Nuprl Lemma : prime-factors3
∀n:{2...}. (∃factors:{m:{2...}| prime(m)} List [(n = Π(factors) ∈ ℤ)])
This theorem is one of freek's list of 100 theorems
Proof
Definitions occuring in Statement :
mul-list: Π(ns) ,
prime: prime(a),
list: T List,
int_upper: {i...},
all: ∀x:A. B[x],
sq_exists: ∃x:A [B[x]],
set: {x:A| B[x]} ,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
member: t ∈ T,
iseg_product_rem: iseg_product_rem(i;j;k),
subtract: n - m,
divide: n ÷ m,
it: ⋅,
nil: [],
cons: [a / b],
so_apply: x[s1;s2],
genrec-ap: genrec-ap,
pi1: fst(t),
prime-factors2,
decidable__proper_divisor,
decidable__le,
decidable__equal_int,
any: any x,
iroot-property,
divisor-in-range,
decidable__and,
decidable__not,
decidable__less_than',
decidable__int_equal,
uniform-comp-nat-induction,
decidable__lt,
rem_bounds_1,
int_seg_properties,
decidable__implies,
decidable__false,
decidable__squash,
decidable_functionality,
squash_elim,
sq_stable_from_decidable,
iff_preserves_decidability,
sq_stable__from_stable,
stable__from_decidable,
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,
so_apply: x[s],
uimplies: b supposing a,
strict4: strict4(F),
and: P ∧ Q,
all: ∀x:A. B[x],
implies: P ⇒ Q,
has-value: (a)↓,
prop: ℙ,
or: P ∨ Q,
squash: ↓T,
so_lambda: λ2x y.t[x; y]
Lemmas referenced :
prime-factors2,
lifting-strict-decide,
istype-void,
strict4-decide,
lifting-strict-less,
lifting-strict-callbyvalue,
lifting-strict-int_eq,
lifting-strict-spread,
has-value_wf_base,
istype-base,
is-exception_wf,
istype-universe,
decidable__proper_divisor,
decidable__le,
decidable__equal_int,
iroot-property,
divisor-in-range,
decidable__and,
decidable__not,
decidable__less_than',
decidable__int_equal,
uniform-comp-nat-induction,
decidable__lt,
rem_bounds_1,
int_seg_properties,
decidable__implies,
decidable__false,
decidable__squash,
decidable_functionality,
squash_elim,
sq_stable_from_decidable,
iff_preserves_decidability,
sq_stable__from_stable,
stable__from_decidable
Rules used in proof :
introduction,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
cut,
instantiate,
extract_by_obid,
hypothesis,
sqequalRule,
thin,
sqequalHypSubstitution,
equalityTransitivity,
equalitySymmetry,
isectElimination,
baseClosed,
isect_memberEquality_alt,
voidElimination,
independent_isectElimination,
independent_pairFormation,
lambdaFormation_alt,
callbyvalueCallbyvalue,
callbyvalueReduce,
universeIsType,
baseApply,
closedConclusion,
hypothesisEquality,
callbyvalueExceptionCases,
inrFormation_alt,
imageMemberEquality,
imageElimination,
exceptionSqequal,
inlFormation_alt
Latex:
\mforall{}n:\{2...\}. (\mexists{}factors:\{m:\{2...\}| prime(m)\} List [(n = \mPi{}(factors) )])
Date html generated:
2019_10_15-AM-11_18_45
Last ObjectModification:
2019_06_26-PM-03_38_54
Theory : general
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