Nuprl Lemma : pcs-to-integer-problem_wf
∀[X:polynomial-constraints()]
(pcs-to-integer-problem(X) ∈ ⋃n:ℕ.({L:ℤ List| ||L|| = (n + 1) ∈ ℤ} List × ({L:ℤ List| ||L|| = (n + 1) ∈ ℤ} List)))
Proof
Definitions occuring in Statement :
pcs-to-integer-problem: pcs-to-integer-problem(X),
polynomial-constraints: polynomial-constraints(),
length: ||as||,
list: T List,
nat: ℕ,
tunion: ⋃x:A.B[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
set: {x:A| B[x]} ,
product: x:A × B[x],
add: n + m,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
pcs-to-integer-problem: pcs-to-integer-problem(X),
all: ∀x:A. B[x],
implies: P ⇒ Q,
polynomial-constraints: polynomial-constraints(),
has-value: (a)↓,
uimplies: b supposing a,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
tunion: ⋃x:A.B[x],
nat: ℕ,
subtract: n - m,
top: Top,
sq_type: SQType(T),
guard: {T},
pi2: snd(t),
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
or: P ∨ Q,
iPolynomial: iPolynomial(),
iMonomial: iMonomial(),
pi1: fst(t),
pcs-mon-vars: pcs-mon-vars(X),
not: ¬A,
false: False,
cons: [a / b],
exists: ∃x:A. B[x],
decidable: Dec(P),
ge: i ≥ j
Lemmas referenced :
reverse_wf,
list_wf,
pcs-mon-vars_wf,
value-type-has-value,
list-value-type,
eager-map_wf,
iPolynomial_wf,
equal-wf-base,
set-value-type,
linearization_wf,
evalall-reduce,
list-valueall-type,
set-valueall-type,
int-valueall-type,
subtype_base_sq,
nat_wf,
set_subtype_base,
le_wf,
int_subtype_base,
add-associates,
add-swap,
length_wf,
add-commutes,
zero-add,
length_wf_nat,
equal-wf-base-T,
equal_wf,
polynomial-constraints_wf,
list_subtype_base,
member-reverse,
nil_wf,
member-pcs-mon-vars,
or_wf,
l_exists_wf,
l_member_wf,
pi1_wf,
iMonomial_wf,
pi2_wf,
list-cases,
length_of_nil_lemma,
null_nil_lemma,
btrue_wf,
member-implies-null-eq-bfalse,
btrue_neq_bfalse,
product_subtype_list,
length_of_cons_lemma,
subtract_wf,
non_neg_length,
nat_properties,
decidable__le
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
intEquality,
hypothesis,
hypothesisEquality,
lambdaFormation,
productElimination,
sqequalRule,
callbyvalueReduce,
independent_isectElimination,
setEquality,
because_Cache,
lambdaEquality,
baseApply,
closedConclusion,
baseClosed,
applyEquality,
imageMemberEquality,
dependent_pairEquality,
independent_pairEquality,
instantiate,
cumulativity,
natural_numberEquality,
isect_memberEquality,
voidElimination,
voidEquality,
dependent_functionElimination,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
productEquality,
addEquality,
setElimination,
rename,
axiomEquality,
inlFormation,
unionElimination,
promote_hyp,
hypothesis_subsumption,
dependent_set_memberEquality,
dependent_pairFormation,
sqequalIntensionalEquality
Latex:
\mforall{}[X:polynomial-constraints()]
(pcs-to-integer-problem(X) \mmember{} \mcup{}n:\mBbbN{}.(\{L:\mBbbZ{} List| ||L|| = (n + 1)\} List
\mtimes{} (\{L:\mBbbZ{} List| ||L|| = (n + 1)\} List)))
Date html generated:
2017_04_14-AM-09_04_40
Last ObjectModification:
2017_02_27-PM-03_44_14
Theory : omega
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