Nuprl Lemma : length-list-diff
∀[T:Type]. ∀eq:EqDecider(T). ∀as,bs:T List. ((||as-bs|| ≤ ||as||) ∧ ||as-bs|| < ||as|| supposing (∃a∈as. (a ∈ bs)))
Proof
Definitions occuring in Statement :
list-diff: as-bs,
l_exists: (∃x∈L. P[x]),
l_member: (x ∈ l),
length: ||as||,
list: T List,
deq: EqDecider(T),
less_than: a < b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
le: A ≤ B,
all: ∀x:A. B[x],
and: P ∧ Q,
universe: Type
Definitions unfolded in proof :
list-diff: as-bs,
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
and: P ∧ Q,
cand: A c∧ B,
uimplies: b supposing a,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
le: A ≤ B,
not: ¬A,
implies: P ⇒ Q,
false: False,
iff: P ⇐⇒ Q,
exists: ∃x:A. B[x],
l_member: (x ∈ l),
l_exists: (∃x∈L. P[x]),
nat: ℕ,
int_seg: {i..j-},
lelt: i ≤ j < k,
guard: {T},
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
less_than: a < b,
squash: ↓T,
rev_implies: P ⇐ Q
Lemmas referenced :
length-filter,
bnot_wf,
deq-member_wf,
l_exists_wf,
l_member_wf,
list_wf,
deq_wf,
less_than'_wf,
length_wf,
filter_wf5,
member-less_than,
length-filter-decreases,
l_exists_iff,
lelt_wf,
not_wf,
assert_wf,
select_wf,
int_seg_properties,
nat_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_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_wf,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
false_wf,
iff_transitivity,
iff_weakening_uiff,
assert_of_bnot,
assert-deq-member
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
lambdaEquality,
cumulativity,
hypothesis,
independent_pairFormation,
setElimination,
rename,
setEquality,
dependent_functionElimination,
productElimination,
independent_pairEquality,
voidElimination,
because_Cache,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
isect_memberEquality,
independent_isectElimination,
universeEquality,
independent_functionElimination,
dependent_pairFormation,
dependent_set_memberEquality,
hyp_replacement,
Error :applyLambdaEquality,
natural_numberEquality,
unionElimination,
int_eqEquality,
intEquality,
voidEquality,
computeAll,
imageElimination,
functionEquality,
addLevel,
impliesFunctionality
Latex:
\mforall{}[T:Type]
\mforall{}eq:EqDecider(T). \mforall{}as,bs:T List.
((||as-bs|| \mleq{} ||as||) \mwedge{} ||as-bs|| < ||as|| supposing (\mexists{}a\mmember{}as. (a \mmember{} bs)))
Date html generated:
2016_10_21-AM-10_42_47
Last ObjectModification:
2016_07_12-AM-05_50_28
Theory : decidable!equality
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