Nuprl Lemma : list-eq-subtype1
∀[A:Type]. ∀[B:A ⟶ ℙ]. ∀[d1,d2:{a:A| B[a]} List]. d1 = d2 ∈ ({a:A| B[a]} List) supposing d1 = d2 ∈ (A List)
Proof
Definitions occuring in Statement :
list: T List,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
so_apply: x[s],
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
uimplies: b supposing a,
prop: ℙ,
implies: P ⇒ Q,
all: ∀x:A. B[x],
or: P ∨ Q,
cons: [a / b],
and: P ∧ Q,
top: Top,
not: ¬A,
false: False,
squash: ↓T,
true: True,
uiff: uiff(P;Q),
guard: {T}
Lemmas referenced :
list_induction,
uall_wf,
list_wf,
isect_wf,
equal_wf,
subtype_rel_list,
equal-wf-base-T,
cons_wf,
set_wf,
list-cases,
nil_wf,
product_subtype_list,
null_nil_lemma,
btrue_wf,
and_wf,
null_wf,
null_cons_lemma,
bfalse_wf,
btrue_neq_bfalse,
squash_wf,
true_wf,
cons_one_one
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
thin,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
setEquality,
cumulativity,
hypothesisEquality,
applyEquality,
functionExtensionality,
hypothesis,
because_Cache,
sqequalRule,
lambdaEquality,
independent_isectElimination,
setElimination,
rename,
independent_functionElimination,
baseClosed,
isect_memberEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
lambdaFormation,
dependent_functionElimination,
functionEquality,
universeEquality,
unionElimination,
promote_hyp,
hypothesis_subsumption,
productElimination,
dependent_set_memberEquality,
independent_pairFormation,
applyLambdaEquality,
voidElimination,
voidEquality,
imageElimination,
natural_numberEquality,
imageMemberEquality
Latex:
\mforall{}[A:Type]. \mforall{}[B:A {}\mrightarrow{} \mBbbP{}]. \mforall{}[d1,d2:\{a:A| B[a]\} List]. d1 = d2 supposing d1 = d2
Date html generated:
2017_04_14-AM-09_27_01
Last ObjectModification:
2017_02_27-PM-04_00_58
Theory : list_1
Home
Index