Nuprl Lemma : rel_path-append
∀[A:Type]. ∀[R:A ⟶ A ⟶ ℙ].
∀L:(a:A × b:A × (R a b)) List. ∀x,y,z:A. ∀r:R y z. rel_path(A;L @ [<y, z, r>];x;z) supposing rel_path(A;L;x;y)
Proof
Definitions occuring in Statement :
rel_path: rel_path(A;L;x;y),
append: as @ bs,
cons: [a / b],
nil: [],
list: T List,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
prop: ℙ,
all: ∀x:A. B[x],
apply: f a,
function: x:A ⟶ B[x],
pair: <a, b>,
product: x:A × B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
uimplies: b supposing a,
prop: ℙ,
so_apply: x[s],
implies: P ⇒ Q,
rel_path: rel_path(A;L;x;y),
so_lambda: so_lambda(x,y,z.t[x; y; z]),
top: Top,
so_apply: x[s1;s2;s3],
append: as @ bs,
pi1: fst(t),
pi2: snd(t),
and: P ∧ Q,
cand: A c∧ B,
sq_stable: SqStable(P),
squash: ↓T
Lemmas referenced :
rel_path_wf,
list_induction,
all_wf,
isect_wf,
append_wf,
cons_wf,
nil_wf,
istype-universe,
list_wf,
subtype_rel_self,
list_ind_nil_lemma,
istype-void,
list_ind_cons_lemma,
sq_stable__rel_path
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
Error :lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
productEquality,
applyEquality,
hypothesis,
because_Cache,
sqequalRule,
Error :lambdaEquality_alt,
cumulativity,
functionExtensionality,
Error :dependent_pairEquality_alt,
Error :universeIsType,
Error :productIsType,
Error :inhabitedIsType,
independent_functionElimination,
instantiate,
universeEquality,
Error :functionIsType,
dependent_functionElimination,
Error :isect_memberEquality_alt,
voidElimination,
equalitySymmetry,
independent_pairFormation,
productElimination,
independent_pairEquality,
axiomEquality,
Error :equalityIsType1,
Error :isectIsType,
imageMemberEquality,
baseClosed,
imageElimination,
independent_isectElimination
Latex:
\mforall{}[A:Type]. \mforall{}[R:A {}\mrightarrow{} A {}\mrightarrow{} \mBbbP{}].
\mforall{}L:(a:A \mtimes{} b:A \mtimes{} (R a b)) List. \mforall{}x,y,z:A. \mforall{}r:R y z.
rel\_path(A;L @ [<y, z, r>];x;z) supposing rel\_path(A;L;x;y)
Date html generated:
2019_06_20-PM-02_01_20
Last ObjectModification:
2018_10_05-AM-11_20_39
Theory : relations2
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