Nuprl Lemma : transitive-closure-contains
∀[A:Type]. ∀[R:A ⟶ A ⟶ ℙ]. R => TC(R)
Proof
Definitions occuring in Statement :
transitive-closure: TC(R),
rel_implies: R1 => R2,
uall: ∀[x:A]. B[x],
prop: ℙ,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
rel_implies: R1 => R2,
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
transitive-closure: TC(R),
infix_ap: x f y,
subtype_rel: A ⊆r B,
prop: ℙ,
and: P ∧ Q,
cand: A c∧ B,
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],
pi1: fst(t),
pi2: snd(t),
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True
Lemmas referenced :
length_wf,
less_than_wf,
rel_path_wf,
and_wf,
length-singleton,
list_ind_nil_lemma,
list_ind_cons_lemma,
nil_wf,
cons_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
rename,
introduction,
sqequalRule,
dependent_set_memberEquality,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
productEquality,
hypothesisEquality,
applyEquality,
hypothesis,
lambdaEquality,
universeEquality,
dependent_pairEquality,
because_Cache,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
natural_numberEquality,
imageMemberEquality,
baseClosed,
functionEquality,
cumulativity
Latex:
\mforall{}[A:Type]. \mforall{}[R:A {}\mrightarrow{} A {}\mrightarrow{} \mBbbP{}]. R => TC(R)
Date html generated:
2016_05_14-PM-03_51_11
Last ObjectModification:
2016_01_14-PM-11_11_18
Theory : relations2
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