Nuprl Lemma : stream-zip_wf2
∀[A,B,C:Type]. ∀[f:A ⟶ B ⟶ C]. ∀[n:ℕ]. ∀[as:primrec(n;Top;λ,T. (A × T))]. ∀[bs:primrec(n;Top;λ,T. (B × T))].
(stream-zip(f;as;bs) ∈ primrec(n;Top;λ,T. (C × T)))
Proof
Definitions occuring in Statement :
stream-zip: stream-zip(f;as;bs),
primrec: primrec(n;b;c),
nat: ℕ,
uall: ∀[x:A]. B[x],
top: Top,
member: t ∈ T,
lambda: λx.A[x],
function: x:A ⟶ B[x],
product: x:A × B[x],
universe: Type
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,
le: A ≤ B,
and: P ∧ Q,
top: Top,
eq_int: (i =z j),
all: ∀x:A. B[x],
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
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,
subtype_rel: A ⊆r B,
decidable: Dec(P),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
less_than': less_than'(a;b),
true: True,
stream-zip: stream-zip(f;as;bs)
Lemmas referenced :
nat_properties,
less_than_transitivity1,
less_than_irreflexivity,
ge_wf,
less_than_wf,
primrec_wf,
top_wf,
int_seg_wf,
primrec-unroll,
btrue_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
eqff_to_assert,
eq_int_wf,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
stream-zip_wf,
le_wf,
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,
nat_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
setElimination,
rename,
sqequalRule,
intWeakElimination,
lambdaFormation,
natural_numberEquality,
independent_isectElimination,
independent_functionElimination,
voidElimination,
lambdaEquality,
dependent_functionElimination,
isect_memberEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
instantiate,
universeEquality,
productEquality,
cumulativity,
because_Cache,
productElimination,
voidEquality,
unionElimination,
equalityElimination,
dependent_pairFormation,
promote_hyp,
functionExtensionality,
applyEquality,
minusEquality,
dependent_set_memberEquality,
independent_pairFormation,
addEquality,
intEquality,
independent_pairEquality,
functionEquality
Latex:
\mforall{}[A,B,C:Type]. \mforall{}[f:A {}\mrightarrow{} B {}\mrightarrow{} C]. \mforall{}[n:\mBbbN{}]. \mforall{}[as:primrec(n;Top;\mlambda{},T. (A \mtimes{} T))].
\mforall{}[bs:primrec(n;Top;\mlambda{},T. (B \mtimes{} T))].
(stream-zip(f;as;bs) \mmember{} primrec(n;Top;\mlambda{},T. (C \mtimes{} T)))
Date html generated:
2017_04_14-AM-07_47_38
Last ObjectModification:
2017_02_27-PM-03_17_57
Theory : co-recursion
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