Nuprl Lemma : HofstadterL_wf
∀n:ℕ
(HofstadterL(n) ∈ {L:(ℤ × ℤ) List|
(||L|| = (n + 1) ∈ ℤ) ∧ (∀i:ℕn + 1. (L[i] = <HofstadterM(n - i), HofstadterF(n - i)> ∈ (ℤ × ℤ)))} )
Proof
Definitions occuring in Statement :
HofstadterL: HofstadterL(n),
HofstadterM: HofstadterM(n),
HofstadterF: HofstadterF(n),
select: L[n],
length: ||as||,
list: T List,
int_seg: {i..j-},
nat: ℕ,
all: ∀x:A. B[x],
and: P ∧ Q,
member: t ∈ T,
set: {x:A| B[x]} ,
pair: <a, b>,
product: x:A × B[x],
subtract: n - m,
add: n + m,
natural_number: $n,
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,
so_apply: x[s],
bool: 𝔹,
subtype_rel: A ⊆r B,
or: P ∨ Q,
decidable: Dec(P),
lelt: i ≤ j < k,
guard: {T},
int_seg: {i..j-},
so_lambda: λ2x.t[x],
cand: A c∧ B,
btrue: tt,
ifthenelse: if b then t else f fi ,
subtract: n - m,
eq_int: (i =z j),
HofstadterL: HofstadterL(n),
less_than': less_than'(a;b),
le: A ≤ B,
HofstadterF: HofstadterF(n),
HofstadterM: HofstadterM(n),
cons: [a / b],
select: L[n],
sq_type: SQType(T),
pi2: snd(t),
pi1: fst(t),
hd: hd(l),
bfalse: ff,
bnot: ¬bb,
lt_int: i <z j,
le_int: i ≤z j,
squash: ↓T,
nequal: a ≠ b ∈ T ,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
has-value: (a)↓,
nil: [],
it: ⋅,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
unit: Unit,
uiff: uiff(P;Q),
less_than: a < b,
assert: ↑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,
istype-less_than,
subtract-1-ge-0,
istype-nat,
lt_int_wf,
HofstadterF_wf,
bool_wf,
le_int_wf,
subtract_wf,
HofstadterM_wf,
int_formula_prop_eq_lemma,
intformeq_wf,
decidable__lt,
int_formula_prop_not_lemma,
intformnot_wf,
decidable__le,
int_seg_properties,
select_wf,
equal_wf,
all_wf,
equal-wf-base,
int_seg_wf,
length-singleton,
nil_wf,
cons_wf,
int_seg_cases,
false_wf,
int_seg_subtype,
int_subtype_base,
subtype_base_sq,
decidable__equal_int,
ifthenelse_wf,
length_of_nil_lemma,
length_of_cons_lemma,
bool_subtype_base,
squash_wf,
true_wf,
istype-universe,
eq_int_eq_false,
bfalse_wf,
subtype_rel_self,
iff_weakening_equal,
value-type-has-value,
int-value-type,
list_wf,
set-value-type,
list-value-type,
istype-false,
subtract-add-cancel,
istype-le,
list-cases,
stuck-spread,
istype-base,
product_subtype_list,
reduce_hd_cons_lemma,
product_subtype_base,
add-commutes,
minus-zero,
add-associates,
add-zero,
eqtt_to_assert,
assert_of_lt_int,
eqff_to_assert,
bool_cases_sqequal,
assert-bnot,
iff_weakening_uiff,
assert_wf,
less_than_wf,
istype-top,
assert_of_le_int,
lelt_wf,
le_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
pi2_wf,
trivial-int-eq1,
bnot_wf,
not_wf,
istype-assert,
bool_cases,
iff_transitivity,
assert_of_bnot,
pi1_wf_top,
base_wf,
subtype_rel_product,
top_wf,
list_subtype_base,
set_subtype_base,
itermAdd_wf,
int_term_value_add_lemma,
select_cons_tl,
minus-add,
minus-minus,
minus-one-mul,
add-swap,
zero-add
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
dependent_functionElimination,
isect_memberEquality_alt,
voidElimination,
sqequalRule,
independent_pairFormation,
universeIsType,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
functionIsTypeImplies,
inhabitedIsType,
because_Cache,
addEquality,
applyEquality,
dependent_pairFormation,
unionElimination,
productElimination,
lambdaEquality,
lambdaFormation,
voidEquality,
isect_memberEquality,
independent_pairEquality,
intEquality,
productEquality,
dependent_set_memberEquality,
sqleReflexivity,
callbyvalueReduce,
hypothesis_subsumption,
cumulativity,
instantiate,
universeEquality,
promote_hyp,
imageElimination,
equalityIsType4,
baseApply,
closedConclusion,
baseClosed,
imageMemberEquality,
setEquality,
equalityIsType1,
dependent_set_memberEquality_alt,
productIsType,
minusEquality,
equalityElimination,
lessCases,
isect_memberFormation_alt,
axiomSqEquality,
isectIsTypeImplies,
sqequalIntensionalEquality,
functionIsType,
multiplyEquality
Latex:
\mforall{}n:\mBbbN{}
(HofstadterL(n) \mmember{} \{L:(\mBbbZ{} \mtimes{} \mBbbZ{}) List|
(||L|| = (n + 1))
\mwedge{} (\mforall{}i:\mBbbN{}n + 1. (L[i] = <HofstadterM(n - i), HofstadterF(n - i)>))\} )
Date html generated:
2019_10_15-AM-11_37_39
Last ObjectModification:
2018_10_18-PM-11_34_51
Theory : general
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