Nuprl Lemma : length-nat-if-has-value
∀l:Base. ((||l||)↓ ⇒ (||l|| ∈ ℕ))
Proof
Definitions occuring in Statement :
length: ||as||,
nat: ℕ,
has-value: (a)↓,
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
base: Base
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
length: ||as||,
list_ind: list_ind,
uall: ∀[x:A]. B[x],
nat: ℕ,
false: False,
ge: i ≥ j ,
guard: {T},
uimplies: b supposing a,
prop: ℙ,
subtype_rel: A ⊆r B,
top: Top,
not: ¬A,
decidable: Dec(P),
or: P ∨ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
subtract: n - m,
le: A ≤ B,
less_than': less_than'(a;b),
true: True,
nat_plus: ℕ+,
has-value: (a)↓,
sq_stable: SqStable(P),
squash: ↓T
Lemmas referenced :
nat_properties,
less_than_transitivity1,
less_than_irreflexivity,
ge_wf,
less_than_wf,
has-value_wf_base,
int_subtype_base,
base_wf,
fun_exp0_lemma,
strictness-apply,
bottom_diverge,
decidable__le,
subtract_wf,
false_wf,
not-ge-2,
less-iff-le,
condition-implies-le,
minus-one-mul,
zero-add,
minus-one-mul-top,
minus-add,
minus-minus,
add-associates,
add-swap,
add-commutes,
add_functionality_wrt_le,
add-zero,
le-add-cancel,
fun_exp_unroll_1,
has-value-implies-dec-ispair-2,
top_wf,
value-type-has-value,
int-value-type,
nat_wf,
add_nat_wf,
le_wf,
sq_stable__le,
equal_wf,
has-value-implies-dec-isaxiom-2
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
sqequalHypSubstitution,
sqequalRule,
hypothesis,
compactness,
thin,
introduction,
extract_by_obid,
isectElimination,
hypothesisEquality,
setElimination,
rename,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
independent_functionElimination,
voidElimination,
lambdaEquality,
dependent_functionElimination,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
baseApply,
closedConclusion,
baseClosed,
applyEquality,
isect_memberEquality,
voidEquality,
unionElimination,
independent_pairFormation,
productElimination,
addEquality,
intEquality,
minusEquality,
because_Cache,
dependent_set_memberEquality,
callbyvalueCallbyvalue,
callbyvalueReduce,
callbyvalueAdd,
imageMemberEquality,
imageElimination
Latex:
\mforall{}l:Base. ((||l||)\mdownarrow{} {}\mRightarrow{} (||l|| \mmember{} \mBbbN{}))
Date html generated:
2017_04_14-AM-08_35_32
Last ObjectModification:
2017_02_27-PM-03_29_02
Theory : list_0
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