Nuprl Lemma : two-factorizations_wf
∀[n:ℕ]. (two-factorizations(n) ∈ {p:ℤ × ℤ| ((1 ≤ (fst(p))) ∧ ((fst(p)) ≤ n)) ∧ (((fst(p)) * (snd(p))) = n ∈ ℤ)} List)
Proof
Definitions occuring in Statement :
two-factorizations: two-factorizations(n),
list: T List,
nat: ℕ,
uall: ∀[x:A]. B[x],
pi1: fst(t),
pi2: snd(t),
le: A ≤ B,
and: P ∧ Q,
member: t ∈ T,
set: {x:A| B[x]} ,
product: x:A × B[x],
multiply: n * m,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
two-factorizations: two-factorizations(n),
nat: ℕ,
and: P ∧ Q,
prop: ℙ,
all: ∀x:A. B[x],
implies: P ⇒ Q,
nequal: a ≠ b ∈ T ,
ge: i ≥ j ,
not: ¬A,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
subtype_rel: A ⊆r B,
pi1: fst(t),
pi2: snd(t),
guard: {T},
so_lambda: λ2x.t[x],
so_apply: x[s],
cand: A c∧ B,
sq_stable: SqStable(P),
squash: ↓T,
decidable: Dec(P),
or: P ∨ Q,
uiff: uiff(P;Q),
int_nzero: ℤ-o,
sq_type: SQType(T)
Lemmas referenced :
from-upto_wf,
list_wf,
le_wf,
less_than_wf,
mapfilter_wf,
eq_int_wf,
nat_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformeq_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
int_formula_prop_and_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
equal-wf-base,
int_subtype_base,
equal_wf,
assert_wf,
equal-wf-T-base,
set_wf,
sq_stable__and,
sq_stable__le,
sq_stable__less_than,
member-less_than,
squash_wf,
decidable__le,
intformnot_wf,
int_formula_prop_not_lemma,
intformless_wf,
itermAdd_wf,
int_formula_prop_less_lemma,
int_term_value_add_lemma,
assert_of_eq_int,
div_rem_sum,
nequal_wf,
subtype_base_sq,
decidable__equal_int,
add-is-int-iff,
multiply-is-int-iff,
itermMultiply_wf,
int_term_value_mul_lemma,
false_wf,
nat_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
natural_numberEquality,
addEquality,
setElimination,
rename,
hypothesisEquality,
hypothesis,
setEquality,
intEquality,
productEquality,
because_Cache,
lambdaFormation,
lambdaEquality,
remainderEquality,
productElimination,
independent_isectElimination,
dependent_pairFormation,
int_eqEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
independent_pairFormation,
computeAll,
applyEquality,
baseClosed,
multiplyEquality,
independent_functionElimination,
dependent_set_memberEquality,
independent_pairEquality,
divideEquality,
imageMemberEquality,
imageElimination,
unionElimination,
instantiate,
cumulativity,
equalityTransitivity,
equalitySymmetry,
pointwiseFunctionality,
promote_hyp,
baseApply,
closedConclusion,
axiomEquality
Latex:
\mforall{}[n:\mBbbN{}]
(two-factorizations(n) \mmember{} \{p:\mBbbZ{} \mtimes{} \mBbbZ{}|
((1 \mleq{} (fst(p))) \mwedge{} ((fst(p)) \mleq{} n)) \mwedge{} (((fst(p)) * (snd(p))) = n)\} List)
Date html generated:
2018_05_21-PM-09_05_51
Last ObjectModification:
2017_07_26-PM-06_28_41
Theory : general
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