Nuprl Lemma : nth-fibs
∀n:ℕ. (s-nth(n;fibs()) = fib(n) ∈ ℤ)
Proof
Definitions occuring in Statement :
fibs: fibs(),
fib: fib(n),
s-nth: s-nth(n;s),
nat: ℕ,
all: ∀x:A. B[x],
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
and: P ∧ Q,
prop: ℙ,
guard: {T},
int_seg: {i..j-},
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_type: SQType(T),
fibs: fibs(),
fib: fib(n),
s-nth: s-nth(n;s),
s-cons: x.s,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff,
bnot: ¬bb,
assert: ↑b,
bor: p ∨bq,
le: A ≤ B,
less_than': less_than'(a;b),
int_upper: {i...},
has-value: (a)↓,
nequal: a ≠ b ∈ T ,
squash: ↓T,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
subtract: n - m
Lemmas referenced :
nat_properties,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
ge_wf,
less_than_wf,
int_seg_wf,
int_seg_properties,
decidable__le,
subtract_wf,
intformnot_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
decidable__equal_int,
subtype_base_sq,
set_subtype_base,
int_subtype_base,
intformeq_wf,
int_formula_prop_eq_lemma,
decidable__lt,
lelt_wf,
subtype_rel_self,
le_wf,
eq_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
testxxx_lemma,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
upper_subtype_nat,
false_wf,
nequal-le-implies,
zero-add,
value-type-has-value,
int-value-type,
int_upper_properties,
itermAdd_wf,
int_term_value_add_lemma,
nat_wf,
nth-stream-zip,
fibs_wf,
stream-subtype,
top_wf,
s-tl_wf,
add-commutes,
fib_wf,
squash_wf,
true_wf,
iff_weakening_equal,
add-associates,
add-swap,
not-le-2,
not-equal-2,
condition-implies-le,
minus-one-mul,
minus-one-mul-top,
minus-add,
minus-minus,
add_functionality_wrt_le,
le-add-cancel,
minus-zero,
add-zero,
le-add-cancel-alt,
s-nth_wf,
s_tl_cons_lemma,
s-cons_wf,
stream-zip_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
independent_pairFormation,
axiomEquality,
because_Cache,
productElimination,
unionElimination,
applyEquality,
instantiate,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
dependent_set_memberEquality,
hypothesis_subsumption,
equalityElimination,
promote_hyp,
cumulativity,
callbyvalueReduce,
addEquality,
imageElimination,
universeEquality,
imageMemberEquality,
baseClosed,
minusEquality
Latex:
\mforall{}n:\mBbbN{}. (s-nth(n;fibs()) = fib(n))
Date html generated:
2018_05_21-PM-00_59_57
Last ObjectModification:
2018_05_19-AM-06_36_41
Theory : num_thy_1
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