Theory "quotient_sum"

Signature

Definitions

SUM_REL_tupled_primitive_def

|- SUM_REL_tupled =
   WFREC (@R. WF R)
     (λSUM_REL_tupled a.
        case a of
          (R1,R2,INL a1,INL a2) => I (R1 a1 a2)
        | (R1,R2,INL a1,INR b2') => I F
        | (R1,R2,INR b1,INL a2') => I F
        | (R1,R2,INR b1,INR b2) => I (R2 b1 b2))

SUM_REL_curried_def

|- ∀x x1 x2 x3. (x +++ x1) x2 x3 ⇔ SUM_REL_tupled (x,x1,x2,x3)

Theorems

SUM_REL_ind

|- ∀P.
     (∀R1 R2 a1 a2. P R1 R2 (INL a1) (INL a2)) ∧
     (∀R1 R2 b1 b2. P R1 R2 (INR b1) (INR b2)) ∧
     (∀R1 R2 a1 b2. P R1 R2 (INL a1) (INR b2)) ∧
     (∀R1 R2 b1 a2. P R1 R2 (INR b1) (INL a2)) ⇒
     ∀v v1 v2 v3. P v v1 v2 v3

SUM_REL_def

|- ((R1 +++ R2) (INL a1) (INL a2) ⇔ R1 a1 a2) ∧
   ((R1 +++ R2) (INR b1) (INR b2) ⇔ R2 b1 b2) ∧
   ((R1 +++ R2) (INL a1) (INR b2) ⇔ F) ∧ ((R1 +++ R2) (INR b1) (INL a2) ⇔ F)

SUM_REL_EQ

|- $= +++ $= = $=

SUM_EQUIV

|- ∀R1 R2. EQUIV R1 ⇒ EQUIV R2 ⇒ EQUIV (R1 +++ R2)

SUM_QUOTIENT

|- ∀R1 abs1 rep1.
     QUOTIENT R1 abs1 rep1 ⇒
     ∀R2 abs2 rep2.
       QUOTIENT R2 abs2 rep2 ⇒
       QUOTIENT (R1 +++ R2) (abs1 ++ abs2) (rep1 ++ rep2)

INL_PRS

|- ∀R1 abs1 rep1.
     QUOTIENT R1 abs1 rep1 ⇒
     ∀R2 abs2 rep2.
       QUOTIENT R2 abs2 rep2 ⇒ ∀a. INL a = (abs1 ++ abs2) (INL (rep1 a))

INL_RSP

|- ∀R1 abs1 rep1.
     QUOTIENT R1 abs1 rep1 ⇒
     ∀R2 abs2 rep2.
       QUOTIENT R2 abs2 rep2 ⇒
       ∀a1 a2. R1 a1 a2 ⇒ (R1 +++ R2) (INL a1) (INL a2)

INR_PRS

|- ∀R1 abs1 rep1.
     QUOTIENT R1 abs1 rep1 ⇒
     ∀R2 abs2 rep2.
       QUOTIENT R2 abs2 rep2 ⇒ ∀b. INR b = (abs1 ++ abs2) (INR (rep2 b))

INR_RSP

|- ∀R1 abs1 rep1.
     QUOTIENT R1 abs1 rep1 ⇒
     ∀R2 abs2 rep2.
       QUOTIENT R2 abs2 rep2 ⇒
       ∀b1 b2. R2 b1 b2 ⇒ (R1 +++ R2) (INR b1) (INR b2)

ISL_PRS

|- ∀R1 abs1 rep1.
     QUOTIENT R1 abs1 rep1 ⇒
     ∀R2 abs2 rep2. QUOTIENT R2 abs2 rep2 ⇒ ∀a. ISL a ⇔ ISL ((rep1 ++ rep2) a)

ISL_RSP

|- ∀R1 abs1 rep1.
     QUOTIENT R1 abs1 rep1 ⇒
     ∀R2 abs2 rep2.
       QUOTIENT R2 abs2 rep2 ⇒ ∀a1 a2. (R1 +++ R2) a1 a2 ⇒ (ISL a1 ⇔ ISL a2)

ISR_PRS

|- ∀R1 abs1 rep1.
     QUOTIENT R1 abs1 rep1 ⇒
     ∀R2 abs2 rep2. QUOTIENT R2 abs2 rep2 ⇒ ∀a. ISR a ⇔ ISR ((rep1 ++ rep2) a)

ISR_RSP

|- ∀R1 abs1 rep1.
     QUOTIENT R1 abs1 rep1 ⇒
     ∀R2 abs2 rep2.
       QUOTIENT R2 abs2 rep2 ⇒ ∀a1 a2. (R1 +++ R2) a1 a2 ⇒ (ISR a1 ⇔ ISR a2)

SUM_MAP_PRS

|- ∀R1 abs1 rep1.
     QUOTIENT R1 abs1 rep1 ⇒
     ∀R2 abs2 rep2.
       QUOTIENT R2 abs2 rep2 ⇒
       ∀R3 abs3 rep3.
         QUOTIENT R3 abs3 rep3 ⇒
         ∀R4 abs4 rep4.
           QUOTIENT R4 abs4 rep4 ⇒
           ∀f g.
             f ++ g =
             ((rep1 ++ rep3) --> (abs2 ++ abs4))
               ((abs1 --> rep2) f ++ (abs3 --> rep4) g)

SUM_MAP_RSP

|- ∀R1 abs1 rep1.
     QUOTIENT R1 abs1 rep1 ⇒
     ∀R2 abs2 rep2.
       QUOTIENT R2 abs2 rep2 ⇒
       ∀R3 abs3 rep3.
         QUOTIENT R3 abs3 rep3 ⇒
         ∀R4 abs4 rep4.
           QUOTIENT R4 abs4 rep4 ⇒
           ∀f1 f2 g1 g2.
             (R1 ===> R2) f1 f2 ∧ (R3 ===> R4) g1 g2 ⇒
             ((R1 +++ R3) ===> (R2 +++ R4)) (f1 ++ g1) (f2 ++ g2)