Nuprl Lemma : map-l-union
∀[T,T':Type]. ∀[f:T ⟶ T']. ∀[eq:EqDecider(T)]. ∀[eq':EqDecider(T')]. ∀[as,bs:T List].
map(f;as ⋃ bs) ~ map(f;as) ⋃ map(f;bs) supposing Inj({x:T| (x ∈ as ⋃ bs)} ;T';f)
Proof
Definitions occuring in Statement :
l-union: as ⋃ bs,
l_member: (x ∈ l),
map: map(f;as),
list: T List,
deq: EqDecider(T),
inject: Inj(A;B;f),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
universe: Type,
sqequal: s ~ t
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
all: ∀x:A. B[x],
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
not: ¬A,
top: Top,
and: P ∧ Q,
prop: ℙ,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
guard: {T},
or: P ∨ Q,
l-union: as ⋃ bs,
cons: [a / b],
colength: colength(L),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
decidable: Dec(P),
nil: [],
it: ⋅,
sq_type: SQType(T),
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
inject: Inj(A;B;f),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
insert: insert(a;L),
has-value: (a)↓,
bool: 𝔹,
unit: Unit,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff,
bnot: ¬bb,
assert: ↑b
Lemmas referenced :
nat_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
ge_wf,
less_than_wf,
inject_wf,
l_member_wf,
l-union_wf,
subtype_rel_dep_function,
set_wf,
equal-wf-T-base,
nat_wf,
colength_wf_list,
less_than_transitivity1,
less_than_irreflexivity,
list-cases,
map_nil_lemma,
reduce_nil_lemma,
product_subtype_list,
spread_cons_lemma,
intformeq_wf,
itermAdd_wf,
int_formula_prop_eq_lemma,
int_term_value_add_lemma,
decidable__le,
intformnot_wf,
int_formula_prop_not_lemma,
le_wf,
equal_wf,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
subtype_base_sq,
set_subtype_base,
int_subtype_base,
decidable__equal_int,
map_cons_lemma,
reduce_cons_lemma,
insert_wf,
list_wf,
deq_wf,
member-insert,
member_wf,
eval_list_sq,
subtype_rel_list,
top_wf,
map_wf,
value-type-has-value,
list-value-type,
deq-member_wf,
bool_wf,
eqtt_to_assert,
assert-deq-member,
eqff_to_assert,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
member-map,
and_wf
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,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
independent_pairFormation,
computeAll,
independent_functionElimination,
sqequalAxiom,
setEquality,
cumulativity,
applyEquality,
because_Cache,
unionElimination,
promote_hyp,
hypothesis_subsumption,
productElimination,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
dependent_set_memberEquality,
addEquality,
baseClosed,
instantiate,
imageElimination,
functionEquality,
universeEquality,
inrFormation,
hyp_replacement,
imageMemberEquality,
functionExtensionality,
callbyvalueReduce,
equalityElimination,
productEquality,
inlFormation
Latex:
\mforall{}[T,T':Type]. \mforall{}[f:T {}\mrightarrow{} T']. \mforall{}[eq:EqDecider(T)]. \mforall{}[eq':EqDecider(T')]. \mforall{}[as,bs:T List].
map(f;as \mcup{} bs) \msim{} map(f;as) \mcup{} map(f;bs) supposing Inj(\{x:T| (x \mmember{} as \mcup{} bs)\} ;T';f)
Date html generated:
2017_04_17-AM-09_09_58
Last ObjectModification:
2017_02_27-PM-05_18_16
Theory : decidable!equality
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