Nuprl Lemma : eager-map-append-sq
∀[T:Type]
∀[A:Type]. ∀[f:A ⟶ T]. ∀[as:A List]. ∀[bs:T List]. (eager-map-append(f;as;bs) ~ map(f;rev(as)) @ bs)
supposing value-type(T)
Proof
Definitions occuring in Statement :
reverse: rev(as),
eager-map-append: eager-map-append(f;as;bs),
map: map(f;as),
append: as @ bs,
list: T List,
value-type: value-type(T),
uimplies: b supposing a,
uall: ∀[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 ,
guard: {T},
prop: ℙ,
subtype_rel: A ⊆r B,
or: P ∨ Q,
top: Top,
append: as @ bs,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
cons: [a / b],
colength: colength(L),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
squash: ↓T,
sq_stable: SqStable(P),
uiff: uiff(P;Q),
and: P ∧ Q,
le: A ≤ B,
not: ¬A,
less_than': less_than'(a;b),
true: True,
decidable: Dec(P),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
subtract: n - m,
nil: [],
it: ⋅,
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_type: SQType(T),
less_than: a < b,
eager-map-append: eager-map-append(f;as;bs),
has-value: (a)↓
Lemmas referenced :
nat_properties,
less_than_transitivity1,
less_than_irreflexivity,
ge_wf,
less_than_wf,
list_wf,
equal-wf-T-base,
nat_wf,
colength_wf_list,
list-cases,
eager_map_append_nil_lemma,
reverse_nil_lemma,
map_nil_lemma,
list_ind_nil_lemma,
product_subtype_list,
spread_cons_lemma,
sq_stable__le,
le_antisymmetry_iff,
add_functionality_wrt_le,
add-associates,
add-zero,
zero-add,
le-add-cancel,
decidable__le,
false_wf,
not-le-2,
condition-implies-le,
minus-add,
minus-one-mul,
minus-one-mul-top,
add-commutes,
le_wf,
equal_wf,
subtract_wf,
not-ge-2,
less-iff-le,
minus-minus,
add-swap,
subtype_base_sq,
set_subtype_base,
int_subtype_base,
reverse-cons,
value-type_wf,
list_accum_cons_lemma,
value-type-has-value,
list-value-type,
cons_wf,
map_append_sq,
map_cons_lemma,
append_assoc_sq,
list_ind_cons_lemma
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,
independent_functionElimination,
voidElimination,
sqequalRule,
lambdaEquality,
dependent_functionElimination,
isect_memberEquality,
sqequalAxiom,
cumulativity,
applyEquality,
because_Cache,
unionElimination,
voidEquality,
promote_hyp,
hypothesis_subsumption,
productElimination,
applyLambdaEquality,
imageMemberEquality,
baseClosed,
imageElimination,
addEquality,
dependent_set_memberEquality,
independent_pairFormation,
minusEquality,
equalityTransitivity,
equalitySymmetry,
intEquality,
instantiate,
functionEquality,
universeEquality,
callbyvalueReduce,
functionExtensionality
Latex:
\mforall{}[T:Type]
\mforall{}[A:Type]. \mforall{}[f:A {}\mrightarrow{} T]. \mforall{}[as:A List]. \mforall{}[bs:T List].
(eager-map-append(f;as;bs) \msim{} map(f;rev(as)) @ bs)
supposing value-type(T)
Date html generated:
2017_04_14-AM-08_55_16
Last ObjectModification:
2017_02_27-PM-03_38_54
Theory : omega
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