### Nuprl Lemma : uniform-continuity-pi-search-prop1

`∀[n:ℕ]. ∀[P:ℕ ⟶ ℙ]. ∀[G:∀m:ℕn. Dec(P[m])]. ∀[x:ℕ].`
`  (uniform-continuity-pi-search(G;n;x) ∈ {k:{x..n + 1-}| P[k] ∧ (∀m:{x..k-}. (¬P[m]))} ) supposing ((x ≤ n) and (∃n:{x..\000Cn + 1-}. P[n]))`

Proof

Definitions occuring in Statement :  uniform-continuity-pi-search: uniform-continuity-pi-search int_seg: `{i..j-}` nat: `ℕ` decidable: `Dec(P)` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` prop: `ℙ` so_apply: `x[s]` le: `A ≤ B` all: `∀x:A. B[x]` exists: `∃x:A. B[x]` not: `¬A` and: `P ∧ Q` member: `t ∈ T` set: `{x:A| B[x]} ` function: `x:A ⟶ B[x]` add: `n + m` natural_number: `\$n`
Definitions unfolded in proof :  member: `t ∈ T` uall: `∀[x:A]. B[x]` nat: `ℕ` so_lambda: `λ2x.t[x]` so_apply: `x[s]` subtype_rel: `A ⊆r B` uimplies: `b supposing a` guard: `{T}` int_seg: `{i..j-}` ge: `i ≥ j ` lelt: `i ≤ j < k` and: `P ∧ Q` all: `∀x:A. B[x]` decidable: `Dec(P)` or: `P ∨ Q` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` false: `False` implies: `P `` Q` not: `¬A` top: `Top` prop: `ℙ` le: `A ≤ B` less_than': `less_than'(a;b)` sq_type: `SQType(T)` uniform-continuity-pi-search: uniform-continuity-pi-search bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` uiff: `uiff(P;Q)` ifthenelse: `if b then t else f fi ` bfalse: `ff` bnot: `¬bb` assert: `↑b` cand: `A c∧ B` isl: `isl(x)` subtract: `n - m` squash: `↓T`
Lemmas referenced :  le_wf exists_wf int_seg_wf nat_wf int_seg_subtype_nat int_seg_properties nat_properties decidable__le satisfiable-full-omega-tt intformand_wf intformnot_wf intformle_wf itermConstant_wf itermVar_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_wf all_wf decidable_wf false_wf subtract_wf itermSubtract_wf int_term_value_subtract_lemma decidable__equal_int intformeq_wf itermAdd_wf int_formula_prop_eq_lemma int_term_value_add_lemma equal_wf subtype_base_sq int_subtype_base intformless_wf int_formula_prop_less_lemma ge_wf less_than_wf less_than_transitivity1 less_than_irreflexivity le_int_wf bool_wf eqtt_to_assert assert_of_le_int eqff_to_assert bool_cases_sqequal bool_subtype_base assert-bnot add-zero decidable__lt lelt_wf not_wf add-commutes add-associates add-swap zero-add int_seg_subtype
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity cut introduction extract_by_obid sqequalHypSubstitution isectElimination thin setElimination rename hypothesisEquality hypothesis because_Cache addEquality natural_numberEquality sqequalRule lambdaEquality applyEquality functionExtensionality independent_isectElimination applyLambdaEquality productElimination dependent_functionElimination unionElimination dependent_pairFormation int_eqEquality intEquality isect_memberEquality voidElimination voidEquality independent_pairFormation computeAll lambdaFormation functionEquality cumulativity universeEquality isect_memberFormation axiomEquality equalityTransitivity equalitySymmetry dependent_set_memberEquality instantiate independent_functionElimination intWeakElimination equalityElimination promote_hyp productEquality imageMemberEquality baseClosed imageElimination hyp_replacement

Latex:
\mforall{}[n:\mBbbN{}].  \mforall{}[P:\mBbbN{}  {}\mrightarrow{}  \mBbbP{}].  \mforall{}[G:\mforall{}m:\mBbbN{}n.  Dec(P[m])].  \mforall{}[x:\mBbbN{}].
(uniform-continuity-pi-search(G;n;x)  \mmember{}  \{k:\{x..n  +  1\msupminus{}\}|  P[k]  \mwedge{}  (\mforall{}m:\{x..k\msupminus{}\}.  (\mneg{}P[m]))\}  )  supposing  (\000C(x  \mleq{}  n)  and  (\mexists{}n:\{x..n  +  1\msupminus{}\}.  P[n]))

Date html generated: 2017_04_17-AM-09_58_47
Last ObjectModification: 2017_02_27-PM-05_52_42

Theory : continuity

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