### Nuprl Lemma : strong-continuity-test-bound-prop4

`∀M:n:ℕ ⟶ (ℕn ⟶ ℕ) ⟶ (ℕn?). ∀n:ℕ. ∀f:ℕn ⟶ ℕ. ∀b:ℕn.`
`  ((↑isr(strong-continuity-test-bound(M;n;f;b)))`
`  `` (∃m:ℕ. (b < m ∧ m < n ∧ (↑isl(M m f)) ∧ (↑isl(strong-continuity-test-bound(M;m;f;b))))))`

Proof

Definitions occuring in Statement :  strong-continuity-test-bound: `strong-continuity-test-bound(M;n;f;b)` int_seg: `{i..j-}` nat: `ℕ` assert: `↑b` isr: `isr(x)` isl: `isl(x)` less_than: `a < b` all: `∀x:A. B[x]` exists: `∃x:A. B[x]` implies: `P `` Q` and: `P ∧ Q` unit: `Unit` apply: `f a` function: `x:A ⟶ B[x]` union: `left + right` natural_number: `\$n`
Definitions unfolded in proof :  all: `∀x:A. B[x]` implies: `P `` Q` uall: `∀[x:A]. B[x]` member: `t ∈ T` guard: `{T}` int_seg: `{i..j-}` lelt: `i ≤ j < k` and: `P ∧ Q` uimplies: `b supposing a` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` false: `False` not: `¬A` top: `Top` prop: `ℙ` nat: `ℕ` decidable: `Dec(P)` or: `P ∨ Q` so_lambda: `λ2x.t[x]` subtype_rel: `A ⊆r B` so_apply: `x[s]` le: `A ≤ B` less_than': `less_than'(a;b)` ge: `i ≥ j ` ifthenelse: `if b then t else f fi ` btrue: `tt` isr: `isr(x)` assert: `↑b` bfalse: `ff` sq_type: `SQType(T)` uiff: `uiff(P;Q)` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` less_than: `a < b` cand: `A c∧ B`
Lemmas referenced :  not-isl-assert-isr decidable__lt decidable__assert assert_of_lt_int iff_weakening_uiff iff_transitivity assert_of_eq_int assert_of_bnot eqff_to_assert eqtt_to_assert bool_subtype_base bool_wf subtype_base_sq bool_cases equal-wf-base-T lt_int_wf int_subtype_base equal-wf-base not_wf bnot_wf int_formula_prop_eq_lemma intformeq_wf eq_int_wf int_seg_subtype_nat subtype_rel_union strong-continuity-test-bound-unroll primrec-wf2 set_wf lelt_wf subtype_rel_self nat_properties false_wf int_seg_subtype subtype_rel_dep_function isl_wf less_than_wf exists_wf int_term_value_subtract_lemma itermSubtract_wf subtract_wf all_wf le_wf int_formula_prop_not_lemma intformnot_wf decidable__le nat_wf strong-continuity-test-bound_wf unit_wf2 int_seg_wf isr_wf assert_wf int_formula_prop_wf int_formula_prop_le_lemma int_term_value_constant_lemma int_term_value_var_lemma int_formula_prop_less_lemma int_formula_prop_and_lemma intformle_wf itermConstant_wf itermVar_wf intformless_wf intformand_wf satisfiable-full-omega-tt int_seg_properties
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation cut thin lemma_by_obid sqequalHypSubstitution isectElimination natural_numberEquality because_Cache hypothesisEquality hypothesis setElimination rename productElimination independent_isectElimination dependent_pairFormation lambdaEquality int_eqEquality intEquality dependent_functionElimination isect_memberEquality voidElimination voidEquality sqequalRule independent_pairFormation computeAll functionExtensionality applyEquality functionEquality dependent_set_memberEquality unionElimination productEquality introduction unionEquality equalityTransitivity equalitySymmetry baseClosed baseApply closedConclusion instantiate cumulativity independent_functionElimination impliesFunctionality

Latex:
\mforall{}M:n:\mBbbN{}  {}\mrightarrow{}  (\mBbbN{}n  {}\mrightarrow{}  \mBbbN{})  {}\mrightarrow{}  (\mBbbN{}n?).  \mforall{}n:\mBbbN{}.  \mforall{}f:\mBbbN{}n  {}\mrightarrow{}  \mBbbN{}.  \mforall{}b:\mBbbN{}n.
((\muparrow{}isr(strong-continuity-test-bound(M;n;f;b)))
{}\mRightarrow{}  (\mexists{}m:\mBbbN{}.  (b  <  m  \mwedge{}  m  <  n  \mwedge{}  (\muparrow{}isl(M  m  f))  \mwedge{}  (\muparrow{}isl(strong-continuity-test-bound(M;m;f;b))))))

Date html generated: 2016_05_19-PM-00_00_03
Last ObjectModification: 2016_05_17-PM-05_52_21

Theory : continuity

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