diff options
Diffstat (limited to 'lib/isabelle/State_lemmas.thy')
| -rw-r--r-- | lib/isabelle/State_lemmas.thy | 138 |
1 files changed, 98 insertions, 40 deletions
diff --git a/lib/isabelle/State_lemmas.thy b/lib/isabelle/State_lemmas.thy index 84b08e6c..cf5e4dbf 100644 --- a/lib/isabelle/State_lemmas.thy +++ b/lib/isabelle/State_lemmas.thy @@ -1,16 +1,18 @@ theory State_lemmas - imports State + imports State State_lifting begin lemma All_liftState_dom: "liftState_dom (r, m)" by (induction m) (auto intro: liftState.domintros) termination liftState using All_liftState_dom by auto -lemma liftState_bind[simp]: +named_theorems liftState_simp + +lemma liftState_bind[liftState_simp]: "liftState r (bind m f) = bindS (liftState r m) (liftState r \<circ> f)" by (induction m f rule: bind.induct) auto -lemma liftState_return[simp]: "liftState r (return a) = returnS a" by (auto simp: return_def) +lemma liftState_return[liftState_simp]: "liftState r (return a) = returnS a" by (auto simp: return_def) lemma Value_liftState_Run: assumes "(Value a, s') \<in> liftState r m s" @@ -19,45 +21,58 @@ lemma Value_liftState_Run: auto simp add: failS_def throwS_def returnS_def simp del: read_regvalS.simps; blast elim: Value_bindS_elim) -lemmas liftState_if_distrib[simp] = if_distrib[where f = "liftState ra" for ra] - -lemma liftState_throw[simp]: "liftState r (throw e) = throwS e" by (auto simp: throw_def) -lemma liftState_assert[simp]: "liftState r (assert_exp c msg) = assert_expS c msg" by (auto simp: assert_exp_def assert_expS_def) -lemma liftState_exit[simp]: "liftState r (exit0 ()) = exitS ()" by (auto simp: exit0_def exitS_def) -lemma liftState_exclResult[simp]: "liftState r (excl_result ()) = excl_resultS ()" by (auto simp: excl_result_def) -lemma liftState_barrier[simp]: "liftState r (barrier bk) = returnS ()" by (auto simp: barrier_def) -lemma liftState_footprint[simp]: "liftState r (footprint ()) = returnS ()" by (auto simp: footprint_def) -lemma liftState_undefined[simp]: "liftState r (undefined_bool ()) = undefined_boolS ()" by (auto simp: undefined_bool_def) -lemma liftState_maybe_fail[simp]: "liftState r (maybe_fail msg x) = maybe_failS msg x" - by (auto simp: maybe_fail_def maybe_failS_def split: option.splits) - -lemma liftState_try_catch[simp]: +lemmas liftState_if_distrib[liftState_simp] = if_distrib[where f = "liftState ra" for ra] + +lemma liftState_throw[liftState_simp]: "liftState r (throw e) = throwS e" + by (auto simp: throw_def) +lemma liftState_assert[liftState_simp]: "liftState r (assert_exp c msg) = assert_expS c msg" + by (auto simp: assert_exp_def assert_expS_def) +lemma liftState_exit[liftState_simp]: "liftState r (exit0 ()) = exitS ()" + by (auto simp: exit0_def exitS_def) +lemma liftState_exclResult[liftState_simp]: "liftState r (excl_result ()) = excl_resultS ()" + by (auto simp: excl_result_def liftState_simp) +lemma liftState_barrier[liftState_simp]: "liftState r (barrier bk) = returnS ()" + by (auto simp: barrier_def) +lemma liftState_footprint[liftState_simp]: "liftState r (footprint ()) = returnS ()" + by (auto simp: footprint_def) +lemma liftState_undefined[liftState_simp]: "liftState r (undefined_bool ()) = undefined_boolS ()" + by (auto simp: undefined_bool_def liftState_simp) +lemma liftState_maybe_fail[liftState_simp]: "liftState r (maybe_fail msg x) = maybe_failS msg x" + by (auto simp: maybe_fail_def maybe_failS_def liftState_simp split: option.splits) +lemma liftState_and_boolM[liftState_simp]: + "liftState r (and_boolM x y) = and_boolS (liftState r x) (liftState r y)" + by (auto simp: and_boolM_def and_boolS_def liftState_simp cong: bindS_cong if_cong) +lemma liftState_or_boolM[liftState_simp]: + "liftState r (or_boolM x y) = or_boolS (liftState r x) (liftState r y)" + by (auto simp: or_boolM_def or_boolS_def liftState_simp cong: bindS_cong if_cong) + +lemma liftState_try_catch[liftState_simp]: "liftState r (try_catch m h) = try_catchS (liftState r m) (liftState r \<circ> h)" by (induction m h rule: try_catch_induct) (auto simp: try_catchS_bindS_no_throw) -lemma liftState_early_return[simp]: +lemma liftState_early_return[liftState_simp]: "liftState r (early_return r) = early_returnS r" - by (auto simp: early_return_def early_returnS_def) + by (auto simp: early_return_def early_returnS_def liftState_simp) -lemma liftState_catch_early_return[simp]: +lemma liftState_catch_early_return[liftState_simp]: "liftState r (catch_early_return m) = catch_early_returnS (liftState r m)" - by (auto simp: catch_early_return_def catch_early_returnS_def sum.case_distrib cong: sum.case_cong) + by (auto simp: catch_early_return_def catch_early_returnS_def sum.case_distrib liftState_simp cong: sum.case_cong) -lemma liftState_liftR[simp]: - "liftState r (liftR m) = liftSR (liftState r m)" - by (auto simp: liftR_def liftSR_def) +lemma liftState_liftR[liftState_simp]: + "liftState r (liftR m) = liftRS (liftState r m)" + by (auto simp: liftR_def liftRS_def liftState_simp) -lemma liftState_try_catchR[simp]: - "liftState r (try_catchR m h) = try_catchSR (liftState r m) (liftState r \<circ> h)" - by (auto simp: try_catchR_def try_catchSR_def sum.case_distrib cong: sum.case_cong) +lemma liftState_try_catchR[liftState_simp]: + "liftState r (try_catchR m h) = try_catchRS (liftState r m) (liftState r \<circ> h)" + by (auto simp: try_catchR_def try_catchRS_def sum.case_distrib liftState_simp cong: sum.case_cong) lemma liftState_read_mem_BC: assumes "unsigned_method BC_bitU_list (bits_of_method BCa a) = unsigned_method BCa a" shows "liftState r (read_mem BCa BCb rk a sz) = read_memS BCa BCb rk a sz" using assms - by (auto simp: read_mem_def read_mem_bytes_def read_memS_def read_mem_bytesS_def maybe_failS_def split: option.splits) + by (auto simp: read_mem_def read_mem_bytes_def read_memS_def read_mem_bytesS_def maybe_failS_def liftState_simp split: option.splits) -lemma liftState_read_mem[simp]: +lemma liftState_read_mem[liftState_simp]: "\<And>a. liftState r (read_mem BC_mword BC_mword rk a sz) = read_memS BC_mword BC_mword rk a sz" "\<And>a. liftState r (read_mem BC_bitU_list BC_bitU_list rk a sz) = read_memS BC_bitU_list BC_bitU_list rk a sz" by (auto simp: liftState_read_mem_BC) @@ -67,14 +82,14 @@ lemma liftState_write_mem_ea_BC: shows "liftState r (write_mem_ea BCa rk a sz) = write_mem_eaS BCa rk a (nat sz)" using assms by (auto simp: write_mem_ea_def write_mem_eaS_def) -lemma liftState_write_mem_ea[simp]: +lemma liftState_write_mem_ea[liftState_simp]: "\<And>a. liftState r (write_mem_ea BC_mword rk a sz) = write_mem_eaS BC_mword rk a (nat sz)" "\<And>a. liftState r (write_mem_ea BC_bitU_list rk a sz) = write_mem_eaS BC_bitU_list rk a (nat sz)" by (auto simp: liftState_write_mem_ea_BC) lemma liftState_write_mem_val: "liftState r (write_mem_val BC v) = write_mem_valS BC v" - by (auto simp: write_mem_val_def write_mem_valS_def split: option.splits) + by (auto simp: write_mem_val_def write_mem_valS_def liftState_simp split: option.splits) lemma liftState_read_reg_readS: assumes "\<And>s. Option.bind (get_regval' (name reg) s) (of_regval reg) = Some (read_from reg s)" @@ -93,22 +108,23 @@ lemma liftState_write_reg_updateS: shows "liftState (get_regval', set_regval') (write_reg reg v) = updateS (regstate_update (write_to reg v))" using assms by (auto simp: write_reg_def updateS_def returnS_def bindS_readS) -lemma liftState_iter_aux[simp]: +lemma liftState_iter_aux[liftState_simp]: shows "liftState r (iter_aux i f xs) = iterS_aux i (\<lambda>i x. liftState r (f i x)) xs" - by (induction i "\<lambda>i x. liftState r (f i x)" xs rule: iterS_aux.induct) (auto cong: bindS_cong) + by (induction i "\<lambda>i x. liftState r (f i x)" xs rule: iterS_aux.induct) + (auto simp: liftState_simp cong: bindS_cong) -lemma liftState_iteri[simp]: +lemma liftState_iteri[liftState_simp]: "liftState r (iteri f xs) = iteriS (\<lambda>i x. liftState r (f i x)) xs" - by (auto simp: iteri_def iteriS_def) + by (auto simp: iteri_def iteriS_def liftState_simp) -lemma liftState_iter[simp]: +lemma liftState_iter[liftState_simp]: "liftState r (iter f xs) = iterS (liftState r \<circ> f) xs" - by (auto simp: iter_def iterS_def) + by (auto simp: iter_def iterS_def liftState_simp) -lemma liftState_foreachM[simp]: +lemma liftState_foreachM[liftState_simp]: "liftState r (foreachM xs vars body) = foreachS xs vars (\<lambda>x vars. liftState r (body x vars))" by (induction xs vars "\<lambda>x vars. liftState r (body x vars)" rule: foreachS.induct) - (auto cong: bindS_cong) + (auto simp: liftState_simp cong: bindS_cong) lemma whileS_dom_step: assumes "whileS_dom (vars, cond, body, s)" @@ -156,7 +172,7 @@ proof (use assms in \<open>induction vars "liftState r \<circ> cond" "liftState qed then show ?case using while while' that IH by auto qed auto - then show ?case by auto + then show ?case by (auto simp: liftState_simp) qed auto qed @@ -194,9 +210,51 @@ proof (use assms in \<open>induction vars "liftState r \<circ> cond" "liftState show "\<exists>t. Run (body vars) t vars'" using k by (auto elim: Value_liftState_Run) show "\<exists>t'. Run (cond vars') t' False" using until that by (auto elim: Value_liftState_Run) qed - then show ?case using k until IH by (auto simp: comp_def) + then show ?case using k until IH by (auto simp: comp_def liftState_simp) qed auto qed auto qed +(* Simplification rules for monadic Boolean connectives *) + +lemma if_return_return[simp]: "(if a then return True else return False) = return a" by auto + +lemma and_boolM_simps[simp]: + "and_boolM (return b) y = (if b then y else return False)" + "and_boolM x (return True) = x" + "and_boolM x (return False) = x \<bind> (\<lambda>_. return False)" + "\<And>x y z. and_boolM (x \<bind> y) z = (x \<bind> (\<lambda>r. and_boolM (y r) z))" + by (auto simp: and_boolM_def) + +lemmas and_boolM_if_distrib[simp] = if_distrib[where f = "\<lambda>x. and_boolM x y" for y] + +lemma or_boolM_simps[simp]: + "or_boolM (return b) y = (if b then return True else y)" + "or_boolM x (return True) = x \<bind> (\<lambda>_. return True)" + "or_boolM x (return False) = x" + "\<And>x y z. or_boolM (x \<bind> y) z = (x \<bind> (\<lambda>r. or_boolM (y r) z))" + by (auto simp: or_boolM_def) + +lemmas or_boolM_if_distrib[simp] = if_distrib[where f = "\<lambda>x. or_boolM x y" for y] + +lemma if_returnS_returnS[simp]: "(if a then returnS True else returnS False) = returnS a" by auto + +lemma and_boolS_simps[simp]: + "and_boolS (returnS b) y = (if b then y else returnS False)" + "and_boolS x (returnS True) = x" + "and_boolS x (returnS False) = bindS x (\<lambda>_. returnS False)" + "\<And>x y z. and_boolS (bindS x y) z = (bindS x (\<lambda>r. and_boolS (y r) z))" + by (auto simp: and_boolS_def) + +lemmas and_boolS_if_distrib[simp] = if_distrib[where f = "\<lambda>x. and_boolS x y" for y] + +lemma or_boolS_simps[simp]: + "or_boolS (returnS b) y = (if b then returnS True else y)" + "or_boolS x (returnS True) = bindS x (\<lambda>_. returnS True)" + "or_boolS x (returnS False) = x" + "\<And>x y z. or_boolS (bindS x y) z = (bindS x (\<lambda>r. or_boolS (y r) z))" + by (auto simp: or_boolS_def) + +lemmas or_boolS_if_distrib[simp] = if_distrib[where f = "\<lambda>x. or_boolS x y" for y] + end |