### Nuprl Lemma : finite-max

`∀[T:Type]. (finite-type(T) `` T `` (∀g:T ⟶ ℤ. ∃x:T. ∀y:T. ((g y) ≤ (g x))))`

Proof

Definitions occuring in Statement :  finite-type: `finite-type(T)` uall: `∀[x:A]. B[x]` le: `A ≤ B` all: `∀x:A. B[x]` exists: `∃x:A. B[x]` implies: `P `` Q` apply: `f a` function: `x:A ⟶ B[x]` int: `ℤ` universe: `Type`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` implies: `P `` Q` all: `∀x:A. B[x]` member: `t ∈ T` iff: `P `⇐⇒` Q` and: `P ∧ Q` exists: `∃x:A. B[x]` or: `P ∨ Q` uimplies: `b supposing a` not: `¬A` false: `False` cons: `[a / b]` top: `Top` nat_plus: `ℕ+` less_than: `a < b` squash: `↓T` less_than': `less_than'(a;b)` true: `True` prop: `ℙ` guard: `{T}` decidable: `Dec(P)` uiff: `uiff(P;Q)` satisfiable_int_formula: `satisfiable_int_formula(fmla)` l_exists: `(∃x∈L. P[x])` int_seg: `{i..j-}` lelt: `i ≤ j < k` le: `A ≤ B` so_lambda: `λ2x.t[x]` so_apply: `x[s]` l_all: `(∀x∈L.P[x])`
Lemmas referenced :  finite-type_wf finite-type-iff-list list-cases length_of_nil_lemma null_nil_lemma btrue_wf member-implies-null-eq-bfalse nil_wf btrue_neq_bfalse product_subtype_list length_of_cons_lemma add_nat_plus length_wf_nat less_than_wf nat_plus_wf nat_plus_properties decidable__lt add-is-int-iff satisfiable-full-omega-tt intformand_wf intformnot_wf intformless_wf itermConstant_wf itermVar_wf itermAdd_wf intformeq_wf int_formula_prop_and_lemma int_formula_prop_not_lemma int_formula_prop_less_lemma int_term_value_constant_lemma int_term_value_var_lemma int_term_value_add_lemma int_formula_prop_eq_lemma int_formula_prop_wf false_wf equal_wf maximal-in-list select_wf int_seg_properties length_wf decidable__le intformle_wf int_formula_prop_le_lemma less_than'_wf all_wf le_wf l_all_iff l_member_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation lambdaFormation functionEquality cumulativity hypothesisEquality intEquality cut introduction extract_by_obid sqequalHypSubstitution isectElimination thin hypothesis universeEquality productElimination independent_functionElimination dependent_functionElimination unionElimination sqequalRule rename because_Cache independent_isectElimination equalityTransitivity equalitySymmetry voidElimination promote_hyp hypothesis_subsumption isect_memberEquality voidEquality dependent_set_memberEquality natural_numberEquality independent_pairFormation imageMemberEquality baseClosed applyLambdaEquality setElimination pointwiseFunctionality baseApply closedConclusion dependent_pairFormation lambdaEquality int_eqEquality computeAll imageElimination independent_pairEquality applyEquality functionExtensionality axiomEquality setEquality

Latex:
\mforall{}[T:Type].  (finite-type(T)  {}\mRightarrow{}  T  {}\mRightarrow{}  (\mforall{}g:T  {}\mrightarrow{}  \mBbbZ{}.  \mexists{}x:T.  \mforall{}y:T.  ((g  y)  \mleq{}  (g  x))))

Date html generated: 2017_04_17-AM-07_45_59
Last ObjectModification: 2017_02_27-PM-04_18_19

Theory : list_1

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