Nuprl Lemma : decidable__proper_divisor
∀n:{2...}. Dec(∃n1:ℤ [(n1 < n ∧ (2 ≤ n1) ∧ (n1 | n))])
Proof
Definitions occuring in Statement :
divides: b | a,
int_upper: {i...},
less_than: a < b,
decidable: Dec(P),
le: A ≤ B,
all: ∀x:A. B[x],
sq_exists: ∃x:A [B[x]],
and: P ∧ Q,
natural_number: $n,
int: ℤ
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
int_upper: {i...},
decidable: Dec(P),
or: P ∨ Q,
uall: ∀[x:A]. B[x],
so_apply: x[s],
so_lambda: λ2x.t[x],
less_than': less_than'(a;b),
le: A ≤ B,
prop: ℙ,
top: Top,
false: False,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
implies: P ⇒ Q,
not: ¬A,
uimplies: b supposing a,
guard: {T},
cand: A c∧ B,
and: P ∧ Q,
sq_exists: ∃x:A [B[x]],
subtype_rel: A ⊆r B,
divides: b | a,
sq_type: SQType(T),
iff: P ⇐⇒ Q,
nequal: a ≠ b ∈ T ,
true: True,
int_nzero: ℤ-o,
nat: ℕ,
nat_plus: ℕ+,
ge: i ≥ j ,
uiff: uiff(P;Q),
squash: ↓T,
less_than: a < b,
subtract: n - m,
primrec: primrec(n;b;c),
exp: i^n,
bfalse: ff,
ifthenelse: if b then t else f fi ,
lt_int: i <z j,
int_seg: {i..j-},
lelt: i ≤ j < k,
rev_uimplies: rev_uimplies(P;Q),
gt: i > j
Lemmas referenced :
decidable__le,
istype-int_upper,
decidable__equal_int,
sq_exists_wf,
divides_wf,
le_wf,
less_than_wf,
istype-false,
int_formula_prop_wf,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_less_lemma,
int_formula_prop_not_lemma,
istype-void,
int_formula_prop_and_lemma,
istype-int,
intformeq_wf,
itermVar_wf,
itermConstant_wf,
intformless_wf,
intformnot_wf,
intformand_wf,
full-omega-unsat,
decidable__lt,
int_upper_properties,
int_subtype_base,
set_subtype_base,
int_term_value_mul_lemma,
itermMultiply_wf,
subtype_base_sq,
int_formula_prop_le_lemma,
intformle_wf,
nequal_wf,
divides_iff_rem_zero,
nat_wf,
iroot_wf,
istype-less_than,
istype-le,
add_nat_wf,
nat_properties,
add-is-int-iff,
itermAdd_wf,
int_term_value_add_lemma,
false_wf,
set-value-type,
equal_wf,
int-value-type,
upper_subtype_nat,
iroot-property,
exp_wf2,
subtract-1-ge-0,
member-less_than,
ge_wf,
int_term_value_subtract_lemma,
itermSubtract_wf,
subtract_wf,
mul-distributes,
mul-distributes-right,
add-associates,
mul-commutes,
one-mul,
add-swap,
add-commutes,
two-mul,
zero-mul,
zero-add,
primrec1_lemma,
primrec-unroll,
mul-associates,
mul-swap,
add-mul-special,
mul_preserves_le,
int_formual_prop_imp_lemma,
intformimplies_wf,
le_witness_for_triv,
primrec-wf2,
not_wf,
divisor-in-range,
less_than_functionality,
add_functionality_wrt_le,
multiply_functionality_wrt_le,
le_weakening,
mul_bounds_1a,
pos_mul_arg_bounds
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
setElimination,
rename,
hypothesisEquality,
hypothesis,
natural_numberEquality,
unionElimination,
isectElimination,
equalitySymmetry,
equalityTransitivity,
inhabitedIsType,
productEquality,
intEquality,
functionIsType,
because_Cache,
productIsType,
universeIsType,
independent_pairFormation,
sqequalRule,
voidElimination,
isect_memberEquality_alt,
int_eqEquality,
lambdaEquality_alt,
dependent_pairFormation_alt,
independent_functionElimination,
approximateComputation,
independent_isectElimination,
dependent_set_memberFormation_alt,
inlFormation_alt,
applyEquality,
baseClosed,
closedConclusion,
baseApply,
equalityIsType4,
cumulativity,
instantiate,
inrFormation_alt,
productElimination,
dependent_set_memberEquality_alt,
promote_hyp,
addEquality,
Error :memTop,
applyLambdaEquality,
pointwiseFunctionality,
equalityIstype,
cutEval,
imageMemberEquality,
multiplyEquality,
equalityIsType1,
imageElimination,
functionIsTypeImplies,
intWeakElimination,
minusEquality,
isect_memberFormation_alt,
isectIsType,
unionIsType,
setIsType,
isectEquality,
unionEquality
Latex:
\mforall{}n:\{2...\}. Dec(\mexists{}n1:\mBbbZ{} [(n1 < n \mwedge{} (2 \mleq{} n1) \mwedge{} (n1 | n))])
Date html generated:
2020_05_20-AM-08_14_07
Last ObjectModification:
2020_01_04-PM-11_10_56
Theory : general
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