Nuprl Lemma : l_tricotomy
∀[T:Type]. ∀x,y:T. ∀L:T List. ((x ∈ L) ⇒ (y ∈ L) ⇒ (((x = y ∈ T) ∨ x before y ∈ L) ∨ y before x ∈ L))
Proof
Definitions occuring in Statement :
l_before: x before y ∈ l,
l_member: (x ∈ l),
list: T List,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
or: P ∨ Q,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
l_before: x before y ∈ l,
l_member: (x ∈ l),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
exists: ∃x:A. B[x],
cand: A c∧ B,
member: t ∈ T,
prop: ℙ,
so_lambda: λ2x.t[x],
nat: ℕ,
uimplies: b supposing a,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
not: ¬A,
top: Top,
and: P ∧ Q,
so_apply: x[s],
squash: ↓T,
sublist: L1 ⊆ L2,
int_seg: {i..j-},
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
lelt: i ≤ j < k,
le: A ≤ B,
bfalse: ff,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
increasing: increasing(f;k),
subtract: n - m,
nequal: a ≠ b ∈ T ,
less_than': less_than'(a;b),
subtype_rel: A ⊆r B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
select: L[n],
cons: [a / b],
true: True
Lemmas referenced :
exists_wf,
nat_wf,
less_than_wf,
length_wf,
equal_wf,
select_wf,
nat_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,
list_wf,
decidable__lt,
sublist_wf,
cons_wf,
nil_wf,
decidable__equal_int,
squash_wf,
le_wf,
intformeq_wf,
int_formula_prop_eq_lemma,
intformless_wf,
int_formula_prop_less_lemma,
or_wf,
length_of_cons_lemma,
length_of_nil_lemma,
eq_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
lelt_wf,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
int_seg_wf,
int_seg_properties,
itermAdd_wf,
int_term_value_add_lemma,
increasing_wf,
false_wf,
all_wf,
non_neg_length,
length_wf_nat,
equal-wf-T-base,
assert_wf,
select-cons-hd,
bnot_wf,
not_wf,
uiff_transitivity,
iff_transitivity,
iff_weakening_uiff,
assert_of_bnot,
int_subtype_base,
true_wf,
select_cons_tl,
length-singleton,
iff_weakening_equal,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
thin,
cut,
introduction,
extract_by_obid,
isectElimination,
hypothesis,
lambdaEquality,
productEquality,
setElimination,
rename,
because_Cache,
cumulativity,
hypothesisEquality,
independent_isectElimination,
dependent_functionElimination,
natural_numberEquality,
unionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
universeEquality,
inlFormation,
inrFormation,
applyEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
imageMemberEquality,
baseClosed,
equalityElimination,
dependent_set_memberEquality,
promote_hyp,
instantiate,
independent_functionElimination,
addEquality,
functionExtensionality,
applyLambdaEquality,
impliesFunctionality
Latex:
\mforall{}[T:Type]. \mforall{}x,y:T. \mforall{}L:T List. ((x \mmember{} L) {}\mRightarrow{} (y \mmember{} L) {}\mRightarrow{} (((x = y) \mvee{} x before y \mmember{} L) \mvee{} y before x \mmember{} L))
Date html generated:
2017_04_14-AM-09_30_01
Last ObjectModification:
2017_02_27-PM-04_02_27
Theory : list_1
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