### Nuprl Lemma : strong-continuity-test-prop2

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

Proof

Definitions occuring in Statement :  strong-continuity-test: `strong-continuity-test(M;n;f;b)` int_seg: `{i..j-}` nat: `ℕ` assert: `↑b` isr: `isr(x)` isl: `isl(x)` less_than: `a < b` uall: `∀[x:A]. B[x]` 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` universe: `Type`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` implies: `P `` Q` member: `t ∈ T` prop: `ℙ` nat: `ℕ` le: `A ≤ B` and: `P ∧ Q` less_than': `less_than'(a;b)` false: `False` not: `¬A` decidable: `Dec(P)` or: `P ∨ Q` uimplies: `b supposing a` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` top: `Top` so_lambda: `λ2x.t[x]` subtype_rel: `A ⊆r B` so_apply: `x[s]` ge: `i ≥ j ` eq_int: `(i =z j)` subtract: `n - m` ifthenelse: `if b then t else f fi ` btrue: `tt` assert: `↑b` bfalse: `ff` sq_type: `SQType(T)` guard: `{T}` uiff: `uiff(P;Q)` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` isr: `isr(x)` cand: `A c∧ B`
Lemmas referenced :  assert_wf isr_wf nat_wf unit_wf2 strong-continuity-test_wf int_seg_wf isl_wf false_wf le_wf decidable__le satisfiable-full-omega-tt intformand_wf intformnot_wf intformle_wf itermConstant_wf itermVar_wf intformless_wf int_formula_prop_and_lemma int_formula_prop_not_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 all_wf subtract_wf itermSubtract_wf int_term_value_subtract_lemma exists_wf less_than_wf subtype_rel_dep_function int_seg_subtype nat_properties set_wf primrec-wf2 strong-continuity-test-unroll isr-not-isl subtype_rel_union top_wf eq_int_wf bnot_wf not_wf equal-wf-base int_subtype_base bool_cases subtype_base_sq bool_wf bool_subtype_base eqtt_to_assert assert_of_eq_int eqff_to_assert iff_transitivity iff_weakening_uiff assert_of_bnot less_than_transitivity1 le_weakening less_than_irreflexivity decidable__assert decidable__lt not-isl-assert-isr
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation lambdaFormation cut introduction extract_by_obid sqequalHypSubstitution isectElimination thin hypothesis cumulativity hypothesisEquality functionExtensionality applyEquality natural_numberEquality setElimination rename unionEquality functionEquality universeEquality because_Cache dependent_set_memberEquality sqequalRule independent_pairFormation dependent_functionElimination unionElimination independent_isectElimination dependent_pairFormation lambdaEquality int_eqEquality intEquality isect_memberEquality voidElimination voidEquality computeAll productEquality independent_functionElimination equalityTransitivity equalitySymmetry baseClosed instantiate productElimination impliesFunctionality

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

Date html generated: 2016_12_12-AM-09_22_17
Last ObjectModification: 2016_11_22-AM-11_48_24

Theory : continuity

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