### Nuprl Lemma : fun_exp-rem

`∀[T:Type]. ∀[f:T ⟶ T]. ∀[x:T]. ∀[n:ℕ+].  ∀[k:ℕ]. ((f^k x) = (f^k rem n x) ∈ T) supposing (f^n x) = x ∈ T`

Proof

Definitions occuring in Statement :  fun_exp: `f^n` nat_plus: `ℕ+` nat: `ℕ` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` apply: `f a` function: `x:A ⟶ B[x]` remainder: `n rem m` universe: `Type` equal: `s = t ∈ T`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` uimplies: `b supposing a` nat: `ℕ` subtype_rel: `A ⊆r B` nat_plus: `ℕ+` int_nzero: `ℤ-o` so_lambda: `λ2x.t[x]` so_apply: `x[s]` prop: `ℙ` all: `∀x:A. B[x]` implies: `P `` Q` nequal: `a ≠ b ∈ T ` not: `¬A` false: `False` guard: `{T}` ge: `i ≥ j ` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` top: `Top` and: `P ∧ Q` le: `A ≤ B` decidable: `Dec(P)` or: `P ∨ Q` uiff: `uiff(P;Q)` sq_type: `SQType(T)` squash: `↓T` true: `True` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` label: `...\$L... t` fun_exp: `f^n` primrec: `primrec(n;b;c)`
Lemmas referenced :  div_rem_sum subtype_rel_sets less_than_wf nequal_wf nat_plus_properties nat_properties satisfiable-full-omega-tt intformand_wf intformeq_wf itermVar_wf itermConstant_wf intformless_wf int_formula_prop_and_lemma int_formula_prop_eq_lemma int_term_value_var_lemma int_term_value_constant_lemma int_formula_prop_less_lemma int_formula_prop_wf equal-wf-base int_subtype_base equal_wf fun_exp_wf nat_plus_subtype_nat nat_plus_wf remainder_wf nat_wf div_bounds_1 mul_bounds_1a divide_wf subtype_base_sq set_subtype_base le_wf decidable__equal_int add-is-int-iff multiply-is-int-iff intformnot_wf itermAdd_wf itermMultiply_wf int_formula_prop_not_lemma int_term_value_add_lemma int_term_value_mul_lemma false_wf decidable__le intformle_wf int_formula_prop_le_lemma fun_exp_add-sq squash_wf true_wf fun_exp-mul iff_weakening_equal fun_exp-fixedpoint
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut extract_by_obid sqequalHypSubstitution isectElimination thin setElimination rename hypothesisEquality hypothesis applyEquality sqequalRule intEquality because_Cache lambdaEquality natural_numberEquality independent_isectElimination setEquality lambdaFormation applyLambdaEquality dependent_pairFormation int_eqEquality dependent_functionElimination isect_memberEquality voidElimination voidEquality independent_pairFormation computeAll baseClosed independent_functionElimination axiomEquality cumulativity functionExtensionality equalityTransitivity equalitySymmetry functionEquality addEquality multiplyEquality divideEquality instantiate productElimination unionElimination pointwiseFunctionality promote_hyp baseApply closedConclusion dependent_set_memberEquality imageElimination universeEquality imageMemberEquality

Latex:
\mforall{}[T:Type].  \mforall{}[f:T  {}\mrightarrow{}  T].  \mforall{}[x:T].  \mforall{}[n:\mBbbN{}\msupplus{}].    \mforall{}[k:\mBbbN{}].  ((f\^{}k  x)  =  (f\^{}k  rem  n  x))  supposing  (f\^{}n  x)  =  x

Date html generated: 2017_04_14-AM-09_16_48
Last ObjectModification: 2017_02_27-PM-03_53_54

Theory : int_2

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