Nuprl Lemma : star-append-iff
∀[T:Type]. ∀[P,Q:(T List) ⟶ ℙ].
∀L:T List
(star-append(T;P;Q) L ⇐⇒ (Q L) ∨ (∃L1,L2:T List. ((L = (L1 @ L2) ∈ (T List)) ∧ (P L1) ∧ (star-append(T;P;Q) L2))))
Proof
Definitions occuring in Statement :
star-append: star-append(T;P;Q),
append: as @ bs,
list: T List,
uall: ∀[x:A]. B[x],
prop: ℙ,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
or: P ∨ Q,
and: P ∧ Q,
apply: f a,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
star-append: star-append(T;P;Q),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
exists: ∃x:A. B[x],
member: t ∈ T,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
rev_implies: P ⇐ Q,
top: Top,
or: P ∨ Q,
concat: concat(ll),
append: as @ bs,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
cons: [a / b],
guard: {T},
cand: A c∧ B,
squash: ↓T,
true: True,
uimplies: b supposing a
Lemmas referenced :
exists_wf,
list_wf,
l_all_wf2,
l_member_wf,
equal_wf,
append_wf,
concat_wf,
or_wf,
length_wf,
length-append,
list-cases,
reduce_nil_lemma,
list_ind_nil_lemma,
product_subtype_list,
reduce_cons_lemma,
append_assoc,
l_all_cons,
nil_wf,
l_all_nil,
cons_wf,
squash_wf,
true_wf,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
lambdaFormation,
independent_pairFormation,
sqequalHypSubstitution,
productElimination,
thin,
cut,
introduction,
extract_by_obid,
isectElimination,
cumulativity,
hypothesisEquality,
hypothesis,
lambdaEquality,
because_Cache,
productEquality,
setElimination,
rename,
applyEquality,
functionExtensionality,
setEquality,
applyLambdaEquality,
isect_memberEquality,
voidElimination,
voidEquality,
functionEquality,
universeEquality,
dependent_functionElimination,
unionElimination,
inlFormation,
hyp_replacement,
equalitySymmetry,
promote_hyp,
hypothesis_subsumption,
inrFormation,
dependent_pairFormation,
independent_functionElimination,
imageElimination,
equalityTransitivity,
equalityUniverse,
levelHypothesis,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_isectElimination
Latex:
\mforall{}[T:Type]. \mforall{}[P,Q:(T List) {}\mrightarrow{} \mBbbP{}].
\mforall{}L:T List
(star-append(T;P;Q) L
\mLeftarrow{}{}\mRightarrow{} (Q L) \mvee{} (\mexists{}L1,L2:T List. ((L = (L1 @ L2)) \mwedge{} (P L1) \mwedge{} (star-append(T;P;Q) L2))))
Date html generated:
2018_05_21-PM-07_33_58
Last ObjectModification:
2017_07_26-PM-05_08_40
Theory : general
Home
Index