Nuprl Lemma : rem-zero-implies-minus
∀x:ℤ. ∀y:ℤ-o. (((x rem y) = 0 ∈ ℤ) ⇒ ((-x rem y) = 0 ∈ ℤ))
Proof
Definitions occuring in Statement :
int_nzero: ℤ-o,
all: ∀x:A. B[x],
implies: P ⇒ Q,
remainder: n rem m,
minus: -n,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
int_nzero: ℤ-o,
nequal: a ≠ b ∈ T ,
decidable: Dec(P),
or: P ∨ Q,
false: False,
uiff: uiff(P;Q),
and: P ∧ Q,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
prop: ℙ,
sq_type: SQType(T),
guard: {T},
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
div_rem_sum,
subtype_base_sq,
int_subtype_base,
int_nzero_properties,
decidable__equal_int,
add-is-int-iff,
multiply-is-int-iff,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformeq_wf,
itermVar_wf,
itermMultiply_wf,
itermAdd_wf,
itermConstant_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_term_value_mul_lemma,
int_term_value_add_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
false_wf,
minus-one-mul,
divide_wfa,
mul-commutes,
mul-swap,
rem-exact,
set_subtype_base,
nequal_wf,
int_nzero_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
instantiate,
cumulativity,
intEquality,
independent_isectElimination,
hypothesis,
setElimination,
rename,
dependent_functionElimination,
because_Cache,
unionElimination,
equalityTransitivity,
equalitySymmetry,
pointwiseFunctionality,
promote_hyp,
sqequalRule,
baseApply,
closedConclusion,
baseClosed,
productElimination,
natural_numberEquality,
approximateComputation,
independent_functionElimination,
Error :dependent_pairFormation_alt,
Error :lambdaEquality_alt,
int_eqEquality,
Error :isect_memberEquality_alt,
voidElimination,
independent_pairFormation,
Error :universeIsType,
multiplyEquality,
minusEquality,
Error :equalityIstype,
Error :inhabitedIsType,
applyEquality,
sqequalBase
Latex:
\mforall{}x:\mBbbZ{}. \mforall{}y:\mBbbZ{}\msupminus{}\msupzero{}. (((x rem y) = 0) {}\mRightarrow{} ((-x rem y) = 0))
Date html generated:
2019_06_20-PM-02_24_48
Last ObjectModification:
2019_03_06-AM-11_06_26
Theory : num_thy_1
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