### Nuprl Lemma : find-xover_wf

`∀[x:ℤ]. ∀[n:{x...}]. ∀[step:ℕ+]. ∀[f:{x...} ⟶ 𝔹].`
`  find-xover(f;x;n;step) ∈ n':{n':ℤ| (n ≤ n') ∧ f n' = tt}  × {x':ℤ| `
`                                  ((n' = n ∈ ℤ) ∧ (x' = x ∈ ℤ))`
`                                  ∨ (((n ≤ x') ∧ f x' = ff) ∧ ((n' = (n + step) ∈ ℤ) ∨ ((n + step) ≤ x')))}  `
`  supposing ∃m:{n...}. ∀k:{m...}. f k = tt`

Proof

Definitions occuring in Statement :  find-xover: `find-xover(f;m;n;step)` int_upper: `{i...}` nat_plus: `ℕ+` bfalse: `ff` btrue: `tt` bool: `𝔹` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` le: `A ≤ B` all: `∀x:A. B[x]` exists: `∃x:A. B[x]` or: `P ∨ Q` and: `P ∧ Q` member: `t ∈ T` set: `{x:A| B[x]} ` apply: `f a` function: `x:A ⟶ B[x]` product: `x:A × B[x]` add: `n + m` int: `ℤ` equal: `s = t ∈ T`
Definitions unfolded in proof :  exists: `∃x:A. B[x]` all: `∀x:A. B[x]` member: `t ∈ T` nat: `ℕ` uall: `∀[x:A]. B[x]` int_upper: `{i...}` guard: `{T}` nat_plus: `ℕ+` decidable: `Dec(P)` or: `P ∨ Q` uimplies: `b supposing a` satisfiable_int_formula: `satisfiable_int_formula(fmla)` false: `False` implies: `P `` Q` not: `¬A` top: `Top` and: `P ∧ Q` prop: `ℙ` int_seg: `{i..j-}` lelt: `i ≤ j < k` so_lambda: `λ2x.t[x]` subtype_rel: `A ⊆r B` so_apply: `x[s]` ge: `i ≥ j ` le: `A ≤ B` less_than': `less_than'(a;b)` less_than: `a < b` find-xover: `find-xover(f;m;n;step)` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` uiff: `uiff(P;Q)` ifthenelse: `if b then t else f fi ` cand: `A c∧ B` assert: `↑b` iff: `P `⇐⇒` Q` true: `True` rev_implies: `P `` Q` sq_stable: `SqStable(P)` squash: `↓T` bfalse: `ff` sq_type: `SQType(T)` bnot: `¬bb` has-value: `(a)↓` subtract: `n - m`
Lemmas referenced :  subtract_wf int_upper_properties nat_plus_properties decidable__le satisfiable-full-omega-tt intformand_wf intformnot_wf intformle_wf itermConstant_wf itermAdd_wf itermSubtract_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_add_lemma int_term_value_subtract_lemma int_term_value_var_lemma int_formula_prop_wf le_wf decidable__lt intformless_wf int_formula_prop_less_lemma lelt_wf all_wf int_upper_wf equal-wf-T-base int_upper_subtype_int_upper int_seg_properties exists_wf bool_wf nat_plus_wf nat_properties ge_wf less_than_wf int_seg_wf less_than_transitivity1 less_than_irreflexivity decidable__equal_int int_seg_subtype false_wf intformeq_wf int_formula_prop_eq_lemma nat_wf eqtt_to_assert iff_imp_equal_bool btrue_wf assert_wf true_wf or_wf equal_wf member_wf equal-wf-base sq_stable__and sq_stable__le sq_stable__equal squash_wf equal-wf-base-T int_subtype_base eqff_to_assert bool_cases_sqequal subtype_base_sq bool_subtype_base assert-bnot value-type-has-value mul_nat_plus subtype_rel_dep_function subtype_rel_self iff_weakening_equal not_assert_elim btrue_neq_bfalse bfalse_wf assert_elim
Rules used in proof :  cut sqequalHypSubstitution sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity productElimination thin hypothesis dependent_functionElimination dependent_set_memberEquality addEquality introduction extract_by_obid isectElimination setElimination rename because_Cache natural_numberEquality hypothesisEquality unionElimination independent_isectElimination dependent_pairFormation lambdaEquality int_eqEquality intEquality isect_memberEquality voidElimination voidEquality sqequalRule independent_pairFormation computeAll applyEquality applyLambdaEquality baseClosed equalityTransitivity equalitySymmetry functionEquality isect_memberFormation axiomEquality lambdaFormation intWeakElimination independent_functionElimination hypothesis_subsumption functionExtensionality equalityElimination dependent_pairEquality productEquality inlFormation imageMemberEquality imageElimination setEquality promote_hyp instantiate cumulativity callbyvalueReduce universeEquality multiplyEquality inrFormation addLevel levelHypothesis

Latex:
\mforall{}[x:\mBbbZ{}].  \mforall{}[n:\{x...\}].  \mforall{}[step:\mBbbN{}\msupplus{}].  \mforall{}[f:\{x...\}  {}\mrightarrow{}  \mBbbB{}].
find-xover(f;x;n;step)  \mmember{}  n':\{n':\mBbbZ{}|  (n  \mleq{}  n')  \mwedge{}  f  n'  =  tt\}    \mtimes{}  \{x':\mBbbZ{}|
((n'  =  n)  \mwedge{}  (x'  =  x))
\mvee{}  (((n  \mleq{}  x')  \mwedge{}  f  x'  =  ff)
\mwedge{}  ((n'  =  (n  +  step))  \mvee{}  ((n  +  step)  \mleq{}  x')))\}
supposing  \mexists{}m:\{n...\}.  \mforall{}k:\{m...\}.  f  k  =  tt

Date html generated: 2017_04_14-AM-09_17_49
Last ObjectModification: 2017_02_27-PM-03_55_27

Theory : int_2

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