Nuprl Lemma : merge-strict-exists
∀[T:Type]. ∀[R:T ⟶ T ⟶ ℙ].
(Trans(T;a,b.R a b)
⇒ (∀sa,sb:T List.
((∀a,b:T. ((a ∈ sa) ⇒ (b ∈ sb) ⇒ ((R a b) ∨ (R b a))))
⇒ sorted-by(R;sa)
⇒ sorted-by(R;sb)
⇒ (∃sc:T List. (sorted-by(R;sc) ∧ sa ⊆ sc ∧ sb ⊆ sc ∧ sc ⊆ sa @ sb)))))
Proof
Definitions occuring in Statement :
sorted-by: sorted-by(R;L),
l_contains: A ⊆ B,
l_member: (x ∈ l),
append: as @ bs,
list: T List,
trans: Trans(T;x,y.E[x; y]),
uall: ∀[x:A]. B[x],
prop: ℙ,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
or: P ∨ Q,
and: P ∧ Q,
apply: f a,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
all: ∀x:A. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
prop: ℙ,
or: P ∨ Q,
so_apply: x[s],
subtype_rel: A ⊆r B,
uimplies: b supposing a,
and: P ∧ Q,
exists: ∃x:A. B[x],
cand: A c∧ B,
top: Top,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
guard: {T},
l_all: (∀x∈L.P[x]),
l_contains: A ⊆ B,
int_seg: {i..j-},
lelt: i ≤ j < k,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
not: ¬A,
less_than: a < b,
squash: ↓T,
l_member: (x ∈ l),
nat: ℕ,
le: A ≤ B,
ge: i ≥ j ,
trans: Trans(T;x,y.E[x; y]),
append: as @ bs,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3]
Lemmas referenced :
list_induction,
all_wf,
list_wf,
l_member_wf,
or_wf,
sorted-by_wf,
subtype_rel_dep_function,
subtype_rel_self,
set_wf,
exists_wf,
l_contains_wf,
append_wf,
l_contains_nil,
l_contains_weakening,
nil-append,
nil_wf,
sorted-by_wf_nil,
trans_wf,
cons_wf,
append_back_nil,
cons_member,
equal_wf,
sorted-by-cons,
l_all_wf2,
member_append,
select_wf,
int_seg_properties,
length_wf,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
int_seg_wf,
lelt_wf,
and_wf,
iff_weakening_equal,
nat_properties,
l_contains_cons,
cons-l_contains,
list_ind_cons_lemma,
l_all_iff
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
sqequalRule,
lambdaEquality,
cumulativity,
hypothesis,
functionEquality,
because_Cache,
applyEquality,
functionExtensionality,
instantiate,
universeEquality,
setEquality,
independent_isectElimination,
setElimination,
rename,
productEquality,
independent_functionElimination,
dependent_pairFormation,
independent_pairFormation,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
productElimination,
inlFormation,
unionElimination,
inrFormation,
addLevel,
impliesFunctionality,
levelHypothesis,
promote_hyp,
natural_numberEquality,
int_eqEquality,
intEquality,
computeAll,
imageElimination,
impliesLevelFunctionality,
dependent_set_memberEquality,
equalitySymmetry,
equalityTransitivity,
hyp_replacement,
Error :applyLambdaEquality
Latex:
\mforall{}[T:Type]. \mforall{}[R:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}].
(Trans(T;a,b.R a b)
{}\mRightarrow{} (\mforall{}sa,sb:T List.
((\mforall{}a,b:T. ((a \mmember{} sa) {}\mRightarrow{} (b \mmember{} sb) {}\mRightarrow{} ((R a b) \mvee{} (R b a))))
{}\mRightarrow{} sorted-by(R;sa)
{}\mRightarrow{} sorted-by(R;sb)
{}\mRightarrow{} (\mexists{}sc:T List. (sorted-by(R;sc) \mwedge{} sa \msubseteq{} sc \mwedge{} sb \msubseteq{} sc \mwedge{} sc \msubseteq{} sa @ sb)))))
Date html generated:
2016_10_25-AM-10_48_44
Last ObjectModification:
2016_07_12-AM-06_59_55
Theory : general
Home
Index