Nuprl Lemma : prec-definition
∀[P:Type]. ∀[a:Atom ⟶ P ⟶ ((P + P + Type) List)]. ∀[A:P ⟶ Type]. ∀[R:i:P ⟶ prec(lbl,p.a[lbl;p];i) ⟶ A[i] ⟶ ℙ].
((∀i:P. ∀x:prec(lbl,p.a[lbl;p];i).
((∀j:P. ∀z:{z:prec(lbl,p.a[lbl;p];j)| prec_sub+(P;lbl,p.a[lbl;p]) <j, z> <i, x>} . (∃a:A[j] [R[j;z;a]])) ⇒ (∃a:A\000C[i] [R[i;x;a]])))
⇒ (∀i:P. ∀x:prec(lbl,p.a[lbl;p];i). (∃a:A[i] [R[i;x;a]])))
Proof
Definitions occuring in Statement :
prec_sub+: prec_sub+(P;lbl,p.a[lbl; p]),
prec: prec(lbl,p.a[lbl; p];i),
list: T List,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2;s3],
so_apply: x[s1;s2],
so_apply: x[s],
all: ∀x:A. B[x],
sq_exists: ∃x:A [B[x]],
implies: P ⇒ Q,
set: {x:A| B[x]} ,
apply: f a,
function: x:A ⟶ B[x],
pair: <a, b>,
union: left + right,
atom: Atom,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
all: ∀x:A. B[x],
so_lambda: λ2x y.t[x; y],
so_apply: x[s],
so_lambda: λ2x.t[x],
so_apply: x[s1;s2;s3],
subtype_rel: A ⊆r B,
prop: ℙ,
so_apply: x[s1;s2],
sq_exists: ∃x:A [B[x]]
Lemmas referenced :
prec-induction-ext,
sq_exists_wf,
subtype-TYPE,
prec_wf,
istype-atom,
prec_sub+_wf,
subtype_rel_self,
list_wf,
istype-universe
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
Error :lambdaFormation_alt,
sqequalRule,
Error :lambdaEquality_alt,
applyEquality,
Error :universeIsType,
Error :inhabitedIsType,
independent_functionElimination,
dependent_functionElimination,
Error :functionIsType,
because_Cache,
Error :setIsType,
Error :dependent_pairEquality_alt,
instantiate,
universeEquality,
setElimination,
rename,
unionEquality,
cumulativity
Latex:
\mforall{}[P:Type]. \mforall{}[a:Atom {}\mrightarrow{} P {}\mrightarrow{} ((P + P + Type) List)]. \mforall{}[A:P {}\mrightarrow{} Type]. \mforall{}[R:i:P
{}\mrightarrow{} prec(lbl,p.a[lbl;p];i)
{}\mrightarrow{} A[i]
{}\mrightarrow{} \mBbbP{}].
((\mforall{}i:P. \mforall{}x:prec(lbl,p.a[lbl;p];i).
((\mforall{}j:P. \mforall{}z:\{z:prec(lbl,p.a[lbl;p];j)| prec\_sub+(P;lbl,p.a[lbl;p]) <j, z> <i, x>\} . (\mexists{}a:A[j] [\000CR[j;z;a]]))
{}\mRightarrow{} (\mexists{}a:A[i] [R[i;x;a]])))
{}\mRightarrow{} (\mforall{}i:P. \mforall{}x:prec(lbl,p.a[lbl;p];i). (\mexists{}a:A[i] [R[i;x;a]])))
Date html generated:
2019_06_20-PM-02_14_32
Last ObjectModification:
2019_04_11-AM-08_44_21
Theory : tuples
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