Nuprl Lemma : small-eqmod
∀m:ℕ+. ∀a:ℤ. ∃b:ℤ. (((2 * |b|) ≤ m) ∧ (b ≡ a mod m))
Proof
Definitions occuring in Statement :
eqmod: a ≡ b mod m,
absval: |i|,
nat_plus: ℕ+,
le: A ≤ B,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
and: P ∧ Q,
multiply: n * m,
natural_number: $n,
int: ℤ
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
decidable: Dec(P),
or: P ∨ Q,
uall: ∀[x:A]. B[x],
nat: ℕ,
subtype_rel: A ⊆r B,
and: P ∧ Q,
nat_plus: ℕ+,
exists: ∃x:A. B[x],
cand: A c∧ B,
squash: ↓T,
prop: ℙ,
true: True,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
less_than: a < b,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
less_than': less_than'(a;b),
bfalse: ff,
sq_type: SQType(T),
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b,
int_nzero: ℤ-o,
nequal: a ≠ b ∈ T ,
eqmod: a ≡ b mod m,
divides: b | a,
so_lambda: λ2x.t[x],
so_apply: x[s],
int_lower: {...i},
ge: i ≥ j ,
gt: i > j
Lemmas referenced :
decidable__le,
istype-int,
nat_plus_wf,
rem_bounds_1,
istype-le,
remainder_wfa,
nat_plus_inc_int_nzero,
le_wf,
squash_wf,
true_wf,
absval_pos,
remainder_wf,
subtype_rel_self,
iff_weakening_equal,
rem-eqmod,
absval_wf,
eqmod_wf,
absval_unfold,
subtract_wf,
lt_int_wf,
eqtt_to_assert,
assert_of_lt_int,
nat_plus_properties,
full-omega-unsat,
intformand_wf,
intformless_wf,
itermConstant_wf,
itermSubtract_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_subtract_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
bool_subtype_base,
assert-bnot,
iff_weakening_uiff,
assert_wf,
less_than_wf,
istype-less_than,
istype-top,
intformeq_wf,
intformnot_wf,
intformle_wf,
itermMultiply_wf,
int_formula_prop_eq_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_mul_lemma,
int_subtype_base,
nequal_wf,
itermMinus_wf,
int_term_value_minus_lemma,
eqmod_functionality_wrt_eqmod,
subtract_functionality_wrt_eqmod,
eqmod_weakening,
decidable__equal_int,
set_subtype_base,
rem_bounds_2,
absval_neg,
itermAdd_wf,
int_term_value_add_lemma,
add_functionality_wrt_eqmod,
add-zero,
eqmod_refl,
eqmod-zero
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
natural_numberEquality,
hypothesisEquality,
hypothesis,
unionElimination,
universeIsType,
isectElimination,
dependent_set_memberEquality_alt,
multiplyEquality,
applyEquality,
sqequalRule,
productElimination,
setElimination,
rename,
dependent_pairFormation_alt,
lambdaEquality_alt,
imageElimination,
equalityTransitivity,
equalitySymmetry,
inhabitedIsType,
imageMemberEquality,
baseClosed,
instantiate,
universeEquality,
independent_isectElimination,
independent_functionElimination,
independent_pairFormation,
because_Cache,
productIsType,
minusEquality,
equalityElimination,
approximateComputation,
int_eqEquality,
Error :memTop,
voidElimination,
lessCases,
isect_memberFormation_alt,
axiomSqEquality,
isect_memberEquality_alt,
isectIsTypeImplies,
equalityIstype,
promote_hyp,
cumulativity,
closedConclusion,
sqequalBase,
intEquality,
baseApply,
addEquality
Latex:
\mforall{}m:\mBbbN{}\msupplus{}. \mforall{}a:\mBbbZ{}. \mexists{}b:\mBbbZ{}. (((2 * |b|) \mleq{} m) \mwedge{} (b \mequiv{} a mod m))
Date html generated:
2020_05_19-PM-10_01_07
Last ObjectModification:
2019_12_31-PM-03_28_17
Theory : num_thy_1
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