### Nuprl Lemma : mu-ge-property2

`∀n:ℤ`
`  ∀[P:{n...} ⟶ ℙ]`
`    ∀d:∀n:{n...}. Dec(P[n]). {P[mu-ge(d;n)] ∧ (∀[i:{n..mu-ge(d;n)-}]. (¬P[i]))} supposing ∃m:{n...}. P[m]`

Proof

Definitions occuring in Statement :  mu-ge: `mu-ge(f;n)` int_upper: `{i...}` int_seg: `{i..j-}` decidable: `Dec(P)` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` prop: `ℙ` guard: `{T}` so_apply: `x[s]` all: `∀x:A. B[x]` exists: `∃x:A. B[x]` not: `¬A` and: `P ∧ Q` function: `x:A ⟶ B[x]` int: `ℤ`
Definitions unfolded in proof :  all: `∀x:A. B[x]` uall: `∀[x:A]. B[x]` uimplies: `b supposing a` member: `t ∈ T` implies: `P `` Q` decidable: `Dec(P)` or: `P ∨ Q` isl: `isl(x)` assert: `↑b` ifthenelse: `if b then t else f fi ` btrue: `tt` iff: `P `⇐⇒` Q` and: `P ∧ Q` prop: `ℙ` rev_implies: `P `` Q` true: `True` bfalse: `ff` false: `False` not: `¬A` so_apply: `x[s]` subtype_rel: `A ⊆r B` exists: `∃x:A. B[x]` squash: `↓T` top: `Top` guard: `{T}` mu-ge: `mu-ge(f;n)` has-value: `(a)↓` strict4: `strict4(F)` so_lambda: `λ2x.t[x]` so_apply: `x[s1;s2;s3;s4]` so_lambda: `so_lambda(x,y,z,w.t[x; y; z; w])` int_upper: `{i...}`
Lemmas referenced :  int_upper_wf true_wf istype-void subtype_rel_self assert_wf btrue_wf bfalse_wf mu-ge_wf2 subtype_rel_union not_wf top_wf decidable_wf istype-int mu-ge-property is-exception_wf base_wf has-value_wf_base equal_wf lifting-strict-decide int_seg_subtype_upper le_reflexive int_seg_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity Error :lambdaFormation_alt,  Error :isect_memberFormation_alt,  cut Error :lambdaEquality_alt,  because_Cache Error :inhabitedIsType,  hypothesis thin sqequalHypSubstitution unionElimination sqequalRule Error :equalityIsType1,  equalityTransitivity equalitySymmetry hypothesisEquality dependent_functionElimination independent_functionElimination Error :universeIsType,  applyEquality functionExtensionality introduction extract_by_obid isectElimination independent_pairFormation natural_numberEquality voidElimination instantiate universeEquality productElimination Error :dependent_pairFormation_alt,  imageMemberEquality baseClosed imageElimination independent_isectElimination Error :isect_memberEquality_alt,  Error :unionIsType,  Error :productIsType,  Error :functionIsType,  inlFormation exceptionSqequal inrFormation decideExceptionCases closedConclusion baseApply sqleReflexivity unionEquality callbyvalueDecide lambdaFormation voidEquality isect_memberEquality promote_hyp setElimination rename

Latex:
\mforall{}n:\mBbbZ{}
\mforall{}[P:\{n...\}  {}\mrightarrow{}  \mBbbP{}]
\mforall{}d:\mforall{}n:\{n...\}.  Dec(P[n])
\{P[mu-ge(d;n)]  \mwedge{}  (\mforall{}[i:\{n..mu-ge(d;n)\msupminus{}\}].  (\mneg{}P[i]))\}  supposing  \mexists{}m:\{n...\}.  P[m]

Date html generated: 2019_06_20-PM-01_16_47
Last ObjectModification: 2018_10_06-AM-11_22_01

Theory : int_2

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