Nuprl Lemma : hd_map

`∀[T,T':Type]. ∀[a:T List+]. ∀[f:T ⟶ T'].  (hd(map(f;a)) = (f hd(a)) ∈ T')`

Proof

Definitions occuring in Statement :  listp: `A List+` map: `map(f;as)` hd: `hd(l)` uall: `∀[x:A]. B[x]` apply: `f a` function: `x:A ⟶ B[x]` universe: `Type` equal: `s = t ∈ T`
Definitions unfolded in proof :  member: `t ∈ T` uall: `∀[x:A]. B[x]` true: `True` less_than': `less_than'(a;b)` so_apply: `x[s]` not: `¬A` false: `False` exists: `∃x:A. B[x]` satisfiable_int_formula: `satisfiable_int_formula(fmla)` and: `P ∧ Q` le: `A ≤ B` or: `P ∨ Q` decidable: `Dec(P)` all: `∀x:A. B[x]` top: `Top` uimplies: `b supposing a` ge: `i ≥ j ` prop: `ℙ` implies: `P `` Q` so_lambda: `λ2x.t[x]` listp: `A List+`
Lemmas referenced :  listp_wf reduce_hd_cons_lemma map_cons_lemma length_of_cons_lemma map_nil_lemma length_of_nil_lemma list_wf less_than_wf int_formula_prop_wf int_formula_prop_le_lemma int_term_value_var_lemma int_term_value_constant_lemma int_formula_prop_less_lemma int_formula_prop_not_lemma int_formula_prop_and_lemma intformle_wf itermVar_wf itermConstant_wf intformless_wf intformnot_wf intformand_wf satisfiable-full-omega-tt decidable__lt map-length map_wf hd_wf equal_wf length_wf ge_wf list_induction listp_properties
Rules used in proof :  functionIsType universeIsType hypothesisEquality sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity functionEquality cut introduction extract_by_obid sqequalHypSubstitution isectElimination thin hypothesis because_Cache inhabitedIsType universeEquality isect_memberFormation_alt sqequalRule isect_memberEquality axiomEquality addEquality lambdaFormation independent_functionElimination dependent_set_memberEquality computeAll independent_pairFormation intEquality int_eqEquality dependent_pairFormation productElimination unionElimination dependent_functionElimination voidEquality voidElimination independent_isectElimination applyEquality functionExtensionality natural_numberEquality cumulativity lambdaEquality rename setElimination

Latex:
\mforall{}[T,T':Type].  \mforall{}[a:T  List\msupplus{}].  \mforall{}[f:T  {}\mrightarrow{}  T'].    (hd(map(f;a))  =  (f  hd(a)))

Date html generated: 2019_10_15-AM-10_53_25
Last ObjectModification: 2018_09_27-AM-10_02_47

Theory : list!

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