Who Cites decl?
decl	Def Decl == LabelType
	Thm* Decl{i} Type{i'}
ioa_mng	Def [[A]] rho de e == mk_sm([[A.da]] rho, [[A.ds]] rho, s.[[A.init]] rho A.ds < > de e s niltrace(), s1,a,s2. (p:pre(). p A.pre p.kind = kind(a) [[p.rel]] rho A.ds dec_lookup(A.da;kind(a)) de e s1 value(a) niltrace()) & (ef:eff(). ef A.eff ef.kind = kind(a) s2.ef.smt.lbl = [[ef.smt.term]] 1of(e) s1 value(a) niltrace() [[ef.smt.typ]] rho) & (fr:frame(). fr A.frame (kind(a) fr.acts) s2.fr.var = s1.fr.var [[fr.typ]] rho))
	Thm* A:ioa{i:l}(), de:sig(), rho:Decl, e:{[[de]] rho}. tc_ioa(A;de) ioa_mentions_trace(A) [[A]] rho de e sm{i:l}()
decls_mng	Def [[ds]] rho == [[d]] rho for d {d:dec()| d ds }
	Thm* ds:Collection(dec()), rho:Decl. [[ds]] rho Decl
ioa	Def ioa{i:l}() == Collection(dec())Collection(dec())Collection(rel())Collection(pre())Collection(eff())Collection(frame())
	Thm* ioa{i:l}() Type{i'}
ioa_all	Def ioa_all(I; i.A(i)) == mk_ioa(i:I. A(i).ds, i:I. A(i).da, i:I. A(i).init, i:I. A(i).pre, i:I. A(i).eff, i:I. A(i).frame)
	Thm* I:Type, A:(Iioa{i:l}()). ioa_all(I; i.A(i)) ioa{i:l}()
ioa_da	Def t.da == 1of(2of(t))
	Thm* t:ioa{i:l}(). t.da Collection(dec())
record_pair	Def {p} == {1of(p)}{2of(p)}
	Thm* p:(DeclDecl). {p} Type
sig	Def sig() == (LabelSimpleType)(Label(SimpleType List))
	Thm* sig() Type
sig_mng	Def [[s]] rho == < op.[[s.fun(op)]] rho,R.[[s.rel(R)]] rho >
	Thm* s:sig(), rho:Decl{i}. sig_mng{i:l}(s; rho) Decl{i}Decl{i'}
sm_action	Def M.action == (M.da)
	Thm* M:sm{i:l}(). M.action Type
sigma	Def (d) == l:Labeldecl_type(d;l)
	Thm* d:Decl. (d) Type
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)
dec	Def dec() == LabelSimpleType
	Thm* dec() Type
frame	Def frame() == LabelSimpleType(Label List)
	Thm* frame() Type
eff	Def eff() == LabelLabelSimpleTypesmt()
	Thm* eff() Type
pre	Def pre() == LabelLabelrel()
	Thm* pre() Type
pred_mng	Def [[p]] rho ds da de e s a tr == r:rel(). r p [[r]] rho ds da de e s a tr
	Thm* p:Fmla, ds,daa:Collection(dec()), da:Collection(SimpleType), de:sig(), rho:Decl, e:{[[de]] rho}, s:{[[ds]] rho}, a:[[da]] rho, tr:trace_env([[daa]] rho). trace_consistent_pred(rho;daa;tr.proj;p) tc_pred(p;ds;da;de) [[p]] rho ds da de e s a tr Prop
rel	Def rel() == relname()(Term List)
	Thm* rel() Type
record	Def {d} == l:Labeldecl_type(d;l)
	Thm* d:Decl. {d} Type
smt	Def smt() == LabelTermSimpleType
	Thm* smt() Type
relname	Def relname() == SimpleType+Label
	Thm* relname() Type
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
dec_mng	Def [[d]] rho == Case(d) Case x : s = > x:[[s]] rho
	Thm* rho:Decl, d:dec(). [[d]] rho Decl
col_union	Def (i:I. C(i))(x) == i:I. x C(i)
	Thm* T,I:Type, C:(ICollection(T)). (i:I. C(i)) 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
dall	Def D(i) for i I(x) == i:I. D(i)(x)
	Thm* I:Type, D:(IDecl). D(i) for i I Decl
col	Def Collection(T) == TProp
	Thm* T:Type{i'}. Collection{i}(T) Type{i'}
ioa_frame	Def t.frame == 2of(2of(2of(2of(2of(t)))))
	Thm* t:ioa{i:l}(). t.frame Collection(frame())
ioa_eff	Def t.eff == 1of(2of(2of(2of(2of(t)))))
	Thm* t:ioa{i:l}(). t.eff Collection(eff())
ioa_pre	Def t.pre == 1of(2of(2of(2of(t))))
	Thm* t:ioa{i:l}(). t.pre Collection(pre())
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
ioa_ds	Def t.ds == 1of(t)
	Thm* t:ioa{i:l}(). t.ds Collection(dec())
mk_ioa	Def mk_ioa(ds, da, init, pre, eff, frame) == < ds,da,init,pre,eff,frame >
	Thm* ds,da:Collection(dec()), init:Collection(rel()), pre:Collection(pre()), eff:Collection(eff()), frame:Collection(frame()). mk_ioa(ds, da, init, pre, eff, frame) ioa{i:l}()
frame_typ	Def t.typ == 1of(2of(t))
	Thm* t:frame(). t.typ SimpleType
frame_acts	Def t.acts == 2of(2of(t))
	Thm* t:frame(). t.acts Label List
value	Def value(a) == 2of(a)
	Thm* d:Decl, a:(d). value(a) d(kind(a))
eff_smt	Def t.smt == 2of(2of(2of(t)))
	Thm* t:eff(). t.smt smt()
smt_term	Def t.term == 1of(2of(t))
	Thm* t:smt(). t.term Term
rel_mng	Def [[r]] rho ds da de e s a tr == list_accum(x,t.x([[t]] 1of(e) s a tr);[[r.name]] rho 2of(e) ;r.args)
	Thm* r:rel(), ds,da:Collection(dec()), de:sig(), rho:Decl, st1:Collection(SimpleType), e:{[[de]] rho}, s:{[[ds]] rho}, a:[[st1]] rho, tr:trace_env([[da]] rho). trace_consistent_rel(rho;da;tr.proj;r) tc(r;ds;st1;de) [[r]] rho ds st1 de e s a tr Prop
	Thm* rho:Decl, ds,daa:Collection(dec()), da1:Collection(SimpleType), de:sig(), s:{[[ds]] rho}, e:{[[de]] rho}, tr:trace_env([[daa]] rho), r:rel(). closed_rel(r) tc(r;ds;da1;de) trace_consistent_rel(rho;daa;tr.proj;r) [[r]] rho ds da1 de e s tr Prop
term_mng	Def [[t]] e s a tr == iterate(statevar x- > s.x statevar x'- > s.x funsymbol f- > e.f freevar x- > a trace(P)- > tr.P x(y)- > x(y) over t)
smt_typ	Def t.typ == 2of(2of(t))
	Thm* t:smt(). t.typ SimpleType
pre_rel	Def t.rel == 2of(2of(t))
	Thm* t:pre(). t.rel rel()
sig_rel	Def t.rel == 2of(t)
	Thm* t:sig(). t.rel Label(SimpleType List)
tproj	Def tre.P == tre.trace | tre.proj(P)
	Thm* d:Decl, tre:trace_env(d), P:Label. tre.P (d) 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
trace_env_proj	Def t.proj == 2of(t)
	Thm* d:Decl, t:trace_env(d). t.proj LabelLabel
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
trace_projection	Def tr | P == filter(x.P(kind(x));tr)
	Thm* d:Decl, tr:(d) List, P:(Label). tr | P (d) List
kind	Def kind(a) == 1of(a)
	Thm* d:Decl, a:(d). kind(a) Label
	Thm* M:sm{i:l}(), a:M.action. kind(a) Label & kind(a) Pattern
smt_lbl	Def t.lbl == 1of(t)
	Thm* t:smt(). t.lbl 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
sig_fun	Def t.fun == 1of(t)
	Thm* t:sig(). t.fun LabelSimpleType
sm_da	Def t.da == 1of(t)
	Thm* t:sm{i:l}(). t.da Decl
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()
trace_env_trace	Def t.trace == 1of(t)
	Thm* d:Decl, t:trace_env(d). t.trace (d) List
pi1	Def 1of(t) == t.1
	Thm* A:Type, B:(AType), p:(a:AB(a)). 1of(p) A
relname_mng	Def [[rn]] rho e == Case(rn) Case eq(Q) = > x,y. x = y [[Q]] rho Case R = > e.R Default = > True
r_select	Def r.l == r(l)
	Thm* d:Decl, r:{d}, l:Label. r.l d(l)
st_list_mng	Def [[l]] rho == reduce(s,m. [[s]] rhom;Prop;l)
	Thm* l:SimpleType List, rho:Decl{i}. [[l]] rho{i} Type{i'}
st_mng	Def [[s]] rho == t_iterate(st_lift(rho);x,y. xy;s)
	Thm* rho:Decl, s:SimpleType. [[s]] rho Type
l_member	Def (x l) == i:. i < ||l|| & x = l[i] T
	Thm* T:Type, x:T, l:T List. (x l) Prop
nat	Def == {i:| 0i }
	Thm* Type
le	Def AB == B < A
	Thm* i,j:. (ij) Prop
not	Def A == A False
	Thm* A:Prop. (A) Prop
niltrace	Def niltrace() == mk_trace_env(nil, P,k. false)
	Thm* d:Decl. niltrace() trace_env(d)
col_none	Def < > (x) == False
	Thm* T:Type. < > Collection(T)
mk_sm	Def mk_sm(da, ds, init, trans) == < da,ds,init,trans >
	Thm* da,ds:Decl, init:({ds}Prop), trans:({ds}(da){ds}Prop). mk_sm(da, ds, init, trans) sm{i:l}()
decl_type	Def decl_type(d;x) == d(x)
	Thm* dec:Decl, x:Label. decl_type(dec;x) 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
dbase	Def x:y(a) == if a = x y else Top fi
	Thm* x:Label, y:Type. x:y Decl
case_mk_dec	Def Case lbl : typ = > body(lbl;typ)(x,z) == x/x2,x1. body(x2;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
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
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
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	Def Case(value) body == body(value,value)
st_lift	Def st_lift(rho)(x) == InjCase(x; x'. rho(x'); a. Top)
	Thm* rho:(LabelType). st_lift(rho) (Label+Unit)Type
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||
mk_trace_env	Def mk_trace_env(trace, proj) == < trace,proj >
	Thm* d:Decl, trace:(d) List, proj:(LabelLabel). mk_trace_env(trace, proj) trace_env(d)
list_accum	Def list_accum(x,a.f(x;a);y;l) == Case of l; nil y ; b.l' list_accum(x,a.f(x;a);f(y;b);l') (recursive)
tree	Def Tree(E) == rec(T.tree_con(E;T))
	Thm* E:Type. Tree(E) Type
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
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_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])
ptn_con	Def ptn_con(T) == Atom++Atom+(TT)
	Thm* T:Type. ptn_con(T) Type
top	Def Top == Void given Void
	Thm* Top Type
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))
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_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_relname_other	Def Case x = > body(x) cont(x1,z) == (x1.inr(x2) = > body(hd([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
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_relname_eq	Def Case eq(x) = > body(x) cont(x1,z) == InjCase(x1; x2. body(x2); _. cont(z,z))
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_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))
le_int	Def ij == j < i
	Thm* i,j:. (ij)
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_ts_var	Def Case ts_var(x) = > body(x) cont(x1,z) == InjCase(x1; x2. body(x2); _. cont(z,z))

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