| | Who Cites rel subst2? |
|
| rel_subst2 |
Def rel_subst2(as;r) == mk_rel(r.name, map( t.term_subst2(as;t);r.args)) |
| | | Thm*  r:rel(), as:(Label Term) List. rel_subst2(as;r)  rel() |
|
| term_subst2 |
Def term_subst2(as;t)
== iterate(statevar 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_subst2(as;t) Term |
|
| 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 |
|
| col |
Def Collection(T) == T  Prop |
| | | Thm* T:Type{i'}. Collection{i}(T) Type{i'} |
|
| tc |
Def tc(r;ds;da;de)
== Case(r.name)
Case eq(Q) = >
||r.args|| = 2 & Q term_types(ds;da;de;r.args[0]) & Q term_types(ds;da;de;r.args[1])
Case R = >
||de.rel(R)|| = ||r.args|| 
& (i:. i < ||r.args||   (de.rel(R))[i] term_types(ds;da;de;r.args[i]))
Default = > False |
| | |
Thm* r:rel(), ds:Collection(dec()), da:Collection(SimpleType), de:sig().
tc(r;ds;da;de) Prop |
|
| term_types |
Def term_types(ds;da;de;t)
== iterate(statevar x- > dec_lookup(ds;x)
statevar x'- > dec_lookup(ds;x)
funsymbol op- > < de.fun(op) >
freevar x- > da
trace(P)- > < lbl_pr( < Trace, P > ) >
c1(c2)- > st_app(c1;c2)
over t) |
| | | Thm* ds:Collection(dec()), da:Collection(SimpleType), de:sig(), t:Term.
term_types(ds;da;de;t) Collection(SimpleType) |
|
| st_app |
Def st_app(c1;c2) == ( s2c2.(s1c1.st_app1(s1;s2))) |
| | | Thm* c1,c2:Collection(SimpleType). st_app(c1;c2) Collection(SimpleType) |
|
| dec_lookup |
Def dec_lookup(ds;x) == < d.typ | d < d ds |  d.lbl = x > > |
| | | Thm* ds:Collection(dec()), x:Label. dec_lookup(ds;x) Collection(SimpleType) |
|
| col_accum |
Def (xc.f(x))(y) ==  x:T. x c & y f(x) |
| | | Thm* T,T':Type, f:(TCollection(T')), c:Collection(T). (xc.f(x)) Collection(T') |
|
| col_filter |
Def < x c | P(x) > (x) == x c & P(x) |
| | | Thm* T:Type, c:Collection(T), Q:(TProp). < i c | Q(i) > Collection(T) |
|
| 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 |
|
| dec |
Def dec() == LabelSimpleType |
| | | Thm* dec() Type |
|
| l_member |
Def (x l) == i:. i < ||l|| & x = l[i] T |
| | | Thm* T:Type, x:T, l:T List. (x l) Prop |
|
| rel |
Def rel() == relname()(Term List) |
| | | Thm* rel() Type |
|
| sig |
Def sig() == (LabelSimpleType)(Label(SimpleType List)) |
| | | Thm* sig() Type |
|
| relname |
Def relname() == SimpleType+Label |
| | | Thm* relname() Type |
|
| st_app1 |
Def st_app1(s1;s2)
== Case(s1)
Case a;b = >
if st_eq(a;s2) < b > else < > fi
Default = > < > |
| | |
Thm* s1,s2:SimpleType. st_app1(s1;s2) Collection(SimpleType) |
|
| st |
Def SimpleType == Tree(Label+Unit) |
| | | Thm* SimpleType Type |
|
| term |
Def Term == Tree(ts()) |
| | | Thm* Term Type |
|
| ts |
Def ts() == Label+Label+Label+Label+Label |
| | | Thm* ts() Type |
|
| lbl |
Def Label == {p:Pattern| ground_ptn(p) } |
| | | Thm* Label Type |
|
| mk_dec |
Def mk_dec(lbl, typ) == < lbl,typ > |
| | | Thm* lbl:Label, typ:SimpleType. mk_dec(lbl, typ) dec() |
|
| rel_primed_vars |
Def rel_primed_vars(r) == reduce(t,vs. term_primed_vars(t) @ vs;nil;r.args) |
| | | Thm* r:rel(). rel_primed_vars(r) Label List |
|
| tvar |
Def l == tree_leaf(ts_var(l)) |
| | | Thm* l:Label. l Term |
|
| rel_name |
Def t.name == 1of(t) |
| | |
Thm* t:rel(). t.name relname() |
|
| sig_fun |
Def t.fun == 1of(t) |
| | |
Thm* t:sig(). t.fun LabelSimpleType |
|
| dec_lbl |
Def t.lbl == 1of(t) |
| | |
Thm* t:dec(). t.lbl Label |
|
| pi1 |
Def 1of(t) == t.1 |
| | | Thm* A:Type, B:(AType), p:(a:AB(a)). 1of(p) A |
|
| st_eq |
Def st_eq(s1;s2)
== Case(s1)
Case a;b = >
Case(s2)
Case a';b' = >
st_eq(a;a')  st_eq(b;b')
Default = > false
Case tree_leaf(x) = >
Case(s2)
Case a';b' = >
false
Case tree_leaf(y) = >
InjCase(x; x'. InjCase(y; y'. x' = y'; b. false); a.
InjCase(y; y'. false; b. true))
Default = > false
Default = > false
(recursive) |
| | |
Thm* s1,s2:SimpleType. st_eq(s1;s2)  |
|
| 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 |
|
| 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 |
|
| rel_args |
Def t.args == 2of(t) |
| | |
Thm* t:rel(). t.args Term List |
|
| sig_rel |
Def t.rel == 2of(t) |
| | |
Thm* t:sig(). t.rel Label(SimpleType List) |
|
| dec_typ |
Def t.typ == 2of(t) |
| | |
Thm* t:dec(). t.typ SimpleType |
|
| 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: . 0 n n < ||l|| l[n] A |
|
| length |
Def ||as|| == Case of as; nil 0 ; a.as' ||as'||+1 (recursive) |
| | |
Thm* A:Type, l:A List. ||l|| |
| | |
Thm* ||nil|| |
|
| nat |
Def == {i:| 0i } |
| | | Thm* Type |
|
| 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 |
|
| term_primed_vars |
Def term_primed_vars(t)
== iterate(statevar v- > nil
statevar v'- > [v]
funsymbol f- > nil
freevar f- > nil
trace(P)- > nil
x(y)- > x @ y
over t) |
| | | Thm* t:Term. term_primed_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 |
|
| 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 |
|
| 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 |
|
| mk_rel |
Def mk_rel(name, args) == < name,args > |
| | | Thm* name:relname(), args:Term List. mk_rel(name, args) rel() |
|
| tree |
Def Tree(E) == rec(T.tree_con(E;T)) |
| | |
Thm* E:Type. Tree(E) Type |
|
| 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 |
|
| 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 |
|
| 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 |
|
| case_relname_other |
Def Case x = > body(x) cont(x1,z)
== (x1.inr(x2) = > body(hd([x2 / tl(x1)])) cont(hd(x1),z))([x1]) |
|
| case_relname_eq |
Def Case eq(x) = > body(x) cont(x1,z)
== InjCase(x1; x2. body(x2); _. cont(z,z)) |
|
| case |
Def Case(value) body == body(value,value) |
|
| clbl |
Def $x == ptn_atom("$x") |
|
| lbl_pair |
Def lbl_pr( < x, y > ) == ptn_pr( < x,y > ) |
| | | Thm* x,y:Pattern. lbl_pr( < x, y > ) Pattern |
| | |
Thm* x,y:Label. lbl_pr( < x, y > ) Label |
|
| typ |
Def t == tree_leaf(inl(t)) |
| | | Thm* t:Label. t SimpleType |
|
| col_singleton |
Def < x > (y) == y = x T |
| | | Thm* T:Type, x:T. < x > Collection(T) |
|
| ts_var |
Def ts_var(x) == inl(x) |
| | | Thm* x:Label. ts_var(x) ts() |
|
| 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 |
|
| 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) |
|
| 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)))) |
|
| eq_atom |
Def x=yAtom == if x=yAtomtrue; false fi |
| | | Thm* x,y:Atom. x=yAtom |
|
| 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]) |
|
| eq_int |
Def i=j == if i=j true ; false fi |
| | | Thm* i,j:. (i=j) |
|
| 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]) |
|
| case_ptn_atom |
Def Case ptn_atom(x) = > body(x) cont(x1,z)
== InjCase(x1; x2. body(x2); _. cont(z,z)) |
|
| 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 |
|
| 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 |
|
| le |
Def AB ==  B < A |
| | | Thm* i,j:. (ij) Prop |
|
| ptn_con |
Def ptn_con(T) == Atom++Atom+(TT) |
| | | Thm* T:Type. ptn_con(T) Type |
|
| tapp |
Def t1 t2 == tree_node( < t1, t2 > ) |
| | | Thm* t1,t2:Term. t1 t2 Term |
|
| tree_con |
Def tree_con(E;T) == E+(TT) |
| | | Thm* E,T:Type. tree_con(E;T) Type |
|
| tl |
Def tl(l) == Case of l; nil nil ; h.t t |
| | |
Thm* A:Type, l:A List. tl(l) A List |
|
| case_inr |
Def inr(x) = > body(x) cont(value,contvalue)
== InjCase(value; _. cont(contvalue,contvalue); x. body(x)) |
|
| ptn_atom |
Def ptn_atom(x) == inl(x) |
| | | Thm* T:Type, x:Atom. ptn_atom(x) ptn_con(T) |
| | |
Thm* x:Atom. ptn_atom(x) Pattern |
| | |
Thm* x:Atom. ptn_atom(x) Label |
|
| ptn_pr |
Def ptn_pr(x) == inr(inr(inr(x))) |
| | | Thm* T:Type, x:(TT). ptn_pr(x) ptn_con(T) |
| | |
Thm* x,y:Pattern. ptn_pr( < x,y > ) Pattern |
|
| case_inl |
Def inl(x) = > body(x) cont(value,contvalue)
== InjCase(value; x. body(x); _. cont(contvalue,contvalue)) |
|
| le_int |
Def ij == j < i |
| | | Thm* i,j:. (ij) |
|
| not |
Def A == A False |
| | | Thm* A:Prop. (A) Prop |
|
| 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() |
|
| 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() |
|
| col_none |
Def < > (x) == False |
| | | Thm* T:Type. < > Collection(T) |
|
| 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)) |
|
| 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) |
|
| case_tree_leaf |
Def Case tree_leaf(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)) |
