Nuprl Lemma : llex-linear
∀[A:Type]. ∀[<:A ⟶ A ⟶ ℙ].
((∀a,b:A. (<[a;b] ∨ (a = b ∈ A) ∨ <[b;a]))
⇒ (∀L1,L2:A List. ((L1 llex(A;a,b.<[a;b]) L2) ∨ (L1 = L2 ∈ (A List)) ∨ (L2 llex(A;a,b.<[a;b]) L1))))
Proof
Definitions occuring in Statement :
llex: llex(A;a,b.<[a; b]),
list: T List,
uall: ∀[x:A]. B[x],
prop: ℙ,
infix_ap: x f y,
so_apply: x[s1;s2],
all: ∀x:A. B[x],
implies: P ⇒ Q,
or: P ∨ Q,
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],
member: t ∈ T,
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
prop: ℙ,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
so_apply: x[s],
or: P ∨ Q,
guard: {T},
infix_ap: x f y,
llex: llex(A;a,b.<[a; b]),
top: Top,
and: P ∧ Q,
int_seg: {i..j-},
uimplies: b supposing a,
lelt: i ≤ j < k,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
ge: i ≥ j ,
le: A ≤ B,
nat: ℕ,
less_than': less_than'(a;b),
cand: A c∧ B,
nat_plus: ℕ+,
less_than: a < b,
squash: ↓T,
true: True,
uiff: uiff(P;Q),
select: L[n],
cons: [a / b],
sq_type: SQType(T),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
subtract: n - m
Lemmas referenced :
list_induction,
all_wf,
list_wf,
or_wf,
infix_ap_wf,
llex_wf,
equal_wf,
nil-llex,
nil_wf,
equal-wf-base-T,
cons_wf,
length_of_cons_lemma,
less_than_wf,
length_wf,
int_seg_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,
non_neg_length,
decidable__lt,
intformless_wf,
itermAdd_wf,
int_formula_prop_less_lemma,
int_term_value_add_lemma,
false_wf,
le_wf,
add_nat_plus,
length_wf_nat,
nat_plus_wf,
nat_plus_properties,
add-is-int-iff,
intformeq_wf,
int_formula_prop_eq_lemma,
nat_properties,
decidable__equal_int,
subtype_base_sq,
int_subtype_base,
exists_wf,
nat_wf,
select-cons-tl,
subtract_wf,
add-subtract-cancel,
select_cons_tl,
iff_weakening_equal,
add-associates,
add-swap,
add-commutes,
zero-add,
and_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
lelt_wf,
squash_wf,
true_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
sqequalRule,
lambdaEquality,
cumulativity,
hypothesis,
instantiate,
because_Cache,
applyEquality,
universeEquality,
functionExtensionality,
independent_functionElimination,
rename,
dependent_functionElimination,
functionEquality,
unionElimination,
inrFormation,
inlFormation,
equalitySymmetry,
baseClosed,
isect_memberEquality,
voidElimination,
voidEquality,
productEquality,
addEquality,
natural_numberEquality,
setElimination,
independent_isectElimination,
productElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
independent_pairFormation,
computeAll,
dependent_set_memberEquality,
imageMemberEquality,
equalityTransitivity,
applyLambdaEquality,
pointwiseFunctionality,
promote_hyp,
baseApply,
closedConclusion,
imageElimination,
hyp_replacement
Latex:
\mforall{}[A:Type]. \mforall{}[<:A {}\mrightarrow{} A {}\mrightarrow{} \mBbbP{}].
((\mforall{}a,b:A. (<[a;b] \mvee{} (a = b) \mvee{} <[b;a]))
{}\mRightarrow{} (\mforall{}L1,L2:A List. ((L1 llex(A;a,b.<[a;b]) L2) \mvee{} (L1 = L2) \mvee{} (L2 llex(A;a,b.<[a;b]) L1))))
Date html generated:
2018_05_21-PM-07_17_47
Last ObjectModification:
2017_07_26-PM-05_04_29
Theory : general
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