Nuprl Lemma : assert-list_eq
∀[A:Type]. ∀[eq:EqDecider(A)]. ∀[as,bs:A List]. uiff(↑list_eq(eq;as;bs);as = bs ∈ (A List))
Proof
Definitions occuring in Statement :
list_eq: list_eq(eq;as;bs),
list: T List,
deq: EqDecider(T),
assert: ↑b,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
deq: EqDecider(T),
so_apply: x[s],
implies: P ⇒ Q,
list_eq: list_eq(eq;as;bs),
ifthenelse: if b then t else f fi ,
btrue: tt,
assert: ↑b,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
prop: ℙ,
true: True,
all: ∀x:A. B[x],
top: Top,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
bfalse: ff,
false: False,
not: ¬A,
bnot: ¬bb,
band: p ∧b q,
cand: A c∧ B,
squash: ↓T,
ge: i ≥ j ,
subtype_rel: A ⊆r B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
eqof: eqof(d),
guard: {T}
Lemmas referenced :
list_induction,
uall_wf,
list_wf,
uiff_wf,
assert_wf,
list_eq_wf,
equal_wf,
nil_wf,
equal-wf-base-T,
null_nil_lemma,
true_wf,
equal-wf-base,
null_cons_lemma,
spread_cons_lemma,
false_wf,
btrue_wf,
and_wf,
null_wf,
bfalse_wf,
btrue_neq_bfalse,
cons_wf,
assert_witness,
equal-wf-T-base,
band_wf,
reduce_hd_cons_lemma,
hd_wf,
squash_wf,
ge_wf,
length_wf,
length_cons_ge_one,
subtype_rel_list,
top_wf,
iff_transitivity,
eqof_wf,
iff_weakening_uiff,
assert_of_band,
safe-assert-deq,
deq_wf,
reduce_tl_cons_lemma,
tl_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
thin,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
sqequalRule,
lambdaEquality,
hypothesis,
setElimination,
rename,
independent_functionElimination,
because_Cache,
baseClosed,
independent_pairFormation,
natural_numberEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
lambdaFormation,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
productElimination,
dependent_set_memberEquality,
applyLambdaEquality,
independent_pairEquality,
applyEquality,
imageElimination,
independent_isectElimination,
imageMemberEquality,
addLevel,
productEquality,
universeEquality,
promote_hyp
Latex:
\mforall{}[A:Type]. \mforall{}[eq:EqDecider(A)]. \mforall{}[as,bs:A List]. uiff(\muparrow{}list\_eq(eq;as;bs);as = bs)
Date html generated:
2018_05_21-PM-00_20_12
Last ObjectModification:
2018_05_19-AM-07_00_23
Theory : list_0
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