Nuprl Lemma : transitive-closure-symmetric
∀[A:Type]. ∀[R:A ⟶ A ⟶ ℙ]. (Sym(A;x,y.R x y) ⇒ Sym(A;x,y.x TC(R) y))
Proof
Definitions occuring in Statement :
transitive-closure: TC(R),
sym: Sym(T;x,y.E[x; y]),
uall: ∀[x:A]. B[x],
prop: ℙ,
infix_ap: x f y,
implies: P ⇒ Q,
apply: f a,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
sym: Sym(T;x,y.E[x; y]),
all: ∀x:A. B[x],
infix_ap: x f y,
transitive-closure: TC(R),
member: t ∈ T,
subtype_rel: A ⊆r B,
prop: ℙ,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
spreadn: spread3,
and: P ∧ Q,
uimplies: b supposing a,
cand: A c∧ B,
top: Top,
so_lambda: λ2x.t[x],
so_apply: x[s],
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],
pi1: fst(t),
pi2: snd(t),
squash: ↓T,
true: True,
guard: {T}
Lemmas referenced :
transitive-closure_wf,
subtype_rel_self,
istype-universe,
sym_wf,
subtype_rel_function,
map_wf,
reverse_wf,
map-length,
istype-void,
length-reverse,
rel_path_wf,
less_than_wf,
length_wf,
list_induction,
all_wf,
list_wf,
list_ind_nil_lemma,
reverse_nil_lemma,
map_nil_lemma,
list_ind_cons_lemma,
reverse-cons,
map_append_sq,
map_cons_lemma,
cons_wf,
squash_wf,
true_wf,
nil_wf,
append_wf,
rel_path-append,
subtype_rel-equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
Error :lambdaFormation_alt,
sqequalHypSubstitution,
rename,
sqequalRule,
Error :universeIsType,
cut,
applyEquality,
introduction,
extract_by_obid,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
instantiate,
universeEquality,
Error :inhabitedIsType,
Error :lambdaEquality_alt,
Error :functionIsType,
setElimination,
productElimination,
Error :dependent_pairEquality_alt,
functionExtensionality,
because_Cache,
functionEquality,
independent_isectElimination,
Error :productIsType,
Error :dependent_set_memberEquality_alt,
productEquality,
promote_hyp,
independent_pairFormation,
Error :isect_memberEquality_alt,
voidElimination,
natural_numberEquality,
independent_functionElimination,
dependent_functionElimination,
equalitySymmetry,
Error :equalityIsType1,
equalityTransitivity,
imageElimination,
imageMemberEquality,
baseClosed,
applyLambdaEquality
Latex:
\mforall{}[A:Type]. \mforall{}[R:A {}\mrightarrow{} A {}\mrightarrow{} \mBbbP{}]. (Sym(A;x,y.R x y) {}\mRightarrow{} Sym(A;x,y.x TC(R) y))
Date html generated:
2019_06_20-PM-02_01_25
Last ObjectModification:
2018_10_07-AM-00_13_27
Theory : relations2
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