Nuprl Lemma : seq-add_wf
∀[T:Type]. ∀[s:sequence(T)]. ∀[x:T]. (seq-add(s;x) ∈ sequence(T))
Proof
Definitions occuring in Statement :
seq-add: seq-add(s;x),
sequence: sequence(T),
uall: ∀[x:A]. B[x],
member: t ∈ T,
universe: Type
Definitions unfolded in proof :
bfalse: ff,
lelt: i ≤ j < k,
ifthenelse: if b then t else f fi ,
btrue: tt,
it: ⋅,
unit: Unit,
bool: 𝔹,
int_seg: {i..j-},
true: True,
less_than': less_than'(a;b),
le: A ≤ B,
top: Top,
subtype_rel: A ⊆r B,
subtract: n - m,
squash: ↓T,
sq_stable: SqStable(P),
uimplies: b supposing a,
uiff: uiff(P;Q),
prop: ℙ,
false: False,
implies: P ⇒ Q,
rev_implies: P ⇐ Q,
not: ¬A,
and: P ∧ Q,
iff: P ⇐⇒ Q,
or: P ∨ Q,
decidable: Dec(P),
all: ∀x:A. B[x],
nat: ℕ,
sequence: sequence(T),
seq-add: seq-add(s;x),
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
sequence_wf,
equal_wf,
less_than_wf,
and_wf,
int_seg_wf,
assert_of_lt_int,
eqtt_to_assert,
bool_wf,
lt_int_wf,
le_wf,
le-add-cancel,
add-zero,
add_functionality_wrt_le,
add-commutes,
add-swap,
add-associates,
minus-one-mul-top,
zero-add,
minus-one-mul,
minus-add,
condition-implies-le,
sq_stable__le,
not-le-2,
false_wf,
decidable__le
Rules used in proof :
universeEquality,
axiomEquality,
functionEquality,
equalitySymmetry,
equalityTransitivity,
functionExtensionality,
equalityElimination,
minusEquality,
intEquality,
voidEquality,
isect_memberEquality,
lambdaEquality,
applyEquality,
imageElimination,
baseClosed,
imageMemberEquality,
isectElimination,
independent_isectElimination,
independent_functionElimination,
voidElimination,
lambdaFormation,
independent_pairFormation,
unionElimination,
hypothesisEquality,
dependent_functionElimination,
extract_by_obid,
natural_numberEquality,
hypothesis,
because_Cache,
rename,
setElimination,
addEquality,
dependent_set_memberEquality,
dependent_pairEquality,
thin,
productElimination,
sqequalHypSubstitution,
sqequalRule,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[T:Type]. \mforall{}[s:sequence(T)]. \mforall{}[x:T]. (seq-add(s;x) \mmember{} sequence(T))
Date html generated:
2018_07_25-PM-01_29_41
Last ObjectModification:
2018_06_19-AM-10_13_15
Theory : arithmetic
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