Structure sumTheory

signature sumTheory =
sig
  type thm = Thm.thm

  (*  Definitions  *)
    val INL_DEF : thm
    val INR_DEF : thm
    val ISL : thm
    val ISR : thm
    val IS_SUM_REP : thm
    val OUTL : thm
    val OUTR : thm
    val SUM_ALL_def : thm
    val SUM_MAP_def : thm
    val sum_ISO_DEF : thm
    val sum_TY_DEF : thm
    val sum_case_def : thm

  (*  Theorems  *)
    val EXISTS_SUM : thm
    val FORALL_SUM : thm
    val INL : thm
    val INL_11 : thm
    val INR : thm
    val INR_11 : thm
    val INR_INL_11 : thm
    val INR_neq_INL : thm
    val ISL_OR_ISR : thm
    val NOT_ISL_ISR : thm
    val NOT_ISR_ISL : thm
    val SUM_ALL_CONG : thm
    val SUM_ALL_MONO : thm
    val SUM_MAP : thm
    val SUM_MAP_CASE : thm
    val SUM_MAP_I : thm
    val cond_sum_expand : thm
    val datatype_sum : thm
    val sum_Axiom : thm
    val sum_CASES : thm
    val sum_INDUCT : thm
    val sum_axiom : thm
    val sum_case_cong : thm
    val sum_distinct : thm
    val sum_distinct1 : thm

  val sum_grammars : type_grammar.grammar * term_grammar.grammar
(*
   [combin] Parent theory of "sum"

   [sat] Parent theory of "sum"

   [INL_DEF]  Definition

      |- ∀e. INL e = ABS_sum (λb x y. (x = e) ∧ b)

   [INR_DEF]  Definition

      |- ∀e. INR e = ABS_sum (λb x y. (y = e) ∧ ¬b)

   [ISL]  Definition

      |- (∀x. ISL (INL x)) ∧ ∀y. ¬ISL (INR y)

   [ISR]  Definition

      |- (∀x. ISR (INR x)) ∧ ∀y. ¬ISR (INL y)

   [IS_SUM_REP]  Definition

      |- ∀f.
           IS_SUM_REP f ⇔
           ∃v1 v2.
             (f = (λb x y. (x = v1) ∧ b)) ∨ (f = (λb x y. (y = v2) ∧ ¬b))

   [OUTL]  Definition

      |- ∀x. OUTL (INL x) = x

   [OUTR]  Definition

      |- ∀x. OUTR (INR x) = x

   [SUM_ALL_def]  Definition

      |- (∀P Q x. SUM_ALL P Q (INL x) ⇔ P x) ∧
         ∀P Q y. SUM_ALL P Q (INR y) ⇔ Q y

   [SUM_MAP_def]  Definition

      |- (∀f g a. (f ++ g) (INL a) = INL (f a)) ∧
         ∀f g b. (f ++ g) (INR b) = INR (g b)

   [sum_ISO_DEF]  Definition

      |- (∀a. ABS_sum (REP_sum a) = a) ∧
         ∀r. IS_SUM_REP r ⇔ (REP_sum (ABS_sum r) = r)

   [sum_TY_DEF]  Definition

      |- ∃rep. TYPE_DEFINITION IS_SUM_REP rep

   [sum_case_def]  Definition

      |- (∀x f f1. sum_CASE (INL x) f f1 = f x) ∧
         ∀y f f1. sum_CASE (INR y) f f1 = f1 y

   [EXISTS_SUM]  Theorem

      |- ∀P. (∃s. P s) ⇔ (∃x. P (INL x)) ∨ ∃y. P (INR y)

   [FORALL_SUM]  Theorem

      |- (∀s. P s) ⇔ (∀x. P (INL x)) ∧ ∀y. P (INR y)

   [INL]  Theorem

      |- ∀x. ISL x ⇒ (INL (OUTL x) = x)

   [INL_11]  Theorem

      |- (INL x = INL y) ⇔ (x = y)

   [INR]  Theorem

      |- ∀x. ISR x ⇒ (INR (OUTR x) = x)

   [INR_11]  Theorem

      |- (INR x = INR y) ⇔ (x = y)

   [INR_INL_11]  Theorem

      |- (∀y x. (INL x = INL y) ⇔ (x = y)) ∧
         ∀y x. (INR x = INR y) ⇔ (x = y)

   [INR_neq_INL]  Theorem

      |- ∀v1 v2. INR v2 ≠ INL v1

   [ISL_OR_ISR]  Theorem

      |- ∀x. ISL x ∨ ISR x

   [NOT_ISL_ISR]  Theorem

      |- ∀x. ¬ISL x ⇔ ISR x

   [NOT_ISR_ISL]  Theorem

      |- ∀x. ¬ISR x ⇔ ISL x

   [SUM_ALL_CONG]  Theorem

      |- ∀s s' P P' Q Q'.
           (s = s') ∧ (∀a. (s' = INL a) ⇒ (P a ⇔ P' a)) ∧
           (∀b. (s' = INR b) ⇒ (Q b ⇔ Q' b)) ⇒
           (SUM_ALL P Q s ⇔ SUM_ALL P' Q' s')

   [SUM_ALL_MONO]  Theorem

      |- (∀x. P x ⇒ P' x) ∧ (∀y. Q y ⇒ Q' y) ⇒
         SUM_ALL P Q s ⇒
         SUM_ALL P' Q' s

   [SUM_MAP]  Theorem

      |- ∀f g z.
           (f ++ g) z =
           if ISL z then INL (f (OUTL z)) else INR (g (OUTR z))

   [SUM_MAP_CASE]  Theorem

      |- ∀f g z. (f ++ g) z = sum_CASE z (INL o f) (INR o g)

   [SUM_MAP_I]  Theorem

      |- I ++ I = I

   [cond_sum_expand]  Theorem

      |- (∀x y z. ((if P then INR x else INL y) = INR z) ⇔ P ∧ (z = x)) ∧
         (∀x y z. ((if P then INR x else INL y) = INL z) ⇔ ¬P ∧ (z = y)) ∧
         (∀x y z. ((if P then INL x else INR y) = INL z) ⇔ P ∧ (z = x)) ∧
         ∀x y z. ((if P then INL x else INR y) = INR z) ⇔ ¬P ∧ (z = y)

   [datatype_sum]  Theorem

      |- DATATYPE (sum INL INR)

   [sum_Axiom]  Theorem

      |- ∀f g. ∃h. (∀x. h (INL x) = f x) ∧ ∀y. h (INR y) = g y

   [sum_CASES]  Theorem

      |- ∀ss. (∃x. ss = INL x) ∨ ∃y. ss = INR y

   [sum_INDUCT]  Theorem

      |- ∀P. (∀x. P (INL x)) ∧ (∀y. P (INR y)) ⇒ ∀s. P s

   [sum_axiom]  Theorem

      |- ∀f g. ∃!h. (h o INL = f) ∧ (h o INR = g)

   [sum_case_cong]  Theorem

      |- ∀M M' f f1.
           (M = M') ∧ (∀x. (M' = INL x) ⇒ (f x = f' x)) ∧
           (∀y. (M' = INR y) ⇒ (f1 y = f1' y)) ⇒
           (sum_CASE M f f1 = sum_CASE M' f' f1')

   [sum_distinct]  Theorem

      |- ∀x y. INL x ≠ INR y

   [sum_distinct1]  Theorem

      |- ∀x y. INR y ≠ INL x


*)
end

HOL 4, Kananaskis-11