Nuprl Lemma : mon_for_append
∀g:IMonoid. ∀A:Type. ∀f:A ⟶ |g|. ∀as,as':A List.
((For{g} x ∈ as @ as'. f[x]) = ((For{g} x ∈ as. f[x]) * (For{g} x ∈ as'. f[x])) ∈ |g|)
Proof
Definitions occuring in Statement :
mon_for: For{g} x ∈ as. f[x],
append: as @ bs,
list: T List,
infix_ap: x f y,
so_apply: x[s],
all: ∀x:A. B[x],
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T,
imon: IMonoid,
grp_op: *,
grp_car: |g|
Definitions unfolded in proof :
mon_for: For{g} x ∈ as. f[x],
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
prop: ℙ,
imon: IMonoid,
so_apply: x[s],
infix_ap: x f y,
implies: P ⇒ Q,
append: as @ bs,
so_lambda: so_lambda3,
so_apply: x[s1;s2;s3],
and: P ∧ Q,
squash: ↓T,
true: True,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
list_induction,
list_wf,
equal_wf,
grp_car_wf,
for_wf,
grp_op_wf,
grp_id_wf,
append_wf,
list_ind_nil_lemma,
for_nil_lemma,
mon_ident,
list_ind_cons_lemma,
for_cons_lemma,
istype-universe,
imon_wf,
squash_wf,
true_wf,
subtype_rel_self,
iff_weakening_equal,
mon_assoc
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
lambdaFormation_alt,
cut,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
lambdaEquality_alt,
functionEquality,
hypothesis,
setElimination,
rename,
because_Cache,
applyEquality,
universeIsType,
independent_functionElimination,
dependent_functionElimination,
Error :memTop,
equalitySymmetry,
productElimination,
functionIsType,
equalityIstype,
inhabitedIsType,
instantiate,
universeEquality,
imageElimination,
equalityTransitivity,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_isectElimination
Latex:
\mforall{}g:IMonoid. \mforall{}A:Type. \mforall{}f:A {}\mrightarrow{} |g|. \mforall{}as,as':A List.
((For\{g\} x \mmember{} as @ as'. f[x]) = ((For\{g\} x \mmember{} as. f[x]) * (For\{g\} x \mmember{} as'. f[x])))
Date html generated:
2020_05_20-AM-09_35_31
Last ObjectModification:
2020_01_08-PM-06_00_20
Theory : list_2
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