Nuprl Lemma : not-prime-mult
∀[n,m:{2...}]. ∀[x:ℤ]. (¬prime((n * m) * x))
Proof
Definitions occuring in Statement :
prime: prime(a),
int_upper: {i...},
uall: ∀[x:A]. B[x],
not: ¬A,
multiply: n * m,
natural_number: $n,
int: ℤ
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
not: ¬A,
implies: P ⇒ Q,
false: False,
all: ∀x:A. B[x],
and: P ∧ Q,
prop: ℙ,
int_upper: {i...},
subtype_rel: A ⊆r B,
uimplies: b supposing a,
le: A ≤ B,
less_than': less_than'(a;b),
guard: {T},
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
atomic: atomic(a),
reducible: reducible(a),
int_nzero: ℤ-o,
so_lambda: λ2x.t[x],
so_apply: x[s],
nequal: a ≠ b ∈ T ,
cand: A c∧ B,
iff: P ⇐⇒ Q
Lemmas referenced :
prime-mult,
prime_wf,
int_upper_wf,
int_upper_properties,
mul_preserves_le,
upper_subtype_nat,
false_wf,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermMultiply_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_mul_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
le_wf,
prime_imp_atomic,
subtype_rel_sets,
nequal_wf,
intformeq_wf,
int_formula_prop_eq_lemma,
equal-wf-base,
int_subtype_base,
assoced_elim,
assoced_wf,
not_wf,
equal_wf,
exists_wf,
int_nzero_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
thin,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
hypothesisEquality,
independent_functionElimination,
hypothesis,
productElimination,
because_Cache,
voidElimination,
isectElimination,
multiplyEquality,
setElimination,
rename,
sqequalRule,
lambdaEquality,
intEquality,
isect_memberEquality,
natural_numberEquality,
applyEquality,
independent_isectElimination,
independent_pairFormation,
dependent_set_memberEquality,
unionElimination,
approximateComputation,
dependent_pairFormation,
int_eqEquality,
voidEquality,
setEquality,
baseClosed,
minusEquality,
productEquality
Latex:
\mforall{}[n,m:\{2...\}]. \mforall{}[x:\mBbbZ{}]. (\mneg{}prime((n * m) * x))
Date html generated:
2019_06_20-PM-02_23_14
Last ObjectModification:
2018_09_22-PM-06_07_09
Theory : num_thy_1
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