### Nuprl Lemma : decidable-exists-int_seg-subtype

`∀[i:ℤ]. ∀[j:{i + 1...}]. ∀[P:{i..j-} ⟶ ℙ].  Dec(∃k:{i + 1..j-}. P[k]) ⊆r Dec(∃k:{i..j-}. P[k]) supposing ¬P[i]`

Proof

Definitions occuring in Statement :  int_upper: `{i...}` int_seg: `{i..j-}` decidable: `Dec(P)` uimplies: `b supposing a` subtype_rel: `A ⊆r B` uall: `∀[x:A]. B[x]` prop: `ℙ` so_apply: `x[s]` exists: `∃x:A. B[x]` not: `¬A` 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` uimplies: `b supposing a` int_upper: `{i...}` so_lambda: `λ2x.t[x]` so_apply: `x[s]` subtype_rel: `A ⊆r B` exists: `∃x:A. B[x]` int_seg: `{i..j-}` lelt: `i ≤ j < k` and: `P ∧ Q` le: `A ≤ B` all: `∀x:A. B[x]` decidable: `Dec(P)` or: `P ∨ Q` iff: `P `⇐⇒` Q` not: `¬A` rev_implies: `P `` Q` implies: `P `` Q` false: `False` prop: `ℙ` uiff: `uiff(P;Q)` subtract: `n - m` less_than': `less_than'(a;b)` true: `True` top: `Top` sq_type: `SQType(T)` guard: `{T}`
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut lemma_by_obid sqequalHypSubstitution isectElimination thin addEquality hypothesisEquality natural_numberEquality setElimination rename hypothesis sqequalRule lambdaEquality applyEquality because_Cache independent_isectElimination productElimination dependent_pairEquality dependent_set_memberEquality independent_pairFormation dependent_functionElimination unionElimination lambdaFormation voidElimination independent_functionElimination minusEquality dependent_pairFormation axiomEquality isect_memberEquality equalityTransitivity equalitySymmetry functionEquality cumulativity universeEquality voidEquality intEquality instantiate

Latex:
\mforall{}[i:\mBbbZ{}].  \mforall{}[j:\{i  +  1...\}].  \mforall{}[P:\{i..j\msupminus{}\}  {}\mrightarrow{}  \mBbbP{}].
Dec(\mexists{}k:\{i  +  1..j\msupminus{}\}.  P[k])  \msubseteq{}r  Dec(\mexists{}k:\{i..j\msupminus{}\}.  P[k])  supposing  \mneg{}P[i]

Date html generated: 2016_05_13-PM-03_47_38
Last ObjectModification: 2015_12_26-AM-09_58_39

Theory : call!by!value_2

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