### Nuprl Lemma : uniform-continuity-pi-dec

`∀T:Type. ∀F:(ℕ ⟶ 𝔹) ⟶ T. ∀n:ℕ.  ((∀x,y:T.  Dec(x = y ∈ T)) `` ucA(T;F;n) `` (∀m:ℕ. (m < n `` Dec(ucA(T;F;m)))))`

Proof

Definitions occuring in Statement :  uniform-continuity-pi: `ucA(T;F;n)` nat: `ℕ` bool: `𝔹` less_than: `a < b` decidable: `Dec(P)` all: `∀x:A. B[x]` implies: `P `` Q` function: `x:A ⟶ B[x]` universe: `Type` equal: `s = t ∈ T`
Definitions unfolded in proof :  all: `∀x:A. B[x]` implies: `P `` Q` uall: `∀[x:A]. B[x]` member: `t ∈ T` nat: `ℕ` prop: `ℙ` ge: `i ≥ j ` decidable: `Dec(P)` or: `P ∨ Q` uimplies: `b supposing a` not: `¬A` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` false: `False` top: `Top` and: `P ∧ Q` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` uniform-continuity-pi2: `ucB(T;F;n)` uniform-continuity-pi: `ucA(T;F;n)` subtype_rel: `A ⊆r B` le: `A ≤ B` less_than': `less_than'(a;b)` int_seg: `{i..j-}` sq_type: `SQType(T)` guard: `{T}` lelt: `i ≤ j < k` assert: `↑b` ifthenelse: `if b then t else f fi ` btrue: `tt` true: `True` ext2Cantor: `ext2Cantor(n;f;d)` bool: `𝔹` unit: `Unit` it: `⋅` uiff: `uiff(P;Q)` bfalse: `ff` so_lambda: `λ2x.t[x]` so_apply: `x[s]` bnot: `¬bb` subtract: `n - m` label: `...\$L... t` squash: `↓T`
Lemmas referenced :  istype-less_than uniform-continuity-pi_wf decidable_wf equal_wf istype-nat bool_wf istype-universe nat_properties decidable__le full-omega-unsat intformand_wf intformnot_wf intformle_wf itermConstant_wf itermAdd_wf itermVar_wf istype-int 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_add_lemma int_term_value_var_lemma int_formula_prop_wf istype-le uniform-continuity-pi2_wf int_seg_wf ext2Cantor_wf btrue_wf bfalse_wf eq_ext2Cantor subtype_rel_function nat_wf int_seg_subtype_nat istype-false subtype_rel_self decidable__equal_bool decidable__equal_int subtype_base_sq int_subtype_base decidable__lt intformless_wf intformeq_wf int_formula_prop_less_lemma int_formula_prop_eq_lemma iff_imp_equal_bool assert_elim bool_subtype_base istype-assert bool_cases lt_int_wf eqtt_to_assert assert_of_lt_int eqff_to_assert set_subtype_base le_wf lelt_wf bool_cases_sqequal assert-bnot iff_weakening_uiff assert_wf less_than_wf int_seg_properties decidable_functionality uniform-continuity-pi2-dec-ext subtract_wf itermSubtract_wf int_term_value_subtract_lemma primrec-wf2 all_wf add-associates add-swap add-commutes zero-add squash_wf true_wf iff_weakening_equal int_seg_subtype not-le-2 condition-implies-le minus-one-mul minus-add minus-minus minus-one-mul-top less-iff-le add_functionality_wrt_le le-add-cancel equal_functionality_wrt_subtype_rel2
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity Error :lambdaFormation_alt,  cut introduction extract_by_obid sqequalHypSubstitution isectElimination thin setElimination rename hypothesisEquality hypothesis Error :inhabitedIsType,  Error :universeIsType,  sqequalRule Error :functionIsType,  because_Cache instantiate universeEquality Error :dependent_set_memberEquality_alt,  addEquality natural_numberEquality dependent_functionElimination unionElimination independent_isectElimination approximateComputation independent_functionElimination Error :dependent_pairFormation_alt,  Error :lambdaEquality_alt,  int_eqEquality Error :isect_memberEquality_alt,  voidElimination independent_pairFormation Error :equalityIsType1,  applyEquality functionExtensionality cumulativity intEquality equalityTransitivity equalitySymmetry productElimination Error :productIsType,  applyLambdaEquality equalityElimination Error :equalityIsType4,  baseApply closedConclusion baseClosed promote_hyp Error :setIsType,  functionEquality Error :inlFormation_alt,  imageElimination imageMemberEquality Error :inrFormation_alt,  minusEquality

Latex:
\mforall{}T:Type.  \mforall{}F:(\mBbbN{}  {}\mrightarrow{}  \mBbbB{})  {}\mrightarrow{}  T.  \mforall{}n:\mBbbN{}.
((\mforall{}x,y:T.    Dec(x  =  y))  {}\mRightarrow{}  ucA(T;F;n)  {}\mRightarrow{}  (\mforall{}m:\mBbbN{}.  (m  <  n  {}\mRightarrow{}  Dec(ucA(T;F;m)))))

Date html generated: 2019_06_20-PM-02_53_17
Last ObjectModification: 2018_10_30-PM-02_45_33

Theory : continuity

Home Index