Nuprl Lemma : modulus_base
∀[m:ℕ+]. ∀[a:ℕm]. (a mod m ~ a)
Proof
Definitions occuring in Statement :
modulus: a mod n
,
int_seg: {i..j-}
,
nat_plus: ℕ+
,
uall: ∀[x:A]. B[x]
,
natural_number: $n
,
sqequal: s ~ t
Definitions unfolded in proof :
uall: ∀[x:A]. B[x]
,
member: t ∈ T
,
uimplies: b supposing a
,
nat: ℕ
,
so_lambda: λ2x.t[x]
,
so_apply: x[s]
,
modulus: a mod n
,
has-value: (a)↓
,
nat_plus: ℕ+
,
int_seg: {i..j-}
,
nequal: a ≠ b ∈ T
,
not: ¬A
,
implies: P
⇒ Q
,
false: False
,
lelt: i ≤ j < k
,
and: P ∧ Q
,
guard: {T}
,
all: ∀x:A. B[x]
,
prop: ℙ
,
le: A ≤ B
,
bool: 𝔹
,
unit: Unit
,
it: ⋅
,
btrue: tt
,
uiff: uiff(P;Q)
,
less_than: a < b
,
less_than': less_than'(a;b)
,
top: Top
,
true: True
,
squash: ↓T
,
bfalse: ff
,
exists: ∃x:A. B[x]
,
or: P ∨ Q
,
sq_type: SQType(T)
,
bnot: ¬bb
,
ifthenelse: if b then t else f fi
,
assert: ↑b
,
iff: P
⇐⇒ Q
,
rev_implies: P
⇐ Q
,
subtype_rel: A ⊆r B
,
int_nzero: ℤ-o
,
decidable: Dec(P)
,
subtract: n - m
Lemmas referenced :
subtype_base_sq,
nat_wf,
set_subtype_base,
le_wf,
int_subtype_base,
value-type-has-value,
int-value-type,
less_than_transitivity1,
le_weakening,
less_than_irreflexivity,
equal_wf,
rem_bounds_1,
lt_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_lt_int,
top_wf,
less_than_wf,
eqff_to_assert,
bool_cases_sqequal,
bool_subtype_base,
iff_transitivity,
assert_wf,
bnot_wf,
not_wf,
iff_weakening_uiff,
assert_of_bnot,
int_seg_wf,
nat_plus_wf,
div_bounds_1,
int_seg_subtype_nat,
false_wf,
div_rem_sum,
subtype_rel_sets,
nequal_wf,
equal-wf-base,
decidable__int_equal,
zero-mul,
zero-add,
decidable__le,
not-le-2,
not-equal-2,
add_functionality_wrt_le,
add-associates,
add-zero,
le-add-cancel,
condition-implies-le,
add-commutes,
minus-add,
minus-zero,
minus-one-mul,
add-swap,
minus-one-mul-top,
le-add-cancel2,
mul_preserves_le,
nat_plus_subtype_nat,
le_reflexive,
multiply-is-int-iff,
add-is-int-iff,
mul-commutes,
one-mul
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
thin,
instantiate,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
cumulativity,
hypothesis,
independent_isectElimination,
sqequalRule,
intEquality,
lambdaEquality,
natural_numberEquality,
hypothesisEquality,
callbyvalueReduce,
remainderEquality,
because_Cache,
setElimination,
rename,
lambdaFormation,
productElimination,
dependent_functionElimination,
independent_functionElimination,
voidElimination,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
lessCases,
sqequalAxiom,
isect_memberEquality,
independent_pairFormation,
voidEquality,
imageMemberEquality,
baseClosed,
imageElimination,
dependent_pairFormation,
promote_hyp,
impliesFunctionality,
applyEquality,
setEquality,
dependent_set_memberEquality,
addEquality,
minusEquality,
multiplyEquality,
baseApply,
closedConclusion
Latex:
\mforall{}[m:\mBbbN{}\msupplus{}]. \mforall{}[a:\mBbbN{}m]. (a mod m \msim{} a)
Date html generated:
2017_04_14-AM-07_19_10
Last ObjectModification:
2017_02_27-PM-02_53_28
Theory : arithmetic
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