Nuprl Lemma : even-implies-two-times
∀n:ℕ. ((↑isEven(n)) ⇒ (∃k:ℕ. (n = (2 * k) ∈ ℤ)))
Proof
Definitions occuring in Statement :
isEven: isEven(n),
nat: ℕ,
assert: ↑b,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
multiply: n * m,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
nat: ℕ,
iff: P ⇐⇒ Q,
and: P ∧ Q,
exists: ∃x:A. B[x],
uall: ∀[x:A]. B[x],
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
not: ¬A,
top: Top,
prop: ℙ
Lemmas referenced :
nat_wf,
isEven_wf,
assert_wf,
equal_wf,
le_wf,
int_formula_prop_wf,
int_term_value_mul_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
itermMultiply_wf,
intformeq_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
decidable__le,
nat_properties,
assert-isEven
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
lemma_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
setElimination,
rename,
hypothesisEquality,
hypothesis,
productElimination,
independent_functionElimination,
dependent_pairFormation,
dependent_set_memberEquality,
isectElimination,
natural_numberEquality,
unionElimination,
independent_isectElimination,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
independent_pairFormation,
computeAll,
multiplyEquality
Latex:
\mforall{}n:\mBbbN{}. ((\muparrow{}isEven(n)) {}\mRightarrow{} (\mexists{}k:\mBbbN{}. (n = (2 * k))))
Date html generated:
2016_05_14-PM-04_23_46
Last ObjectModification:
2016_01_14-PM-11_38_55
Theory : num_thy_1
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