Nuprl Lemma : member-and-poly-constraints
∀L1,L2:polynomial-constraints() List. ∀X:polynomial-constraints().
((X ∈ and-poly-constraints(L1;L2)) ⇐⇒ (∃A∈L1. (∃B∈L2. X = combine-pcs(A;B) ∈ polynomial-constraints())))
Proof
Definitions occuring in Statement :
and-poly-constraints: and-poly-constraints(Xs;Ys),
combine-pcs: combine-pcs(X;Y),
polynomial-constraints: polynomial-constraints(),
l_exists: (∃x∈L. P[x]),
l_member: (x ∈ l),
list: T List,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
equal: s = t ∈ T
Definitions unfolded in proof :
and-poly-constraints: and-poly-constraints(Xs;Ys),
all: ∀x:A. B[x],
member: t ∈ T,
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
or: P ∨ Q,
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
not: ¬A,
false: False,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
rev_implies: P ⇐ Q,
guard: {T},
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
top: Top,
subtype_rel: A ⊆r B,
l_exists: (∃x∈L. P[x]),
exists: ∃x:A. B[x],
polynomial-constraints: polynomial-constraints(),
iPolynomial: iPolynomial(),
int_seg: {i..j-},
sq_stable: SqStable(P),
lelt: i ≤ j < k,
squash: ↓T,
iMonomial: iMonomial(),
int_nzero: ℤ-o,
sq_type: SQType(T)
Lemmas referenced :
null_nil_lemma,
btrue_wf,
member-implies-null-eq-bfalse,
polynomial-constraints_wf,
nil_wf,
btrue_neq_bfalse,
or_wf,
l_member_wf,
l_exists_wf,
equal_wf,
combine-pcs_wf,
list_accum_wf,
list_wf,
cons_wf,
all_wf,
iff_wf,
list_induction,
list_accum_nil_lemma,
false_wf,
list_accum_cons_lemma,
l_exists_nil,
l_exists_wf_nil,
l_exists_functionality,
set_wf,
l_exists_cons,
subtype_base_sq,
list_subtype_base,
product_subtype_base,
iPolynomial_wf,
set_subtype_base,
iMonomial_wf,
int_seg_wf,
length_wf,
imonomial-less_wf,
select_wf,
sq_stable__le,
less_than_transitivity2,
le_weakening2,
int_nzero_wf,
sorted_wf,
subtype_rel_self,
nequal_wf,
int_subtype_base,
cons_member,
exists_wf,
l_exists_iff,
member_singleton,
and_wf
Rules used in proof :
cut,
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
lambdaFormation,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
independent_pairFormation,
unionElimination,
introduction,
extract_by_obid,
isectElimination,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
voidElimination,
lambdaEquality,
setElimination,
rename,
setEquality,
inrFormation,
because_Cache,
addLevel,
allFunctionality,
productElimination,
impliesFunctionality,
isect_memberEquality,
voidEquality,
inlFormation,
orFunctionality,
applyEquality,
instantiate,
cumulativity,
natural_numberEquality,
imageMemberEquality,
baseClosed,
imageElimination,
intEquality,
functionEquality,
dependent_pairFormation,
dependent_set_memberEquality,
applyLambdaEquality,
levelHypothesis,
promote_hyp,
productEquality
Latex:
\mforall{}L1,L2:polynomial-constraints() List. \mforall{}X:polynomial-constraints().
((X \mmember{} and-poly-constraints(L1;L2)) \mLeftarrow{}{}\mRightarrow{} (\mexists{}A\mmember{}L1. (\mexists{}B\mmember{}L2. X = combine-pcs(A;B))))
Date html generated:
2017_04_14-AM-09_02_15
Last ObjectModification:
2017_02_27-PM-03_44_36
Theory : omega
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