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Who Cites vc mng?
vc_mngDef vc_mng(v;rho;ds;da;de;e;s;tr) == Case(v) Case vc_imp(hc) = > [[hc.hyp]] rho ds < > de e s mk_trace_env(nil, tr.proj) [[hc.concl]] rho ds < > de e s mk_trace_env(nil, tr.proj) Case vc_qimp(qhc) = > v:[[dec_lookup(da;qhc.lbl)]] rho. [[qhc.hyp]] rho ds dec_lookup(da;qhc.lbl) de e s v tr [[qhc.concl]] rho ds dec_lookup(da;qhc.lbl) de e s v tappend(tr; < qhc.lbl,v > ) Default = > False
Thm* v:vc{i:l}(), ds,da:Collection(dec()), de:sig(), rho:Decl, e:{[[de]] rho}, s:{[[ds]] rho}, tr:trace_env([[da]] rho). tc_vc(v;ds;da;de) trace_consistent_vc(rho;da;tr.proj;v) vc_mng(v;rho;ds;da;de;e;s;tr) 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)
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
sts_mng Def [[sts]] rho == x:{x:SimpleType| x sts }. [[x]] rho
Thm* sts:Collection(SimpleType), rho:Decl. [[sts]] rho Type
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
dec Def dec() == LabelSimpleType
Thm* dec() Type
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
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
relname_mng Def [[rn]] rho e == Case(rn) Case eq(Q) = > x,y. x = y [[Q]] rho Case R = > e.R Default = > True
st_mng Def [[s]] rho == t_iterate(st_lift(rho);x,y. xy;s)
Thm* rho:Decl, s:SimpleType. [[s]] rho 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
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)
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
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)
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_default Def Default = > body(value,value) == body
qimp_lbl Def t.lbl == 1of(t)
Thm* t:qimp{i:l}(). t.lbl Label
tappend Def tappend(tr;a) == mk_trace_env(tr.trace @ [a], tr.proj)
Thm* d:Decl, tr:trace_env(d), a:(d). tappend(tr;a) trace_env(d)
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
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])
tproj Def tre.P == tre.trace | tre.proj(P)
Thm* d:Decl, tre:trace_env(d), P:Label. tre.P (d) List
trace_env_proj Def t.proj == 2of(t)
Thm* d:Decl, t:trace_env(d). t.proj LabelLabel
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)
col_none Def < > (x) == False
Thm* T:Type. < > Collection(T)
imp_concl Def t.concl == 2of(t)
Thm* t:imp{i:l}(). t.concl Fmla
imp_hyp Def t.hyp == 1of(t)
Thm* t:imp{i:l}(). t.hyp Fmla
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)
trace_env_trace Def t.trace == 1of(t)
Thm* d:Decl, t:trace_env(d). t.trace (d) List
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_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
pi1 Def 1of(t) == t.1
Thm* A:Type, B:(AType), p:(a:AB(a)). 1of(p) A
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
dec_typ Def t.typ == 2of(t)
Thm* t:dec(). t.typ SimpleType
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')
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))
col_member Def x c == c(x)
Thm* T:Type, x:T, c:Collection(T). x c Prop
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_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
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))
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))
tree Def Tree(E) == rec(T.tree_con(E;T))
Thm* E:Type. Tree(E) Type
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)
st_lift Def st_lift(rho)(x) == InjCase(x; x'. rho(x'); a. Top)
Thm* rho:(LabelType). st_lift(rho) (Label+Unit)Type
case_inl Def inl(x) = > body(x) cont(value,contvalue) == InjCase(value; x. body(x); _. cont(contvalue,contvalue))
ptn Def Pattern == rec(T.ptn_con(T))
Thm* Pattern Type
tree_con Def tree_con(E;T) == E+(TT)
Thm* E,T:Type. tree_con(E;T) Type
r_select Def r.l == r(l)
Thm* d:Decl, r:{d}, l:Label. r.l d(l)
case_relname_eq Def Case eq(x) = > body(x) cont(x1,z) == InjCase(x1; x2. body(x2); _. cont(z,z))
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))
ptn_con Def ptn_con(T) == Atom++Atom+(TT)
Thm* T:Type. ptn_con(T) 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_ts_var Def Case ts_var(x) = > body(x) cont(x1,z) == InjCase(x1; x2. body(x2); _. cont(z,z))

Syntax:vc_mng(v;rho;ds;da;de;e;s;tr) has structure: vc_mng(v; rho; ds; da; de; e; s; tr)

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