Who Cites ioa trans all?
ioa_trans_all	Def ioa_trans_all{i}(A;I) == < ioa_trans(A;a.lbl;I) | a A.da >
	Thm* A:ioa{i:l}(), I:Fmla. ioa_trans_all{i}(A;I) VCs
dec_lbl	Def t.lbl == 1of(t)
	Thm* t:dec(). t.lbl Label
ioa_trans	Def ioa_trans(A;a;I) == vc_qimp(mk_qimp(a, I action_pre(a;A.pre), smts_eff_pred(action_effect(a;A.eff;A.frame);I)))
	Thm* A:ioa{i:l}(), a:Label, I:Fmla. ioa_trans(A;a;I) vc{i:l}()
ioa_da	Def t.da == 1of(2of(t))
	Thm* t:ioa{i:l}(). t.da Collection(dec())
vc	Def vc{i:l}() == imp{i:l}()+qimp{i:l}()
	Thm* vc{i:l}() Type{i'}
dec	Def dec() == LabelSimpleType
	Thm* dec() Type
action_effect	Def action_effect(a;es;fs) == < e.smt | e < e es | e.kind = a > > + < mk_smt(f.var, f.var, f.typ) | f < f fs | a f.acts > >
	Thm* a:Label, es:Collection(eff()), fs:Collection(frame()). action_effect(a;es;fs) Collection(smt())
smts_eff_pred	Def smts_eff_pred(ss;p) == (rp.smts_eff_rel(ss;r))
	Thm* p:Fmla, ss:Collection(smt()). smts_eff_pred(ss;p) Fmla
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
smts_eff_rel	Def smts_eff_rel(ss;r) == col_subst(x.smts_eff(ss;x);r)
	Thm* r:rel(), ss:Collection(smt()). smts_eff_rel(ss;r) Fmla
smts_eff	Def smts_eff(ss;x) == smt_terms( < s ss | s.lbl = x > )
	Thm* ss:Collection(smt()), x:Label. smts_eff(ss;x) Collection(Term)
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())
smt_terms	Def smt_terms(c) == < s.term | s c >
	Thm* c:Collection(smt()). smt_terms(c) Collection(Term)
col_map_subst	Def col_map_subst(x.f(x);c) == < f(x) | x c >
	Thm* f:(((LabelTerm) 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')
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())
frame_typ	Def t.typ == 1of(2of(t))
	Thm* t:frame(). t.typ SimpleType
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
smt_lbl	Def t.lbl == 1of(t)
	Thm* t:smt(). t.lbl Label
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()
smt_term	Def t.term == 1of(2of(t))
	Thm* t:smt(). t.term Term
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
ioa_frame	Def t.frame == 2of(2of(2of(2of(2of(t)))))
	Thm* t:ioa{i:l}(). t.frame Collection(frame())
pred_and	Def P Q == P + Q
	Thm* P,Q:Fmla. (P Q) Fmla
mk_qimp	Def mk_qimp(lbl, hyp, concl) == < lbl,hyp,concl >
	Thm* lbl:Label, hyp,concl:Fmla. mk_qimp(lbl, hyp, concl) qimp{i:l}()
vc_qimp	Def vc_qimp(x) == inr(x)
	Thm* x:qimp{i:l}(). vc_qimp(x) vc{i:l}()
frame_acts	Def t.acts == 2of(2of(t))
	Thm* t:frame(). t.acts Label List
eff_smt	Def t.smt == 2of(2of(2of(t)))
	Thm* t:eff(). t.smt smt()
pre_rel	Def t.rel == 2of(2of(t))
	Thm* t:pre(). t.rel rel()
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))
qimp	Def qimp{i:l}() == LabelFmlaFmla
	Thm* qimp{i:l}() Type{i'}
imp	Def imp{i:l}() == FmlaFmla
	Thm* imp{i:l}() Type{i'}
eff	Def eff() == LabelLabelSimpleTypesmt()
	Thm* eff() Type
smt	Def smt() == LabelTermSimpleType
	Thm* smt() Type
frame	Def frame() == LabelSimpleType(Label List)
	Thm* frame() Type
pre	Def pre() == LabelLabelrel()
	Thm* pre() Type
pred	Def Fmla == Collection(rel())
	Thm* Fmla{i} Type{i'}
rel	Def rel() == relname()(Term List)
	Thm* rel() 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
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_add	Def (a + b)(x) == x a x b
	Thm* T:Type, a,b:Collection(T). (a + b) 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_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_member	Def x c == c(x)
	Thm* T:Type, x:T, c:Collection(T). x c Prop
tvar	Def l == tree_leaf(ts_var(l))
	Thm* l:Label. l Term
mk_smt	Def mk_smt(lbl, term, typ) == < lbl,term,typ >
	Thm* lbl:Label, term:Term, typ:SimpleType. mk_smt(lbl, term, typ) smt()
lbls_member	Def x ls == reduce(a,b. x = a b;false;ls)
	Thm* x:Label, ls:Label List. x ls
select	Def l[i] == hd(nth_tl(i;l))
	Thm* A:Type, l:A List, n:. 0n n < ||l|| l[n] 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
le_int	Def ij == j < i
	Thm* i,j:. (ij)
bnot	Def b == if b false else true fi
	Thm* b:. b
assert	Def b == if b True else False fi
	Thm* b:. b Prop
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
tree	Def Tree(E) == rec(T.tree_con(E;T))
	Thm* E:Type. Tree(E) 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)
ptn	Def Pattern == rec(T.ptn_con(T))
	Thm* Pattern Type
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)
bor	Def p q == if p true else q fi
	Thm* p,q:. (p q)
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
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
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))
col	Def Collection(T) == TProp
	Thm* T:Type{i'}. Collection{i}(T) Type{i'}
tree_con	Def tree_con(E;T) == E+(TT)
	Thm* E,T:Type. tree_con(E;T) Type
ptn_con	Def ptn_con(T) == Atom++Atom+(TT)
	Thm* T:Type. ptn_con(T) Type
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
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))
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
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
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
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
mk_rel	Def mk_rel(name, args) == < name,args >
	Thm* name:relname(), args:Term List. mk_rel(name, args) rel()
le	Def AB == B < A
	Thm* i,j:. (ij) Prop
tapp	Def t1 t2 == tree_node( < t1, t2 > )
	Thm* t1,t2:Term. t1 t2 Term
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()
lt_int	Def i < j == if i < j true ; false fi
	Thm* i,j:. (i < j)
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_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))

Syntax:

ioa_trans_all{i}(A;I)

has structure:

ioa_trans_all{i:l}(A; I)

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