Nuprl Lemma : transitive-closure-map
∀[A:Type]. ∀[R:A ⟶ A ⟶ ℙ].
∀f:A ⟶ A. ((∀x,y:A. ((R x y) ⇒ (R (f x) (f y)))) ⇒ (∀x,y:A. ((TC(R) x y) ⇒ (TC(R) (f x) (f y)))))
Proof
Definitions occuring in Statement :
transitive-closure: TC(R),
uall: ∀[x:A]. B[x],
prop: ℙ,
all: ∀x:A. B[x],
implies: P ⇒ Q,
apply: f a,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
transitive-closure: TC(R),
member: t ∈ T,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
and: P ∧ Q,
subtype_rel: A ⊆r B,
spreadn: spread3,
cand: A c∧ B,
squash: ↓T,
label: ...$L... t,
top: Top,
uimplies: b supposing a,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
pi1: fst(t),
pi2: snd(t),
rel_path: rel_path(A;L;x;y),
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3]
Lemmas referenced :
transitive-closure_wf,
all_wf,
map_wf,
less_than_wf,
squash_wf,
true_wf,
equal_wf,
length-map-sq,
subtype_rel_list,
top_wf,
length_wf,
iff_weakening_equal,
rel_path_wf,
list_induction,
list_wf,
list_ind_nil_lemma,
map_nil_lemma,
list_ind_cons_lemma,
map_cons_lemma,
and_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
rename,
sqequalHypSubstitution,
sqequalRule,
applyEquality,
cut,
introduction,
extract_by_obid,
isectElimination,
thin,
cumulativity,
hypothesisEquality,
functionExtensionality,
hypothesis,
lambdaEquality,
functionEquality,
universeEquality,
setElimination,
dependent_set_memberEquality,
productElimination,
productEquality,
because_Cache,
dependent_pairEquality,
independent_pairFormation,
imageElimination,
equalityTransitivity,
equalitySymmetry,
intEquality,
natural_numberEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_isectElimination,
imageMemberEquality,
baseClosed,
independent_functionElimination,
dependent_functionElimination,
applyLambdaEquality
Latex:
\mforall{}[A:Type]. \mforall{}[R:A {}\mrightarrow{} A {}\mrightarrow{} \mBbbP{}].
\mforall{}f:A {}\mrightarrow{} A
((\mforall{}x,y:A. ((R x y) {}\mRightarrow{} (R (f x) (f y)))) {}\mRightarrow{} (\mforall{}x,y:A. ((TC(R) x y) {}\mRightarrow{} (TC(R) (f x) (f y)))))
Date html generated:
2017_04_17-AM-09_25_42
Last ObjectModification:
2017_02_27-PM-05_26_24
Theory : relations2
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