Nuprl Lemma : assert-deq-all-disjoint
∀[A:Type]. ∀[eq:EqDecider(A)]. ∀[ass:A List List]. ∀[bs:A List].
uiff(↑deq-all-disjoint(eq;ass;bs);(∀as∈ass.l_disjoint(A;as;bs)))
Proof
Definitions occuring in Statement :
deq-all-disjoint: deq-all-disjoint(eq;ass;bs),
l_disjoint: l_disjoint(T;l1;l2),
l_all: (∀x∈L.P[x]),
list: T List,
deq: EqDecider(T),
assert: ↑b,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
universe: Type
Definitions unfolded in proof :
deq-all-disjoint: deq-all-disjoint(eq;ass;bs),
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
member: t ∈ T,
l_all: (∀x∈L.P[x]),
all: ∀x:A. B[x],
l_disjoint: l_disjoint(T;l1;l2),
not: ¬A,
implies: P ⇒ Q,
false: False,
uall: ∀[x:A]. B[x],
int_seg: {i..j-},
guard: {T},
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
prop: ℙ,
less_than: a < b,
squash: ↓T,
so_lambda: λ2x.t[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
assert-deq-disjoint,
l_all_functionality,
assert-bl-all,
iff_weakening_uiff,
iff_transitivity,
assert_witness,
deq_wf,
deq-all-disjoint_wf,
uiff_wf,
deq-disjoint_wf,
bl-all_wf,
assert_wf,
l_disjoint_wf,
l_all_wf,
int_seg_wf,
int_formula_prop_less_lemma,
intformless_wf,
decidable__lt,
int_formula_prop_wf,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
itermVar_wf,
itermConstant_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
decidable__le,
length_wf,
int_seg_properties,
list_wf,
select_wf,
l_member_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
independent_pairFormation,
isect_memberFormation,
introduction,
cut,
hypothesis,
sqequalRule,
sqequalHypSubstitution,
lambdaEquality,
dependent_functionElimination,
thin,
hypothesisEquality,
voidElimination,
productEquality,
lemma_by_obid,
isectElimination,
cumulativity,
because_Cache,
setElimination,
rename,
independent_isectElimination,
natural_numberEquality,
productElimination,
unionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidEquality,
computeAll,
imageElimination,
setEquality,
applyEquality,
universeEquality,
independent_pairEquality,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
addLevel,
lambdaFormation
Latex:
\mforall{}[A:Type]. \mforall{}[eq:EqDecider(A)]. \mforall{}[ass:A List List]. \mforall{}[bs:A List].
uiff(\muparrow{}deq-all-disjoint(eq;ass;bs);(\mforall{}as\mmember{}ass.l\_disjoint(A;as;bs)))
Date html generated:
2016_05_14-PM-03_24_08
Last ObjectModification:
2016_01_14-PM-11_22_44
Theory : decidable!equality
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