Nuprl Lemma : rel-pre-preserving-compose

∀es:EO
  ∀[P,Q:E ⟶ ℙ]. ∀[R:E ⟶ E ⟶ ℙ].
    ∀f1:{e:E| P e}  ⟶ {e:E| Q e} . ∀f2:{e:E| Q e}  ⟶ E.
      ((f1 is R-pre-preserving on P ∧ f2 is R-pre-preserving on Q) ⇒ f2 o f1 is R-pre-preserving on P)

Proof

Definitions occuring in Statement : rel-pre-preserving: f is R-pre-preserving on P, es-E: E, event_ordering: EO, compose: f o g, uall: ∀[x:A]. B[x], prop: ℙ, all: ∀x:A. B[x], implies: P ⇒ Q, and: P ∧ Q, set: {x:A| B[x]} , apply: f a, function: x:A ⟶ B[x]
Definitions unfolded in proof : rel-pre-preserving: f is R-pre-preserving on P, all: ∀x:A. B[x], uall: ∀[x:A]. B[x], implies: P ⇒ Q, and: P ∧ Q, member: t ∈ T, prop: ℙ, subtype_rel: A ⊆r B, so_lambda: λ₂x.t[x], so_apply: x[s], uimplies: b supposing a

Latex:
\mforall{}es:EO \mforall{}[P,Q:E {}\mrightarrow{} \mBbbP{}]. \mforall{}[R:E {}\mrightarrow{} E {}\mrightarrow{} \mBbbP{}]. \mforall{}f1:\{e:E| P e\} {}\mrightarrow{} \{e:E| Q e\} . \mforall{}f2:\{e:E| Q e\} {}\mrightarrow{} E. ((f1 is R-pre-preserving on P \mwedge{} f2 is R-pre-preserving on Q) {}\mRightarrow{} f2 o f1 is R-pre-preserving on P) Date html generated: 2016_05_16-AM-10_20_14 Last ObjectModification: 2015_12_28-PM-09_22_31 Theory : new!event-ordering Home Index