Nuprl Lemma : genfact-unbounded
∀f:ℕ+ ⟶ ℤ. ∀b:ℕ+. ∀N:ℤ. (∃n:ℕ [(N ≤ genfact(n;b;m.f[m]))]) supposing ∀m:ℕ+. 1 < f[m]
Proof
Definitions occuring in Statement :
genfact: genfact(n;b;m.f[m]),
nat_plus: ℕ+,
nat: ℕ,
less_than: a < b,
uimplies: b supposing a,
so_apply: x[s],
le: A ≤ B,
all: ∀x:A. B[x],
sq_exists: ∃x:A [B[x]],
function: x:A ⟶ B[x],
natural_number: $n,
int: ℤ
Definitions unfolded in proof :
genfact: genfact(n;b;m.f[m]),
lelt: i ≤ j < k,
less_than: a < b,
subtract: n - m,
lt_int: i <z j,
assert: ↑b,
bnot: ¬bb,
bfalse: ff,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
btrue: tt,
it: ⋅,
unit: Unit,
bool: 𝔹,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
true: True,
top: Top,
false: False,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
not: ¬A,
ge: i ≥ j ,
squash: ↓T,
sq_stable: SqStable(P),
or: P ∨ Q,
decidable: Dec(P),
sq_exists: ∃x:A [B[x]],
int_seg: {i..j-},
and: P ∧ Q,
le: A ≤ B,
nat_plus: ℕ+,
nat: ℕ,
subtype_rel: A ⊆r B,
prop: ℙ,
guard: {T},
implies: P ⇒ Q,
sq_type: SQType(T),
so_lambda: λ2x.t[x],
so_apply: x[s],
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
all: ∀x:A. B[x]
Lemmas referenced :
imax_unfold,
add_functionality_wrt_eq,
assert_of_le_int,
le_int_wf,
ifthenelse_wf,
imax_nat,
imax_wf,
false_wf,
int_term_value_mul_lemma,
itermMultiply_wf,
multiply-is-int-iff,
subtract_wf,
mul_preserves_lt,
subtract-1-ge-0,
btrue_wf,
ge_wf,
int_term_value_subtract_lemma,
itermSubtract_wf,
decidable__equal_int,
less_than_wf,
assert_wf,
iff_weakening_uiff,
assert-bnot,
bool_subtype_base,
bool_wf,
bool_cases_sqequal,
eqff_to_assert,
assert_of_lt_int,
eqtt_to_assert,
lt_int_wf,
iff_weakening_equal,
subtype_rel_self,
genfact-step,
istype-universe,
true_wf,
squash_wf,
int_term_value_add_lemma,
int_term_value_constant_lemma,
int_formula_prop_less_lemma,
itermAdd_wf,
itermConstant_wf,
intformless_wf,
decidable__lt,
int_formula_prop_wf,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
istype-void,
int_formula_prop_and_lemma,
intformeq_wf,
itermVar_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
full-omega-unsat,
nat_plus_properties,
nat_properties,
sq_stable__equal,
decidable__le,
int_seg_wf,
istype-le,
le_witness_for_triv,
istype-nat,
sq_exists_wf,
le_wf,
genfact_wf,
equal-wf-base,
nat_wf,
uniform-comp-nat-induction,
istype-less_than,
nat_plus_wf,
istype-int,
int_subtype_base,
subtype_base_sq,
int-value-type,
equal_wf,
set-value-type,
member-less_than
Rules used in proof :
Error :productIsType,
applyLambdaEquality,
sqequalIntensionalEquality,
closedConclusion,
baseApply,
pointwiseFunctionality,
intWeakElimination,
promote_hyp,
equalityElimination,
universeEquality,
multiplyEquality,
independent_pairFormation,
voidElimination,
Error :isect_memberEquality_alt,
int_eqEquality,
Error :dependent_pairFormation_alt,
approximateComputation,
imageElimination,
baseClosed,
imageMemberEquality,
Error :dependent_set_memberFormation_alt,
unionElimination,
Error :isectIsType,
sqequalBase,
because_Cache,
Error :setIsType,
productElimination,
addEquality,
isectEquality,
setEquality,
functionEquality,
Error :functionIsType,
independent_functionElimination,
cumulativity,
instantiate,
setElimination,
Error :universeIsType,
equalitySymmetry,
equalityTransitivity,
Error :equalityIstype,
Error :dependent_set_memberEquality_alt,
cutEval,
intEquality,
rename,
Error :inhabitedIsType,
Error :functionIsTypeImplies,
hypothesis,
independent_isectElimination,
applyEquality,
natural_numberEquality,
isectElimination,
extract_by_obid,
hypothesisEquality,
thin,
dependent_functionElimination,
Error :lambdaEquality_alt,
sqequalHypSubstitution,
sqequalRule,
introduction,
cut,
Error :isect_memberFormation_alt,
Error :lambdaFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}f:\mBbbN{}\msupplus{} {}\mrightarrow{} \mBbbZ{}. \mforall{}b:\mBbbN{}\msupplus{}. \mforall{}N:\mBbbZ{}. (\mexists{}n:\mBbbN{} [(N \mleq{} genfact(n;b;m.f[m]))]) supposing \mforall{}m:\mBbbN{}\msupplus{}. 1 < f[m]
Date html generated:
2019_06_20-PM-02_25_50
Last ObjectModification:
2019_06_19-PM-00_08_06
Theory : num_thy_1
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