Who Cites closed pred?
closed_pred	Def closed_pred(p) == r:rel(). r p closed_rel(r)
	Thm* p:Fmla. closed_pred(p) Prop
tc_ioa	Def tc_ioa(A;de) == tc_pred(A.init;A.ds; < > ;de) & (p:pre(). p A.pre tc(p.rel;A.ds;dec_lookup(A.da;p.kind);de)) & (ef:eff(). ef A.eff mk_dec(ef.kind, ef.typ) A.da & tc_eff(ef;A.ds;de)) & (f:frame(). f A.frame mk_dec(f.var, f.typ) A.ds)
	Thm* A:ioa{i:l}(), de:sig(). tc_ioa(A;de) Prop
tc_vc	Def tc_vc(v;ds;da;de) == Case(v) Case vc_imp(hc) = > tc_pred(hc.hyp;ds; < > ;de) & tc_pred(hc.concl;ds; < > ;de) Case vc_qimp(qhc) = > tc_pred(qhc.hyp;ds;dec_lookup(da;qhc.lbl);de) & tc_pred(qhc.concl;ds;dec_lookup(da;qhc.lbl);de) Default = > False
	Thm* v:vc{i:l}(), ds,da:Collection(dec()), de:sig(). tc_vc(v;ds;da;de) Prop
tc_pred	Def tc_pred(P;ds;da;de) == r:rel(). r P tc(r;ds;da;de)
	Thm* P:Fmla, ds:Collection(dec()), da:Collection(SimpleType), de:sig(). tc_pred(P;ds;da;de) Prop
tc_eff	Def tc_eff(ef;ds;de) == tc_smt(ef.smt;ds; < ef.typ > ;de)
	Thm* ef:eff(), ds:Collection(dec()), de:sig(). tc_eff(ef;ds;de) Prop
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
tc_smt	Def tc_smt(s;ds;da;de) == mk_dec(s.lbl, s.typ) ds & s.typ term_types(ds;da;de;s.term)
	Thm* s:smt(), ds:Collection(dec()), da:Collection(SimpleType), de:sig(). tc_smt(s;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)
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)
col_none	Def < > (x) == False
	Thm* T:Type. < > Collection(T)
covers_pred	Def covers_pred(A;p) == x:Label. pred_mentions(p;x) covers_var(A;x)
	Thm* A:ioa{i:l}(), p:Fmla. covers_pred(A;p) Prop
ioa	Def ioa{i:l}() == Collection(dec())Collection(dec())Collection(rel())Collection(pre())Collection(eff())Collection(frame())
	Thm* ioa{i:l}() Type{i'}
ioa_da	Def t.da == 1of(2of(t))
	Thm* t:ioa{i:l}(). t.da Collection(dec())
ioa_ds	Def t.ds == 1of(t)
	Thm* t:ioa{i:l}(). t.ds Collection(dec())
ioa_init	Def t.init == 1of(2of(2of(t)))
	Thm* t:ioa{i:l}(). t.init Collection(rel())
	Thm* t:ioa{i:l}(). t.init Fmla
mk_imp	Def mk_imp(hyp, concl) == < hyp,concl >
	Thm* hyp,concl:Fmla. mk_imp(hyp, concl) imp{i:l}()
vc	Def vc{i:l}() == imp{i:l}()+qimp{i:l}()
	Thm* vc{i:l}() Type{i'}
qimp	Def qimp{i:l}() == LabelFmlaFmla
	Thm* qimp{i:l}() Type{i'}
imp	Def imp{i:l}() == FmlaFmla
	Thm* imp{i:l}() Type{i'}
pred	Def Fmla == Collection(rel())
	Thm* Fmla{i} Type{i'}
sig	Def sig() == (LabelSimpleType)(Label(SimpleType List))
	Thm* sig() Type
single_valued_decls	Def single_valued_decls(c) == x:Label, t1,t2:SimpleType. mk_dec(x, t1) c mk_dec(x, t2) c t1 = t2
	Thm* c:Collection(dec()). single_valued_decls(c) Prop
vc_imp	Def vc_imp(x) == inl(x)
	Thm* x:imp{i:l}(). vc_imp(x) vc{i:l}()
closed_rel	Def closed_rel(r) == rel_free_vars(r) = nil
	Thm* r:rel(). closed_rel(r) Prop
covers_var	Def covers_var(A;x) == fr:frame(). fr < fr A.frame | fr.var = x > & (a:Label. (a fr.acts) (ef:eff(). ef < ef A.eff | ef.kind = a & ef.smt.lbl = x > ))
	Thm* A:ioa{i:l}(), x:Label. covers_var(A;x) Prop
pred_mentions	Def pred_mentions(p;x) == r:rel(). r p & rel_mentions(r;x)
	Thm* p:Fmla, x:Label. pred_mentions(p;x) Prop
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_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_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_member	Def x c == c(x)
	Thm* T:Type, x:T, c:Collection(T). x c Prop
pre	Def pre() == LabelLabelrel()
	Thm* pre() Type
rel	Def rel() == relname()(Term List)
	Thm* rel() Type
frame	Def frame() == LabelSimpleType(Label List)
	Thm* frame() Type
eff	Def eff() == LabelLabelSimpleTypesmt()
	Thm* eff() Type
dec	Def dec() == LabelSimpleType
	Thm* dec() Type
relname	Def relname() == SimpleType+Label
	Thm* relname() Type
smt	Def smt() == LabelTermSimpleType
	Thm* smt() Type
st	Def SimpleType == Tree(Label+Unit)
	Thm* SimpleType Type
term	Def Term == Tree(ts())
	Thm* Term Type
rel_mentions	Def rel_mentions(r;x) == i:. i < ||r.args|| & (x term_vars(r.args[i]))
	Thm* r:rel(), x:Label. rel_mentions(r;x) Prop
ts	Def ts() == Label+Label+Label+Label+Label
	Thm* ts() Type
lbl	Def Label == {p:Pattern| ground_ptn(p) }
	Thm* Label Type
col	Def Collection(T) == TProp
	Thm* T:Type{i'}. Collection{i}(T) Type{i'}
frame_typ	Def t.typ == 1of(2of(t))
	Thm* t:frame(). t.typ SimpleType
ioa_frame	Def t.frame == 2of(2of(2of(2of(2of(t)))))
	Thm* t:ioa{i:l}(). t.frame Collection(frame())
eff_typ	Def t.typ == 1of(2of(2of(t)))
	Thm* t:eff(). t.typ SimpleType
ioa_eff	Def t.eff == 1of(2of(2of(2of(2of(t)))))
	Thm* t:ioa{i:l}(). t.eff Collection(eff())
pre_rel	Def t.rel == 2of(2of(t))
	Thm* t:pre(). t.rel rel()
ioa_pre	Def t.pre == 1of(2of(2of(2of(t))))
	Thm* t:ioa{i:l}(). t.pre Collection(pre())
qimp_concl	Def t.concl == 2of(2of(t))
	Thm* t:qimp{i:l}(). t.concl Fmla
qimp_hyp	Def t.hyp == 1of(2of(t))
	Thm* t:qimp{i:l}(). t.hyp Fmla
imp_concl	Def t.concl == 2of(t)
	Thm* t:imp{i:l}(). t.concl Fmla
rel_free_vars	Def rel_free_vars(r) == reduce(t,vs. term_free_vars(t) @ vs;nil;r.args)
	Thm* r:rel(). rel_free_vars(r) Label List
eff_smt	Def t.smt == 2of(2of(2of(t)))
	Thm* t:eff(). t.smt smt()
frame_acts	Def t.acts == 2of(2of(t))
	Thm* t:frame(). t.acts Label List
dec_typ	Def t.typ == 2of(t)
	Thm* t:dec(). t.typ SimpleType
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)
smt_term	Def t.term == 1of(2of(t))
	Thm* t:smt(). t.term Term
smt_typ	Def t.typ == 2of(2of(t))
	Thm* t:smt(). t.typ SimpleType
pi2	Def 2of(t) == t.2
	Thm* A:Type, B:(AType), p:(a:AB(a)). 2of(p) B(1of(p))
frame_var	Def t.var == 1of(t)
	Thm* t:frame(). t.var Label
eff_kind	Def t.kind == 1of(t)
	Thm* t:eff(). t.kind Label
pre_kind	Def t.kind == 1of(t)
	Thm* t:pre(). t.kind Label
qimp_lbl	Def t.lbl == 1of(t)
	Thm* t:qimp{i:l}(). t.lbl Label
imp_hyp	Def t.hyp == 1of(t)
	Thm* t:imp{i:l}(). t.hyp Fmla
smt_lbl	Def t.lbl == 1of(t)
	Thm* t:smt(). t.lbl Label
dec_lbl	Def t.lbl == 1of(t)
	Thm* t:dec(). t.lbl Label
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
pi1	Def 1of(t) == t.1
	Thm* A:Type, B:(AType), p:(a:AB(a)). 1of(p) A
mk_dec	Def mk_dec(lbl, typ) == < lbl,typ >
	Thm* lbl:Label, typ:SimpleType. mk_dec(lbl, typ) dec()
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
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)
term_free_vars	Def term_free_vars(t) == term_iterate(f.nil;f.nil;f.nil;v.[v];P.nil;x,y. x @ y;t)
	Thm* t:Term. term_free_vars(t) Label List
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
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_vc_qimp	Def Case vc_qimp(x) = > body(x) cont(x1,z) == (x1.inr(x2) = > body(hd([x2 / tl(x1)])) cont(hd(x1),z))([x1])
case_vc_imp	Def Case vc_imp(x) = > body(x) cont(x1,z) == InjCase(x1; x2. body(x2); _. cont(z,z))
case	Def Case(value) body == body(value,value)
assert	Def b == if b True else False fi
	Thm* b:. b Prop
l_member	Def (x l) == i:. i < ||l|| & x = l[i] T
	Thm* T:Type, x:T, l:T List. (x l) Prop
ptn	Def Pattern == rec(T.ptn_con(T))
	Thm* Pattern Type
tree	Def Tree(E) == rec(T.tree_con(E;T))
	Thm* E:Type. Tree(E) Type
col_singleton	Def < x > (y) == y = x T
	Thm* T:Type, x:T. < x > Collection(T)
select	Def l[i] == hd(nth_tl(i;l))
	Thm* A:Type, l:A List, n:. 0n 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
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_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])
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_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
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
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))
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
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
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
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))
ptn_con	Def ptn_con(T) == Atom++Atom+(TT)
	Thm* T:Type. ptn_con(T) Type
tree_con	Def tree_con(E;T) == E+(TT)
	Thm* E,T:Type. tree_con(E;T) Type
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
le	Def AB == B < A
	Thm* i,j:. (ij) Prop
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)
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
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)
not	Def A == A False
	Thm* A:Prop. (A) Prop
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
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))
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))

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