### Nuprl Lemma : strict-majority-or-max-property

`∀t:ℕ. ∀L:ℤ List.`
`  ((∀v:ℤ`
`      (strict-majority-or-max(L) = v ∈ ℤ) supposing `
`         (((t + 1) ≤ ||filter(λx.(x =z v);L)||) and `
`         (||L|| = ((2 * t) + 1) ∈ ℤ)))`
`  ∧ (strict-majority-or-max(L) ∈ L) supposing ||L|| ≥ 1 )`

Proof

Definitions occuring in Statement :  strict-majority-or-max: `strict-majority-or-max(L)` l_member: `(x ∈ l)` length: `||as||` filter: `filter(P;l)` list: `T List` nat: `ℕ` eq_int: `(i =z j)` uimplies: `b supposing a` ge: `i ≥ j ` le: `A ≤ B` all: `∀x:A. B[x]` and: `P ∧ Q` lambda: `λx.A[x]` multiply: `n * m` add: `n + m` natural_number: `\$n` int: `ℤ` equal: `s = t ∈ T`
Definitions unfolded in proof :  all: `∀x:A. B[x]` strict-majority-or-max: `strict-majority-or-max(L)` and: `P ∧ Q` cand: `A c∧ B` uimplies: `b supposing a` member: `t ∈ T` prop: `ℙ` uall: `∀[x:A]. B[x]` nat: `ℕ` subtype_rel: `A ⊆r B` ge: `i ≥ j ` le: `A ≤ B` not: `¬A` implies: `P `` Q` false: `False` uiff: `uiff(P;Q)` int-deq: `IntDeq` decidable: `Dec(P)` or: `P ∨ Q` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` top: `Top` sq_type: `SQType(T)` guard: `{T}` l_all: `(∀x∈L.P[x])` int_seg: `{i..j-}` lelt: `i ≤ j < k` less_than: `a < b` squash: `↓T` rev_uimplies: `rev_uimplies(P;Q)` nequal: `a ≠ b ∈ T ` l_member: `(x ∈ l)` less_than': `less_than'(a;b)` strict-majority: `strict-majority(eq;L)` let: let ifthenelse: `if b then t else f fi ` null: `null(as)` filter: `filter(P;l)` reduce: `reduce(f;k;as)` list_ind: list_ind count-repeats: `count-repeats(L,eq)` list_accum: list_accum nil: `[]` it: `⋅` btrue: `tt` true: `True` cons: `[a / b]` bfalse: `ff`
Lemmas referenced :  le_wf length_wf filter_wf5 eq_int_wf l_member_wf equal-wf-base-T list_subtype_base int_subtype_base less_than'_wf ge_wf list_wf nat_wf strict-majority-property int-deq_wf nat_properties decidable__lt satisfiable-full-omega-tt intformand_wf intformnot_wf intformless_wf itermVar_wf itermMultiply_wf itermConstant_wf intformle_wf itermAdd_wf intformeq_wf int_formula_prop_and_lemma int_formula_prop_not_lemma int_formula_prop_less_lemma int_term_value_var_lemma int_term_value_mul_lemma int_term_value_constant_lemma int_formula_prop_le_lemma int_term_value_add_lemma int_formula_prop_eq_lemma int_formula_prop_wf subtype_base_sq unit_wf2 union_subtype_base unit_subtype_base strict-majority_wf equal_wf decidable__l_member decidable__equal_int filter_is_nil assert_wf select_wf int_seg_properties decidable__le int_seg_wf length_of_nil_lemma neg_assert_of_eq_int int_seg_subtype_nat false_wf less_than_wf equal-wf-T-base list-cases filter_nil_lemma null_nil_lemma product_subtype_list length_of_cons_lemma filter_cons_lemma null_cons_lemma imax-list-member cons_wf add-is-int-iff
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation cut isect_memberFormation introduction hypothesis extract_by_obid sqequalHypSubstitution isectElimination thin addEquality setElimination rename hypothesisEquality natural_numberEquality intEquality lambdaEquality setEquality sqequalRule isect_memberEquality axiomEquality because_Cache equalityTransitivity equalitySymmetry baseApply closedConclusion baseClosed applyEquality independent_isectElimination multiplyEquality independent_pairFormation productElimination independent_pairEquality dependent_functionElimination voidElimination unionElimination dependent_pairFormation int_eqEquality voidEquality computeAll instantiate cumulativity unionEquality independent_functionElimination imageElimination productEquality promote_hyp hypothesis_subsumption pointwiseFunctionality

Latex:
\mforall{}t:\mBbbN{}.  \mforall{}L:\mBbbZ{}  List.
((\mforall{}v:\mBbbZ{}
(strict-majority-or-max(L)  =  v)  supposing
(((t  +  1)  \mleq{}  ||filter(\mlambda{}x.(x  =\msubz{}  v);L)||)  and
(||L||  =  ((2  *  t)  +  1))))
\mwedge{}  (strict-majority-or-max(L)  \mmember{}  L)  supposing  ||L||  \mgeq{}  1  )

Date html generated: 2017_04_17-AM-09_09_44
Last ObjectModification: 2017_02_27-PM-05_17_59

Theory : decidable!equality

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