Nuprl Lemma : sq_stable__sorted-by
∀[T:Type]
∀L:T List
∀[R:{x:T| (x ∈ L)} ⟶ {x:T| (x ∈ L)} ⟶ ℙ]. ((∀x,y:{x:T| (x ∈ L)} . SqStable(R x y)) ⇒ SqStable(sorted-by(R;L)))
Proof
Definitions occuring in Statement :
sorted-by: sorted-by(R;L),
l_member: (x ∈ l),
list: T List,
sq_stable: SqStable(P),
uall: ∀[x:A]. B[x],
prop: ℙ,
all: ∀x:A. B[x],
implies: P ⇒ Q,
set: {x:A| B[x]} ,
apply: f a,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
sorted-by: sorted-by(R;L),
int_seg: {i..j-},
uimplies: b supposing a,
guard: {T},
lelt: i ≤ j < k,
and: P ∧ Q,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top,
less_than: a < b,
squash: ↓T
Lemmas referenced :
list-subtype,
list_wf,
l_member_wf,
equal_wf,
all_wf,
sq_stable_wf,
sq_stable__all,
int_seg_wf,
length_wf,
select_wf,
int_seg_properties,
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
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
cumulativity,
hypothesisEquality,
equalityTransitivity,
hypothesis,
equalitySymmetry,
setEquality,
dependent_functionElimination,
independent_functionElimination,
sqequalRule,
lambdaEquality,
applyEquality,
functionExtensionality,
setElimination,
rename,
dependent_set_memberEquality,
because_Cache,
functionEquality,
universeEquality,
natural_numberEquality,
independent_isectElimination,
productElimination,
unionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
imageElimination
Latex:
\mforall{}[T:Type]
\mforall{}L:T List
\mforall{}[R:\{x:T| (x \mmember{} L)\} {}\mrightarrow{} \{x:T| (x \mmember{} L)\} {}\mrightarrow{} \mBbbP{}]
((\mforall{}x,y:\{x:T| (x \mmember{} L)\} . SqStable(R x y)) {}\mRightarrow{} SqStable(sorted-by(R;L)))
Date html generated:
2018_05_21-PM-06_51_06
Last ObjectModification:
2017_07_26-PM-04_57_42
Theory : general
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