Nuprl Lemma : llex-append1
∀[A:Type]. ∀[<:A ⟶ A ⟶ ℙ].
∀L1,L2:A List. ∀a:A.
((L1 llex(A;a,b.<[a;b]) (L2 @ [a]))
⇒ ((L1 llex(A;a,b.<[a;b]) L2) ∨ (∃L3:A List. ((L1 = (L2 @ L3) ∈ (A List)) ∧ <[hd(L3);a] supposing 0 < ||L3||))))
Proof
Definitions occuring in Statement :
llex: llex(A;a,b.<[a; b]),
length: ||as||,
append: as @ bs,
hd: hd(l),
cons: [a / b],
nil: [],
list: T List,
less_than: a < b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
prop: ℙ,
infix_ap: x f y,
so_apply: x[s1;s2],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
or: P ∨ Q,
and: P ∧ Q,
function: x:A ⟶ B[x],
natural_number: $n,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
llex: llex(A;a,b.<[a; b]),
infix_ap: x f y,
or: P ∨ Q,
member: t ∈ T,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
subtype_rel: A ⊆r B,
prop: ℙ,
decidable: Dec(P),
and: P ∧ Q,
exists: ∃x:A. B[x],
uimplies: b supposing a,
less_than: a < b,
squash: ↓T,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
cand: A c∧ B,
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
nat: ℕ,
ge: i ≥ j ,
top: Top,
less_than': less_than'(a;b),
sq_type: SQType(T),
iseg: l1 ≤ l2,
cons: [a / b],
select: L[n],
subtract: n - m,
so_apply: x[s],
so_lambda: λ2x.t[x]
Lemmas referenced :
llex_wf,
append_wf,
cons_wf,
nil_wf,
subtype_rel_self,
list_wf,
istype-universe,
decidable__lt,
length_wf,
length-append,
istype-less_than,
length_of_cons_lemma,
length_of_nil_lemma,
full-omega-unsat,
intformand_wf,
intformless_wf,
itermConstant_wf,
itermVar_wf,
intformeq_wf,
itermAdd_wf,
istype-int,
int_formula_prop_and_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_eq_lemma,
int_term_value_add_lemma,
int_formula_prop_wf,
equal_wf,
select_wf,
int_seg_properties,
decidable__le,
intformnot_wf,
intformle_wf,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
iff_weakening_equal,
select_append_front,
istype-le,
int_seg_wf,
istype-nat,
nat_properties,
infix_ap_wf,
istype-void,
hd_wf,
member-less_than,
top_wf,
subtype_rel_list,
append-nil,
true_wf,
squash_wf,
decidable__equal_int,
length_append,
length_cons,
non_neg_length,
length_nil,
list_extensionality,
int_subtype_base,
subtype_base_sq,
iseg_select,
select_append_back,
int_term_value_subtract_lemma,
itermSubtract_wf,
less_than_wf,
trivial-equal,
select0,
zero-mul,
add-mul-special,
minus-one-mul,
le_wf,
set_subtype_base,
length_wf_nat
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
lambdaFormation_alt,
sqequalHypSubstitution,
sqequalRule,
unionElimination,
thin,
universeIsType,
cut,
applyEquality,
introduction,
extract_by_obid,
isectElimination,
hypothesisEquality,
lambdaEquality_alt,
inhabitedIsType,
hypothesis,
instantiate,
universeEquality,
functionIsType,
because_Cache,
dependent_functionElimination,
inlFormation_alt,
productElimination,
productIsType,
equalityIstype,
applyLambdaEquality,
Error :memTop,
isectIsType,
natural_numberEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
int_eqEquality,
independent_pairFormation,
voidElimination,
setElimination,
rename,
imageMemberEquality,
baseClosed,
dependent_set_memberEquality_alt,
cumulativity,
isect_memberEquality_alt,
equalityIsType1,
inrFormation_alt,
voidEquality,
intEquality,
promote_hyp,
addEquality,
hyp_replacement,
closedConclusion,
baseApply,
equalityIsType4
Latex:
\mforall{}[A:Type]. \mforall{}[<:A {}\mrightarrow{} A {}\mrightarrow{} \mBbbP{}].
\mforall{}L1,L2:A List. \mforall{}a:A.
((L1 llex(A;a,b.<[a;b]) (L2 @ [a]))
{}\mRightarrow{} ((L1 llex(A;a,b.<[a;b]) L2)
\mvee{} (\mexists{}L3:A List. ((L1 = (L2 @ L3)) \mwedge{} <[hd(L3);a] supposing 0 < ||L3||))))
Date html generated:
2020_05_20-AM-08_07_51
Last ObjectModification:
2019_12_26-PM-04_06_53
Theory : general
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