### Nuprl Lemma : strong-continuity-test-bound-prop1

`∀[T:Type]. ∀[M:n:ℕ ⟶ (ℕn ⟶ T) ⟶ (ℕn?)]. ∀[n:ℕ]. ∀[f:ℕn ⟶ T]. ∀[b:ℕn].`
`  ((↑isl(strong-continuity-test-bound(M;n;f;b))) `` (strong-continuity-test-bound(M;n;f;b) = (inl b) ∈ (ℕn?)))`

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` isl: `isl(x)` uall: `∀[x:A]. B[x]` implies: `P `` Q` unit: `Unit` function: `x:A ⟶ B[x]` inl: `inl x` union: `left + right` natural_number: `\$n` universe: `Type` equal: `s = t ∈ T`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` nat: `ℕ` implies: `P `` Q` false: `False` ge: `i ≥ j ` uimplies: `b supposing a` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` not: `¬A` all: `∀x:A. B[x]` top: `Top` and: `P ∧ Q` prop: `ℙ` le: `A ≤ B` guard: `{T}` int_seg: `{i..j-}` lelt: `i ≤ j < k` decidable: `Dec(P)` or: `P ∨ Q` subtype_rel: `A ⊆r B` so_lambda: `λ2x.t[x]` so_apply: `x[s]` less_than': `less_than'(a;b)` eq_int: `(i =z j)` subtract: `n - m` ifthenelse: `if b then t else f fi ` btrue: `tt` isl: `isl(x)` assert: `↑b` bfalse: `ff` sq_type: `SQType(T)` uiff: `uiff(P;Q)` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` exposed-it: `exposed-it` bool: `𝔹` unit: `Unit` it: `⋅` bnot: `¬bb` nequal: `a ≠ b ∈ T ` less_than: `a < b` squash: `↓T` true: `True`
Lemmas referenced :  nat_properties satisfiable-full-omega-tt intformand_wf intformle_wf itermConstant_wf itermVar_wf intformless_wf int_formula_prop_and_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 ge_wf less_than_wf assert_wf isl_wf int_seg_wf unit_wf2 strong-continuity-test-bound_wf nat_wf int_seg_properties decidable__le subtract_wf intformnot_wf itermSubtract_wf int_formula_prop_not_lemma int_term_value_subtract_lemma le_wf strong-continuity-test-bound-unroll subtype_rel_dep_function subtype_rel_union int_seg_subtype_nat false_wf eq_int_wf bnot_wf not_wf equal-wf-base int_subtype_base lt_int_wf equal-wf-base-T 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 assert_of_lt_int intformeq_wf int_formula_prop_eq_lemma equal_wf bool_cases_sqequal assert-bnot neg_assert_of_eq_int int_seg_subtype decidable__lt lelt_wf iff_weakening_equal
Rules used in proof :  cut thin introduction extract_by_obid sqequalHypSubstitution sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isectElimination hypothesisEquality hypothesis setElimination rename intWeakElimination lambdaFormation natural_numberEquality independent_isectElimination dependent_pairFormation lambdaEquality int_eqEquality intEquality dependent_functionElimination isect_memberEquality voidElimination voidEquality sqequalRule independent_pairFormation computeAll independent_functionElimination axiomEquality productElimination cumulativity functionExtensionality applyEquality because_Cache functionEquality unionElimination dependent_set_memberEquality unionEquality universeEquality isect_memberFormation equalityTransitivity equalitySymmetry baseClosed baseApply closedConclusion instantiate impliesFunctionality equalityElimination promote_hyp inlEquality imageElimination imageMemberEquality

Latex:
\mforall{}[T:Type].  \mforall{}[M:n:\mBbbN{}  {}\mrightarrow{}  (\mBbbN{}n  {}\mrightarrow{}  T)  {}\mrightarrow{}  (\mBbbN{}n?)].  \mforall{}[n:\mBbbN{}].  \mforall{}[f:\mBbbN{}n  {}\mrightarrow{}  T].  \mforall{}[b:\mBbbN{}n].
((\muparrow{}isl(strong-continuity-test-bound(M;n;f;b)))
{}\mRightarrow{}  (strong-continuity-test-bound(M;n;f;b)  =  (inl  b)))

Date html generated: 2017_04_17-AM-10_00_44
Last ObjectModification: 2017_02_27-PM-05_53_24

Theory : continuity

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