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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