Nuprl Lemma : modulus_base_neg
∀[m:ℕ+]. ∀[a:{-m..0-}]. (a mod m ~ m + a)
Proof
Definitions occuring in Statement :
modulus: a mod n,
int_seg: {i..j-},
nat_plus: ℕ+,
uall: ∀[x:A]. B[x],
add: n + m,
minus: -n,
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+,
nequal: a ≠ b ∈ T ,
guard: {T},
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
not: ¬A,
implies: P ⇒ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
all: ∀x:A. B[x],
top: Top,
prop: ℙ,
subtype_rel: A ⊆r B,
int_lower: {...i},
decidable: Dec(P),
or: P ∨ Q,
sq_type: SQType(T),
gt: i > j,
ge: i ≥ j ,
less_than: a < b,
squash: ↓T,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
less_than': less_than'(a;b),
true: True,
bfalse: ff,
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b,
le: A ≤ B
Lemmas referenced :
subtype_base_sq,
nat_wf,
set_subtype_base,
le_wf,
int_subtype_base,
value-type-has-value,
int-value-type,
int_seg_properties,
nat_plus_properties,
full-omega-unsat,
intformand_wf,
intformeq_wf,
itermVar_wf,
itermConstant_wf,
intformless_wf,
intformle_wf,
itermMinus_wf,
int_formula_prop_and_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_less_lemma,
int_formula_prop_le_lemma,
int_term_value_minus_lemma,
int_formula_prop_wf,
rem_bounds_2,
subtype_rel_sets,
lelt_wf,
decidable__le,
intformnot_wf,
int_formula_prop_not_lemma,
div_bounds_2,
div_rem_sum,
absval-non-neg,
absval_pos,
nat_plus_subtype_nat,
int_seg_wf,
nat_plus_wf,
decidable__equal_int,
equal-wf-base,
lt_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_lt_int,
top_wf,
less_than_wf,
itermAdd_wf,
itermMultiply_wf,
int_term_value_add_lemma,
int_term_value_mul_lemma,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
mul_preserves_le
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,
minusEquality,
productElimination,
lambdaFormation,
approximateComputation,
independent_functionElimination,
dependent_pairFormation,
int_eqEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
applyEquality,
setEquality,
unionElimination,
dependent_set_memberEquality,
equalityTransitivity,
equalitySymmetry,
sqequalAxiom,
divideEquality,
imageElimination,
baseClosed,
equalityElimination,
lessCases,
imageMemberEquality,
addEquality,
promote_hyp
Latex:
\mforall{}[m:\mBbbN{}\msupplus{}]. \mforall{}[a:\{-m..0\msupminus{}\}]. (a mod m \msim{} m + a)
Date html generated:
2018_05_21-PM-00_25_29
Last ObjectModification:
2018_05_19-AM-06_52_15
Theory : int_2
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