Nuprl Lemma : iterate-rotate-by
∀[n,i,k:ℕ]. (rotate-by(n;i)^k = rotate-by(n;k * i) ∈ (ℕn ⟶ ℕn))
Proof
Definitions occuring in Statement :
rotate-by: rotate-by(n;i),
fun_exp: f^n,
int_seg: {i..j-},
nat: ℕ,
uall: ∀[x:A]. B[x],
function: x:A ⟶ B[x],
multiply: n * m,
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,
squash: ↓T,
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
less_than: a < b,
decidable: Dec(P),
or: P ∨ Q,
true: True,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
nat_plus: ℕ+,
so_lambda: λ2x.t[x],
so_apply: x[s],
compose: f o g
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,
fun_exp0_lemma,
subtract-1-ge-0,
istype-nat,
equal_wf,
squash_wf,
true_wf,
istype-universe,
int_seg_wf,
rotate-by-is-id,
decidable__le,
intformnot_wf,
itermMultiply_wf,
int_formula_prop_not_lemma,
int_term_value_mul_lemma,
istype-le,
zero-mul,
any_divs_zero,
subtype_rel_self,
iff_weakening_equal,
fun_exp_unroll_1,
decidable__lt,
equal-wf-T-base,
rotate-by_wf,
set_subtype_base,
le_wf,
int_subtype_base,
compose-rotate-by,
subtract_wf,
mul_bounds_1a,
itermSubtract_wf,
int_term_value_subtract_lemma,
multiply_nat_wf,
decidable__equal_int,
intformeq_wf,
itermAdd_wf,
int_formula_prop_eq_lemma,
int_term_value_add_lemma,
add_nat_wf,
subtract_nat_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
introduction,
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,
because_Cache,
applyEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
instantiate,
universeEquality,
functionEquality,
productElimination,
dependent_set_memberEquality_alt,
multiplyEquality,
unionElimination,
imageMemberEquality,
baseClosed,
hyp_replacement,
applyLambdaEquality,
baseApply,
closedConclusion,
intEquality,
addEquality,
equalityIstype
Latex:
\mforall{}[n,i,k:\mBbbN{}]. (rotate-by(n;i)\^{}k = rotate-by(n;k * i))
Date html generated:
2020_05_20-AM-08_15_29
Last ObjectModification:
2020_01_09-AM-00_17_41
Theory : general
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