Nuprl Lemma : less-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],
set: {x:A| B[x]} ,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
implies: P ⇒ Q,
subtype_rel: A ⊆r B,
nat: ℕ,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
prop: ℙ,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
and: P ∧ Q,
guard: {T},
le: A ≤ B,
subtract: n - m,
squash: ↓T,
true: True,
sq_type: SQType(T),
fib: fib(n),
eq_int: (i =z j),
ifthenelse: if b then t else f fi ,
btrue: tt,
bor: p ∨bq,
bfalse: ff,
less_than': less_than'(a;b),
nequal: a ≠ b ∈ T ,
int_upper: {i...},
bool: 𝔹,
unit: Unit,
it: ⋅,
uiff: uiff(P;Q),
bnot: ¬bb,
assert: ↑b,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
less_than: a < b
Lemmas referenced :
nat_wf,
set_subtype_base,
le_wf,
int_subtype_base,
istype-less_than,
istype-int,
primrec-wf2,
all_wf,
set_wf,
equal-wf-base,
less_than_wf,
nat_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
zero-add,
itermAdd_wf,
int_term_value_add_lemma,
add-subtract-cancel,
decidable__equal_int,
intformeq_wf,
int_formula_prop_eq_lemma,
add-associates,
add-swap,
add-commutes,
fib_wf,
squash_wf,
true_wf,
intformless_wf,
int_formula_prop_less_lemma,
subtype_base_sq,
testxxx_lemma,
upper_subtype_nat,
istype-false,
nequal-le-implies,
eq_int_wf,
eqtt_to_assert,
assert_of_eq_int,
int_upper_properties,
eqff_to_assert,
bool_subtype_base,
bool_cases_sqequal,
bool_wf,
assert-bnot,
neg_assert_of_eq_int,
equal_wf,
istype-universe,
add-comm,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
subtype_rel_self,
iff_weakening_equal,
add_functionality_wrt_eq,
decidable__lt,
equal-wf-base-T,
false_wf,
subtype_rel_sets,
add-zero
Rules used in proof :
cut,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :lambdaFormation_alt,
thin,
Error :inhabitedIsType,
hypothesisEquality,
Error :universeIsType,
because_Cache,
rename,
setElimination,
sqequalRule,
Error :functionIsType,
introduction,
extract_by_obid,
hypothesis,
Error :setIsType,
Error :equalityIsType4,
baseApply,
closedConclusion,
baseClosed,
applyEquality,
sqequalHypSubstitution,
isectElimination,
intEquality,
Error :lambdaEquality_alt,
natural_numberEquality,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
functionEquality,
Error :dependent_set_memberEquality_alt,
dependent_functionElimination,
unionElimination,
approximateComputation,
independent_functionElimination,
Error :dependent_pairFormation_alt,
int_eqEquality,
Error :isect_memberEquality_alt,
voidElimination,
independent_pairFormation,
addEquality,
Error :dependent_set_memberFormation_alt,
applyLambdaEquality,
productElimination,
imageElimination,
imageMemberEquality,
instantiate,
cumulativity,
hypothesis_subsumption,
equalityElimination,
promote_hyp,
Error :equalityIsType1,
universeEquality,
minusEquality,
lambdaEquality,
setEquality,
voidEquality,
isect_memberEquality,
dependent_pairFormation,
lambdaFormation,
dependent_set_memberEquality,
functionExtensionality
Latex:
\mforall{}n:\mBbbN{}. \{m:\mBbbN{}| m = fib(n)\}
Date html generated:
2019_06_20-PM-02_25_23
Last ObjectModification:
2018_10_17-AM-10_43_29
Theory : num_thy_1
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