Nuprl Lemma : combine-list-flip
∀[A:Type]. ∀[f:A ⟶ A ⟶ A].
(∀[as:A List]. ∀[a1,a2:A].
(combine-list(x,y.f[x;y];[a1; [a2 / as]]) = combine-list(x,y.f[x;y];[a2; [a1 / as]]) ∈ A)) 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),
cons: [a / b],
list: T List,
comm: Comm(T;op),
assoc: Assoc(T;op),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s1;s2],
lambda: λx.A[x],
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
combine-list: combine-list(x,y.f[x; y];L),
all: ∀x:A. B[x],
top: Top,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
prop: ℙ,
comm: Comm(T;op),
infix_ap: x f y,
true: True,
squash: ↓T
Lemmas referenced :
list_wf,
true_wf,
squash_wf,
list_accum_wf,
assoc_wf,
comm_wf,
list_accum_cons_lemma,
reduce_tl_cons_lemma,
reduce_hd_cons_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
lemma_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
isect_memberEquality,
voidElimination,
voidEquality,
hypothesis,
because_Cache,
isectElimination,
hypothesisEquality,
axiomEquality,
lambdaEquality,
applyEquality,
equalityTransitivity,
equalitySymmetry,
natural_numberEquality,
imageElimination,
functionEquality,
universeEquality,
imageMemberEquality,
baseClosed
Latex:
\mforall{}[A:Type]. \mforall{}[f:A {}\mrightarrow{} A {}\mrightarrow{} A].
(\mforall{}[as:A List]. \mforall{}[a1,a2:A].
(combine-list(x,y.f[x;y];[a1; [a2 / as]])
= combine-list(x,y.f[x;y];[a2; [a1 / as]]))) supposing
(Comm(A;\mlambda{}x,y. f[x;y]) and
Assoc(A;\mlambda{}x,y. f[x;y]))
Date html generated:
2016_05_14-PM-01_40_59
Last ObjectModification:
2016_01_15-AM-08_23_58
Theory : list_1
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