Nuprl Lemma : rotate-by-transitive
∀n,b:ℕ. (gcd(b;n) = 1 ∈ ℤ supposing 0 < n ⇐⇒ ∀x,y:ℕn. ∃k:ℕ. ((rotate-by(n;b)^k x) = y ∈ ℤ))
Proof
Definitions occuring in Statement :
rotate-by: rotate-by(n;i),
fun_exp: f^n,
gcd: gcd(a;b),
int_seg: {i..j-},
nat: ℕ,
less_than: a < b,
uimplies: b supposing a,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
apply: f a,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
nat: ℕ,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
rev_implies: P ⇐ Q,
exists: ∃x:A. B[x],
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
less_than: a < b,
squash: ↓T,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
top: Top,
prop: ℙ,
sq_type: SQType(T),
guard: {T},
modulus: a mod n,
has-value: (a)↓,
int_nzero: ℤ-o,
nequal: a ≠ b ∈ 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,
nat_plus: ℕ+,
gt: i > j,
divides: b | a,
rotate-by: rotate-by(n;i),
remainder: n rem m,
gcd: gcd(a;b),
eq_int: (i =z j),
gcd_p: GCD(a;b;y),
cand: A c∧ B
Lemmas referenced :
int_seg_wf,
istype-less_than,
istype-int,
set_subtype_base,
le_wf,
int_subtype_base,
lelt_wf,
istype-nat,
int_seg_properties,
nat_properties,
decidable__lt,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermVar_wf,
intformle_wf,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
bezout_ident_n,
gcd_sat_gcd_p,
gcd_unique,
subtype_base_sq,
assoced_elim,
decidable__equal_int,
intformeq_wf,
itermAdd_wf,
itermMultiply_wf,
itermMinus_wf,
int_formula_prop_eq_lemma,
int_term_value_add_lemma,
int_term_value_mul_lemma,
int_term_value_minus_lemma,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
value-type-has-value,
int-value-type,
remainder_wfa,
nequal_wf,
lt_int_wf,
eqtt_to_assert,
assert_of_lt_int,
istype-top,
eqff_to_assert,
bool_cases_sqequal,
bool_wf,
bool_subtype_base,
assert-bnot,
iff_weakening_uiff,
assert_wf,
less_than_wf,
equal_wf,
squash_wf,
true_wf,
istype-universe,
rem_base_case,
int_seg_subtype_nat,
istype-false,
subtype_rel_self,
iff_weakening_equal,
absval_wf,
add_functionality_wrt_eq,
rem_bounds_1,
decidable__le,
istype-le,
div_rem_sum,
divide_wfa,
add-is-int-iff,
multiply-is-int-iff,
false_wf,
pos_mul_arg_bounds,
modulus-equal,
modulus-is-rem,
add_nat_wf,
multiply_nat_wf,
nat_wf,
equal-wf-base,
iterate-rotate-by,
rem-one,
zero-add,
one_divs_any,
divides_wf,
gcd_wf,
gcd_sat_pred,
mul-commutes,
zero-mul,
divisor_bound,
gcd-positive
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
independent_pairFormation,
inhabitedIsType,
hypothesisEquality,
universeIsType,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
natural_numberEquality,
setElimination,
rename,
hypothesis,
sqequalRule,
isectIsType,
equalityIstype,
baseApply,
closedConclusion,
baseClosed,
applyEquality,
intEquality,
lambdaEquality_alt,
independent_isectElimination,
sqequalBase,
equalitySymmetry,
isect_memberFormation_alt,
because_Cache,
functionIsType,
productIsType,
productElimination,
imageElimination,
dependent_functionElimination,
unionElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
addEquality,
multiplyEquality,
equalityTransitivity,
instantiate,
cumulativity,
minusEquality,
applyLambdaEquality,
hyp_replacement,
callbyvalueReduce,
dependent_set_memberEquality_alt,
equalityElimination,
lessCases,
axiomSqEquality,
isectIsTypeImplies,
imageMemberEquality,
promote_hyp,
universeEquality,
pointwiseFunctionality,
productEquality
Latex:
\mforall{}n,b:\mBbbN{}. (gcd(b;n) = 1 supposing 0 < n \mLeftarrow{}{}\mRightarrow{} \mforall{}x,y:\mBbbN{}n. \mexists{}k:\mBbbN{}. ((rotate-by(n;b)\^{}k x) = y))
Date html generated:
2019_10_15-AM-11_20_13
Last ObjectModification:
2019_06_25-PM-01_30_45
Theory : general
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