Nuprl Lemma : combine-list-append
∀[A:Type]. ∀[f:A ⟶ A ⟶ A].
(∀[as,bs:A List].
combine-list(x,y.f[x;y];as @ bs) = combine-list(x,y.f[x;y];bs @ as) ∈ A supposing 0 < ||as @ bs||) supposing
(Comm(A;λx,y. f[x;y]) and
Assoc(A;λx,y. f[x;y]))
Proof
Definitions occuring in Statement :
combine-list: combine-list(x,y.f[x; y];L),
length: ||as||,
append: as @ bs,
list: T List,
comm: Comm(T;op),
assoc: Assoc(T;op),
less_than: a < b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s1;s2],
lambda: λx.A[x],
function: x:A ⟶ B[x],
natural_number: $n,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
all: ∀x:A. B[x],
or: P ∨ Q,
append: as @ bs,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
top: Top,
so_apply: x[s1;s2;s3],
squash: ↓T,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
cons: [a / b],
combine-list: combine-list(x,y.f[x; y];L),
prop: ℙ,
so_apply: x[s],
true: True,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
comm: Comm(T;op),
infix_ap: x f y
Lemmas referenced :
list-cases,
list_ind_nil_lemma,
combine-list_wf,
append_back_nil,
product_subtype_list,
list_ind_cons_lemma,
length_of_cons_lemma,
cons_wf,
reduce_hd_cons_lemma,
reduce_tl_cons_lemma,
less_than_wf,
length_wf,
append_wf,
comm_wf,
assoc_wf,
equal_wf,
squash_wf,
true_wf,
iff_weakening_equal,
list_accum_cons_lemma,
list_accum_wf,
list_accum_permute
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
hypothesisEquality,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesis,
dependent_functionElimination,
unionElimination,
sqequalRule,
isect_memberEquality,
voidElimination,
voidEquality,
applyEquality,
lambdaEquality,
imageElimination,
because_Cache,
independent_isectElimination,
functionExtensionality,
imageMemberEquality,
baseClosed,
promote_hyp,
hypothesis_subsumption,
productElimination,
natural_numberEquality,
cumulativity,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
universeEquality,
independent_functionElimination,
hyp_replacement,
applyLambdaEquality
Latex:
\mforall{}[A:Type]. \mforall{}[f:A {}\mrightarrow{} A {}\mrightarrow{} A].
(\mforall{}[as,bs:A List].
combine-list(x,y.f[x;y];as @ bs) = combine-list(x,y.f[x;y];bs @ as)
supposing 0 < ||as @ bs||) supposing
(Comm(A;\mlambda{}x,y. f[x;y]) and
Assoc(A;\mlambda{}x,y. f[x;y]))
Date html generated:
2017_04_17-AM-07_38_52
Last ObjectModification:
2017_02_27-PM-04_12_50
Theory : list_1
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