Nuprl Lemma : seq-cons_wf
∀[T:Type]. ∀[a:T]. ∀[s:sequence(T)]. (seq-cons(a;s) ∈ sequence(T))
Proof
Definitions occuring in Statement :
seq-cons: seq-cons(a;s),
sequence: sequence(T),
uall: ∀[x:A]. B[x],
member: t ∈ T,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
seq-cons: seq-cons(a;s),
sequence: sequence(T),
nat: ℕ,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
not: ¬A,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
false: False,
prop: ℙ,
uiff: uiff(P;Q),
uimplies: b supposing a,
sq_stable: SqStable(P),
squash: ↓T,
subtract: n - m,
subtype_rel: A ⊆r B,
top: Top,
le: A ≤ B,
less_than': less_than'(a;b),
true: True,
int_seg: {i..j-},
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
exists: ∃x:A. B[x],
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
lelt: i ≤ j < k
Lemmas referenced :
decidable__le,
false_wf,
not-le-2,
sq_stable__le,
condition-implies-le,
minus-add,
minus-one-mul,
zero-add,
minus-one-mul-top,
add-associates,
add-swap,
add-commutes,
add_functionality_wrt_le,
add-zero,
le-add-cancel,
le_wf,
eq_int_wf,
bool_wf,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
iff_transitivity,
assert_wf,
bnot_wf,
not_wf,
equal-wf-T-base,
iff_weakening_uiff,
assert_of_bnot,
assert_of_eq_int,
int_seg_wf,
subtract_wf,
not-equal-2,
minus-zero,
minus-minus,
decidable__lt,
not-lt-2,
less-iff-le,
and_wf,
less_than_wf,
sequence_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
sqequalHypSubstitution,
productElimination,
thin,
dependent_pairEquality,
dependent_set_memberEquality,
addEquality,
setElimination,
rename,
because_Cache,
hypothesis,
natural_numberEquality,
extract_by_obid,
dependent_functionElimination,
hypothesisEquality,
unionElimination,
independent_pairFormation,
lambdaFormation,
voidElimination,
independent_functionElimination,
independent_isectElimination,
isectElimination,
imageMemberEquality,
baseClosed,
imageElimination,
applyEquality,
lambdaEquality,
isect_memberEquality,
voidEquality,
intEquality,
minusEquality,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
dependent_pairFormation,
promote_hyp,
instantiate,
cumulativity,
impliesFunctionality,
functionExtensionality,
functionEquality,
axiomEquality,
universeEquality
Latex:
\mforall{}[T:Type]. \mforall{}[a:T]. \mforall{}[s:sequence(T)]. (seq-cons(a;s) \mmember{} sequence(T))
Date html generated:
2018_07_25-PM-01_28_57
Last ObjectModification:
2018_06_12-PM-10_26_52
Theory : arithmetic
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