signature TestTheory = sig type thm = Thm.thm (* Axioms *) val VEC_TO_LIST_BOUNDS : thm val i32_to_int_bounds : thm val insert_def : thm val int_to_i32_id : thm val int_to_u32_id : thm val u32_to_int_bounds : thm (* Definitions *) val distinct_keys_def : thm val error_BIJ : thm val error_CASE : thm val error_TY_DEF : thm val error_size_def : thm val i32_add_def : thm val i32_max_def : thm val i32_min_def : thm val int1_def : thm val int2_def : thm val is_loop_def : thm val is_true_def : thm val list_t_TY_DEF : thm val list_t_case_def : thm val list_t_size_def : thm val list_t_v_def : thm val lookup_def : thm val mk_i32_def : thm val mk_u32_def : thm val nth_expand_def : thm val nth_fuel_P_def : thm val result_TY_DEF : thm val result_case_def : thm val result_size_def : thm val st_ex_bind_def : thm val st_ex_return_def : thm val test1_def : thm val test2_TY_DEF : thm val test2_case_def : thm val test2_size_def : thm val test_TY_DEF : thm val test_case_def : thm val test_monad2_def : thm val test_monad3_def : thm val test_monad_def : thm val test_size_def : thm val u32_add_def : thm val u32_max_def : thm val u32_sub_def : thm val usize_max_def : thm val vec_len_def : thm (* Theorems *) val I32_ADD_EQ : thm val INT_OF_NUM_INJ : thm val INT_THM1 : thm val INT_THM2 : thm val MK_I32_SUCCESS : thm val MK_U32_SUCCESS : thm val NAT_THM1 : thm val NAT_THM2 : thm val NUM_SUB_1_EQ : thm val NUM_SUB_1_EQ1 : thm val NUM_SUB_EQ : thm val U32_ADD_EQ : thm val U32_SUB_EQ : thm val VEC_TO_LIST_INT_BOUNDS : thm val datatype_error : thm val datatype_list_t : thm val datatype_result : thm val datatype_test : thm val datatype_test2 : thm val error2num_11 : thm val error2num_ONTO : thm val error2num_num2error : thm val error2num_thm : thm val error_Axiom : thm val error_EQ_error : thm val error_case_cong : thm val error_case_def : thm val error_case_eq : thm val error_induction : thm val error_nchotomy : thm val insert_lem : thm val list_nth_mut_loop_loop_fwd_def : thm val list_nth_mut_loop_loop_fwd_ind : thm val list_t_11 : thm val list_t_Axiom : thm val list_t_case_cong : thm val list_t_case_eq : thm val list_t_distinct : thm val list_t_induction : thm val list_t_nchotomy : thm val lookup_raw_def : thm val lookup_raw_ind : thm val nth_def : thm val nth_def_loop : thm val nth_def_terminates : thm val nth_fuel_P_mono : thm val nth_fuel_def : thm val nth_fuel_ind : thm val nth_fuel_least_fail_mono : thm val nth_fuel_least_success_mono : thm val nth_fuel_mono : thm val num2error_11 : thm val num2error_ONTO : thm val num2error_error2num : thm val num2error_thm : thm val result_11 : thm val result_Axiom : thm val result_case_cong : thm val result_case_eq : thm val result_distinct : thm val result_induction : thm val result_nchotomy : thm val test2_11 : thm val test2_Axiom : thm val test2_case_cong : thm val test2_case_eq : thm val test2_distinct : thm val test2_induction : thm val test2_nchotomy : thm val test_11 : thm val test_Axiom : thm val test_case_cong : thm val test_case_eq : thm val test_distinct : thm val test_induction : thm val test_nchotomy : thm val Test_grammars : type_grammar.grammar * term_grammar.grammar (* [Omega] Parent theory of "Test" [int_arith] Parent theory of "Test" [words] Parent theory of "Test" [insert_def] Axiom [oracles: ] [axioms: insert_def] [] ⊢ insert key value ls = case ls of ListCons (ckey,cvalue) tl => if ckey = key then Return (ListCons (ckey,value) tl) else do tl0 <- insert key value tl; Return (ListCons (ckey,cvalue) tl0) od | ListNil => Return (ListCons (key,value) ListNil) [u32_to_int_bounds] Axiom [oracles: ] [axioms: u32_to_int_bounds] [] ⊢ ∀n. 0 ≤ u32_to_int n ∧ u32_to_int n ≤ u32_max [i32_to_int_bounds] Axiom [oracles: ] [axioms: i32_to_int_bounds] [] ⊢ ∀n. i32_min ≤ i32_to_int n ∧ i32_to_int n ≤ i32_max [int_to_u32_id] Axiom [oracles: ] [axioms: int_to_u32_id] [] ⊢ ∀n. 0 ≤ n ∧ n ≤ u32_max ⇒ u32_to_int (int_to_u32 n) = n [int_to_i32_id] Axiom [oracles: ] [axioms: int_to_i32_id] [] ⊢ ∀n. i32_min ≤ n ∧ n ≤ i32_max ⇒ i32_to_int (int_to_i32 n) = n [VEC_TO_LIST_BOUNDS] Axiom [oracles: ] [axioms: VEC_TO_LIST_BOUNDS] [] ⊢ ∀v. (let l = LENGTH (vec_to_list v) in 0 ≤ l ∧ l ≤ 4294967295) [distinct_keys_def] Definition ⊢ ∀ls. distinct_keys ls ⇔ ∀i j. i < LENGTH ls ⇒ j < LENGTH ls ⇒ FST (EL i ls) = FST (EL j ls) ⇒ i = j [error_BIJ] Definition ⊢ (∀a. num2error (error2num a) = a) ∧ ∀r. (λn. n < 1) r ⇔ error2num (num2error r) = r [error_CASE] Definition ⊢ ∀x v0. (case x of Failure => v0) = (λm. v0) (error2num x) [error_TY_DEF] Definition ⊢ ∃rep. TYPE_DEFINITION (λn. n < 1) rep [error_size_def] Definition ⊢ ∀x. error_size x = 0 [i32_add_def] Definition ⊢ ∀x y. i32_add x y = mk_i32 (i32_to_int x + i32_to_int y) [i32_max_def] Definition ⊢ i32_max = 2147483647 [i32_min_def] Definition ⊢ i32_min = -2147483648 [int1_def] Definition ⊢ int1 = 32 [int2_def] Definition ⊢ int2 = -32 [is_loop_def] Definition ⊢ ∀r. is_loop r ⇔ case r of Return v2 => F | Fail v3 => F | Loop => T [is_true_def] Definition ⊢ ∀x. is_true x = if x then Return () else Fail Failure [list_t_TY_DEF] Definition ⊢ ∃rep. TYPE_DEFINITION (λa0'. ∀ $var$('list_t'). (∀a0'. (∃a0 a1. a0' = (λa0 a1. ind_type$CONSTR 0 a0 (ind_type$FCONS a1 (λn. ind_type$BOTTOM))) a0 a1 ∧ $var$('list_t') a1) ∨ a0' = ind_type$CONSTR (SUC 0) ARB (λn. ind_type$BOTTOM) ⇒ $var$('list_t') a0') ⇒ $var$('list_t') a0') rep [list_t_case_def] Definition ⊢ (∀a0 a1 f v. list_t_CASE (ListCons a0 a1) f v = f a0 a1) ∧ ∀f v. list_t_CASE ListNil f v = v [list_t_size_def] Definition ⊢ (∀f a0 a1. list_t_size f (ListCons a0 a1) = 1 + (f a0 + list_t_size f a1)) ∧ ∀f. list_t_size f ListNil = 0 [list_t_v_def] Definition ⊢ list_t_v ListNil = [] ∧ ∀x tl. list_t_v (ListCons x tl) = x::list_t_v tl [lookup_def] Definition ⊢ ∀key ls. lookup key ls = lookup_raw key (list_t_v ls) [mk_i32_def] Definition ⊢ ∀n. mk_i32 n = if i32_min ≤ n ∧ n ≤ i32_max then Return (int_to_i32 n) else Fail Failure [mk_u32_def] Definition ⊢ ∀n. mk_u32 n = if 0 ≤ n ∧ n ≤ u32_max then Return (int_to_u32 n) else Fail Failure [nth_expand_def] Definition ⊢ ∀nth ls i. nth_expand nth ls i = case ls of ListCons x tl => if u32_to_int i = 0 then Return x else do i0 <- u32_sub i (int_to_u32 1); nth tl i0 od | ListNil => Fail Failure [nth_fuel_P_def] Definition ⊢ ∀ls i n. nth_fuel_P ls i n ⇔ ¬is_loop (nth_fuel n ls i) [result_TY_DEF] Definition ⊢ ∃rep. TYPE_DEFINITION (λa0. ∀ $var$('result'). (∀a0. (∃a. a0 = (λa. ind_type$CONSTR 0 (a,ARB) (λn. ind_type$BOTTOM)) a) ∨ (∃a. a0 = (λa. ind_type$CONSTR (SUC 0) (ARB,a) (λn. ind_type$BOTTOM)) a) ∨ a0 = ind_type$CONSTR (SUC (SUC 0)) (ARB,ARB) (λn. ind_type$BOTTOM) ⇒ $var$('result') a0) ⇒ $var$('result') a0) rep [result_case_def] Definition ⊢ (∀a f f1 v. result_CASE (Return a) f f1 v = f a) ∧ (∀a f f1 v. result_CASE (Fail a) f f1 v = f1 a) ∧ ∀f f1 v. result_CASE Loop f f1 v = v [result_size_def] Definition ⊢ (∀f a. result_size f (Return a) = 1 + f a) ∧ (∀f a. result_size f (Fail a) = 1 + error_size a) ∧ ∀f. result_size f Loop = 0 [st_ex_bind_def] Definition ⊢ ∀x f. monad_bind x f = case x of Return y => f y | Fail e => Fail e | Loop => Loop [st_ex_return_def] Definition ⊢ ∀x. st_ex_return x = Return x [test1_def] Definition ⊢ ∀x. test1 x = Return x [test2_TY_DEF] Definition ⊢ ∃rep. TYPE_DEFINITION (λa0. ∀ $var$('test2'). (∀a0. (∃a. a0 = (λa. ind_type$CONSTR 0 (a,ARB) (λn. ind_type$BOTTOM)) a) ∨ (∃a. a0 = (λa. ind_type$CONSTR (SUC 0) (ARB,a) (λn. ind_type$BOTTOM)) a) ⇒ $var$('test2') a0) ⇒ $var$('test2') a0) rep [test2_case_def] Definition ⊢ (∀a f f1. test2_CASE (Variant1_2 a) f f1 = f a) ∧ ∀a f f1. test2_CASE (Variant2_2 a) f f1 = f1 a [test2_size_def] Definition ⊢ (∀f f1 a. test2_size f f1 (Variant1_2 a) = 1 + f a) ∧ ∀f f1 a. test2_size f f1 (Variant2_2 a) = 1 + f1 a [test_TY_DEF] Definition ⊢ ∃rep. TYPE_DEFINITION (λa0. ∀ $var$('test'). (∀a0. (∃a. a0 = (λa. ind_type$CONSTR 0 (a,ARB) (λn. ind_type$BOTTOM)) a) ∨ (∃a. a0 = (λa. ind_type$CONSTR (SUC 0) (ARB,a) (λn. ind_type$BOTTOM)) a) ⇒ $var$('test') a0) ⇒ $var$('test') a0) rep [test_case_def] Definition ⊢ (∀a f f1. test_CASE (Variant1 a) f f1 = f a) ∧ ∀a f f1. test_CASE (Variant2 a) f f1 = f1 a [test_monad2_def] Definition ⊢ test_monad2 = do x <- Return T; Return x od [test_monad3_def] Definition ⊢ ∀x. test_monad3 x = monad_ignore_bind (is_true x) (Return x) [test_monad_def] Definition ⊢ ∀v. test_monad v = do x <- Return v; Return x od [test_size_def] Definition ⊢ (∀f f1 a. test_size f f1 (Variant1 a) = 1 + f1 a) ∧ ∀f f1 a. test_size f f1 (Variant2 a) = 1 + f a [u32_add_def] Definition ⊢ ∀x y. u32_add x y = mk_u32 (u32_to_int x + u32_to_int y) [u32_max_def] Definition ⊢ u32_max = 4294967295 [u32_sub_def] Definition ⊢ ∀x y. u32_sub x y = mk_u32 (u32_to_int x − u32_to_int y) [usize_max_def] Definition ⊢ usize_max = 4294967295 [vec_len_def] Definition ⊢ ∀v. vec_len v = int_to_u32 (&LENGTH (vec_to_list v)) [I32_ADD_EQ] Theorem [oracles: DISK_THM] [axioms: int_to_i32_id] [] ⊢ ∀x y. i32_min ≤ i32_to_int x + i32_to_int y ⇒ i32_to_int x + i32_to_int y ≤ i32_max ⇒ ∃z. i32_add x y = Return z ∧ i32_to_int z = i32_to_int x + i32_to_int y [INT_OF_NUM_INJ] Theorem ⊢ ∀n m. &n = &m ⇒ n = m [INT_THM1] Theorem ⊢ ∀x y. x > 0 ⇒ y > 0 ⇒ x + y > 0 [INT_THM2] Theorem ⊢ ∀x. T [MK_I32_SUCCESS] Theorem ⊢ ∀n. i32_min ≤ n ∧ n ≤ i32_max ⇒ mk_i32 n = Return (int_to_i32 n) [MK_U32_SUCCESS] Theorem ⊢ ∀n. 0 ≤ n ∧ n ≤ u32_max ⇒ mk_u32 n = Return (int_to_u32 n) [NAT_THM1] Theorem ⊢ ∀n. n < n + 1 [NAT_THM2] Theorem ⊢ ∀n. n < n + 1 [NUM_SUB_1_EQ] Theorem ⊢ ∀x y. x = y − 1 ⇒ 0 ≤ x ⇒ Num y = SUC (Num x) [NUM_SUB_1_EQ1] Theorem ⊢ ∀i. 0 ≤ i − 1 ⇒ Num i = SUC (Num (i − 1)) [NUM_SUB_EQ] Theorem ⊢ ∀x y z. x = y − z ⇒ 0 ≤ x ⇒ 0 ≤ z ⇒ Num y = Num z + Num x [U32_ADD_EQ] Theorem [oracles: DISK_THM] [axioms: int_to_u32_id, u32_to_int_bounds] [] ⊢ ∀x y. u32_to_int x + u32_to_int y ≤ u32_max ⇒ ∃z. u32_add x y = Return z ∧ u32_to_int z = u32_to_int x + u32_to_int y [U32_SUB_EQ] Theorem [oracles: DISK_THM] [axioms: int_to_u32_id, u32_to_int_bounds] [] ⊢ ∀x y. 0 ≤ u32_to_int x − u32_to_int y ⇒ ∃z. u32_sub x y = Return z ∧ u32_to_int z = u32_to_int x − u32_to_int y [VEC_TO_LIST_INT_BOUNDS] Theorem [oracles: DISK_THM] [axioms: VEC_TO_LIST_BOUNDS] [] ⊢ ∀v. (let l = &LENGTH (vec_to_list v) in 0 ≤ l ∧ l ≤ u32_max) [datatype_error] Theorem ⊢ DATATYPE (error Failure) [datatype_list_t] Theorem ⊢ DATATYPE (list_t ListCons ListNil) [datatype_result] Theorem ⊢ DATATYPE (result Return Fail Loop) [datatype_test] Theorem ⊢ DATATYPE (test Variant1 Variant2) [datatype_test2] Theorem ⊢ DATATYPE (test2 Variant1_2 Variant2_2) [error2num_11] Theorem ⊢ ∀a a'. error2num a = error2num a' ⇔ a = a' [error2num_ONTO] Theorem ⊢ ∀r. r < 1 ⇔ ∃a. r = error2num a [error2num_num2error] Theorem ⊢ ∀r. r < 1 ⇔ error2num (num2error r) = r [error2num_thm] Theorem ⊢ error2num Failure = 0 [error_Axiom] Theorem ⊢ ∀x0. ∃f. f Failure = x0 [error_EQ_error] Theorem ⊢ ∀a a'. a = a' ⇔ error2num a = error2num a' [error_case_cong] Theorem ⊢ ∀M M' v0. M = M' ∧ (M' = Failure ⇒ v0 = v0') ⇒ (case M of Failure => v0) = case M' of Failure => v0' [error_case_def] Theorem ⊢ ∀v0. (case Failure of Failure => v0) = v0 [error_case_eq] Theorem ⊢ (case x of Failure => v0) = v ⇔ x = Failure ∧ v0 = v [error_induction] Theorem ⊢ ∀P. P Failure ⇒ ∀a. P a [error_nchotomy] Theorem ⊢ ∀a. a = Failure [insert_lem] Theorem [oracles: DISK_THM] [axioms: u32_to_int_bounds, insert_def] [] ⊢ ∀ls key value. distinct_keys (list_t_v ls) ⇒ case insert key value ls of Return ls1 => lookup key ls1 = SOME value ∧ ∀k. k ≠ key ⇒ lookup k ls = lookup k ls1 | Fail v1 => F | Loop => F [list_nth_mut_loop_loop_fwd_def] Theorem ⊢ ∀ls i. list_nth_mut_loop_loop_fwd ls i = case ls of ListCons x tl => if u32_to_int i = 0 then Return x else do i0 <- u32_sub i (int_to_u32 1); list_nth_mut_loop_loop_fwd tl i0 od | ListNil => Fail Failure [list_nth_mut_loop_loop_fwd_ind] Theorem ⊢ ∀P. (∀ls i. (∀x tl i0. ls = ListCons x tl ∧ u32_to_int i ≠ 0 ⇒ P tl i0) ⇒ P ls i) ⇒ ∀v v1. P v v1 [list_t_11] Theorem ⊢ ∀a0 a1 a0' a1'. ListCons a0 a1 = ListCons a0' a1' ⇔ a0 = a0' ∧ a1 = a1' [list_t_Axiom] Theorem ⊢ ∀f0 f1. ∃fn. (∀a0 a1. fn (ListCons a0 a1) = f0 a0 a1 (fn a1)) ∧ fn ListNil = f1 [list_t_case_cong] Theorem ⊢ ∀M M' f v. M = M' ∧ (∀a0 a1. M' = ListCons a0 a1 ⇒ f a0 a1 = f' a0 a1) ∧ (M' = ListNil ⇒ v = v') ⇒ list_t_CASE M f v = list_t_CASE M' f' v' [list_t_case_eq] Theorem ⊢ list_t_CASE x f v = v' ⇔ (∃t l. x = ListCons t l ∧ f t l = v') ∨ x = ListNil ∧ v = v' [list_t_distinct] Theorem ⊢ ∀a1 a0. ListCons a0 a1 ≠ ListNil [list_t_induction] Theorem ⊢ ∀P. (∀l. P l ⇒ ∀t. P (ListCons t l)) ∧ P ListNil ⇒ ∀l. P l [list_t_nchotomy] Theorem ⊢ ∀ll. (∃t l. ll = ListCons t l) ∨ ll = ListNil [lookup_raw_def] Theorem ⊢ (∀key. lookup_raw key [] = NONE) ∧ ∀v ls key k. lookup_raw key ((k,v)::ls) = if k = key then SOME v else lookup_raw key ls [lookup_raw_ind] Theorem ⊢ ∀P. (∀key. P key []) ∧ (∀key k v ls. (k ≠ key ⇒ P key ls) ⇒ P key ((k,v)::ls)) ⇒ ∀v v1. P v v1 [nth_def] Theorem ⊢ ∀ls i. nth ls i = case ls of ListCons x tl => if u32_to_int i = 0 then Return x else do i0 <- u32_sub i (int_to_u32 1); nth tl i0 od | ListNil => Fail Failure [nth_def_loop] Theorem ⊢ ∀ls i. (∀n. ¬nth_fuel_P ls i n) ⇒ nth ls i = nth_expand nth ls i [nth_def_terminates] Theorem ⊢ ∀ls i. (∃n. nth_fuel_P ls i n) ⇒ nth ls i = nth_expand nth ls i [nth_fuel_P_mono] Theorem ⊢ ∀n m ls i. n ≤ m ⇒ nth_fuel_P ls i n ⇒ nth_fuel n ls i = nth_fuel m ls i [nth_fuel_def] Theorem ⊢ ∀n ls i. nth_fuel n ls i = case n of 0 => Loop | SUC n' => case ls of ListCons x tl => if u32_to_int i = 0 then Return x else do i0 <- u32_sub i (int_to_u32 1); nth_fuel n' tl i0 od | ListNil => Fail Failure [nth_fuel_ind] Theorem ⊢ ∀P. (∀n ls i. (∀n' x tl i0. n = SUC n' ∧ ls = ListCons x tl ∧ u32_to_int i ≠ 0 ⇒ P n' tl i0) ⇒ P n ls i) ⇒ ∀v v1 v2. P v v1 v2 [nth_fuel_least_fail_mono] Theorem ⊢ ∀n ls i. n < $LEAST (nth_fuel_P ls i) ⇒ nth_fuel n ls i = Loop [nth_fuel_least_success_mono] Theorem ⊢ ∀n ls i. $LEAST (nth_fuel_P ls i) ≤ n ⇒ nth_fuel n ls i = nth_fuel ($LEAST (nth_fuel_P ls i)) ls i [nth_fuel_mono] Theorem ⊢ ∀n m ls i. n ≤ m ⇒ if is_loop (nth_fuel n ls i) then T else nth_fuel n ls i = nth_fuel m ls i [num2error_11] Theorem ⊢ ∀r r'. r < 1 ⇒ r' < 1 ⇒ (num2error r = num2error r' ⇔ r = r') [num2error_ONTO] Theorem ⊢ ∀a. ∃r. a = num2error r ∧ r < 1 [num2error_error2num] Theorem ⊢ ∀a. num2error (error2num a) = a [num2error_thm] Theorem ⊢ num2error 0 = Failure [result_11] Theorem ⊢ (∀a a'. Return a = Return a' ⇔ a = a') ∧ ∀a a'. Fail a = Fail a' ⇔ a = a' [result_Axiom] Theorem ⊢ ∀f0 f1 f2. ∃fn. (∀a. fn (Return a) = f0 a) ∧ (∀a. fn (Fail a) = f1 a) ∧ fn Loop = f2 [result_case_cong] Theorem ⊢ ∀M M' f f1 v. M = M' ∧ (∀a. M' = Return a ⇒ f a = f' a) ∧ (∀a. M' = Fail a ⇒ f1 a = f1' a) ∧ (M' = Loop ⇒ v = v') ⇒ result_CASE M f f1 v = result_CASE M' f' f1' v' [result_case_eq] Theorem ⊢ result_CASE x f f1 v = v' ⇔ (∃a. x = Return a ∧ f a = v') ∨ (∃e. x = Fail e ∧ f1 e = v') ∨ x = Loop ∧ v = v' [result_distinct] Theorem ⊢ (∀a' a. Return a ≠ Fail a') ∧ (∀a. Return a ≠ Loop) ∧ ∀a. Fail a ≠ Loop [result_induction] Theorem ⊢ ∀P. (∀a. P (Return a)) ∧ (∀e. P (Fail e)) ∧ P Loop ⇒ ∀r. P r [result_nchotomy] Theorem ⊢ ∀rr. (∃a. rr = Return a) ∨ (∃e. rr = Fail e) ∨ rr = Loop [test2_11] Theorem ⊢ (∀a a'. Variant1_2 a = Variant1_2 a' ⇔ a = a') ∧ ∀a a'. Variant2_2 a = Variant2_2 a' ⇔ a = a' [test2_Axiom] Theorem ⊢ ∀f0 f1. ∃fn. (∀a. fn (Variant1_2 a) = f0 a) ∧ ∀a. fn (Variant2_2 a) = f1 a [test2_case_cong] Theorem ⊢ ∀M M' f f1. M = M' ∧ (∀a. M' = Variant1_2 a ⇒ f a = f' a) ∧ (∀a. M' = Variant2_2 a ⇒ f1 a = f1' a) ⇒ test2_CASE M f f1 = test2_CASE M' f' f1' [test2_case_eq] Theorem ⊢ test2_CASE x f f1 = v ⇔ (∃T'. x = Variant1_2 T' ∧ f T' = v) ∨ ∃T'. x = Variant2_2 T' ∧ f1 T' = v [test2_distinct] Theorem ⊢ ∀a' a. Variant1_2 a ≠ Variant2_2 a' [test2_induction] Theorem ⊢ ∀P. (∀T. P (Variant1_2 T)) ∧ (∀T. P (Variant2_2 T)) ⇒ ∀t. P t [test2_nchotomy] Theorem ⊢ ∀tt. (∃T. tt = Variant1_2 T) ∨ ∃T. tt = Variant2_2 T [test_11] Theorem ⊢ (∀a a'. Variant1 a = Variant1 a' ⇔ a = a') ∧ ∀a a'. Variant2 a = Variant2 a' ⇔ a = a' [test_Axiom] Theorem ⊢ ∀f0 f1. ∃fn. (∀a. fn (Variant1 a) = f0 a) ∧ ∀a. fn (Variant2 a) = f1 a [test_case_cong] Theorem ⊢ ∀M M' f f1. M = M' ∧ (∀a. M' = Variant1 a ⇒ f a = f' a) ∧ (∀a. M' = Variant2 a ⇒ f1 a = f1' a) ⇒ test_CASE M f f1 = test_CASE M' f' f1' [test_case_eq] Theorem ⊢ test_CASE x f f1 = v ⇔ (∃b. x = Variant1 b ∧ f b = v) ∨ ∃a. x = Variant2 a ∧ f1 a = v [test_distinct] Theorem ⊢ ∀a' a. Variant1 a ≠ Variant2 a' [test_induction] Theorem ⊢ ∀P. (∀b. P (Variant1 b)) ∧ (∀a. P (Variant2 a)) ⇒ ∀t. P t [test_nchotomy] Theorem ⊢ ∀tt. (∃b. tt = Variant1 b) ∨ ∃a. tt = Variant2 a *) end