### Nuprl Lemma : hd-map

`∀[f:Top]. ∀[L:Top List].  (hd(map(f;L)) ~ if null(L) then ⊥ else f hd(L) fi )`

Proof

Definitions occuring in Statement :  hd: `hd(l)` null: `null(as)` map: `map(f;as)` list: `T List` bottom: `⊥` ifthenelse: `if b then t else f fi ` uall: `∀[x:A]. B[x]` top: `Top` apply: `f a` sqequal: `s ~ t`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` top: `Top` all: `∀x:A. B[x]` or: `P ∨ Q` cons: `[a / b]` select: `L[n]` uimplies: `b supposing a` nil: `[]` it: `⋅` so_lambda: `λ2x y.t[x; y]` so_apply: `x[s1;s2]` ifthenelse: `if b then t else f fi ` btrue: `tt` int_seg: `{i..j-}` lelt: `i ≤ j < k` and: `P ∧ Q` le: `A ≤ B` less_than': `less_than'(a;b)` false: `False` not: `¬A` implies: `P `` Q` prop: `ℙ` nat_plus: `ℕ+` less_than: `a < b` squash: `↓T` true: `True` guard: `{T}` decidable: `Dec(P)` uiff: `uiff(P;Q)` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` bool: `𝔹` unit: `Unit` bfalse: `ff` sq_type: `SQType(T)` bnot: `¬bb` assert: `↑b`
Lemmas referenced :  select0 list_wf top_wf list-cases product_subtype_list map_nil_lemma null_nil_lemma stuck-spread base_wf select-map cons_wf length_of_cons_lemma false_wf 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 equal_wf lelt_wf length_wf null_wf bool_wf eqtt_to_assert assert_of_null btrue_wf and_wf null_cons_lemma bfalse_wf btrue_neq_bfalse eqff_to_assert bool_cases_sqequal subtype_base_sq bool_subtype_base assert-bnot equal-wf-T-base
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut sqequalRule extract_by_obid sqequalHypSubstitution isectElimination thin isect_memberEquality voidElimination voidEquality hypothesis sqequalAxiom hypothesisEquality because_Cache dependent_functionElimination unionElimination promote_hyp hypothesis_subsumption productElimination baseClosed independent_isectElimination lambdaFormation dependent_set_memberEquality natural_numberEquality independent_pairFormation imageMemberEquality equalityTransitivity equalitySymmetry applyLambdaEquality setElimination rename pointwiseFunctionality baseApply closedConclusion dependent_pairFormation lambdaEquality int_eqEquality intEquality computeAll independent_functionElimination addEquality equalityElimination instantiate cumulativity

Latex:
\mforall{}[f:Top].  \mforall{}[L:Top  List].    (hd(map(f;L))  \msim{}  if  null(L)  then  \mbot{}  else  f  hd(L)  fi  )

Date html generated: 2017_04_14-AM-09_26_41
Last ObjectModification: 2017_02_27-PM-04_00_39

Theory : list_1

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