Nuprl Lemma : rem_add1
∀[i:ℕ]. ∀[n:ℕ+]. ((i + 1 rem n) = if (i rem n =z n - 1) then 0 else (i rem n) + 1 fi ∈ ℤ)
Proof
Definitions occuring in Statement :
nat_plus: ℕ+,
nat: ℕ,
ifthenelse: if b then t else f fi ,
eq_int: (i =z j),
uall: ∀[x:A]. B[x],
remainder: n rem m,
subtract: n - m,
add: n + m,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
nat_plus: ℕ+,
decidable: Dec(P),
or: P ∨ Q,
subtype_rel: A ⊆r B,
top: Top,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
ge: i ≥ j ,
nequal: a ≠ b ∈ T ,
uimplies: b supposing a,
prop: ℙ,
implies: P ⇒ Q,
not: ¬A,
false: False,
less_than': less_than'(a;b),
and: P ∧ Q,
le: A ≤ B,
nat: ℕ,
assert: ↑b,
bnot: ¬bb,
guard: {T},
sq_type: SQType(T),
bfalse: ff,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
btrue: tt,
it: ⋅,
unit: Unit,
bool: 𝔹,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
true: True,
squash: ↓T,
less_than: a < b,
subtract: n - m
Lemmas referenced :
decidable__lt,
nat_plus_wf,
istype-nat,
int_subtype_base,
equal-wf-base,
int_formula_prop_wf,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_and_lemma,
intformless_wf,
itermConstant_wf,
itermVar_wf,
intformeq_wf,
intformand_wf,
satisfiable-full-omega-tt,
nat_properties,
nat_plus_properties,
equal_wf,
one-rem,
le_wf,
false_wf,
rem_addition,
not_wf,
bnot_wf,
assert_wf,
equal-wf-T-base,
neg_assert_of_eq_int,
assert-bnot,
bool_subtype_base,
subtype_base_sq,
bool_cases_sqequal,
eqff_to_assert,
assert_of_eq_int,
eqtt_to_assert,
bool_wf,
subtract_wf,
eq_int_wf,
equal-wf-base-T,
uiff_transitivity,
iff_transitivity,
iff_weakening_uiff,
assert_of_bnot,
decidable__equal_int,
int_term_value_subtract_lemma,
itermSubtract_wf,
rem_base_case,
iff_weakening_equal,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
intformle_wf,
intformnot_wf,
decidable__le,
subtract-add-cancel,
rem_rec_case,
true_wf,
squash_wf,
int_term_value_add_lemma,
itermAdd_wf,
rem_bounds_1,
less_than_wf,
rem-1,
ifthenelse_wf,
add_functionality_wrt_eq
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
natural_numberEquality,
setElimination,
rename,
hypothesisEquality,
hypothesis,
unionElimination,
universeIsType,
sqequalRule,
isect_memberEquality_alt,
isectElimination,
axiomEquality,
isectIsTypeImplies,
inhabitedIsType,
equalityTransitivity,
baseClosed,
applyEquality,
computeAll,
voidEquality,
voidElimination,
isect_memberEquality,
int_eqEquality,
lambdaEquality,
dependent_pairFormation,
because_Cache,
addEquality,
remainderEquality,
intEquality,
applyLambdaEquality,
hyp_replacement,
equalitySymmetry,
independent_isectElimination,
lambdaFormation,
independent_pairFormation,
dependent_set_memberEquality,
cumulativity,
independent_functionElimination,
instantiate,
promote_hyp,
productElimination,
equalityElimination,
impliesFunctionality,
imageMemberEquality,
universeEquality,
imageElimination,
lambdaFormation_alt,
addLevel
Latex:
\mforall{}[i:\mBbbN{}]. \mforall{}[n:\mBbbN{}\msupplus{}]. ((i + 1 rem n) = if (i rem n =\msubz{} n - 1) then 0 else (i rem n) + 1 fi )
Date html generated:
2020_05_19-PM-09_41_27
Last ObjectModification:
2019_12_31-PM-00_59_49
Theory : int_2
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