Nuprl Lemma : assert-exists-l_subset
∀[T:Type]
((∀x,y:T. Dec(x = y ∈ T))
⇒ (∀P:(T List) ⟶ 𝔹
((∀L1,L2:T List. (set-equal(T;L1;L2) ⇒ (↑(P L1) ⇐⇒ ↑(P L2))))
⇒ (∀L:T List. (↑exists_sublist(L;P) ⇐⇒ ∃LL:T List. (l_subset(T;LL;L) ∧ (↑(P LL))))))))
Proof
Definitions occuring in Statement :
set-equal: set-equal(T;x;y),
l_subset: l_subset(T;as;bs),
exists_sublist: exists_sublist(L;P),
list: T List,
assert: ↑b,
bool: 𝔹,
decidable: Dec(P),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
and: P ∧ Q,
apply: f a,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
member: t ∈ T,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
rev_implies: P ⇐ Q,
exists: ∃x:A. B[x],
cand: A c∧ B,
l_subset: l_subset(T;as;bs),
guard: {T},
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
false: False,
set-equal: set-equal(T;x;y)
Lemmas referenced :
exists_wf,
list_wf,
sublist_wf,
assert_wf,
l_subset_wf,
assert-exists_sublist,
exists_sublist_wf,
all_wf,
iff_wf,
set-equal_wf,
bool_wf,
decidable_wf,
equal_wf,
l_member_wf,
member_sublist,
list_induction,
nil_wf,
cons_wf,
l_subset_nil_right,
sublist_nil,
set-equal-reflex,
decidable__l_member,
deq-exists,
list-diff_wf,
cons_member,
member_singleton,
not_wf,
member-list-diff,
cons_sublist_cons,
or_wf,
and_wf,
sublist_tl2
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
independent_pairFormation,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
cumulativity,
hypothesisEquality,
hypothesis,
sqequalRule,
lambdaEquality,
productEquality,
applyEquality,
functionExtensionality,
addLevel,
allFunctionality,
productElimination,
impliesFunctionality,
dependent_functionElimination,
independent_functionElimination,
because_Cache,
functionEquality,
universeEquality,
dependent_pairFormation,
promote_hyp,
rename,
unionElimination,
andLevelFunctionality,
impliesLevelFunctionality,
voidElimination,
inlFormation,
levelHypothesis,
hyp_replacement,
equalitySymmetry,
dependent_set_memberEquality,
applyLambdaEquality,
setElimination,
inrFormation
Latex:
\mforall{}[T:Type]
((\mforall{}x,y:T. Dec(x = y))
{}\mRightarrow{} (\mforall{}P:(T List) {}\mrightarrow{} \mBbbB{}
((\mforall{}L1,L2:T List. (set-equal(T;L1;L2) {}\mRightarrow{} (\muparrow{}(P L1) \mLeftarrow{}{}\mRightarrow{} \muparrow{}(P L2))))
{}\mRightarrow{} (\mforall{}L:T List. (\muparrow{}exists\_sublist(L;P) \mLeftarrow{}{}\mRightarrow{} \mexists{}LL:T List. (l\_subset(T;LL;L) \mwedge{} (\muparrow{}(P LL))))))))
Date html generated:
2018_05_21-PM-00_51_28
Last ObjectModification:
2017_10_12-AM-10_50_37
Theory : decidable!equality
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