Nuprl Lemma : list_eq_wf
∀[A:Type]. ∀[eq:A ⟶ A ⟶ 𝔹]. ∀[as,bs:A List]. (list_eq(eq;as;bs) ∈ 𝔹)
Proof
Definitions occuring in Statement :
list_eq: list_eq(eq;as;bs),
list: T List,
bool: 𝔹,
uall: ∀[x:A]. B[x],
member: t ∈ T,
function: x:A ⟶ B[x],
universe: Type
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,
cons: [a / b],
colength: colength(L),
so_lambda: λ2x y.t[x; y],
top: Top,
so_apply: x[s1;s2],
nil: [],
it: ⋅,
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_type: SQType(T),
squash: ↓T,
sq_stable: SqStable(P),
uiff: uiff(P;Q),
and: P ∧ Q,
le: A ≤ B,
not: ¬A,
less_than': less_than'(a;b),
true: True,
decidable: Dec(P),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
subtract: n - m,
less_than: a < b,
bfalse: ff,
bnot: ¬bb,
band: p ∧b q
Lemmas referenced :
nat_properties,
less_than_transitivity1,
less_than_irreflexivity,
ge_wf,
less_than_wf,
equal-wf-T-base,
nat_wf,
colength_wf_list,
int_subtype_base,
list-cases,
null_nil_lemma,
btrue_wf,
product_subtype_list,
spread_cons_lemma,
equal_wf,
subtype_base_sq,
set_subtype_base,
le_wf,
null_cons_lemma,
bfalse_wf,
sq_stable__le,
le_antisymmetry_iff,
add_functionality_wrt_le,
add-associates,
add-zero,
zero-add,
le-add-cancel,
decidable__le,
false_wf,
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,
band_wf,
list_wf,
bool_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
thin,
lambdaFormation,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
hypothesis,
setElimination,
rename,
sqequalRule,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
independent_functionElimination,
voidElimination,
lambdaEquality,
dependent_functionElimination,
isect_memberEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
because_Cache,
applyEquality,
unionElimination,
promote_hyp,
hypothesis_subsumption,
productElimination,
voidEquality,
baseClosed,
instantiate,
cumulativity,
intEquality,
applyLambdaEquality,
imageMemberEquality,
imageElimination,
addEquality,
dependent_set_memberEquality,
independent_pairFormation,
minusEquality,
functionEquality,
universeEquality
Latex:
\mforall{}[A:Type]. \mforall{}[eq:A {}\mrightarrow{} A {}\mrightarrow{} \mBbbB{}]. \mforall{}[as,bs:A List]. (list\_eq(eq;as;bs) \mmember{} \mBbbB{})
Date html generated:
2018_05_21-PM-00_20_01
Last ObjectModification:
2018_05_19-AM-07_00_15
Theory : list_0
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