Nuprl Lemma : test-exact-eq-constraint_wf
∀[L:ℤ List]. (test-exact-eq-constraint(L) ∈ ∃i:ℕ||L|| [(|L[i]| = 1 ∈ ℤ)]?)
Proof
Definitions occuring in Statement :
test-exact-eq-constraint: test-exact-eq-constraint(L),
select: L[n],
length: ||as||,
list: T List,
absval: |i|,
int_seg: {i..j-},
uall: ∀[x:A]. B[x],
sq_exists: ∃x:A [B[x]],
unit: Unit,
member: t ∈ T,
union: left + right,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
test-exact-eq-constraint: test-exact-eq-constraint(L),
sq_exists: ∃x:A [B[x]],
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
guard: {T},
uimplies: b supposing a,
prop: ℙ,
or: P ∨ Q,
select: L[n],
nil: [],
it: ⋅,
so_lambda: λ2x y.t[x; y],
top: Top,
so_apply: x[s1;s2],
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
cons: [a / b],
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
not: ¬A,
colength: colength(L),
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_type: SQType(T),
less_than: a < b,
squash: ↓T,
sq_stable: SqStable(P),
subtract: n - m,
subtype_rel: A ⊆r B,
exposed-bfalse: exposed-bfalse,
bool: 𝔹,
unit: Unit,
btrue: tt,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
int_seg: {i..j-},
lelt: i ≤ j < k,
nat_plus: ℕ+,
true: True,
bfalse: ff,
exists: ∃x:A. B[x],
bnot: ¬bb,
assert: ↑b,
nequal: a ≠ b ∈ T ,
decidable: Dec(P),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
nat_properties,
less_than_transitivity1,
less_than_irreflexivity,
ge_wf,
istype-less_than,
list-cases,
length_of_nil_lemma,
stuck-spread,
istype-base,
istype-void,
list_ind_nil_lemma,
it_wf,
int_seg_wf,
istype-int,
product_subtype_list,
colength-cons-not-zero,
istype-nat,
colength_wf_list,
istype-false,
istype-le,
list_wf,
subtract-1-ge-0,
subtype_base_sq,
nat_wf,
set_subtype_base,
le_wf,
int_subtype_base,
spread_cons_lemma,
sq_stable__le,
add-associates,
add-commutes,
add-swap,
zero-add,
length_of_cons_lemma,
list_ind_cons_lemma,
absval_wf,
eq_int_wf,
eqtt_to_assert,
assert_of_eq_int,
add_nat_plus,
length_wf_nat,
length_wf,
select-cons-hd,
list_subtype_base,
lelt_wf,
unit_wf2,
eqff_to_assert,
bool_subtype_base,
bool_cases_sqequal,
bool_wf,
assert-bnot,
neg_assert_of_eq_int,
add-member-int_seg2,
decidable__le,
subtract_wf,
not-le-2,
not-equal-2,
condition-implies-le,
minus-add,
minus-one-mul,
minus-one-mul-top,
add_functionality_wrt_le,
add-zero,
le-add-cancel2,
select-cons-tl,
decidable__lt,
not-lt-2,
le-add-cancel,
le_weakening2
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
Error :isect_memberFormation_alt,
introduction,
cut,
thin,
Error :lambdaFormation_alt,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
independent_functionElimination,
voidElimination,
Error :universeIsType,
Error :lambdaEquality_alt,
dependent_functionElimination,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
Error :functionIsTypeImplies,
Error :inhabitedIsType,
intEquality,
unionElimination,
baseClosed,
Error :isect_memberEquality_alt,
Error :inrEquality_alt,
closedConclusion,
Error :setIsType,
Error :equalityIsType4,
promote_hyp,
hypothesis_subsumption,
productElimination,
Error :equalityIsType1,
Error :dependent_set_memberEquality_alt,
because_Cache,
independent_pairFormation,
instantiate,
cumulativity,
imageElimination,
imageMemberEquality,
applyLambdaEquality,
applyEquality,
minusEquality,
baseApply,
equalityElimination,
Error :inlEquality_alt,
Error :productIsType,
addEquality,
Error :dependent_pairFormation_alt
Latex:
\mforall{}[L:\mBbbZ{} List]. (test-exact-eq-constraint(L) \mmember{} \mexists{}i:\mBbbN{}||L|| [(|L[i]| = 1)]?)
Date html generated:
2019_06_20-PM-00_50_45
Last ObjectModification:
2018_10_18-PM-01_14_44
Theory : omega
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