Nuprl Lemma : loop-class-state-program_wf
∀[Info,B:Type].
∀[X:EClass(B ⟶ B)]. ∀[init:Id ⟶ bag(B)]. ∀[pr:LocalClass(X)].
(loop-class-state-program(pr;init) ∈ LocalClass(loop-class-state(X;init)))
supposing valueall-type(B)
Proof
Definitions occuring in Statement :
loop-class-state-program: loop-class-state-program(pr;init),
loop-class-state: loop-class-state(X;init),
local-class: LocalClass(X),
eclass: EClass(A[eo; e]),
Id: Id,
valueall-type: valueall-type(T),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
member: t ∈ T,
function: x:A ⟶ B[x],
universe: Type,
bag: bag(T)
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
local-class: LocalClass(X),
sq_exists: ∃x:{A| B[x]},
loop-class-state-program: loop-class-state-program(pr;init),
all: ∀x:A. B[x],
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
implies: P ⇒ Q,
so_apply: x[s],
prop: ℙ,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
strongwellfounded: SWellFounded(R[x; y]),
exists: ∃x:A. B[x],
nat: ℕ,
false: False,
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
not: ¬A,
top: Top,
and: P ∧ Q,
guard: {T},
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
less_than': less_than'(a;b),
decidable: Dec(P),
or: P ∨ Q,
less_than: a < b,
squash: ↓T,
class-ap: X(e),
es-before: before(e),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff,
true: True,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
iff: P ⇐⇒ Q,
hdf-state: hdf-state(X;bs),
mk-hdf: mk-hdf(s,m.G[s; m];st.H[st];s0),
hdf-ap: X(a),
hdf-run: hdf-run(P),
Id: Id,
es-E: E,
es-base-E: es-base-E(es),
rev_implies: P ⇐ Q,
ext-eq: A ≡ B,
pi1: fst(t),
callbyvalueall: callbyvalueall,
evalall: evalall(t),
empty-bag: {},
nil: [],
has-value: (a)↓,
has-valueall: has-valueall(a),
loop-class-state: loop-class-state(X;init),
eclass-cond: eclass-cond(X;Y),
eclass3: eclass3(X;Y),
member-eclass: e ∈b X,
pi2: snd(t),
hdf-halt: hdf-halt(),
eq_int: (i =z j),
bag-map: bag-map(f;bs),
bag-combine: ⋃x∈bs.f[x],
bag-null: bag-null(bs),
map: map(f;as),
bag-union: bag-union(bbs),
null: null(as),
cons: [a / b],
list_ind: list_ind,
concat: concat(ll),
bottom: ⊥,
append: as @ bs,
reduce: reduce(f;k;as),
so_lambda: so_lambda(x,y,z,w.t[x; y; z; w]),
so_apply: x[s1;s2;s3;s4]
Latex:
\mforall{}[Info,B:Type].
\mforall{}[X:EClass(B {}\mrightarrow{} B)]. \mforall{}[init:Id {}\mrightarrow{} bag(B)]. \mforall{}[pr:LocalClass(X)].
(loop-class-state-program(pr;init) \mmember{} LocalClass(loop-class-state(X;init)))
supposing valueall-type(B)
Date html generated:
2016_05_17-AM-09_07_28
Last ObjectModification:
2016_01_17-PM-09_17_22
Theory : local!classes
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