Nuprl Lemma : iterate-rotate
∀[n,k:ℕ]. (rot(n)^k = (λx.(x + k rem n)) ∈ (ℕn ⟶ ℕn))
Proof
Definitions occuring in Statement :
rotate: rot(n),
fun_exp: f^n,
int_seg: {i..j-},
nat: ℕ,
uall: ∀[x:A]. B[x],
lambda: λx.A[x],
function: x:A ⟶ B[x],
remainder: n rem m,
add: n + m,
natural_number: $n,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat: ℕ,
all: ∀x:A. B[x],
le: A ≤ B,
nat_plus: ℕ+,
prop: ℙ,
top: Top,
not: ¬A,
implies: P ⇒ Q,
false: False,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
uimplies: b supposing a,
or: P ∨ Q,
decidable: Dec(P),
and: P ∧ Q,
lelt: i ≤ j < k,
ge: i ≥ j ,
guard: {T},
int_seg: {i..j-},
fun_exp: f^n,
lt_int: i <z j,
subtract: n - m,
ifthenelse: if b then t else f fi ,
btrue: tt,
less_than: a < b,
squash: ↓T,
subtype_rel: A ⊆r B,
less_than': less_than'(a;b),
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
uiff: uiff(P;Q),
bfalse: ff,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
so_apply: x[s],
so_lambda: λ2x.t[x],
compose: f o g,
rotate: rot(n),
nequal: a ≠ b ∈ T
Lemmas referenced :
istype-nat,
int_seg_wf,
nat_wf,
lelt_wf,
less_than_wf,
less_than_transitivity2,
le_wf,
int_formula_prop_wf,
int_term_value_var_lemma,
int_term_value_add_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
itermVar_wf,
itermAdd_wf,
itermConstant_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
decidable__le,
int_seg_properties,
nat_properties,
rem_bounds_1,
full-omega-unsat,
intformless_wf,
istype-int,
istype-void,
int_formula_prop_less_lemma,
ge_wf,
istype-less_than,
primrec-unroll,
decidable__lt,
add-zero,
equal_wf,
rem_base_case,
int_seg_subtype_nat,
istype-false,
iff_weakening_equal,
trivial-equal,
istype-le,
subtract-1-ge-0,
lt_int_wf,
eqtt_to_assert,
assert_of_lt_int,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
bool_subtype_base,
assert-bnot,
iff_weakening_uiff,
assert_wf,
int_subtype_base,
set_subtype_base,
rotate_wf,
compose_wf,
equal-wf-T-base,
subtract_wf,
int_term_value_subtract_lemma,
int_formula_prop_eq_lemma,
itermSubtract_wf,
intformeq_wf,
decidable__equal_int,
rem_add1
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
introduction,
cut,
sqequalRule,
sqequalHypSubstitution,
Error :isect_memberEquality_alt,
isectElimination,
thin,
hypothesisEquality,
axiomEquality,
hypothesis,
Error :isectIsTypeImplies,
Error :inhabitedIsType,
extract_by_obid,
rename,
setElimination,
natural_numberEquality,
lambdaFormation,
because_Cache,
computeAll,
independent_pairFormation,
voidEquality,
voidElimination,
isect_memberEquality,
intEquality,
int_eqEquality,
lambdaEquality,
dependent_pairFormation,
independent_isectElimination,
unionElimination,
dependent_functionElimination,
productElimination,
addEquality,
dependent_set_memberEquality,
intWeakElimination,
Error :lambdaFormation_alt,
approximateComputation,
independent_functionElimination,
Error :dependent_pairFormation_alt,
Error :lambdaEquality_alt,
Error :universeIsType,
Error :functionIsTypeImplies,
Error :functionExtensionality_alt,
imageElimination,
Error :dependent_set_memberEquality_alt,
applyEquality,
imageMemberEquality,
baseClosed,
equalityTransitivity,
equalitySymmetry,
Error :productIsType,
equalityElimination,
Error :equalityIstype,
promote_hyp,
instantiate,
cumulativity,
closedConclusion,
baseApply,
functionEquality,
applyLambdaEquality,
hyp_replacement,
remainderEquality
Latex:
\mforall{}[n,k:\mBbbN{}]. (rot(n)\^{}k = (\mlambda{}x.(x + k rem n)))
Date html generated:
2019_06_20-PM-01_37_04
Last ObjectModification:
2019_03_06-AM-10_53_57
Theory : list_1
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