Nuprl Lemma : regext-loopset-empty
Now we do get ⌜(loopset() ⊆ co-regext(loopset()))⌝, because the type
⌜coW(Unit;x.Unit)⌝ has a single member (while the corresponding W type
⌜W(Unit;x.Unit)⌝ is empty). So the needed property
⌜∀a:coSet{i:l}. (transitive-set(a) ⇒ (a ⊆ co-regext(a)))⌝ could be true??..⋅
seteq(regext(loopset());{})
Proof
Definitions occuring in Statement :
regext: regext(a),
emptyset: {},
loopset: loopset(),
seteq: seteq(s1;s2)
Definitions unfolded in proof :
sq_stable: SqStable(P),
le: A ≤ B,
sq_type: SQType(T),
guard: {T},
cand: A c∧ B,
pcw-step-agree: StepAgree(s;p1;w),
isl: isl(x),
pi2: snd(t),
pcw-steprel: StepRel(s1;s2),
param-W-rel: param-W-rel(P;p.A[p];p,a.B[p; a];p,a,b.C[p; a; b];par;w),
W-rel: W-rel(A;a.B[a];w),
nat_plus: ℕ+,
ext-family: F ≡ G,
so_apply: x[s1;s2;s3],
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2],
so_lambda: λ2x y.t[x; y],
it: ⋅,
ext-eq: A ≡ B,
btrue: tt,
bfalse: ff,
ifthenelse: if b then t else f fi ,
assert: ↑b,
isr: isr(x),
squash: ↓T,
true: True,
less_than': less_than'(a;b),
less_than: a < b,
spreadn: spread3,
pcw-step: pcw-step(P;p.A[p];p,a.B[p; a];p,a,b.C[p; a; b]),
cw-step: cw-step(A;a.B[a]),
satisfiable_int_formula: satisfiable_int_formula(fmla),
not: ¬A,
uimplies: b supposing a,
or: P ∨ Q,
decidable: Dec(P),
ge: i ≥ j ,
lelt: i ≤ j < k,
int_seg: {i..j-},
nat: ℕ,
pcw-pp-barred: Barred(pp),
so_apply: x[s],
so_lambda: λ2x.t[x],
exists: ∃x:A. B[x],
top: Top,
prop: ℙ,
false: False,
implies: P ⇒ Q,
rev_implies: P ⇐ Q,
and: P ∧ Q,
iff: P ⇐⇒ Q,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
member: t ∈ T,
unit: Unit,
pi1: fst(t),
set-dom: set-dom(s),
mk-coset: mk-coset(T;f),
Wsup: Wsup(a;b),
mk-set: f"(T),
loopset: loopset(),
regext: regext(a)
Lemmas referenced :
sq_stable__le,
int_seg_subtype,
subtype_rel_function,
int_formula_prop_eq_lemma,
intformeq_wf,
decidable__equal_int,
int_subtype_base,
le_wf,
set_subtype_base,
nat_wf,
subtype_base_sq,
subtype_rel_dep_function,
pcw-steprel_wf,
param-co-W_wf,
it_wf,
param-co-W-ext,
W-ext,
int_term_value_add_lemma,
itermAdd_wf,
add-subtract-cancel,
equal_wf,
true_wf,
less_than_wf,
top_wf,
lelt_wf,
decidable__lt,
int_formula_prop_wf,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_subtract_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
intformless_wf,
itermVar_wf,
itermSubtract_wf,
itermConstant_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
full-omega-unsat,
decidable__le,
nat_properties,
subtract_wf,
int_seg_wf,
subtype_rel_self,
unit_wf2,
W-elimination-facts,
setmem-mk-coset,
coSet_wf,
iff_wf,
setmem-emptyset,
false_wf,
setmem_wf,
set-subtype-coSet,
emptyset_wf,
loopset_wf,
regext_wf,
co-seteq-iff
Rules used in proof :
cumulativity,
independent_pairEquality,
inrEquality,
applyLambdaEquality,
hyp_replacement,
productEquality,
unionEquality,
inlEquality,
dependent_pairEquality,
equalityElimination,
hypothesis_subsumption,
promote_hyp,
int_eqReduceTrueSq,
addEquality,
axiomEquality,
imageElimination,
baseClosed,
imageMemberEquality,
sqequalAxiom,
isect_memberFormation,
lessCases,
intEquality,
int_eqEquality,
dependent_pairFormation,
approximateComputation,
independent_isectElimination,
unionElimination,
because_Cache,
dependent_set_memberEquality,
rename,
setElimination,
natural_numberEquality,
functionExtensionality,
universeEquality,
instantiate,
equalitySymmetry,
equalityTransitivity,
strong_bar_Induction,
lambdaEquality,
voidEquality,
isect_memberEquality,
impliesFunctionality,
addLevel,
voidElimination,
hypothesisEquality,
independent_pairFormation,
lambdaFormation,
independent_functionElimination,
productElimination,
applyEquality,
hypothesis,
isectElimination,
thin,
dependent_functionElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalRule,
sqequalSubstitution,
cut
Latex:
seteq(regext(loopset());\{\})
Date html generated:
2018_07_29-AM-10_07_50
Last ObjectModification:
2018_07_21-PM-00_46_11
Theory : constructive!set!theory
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