Nuprl Lemma : list_eq-sq-list-deq
∀[eq:Top]. ∀[as,bs:Top List]. (list_eq(eq;as;bs) ~ list-deq(eq) as bs)
Proof
Definitions occuring in Statement :
list_eq: list_eq(eq;as;bs),
list-deq: list-deq(eq),
list: T List,
uall: ∀[x:A]. B[x],
top: Top,
apply: f a,
sqequal: s ~ t
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
guard: {T},
uimplies: b supposing a,
prop: ℙ,
subtype_rel: A ⊆r B,
or: P ∨ Q,
list_eq: list_eq(eq;as;bs),
ifthenelse: if b then t else f fi ,
btrue: tt,
list-deq: list-deq(eq),
list_ind: list_ind,
nil: [],
it: ⋅,
null: null(as),
assert: ↑b,
iff: P ⇐⇒ Q,
and: P ∧ Q,
true: True,
rev_implies: P ⇐ Q,
sq_type: SQType(T),
cons: [a / b],
colength: colength(L),
so_lambda: λ2x y.t[x; y],
top: Top,
so_apply: x[s1;s2],
so_lambda: λ2x.t[x],
so_apply: x[s],
bfalse: ff,
squash: ↓T,
sq_stable: SqStable(P),
uiff: uiff(P;Q),
le: A ≤ B,
not: ¬A,
less_than': less_than'(a;b),
decidable: Dec(P),
subtract: n - m,
less_than: a < b,
bnot: ¬bb,
band: p ∧b q,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3]
Lemmas referenced :
nat_properties,
less_than_transitivity1,
less_than_irreflexivity,
ge_wf,
less_than_wf,
equal-wf-T-base,
nat_wf,
colength_wf_list,
top_wf,
int_subtype_base,
list-cases,
null_nil_lemma,
btrue_wf,
subtype_base_sq,
bool_wf,
bool_subtype_base,
iff_imp_equal_bool,
true_wf,
assert_wf,
product_subtype_list,
spread_cons_lemma,
equal_wf,
set_subtype_base,
le_wf,
null_cons_lemma,
bfalse_wf,
false_wf,
sq_stable__le,
le_antisymmetry_iff,
add_functionality_wrt_le,
add-associates,
add-zero,
zero-add,
le-add-cancel,
decidable__le,
not-le-2,
condition-implies-le,
minus-add,
minus-one-mul,
minus-one-mul-top,
add-commutes,
subtract_wf,
not-ge-2,
less-iff-le,
minus-minus,
add-swap,
list_wf,
list_ind_cons_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
thin,
lambdaFormation,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
independent_functionElimination,
voidElimination,
sqequalRule,
lambdaEquality,
dependent_functionElimination,
isect_memberEquality,
sqequalAxiom,
because_Cache,
applyEquality,
unionElimination,
instantiate,
cumulativity,
independent_pairFormation,
equalityTransitivity,
equalitySymmetry,
promote_hyp,
hypothesis_subsumption,
productElimination,
voidEquality,
baseClosed,
intEquality,
applyLambdaEquality,
imageMemberEquality,
imageElimination,
addEquality,
dependent_set_memberEquality,
minusEquality
Latex:
\mforall{}[eq:Top]. \mforall{}[as,bs:Top List]. (list\_eq(eq;as;bs) \msim{} list-deq(eq) as bs)
Date html generated:
2018_05_21-PM-00_20_28
Last ObjectModification:
2018_05_19-AM-07_00_30
Theory : list_0
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