Nuprl Lemma : Hofstadter_wf
∀n:ℤ. ((HofstadterF(n) ∈ if 0 <z n then ℕn + 1 else ℕ+2 fi ) ∧ (HofstadterM(n) ∈ if 0 ≤z n then ℕn + 1 else ℕ1 fi ))
Proof
Definitions occuring in Statement :
HofstadterM: HofstadterM(n),
HofstadterF: HofstadterF(n),
int_seg: {i..j-},
le_int: i ≤z j,
ifthenelse: if b then t else f fi ,
lt_int: i <z j,
all: ∀x:A. B[x],
and: P ∧ Q,
member: t ∈ T,
add: n + m,
natural_number: $n,
int: ℤ
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),
cand: A c∧ B,
HofstadterF: HofstadterF(n),
less_than: a < b,
less_than': less_than'(a;b),
true: True,
squash: ↓T,
HofstadterM: HofstadterM(n),
le: A ≤ B,
subtract: n - m,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
has-value: (a)↓,
assert: ↑b,
bnot: ¬bb,
bfalse: ff,
ifthenelse: if b then t else f fi ,
btrue: tt,
it: ⋅,
unit: Unit,
bool: 𝔹
Lemmas referenced :
nat_properties,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
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_properties,
int_seg_wf,
subtract-1-ge-0,
decidable__equal_int,
subtract_wf,
subtype_base_sq,
set_subtype_base,
int_subtype_base,
intformnot_wf,
intformeq_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_subtract_lemma,
decidable__le,
decidable__lt,
le_wf,
subtype_rel_self,
istype-top,
nat_wf,
istype-false,
itermAdd_wf,
int_term_value_add_lemma,
subtract-add-cancel,
int_seg_subtype,
not-le-2,
condition-implies-le,
minus-add,
minus-one-mul,
add-swap,
minus-one-mul-top,
add-mul-special,
zero-mul,
add-zero,
add-associates,
add-commutes,
le-add-cancel,
value-type-has-value,
int-value-type,
set-value-type,
lelt_wf,
subtract_nat_wf,
assert-bnot,
bool_subtype_base,
bool_cases_sqequal,
equal_wf,
eqff_to_assert,
assert_of_lt_int,
eqtt_to_assert,
bool_wf,
lt_int_wf,
false_wf,
assert_of_le_int,
le_int_wf,
ifthenelse_wf,
subtype_rel-equal,
top_wf
Rules used in proof :
cut,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
hypothesis,
setElimination,
rename,
sqequalRule,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
dependent_functionElimination,
isect_memberEquality_alt,
voidElimination,
independent_pairFormation,
universeIsType,
productElimination,
independent_pairEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
functionIsTypeImplies,
inhabitedIsType,
unionElimination,
applyEquality,
instantiate,
because_Cache,
applyLambdaEquality,
dependent_set_memberEquality_alt,
productIsType,
hypothesis_subsumption,
lessCases,
isect_memberFormation_alt,
axiomSqEquality,
imageMemberEquality,
baseClosed,
imageElimination,
cumulativity,
intEquality,
closedConclusion,
addEquality,
callbyvalueReduce,
sqleReflexivity,
minusEquality,
multiplyEquality,
equalityIsType1,
lambdaFormation,
dependent_set_memberEquality,
promote_hyp,
dependent_pairFormation,
equalityElimination,
voidEquality,
isect_memberEquality,
lambdaEquality,
universeEquality,
isect_memberFormation
Latex:
\mforall{}n:\mBbbZ{}
((HofstadterF(n) \mmember{} if 0 <z n then \mBbbN{}n + 1 else \mBbbN{}\msupplus{}2 fi )
\mwedge{} (HofstadterM(n) \mmember{} if 0 \mleq{}z n then \mBbbN{}n + 1 else \mBbbN{}1 fi ))
Date html generated:
2019_10_15-AM-11_37_13
Last ObjectModification:
2018_10_11-PM-10_29_16
Theory : general
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