Nuprl Lemma : genfact-inv_wf
∀[f:ℕ+ ⟶ ℤ]. ∀[b:ℕ+]. ∀[N:ℤ]. (genfact-inv(N;b;m.f[m]) ∈ {n:ℕ| N ≤ genfact(n;b;m.f[m])} ) supposing ∀m:ℕ+. 1 < f[m]
Proof
Definitions occuring in Statement :
genfact-inv: genfact-inv(N;b;m.f[m]),
genfact: genfact(n;b;m.f[m]),
nat_plus: ℕ+,
nat: ℕ,
less_than: a < b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
le: A ≤ B,
all: ∀x:A. B[x],
member: t ∈ T,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
natural_number: $n,
int: ℤ
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
sq_exists: ∃x:A [B[x]],
genfact-inv: genfact-inv(N;b;m.f[m]),
genfact-unbounded-ext,
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
subtype_rel: A ⊆r B,
guard: {T},
so_lambda: λ2x.t[x],
nat: ℕ,
nat_plus: ℕ+,
prop: ℙ
Lemmas referenced :
genfact-unbounded-ext,
uimplies_subtype,
nat_plus_wf,
sq_exists_wf,
nat_wf,
le_wf,
genfact_wf,
istype-nat,
less_than_wf,
subtype_rel_self,
istype-int,
istype-less_than
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
introduction,
cut,
sqequalRule,
thin,
instantiate,
extract_by_obid,
hypothesis,
Error :inhabitedIsType,
Error :lambdaFormation_alt,
sqequalHypSubstitution,
dependent_functionElimination,
hypothesisEquality,
rename,
isectElimination,
applyEquality,
equalityTransitivity,
equalitySymmetry,
functionEquality,
intEquality,
Error :lambdaEquality_alt,
setElimination,
because_Cache,
natural_numberEquality,
independent_isectElimination,
Error :equalityIstype,
independent_functionElimination,
axiomEquality,
Error :isect_memberEquality_alt,
Error :isectIsTypeImplies,
Error :universeIsType,
Error :functionIsType
Latex:
\mforall{}[f:\mBbbN{}\msupplus{} {}\mrightarrow{} \mBbbZ{}]. \mforall{}[b:\mBbbN{}\msupplus{}]. \mforall{}[N:\mBbbZ{}]. (genfact-inv(N;b;m.f[m]) \mmember{} \{n:\mBbbN{}| N \mleq{} genfact(n;b;m.f[m])\} ) supposin\000Cg \mforall{}m:\mBbbN{}\msupplus{}. 1 < f[m]
Date html generated:
2019_06_20-PM-02_26_01
Last ObjectModification:
2019_02_11-PM-00_14_07
Theory : num_thy_1
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