Nuprl Lemma : decidable__equal_list
∀[T:Type]. ((∀x,y:T. Dec(x = y ∈ T)) ⇒ (∀xs,ys:T List. Dec(xs = ys ∈ (T List))))
Proof
Definitions occuring in Statement :
list: T List,
decidable: Dec(P),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
all: ∀x:A. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
so_apply: x[s],
prop: ℙ,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
and: P ∧ Q,
top: Top,
false: False,
squash: ↓T,
true: True,
cand: A c∧ B,
uimplies: b supposing a,
ge: i ≥ j ,
subtype_rel: A ⊆r B
Lemmas referenced :
tl_wf,
reduce_tl_cons_lemma,
top_wf,
subtype_rel_list,
length_cons_ge_one,
length_wf,
ge_wf,
squash_wf,
hd_wf,
reduce_hd_cons_lemma,
decidable__and2,
cons_neq_nil,
btrue_neq_bfalse,
bfalse_wf,
null_cons_lemma,
null_wf,
and_wf,
btrue_wf,
null_nil_lemma,
not_wf,
cons_wf,
nil_wf,
equal_wf,
decidable_wf,
list_wf,
all_wf,
list_induction
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
thin,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
sqequalRule,
lambdaEquality,
hypothesis,
independent_functionElimination,
because_Cache,
rename,
dependent_functionElimination,
universeEquality,
inlFormation,
inrFormation,
dependent_set_memberEquality,
independent_pairFormation,
equalityTransitivity,
equalitySymmetry,
applyEquality,
setElimination,
productElimination,
setEquality,
isect_memberEquality,
voidElimination,
voidEquality,
unionElimination,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
introduction,
independent_isectElimination
Latex:
\mforall{}[T:Type]. ((\mforall{}x,y:T. Dec(x = y)) {}\mRightarrow{} (\mforall{}xs,ys:T List. Dec(xs = ys)))
Date html generated:
2016_05_14-AM-07_39_53
Last ObjectModification:
2016_01_15-AM-08_36_41
Theory : list_1
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