Nuprl Lemma : nth-stream-zip
∀[f:Top]. ∀[n:ℕ]. ∀[as,bs:stream(Top)]. (s-nth(n;stream-zip(f;as;bs)) ~ f s-nth(n;as) s-nth(n;bs))
Proof
Definitions occuring in Statement :
stream-zip: stream-zip(f;as;bs),
s-nth: s-nth(n;s),
stream: stream(A),
nat: ℕ,
uall: ∀[x:A]. B[x],
top: Top,
apply: f a,
sqequal: s ~ t
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
guard: {T},
uimplies: b supposing a,
prop: ℙ,
s-nth: s-nth(n;s),
stream-zip: stream-zip(f;as;bs),
eq_int: (i =z j),
all: ∀x:A. B[x],
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
ifthenelse: if b then t else f fi ,
subtype_rel: A ⊆r B,
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
subtract: n - m,
nequal: a ≠ b ∈ T ,
not: ¬A,
decidable: Dec(P),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
top: Top,
le: A ≤ B,
less_than': less_than'(a;b),
true: True,
has-value: (a)↓
Lemmas referenced :
nat_properties,
less_than_transitivity1,
less_than_irreflexivity,
ge_wf,
less_than_wf,
stream_wf,
top_wf,
btrue_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
stream-ext,
subtype_rel_weakening,
equal_wf,
eqff_to_assert,
eq_int_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
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,
le_weakening,
nat_wf,
value-type-has-value,
int-value-type
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
lambdaFormation,
natural_numberEquality,
independent_isectElimination,
independent_functionElimination,
voidElimination,
sqequalRule,
lambdaEquality,
dependent_functionElimination,
isect_memberEquality,
sqequalAxiom,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
productElimination,
because_Cache,
applyEquality,
productEquality,
dependent_pairFormation,
promote_hyp,
instantiate,
cumulativity,
callbyvalueReduce,
sqleReflexivity,
independent_pairFormation,
addEquality,
voidEquality,
intEquality,
minusEquality
Latex:
\mforall{}[f:Top]. \mforall{}[n:\mBbbN{}]. \mforall{}[as,bs:stream(Top)]. (s-nth(n;stream-zip(f;as;bs)) \msim{} f s-nth(n;as) s-nth(n;bs))
Date html generated:
2017_04_14-AM-07_47_44
Last ObjectModification:
2017_02_27-PM-03_17_46
Theory : co-recursion
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