### Nuprl Lemma : reg-seq-list-add_functionality_wrt_permutation

`∀[L,L':ℝ List].`
`  reg-seq-list-add(L) = reg-seq-list-add(L') ∈ {f:ℕ+ ⟶ ℤ| ||L||-regular-seq(f)}  supposing permutation(ℝ;L;L')`

Proof

Definitions occuring in Statement :  reg-seq-list-add: `reg-seq-list-add(L)` real: `ℝ` regular-int-seq: `k-regular-seq(f)` permutation: `permutation(T;L1;L2)` length: `||as||` list: `T List` nat_plus: `ℕ+` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` set: `{x:A| B[x]} ` function: `x:A ⟶ B[x]` int: `ℤ` equal: `s = t ∈ T`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` uimplies: `b supposing a` all: `∀x:A. B[x]` decidable: `Dec(P)` or: `P ∨ Q` prop: `ℙ` squash: `↓T` cons: `[a / b]` top: `Top` ge: `i ≥ j ` le: `A ≤ B` and: `P ∧ Q` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` false: `False` implies: `P `` Q` not: `¬A` true: `True` nat_plus: `ℕ+` guard: `{T}` nat: `ℕ` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` uiff: `uiff(P;Q)` subtract: `n - m` subtype_rel: `A ⊆r B` less_than': `less_than'(a;b)` sq_stable: `SqStable(P)` real: `ℝ`
Lemmas referenced :  decidable__equal_int length_wf real_wf permutation_wf list_wf reg-seq-list-add_wf squash_wf true_wf permutation-length list-cases length_of_nil_lemma nil_wf product_subtype_list length_of_cons_lemma non_neg_length satisfiable-full-omega-tt intformand_wf intformle_wf itermConstant_wf itermVar_wf intformeq_wf itermAdd_wf int_formula_prop_and_lemma int_formula_prop_le_lemma int_term_value_constant_lemma int_term_value_var_lemma int_formula_prop_eq_lemma int_term_value_add_lemma int_formula_prop_wf sq_stable__regular-int-seq length_wf_nat nat_wf decidable__lt false_wf not-lt-2 condition-implies-le minus-add minus-one-mul zero-add minus-one-mul-top add-commutes add_functionality_wrt_le add-associates add-zero le-add-cancel equal_wf less_than_wf regular-int-seq_wf nat_plus_wf reg-seq-list-add-as-l_sum subtype_rel_list iff_weakening_equal l_sum_functionality_wrt_permutation map_wf permutation-map
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut extract_by_obid sqequalHypSubstitution dependent_functionElimination thin isectElimination hypothesis hypothesisEquality natural_numberEquality unionElimination sqequalRule isect_memberEquality axiomEquality because_Cache equalityTransitivity equalitySymmetry applyEquality lambdaEquality imageElimination independent_isectElimination promote_hyp hypothesis_subsumption productElimination voidElimination voidEquality dependent_pairFormation int_eqEquality intEquality independent_pairFormation computeAll imageMemberEquality baseClosed dependent_set_memberEquality lambdaFormation setElimination rename independent_functionElimination addEquality minusEquality universeEquality functionEquality

Latex:
\mforall{}[L,L':\mBbbR{}  List].    reg-seq-list-add(L)  =  reg-seq-list-add(L')  supposing  permutation(\mBbbR{};L;L')

Date html generated: 2017_10_02-PM-07_13_58
Last ObjectModification: 2017_07_28-AM-07_20_07

Theory : reals

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