Nuprl Lemma : subtype_rel_list-iff
∀[A,B:Type]. uiff((A List) ⊆r (B List);A ⊆r B)
Proof
Definitions occuring in Statement :
list: T List,
uiff: uiff(P;Q),
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
all: ∀x:A. B[x],
implies: P ⇒ Q,
prop: ℙ,
nat: ℕ,
false: False,
ge: i ≥ j ,
guard: {T},
or: P ∨ Q,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
top: Top,
so_apply: x[s1;s2;s3],
not: ¬A,
cons: [a / b],
colength: colength(L),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
squash: ↓T,
sq_stable: SqStable(P),
le: A ≤ B,
less_than': less_than'(a;b),
true: True,
decidable: Dec(P),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
subtract: n - m,
nil: [],
it: ⋅,
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_type: SQType(T),
less_than: a < b
Lemmas referenced :
subtype_rel_wf,
list_wf,
subtype_rel_list,
cons_wf,
nil_wf,
equal_wf,
nat_properties,
less_than_transitivity1,
less_than_irreflexivity,
ge_wf,
less_than_wf,
equal-wf-T-base,
nat_wf,
colength_wf_list,
list-cases,
list_ind_nil_lemma,
null_nil_lemma,
btrue_wf,
null_cons_lemma,
bfalse_wf,
and_wf,
null_wf,
btrue_neq_bfalse,
product_subtype_list,
spread_cons_lemma,
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,
le_wf,
subtract_wf,
not-ge-2,
less-iff-le,
minus-minus,
add-swap,
subtype_base_sq,
set_subtype_base,
int_subtype_base,
list_ind_cons_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
independent_pairFormation,
sqequalRule,
axiomEquality,
hypothesis,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
cumulativity,
hypothesisEquality,
independent_isectElimination,
productElimination,
independent_pairEquality,
isect_memberEquality,
because_Cache,
equalityTransitivity,
equalitySymmetry,
universeEquality,
lambdaEquality,
applyEquality,
lambdaFormation,
dependent_functionElimination,
independent_functionElimination,
setElimination,
rename,
intWeakElimination,
natural_numberEquality,
voidElimination,
unionElimination,
voidEquality,
dependent_set_memberEquality,
applyLambdaEquality,
baseClosed,
promote_hyp,
hypothesis_subsumption,
imageMemberEquality,
imageElimination,
addEquality,
minusEquality,
intEquality,
instantiate
Latex:
\mforall{}[A,B:Type]. uiff((A List) \msubseteq{}r (B List);A \msubseteq{}r B)
Date html generated:
2017_04_14-AM-08_34_55
Last ObjectModification:
2017_02_27-PM-03_22_36
Theory : list_0
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