Who Cites action pre?
action_pre	Def action_pre(a;ps) == < p.rel | p < p ps | p.kind = a > >
	Thm* a:Label, ps:Collection(pre()). action_pre(a;ps) Fmla
decl	Def Decl == LabelType
	Thm* Decl{i} Type{i'}
ioa	Def ioa{i:l}() == Collection(dec())Collection(dec())Collection(rel())Collection(pre())Collection(eff())Collection(frame())
	Thm* ioa{i:l}() Type{i'}
ioa_mentions_trace	Def ioa_mentions_trace(A) == (e:eff(). e A.eff & mentions_trace(e.smt.term)) (p:pre(). p A.pre & rel_mentions_trace(p.rel)) (r:rel(). r A.init & rel_mentions_trace(r))
	Thm* A:ioa{i:l}(). ioa_mentions_trace(A) Prop
pre	Def pre() == LabelLabelrel()
	Thm* pre() Type
rel	Def rel() == relname()(Term List)
	Thm* rel() Type
trace_consistent_rel	Def trace_consistent_rel(rho;da;R;r) == i:||r.args||. trace_consistent(rho;da;R;r.args[i])
	Thm* rho:Decl, r:rel(), da:Collection(dec()), R:(LabelLabel). trace_consistent_rel(rho;da;R;r) Prop
frame	Def frame() == LabelSimpleType(Label List)
	Thm* frame() Type
eff	Def eff() == LabelLabelSimpleTypesmt()
	Thm* eff() Type
trace_consistent	Def trace_consistent(rho;da;R;t) == g:Label. term_mentions_guard(g;t) subtype_rel(({a:([[da]] rho)| (R(g,kind(a))) } List); (rho(lbl_pr( < Trace, g > ))))
	Thm* rho:Decl, t:Term, da:Collection(dec()), R:(LabelLabel). trace_consistent(rho;da;R;t) Prop
decls_mng	Def [[ds]] rho == [[d]] rho for d {d:dec()| d ds }
	Thm* ds:Collection(dec()), rho:Decl. [[ds]] rho Decl
dec	Def dec() == LabelSimpleType
	Thm* dec() Type
smt	Def smt() == LabelTermSimpleType
	Thm* smt() Type
term	Def Term == Tree(ts())
	Thm* Term Type
relname	Def relname() == SimpleType+Label
	Thm* relname() Type
st	Def SimpleType == Tree(Label+Unit)
	Thm* SimpleType Type
ts	Def ts() == Label+Label+Label+Label+Label
	Thm* ts() Type
sigma	Def (d) == l:Labeldecl_type(d;l)
	Thm* d:Decl. (d) Type
lbl	Def Label == {p:Pattern| ground_ptn(p) }
	Thm* Label Type
assert	Def b == if b True else False fi
	Thm* b:. b Prop
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
ioa_da	Def t.da == 1of(2of(t))
	Thm* t:ioa{i:l}(). t.da Collection(dec())
ioa_pre	Def t.pre == 1of(2of(2of(2of(t))))
	Thm* t:ioa{i:l}(). t.pre Collection(pre())
int_seg	Def {i..j} == {k:| i k < j }
	Thm* m,n:. {m..n} Type
lelt	Def i j < k == ij & j < k
le	Def AB == B < A
	Thm* i,j:. (ij) Prop
not	Def A == A False
	Thm* A:Prop. (A) Prop
pre_rel	Def t.rel == 2of(2of(t))
	Thm* t:pre(). t.rel rel()
rel_mentions_trace	Def rel_mentions_trace(r) == reduce(x,y. mentions_trace(x) y;false;r.args)
	Thm* r:rel(). rel_mentions_trace(r)
pre_kind	Def t.kind == 1of(t)
	Thm* t:pre(). t.kind Label
term_mentions_guard	Def term_mentions_guard(g;t) == term_iterate(x.false;x.false;x.false;x.false;x.x = g;x,y. x y;t)
	Thm* t:Term, g:Label. term_mentions_guard(g;t)
dec_mng	Def [[d]] rho == Case(d) Case x : s = > x:[[s]] rho
	Thm* rho:Decl, d:dec(). [[d]] rho Decl
dbase	Def x:y(a) == if a = x y else Top fi
	Thm* x:Label, y:Type. x:y Decl
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
col	Def Collection(T) == TProp
	Thm* T:Type{i'}. Collection{i}(T) Type{i'}
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
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
ioa_eff	Def t.eff == 1of(2of(2of(2of(2of(t)))))
	Thm* t:ioa{i:l}(). t.eff Collection(eff())
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))
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
pi1	Def 1of(t) == t.1
	Thm* A:Type, B:(AType), p:(a:AB(a)). 1of(p) A
mentions_trace	Def mentions_trace(t) == iterate(statevar x- > false statevar x'- > false funsymbol x- > false freevar x- > false trace(P)- > true x(y)- > x y over t)
	Thm* t:Term. mentions_trace(t)
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)
ptn	Def Pattern == rec(T.ptn_con(T))
	Thm* Pattern Type
bor	Def p q == if p true else q fi
	Thm* p,q:. (p q)
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
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||
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
st_mng	Def [[s]] rho == t_iterate(st_lift(rho);x,y. xy;s)
	Thm* rho:Decl, s:SimpleType. [[s]] rho 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)
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))
ptn_con	Def ptn_con(T) == Atom++Atom+(TT)
	Thm* T:Type. ptn_con(T) Type
tree	Def Tree(E) == rec(T.tree_con(E;T))
	Thm* E:Type. Tree(E) Type
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
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
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))
tree_con	Def tree_con(E;T) == E+(TT)
	Thm* E,T:Type. tree_con(E;T) Type
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
dall	Def D(i) for i I(x) == i:I. D(i)(x)
	Thm* I:Type, D:(IDecl). D(i) for i I Decl
decl_type	Def decl_type(d;x) == d(x)
	Thm* dec:Decl, x:Label. decl_type(dec;x) Type
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_mk_dec	Def Case lbl : typ = > body(lbl;typ)(x,z) == x/x2,x1. body(x2;x1)
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))
st_lift	Def st_lift(rho)(x) == InjCase(x; x'. rho(x'); a. Top)
	Thm* rho:(LabelType). st_lift(rho) (Label+Unit)Type
top	Def Top == Void given Void
	Thm* Top Type

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