Nuprl Lemma : sublist_map_inj
∀[A,B:Type]. ∀f:A ⟶ B. ∀as,bs:A List. (Inj(A;B;f) ⇒ (as ⊆ bs ⇐⇒ map(f;as) ⊆ map(f;bs)))
Proof
Definitions occuring in Statement :
sublist: L1 ⊆ L2,
map: map(f;as),
list: T List,
inject: Inj(A;B;f),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
sublist: L1 ⊆ L2,
exists: ∃x:A. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
prop: ℙ,
squash: ↓T,
true: True,
guard: {T},
rev_implies: P ⇐ Q,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
int_seg: {i..j-},
lelt: i ≤ j < k,
cand: A c∧ B,
less_than: a < b,
ge: i ≥ j ,
nat: ℕ,
inject: Inj(A;B;f)
Lemmas referenced :
subtype_rel_dep_function,
int_seg_wf,
length_wf,
map_wf,
int_seg_subtype,
false_wf,
le_wf,
map_length_nat,
iff_weakening_equal,
decidable__le,
satisfiable-full-omega-tt,
intformnot_wf,
intformle_wf,
itermVar_wf,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
int_seg_properties,
increasing_wf,
length_wf_nat,
all_wf,
equal_wf,
select_wf,
intformand_wf,
itermConstant_wf,
int_formula_prop_and_lemma,
int_term_value_constant_lemma,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
non_neg_length,
map_length,
lelt_wf,
nat_properties,
map-length,
intformeq_wf,
int_formula_prop_eq_lemma,
sublist_wf,
inject_wf,
list_wf,
select-map,
subtype_rel_list,
top_wf,
squash_wf,
true_wf,
map_select,
length-map
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
independent_pairFormation,
sqequalHypSubstitution,
productElimination,
thin,
dependent_pairFormation,
cut,
hypothesisEquality,
applyEquality,
introduction,
extract_by_obid,
isectElimination,
natural_numberEquality,
cumulativity,
hypothesis,
sqequalRule,
lambdaEquality,
functionExtensionality,
independent_isectElimination,
because_Cache,
imageElimination,
imageMemberEquality,
baseClosed,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
dependent_functionElimination,
unionElimination,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
setElimination,
rename,
productEquality,
dependent_set_memberEquality,
applyLambdaEquality,
functionEquality,
universeEquality
Latex:
\mforall{}[A,B:Type]. \mforall{}f:A {}\mrightarrow{} B. \mforall{}as,bs:A List. (Inj(A;B;f) {}\mRightarrow{} (as \msubseteq{} bs \mLeftarrow{}{}\mRightarrow{} map(f;as) \msubseteq{} map(f;bs)))
Date html generated:
2017_04_14-AM-09_29_53
Last ObjectModification:
2017_02_27-PM-04_02_11
Theory : list_1
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