Nuprl Lemma : satisfies-combine-pcs
∀f:ℤ ⟶ ℤ. ∀A,B:polynomial-constraints().
(satisfies-poly-constraints(f;combine-pcs(A;B)) ⇐⇒ satisfies-poly-constraints(f;A) ∧ satisfies-poly-constraints(f;B))
Proof
Definitions occuring in Statement :
combine-pcs: combine-pcs(X;Y),
satisfies-poly-constraints: satisfies-poly-constraints(f;X),
polynomial-constraints: polynomial-constraints(),
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],
polynomial-constraints: polynomial-constraints(),
satisfies-poly-constraints: satisfies-poly-constraints(f;X),
combine-pcs: combine-pcs(X;Y),
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
iPolynomial: iPolynomial(),
so_apply: x[s],
rev_implies: P ⇐ Q
Lemmas referenced :
polynomial-constraints_wf,
iff_wf,
append_wf,
equal_wf,
l_all_append,
le_wf,
ipolynomial-term_wf,
int_term_value_wf,
equal-wf-T-base,
l_member_wf,
iPolynomial_wf,
l_all_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
thin,
sqequalRule,
cut,
independent_pairFormation,
hypothesis,
productEquality,
lemma_by_obid,
isectElimination,
hypothesisEquality,
lambdaEquality,
setElimination,
rename,
intEquality,
baseClosed,
setEquality,
because_Cache,
natural_numberEquality,
addLevel,
impliesFunctionality,
dependent_functionElimination,
independent_functionElimination,
andLevelFunctionality,
functionEquality
Latex:
\mforall{}f:\mBbbZ{} {}\mrightarrow{} \mBbbZ{}. \mforall{}A,B:polynomial-constraints().
(satisfies-poly-constraints(f;combine-pcs(A;B))
\mLeftarrow{}{}\mRightarrow{} satisfies-poly-constraints(f;A) \mwedge{} satisfies-poly-constraints(f;B))
Date html generated:
2016_05_14-AM-07_08_11
Last ObjectModification:
2016_01_14-PM-08_41_20
Theory : omega
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