### Nuprl Lemma : imax-list-ub

`∀L:ℤ List. ∀a:ℤ.  a ≤ imax-list(L) `⇐⇒` (∃b∈L. a ≤ b) supposing 0 < ||L||`

Proof

Definitions occuring in Statement :  imax-list: `imax-list(L)` l_exists: `(∃x∈L. P[x])` length: `||as||` list: `T List` less_than: `a < b` uimplies: `b supposing a` le: `A ≤ B` all: `∀x:A. B[x]` iff: `P `⇐⇒` Q` natural_number: `\$n` int: `ℤ`
Definitions unfolded in proof :  all: `∀x:A. B[x]` uimplies: `b supposing a` member: `t ∈ T` uall: `∀[x:A]. B[x]` so_lambda: `λ2x y.t[x; y]` so_apply: `x[s1;s2]` implies: `P `` Q` iff: `P `⇐⇒` Q` and: `P ∧ Q` prop: `ℙ` rev_implies: `P `` Q` so_lambda: `λ2x.t[x]` so_apply: `x[s]` or: `P ∨ Q` cand: `A c∧ B` assoc: `Assoc(T;op)` infix_ap: `x f y` squash: `↓T` true: `True` subtype_rel: `A ⊆r B` guard: `{T}` imax-list: `imax-list(L)`
Lemmas referenced :  member-less_than length_wf combine-list-rel-or imax_wf le_wf imax_ub all_wf iff_wf or_wf equal_wf squash_wf true_wf imax_assoc iff_weakening_equal less_than_wf list_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation isect_memberFormation cut introduction extract_by_obid sqequalHypSubstitution isectElimination thin natural_numberEquality intEquality hypothesisEquality hypothesis independent_isectElimination rename because_Cache dependent_functionElimination sqequalRule lambdaEquality independent_functionElimination independent_pairFormation addLevel allFunctionality productElimination impliesFunctionality applyEquality imageElimination equalityTransitivity equalitySymmetry universeEquality imageMemberEquality baseClosed isect_memberEquality axiomEquality

Latex:
\mforall{}L:\mBbbZ{}  List.  \mforall{}a:\mBbbZ{}.    a  \mleq{}  imax-list(L)  \mLeftarrow{}{}\mRightarrow{}  (\mexists{}b\mmember{}L.  a  \mleq{}  b)  supposing  0  <  ||L||

Date html generated: 2017_04_14-AM-09_23_47
Last ObjectModification: 2017_02_27-PM-03_58_27

Theory : list_1

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