Nuprl Lemma : satisfies-and-poly-constraints
∀f:ℤ ⟶ ℤ. ∀L2,L1:polynomial-constraints() List.
((∃X∈L1. satisfies-poly-constraints(f;X)) ∧ (∃X∈L2. satisfies-poly-constraints(f;X))
⇐⇒ (∃X∈and-poly-constraints(L1;L2). satisfies-poly-constraints(f;X)))
Proof
Definitions occuring in Statement :
and-poly-constraints: and-poly-constraints(Xs;Ys),
satisfies-poly-constraints: satisfies-poly-constraints(f;X),
polynomial-constraints: polynomial-constraints(),
l_exists: (∃x∈L. P[x]),
list: T List,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
function: x:A ⟶ B[x],
int: ℤ
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
exists: ∃x:A. B[x],
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
rev_implies: P ⇐ Q,
cand: A c∧ B,
uimplies: b supposing a,
polynomial-constraints: polynomial-constraints(),
iPolynomial: iPolynomial(),
int_seg: {i..j-},
sq_stable: SqStable(P),
lelt: i ≤ j < k,
squash: ↓T,
guard: {T},
iMonomial: iMonomial(),
int_nzero: ℤ-o,
sq_type: SQType(T)
Lemmas referenced :
list_wf,
polynomial-constraints_wf,
exists_wf,
l_member_wf,
satisfies-poly-constraints_wf,
equal_wf,
combine-pcs_wf,
l_exists_wf,
and-poly-constraints_wf,
iff_wf,
l_exists_iff,
member-and-poly-constraints,
satisfies-combine-pcs,
subtype_base_sq,
product_subtype_base,
iPolynomial_wf,
list_subtype_base,
set_subtype_base,
iMonomial_wf,
all_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
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesis,
functionEquality,
intEquality,
independent_pairFormation,
productElimination,
productEquality,
sqequalRule,
lambdaEquality,
hypothesisEquality,
functionExtensionality,
applyEquality,
because_Cache,
setElimination,
rename,
setEquality,
addLevel,
impliesFunctionality,
dependent_functionElimination,
independent_functionElimination,
existsFunctionality,
andLevelFunctionality,
levelHypothesis,
existsLevelFunctionality,
dependent_pairFormation,
instantiate,
cumulativity,
independent_isectElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
imageElimination,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}f:\mBbbZ{} {}\mrightarrow{} \mBbbZ{}. \mforall{}L2,L1:polynomial-constraints() List.
((\mexists{}X\mmember{}L1. satisfies-poly-constraints(f;X)) \mwedge{} (\mexists{}X\mmember{}L2. satisfies-poly-constraints(f;X))
\mLeftarrow{}{}\mRightarrow{} (\mexists{}X\mmember{}and-poly-constraints(L1;L2). satisfies-poly-constraints(f;X)))
Date html generated:
2017_04_14-AM-09_02_24
Last ObjectModification:
2017_02_27-PM-03_43_49
Theory : omega
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