Nuprl Lemma : fast-fib
∀n:ℕ. (∃m:ℕ [(m = fib(n) ∈ ℕ)])
Proof
Definitions occuring in Statement :
fib: fib(n),
nat: ℕ,
all: ∀x:A. B[x],
sq_exists: ∃x:A [B[x]],
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
nat: ℕ,
decidable: Dec(P),
or: P ∨ Q,
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
sq_type: SQType(T),
guard: {T},
prop: ℙ,
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
int_seg: {i..j-},
ge: i ≥ j ,
lelt: i ≤ j < k,
and: P ∧ Q,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
bfalse: ff,
so_apply: x[s],
sq_exists: ∃x:A [B[x]],
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
le_int: i ≤z j,
lt_int: i <z j,
bnot: ¬bb,
le: A ≤ B,
less_than': less_than'(a;b),
nequal: a ≠ b ∈ T ,
int_upper: {i...},
fib: fib(n),
assert: ↑b,
bor: p ∨bq,
subtract: n - m,
squash: ↓T,
true: True,
eq_int: (i =z j),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
decidable__equal_int,
subtype_base_sq,
int_subtype_base,
all_wf,
int_seg_wf,
sq_exists_wf,
nat_wf,
equal-wf-base-T,
fib_wf,
nat_properties,
int_seg_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermAdd_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
le_wf,
le_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_le_int,
equal_wf,
natrec_wf,
zero-add,
set-value-type,
int-value-type,
subtract_wf,
intformeq_wf,
itermSubtract_wf,
int_formula_prop_eq_lemma,
int_term_value_subtract_lemma,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
lelt_wf,
upper_subtype_nat,
false_wf,
nequal-le-implies,
int_upper_properties,
eq_int_wf,
assert_of_eq_int,
testxxx_lemma,
eqff_to_assert,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
add-associates,
add-swap,
add-commutes,
squash_wf,
true_wf,
assert_wf,
bnot_wf,
not_wf,
add-zero,
bool_cases,
iff_transitivity,
iff_weakening_uiff,
assert_of_bnot
Rules used in proof :
cut,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
setElimination,
rename,
hypothesisEquality,
hypothesis,
natural_numberEquality,
unionElimination,
instantiate,
isectElimination,
cumulativity,
intEquality,
independent_isectElimination,
because_Cache,
independent_functionElimination,
sqequalRule,
lambdaEquality,
functionEquality,
applyEquality,
dependent_set_memberEquality,
addEquality,
productElimination,
approximateComputation,
dependent_pairFormation,
int_eqEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
functionExtensionality,
cutEval,
hypothesis_subsumption,
promote_hyp,
imageElimination,
imageMemberEquality,
baseClosed,
applyLambdaEquality,
impliesFunctionality
Latex:
\mforall{}n:\mBbbN{}. (\mexists{}m:\mBbbN{} [(m = fib(n))])
Date html generated:
2018_05_21-PM-00_56_13
Last ObjectModification:
2018_05_19-AM-06_34_43
Theory : num_thy_1
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