Nuprl Lemma : member-pcs-mon-vars
∀X:polynomial-constraints(). ∀vs:ℤ List.
((vs ∈ pcs-mon-vars(X))
⇐⇒ (vs = [] ∈ (ℤ List))
∨ (∃p∈fst(X). (∃m∈p. vs = (snd(m)) ∈ (ℤ List)))
∨ (∃p∈snd(X). (∃m∈p. vs = (snd(m)) ∈ (ℤ List))))
Proof
Definitions occuring in Statement :
pcs-mon-vars: pcs-mon-vars(X),
polynomial-constraints: polynomial-constraints(),
l_exists: (∃x∈L. P[x]),
l_member: (x ∈ l),
nil: [],
list: T List,
pi1: fst(t),
pi2: snd(t),
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
or: P ∨ Q,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
polynomial-constraints: polynomial-constraints(),
pcs-mon-vars: pcs-mon-vars(X),
pi1: fst(t),
pi2: snd(t),
member: t ∈ T,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
uimplies: b supposing a,
iPolynomial: iPolynomial(),
int_seg: {i..j-},
sq_stable: SqStable(P),
implies: P ⇒ Q,
lelt: i ≤ j < k,
and: P ∧ Q,
squash: ↓T,
guard: {T},
so_apply: x[s],
iMonomial: iMonomial(),
prop: ℙ,
top: Top,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
iff: P ⇐⇒ Q,
or: P ∨ Q,
rev_implies: P ⇐ Q,
false: False
Lemmas referenced :
list_wf,
polynomial-constraints_wf,
list_induction,
iPolynomial_wf,
all_wf,
iff_wf,
l_member_wf,
list_accum_wf,
top_wf,
subtype_rel_list,
subtype_rel_set,
iMonomial_wf,
int_seg_wf,
length_wf,
imonomial-less_wf,
select_wf,
sq_stable__le,
less_than_transitivity2,
le_weakening2,
subtype_rel_product,
int_nzero_wf,
sorted_wf,
subtype_rel_self,
polynomial-mon-vars_wf,
or_wf,
l_exists_wf,
equal-wf-base-T,
list_subtype_base,
int_subtype_base,
list_accum_nil_lemma,
list_accum_cons_lemma,
false_wf,
l_exists_nil,
l_exists_wf_nil,
member-polynomial-mon-vars,
l_exists_cons,
cons_wf,
equal-wf-base,
nil_wf,
member_singleton
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
thin,
sqequalRule,
cut,
introduction,
extract_by_obid,
isectElimination,
intEquality,
hypothesis,
lambdaEquality,
because_Cache,
hypothesisEquality,
productEquality,
applyEquality,
independent_isectElimination,
natural_numberEquality,
setElimination,
rename,
independent_functionElimination,
imageMemberEquality,
baseClosed,
imageElimination,
dependent_functionElimination,
setEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
inlFormation,
unionElimination,
addLevel,
allFunctionality,
impliesFunctionality,
orFunctionality,
inrFormation,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}X:polynomial-constraints(). \mforall{}vs:\mBbbZ{} List.
((vs \mmember{} pcs-mon-vars(X))
\mLeftarrow{}{}\mRightarrow{} (vs = []) \mvee{} (\mexists{}p\mmember{}fst(X). (\mexists{}m\mmember{}p. vs = (snd(m)))) \mvee{} (\mexists{}p\mmember{}snd(X). (\mexists{}m\mmember{}p. vs = (snd(m)))))
Date html generated:
2017_04_14-AM-09_03_31
Last ObjectModification:
2017_02_27-PM-03_44_58
Theory : omega
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