)) fi (recursive) |
| | |
Thm* A:Type, as:A List, i:. nth_tl(i;as) A List |
|
| eq_lbl |
Def l1 = l2
== Case(l1)
Case ptn_atom(x) = >
Case(l2)
Case ptn_atom(y) = >
x=yAtom
Default = > false
Case ptn_int(x) = >
Case(l2)
Case ptn_int(y) = >
x=y
Default = > false
Case ptn_var(x) = >
Case(l2)
Case ptn_var(y) = >
x=yAtom
Default = > false
Case ptn_pr( < x, y > ) = >
Case(l2)
Case ptn_pr( < u, v > ) = >
x = uy = v
Default = > false
Default = > false
(recursive) |
| | |
Thm* l1,l2:Pattern. l1 = l2 |
|
| case_ptn_var |
Def Case ptn_var(x) = > body(x) cont(x1,z)
== (x1.inr(x2) = >
(x1.inr(x2) = >
(x1.inl(x2) = > body(hd([x2 / tl(x1)])) cont(hd(x1),z))([x2 / tl(x1)])
cont
(hd(x1)
,z))
([x2 / tl(x1)])
cont
(hd(x1)
,z))
([x1]) |
|
| term_iter |
Def iterate(statevar x- > v(x)
statevar x''- > v'(x')
funsymbol op- > opr(op)
freevar f- > fvar(f)
trace(tr)- > trace(tr)
a(b)- > comb(a;b)
over t)
== term_iterate(x.v(x);
x'.v'(x');
op.opr(op);
f.fvar(f);
tr.trace(tr);
a,b. comb(a;b);
t) |
| | | Thm* A:Type, v,v',opr,fvar,trace:(LabelA), comb:(AAA), t:Term.
iterate(statevar x- > v(x)
statevar x''- > v'(x')
funsymbol op- > opr(op)
freevar f- > fvar(f)
trace(tr)- > trace(tr)
a(b)- > comb(a,b)
over t)
A |
|
| term_iterate |
Def term_iterate(v;
p;
op;
f;
tr;
a;
t)
== t_iterate(x.ts_case(x)
var(a)= > v(a)
var'(b)= > p(b)
opr(c)= > op(c)
fvar(d)= > f(d)
trace(P)= > tr(P)
end_ts_case ;a;t) |
| | | Thm* A:Type, v,op,f,p,tr:(LabelA), a:(AAA), t:Term. term_iterate(v;p;op;f;tr;a;t) A |
|
| case_ptn_int |
Def Case ptn_int(x) = > body(x) cont(x1,z)
== (x1.inr(x2) = >
(x1.inl(x2) = > body(hd([x2 / tl(x1)])) cont(hd(x1),z))([x2 / tl(x1)])
cont
(hd(x1)
,z))
([x1]) |
|
| ts_case |
Def ts_case(x)
var(a)= > v(a)
var'(b)= > p(b)
opr(f)= > op(f)
fvar(x)= > f(x)
trace(P)= > t(P)
end_ts_case
== Case(x)
Case ts_var(a) = >
v(a)
Case ts_pvar(b) = >
p(b)
Case ts_op(f) = >
op(f)
Case ts_fvar(x) = >
f(x)
Case ts_trace(P) = >
t(P)
Default = >  |
| | |
Thm* A:Type, v,op,f,p,t:(LabelA), x:ts().
ts_case(x)var(a)= > v(a)var'(b)= > p(b)opr(f)= > op(f)fvar(y)= > f(y)trace(P)= > t(P)end_ts_case A |
|
| case_ts_trace |
Def Case ts_trace(x) = > body(x) cont(x1,z)
== (x1.inr(x2) = >
(x1.inr(x2) = >
(x1.inr(x2) = >
(x1.inr(x2) = > body(hd([x2 / tl(x1)])) cont(hd(x1),z))([x2 / tl(x1)])
cont
(hd(x1)
,z))
([x2 / tl(x1)])
cont
(hd(x1)
,z))
([x2 / tl(x1)])
cont
(hd(x1)
,z))
([x1]) |
|
| case_ts_fvar |
Def Case ts_fvar(x) = > body(x) cont(x1,z)
== (x1.inr(x2) = >
(x1.inr(x2) = >
(x1.inr(x2) = >
(x1.inl(x2) = > body(hd([x2 / tl(x1)])) cont(hd(x1),z))([x2 / tl(x1)])
cont
(hd(x1)
,z))
([x2 / tl(x1)])
cont
(hd(x1)
,z))
([x2 / tl(x1)])
cont
(hd(x1)
,z))
([x1]) |
|
| case_ts_op |
Def Case ts_op(x) = > body(x) cont(x1,z)
== (x1.inr(x2) = >
(x1.inr(x2) = >
(x1.inl(x2) = > body(hd([x2 / tl(x1)])) cont(hd(x1),z))([x2 / tl(x1)])
cont
(hd(x1)
,z))
([x2 / tl(x1)])
cont
(hd(x1)
,z))
([x1]) |
|
| case_ts_pvar |
Def Case ts_pvar(x) = > body(x) cont(x1,z)
== (x1.inr(x2) = >
(x1.inl(x2) = > body(hd([x2 / tl(x1)])) cont(hd(x1),z))([x2 / tl(x1)])
cont
(hd(x1)
,z))
([x1]) |
|
| hd |
Def hd(l) == Case of l; nil "?" ; h.t h |
| | |
Thm* A:Type, l:A List. ||l|| 1 hd(l) A |
| | |
Thm* A:Type, l:A List. hd(l) A |
|
| tree |
Def Tree(E) == rec(T.tree_con(E;T)) |
| | |
Thm* E:Type. Tree(E) Type |
|
| t_iterate |
Def t_iterate(l;n;t)
== Case(t)
Case x;y = >
n(t_iterate(l;n;x),t_iterate(l;n;y))
Case tree_leaf(x) = >
l(x)
Default = > True
(recursive) |
| | |
Thm* E,A:Type, l:(EA), n:(AAA), t:Tree(E). t_iterate(l;n;t) A |
|
| case_default |
Def Default = > body(value,value) == body |
|
| band |
Def pq == if p q else false fi |
| | | Thm* p,q:. (pq) |
|
| case_lbl_pair |
Def Case ptn_pr( < x, y > ) = > body(x;y) cont(x1,z)
== InjCase(x1; _. cont(z,z); x2.
InjCase(x2; _. cont(z,z); x2@0. InjCase(x2@0; _. cont(z,z); x2@1. x2@1/x3,x2@2. body(x3;x2@2)))) |
|
| case |
Def Case(value) body == body(value,value) |
|
| ptn_con |
Def ptn_con(T) == Atom++Atom+(TT) |
| | | Thm* T:Type. ptn_con(T) Type |
|
| not |
Def A == A False |
| | | Thm* A:Prop. (A) Prop |
|
| tapp |
Def t1 t2 == tree_node( < t1, t2 > ) |
| | | Thm* t1,t2:Term. t1 t2 Term |
|
| ttrace |
Def trace(l) == tree_leaf(ts_trace(l)) |
| | | Thm* l:Label. trace(l) Term |
|
| tfvar |
Def l == tree_leaf(ts_fvar(l)) |
| | | Thm* l:Label. l Term |
|
| topr |
Def f == tree_leaf(ts_op(f)) |
| | | Thm* f:Label. f Term |
|
| tpvar |
Def l' == tree_leaf(ts_pvar(l)) |
| | | Thm* l:Label. l' Term |
|
| tvar |
Def l == tree_leaf(ts_var(l)) |
| | | Thm* l:Label. l Term |
|
| tl |
Def tl(l) == Case of l; nil nil ; h.t t |
| | |
Thm* A:Type, l:A List. tl(l) A List |
|
| le_int |
Def ij == j < i |
| | | Thm* i,j:. (ij) |
|
| tree_con |
Def tree_con(E;T) == E+(TT) |
| | | Thm* E,T:Type. tree_con(E;T) Type |
|
| case_inl |
Def inl(x) = > body(x) cont(value,contvalue)
== InjCase(value; x. body(x); _. cont(contvalue,contvalue)) |
|
| case_inr |
Def inr(x) = > body(x) cont(value,contvalue)
== InjCase(value; _. cont(contvalue,contvalue); x. body(x)) |
|
| node |
Def tree_node( < x, y > ) == tree_node( < x,y > ) |
| | | Thm* E:Type, x,y:Tree(E). tree_node( < x, y > ) Tree(E) |
|
| ts_trace |
Def ts_trace(x) == inr(inr(inr(inr(x)))) |
| | | Thm* x:Label. ts_trace(x) ts() |
|
| tree_leaf |
Def tree_leaf(x) == inl(x) |
| | | Thm* E,T:Type, x:E. tree_leaf(x) tree_con(E;T) |
| | |
Thm* E:Type, x:E. tree_leaf(x) Tree(E) |
|
| ts_fvar |
Def ts_fvar(x) == inr(inr(inr(inl(x)))) |
| | | Thm* x:Label. ts_fvar(x) ts() |
|
| ts_op |
Def ts_op(x) == inr(inr(inl(x))) |
| | | Thm* x:Label. ts_op(x) ts() |
|
| ts_pvar |
Def ts_pvar(x) == inr(inl(x)) |
| | | Thm* x:Label. ts_pvar(x) ts() |
|
| ts_var |
Def ts_var(x) == inl(x) |
| | | Thm* x:Label. ts_var(x) ts() |
|
| lt_int |
Def i < j == if i < j true ; false fi |
| | | Thm* i,j:. (i < j) |
|
| bnot |
Def b == if b false else true fi |
| | | Thm* b:. b |
|
| tree_node |
Def tree_node(x) == inr(x) |
| | | Thm* E,T:Type, x:(TT). tree_node(x) tree_con(E;T) |
| | |
Thm* E:Type, x,y:Tree(E). tree_node( < x,y > ) Tree(E) |
|
| eq_atom |
Def x=yAtom == if x=yAtomtrue; false fi |
| | | Thm* x,y:Atom. x=yAtom |
|
| eq_int |
Def i=j == if i=j true ; false fi |
| | | Thm* i,j:. (i=j) |
|
| case_ptn_atom |
Def Case ptn_atom(x) = > body(x) cont(x1,z)
== InjCase(x1; x2. body(x2); _. cont(z,z)) |
|
| case_ts_var |
Def Case ts_var(x) = > body(x) cont(x1,z)
== InjCase(x1; x2. body(x2); _. cont(z,z)) |
|
| case_tree_leaf |
Def Case tree_leaf(x) = > body(x) cont(x1,z)
== InjCase(x1; x2. body(x2); _. cont(z,z)) |
|
| case_node |
Def Case x;y = > body(x;y) cont(x1,z)
== InjCase(x1; _. cont(z,z); x2. x2/x3,x2@0. body(x3;x2@0)) |
