### Nuprl Lemma : list-max-map

`∀[T,A:Type]. ∀[g:A ⟶ T]. ∀[f:T ⟶ ℤ]. ∀[L:A List].`
`  list-max(x.f[x];map(g;L)) = ((λp.<fst(p), g (snd(p))>) list-max(x.f[g x];L)) ∈ (i:ℤ × {x:T| f[x] = i ∈ ℤ} ) `
`  supposing 0 < ||L||`

Proof

Definitions occuring in Statement :  list-max: `list-max(x.f[x];L)` length: `||as||` map: `map(f;as)` list: `T List` less_than: `a < b` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` so_apply: `x[s]` pi1: `fst(t)` pi2: `snd(t)` set: `{x:A| B[x]} ` apply: `f a` lambda: `λx.A[x]` function: `x:A ⟶ B[x]` pair: `<a, b>` product: `x:A × B[x]` natural_number: `\$n` int: `ℤ` universe: `Type` equal: `s = t ∈ T`
Definitions unfolded in proof :  member: `t ∈ T` uall: `∀[x:A]. B[x]` uimplies: `b supposing a` prop: `ℙ` list-max: `list-max(x.f[x];L)` list-max-aux: `list-max-aux(x.f[x];L)` top: `Top` so_lambda: `λ2x y.t[x; y]` so_apply: `x[s1;s2]` all: `∀x:A. B[x]` or: `P ∨ Q` less_than: `a < b` squash: `↓T` less_than': `less_than'(a;b)` false: `False` and: `P ∧ Q` cons: `[a / b]` so_apply: `x[s]` outl: `outl(x)` pi1: `fst(t)` pi2: `snd(t)` assert: `↑b` ifthenelse: `if b then t else f fi ` isl: `isl(x)` btrue: `tt` true: `True` so_lambda: `λ2x.t[x]` implies: `P `` Q` nat: `ℕ` ge: `i ≥ j ` not: `¬A` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` subtype_rel: `A ⊆r B` bfalse: `ff` colength: `colength(L)` guard: `{T}` decidable: `Dec(P)` nil: `[]` it: `⋅` sq_type: `SQType(T)` has-value: `(a)↓` bool: `𝔹` unit: `Unit` uiff: `uiff(P;Q)` bnot: `¬bb` isr: `isr(x)`
Lemmas referenced :  less_than_wf length_wf list_wf list_accum-map list-cases length_of_nil_lemma product_subtype_list length_of_cons_lemma list_accum_cons_lemma value-type-has-value int-value-type top_wf assert_wf isl_wf set_wf nat_properties full-omega-unsat intformand_wf intformle_wf itermConstant_wf itermVar_wf intformless_wf int_formula_prop_and_lemma int_formula_prop_le_lemma int_term_value_constant_lemma int_term_value_var_lemma int_formula_prop_less_lemma int_formula_prop_wf ge_wf equal-wf-T-base nat_wf colength_wf_list int_subtype_base list_accum_nil_lemma spread_cons_lemma intformeq_wf itermAdd_wf int_formula_prop_eq_lemma int_term_value_add_lemma decidable__le intformnot_wf int_formula_prop_not_lemma le_wf equal_wf subtract_wf itermSubtract_wf int_term_value_subtract_lemma subtype_base_sq set_subtype_base decidable__equal_int lt_int_wf pi1_wf bool_wf eqtt_to_assert assert_of_lt_int eqff_to_assert bool_cases_sqequal bool_subtype_base assert-bnot ifthenelse_wf squash_wf true_wf list_accum_wf not-isr-isl isr_wf pi2_wf list-max_wf map_wf map-length list-max-property2
Rules used in proof :  cut sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity introduction extract_by_obid sqequalHypSubstitution isectElimination thin natural_numberEquality hypothesisEquality hypothesis because_Cache functionEquality intEquality universeEquality isect_memberFormation sqequalRule isect_memberEquality axiomEquality equalityTransitivity equalitySymmetry voidElimination voidEquality dependent_functionElimination unionElimination imageElimination productElimination promote_hyp hypothesis_subsumption independent_isectElimination applyEquality dependent_set_memberEquality inlEquality independent_pairEquality productEquality unionEquality lambdaEquality lambdaFormation setElimination rename intWeakElimination approximateComputation independent_functionElimination dependent_pairFormation int_eqEquality independent_pairFormation setEquality dependent_pairEquality functionExtensionality cumulativity applyLambdaEquality addEquality baseClosed instantiate callbyvalueReduce equalityElimination hyp_replacement imageMemberEquality

Latex:
\mforall{}[T,A:Type].  \mforall{}[g:A  {}\mrightarrow{}  T].  \mforall{}[f:T  {}\mrightarrow{}  \mBbbZ{}].  \mforall{}[L:A  List].
list-max(x.f[x];map(g;L))  =  ((\mlambda{}p.<fst(p),  g  (snd(p))>)  list-max(x.f[g  x];L))  supposing  0  <  ||L||

Date html generated: 2019_06_20-PM-01_30_49
Last ObjectModification: 2018_08_21-PM-01_55_35

Theory : list_1

Home Index