### Nuprl Lemma : next_wf_bound

`∀[b:ℕ]. ∀[k:ℤ]. ∀[p:{i:ℤ| k < i}  ⟶ 𝔹].`
`  (next i > k s.t. ↑p[i]) ∈ {i:ℤ| (k < i ∧ (i ≤ (k + b))) ∧ (↑p[i]) ∧ (∀j:{k + 1..i-}. (¬↑p[j]))}  `
`  supposing ∃n:{i:ℤ| k < i ∧ (i ≤ (k + b))} . (↑p[n])`

Proof

Definitions occuring in Statement :  next: `(next i > k s.t. ↑p[i])` int_seg: `{i..j-}` nat: `ℕ` assert: `↑b` bool: `𝔹` less_than: `a < b` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` 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` int: `ℤ`
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` so_lambda: `λ2x.t[x]` so_apply: `x[s]` decidable: `Dec(P)` or: `P ∨ Q` next: `(next i > k s.t. ↑p[i])` has-value: `(a)↓` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` uiff: `uiff(P;Q)` ifthenelse: `if b then t else f fi ` bfalse: `ff` cand: `A c∧ B` guard: `{T}` int_seg: `{i..j-}` lelt: `i ≤ j < k` subtype_rel: `A ⊆r B` le: `A ≤ B` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` subtract: `n - m` less_than': `less_than'(a;b)` true: `True` sq_type: `SQType(T)` sq_stable: `SqStable(P)` squash: `↓T` label: `...\$L... t` less_than: `a < b`
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 exists_wf le_wf assert_wf bool_wf decidable__le subtract_wf intformnot_wf itermSubtract_wf int_formula_prop_not_lemma int_term_value_subtract_lemma nat_wf itermAdd_wf int_term_value_add_lemma decidable__lt equal-wf-T-base bnot_wf not_wf eqtt_to_assert uiff_transitivity eqff_to_assert assert_of_bnot equal_wf value-type-has-value int-value-type int_seg_properties subtype_rel_sets lelt_wf less_than_transitivity1 less_than_irreflexivity int_seg_wf all_wf false_wf not-lt-2 condition-implies-le minus-add minus-one-mul add-swap minus-one-mul-top add-commutes add_functionality_wrt_le le-add-cancel subtype_rel_dep_function less-iff-le add-associates subtype_rel_self set_wf add-is-int-iff int_subtype_base decidable__equal_int assert_elim subtype_base_sq not_assert_elim btrue_neq_bfalse intformeq_wf int_formula_prop_eq_lemma sq_stable__and sq_stable__less_than sq_stable__le less_than'_wf squash_wf assert_functionality_wrt_uiff
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut extract_by_obid sqequalHypSubstitution isectElimination thin hypothesisEquality hypothesis setElimination rename sqequalRule intWeakElimination lambdaFormation natural_numberEquality independent_isectElimination dependent_pairFormation lambdaEquality int_eqEquality intEquality dependent_functionElimination isect_memberEquality voidElimination voidEquality independent_pairFormation computeAll independent_functionElimination axiomEquality equalityTransitivity equalitySymmetry setEquality productEquality addEquality applyEquality functionExtensionality dependent_set_memberEquality productElimination functionEquality unionElimination because_Cache callbyvalueReduce baseClosed equalityElimination minusEquality baseApply closedConclusion instantiate cumulativity independent_pairEquality imageMemberEquality imageElimination addLevel impliesFunctionality levelHypothesis

Latex:
\mforall{}[b:\mBbbN{}].  \mforall{}[k:\mBbbZ{}].  \mforall{}[p:\{i:\mBbbZ{}|  k  <  i\}    {}\mrightarrow{}  \mBbbB{}].
(next  i  >  k  s.t.  \muparrow{}p[i])  \mmember{}  \{i:\mBbbZ{}|  (k  <  i  \mwedge{}  (i  \mleq{}  (k  +  b)))  \mwedge{}  (\muparrow{}p[i])  \mwedge{}  (\mforall{}j:\{k  +  1..i\msupminus{}\}.  (\mneg{}\muparrow{}p[j]))\}
supposing  \mexists{}n:\{i:\mBbbZ{}|  k  <  i  \mwedge{}  (i  \mleq{}  (k  +  b))\}  .  (\muparrow{}p[n])

Date html generated: 2018_05_21-PM-06_54_54
Last ObjectModification: 2017_07_26-PM-04_59_16

Theory : general

Home Index