Nuprl Lemma : add-one-mod-2
∀[x:ℤ]. (((x + 1) mod 2) = (1 - x mod 2) ∈ ℤ)
Proof
Definitions occuring in Statement :
modulus: a mod n,
uall: ∀[x:A]. B[x],
subtract: n - m,
add: n + m,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat_plus: ℕ+,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
and: P ∧ Q,
prop: ℙ,
all: ∀x:A. B[x],
int_nzero: ℤ-o,
nequal: a ≠ b ∈ T ,
not: ¬A,
implies: P ⇒ Q,
uimplies: b supposing a,
sq_type: SQType(T),
guard: {T},
false: False,
top: Top,
nat: ℕ,
iff: P ⇐⇒ Q,
subtract: n - m,
rev_implies: P ⇐ Q,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x]
Lemmas referenced :
mod_bounds,
less_than_wf,
modulus-equal,
modulus_wf,
subtype_base_sq,
int_subtype_base,
equal-wf-base,
true_wf,
nequal_wf,
nat_wf,
equal_wf,
iff_wf,
divides_wf,
le_wf,
add-associates,
minus-one-mul,
add-swap,
add-mul-special,
zero-mul,
add-zero,
divides_iff_rem_zero,
nat_properties,
decidable__equal_int,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformeq_wf,
itermVar_wf,
itermAdd_wf,
itermConstant_wf,
itermMultiply_wf,
intformimplies_wf,
intformless_wf,
intformle_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_term_value_add_lemma,
int_term_value_constant_lemma,
int_term_value_mul_lemma,
int_formual_prop_imp_lemma,
int_formula_prop_less_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
dependent_set_memberEquality,
natural_numberEquality,
sqequalRule,
independent_pairFormation,
imageMemberEquality,
baseClosed,
hypothesis,
addEquality,
because_Cache,
dependent_functionElimination,
addLevel,
lambdaFormation,
instantiate,
cumulativity,
intEquality,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
voidElimination,
isect_memberEquality,
voidEquality,
setElimination,
rename,
productElimination,
productEquality,
impliesFunctionality,
levelHypothesis,
promote_hyp,
applyLambdaEquality,
unionElimination,
minusEquality,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
computeAll,
impliesLevelFunctionality
Latex:
\mforall{}[x:\mBbbZ{}]. (((x + 1) mod 2) = (1 - x mod 2))
Date html generated:
2017_04_17-AM-09_43_07
Last ObjectModification:
2017_02_27-PM-05_37_48
Theory : num_thy_1
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