Nuprl Lemma : assert-isOdd
∀n:ℤ. (↑isOdd(n) ⇐⇒ ∃k:ℤ. (n = ((2 * k) + 1) ∈ ℤ))
Proof
Definitions occuring in Statement :
isOdd: isOdd(n),
assert: ↑b,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
multiply: n * m,
add: n + m,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
rev_implies: P ⇐ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
int_nzero: ℤ-o,
true: True,
nequal: a ≠ b ∈ T ,
not: ¬A,
uimplies: b supposing a,
sq_type: SQType(T),
guard: {T},
false: False,
decidable: Dec(P),
or: P ∨ Q,
nat: ℕ,
nat_plus: ℕ+,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
le: A ≤ B,
subtype_rel: A ⊆r B,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
isOdd: isOdd(n),
modulus: a mod n,
absval: |i|,
eq_int: (i =z j),
assert: ↑b,
ifthenelse: if b then t else f fi ,
bfalse: ff,
int_lower: {...i},
ge: i ≥ j ,
gt: i > j,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
eqmod: a ≡ b mod m,
divides: b | a
Lemmas referenced :
modulus_wf,
assert_of_eq_int,
modulus-equal-iff-eqmod,
int_term_value_subtract_lemma,
itermSubtract_wf,
subtract_wf,
false_wf,
int_term_value_mul_lemma,
int_term_value_add_lemma,
itermMultiply_wf,
itermAdd_wf,
multiply-is-int-iff,
add-is-int-iff,
rem_bounds_2,
int_formula_prop_wf,
int_formula_prop_le_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
intformle_wf,
intformless_wf,
itermConstant_wf,
itermVar_wf,
intformeq_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
decidable__equal_int,
equal-wf-base,
less_than_wf,
le_wf,
rem_bounds_1,
decidable__le,
nequal_wf,
true_wf,
int_subtype_base,
subtype_base_sq,
div_rem_sum,
equal_wf,
exists_wf,
isOdd_wf,
assert_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
independent_pairFormation,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
intEquality,
sqequalRule,
lambdaEquality,
addEquality,
multiplyEquality,
natural_numberEquality,
dependent_set_memberEquality,
addLevel,
instantiate,
cumulativity,
independent_isectElimination,
dependent_functionElimination,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
voidElimination,
unionElimination,
introduction,
imageMemberEquality,
baseClosed,
productElimination,
inlFormation,
baseApply,
closedConclusion,
applyEquality,
because_Cache,
imageElimination,
dependent_pairFormation,
int_eqEquality,
isect_memberEquality,
voidEquality,
computeAll,
inrFormation,
minusEquality,
divideEquality,
pointwiseFunctionality,
rename,
promote_hyp,
callbyvalueReduce,
sqleReflexivity
Latex:
\mforall{}n:\mBbbZ{}. (\muparrow{}isOdd(n) \mLeftarrow{}{}\mRightarrow{} \mexists{}k:\mBbbZ{}. (n = ((2 * k) + 1)))
Date html generated:
2016_05_14-PM-04_23_29
Last ObjectModification:
2016_01_14-PM-11_39_46
Theory : num_thy_1
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