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]))


Definitions occuring in Statement :  uniform-continuity-pi-search: uniform-continuity-pi-search int_seg: {i..j-} nat: decidable: Dec(P) uimplies: supposing a uall: [x:A]. B[x] prop: 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: m natural_number: $n
Definitions unfolded in proof :  member: t ∈ T uall: [x:A]. B[x] nat: so_lambda: λ2x.t[x] so_apply: x[s] subtype_rel: A ⊆B uimplies: supposing a guard: {T} int_seg: {i..j-} ge: i ≥  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:  Q not: ¬A top: Top prop: 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 then else fi  bfalse: ff bnot: ¬bb assert: b cand: c∧ B isl: isl(x) subtract: 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

\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

Home Index