# Structure quotient_pairTheory

Source File Identifier index Theory binding index

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

(*  Definitions  *)
val PAIR_REL : thm

(*  Theorems  *)
val COMMA_PRS : thm
val COMMA_RSP : thm
val CURRY_PRS : thm
val CURRY_RSP : thm
val FST_PRS : thm
val FST_RSP : thm
val PAIR_EQUIV : thm
val PAIR_MAP_I : thm
val PAIR_MAP_PRS : thm
val PAIR_MAP_RSP : thm
val PAIR_QUOTIENT : thm
val PAIR_REL_EQ : thm
val PAIR_REL_REFL : thm
val PAIR_REL_THM : thm
val SND_PRS : thm
val SND_RSP : thm
val UNCURRY_PRS : thm
val UNCURRY_RSP : thm

val quotient_pair_grammars : type_grammar.grammar * term_grammar.grammar
(*
[quotient] Parent theory of "quotient_pair"

[PAIR_REL]  Definition

⊢ ∀R1 R2. R1 ### R2 = (λ(a,b) (c,d). R1 a c ∧ R2 b d)

[COMMA_PRS]  Theorem

⊢ ∀R1 abs1 rep1.
QUOTIENT R1 abs1 rep1 ⇒
∀R2 abs2 rep2.
QUOTIENT R2 abs2 rep2 ⇒
∀a b. (a,b) = (abs1 ## abs2) (rep1 a,rep2 b)

[COMMA_RSP]  Theorem

⊢ ∀R1 abs1 rep1.
QUOTIENT R1 abs1 rep1 ⇒
∀R2 abs2 rep2.
QUOTIENT R2 abs2 rep2 ⇒
∀a1 a2 b1 b2. R1 a1 b1 ∧ R2 a2 b2 ⇒ (R1 ### R2) (a1,a2) (b1,b2)

[CURRY_PRS]  Theorem

⊢ ∀R1 abs1 rep1.
QUOTIENT R1 abs1 rep1 ⇒
∀R2 abs2 rep2.
QUOTIENT R2 abs2 rep2 ⇒
∀R3 abs3 rep3.
QUOTIENT R3 abs3 rep3 ⇒
∀f a b.
CURRY f a b =
abs3
(CURRY (((abs1 ## abs2) --> rep3) f) (rep1 a) (rep2 b))

[CURRY_RSP]  Theorem

⊢ ∀R1 abs1 rep1.
QUOTIENT R1 abs1 rep1 ⇒
∀R2 abs2 rep2.
QUOTIENT R2 abs2 rep2 ⇒
∀R3 abs3 rep3.
QUOTIENT R3 abs3 rep3 ⇒
∀f1 f2.
((R1 ### R2) ===> R3) f1 f2 ⇒
(R1 ===> R2 ===> R3) (CURRY f1) (CURRY f2)

[FST_PRS]  Theorem

⊢ ∀R1 abs1 rep1.
QUOTIENT R1 abs1 rep1 ⇒
∀R2 abs2 rep2.
QUOTIENT R2 abs2 rep2 ⇒
∀p. FST p = abs1 (FST ((rep1 ## rep2) p))

[FST_RSP]  Theorem

⊢ ∀R1 abs1 rep1.
QUOTIENT R1 abs1 rep1 ⇒
∀R2 abs2 rep2.
QUOTIENT R2 abs2 rep2 ⇒
∀p1 p2. (R1 ### R2) p1 p2 ⇒ R1 (FST p1) (FST p2)

[PAIR_EQUIV]  Theorem

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

[PAIR_MAP_I]  Theorem

⊢ I ## I = I

[PAIR_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.
f ## g =
((rep1 ## rep3) --> (abs2 ## abs4))
((abs1 --> rep2) f ## (abs3 --> rep4) g)

[PAIR_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) (f1 ## g1) (f2 ## g2)

[PAIR_QUOTIENT]  Theorem

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

[PAIR_REL_EQ]  Theorem

⊢ \$= ### \$= = \$=

[PAIR_REL_REFL]  Theorem

⊢ ∀R1 R2.
(∀x y. R1 x y ⇔ R1 x = R1 y) ∧ (∀x y. R2 x y ⇔ R2 x = R2 y) ⇒
∀x. (R1 ### R2) x x

[PAIR_REL_THM]  Theorem

⊢ ∀R1 R2 a b c d. (R1 ### R2) (a,b) (c,d) ⇔ R1 a c ∧ R2 b d

[SND_PRS]  Theorem

⊢ ∀R1 abs1 rep1.
QUOTIENT R1 abs1 rep1 ⇒
∀R2 abs2 rep2.
QUOTIENT R2 abs2 rep2 ⇒
∀p. SND p = abs2 (SND ((rep1 ## rep2) p))

[SND_RSP]  Theorem

⊢ ∀R1 abs1 rep1.
QUOTIENT R1 abs1 rep1 ⇒
∀R2 abs2 rep2.
QUOTIENT R2 abs2 rep2 ⇒
∀p1 p2. (R1 ### R2) p1 p2 ⇒ R2 (SND p1) (SND p2)

[UNCURRY_PRS]  Theorem

⊢ ∀R1 abs1 rep1.
QUOTIENT R1 abs1 rep1 ⇒
∀R2 abs2 rep2.
QUOTIENT R2 abs2 rep2 ⇒
∀R3 abs3 rep3.
QUOTIENT R3 abs3 rep3 ⇒
∀f p.
UNCURRY f p =
abs3
(UNCURRY ((abs1 --> abs2 --> rep3) f) ((rep1 ## rep2) p))

[UNCURRY_RSP]  Theorem

⊢ ∀R1 abs1 rep1.
QUOTIENT R1 abs1 rep1 ⇒
∀R2 abs2 rep2.
QUOTIENT R2 abs2 rep2 ⇒
∀R3 abs3 rep3.
QUOTIENT R3 abs3 rep3 ⇒
∀f1 f2.
(R1 ===> R2 ===> R3) f1 f2 ⇒
((R1 ### R2) ===> R3) (UNCURRY f1) (UNCURRY f2)

*)
end

```

Source File Identifier index Theory binding index