Nuprl Lemma : trans_rel_func_wrt_sym_self
∀[T:Type]. ∀[R:T ⟶ T ⟶ ℙ].
(Trans(T;x,y.R[x;y])
⇒ {∀a,a',b,b':T. (Symmetrize(x,y.R[x;y];a;b) ⇒ Symmetrize(x,y.R[x;y];a';b') ⇒ (R[a;a'] ⇐⇒ R[b;b']))})
Proof
Definitions occuring in Statement :
symmetrize: Symmetrize(x,y.R[x; y];a;b),
trans: Trans(T;x,y.E[x; y]),
uall: ∀[x:A]. B[x],
prop: ℙ,
guard: {T},
so_apply: x[s1;s2],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
symmetrize: Symmetrize(x,y.R[x; y];a;b),
guard: {T},
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
member: t ∈ T,
prop: ℙ,
so_apply: x[s1;s2],
rev_implies: P ⇐ Q,
subtype_rel: A ⊆r B,
so_lambda: λ2x y.t[x; y]
Lemmas referenced :
subtype_rel_self,
trans_wf,
trans_rel_self_functionality
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
Error :isect_memberFormation_alt,
lambdaFormation,
independent_pairFormation,
sqequalHypSubstitution,
productElimination,
thin,
applyEquality,
hypothesisEquality,
productEquality,
cut,
hypothesis,
instantiate,
introduction,
extract_by_obid,
isectElimination,
universeEquality,
because_Cache,
lambdaEquality,
Error :functionIsType,
Error :universeIsType,
Error :inhabitedIsType,
independent_functionElimination,
dependent_functionElimination
Latex:
\mforall{}[T:Type]. \mforall{}[R:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}].
(Trans(T;x,y.R[x;y])
{}\mRightarrow{} \{\mforall{}a,a',b,b':T.
(Symmetrize(x,y.R[x;y];a;b) {}\mRightarrow{} Symmetrize(x,y.R[x;y];a';b') {}\mRightarrow{} (R[a;a'] \mLeftarrow{}{}\mRightarrow{} R[b;b']))\})
Date html generated:
2019_06_20-PM-00_29_00
Last ObjectModification:
2018_09_26-AM-11_46_40
Theory : rel_1
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