Nuprl Lemma : mon_for_swap
∀g:IAbMonoid. ∀A,B:Type. ∀f:A ⟶ B ⟶ |g|. ∀as:A List. ∀bs:B List.
((For{g} x ∈ as. For{g} y ∈ bs. f[x;y]) = (For{g} y ∈ bs. For{g} x ∈ as. f[x;y]) ∈ |g|)
Proof
Definitions occuring in Statement :
mon_for: For{g} x ∈ as. f[x],
list: T List,
so_apply: x[s1;s2],
all: ∀x:A. B[x],
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T,
iabmonoid: IAbMonoid,
grp_car: |g|
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
iabmonoid: IAbMonoid,
imon: IMonoid,
so_apply: x[s1;s2],
so_apply: x[s],
implies: P ⇒ Q,
top: Top,
squash: ↓T,
prop: ℙ,
true: True,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
infix_ap: x f y
Lemmas referenced :
list_induction,
all_wf,
list_wf,
equal_wf,
grp_car_wf,
mon_for_wf,
mon_for_nil_lemma,
squash_wf,
true_wf,
grp_id_wf,
mon_for_of_id,
iff_weakening_equal,
mon_for_cons_lemma,
iabmonoid_wf,
grp_op_wf,
infix_ap_wf,
mon_for_of_op
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
sqequalRule,
lambdaEquality,
cumulativity,
hypothesis,
setElimination,
rename,
because_Cache,
dependent_functionElimination,
applyEquality,
functionExtensionality,
independent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_isectElimination,
productElimination,
functionEquality
Latex:
\mforall{}g:IAbMonoid. \mforall{}A,B:Type. \mforall{}f:A {}\mrightarrow{} B {}\mrightarrow{} |g|. \mforall{}as:A List. \mforall{}bs:B List.
((For\{g\} x \mmember{} as. For\{g\} y \mmember{} bs. f[x;y]) = (For\{g\} y \mmember{} bs. For\{g\} x \mmember{} as. f[x;y]))
Date html generated:
2017_10_01-AM-09_55_31
Last ObjectModification:
2017_03_03-PM-00_50_33
Theory : list_2
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