Nuprl Lemma : sq_stable__rel_path
∀[A:Type]. ∀[R:A ⟶ A ⟶ ℙ]. ∀L:(a:A × b:A × (R a b)) List. ∀[x,y:A]. SqStable(rel_path(A;L;x;y))
Proof
Definitions occuring in Statement :
rel_path: rel_path(A;L;x;y),
list: T List,
sq_stable: SqStable(P),
uall: ∀[x:A]. B[x],
prop: ℙ,
all: ∀x:A. B[x],
apply: f a,
function: x:A ⟶ B[x],
product: x:A × B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
rel_path: rel_path(A;L;x;y),
member: t ∈ T,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
prop: ℙ,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
and: P ∧ Q,
pi1: fst(t),
pi2: snd(t),
so_apply: x[s1;s2;s3],
so_apply: x[s],
implies: P ⇒ Q,
top: Top,
sq_stable: SqStable(P)
Lemmas referenced :
list_induction,
uall_wf,
sq_stable_wf,
list_ind_wf,
equal_wf,
list_wf,
list_ind_nil_lemma,
sq_stable__equal,
squash_wf,
list_ind_cons_lemma,
sq_stable__and,
pi1_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
sqequalRule,
cut,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
productEquality,
cumulativity,
hypothesisEquality,
applyEquality,
functionExtensionality,
hypothesis,
because_Cache,
lambdaEquality,
instantiate,
functionEquality,
universeEquality,
productElimination,
independent_functionElimination,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
axiomEquality,
rename,
dependent_pairEquality
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:A]. SqStable(rel\_path(A;L;x;y))
Date html generated:
2017_04_17-AM-09_25_37
Last ObjectModification:
2017_02_27-PM-05_26_16
Theory : relations2
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