Nuprl Lemma : posint_fact_exists
∀i:ℕ+. (∃ps:{j:ℕ+| prime(j)} List [(i = (Π ps) ∈ ℤ)])
Proof
Definitions occuring in Statement :
posint_mul_mon: <ℤ+,*>,
mon_reduce: mon_reduce,
prime: prime(a),
list: T List,
nat_plus: ℕ+,
all: ∀x:A. B[x],
sq_exists: ∃x:A [B[x]],
set: {x:A| B[x]} ,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
implies: P ⇒ Q,
uall: ∀[x:A]. B[x],
guard: {T},
uimplies: b supposing a,
posint_mul_mon: <ℤ+,*>,
grp_car: |g|,
pi1: fst(t),
exists: ∃x:A. B[x],
massoc: a ~ b,
mdivides: b | a,
symmetrize: Symmetrize(x,y.R[x; y];a;b),
grp_op: *,
pi2: snd(t),
infix_ap: x f y,
and: P ∧ Q,
matom_ty: Atom{g},
nat_plus: ℕ+,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
assoced: a ~ b,
sq_exists: ∃x:A [B[x]],
so_lambda: λ2x.t[x],
so_apply: x[s],
prop: ℙ,
matomic: Atomic(a),
mreducible: Reducible(a),
munit: g-unit(u),
grp_id: e,
not: ¬A,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
atomic: atomic(a),
false: False,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
reducible: reducible(a),
gt: i > j,
int_nzero: ℤ-o,
nequal: a ≠ b ∈ T ,
decidable: Dec(P),
or: P ∨ Q,
cand: A c∧ B
Lemmas referenced :
nat_plus_wf,
mfact_exists_a,
posint_mul_mon_wf,
abmonoid_subtype_iabmonoid,
posint_cancel,
posint_well_fnd,
posint_reduc_dec,
grp_car_wf,
mon_subtype_grp_sig,
abmonoid_subtype_mon,
subtype_rel_transitivity,
abmonoid_wf,
mon_wf,
grp_sig_wf,
posint_unit_dec,
divides_nchar,
mon_reduce_wf,
iabmonoid_subtype_imon,
iabmonoid_wf,
imon_wf,
subtype_rel_list,
matom_ty_wf,
subtype_rel_self,
assoced_nelim,
nat_plus_subtype_nat,
set_subtype_base,
less_than_wf,
int_subtype_base,
list_subtype_base,
prime_wf,
matomic_wf,
subtype_rel_sets,
divides_wf,
not_wf,
exists_wf,
equal-wf-base,
atomic_imp_prime,
nat_plus_properties,
full-omega-unsat,
intformand_wf,
intformeq_wf,
itermVar_wf,
itermConstant_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
assoced_wf,
reducible_wf,
int_nzero_properties,
decidable__lt,
intformnot_wf,
itermMultiply_wf,
int_formula_prop_not_lemma,
int_term_value_mul_lemma,
pos_mul_arg_bounds,
unit_chars,
one_divs_any,
decidable__equal_int,
itermMinus_wf,
int_term_value_minus_lemma,
divides_invar_1
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
universeIsType,
cut,
introduction,
extract_by_obid,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
applyEquality,
sqequalRule,
independent_functionElimination,
hypothesisEquality,
isectElimination,
instantiate,
independent_isectElimination,
because_Cache,
productElimination,
lambdaEquality_alt,
setElimination,
rename,
independent_pairFormation,
promote_hyp,
dependent_set_memberFormation_alt,
equalityIsType4,
inhabitedIsType,
equalityTransitivity,
equalitySymmetry,
intEquality,
natural_numberEquality,
baseApply,
closedConclusion,
baseClosed,
setEquality,
setIsType,
dependent_set_memberEquality_alt,
imageMemberEquality,
dependent_pairFormation_alt,
productIsType,
multiplyEquality,
voidElimination,
approximateComputation,
int_eqEquality,
isect_memberEquality_alt,
unionElimination,
minusEquality
Latex:
\mforall{}i:\mBbbN{}\msupplus{}. (\mexists{}ps:\{j:\mBbbN{}\msupplus{}| prime(j)\} List [(i = (\mPi{} ps))])
Date html generated:
2019_10_16-PM-01_06_22
Last ObjectModification:
2018_10_08-AM-10_50_48
Theory : factor_1
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