### Nuprl Lemma : exact-xover_wf

`∀[n:ℤ]. ∀[f:{n...} ⟶ 𝔹].`
`  exact-xover(f;n) ∈ {x:ℤ| (n ≤ x) ∧ f x = ff ∧ f (x + 1) = tt}  `
`  supposing (∃m:{n...}. ((∀k:{n..m-}. f k = ff) ∧ (∀k:{m...}. f k = tt))) ∧ f n = ff`

Proof

Definitions occuring in Statement :  exact-xover: `exact-xover(f;n)` int_upper: `{i...}` int_seg: `{i..j-}` 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]` and: `P ∧ Q` member: `t ∈ T` set: `{x:A| B[x]} ` apply: `f a` function: `x:A ⟶ B[x]` add: `n + m` natural_number: `\$n` int: `ℤ` equal: `s = t ∈ T`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` uimplies: `b supposing a` and: `P ∧ Q` exists: `∃x:A. B[x]` all: `∀x:A. B[x]` nat: `ℕ` int_upper: `{i...}` decidable: `Dec(P)` or: `P ∨ Q` satisfiable_int_formula: `satisfiable_int_formula(fmla)` false: `False` implies: `P `` Q` not: `¬A` top: `Top` prop: `ℙ` cand: `A c∧ B` int_seg: `{i..j-}` lelt: `i ≤ j < k` so_lambda: `λ2x.t[x]` subtype_rel: `A ⊆r B` so_apply: `x[s]` guard: `{T}` ge: `i ≥ j ` le: `A ≤ B` less_than': `less_than'(a;b)` exact-xover: `exact-xover(f;n)` less_than: `a < b` nat_plus: `ℕ+` squash: `↓T` true: `True` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` uiff: `uiff(P;Q)` bfalse: `ff` sq_type: `SQType(T)` bnot: `¬bb` ifthenelse: `if b then t else f fi ` assert: `↑b` nequal: `a ≠ b ∈ T ` label: `...\$L... t` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` sq_stable: `SqStable(P)`
Lemmas referenced :  subtract_wf int_upper_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_seg_wf equal-wf-T-base subtype_rel_sets int_upper_wf bool_wf int_upper_subtype_int_upper int_seg_properties exists_wf nat_properties ge_wf less_than_wf less_than_transitivity1 less_than_irreflexivity decidable__equal_int int_seg_subtype false_wf intformeq_wf int_formula_prop_eq_lemma nat_wf find-xover_wf or_wf equal-wf-base int_subtype_base equal_wf eq_int_wf eqtt_to_assert assert_of_eq_int eqff_to_assert bool_cases_sqequal subtype_base_sq bool_subtype_base assert-bnot neg_assert_of_eq_int squash_wf true_wf iff_weakening_equal not_wf subtype_rel_dep_function subtype_rel_self sq_stable__and sq_stable__le sq_stable__equal btrue_neq_bfalse intformor_wf int_formula_prop_or_lemma
Rules used in proof :  cut sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction sqequalHypSubstitution productElimination thin hypothesis dependent_functionElimination dependent_set_memberEquality addEquality extract_by_obid isectElimination setElimination rename because_Cache hypothesisEquality natural_numberEquality unionElimination independent_isectElimination dependent_pairFormation lambdaEquality int_eqEquality intEquality isect_memberEquality voidElimination voidEquality sqequalRule independent_pairFormation computeAll productEquality applyEquality setEquality lambdaFormation baseClosed functionExtensionality applyLambdaEquality equalityTransitivity equalitySymmetry axiomEquality functionEquality intWeakElimination independent_functionElimination hypothesis_subsumption imageMemberEquality baseApply closedConclusion equalityElimination int_eqReduceTrueSq promote_hyp instantiate cumulativity int_eqReduceFalseSq imageElimination universeEquality equalityUniverse levelHypothesis independent_pairEquality

Latex:
\mforall{}[n:\mBbbZ{}].  \mforall{}[f:\{n...\}  {}\mrightarrow{}  \mBbbB{}].
exact-xover(f;n)  \mmember{}  \{x:\mBbbZ{}|  (n  \mleq{}  x)  \mwedge{}  f  x  =  ff  \mwedge{}  f  (x  +  1)  =  tt\}
supposing  (\mexists{}m:\{n...\}.  ((\mforall{}k:\{n..m\msupminus{}\}.  f  k  =  ff)  \mwedge{}  (\mforall{}k:\{m...\}.  f  k  =  tt)))  \mwedge{}  f  n  =  ff

Date html generated: 2017_04_14-AM-09_18_02
Last ObjectModification: 2017_02_27-PM-03_55_16

Theory : int_2

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