Nuprl Lemma : agree_on_common_append
∀[T:Type]
∀as,bs,cs,ds:T List.
(agree_on_common(T;as;cs) ⇒ agree_on_common(T;bs;ds) ⇒ agree_on_common(T;as @ bs;cs @ ds)) supposing
((∀x∈as.¬(x ∈ ds)) and
(∀x∈cs.¬(x ∈ bs)))
Proof
Definitions occuring in Statement :
agree_on_common: agree_on_common(T;as;bs),
l_all: (∀x∈L.P[x]),
l_member: (x ∈ l),
append: as @ bs,
list: T List,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
universe: Type
Definitions unfolded in proof :
so_apply: x[s1;s2;s3],
top: Top,
so_lambda: so_lambda3,
append: as @ bs,
implies: P ⇒ Q,
so_apply: x[s],
prop: ℙ,
uimplies: b supposing a,
so_lambda: λ2x.t[x],
member: t ∈ T,
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
false: False,
not: ¬A,
l_all: (∀x∈L.P[x]),
subtype_rel: A ⊆r B,
true: True,
agree_on_common: agree_on_common(T;as;bs),
and: P ∧ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
or: P ∨ Q,
exists: ∃x:A. B[x],
l_member: (x ∈ l),
cand: A c∧ B
Lemmas referenced :
list_ind_cons_lemma,
istype-void,
list_ind_nil_lemma,
append_wf,
agree_on_common_wf,
not_wf,
istype-universe,
l_member_wf,
l_all_wf,
isect_wf,
list_wf,
all_wf,
list_induction,
cons_wf,
nil_wf,
l_all_wf_nil,
l_all_cons,
agree_on_common_cons2,
agree_on_common_nil,
member_append,
cons_member
Rules used in proof :
universeEquality,
isectIsType,
functionIsType,
voidElimination,
isect_memberEquality_alt,
dependent_functionElimination,
independent_functionElimination,
inhabitedIsType,
functionEquality,
universeIsType,
setIsType,
rename,
setElimination,
because_Cache,
hypothesis,
lambdaEquality_alt,
sqequalRule,
hypothesisEquality,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
thin,
cut,
lambdaFormation_alt,
isect_memberFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
functionIsTypeImplies,
applyEquality,
natural_numberEquality,
productElimination,
independent_isectElimination,
inrFormation_alt,
equalityIsType1,
unionIsType,
promote_hyp,
independent_pairFormation,
inlFormation_alt,
unionElimination,
productIsType,
equalityIstype
Latex:
\mforall{}[T:Type]
\mforall{}as,bs,cs,ds:T List.
(agree\_on\_common(T;as;cs)
{}\mRightarrow{} agree\_on\_common(T;bs;ds)
{}\mRightarrow{} agree\_on\_common(T;as @ bs;cs @ ds)) supposing
((\mforall{}x\mmember{}as.\mneg{}(x \mmember{} ds)) and
(\mforall{}x\mmember{}cs.\mneg{}(x \mmember{} bs)))
Date html generated:
2020_05_20-AM-07_48_15
Last ObjectModification:
2020_01_22-PM-05_26_07
Theory : list!
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