Nuprl Lemma : l_before_map
∀[A,B:Type].
∀l:A List. ∀f:A ⟶ B. ∀x,y:B.
(x before y ∈ map(f;l) ⇐⇒ ∃u,v:A. ((x = (f u) ∈ B) ∧ (y = (f v) ∈ B) ∧ u before v ∈ l))
Proof
Definitions occuring in Statement :
l_before: x before y ∈ l,
map: map(f;as),
list: T List,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
apply: f a,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
so_apply: x[s],
so_lambda: λ2x.t[x],
rev_implies: P ⇐ Q,
prop: ℙ,
exists: ∃x:A. B[x],
top: Top,
member: t ∈ T,
sublist: L1 ⊆ L2,
l_before: x before y ∈ l,
implies: P ⇒ Q,
and: P ∧ Q,
iff: P ⇐⇒ Q,
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
sq_type: SQType(T),
cand: A c∧ B,
nat: ℕ,
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
or: P ∨ Q,
decidable: Dec(P),
guard: {T},
uimplies: b supposing a,
subtype_rel: A ⊆r B,
subtract: n - m,
cons: [a / b],
select: L[n],
true: True,
squash: ↓T,
less_than: a < b,
not: ¬A,
false: False,
less_than': less_than'(a;b),
le: A ≤ B,
lelt: i ≤ j < k,
int_seg: {i..j-}
Lemmas referenced :
list_wf,
equal_wf,
exists_wf,
map_wf,
l_before_wf,
length_of_nil_lemma,
length_of_cons_lemma,
int_term_value_add_lemma,
itermAdd_wf,
nil_wf,
cons_wf,
all_wf,
increasing_wf,
int_seg_cases,
int_subtype_base,
subtype_base_sq,
decidable__equal_int,
int_formula_prop_eq_lemma,
int_formula_prop_less_lemma,
intformeq_wf,
intformless_wf,
decidable__lt,
int_term_value_constant_lemma,
int_formula_prop_and_lemma,
itermConstant_wf,
intformand_wf,
nat_properties,
length_wf_nat,
map_length,
non_neg_length,
select_wf,
int_formula_prop_wf,
int_term_value_var_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
itermVar_wf,
intformle_wf,
intformnot_wf,
satisfiable-full-omega-tt,
decidable__le,
int_seg_properties,
iff_weakening_equal,
map_length_nat,
le_wf,
length_wf,
int_seg_subtype,
int_seg_wf,
top_wf,
subtype_rel_list,
select-map,
lelt_wf,
false_wf,
istype-false,
full-omega-unsat,
istype-int,
istype-void,
istype-le,
istype-less_than,
int_seg_subtype_special,
map-length,
squash_wf,
true_wf,
istype-universe,
subtype_rel_self
Rules used in proof :
universeEquality,
functionEquality,
productEquality,
lambdaEquality,
applyEquality,
functionExtensionality,
hypothesisEquality,
cumulativity,
isectElimination,
productElimination,
hypothesis,
voidEquality,
voidElimination,
isect_memberEquality,
thin,
dependent_functionElimination,
extract_by_obid,
introduction,
cut,
sqequalRule,
sqequalHypSubstitution,
independent_pairFormation,
lambdaFormation,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
addEquality,
hypothesis_subsumption,
instantiate,
computeAll,
intEquality,
int_eqEquality,
dependent_pairFormation,
unionElimination,
rename,
setElimination,
applyLambdaEquality,
independent_functionElimination,
equalitySymmetry,
equalityTransitivity,
imageElimination,
independent_isectElimination,
because_Cache,
baseClosed,
imageMemberEquality,
natural_numberEquality,
dependent_set_memberEquality,
Error :dependent_pairFormation_alt,
Error :lambdaFormation_alt,
approximateComputation,
Error :lambdaEquality_alt,
Error :isect_memberEquality_alt,
Error :universeIsType,
closedConclusion,
Error :inhabitedIsType,
Error :dependent_set_memberEquality_alt,
Error :productIsType,
Error :functionIsType,
Error :equalityIstype
Latex:
\mforall{}[A,B:Type].
\mforall{}l:A List. \mforall{}f:A {}\mrightarrow{} B. \mforall{}x,y:B.
(x before y \mmember{} map(f;l) \mLeftarrow{}{}\mRightarrow{} \mexists{}u,v:A. ((x = (f u)) \mwedge{} (y = (f v)) \mwedge{} u before v \mmember{} l))
Date html generated:
2019_06_20-PM-01_25_44
Last ObjectModification:
2019_01_01-AM-09_13_53
Theory : list_1
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