Nuprl Lemma : rem_eq_args_z
∀[a:ℤ]. ∀[b:ℤ-o]. (a rem b) = 0 ∈ ℤ supposing |a| = |b| ∈ ℤ
Proof
Definitions occuring in Statement :
absval: |i|,
int_nzero: ℤ-o,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
remainder: n rem m,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
nat_plus: ℕ+,
uimplies: b supposing a,
int_nzero: ℤ-o,
nat: ℕ,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
less_than: a < b,
less_than': less_than'(a;b),
top: Top,
true: True,
squash: ↓T,
not: ¬A,
false: False,
nequal: a ≠ b ∈ T ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
bfalse: ff,
or: P ∨ Q,
sq_type: SQType(T),
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b,
decidable: Dec(P),
int_lower: {...i}
Lemmas referenced :
equal-wf-base-T,
int_subtype_base,
nat_plus_wf,
absval_wf,
nat_wf,
int_nzero_wf,
absval_unfold,
lt_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_lt_int,
top_wf,
less_than_wf,
equal_wf,
squash_wf,
true_wf,
nat_plus_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformeq_wf,
itermVar_wf,
itermConstant_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
equal-wf-base,
iff_weakening_equal,
rem_eq_args,
less_than_transitivity1,
le_weakening,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
minus_mono_wrt_eq,
itermMinus_wf,
int_term_value_minus_lemma,
rem_antisym,
nequal_wf,
minus-zero,
decidable__lt,
absval_pos,
int_nzero_properties,
decidable__le,
intformnot_wf,
intformle_wf,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
le_wf,
rem_sym,
absval_neg
Rules used in proof :
cut,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
hypothesis,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
intEquality,
sqequalRule,
baseApply,
closedConclusion,
baseClosed,
hypothesisEquality,
applyEquality,
setElimination,
rename,
isect_memberFormation,
because_Cache,
lambdaEquality,
isect_memberEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
minusEquality,
natural_numberEquality,
unionElimination,
equalityElimination,
productElimination,
independent_isectElimination,
lessCases,
sqequalAxiom,
independent_pairFormation,
voidElimination,
voidEquality,
imageMemberEquality,
imageElimination,
independent_functionElimination,
universeEquality,
equalityUniverse,
levelHypothesis,
remainderEquality,
dependent_pairFormation,
int_eqEquality,
dependent_functionElimination,
computeAll,
dependent_set_memberEquality,
promote_hyp,
instantiate,
cumulativity
Latex:
\mforall{}[a:\mBbbZ{}]. \mforall{}[b:\mBbbZ{}\msupminus{}\msupzero{}]. (a rem b) = 0 supposing |a| = |b|
Date html generated:
2017_04_14-AM-09_16_42
Last ObjectModification:
2017_02_27-PM-03_53_50
Theory : int_2
Home
Index