### 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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