Nuprl Lemma : small-eqmod-odd
∀m:ℕ+. ((↑isOdd(m)) ⇒ (∀a:ℤ. ∃b:ℤ. (2 * |b| < m ∧ (b ≡ a mod m))))
Proof
Definitions occuring in Statement :
isOdd: isOdd(n),
eqmod: a ≡ b mod m,
absval: |i|,
nat_plus: ℕ+,
assert: ↑b,
less_than: a < b,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
multiply: n * m,
natural_number: $n,
int: ℤ
Definitions unfolded in proof :
sq_type: SQType(T),
guard: {T},
prop: ℙ,
top: Top,
satisfiable_int_formula: satisfiable_int_formula(fmla),
not: ¬A,
uimplies: b supposing a,
nat: ℕ,
le: A ≤ B,
false: False,
or: P ∨ Q,
decidable: Dec(P),
nat_plus: ℕ+,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
cand: A c∧ B,
and: P ∧ Q,
exists: ∃x:A. B[x],
member: t ∈ T,
implies: P ⇒ Q,
all: ∀x:A. B[x],
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q
Lemmas referenced :
nat_plus_wf,
isOdd_wf,
assert_wf,
eqmod_wf,
less_than_wf,
int_subtype_base,
subtype_base_sq,
odd-implies,
false_wf,
int_formula_prop_wf,
int_formula_prop_le_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_mul_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,
itermMultiply_wf,
itermVar_wf,
intformeq_wf,
intformnot_wf,
intformand_wf,
full-omega-unsat,
nat_wf,
decidable__equal_int,
nat_plus_properties,
absval_wf,
decidable__lt,
small-eqmod,
equal-wf-base,
assert-isEven
Rules used in proof :
productEquality,
cumulativity,
instantiate,
independent_pairFormation,
voidEquality,
voidElimination,
isect_memberEquality,
intEquality,
int_eqEquality,
independent_functionElimination,
approximateComputation,
independent_isectElimination,
lambdaEquality,
promote_hyp,
equalitySymmetry,
equalityTransitivity,
pointwiseFunctionality,
unionElimination,
rename,
setElimination,
sqequalRule,
because_Cache,
applyEquality,
hypothesis,
isectElimination,
natural_numberEquality,
multiplyEquality,
dependent_pairFormation,
productElimination,
hypothesisEquality,
thin,
dependent_functionElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
baseClosed,
closedConclusion,
baseApply
Latex:
\mforall{}m:\mBbbN{}\msupplus{}. ((\muparrow{}isOdd(m)) {}\mRightarrow{} (\mforall{}a:\mBbbZ{}. \mexists{}b:\mBbbZ{}. (2 * |b| < m \mwedge{} (b \mequiv{} a mod m))))
Date html generated:
2018_05_21-PM-00_56_07
Last ObjectModification:
2017_12_31-PM-07_41_57
Theory : num_thy_1
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