Nuprl Lemma : adjacent-reverse
∀[T:Type]. ∀L:T List. ∀x,y:T. (adjacent(T;rev(L);x;y) ⇐⇒ adjacent(T;L;y;x))
Proof
Definitions occuring in Statement :
adjacent: adjacent(T;L;x;y),
reverse: rev(as),
list: T List,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
so_apply: x[s],
implies: P ⇒ Q,
top: Top,
prop: ℙ,
iff: P ⇐⇒ Q,
and: P ∧ Q,
uimplies: b supposing a,
false: False,
rev_implies: P ⇐ Q,
decidable: Dec(P),
or: P ∨ Q,
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
cons: [a / b],
bfalse: ff,
not: ¬A,
guard: {T},
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
cand: A c∧ B,
true: True,
append: as @ bs,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
le: A ≤ B
Lemmas referenced :
list_induction,
all_wf,
iff_wf,
adjacent_wf,
reverse_wf,
list_wf,
reverse_nil_lemma,
reverse-cons,
adjacent-nil,
nil_wf,
decidable__lt,
length_wf,
length-reverse,
list-cases,
null_nil_lemma,
length_of_nil_lemma,
product_subtype_list,
null_cons_lemma,
length_of_cons_lemma,
false_wf,
less_than_wf,
equal_wf,
reduce_hd_cons_lemma,
hd_wf,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_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_less_lemma,
int_formula_prop_wf,
last-reverse,
or_wf,
last_wf,
cons_wf,
adjacent-append,
append_wf,
itermAdd_wf,
int_term_value_add_lemma,
adjacent-cons,
adjacent-singleton,
list_ind_nil_lemma,
non_neg_length
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
sqequalRule,
lambdaEquality,
cumulativity,
hypothesis,
independent_functionElimination,
rename,
isect_memberEquality,
voidElimination,
voidEquality,
because_Cache,
dependent_functionElimination,
universeEquality,
independent_pairFormation,
independent_isectElimination,
natural_numberEquality,
unionElimination,
productElimination,
imageElimination,
promote_hyp,
hypothesis_subsumption,
equalityTransitivity,
equalitySymmetry,
inrFormation,
productEquality,
dependent_pairFormation,
int_eqEquality,
intEquality,
computeAll,
inlFormation,
imageMemberEquality,
baseClosed,
addLevel,
impliesFunctionality,
addEquality
Latex:
\mforall{}[T:Type]. \mforall{}L:T List. \mforall{}x,y:T. (adjacent(T;rev(L);x;y) \mLeftarrow{}{}\mRightarrow{} adjacent(T;L;y;x))
Date html generated:
2018_05_21-PM-06_39_13
Last ObjectModification:
2017_07_26-PM-04_53_18
Theory : general
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