Nuprl Lemma : rem_sym
∀[a:ℤ]. ∀[b:ℤ-o]. ((a rem -b) = (a rem b) ∈ ℤ)
Proof
Definitions occuring in Statement :
int_nzero: ℤ-o,
uall: ∀[x:A]. B[x],
remainder: n rem m,
minus: -n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
or: P ∨ Q,
decidable: Dec(P),
int_nzero: ℤ-o,
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
nat_plus: ℕ+,
le: A ≤ B,
and: P ∧ Q,
nequal: a ≠ b ∈ T ,
iff: P ⇐⇒ Q,
not: ¬A,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
false: False,
uiff: uiff(P;Q),
uimplies: b supposing a,
less_than': less_than'(a;b),
true: True,
subtract: n - m,
subtype_rel: A ⊆r B,
top: Top,
prop: ℙ,
int_lower: {...i},
guard: {T},
so_lambda: λ2x.t[x],
so_apply: x[s],
cand: A c∧ B,
rev_uimplies: rev_uimplies(P;Q),
nat: ℕ,
squash: ↓T,
less_than: a < b,
sq_type: SQType(T)
Lemmas referenced :
decidable__le,
decidable__lt,
istype-false,
not-lt-2,
not-equal-2,
add_functionality_wrt_le,
zero-add,
add-zero,
le-add-cancel,
condition-implies-le,
add-commutes,
istype-void,
minus-add,
minus-zero,
less_than_wf,
le_wf,
not-le-2,
minus-one-mul,
add-swap,
minus-one-mul-top,
add-associates,
le-add-cancel2,
subtract_wf,
int_nzero_wf,
le_reflexive,
minus-minus,
add-mul-special,
one-mul,
subtype_rel_sets_simple,
nequal_wf,
istype-le,
int_subtype_base,
div_4_to_1,
divide_wfa,
mul-associates,
mul-swap,
mul-commutes,
equal_wf,
squash_wf,
true_wf,
istype-universe,
rem_to_div,
subtype_rel_self,
iff_weakening_equal,
istype-int,
rem_2_to_1,
rem_3_to_1,
false_wf,
istype-less_than,
zero-mul,
add_functionality_wrt_lt,
less_than_transitivity1,
le_weakening,
less_than_irreflexivity,
subtype_base_sq,
remainder_wfa
Rules used in proof :
inhabitedIsType,
isectIsTypeImplies,
axiomEquality,
isectElimination,
isect_memberEquality_alt,
sqequalRule,
because_Cache,
unionElimination,
hypothesis,
hypothesisEquality,
rename,
setElimination,
natural_numberEquality,
thin,
dependent_functionElimination,
sqequalHypSubstitution,
extract_by_obid,
cut,
introduction,
isect_memberFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
dependent_set_memberEquality_alt,
productElimination,
independent_pairFormation,
lambdaFormation_alt,
voidElimination,
independent_functionElimination,
independent_isectElimination,
addEquality,
minusEquality,
applyEquality,
lambdaEquality_alt,
universeIsType,
intEquality,
equalityTransitivity,
equalitySymmetry,
multiplyEquality,
inlFormation_alt,
inrFormation_alt,
Error :memTop,
equalityIstype,
baseClosed,
sqequalBase,
imageElimination,
instantiate,
universeEquality,
imageMemberEquality,
remainderEquality,
equalityIsType3,
voidEquality,
isect_memberEquality,
lambdaEquality,
lambdaFormation,
cumulativity
Latex:
\mforall{}[a:\mBbbZ{}]. \mforall{}[b:\mBbbZ{}\msupminus{}\msupzero{}]. ((a rem -b) = (a rem b))
Date html generated:
2020_05_19-PM-09_35_34
Last ObjectModification:
2019_12_31-PM-01_07_17
Theory : arithmetic
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