Nuprl Lemma : sublist-sorted-by
∀[T:Type]. ∀[R:T ⟶ T ⟶ ℙ]. ∀sa,sb:T List. (sa ⊆ sb ⇒ sorted-by(R;sb) ⇒ sorted-by(R;sa))
Proof
Definitions occuring in Statement :
sorted-by: sorted-by(R;L),
sublist: L1 ⊆ L2,
list: T List,
uall: ∀[x:A]. B[x],
prop: ℙ,
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
sublist: L1 ⊆ L2,
exists: ∃x:A. B[x],
sorted-by: sorted-by(R;L),
and: P ∧ Q,
member: t ∈ T,
int_seg: {i..j-},
lelt: i ≤ j < k,
guard: {T},
decidable: Dec(P),
or: P ∨ Q,
less_than: a < b,
squash: ↓T,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
not: ¬A,
top: Top,
prop: ℙ,
le: A ≤ B,
subtype_rel: A ⊆r B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
ge: i ≥ j ,
nat: ℕ
Lemmas referenced :
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
itermConstant_wf,
intformle_wf,
le_wf,
nat_properties,
decidable__le,
non_neg_length,
length_wf_nat,
increasing_implies,
list_wf,
sublist_wf,
set_wf,
subtype_rel_self,
l_member_wf,
subtype_rel_dep_function,
sorted-by_wf,
int_seg_wf,
iff_weakening_equal,
lelt_wf,
int_formula_prop_wf,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
itermVar_wf,
intformless_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
decidable__lt,
length_wf,
int_seg_properties
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
thin,
cut,
applyEquality,
hypothesisEquality,
hypothesis,
dependent_functionElimination,
setElimination,
rename,
dependent_set_memberEquality,
independent_pairFormation,
lemma_by_obid,
isectElimination,
natural_numberEquality,
unionElimination,
imageElimination,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
computeAll,
cumulativity,
universeEquality,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
instantiate,
functionEquality,
setEquality,
because_Cache
Latex:
\mforall{}[T:Type]. \mforall{}[R:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}]. \mforall{}sa,sb:T List. (sa \msubseteq{} sb {}\mRightarrow{} sorted-by(R;sb) {}\mRightarrow{} sorted-by(R;sa))
Date html generated:
2016_05_14-PM-01_48_11
Last ObjectModification:
2016_01_15-AM-08_23_42
Theory : list_1
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