Nuprl Lemma : nth-better-fibs

`∀n:ℕ. (s-nth(n;better-fibs()) = fib(n) ∈ ℤ)`

Proof

Definitions occuring in Statement :  better-fibs: `better-fibs()` fib: `fib(n)` s-nth: `s-nth(n;s)` nat: `ℕ` all: `∀x:A. B[x]` int: `ℤ` equal: `s = t ∈ T`
Definitions unfolded in proof :  all: `∀x:A. B[x]` better-fibs: `better-fibs()` uall: `∀[x:A]. B[x]` member: `t ∈ T` top: `Top` has-value: `(a)↓` uimplies: `b supposing a` nat: `ℕ` ge: `i ≥ j ` decidable: `Dec(P)` or: `P ∨ Q` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` false: `False` implies: `P `` Q` not: `¬A` and: `P ∧ Q` prop: `ℙ` le: `A ≤ B` less_than': `less_than'(a;b)` so_lambda: `λ2x.t[x]` so_apply: `x[s]` subtype_rel: `A ⊆r B` fib: `fib(n)` eq_int: `(i =z j)` subtract: `n - m` ifthenelse: `if b then t else f fi ` btrue: `tt` bor: `p ∨bq` bfalse: `ff` s-nth: `s-nth(n;s)` mk-stream: `mk-stream(f;x)` bool: `𝔹` unit: `Unit` it: `⋅` uiff: `uiff(P;Q)` squash: `↓T` true: `True` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` sq_type: `SQType(T)` guard: `{T}` bnot: `¬bb` assert: `↑b` nequal: `a ≠ b ∈ T ` pi1: `fst(t)` callbyvalueall: callbyvalueall has-valueall: `has-valueall(a)`
Lemmas referenced :  nth-stream-map nat_wf mk-stream_wf value-type-has-value int-value-type nat_properties decidable__le satisfiable-full-omega-tt intformand_wf intformnot_wf intformle_wf itermConstant_wf itermAdd_wf itermVar_wf int_formula_prop_and_lemma int_formula_prop_not_lemma int_formula_prop_le_lemma int_term_value_constant_lemma int_term_value_add_lemma int_term_value_var_lemma int_formula_prop_wf le_wf false_wf product-valueall-type set-valueall-type int-valueall-type stream-subtype top_wf testxxx_lemma intformless_wf int_formula_prop_less_lemma ge_wf less_than_wf eq_int_wf bool_wf eqtt_to_assert assert_of_eq_int fib_wf squash_wf true_wf add-zero add-commutes add-associates zero-add not-le-2 condition-implies-le minus-add minus-one-mul minus-one-mul-top add-swap add_functionality_wrt_le le-add-cancel eqff_to_assert equal_wf bool_cases_sqequal subtype_base_sq bool_subtype_base assert-bnot neg_assert_of_eq_int intformeq_wf int_formula_prop_eq_lemma subtract_wf itermSubtract_wf int_term_value_subtract_lemma product_subtype_base int_subtype_base set_subtype_base decidable__equal_int valueall-type-has-valueall evalall-reduce iff_weakening_equal
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation cut sqequalRule introduction extract_by_obid sqequalHypSubstitution isectElimination thin isect_memberEquality voidElimination voidEquality hypothesisEquality hypothesis productEquality because_Cache lambdaEquality productElimination callbyvalueReduce intEquality independent_isectElimination addEquality setElimination rename independent_pairEquality dependent_set_memberEquality dependent_functionElimination natural_numberEquality unionElimination dependent_pairFormation int_eqEquality independent_pairFormation computeAll independent_functionElimination applyEquality intWeakElimination axiomEquality equalityElimination sqleReflexivity imageElimination equalityTransitivity equalitySymmetry imageMemberEquality baseClosed minusEquality promote_hyp instantiate cumulativity universeEquality

Latex:
\mforall{}n:\mBbbN{}.  (s-nth(n;better-fibs())  =  fib(n))

Date html generated: 2017_04_17-AM-09_49_23
Last ObjectModification: 2017_02_27-PM-05_45_45

Theory : num_thy_1

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