Nuprl Lemma : rem_antisym
∀[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 :
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
decidable: Dec(P),
or: P ∨ Q,
squash: ↓T,
prop: ℙ,
nat_plus: ℕ+,
le: A ≤ B,
and: P ∧ Q,
nequal: a ≠ b ∈ T ,
not: ¬A,
implies: P ⇒ Q,
false: False,
guard: {T},
uimplies: b supposing a,
true: True,
subtype_rel: A ⊆r B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
top: Top,
int_lower: {...i},
uiff: uiff(P;Q),
subtract: n - m,
less_than': less_than'(a;b),
int_nzero: ℤ-o,
so_apply: x[s],
so_lambda: λ2x.t[x]
Lemmas referenced :
nat_plus_wf,
int_nzero_wf,
decidable__le,
equal_wf,
squash_wf,
true_wf,
rem_2_to_1,
less_than_transitivity1,
le_weakening,
less_than_irreflexivity,
subtype_rel_self,
iff_weakening_equal,
minus-minus,
minus-one-mul,
minus-one-mul-top,
le_wf,
add_functionality_wrt_le,
le_reflexive,
zero-add,
add-mul-special,
zero-mul,
false_wf,
not-le-2,
condition-implies-le,
minus-add,
add-commutes,
minus-zero,
add-associates,
add-zero,
le-add-cancel,
less_than_wf,
not-equal-2,
not-lt-2,
decidable__lt,
rem_sym,
le-add-cancel2,
one-mul,
mul-associates,
add-swap,
int_subtype_base,
nequal_wf,
set_subtype_base,
add-is-int-iff,
subtract_wf
Rules used in proof :
cut,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
hypothesis,
introduction,
extract_by_obid,
intEquality,
Error :isect_memberFormation_alt,
Error :universeIsType,
sqequalRule,
sqequalHypSubstitution,
isect_memberEquality,
isectElimination,
thin,
hypothesisEquality,
axiomEquality,
because_Cache,
dependent_functionElimination,
natural_numberEquality,
unionElimination,
applyEquality,
lambdaEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
minusEquality,
remainderEquality,
setElimination,
rename,
productElimination,
independent_isectElimination,
independent_functionElimination,
voidElimination,
imageMemberEquality,
baseClosed,
instantiate,
voidEquality,
dependent_set_memberEquality,
multiplyEquality,
independent_pairFormation,
addEquality,
closedConclusion,
baseApply
Latex:
\mforall{}[a:\mBbbZ{}]. \mforall{}[b:\mBbbZ{}\msupminus{}\msupzero{}]. ((-a rem b) = (-(a rem b)))
Date html generated:
2019_06_20-AM-11_25_22
Last ObjectModification:
2018_09_26-AM-10_58_28
Theory : arithmetic
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