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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