Nuprl Lemma : satisfiable-pcs-to-integer-problem
∀X:polynomial-constraints()
(satisfiable_polynomial_constraints(X) ⇒ satisfiable(fst(pcs-to-integer-problem(X));snd(pcs-to-integer-problem(X))))
Proof
Definitions occuring in Statement :
satisfiable-integer-problem: satisfiable(eqs;ineqs),
pcs-to-integer-problem: pcs-to-integer-problem(X),
satisfiable_polynomial_constraints: satisfiable_polynomial_constraints(X),
polynomial-constraints: polynomial-constraints(),
pi1: fst(t),
pi2: snd(t),
all: ∀x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
satisfiable_polynomial_constraints: satisfiable_polynomial_constraints(X),
exists: ∃x:A. B[x],
pcs-to-integer-problem: pcs-to-integer-problem(X),
member: t ∈ T,
uall: ∀[x:A]. B[x],
prop: ℙ,
uimplies: b supposing a,
polynomial-constraints: polynomial-constraints(),
satisfies-poly-constraints: satisfies-poly-constraints(f;X),
and: P ∧ Q,
subtype_rel: A ⊆r B,
pi1: fst(t),
pi2: snd(t),
has-value: (a)↓,
sq_type: SQType(T),
guard: {T},
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
or: P ∨ Q,
squash: ↓T,
iPolynomial: iPolynomial(),
cand: A c∧ B,
l_all: (∀x∈L.P[x]),
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
int_seg: {i..j-},
sq_stable: SqStable(P),
lelt: i ≤ j < k,
iMonomial: iMonomial(),
pcs-mon-vars: pcs-mon-vars(X),
so_lambda: λ2x.t[x],
so_apply: x[s],
l_member: (x ∈ l),
l_exists: (∃x∈L. P[x]),
nat: ℕ,
le: A ≤ B,
less_than: a < b,
int_nzero: ℤ-o,
satisfiable-integer-problem: satisfiable(eqs;ineqs),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
satisfies-integer-problem: satisfies-integer-problem(eqs;ineqs;xs),
top: Top,
satisfies-integer-equality: xs ⋅ as =0,
linearization: linearization(p;L),
subtract: n - m,
less_than': less_than'(a;b),
false: False,
cons: [a / b],
not: ¬A,
satisfies-integer-inequality: xs ⋅ as ≥0
Lemmas referenced :
hd-rev-pcs-mon-vars,
reverse_wf,
list_wf,
pcs-mon-vars_wf,
equal_wf,
satisfiable_polynomial_constraints_wf,
polynomial-constraints_wf,
value-type-has-value,
list-value-type,
iPolynomial_wf,
linearization_wf,
equal-wf-base,
list_subtype_base,
int_subtype_base,
less_than_wf,
length_wf,
map_wf,
list-valueall-type,
int-valueall-type,
eager-map-is-map,
evalall-reduce,
subtype_base_sq,
no_repeats_reverse,
no_repeats-pcs-mon-vars,
or_wf,
l_member_wf,
squash_wf,
true_wf,
int_term_value_wf,
ipolynomial-term_wf,
linearization-value,
int_seg_wf,
iMonomial_wf,
iff_weakening_equal,
member-pcs-mon-vars,
select_wf,
sq_stable__le,
member-reverse,
equal-wf-T-base,
l_exists_wf,
pi1_wf,
pi2_wf,
lelt_wf,
set_subtype_base,
product_subtype_base,
list_accum_wf,
satisfies-integer-problem_wf,
select-map,
subtype_rel_list,
top_wf,
map-length,
length_wf_nat,
and_wf,
nat_wf,
map_length,
poly-coeff-of_wf,
non_neg_length,
le_wf,
subtract_wf,
minus-one-mul,
add-swap,
add-commutes,
add-associates,
add-mul-special,
two-mul,
mul-distributes-right,
zero-mul,
zero-add,
one-mul,
int_seg_properties,
nat_properties,
list-cases,
map_nil_lemma,
length_of_nil_lemma,
product_subtype_list,
map_cons_lemma,
length_of_cons_lemma,
reduce_hd_cons_lemma,
list_accum_nil_lemma,
list_accum_cons_lemma,
null_nil_lemma,
btrue_wf,
null_wf,
null_cons_lemma,
bfalse_wf,
btrue_neq_bfalse,
length-map,
select_member,
ge_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
thin,
cut,
introduction,
extract_by_obid,
dependent_functionElimination,
hypothesisEquality,
isectElimination,
intEquality,
hypothesis,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
independent_isectElimination,
sqequalRule,
lambdaEquality,
applyEquality,
setElimination,
rename,
setEquality,
baseApply,
closedConclusion,
baseClosed,
because_Cache,
productEquality,
natural_numberEquality,
callbyvalueReduce,
instantiate,
cumulativity,
independent_pairEquality,
isect_memberFormation,
functionEquality,
imageElimination,
universeEquality,
functionExtensionality,
independent_pairFormation,
imageMemberEquality,
axiomEquality,
addLevel,
inrFormation,
unionElimination,
inlFormation,
dependent_pairFormation,
dependent_set_memberEquality,
multiplyEquality,
isect_memberEquality,
voidElimination,
voidEquality,
hyp_replacement,
applyLambdaEquality,
levelHypothesis,
sqequalIntensionalEquality,
promote_hyp,
addEquality,
hypothesis_subsumption
Latex:
\mforall{}X:polynomial-constraints()
(satisfiable\_polynomial\_constraints(X)
{}\mRightarrow{} satisfiable(fst(pcs-to-integer-problem(X));snd(pcs-to-integer-problem(X))))
Date html generated:
2017_04_14-AM-09_04_51
Last ObjectModification:
2017_02_27-PM-03_45_39
Theory : omega
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