100 lines
3.4 KiB
Plaintext
100 lines
3.4 KiB
Plaintext
(*
|
|
* Copyright 2020, Data61, CSIRO (ABN 41 687 119 230)
|
|
*
|
|
* SPDX-License-Identifier: BSD-2-Clause
|
|
*)
|
|
|
|
theory word_abs_cases imports
|
|
"AutoCorres.AutoCorres"
|
|
begin
|
|
|
|
(* extra optimization for word_abs (currently does not work) *)
|
|
lemma INT_MIN_MAX_smod: "\<And>a b :: int. \<lbrakk> INT_MIN \<le> b; b \<le> INT_MAX; b \<noteq> 0 \<rbrakk> \<Longrightarrow> INT_MIN \<le> a smod b \<and> a smod b \<le> INT_MAX"
|
|
apply (case_tac "b > 0")
|
|
apply (rule conjI)
|
|
apply (case_tac "a \<ge> 0")
|
|
apply (frule_tac a = a and b = b in smod_int_compares(2))
|
|
apply assumption
|
|
apply (simp add: INT_MIN_def)
|
|
apply (subgoal_tac "0 \<ge> a")
|
|
apply (frule_tac a = a and b = b in smod_int_compares(3))
|
|
apply assumption
|
|
apply (simp add: INT_MIN_def INT_MAX_def)
|
|
apply arith
|
|
apply (case_tac "a \<ge> 0")
|
|
apply (frule_tac a = a and b = b in smod_int_compares(1))
|
|
apply assumption
|
|
apply arith
|
|
apply (subgoal_tac "0 \<ge> a")
|
|
apply (frule_tac a = a and b = b in smod_int_compares(4))
|
|
apply assumption
|
|
apply arith
|
|
apply arith
|
|
apply (subgoal_tac "0 > b")
|
|
apply (rule conjI)
|
|
apply (case_tac "a \<ge> 0")
|
|
apply (frule_tac a = a and b = b in smod_int_compares(6))
|
|
apply assumption
|
|
apply simp
|
|
apply (subgoal_tac "0 \<ge> a")
|
|
apply (frule_tac a = a and b = b in smod_int_compares(8))
|
|
apply assumption
|
|
apply (simp add: INT_MIN_def)
|
|
apply arith
|
|
apply (case_tac "a \<ge> 0")
|
|
apply (frule_tac a = a and b = b in smod_int_compares(5))
|
|
apply assumption
|
|
apply (simp add: INT_MIN_def INT_MAX_def)
|
|
apply (subgoal_tac "0 \<ge> a")
|
|
apply (frule_tac a = a and b = b in smod_int_compares(7))
|
|
apply assumption
|
|
apply (simp add: INT_MAX_def)
|
|
apply arith
|
|
apply arith
|
|
done
|
|
lemma [unfolded INT_MIN_def INT_MAX_def, simplified, L2flow]:
|
|
"simp_expr (INT_MIN \<le> b \<and> b \<le> INT_MAX \<and> b \<noteq> 0) (INT_MIN \<le> a smod b) True"
|
|
by (simp add: simp_expr_def INT_MIN_MAX_smod)
|
|
lemma [unfolded INT_MIN_def INT_MAX_def, simplified, L2flow]:
|
|
"simp_expr (INT_MIN \<le> b \<and> b \<le> INT_MAX \<and> b \<noteq> 0) (a smod b \<le> INT_MAX) True"
|
|
by (simp add: simp_expr_def INT_MIN_MAX_smod)
|
|
|
|
external_file "word_abs_cases.c"
|
|
install_C_file "word_abs_cases.c"
|
|
autocorres [
|
|
unsigned_word_abs = callee_flat_u_abs
|
|
caller_flat_u_aa
|
|
caller_flat_u_an
|
|
] "word_abs_cases.c"
|
|
|
|
context word_abs_cases begin
|
|
|
|
declare K_bind_def[simp del]
|
|
|
|
thm callee_flat_s'_def caller_flat_s'_def
|
|
|
|
thm callee_flat_u_abs'_def callee_flat_u_noabs'_def
|
|
thm caller_flat_u_aa'_def caller_flat_u_an'_def caller_flat_u_na'_def caller_flat_u_nn'_def
|
|
|
|
thm callee_deep_s'.simps caller_deep_s'_def
|
|
|
|
thm mutual_s1'.simps mutual_s2'.simps
|
|
|
|
thm cross'_def gcd_s_rec'.simps gcd_s_loop'_def
|
|
|
|
lemma int_max: "2147483647 \<le> a \<Longrightarrow> sint (n :: sword32) \<le> a"
|
|
using INT_MIN_MAX_lemmas(10)[where s = n, unfolded INT_MAX_def]
|
|
by arith
|
|
lemma int_min: "a \<le> -2147483648 \<Longrightarrow> a \<le> sint (n :: sword32)"
|
|
using INT_MIN_MAX_lemmas(11)[where s = n, unfolded INT_MIN_def]
|
|
by arith
|
|
lemma oguard_True: "(\<And>s. P s) \<Longrightarrow> DO oguard P; f OD = f"
|
|
by (monad_eq simp: oguard_def obind_def K_bind_def)
|
|
|
|
thm gcd_s_loop'_def gcd_s_loop'_def [simplified int_min int_max oguard_True]
|
|
|
|
thm sum'_def sum'_def[simplified, folded INT_MAX_def[simplified] INT_MIN_def[simplified]]
|
|
end
|
|
|
|
end
|