Nuprl Lemma : rem-1
∀a:ℤ. (a rem 1 ~ 0)
Proof
Definitions occuring in Statement :
all: ∀x:A. B[x],
remainder: n rem m,
natural_number: $n,
int: ℤ,
sqequal: s ~ t
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
int_nzero: ℤ-o,
true: True,
nequal: a ≠ b ∈ T ,
not: ¬A,
implies: P ⇒ Q,
sq_type: SQType(T),
guard: {T},
false: False,
prop: ℙ,
absval: |i|,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
less_than: a < b,
less_than': less_than'(a;b),
squash: ↓T,
cand: A c∧ B,
decidable: Dec(P),
or: P ∨ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
le: A ≤ B,
subtract: n - m,
bfalse: ff,
exists: ∃x:A. B[x],
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b
Lemmas referenced :
subtype_base_sq,
int_subtype_base,
rem_bounds_absval,
nequal_wf,
remainder_wfa,
istype-int,
absval_unfold,
lt_int_wf,
eqtt_to_assert,
assert_of_lt_int,
istype-top,
decidable__int_equal,
istype-false,
not-equal-2,
less-iff-le,
add_functionality_wrt_le,
add-associates,
add-swap,
add-zero,
le-add-cancel,
condition-implies-le,
add-commutes,
minus-add,
minus-zero,
le-add-cancel2,
istype-less_than,
eqff_to_assert,
bool_cases_sqequal,
bool_wf,
bool_subtype_base,
iff_transitivity,
assert_wf,
bnot_wf,
not_wf,
less_than_wf,
iff_weakening_uiff,
assert_of_bnot,
istype-assert,
istype-void,
add_functionality_wrt_lt,
le_reflexive,
not-lt-2,
zero-add,
minus-minus,
minus-one-mul,
minus-one-mul-top,
two-mul,
mul-distributes-right,
one-mul
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
thin,
instantiate,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
cumulativity,
intEquality,
independent_isectElimination,
hypothesis,
dependent_functionElimination,
dependent_set_memberEquality_alt,
natural_numberEquality,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
voidElimination,
equalityIstype,
inhabitedIsType,
hypothesisEquality,
baseClosed,
sqequalBase,
universeIsType,
because_Cache,
sqequalRule,
minusEquality,
unionElimination,
equalityElimination,
productElimination,
lessCases,
isect_memberFormation_alt,
axiomSqEquality,
isect_memberEquality_alt,
isectIsTypeImplies,
independent_pairFormation,
Error :memTop,
imageMemberEquality,
imageElimination,
addEquality,
dependent_pairFormation_alt,
promote_hyp,
functionIsType,
multiplyEquality
Latex:
\mforall{}a:\mBbbZ{}. (a rem 1 \msim{} 0)
Date html generated:
2020_05_19-PM-09_35_25
Last ObjectModification:
2019_12_26-PM-09_49_43
Theory : arithmetic
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