Nuprl Lemma : adjacent-append
∀[T:Type]
∀x,y:T. ∀L1,L2:T List.
(adjacent(T;L1 @ L2;x;y)
⇐⇒ adjacent(T;L1;x;y) ∨ (0 < ||L1|| ∧ 0 < ||L2|| ∧ (x = last(L1) ∈ T) ∧ (y = hd(L2) ∈ T)) ∨ adjacent(T;L2;x;y))
Proof
Definitions occuring in Statement :
adjacent: adjacent(T;L;x;y),
last: last(L),
length: ||as||,
append: as @ bs,
hd: hd(l),
list: T List,
less_than: a < b,
uall: ∀[x:A]. B[x],
all: ∀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 :
adjacent: adjacent(T;L;x;y),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
exists: ∃x:A. B[x],
member: t ∈ T,
int_seg: {i..j-},
uimplies: b supposing a,
top: Top,
guard: {T},
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
prop: ℙ,
less_than: a < b,
squash: ↓T,
uiff: uiff(P;Q),
rev_implies: P ⇐ Q,
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
less_than': less_than'(a;b),
cons: [a / b],
bfalse: ff,
ge: i ≥ j ,
subtract: n - m,
append: as @ bs,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
true: True,
subtype_rel: A ⊆r B,
cand: A c∧ B,
le: A ≤ B,
last: last(L),
select: L[n],
nil: [],
it: ⋅,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
sq_type: SQType(T)
Lemmas referenced :
int_seg_wf,
subtract_wf,
length_wf,
append_wf,
select_wf,
length-append,
istype-void,
int_seg_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
istype-int,
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,
subtract-is-int-iff,
add-is-int-iff,
intformless_wf,
itermAdd_wf,
itermSubtract_wf,
int_formula_prop_less_lemma,
int_term_value_add_lemma,
int_term_value_subtract_lemma,
false_wf,
istype-less_than,
last_wf,
list-cases,
null_nil_lemma,
length_of_nil_lemma,
product_subtype_list,
null_cons_lemma,
length_of_cons_lemma,
hd_wf,
list_wf,
istype-universe,
list_ind_nil_lemma,
equal_wf,
squash_wf,
true_wf,
select_append_front,
istype-le,
subtype_rel_self,
iff_weakening_equal,
add-member-int_seg2,
select_append_back,
non_neg_length,
length_append,
subtype_rel_list,
top_wf,
le_wf,
less_than_wf,
decidable__equal_int,
intformeq_wf,
int_formula_prop_eq_lemma,
stuck-spread,
istype-base,
subtype_base_sq,
int_subtype_base,
reduce_hd_cons_lemma,
general_arith_equation1,
add-associates,
add-swap,
add-commutes,
zero-add,
select-nthtl,
length_wf_nat,
nth_tl_append,
add-member-int_seg1,
minus-one-mul,
add-mul-special,
zero-mul,
add-zero
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation_alt,
lambdaFormation_alt,
independent_pairFormation,
sqequalHypSubstitution,
productElimination,
thin,
productIsType,
universeIsType,
cut,
introduction,
extract_by_obid,
isectElimination,
natural_numberEquality,
hypothesisEquality,
hypothesis,
equalityIsType1,
inhabitedIsType,
because_Cache,
setElimination,
rename,
independent_isectElimination,
isect_memberEquality_alt,
voidElimination,
addEquality,
dependent_functionElimination,
unionElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
pointwiseFunctionality,
equalityTransitivity,
equalitySymmetry,
promote_hyp,
imageElimination,
baseApply,
closedConclusion,
baseClosed,
unionIsType,
hypothesis_subsumption,
instantiate,
universeEquality,
inlFormation_alt,
applyEquality,
dependent_set_memberEquality_alt,
imageMemberEquality,
minusEquality,
inrFormation_alt,
productEquality,
hyp_replacement,
applyLambdaEquality,
cumulativity,
intEquality
Latex:
\mforall{}[T:Type]
\mforall{}x,y:T. \mforall{}L1,L2:T List.
(adjacent(T;L1 @ L2;x;y)
\mLeftarrow{}{}\mRightarrow{} adjacent(T;L1;x;y)
\mvee{} (0 < ||L1|| \mwedge{} 0 < ||L2|| \mwedge{} (x = last(L1)) \mwedge{} (y = hd(L2)))
\mvee{} adjacent(T;L2;x;y))
Date html generated:
2019_10_15-AM-11_08_21
Last ObjectModification:
2018_10_18-PM-11_52_15
Theory : general
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