Nuprl Lemma : iseg_append_iff
∀[T:Type]
∀l1,l2,l3:T List. (l1 ≤ l2 @ l3 ⇐⇒ l1 ≤ l2 ∨ (∃l:T List. (0 < ||l|| ∧ (l1 = (l2 @ l) ∈ (T List)) ∧ l ≤ l3)))
Proof
Definitions occuring in Statement :
iseg: l1 ≤ l2,
length: ||as||,
append: as @ bs,
list: T List,
less_than: a < b,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
or: P ∨ Q,
and: P ∧ Q,
natural_number: $n,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
prop: ℙ,
and: P ∧ Q,
top: Top,
so_apply: x[s],
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
or: P ∨ Q,
rev_implies: P ⇐ Q,
append: as @ bs,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
guard: {T},
exists: ∃x:A. B[x],
cand: A c∧ B,
ge: i ≥ j ,
decidable: Dec(P),
le: A ≤ B,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
not: ¬A,
assert: ↑b,
ifthenelse: if b then t else f fi ,
bfalse: ff,
squash: ↓T,
true: True
Lemmas referenced :
list_induction,
all_wf,
list_wf,
iff_wf,
iseg_wf,
append_wf,
or_wf,
exists_wf,
less_than_wf,
length_wf,
equal_wf,
length-append,
nil_iseg,
equal-wf-base-T,
length_of_nil_lemma,
nil_wf,
cons_wf,
length_of_cons_lemma,
list_ind_nil_lemma,
non_neg_length,
decidable__lt,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermAdd_wf,
itermVar_wf,
intformle_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
iseg_nil,
null_cons_lemma,
list_ind_cons_lemma,
cons_iseg,
iseg_append,
reduce_tl_cons_lemma,
and_wf,
tl_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,
because_Cache,
productEquality,
natural_numberEquality,
applyLambdaEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_functionElimination,
rename,
dependent_functionElimination,
universeEquality,
independent_pairFormation,
inlFormation,
baseClosed,
inrFormation,
dependent_pairFormation,
addEquality,
unionElimination,
productElimination,
independent_isectElimination,
int_eqEquality,
intEquality,
computeAll,
hyp_replacement,
equalitySymmetry,
applyEquality,
imageElimination,
imageMemberEquality,
addLevel,
orFunctionality,
dependent_set_memberEquality,
equalityTransitivity,
setElimination
Latex:
\mforall{}[T:Type]
\mforall{}l1,l2,l3:T List.
(l1 \mleq{} l2 @ l3 \mLeftarrow{}{}\mRightarrow{} l1 \mleq{} l2 \mvee{} (\mexists{}l:T List. (0 < ||l|| \mwedge{} (l1 = (l2 @ l)) \mwedge{} l \mleq{} l3)))
Date html generated:
2017_04_17-AM-08_45_52
Last ObjectModification:
2017_02_27-PM-05_05_05
Theory : list_1
Home
Index