Nuprl Lemma : genfact-unbounded-ext
∀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 :
member: t ∈ T,
so_apply: x[s],
genrec-ap: genrec-ap,
genfact-unbounded,
uniform-comp-nat-induction,
decidable__le,
decidable__and,
decidable__not,
decidable__less_than',
decidable__implies,
decidable__false,
any: any x,
uall: ∀[x:A]. B[x],
so_lambda: so_lambda(x,y,z,w.t[x; y; z; w]),
so_apply: x[s1;s2;s3;s4],
so_lambda: λ2x.t[x],
top: Top,
uimplies: b supposing a
Lemmas referenced :
genfact-unbounded,
lifting-strict-decide,
istype-void,
strict4-decide,
lifting-strict-less,
uniform-comp-nat-induction,
decidable__le,
decidable__and,
decidable__not,
decidable__less_than',
decidable__implies,
decidable__false
Rules used in proof :
introduction,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
cut,
instantiate,
extract_by_obid,
hypothesis,
sqequalRule,
thin,
sqequalHypSubstitution,
equalityTransitivity,
equalitySymmetry,
isectElimination,
baseClosed,
Error :isect_memberEquality_alt,
voidElimination,
independent_isectElimination
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_54
Last ObjectModification:
2019_03_26-AM-07_43_38
Theory : num_thy_1
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