src/HOL/ex/Simps_Case_Conv_Examples.thy
 author wenzelm Wed Jun 22 10:09:20 2016 +0200 (2016-06-22) changeset 63343 fb5d8a50c641 parent 61343 5b5656a63bd6 child 66453 cc19f7ca2ed6 permissions -rw-r--r--
bundle lifting_syntax;
```     1 theory Simps_Case_Conv_Examples imports
```
```     2   "~~/src/HOL/Library/Simps_Case_Conv"
```
```     3 begin
```
```     4
```
```     5 section \<open>Tests for the Simps<->Case conversion tools\<close>
```
```     6
```
```     7 fun foo where
```
```     8   "foo (x # xs) Nil = 0" |
```
```     9   "foo (x # xs) (y # ys) = foo [] []" |
```
```    10   "foo Nil (y # ys) = 1" |
```
```    11   "foo Nil Nil = 3"
```
```    12
```
```    13 fun bar where
```
```    14   "bar x 0 y = 0 + x" |
```
```    15   "bar x (Suc n) y = n + x"
```
```    16
```
```    17 definition
```
```    18   split_rule_test :: "((nat => 'a) + ('b * (('b => 'a) option))) => ('a => nat) => nat"
```
```    19 where
```
```    20  "split_rule_test x f = f (case x of Inl af \<Rightarrow> af 1
```
```    21     | Inr (b, None) => inv f 0
```
```    22     | Inr (b, Some g) => g b)"
```
```    23
```
```    24 definition test where "test x y = (case x of None => (case y of [] => 1 | _ # _ => 2) | Some x => x)"
```
```    25
```
```    26 definition nosplit where "nosplit x = x @ (case x of [] \<Rightarrow>  | xs \<Rightarrow> xs)"
```
```    27
```
```    28
```
```    29 text \<open>Function with complete, non-overlapping patterns\<close>
```
```    30 case_of_simps foo_cases1: foo.simps
```
```    31 lemma
```
```    32   fixes xs :: "'a list" and ys :: "'b list"
```
```    33   shows "foo xs ys = (case (xs, ys) of
```
```    34     ( [], []) \<Rightarrow> 3
```
```    35     | ([], y # ys) \<Rightarrow> 1
```
```    36     | (x # xs, []) \<Rightarrow> 0
```
```    37     | (x # xs, y # ys) \<Rightarrow> foo ([] :: 'a list) ([] :: 'b list))"
```
```    38   by (fact foo_cases1)
```
```    39
```
```    40 text \<open>Redundant equations are ignored\<close>
```
```    41 case_of_simps foo_cases2: foo.simps foo.simps
```
```    42 lemma
```
```    43   fixes xs :: "'a list" and ys :: "'b list"
```
```    44   shows "foo xs ys = (case (xs, ys) of
```
```    45     ( [], []) \<Rightarrow> 3
```
```    46     | ([], y # ys) \<Rightarrow> 1
```
```    47     | (x # xs, []) \<Rightarrow> 0
```
```    48     | (x # xs, y # ys) \<Rightarrow> foo ([] :: 'a list) ([] :: 'b list))"
```
```    49   by (fact foo_cases2)
```
```    50
```
```    51 text \<open>Variable patterns\<close>
```
```    52 case_of_simps bar_cases: bar.simps
```
```    53 print_theorems
```
```    54
```
```    55 text \<open>Case expression not at top level\<close>
```
```    56 simps_of_case split_rule_test_simps: split_rule_test_def
```
```    57 lemma
```
```    58   "split_rule_test (Inl x) f = f (x 1)"
```
```    59   "split_rule_test (Inr (x, None)) f = f (inv f 0)"
```
```    60   "split_rule_test (Inr (x, Some y)) f = f (y x)"
```
```    61   by (fact split_rule_test_simps)+
```
```    62
```
```    63 text \<open>Argument occurs both as case parameter and seperately\<close>
```
```    64 simps_of_case nosplit_simps1: nosplit_def
```
```    65 lemma
```
```    66   "nosplit [] = [] @ "
```
```    67   "nosplit (x # xs) = (x # xs) @ x # xs"
```
```    68   by (fact nosplit_simps1)+
```
```    69
```
```    70 text \<open>Nested case expressions\<close>
```
```    71 simps_of_case test_simps1: test_def
```
```    72 lemma
```
```    73   "test None [] = 1"
```
```    74   "test None (x # xs) = 2"
```
```    75   "test (Some x) y = x"
```
```    76   by (fact test_simps1)+
```
```    77
```
```    78 text \<open>Single-constructor patterns\<close>
```
```    79 case_of_simps fst_conv_simps: fst_conv
```
```    80 lemma "fst x = (case x of (a,b) \<Rightarrow> a)"
```
```    81   by (fact fst_conv_simps)
```
```    82
```
```    83 text \<open>Partial split of case\<close>
```
```    84 simps_of_case nosplit_simps2: nosplit_def (splits: list.split)
```
```    85 lemma
```
```    86   "nosplit [] = [] @ "
```
```    87   "nosplit (x # xs) = (x # xs) @ x # xs"
```
```    88   by (fact nosplit_simps1)+
```
```    89
```
```    90 simps_of_case test_simps2: test_def (splits: option.split)
```
```    91 lemma
```
```    92   "test None y = (case y of [] \<Rightarrow> 1 | x # xs \<Rightarrow> 2)"
```
```    93   "test (Some x) y = x"
```
```    94   by (fact test_simps2)+
```
```    95
```
```    96 text \<open>Reversal\<close>
```
```    97 case_of_simps test_def1: test_simps1
```
```    98 lemma
```
```    99   "test x y = (case (x,y) of
```
```   100     (None, []) \<Rightarrow> 1
```
```   101   | (None, _#_) \<Rightarrow> 2
```
```   102   | (Some x, _) \<Rightarrow> x)"
```
```   103   by (fact test_def1)
```
```   104
```
```   105 text \<open>Case expressions on RHS\<close>
```
```   106 case_of_simps test_def2: test_simps2
```
```   107 lemma "test xs y = (case (xs, y) of (None, []) \<Rightarrow> 1 | (None, x # xa) \<Rightarrow> 2 | (Some x, y) \<Rightarrow> x)"
```
```   108   by (fact test_def2)
```
```   109
```
```   110 text \<open>Partial split of simps\<close>
```
```   111 case_of_simps foo_cons_def: foo.simps(1,2)
```
```   112 lemma
```
```   113   fixes xs :: "'a list" and ys :: "'b list"
```
```   114   shows "foo (x # xs) ys = (case (x,xs,ys) of
```
```   115       (_,_,[]) \<Rightarrow> 0
```
```   116     | (_,_,_ # _) \<Rightarrow> foo ([] :: 'a list) ([] :: 'b list))"
```
```   117   by (fact foo_cons_def)
```
```   118
```
```   119 end
```