Nuprl Lemma : even-implies
∀n:ℤ. ((↑isEven(n)) ⇒ {((↑isOdd(n - 1)) ∧ (¬↑isOdd(n))) ∧ (↑isOdd(n + 1))})
Proof
Definitions occuring in Statement :
isEven: isEven(n),
isOdd: isOdd(n),
assert: ↑b,
guard: {T},
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
and: P ∧ Q,
subtract: n - m,
add: n + m,
natural_number: $n,
int: ℤ
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
guard: {T},
and: P ∧ Q,
cand: A c∧ B,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
exists: ∃x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
not: ¬A,
top: Top,
isEven: isEven(n),
int_nzero: ℤ-o,
true: True,
nequal: a ≠ b ∈ T ,
sq_type: SQType(T),
subtype_rel: A ⊆r B,
uiff: uiff(P;Q),
isOdd: isOdd(n),
eq_int: (i =z j),
assert: ↑b,
ifthenelse: if b then t else f fi ,
bfalse: ff
Lemmas referenced :
false_wf,
nequal_wf,
true_wf,
int_subtype_base,
subtype_base_sq,
modulus_wf,
assert_of_eq_int,
equal_wf,
int_formula_prop_wf,
int_term_value_mul_lemma,
int_term_value_add_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_term_value_subtract_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
itermMultiply_wf,
itermAdd_wf,
itermConstant_wf,
itermVar_wf,
itermSubtract_wf,
intformeq_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
decidable__equal_int,
subtract_wf,
assert-isOdd,
assert-isEven,
isEven_wf,
assert_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
independent_pairFormation,
cut,
hypothesis,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
intEquality,
dependent_functionElimination,
productElimination,
independent_functionElimination,
natural_numberEquality,
dependent_pairFormation,
because_Cache,
unionElimination,
independent_isectElimination,
lambdaEquality,
int_eqEquality,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
computeAll,
addEquality,
multiplyEquality,
dependent_set_memberEquality,
addLevel,
instantiate,
cumulativity,
equalityTransitivity,
equalitySymmetry,
applyEquality
Latex:
\mforall{}n:\mBbbZ{}. ((\muparrow{}isEven(n)) {}\mRightarrow{} \{((\muparrow{}isOdd(n - 1)) \mwedge{} (\mneg{}\muparrow{}isOdd(n))) \mwedge{} (\muparrow{}isOdd(n + 1))\})
Date html generated:
2016_05_14-PM-04_24_01
Last ObjectModification:
2016_01_14-PM-11_37_55
Theory : num_thy_1
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