Nuprl Lemma : adjacent-to-last
∀[T:Type]. ∀L:T List. (∀a:T. (adjacent(T;L;last(L);a) ⇐⇒ False)) supposing (no_repeats(T;L) and 0 < ||L||)
Proof
Definitions occuring in Statement :
adjacent: adjacent(T;L;x;y),
last: last(L),
no_repeats: no_repeats(T;l),
length: ||as||,
list: T List,
less_than: a < b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
false: False,
natural_number: $n,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
uimplies: b supposing a,
prop: 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),
false: False,
and: P ∧ Q,
cons: [a / b],
top: Top,
bfalse: ff,
not: ¬A,
implies: P ⇒ Q,
so_apply: x[s],
last: last(L),
subtract: n - m,
select: L[n],
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
subtype_rel: A ⊆r B,
true: True,
guard: {T},
no_repeats: no_repeats(T;l),
nat: ℕ,
le: A ≤ B,
ge: i ≥ j ,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
uiff: uiff(P;Q),
sq_type: SQType(T)
Lemmas referenced :
list_induction,
isect_wf,
less_than_wf,
length_wf,
no_repeats_wf,
all_wf,
iff_wf,
adjacent_wf,
last_wf,
list-cases,
null_nil_lemma,
length_of_nil_lemma,
product_subtype_list,
null_cons_lemma,
length_of_cons_lemma,
false_wf,
list_wf,
member-less_than,
no_repeats_witness,
cons_wf,
nil_wf,
adjacent-singleton,
assert_elim,
null_wf3,
subtype_rel_list,
top_wf,
bfalse_wf,
btrue_neq_bfalse,
assert_wf,
squash_wf,
true_wf,
last_cons,
iff_weakening_equal,
adjacent-cons,
le_wf,
subtract_wf,
le_weakening2,
non_neg_length,
decidable__lt,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermSubtract_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_subtract_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
add-is-int-iff,
nat_properties,
intformeq_wf,
int_formula_prop_eq_lemma,
equal_wf,
nat_wf,
decidable__le,
select_cons_tl,
add-subtract-cancel,
subtype_base_sq,
int_subtype_base,
decidable__equal_int,
no_repeats_cons
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
sqequalRule,
lambdaEquality,
natural_numberEquality,
cumulativity,
hypothesis,
because_Cache,
independent_isectElimination,
dependent_functionElimination,
unionElimination,
imageElimination,
productElimination,
voidElimination,
promote_hyp,
hypothesis_subsumption,
isect_memberEquality,
voidEquality,
independent_functionElimination,
rename,
universeEquality,
independent_pairFormation,
addEquality,
addLevel,
applyEquality,
levelHypothesis,
equalityTransitivity,
equalitySymmetry,
imageMemberEquality,
baseClosed,
dependent_set_memberEquality,
dependent_pairFormation,
int_eqEquality,
intEquality,
computeAll,
pointwiseFunctionality,
baseApply,
closedConclusion,
applyLambdaEquality,
setElimination,
instantiate
Latex:
\mforall{}[T:Type]
\mforall{}L:T List. (\mforall{}a:T. (adjacent(T;L;last(L);a) \mLeftarrow{}{}\mRightarrow{} False)) supposing (no\_repeats(T;L) and 0 < ||L||)
Date html generated:
2018_05_21-PM-06_40_06
Last ObjectModification:
2017_07_26-PM-04_53_36
Theory : general
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