# Structure quotient_sumTheory

Source File Identifier index Theory binding index

```signature quotient_sumTheory =
sig
type thm = Thm.thm

(*  Theorems  *)
val INL_PRS : thm
val INL_RSP : thm
val INR_PRS : thm
val INR_RSP : thm
val ISL_PRS : thm
val ISL_RSP : thm
val ISR_PRS : thm
val ISR_RSP : thm
val SUM_EQUIV : thm
val SUM_MAP_PRS : thm
val SUM_MAP_RSP : thm
val SUM_QUOTIENT : thm
val SUM_REL_EQ : thm
val SUM_REL_def : thm
val SUM_REL_ind : thm

val quotient_sum_grammars : type_grammar.grammar * term_grammar.grammar
(*
[quotient] Parent theory of "quotient_sum"

[INL_PRS]  Theorem

⊢ ∀R1 abs1 rep1.
QUOTIENT R1 abs1 rep1 ⇒
∀R2 abs2 rep2.
QUOTIENT R2 abs2 rep2 ⇒
∀a. INL a = SUM_MAP abs1 abs2 (INL (rep1 a))

[INL_RSP]  Theorem

⊢ ∀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]  Theorem

⊢ ∀R1 abs1 rep1.
QUOTIENT R1 abs1 rep1 ⇒
∀R2 abs2 rep2.
QUOTIENT R2 abs2 rep2 ⇒
∀b. INR b = SUM_MAP abs1 abs2 (INR (rep2 b))

[INR_RSP]  Theorem

⊢ ∀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]  Theorem

⊢ ∀R1 abs1 rep1.
QUOTIENT R1 abs1 rep1 ⇒
∀R2 abs2 rep2.
QUOTIENT R2 abs2 rep2 ⇒ ∀a. ISL a ⇔ ISL (SUM_MAP rep1 rep2 a)

[ISL_RSP]  Theorem

⊢ ∀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]  Theorem

⊢ ∀R1 abs1 rep1.
QUOTIENT R1 abs1 rep1 ⇒
∀R2 abs2 rep2.
QUOTIENT R2 abs2 rep2 ⇒ ∀a. ISR a ⇔ ISR (SUM_MAP rep1 rep2 a)

[ISR_RSP]  Theorem

⊢ ∀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_EQUIV]  Theorem

⊢ ∀R1 R2. EQUIV R1 ⇒ EQUIV R2 ⇒ EQUIV (R1 +++ R2)

[SUM_MAP_PRS]  Theorem

⊢ ∀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.
SUM_MAP f g =
(SUM_MAP rep1 rep3 --> SUM_MAP abs2 abs4)
(SUM_MAP ((abs1 --> rep2) f) ((abs3 --> rep4) g))

[SUM_MAP_RSP]  Theorem

⊢ ∀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)) (SUM_MAP f1 g1)
(SUM_MAP f2 g2)

[SUM_QUOTIENT]  Theorem

⊢ ∀R1 abs1 rep1.
QUOTIENT R1 abs1 rep1 ⇒
∀R2 abs2 rep2.
QUOTIENT R2 abs2 rep2 ⇒
QUOTIENT (R1 +++ R2) (SUM_MAP abs1 abs2) (SUM_MAP rep1 rep2)

[SUM_REL_EQ]  Theorem

⊢ \$= +++ \$= = \$=

[SUM_REL_def]  Theorem

⊢ ((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_ind]  Theorem

⊢ ∀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

*)
end

```

Source File Identifier index Theory binding index