### Nuprl Lemma : less-fast-fib

`∀n:ℕ. {m:ℕ| m = fib(n) ∈ ℕ} `

Proof

Definitions occuring in Statement :  fib: `fib(n)` nat: `ℕ` all: `∀x:A. B[x]` set: `{x:A| B[x]} ` equal: `s = t ∈ T`
Definitions unfolded in proof :  all: `∀x:A. B[x]` member: `t ∈ T` implies: `P `` Q` subtype_rel: `A ⊆r B` nat: `ℕ` uall: `∀[x:A]. B[x]` so_lambda: `λ2x.t[x]` so_apply: `x[s]` uimplies: `b supposing a` prop: `ℙ` ge: `i ≥ j ` decidable: `Dec(P)` or: `P ∨ Q` not: `¬A` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` false: `False` top: `Top` and: `P ∧ Q` guard: `{T}` le: `A ≤ B` subtract: `n - m` squash: `↓T` true: `True` sq_type: `SQType(T)` fib: `fib(n)` eq_int: `(i =z j)` ifthenelse: `if b then t else f fi ` btrue: `tt` bor: `p ∨bq` bfalse: `ff` less_than': `less_than'(a;b)` nequal: `a ≠ b ∈ T ` int_upper: `{i...}` bool: `𝔹` unit: `Unit` it: `⋅` uiff: `uiff(P;Q)` bnot: `¬bb` assert: `↑b` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` less_than: `a < b`
Lemmas referenced :  nat_wf set_subtype_base le_wf int_subtype_base istype-less_than istype-int primrec-wf2 all_wf set_wf equal-wf-base less_than_wf nat_properties decidable__le full-omega-unsat intformand_wf intformnot_wf intformle_wf itermConstant_wf itermVar_wf int_formula_prop_and_lemma istype-void int_formula_prop_not_lemma int_formula_prop_le_lemma int_term_value_constant_lemma int_term_value_var_lemma int_formula_prop_wf zero-add itermAdd_wf int_term_value_add_lemma add-subtract-cancel decidable__equal_int intformeq_wf int_formula_prop_eq_lemma add-associates add-swap add-commutes fib_wf squash_wf true_wf intformless_wf int_formula_prop_less_lemma subtype_base_sq testxxx_lemma upper_subtype_nat istype-false nequal-le-implies eq_int_wf eqtt_to_assert assert_of_eq_int int_upper_properties eqff_to_assert bool_subtype_base bool_cases_sqequal bool_wf assert-bnot neg_assert_of_eq_int equal_wf istype-universe add-comm subtract_wf itermSubtract_wf int_term_value_subtract_lemma subtype_rel_self iff_weakening_equal add_functionality_wrt_eq decidable__lt equal-wf-base-T false_wf subtype_rel_sets add-zero
Rules used in proof :  cut sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity Error :lambdaFormation_alt,  thin Error :inhabitedIsType,  hypothesisEquality Error :universeIsType,  because_Cache rename setElimination sqequalRule Error :functionIsType,  introduction extract_by_obid hypothesis Error :setIsType,  Error :equalityIsType4,  baseApply closedConclusion baseClosed applyEquality sqequalHypSubstitution isectElimination intEquality Error :lambdaEquality_alt,  natural_numberEquality independent_isectElimination equalityTransitivity equalitySymmetry functionEquality Error :dependent_set_memberEquality_alt,  dependent_functionElimination unionElimination approximateComputation independent_functionElimination Error :dependent_pairFormation_alt,  int_eqEquality Error :isect_memberEquality_alt,  voidElimination independent_pairFormation addEquality Error :dependent_set_memberFormation_alt,  applyLambdaEquality productElimination imageElimination imageMemberEquality instantiate cumulativity hypothesis_subsumption equalityElimination promote_hyp Error :equalityIsType1,  universeEquality minusEquality lambdaEquality setEquality voidEquality isect_memberEquality dependent_pairFormation lambdaFormation dependent_set_memberEquality functionExtensionality

Latex:
\mforall{}n:\mBbbN{}.  \{m:\mBbbN{}|  m  =  fib(n)\}

Date html generated: 2019_06_20-PM-02_25_23
Last ObjectModification: 2018_10_17-AM-10_43_29

Theory : num_thy_1

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