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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