### Nuprl Lemma : finite-nat-seq-to-list-prop1

`∀[f:finite-nat-seq()]`
`  ((||finite-nat-seq-to-list(f)|| = (fst(f)) ∈ ℕ) ∧ (∀i:ℕfst(f). (finite-nat-seq-to-list(f)[i] = ((snd(f)) i) ∈ ℕ)))`

Proof

Definitions occuring in Statement :  finite-nat-seq-to-list: `finite-nat-seq-to-list(f)` finite-nat-seq: `finite-nat-seq()` select: `L[n]` length: `||as||` int_seg: `{i..j-}` nat: `ℕ` uall: `∀[x:A]. B[x]` pi1: `fst(t)` pi2: `snd(t)` all: `∀x:A. B[x]` and: `P ∧ Q` apply: `f a` natural_number: `\$n` equal: `s = t ∈ T`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` finite-nat-seq: `finite-nat-seq()` finite-nat-seq-to-list: `finite-nat-seq-to-list(f)` pi1: `fst(t)` pi2: `snd(t)` nat: `ℕ` implies: `P `` Q` false: `False` ge: `i ≥ j ` uimplies: `b supposing a` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` not: `¬A` all: `∀x:A. B[x]` top: `Top` and: `P ∧ Q` prop: `ℙ` select: `L[n]` nil: `[]` it: `⋅` so_lambda: `λ2x y.t[x; y]` so_apply: `x[s1;s2]` decidable: `Dec(P)` or: `P ∨ Q` cand: `A c∧ B` le: `A ≤ B` less_than': `less_than'(a;b)` guard: `{T}` int_seg: `{i..j-}` lelt: `i ≤ j < k` subtype_rel: `A ⊆r B` so_lambda: `λ2x.t[x]` so_apply: `x[s]` bool: `𝔹` unit: `Unit` btrue: `tt` uiff: `uiff(P;Q)` ifthenelse: `if b then t else f fi ` bfalse: `ff` sq_type: `SQType(T)` bnot: `¬bb` assert: `↑b` nequal: `a ≠ b ∈ T ` squash: `↓T` less_than: `a < b` true: `True` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` cons: `[a / b]`
Lemmas referenced :  nat_properties satisfiable-full-omega-tt intformand_wf intformle_wf itermConstant_wf itermVar_wf intformless_wf 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 less_than_wf int_seg_wf nat_wf primrec0_lemma length_of_nil_lemma stuck-spread base_wf decidable__le subtract_wf intformnot_wf itermSubtract_wf int_formula_prop_not_lemma int_term_value_subtract_lemma finite-nat-seq_wf decidable__equal_int intformeq_wf int_formula_prop_eq_lemma false_wf le_wf int_seg_properties lelt_wf subtype_rel_dep_function int_seg_subtype subtype_rel_self primrec-unroll eq_int_wf bool_wf eqtt_to_assert assert_of_eq_int eqff_to_assert equal_wf bool_cases_sqequal subtype_base_sq bool_subtype_base assert-bnot neg_assert_of_eq_int length-append length_of_cons_lemma itermAdd_wf int_term_value_add_lemma decidable__lt squash_wf true_wf select_append_front primrec_wf list_wf nil_wf append_wf cons_wf iff_weakening_equal non_neg_length select_wf length_wf_nat set_subtype_base int_subtype_base select_append_back length-singleton length_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut sqequalHypSubstitution productElimination thin sqequalRule extract_by_obid isectElimination hypothesisEquality hypothesis setElimination rename intWeakElimination lambdaFormation natural_numberEquality independent_isectElimination dependent_pairFormation lambdaEquality int_eqEquality intEquality dependent_functionElimination isect_memberEquality voidElimination voidEquality independent_pairFormation computeAll independent_functionElimination independent_pairEquality axiomEquality functionEquality baseClosed unionElimination because_Cache dependent_set_memberEquality equalityTransitivity equalitySymmetry applyLambdaEquality applyEquality functionExtensionality equalityElimination promote_hyp instantiate cumulativity addEquality imageElimination universeEquality imageMemberEquality minusEquality productEquality

Latex:
\mforall{}[f:finite-nat-seq()]
((||finite-nat-seq-to-list(f)||  =  (fst(f)))
\mwedge{}  (\mforall{}i:\mBbbN{}fst(f).  (finite-nat-seq-to-list(f)[i]  =  ((snd(f))  i))))

Date html generated: 2017_04_20-AM-07_29_14
Last ObjectModification: 2017_02_27-PM-06_01_37

Theory : continuity

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