Nuprl Lemma : iterate-rotate-rotate-by
∀[n,i:ℕ]. (rot(n)^i = rotate-by(n;i) ∈ (ℕn ⟶ ℕn))
Proof
Definitions occuring in Statement :
rotate-by: rotate-by(n;i),
rotate: rot(n),
fun_exp: f^n,
int_seg: {i..j-},
nat: ℕ,
uall: ∀[x:A]. B[x],
function: x:A ⟶ B[x],
natural_number: $n,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
all: ∀x:A. B[x],
and: P ∧ Q,
prop: ℙ,
fun_exp: f^n,
lt_int: i <z j,
subtract: n - m,
ifthenelse: if b then t else f fi ,
btrue: tt,
bool: 𝔹,
unit: Unit,
it: ⋅,
uiff: uiff(P;Q),
bfalse: ff,
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
less_than: a < b,
squash: ↓T,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
rotate-by: rotate-by(n;i),
rotate: rot(n),
compose: f o g,
decidable: Dec(P),
less_than': less_than'(a;b),
nat_plus: ℕ+,
true: True,
int_nzero: ℤ-o,
nequal: a ≠ b ∈ T ,
remainder: n rem m
Lemmas referenced :
nat_properties,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
ge_wf,
istype-less_than,
primrec-unroll,
rotate-by-zero,
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,
less_than_wf,
istype-nat,
equal-wf-T-base,
int_seg_wf,
compose_wf,
rotate_wf,
int_subtype_base,
set_subtype_base,
le_wf,
rem_addition,
subtract_wf,
int_seg_properties,
decidable__le,
intformnot_wf,
itermAdd_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_term_value_add_lemma,
int_term_value_subtract_lemma,
istype-le,
istype-void,
decidable__lt,
equal_wf,
squash_wf,
true_wf,
istype-universe,
rem_bounds_1,
decidable__equal_int,
remainder_wfa,
nequal_wf,
eq_int_wf,
assert_of_eq_int,
neg_assert_of_eq_int,
intformeq_wf,
int_formula_prop_eq_lemma,
rem-1,
add-swap,
add-commutes,
rem_add1,
remainder_wf,
iff_weakening_equal,
ifthenelse_wf,
add_functionality_wrt_eq,
rem_rem_to_rem,
lelt_wf,
one-rem,
int_nzero_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
lambdaFormation_alt,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
dependent_functionElimination,
Error :memTop,
sqequalRule,
independent_pairFormation,
universeIsType,
voidElimination,
axiomEquality,
functionIsTypeImplies,
inhabitedIsType,
equalitySymmetry,
because_Cache,
unionElimination,
equalityElimination,
equalityTransitivity,
productElimination,
equalityIstype,
promote_hyp,
instantiate,
cumulativity,
isect_memberEquality_alt,
isectIsTypeImplies,
hyp_replacement,
applyLambdaEquality,
functionEquality,
imageElimination,
baseApply,
closedConclusion,
baseClosed,
applyEquality,
intEquality,
functionExtensionality_alt,
dependent_set_memberEquality_alt,
addEquality,
universeEquality,
imageMemberEquality,
productIsType,
sqequalBase,
minusEquality
Latex:
\mforall{}[n,i:\mBbbN{}]. (rot(n)\^{}i = rotate-by(n;i))
Date html generated:
2020_05_20-AM-08_15_16
Last ObjectModification:
2019_12_31-PM-08_42_31
Theory : general
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