Nuprl Lemma : assert-isEven
∀n:ℤ. (↑isEven(n) ⇐⇒ ∃k:ℤ. (n = (2 * k) ∈ ℤ))
Proof
Definitions occuring in Statement :
isEven: isEven(n),
assert: ↑b,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
multiply: n * m,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
so_apply: x[s],
so_lambda: λ2x.t[x],
rev_implies: P ⇐ Q,
uall: ∀[x:A]. B[x],
prop: ℙ,
member: t ∈ T,
implies: P ⇒ Q,
and: P ∧ Q,
iff: P ⇐⇒ Q,
all: ∀x:A. B[x],
false: False,
guard: {T},
sq_type: SQType(T),
uimplies: b supposing a,
not: ¬A,
nequal: a ≠ b ∈ T ,
true: True,
int_nzero: ℤ-o,
or: P ∨ Q,
decidable: Dec(P),
le: A ≤ B,
less_than': less_than'(a;b),
squash: ↓T,
less_than: a < b,
nat_plus: ℕ+,
nat: ℕ,
bfalse: ff,
ifthenelse: if b then t else f fi ,
assert: ↑b,
eq_int: (i =z j),
top: Top,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
absval: |i|,
modulus: a mod n,
isEven: isEven(n),
int_lower: {...i},
gt: i > j,
ge: i ≥ j ,
uiff: uiff(P;Q),
subtype_rel: A ⊆r B,
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
subtype_base_sq,
int_subtype_base,
modulus-equal-iff-eqmod,
decidable__lt,
full-omega-unsat,
intformnot_wf,
intformless_wf,
itermConstant_wf,
istype-int,
int_formula_prop_not_lemma,
istype-void,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
istype-less_than,
assert_of_eq_int,
modulus_wf,
nequal_wf,
mod2-2n,
add-is-int-iff,
multiply-is-int-iff,
itermMultiply_wf,
itermAdd_wf,
int_term_value_mul_lemma,
int_term_value_add_lemma,
false_wf,
rem_bounds_2,
decidable__equal_int,
satisfiable-full-omega-tt,
intformand_wf,
intformeq_wf,
itermVar_wf,
intformle_wf,
int_formula_prop_and_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_formula_prop_le_lemma,
rem_bounds_1,
le_wf,
less_than_wf,
decidable__le,
div_rem_sum,
true_wf,
assert_wf,
isEven_wf,
exists_wf,
equal_wf
Rules used in proof :
natural_numberEquality,
multiplyEquality,
lambdaEquality,
sqequalRule,
intEquality,
hypothesis,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
lemma_by_obid,
cut,
independent_pairFormation,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
voidElimination,
independent_functionElimination,
equalitySymmetry,
equalityTransitivity,
dependent_functionElimination,
independent_isectElimination,
cumulativity,
instantiate,
addLevel,
dependent_set_memberEquality,
unionElimination,
productElimination,
baseClosed,
imageMemberEquality,
introduction,
computeAll,
voidEquality,
isect_memberEquality,
int_eqEquality,
dependent_pairFormation,
imageElimination,
because_Cache,
sqleReflexivity,
callbyvalueReduce,
minusEquality,
closedConclusion,
baseApply,
promote_hyp,
rename,
pointwiseFunctionality,
divideEquality,
extract_by_obid,
Error :dependent_set_memberEquality_alt,
approximateComputation,
Error :dependent_pairFormation_alt,
Error :lambdaEquality_alt,
Error :isect_memberEquality_alt,
Error :universeIsType,
Error :lambdaFormation_alt,
Error :equalityIstype,
Error :inhabitedIsType,
sqequalBase,
applyEquality
Latex:
\mforall{}n:\mBbbZ{}. (\muparrow{}isEven(n) \mLeftarrow{}{}\mRightarrow{} \mexists{}k:\mBbbZ{}. (n = (2 * k)))
Date html generated:
2019_06_20-PM-02_24_42
Last ObjectModification:
2019_05_02-PM-06_05_44
Theory : num_thy_1
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