### Nuprl Lemma : select-map

`∀[f:Top]. ∀[L:Top List]. ∀[i:ℕ||L||].  (map(f;L)[i] ~ f L[i])`

Proof

Definitions occuring in Statement :  select: `L[n]` length: `||as||` map: `map(f;as)` list: `T List` int_seg: `{i..j-}` uall: `∀[x:A]. B[x]` top: `Top` apply: `f a` natural_number: `\$n` sqequal: `s ~ t`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` all: `∀x:A. B[x]` nat: `ℕ` implies: `P `` Q` false: `False` ge: `i ≥ j ` guard: `{T}` uimplies: `b supposing a` prop: `ℙ` subtype_rel: `A ⊆r B` or: `P ∨ Q` top: `Top` select: `L[n]` nil: `[]` it: `⋅` so_lambda: `λ2x y.t[x; y]` so_apply: `x[s1;s2]` int_seg: `{i..j-}` lelt: `i ≤ j < k` and: `P ∧ Q` cons: `[a / b]` colength: `colength(L)` squash: `↓T` sq_stable: `SqStable(P)` uiff: `uiff(P;Q)` le: `A ≤ B` not: `¬A` less_than': `less_than'(a;b)` true: `True` decidable: `Dec(P)` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` subtract: `n - m` so_lambda: `λ2x.t[x]` so_apply: `x[s]` sq_type: `SQType(T)` less_than: `a < b` bool: `𝔹` unit: `Unit` btrue: `tt` ifthenelse: `if b then t else f fi ` bfalse: `ff`
Lemmas referenced :  nat_properties less_than_transitivity1 less_than_irreflexivity ge_wf less_than_wf int_seg_wf length_wf top_wf equal-wf-T-base nat_wf colength_wf_list list-cases length_of_nil_lemma map_nil_lemma stuck-spread base_wf product_subtype_list spread_cons_lemma sq_stable__le le_antisymmetry_iff add_functionality_wrt_le add-associates add-zero zero-add le-add-cancel decidable__le false_wf not-le-2 condition-implies-le minus-add minus-one-mul minus-one-mul-top add-commutes le_wf equal_wf subtract_wf not-ge-2 less-iff-le minus-minus add-swap subtype_base_sq set_subtype_base int_subtype_base length_of_cons_lemma map_cons_lemma list_wf le_int_wf bool_wf assert_wf lt_int_wf bnot_wf decidable__lt not-lt-2 lelt_wf select-cons uiff_transitivity eqtt_to_assert assert_of_le_int eqff_to_assert assert_functionality_wrt_uiff bnot_of_le_int assert_of_lt_int
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut thin lambdaFormation extract_by_obid sqequalHypSubstitution isectElimination hypothesisEquality hypothesis setElimination rename intWeakElimination natural_numberEquality independent_isectElimination independent_functionElimination voidElimination sqequalRule lambdaEquality dependent_functionElimination isect_memberEquality sqequalAxiom applyEquality because_Cache unionElimination voidEquality baseClosed productElimination promote_hyp hypothesis_subsumption applyLambdaEquality imageMemberEquality imageElimination addEquality dependent_set_memberEquality independent_pairFormation minusEquality equalityTransitivity equalitySymmetry intEquality instantiate cumulativity equalityElimination

Latex:
\mforall{}[f:Top].  \mforall{}[L:Top  List].  \mforall{}[i:\mBbbN{}||L||].    (map(f;L)[i]  \msim{}  f  L[i])

Date html generated: 2017_04_14-AM-08_38_28
Last ObjectModification: 2017_02_27-PM-03_29_35

Theory : list_0

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