### Nuprl Lemma : gamma-neighbourhood-prop3

`∀beta:ℕ ⟶ ℕ. ∀n,m:ℕ.`
`  (((beta 0) = 0 ∈ ℤ)`
`  `` (↑isl(gamma-neighbourhood(beta;0s^(n)) 0s^(m)))`
`  `` (n < m ∧ ((gamma-neighbourhood(beta;0s^(n)) 0s^(m)) = (inl 0) ∈ (ℕ?))))`

Proof

Definitions occuring in Statement :  gamma-neighbourhood: `gamma-neighbourhood(beta;n0)` mk-finite-nat-seq: `f^(n)` zero-seq: `0s` nat: `ℕ` assert: `↑b` isl: `isl(x)` less_than: `a < b` all: `∀x:A. B[x]` implies: `P `` Q` and: `P ∧ Q` unit: `Unit` apply: `f a` function: `x:A ⟶ B[x]` inl: `inl x` union: `left + right` natural_number: `\$n` int: `ℤ` equal: `s = t ∈ T`
Definitions unfolded in proof :  all: `∀x:A. B[x]` implies: `P `` Q` uall: `∀[x:A]. B[x]` member: `t ∈ T` subtype_rel: `A ⊆r B` nat: `ℕ` uimplies: `b supposing a` le: `A ≤ B` and: `P ∧ Q` less_than': `less_than'(a;b)` false: `False` not: `¬A` isl: `isl(x)` decidable: `Dec(P)` or: `P ∨ Q` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` top: `Top` prop: `ℙ` so_lambda: `λ2x.t[x]` so_apply: `x[s]` gamma-neighbourhood: `gamma-neighbourhood(beta;n0)` exposed-bfalse: `exposed-bfalse` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` uiff: `uiff(P;Q)` ifthenelse: `if b then t else f fi ` assert: `↑b` bfalse: `ff` sq_type: `SQType(T)` guard: `{T}` bnot: `¬bb` ge: `i ≥ j ` iff: `P `⇐⇒` Q` zero-seq: `0s` mk-finite-nat-seq: `f^(n)` append-finite-nat-seq: `f**g` pi1: `fst(t)` pi2: `snd(t)` int_seg: `{i..j-}` lelt: `i ≤ j < k` less_than: `a < b` true: `True` squash: `↓T` cand: `A c∧ B` rev_implies: `P `` Q` finite-nat-seq: `finite-nat-seq()`
Lemmas referenced :  istype-assert gamma-neighbourhood_wf mk-finite-nat-seq_wf zero-seq_wf subtype_rel_function nat_wf int_seg_wf int_seg_subtype_nat istype-false subtype_rel_self btrue_wf bfalse_wf istype-int decidable__le full-omega-unsat intformnot_wf intformle_wf itermConstant_wf int_formula_prop_not_lemma istype-void int_formula_prop_le_lemma int_term_value_constant_lemma int_formula_prop_wf istype-le set_subtype_base le_wf int_subtype_base istype-nat init-seg-nat-seq_wf eqtt_to_assert eqff_to_assert bool_cases_sqequal subtype_base_sq bool_wf bool_subtype_base assert-bnot extend-seq1-all-dec finite-nat-seq_wf decidable_wf assert_wf append-finite-nat-seq_wf not_wf equal-wf-base true_wf nat_properties intformand_wf intformeq_wf itermVar_wf int_formula_prop_and_lemma int_formula_prop_eq_lemma int_term_value_var_lemma assert-init-seg-nat-seq2 decidable__lt intformless_wf itermAdd_wf int_formula_prop_less_lemma int_term_value_add_lemma istype-less_than istype-top less_than_anti-reflexive less_than_wf decidable__equal_int unit_wf2
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity Error :lambdaFormation_alt,  cut introduction extract_by_obid sqequalHypSubstitution isectElimination thin applyEquality hypothesisEquality hypothesis because_Cache natural_numberEquality setElimination rename independent_isectElimination sqequalRule independent_pairFormation Error :inhabitedIsType,  unionElimination Error :equalityIstype,  equalityTransitivity equalitySymmetry dependent_functionElimination independent_functionElimination Error :dependent_set_memberEquality_alt,  approximateComputation Error :dependent_pairFormation_alt,  Error :lambdaEquality_alt,  Error :isect_memberEquality_alt,  voidElimination Error :universeIsType,  intEquality baseClosed sqequalBase Error :functionIsType,  equalityElimination productElimination promote_hyp instantiate cumulativity functionEquality productEquality int_eqEquality hyp_replacement applyLambdaEquality addEquality Error :productIsType,  baseApply closedConclusion lessCases Error :isect_memberFormation_alt,  axiomSqEquality Error :isectIsTypeImplies,  imageMemberEquality imageElimination Error :inlEquality_alt,  Error :functionExtensionality_alt

Latex:
\mforall{}beta:\mBbbN{}  {}\mrightarrow{}  \mBbbN{}.  \mforall{}n,m:\mBbbN{}.
(((beta  0)  =  0)
{}\mRightarrow{}  (\muparrow{}isl(gamma-neighbourhood(beta;0s\^{}(n))  0s\^{}(m)))
{}\mRightarrow{}  (n  <  m  \mwedge{}  ((gamma-neighbourhood(beta;0s\^{}(n))  0s\^{}(m))  =  (inl  0))))

Date html generated: 2019_06_20-PM-03_04_12
Last ObjectModification: 2018_12_06-PM-11_34_57

Theory : continuity

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