--- a/abacus.thy Wed Feb 06 02:25:00 2013 +0000
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,6925 +0,0 @@
-header {*
- {\em abacus} a kind of register machine
-*}
-
-theory abacus
-imports Main turing_basic
-begin
-
-text {*
- {\em Abacus} instructions:
-*}
-
-datatype abc_inst =
- -- {* @{text "Inc n"} increments the memory cell (or register) with address @{text "n"} by one.
- *}
- Inc nat
- -- {*
- @{text "Dec n label"} decrements the memory cell with address @{text "n"} by one.
- If cell @{text "n"} is already zero, no decrements happens and the executio jumps to
- the instruction labeled by @{text "label"}.
- *}
- | Dec nat nat
- -- {*
- @{text "Goto label"} unconditionally jumps to the instruction labeled by @{text "label"}.
- *}
- | Goto nat
-
-
-text {*
- Abacus programs are defined as lists of Abacus instructions.
-*}
-type_synonym abc_prog = "abc_inst list"
-
-section {*
- Sample Abacus programs
- *}
-
-text {*
- Abacus for addition and clearance.
-*}
-fun plus_clear :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> abc_prog"
- where
- "plus_clear m n e = [Dec m e, Inc n, Goto 0]"
-
-text {*
- Abacus for clearing memory untis.
-*}
-fun clear :: "nat \<Rightarrow> nat \<Rightarrow> abc_prog"
- where
- "clear n e = [Dec n e, Goto 0]"
-
-fun plus:: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> abc_prog"
- where
- "plus m n p e = [Dec m 4, Inc n, Inc p,
- Goto 0, Dec p e, Inc m, Goto 4]"
-
-fun mult :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> abc_prog"
- where
- "mult m1 m2 n p e = [Dec m1 e]@ plus m1 m2 p 1"
-
-fun expo :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> abc_prog"
- where
- "expo n m1 m2 p e = [Inc n, Dec m1 e] @ mult m2 n n p 2"
-
-
-text {*
- The state of Abacus machine.
- *}
-type_synonym abc_state = nat
-
-(* text {*
- The memory of Abacus machine is defined as a function from address to contents.
-*}
-type_synonym abc_mem = "nat \<Rightarrow> nat" *)
-
-text {*
- The memory of Abacus machine is defined as a list of contents, with
- every units addressed by index into the list.
- *}
-type_synonym abc_lm = "nat list"
-
-text {*
- Fetching contents out of memory. Units not represented by list elements are considered
- as having content @{text "0"}.
-*}
-fun abc_lm_v :: "abc_lm \<Rightarrow> nat \<Rightarrow> nat"
- where
- "abc_lm_v lm n = (if (n < length lm) then (lm!n) else 0)"
-
-(*
-fun abc_l2m :: "abc_lm \<Rightarrow> abc_mem"
- where
- "abc_l2m lm = abc_lm_v lm"
-*)
-
-text {*
- Set the content of memory unit @{text "n"} to value @{text "v"}.
- @{text "am"} is the Abacus memory before setting.
- If address @{text "n"} is outside to scope of @{text "am"}, @{text "am"}
- is extended so that @{text "n"} becomes in scope.
-*}
-fun abc_lm_s :: "abc_lm \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> abc_lm"
- where
- "abc_lm_s am n v = (if (n < length am) then (am[n:=v]) else
- am@ (replicate (n - length am) 0) @ [v])"
-
-
-text {*
- The configuration of Abaucs machines consists of its current state and its
- current memory:
-*}
-type_synonym abc_conf_l = "abc_state \<times> abc_lm"
-
-text {*
- Fetch instruction out of Abacus program:
-*}
-
-fun abc_fetch :: "nat \<Rightarrow> abc_prog \<Rightarrow> abc_inst option"
- where
- "abc_fetch s p = (if (s < length p) then Some (p ! s)
- else None)"
-
-text {*
- Single step execution of Abacus machine. If no instruction is feteched,
- configuration does not change.
-*}
-fun abc_step_l :: "abc_conf_l \<Rightarrow> abc_inst option \<Rightarrow> abc_conf_l"
- where
- "abc_step_l (s, lm) a = (case a of
- None \<Rightarrow> (s, lm) |
- Some (Inc n) \<Rightarrow> (let nv = abc_lm_v lm n in
- (s + 1, abc_lm_s lm n (nv + 1))) |
- Some (Dec n e) \<Rightarrow> (let nv = abc_lm_v lm n in
- if (nv = 0) then (e, abc_lm_s lm n 0)
- else (s + 1, abc_lm_s lm n (nv - 1))) |
- Some (Goto n) \<Rightarrow> (n, lm)
- )"
-
-text {*
- Multi-step execution of Abacus machine.
-*}
-fun abc_steps_l :: "abc_conf_l \<Rightarrow> abc_prog \<Rightarrow> nat \<Rightarrow> abc_conf_l"
- where
- "abc_steps_l (s, lm) p 0 = (s, lm)" |
- "abc_steps_l (s, lm) p (Suc n) = abc_steps_l (abc_step_l (s, lm) (abc_fetch s p)) p n"
-
-section {*
- Compiling Abacus machines into Truing machines
-*}
-
-
-subsection {*
- Compiling functions
-*}
-
-text {*
- @{text "findnth n"} returns the TM which locates the represention of
- memory cell @{text "n"} on the tape and changes representation of zero
- on the way.
-*}
-
-fun findnth :: "nat \<Rightarrow> tprog"
- where
- "findnth 0 = []" |
- "findnth (Suc n) = (findnth n @ [(W1, 2 * n + 1),
- (R, 2 * n + 2), (R, 2 * n + 3), (R, 2 * n + 2)])"
-
-text {*
- @{text "tinc_b"} returns the TM which increments the representation
- of the memory cell under rw-head by one and move the representation
- of cells afterwards to the right accordingly.
- *}
-
-definition tinc_b :: "tprog"
- where
- "tinc_b \<equiv> [(W1, 1), (R, 2), (W1, 3), (R, 2), (W1, 3), (R, 4),
- (L, 7), (W0, 5), (R, 6), (W0, 5), (W1, 3), (R, 6),
- (L, 8), (L, 7), (R, 9), (L, 7), (R, 10), (W0, 9)]"
-
-(* FIXME: doubly defined
-text {*
- @{text "tshift tm off"} shifts @{text "tm"} by offset @{text "off"}, leaving
- instructions concerning state @{text "0"} unchanged, because state @{text "0"}
- is the end state, which needs not be changed with shift operation.
- *}
-
-fun tshift :: "tprog \<Rightarrow> nat \<Rightarrow> tprog"
- where
- "tshift tp off = (map (\<lambda> (action, state).
- (action, (if state = 0 then 0
- else state + off))) tp)"
-*)
-
-text {*
- @{text "tinc ss n"} returns the TM which simulates the execution of
- Abacus instruction @{text "Inc n"}, assuming that TM is located at
- location @{text "ss"} in the final TM complied from the whole
- Abacus program.
-*}
-
-fun tinc :: "nat \<Rightarrow> nat \<Rightarrow> tprog"
- where
- "tinc ss n = tshift (findnth n @ tshift tinc_b (2 * n)) (ss - 1)"
-
-text {*
- @{text "tinc_b"} returns the TM which decrements the representation
- of the memory cell under rw-head by one and move the representation
- of cells afterwards to the left accordingly.
- *}
-
-definition tdec_b :: "tprog"
- where
- "tdec_b \<equiv> [(W1, 1), (R, 2), (L, 14), (R, 3), (L, 4), (R, 3),
- (R, 5), (W0, 4), (R, 6), (W0, 5), (L, 7), (L, 8),
- (L, 11), (W0, 7), (W1, 8), (R, 9), (L, 10), (R, 9),
- (R, 5), (W0, 10), (L, 12), (L, 11), (R, 13), (L, 11),
- (R, 17), (W0, 13), (L, 15), (L, 14), (R, 16), (L, 14),
- (R, 0), (W0, 16)]"
-
-text {*
- @{text "sete tp e"} attaches the termination edges (edges leading to state @{text "0"})
- of TM @{text "tp"} to the intruction labelled by @{text "e"}.
- *}
-
-fun sete :: "tprog \<Rightarrow> nat \<Rightarrow> tprog"
- where
- "sete tp e = map (\<lambda> (action, state). (action, if state = 0 then e else state)) tp"
-
-text {*
- @{text "tdec ss n label"} returns the TM which simulates the execution of
- Abacus instruction @{text "Dec n label"}, assuming that TM is located at
- location @{text "ss"} in the final TM complied from the whole
- Abacus program.
-*}
-
-fun tdec :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> tprog"
- where
- "tdec ss n e = sete (tshift (findnth n @ tshift tdec_b (2 * n))
- (ss - 1)) e"
-
-text {*
- @{text "tgoto f(label)"} returns the TM simulating the execution of Abacus instruction
- @{text "Goto label"}, where @{text "f(label)"} is the corresponding location of
- @{text "label"} in the final TM compiled from the overall Abacus program.
-*}
-
-fun tgoto :: "nat \<Rightarrow> tprog"
- where
- "tgoto n = [(Nop, n), (Nop, n)]"
-
-text {*
- The layout of the final TM compiled from an Abacus program is represented
- as a list of natural numbers, where the list element at index @{text "n"} represents the
- starting state of the TM simulating the execution of @{text "n"}-th instruction
- in the Abacus program.
-*}
-
-type_synonym layout = "nat list"
-
-text {*
- @{text "length_of i"} is the length of the
- TM simulating the Abacus instruction @{text "i"}.
-*}
-fun length_of :: "abc_inst \<Rightarrow> nat"
- where
- "length_of i = (case i of
- Inc n \<Rightarrow> 2 * n + 9 |
- Dec n e \<Rightarrow> 2 * n + 16 |
- Goto n \<Rightarrow> 1)"
-
-text {*
- @{text "layout_of ap"} returns the layout of Abacus program @{text "ap"}.
-*}
-fun layout_of :: "abc_prog \<Rightarrow> layout"
- where "layout_of ap = map length_of ap"
-
-
-text {*
- @{text "start_of layout n"} looks out the starting state of @{text "n"}-th
- TM in the finall TM.
-*}
-
-fun start_of :: "nat list \<Rightarrow> nat \<Rightarrow> nat"
- where
- "start_of ly 0 = Suc 0" |
- "start_of ly (Suc as) =
- (if as < length ly then start_of ly as + (ly ! as)
- else start_of ly as)"
-
-text {*
- @{text "ci lo ss i"} complies Abacus instruction @{text "i"}
- assuming the TM of @{text "i"} starts from state @{text "ss"}
- within the overal layout @{text "lo"}.
-*}
-
-fun ci :: "layout \<Rightarrow> nat \<Rightarrow> abc_inst \<Rightarrow> tprog"
- where
- "ci ly ss i = (case i of
- Inc n \<Rightarrow> tinc ss n |
- Dec n e \<Rightarrow> tdec ss n (start_of ly e) |
- Goto n \<Rightarrow> tgoto (start_of ly n))"
-
-text {*
- @{text "tpairs_of ap"} transfroms Abacus program @{text "ap"} pairing
- every instruction with its starting state.
-*}
-fun tpairs_of :: "abc_prog \<Rightarrow> (nat \<times> abc_inst) list"
- where "tpairs_of ap = (zip (map (start_of (layout_of ap))
- [0..<(length ap)]) ap)"
-
-
-text {*
- @{text "tms_of ap"} returns the list of TMs, where every one of them simulates
- the corresponding Abacus intruction in @{text "ap"}.
-*}
-
-fun tms_of :: "abc_prog \<Rightarrow> tprog list"
- where "tms_of ap = map (\<lambda> (n, tm). ci (layout_of ap) n tm)
- (tpairs_of ap)"
-
-text {*
- @{text "tm_of ap"} returns the final TM machine compiled from Abacus program @{text "ap"}.
-*}
-fun tm_of :: "abc_prog \<Rightarrow> tprog"
- where "tm_of ap = concat (tms_of ap)"
-
-text {*
- The following two functions specify the well-formedness of complied TM.
-*}
-fun t_ncorrect :: "tprog \<Rightarrow> bool"
- where
- "t_ncorrect tp = (length tp mod 2 = 0)"
-
-fun abc2t_correct :: "abc_prog \<Rightarrow> bool"
- where
- "abc2t_correct ap = list_all (\<lambda> (n, tm).
- t_ncorrect (ci (layout_of ap) n tm)) (tpairs_of ap)"
-
-lemma findnth_length: "length (findnth n) div 2 = 2 * n"
-apply(induct n, simp, simp)
-done
-
-lemma ci_length : "length (ci ns n ai) div 2 = length_of ai"
-apply(auto simp: ci.simps tinc_b_def tdec_b_def findnth_length
- split: abc_inst.splits)
-done
-
-subsection {*
- Representation of Abacus memory by TM tape
-*}
-
-consts tape_of :: "'a \<Rightarrow> block list" ("<_>" 100)
-
-text {*
- @{text "tape_of_nat_list am"} returns the TM tape representing
- Abacus memory @{text "am"}.
- *}
-
-fun tape_of_nat_list :: "nat list \<Rightarrow> block list"
- where
- "tape_of_nat_list [] = []" |
- "tape_of_nat_list [n] = Oc\<^bsup>n+1\<^esup>" |
- "tape_of_nat_list (n#ns) = (Oc\<^bsup>n+1\<^esup>) @ [Bk] @ (tape_of_nat_list ns)"
-
-defs (overloaded)
- tape_of_nl_abv: "<am> \<equiv> tape_of_nat_list am"
- tape_of_nat_abv : "<(n::nat)> \<equiv> Oc\<^bsup>n+1\<^esup>"
-
-text {*
- @{text "crsp_l acf tcf"} meams the abacus configuration @{text "acf"}
- is corretly represented by the TM configuration @{text "tcf"}.
-*}
-
-fun crsp_l :: "layout \<Rightarrow> abc_conf_l \<Rightarrow> t_conf \<Rightarrow> block list \<Rightarrow> bool"
- where
- "crsp_l ly (as, lm) (ts, (l, r)) inres =
- (ts = start_of ly as \<and> (\<exists> rn. r = <lm> @ Bk\<^bsup>rn\<^esup>)
- \<and> l = Bk # Bk # inres)"
-
-declare crsp_l.simps[simp del]
-
-subsection {*
- A more general definition of TM execution.
-*}
-
-(*
-fun nnth_of :: "(taction \<times> nat) list \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> (taction \<times> nat)"
- where
- "nnth_of p s b = (if 2*s < length p
- then (p ! (2*s + b)) else (Nop, 0))"
-
-thm nth_of.simps
-
-fun nfetch :: "tprog \<Rightarrow> nat \<Rightarrow> block \<Rightarrow> taction \<times> nat"
- where
- "nfetch p 0 b = (Nop, 0)" |
- "nfetch p (Suc s) b =
- (case b of
- Bk \<Rightarrow> nnth_of p s 0 |
- Oc \<Rightarrow> nnth_of p s 1)"
-*)
-
-text {*
- @{text "t_step tcf (tp, ss)"} returns the result of one step exection of TM @{text "tp"}
- assuming @{text "tp"} starts from instial state @{text "ss"}.
-*}
-
-fun t_step :: "t_conf \<Rightarrow> (tprog \<times> nat) \<Rightarrow> t_conf"
- where
- "t_step c (p, off) =
- (let (state, leftn, rightn) = c in
- let (action, next_state) = fetch p (state-off)
- (case rightn of
- [] \<Rightarrow> Bk |
- Bk # xs \<Rightarrow> Bk |
- Oc # xs \<Rightarrow> Oc
- )
- in
- (next_state, new_tape action (leftn, rightn)))"
-
-
-text {*
- @{text "t_steps tcf (tp, ss) n"} returns the result of @{text "n"}-step exection
- of TM @{text "tp"} assuming @{text "tp"} starts from instial state @{text "ss"}.
-*}
-
-fun t_steps :: "t_conf \<Rightarrow> (tprog \<times> nat) \<Rightarrow> nat \<Rightarrow> t_conf"
- where
- "t_steps c (p, off) 0 = c" |
- "t_steps c (p, off) (Suc n) = t_steps
- (t_step c (p, off)) (p, off) n"
-
-lemma stepn: "t_steps c (p, off) (Suc n) =
- t_step (t_steps c (p, off) n) (p, off)"
-apply(induct n arbitrary: c, simp add: t_steps.simps)
-apply(simp add: t_steps.simps)
-done
-
-text {*
- The type of invarints expressing correspondence between
- Abacus configuration and TM configuration.
-*}
-
-type_synonym inc_inv_t = "abc_conf_l \<Rightarrow> t_conf \<Rightarrow> block list \<Rightarrow> bool"
-
-declare tms_of.simps[simp del] tm_of.simps[simp del]
- layout_of.simps[simp del] abc_fetch.simps [simp del]
- t_step.simps[simp del] t_steps.simps[simp del]
- tpairs_of.simps[simp del] start_of.simps[simp del]
- fetch.simps [simp del] t_ncorrect.simps[simp del]
- new_tape.simps [simp del] ci.simps [simp del] length_of.simps[simp del]
- layout_of.simps[simp del] crsp_l.simps[simp del]
- abc2t_correct.simps[simp del]
-
-lemma tct_div2: "t_ncorrect tp \<Longrightarrow> (length tp) mod 2 = 0"
-apply(simp add: t_ncorrect.simps)
-done
-
-lemma t_shift_fetch:
- "\<lbrakk>t_ncorrect tp1; t_ncorrect tp;
- length tp1 div 2 < a \<and> a \<le> length tp1 div 2 + length tp div 2\<rbrakk>
- \<Longrightarrow> fetch tp (a - length tp1 div 2) b =
- fetch (tp1 @ tp @ tp2) a b"
-apply(subgoal_tac "\<exists> x. a = length tp1 div 2 + x", erule exE, simp)
-apply(case_tac x, simp)
-apply(subgoal_tac "length tp1 div 2 + Suc nat =
- Suc (length tp1 div 2 + nat)")
-apply(simp only: fetch.simps nth_of.simps, auto)
-apply(case_tac b, simp)
-apply(subgoal_tac "2 * (length tp1 div 2) = length tp1", simp)
-apply(subgoal_tac "2 * nat < length tp", simp add: nth_append, simp)
-apply(simp add: t_ncorrect.simps, auto)
-apply(subgoal_tac "2 * (length tp1 div 2) = length tp1", simp)
-apply(subgoal_tac "2 * nat < length tp", simp add: nth_append, auto)
-apply(simp add: t_ncorrect.simps, auto)
-apply(rule_tac x = "a - length tp1 div 2" in exI, simp)
-done
-
-lemma t_shift_in_step:
- "\<lbrakk>t_step (a, aa, ba) (tp, length tp1 div 2) = (s, l, r);
- t_ncorrect tp1; t_ncorrect tp;
- length tp1 div 2 < a \<and> a \<le> length tp1 div 2 + length tp div 2\<rbrakk>
- \<Longrightarrow> t_step (a, aa, ba) (tp1 @ tp @ tp2, 0) = (s, l, r)"
-apply(simp add: t_step.simps)
-apply(subgoal_tac "fetch tp (a - length tp1 div 2) (case ba of [] \<Rightarrow>
- Bk | x # xs \<Rightarrow> x)
- = fetch (tp1 @ tp @ tp2) a (case ba of [] \<Rightarrow> Bk | x # xs
- \<Rightarrow> x)")
-apply(case_tac "fetch tp (a - length tp1 div 2) (case ba of [] \<Rightarrow> Bk
- | x # xs \<Rightarrow> x)")
-apply(auto intro: t_shift_fetch)
-apply(case_tac ba, simp, simp)
-apply(case_tac aaa, simp, simp)
-done
-
-declare add_Suc_right[simp del]
-lemma t_step_add: "t_steps c (p, off) (m + n) =
- t_steps (t_steps c (p, off) m) (p, off) n"
-apply(induct m arbitrary: n, simp add: t_steps.simps, simp)
-apply(subgoal_tac "t_steps c (p, off) (Suc (m + n)) =
- t_steps c (p, off) (m + Suc n)", simp)
-apply(subgoal_tac "t_steps (t_steps c (p, off) m) (p, off) (Suc n) =
- t_steps (t_step (t_steps c (p, off) m) (p, off))
- (p, off) n")
-apply(simp, simp add: stepn)
-apply(simp only: t_steps.simps)
-apply(simp only: add_Suc_right)
-done
-declare add_Suc_right[simp]
-
-lemma s_out_fetch: "\<lbrakk>t_ncorrect tp;
- \<not> (length tp1 div 2 < a \<and> a \<le> length tp1 div 2 +
- length tp div 2)\<rbrakk>
- \<Longrightarrow> fetch tp (a - length tp1 div 2) b = (Nop, 0)"
-apply(auto)
-apply(simp add: fetch.simps)
-apply(subgoal_tac "\<exists> x. a - length tp1 div 2 = length tp div 2 + x")
-apply(erule exE, simp)
-apply(case_tac x, simp)
-apply(auto simp add: fetch.simps)
-apply(subgoal_tac "2 * (length tp div 2) = length tp")
-apply(auto simp: t_ncorrect.simps split: block.splits)
-apply(rule_tac x = "a - length tp1 div 2 - length tp div 2" in exI
- , simp)
-done
-
-lemma conf_keep_step:
- "\<lbrakk>t_ncorrect tp;
- \<not> (length tp1 div 2 < a \<and> a \<le> length tp1 div 2 +
- length tp div 2)\<rbrakk>
- \<Longrightarrow> t_step (a, aa, ba) (tp, length tp1 div 2) = (0, aa, ba)"
-apply(simp add: t_step.simps)
-apply(subgoal_tac "fetch tp (a - length tp1 div 2) (case ba of [] \<Rightarrow>
- Bk | Bk # xs \<Rightarrow> Bk | Oc # xs \<Rightarrow> Oc) = (Nop, 0)")
-apply(simp add: new_tape.simps)
-apply(rule s_out_fetch, simp, simp)
-done
-
-lemma conf_keep:
- "\<lbrakk>t_ncorrect tp;
- \<not> (length tp1 div 2 < a \<and>
- a \<le> length tp1 div 2 + length tp div 2); n > 0\<rbrakk>
- \<Longrightarrow> t_steps (a, aa, ba) (tp, length tp1 div 2) n = (0, aa, ba)"
-apply(induct n, simp)
-apply(case_tac n, simp add: t_steps.simps)
-apply(rule_tac conf_keep_step, simp+)
-apply(subgoal_tac " t_steps (a, aa, ba)
- (tp, length tp1 div 2) (Suc (Suc nat))
- = t_step (t_steps (a, aa, ba)
- (tp, length tp1 div 2) (Suc nat)) (tp, length tp1 div 2)")
-apply(simp)
-apply(rule_tac conf_keep_step, simp, simp)
-apply(rule stepn)
-done
-
-lemma state_bef_inside:
- "\<lbrakk>t_ncorrect tp1; t_ncorrect tp;
- t_steps (s0, l0, r0) (tp, length tp1 div 2) stp = (s, l, r);
- length tp1 div 2 < s0 \<and>
- s0 \<le> length tp1 div 2 + length tp div 2;
- length tp1 div 2 < s \<and> s \<le> length tp1 div 2 + length tp div 2;
- n < stp; t_steps (s0, l0, r0) (tp, length tp1 div 2) n =
- (a, aa, ba)\<rbrakk>
- \<Longrightarrow> length tp1 div 2 < a \<and>
- a \<le> length tp1 div 2 + length tp div 2"
-apply(subgoal_tac "\<exists> x. stp = n + x", erule exE)
-apply(simp only: t_step_add)
-apply(rule classical)
-apply(subgoal_tac "t_steps (a, aa, ba)
- (tp, length tp1 div 2) x = (0, aa, ba)")
-apply(simp)
-apply(rule conf_keep, simp, simp, simp)
-apply(rule_tac x = "stp - n" in exI, simp)
-done
-
-lemma turing_shift_inside:
- "\<lbrakk>t_steps (s0, l0, r0) (tp, length tp1 div 2) stp = (s, l, r);
- length tp1 div 2 < s0 \<and>
- s0 \<le> length tp1 div 2 + length tp div 2;
- t_ncorrect tp1; t_ncorrect tp;
- length tp1 div 2 < s \<and>
- s \<le> length tp1 div 2 + length tp div 2\<rbrakk>
- \<Longrightarrow> t_steps (s0, l0, r0) (tp1 @ tp @ tp2, 0) stp = (s, l, r)"
-apply(induct stp arbitrary: s l r)
-apply(simp add: t_steps.simps)
-apply(subgoal_tac " t_steps (s0, l0, r0)
- (tp, length tp1 div 2) (Suc stp)
- = t_step (t_steps (s0, l0, r0)
- (tp, length tp1 div 2) stp) (tp, length tp1 div 2)")
-apply(case_tac "t_steps (s0, l0, r0) (tp, length tp1 div 2) stp")
-apply(subgoal_tac "length tp1 div 2 < a \<and>
- a \<le> length tp1 div 2 + length tp div 2")
-apply(subgoal_tac "t_steps (s0, l0, r0)
- (tp1 @ tp @ tp2, 0) stp = (a, b, c)")
-apply(simp only: stepn, simp)
-apply(rule_tac t_shift_in_step, simp+)
-defer
-apply(rule stepn)
-apply(rule_tac n = stp and stp = "Suc stp" and a = a
- and aa = b and ba = c in state_bef_inside, simp+)
-done
-
-lemma take_Suc_last[elim]: "Suc as \<le> length xs \<Longrightarrow>
- take (Suc as) xs = take as xs @ [xs ! as]"
-apply(induct xs arbitrary: as, simp, simp)
-apply(case_tac as, simp, simp)
-done
-
-lemma concat_suc: "Suc as \<le> length xs \<Longrightarrow>
- concat (take (Suc as) xs) = concat (take as xs) @ xs! as"
-apply(subgoal_tac "take (Suc as) xs = take as xs @ [xs ! as]", simp)
-by auto
-
-lemma concat_take_suc_iff: "Suc n \<le> length tps \<Longrightarrow>
- concat (take n tps) @ (tps ! n) = concat (take (Suc n) tps)"
-apply(drule_tac concat_suc, simp)
-done
-
-lemma concat_drop_suc_iff:
- "Suc n < length tps \<Longrightarrow> concat (drop (Suc n) tps) =
- tps ! Suc n @ concat (drop (Suc (Suc n)) tps)"
-apply(induct tps arbitrary: n, simp, simp)
-apply(case_tac tps, simp, simp)
-apply(case_tac n, simp, simp)
-done
-
-declare append_assoc[simp del]
-
-lemma tm_append: "\<lbrakk>n < length tps; tp = tps ! n\<rbrakk> \<Longrightarrow>
- \<exists> tp1 tp2. concat tps = tp1 @ tp @ tp2 \<and> tp1 =
- concat (take n tps) \<and> tp2 = concat (drop (Suc n) tps)"
-apply(rule_tac x = "concat (take n tps)" in exI)
-apply(rule_tac x = "concat (drop (Suc n) tps)" in exI)
-apply(auto)
-apply(induct n, simp)
-apply(case_tac tps, simp, simp, simp)
-apply(subgoal_tac "concat (take n tps) @ (tps ! n) =
- concat (take (Suc n) tps)")
-apply(simp only: append_assoc[THEN sym], simp only: append_assoc)
-apply(subgoal_tac " concat (drop (Suc n) tps) = tps ! Suc n @
- concat (drop (Suc (Suc n)) tps)", simp)
-apply(rule_tac concat_drop_suc_iff, simp)
-apply(rule_tac concat_take_suc_iff, simp)
-done
-
-declare append_assoc[simp]
-
-lemma map_of: "n < length xs \<Longrightarrow> (map f xs) ! n = f (xs ! n)"
-by(auto)
-
-lemma [simp]: "length (tms_of aprog) = length aprog"
-apply(auto simp: tms_of.simps tpairs_of.simps)
-done
-
-lemma ci_nth: "\<lbrakk>ly = layout_of aprog; as < length aprog;
- abc_fetch as aprog = Some ins\<rbrakk>
- \<Longrightarrow> ci ly (start_of ly as) ins = tms_of aprog ! as"
-apply(simp add: tms_of.simps tpairs_of.simps
- abc_fetch.simps map_of del: map_append)
-done
-
-lemma t_split:"\<lbrakk>
- ly = layout_of aprog;
- as < length aprog; abc_fetch as aprog = Some ins\<rbrakk>
- \<Longrightarrow> \<exists> tp1 tp2. concat (tms_of aprog) =
- tp1 @ (ci ly (start_of ly as) ins) @ tp2
- \<and> tp1 = concat (take as (tms_of aprog)) \<and>
- tp2 = concat (drop (Suc as) (tms_of aprog))"
-apply(insert tm_append[of "as" "tms_of aprog"
- "ci ly (start_of ly as) ins"], simp)
-apply(subgoal_tac "ci ly (start_of ly as) ins = (tms_of aprog) ! as")
-apply(subgoal_tac "length (tms_of aprog) = length aprog", simp, simp)
-apply(rule_tac ci_nth, auto)
-done
-
-lemma math_sub: "\<lbrakk>x >= Suc 0; x - 1 = z\<rbrakk> \<Longrightarrow> x + y - Suc 0 = z + y"
-by auto
-
-lemma start_more_one: "as \<noteq> 0 \<Longrightarrow> start_of ly as >= Suc 0"
-apply(induct as, simp add: start_of.simps)
-apply(case_tac as, auto simp: start_of.simps)
-done
-
-lemma tm_ct: "\<lbrakk>abc2t_correct aprog; tp \<in> set (tms_of aprog)\<rbrakk> \<Longrightarrow>
- t_ncorrect tp"
-apply(simp add: abc2t_correct.simps tms_of.simps)
-apply(auto)
-apply(simp add:list_all_iff, auto)
-done
-
-lemma div_apart: "\<lbrakk>x mod (2::nat) = 0; y mod 2 = 0\<rbrakk>
- \<Longrightarrow> (x + y) div 2 = x div 2 + y div 2"
-apply(drule mod_eqD)+
-apply(auto)
-done
-
-lemma div_apart_iff: "\<lbrakk>x mod (2::nat) = 0; y mod 2 = 0\<rbrakk> \<Longrightarrow>
- (x + y) mod 2 = 0"
-apply(auto)
-done
-
-lemma tms_ct: "\<lbrakk>abc2t_correct aprog; n < length aprog\<rbrakk> \<Longrightarrow>
- t_ncorrect (concat (take n (tms_of aprog)))"
-apply(induct n, simp add: t_ncorrect.simps, simp)
-apply(subgoal_tac "concat (take (Suc n) (tms_of aprog)) =
- concat (take n (tms_of aprog)) @ (tms_of aprog ! n)", simp)
-apply(simp add: t_ncorrect.simps)
-apply(rule_tac div_apart_iff, simp)
-apply(subgoal_tac "t_ncorrect (tms_of aprog ! n)",
- simp add: t_ncorrect.simps)
-apply(rule_tac tm_ct, simp)
-apply(rule_tac nth_mem, simp add: tms_of.simps tpairs_of.simps)
-apply(rule_tac concat_suc, simp add: tms_of.simps tpairs_of.simps)
-done
-
-lemma tcorrect_div2: "\<lbrakk>abc2t_correct aprog; Suc as < length aprog\<rbrakk>
- \<Longrightarrow> (length (concat (take as (tms_of aprog))) + length (tms_of aprog
- ! as)) div 2 = length (concat (take as (tms_of aprog))) div 2 +
- length (tms_of aprog ! as) div 2"
-apply(subgoal_tac "t_ncorrect (tms_of aprog ! as)")
-apply(subgoal_tac "t_ncorrect (concat (take as (tms_of aprog)))")
-apply(rule_tac div_apart)
-apply(rule tct_div2, simp)+
-apply(erule_tac tms_ct, simp)
-apply(rule_tac tm_ct, simp)
-apply(rule_tac nth_mem)
-apply(simp add: tms_of.simps tpairs_of.simps)
-done
-
-lemma [simp]: "length (layout_of aprog) = length aprog"
-apply(auto simp: layout_of.simps)
-done
-
-lemma start_of_ind: "\<lbrakk>as < length aprog; ly = layout_of aprog\<rbrakk> \<Longrightarrow>
- start_of ly (Suc as) = start_of ly as +
- length ((tms_of aprog) ! as) div 2"
-apply(simp only: start_of.simps, simp)
-apply(auto simp: start_of.simps tms_of.simps layout_of.simps
- tpairs_of.simps)
-apply(simp add: ci_length)
-done
-
-lemma concat_take_suc: "Suc n \<le> length xs \<Longrightarrow>
- concat (take (Suc n) xs) = concat (take n xs) @ (xs ! n)"
-apply(subgoal_tac "take (Suc n) xs =
- take n xs @ [xs ! n]")
-apply(auto)
-done
-
-lemma ci_length_not0: "Suc 0 <= length (ci ly as i) div 2"
-apply(subgoal_tac "length (ci ly as i) div 2 = length_of i")
-apply(simp add: length_of.simps split: abc_inst.splits)
-apply(rule ci_length)
-done
-
-lemma findnth_length2: "length (findnth n) = 4 * n"
-apply(induct n, simp)
-apply(simp)
-done
-
-lemma ci_length2: "length (ci ly as i) = 2 * (length_of i)"
-apply(simp add: ci.simps length_of.simps tinc_b_def tdec_b_def
- split: abc_inst.splits, auto)
-apply(simp add: findnth_length2)+
-done
-
-lemma tm_mod2: "as < length aprog \<Longrightarrow>
- length (tms_of aprog ! as) mod 2 = 0"
-apply(simp add: tms_of.simps)
-apply(subgoal_tac "map (\<lambda>(x, y). ci (layout_of aprog) x y)
- (tpairs_of aprog) ! as
- = (\<lambda>(x, y). ci (layout_of aprog) x y)
- ((tpairs_of aprog) ! as)", simp)
-apply(case_tac "(tpairs_of aprog ! as)", simp)
-apply(subgoal_tac "length (ci (layout_of aprog) a b) =
- 2 * (length_of b)", simp)
-apply(rule ci_length2)
-apply(rule map_of, simp add: tms_of.simps tpairs_of.simps)
-done
-
-lemma tms_mod2: "as \<le> length aprog \<Longrightarrow>
- length (concat (take as (tms_of aprog))) mod 2 = 0"
-apply(induct as, simp, simp)
-apply(subgoal_tac "concat (take (Suc as) (tms_of aprog))
- = concat (take as (tms_of aprog)) @
- (tms_of aprog ! as)", auto)
-apply(rule div_apart_iff, simp, rule tm_mod2, simp)
-apply(rule concat_take_suc, simp add: tms_of.simps tpairs_of.simps)
-done
-
-lemma [simp]: "\<lbrakk>as < length aprog; (abc_fetch as aprog) = Some ins\<rbrakk>
- \<Longrightarrow> ci (layout_of aprog)
- (start_of (layout_of aprog) as) (ins) \<in> set (tms_of aprog)"
-apply(insert ci_nth[of "layout_of aprog" aprog as], simp)
-done
-
-lemma startof_not0: "start_of ly as > 0"
-apply(induct as, simp add: start_of.simps)
-apply(case_tac as, auto simp: start_of.simps)
-done
-
-declare abc_step_l.simps[simp del]
-lemma pre_lheq: "\<lbrakk>tp = concat (take as (tms_of aprog));
- abc2t_correct aprog; as \<le> length aprog\<rbrakk> \<Longrightarrow>
- start_of (layout_of aprog) as - Suc 0 = length tp div 2"
-apply(induct as arbitrary: tp, simp add: start_of.simps, simp)
-proof -
- fix as tp
- assume h1: "\<And>tp. tp = concat (take as (tms_of aprog)) \<Longrightarrow>
- start_of (layout_of aprog) as - Suc 0 =
- length (concat (take as (tms_of aprog))) div 2"
- and h2: " abc2t_correct aprog" "Suc as \<le> length aprog"
- from h2 show "start_of (layout_of aprog) (Suc as) - Suc 0 =
- length (concat (take (Suc as) (tms_of aprog))) div 2"
- apply(insert h1[of "concat (take as (tms_of aprog))"], simp)
- apply(insert start_of_ind[of as aprog "layout_of aprog"], simp)
- apply(subgoal_tac "(take (Suc as) (tms_of aprog)) =
- take as (tms_of aprog) @ [(tms_of aprog) ! as]", simp)
- apply(subgoal_tac "(length (concat (take as (tms_of aprog))) +
- length (tms_of aprog ! as)) div 2
- = length (concat (take as (tms_of aprog))) div 2 +
- length (tms_of aprog ! as) div 2", simp)
- apply(subgoal_tac "start_of (layout_of aprog) as =
- length (concat (take as (tms_of aprog))) div 2 + Suc 0", simp)
- apply(subgoal_tac "start_of (layout_of aprog) as > 0", simp,
- rule_tac startof_not0)
- apply(insert tm_mod2[of as aprog], simp)
- apply(insert tms_mod2[of as aprog], simp, arith)
- apply(rule take_Suc_last, simp)
- done
-qed
-
-lemma crsp2stateq:
- "\<lbrakk>as < length aprog; abc2t_correct aprog;
- crsp_l (layout_of aprog) (as, am) (a, aa, ba) inres\<rbrakk> \<Longrightarrow>
- a = length (concat (take as (tms_of aprog))) div 2 + 1"
-apply(simp add: crsp_l.simps)
-apply(insert pre_lheq[of "(concat (take as (tms_of aprog)))" as aprog]
-, simp)
-apply(subgoal_tac "start_of (layout_of aprog) as > 0",
- auto intro: startof_not0)
-done
-
-lemma turing_shift_outside:
- "\<lbrakk>t_steps (s0, l0, r0) (tp, length tp1 div 2) stp = (s, l, r);
- s \<noteq> 0; stp > 0;
- length tp1 div 2 < s0 \<and>
- s0 \<le> length tp1 div 2 + length tp div 2;
- t_ncorrect tp1; t_ncorrect tp;
- \<not> (length tp1 div 2 < s \<and>
- s \<le> length tp1 div 2 + length tp div 2)\<rbrakk>
- \<Longrightarrow> \<exists>stp' > 0. t_steps (s0, l0, r0) (tp1 @ tp @ tp2, 0) stp'
- = (s, l, r)"
-apply(rule_tac x = stp in exI)
-apply(case_tac stp, simp add: t_steps.simps)
-apply(simp only: stepn)
-apply(case_tac "t_steps (s0, l0, r0) (tp, length tp1 div 2) nat")
-apply(subgoal_tac "length tp1 div 2 < a \<and>
- a \<le> length tp1 div 2 + length tp div 2")
-apply(subgoal_tac "t_steps (s0, l0, r0) (tp1 @ tp @ tp2, 0) nat
- = (a, b, c)", simp)
-apply(rule_tac t_shift_in_step, simp+)
-apply(rule_tac turing_shift_inside, simp+)
-apply(rule classical)
-apply(subgoal_tac "t_step (a,b,c)
- (tp, length tp1 div 2) = (0, b, c)", simp)
-apply(rule_tac conf_keep_step, simp+)
-done
-
-lemma turing_shift:
- "\<lbrakk>t_steps (s0, (l0, r0)) (tp, (length tp1 div 2)) stp
- = (s, (l, r)); s \<noteq> 0; stp > 0;
- (length tp1 div 2 < s0 \<and> s0 <= length tp1 div 2 + length tp div 2);
- t_ncorrect tp1; t_ncorrect tp\<rbrakk> \<Longrightarrow>
- \<exists> stp' > 0. t_steps (s0, (l0, r0)) (tp1 @ tp @ tp2, 0) stp' =
- (s, (l, r))"
-apply(case_tac "s > length tp1 div 2 \<and>
- s <= length tp1 div 2 + length tp div 2")
-apply(subgoal_tac " t_steps (s0, l0, r0) (tp1 @ tp @ tp2, 0) stp =
- (s, l, r)")
-apply(rule_tac x = stp in exI, simp)
-apply(rule_tac turing_shift_inside, simp+)
-apply(rule_tac turing_shift_outside, simp+)
-done
-
-lemma inc_startof_not0: "start_of ly as \<ge> Suc 0"
-apply(induct as, simp add: start_of.simps)
-apply(simp add: start_of.simps)
-done
-
-lemma s_crsp:
- "\<lbrakk>as < length aprog; abc_fetch as aprog = Some ins;
- abc2t_correct aprog;
- crsp_l (layout_of aprog) (as, am) (a, aa, ba) inres\<rbrakk> \<Longrightarrow>
- length (concat (take as (tms_of aprog))) div 2 < a
- \<and> a \<le> length (concat (take as (tms_of aprog))) div 2 +
- length (ci (layout_of aprog) (start_of (layout_of aprog) as)
- ins) div 2"
-apply(subgoal_tac "a = length (concat (take as (tms_of aprog))) div
- 2 + 1", simp)
-apply(rule_tac ci_length_not0)
-apply(rule crsp2stateq, simp+)
-done
-
-lemma tms_out_ex:
- "\<lbrakk>ly = layout_of aprog; tprog = tm_of aprog;
- abc2t_correct aprog;
- crsp_l ly (as, am) tc inres; as < length aprog;
- abc_fetch as aprog = Some ins;
- t_steps tc (ci ly (start_of ly as) ins,
- (start_of ly as) - 1) n = (s, l, r);
- n > 0;
- abc_step_l (as, am) (abc_fetch as aprog) = (as', am');
- s = start_of ly as'
- \<rbrakk>
- \<Longrightarrow> \<exists> stp > 0. (t_steps tc (tprog, 0) stp = (s, (l, r)))"
-apply(simp only: tm_of.simps)
-apply(subgoal_tac "\<exists> tp1 tp2. concat (tms_of aprog) =
- tp1 @ (ci ly (start_of ly as) ins) @ tp2
- \<and> tp1 = concat (take as (tms_of aprog)) \<and>
- tp2 = concat (drop (Suc as) (tms_of aprog))")
-apply(erule exE, erule exE, erule conjE, erule conjE,
- case_tac tc, simp)
-apply(rule turing_shift)
-apply(subgoal_tac "start_of (layout_of aprog) as - Suc 0
- = length tp1 div 2", simp)
-apply(rule_tac pre_lheq, simp, simp, simp)
-apply(simp add: startof_not0, simp)
-apply(rule_tac s_crsp, simp, simp, simp, simp)
-apply(rule tms_ct, simp, simp)
-apply(rule tm_ct, simp)
-apply(subgoal_tac "ci (layout_of aprog)
- (start_of (layout_of aprog) as) ins
- = (tms_of aprog ! as)", simp)
-apply(simp add: tms_of.simps tpairs_of.simps)
-apply(simp add: tms_of.simps tpairs_of.simps abc_fetch.simps)
-apply(erule_tac t_split, auto simp: tm_of.simps)
-done
-
-(*
-subsection {* The compilation of @{text "Inc n"} *}
-*)
-
-text {*
- The lemmas in this section lead to the correctness of
- the compilation of @{text "Inc n"} instruction.
-*}
-
-fun at_begin_fst_bwtn :: "inc_inv_t"
- where
- "at_begin_fst_bwtn (as, lm) (s, l, r) ires =
- (\<exists> lm1 tn rn. lm1 = (lm @ (0\<^bsup>tn\<^esup>)) \<and> length lm1 = s \<and>
- (if lm1 = [] then l = Bk # Bk # ires
- else l = [Bk]@<rev lm1>@Bk#Bk#ires) \<and> r = (Bk\<^bsup>rn\<^esup>))"
-
-
-fun at_begin_fst_awtn :: "inc_inv_t"
- where
- "at_begin_fst_awtn (as, lm) (s, l, r) ires =
- (\<exists> lm1 tn rn. lm1 = (lm @ (0\<^bsup>tn\<^esup>)) \<and> length lm1 = s \<and>
- (if lm1 = [] then l = Bk # Bk # ires
- else l = [Bk]@<rev lm1>@Bk#Bk#ires) \<and> r = [Oc]@Bk\<^bsup>rn\<^esup>
- )"
-
-fun at_begin_norm :: "inc_inv_t"
- where
- "at_begin_norm (as, lm) (s, l, r) ires=
- (\<exists> lm1 lm2 rn. lm = lm1 @ lm2 \<and> length lm1 = s \<and>
- (if lm1 = [] then l = Bk # Bk # ires
- else l = Bk # <rev lm1> @ Bk# Bk # ires ) \<and> r = <lm2> @ (Bk\<^bsup>rn\<^esup>))"
-
-fun in_middle :: "inc_inv_t"
- where
- "in_middle (as, lm) (s, l, r) ires =
- (\<exists> lm1 lm2 tn m ml mr rn. lm @ 0\<^bsup>tn\<^esup> = lm1 @ [m] @ lm2
- \<and> length lm1 = s \<and> m + 1 = ml + mr \<and>
- ml \<noteq> 0 \<and> tn = s + 1 - length lm \<and>
- (if lm1 = [] then l = Oc\<^bsup>ml\<^esup> @ Bk # Bk # ires
- else l = (Oc\<^bsup>ml\<^esup>)@[Bk]@<rev lm1>@
- Bk # Bk # ires) \<and> (r = (Oc\<^bsup>mr\<^esup>) @ [Bk] @ <lm2>@ (Bk\<^bsup>rn\<^esup>) \<or>
- (lm2 = [] \<and> r = (Oc\<^bsup>mr\<^esup>)))
- )"
-
-fun inv_locate_a :: "inc_inv_t"
- where "inv_locate_a (as, lm) (s, l, r) ires =
- (at_begin_norm (as, lm) (s, l, r) ires \<or>
- at_begin_fst_bwtn (as, lm) (s, l, r) ires \<or>
- at_begin_fst_awtn (as, lm) (s, l, r) ires
- )"
-
-fun inv_locate_b :: "inc_inv_t"
- where "inv_locate_b (as, lm) (s, l, r) ires =
- (in_middle (as, lm) (s, l, r)) ires "
-
-fun inv_after_write :: "inc_inv_t"
- where "inv_after_write (as, lm) (s, l, r) ires =
- (\<exists> rn m lm1 lm2. lm = lm1 @ m # lm2 \<and>
- (if lm1 = [] then l = Oc\<^bsup>m\<^esup> @ Bk # Bk # ires
- else Oc # l = Oc\<^bsup>Suc m \<^esup>@ Bk # <rev lm1> @
- Bk # Bk # ires) \<and> r = [Oc] @ <lm2> @ (Bk\<^bsup>rn\<^esup>))"
-
-fun inv_after_move :: "inc_inv_t"
- where "inv_after_move (as, lm) (s, l, r) ires =
- (\<exists> rn m lm1 lm2. lm = lm1 @ m # lm2 \<and>
- (if lm1 = [] then l = Oc\<^bsup>Suc m\<^esup> @ Bk # Bk # ires
- else l = Oc\<^bsup>Suc m\<^esup>@ Bk # <rev lm1> @ Bk # Bk # ires) \<and>
- r = <lm2> @ (Bk\<^bsup>rn\<^esup>))"
-
-fun inv_after_clear :: "inc_inv_t"
- where "inv_after_clear (as, lm) (s, l, r) ires =
- (\<exists> rn m lm1 lm2 r'. lm = lm1 @ m # lm2 \<and>
- (if lm1 = [] then l = Oc\<^bsup>Suc m\<^esup> @ Bk # Bk # ires
- else l = Oc\<^bsup>Suc m\<^esup>@ Bk # <rev lm1> @ Bk # Bk # ires) \<and>
- r = Bk # r' \<and> Oc # r' = <lm2> @ (Bk\<^bsup>rn\<^esup>))"
-
-fun inv_on_right_moving :: "inc_inv_t"
- where "inv_on_right_moving (as, lm) (s, l, r) ires =
- (\<exists> lm1 lm2 m ml mr rn. lm = lm1 @ [m] @ lm2 \<and>
- ml + mr = m \<and>
- (if lm1 = [] then l = Oc\<^bsup>ml\<^esup> @ Bk # Bk # ires
- else l = (Oc\<^bsup>ml\<^esup>) @ [Bk] @ <rev lm1> @ Bk # Bk # ires) \<and>
- ((r = (Oc\<^bsup>mr\<^esup>) @ [Bk] @ <lm2> @ (Bk\<^bsup>rn\<^esup>)) \<or>
- (r = (Oc\<^bsup>mr\<^esup>) \<and> lm2 = [])))"
-
-fun inv_on_left_moving_norm :: "inc_inv_t"
- where "inv_on_left_moving_norm (as, lm) (s, l, r) ires =
- (\<exists> lm1 lm2 m ml mr rn. lm = lm1 @ [m] @ lm2 \<and>
- ml + mr = Suc m \<and> mr > 0 \<and> (if lm1 = [] then l = Oc\<^bsup>ml\<^esup> @ Bk # Bk # ires
- else l = (Oc\<^bsup>ml\<^esup>) @ Bk # <rev lm1> @ Bk # Bk # ires)
- \<and> (r = (Oc\<^bsup>mr\<^esup>) @ Bk # <lm2> @ (Bk\<^bsup>rn\<^esup>) \<or>
- (lm2 = [] \<and> r = Oc\<^bsup>mr\<^esup>)))"
-
-fun inv_on_left_moving_in_middle_B:: "inc_inv_t"
- where "inv_on_left_moving_in_middle_B (as, lm) (s, l, r) ires =
- (\<exists> lm1 lm2 rn. lm = lm1 @ lm2 \<and>
- (if lm1 = [] then l = Bk # ires
- else l = <rev lm1> @ Bk # Bk # ires) \<and>
- r = Bk # <lm2> @ (Bk\<^bsup>rn\<^esup>))"
-
-fun inv_on_left_moving :: "inc_inv_t"
- where "inv_on_left_moving (as, lm) (s, l, r) ires =
- (inv_on_left_moving_norm (as, lm) (s, l, r) ires \<or>
- inv_on_left_moving_in_middle_B (as, lm) (s, l, r) ires)"
-
-
-fun inv_check_left_moving_on_leftmost :: "inc_inv_t"
- where "inv_check_left_moving_on_leftmost (as, lm) (s, l, r) ires =
- (\<exists> rn. l = ires \<and> r = [Bk, Bk] @ <lm> @ (Bk\<^bsup>rn\<^esup>))"
-
-fun inv_check_left_moving_in_middle :: "inc_inv_t"
- where "inv_check_left_moving_in_middle (as, lm) (s, l, r) ires =
-
- (\<exists> lm1 lm2 r' rn. lm = lm1 @ lm2 \<and>
- (Oc # l = <rev lm1> @ Bk # Bk # ires) \<and> r = Oc # Bk # r' \<and>
- r' = <lm2> @ (Bk\<^bsup>rn\<^esup>))"
-
-fun inv_check_left_moving :: "inc_inv_t"
- where "inv_check_left_moving (as, lm) (s, l, r) ires =
- (inv_check_left_moving_on_leftmost (as, lm) (s, l, r) ires \<or>
- inv_check_left_moving_in_middle (as, lm) (s, l, r) ires)"
-
-fun inv_after_left_moving :: "inc_inv_t"
- where "inv_after_left_moving (as, lm) (s, l, r) ires=
- (\<exists> rn. l = Bk # ires \<and> r = Bk # <lm> @ (Bk\<^bsup>rn\<^esup>))"
-
-fun inv_stop :: "inc_inv_t"
- where "inv_stop (as, lm) (s, l, r) ires=
- (\<exists> rn. l = Bk # Bk # ires \<and> r = <lm> @ (Bk\<^bsup>rn\<^esup>))"
-
-
-fun inc_inv :: "layout \<Rightarrow> nat \<Rightarrow> inc_inv_t"
- where
- "inc_inv ly n (as, lm) (s, l, r) ires =
- (let ss = start_of ly as in
- let lm' = abc_lm_s lm n ((abc_lm_v lm n)+1) in
- if s = 0 then False
- else if s < ss then False
- else if s < ss + 2 * n then
- if (s - ss) mod 2 = 0 then
- inv_locate_a (as, lm) ((s - ss) div 2, l, r) ires
- else inv_locate_b (as, lm) ((s - ss) div 2, l, r) ires
- else if s = ss + 2 * n then
- inv_locate_a (as, lm) (n, l, r) ires
- else if s = ss + 2 * n + 1 then
- inv_locate_b (as, lm) (n, l, r) ires
- else if s = ss + 2 * n + 2 then
- inv_after_write (as, lm') (s - ss, l, r) ires
- else if s = ss + 2 * n + 3 then
- inv_after_move (as, lm') (s - ss, l, r) ires
- else if s = ss + 2 * n + 4 then
- inv_after_clear (as, lm') (s - ss, l, r) ires
- else if s = ss + 2 * n + 5 then
- inv_on_right_moving (as, lm') (s - ss, l, r) ires
- else if s = ss + 2 * n + 6 then
- inv_on_left_moving (as, lm') (s - ss, l, r) ires
- else if s = ss + 2 * n + 7 then
- inv_check_left_moving (as, lm') (s - ss, l, r) ires
- else if s = ss + 2 * n + 8 then
- inv_after_left_moving (as, lm') (s - ss, l, r) ires
- else if s = ss + 2 * n + 9 then
- inv_stop (as, lm') (s - ss, l, r) ires
- else False) "
-
-lemma fetch_intro:
- "\<lbrakk>\<And>xs.\<lbrakk>ba = Oc # xs\<rbrakk> \<Longrightarrow> P (fetch prog i Oc);
- \<And>xs.\<lbrakk>ba = Bk # xs\<rbrakk> \<Longrightarrow> P (fetch prog i Bk);
- ba = [] \<Longrightarrow> P (fetch prog i Bk)
- \<rbrakk> \<Longrightarrow> P (fetch prog i
- (case ba of [] \<Rightarrow> Bk | Bk # xs \<Rightarrow> Bk | Oc # xs \<Rightarrow> Oc))"
-by (auto split:list.splits block.splits)
-
-lemma length_findnth[simp]: "length (findnth n) = 4 * n"
-apply(induct n, simp)
-apply(simp)
-done
-
-declare tshift.simps[simp del]
-declare findnth.simps[simp del]
-
-lemma findnth_nth:
- "\<lbrakk>n > q; x < 4\<rbrakk> \<Longrightarrow>
- (findnth n) ! (4 * q + x) = (findnth (Suc q) ! (4 * q + x))"
-apply(induct n, simp)
-apply(case_tac "q < n", simp add: findnth.simps, auto)
-apply(simp add: nth_append)
-apply(subgoal_tac "q = n", simp)
-apply(arith)
-done
-
-lemma Suc_pre[simp]: "\<not> a < start_of ly as \<Longrightarrow>
- (Suc a - start_of ly as) = Suc (a - start_of ly as)"
-apply(arith)
-done
-
-lemma fetch_locate_a_o: "
-\<And>a q xs.
- \<lbrakk>\<not> a < start_of (layout_of aprog) as;
- a < start_of (layout_of aprog) as + 2 * n;
- a - start_of (layout_of aprog) as = 2 * q;
- start_of (layout_of aprog) as > 0\<rbrakk>
- \<Longrightarrow> (fetch (ci (layout_of aprog) (start_of (layout_of aprog) as)
- (Inc n)) (Suc (2 * q)) Oc) = (R, a+1)"
-apply(auto simp: ci.simps findnth.simps fetch.simps
- nth_of.simps tshift.simps nth_append Suc_pre)
-apply(subgoal_tac "(findnth n ! Suc (4 * q)) =
- findnth (Suc q) ! (4 * q + 1)")
-apply(simp add: findnth.simps nth_append)
-apply(subgoal_tac " findnth n !(4 * q + 1) =
- findnth (Suc q) ! (4 * q + 1)", simp)
-apply(rule_tac findnth_nth, auto)
-done
-
-lemma fetch_locate_a_b: "
-\<And>a q xs.
- \<lbrakk>abc_fetch as aprog = Some (Inc n);
- \<not> a < start_of (layout_of aprog) as;
- a < start_of (layout_of aprog) as + 2 * n;
- a - start_of (layout_of aprog) as = 2 * q;
- start_of (layout_of aprog) as > 0\<rbrakk>
- \<Longrightarrow> (fetch (ci (layout_of aprog)
- (start_of (layout_of aprog) as) (Inc n)) (Suc (2 * q)) Bk)
- = (W1, a)"
-apply(auto simp: ci.simps findnth.simps fetch.simps
- tshift.simps nth_append)
-apply(subgoal_tac "(findnth n ! (4 * q)) =
- findnth (Suc q) ! (4 * q )")
-apply(simp add: findnth.simps nth_append)
-apply(subgoal_tac " findnth n !(4 * q + 0) =
- findnth (Suc q) ! (4 * q + 0)", simp)
-apply(rule_tac findnth_nth, auto)
-done
-
-lemma [intro]: "x mod 2 = Suc 0 \<Longrightarrow> \<exists> q. x = Suc (2 * q)"
-apply(drule mod_eqD, auto)
-done
-
-lemma add3_Suc: "x + 3 = Suc (Suc (Suc x))"
-apply(arith)
-done
-
-declare start_of.simps[simp]
-(*
-lemma layout_not0: "start_of ly as > 0"
-by(induct as, auto)
-*)
-lemma [simp]:
- "\<lbrakk>\<not> a < start_of (layout_of aprog) as;
- a - start_of (layout_of aprog) as = Suc (2 * q);
- abc_fetch as aprog = Some (Inc n);
- start_of (layout_of aprog) as > 0\<rbrakk>
- \<Longrightarrow> Suc (Suc (2 * q + start_of (layout_of aprog) as - Suc 0)) = a"
-apply(subgoal_tac
-"Suc (Suc (2 * q + start_of (layout_of aprog) as - Suc 0))
- = 2 + 2 * q + start_of (layout_of aprog) as - Suc 0",
- simp, simp add: inc_startof_not0)
-done
-
-lemma fetch_locate_b_o: "
-\<And>a xs.
- \<lbrakk>0 < a; \<not> a < start_of (layout_of aprog) as;
- a < start_of (layout_of aprog) as + 2 * n;
- (a - start_of (layout_of aprog) as) mod 2 = Suc 0;
- start_of (layout_of aprog) as > 0\<rbrakk>
- \<Longrightarrow> (fetch (ci (layout_of aprog) (start_of (layout_of aprog) as)
- (Inc n)) (Suc (a - start_of (layout_of aprog) as)) Oc) = (R, a)"
-apply(auto simp: ci.simps findnth.simps fetch.simps
- nth_of.simps tshift.simps nth_append)
-apply(subgoal_tac "\<exists> q. (a - start_of (layout_of aprog) as) =
- 2 * q + 1", auto)
-apply(subgoal_tac "(findnth n ! Suc (Suc (Suc (4 * q))))
- = findnth (Suc q) ! (4 * q + 3)")
-apply(simp add: findnth.simps nth_append)
-apply(subgoal_tac " findnth n ! (4 * q + 3) =
- findnth (Suc q) ! (4 * q + 3)", simp add: add3_Suc)
-apply(rule_tac findnth_nth, auto)
-done
-
-lemma fetch_locate_b_b: "
-\<And>a xs.
- \<lbrakk>0 < a; \<not> a < start_of (layout_of aprog) as;
- a < start_of (layout_of aprog) as + 2 * n;
- (a - start_of (layout_of aprog) as) mod 2 = Suc 0;
- start_of (layout_of aprog) as > 0\<rbrakk>
- \<Longrightarrow> (fetch (ci (layout_of aprog) (start_of (layout_of aprog) as)
- (Inc n)) (Suc (a - start_of (layout_of aprog) as)) Bk)
- = (R, a + 1)"
-apply(auto simp: ci.simps findnth.simps fetch.simps
- nth_of.simps tshift.simps nth_append)
-apply(subgoal_tac "\<exists> q. (a - start_of (layout_of aprog) as) =
- 2 * q + 1", auto)
-apply(subgoal_tac "(findnth n ! Suc ((Suc (4 * q)))) =
- findnth (Suc q) ! (4 * q + 2)")
-apply(simp add: findnth.simps nth_append)
-apply(subgoal_tac " findnth n ! (4 * q + 2) =
- findnth (Suc q) ! (4 * q + 2)", simp)
-apply(rule_tac findnth_nth, auto)
-done
-
-lemma fetch_locate_n_a_o:
- "start_of (layout_of aprog) as > 0
- \<Longrightarrow> (fetch (ci (layout_of aprog)
- (start_of (layout_of aprog) as) (Inc n)) (Suc (2 * n)) Oc) =
- (R, start_of (layout_of aprog) as + 2 * n + 1)"
-apply(auto simp: ci.simps findnth.simps fetch.simps
- nth_of.simps tshift.simps nth_append tinc_b_def)
-done
-
-lemma fetch_locate_n_a_b: "
- start_of (layout_of aprog) as > 0
- \<Longrightarrow> (fetch (ci (layout_of aprog)
- (start_of (layout_of aprog) as) (Inc n)) (Suc (2 * n)) Bk)
- = (W1, start_of (layout_of aprog) as + 2 * n)"
-apply(auto simp: ci.simps findnth.simps fetch.simps
- nth_of.simps tshift.simps nth_append tinc_b_def)
-done
-
-lemma fetch_locate_n_b_o: "
- start_of (layout_of aprog) as > 0
- \<Longrightarrow> (fetch (ci (layout_of aprog) (start_of (layout_of aprog) as)
- (Inc n)) (Suc (Suc (2 * n))) Oc) =
- (R, start_of (layout_of aprog) as + 2 * n + 1)"
-apply(auto simp: ci.simps findnth.simps fetch.simps
- nth_of.simps tshift.simps nth_append tinc_b_def)
-done
-
-lemma fetch_locate_n_b_b: "
- start_of (layout_of aprog) as > 0
- \<Longrightarrow> (fetch (ci (layout_of aprog) (start_of (layout_of aprog) as)
- (Inc n)) (Suc (Suc (2 * n))) Bk) =
- (W1, start_of (layout_of aprog) as + 2 * n + 2)"
-apply(auto simp: ci.simps findnth.simps fetch.simps
- nth_of.simps tshift.simps nth_append tinc_b_def)
-done
-
-lemma fetch_after_write_o: "
- start_of (layout_of aprog) as > 0
- \<Longrightarrow> (fetch (ci (layout_of aprog) (start_of (layout_of aprog) as)
- (Inc n)) (Suc (Suc (Suc (2 * n)))) Oc) =
- (R, start_of (layout_of aprog) as + 2*n + 3)"
-apply(auto simp: ci.simps findnth.simps fetch.simps
- nth_of.simps tshift.simps nth_append tinc_b_def)
-done
-
-lemma fetch_after_move_o: "
- start_of (layout_of aprog) as > 0
- \<Longrightarrow> (fetch (ci (layout_of aprog)
- (start_of (layout_of aprog) as) (Inc n)) (4 + 2 * n) Oc)
- = (W0, start_of (layout_of aprog) as + 2 * n + 4)"
-apply(auto simp: ci.simps findnth.simps tshift.simps
- tinc_b_def add3_Suc)
-apply(subgoal_tac "4 + 2*n = Suc (2*n + 3)", simp only: fetch.simps)
-apply(auto simp: nth_of.simps nth_append)
-done
-
-lemma fetch_after_move_b: "
- start_of (layout_of aprog) as > 0
- \<Longrightarrow>(fetch (ci (layout_of aprog)
- (start_of (layout_of aprog) as) (Inc n)) (4 + 2 * n) Bk)
- = (L, start_of (layout_of aprog) as + 2 * n + 6)"
-apply(auto simp: ci.simps findnth.simps tshift.simps
- tinc_b_def add3_Suc)
-apply(subgoal_tac "4 + 2*n = Suc (2*n + 3)", simp only: fetch.simps)
-apply(auto simp: nth_of.simps nth_append)
-done
-
-lemma fetch_clear_b: "
- start_of (layout_of aprog) as > 0
- \<Longrightarrow> (fetch (ci (layout_of aprog)
- (start_of (layout_of aprog) as) (Inc n)) (5 + 2 * n) Bk)
- = (R, start_of (layout_of aprog) as + 2 * n + 5)"
-apply(auto simp: ci.simps findnth.simps
- tshift.simps tinc_b_def add3_Suc)
-apply(subgoal_tac "5 + 2*n = Suc (2*n + 4)", simp only: fetch.simps)
-apply(auto simp: nth_of.simps nth_append)
-done
-
-lemma fetch_right_move_o: "
- start_of (layout_of aprog) as > 0
- \<Longrightarrow> (fetch (ci (layout_of aprog)
- (start_of (layout_of aprog) as) (Inc n)) (6 + 2*n) Oc)
- = (R, start_of (layout_of aprog) as + 2 * n + 5)"
-apply(auto simp: ci.simps findnth.simps tshift.simps
- tinc_b_def add3_Suc)
-apply(subgoal_tac "6 + 2*n = Suc (2*n + 5)", simp only: fetch.simps)
-apply(auto simp: nth_of.simps nth_append)
-done
-
-lemma fetch_right_move_b: "
- start_of (layout_of aprog) as > 0
- \<Longrightarrow> (fetch (ci (layout_of aprog)
- (start_of (layout_of aprog) as) (Inc n)) (6 + 2*n) Bk)
- = (W1, start_of (layout_of aprog) as + 2 * n + 2)"
-apply(auto simp: ci.simps findnth.simps
- tshift.simps tinc_b_def add3_Suc)
-apply(subgoal_tac "6 + 2*n = Suc (2*n + 5)", simp only: fetch.simps)
-apply(auto simp: nth_of.simps nth_append)
-done
-
-lemma fetch_left_move_o: "
- start_of (layout_of aprog) as > 0
- \<Longrightarrow> (fetch (ci (layout_of aprog)
- (start_of (layout_of aprog) as) (Inc n)) (7 + 2*n) Oc)
- = (L, start_of (layout_of aprog) as + 2 * n + 6)"
-apply(auto simp: ci.simps findnth.simps tshift.simps
- tinc_b_def add3_Suc)
-apply(subgoal_tac "7 + 2*n = Suc (2*n + 6)", simp only: fetch.simps)
-apply(auto simp: nth_of.simps nth_append)
-done
-
-lemma fetch_left_move_b: "
- start_of (layout_of aprog) as > 0
- \<Longrightarrow> (fetch (ci (layout_of aprog)
- (start_of (layout_of aprog) as) (Inc n)) (7 + 2*n) Bk)
- = (L, start_of (layout_of aprog) as + 2 * n + 7)"
-apply(auto simp: ci.simps findnth.simps
- tshift.simps tinc_b_def add3_Suc)
-apply(subgoal_tac "7 + 2*n = Suc (2*n + 6)", simp only: fetch.simps)
-apply(auto simp: nth_of.simps nth_append)
-done
-
-lemma fetch_check_left_move_o: "
- start_of (layout_of aprog) as > 0
- \<Longrightarrow> (fetch (ci (layout_of aprog)
- (start_of (layout_of aprog) as) (Inc n)) (8 + 2*n) Oc)
- = (L, start_of (layout_of aprog) as + 2 * n + 6)"
-apply(auto simp: ci.simps findnth.simps tshift.simps tinc_b_def)
-apply(subgoal_tac "8 + 2 * n = Suc (2 * n + 7)",
- simp only: fetch.simps)
-apply(auto simp: nth_of.simps nth_append)
-done
-
-lemma fetch_check_left_move_b: "
- start_of (layout_of aprog) as > 0
- \<Longrightarrow> (fetch (ci (layout_of aprog)
- (start_of (layout_of aprog) as) (Inc n)) (8 + 2*n) Bk)
- = (R, start_of (layout_of aprog) as + 2 * n + 8) "
-apply(auto simp: ci.simps findnth.simps
- tshift.simps tinc_b_def add3_Suc)
-apply(subgoal_tac "8 + 2*n= Suc (2*n + 7)", simp only: fetch.simps)
-apply(auto simp: nth_of.simps nth_append)
-done
-
-lemma fetch_after_left_move: "
- start_of (layout_of aprog) as > 0
- \<Longrightarrow> (fetch (ci (layout_of aprog)
- (start_of (layout_of aprog) as) (Inc n)) (9 + 2*n) Bk)
- = (R, start_of (layout_of aprog) as + 2 * n + 9)"
-apply(auto simp: ci.simps findnth.simps fetch.simps
- nth_of.simps tshift.simps nth_append tinc_b_def)
-done
-
-lemma fetch_stop: "
- start_of (layout_of aprog) as > 0
- \<Longrightarrow> (fetch (ci (layout_of aprog)
- (start_of (layout_of aprog) as) (Inc n)) (10 + 2 *n) b)
- = (Nop, 0)"
-apply(auto simp: ci.simps findnth.simps fetch.simps
- nth_of.simps tshift.simps nth_append tinc_b_def
- split: block.splits)
-done
-
-lemma fetch_state0: "
- (fetch (ci (layout_of aprog)
- (start_of (layout_of aprog) as) (Inc n)) 0 b)
- = (Nop, 0)"
-apply(auto simp: ci.simps findnth.simps fetch.simps
- nth_of.simps tshift.simps nth_append tinc_b_def)
-done
-
-lemmas fetch_simps =
- fetch_locate_a_o fetch_locate_a_b fetch_locate_b_o fetch_locate_b_b
- fetch_locate_n_a_b fetch_locate_n_a_o fetch_locate_n_b_o
- fetch_locate_n_b_b fetch_after_write_o fetch_after_move_o
- fetch_after_move_b fetch_clear_b fetch_right_move_o
- fetch_right_move_b fetch_left_move_o fetch_left_move_b
- fetch_after_left_move fetch_check_left_move_o fetch_stop
- fetch_state0 fetch_check_left_move_b
-
-text {* *}
-declare exponent_def[simp del] tape_of_nat_list.simps[simp del]
- at_begin_norm.simps[simp del] at_begin_fst_bwtn.simps[simp del]
- at_begin_fst_awtn.simps[simp del] in_middle.simps[simp del]
- abc_lm_s.simps[simp del] abc_lm_v.simps[simp del]
- ci.simps[simp del] t_step.simps[simp del]
- inv_after_move.simps[simp del]
- inv_on_left_moving_norm.simps[simp del]
- inv_on_left_moving_in_middle_B.simps[simp del]
- inv_after_clear.simps[simp del]
- inv_after_write.simps[simp del] inv_on_left_moving.simps[simp del]
- inv_on_right_moving.simps[simp del]
- inv_check_left_moving.simps[simp del]
- inv_check_left_moving_in_middle.simps[simp del]
- inv_check_left_moving_on_leftmost.simps[simp del]
- inv_after_left_moving.simps[simp del]
- inv_stop.simps[simp del] inv_locate_a.simps[simp del]
- inv_locate_b.simps[simp del]
-declare tms_of.simps[simp del] tm_of.simps[simp del]
- layout_of.simps[simp del] abc_fetch.simps [simp del]
- t_step.simps[simp del] t_steps.simps[simp del]
- tpairs_of.simps[simp del] start_of.simps[simp del]
- fetch.simps [simp del] new_tape.simps [simp del]
- nth_of.simps [simp del] ci.simps [simp del]
- length_of.simps[simp del]
-
-(*! Start point *)
-lemma [simp]: "Suc (2 * q) mod 2 = Suc 0"
-by arith
-
-lemma [simp]: "Suc (2 * q) div 2 = q"
-by arith
-
-lemma [simp]: "\<lbrakk> \<not> a < start_of ly as;
- a < start_of ly as + 2 * n; a - start_of ly as = 2 * q\<rbrakk>
- \<Longrightarrow> Suc a < start_of ly as + 2 * n"
-apply(arith)
-done
-
-lemma [simp]: "x mod 2 = Suc 0 \<Longrightarrow> (Suc x) mod 2 = 0"
-by arith
-
-lemma [simp]: "x mod 2 = Suc 0 \<Longrightarrow> (Suc x) div 2 = Suc (x div 2)"
-by arith
-lemma exp_def[simp]: "a\<^bsup>Suc n \<^esup>= a # a\<^bsup>n\<^esup>"
-by(simp add: exponent_def)
-lemma [intro]: "Bk # r = Oc\<^bsup>mr\<^esup> @ r' \<Longrightarrow> mr = 0"
-by(case_tac mr, auto simp: exponent_def)
-
-lemma [intro]: "Bk # r = replicate mr Oc \<Longrightarrow> mr = 0"
-by(case_tac mr, auto)
-lemma tape_of_nl_abv_cons[simp]: "xs \<noteq> [] \<Longrightarrow>
- <x # xs> = Oc\<^bsup>Suc x\<^esup>@ Bk # <xs>"
-apply(simp add: tape_of_nl_abv tape_of_nat_list.simps)
-apply(case_tac xs, simp, simp add: tape_of_nat_list.simps)
-done
-
-lemma [simp]: "<[]::nat list> = []"
-by(auto simp: tape_of_nl_abv tape_of_nat_list.simps)
-lemma [simp]: "Oc # r = <(lm::nat list)> @ Bk\<^bsup>rn\<^esup>\<Longrightarrow> lm \<noteq> []"
-apply(auto simp: tape_of_nl_abv tape_of_nat_list.simps)
-apply(case_tac rn, auto simp: exponent_def)
-done
-lemma BkCons_nil: "Bk # xs = <lm::nat list> @ Bk\<^bsup>rn\<^esup>\<Longrightarrow> lm = []"
-apply(case_tac lm, simp)
-apply(case_tac list, auto simp: tape_of_nl_abv tape_of_nat_list.simps)
-done
-lemma BkCons_nil': "Bk # xs = <lm::nat list> @ Bk\<^bsup>ln\<^esup>\<Longrightarrow> lm = []"
-by(auto intro: BkCons_nil)
-
-lemma hd_tl_tape_of_nat_list:
- "tl (lm::nat list) \<noteq> [] \<Longrightarrow> <lm> = <hd lm> @ Bk # <tl lm>"
-apply(frule tape_of_nl_abv_cons[of "tl lm" "hd lm"])
-apply(simp add: tape_of_nat_abv Bk_def del: tape_of_nl_abv_cons)
-apply(subgoal_tac "lm = hd lm # tl lm", auto)
-apply(case_tac lm, auto)
-done
-lemma [simp]: "Oc # xs = Oc\<^bsup>mr\<^esup> @ Bk # <lm2> @ Bk\<^bsup>rn\<^esup>\<Longrightarrow> mr > 0"
-apply(case_tac mr, auto simp: exponent_def)
-done
-
-lemma tape_of_nat_list_cons: "xs \<noteq> [] \<Longrightarrow> tape_of_nat_list (x # xs) =
- replicate (Suc x) Oc @ Bk # tape_of_nat_list xs"
-apply(drule tape_of_nl_abv_cons[of xs x])
-apply(auto simp: tape_of_nl_abv tape_of_nat_abv Oc_def Bk_def exponent_def)
-done
-
-lemma rev_eq: "rev xs = rev ys \<Longrightarrow> xs = ys"
-by simp
-
-lemma tape_of_nat_list_eq: " xs = ys \<Longrightarrow>
- tape_of_nat_list xs = tape_of_nat_list ys"
-by simp
-
-lemma tape_of_nl_nil_eq: "<(lm::nat list)> = [] = (lm = [])"
-apply(simp add: tape_of_nl_abv tape_of_nat_list.simps)
-apply(case_tac lm, simp add: tape_of_nat_list.simps)
-apply(case_tac "list")
-apply(auto simp: tape_of_nat_list.simps)
-done
-
-lemma rep_ind: "replicate (Suc n) a = replicate n a @ [a]"
-apply(induct n, simp, simp)
-done
-
-lemma [simp]: "Oc # r = <lm::nat list> @ replicate rn Bk \<Longrightarrow> Suc 0 \<le> length lm"
-apply(rule_tac classical, auto)
-apply(case_tac lm, simp, case_tac rn, auto)
-done
-lemma Oc_Bk_Cons: "Oc # Bk # list = <lm::nat list> @ Bk\<^bsup>ln\<^esup> \<Longrightarrow>
- lm \<noteq> [] \<and> hd lm = 0"
-apply(case_tac lm, simp, case_tac ln, simp add: exponent_def, simp add: exponent_def, simp)
-apply(case_tac lista, auto simp: tape_of_nl_abv tape_of_nat_list.simps)
-done
-(*lemma Oc_Oc_Cons: "Oc # Oc # list = <lm::nat list> @ Bk\<^bsup>ln\<^esup> \<Longrightarrow>
- lm \<noteq> [] \<and> hd lm > 0"
-apply(case_tac lm, simp add: exponent_def, case_tac ln, simp, simp)
-apply(case_tac lista,
- auto simp: tape_of_nl_abv tape_of_nat_list.simps exponent_def)
-apply(case_tac [!] a, auto)
-apply(case_tac ln, auto)
-done
-*)
-lemma Oc_nil_zero[simp]: "[Oc] = <lm::nat list> @ Bk\<^bsup>ln\<^esup>
- \<Longrightarrow> lm = [0] \<and> ln = 0"
-apply(case_tac lm, simp)
-apply(case_tac ln, auto simp: exponent_def)
-apply(case_tac [!] list,
- auto simp: tape_of_nl_abv tape_of_nat_list.simps)
-done
-
-lemma [simp]: "Oc # r = <lm2> @ replicate rn Bk \<Longrightarrow>
- (\<exists>rn. r = replicate (hd lm2) Oc @ Bk # <tl lm2> @
- replicate rn Bk) \<or>
- tl lm2 = [] \<and> r = replicate (hd lm2) Oc"
-apply(rule_tac disjCI, simp)
-apply(case_tac "tl lm2 = []", simp)
-apply(case_tac lm2, simp add: tape_of_nl_abv tape_of_nat_list.simps)
-apply(case_tac rn, simp, simp, simp)
-apply(simp add: tape_of_nl_abv tape_of_nat_list.simps exponent_def)
-apply(case_tac rn, simp, simp)
-apply(rule_tac x = rn in exI)
-apply(simp add: hd_tl_tape_of_nat_list)
-apply(simp add: tape_of_nat_abv Oc_def exponent_def)
-done
-
-(*inv: from locate_a to locate_b*)
-lemma [simp]:
- "inv_locate_a (as, lm) (q, l, Oc # r) ires
- \<Longrightarrow> inv_locate_b (as, lm) (q, Oc # l, r) ires"
-apply(simp only: inv_locate_a.simps inv_locate_b.simps in_middle.simps
- at_begin_norm.simps at_begin_fst_bwtn.simps
- at_begin_fst_awtn.simps)
-apply(erule disjE, erule exE, erule exE, erule exE)
-apply(rule_tac x = lm1 in exI, rule_tac x = "tl lm2" in exI, simp)
-apply(rule_tac x = "0" in exI, rule_tac x = "hd lm2" in exI,
- auto simp: exponent_def)
-apply(rule_tac x = "Suc 0" in exI, simp add:exponent_def)
-apply(rule_tac x = "lm @ replicate tn 0" in exI,
- rule_tac x = "[]" in exI,
- rule_tac x = "Suc tn" in exI, rule_tac x = 0 in exI)
-apply(simp only: rep_ind, simp)
-apply(rule_tac x = "Suc 0" in exI, auto)
-apply(case_tac [1-3] rn, simp_all )
-apply(rule_tac x = "lm @ replicate tn 0" in exI,
- rule_tac x = "[]" in exI,
- rule_tac x = "Suc tn" in exI,
- rule_tac x = 0 in exI, simp add: rep_ind del: replicate_Suc split:if_splits)
-apply(rule_tac x = "Suc 0" in exI, auto)
-apply(case_tac rn, simp, simp)
-apply(rule_tac [!] x = "Suc 0" in exI, auto)
-apply(case_tac [!] rn, simp_all)
-done
-
-(*inv: from locate_a to _locate_a*)
-lemma locate_a_2_locate_a[simp]: "inv_locate_a (as, am) (q, aaa, Bk # xs) ires
- \<Longrightarrow> inv_locate_a (as, am) (q, aaa, Oc # xs) ires"
-apply(simp only: inv_locate_a.simps at_begin_norm.simps
- at_begin_fst_bwtn.simps at_begin_fst_awtn.simps)
-apply(erule_tac disjE, erule exE, erule exE, erule exE,
- rule disjI2, rule disjI2)
-defer
-apply(erule_tac disjE, erule exE, erule exE,
- erule exE, rule disjI2, rule disjI2)
-prefer 2
-apply(simp)
-proof-
- fix lm1 tn rn
- assume k: "lm1 = am @ 0\<^bsup>tn\<^esup> \<and> length lm1 = q \<and> (if lm1 = [] then aaa = Bk # Bk #
- ires else aaa = [Bk] @ <rev lm1> @ Bk # Bk # ires) \<and> Bk # xs = Bk\<^bsup>rn\<^esup>"
- thus "\<exists>lm1 tn rn. lm1 = am @ 0\<^bsup>tn\<^esup> \<and> length lm1 = q \<and> (if lm1 = [] then
- aaa = Bk # Bk # ires else aaa = [Bk] @ <rev lm1> @ Bk # Bk # ires) \<and> Oc # xs = [Oc] @ Bk\<^bsup>rn\<^esup>"
- (is "\<exists>lm1 tn rn. ?P lm1 tn rn")
- proof -
- from k have "?P lm1 tn (rn - 1)"
- apply(auto simp: Oc_def)
- by(case_tac [!] "rn::nat", auto simp: exponent_def)
- thus ?thesis by blast
- qed
-next
- fix lm1 lm2 rn
- assume h1: "am = lm1 @ lm2 \<and> length lm1 = q \<and> (if lm1 = []
- then aaa = Bk # Bk # ires else aaa = Bk # <rev lm1> @ Bk # Bk # ires) \<and>
- Bk # xs = <lm2> @ Bk\<^bsup>rn\<^esup>"
- from h1 have h2: "lm2 = []"
- proof(rule_tac xs = xs and rn = rn in BkCons_nil, simp)
- qed
- from h1 and h2 show "\<exists>lm1 tn rn. lm1 = am @ 0\<^bsup>tn\<^esup> \<and> length lm1 = q \<and>
- (if lm1 = [] then aaa = Bk # Bk # ires else aaa = [Bk] @ <rev lm1> @ Bk # Bk # ires) \<and>
- Oc # xs = [Oc] @ Bk\<^bsup>rn\<^esup>"
- (is "\<exists>lm1 tn rn. ?P lm1 tn rn")
- proof -
- from h1 and h2 have "?P lm1 0 (rn - 1)"
- apply(auto simp: Oc_def exponent_def
- tape_of_nl_abv tape_of_nat_list.simps)
- by(case_tac "rn::nat", simp, simp)
- thus ?thesis by blast
- qed
-qed
-
-lemma [intro]: "\<exists>rn. [a] = a\<^bsup>rn\<^esup>"
-by(rule_tac x = "Suc 0" in exI, simp add: exponent_def)
-
-lemma [intro]: "\<exists>tn. [] = a\<^bsup>tn\<^esup>"
-apply(rule_tac x = 0 in exI, simp add: exponent_def)
-done
-
-lemma [intro]: "at_begin_norm (as, am) (q, aaa, []) ires
- \<Longrightarrow> at_begin_norm (as, am) (q, aaa, [Bk]) ires"
-apply(simp add: at_begin_norm.simps, erule_tac exE, erule_tac exE)
-apply(rule_tac x = lm1 in exI, simp, auto)
-done
-
-lemma [intro]: "at_begin_fst_bwtn (as, am) (q, aaa, []) ires
- \<Longrightarrow> at_begin_fst_bwtn (as, am) (q, aaa, [Bk]) ires"
-apply(simp only: at_begin_fst_bwtn.simps, erule_tac exE, erule_tac exE, erule_tac exE)
-apply(rule_tac x = "am @ 0\<^bsup>tn\<^esup>" in exI, auto)
-done
-
-lemma [intro]: "at_begin_fst_awtn (as, am) (q, aaa, []) ires
- \<Longrightarrow> at_begin_fst_awtn (as, am) (q, aaa, [Bk]) ires"
-apply(auto simp: at_begin_fst_awtn.simps)
-done
-
-lemma [intro]: "inv_locate_a (as, am) (q, aaa, []) ires
- \<Longrightarrow> inv_locate_a (as, am) (q, aaa, [Bk]) ires"
-apply(simp only: inv_locate_a.simps)
-apply(erule disj_forward)
-defer
-apply(erule disj_forward, auto)
-done
-
-lemma [simp]: "inv_locate_a (as, am) (q, aaa, []) ires \<Longrightarrow>
- inv_locate_a (as, am) (q, aaa, [Oc]) ires"
-apply(insert locate_a_2_locate_a [of as am q aaa "[]"])
-apply(subgoal_tac "inv_locate_a (as, am) (q, aaa, [Bk]) ires", auto)
-done
-
-(*inv: from locate_b to locate_b*)
-lemma [simp]: "inv_locate_b (as, am) (q, aaa, Oc # xs) ires
- \<Longrightarrow> inv_locate_b (as, am) (q, Oc # aaa, xs) ires"
-apply(simp only: inv_locate_b.simps in_middle.simps)
-apply(erule exE)+
-apply(rule_tac x = lm1 in exI, rule_tac x = lm2 in exI,
- rule_tac x = tn in exI, rule_tac x = m in exI)
-apply(rule_tac x = "Suc ml" in exI, rule_tac x = "mr - 1" in exI,
- rule_tac x = rn in exI)
-apply(case_tac mr, simp_all add: exponent_def, auto)
-done
-lemma zero_and_nil[intro]: "(Bk # Bk\<^bsup>n\<^esup> = Oc\<^bsup>mr\<^esup> @ Bk # <lm::nat list> @
- Bk\<^bsup>rn \<^esup>) \<or> (lm2 = [] \<and> Bk # Bk\<^bsup>n\<^esup> = Oc\<^bsup>mr\<^esup>)
- \<Longrightarrow> mr = 0 \<and> lm = []"
-apply(rule context_conjI)
-apply(case_tac mr, auto simp:exponent_def)
-apply(insert BkCons_nil[of "replicate (n - 1) Bk" lm rn])
-apply(case_tac n, auto simp: exponent_def Bk_def tape_of_nl_nil_eq)
-done
-
-lemma tape_of_nat_def: "<[m::nat]> = Oc # Oc\<^bsup>m\<^esup>"
-apply(simp add: tape_of_nl_abv tape_of_nat_list.simps)
-done
-lemma [simp]: "\<lbrakk>inv_locate_b (as, am) (q, aaa, Bk # xs) ires; \<exists>n. xs = Bk\<^bsup>n\<^esup>\<rbrakk>
- \<Longrightarrow> inv_locate_a (as, am) (Suc q, Bk # aaa, xs) ires"
-apply(simp add: inv_locate_b.simps inv_locate_a.simps)
-apply(rule_tac disjI2, rule_tac disjI1)
-apply(simp only: in_middle.simps at_begin_fst_bwtn.simps)
-apply(erule_tac exE)+
-apply(rule_tac x = "lm1 @ [m]" in exI, rule_tac x = tn in exI, simp)
-apply(subgoal_tac "mr = 0 \<and> lm2 = []")
-defer
-apply(rule_tac n = n and mr = mr and lm = "lm2"
- and rn = rn and n = n in zero_and_nil)
-apply(auto simp: exponent_def)
-apply(case_tac "lm1 = []", auto simp: tape_of_nat_def)
-done
-
-lemma length_equal: "xs = ys \<Longrightarrow> length xs = length ys"
-by auto
-lemma [simp]: "a\<^bsup>0\<^esup> = []"
-by(simp add: exp_zero)
-(*inv: from locate_b to locate_a*)
-lemma [simp]: "length (a\<^bsup>b\<^esup>) = b"
-apply(simp add: exponent_def)
-done
-
-lemma [simp]: "\<lbrakk>inv_locate_b (as, am) (q, aaa, Bk # xs) ires;
- \<not> (\<exists>n. xs = Bk\<^bsup>n\<^esup>)\<rbrakk>
- \<Longrightarrow> inv_locate_a (as, am) (Suc q, Bk # aaa, xs) ires"
-apply(simp add: inv_locate_b.simps inv_locate_a.simps)
-apply(rule_tac disjI1)
-apply(simp only: in_middle.simps at_begin_norm.simps)
-apply(erule_tac exE)+
-apply(rule_tac x = "lm1 @ [m]" in exI, rule_tac x = lm2 in exI, simp)
-apply(subgoal_tac "tn = 0", simp add: exponent_def , auto split: if_splits)
-apply(case_tac [!] mr, simp_all add: tape_of_nat_def, auto)
-apply(case_tac lm2, simp, erule_tac x = rn in allE, simp)
-apply(case_tac am, simp, simp)
-apply(case_tac lm2, simp, erule_tac x = rn in allE, simp)
-apply(drule_tac length_equal, simp)
-done
-
-lemma locate_b_2_a[intro]:
- "inv_locate_b (as, am) (q, aaa, Bk # xs) ires
- \<Longrightarrow> inv_locate_a (as, am) (Suc q, Bk # aaa, xs) ires"
-apply(case_tac "\<exists> n. xs = Bk\<^bsup>n\<^esup>", simp, simp)
-done
-
-lemma locate_b_2_locate_a[simp]:
- "\<lbrakk>\<not> a < start_of ly as;
- a < start_of ly as + 2 * n;
- (a - start_of ly as) mod 2 = Suc 0;
- inv_locate_b (as, am) ((a - start_of ly as) div 2, aaa, Bk # xs) ires\<rbrakk>
- \<Longrightarrow> (Suc a < start_of ly as + 2 * n \<longrightarrow> inv_locate_a (as, am)
- (Suc ((a - start_of ly as) div 2), Bk # aaa, xs) ires) \<and>
- (\<not> Suc a < start_of ly as + 2 * n \<longrightarrow>
- inv_locate_a (as, am) (n, Bk # aaa, xs) ires)"
-apply(auto)
-apply(subgoal_tac "n > 0")
-apply(subgoal_tac "(a - start_of ly as) div 2 = n - 1")
-apply(insert locate_b_2_a [of as am "n - 1" aaa xs], simp)
-apply(arith)
-apply(case_tac n, simp, simp)
-done
-
-lemma [simp]: "inv_locate_b (as, am) (q, l, []) ires
- \<Longrightarrow> inv_locate_b (as, am) (q, l, [Bk]) ires"
-apply(simp only: inv_locate_b.simps in_middle.simps)
-apply(erule exE)+
-apply(rule_tac x = lm1 in exI, rule_tac x = lm2 in exI,
- rule_tac x = tn in exI, rule_tac x = m in exI,
- rule_tac x = ml in exI, rule_tac x = mr in exI)
-apply(auto)
-done
-
-lemma locate_b_2_locate_a_B[simp]:
- "\<lbrakk>\<not> a < start_of ly as;
- a < start_of ly as + 2 * n;
- (a - start_of ly as) mod 2 = Suc 0;
- inv_locate_b (as, am) ((a - start_of ly as) div 2, aaa, []) ires\<rbrakk>
- \<Longrightarrow> (Suc a < start_of ly as + 2 * n \<longrightarrow>
- inv_locate_a (as, am)
- (Suc ((a - start_of ly as) div 2), Bk # aaa, []) ires)
- \<and> (\<not> Suc a < start_of ly as + 2 * n \<longrightarrow>
- inv_locate_a (as, am) (n, Bk # aaa, []) ires)"
-apply(insert locate_b_2_locate_a [of a ly as n am aaa "[]"], simp)
-done
-
-(*inv: from locate_b to after_write*)
-lemma inv_locate_b_2_after_write[simp]:
- "inv_locate_b (as, am) (n, aaa, Bk # xs) ires
- \<Longrightarrow> inv_after_write (as, abc_lm_s am n (Suc (abc_lm_v am n)))
- (Suc (Suc (2 * n)), aaa, Oc # xs) ires"
-apply(auto simp: in_middle.simps inv_after_write.simps
- abc_lm_v.simps abc_lm_s.simps inv_locate_b.simps)
-apply(subgoal_tac [!] "mr = 0", auto simp: exponent_def split: if_splits)
-apply(subgoal_tac "lm2 = []", simp)
-apply(rule_tac x = rn in exI, rule_tac x = "Suc m" in exI,
- rule_tac x = "lm1" in exI, simp, rule_tac x = "[]" in exI, simp)
-apply(case_tac "Suc (length lm1) - length am", simp, simp only: rep_ind, simp)
-apply(subgoal_tac "length lm1 - length am = nat", simp, arith)
-apply(drule_tac length_equal, simp)
-done
-
-lemma [simp]: "inv_locate_b (as, am) (n, aaa, []) ires \<Longrightarrow>
- inv_after_write (as, abc_lm_s am n (Suc (abc_lm_v am n)))
- (Suc (Suc (2 * n)), aaa, [Oc]) ires"
-apply(insert inv_locate_b_2_after_write [of as am n aaa "[]"])
-by(simp)
-
-(*inv: from after_write to after_move*)
-lemma [simp]: "inv_after_write (as, lm) (Suc (Suc (2 * n)), l, Oc # r) ires
- \<Longrightarrow> inv_after_move (as, lm) (2 * n + 3, Oc # l, r) ires"
-apply(auto simp:inv_after_move.simps inv_after_write.simps split: if_splits)
-done
-
-lemma [simp]: "inv_after_write (as, abc_lm_s am n (Suc (abc_lm_v am n)
- )) (Suc (Suc (2 * n)), aaa, Bk # xs) ires = False"
-apply(simp add: inv_after_write.simps )
-done
-
-lemma [simp]:
- "inv_after_write (as, abc_lm_s am n (Suc (abc_lm_v am n)))
- (Suc (Suc (2 * n)), aaa, []) ires = False"
-apply(simp add: inv_after_write.simps )
-done
-
-(*inv: from after_move to after_clear*)
-lemma [simp]: "inv_after_move (as, lm) (s, l, Oc # r) ires
- \<Longrightarrow> inv_after_clear (as, lm) (s', l, Bk # r) ires"
-apply(auto simp: inv_after_move.simps inv_after_clear.simps split: if_splits)
-done
-
-(*inv: from after_move to on_leftmoving*)
-lemma inv_after_move_2_inv_on_left_moving[simp]:
- "inv_after_move (as, lm) (s, l, Bk # r) ires
- \<Longrightarrow> (l = [] \<longrightarrow>
- inv_on_left_moving (as, lm) (s', [], Bk # Bk # r) ires) \<and>
- (l \<noteq> [] \<longrightarrow>
- inv_on_left_moving (as, lm) (s', tl l, hd l # Bk # r) ires)"
-apply(simp only: inv_after_move.simps inv_on_left_moving.simps)
-apply(subgoal_tac "l \<noteq> []", rule conjI, simp, rule impI,
- rule disjI1, simp only: inv_on_left_moving_norm.simps)
-apply(erule exE)+
-apply(subgoal_tac "lm2 = []")
-apply(rule_tac x = lm1 in exI, rule_tac x = lm2 in exI,
- rule_tac x = m in exI, rule_tac x = m in exI,
- rule_tac x = 1 in exI,
- rule_tac x = "rn - 1" in exI, simp, case_tac rn)
-apply(auto simp: exponent_def intro: BkCons_nil split: if_splits)
-done
-
-lemma [elim]: "[] = <lm::nat list> \<Longrightarrow> lm = []"
-using tape_of_nl_nil_eq[of lm]
-by simp
-
-lemma inv_after_move_2_inv_on_left_moving_B[simp]:
- "inv_after_move (as, lm) (s, l, []) ires
- \<Longrightarrow> (l = [] \<longrightarrow> inv_on_left_moving (as, lm) (s', [], [Bk]) ires) \<and>
- (l \<noteq> [] \<longrightarrow> inv_on_left_moving (as, lm) (s', tl l, [hd l]) ires)"
-apply(simp only: inv_after_move.simps inv_on_left_moving.simps)
-apply(subgoal_tac "l \<noteq> []", rule conjI, simp, rule impI, rule disjI1,
- simp only: inv_on_left_moving_norm.simps)
-apply(erule exE)+
-apply(subgoal_tac "lm2 = []")
-apply(rule_tac x = lm1 in exI, rule_tac x = lm2 in exI,
- rule_tac x = m in exI, rule_tac x = m in exI,
- rule_tac x = 1 in exI, rule_tac x = "rn - 1" in exI, simp, case_tac rn)
-apply(auto simp: exponent_def tape_of_nl_nil_eq intro: BkCons_nil split: if_splits)
-done
-
-(*inv: from after_clear to on_right_moving*)
-lemma [simp]: "Oc # r = replicate rn Bk = False"
-apply(case_tac rn, simp, simp)
-done
-
-lemma inv_after_clear_2_inv_on_right_moving[simp]:
- "inv_after_clear (as, lm) (2 * n + 4, l, Bk # r) ires
- \<Longrightarrow> inv_on_right_moving (as, lm) (2 * n + 5, Bk # l, r) ires"
-apply(auto simp: inv_after_clear.simps inv_on_right_moving.simps )
-apply(subgoal_tac "lm2 \<noteq> []")
-apply(rule_tac x = "lm1 @ [m]" in exI, rule_tac x = "tl lm2" in exI,
- rule_tac x = "hd lm2" in exI, simp)
-apply(rule_tac x = 0 in exI, rule_tac x = "hd lm2" in exI)
-apply(simp add: exponent_def, rule conjI)
-apply(case_tac [!] "lm2::nat list", auto simp: exponent_def)
-apply(case_tac rn, auto split: if_splits simp: tape_of_nat_def)
-apply(case_tac list,
- simp add: tape_of_nl_abv tape_of_nat_list.simps exponent_def)
-apply(erule_tac x = "rn - 1" in allE,
- case_tac rn, auto simp: exponent_def)
-apply(case_tac list,
- simp add: tape_of_nl_abv tape_of_nat_list.simps exponent_def)
-apply(erule_tac x = "rn - 1" in allE,
- case_tac rn, auto simp: exponent_def)
-done
-
-
-lemma [simp]: "inv_after_clear (as, lm) (2 * n + 4, l, []) ires\<Longrightarrow>
- inv_after_clear (as, lm) (2 * n + 4, l, [Bk]) ires"
-by(auto simp: inv_after_clear.simps)
-
-lemma [simp]: "inv_after_clear (as, lm) (2 * n + 4, l, []) ires
- \<Longrightarrow> inv_on_right_moving (as, lm) (2 * n + 5, Bk # l, []) ires"
-by(insert
- inv_after_clear_2_inv_on_right_moving[of as lm n l "[]"], simp)
-
-(*inv: from on_right_moving to on_right_movign*)
-lemma [simp]: "inv_on_right_moving (as, lm) (2 * n + 5, l, Oc # r) ires
- \<Longrightarrow> inv_on_right_moving (as, lm) (2 * n + 5, Oc # l, r) ires"
-apply(auto simp: inv_on_right_moving.simps)
-apply(rule_tac x = lm1 in exI, rule_tac x = lm2 in exI,
- rule_tac x = "ml + mr" in exI, simp)
-apply(rule_tac x = "Suc ml" in exI,
- rule_tac x = "mr - 1" in exI, simp)
-apply(case_tac mr, auto simp: exponent_def )
-apply(rule_tac x = lm1 in exI, rule_tac x = "[]" in exI,
- rule_tac x = "ml + mr" in exI, simp)
-apply(rule_tac x = "Suc ml" in exI,
- rule_tac x = "mr - 1" in exI, simp)
-apply(case_tac mr, auto split: if_splits simp: exponent_def)
-done
-
-lemma inv_on_right_moving_2_inv_on_right_moving[simp]:
- "inv_on_right_moving (as, lm) (2 * n + 5, l, Bk # r) ires
- \<Longrightarrow> inv_after_write (as, lm) (Suc (Suc (2 * n)), l, Oc # r) ires"
-apply(auto simp: inv_on_right_moving.simps inv_after_write.simps )
-apply(case_tac mr, auto simp: exponent_def split: if_splits)
-apply(case_tac [!] mr, simp_all)
-done
-
-lemma [simp]: "inv_on_right_moving (as, lm) (2 * n + 5, l, []) ires\<Longrightarrow>
- inv_on_right_moving (as, lm) (2 * n + 5, l, [Bk]) ires"
-apply(auto simp: inv_on_right_moving.simps exponent_def)
-apply(rule_tac x = lm1 in exI, rule_tac x = "[]" in exI, simp)
-apply (rule_tac x = m in exI, auto split: if_splits simp: exponent_def)
-done
-
-(*inv: from on_right_moving to after_write*)
-lemma [simp]: "inv_on_right_moving (as, lm) (2 * n + 5, l, []) ires
- \<Longrightarrow> inv_after_write (as, lm) (Suc (Suc (2 * n)), l, [Oc]) ires"
-apply(rule_tac inv_on_right_moving_2_inv_on_right_moving, simp)
-done
-
-(*inv: from on_left_moving to on_left_moving*)
-lemma [simp]: "inv_on_left_moving_in_middle_B (as, lm)
- (s, l, Oc # r) ires = False"
-apply(auto simp: inv_on_left_moving_in_middle_B.simps )
-done
-
-lemma [simp]: "inv_on_left_moving_norm (as, lm) (s, l, Bk # r) ires
- = False"
-apply(auto simp: inv_on_left_moving_norm.simps)
-apply(case_tac [!] mr, auto simp: )
-done
-
-lemma [intro]: "\<exists>rna. Oc # Oc\<^bsup>m\<^esup> @ Bk # <lm> @ Bk\<^bsup>rn\<^esup> = <m # lm> @ Bk\<^bsup>rna\<^esup>"
-apply(case_tac lm, simp add: tape_of_nl_abv tape_of_nat_list.simps)
-apply(rule_tac x = "Suc rn" in exI, simp)
-apply(case_tac list, simp_all add: tape_of_nl_abv tape_of_nat_list.simps, auto)
-done
-
-
-lemma [simp]:
- "\<lbrakk>inv_on_left_moving_norm (as, lm) (s, l, Oc # r) ires;
- hd l = Bk; l \<noteq> []\<rbrakk> \<Longrightarrow>
- inv_on_left_moving_in_middle_B (as, lm) (s, tl l, Bk # Oc # r) ires"
-apply(case_tac l, simp, simp)
-apply(simp only: inv_on_left_moving_norm.simps
- inv_on_left_moving_in_middle_B.simps)
-apply(erule_tac exE)+
-apply(rule_tac x = lm1 in exI, rule_tac x = "m # lm2" in exI, auto)
-apply(case_tac [!] ml, auto)
-apply(rule_tac [!] x = 0 in exI, simp_all add: tape_of_nl_abv tape_of_nat_list.simps)
-done
-
-lemma [simp]: "\<lbrakk>inv_on_left_moving_norm (as, lm) (s, l, Oc # r) ires;
- hd l = Oc; l \<noteq> []\<rbrakk>
- \<Longrightarrow> inv_on_left_moving_norm (as, lm)
- (s, tl l, Oc # Oc # r) ires"
-apply(simp only: inv_on_left_moving_norm.simps)
-apply(erule exE)+
-apply(rule_tac x = lm1 in exI, rule_tac x = lm2 in exI,
- rule_tac x = m in exI, rule_tac x = "ml - 1" in exI,
- rule_tac x = "Suc mr" in exI, rule_tac x = rn in exI, simp)
-apply(case_tac ml, auto simp: exponent_def split: if_splits)
-done
-
-lemma [simp]: "inv_on_left_moving_norm (as, lm) (s, [], Oc # r) ires
- \<Longrightarrow> inv_on_left_moving_in_middle_B (as, lm) (s, [], Bk # Oc # r) ires"
-apply(auto simp: inv_on_left_moving_norm.simps
- inv_on_left_moving_in_middle_B.simps split: if_splits)
-done
-
-lemma [simp]:"inv_on_left_moving (as, lm) (s, l, Oc # r) ires
- \<Longrightarrow> (l = [] \<longrightarrow> inv_on_left_moving (as, lm) (s, [], Bk # Oc # r) ires)
- \<and> (l \<noteq> [] \<longrightarrow> inv_on_left_moving (as, lm) (s, tl l, hd l # Oc # r) ires)"
-apply(simp add: inv_on_left_moving.simps)
-apply(case_tac "l \<noteq> []", rule conjI, simp, simp)
-apply(case_tac "hd l", simp, simp, simp)
-done
-
-(*inv: from on_left_moving to check_left_moving*)
-lemma [simp]: "inv_on_left_moving_in_middle_B (as, lm)
- (s, Bk # list, Bk # r) ires
- \<Longrightarrow> inv_check_left_moving_on_leftmost (as, lm)
- (s', list, Bk # Bk # r) ires"
-apply(auto simp: inv_on_left_moving_in_middle_B.simps
- inv_check_left_moving_on_leftmost.simps split: if_splits)
-apply(case_tac [!] "rev lm1", simp_all)
-apply(case_tac [!] lista, simp_all add: tape_of_nl_abv tape_of_nat_list.simps)
-done
-
-lemma [simp]:
- "inv_check_left_moving_in_middle (as, lm) (s, l, Bk # r) ires= False"
-by(auto simp: inv_check_left_moving_in_middle.simps )
-
-lemma [simp]:
- "inv_on_left_moving_in_middle_B (as, lm) (s, [], Bk # r) ires\<Longrightarrow>
- inv_check_left_moving_on_leftmost (as, lm) (s', [], Bk # Bk # r) ires"
-apply(auto simp: inv_on_left_moving_in_middle_B.simps
- inv_check_left_moving_on_leftmost.simps split: if_splits)
-done
-
-
-lemma [simp]: "inv_check_left_moving_on_leftmost (as, lm)
- (s, list, Oc # r) ires= False"
-by(auto simp: inv_check_left_moving_on_leftmost.simps split: if_splits)
-
-lemma [simp]: "inv_on_left_moving_in_middle_B (as, lm)
- (s, Oc # list, Bk # r) ires
- \<Longrightarrow> inv_check_left_moving_in_middle (as, lm) (s', list, Oc # Bk # r) ires"
-apply(auto simp: inv_on_left_moving_in_middle_B.simps
- inv_check_left_moving_in_middle.simps split: if_splits)
-done
-
-lemma inv_on_left_moving_2_check_left_moving[simp]:
- "inv_on_left_moving (as, lm) (s, l, Bk # r) ires
- \<Longrightarrow> (l = [] \<longrightarrow> inv_check_left_moving (as, lm) (s', [], Bk # Bk # r) ires)
- \<and> (l \<noteq> [] \<longrightarrow>
- inv_check_left_moving (as, lm) (s', tl l, hd l # Bk # r) ires)"
-apply(simp add: inv_on_left_moving.simps inv_check_left_moving.simps)
-apply(case_tac l, simp, simp)
-apply(case_tac a, simp, simp)
-done
-
-lemma [simp]: "inv_on_left_moving_norm (as, lm) (s, l, []) ires = False"
-apply(auto simp: inv_on_left_moving_norm.simps)
-by(case_tac [!] mr, auto)
-
-lemma [simp]: "inv_on_left_moving (as, lm) (s, l, []) ires\<Longrightarrow>
- inv_on_left_moving (as, lm) (6 + 2 * n, l, [Bk]) ires"
-apply(simp add: inv_on_left_moving.simps)
-apply(auto simp: inv_on_left_moving_in_middle_B.simps)
-done
-
-lemma [simp]: "inv_on_left_moving (as, lm) (s, l, []) ires = False"
-apply(simp add: inv_on_left_moving.simps)
-apply(simp add: inv_on_left_moving_in_middle_B.simps)
-done
-
-lemma [simp]: "inv_on_left_moving (as, lm) (s, l, []) ires
- \<Longrightarrow> (l = [] \<longrightarrow> inv_check_left_moving (as, lm) (s', [], [Bk]) ires) \<and>
- (l \<noteq> [] \<longrightarrow> inv_check_left_moving (as, lm) (s', tl l, [hd l]) ires)"
-by simp
-
-lemma Oc_Bk_Cons_ex[simp]:
- "Oc # Bk # list = <lm::nat list> @ Bk\<^bsup>ln\<^esup> \<Longrightarrow>
- \<exists>ln. list = <tl (lm)> @ Bk\<^bsup>ln\<^esup>"
-apply(case_tac "lm", simp)
-apply(case_tac ln, simp_all add: exponent_def)
-apply(case_tac lista,
- auto simp: tape_of_nl_abv tape_of_nat_list.simps exponent_def)
-apply(case_tac [!] a, auto simp: )
-apply(case_tac ln, simp, rule_tac x = nat in exI, simp)
-done
-
-lemma [simp]:
- "Oc # Bk # list = <rev lm1::nat list> @ Bk\<^bsup>ln\<^esup> \<Longrightarrow>
- \<exists>rna. Oc # Bk # <lm2> @ Bk\<^bsup>rn\<^esup> = <hd (rev lm1) # lm2> @ Bk\<^bsup>rna\<^esup>"
-apply(frule Oc_Bk_Cons, simp)
-apply(case_tac lm2,
- auto simp: tape_of_nl_abv tape_of_nat_list.simps exponent_def )
-apply(rule_tac x = "Suc rn" in exI, simp)
-done
-
-(*inv: from check_left_moving to on_left_moving*)
-lemma [intro]: "\<exists>rna. a # a\<^bsup>rn\<^esup> = a\<^bsup>rna\<^esup>"
-apply(rule_tac x = "Suc rn" in exI, simp)
-done
-
-lemma
-inv_check_left_moving_in_middle_2_on_left_moving_in_middle_B[simp]:
-"inv_check_left_moving_in_middle (as, lm) (s, Bk # list, Oc # r) ires
- \<Longrightarrow> inv_on_left_moving_in_middle_B (as, lm) (s', list, Bk # Oc # r) ires"
-apply(simp only: inv_check_left_moving_in_middle.simps
- inv_on_left_moving_in_middle_B.simps)
-apply(erule_tac exE)+
-apply(rule_tac x = "rev (tl (rev lm1))" in exI,
- rule_tac x = "[hd (rev lm1)] @ lm2" in exI, auto)
-apply(case_tac [!] "rev lm1",simp_all add: tape_of_nl_abv tape_of_nat_list.simps)
-apply(case_tac [!] a, simp_all)
-apply(case_tac [1] lm2, simp_all add: tape_of_nat_list.simps, auto)
-apply(case_tac [3] lm2, simp_all add: tape_of_nat_list.simps, auto)
-apply(case_tac [!] lista, simp_all add: tape_of_nat_list.simps)
-done
-
-lemma [simp]:
- "inv_check_left_moving_in_middle (as, lm) (s, [], Oc # r) ires\<Longrightarrow>
- inv_check_left_moving_in_middle (as, lm) (s', [Bk], Oc # r) ires"
-apply(auto simp: inv_check_left_moving_in_middle.simps )
-done
-
-lemma [simp]:
- "inv_check_left_moving_in_middle (as, lm) (s, [], Oc # r) ires
- \<Longrightarrow> inv_on_left_moving_in_middle_B (as, lm) (s', [], Bk # Oc # r) ires"
-apply(insert
-inv_check_left_moving_in_middle_2_on_left_moving_in_middle_B[of
- as lm n "[]" r], simp)
-done
-
-lemma [simp]: "a\<^bsup>0\<^esup> = []"
-apply(simp add: exponent_def)
-done
-
-lemma [simp]: "inv_check_left_moving_in_middle (as, lm)
- (s, Oc # list, Oc # r) ires
- \<Longrightarrow> inv_on_left_moving_norm (as, lm) (s', list, Oc # Oc # r) ires"
-apply(auto simp: inv_check_left_moving_in_middle.simps
- inv_on_left_moving_norm.simps)
-apply(rule_tac x = "rev (tl (rev lm1))" in exI,
- rule_tac x = lm2 in exI, rule_tac x = "hd (rev lm1)" in exI)
-apply(rule_tac conjI)
-apply(case_tac "rev lm1", simp, simp)
-apply(rule_tac x = "hd (rev lm1) - 1" in exI, auto)
-apply(rule_tac [!] x = "Suc (Suc 0)" in exI, simp)
-apply(case_tac [!] "rev lm1", simp_all)
-apply(case_tac [!] a, simp_all add: tape_of_nl_abv tape_of_nat_list.simps, auto)
-done
-
-lemma [simp]: "inv_check_left_moving (as, lm) (s, l, Oc # r) ires
-\<Longrightarrow> (l = [] \<longrightarrow> inv_on_left_moving (as, lm) (s', [], Bk # Oc # r) ires) \<and>
- (l \<noteq> [] \<longrightarrow> inv_on_left_moving (as, lm) (s', tl l, hd l # Oc # r) ires)"
-apply(case_tac l,
- auto simp: inv_check_left_moving.simps inv_on_left_moving.simps)
-apply(case_tac a, simp, simp)
-done
-
-(*inv: check_left_moving to after_left_moving*)
-lemma [simp]: "inv_check_left_moving (as, lm) (s, l, Bk # r) ires
- \<Longrightarrow> inv_after_left_moving (as, lm) (s', Bk # l, r) ires"
-apply(auto simp: inv_check_left_moving.simps
- inv_check_left_moving_on_leftmost.simps inv_after_left_moving.simps)
-done
-
-
-lemma [simp]:"inv_check_left_moving (as, lm) (s, l, []) ires
- \<Longrightarrow> inv_after_left_moving (as, lm) (s', Bk # l, []) ires"
-by(simp add: inv_check_left_moving.simps
-inv_check_left_moving_in_middle.simps
-inv_check_left_moving_on_leftmost.simps)
-
-(*inv: after_left_moving to inv_stop*)
-lemma [simp]: "inv_after_left_moving (as, lm) (s, l, Bk # r) ires
- \<Longrightarrow> inv_stop (as, lm) (s', Bk # l, r) ires"
-apply(auto simp: inv_after_left_moving.simps inv_stop.simps)
-done
-
-lemma [simp]: "inv_after_left_moving (as, lm) (s, l, []) ires
- \<Longrightarrow> inv_stop (as, lm) (s', Bk # l, []) ires"
-by(auto simp: inv_after_left_moving.simps)
-
-(*inv: stop to stop*)
-lemma [simp]: "inv_stop (as, lm) (x, l, r) ires \<Longrightarrow>
- inv_stop (as, lm) (y, l, r) ires"
-apply(simp add: inv_stop.simps)
-done
-
-lemma [simp]: "inv_after_clear (as, lm) (s, aaa, Oc # xs) ires= False"
-apply(auto simp: inv_after_clear.simps )
-done
-
-lemma [simp]:
- "inv_after_left_moving (as, lm) (s, aaa, Oc # xs) ires = False"
-by(auto simp: inv_after_left_moving.simps )
-
-lemma start_of_not0: "as \<noteq> 0 \<Longrightarrow> start_of ly as > 0"
-apply(rule startof_not0)
-done
-
-text {*
- The single step currectness of the TM complied from Abacus instruction @{text "Inc n"}.
- It shows every single step execution of this TM keeps the invariant.
-*}
-
-lemma inc_inv_step:
- assumes
- -- {* Invariant holds on the start *}
- h11: "inc_inv ly n (as, am) tc ires"
- -- {* The layout of Abacus program @{text "aprog"} is @{text "ly"} *}
- and h12: "ly = layout_of aprog"
- -- {* The instruction at position @{text "as"} is @{text "Inc n"} *}
- and h21: "abc_fetch as aprog = Some (Inc n)"
- -- {* TM not yet reach the final state, where @{text "start_of ly as + 2*n + 9"} is the state
- where the current TM stops and the next TM starts. *}
- and h22: "(\<lambda> (s, l, r). s \<noteq> start_of ly as + 2*n + 9) tc"
- shows
- -- {*
- Single step execution of the TM keeps the invaraint, where
- the TM compiled from @{text "Inc n"} is @{text "(ci ly (start_of ly as) (Inc n))"}
- @{text "start_of ly as - Suc 0)"} is the offset used to execute this {\em shifted}
- TM.
- *}
- "inc_inv ly n (as, am) (t_step tc (ci ly (start_of ly as) (Inc n), start_of ly as - Suc 0)) ires"
-proof -
- from h21 h22 have h3 : "start_of (layout_of aprog) as > 0"
- apply(case_tac as, simp add: start_of.simps abc_fetch.simps)
- apply(insert start_of_not0[of as "layout_of aprog"], simp)
- done
- from h11 h12 and h21 h22 and this show ?thesis
- apply(case_tac tc, simp)
- apply(case_tac "a = 0",
- auto split:if_splits simp add:t_step.simps,
- tactic {* ALLGOALS (resolve_tac [@{thm fetch_intro}]) *})
- apply (simp_all add:fetch_simps new_tape.simps)
- done
-qed
-
-
-lemma t_steps_ind: "t_steps tc (p, off) (Suc n)
- = t_step (t_steps tc (p, off) n) (p, off)"
-apply(induct n arbitrary: tc)
-apply(simp add: t_steps.simps)
-apply(simp add: t_steps.simps)
-done
-
-definition lex_pair :: "((nat \<times> nat) \<times> (nat \<times> nat)) set"
- where
- "lex_pair \<equiv> less_than <*lex*> less_than"
-
-definition lex_triple ::
- "((nat \<times> (nat \<times> nat)) \<times> (nat \<times> (nat \<times> nat))) set"
- where "lex_triple \<equiv> less_than <*lex*> lex_pair"
-
-definition lex_square ::
- "((nat \<times> nat \<times> nat \<times> nat) \<times> (nat \<times> nat \<times> nat \<times> nat)) set"
- where "lex_square \<equiv> less_than <*lex*> lex_triple"
-
-fun abc_inc_stage1 :: "t_conf \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat"
- where
- "abc_inc_stage1 (s, l, r) ss n =
- (if s = 0 then 0
- else if s \<le> ss+2*n+1 then 5
- else if s\<le> ss+2*n+5 then 4
- else if s \<le> ss+2*n+7 then 3
- else if s = ss+2*n+8 then 2
- else 1)"
-
-fun abc_inc_stage2 :: "t_conf \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat"
- where
- "abc_inc_stage2 (s, l, r) ss n =
- (if s \<le> ss + 2*n + 1 then 0
- else if s = ss + 2*n + 2 then length r
- else if s = ss + 2*n + 3 then length r
- else if s = ss + 2*n + 4 then length r
- else if s = ss + 2*n + 5 then
- if r \<noteq> [] then length r
- else 1
- else if s = ss+2*n+6 then length l
- else if s = ss+2*n+7 then length l
- else 0)"
-
-fun abc_inc_stage3 :: "t_conf \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> block list \<Rightarrow> nat"
- where
- "abc_inc_stage3 (s, l, r) ss n ires = (
- if s = ss + 2*n + 3 then 4
- else if s = ss + 2*n + 4 then 3
- else if s = ss + 2*n + 5 then
- if r \<noteq> [] \<and> hd r = Oc then 2
- else 1
- else if s = ss + 2*n + 2 then 0
- else if s = ss + 2*n + 6 then
- if l = Bk # ires \<and> r \<noteq> [] \<and> hd r = Oc then 2
- else 1
- else if s = ss + 2*n + 7 then
- if r \<noteq> [] \<and> hd r = Oc then 3
- else 0
- else ss+2*n+9 - s)"
-
-fun abc_inc_stage4 :: "t_conf \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> block list \<Rightarrow> nat"
- where
- "abc_inc_stage4 (s, l, r) ss n ires =
- (if s \<le> ss+2*n+1 \<and> (s - ss) mod 2 = 0 then
- if (r\<noteq>[] \<and> hd r = Oc) then 0
- else 1
- else if (s \<le> ss+2*n+1 \<and> (s - ss) mod 2 = Suc 0)
- then length r
- else if s = ss + 2*n + 6 then
- if l = Bk # ires \<and> hd r = Bk then 0
- else Suc (length l)
- else 0)"
-
-fun abc_inc_measure :: "(t_conf \<times> nat \<times> nat \<times> block list) \<Rightarrow>
- (nat \<times> nat \<times> nat \<times> nat)"
- where
- "abc_inc_measure (c, ss, n, ires) =
- (abc_inc_stage1 c ss n, abc_inc_stage2 c ss n,
- abc_inc_stage3 c ss n ires, abc_inc_stage4 c ss n ires)"
-
-definition abc_inc_LE :: "(((nat \<times> block list \<times> block list) \<times> nat \<times>
- nat \<times> block list) \<times> ((nat \<times> block list \<times> block list) \<times> nat \<times> nat \<times> block list)) set"
- where "abc_inc_LE \<equiv> (inv_image lex_square abc_inc_measure)"
-
-lemma wf_lex_triple: "wf lex_triple"
-by (auto intro:wf_lex_prod simp:lex_triple_def lex_pair_def)
-
-lemma wf_lex_square: "wf lex_square"
-by (auto intro:wf_lex_triple simp:lex_triple_def lex_square_def lex_pair_def)
-
-lemma wf_abc_inc_le[intro]: "wf abc_inc_LE"
-by(auto intro:wf_inv_image wf_lex_square simp:abc_inc_LE_def)
-
-(********************************************************************)
-declare inc_inv.simps[simp del]
-
-lemma halt_lemma2':
- "\<lbrakk>wf LE; \<forall> n. ((\<not> P (f n) \<and> Q (f n)) \<longrightarrow>
- (Q (f (Suc n)) \<and> (f (Suc n), (f n)) \<in> LE)); Q (f 0)\<rbrakk>
- \<Longrightarrow> \<exists> n. P (f n)"
-apply(intro exCI, simp)
-apply(subgoal_tac "\<forall> n. Q (f n)", simp)
-apply(drule_tac f = f in wf_inv_image)
-apply(simp add: inv_image_def)
-apply(erule wf_induct, simp)
-apply(erule_tac x = x in allE)
-apply(erule_tac x = n in allE, erule_tac x = n in allE)
-apply(erule_tac x = "Suc x" in allE, simp)
-apply(rule_tac allI)
-apply(induct_tac n, simp)
-apply(erule_tac x = na in allE, simp)
-done
-
-lemma halt_lemma2'':
- "\<lbrakk>P (f n); \<not> P (f (0::nat))\<rbrakk> \<Longrightarrow>
- \<exists> n. (P (f n) \<and> (\<forall> i < n. \<not> P (f i)))"
-apply(induct n rule: nat_less_induct, auto)
-done
-
-lemma halt_lemma2''':
- "\<lbrakk>\<forall>n. \<not> P (f n) \<and> Q (f n) \<longrightarrow> Q (f (Suc n)) \<and> (f (Suc n), f n) \<in> LE;
- Q (f 0); \<forall>i<na. \<not> P (f i)\<rbrakk> \<Longrightarrow> Q (f na)"
-apply(induct na, simp, simp)
-done
-
-lemma halt_lemma2:
- "\<lbrakk>wf LE;
- \<forall> n. ((\<not> P (f n) \<and> Q (f n)) \<longrightarrow> (Q (f (Suc n)) \<and> (f (Suc n), (f n)) \<in> LE));
- Q (f 0); \<not> P (f 0)\<rbrakk>
- \<Longrightarrow> \<exists> n. P (f n) \<and> Q (f n)"
-apply(insert halt_lemma2' [of LE P f Q], simp, erule_tac exE)
-apply(subgoal_tac "\<exists> n. (P (f n) \<and> (\<forall> i < n. \<not> P (f i)))")
-apply(erule_tac exE)+
-apply(rule_tac x = na in exI, auto)
-apply(rule halt_lemma2''', simp, simp, simp)
-apply(erule_tac halt_lemma2'', simp)
-done
-
-lemma [simp]:
- "\<lbrakk>ly = layout_of aprog; abc_fetch as aprog = Some (Inc n)\<rbrakk>
- \<Longrightarrow> start_of ly (Suc as) = start_of ly as + 2*n +9"
-apply(case_tac as, auto simp: abc_fetch.simps start_of.simps
- layout_of.simps length_of.simps split: if_splits)
-done
-
-lemma inc_inv_init:
- "\<lbrakk>abc_fetch as aprog = Some (Inc n);
- crsp_l ly (as, am) (start_of ly as, l, r) ires; ly = layout_of aprog\<rbrakk>
- \<Longrightarrow> inc_inv ly n (as, am) (start_of ly as, l, r) ires"
-apply(auto simp: crsp_l.simps inc_inv.simps
- inv_locate_a.simps at_begin_fst_bwtn.simps
- at_begin_fst_awtn.simps at_begin_norm.simps )
-apply(auto intro: startof_not0)
-done
-
-lemma inc_inv_stop_pre[simp]:
- "\<lbrakk>ly = layout_of aprog; inc_inv ly n (as, am) (s, l, r) ires;
- s = start_of ly as; abc_fetch as aprog = Some (Inc n)\<rbrakk>
- \<Longrightarrow> (\<forall>na. \<not> (\<lambda>((s, l, r), ss, n', ires'). s = start_of ly (Suc as))
- (t_steps (s, l, r) (ci ly (start_of ly as)
- (Inc n), start_of ly as - Suc 0) na, s, n, ires) \<and>
- (\<lambda>((s, l, r), ss, n', ires'). inc_inv ly n (as, am) (s, l, r) ires')
- (t_steps (s, l, r) (ci ly (start_of ly as)
- (Inc n), start_of ly as - Suc 0) na, s, n, ires) \<longrightarrow>
- (\<lambda>((s, l, r), ss, n', ires'). inc_inv ly n (as, am) (s, l, r) ires')
- (t_steps (s, l, r) (ci ly (start_of ly as)
- (Inc n), start_of ly as - Suc 0) (Suc na), s, n, ires) \<and>
- ((t_steps (s, l, r) (ci ly (start_of ly as) (Inc n),
- start_of ly as - Suc 0) (Suc na), s, n, ires),
- t_steps (s, l, r) (ci ly (start_of ly as)
- (Inc n), start_of ly as - Suc 0) na, s, n, ires) \<in> abc_inc_LE)"
-apply(rule allI, rule impI, simp add: t_steps_ind,
- rule conjI, erule_tac conjE)
-apply(rule_tac inc_inv_step, simp, simp, simp)
-apply(case_tac "t_steps (start_of (layout_of aprog) as, l, r) (ci (layout_of aprog)
- (start_of (layout_of aprog) as) (Inc n), start_of (layout_of aprog) as - Suc 0) na", simp)
-proof -
- fix na
- assume h1: "abc_fetch as aprog = Some (Inc n)"
- "\<not> (\<lambda>(s, l, r) (ss, n', ires'). s = start_of (layout_of aprog) as + 2 * n + 9)
- (t_steps (start_of (layout_of aprog) as, l, r) (ci (layout_of aprog)
- (start_of (layout_of aprog) as) (Inc n), start_of (layout_of aprog) as - Suc 0) na)
- (start_of (layout_of aprog) as, n, ires) \<and>
- inc_inv (layout_of aprog) n (as, am) (t_steps (start_of (layout_of aprog) as, l, r)
- (ci (layout_of aprog) (start_of (layout_of aprog) as) (Inc n), start_of (layout_of aprog) as - Suc 0) na) ires"
- from h1 have h2: "start_of (layout_of aprog) as > 0"
- apply(rule_tac startof_not0)
- done
- from h1 and h2 show "((t_step (t_steps (start_of (layout_of aprog) as, l, r) (ci (layout_of aprog)
- (start_of (layout_of aprog) as) (Inc n), start_of (layout_of aprog) as - Suc 0) na)
- (ci (layout_of aprog) (start_of (layout_of aprog) as) (Inc n), start_of (layout_of aprog) as - Suc 0),
- start_of (layout_of aprog) as, n, ires),
- t_steps (start_of (layout_of aprog) as, l, r)
- (ci (layout_of aprog) (start_of (layout_of aprog) as) (Inc n), start_of (layout_of aprog) as - Suc 0) na,
- start_of (layout_of aprog) as, n, ires)
- \<in> abc_inc_LE"
- apply(case_tac "(t_steps (start_of (layout_of aprog) as, l, r)
- (ci (layout_of aprog)
- (start_of (layout_of aprog) as) (Inc n),
- start_of (layout_of aprog) as - Suc 0) na)", simp)
- apply(case_tac "a = 0",
- auto split:if_splits simp add:t_step.simps inc_inv.simps,
- tactic {* ALLGOALS (resolve_tac [@{thm fetch_intro}]) *})
- apply(simp_all add:fetch_simps new_tape.simps)
- apply(auto simp add: abc_inc_LE_def
- lex_square_def lex_triple_def lex_pair_def
- inv_after_write.simps inv_after_move.simps inv_after_clear.simps
- inv_on_left_moving.simps inv_on_left_moving_norm.simps split: if_splits)
- done
-qed
-
-lemma inc_inv_stop_pre1:
- "\<lbrakk>
- ly = layout_of aprog;
- abc_fetch as aprog = Some (Inc n);
- s = start_of ly as;
- inc_inv ly n (as, am) (s, l, r) ires
- \<rbrakk> \<Longrightarrow>
- (\<exists> stp > 0. (\<lambda> (s', l', r').
- s' = start_of ly (Suc as) \<and>
- inc_inv ly n (as, am) (s', l', r') ires)
- (t_steps (s, l, r) (ci ly (start_of ly as) (Inc n),
- start_of ly as - Suc 0) stp))"
-apply(insert halt_lemma2[of abc_inc_LE
- "\<lambda> ((s, l, r), ss, n', ires'). s = start_of ly (Suc as)"
- "(\<lambda> stp. (t_steps (s, l, r)
- (ci ly (start_of ly as) (Inc n),
- start_of ly as - Suc 0) stp, s, n, ires))"
- "\<lambda> ((s, l, r), ss, n'). inc_inv ly n (as, am) (s, l, r) ires"])
-apply(insert wf_abc_inc_le)
-apply(insert inc_inv_stop_pre[of ly aprog n as am s l r ires], simp)
-apply(simp only: t_steps.simps, auto)
-apply(rule_tac x = na in exI)
-apply(case_tac "(t_steps (start_of (layout_of aprog) as, l, r)
- (ci (layout_of aprog) (start_of (layout_of aprog) as)
- (Inc n), start_of (layout_of aprog) as - Suc 0) na)", simp)
-apply(case_tac na, simp add: t_steps.simps, simp)
-done
-
-lemma inc_inv_stop:
- assumes program_and_layout:
- -- {* There is an Abacus program @{text "aprog"} and its layout is @{text "ly"}: *}
- "ly = layout_of aprog"
- and an_instruction:
- -- {* There is an instruction @{text "Inc n"} at postion @{text "as"} of @{text "aprog"} *}
- "abc_fetch as aprog = Some (Inc n)"
- and the_start_state:
- -- {* According to @{text "ly"} and @{text "as"},
- the start state of the TM compiled from this
- @{text "Inc n"} instruction should be @{text "s"}:
- *}
- "s = start_of ly as"
- and inv:
- -- {* Invariant holds on configuration @{text "(s, l, r)"} *}
- "inc_inv ly n (as, am) (s, l, r) ires"
- shows -- {* After @{text "stp"} steps of execution, the compiled
- TM reaches the start state of next compiled TM and the invariant
- still holds.
- *}
- "(\<exists> stp > 0. (\<lambda> (s', l', r').
- s' = start_of ly (Suc as) \<and>
- inc_inv ly n (as, am) (s', l', r') ires)
- (t_steps (s, l, r) (ci ly (start_of ly as) (Inc n),
- start_of ly as - Suc 0) stp))"
-proof -
- from inc_inv_stop_pre1 [OF program_and_layout an_instruction the_start_state inv]
- show ?thesis .
-qed
-
-lemma inc_inv_stop_cond:
- "\<lbrakk>ly = layout_of aprog;
- s' = start_of ly (as + 1);
- inc_inv ly n (as, lm) (s', (l', r')) ires;
- abc_fetch as aprog = Some (Inc n)\<rbrakk> \<Longrightarrow>
- crsp_l ly (Suc as, abc_lm_s lm n (Suc (abc_lm_v lm n)))
- (s', l', r') ires"
-apply(subgoal_tac "s' = start_of ly as + 2*n + 9", simp)
-apply(auto simp: inc_inv.simps inv_stop.simps crsp_l.simps )
-done
-
-lemma inc_crsp_ex_pre:
- "\<lbrakk>ly = layout_of aprog;
- crsp_l ly (as, am) tc ires;
- abc_fetch as aprog = Some (Inc n)\<rbrakk>
- \<Longrightarrow> \<exists>stp > 0. crsp_l ly (abc_step_l (as, am) (Some (Inc n)))
- (t_steps tc (ci ly (start_of ly as) (Inc n),
- start_of ly as - Suc 0) stp) ires"
-proof(case_tac tc, simp add: abc_step_l.simps)
- fix a b c
- assume h1: "ly = layout_of aprog"
- "crsp_l (layout_of aprog) (as, am) (a, b, c) ires"
- "abc_fetch as aprog = Some (Inc n)"
- hence h2: "a = start_of ly as"
- by(auto simp: crsp_l.simps)
- from h1 and h2 have h3:
- "inc_inv ly n (as, am) (start_of ly as, b, c) ires"
- by(rule_tac inc_inv_init, simp, simp, simp)
- from h1 and h2 and h3 have h4:
- "(\<exists> stp > 0. (\<lambda> (s', l', r'). s' =
- start_of ly (Suc as) \<and> inc_inv ly n (as, am) (s', l', r') ires)
- (t_steps (a, b, c) (ci ly (start_of ly as)
- (Inc n), start_of ly as - Suc 0) stp))"
- apply(rule_tac inc_inv_stop, auto)
- done
- from h1 and h2 and h3 and h4 show
- "\<exists>stp > 0. crsp_l (layout_of aprog)
- (Suc as, abc_lm_s am n (Suc (abc_lm_v am n)))
- (t_steps (a, b, c) (ci (layout_of aprog)
- (start_of (layout_of aprog) as) (Inc n),
- start_of (layout_of aprog) as - Suc 0) stp) ires"
- apply(erule_tac exE)
- apply(rule_tac x = stp in exI)
- apply(case_tac "(t_steps (a, b, c) (ci (layout_of aprog)
- (start_of (layout_of aprog) as) (Inc n),
- start_of (layout_of aprog) as - Suc 0) stp)", simp)
- apply(rule_tac inc_inv_stop_cond, auto)
- done
-qed
-
-text {*
- The total correctness of the compilaton of @{text "Inc n"} instruction.
-*}
-
-lemma inc_crsp_ex:
- assumes layout:
- -- {* For any Abacus program @{text "aprog"}, assuming its layout is @{text "ly"} *}
- "ly = layout_of aprog"
- and corresponds:
- -- {* Abacus configuration @{text "(as, am)"} is in correspondence with
- TM configuration @{text "tc"} *}
- "crsp_l ly (as, am) tc ires"
- and inc:
- -- {* There is an instruction @{text "Inc n"} at postion @{text "as"} of @{text "aprog"} *}
- "abc_fetch as aprog = Some (Inc n)"
- shows
- -- {*
- After @{text "stp"} steps of execution, the TM compiled from this @{text "Inc n"}
- stops with a configuration which corresponds to the Abacus configuration obtained
- from the execution of this @{text "Inc n"} instruction.
- *}
- "\<exists>stp > 0. crsp_l ly (abc_step_l (as, am) (Some (Inc n)))
- (t_steps tc (ci ly (start_of ly as) (Inc n),
- start_of ly as - Suc 0) stp) ires"
-proof -
- from inc_crsp_ex_pre [OF layout corresponds inc] show ?thesis .
-qed
-
-(*
-subsection {* The compilation of @{text "Dec n e"} *}
-*)
-
-text {*
- The lemmas in this section lead to the correctness of the compilation
- of @{text "Dec n e"} instruction using the same techniques as
- @{text "Inc n"}.
-*}
-
-type_synonym dec_inv_t = "(nat * nat list) \<Rightarrow> t_conf \<Rightarrow> block list \<Rightarrow> bool"
-
-fun dec_first_on_right_moving :: "nat \<Rightarrow> dec_inv_t"
- where
- "dec_first_on_right_moving n (as, lm) (s, l, r) ires =
- (\<exists> lm1 lm2 m ml mr rn. lm = lm1 @ [m] @ lm2 \<and>
- ml + mr = Suc m \<and> length lm1 = n \<and> ml > 0 \<and> m > 0 \<and>
- (if lm1 = [] then l = Oc\<^bsup>ml\<^esup> @ Bk # Bk # ires
- else l = (Oc\<^bsup>ml\<^esup>) @ [Bk] @ <rev lm1> @ Bk # Bk # ires) \<and>
- ((r = (Oc\<^bsup>mr\<^esup>) @ [Bk] @ <lm2> @ (Bk\<^bsup>rn\<^esup>)) \<or> (r = (Oc\<^bsup>mr\<^esup>) \<and> lm2 = [])))"
-
-fun dec_on_right_moving :: "dec_inv_t"
- where
- "dec_on_right_moving (as, lm) (s, l, r) ires =
- (\<exists> lm1 lm2 m ml mr rn. lm = lm1 @ [m] @ lm2 \<and>
- ml + mr = Suc (Suc m) \<and>
- (if lm1 = [] then l = Oc\<^bsup>ml\<^esup> @ Bk # Bk # ires
- else l = (Oc\<^bsup>ml\<^esup>) @ [Bk] @ <rev lm1> @ Bk # Bk # ires) \<and>
- ((r = (Oc\<^bsup>mr\<^esup>) @ [Bk] @ <lm2> @ (Bk\<^bsup>rn\<^esup>)) \<or> (r = (Oc\<^bsup>mr\<^esup>) \<and> lm2 = [])))"
-
-fun dec_after_clear :: "dec_inv_t"
- where
- "dec_after_clear (as, lm) (s, l, r) ires =
- (\<exists> lm1 lm2 m ml mr rn. lm = lm1 @ [m] @ lm2 \<and>
- ml + mr = Suc m \<and> ml = Suc m \<and> r \<noteq> [] \<and> r \<noteq> [] \<and>
- (if lm1 = [] then l = Oc\<^bsup>ml\<^esup> @ Bk # Bk # ires
- else l = (Oc\<^bsup>ml \<^esup>) @ [Bk] @ <rev lm1> @ Bk # Bk # ires) \<and>
- (tl r = Bk # <lm2> @ (Bk\<^bsup>rn\<^esup>) \<or> tl r = [] \<and> lm2 = []))"
-
-fun dec_after_write :: "dec_inv_t"
- where
- "dec_after_write (as, lm) (s, l, r) ires =
- (\<exists> lm1 lm2 m ml mr rn. lm = lm1 @ [m] @ lm2 \<and>
- ml + mr = Suc m \<and> ml = Suc m \<and> lm2 \<noteq> [] \<and>
- (if lm1 = [] then l = Bk # Oc\<^bsup>ml\<^esup> @ Bk # Bk # ires
- else l = Bk # (Oc\<^bsup>ml \<^esup>) @ [Bk] @ <rev lm1> @ Bk # Bk # ires) \<and>
- tl r = <lm2> @ (Bk\<^bsup>rn\<^esup>))"
-
-fun dec_right_move :: "dec_inv_t"
- where
- "dec_right_move (as, lm) (s, l, r) ires =
- (\<exists> lm1 lm2 m ml mr rn. lm = lm1 @ [m] @ lm2
- \<and> ml = Suc m \<and> mr = (0::nat) \<and>
- (if lm1 = [] then l = Bk # Oc\<^bsup>ml\<^esup> @ Bk # Bk # ires
- else l = Bk # Oc\<^bsup>ml\<^esup>@ [Bk] @ <rev lm1> @ Bk # Bk # ires)
- \<and> (r = Bk # <lm2> @ Bk\<^bsup>rn\<^esup>\<or> r = [] \<and> lm2 = []))"
-
-fun dec_check_right_move :: "dec_inv_t"
- where
- "dec_check_right_move (as, lm) (s, l, r) ires =
- (\<exists> lm1 lm2 m ml mr rn. lm = lm1 @ [m] @ lm2 \<and>
- ml = Suc m \<and> mr = (0::nat) \<and>
- (if lm1 = [] then l = Bk # Bk # Oc\<^bsup>ml\<^esup> @ Bk # Bk # ires
- else l = Bk # Bk # Oc\<^bsup>ml \<^esup>@ [Bk] @ <rev lm1> @ Bk # Bk # ires) \<and>
- r = <lm2> @ Bk\<^bsup>rn\<^esup>)"
-
-fun dec_left_move :: "dec_inv_t"
- where
- "dec_left_move (as, lm) (s, l, r) ires =
- (\<exists> lm1 m rn. (lm::nat list) = lm1 @ [m::nat] \<and>
- rn > 0 \<and>
- (if lm1 = [] then l = Bk # Oc\<^bsup>Suc m\<^esup> @ Bk # Bk # ires
- else l = Bk # Oc\<^bsup>Suc m\<^esup> @ Bk # <rev lm1> @ Bk # Bk # ires) \<and> r = Bk\<^bsup>rn\<^esup>)"
-
-declare
- dec_on_right_moving.simps[simp del] dec_after_clear.simps[simp del]
- dec_after_write.simps[simp del] dec_left_move.simps[simp del]
- dec_check_right_move.simps[simp del] dec_right_move.simps[simp del]
- dec_first_on_right_moving.simps[simp del]
-
-fun inv_locate_n_b :: "inc_inv_t"
- where
- "inv_locate_n_b (as, lm) (s, l, r) ires=
- (\<exists> lm1 lm2 tn m ml mr rn. lm @ 0\<^bsup>tn\<^esup> = lm1 @ [m] @ lm2 \<and>
- length lm1 = s \<and> m + 1 = ml + mr \<and>
- ml = 1 \<and> tn = s + 1 - length lm \<and>
- (if lm1 = [] then l = Oc\<^bsup>ml\<^esup> @ Bk # Bk # ires
- else l = Oc\<^bsup>ml\<^esup>@Bk#<rev lm1>@Bk#Bk#ires) \<and>
- (r = (Oc\<^bsup>mr\<^esup>) @ [Bk] @ <lm2>@ (Bk\<^bsup>rn\<^esup>) \<or> (lm2 = [] \<and> r = (Oc\<^bsup>mr\<^esup>)))
- )"
-
-fun dec_inv_1 :: "layout \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> dec_inv_t"
- where
- "dec_inv_1 ly n e (as, am) (s, l, r) ires =
- (let ss = start_of ly as in
- let am' = abc_lm_s am n (abc_lm_v am n - Suc 0) in
- let am'' = abc_lm_s am n (abc_lm_v am n) in
- if s = start_of ly e then inv_stop (as, am'') (s, l, r) ires
- else if s = ss then False
- else if ss \<le> s \<and> s < ss + 2*n then
- if (s - ss) mod 2 = 0 then
- inv_locate_a (as, am) ((s - ss) div 2, l, r) ires
- \<or> inv_locate_a (as, am'') ((s - ss) div 2, l, r) ires
- else
- inv_locate_b (as, am) ((s - ss) div 2, l, r) ires
- \<or> inv_locate_b (as, am'') ((s - ss) div 2, l, r) ires
- else if s = ss + 2 * n then
- inv_locate_a (as, am) (n, l, r) ires
- \<or> inv_locate_a (as, am'') (n, l, r) ires
- else if s = ss + 2 * n + 1 then
- inv_locate_b (as, am) (n, l, r) ires
- else if s = ss + 2 * n + 13 then
- inv_on_left_moving (as, am'') (s, l, r) ires
- else if s = ss + 2 * n + 14 then
- inv_check_left_moving (as, am'') (s, l, r) ires
- else if s = ss + 2 * n + 15 then
- inv_after_left_moving (as, am'') (s, l, r) ires
- else False)"
-
-fun dec_inv_2 :: "layout \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> dec_inv_t"
- where
- "dec_inv_2 ly n e (as, am) (s, l, r) ires =
- (let ss = start_of ly as in
- let am' = abc_lm_s am n (abc_lm_v am n - Suc 0) in
- let am'' = abc_lm_s am n (abc_lm_v am n) in
- if s = 0 then False
- else if s = ss then False
- else if ss \<le> s \<and> s < ss + 2*n then
- if (s - ss) mod 2 = 0 then
- inv_locate_a (as, am) ((s - ss) div 2, l, r) ires
- else inv_locate_b (as, am) ((s - ss) div 2, l, r) ires
- else if s = ss + 2 * n then
- inv_locate_a (as, am) (n, l, r) ires
- else if s = ss + 2 * n + 1 then
- inv_locate_n_b (as, am) (n, l, r) ires
- else if s = ss + 2 * n + 2 then
- dec_first_on_right_moving n (as, am'') (s, l, r) ires
- else if s = ss + 2 * n + 3 then
- dec_after_clear (as, am') (s, l, r) ires
- else if s = ss + 2 * n + 4 then
- dec_right_move (as, am') (s, l, r) ires
- else if s = ss + 2 * n + 5 then
- dec_check_right_move (as, am') (s, l, r) ires
- else if s = ss + 2 * n + 6 then
- dec_left_move (as, am') (s, l, r) ires
- else if s = ss + 2 * n + 7 then
- dec_after_write (as, am') (s, l, r) ires
- else if s = ss + 2 * n + 8 then
- dec_on_right_moving (as, am') (s, l, r) ires
- else if s = ss + 2 * n + 9 then
- dec_after_clear (as, am') (s, l, r) ires
- else if s = ss + 2 * n + 10 then
- inv_on_left_moving (as, am') (s, l, r) ires
- else if s = ss + 2 * n + 11 then
- inv_check_left_moving (as, am') (s, l, r) ires
- else if s = ss + 2 * n + 12 then
- inv_after_left_moving (as, am') (s, l, r) ires
- else if s = ss + 2 * n + 16 then
- inv_stop (as, am') (s, l, r) ires
- else False)"
-
-(*begin: dec_fetch lemmas*)
-
-lemma dec_fetch_locate_a_o:
- "\<lbrakk>start_of ly as \<le> a;
- a < start_of ly as + 2 * n; start_of ly as > 0;
- a - start_of ly as = 2 * q\<rbrakk>
- \<Longrightarrow> fetch (ci (layout_of aprog)
- (start_of ly as) (Dec n e)) (Suc (2 * q)) Oc = (R, a + 1)"
-apply(auto simp: ci.simps findnth.simps fetch.simps
- nth_of.simps tshift.simps nth_append Suc_pre)
-apply(subgoal_tac "(findnth n ! Suc (4 * q)) =
- findnth (Suc q) ! (4 * q + 1)")
-apply(simp add: findnth.simps nth_append)
-apply(subgoal_tac " findnth n !(4 * q + 1) =
- findnth (Suc q) ! (4 * q + 1)", simp)
-apply(rule_tac findnth_nth, auto)
-done
-
-lemma dec_fetch_locate_a_b:
- "\<lbrakk>start_of ly as \<le> a;
- a < start_of ly as + 2 * n;
- start_of ly as > 0;
- a - start_of ly as = 2 * q\<rbrakk>
- \<Longrightarrow> fetch (ci (layout_of aprog) (start_of ly as) (Dec n e))
- (Suc (2 * q)) Bk = (W1, a)"
-apply(auto simp: ci.simps findnth.simps fetch.simps
- nth_of.simps tshift.simps nth_append)
-apply(subgoal_tac "(findnth n ! (4 * q)) =
- findnth (Suc q) ! (4 * q )")
-apply(simp add: findnth.simps nth_append)
-apply(subgoal_tac " findnth n !(4 * q + 0) =
- findnth (Suc q) ! (4 * q + 0)", simp)
-apply(rule_tac findnth_nth, auto)
-done
-
-lemma dec_fetch_locate_b_o:
- "\<lbrakk>start_of ly as \<le> a;
- a < start_of ly as + 2 * n;
- (a - start_of ly as) mod 2 = Suc 0;
- start_of ly as> 0\<rbrakk>
- \<Longrightarrow> fetch (ci (layout_of aprog) (start_of ly as) (Dec n e))
- (Suc (a - start_of ly as)) Oc = (R, a)"
-apply(auto simp: ci.simps findnth.simps fetch.simps
- nth_of.simps tshift.simps nth_append)
-apply(subgoal_tac "\<exists> q. (a - start_of ly as) = 2 * q + 1", auto)
-apply(subgoal_tac "(findnth n ! Suc (Suc (Suc (4 * q)))) =
- findnth (Suc q) ! (4 * q + 3)")
-apply(simp add: findnth.simps nth_append)
-apply(subgoal_tac " findnth n ! (4 * q + 3) =
- findnth (Suc q) ! (4 * q + 3)", simp add: add3_Suc)
-apply(rule_tac findnth_nth, auto)
-done
-
-lemma dec_fetch_locate_b_b:
- "\<lbrakk>\<not> a < start_of ly as;
- a < start_of ly as + 2 * n;
- (a - start_of ly as) mod 2 = Suc 0;
- start_of ly as > 0\<rbrakk>
- \<Longrightarrow> fetch (ci (layout_of aprog) (start_of ly as) (Dec n e))
- (Suc (a - start_of ly as)) Bk = (R, a + 1)"
-apply(auto simp: ci.simps findnth.simps fetch.simps
- nth_of.simps tshift.simps nth_append)
-apply(subgoal_tac "\<exists> q. (a - start_of ly as) = 2 * q + 1", auto)
-apply(subgoal_tac "(findnth n ! Suc ((Suc (4 * q)))) =
- findnth (Suc q) ! (4 * q + 2)")
-apply(simp add: findnth.simps nth_append)
-apply(subgoal_tac " findnth n ! (4 * q + 2) =
- findnth (Suc q) ! (4 * q + 2)", simp)
-apply(rule_tac findnth_nth, auto)
-done
-
-lemma dec_fetch_locate_n_a_o:
- "start_of ly as > 0 \<Longrightarrow> fetch (ci (layout_of aprog)
- (start_of ly as) (Dec n e)) (Suc (2 * n)) Oc
- = (R, start_of ly as + 2*n + 1)"
-apply(auto simp: ci.simps findnth.simps fetch.simps
- nth_of.simps tshift.simps nth_append tdec_b_def)
-done
-
-lemma dec_fetch_locate_n_a_b:
- "start_of ly as > 0 \<Longrightarrow> fetch (ci (layout_of aprog)
- (start_of ly as) (Dec n e)) (Suc (2 * n)) Bk
- = (W1, start_of ly as + 2*n)"
-apply(auto simp: ci.simps findnth.simps fetch.simps
- nth_of.simps tshift.simps nth_append tdec_b_def)
-done
-
-lemma dec_fetch_locate_n_b_o:
- "start_of ly as > 0 \<Longrightarrow>
- fetch (ci (layout_of aprog)
- (start_of ly as) (Dec n e)) (Suc (Suc (2 * n))) Oc
- = (R, start_of ly as + 2*n + 2)"
-apply(auto simp: ci.simps findnth.simps fetch.simps
- nth_of.simps tshift.simps nth_append tdec_b_def)
-done
-
-
-lemma dec_fetch_locate_n_b_b:
- "start_of ly as > 0 \<Longrightarrow>
- fetch (ci (layout_of aprog)
- (start_of ly as) (Dec n e)) (Suc (Suc (2 * n))) Bk
- = (L, start_of ly as + 2*n + 13)"
-apply(auto simp: ci.simps findnth.simps fetch.simps
- nth_of.simps tshift.simps nth_append tdec_b_def)
-done
-
-lemma dec_fetch_first_on_right_move_o:
- "start_of ly as > 0 \<Longrightarrow>
- fetch (ci (layout_of aprog)
- (start_of ly as) (Dec n e)) (Suc (Suc (Suc (2 * n)))) Oc
- = (R, start_of ly as + 2*n + 2)"
-apply(auto simp: ci.simps findnth.simps fetch.simps
- nth_of.simps tshift.simps nth_append tdec_b_def)
-done
-
-lemma dec_fetch_first_on_right_move_b:
- "start_of ly as > 0 \<Longrightarrow>
- fetch (ci (layout_of aprog) (start_of ly as) (Dec n e))
- (Suc (Suc (Suc (2 * n)))) Bk
- = (L, start_of ly as + 2*n + 3)"
-apply(auto simp: ci.simps findnth.simps fetch.simps
- nth_of.simps tshift.simps nth_append tdec_b_def)
-done
-
-lemma [simp]: "fetch x (a + 2 * n) b = fetch x (2 * n + a) b"
-thm arg_cong
-apply(rule_tac x = "a + 2*n" and y = "2*n + a" in arg_cong, simp)
-done
-
-lemma dec_fetch_first_after_clear_o:
- "start_of ly as > 0 \<Longrightarrow> fetch (ci (layout_of aprog)
- (start_of ly as) (Dec n e)) (2 * n + 4) Oc
- = (W0, start_of ly as + 2*n + 3)"
-apply(auto simp: ci.simps findnth.simps tshift.simps
- tdec_b_def add3_Suc)
-apply(subgoal_tac "2*n + 4 = Suc (2*n + 3)", simp only: fetch.simps)
-apply(auto simp: nth_of.simps nth_append)
-done
-
-lemma dec_fetch_first_after_clear_b:
- "start_of ly as > 0 \<Longrightarrow>
- fetch (ci (layout_of aprog)
- (start_of ly as) (Dec n e)) (2 * n + 4) Bk
- = (R, start_of ly as + 2*n + 4)"
-apply(auto simp: ci.simps findnth.simps
- tshift.simps tdec_b_def add3_Suc)
-apply(subgoal_tac "2*n + 4= Suc (2*n + 3)", simp only: fetch.simps)
-apply(auto simp: nth_of.simps nth_append)
-done
-
-lemma dec_fetch_right_move_b:
- "start_of ly as > 0 \<Longrightarrow> fetch (ci (layout_of aprog)
- (start_of ly as) (Dec n e)) (2 * n + 5) Bk
- = (R, start_of ly as + 2*n + 5)"
-apply(auto simp: ci.simps findnth.simps
- tshift.simps tdec_b_def add3_Suc)
-apply(subgoal_tac "2*n + 5= Suc (2*n + 4)", simp only: fetch.simps)
-apply(auto simp: nth_of.simps nth_append)
-done
-
-lemma dec_fetch_check_right_move_b:
- "start_of ly as > 0 \<Longrightarrow>
- fetch (ci (layout_of aprog)
- (start_of ly as) (Dec n e)) (2 * n + 6) Bk
- = (L, start_of ly as + 2*n + 6)"
-apply(auto simp: ci.simps findnth.simps
- tshift.simps tdec_b_def add3_Suc)
-apply(subgoal_tac "2*n + 6 = Suc (2*n + 5)", simp only: fetch.simps)
-apply(auto simp: nth_of.simps nth_append)
-done
-
-lemma dec_fetch_check_right_move_o:
- "start_of ly as > 0 \<Longrightarrow>
- fetch (ci (layout_of aprog) (start_of ly as)
- (Dec n e)) (2 * n + 6) Oc
- = (L, start_of ly as + 2*n + 7)"
-apply(auto simp: ci.simps findnth.simps
- tshift.simps tdec_b_def add3_Suc)
-apply(subgoal_tac "2*n + 6 = Suc (2*n + 5)", simp only: fetch.simps)
-apply(auto simp: nth_of.simps nth_append)
-done
-
-lemma dec_fetch_left_move_b:
- "start_of ly as > 0 \<Longrightarrow>
- fetch (ci (layout_of aprog)
- (start_of ly as) (Dec n e)) (2 * n + 7) Bk
- = (L, start_of ly as + 2*n + 10)"
-apply(auto simp: ci.simps findnth.simps
- tshift.simps tdec_b_def add3_Suc)
-apply(subgoal_tac "2*n + 7 = Suc (2*n + 6)", simp only: fetch.simps)
-apply(auto simp: nth_of.simps nth_append)
-done
-
-lemma dec_fetch_after_write_b:
- "start_of ly as > 0 \<Longrightarrow>
- fetch (ci (layout_of aprog)
- (start_of ly as) (Dec n e)) (2 * n + 8) Bk
- = (W1, start_of ly as + 2*n + 7)"
-apply(auto simp: ci.simps findnth.simps
- tshift.simps tdec_b_def add3_Suc)
-apply(subgoal_tac "2*n + 8 = Suc (2*n + 7)", simp only: fetch.simps)
-apply(auto simp: nth_of.simps nth_append)
-done
-
-lemma dec_fetch_after_write_o:
- "start_of ly as > 0 \<Longrightarrow>
- fetch (ci (layout_of aprog)
- (start_of ly as) (Dec n e)) (2 * n + 8) Oc
- = (R, start_of ly as + 2*n + 8)"
-apply(auto simp: ci.simps findnth.simps
- tshift.simps tdec_b_def add3_Suc)
-apply(subgoal_tac "2*n + 8 = Suc (2*n + 7)", simp only: fetch.simps)
-apply(auto simp: nth_of.simps nth_append)
-done
-
-lemma dec_fetch_on_right_move_b:
- "start_of ly as > 0 \<Longrightarrow>
- fetch (ci (layout_of aprog)
- (start_of ly as) (Dec n e)) (2 * n + 9) Bk
- = (L, start_of ly as + 2*n + 9)"
-apply(auto simp: ci.simps findnth.simps
- tshift.simps tdec_b_def add3_Suc)
-apply(subgoal_tac "2*n + 9 = Suc (2*n + 8)", simp only: fetch.simps)
-apply(auto simp: nth_of.simps nth_append)
-done
-
-lemma dec_fetch_on_right_move_o:
- "start_of ly as > 0 \<Longrightarrow>
- fetch (ci (layout_of aprog)
- (start_of ly as) (Dec n e)) (2 * n + 9) Oc
- = (R, start_of ly as + 2*n + 8)"
-apply(auto simp: ci.simps findnth.simps
- tshift.simps tdec_b_def add3_Suc)
-apply(subgoal_tac "2*n + 9 = Suc (2*n + 8)", simp only: fetch.simps)
-apply(auto simp: nth_of.simps nth_append)
-done
-
-lemma dec_fetch_after_clear_b:
- "start_of ly as > 0 \<Longrightarrow>
- fetch (ci (layout_of aprog)
- (start_of ly as) (Dec n e)) (2 * n + 10) Bk
- = (R, start_of ly as + 2*n + 4)"
-apply(auto simp: ci.simps findnth.simps
- tshift.simps tdec_b_def add3_Suc)
-apply(subgoal_tac "2*n + 10 = Suc (2*n + 9)", simp only: fetch.simps)
-apply(auto simp: nth_of.simps nth_append)
-done
-
-lemma dec_fetch_after_clear_o:
- "start_of ly as > 0 \<Longrightarrow>
- fetch (ci (layout_of aprog)
- (start_of ly as) (Dec n e)) (2 * n + 10) Oc
- = (W0, start_of ly as + 2*n + 9)"
-apply(auto simp: ci.simps findnth.simps
- tshift.simps tdec_b_def add3_Suc)
-apply(subgoal_tac "2*n + 10= Suc (2*n + 9)", simp only: fetch.simps)
-apply(auto simp: nth_of.simps nth_append)
-done
-
-lemma dec_fetch_on_left_move1_o:
- "start_of ly as > 0 \<Longrightarrow>
- fetch (ci (layout_of aprog)
- (start_of ly as) (Dec n e)) (2 * n + 11) Oc
- = (L, start_of ly as + 2*n + 10)"
-apply(auto simp: ci.simps findnth.simps
- tshift.simps tdec_b_def add3_Suc)
-apply(subgoal_tac "2*n + 11= Suc (2*n + 10)", simp only: fetch.simps)
-apply(auto simp: nth_of.simps nth_append)
-done
-
-lemma dec_fetch_on_left_move1_b:
- "start_of ly as > 0 \<Longrightarrow>
- fetch (ci (layout_of aprog)
- (start_of ly as) (Dec n e)) (2 * n + 11) Bk
- = (L, start_of ly as + 2*n + 11)"
-apply(auto simp: ci.simps findnth.simps
- tshift.simps tdec_b_def add3_Suc)
-apply(subgoal_tac "2*n + 11 = Suc (2*n + 10)",
- simp only: fetch.simps)
-apply(auto simp: nth_of.simps nth_append)
-done
-
-lemma dec_fetch_check_left_move1_o:
- "start_of ly as > 0 \<Longrightarrow>
- fetch (ci (layout_of aprog)
- (start_of ly as) (Dec n e)) (2 * n + 12) Oc
- = (L, start_of ly as + 2*n + 10)"
-apply(auto simp: ci.simps findnth.simps
- tshift.simps tdec_b_def add3_Suc)
-apply(subgoal_tac "2*n + 12= Suc (2*n + 11)", simp only: fetch.simps)
-apply(auto simp: nth_of.simps nth_append)
-done
-
-lemma dec_fetch_check_left_move1_b:
- "start_of ly as > 0 \<Longrightarrow>
- fetch (ci (layout_of aprog)
- (start_of ly as) (Dec n e)) (2 * n + 12) Bk
- = (R, start_of ly as + 2*n + 12)"
-apply(auto simp: ci.simps findnth.simps
- tshift.simps tdec_b_def add3_Suc)
-apply(subgoal_tac "2*n + 12 = Suc (2*n + 11)",
- simp only: fetch.simps)
-apply(auto simp: nth_of.simps nth_append)
-done
-
-lemma dec_fetch_after_left_move1_b:
- "start_of ly as > 0 \<Longrightarrow>
- fetch (ci (layout_of aprog)
- (start_of ly as) (Dec n e)) (2 * n + 13) Bk
- = (R, start_of ly as + 2*n + 16)"
-apply(auto simp: ci.simps findnth.simps
- tshift.simps tdec_b_def add3_Suc)
-apply(subgoal_tac "2*n + 13 = Suc (2*n + 12)",
- simp only: fetch.simps)
-apply(auto simp: nth_of.simps nth_append)
-done
-
-lemma dec_fetch_on_left_move2_o:
- "start_of ly as > 0 \<Longrightarrow>
- fetch (ci (layout_of aprog)
- (start_of ly as) (Dec n e)) (2 * n + 14) Oc
- = (L, start_of ly as + 2*n + 13)"
-apply(auto simp: ci.simps findnth.simps
- tshift.simps tdec_b_def add3_Suc)
-apply(subgoal_tac "2*n + 14 = Suc (2*n + 13)",
- simp only: fetch.simps)
-apply(auto simp: nth_of.simps nth_append)
-done
-
-lemma dec_fetch_on_left_move2_b:
- "start_of ly as > 0 \<Longrightarrow>
- fetch (ci (layout_of aprog)
- (start_of ly as) (Dec n e)) (2 * n + 14) Bk
- = (L, start_of ly as + 2*n + 14)"
-apply(auto simp: ci.simps findnth.simps
- tshift.simps tdec_b_def add3_Suc)
-apply(subgoal_tac "2*n + 14 = Suc (2*n + 13)",
- simp only: fetch.simps)
-apply(auto simp: nth_of.simps nth_append)
-done
-
-lemma dec_fetch_check_left_move2_o:
- "start_of ly as > 0 \<Longrightarrow>
- fetch (ci (layout_of aprog)
- (start_of ly as) (Dec n e)) (2 * n + 15) Oc
- = (L, start_of ly as + 2*n + 13)"
-apply(auto simp: ci.simps findnth.simps
- tshift.simps tdec_b_def add3_Suc)
-apply(subgoal_tac "2*n + 15 = Suc (2*n + 14)",
- simp only: fetch.simps)
-apply(auto simp: nth_of.simps nth_append)
-done
-
-lemma dec_fetch_check_left_move2_b:
- "start_of ly as > 0 \<Longrightarrow>
- fetch (ci (layout_of aprog)
- (start_of ly as) (Dec n e)) (2 * n + 15) Bk
- = (R, start_of ly as + 2*n + 15)"
-apply(auto simp: ci.simps findnth.simps
- tshift.simps tdec_b_def add3_Suc)
-apply(subgoal_tac "2*n + 15= Suc (2*n + 14)", simp only: fetch.simps)
-apply(auto simp: nth_of.simps nth_append)
-done
-
-lemma dec_fetch_after_left_move2_b:
- "\<lbrakk>ly = layout_of aprog;
- abc_fetch as aprog = Some (Dec n e);
- start_of ly as > 0\<rbrakk> \<Longrightarrow>
- fetch (ci (layout_of aprog) (start_of ly as)
- (Dec n e)) (2 * n + 16) Bk
- = (R, start_of ly e)"
-apply(auto simp: ci.simps findnth.simps
- tshift.simps tdec_b_def add3_Suc)
-apply(subgoal_tac "2*n + 16 = Suc (2*n + 15)",
- simp only: fetch.simps)
-apply(auto simp: nth_of.simps nth_append)
-done
-
-lemma dec_fetch_next_state:
- "start_of ly as > 0 \<Longrightarrow>
- fetch (ci (layout_of aprog)
- (start_of ly as) (Dec n e)) (2* n + 17) b
- = (Nop, 0)"
-apply(case_tac b)
-apply(auto simp: ci.simps findnth.simps
- tshift.simps tdec_b_def add3_Suc)
-apply(subgoal_tac [!] "2*n + 17 = Suc (2*n + 16)",
- simp_all only: fetch.simps)
-apply(auto simp: nth_of.simps nth_append)
-done
-
-(*End: dec_fetch lemmas*)
-lemmas dec_fetch_simps =
- dec_fetch_locate_a_o dec_fetch_locate_a_b dec_fetch_locate_b_o
- dec_fetch_locate_b_b dec_fetch_locate_n_a_o
- dec_fetch_locate_n_a_b dec_fetch_locate_n_b_o
- dec_fetch_locate_n_b_b dec_fetch_first_on_right_move_o
- dec_fetch_first_on_right_move_b dec_fetch_first_after_clear_b
- dec_fetch_first_after_clear_o dec_fetch_right_move_b
- dec_fetch_on_right_move_b dec_fetch_on_right_move_o
- dec_fetch_after_clear_b dec_fetch_after_clear_o
- dec_fetch_check_right_move_b dec_fetch_check_right_move_o
- dec_fetch_left_move_b dec_fetch_on_left_move1_b
- dec_fetch_on_left_move1_o dec_fetch_check_left_move1_b
- dec_fetch_check_left_move1_o dec_fetch_after_left_move1_b
- dec_fetch_on_left_move2_b dec_fetch_on_left_move2_o
- dec_fetch_check_left_move2_o dec_fetch_check_left_move2_b
- dec_fetch_after_left_move2_b dec_fetch_after_write_b
- dec_fetch_after_write_o dec_fetch_next_state
-
-lemma [simp]:
- "\<lbrakk>start_of ly as \<le> a;
- a < start_of ly as + 2 * n;
- (a - start_of ly as) mod 2 = Suc 0;
- inv_locate_b (as, am) ((a - start_of ly as) div 2, aaa, Bk # xs) ires\<rbrakk>
- \<Longrightarrow> \<not> Suc a < start_of ly as + 2 * n \<longrightarrow>
- inv_locate_a (as, am) (n, Bk # aaa, xs) ires"
-apply(insert locate_b_2_locate_a[of a ly as n am aaa xs], simp)
-done
-
-lemma [simp]:
- "\<lbrakk>start_of ly as \<le> a;
- a < start_of ly as + 2 * n;
- (a - start_of ly as) mod 2 = Suc 0;
- inv_locate_b (as, am) ((a - start_of ly as) div 2, aaa, []) ires\<rbrakk>
- \<Longrightarrow> \<not> Suc a < start_of ly as + 2 * n \<longrightarrow>
- inv_locate_a (as, am) (n, Bk # aaa, []) ires"
-apply(insert locate_b_2_locate_a_B[of a ly as n am aaa], simp)
-done
-
-(*
-lemma [simp]: "a\<^bsup>0\<^esup>=[]"
-apply(simp add: exponent_def)
-done
-*)
-
-lemma exp_ind: "a\<^bsup>Suc b\<^esup> = a\<^bsup>b\<^esup> @ [a]"
-apply(simp only: exponent_def rep_ind)
-done
-
-lemma [simp]:
- "inv_locate_b (as, am) (n, l, Oc # r) ires
- \<Longrightarrow> dec_first_on_right_moving n (as, abc_lm_s am n (abc_lm_v am n))
- (Suc (Suc (start_of ly as + 2 * n)), Oc # l, r) ires"
-apply(simp only: inv_locate_b.simps
- dec_first_on_right_moving.simps in_middle.simps
- abc_lm_s.simps abc_lm_v.simps)
-apply(erule_tac exE)+
-apply(erule conjE)+
-apply(case_tac "n < length am", simp)
-apply(rule_tac x = lm1 in exI, rule_tac x = lm2 in exI,
- rule_tac x = m in exI, simp)
-apply(rule_tac x = "Suc ml" in exI, rule_tac conjI, rule_tac [1-2] impI)
-prefer 3
-apply(rule_tac x = lm1 in exI, rule_tac x = lm2 in exI,
- rule_tac x = m in exI, simp)
-apply(subgoal_tac "Suc n - length am = Suc (n - length am)",
- simp only:exponent_def rep_ind, simp)
-apply(rule_tac x = "Suc ml" in exI, simp_all)
-apply(rule_tac [!] x = "mr - 1" in exI, simp_all)
-apply(case_tac [!] mr, auto)
-done
-
-lemma [simp]:
- "\<lbrakk>inv_locate_b (as, am) (n, l, r) ires; l \<noteq> []\<rbrakk> \<Longrightarrow>
- \<not> inv_on_left_moving_in_middle_B (as, abc_lm_s am n (abc_lm_v am n))
- (s, tl l, hd l # r) ires"
-apply(auto simp: inv_locate_b.simps
- inv_on_left_moving_in_middle_B.simps in_middle.simps)
-apply(case_tac [!] ml, auto split: if_splits)
-done
-
-lemma [simp]: "inv_locate_b (as, am) (n, l, r) ires \<Longrightarrow> l \<noteq> []"
-apply(auto simp: inv_locate_b.simps in_middle.simps split: if_splits)
-done
-
-lemma [simp]: "\<lbrakk>inv_locate_b (as, am) (n, l, Bk # r) ires; n < length am\<rbrakk>
- \<Longrightarrow> inv_on_left_moving_norm (as, am) (s, tl l, hd l # Bk # r) ires"
-apply(simp only: inv_locate_b.simps inv_on_left_moving_norm.simps
- in_middle.simps)
-apply(erule_tac exE)+
-apply(erule_tac conjE)+
-apply(rule_tac x = lm1 in exI, rule_tac x = lm2 in exI,
- rule_tac x = m in exI, simp)
-apply(rule_tac x = "ml - 1" in exI, auto)
-apply(rule_tac [!] x = "Suc mr" in exI)
-apply(case_tac [!] mr, auto)
-done
-
-lemma [simp]: "\<lbrakk>inv_locate_b (as, am) (n, l, Bk # r) ires; \<not> n < length am\<rbrakk>
- \<Longrightarrow> inv_on_left_moving_norm (as, am @
- replicate (n - length am) 0 @ [0]) (s, tl l, hd l # Bk # r) ires"
-apply(simp only: inv_locate_b.simps inv_on_left_moving_norm.simps
- in_middle.simps)
-apply(erule_tac exE)+
-apply(erule_tac conjE)+
-apply(rule_tac x = lm1 in exI, rule_tac x = lm2 in exI,
- rule_tac x = m in exI, simp)
-apply(subgoal_tac "Suc n - length am = Suc (n - length am)", simp only: rep_ind exponent_def, simp_all)
-apply(rule_tac x = "Suc mr" in exI, auto)
-done
-
-lemma inv_locate_b_2_on_left_moving[simp]:
- "\<lbrakk>inv_locate_b (as, am) (n, l, Bk # r) ires\<rbrakk>
- \<Longrightarrow> (l = [] \<longrightarrow> inv_on_left_moving (as,
- abc_lm_s am n (abc_lm_v am n)) (s, [], Bk # Bk # r) ires) \<and>
- (l \<noteq> [] \<longrightarrow> inv_on_left_moving (as,
- abc_lm_s am n (abc_lm_v am n)) (s, tl l, hd l # Bk # r) ires)"
-apply(subgoal_tac "l\<noteq>[]")
-apply(subgoal_tac "\<not> inv_on_left_moving_in_middle_B
- (as, abc_lm_s am n (abc_lm_v am n)) (s, tl l, hd l # Bk # r) ires")
-apply(simp add:inv_on_left_moving.simps
- abc_lm_s.simps abc_lm_v.simps split: if_splits, auto)
-done
-
-lemma [simp]:
- "inv_locate_b (as, am) (n, l, []) ires \<Longrightarrow>
- inv_locate_b (as, am) (n, l, [Bk]) ires"
-apply(auto simp: inv_locate_b.simps in_middle.simps)
-apply(rule_tac x = lm1 in exI, rule_tac x = "[]" in exI,
- rule_tac x = "Suc (length lm1) - length am" in exI,
- rule_tac x = m in exI, simp)
-apply(rule_tac x = ml in exI, rule_tac x = mr in exI)
-apply(auto)
-done
-
-lemma nil_2_nil: "<lm::nat list> = [] \<Longrightarrow> lm = []"
-apply(auto simp: tape_of_nl_abv)
-apply(case_tac lm, simp)
-apply(case_tac list, auto simp: tape_of_nat_list.simps)
-done
-
-lemma inv_locate_b_2_on_left_moving_b[simp]:
- "inv_locate_b (as, am) (n, l, []) ires
- \<Longrightarrow> (l = [] \<longrightarrow> inv_on_left_moving (as,
- abc_lm_s am n (abc_lm_v am n)) (s, [], [Bk]) ires) \<and>
- (l \<noteq> [] \<longrightarrow> inv_on_left_moving (as, abc_lm_s am n
- (abc_lm_v am n)) (s, tl l, [hd l]) ires)"
-apply(insert inv_locate_b_2_on_left_moving[of as am n l "[]" ires s])
-apply(simp only: inv_on_left_moving.simps, simp)
-apply(subgoal_tac "\<not> inv_on_left_moving_in_middle_B
- (as, abc_lm_s am n (abc_lm_v am n)) (s, tl l, [hd l]) ires", simp)
-apply(simp only: inv_on_left_moving_norm.simps)
-apply(erule_tac exE)+
-apply(erule_tac conjE)+
-apply(rule_tac x = lm1 in exI, rule_tac x = lm2 in exI,
- rule_tac x = m in exI, rule_tac x = ml in exI,
- rule_tac x = mr in exI, simp)
-apply(case_tac mr, simp, simp, case_tac nat, auto intro: nil_2_nil)
-done
-
-lemma [simp]:
- "\<lbrakk>dec_first_on_right_moving n (as, am) (s, aaa, Oc # xs) ires\<rbrakk>
- \<Longrightarrow> dec_first_on_right_moving n (as, am) (s', Oc # aaa, xs) ires"
-apply(simp only: dec_first_on_right_moving.simps)
-apply(erule exE)+
-apply(rule_tac x = lm1 in exI, rule_tac x = lm2 in exI,
- rule_tac x = m in exI, simp)
-apply(rule_tac x = "Suc ml" in exI,
- rule_tac x = "mr - 1" in exI, auto)
-apply(case_tac [!] mr, auto)
-done
-
-lemma [simp]:
- "dec_first_on_right_moving n (as, am) (s, l, Bk # xs) ires \<Longrightarrow> l \<noteq> []"
-apply(auto simp: dec_first_on_right_moving.simps split: if_splits)
-done
-
-lemma [elim]:
- "\<lbrakk>\<not> length lm1 < length am;
- am @ replicate (length lm1 - length am) 0 @ [0::nat] =
- lm1 @ m # lm2;
- 0 < m\<rbrakk>
- \<Longrightarrow> RR"
-apply(subgoal_tac "lm2 = []", simp)
-apply(drule_tac length_equal, simp)
-done
-
-lemma [simp]:
- "\<lbrakk>dec_first_on_right_moving n (as,
- abc_lm_s am n (abc_lm_v am n)) (s, l, Bk # xs) ires\<rbrakk>
-\<Longrightarrow> dec_after_clear (as, abc_lm_s am n
- (abc_lm_v am n - Suc 0)) (s', tl l, hd l # Bk # xs) ires"
-apply(simp only: dec_first_on_right_moving.simps
- dec_after_clear.simps abc_lm_s.simps abc_lm_v.simps)
-apply(erule_tac exE)+
-apply(case_tac "n < length am")
-apply(rule_tac x = lm1 in exI, rule_tac x = lm2 in exI,
- rule_tac x = "m - 1" in exI, auto simp: )
-apply(case_tac [!] mr, auto)
-done
-
-lemma [simp]:
- "\<lbrakk>dec_first_on_right_moving n (as,
- abc_lm_s am n (abc_lm_v am n)) (s, l, []) ires\<rbrakk>
-\<Longrightarrow> (l = [] \<longrightarrow> dec_after_clear (as,
- abc_lm_s am n (abc_lm_v am n - Suc 0)) (s', [], [Bk]) ires) \<and>
- (l \<noteq> [] \<longrightarrow> dec_after_clear (as, abc_lm_s am n
- (abc_lm_v am n - Suc 0)) (s', tl l, [hd l]) ires)"
-apply(subgoal_tac "l \<noteq> []",
- simp only: dec_first_on_right_moving.simps
- dec_after_clear.simps abc_lm_s.simps abc_lm_v.simps)
-apply(erule_tac exE)+
-apply(case_tac "n < length am", simp)
-apply(rule_tac x = lm1 in exI, rule_tac x = "m - 1" in exI, auto)
-apply(case_tac [1-2] mr, auto)
-apply(case_tac [1-2] m, auto simp: dec_first_on_right_moving.simps split: if_splits)
-done
-
-lemma [simp]: "\<lbrakk>dec_after_clear (as, am) (s, l, Oc # r) ires\<rbrakk>
- \<Longrightarrow> dec_after_clear (as, am) (s', l, Bk # r) ires"
-apply(auto simp: dec_after_clear.simps)
-done
-
-lemma [simp]: "\<lbrakk>dec_after_clear (as, am) (s, l, Bk # r) ires\<rbrakk>
- \<Longrightarrow> dec_right_move (as, am) (s', Bk # l, r) ires"
-apply(auto simp: dec_after_clear.simps dec_right_move.simps split: if_splits)
-done
-
-lemma [simp]: "\<lbrakk>dec_after_clear (as, am) (s, l, []) ires\<rbrakk>
- \<Longrightarrow> dec_right_move (as, am) (s', Bk # l, []) ires"
-apply(auto simp: dec_after_clear.simps dec_right_move.simps )
-done
-
-lemma [simp]: "\<exists>rn. a::block\<^bsup>rn\<^esup> = []"
-apply(rule_tac x = 0 in exI, simp)
-done
-
-lemma [simp]: "\<lbrakk>dec_after_clear (as, am) (s, l, []) ires\<rbrakk>
- \<Longrightarrow> dec_right_move (as, am) (s', Bk # l, [Bk]) ires"
-apply(auto simp: dec_after_clear.simps dec_right_move.simps split: if_splits)
-done
-
-lemma [simp]:"dec_right_move (as, am) (s, l, Oc # r) ires = False"
-apply(auto simp: dec_right_move.simps)
-done
-
-lemma dec_right_move_2_check_right_move[simp]:
- "\<lbrakk>dec_right_move (as, am) (s, l, Bk # r) ires\<rbrakk>
- \<Longrightarrow> dec_check_right_move (as, am) (s', Bk # l, r) ires"
-apply(auto simp: dec_right_move.simps dec_check_right_move.simps split: if_splits)
-done
-
-lemma [simp]:
- "dec_right_move (as, am) (s, l, []) ires=
- dec_right_move (as, am) (s, l, [Bk]) ires"
-apply(simp add: dec_right_move.simps)
-apply(rule_tac iffI)
-apply(erule_tac [!] exE)+
-apply(erule_tac [2] exE)
-apply(rule_tac [!] x = lm1 in exI, rule_tac x = "[]" in exI,
- rule_tac [!] x = m in exI, auto)
-apply(auto intro: nil_2_nil)
-done
-
-lemma [simp]: "\<lbrakk>dec_right_move (as, am) (s, l, []) ires\<rbrakk>
- \<Longrightarrow> dec_check_right_move (as, am) (s, Bk # l, []) ires"
-apply(insert dec_right_move_2_check_right_move[of as am s l "[]" s'],
- simp)
-done
-
-lemma [simp]: "dec_check_right_move (as, am) (s, l, r) ires\<Longrightarrow> l \<noteq> []"
-apply(auto simp: dec_check_right_move.simps split: if_splits)
-done
-
-lemma [simp]: "\<lbrakk>dec_check_right_move (as, am) (s, l, Oc # r) ires\<rbrakk>
- \<Longrightarrow> dec_after_write (as, am) (s', tl l, hd l # Oc # r) ires"
-apply(auto simp: dec_check_right_move.simps dec_after_write.simps)
-apply(rule_tac x = lm1 in exI, rule_tac x = lm2 in exI,
- rule_tac x = m in exI, auto)
-done
-
-lemma [simp]: "\<lbrakk>dec_check_right_move (as, am) (s, l, Bk # r) ires\<rbrakk>
- \<Longrightarrow> dec_left_move (as, am) (s', tl l, hd l # Bk # r) ires"
-apply(auto simp: dec_check_right_move.simps
- dec_left_move.simps inv_after_move.simps)
-apply(rule_tac x = lm1 in exI, rule_tac x = m in exI, auto)
-apply(auto intro: BkCons_nil nil_2_nil dest: BkCons_nil)
-apply(rule_tac x = "Suc rn" in exI)
-apply(auto intro: BkCons_nil nil_2_nil dest: BkCons_nil)
-done
-
-lemma [simp]: "\<lbrakk>dec_check_right_move (as, am) (s, l, []) ires\<rbrakk>
- \<Longrightarrow> dec_left_move (as, am) (s', tl l, [hd l]) ires"
-apply(auto simp: dec_check_right_move.simps
- dec_left_move.simps inv_after_move.simps)
-apply(rule_tac x = lm1 in exI, rule_tac x = m in exI, auto)
-apply(auto intro: BkCons_nil nil_2_nil dest: BkCons_nil)
-done
-
-lemma [simp]: "dec_left_move (as, am) (s, aaa, Oc # xs) ires = False"
-apply(auto simp: dec_left_move.simps inv_after_move.simps)
-apply(case_tac [!] rn, auto)
-done
-
-lemma [simp]: "dec_left_move (as, am) (s, l, r) ires
- \<Longrightarrow> l \<noteq> []"
-apply(auto simp: dec_left_move.simps split: if_splits)
-done
-
-lemma tape_of_nl_abv_cons_ex[simp]:
- "\<exists>lna. Oc # Oc\<^bsup>m\<^esup> @ Bk # <rev lm1> @ Bk\<^bsup>ln\<^esup> = <m # rev lm1> @ Bk\<^bsup>lna\<^esup>"
-apply(case_tac "lm1=[]", auto simp: tape_of_nl_abv
- tape_of_nat_list.simps)
-apply(rule_tac x = "ln" in exI, simp)
-apply(simp add: tape_of_nat_list_cons exponent_def)
-done
-
-(*
-lemma [simp]: "inv_on_left_moving_in_middle_B (as, lm1 @ [m])
- (s', Oc # Oc\<^bsup>m\<^esup> @ Bk # <rev lm1> @ Bk\<^bsup>ln\<^esup>, Bk # Bk\<^bsup>rn\<^esup>) ires"
-apply(simp only: inv_on_left_moving_in_middle_B.simps)
-apply(rule_tac x = "lm1 @ [m]" in exI, rule_tac x = "[]" in exI, auto)
-done
-*)
-lemma [simp]: "inv_on_left_moving_in_middle_B (as, [m])
- (s', Oc # Oc\<^bsup>m\<^esup> @ Bk # Bk # ires, Bk # Bk\<^bsup>rn\<^esup>) ires"
-apply(simp add: inv_on_left_moving_in_middle_B.simps)
-apply(rule_tac x = "[m]" in exI, simp, auto simp: tape_of_nat_def)
-done
-
-lemma [simp]: "inv_on_left_moving_in_middle_B (as, [m])
- (s', Oc # Oc\<^bsup>m\<^esup> @ Bk # Bk # ires, [Bk]) ires"
-apply(simp add: inv_on_left_moving_in_middle_B.simps)
-apply(rule_tac x = "[m]" in exI, simp, auto simp: tape_of_nat_def)
-done
-
-lemma [simp]: "lm1 \<noteq> [] \<Longrightarrow>
- inv_on_left_moving_in_middle_B (as, lm1 @ [m]) (s',
- Oc # Oc\<^bsup>m\<^esup> @ Bk # <rev lm1> @ Bk # Bk # ires, Bk # Bk\<^bsup>rn\<^esup>) ires"
-apply(simp only: inv_on_left_moving_in_middle_B.simps)
-apply(rule_tac x = "lm1 @ [m ]" in exI, rule_tac x = "[]" in exI, simp, auto)
-done
-
-lemma [simp]: "lm1 \<noteq> [] \<Longrightarrow>
- inv_on_left_moving_in_middle_B (as, lm1 @ [m]) (s',
- Oc # Oc\<^bsup>m\<^esup> @ Bk # <rev lm1> @ Bk # Bk # ires, [Bk]) ires"
-apply(simp only: inv_on_left_moving_in_middle_B.simps)
-apply(rule_tac x = "lm1 @ [m ]" in exI, rule_tac x = "[]" in exI, simp, auto)
-done
-
-lemma [simp]: "dec_left_move (as, am) (s, l, Bk # r) ires
- \<Longrightarrow> inv_on_left_moving (as, am) (s', tl l, hd l # Bk # r) ires"
-apply(auto simp: dec_left_move.simps inv_on_left_moving.simps split: if_splits)
-done
-
-(*
-lemma [simp]: "inv_on_left_moving_in_middle_B (as, lm1 @ [m])
- (s', Oc # Oc\<^bsup>m\<^esup> @ Bk # <rev lm1> @ Bk\<^bsup>ln\<^esup>, [Bk]) ires"
-apply(auto simp: inv_on_left_moving_in_middle_B.simps)
-apply(rule_tac x = "lm1 @ [m]" in exI, rule_tac x = "[]" in exI, auto)
-done
-*)
-
-lemma [simp]: "dec_left_move (as, am) (s, l, []) ires
- \<Longrightarrow> inv_on_left_moving (as, am) (s', tl l, [hd l]) ires"
-apply(auto simp: dec_left_move.simps inv_on_left_moving.simps split: if_splits)
-done
-
-lemma [simp]: "dec_after_write (as, am) (s, l, Oc # r) ires
- \<Longrightarrow> dec_on_right_moving (as, am) (s', Oc # l, r) ires"
-apply(auto simp: dec_after_write.simps dec_on_right_moving.simps)
-apply(rule_tac x = "lm1 @ [m]" in exI, rule_tac x = "tl lm2" in exI,
- rule_tac x = "hd lm2" in exI, simp)
-apply(rule_tac x = "Suc 0" in exI,rule_tac x = "Suc (hd lm2)" in exI)
-apply(case_tac lm2, simp, simp)
-apply(case_tac "list = []",
- auto simp: tape_of_nl_abv tape_of_nat_list.simps split: if_splits )
-apply(case_tac rn, auto)
-apply(case_tac "rev lm1", simp, simp add: tape_of_nat_list.simps)
-apply(case_tac rn, auto)
-apply(case_tac list, simp_all add: tape_of_nat_list.simps, auto)
-apply(case_tac "rev lm1", simp, simp add: tape_of_nat_list.simps)
-apply(case_tac list, simp_all add: tape_of_nat_list.simps, auto)
-done
-
-lemma [simp]: "dec_after_write (as, am) (s, l, Bk # r) ires
- \<Longrightarrow> dec_after_write (as, am) (s', l, Oc # r) ires"
-apply(auto simp: dec_after_write.simps)
-done
-
-lemma [simp]: "dec_after_write (as, am) (s, aaa, []) ires
- \<Longrightarrow> dec_after_write (as, am) (s', aaa, [Oc]) ires"
-apply(auto simp: dec_after_write.simps)
-done
-
-lemma [simp]: "dec_on_right_moving (as, am) (s, l, Oc # r) ires
- \<Longrightarrow> dec_on_right_moving (as, am) (s', Oc # l, r) ires"
-apply(simp only: dec_on_right_moving.simps)
-apply(erule_tac exE)+
-apply(erule conjE)+
-apply(rule_tac x = lm1 in exI, rule_tac x = lm2 in exI,
- rule_tac x = "m" in exI, rule_tac x = "Suc ml" in exI,
- rule_tac x = "mr - 1" in exI, simp)
-apply(case_tac mr, auto)
-done
-
-lemma [simp]: "dec_on_right_moving (as, am) (s, l, r) ires\<Longrightarrow> l \<noteq> []"
-apply(auto simp: dec_on_right_moving.simps split: if_splits)
-done
-
-lemma [simp]: "dec_on_right_moving (as, am) (s, l, Bk # r) ires
- \<Longrightarrow> dec_after_clear (as, am) (s', tl l, hd l # Bk # r) ires"
-apply(auto simp: dec_on_right_moving.simps dec_after_clear.simps)
-apply(case_tac [!] mr, auto split: if_splits)
-done
-
-lemma [simp]: "dec_on_right_moving (as, am) (s, l, []) ires
- \<Longrightarrow> dec_after_clear (as, am) (s', tl l, [hd l]) ires"
-apply(auto simp: dec_on_right_moving.simps dec_after_clear.simps)
-apply(case_tac mr, simp_all split: if_splits)
-apply(rule_tac x = lm1 in exI, simp)
-done
-
-lemma start_of_le: "a < b \<Longrightarrow> start_of ly a \<le> start_of ly b"
-proof(induct b arbitrary: a, simp, case_tac "a = b", simp)
- fix b a
- show "start_of ly b \<le> start_of ly (Suc b)"
- apply(case_tac "b::nat",
- simp add: start_of.simps, simp add: start_of.simps)
- done
-next
- fix b a
- assume h1: "\<And>a. a < b \<Longrightarrow> start_of ly a \<le> start_of ly b"
- "a < Suc b" "a \<noteq> b"
- hence "a < b"
- by(simp)
- from h1 and this have h2: "start_of ly a \<le> start_of ly b"
- by(drule_tac h1, simp)
- from h2 show "start_of ly a \<le> start_of ly (Suc b)"
- proof -
- have "start_of ly b \<le> start_of ly (Suc b)"
- apply(case_tac "b::nat",
- simp add: start_of.simps, simp add: start_of.simps)
- done
- from h2 and this show "start_of ly a \<le> start_of ly (Suc b)"
- by simp
- qed
-qed
-
-lemma start_of_dec_length[simp]:
- "\<lbrakk>abc_fetch a aprog = Some (Dec n e)\<rbrakk> \<Longrightarrow>
- start_of (layout_of aprog) (Suc a)
- = start_of (layout_of aprog) a + 2*n + 16"
-apply(case_tac a, auto simp: abc_fetch.simps start_of.simps
- layout_of.simps length_of.simps
- split: if_splits)
-done
-
-lemma start_of_ge:
- "\<lbrakk>abc_fetch a aprog = Some (Dec n e); a < e\<rbrakk> \<Longrightarrow>
- start_of (layout_of aprog) e >
- start_of (layout_of aprog) a + 2*n + 15"
-apply(case_tac "e = Suc a",
- simp add: start_of.simps abc_fetch.simps layout_of.simps
- length_of.simps split: if_splits)
-apply(subgoal_tac "Suc a < e", drule_tac a = "Suc a"
- and ly = "layout_of aprog" in start_of_le)
-apply(subgoal_tac "start_of (layout_of aprog) (Suc a)
- = start_of (layout_of aprog) a + 2*n + 16", simp)
-apply(rule_tac start_of_dec_length, simp)
-apply(arith)
-done
-
-lemma starte_not_equal[simp]:
- "\<lbrakk>abc_fetch as aprog = Some (Dec n e); ly = layout_of aprog\<rbrakk>
- \<Longrightarrow> (start_of ly e \<noteq> Suc (Suc (start_of ly as + 2 * n)) \<and>
- start_of ly e \<noteq> start_of ly as + 2 * n + 3 \<and>
- start_of ly e \<noteq> start_of ly as + 2 * n + 4 \<and>
- start_of ly e \<noteq> start_of ly as + 2 * n + 5 \<and>
- start_of ly e \<noteq> start_of ly as + 2 * n + 6 \<and>
- start_of ly e \<noteq> start_of ly as + 2 * n + 7 \<and>
- start_of ly e \<noteq> start_of ly as + 2 * n + 8 \<and>
- start_of ly e \<noteq> start_of ly as + 2 * n + 9 \<and>
- start_of ly e \<noteq> start_of ly as + 2 * n + 10 \<and>
- start_of ly e \<noteq> start_of ly as + 2 * n + 11 \<and>
- start_of ly e \<noteq> start_of ly as + 2 * n + 12 \<and>
- start_of ly e \<noteq> start_of ly as + 2 * n + 13 \<and>
- start_of ly e \<noteq> start_of ly as + 2 * n + 14 \<and>
- start_of ly e \<noteq> start_of ly as + 2 * n + 15)"
-apply(case_tac "e = as", simp)
-apply(case_tac "e < as")
-apply(drule_tac a = e and b = as and ly = ly in start_of_le, simp)
-apply(drule_tac a = as and e = e in start_of_ge, simp, simp)
-done
-
-lemma [simp]: "\<lbrakk>abc_fetch as aprog = Some (Dec n e); ly = layout_of aprog\<rbrakk>
- \<Longrightarrow> (Suc (Suc (start_of ly as + 2 * n)) \<noteq> start_of ly e \<and>
- start_of ly as + 2 * n + 3 \<noteq> start_of ly e \<and>
- start_of ly as + 2 * n + 4 \<noteq> start_of ly e \<and>
- start_of ly as + 2 * n + 5 \<noteq>start_of ly e \<and>
- start_of ly as + 2 * n + 6 \<noteq> start_of ly e \<and>
- start_of ly as + 2 * n + 7 \<noteq> start_of ly e \<and>
- start_of ly as + 2 * n + 8 \<noteq> start_of ly e \<and>
- start_of ly as + 2 * n + 9 \<noteq> start_of ly e \<and>
- start_of ly as + 2 * n + 10 \<noteq> start_of ly e \<and>
- start_of ly as + 2 * n + 11 \<noteq> start_of ly e \<and>
- start_of ly as + 2 * n + 12 \<noteq> start_of ly e \<and>
- start_of ly as + 2 * n + 13 \<noteq> start_of ly e \<and>
- start_of ly as + 2 * n + 14 \<noteq> start_of ly e \<and>
- start_of ly as + 2 * n + 15 \<noteq> start_of ly e)"
-apply(insert starte_not_equal[of as aprog n e ly],
- simp del: starte_not_equal)
-apply(erule_tac conjE)+
-apply(rule_tac conjI, simp del: starte_not_equal)+
-apply(rule not_sym, simp)
-done
-
-lemma [simp]: "start_of (layout_of aprog) as > 0 \<Longrightarrow>
- fetch (ci (layout_of aprog) (start_of (layout_of aprog) as)
- (Dec n as)) (Suc 0) Oc =
- (R, Suc (start_of (layout_of aprog) as))"
-
-apply(auto simp: ci.simps findnth.simps fetch.simps
- nth_of.simps tshift.simps nth_append
- Suc_pre tdec_b_def)
-apply(insert findnth_nth[of 0 n "Suc 0"], simp)
-apply(simp add: findnth.simps)
-done
-
-lemma start_of_inj[simp]:
- "\<lbrakk>abc_fetch as aprog = Some (Dec n e); e \<noteq> as; ly = layout_of aprog\<rbrakk>
- \<Longrightarrow> start_of ly as \<noteq> start_of ly e"
-apply(case_tac "e < as")
-apply(case_tac "as", simp, simp)
-apply(case_tac "e = nat", simp add: start_of.simps
- layout_of.simps length_of.simps)
-apply(subgoal_tac "e < length aprog", simp add: length_of.simps
- split: abc_inst.splits)
-apply(simp add: abc_fetch.simps split: if_splits)
-apply(subgoal_tac "e < nat", drule_tac a = e and b = nat
- and ly =ly in start_of_le, simp)
-apply(subgoal_tac "start_of ly nat < start_of ly (Suc nat)",
- simp, simp add: start_of.simps layout_of.simps)
-apply(subgoal_tac "nat < length aprog", simp)
-apply(case_tac "aprog ! nat", auto simp: length_of.simps)
-apply(simp add: abc_fetch.simps split: if_splits)
-apply(subgoal_tac "e > as", drule_tac start_of_ge, auto)
-done
-
-lemma [simp]: "\<lbrakk>abc_fetch as aprog = Some (Dec n e); e < as\<rbrakk>
- \<Longrightarrow> Suc (start_of (layout_of aprog) e) -
- start_of (layout_of aprog) as = 0"
-apply(frule_tac ly = "layout_of aprog" in start_of_le, simp)
-apply(subgoal_tac "start_of (layout_of aprog) as \<noteq>
- start_of (layout_of aprog) e", arith)
-apply(rule start_of_inj, auto)
-done
-
-lemma [simp]:
- "\<lbrakk>abc_fetch as aprog = Some (Dec n e);
- 0 < start_of (layout_of aprog) as\<rbrakk>
- \<Longrightarrow> (fetch (ci (layout_of aprog) (start_of (layout_of aprog) as)
- (Dec n e)) (Suc (start_of (layout_of aprog) e) -
- start_of (layout_of aprog) as) Oc)
- = (if e = as then (R, start_of (layout_of aprog) as + 1)
- else (Nop, 0))"
-apply(auto split: if_splits)
-apply(case_tac "e < as", simp add: fetch.simps)
-apply(subgoal_tac " e > as")
-apply(drule start_of_ge, simp,
- auto simp: fetch.simps ci_length nth_of.simps)
-apply(subgoal_tac
- "length (ci (layout_of aprog) (start_of (layout_of aprog) as)
- (Dec n e)) div 2= length_of (Dec n e)")
-defer
-apply(simp add: ci_length)
-apply(subgoal_tac
- "length (ci (layout_of aprog) (start_of (layout_of aprog) as)
- (Dec n e)) mod 2 = 0", auto simp: length_of.simps)
-done
-
-lemma [simp]:
- "start_of (layout_of aprog) as > 0 \<Longrightarrow>
- fetch (ci (layout_of aprog) (start_of (layout_of aprog) as)
- (Dec n as)) (Suc 0) Bk
- = (W1, start_of (layout_of aprog) as)"
-apply(auto simp: ci.simps findnth.simps fetch.simps nth_of.simps
- tshift.simps nth_append Suc_pre tdec_b_def)
-apply(insert findnth_nth[of 0 n "0"], simp)
-apply(simp add: findnth.simps)
-done
-
-lemma [simp]:
- "\<lbrakk>abc_fetch as aprog = Some (Dec n e);
- 0 < start_of (layout_of aprog) as\<rbrakk>
-\<Longrightarrow> (fetch (ci (layout_of aprog) (start_of (layout_of aprog) as)
- (Dec n e)) (Suc (start_of (layout_of aprog) e) -
- start_of (layout_of aprog) as) Bk)
- = (if e = as then (W1, start_of (layout_of aprog) as)
- else (Nop, 0))"
-apply(auto split: if_splits)
-apply(case_tac "e < as", simp add: fetch.simps)
-apply(subgoal_tac " e > as")
-apply(drule start_of_ge, simp, auto simp: fetch.simps
- ci_length nth_of.simps)
-apply(subgoal_tac
- "length (ci (layout_of aprog) (start_of (layout_of aprog) as)
- (Dec n e)) div 2= length_of (Dec n e)")
-defer
-apply(simp add: ci_length)
-apply(subgoal_tac
- "length (ci (layout_of aprog) (start_of (layout_of aprog) as)
- (Dec n e)) mod 2 = 0", auto simp: length_of.simps)
-apply(simp add: ci.simps tshift.simps tdec_b_def)
-done
-
-lemma [simp]:
- "inv_stop (as, abc_lm_s am n (abc_lm_v am n)) (s, l, r) ires \<Longrightarrow> l \<noteq> []"
-apply(auto simp: inv_stop.simps)
-done
-
-lemma [simp]:
- "\<lbrakk>abc_fetch as aprog = Some (Dec n e); e \<noteq> as; ly = layout_of aprog\<rbrakk>
- \<Longrightarrow> (\<not> (start_of ly as \<le> start_of ly e \<and>
- start_of ly e < start_of ly as + 2 * n))
- \<and> start_of ly e \<noteq> start_of ly as + 2*n \<and>
- start_of ly e \<noteq> Suc (start_of ly as + 2*n) "
-apply(case_tac "e < as")
-apply(drule_tac ly = ly in start_of_le, simp)
-apply(case_tac n, simp, drule start_of_inj, simp, simp, simp, simp)
-apply(drule_tac start_of_ge, simp, simp)
-done
-
-lemma [simp]:
- "\<lbrakk>abc_fetch as aprog = Some (Dec n e); start_of ly as \<le> s;
- s < start_of ly as + 2 * n; ly = layout_of aprog\<rbrakk>
- \<Longrightarrow> Suc s \<noteq> start_of ly e "
-apply(case_tac "e = as", simp)
-apply(case_tac "e < as")
-apply(drule_tac a = e and b = as and ly = ly in start_of_le, simp)
-apply(drule_tac start_of_ge, auto)
-done
-
-lemma [simp]: "\<lbrakk>abc_fetch as aprog = Some (Dec n e);
- ly = layout_of aprog\<rbrakk>
- \<Longrightarrow> Suc (start_of ly as + 2 * n) \<noteq> start_of ly e"
-apply(case_tac "e = as", simp)
-apply(case_tac "e < as")
-apply(drule_tac a = e and b = as and ly = ly in start_of_le, simp)
-apply(drule_tac start_of_ge, auto)
-done
-
-lemma dec_false_1[simp]:
- "\<lbrakk>abc_lm_v am n = 0; inv_locate_b (as, am) (n, aaa, Oc # xs) ires\<rbrakk>
- \<Longrightarrow> False"
-apply(auto simp: inv_locate_b.simps in_middle.simps exponent_def)
-apply(case_tac "length lm1 \<ge> length am", auto)
-apply(subgoal_tac "lm2 = []", simp, subgoal_tac "m = 0", simp)
-apply(case_tac mr, auto simp: )
-apply(subgoal_tac "Suc (length lm1) - length am =
- Suc (length lm1 - length am)",
- simp add: rep_ind del: replicate.simps, simp)
-apply(drule_tac xs = "am @ replicate (Suc (length lm1) - length am) 0"
- and ys = "lm1 @ m # lm2" in length_equal, simp)
-apply(case_tac mr, auto simp: abc_lm_v.simps)
-apply(case_tac "mr = 0", simp_all add: exponent_def split: if_splits)
-apply(subgoal_tac "Suc (length lm1) - length am =
- Suc (length lm1 - length am)",
- simp add: rep_ind del: replicate.simps, simp)
-done
-
-lemma [simp]:
- "\<lbrakk>inv_locate_b (as, am) (n, aaa, Bk # xs) ires;
- abc_lm_v am n = 0\<rbrakk>
- \<Longrightarrow> inv_on_left_moving (as, abc_lm_s am n 0)
- (s, tl aaa, hd aaa # Bk # xs) ires"
-apply(insert inv_locate_b_2_on_left_moving[of as am n aaa xs ires s], simp)
-done
-
-lemma [simp]:
- "\<lbrakk>abc_lm_v am n = 0; inv_locate_b (as, am) (n, aaa, []) ires\<rbrakk>
- \<Longrightarrow> inv_on_left_moving (as, abc_lm_s am n 0) (s, tl aaa, [hd aaa]) ires"
-apply(insert inv_locate_b_2_on_left_moving_b[of as am n aaa ires s], simp)
-done
-
-lemma [simp]: "\<lbrakk>am ! n = (0::nat); n < length am\<rbrakk> \<Longrightarrow> am[n := 0] = am"
-apply(simp add: list_update_same_conv)
-done
-
-lemma [simp]: "\<lbrakk>abc_lm_v am n = 0;
- inv_locate_b (as, abc_lm_s am n 0) (n, Oc # aaa, xs) ires\<rbrakk>
- \<Longrightarrow> inv_locate_b (as, am) (n, Oc # aaa, xs) ires"
-apply(simp only: inv_locate_b.simps in_middle.simps abc_lm_s.simps
- abc_lm_v.simps)
-apply(erule_tac exE)+
-apply(rule_tac x = lm1 in exI, rule_tac x = lm2 in exI, simp)
-apply(case_tac "n < length am", simp_all)
-apply(erule_tac conjE)+
-apply(rule_tac x = tn in exI, rule_tac x = m in exI, simp)
-apply(rule_tac x = ml in exI, rule_tac x = mr in exI, simp)
-defer
-apply(rule_tac x = "Suc n - length am" in exI, rule_tac x = m in exI)
-apply(subgoal_tac "Suc n - length am = Suc (n - length am)")
-apply(simp add: exponent_def rep_ind del: replicate.simps, auto)
-done
-
-lemma [intro]: "\<lbrakk>abc_lm_v (a # list) 0 = 0\<rbrakk> \<Longrightarrow> a = 0"
-apply(simp add: abc_lm_v.simps split: if_splits)
-done
-
-lemma [simp]:
- "inv_stop (as, abc_lm_s am n 0)
- (start_of (layout_of aprog) e, aaa, Oc # xs) ires
- \<Longrightarrow> inv_locate_a (as, abc_lm_s am n 0) (0, aaa, Oc # xs) ires"
-apply(simp add: inv_locate_a.simps)
-apply(rule disjI1)
-apply(auto simp: inv_stop.simps at_begin_norm.simps)
-done
-
-lemma [simp]:
- "\<lbrakk>abc_lm_v am 0 = 0;
- inv_stop (as, abc_lm_s am 0 0)
- (start_of (layout_of aprog) e, aaa, Oc # xs) ires\<rbrakk> \<Longrightarrow>
- inv_locate_b (as, am) (0, Oc # aaa, xs) ires"
-apply(auto simp: inv_stop.simps inv_locate_b.simps
- in_middle.simps abc_lm_s.simps)
-apply(case_tac "am = []", simp)
-apply(rule_tac x = "[]" in exI, rule_tac x = "Suc 0" in exI,
- rule_tac x = 0 in exI, simp)
-apply(rule_tac x = "Suc 0" in exI, rule_tac x = 0 in exI,
- simp add: tape_of_nl_abv tape_of_nat_list.simps, auto)
-apply(case_tac rn, auto)
-apply(rule_tac x = "tl am" in exI, rule_tac x = 0 in exI,
- rule_tac x = "hd am" in exI, simp)
-apply(rule_tac x = "Suc 0" in exI, rule_tac x = "hd am" in exI, simp)
-apply(case_tac am, simp, simp)
-apply(subgoal_tac "a = 0", case_tac list,
- auto simp: tape_of_nat_list.simps tape_of_nl_abv)
-apply(case_tac rn, auto)
-done
-
-lemma [simp]:
- "\<lbrakk>inv_stop (as, abc_lm_s am n 0)
- (start_of (layout_of aprog) e, aaa, Oc # xs) ires\<rbrakk>
- \<Longrightarrow> inv_locate_b (as, am) (0, Oc # aaa, xs) ires \<or>
- inv_locate_b (as, abc_lm_s am n 0) (0, Oc # aaa, xs) ires"
-apply(simp)
-done
-
-lemma [simp]:
-"\<lbrakk>abc_lm_v am n = 0;
- inv_stop (as, abc_lm_s am n 0)
- (start_of (layout_of aprog) e, aaa, Oc # xs) ires\<rbrakk>
- \<Longrightarrow> \<not> Suc 0 < 2 * n \<longrightarrow> e = as \<longrightarrow>
- inv_locate_b (as, am) (n, Oc # aaa, xs) ires"
-apply(case_tac n, simp, simp)
-done
-
-lemma dec_false2:
- "inv_stop (as, abc_lm_s am n 0)
- (start_of (layout_of aprog) e, aaa, Bk # xs) ires = False"
-apply(auto simp: inv_stop.simps abc_lm_s.simps)
-apply(case_tac "am", simp, case_tac n, simp add: tape_of_nl_abv)
-apply(case_tac list, simp add: tape_of_nat_list.simps )
-apply(simp add: tape_of_nat_list.simps , simp)
-apply(case_tac "list[nat := 0]",
- simp add: tape_of_nat_list.simps tape_of_nl_abv)
-apply(simp add: tape_of_nat_list.simps )
-apply(case_tac "am @ replicate (n - length am) 0 @ [0]", simp)
-apply(case_tac list, auto simp: tape_of_nl_abv
- tape_of_nat_list.simps )
-done
-
-lemma dec_false3:
- "inv_stop (as, abc_lm_s am n 0)
- (start_of (layout_of aprog) e, aaa, []) ires = False"
-apply(auto simp: inv_stop.simps abc_lm_s.simps)
-apply(case_tac "am", case_tac n, auto)
-apply(case_tac n, auto simp: tape_of_nl_abv)
-apply(case_tac "list::nat list",
- simp add: tape_of_nat_list.simps tape_of_nat_list.simps)
-apply(simp add: tape_of_nat_list.simps)
-apply(case_tac "list[nat := 0]",
- simp add: tape_of_nat_list.simps tape_of_nat_list.simps)
-apply(simp add: tape_of_nat_list.simps)
-apply(case_tac "(am @ replicate (n - length am) 0 @ [0])", simp)
-apply(case_tac list, auto simp: tape_of_nat_list.simps)
-done
-
-lemma [simp]:
- "fetch (ci (layout_of aprog)
- (start_of (layout_of aprog) as) (Dec n e)) 0 b = (Nop, 0)"
-by(simp add: fetch.simps)
-
-declare dec_inv_1.simps[simp del]
-
-declare inv_locate_n_b.simps [simp del]
-
-lemma [simp]:
-"\<lbrakk>0 < abc_lm_v am n; 0 < n;
- at_begin_norm (as, am) (n, aaa, Oc # xs) ires\<rbrakk>
- \<Longrightarrow> inv_locate_n_b (as, am) (n, Oc # aaa, xs) ires"
-apply(simp only: at_begin_norm.simps inv_locate_n_b.simps)
-apply(erule_tac exE)+
-apply(rule_tac x = lm1 in exI, simp)
-apply(case_tac "length lm2", simp)
-apply(case_tac rn, simp, simp)
-apply(rule_tac x = "tl lm2" in exI, rule_tac x = "hd lm2" in exI, simp)
-apply(rule conjI)
-apply(case_tac "lm2", simp, simp)
-apply(case_tac "lm2", auto simp: tape_of_nl_abv tape_of_nat_list.simps)
-apply(case_tac [!] "list", auto simp: tape_of_nl_abv tape_of_nat_list.simps)
-apply(case_tac rn, auto)
-done
-lemma [simp]: "(\<exists>rn. Oc # xs = Bk\<^bsup>rn\<^esup>) = False"
-apply(auto)
-apply(case_tac rn, auto simp: )
-done
-
-lemma [simp]:
- "\<lbrakk>0 < abc_lm_v am n; 0 < n;
- at_begin_fst_bwtn (as, am) (n, aaa, Oc # xs) ires\<rbrakk>
- \<Longrightarrow> inv_locate_n_b (as, am) (n, Oc # aaa, xs) ires"
-apply(simp add: at_begin_fst_bwtn.simps inv_locate_n_b.simps )
-done
-
-lemma Suc_minus:"length am + tn = n
- \<Longrightarrow> Suc tn = Suc n - length am "
-apply(arith)
-done
-
-lemma [simp]:
- "\<lbrakk>0 < abc_lm_v am n; 0 < n;
- at_begin_fst_awtn (as, am) (n, aaa, Oc # xs) ires\<rbrakk>
- \<Longrightarrow> inv_locate_n_b (as, am) (n, Oc # aaa, xs) ires"
-apply(simp only: at_begin_fst_awtn.simps inv_locate_n_b.simps )
-apply(erule exE)+
-apply(erule conjE)+
-apply(rule_tac x = lm1 in exI, rule_tac x = "[]" in exI,
- rule_tac x = "Suc tn" in exI, rule_tac x = 0 in exI)
-apply(simp add: exponent_def rep_ind del: replicate.simps)
-apply(rule conjI)+
-apply(auto)
-apply(case_tac [!] rn, auto)
-done
-
-lemma [simp]:
- "\<lbrakk>0 < abc_lm_v am n; 0 < n; inv_locate_a (as, am) (n, aaa, Oc # xs) ires\<rbrakk>
- \<Longrightarrow> inv_locate_n_b (as, am) (n, Oc#aaa, xs) ires"
-apply(auto simp: inv_locate_a.simps)
-done
-
-lemma [simp]:
- "\<lbrakk>inv_locate_n_b (as, am) (n, aaa, Oc # xs) ires\<rbrakk>
- \<Longrightarrow> dec_first_on_right_moving n (as, abc_lm_s am n (abc_lm_v am n))
- (s, Oc # aaa, xs) ires"
-apply(auto simp: inv_locate_n_b.simps dec_first_on_right_moving.simps
- abc_lm_s.simps abc_lm_v.simps)
-apply(rule_tac x = lm1 in exI, rule_tac x = lm2 in exI,
- rule_tac x = m in exI, simp)
-apply(rule_tac x = "Suc (Suc 0)" in exI,
- rule_tac x = "m - 1" in exI, simp)
-apply(case_tac m, auto simp: exponent_def)
-apply(rule_tac x = lm1 in exI, rule_tac x = lm2 in exI,
- rule_tac x = m in exI,
- simp add: Suc_diff_le rep_ind del: replicate.simps)
-apply(rule_tac x = "Suc (Suc 0)" in exI,
- rule_tac x = "m - 1" in exI, simp)
-apply(case_tac m, auto simp: exponent_def)
-apply(rule_tac x = lm1 in exI, rule_tac x = "[]" in exI,
- rule_tac x = m in exI, simp)
-apply(rule_tac x = "Suc (Suc 0)" in exI,
- rule_tac x = "m - 1" in exI, simp)
-apply(case_tac m, auto)
-apply(rule_tac x = lm1 in exI, rule_tac x = lm2 in exI,
- rule_tac x = m in exI,
- simp add: Suc_diff_le rep_ind del: replicate.simps, simp)
-done
-
-lemma dec_false_2:
- "\<lbrakk>0 < abc_lm_v am n; inv_locate_n_b (as, am) (n, aaa, Bk # xs) ires\<rbrakk>
- \<Longrightarrow> False"
-apply(auto simp: inv_locate_n_b.simps abc_lm_v.simps split: if_splits)
-apply(case_tac [!] m, auto)
-done
-
-lemma dec_false_2_b:
- "\<lbrakk>0 < abc_lm_v am n; inv_locate_n_b (as, am)
- (n, aaa, []) ires\<rbrakk> \<Longrightarrow> False"
-apply(auto simp: inv_locate_n_b.simps abc_lm_v.simps split: if_splits)
-apply(case_tac [!] m, auto simp: )
-done
-
-
-(*begin: dec halt1 lemmas*)
-thm abc_inc_stage1.simps
-fun abc_dec_1_stage1:: "t_conf \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat"
- where
- "abc_dec_1_stage1 (s, l, r) ss n =
- (if s > ss \<and> s \<le> ss + 2*n + 1 then 4
- else if s = ss + 2 * n + 13 \<or> s = ss + 2*n + 14 then 3
- else if s = ss + 2*n + 15 then 2
- else 0)"
-
-fun abc_dec_1_stage2:: "t_conf \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat"
- where
- "abc_dec_1_stage2 (s, l, r) ss n =
- (if s \<le> ss + 2 * n + 1 then (ss + 2 * n + 16 - s)
- else if s = ss + 2*n + 13 then length l
- else if s = ss + 2*n + 14 then length l
- else 0)"
-
-fun abc_dec_1_stage3 :: "t_conf \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> block list \<Rightarrow> nat"
- where
- "abc_dec_1_stage3 (s, l, r) ss n ires =
- (if s \<le> ss + 2*n + 1 then
- if (s - ss) mod 2 = 0 then
- if r \<noteq> [] \<and> hd r = Oc then 0 else 1
- else length r
- else if s = ss + 2 * n + 13 then
- if l = Bk # ires \<and> r \<noteq> [] \<and> hd r = Oc then 2
- else 1
- else if s = ss + 2 * n + 14 then
- if r \<noteq> [] \<and> hd r = Oc then 3 else 0
- else 0)"
-
-fun abc_dec_1_measure :: "(t_conf \<times> nat \<times> nat \<times> block list) \<Rightarrow> (nat \<times> nat \<times> nat)"
- where
- "abc_dec_1_measure (c, ss, n, ires) = (abc_dec_1_stage1 c ss n,
- abc_dec_1_stage2 c ss n, abc_dec_1_stage3 c ss n ires)"
-
-definition abc_dec_1_LE ::
- "(((nat \<times> block list \<times> block list) \<times> nat \<times>
- nat \<times> block list) \<times> ((nat \<times> block list \<times> block list) \<times> nat \<times> nat \<times> block list)) set"
- where "abc_dec_1_LE \<equiv> (inv_image lex_triple abc_dec_1_measure)"
-
-lemma wf_dec_le: "wf abc_dec_1_LE"
-by(auto intro:wf_inv_image wf_lex_triple simp:abc_dec_1_LE_def)
-
-declare dec_inv_1.simps[simp del] dec_inv_2.simps[simp del]
-
-lemma [elim]:
- "\<lbrakk>abc_fetch as aprog = Some (Dec n e);
- start_of (layout_of aprog) as < start_of (layout_of aprog) e;
- start_of (layout_of aprog) e \<le>
- Suc (start_of (layout_of aprog) as + 2 * n)\<rbrakk> \<Longrightarrow> False"
-apply(case_tac "e = as", simp)
-apply(case_tac "e < as")
-apply(drule_tac a = e and b = as and ly = "layout_of aprog" in
- start_of_le, simp)
-apply(drule_tac start_of_ge, auto)
-done
-
-lemma [elim]: "\<lbrakk>abc_fetch as aprog = Some (Dec n e);
- start_of (layout_of aprog) e
- = start_of (layout_of aprog) as + 2 * n + 13\<rbrakk>
- \<Longrightarrow> False"
-apply(insert starte_not_equal[of as aprog n e "layout_of aprog"],
- simp)
-done
-
-lemma [elim]: "\<lbrakk>abc_fetch as aprog = Some (Dec n e);
- start_of (layout_of aprog) e =
- start_of (layout_of aprog) as + 2 * n + 14\<rbrakk>
- \<Longrightarrow> False"
-apply(insert starte_not_equal[of as aprog n e "layout_of aprog"],
- simp)
-done
-
-lemma [elim]:
- "\<lbrakk>abc_fetch as aprog = Some (Dec n e);
- start_of (layout_of aprog) as < start_of (layout_of aprog) e;
- start_of (layout_of aprog) e \<le>
- Suc (start_of (layout_of aprog) as + 2 * n)\<rbrakk>
- \<Longrightarrow> False"
-apply(case_tac "e = as", simp)
-apply(case_tac "e < as")
-apply(drule_tac a = e and b = as and ly = "layout_of aprog" in
- start_of_le, simp)
-apply(drule_tac start_of_ge, auto)
-done
-
-lemma [elim]:
- "\<lbrakk>abc_fetch as aprog = Some (Dec n e);
- start_of (layout_of aprog) e =
- start_of (layout_of aprog) as + 2 * n + 13\<rbrakk>
- \<Longrightarrow> False"
-apply(insert starte_not_equal[of as aprog n e "layout_of aprog"],
- simp)
-done
-
-lemma [simp]:
- "abc_fetch as aprog = Some (Dec n e) \<Longrightarrow>
- Suc (start_of (layout_of aprog) as) \<noteq> start_of (layout_of aprog) e"
-apply(case_tac "e = as", simp)
-apply(case_tac "e < as")
-apply(drule_tac a = e and b = as and ly = "(layout_of aprog)" in
- start_of_le, simp)
-apply(drule_tac a = as and e = e in start_of_ge, simp, simp)
-done
-
-lemma [simp]: "inv_on_left_moving (as, am) (s, [], r) ires
- = False"
-apply(simp add: inv_on_left_moving.simps inv_on_left_moving_norm.simps
- inv_on_left_moving_in_middle_B.simps)
-done
-
-lemma [simp]:
- "inv_check_left_moving (as, abc_lm_s am n 0)
- (start_of (layout_of aprog) as + 2 * n + 14, [], Oc # xs) ires
- = False"
-apply(simp add: inv_check_left_moving.simps inv_check_left_moving_in_middle.simps)
-done
-
-lemma dec_inv_stop1_pre:
- "\<lbrakk>abc_fetch as aprog = Some (Dec n e); abc_lm_v am n = 0;
- start_of (layout_of aprog) as > 0\<rbrakk>
- \<Longrightarrow> \<forall>na. \<not> (\<lambda>(s, l, r) (ss, n', ires'). s = start_of (layout_of aprog) e)
- (t_steps (Suc (start_of (layout_of aprog) as), l, r)
- (ci (layout_of aprog) (start_of (layout_of aprog) as)
- (Dec n e), start_of (layout_of aprog) as - Suc 0) na)
- (start_of (layout_of aprog) as, n, ires) \<and>
- dec_inv_1 (layout_of aprog) n e (as, am)
- (t_steps (Suc (start_of (layout_of aprog) as), l, r)
- (ci (layout_of aprog) (start_of (layout_of aprog) as)
- (Dec n e), start_of (layout_of aprog) as - Suc 0) na) ires
- \<longrightarrow> dec_inv_1 (layout_of aprog) n e (as, am)
- (t_steps (Suc (start_of (layout_of aprog) as), l, r)
- (ci (layout_of aprog) (start_of (layout_of aprog) as)
- (Dec n e), start_of (layout_of aprog) as - Suc 0)
- (Suc na)) ires \<and>
- ((t_steps (Suc (start_of (layout_of aprog) as), l, r)
- (ci (layout_of aprog) (start_of (layout_of aprog) as)
- (Dec n e), start_of (layout_of aprog) as - Suc 0) (Suc na),
- start_of (layout_of aprog) as, n, ires),
- t_steps (Suc (start_of (layout_of aprog) as), l, r)
- (ci (layout_of aprog) (start_of (layout_of aprog) as)
- (Dec n e), start_of (layout_of aprog) as - Suc 0) na,
- start_of (layout_of aprog) as, n, ires)
- \<in> abc_dec_1_LE"
-apply(rule allI, rule impI, simp add: t_steps_ind)
-apply(case_tac "(t_steps (Suc (start_of (layout_of aprog) as), l, r)
-(ci (layout_of aprog) (start_of (layout_of aprog) as) (Dec n e),
-start_of (layout_of aprog) as - Suc 0) na)", simp)
-apply(auto split:if_splits simp add:t_step.simps dec_inv_1.simps,
- tactic {* ALLGOALS (resolve_tac [@{thm fetch_intro}]) *})
-apply(simp_all add:dec_fetch_simps new_tape.simps dec_inv_1.simps)
-apply(auto simp add: abc_dec_1_LE_def lex_square_def
- lex_triple_def lex_pair_def
- split: if_splits)
-apply(rule dec_false_1, simp, simp)
-done
-
-lemma dec_inv_stop1:
- "\<lbrakk>ly = layout_of aprog;
- dec_inv_1 ly n e (as, am) (start_of ly as + 1, l, r) ires;
- abc_fetch as aprog = Some (Dec n e); abc_lm_v am n = 0\<rbrakk> \<Longrightarrow>
- (\<exists> stp. (\<lambda> (s', l', r'). s' = start_of ly e \<and>
- dec_inv_1 ly n e (as, am) (s', l' , r') ires)
- (t_steps (start_of ly as + 1, l, r)
- (ci ly (start_of ly as) (Dec n e), start_of ly as - Suc 0) stp))"
-apply(insert halt_lemma2[of abc_dec_1_LE
- "\<lambda> ((s, l, r), ss, n', ires'). s = start_of ly e"
- "(\<lambda> stp. (t_steps (start_of ly as + 1, l, r)
- (ci ly (start_of ly as) (Dec n e), start_of ly as - Suc 0)
- stp, start_of ly as, n, ires))"
- "\<lambda> ((s, l, r), ss, n, ires'). dec_inv_1 ly n e (as, am) (s, l, r) ires'"],
- simp)
-apply(insert wf_dec_le, simp)
-apply(insert dec_inv_stop1_pre[of as aprog n e am l r], simp)
-apply(subgoal_tac "start_of (layout_of aprog) as > 0",
- simp add: t_steps.simps)
-apply(erule_tac exE, rule_tac x = na in exI)
-apply(case_tac
- "(t_steps (Suc (start_of (layout_of aprog) as), l, r)
- (ci (layout_of aprog) (start_of (layout_of aprog) as)
- (Dec n e), start_of (layout_of aprog) as - Suc 0) na)",
- case_tac b, auto)
-apply(rule startof_not0)
-done
-
-(*begin: dec halt2 lemmas*)
-
-lemma [simp]:
- "\<lbrakk>abc_fetch as aprog = Some (Dec n e);
- ly = layout_of aprog\<rbrakk> \<Longrightarrow>
- start_of ly (Suc as) = start_of ly as + 2*n + 16"
-by simp
-
-fun abc_dec_2_stage1 :: "t_conf \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat"
- where
- "abc_dec_2_stage1 (s, l, r) ss n =
- (if s \<le> ss + 2*n + 1 then 7
- else if s = ss + 2*n + 2 then 6
- else if s = ss + 2*n + 3 then 5
- else if s \<ge> ss + 2*n + 4 \<and> s \<le> ss + 2*n + 9 then 4
- else if s = ss + 2*n + 6 then 3
- else if s = ss + 2*n + 10 \<or> s = ss + 2*n + 11 then 2
- else if s = ss + 2*n + 12 then 1
- else 0)"
-
-thm new_tape.simps
-
-fun abc_dec_2_stage2 :: "t_conf \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat"
- where
- "abc_dec_2_stage2 (s, l, r) ss n =
- (if s \<le> ss + 2 * n + 1 then (ss + 2 * n + 16 - s)
- else if s = ss + 2*n + 10 then length l
- else if s = ss + 2*n + 11 then length l
- else if s = ss + 2*n + 4 then length r - 1
- else if s = ss + 2*n + 5 then length r
- else if s = ss + 2*n + 7 then length r - 1
- else if s = ss + 2*n + 8 then
- length r + length (takeWhile (\<lambda> a. a = Oc) l) - 1
- else if s = ss + 2*n + 9 then
- length r + length (takeWhile (\<lambda> a. a = Oc) l) - 1
- else 0)"
-
-fun abc_dec_2_stage3 :: "t_conf \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> block list \<Rightarrow> nat"
- where
- "abc_dec_2_stage3 (s, l, r) ss n ires =
- (if s \<le> ss + 2*n + 1 then
- if (s - ss) mod 2 = 0 then if r \<noteq> [] \<and>
- hd r = Oc then 0 else 1
- else length r
- else if s = ss + 2 * n + 10 then
- if l = Bk # ires \<and> r \<noteq> [] \<and> hd r = Oc then 2
- else 1
- else if s = ss + 2 * n + 11 then
- if r \<noteq> [] \<and> hd r = Oc then 3
- else 0
- else (ss + 2 * n + 16 - s))"
-
-fun abc_dec_2_stage4 :: "t_conf \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat"
- where
- "abc_dec_2_stage4 (s, l, r) ss n =
- (if s = ss + 2*n + 2 then length r
- else if s = ss + 2*n + 8 then length r
- else if s = ss + 2*n + 3 then
- if r \<noteq> [] \<and> hd r = Oc then 1
- else 0
- else if s = ss + 2*n + 7 then
- if r \<noteq> [] \<and> hd r = Oc then 0
- else 1
- else if s = ss + 2*n + 9 then
- if r \<noteq> [] \<and> hd r = Oc then 1
- else 0
- else 0)"
-
-fun abc_dec_2_measure :: "(t_conf \<times> nat \<times> nat \<times> block list) \<Rightarrow>
- (nat \<times> nat \<times> nat \<times> nat)"
- where
- "abc_dec_2_measure (c, ss, n, ires) =
- (abc_dec_2_stage1 c ss n, abc_dec_2_stage2 c ss n,
- abc_dec_2_stage3 c ss n ires, abc_dec_2_stage4 c ss n)"
-
-definition abc_dec_2_LE ::
- "(((nat \<times> block list \<times> block list) \<times> nat \<times> nat \<times> block list) \<times>
- ((nat \<times> block list \<times> block list) \<times> nat \<times> nat \<times> block list)) set"
- where "abc_dec_2_LE \<equiv> (inv_image lex_square abc_dec_2_measure)"
-
-lemma wf_dec_2_le: "wf abc_dec_2_LE"
-by(auto intro:wf_inv_image wf_lex_triple wf_lex_square
- simp:abc_dec_2_LE_def)
-
-lemma [simp]: "dec_after_write (as, am) (s, aa, r) ires
- \<Longrightarrow> takeWhile (\<lambda>a. a = Oc) aa = []"
-apply(simp only : dec_after_write.simps)
-apply(erule exE)+
-apply(erule_tac conjE)+
-apply(case_tac aa, simp)
-apply(case_tac a, simp only: takeWhile.simps , simp, simp split: if_splits)
-done
-
-lemma [simp]:
- "\<lbrakk>dec_on_right_moving (as, lm) (s, aa, []) ires;
- length (takeWhile (\<lambda>a. a = Oc) (tl aa))
- \<noteq> length (takeWhile (\<lambda>a. a = Oc) aa) - Suc 0\<rbrakk>
- \<Longrightarrow> length (takeWhile (\<lambda>a. a = Oc) (tl aa)) <
- length (takeWhile (\<lambda>a. a = Oc) aa) - Suc 0"
-apply(simp only: dec_on_right_moving.simps)
-apply(erule_tac exE)+
-apply(erule_tac conjE)+
-apply(case_tac mr, auto split: if_splits)
-done
-
-lemma [simp]:
- "dec_after_clear (as, abc_lm_s am n (abc_lm_v am n - Suc 0))
- (start_of (layout_of aprog) as + 2 * n + 9, aa, Bk # xs) ires
- \<Longrightarrow> length xs - Suc 0 < length xs +
- length (takeWhile (\<lambda>a. a = Oc) aa)"
-apply(simp only: dec_after_clear.simps)
-apply(erule_tac exE)+
-apply(erule conjE)+
-apply(simp split: if_splits )
-done
-
-lemma [simp]:
- "\<lbrakk>dec_after_clear (as, abc_lm_s am n (abc_lm_v am n - Suc 0))
- (start_of (layout_of aprog) as + 2 * n + 9, aa, []) ires\<rbrakk>
- \<Longrightarrow> Suc 0 < length (takeWhile (\<lambda>a. a = Oc) aa)"
-apply(simp add: dec_after_clear.simps split: if_splits)
-done
-
-lemma [simp]:
- "\<lbrakk>dec_on_right_moving (as, am) (s, aa, Bk # xs) ires;
- Suc (length (takeWhile (\<lambda>a. a = Oc) (tl aa)))
- \<noteq> length (takeWhile (\<lambda>a. a = Oc) aa)\<rbrakk>
- \<Longrightarrow> Suc (length (takeWhile (\<lambda>a. a = Oc) (tl aa)))
- < length (takeWhile (\<lambda>a. a = Oc) aa)"
-apply(simp only: dec_on_right_moving.simps)
-apply(erule exE)+
-apply(erule conjE)+
-apply(case_tac ml, auto split: if_splits )
-done
-
-(*
-lemma abc_dec_2_wf:
- "\<lbrakk>ly = layout_of aprog; dec_inv_2 ly n e (as, am) (start_of ly as + 1, l, r); abc_fetch as aprog = Dec n e; abc_lm_v am n > 0\<rbrakk>
- \<Longrightarrow> \<forall>na. \<not> (\<lambda>(s, l, r) (ss, n'). s = start_of (layout_of aprog) as + 2*n + 16)
- (t_steps (Suc (start_of (layout_of aprog) as), l, r) (ci (layout_of aprog) (start_of (layout_of aprog) as) (Dec n e), start_of (layout_of aprog) as - Suc 0) na)
- (start_of (layout_of aprog) as, n) \<longrightarrow>
- ((t_steps (Suc (start_of (layout_of aprog) as), l, r) (ci (layout_of aprog) (start_of (layout_of aprog) as) (Dec n e), start_of (layout_of aprog) as - Suc 0) (Suc na),
- start_of (layout_of aprog) as, n),
- t_steps (Suc (start_of (layout_of aprog) as), l, r) (ci (layout_of aprog) (start_of (layout_of aprog) as) (Dec n e), start_of (layout_of aprog) as - Suc 0) na,
- start_of (layout_of aprog) as, n)
- \<in> abc_dec_2_LE"
-proof(rule allI, rule impI, simp add: t_steps_ind)
- fix na
- assume h1 :"ly = layout_of aprog" "dec_inv_2 (layout_of aprog) n e (as, am) (Suc (start_of (layout_of aprog) as), l, r)"
- "abc_fetch as aprog = Dec n e" "abc_lm_v am n > 0"
- "\<not> (\<lambda>(s, l, r) (ss, n'). s = start_of (layout_of aprog) as + 2*n + 16)
- (t_steps (Suc (start_of (layout_of aprog) as), l, r) (ci (layout_of aprog) (start_of (layout_of aprog) as) (Dec n e), start_of (layout_of aprog) as - Suc 0) na)
- (start_of (layout_of aprog) as, n)"
- thus "((t_step (t_steps (Suc (start_of (layout_of aprog) as), l, r) (ci (layout_of aprog) (start_of (layout_of aprog) as) (Dec n e), start_of (layout_of aprog) as - Suc 0) na)
- (ci (layout_of aprog) (start_of (layout_of aprog) as) (Dec n e), start_of (layout_of aprog) as - Suc 0),
- start_of (layout_of aprog) as, n),
- t_steps (Suc (start_of (layout_of aprog) as), l, r) (ci (layout_of aprog) (start_of (layout_of aprog) as) (Dec n e), start_of (layout_of aprog) as - Suc 0) na,
- start_of (layout_of aprog) as, n)
- \<in> abc_dec_2_LE"
- proof(insert dec_inv_2_steps[of "layout_of aprog" n e as am "(start_of (layout_of aprog) as + 1, l, r)" aprog na],
- case_tac "(t_steps (start_of (layout_of aprog) as + 1, l, r) (ci (layout_of aprog) (start_of (layout_of aprog) as) (Dec n e), start_of (layout_of aprog) as - Suc 0) na)", case_tac b, simp)
- fix a b aa ba
- assume "dec_inv_2 (layout_of aprog) n e (as, am) (a, aa, ba)" " a \<noteq> start_of (layout_of aprog) as + 2*n + 16" "abc_lm_v am n > 0" "abc_fetch as aprog = Dec n e "
- thus "((t_step (a, aa, ba) (ci (layout_of aprog) (start_of (layout_of aprog) as) (Dec n e), start_of (layout_of aprog) as - Suc 0), start_of (layout_of aprog) as, n), (a, aa, ba),
- start_of (layout_of aprog) as, n)
- \<in> abc_dec_2_LE"
- apply(case_tac "a = 0", auto split:if_splits simp add:t_step.simps dec_inv_2.simps,
- tactic {* ALLGOALS (resolve_tac (thms "fetch_intro")) *})
- apply(simp_all add:dec_fetch_simps new_tape.simps)
- apply(auto simp add: abc_dec_2_LE_def lex_square_def lex_triple_def lex_pair_def
- split: if_splits)
-
- done
- qed
-qed
-*)
-
-lemma [simp]: "inv_check_left_moving (as, abc_lm_s am n (abc_lm_v am n - Suc 0))
- (start_of (layout_of aprog) as + 2 * n + 11, [], Oc # xs) ires = False"
-apply(simp add: inv_check_left_moving.simps inv_check_left_moving_in_middle.simps)
-done
-
-lemma dec_inv_stop2_pre:
- "\<lbrakk>abc_fetch as aprog = Some (Dec n e); abc_lm_v am n > 0\<rbrakk> \<Longrightarrow>
- \<forall>na. \<not> (\<lambda>(s, l, r) (ss, n', ires').
- s = start_of (layout_of aprog) as + 2 * n + 16)
- (t_steps (Suc (start_of (layout_of aprog) as), l, r)
- (ci (layout_of aprog) (start_of (layout_of aprog) as) (Dec n e),
- start_of (layout_of aprog) as - Suc 0) na)
- (start_of (layout_of aprog) as, n, ires) \<and>
- dec_inv_2 (layout_of aprog) n e (as, am)
- (t_steps (Suc (start_of (layout_of aprog) as), l, r)
- (ci (layout_of aprog) (start_of (layout_of aprog) as) (Dec n e),
- start_of (layout_of aprog) as - Suc 0) na) ires
- \<longrightarrow>
- dec_inv_2 (layout_of aprog) n e (as, am)
- (t_steps (Suc (start_of (layout_of aprog) as), l, r)
- (ci (layout_of aprog) (start_of (layout_of aprog) as) (Dec n e),
- start_of (layout_of aprog) as - Suc 0) (Suc na)) ires \<and>
- ((t_steps (Suc (start_of (layout_of aprog) as), l, r)
- (ci (layout_of aprog) (start_of (layout_of aprog) as) (Dec n e),
- start_of (layout_of aprog) as - Suc 0) (Suc na),
- start_of (layout_of aprog) as, n, ires),
- t_steps (Suc (start_of (layout_of aprog) as), l, r)
- (ci (layout_of aprog) (start_of (layout_of aprog) as) (Dec n e),
- start_of (layout_of aprog) as - Suc 0) na,
- start_of (layout_of aprog) as, n, ires)
- \<in> abc_dec_2_LE"
-apply(subgoal_tac "start_of (layout_of aprog) as > 0")
-apply(rule allI, rule impI, simp add: t_steps_ind)
-apply(case_tac "(t_steps (Suc (start_of (layout_of aprog) as), l, r)
- (ci (layout_of aprog) (start_of (layout_of aprog) as) (Dec n e),
- start_of (layout_of aprog) as - Suc 0) na)", simp)
-apply(auto split:if_splits simp add:t_step.simps dec_inv_2.simps,
- tactic {* ALLGOALS (resolve_tac [@{thm fetch_intro}]) *})
-apply(simp_all add:dec_fetch_simps new_tape.simps dec_inv_2.simps)
-apply(auto simp add: abc_dec_2_LE_def lex_square_def lex_triple_def
- lex_pair_def split: if_splits)
-apply(auto intro: dec_false_2_b dec_false_2)
-apply(rule startof_not0)
-done
-
-lemma dec_stop2:
- "\<lbrakk>ly = layout_of aprog;
- dec_inv_2 ly n e (as, am) (start_of ly as + 1, l, r) ires;
- abc_fetch as aprog = Some (Dec n e);
- abc_lm_v am n > 0\<rbrakk> \<Longrightarrow>
- (\<exists> stp. (\<lambda> (s', l', r'). s' = start_of ly (Suc as) \<and>
- dec_inv_2 ly n e (as, am) (s', l', r') ires)
- (t_steps (start_of ly as+1, l, r) (ci ly (start_of ly as)
- (Dec n e), start_of ly as - Suc 0) stp))"
-apply(insert halt_lemma2[of abc_dec_2_LE
- "\<lambda> ((s, l, r), ss, n', ires'). s = start_of ly (Suc as)"
- "(\<lambda> stp. (t_steps (start_of ly as + 1, l, r)
- (ci ly (start_of ly as) (Dec n e), start_of ly as - Suc 0) stp,
- start_of ly as, n, ires))"
- "(\<lambda> ((s, l, r), ss, n, ires'). dec_inv_2 ly n e (as, am) (s, l, r) ires')"])
-apply(insert wf_dec_2_le, simp)
-apply(insert dec_inv_stop2_pre[of as aprog n e am l r],
- simp add: t_steps.simps)
-apply(erule_tac exE)
-apply(rule_tac x = na in exI)
-apply(case_tac "(t_steps (Suc (start_of (layout_of aprog) as), l, r)
-(ci (layout_of aprog) (start_of (layout_of aprog) as) (Dec n e),
- start_of (layout_of aprog) as - Suc 0) na)",
- case_tac b, auto)
-done
-
-lemma dec_inv_stop_cond1:
- "\<lbrakk>ly = layout_of aprog;
- dec_inv_1 ly n e (as, lm) (s, (l, r)) ires; s = start_of ly e;
- abc_fetch as aprog = Some (Dec n e); abc_lm_v lm n = 0\<rbrakk>
- \<Longrightarrow> crsp_l ly (e, abc_lm_s lm n 0) (s, l, r) ires"
-apply(simp add: dec_inv_1.simps split: if_splits)
-apply(auto simp: crsp_l.simps inv_stop.simps )
-done
-
-lemma dec_inv_stop_cond2:
- "\<lbrakk>ly = layout_of aprog; s = start_of ly (Suc as);
- dec_inv_2 ly n e (as, lm) (s, (l, r)) ires;
- abc_fetch as aprog = Some (Dec n e);
- abc_lm_v lm n > 0\<rbrakk>
- \<Longrightarrow> crsp_l ly (Suc as,
- abc_lm_s lm n (abc_lm_v lm n - Suc 0)) (s, l, r) ires"
-apply(simp add: dec_inv_2.simps split: if_splits)
-apply(auto simp: crsp_l.simps inv_stop.simps )
-done
-
-lemma [simp]: "(case Bk\<^bsup>rn\<^esup> of [] \<Rightarrow> Bk |
- Bk # xs \<Rightarrow> Bk | Oc # xs \<Rightarrow> Oc) = Bk"
-apply(case_tac rn, auto)
-done
-
-lemma [simp]: "t_steps tc (p,off) (m + n) =
- t_steps (t_steps tc (p, off) m) (p, off) n"
-apply(induct m arbitrary: n)
-apply(simp add: t_steps.simps)
-proof -
- fix m n
- assume h1: "\<And>n. t_steps tc (p, off) (m + n) =
- t_steps (t_steps tc (p, off) m) (p, off) n"
- hence h2: "t_steps tc (p, off) (Suc m + n) =
- t_steps tc (p, off) (m + Suc n)"
- by simp
- from h1 and this show
- "t_steps tc (p, off) (Suc m + n) =
- t_steps (t_steps tc (p, off) (Suc m)) (p, off) n"
- proof(simp only: h2, simp add: t_steps.simps)
- have h3: "(t_step (t_steps tc (p, off) m) (p, off)) =
- (t_steps (t_step tc (p, off)) (p, off) m)"
- apply(simp add: t_steps.simps[THEN sym] t_steps_ind[THEN sym])
- done
- from h3 show
- "t_steps (t_step (t_steps tc (p, off) m) (p, off)) (p, off) n = t_steps (t_steps (t_step tc (p, off)) (p, off) m) (p, off) n"
- by simp
- qed
-qed
-
-lemma [simp]: " abc_fetch as aprog = Some (Dec n e) \<Longrightarrow>
- Suc (start_of (layout_of aprog) as) \<noteq>
- start_of (layout_of aprog) e"
-apply(case_tac "e = as", simp)
-apply(case_tac "e < as")
-apply(drule_tac a = e and b = as and ly = "layout_of aprog"
- in start_of_le, simp)
-apply(drule_tac start_of_ge, auto)
-done
-
-lemma [simp]: "inv_locate_b (as, []) (0, Oc # Bk # Bk # ires, Bk\<^bsup>rn - Suc 0\<^esup>) ires"
-apply(auto simp: inv_locate_b.simps in_middle.simps)
-apply(rule_tac x = "[]" in exI, rule_tac x = "Suc 0" in exI,
- rule_tac x = 0 in exI, simp)
-apply(rule_tac x = "Suc 0" in exI, rule_tac x = 0 in exI, auto)
-apply(case_tac rn, simp, case_tac nat, auto)
-done
-
-lemma [simp]:
- "inv_locate_n_b (as, []) (0, Oc # Bk # Bk # ires, Bk\<^bsup>rn - Suc 0\<^esup>) ires"
-apply(auto simp: inv_locate_n_b.simps in_middle.simps)
-apply(case_tac rn, simp, case_tac nat, auto)
-done
-
-lemma [simp]:
-"abc_fetch as aprog = Some (Dec n e) \<Longrightarrow>
- dec_inv_1 (layout_of aprog) n e (as, [])
- (Suc (start_of (layout_of aprog) as), Oc # Bk # Bk # ires, Bk\<^bsup>rn - Suc 0\<^esup>) ires
-\<and>
- dec_inv_2 (layout_of aprog) n e (as, [])
- (Suc (start_of (layout_of aprog) as), Oc # Bk # Bk # ires, Bk\<^bsup>rn - Suc 0\<^esup>) ires"
-apply(simp add: dec_inv_1.simps dec_inv_2.simps)
-apply(case_tac n, auto)
-done
-
-lemma [simp]:
- "\<lbrakk>am \<noteq> []; <am> = Oc # r';
- abc_fetch as aprog = Some (Dec n e)\<rbrakk>
- \<Longrightarrow> inv_locate_b (as, am) (0, Oc # Bk # Bk # ires, r' @ Bk\<^bsup>rn\<^esup>) ires"
-apply(auto simp: inv_locate_b.simps in_middle.simps)
-apply(rule_tac x = "tl am" in exI, rule_tac x = 0 in exI,
- rule_tac x = "hd am" in exI, simp)
-apply(rule_tac x = "Suc 0" in exI)
-apply(rule_tac x = "hd am" in exI, simp)
-apply(case_tac am, simp, case_tac list, auto simp: tape_of_nl_abv tape_of_nat_list.simps)
-apply(case_tac rn, auto)
-done
-
-lemma [simp]:
- "\<lbrakk><am> = Oc # r'; abc_fetch as aprog = Some (Dec n e)\<rbrakk> \<Longrightarrow>
- inv_locate_n_b (as, am) (0, Oc # Bk # Bk # ires, r' @ Bk\<^bsup>rn\<^esup>) ires"
-apply(auto simp: inv_locate_n_b.simps)
-apply(rule_tac x = "tl am" in exI, rule_tac x = "hd am" in exI, auto)
-apply(case_tac [!] am, auto simp: tape_of_nl_abv tape_of_nat_list.simps )
-apply(case_tac [!]list, auto simp: tape_of_nl_abv tape_of_nat_list.simps)
-apply(case_tac rn, simp, simp)
-apply(erule_tac x = nat in allE, simp)
-done
-
-lemma [simp]:
- "\<lbrakk>am \<noteq> [];
- <am> = Oc # r';
- abc_fetch as aprog = Some (Dec n e)\<rbrakk> \<Longrightarrow>
- dec_inv_1 (layout_of aprog) n e (as, am)
- (Suc (start_of (layout_of aprog) as),
- Oc # Bk # Bk # ires, r' @ Bk\<^bsup>rn\<^esup>) ires \<and>
- dec_inv_2 (layout_of aprog) n e (as, am)
- (Suc (start_of (layout_of aprog) as),
- Oc # Bk # Bk # ires, r' @ Bk\<^bsup>rn\<^esup>) ires"
-apply(simp add: dec_inv_1.simps dec_inv_2.simps)
-apply(case_tac n, auto)
-done
-
-lemma [simp]: "am \<noteq> [] \<Longrightarrow> \<exists>r'. <am::nat list> = Oc # r'"
-apply(case_tac am, simp, case_tac list)
-apply(auto simp: tape_of_nl_abv tape_of_nat_list.simps )
-done
-
-lemma [simp]: "start_of (layout_of aprog) as > 0 \<Longrightarrow>
- (fetch (ci (layout_of aprog)
- (start_of (layout_of aprog) as) (Dec n e)) (Suc 0) Bk)
- = (W1, start_of (layout_of aprog) as)"
-apply(auto simp: ci.simps findnth.simps fetch.simps
- nth_of.simps tshift.simps nth_append Suc_pre tdec_b_def)
-thm findnth_nth
-apply(insert findnth_nth[of 0 n 0], simp)
-apply(simp add: findnth.simps)
-done
-
-lemma [simp]:
- "start_of (layout_of aprog) as > 0
- \<Longrightarrow> (t_step (start_of (layout_of aprog) as, Bk # Bk # ires, Bk\<^bsup>rn\<^esup>)
- (ci (layout_of aprog) (start_of (layout_of aprog) as) (Dec n e),
- start_of (layout_of aprog) as - Suc 0))
- = (start_of (layout_of aprog) as, Bk # Bk # ires, Oc # Bk\<^bsup>rn- Suc 0\<^esup>)"
-apply(simp add: t_step.simps)
-apply(case_tac "start_of (layout_of aprog) as",
- auto simp: new_tape.simps)
-apply(case_tac rn, auto)
-done
-
-lemma [simp]: "start_of (layout_of aprog) as > 0 \<Longrightarrow>
- (fetch (ci (layout_of aprog) (start_of (layout_of aprog) as)
- (Dec n e)) (Suc 0) Oc)
- = (R, Suc (start_of (layout_of aprog) as))"
-
-apply(auto simp: ci.simps findnth.simps fetch.simps
- nth_of.simps tshift.simps nth_append
- Suc_pre tdec_b_def)
-apply(insert findnth_nth[of 0 n "Suc 0"], simp)
-apply(simp add: findnth.simps)
-done
-
-lemma [simp]: "start_of (layout_of aprog) as > 0 \<Longrightarrow>
- (t_step (start_of (layout_of aprog) as, Bk # Bk # ires, Oc # Bk\<^bsup>rn - Suc 0\<^esup>)
- (ci (layout_of aprog) (start_of (layout_of aprog) as) (Dec n e),
- start_of (layout_of aprog) as - Suc 0)) =
- (Suc (start_of (layout_of aprog) as), Oc # Bk # Bk # ires, Bk\<^bsup>rn-Suc 0\<^esup>)"
-apply(simp add: t_step.simps)
-apply(case_tac "start_of (layout_of aprog) as",
- auto simp: new_tape.simps)
-done
-
-lemma [simp]: "start_of (layout_of aprog) as > 0 \<Longrightarrow>
- t_step (start_of (layout_of aprog) as, Bk # Bk # ires, Oc # r' @ Bk\<^bsup>rn\<^esup>)
- (ci (layout_of aprog) (start_of (layout_of aprog) as) (Dec n e),
- start_of (layout_of aprog) as - Suc 0) =
- (Suc (start_of (layout_of aprog) as), Oc # Bk # Bk # ires, r' @ Bk\<^bsup>rn\<^esup>)"
-apply(simp add: t_step.simps)
-apply(case_tac "start_of (layout_of aprog) as",
- auto simp: new_tape.simps)
-done
-
-lemma crsp_next_state:
- "\<lbrakk>crsp_l (layout_of aprog) (as, am) tc ires;
- abc_fetch as aprog = Some (Dec n e)\<rbrakk>
- \<Longrightarrow> \<exists> stp' > 0. (\<lambda> (s, l, r).
- (s = Suc (start_of (layout_of aprog) as)
- \<and> (dec_inv_1 (layout_of aprog) n e (as, am) (s, l, r) ires)
- \<and> (dec_inv_2 (layout_of aprog) n e (as, am) (s, l, r)) ires))
- (t_steps tc (ci (layout_of aprog) (start_of (layout_of aprog) as)
- (Dec n e), start_of (layout_of aprog) as - Suc 0) stp')"
-apply(subgoal_tac "start_of (layout_of aprog) as > 0")
-apply(case_tac tc, case_tac b, auto simp: crsp_l.simps)
-apply(case_tac "am = []", simp)
-apply(rule_tac x = "Suc (Suc 0)" in exI, simp add: t_steps.simps)
-proof-
- fix rn
- assume h1: "am \<noteq> []" "abc_fetch as aprog = Some (Dec n e)"
- "start_of (layout_of aprog) as > 0"
- hence h2: "\<exists> r'. <am> = Oc # r'"
- by simp
- from h1 and h2 show
- "\<exists>stp'>0. case t_steps (start_of (layout_of aprog) as, Bk # Bk # ires, <am> @ Bk\<^bsup>rn\<^esup>)
- (ci (layout_of aprog) (start_of (layout_of aprog) as) (Dec n e),
- start_of (layout_of aprog) as - Suc 0) stp' of
- (s, ab) \<Rightarrow> s = Suc (start_of (layout_of aprog) as) \<and>
- dec_inv_1 (layout_of aprog) n e (as, am) (s, ab) ires \<and>
- dec_inv_2 (layout_of aprog) n e (as, am) (s, ab) ires"
- proof(erule_tac exE, simp, rule_tac x = "Suc 0" in exI,
- simp add: t_steps.simps)
- qed
-next
- assume "abc_fetch as aprog = Some (Dec n e)"
- thus "0 < start_of (layout_of aprog) as"
- apply(insert startof_not0[of "layout_of aprog" as], simp)
- done
-qed
-
-lemma dec_crsp_ex1:
- "\<lbrakk>crsp_l (layout_of aprog) (as, am) tc ires;
- abc_fetch as aprog = Some (Dec n e);
- abc_lm_v am n = 0\<rbrakk>
- \<Longrightarrow> \<exists>stp > 0. crsp_l (layout_of aprog) (e, abc_lm_s am n 0)
- (t_steps tc (ci (layout_of aprog) (start_of (layout_of aprog) as)
- (Dec n e), start_of (layout_of aprog) as - Suc 0) stp) ires"
-proof -
- assume h1: "crsp_l (layout_of aprog) (as, am) tc ires"
- "abc_fetch as aprog = Some (Dec n e)" "abc_lm_v am n = 0"
- hence h2: "\<exists> stp' > 0. (\<lambda> (s, l, r).
- (s = Suc (start_of (layout_of aprog) as) \<and>
- (dec_inv_1 (layout_of aprog) n e (as, am) (s, l, r)) ires))
- (t_steps tc (ci (layout_of aprog) (start_of (layout_of aprog) as)
- (Dec n e), start_of (layout_of aprog) as - Suc 0) stp')"
- apply(insert crsp_next_state[of aprog as am tc ires n e], auto)
- done
- from h1 and h2 show
- "\<exists>stp > 0. crsp_l (layout_of aprog) (e, abc_lm_s am n 0)
- (t_steps tc (ci (layout_of aprog) (start_of
- (layout_of aprog) as) (Dec n e),
- start_of (layout_of aprog) as - Suc 0) stp) ires"
- proof(erule_tac exE, case_tac "(t_steps tc (ci (layout_of aprog)
- (start_of (layout_of aprog) as) (Dec n e), start_of
- (layout_of aprog) as - Suc 0) stp')", simp)
- fix stp' a b c
- assume h3: "stp' > 0 \<and> a = Suc (start_of (layout_of aprog) as) \<and>
- dec_inv_1 (layout_of aprog) n e (as, am) (a, b, c) ires"
- "abc_fetch as aprog = Some (Dec n e)" "abc_lm_v am n = 0"
- "t_steps tc (ci (layout_of aprog) (start_of (layout_of aprog) as)
- (Dec n e), start_of (layout_of aprog) as - Suc 0) stp'
- = (Suc (start_of (layout_of aprog) as), b, c)"
- thus "\<exists>stp > 0. crsp_l (layout_of aprog) (e, abc_lm_s am n 0)
- (t_steps tc (ci (layout_of aprog) (start_of (layout_of aprog) as)
- (Dec n e), start_of (layout_of aprog) as - Suc 0) stp) ires"
- proof(erule_tac conjE, simp)
- assume "dec_inv_1 (layout_of aprog) n e (as, am)
- (Suc (start_of (layout_of aprog) as), b, c) ires"
- "abc_fetch as aprog = Some (Dec n e)"
- "abc_lm_v am n = 0"
- " t_steps tc (ci (layout_of aprog)
- (start_of (layout_of aprog) as) (Dec n e),
- start_of (layout_of aprog) as - Suc 0) stp'
- = (Suc (start_of (layout_of aprog) as), b, c)"
- hence h4: "\<exists>stp. (\<lambda>(s', l', r'). s' =
- start_of (layout_of aprog) e \<and>
- dec_inv_1 (layout_of aprog) n e (as, am) (s', l', r') ires)
- (t_steps (start_of (layout_of aprog) as + 1, b, c)
- (ci (layout_of aprog)
- (start_of (layout_of aprog) as) (Dec n e),
- start_of (layout_of aprog) as - Suc 0) stp)"
- apply(rule_tac dec_inv_stop1, auto)
- done
- from h3 and h4 show ?thesis
- apply(erule_tac exE)
- apply(rule_tac x = "stp' + stp" in exI, simp)
- apply(case_tac "(t_steps (Suc (start_of (layout_of aprog) as),
- b, c) (ci (layout_of aprog)
- (start_of (layout_of aprog) as) (Dec n e),
- start_of (layout_of aprog) as - Suc 0) stp)",
- simp)
- apply(rule_tac dec_inv_stop_cond1, auto)
- done
- qed
- qed
-qed
-
-lemma dec_crsp_ex2:
- "\<lbrakk>crsp_l (layout_of aprog) (as, am) tc ires;
- abc_fetch as aprog = Some (Dec n e);
- 0 < abc_lm_v am n\<rbrakk>
- \<Longrightarrow> \<exists>stp > 0. crsp_l (layout_of aprog)
- (Suc as, abc_lm_s am n (abc_lm_v am n - Suc 0))
- (t_steps tc (ci (layout_of aprog) (start_of (layout_of aprog) as)
- (Dec n e), start_of (layout_of aprog) as - Suc 0) stp) ires"
-proof -
- assume h1:
- "crsp_l (layout_of aprog) (as, am) tc ires"
- "abc_fetch as aprog = Some (Dec n e)"
- "abc_lm_v am n > 0"
- hence h2:
- "\<exists> stp' > 0. (\<lambda> (s, l, r). (s = Suc (start_of (layout_of aprog) as)
- \<and> (dec_inv_2 (layout_of aprog) n e (as, am) (s, l, r)) ires))
-(t_steps tc (ci (layout_of aprog) (start_of (layout_of aprog) as)
- (Dec n e), start_of (layout_of aprog) as - Suc 0) stp')"
- apply(insert crsp_next_state[of aprog as am tc ires n e], auto)
- done
- from h1 and h2 show
- "\<exists>stp >0. crsp_l (layout_of aprog)
- (Suc as, abc_lm_s am n (abc_lm_v am n - Suc 0))
- (t_steps tc (ci (layout_of aprog) (start_of (layout_of aprog) as)
- (Dec n e), start_of (layout_of aprog) as - Suc 0) stp) ires"
- proof(erule_tac exE,
- case_tac
- "(t_steps tc (ci (layout_of aprog) (start_of (layout_of aprog) as)
- (Dec n e), start_of (layout_of aprog) as - Suc 0) stp')", simp)
- fix stp' a b c
- assume h3: "0 < stp' \<and> a = Suc (start_of (layout_of aprog) as) \<and>
- dec_inv_2 (layout_of aprog) n e (as, am) (a, b, c) ires"
- "abc_fetch as aprog = Some (Dec n e)"
- "abc_lm_v am n > 0"
- "t_steps tc (ci (layout_of aprog)
- (start_of (layout_of aprog) as) (Dec n e),
- start_of (layout_of aprog) as - Suc 0) stp'
- = (Suc (start_of (layout_of aprog) as), b, c)"
- thus "?thesis"
- proof(erule_tac conjE, simp)
- assume
- "dec_inv_2 (layout_of aprog) n e (as, am)
- (Suc (start_of (layout_of aprog) as), b, c) ires"
- "abc_fetch as aprog = Some (Dec n e)" "abc_lm_v am n > 0"
- "t_steps tc (ci (layout_of aprog) (start_of (layout_of aprog) as)
- (Dec n e), start_of (layout_of aprog) as - Suc 0) stp'
- = (Suc (start_of (layout_of aprog) as), b, c)"
- hence h4:
- "\<exists>stp. (\<lambda>(s', l', r'). s' = start_of (layout_of aprog) (Suc as) \<and>
- dec_inv_2 (layout_of aprog) n e (as, am) (s', l', r') ires)
- (t_steps (start_of (layout_of aprog) as + 1, b, c)
- (ci (layout_of aprog) (start_of (layout_of aprog) as)
- (Dec n e), start_of (layout_of aprog) as - Suc 0) stp)"
- apply(rule_tac dec_stop2, auto)
- done
- from h3 and h4 show ?thesis
- apply(erule_tac exE)
- apply(rule_tac x = "stp' + stp" in exI, simp)
- apply(case_tac
- "(t_steps (Suc (start_of (layout_of aprog) as), b, c)
- (ci (layout_of aprog) (start_of (layout_of aprog) as)
- (Dec n e), start_of (layout_of aprog) as - Suc 0) stp)"
- ,simp)
- apply(rule_tac dec_inv_stop_cond2, auto)
- done
- qed
- qed
-qed
-
-lemma dec_crsp_ex_pre:
- "\<lbrakk>ly = layout_of aprog; crsp_l ly (as, am) tc ires;
- abc_fetch as aprog = Some (Dec n e)\<rbrakk>
- \<Longrightarrow> \<exists>stp > 0. crsp_l ly (abc_step_l (as, am) (Some (Dec n e)))
- (t_steps tc (ci (layout_of aprog) (start_of ly as) (Dec n e),
- start_of ly as - Suc 0) stp) ires"
-apply(auto simp: abc_step_l.simps intro: dec_crsp_ex2 dec_crsp_ex1)
-done
-
-lemma dec_crsp_ex:
- assumes layout: -- {* There is an Abacus program @{text "aprog"} with layout @{text "ly"} *}
- "ly = layout_of aprog"
- and dec: -- {* There is an @{text "Dec n e"} instruction at postion @{text "as"} of @{text "aprog"} *}
- "abc_fetch as aprog = Some (Dec n e)"
- and correspond:
- -- {* Abacus configuration @{text "(as, am)"} is in correspondence with TM
- configuration @{text "tc"}
- *}
- "crsp_l ly (as, am) tc ires"
-shows
- "\<exists>stp > 0. crsp_l ly (abc_step_l (as, am) (Some (Dec n e)))
- (t_steps tc (ci (layout_of aprog) (start_of ly as) (Dec n e),
- start_of ly as - Suc 0) stp) ires"
-proof -
- from dec_crsp_ex_pre layout dec correspond show ?thesis by blast
-qed
-
-(*
-subsection {* Compilation of @{text "Goto n"}*}
-*)
-
-lemma goto_fetch:
- "fetch (ci (layout_of aprog)
- (start_of (layout_of aprog) as) (Goto n)) (Suc 0) b
- = (Nop, start_of (layout_of aprog) n)"
-apply(auto simp: ci.simps fetch.simps nth_of.simps
- split: block.splits)
-done
-
-text {*
- Correctness of complied @{text "Goto n"}
- *}
-
-lemma goto_crsp_ex_pre:
- "\<lbrakk>ly = layout_of aprog;
- crsp_l ly (as, am) tc ires;
- abc_fetch as aprog = Some (Goto n)\<rbrakk>
- \<Longrightarrow> \<exists>stp > 0. crsp_l ly (abc_step_l (as, am) (Some (Goto n)))
- (t_steps tc (ci (layout_of aprog) (start_of ly as) (Goto n),
- start_of ly as - Suc 0) stp) ires"
-apply(rule_tac x = 1 in exI)
-apply(simp add: abc_step_l.simps t_steps.simps t_step.simps)
-apply(case_tac tc, simp)
-apply(subgoal_tac "a = start_of (layout_of aprog) as", auto)
-apply(subgoal_tac "start_of (layout_of aprog) as > 0", simp)
-apply(auto simp: goto_fetch new_tape.simps crsp_l.simps)
-apply(rule startof_not0)
-done
-
-lemma goto_crsp_ex:
- assumes layout: "ly = layout_of aprog"
- and goto: "abc_fetch as aprog = Some (Goto n)"
- and correspondence: "crsp_l ly (as, am) tc ires"
- shows "\<exists>stp>0. crsp_l ly (abc_step_l (as, am) (Some (Goto n)))
- (t_steps tc (ci (layout_of aprog) (start_of ly as) (Goto n),
- start_of ly as - Suc 0) stp) ires"
-proof -
- from goto_crsp_ex_pre and layout goto correspondence show "?thesis" by blast
-qed
-
-subsection {*
- The correctness of the compiler
- *}
-
-declare abc_step_l.simps[simp del]
-
-lemma tm_crsp_ex:
- "\<lbrakk>ly = layout_of aprog;
- crsp_l ly (as, am) tc ires;
- as < length aprog;
- abc_fetch as aprog = Some ins\<rbrakk>
- \<Longrightarrow> \<exists> n > 0. crsp_l ly (abc_step_l (as,am) (Some ins))
- (t_steps tc (ci (layout_of aprog) (start_of ly as)
- (ins), (start_of ly as) - (Suc 0)) n) ires"
-apply(case_tac "ins", simp)
-apply(auto intro: inc_crsp_ex_pre dec_crsp_ex goto_crsp_ex)
-done
-
-lemma start_of_pre:
- "n < length aprog \<Longrightarrow> start_of (layout_of aprog) n
- = start_of (layout_of (butlast aprog)) n"
-apply(induct n, simp add: start_of.simps, simp)
-apply(simp add: layout_of.simps start_of.simps)
-apply(subgoal_tac "n < length aprog - Suc 0", simp)
-apply(subgoal_tac "(aprog ! n) = (butlast aprog ! n)", simp)
-proof -
- fix n
- assume h1: "Suc n < length aprog"
- thus "aprog ! n = butlast aprog ! n"
- apply(case_tac "length aprog", simp, simp)
- apply(insert nth_append[of "butlast aprog" "[last aprog]" n])
- apply(subgoal_tac "(butlast aprog @ [last aprog]) = aprog")
- apply(simp split: if_splits)
- apply(rule append_butlast_last_id, case_tac aprog, simp, simp)
- done
-next
- fix n
- assume "Suc n < length aprog"
- thus "n < length aprog - Suc 0"
- apply(case_tac aprog, simp, simp)
- done
-qed
-
-lemma zip_eq: "xs = ys \<Longrightarrow> zip xs zs = zip ys zs"
-by simp
-
-lemma tpairs_of_append_iff: "length aprog = Suc n \<Longrightarrow>
- tpairs_of aprog = tpairs_of (butlast aprog) @
- [(start_of (layout_of aprog) n, aprog ! n)]"
-apply(simp add: tpairs_of.simps)
-apply(insert zip_append[of "map (start_of (layout_of aprog)) [0..<n]"
- "butlast aprog" "[start_of (layout_of aprog) n]" "[last aprog]"])
-apply(simp del: zip_append)
-apply(subgoal_tac "(butlast aprog @ [last aprog]) = aprog", auto)
-apply(rule_tac zip_eq, auto)
-apply(rule_tac start_of_pre, simp)
-apply(insert last_conv_nth[of aprog], case_tac aprog, simp, simp)
-apply(rule append_butlast_last_id, case_tac aprog, simp, simp)
-done
-
-lemma [simp]: "list_all (\<lambda>(n, tm). abacus.t_ncorrect (ci layout n tm))
- (zip (map (start_of layout) [0..<length aprog]) aprog)"
-proof(induct "length aprog" arbitrary: aprog, simp)
- fix x aprog
- assume ind: "\<And>aprog. x = length aprog \<Longrightarrow>
- list_all (\<lambda>(n, tm). abacus.t_ncorrect (ci layout n tm))
- (zip (map (start_of layout) [0..<length aprog]) aprog)"
- and h: "Suc x = length (aprog::abc_inst list)"
- have g1: "list_all (\<lambda>(n, tm). abacus.t_ncorrect (ci layout n tm))
- (zip (map (start_of layout) [0..<length (butlast aprog)])
- (butlast aprog))"
- using h
- apply(rule_tac ind, auto)
- done
- have g2: "(map (start_of layout) [0..<length aprog]) =
- map (start_of layout) ([0..<length aprog - 1]
- @ [length aprog - 1])"
- using h
- apply(case_tac aprog, simp, simp)
- done
- have "\<exists> xs a. aprog = xs @ [a]"
- using h
- apply(rule_tac x = "butlast aprog" in exI,
- rule_tac x = "last aprog" in exI)
- apply(case_tac "aprog = []", simp, simp)
- done
- from this obtain xs where "\<exists> a. aprog = xs @ [a]" ..
- from this obtain a where g3: "aprog = xs @ [a]" ..
- from g1 and g2 and g3 show "list_all (\<lambda>(n, tm).
- abacus.t_ncorrect (ci layout n tm))
- (zip (map (start_of layout) [0..<length aprog]) aprog)"
- apply(simp)
- apply(auto simp: t_ncorrect.simps ci.simps tshift.simps
- tinc_b_def tdec_b_def split: abc_inst.splits)
- apply arith+
- done
-qed
-
-lemma [intro]: "abc2t_correct aprog"
-apply(simp add: abc2t_correct.simps tpairs_of.simps
- split: abc_inst.splits)
-done
-
-lemma as_out: "\<lbrakk>ly = layout_of aprog; tprog = tm_of aprog;
- crsp_l ly (as, am) tc ires; length aprog \<le> as\<rbrakk>
- \<Longrightarrow> abc_step_l (as, am) (abc_fetch as aprog) = (as, am)"
-apply(simp add: abc_fetch.simps abc_step_l.simps)
-done
-
-lemma tm_merge_ex:
- "\<lbrakk>crsp_l (layout_of aprog) (as, am) tc ires;
- as < length aprog;
- abc_fetch as aprog = Some a;
- abc2t_correct aprog;
- crsp_l (layout_of aprog) (abc_step_l (as, am) (Some a))
- (t_steps tc (ci (layout_of aprog) (start_of (layout_of aprog) as)
- a, start_of (layout_of aprog) as - Suc 0) n) ires;
- n > 0\<rbrakk>
- \<Longrightarrow> \<exists>stp > 0. crsp_l (layout_of aprog) (abc_step_l (as, am)
- (Some a)) (t_steps tc (tm_of aprog, 0) stp) ires"
-apply(case_tac "(t_steps tc (ci (layout_of aprog)
- (start_of (layout_of aprog) as) a,
- start_of (layout_of aprog) as - Suc 0) n)", simp)
-apply(case_tac "(abc_step_l (as, am) (Some a))", simp)
-proof -
- fix aa b c aaa ba
- assume h:
- "crsp_l (layout_of aprog) (as, am) tc ires"
- "as < length aprog"
- "abc_fetch as aprog = Some a"
- "crsp_l (layout_of aprog) (aaa, ba) (aa, b, c) ires"
- "abc2t_correct aprog"
- "n>0"
- "t_steps tc (ci (layout_of aprog) (start_of (layout_of aprog) as) a,
- start_of (layout_of aprog) as - Suc 0) n = (aa, b, c)"
- "abc_step_l (as, am) (Some a) = (aaa, ba)"
- hence "aa = start_of (layout_of aprog) aaa"
- apply(simp add: crsp_l.simps)
- done
- from this and h show
- "\<exists>stp > 0. crsp_l (layout_of aprog) (aaa, ba)
- (t_steps tc (tm_of aprog, 0) stp) ires"
- apply(insert tms_out_ex[of "layout_of aprog" aprog
- "tm_of aprog" as am tc ires a n aa b c aaa ba], auto)
- done
-qed
-
-lemma crsp_inside:
- "\<lbrakk>ly = layout_of aprog;
- tprog = tm_of aprog;
- crsp_l ly (as, am) tc ires;
- as < length aprog\<rbrakk> \<Longrightarrow>
- (\<exists> stp > 0. crsp_l ly (abc_step_l (as,am) (abc_fetch as aprog))
- (t_steps tc (tprog, 0) stp) ires)"
-apply(case_tac "abc_fetch as aprog", simp add: abc_fetch.simps)
-proof -
- fix a
- assume "ly = layout_of aprog"
- "tprog = tm_of aprog"
- "crsp_l ly (as, am) tc ires"
- "as < length aprog"
- "abc_fetch as aprog = Some a"
- thus "\<exists>stp > 0. crsp_l ly (abc_step_l (as, am)
- (abc_fetch as aprog)) (t_steps tc (tprog, 0) stp) ires"
- proof(insert tm_crsp_ex[of ly aprog as am tc ires a],
- auto intro: tm_merge_ex)
- qed
-qed
-
-lemma crsp_outside:
- "\<lbrakk>ly = layout_of aprog; tprog = tm_of aprog;
- crsp_l ly (as, am) tc ires; as \<ge> length aprog\<rbrakk>
- \<Longrightarrow> (\<exists> stp. crsp_l ly (abc_step_l (as,am) (abc_fetch as aprog))
- (t_steps tc (tprog, 0) stp) ires)"
-apply(subgoal_tac "abc_step_l (as, am) (abc_fetch as aprog)
- = (as, am)", simp)
-apply(rule_tac x = 0 in exI, simp add: t_steps.simps)
-apply(rule as_out, simp+)
-done
-
-text {*
- Single-step correntess of the compiler.
-*}
-lemma astep_crsp_pre:
- "\<lbrakk>ly = layout_of aprog;
- tprog = tm_of aprog;
- crsp_l ly (as, am) tc ires\<rbrakk>
- \<Longrightarrow> (\<exists> stp. crsp_l ly (abc_step_l (as,am)
- (abc_fetch as aprog)) (t_steps tc (tprog, 0) stp) ires)"
-apply(case_tac "as < length aprog")
-apply(drule_tac crsp_inside, auto)
-apply(rule_tac crsp_outside, simp+)
-done
-
-text {*
- Single-step correntess of the compiler.
-*}
-lemma astep_crsp_pre1:
- "\<lbrakk>ly = layout_of aprog;
- tprog = tm_of aprog;
- crsp_l ly (as, am) tc ires\<rbrakk>
- \<Longrightarrow> (\<exists> stp. crsp_l ly (abc_step_l (as,am)
- (abc_fetch as aprog)) (t_steps tc (tprog, 0) stp) ires)"
-apply(case_tac "as < length aprog")
-apply(drule_tac crsp_inside, auto)
-apply(rule_tac crsp_outside, simp+)
-done
-
-lemma astep_crsp:
- assumes layout:
- -- {* There is a Abacus program @{text "aprog"} with layout @{text "ly"} *}
- "ly = layout_of aprog"
- and compiled:
- -- {* @{text "tprog"} is the TM compiled from @{text "aprog"} *}
- "tprog = tm_of aprog"
- and corresponds:
- -- {* Abacus configuration @{text "(as, am)"} is in correspondence with TM configuration
- @{text "tc"} *}
- "crsp_l ly (as, am) tc ires"
- -- {* One step execution of @{text "aprog"} can be simulated by multi-step execution
- of @{text "tprog"} *}
- shows "(\<exists> stp. crsp_l ly (abc_step_l (as,am)
- (abc_fetch as aprog)) (t_steps tc (tprog, 0) stp) ires)"
-proof -
- from astep_crsp_pre1 [OF layout compiled corresponds] show ?thesis .
-qed
-
-lemma steps_crsp_pre:
- "\<lbrakk>ly = layout_of aprog; tprog = tm_of aprog;
- crsp_l ly ac tc ires; ac' = abc_steps_l ac aprog n\<rbrakk> \<Longrightarrow>
- (\<exists> n'. crsp_l ly ac' (t_steps tc (tprog, 0) n') ires)"
-apply(induct n arbitrary: ac' ac tc, simp add: abc_steps_l.simps)
-apply(rule_tac x = 0 in exI)
-apply(case_tac ac, simp add: abc_steps_l.simps t_steps.simps)
-apply(case_tac ac, simp add: abc_steps_l.simps)
-apply(subgoal_tac
- "(\<exists> stp. crsp_l ly (abc_step_l (a, b)
- (abc_fetch a aprog)) (t_steps tc (tprog, 0) stp) ires)")
-apply(erule exE)
-apply(subgoal_tac
- "\<exists>n'. crsp_l (layout_of aprog)
- (abc_steps_l (abc_step_l (a, b) (abc_fetch a aprog)) aprog n)
- (t_steps ((t_steps tc (tprog, 0) stp)) (tm_of aprog, 0) n') ires")
-apply(erule exE)
-apply(subgoal_tac
- "t_steps (t_steps tc (tprog, 0) stp) (tm_of aprog, 0) n' =
- t_steps tc (tprog, 0) (stp + n')")
-apply(rule_tac x = "stp + n'" in exI, simp)
-apply(auto intro: astep_crsp simp: t_step_add)
-done
-
-text {*
- Multi-step correctess of the compiler.
-*}
-
-lemma steps_crsp:
- assumes layout:
- -- {* There is an Abacus program @{text "aprog"} with layout @{text "ly"} *}
- "ly = layout_of aprog"
- and compiled:
- -- {* @{text "tprog"} is the TM compiled from @{text "aprog"} *}
- "tprog = tm_of aprog"
- and correspond:
- -- {* Abacus configuration @{text "ac"} is in correspondence with TM configuration @{text "tc"} *}
- "crsp_l ly ac tc ires"
- and execution:
- -- {* @{text "ac'"} is the configuration obtained from @{text "n"}-step execution
- of @{text "aprog"} starting from configuration @{text "ac"} *}
- "ac' = abc_steps_l ac aprog n"
- -- {* There exists steps @{text "n'"} steps, after these steps of execution,
- the Turing configuration such obtained is in correspondence with @{text "ac'"} *}
- shows "(\<exists> n'. crsp_l ly ac' (t_steps tc (tprog, 0) n') ires)"
-proof -
- from steps_crsp_pre [OF layout compiled correspond execution] show ?thesis .
-qed
-
-subsection {* The Mop-up machine *}
-
-fun mop_bef :: "nat \<Rightarrow> tprog"
- where
- "mop_bef 0 = []" |
- "mop_bef (Suc n) = mop_bef n @
- [(R, 2*n + 3), (W0, 2*n + 2), (R, 2*n + 1), (W1, 2*n + 2)]"
-
-definition mp_up :: "tprog"
- where
- "mp_up \<equiv> [(R, 2), (R, 1), (L, 5), (W0, 3), (R, 4), (W0, 3),
- (R, 2), (W0, 3), (L, 5), (L, 6), (R, 0), (L, 6)]"
-
-fun tMp :: "nat \<Rightarrow> nat \<Rightarrow> tprog"
- where
- "tMp n off = tshift (mop_bef n @ tshift mp_up (2*n)) off"
-
-declare mp_up_def[simp del] tMp.simps[simp del] mop_bef.simps[simp del]
-(**********Begin: equiv among aba and turing***********)
-
-lemma tm_append_step:
- "\<lbrakk>t_ncorrect tp1; t_step tc (tp1, 0) = (s, l, r); s \<noteq> 0\<rbrakk>
- \<Longrightarrow> t_step tc (tp1 @ tp2, 0) = (s, l, r)"
-apply(simp add: t_step.simps)
-apply(case_tac tc, simp)
-apply(case_tac
- "(fetch tp1 a (case c of [] \<Rightarrow> Bk |
- Bk # xs \<Rightarrow> Bk | Oc # xs \<Rightarrow> Oc))", simp)
-apply(case_tac a, simp add: fetch.simps)
-apply(simp add: fetch.simps)
-apply(case_tac c, simp)
-apply(case_tac [!] "ab::block")
-apply(auto simp: nth_of.simps nth_append t_ncorrect.simps
- split: if_splits)
-done
-
-lemma state0_ind: "t_steps (0, l, r) (tp, 0) stp = (0, l, r)"
-apply(induct stp, simp add: t_steps.simps)
-apply(simp add: t_steps.simps t_step.simps fetch.simps new_tape.simps)
-done
-
-lemma tm_append_steps:
- "\<lbrakk>t_ncorrect tp1; t_steps tc (tp1, 0) stp = (s, l ,r); s \<noteq> 0\<rbrakk>
- \<Longrightarrow> t_steps tc (tp1 @ tp2, 0) stp = (s, l, r)"
-apply(induct stp arbitrary: tc s l r)
-apply(case_tac tc, simp)
-apply(simp add: t_steps.simps)
-proof -
- fix stp tc s l r
- assume h1: "\<And>tc s l r. \<lbrakk>t_ncorrect tp1; t_steps tc (tp1, 0) stp =
- (s, l, r); s \<noteq> 0\<rbrakk> \<Longrightarrow> t_steps tc (tp1 @ tp2, 0) stp = (s, l, r)"
- and h2: "t_steps tc (tp1, 0) (Suc stp) = (s, l, r)" "s \<noteq> 0"
- "t_ncorrect tp1"
- thus "t_steps tc (tp1 @ tp2, 0) (Suc stp) = (s, l, r)"
- apply(simp add: t_steps.simps)
- apply(case_tac "(t_step tc (tp1, 0))", simp)
- proof-
- fix a b c
- assume g1: "\<And>tc s l r. \<lbrakk>t_steps tc (tp1, 0) stp = (s, l, r);
- 0 < s\<rbrakk> \<Longrightarrow> t_steps tc (tp1 @ tp2, 0) stp = (s, l, r)"
- and g2: "t_step tc (tp1, 0) = (a, b, c)"
- "t_steps (a, b, c) (tp1, 0) stp = (s, l, r)"
- "0 < s"
- "t_ncorrect tp1"
- hence g3: "a > 0"
- apply(case_tac "a::nat", auto simp: t_steps.simps)
- apply(simp add: state0_ind)
- done
- from g1 and g2 and this have g4:
- "(t_step tc (tp1 @ tp2, 0)) = (a, b, c)"
- apply(rule_tac tm_append_step, simp, simp, simp)
- done
- from g1 and g2 and g3 and g4 show
- "t_steps (t_step tc (tp1 @ tp2, 0)) (tp1 @ tp2, 0) stp
- = (s, l, r)"
- apply(simp)
- done
- qed
-qed
-
-lemma shift_fetch:
- "\<lbrakk>n < length tp;
- (tp:: (taction \<times> nat) list) ! n = (aa, ba);
- ba \<noteq> 0\<rbrakk>
- \<Longrightarrow> (tshift tp (length tp div 2)) ! n =
- (aa , ba + length tp div 2)"
-apply(simp add: tshift.simps)
-done
-
-lemma tshift_length_equal: "length (tshift tp q) = length tp"
-apply(auto simp: tshift.simps)
-done
-
-thm nth_of.simps
-
-lemma [simp]: "t_ncorrect tp \<Longrightarrow> 2 * (length tp div 2) = length tp"
-apply(auto simp: t_ncorrect.simps)
-done
-
-lemma tm_append_step_equal':
- "\<lbrakk>t_ncorrect tp; t_ncorrect tp'; off = length tp div 2\<rbrakk> \<Longrightarrow>
- (\<lambda> (s, l, r). ((\<lambda> (s', l', r').
- (s'\<noteq> 0 \<longrightarrow> (s = s' + off \<and> l = l' \<and> r = r')))
- (t_step (a, b, c) (tp', 0))))
- (t_step (a + off, b, c) (tp @ tshift tp' off, 0))"
-apply(simp add: t_step.simps)
-apply(case_tac a, simp add: fetch.simps)
-apply(case_tac
-"(fetch tp' a (case c of [] \<Rightarrow> Bk | Bk # xs \<Rightarrow> Bk | Oc # xs \<Rightarrow> Oc))",
- simp)
-apply(case_tac
-"(fetch (tp @ tshift tp' (length tp div 2))
- (Suc (nat + length tp div 2))
- (case c of [] \<Rightarrow> Bk | Bk # xs \<Rightarrow> Bk | Oc # xs \<Rightarrow> Oc))",
- simp)
-apply(case_tac "(new_tape aa (b, c))",
- case_tac "(new_tape aaa (b, c))", simp,
- rule impI, simp add: fetch.simps split: block.splits option.splits)
-apply (auto simp: nth_of.simps t_ncorrect.simps
- nth_append tshift_length_equal tshift.simps split: if_splits)
-done
-
-
-lemma tm_append_step_equal:
- "\<lbrakk>t_ncorrect tp; t_ncorrect tp'; off = length tp div 2;
- t_step (a, b, c) (tp', 0) = (aa, ab, bb); aa \<noteq> 0\<rbrakk>
- \<Longrightarrow> t_step (a + length tp div 2, b, c)
- (tp @ tshift tp' (length tp div 2), 0)
- = (aa + length tp div 2, ab, bb)"
-apply(insert tm_append_step_equal'[of tp tp' off a b c], simp)
-apply(case_tac "(t_step (a + length tp div 2, b, c)
- (tp @ tshift tp' (length tp div 2), 0))", simp)
-done
-
-lemma tm_append_steps_equal:
- "\<lbrakk>t_ncorrect tp; t_ncorrect tp'; off = length tp div 2\<rbrakk> \<Longrightarrow>
- (\<lambda> (s, l, r). ((\<lambda> (s', l', r'). ((s'\<noteq> 0 \<longrightarrow> s = s' + off \<and> l = l'
- \<and> r = r'))) (t_steps (a, b, c) (tp', 0) stp)))
- (t_steps (a + off, b, c) (tp @ tshift tp' off, 0) stp)"
-apply(induct stp arbitrary : a b c, simp add: t_steps.simps)
-apply(simp add: t_steps.simps)
-apply(case_tac "(t_step (a, b, c) (tp', 0))", simp)
-apply(case_tac "aa = 0", simp add: state0_ind)
-apply(subgoal_tac "(t_step (a + length tp div 2, b, c)
- (tp @ tshift tp' (length tp div 2), 0))
- = (aa + length tp div 2, ba, ca)", simp)
-apply(rule tm_append_step_equal, auto)
-done
-
-(*********Begin: mop_up***************)
-type_synonym mopup_type = "t_conf \<Rightarrow> nat list \<Rightarrow> nat \<Rightarrow> block list \<Rightarrow> bool"
-
-fun mopup_stop :: "mopup_type"
- where
- "mopup_stop (s, l, r) lm n ires=
- (\<exists> ln rn. l = Bk\<^bsup>ln\<^esup> @ Bk # Bk # ires \<and> r = <abc_lm_v lm n> @ Bk\<^bsup>rn\<^esup>)"
-
-fun mopup_bef_erase_a :: "mopup_type"
- where
- "mopup_bef_erase_a (s, l, r) lm n ires=
- (\<exists> ln m rn. l = Bk \<^bsup>ln\<^esup> @ Bk # Bk # ires \<and>
- r = Oc\<^bsup>m \<^esup>@ Bk # <(drop ((s + 1) div 2) lm)> @ Bk\<^bsup>rn\<^esup>)"
-
-fun mopup_bef_erase_b :: "mopup_type"
- where
- "mopup_bef_erase_b (s, l, r) lm n ires =
- (\<exists> ln m rn. l = Bk\<^bsup>ln\<^esup> @ Bk # Bk # ires \<and> r = Bk # Oc\<^bsup>m\<^esup> @ Bk #
- <(drop (s div 2) lm)> @ Bk\<^bsup>rn\<^esup>)"
-
-
-fun mopup_jump_over1 :: "mopup_type"
- where
- "mopup_jump_over1 (s, l, r) lm n ires =
- (\<exists> ln m1 m2 rn. m1 + m2 = Suc (abc_lm_v lm n) \<and>
- l = Oc\<^bsup>m1\<^esup> @ Bk\<^bsup>ln\<^esup> @ Bk # Bk # ires \<and>
- (r = Oc\<^bsup>m2\<^esup> @ Bk # <(drop (Suc n) lm)> @ Bk\<^bsup>rn \<^esup>\<or>
- (r = Oc\<^bsup>m2\<^esup> \<and> (drop (Suc n) lm) = [])))"
-
-fun mopup_aft_erase_a :: "mopup_type"
- where
- "mopup_aft_erase_a (s, l, r) lm n ires =
- (\<exists> lnl lnr rn (ml::nat list) m.
- m = Suc (abc_lm_v lm n) \<and> l = Bk\<^bsup>lnr \<^esup>@ Oc\<^bsup>m \<^esup>@ Bk\<^bsup>lnl\<^esup> @ Bk # Bk # ires \<and>
- (r = <ml> @ Bk\<^bsup>rn\<^esup>))"
-
-fun mopup_aft_erase_b :: "mopup_type"
- where
- "mopup_aft_erase_b (s, l, r) lm n ires=
- (\<exists> lnl lnr rn (ml::nat list) m.
- m = Suc (abc_lm_v lm n) \<and>
- l = Bk\<^bsup>lnr \<^esup>@ Oc\<^bsup>m \<^esup>@ Bk\<^bsup>lnl\<^esup> @ Bk # Bk # ires \<and>
- (r = Bk # <ml> @ Bk\<^bsup>rn \<^esup>\<or>
- r = Bk # Bk # <ml> @ Bk\<^bsup>rn\<^esup>))"
-
-fun mopup_aft_erase_c :: "mopup_type"
- where
- "mopup_aft_erase_c (s, l, r) lm n ires =
- (\<exists> lnl lnr rn (ml::nat list) m.
- m = Suc (abc_lm_v lm n) \<and>
- l = Bk\<^bsup>lnr \<^esup>@ Oc\<^bsup>m \<^esup>@ Bk\<^bsup>lnl\<^esup> @ Bk # Bk # ires \<and>
- (r = <ml> @ Bk\<^bsup>rn \<^esup>\<or> r = Bk # <ml> @ Bk\<^bsup>rn\<^esup>))"
-
-fun mopup_left_moving :: "mopup_type"
- where
- "mopup_left_moving (s, l, r) lm n ires =
- (\<exists> lnl lnr rn m.
- m = Suc (abc_lm_v lm n) \<and>
- ((l = Bk\<^bsup>lnr \<^esup>@ Oc\<^bsup>m \<^esup>@ Bk\<^bsup>lnl\<^esup> @ Bk # Bk # ires \<and> r = Bk\<^bsup>rn\<^esup>) \<or>
- (l = Oc\<^bsup>m - 1\<^esup> @ Bk\<^bsup>lnl\<^esup> @ Bk # Bk # ires \<and> r = Oc # Bk\<^bsup>rn\<^esup>)))"
-
-fun mopup_jump_over2 :: "mopup_type"
- where
- "mopup_jump_over2 (s, l, r) lm n ires =
- (\<exists> ln rn m1 m2.
- m1 + m2 = Suc (abc_lm_v lm n)
- \<and> r \<noteq> []
- \<and> (hd r = Oc \<longrightarrow> (l = Oc\<^bsup>m1\<^esup> @ Bk\<^bsup>ln\<^esup> @ Bk # Bk # ires \<and> r = Oc\<^bsup>m2\<^esup> @ Bk\<^bsup>rn\<^esup>))
- \<and> (hd r = Bk \<longrightarrow> (l = Bk\<^bsup>ln\<^esup> @ Bk # ires \<and> r = Bk # Oc\<^bsup>m1 + m2\<^esup> @ Bk\<^bsup>rn\<^esup>)))"
-
-
-fun mopup_inv :: "mopup_type"
- where
- "mopup_inv (s, l, r) lm n ires =
- (if s = 0 then mopup_stop (s, l, r) lm n ires
- else if s \<le> 2*n then
- if s mod 2 = 1 then mopup_bef_erase_a (s, l, r) lm n ires
- else mopup_bef_erase_b (s, l, r) lm n ires
- else if s = 2*n + 1 then
- mopup_jump_over1 (s, l, r) lm n ires
- else if s = 2*n + 2 then mopup_aft_erase_a (s, l, r) lm n ires
- else if s = 2*n + 3 then mopup_aft_erase_b (s, l, r) lm n ires
- else if s = 2*n + 4 then mopup_aft_erase_c (s, l, r) lm n ires
- else if s = 2*n + 5 then mopup_left_moving (s, l, r) lm n ires
- else if s = 2*n + 6 then mopup_jump_over2 (s, l, r) lm n ires
- else False)"
-
-declare
- mopup_jump_over2.simps[simp del] mopup_left_moving.simps[simp del]
- mopup_aft_erase_c.simps[simp del] mopup_aft_erase_b.simps[simp del]
- mopup_aft_erase_a.simps[simp del] mopup_jump_over1.simps[simp del]
- mopup_bef_erase_a.simps[simp del] mopup_bef_erase_b.simps[simp del]
- mopup_stop.simps[simp del]
-
-lemma mopup_fetch_0[simp]:
- "(fetch (mop_bef n @ tshift mp_up (2 * n)) 0 b) = (Nop, 0)"
-by(simp add: fetch.simps)
-
-lemma mop_bef_length[simp]: "length (mop_bef n) = 4 * n"
-apply(induct n, simp add: mop_bef.simps, simp add: mop_bef.simps)
-done
-
-thm findnth_nth
-lemma mop_bef_nth:
- "\<lbrakk>q < n; x < 4\<rbrakk> \<Longrightarrow> mop_bef n ! (4 * q + x) =
- mop_bef (Suc q) ! ((4 * q) + x)"
-apply(induct n, simp)
-apply(case_tac "q < n", simp add: mop_bef.simps, auto)
-apply(simp add: nth_append)
-apply(subgoal_tac "q = n", simp)
-apply(arith)
-done
-
-lemma fetch_bef_erase_a_o[simp]:
- "\<lbrakk>0 < s; s \<le> 2 * n; s mod 2 = Suc 0\<rbrakk>
- \<Longrightarrow> (fetch (mop_bef n @ tshift mp_up (2 * n)) s Oc) = (W0, s + 1)"
-apply(subgoal_tac "\<exists> q. s = 2*q + 1", auto)
-apply(subgoal_tac "length (mop_bef n) = 4*n")
-apply(auto simp: fetch.simps nth_of.simps nth_append)
-apply(subgoal_tac "mop_bef n ! (4 * q + 1) =
- mop_bef (Suc q) ! ((4 * q) + 1)",
- simp add: mop_bef.simps nth_append)
-apply(rule mop_bef_nth, auto)
-done
-
-lemma fetch_bef_erase_a_b[simp]:
- "\<lbrakk>n < length lm; 0 < s; s \<le> 2 * n; s mod 2 = Suc 0\<rbrakk>
- \<Longrightarrow> (fetch (mop_bef n @ tshift mp_up (2 * n)) s Bk) = (R, s + 2)"
-apply(subgoal_tac "\<exists> q. s = 2*q + 1", auto)
-apply(subgoal_tac "length (mop_bef n) = 4*n")
-apply(auto simp: fetch.simps nth_of.simps nth_append)
-apply(subgoal_tac "mop_bef n ! (4 * q + 0) =
- mop_bef (Suc q) ! ((4 * q + 0))",
- simp add: mop_bef.simps nth_append)
-apply(rule mop_bef_nth, auto)
-done
-
-lemma fetch_bef_erase_b_b:
- "\<lbrakk>n < length lm; 0 < s; s \<le> 2 * n; s mod 2 = 0\<rbrakk> \<Longrightarrow>
- (fetch (mop_bef n @ tshift mp_up (2 * n)) s Bk) = (R, s - 1)"
-apply(subgoal_tac "\<exists> q. s = 2 * q", auto)
-apply(case_tac qa, simp, simp)
-apply(auto simp: fetch.simps nth_of.simps nth_append)
-apply(subgoal_tac "mop_bef n ! (4 * nat + 2) =
- mop_bef (Suc nat) ! ((4 * nat) + 2)",
- simp add: mop_bef.simps nth_append)
-apply(rule mop_bef_nth, auto)
-done
-
-lemma fetch_jump_over1_o:
- "fetch (mop_bef n @ tshift mp_up (2 * n)) (Suc (2 * n)) Oc
- = (R, Suc (2 * n))"
-apply(subgoal_tac "length (mop_bef n) = 4 * n")
-apply(auto simp: fetch.simps nth_of.simps mp_up_def nth_append
- tshift.simps)
-done
-
-lemma fetch_jump_over1_b:
- "fetch (mop_bef n @ tshift mp_up (2 * n)) (Suc (2 * n)) Bk
- = (R, Suc (Suc (2 * n)))"
-apply(subgoal_tac "length (mop_bef n) = 4 * n")
-apply(auto simp: fetch.simps nth_of.simps mp_up_def
- nth_append tshift.simps)
-done
-
-lemma fetch_aft_erase_a_o:
- "fetch (mop_bef n @ tshift mp_up (2 * n)) (Suc (Suc (2 * n))) Oc
- = (W0, Suc (2 * n + 2))"
-apply(subgoal_tac "length (mop_bef n) = 4 * n")
-apply(auto simp: fetch.simps nth_of.simps mp_up_def
- nth_append tshift.simps)
-done
-
-lemma fetch_aft_erase_a_b:
- "fetch (mop_bef n @ tshift mp_up (2 * n)) (Suc (Suc (2 * n))) Bk
- = (L, Suc (2 * n + 4))"
-apply(subgoal_tac "length (mop_bef n) = 4 * n")
-apply(auto simp: fetch.simps nth_of.simps mp_up_def
- nth_append tshift.simps)
-done
-
-lemma fetch_aft_erase_b_b:
- "fetch (mop_bef n @ tshift mp_up (2 * n)) (2*n + 3) Bk
- = (R, Suc (2 * n + 3))"
-apply(subgoal_tac "length (mop_bef n) = 4 * n")
-apply(subgoal_tac "2*n + 3 = Suc (2*n + 2)", simp only: fetch.simps)
-apply(auto simp: nth_of.simps mp_up_def nth_append tshift.simps)
-done
-
-lemma fetch_aft_erase_c_o:
- "fetch (mop_bef n @ tshift mp_up (2 * n)) (2 * n + 4) Oc
- = (W0, Suc (2 * n + 2))"
-apply(subgoal_tac "length (mop_bef n) = 4 * n")
-apply(subgoal_tac "2*n + 4 = Suc (2*n + 3)", simp only: fetch.simps)
-apply(auto simp: nth_of.simps mp_up_def nth_append tshift.simps)
-done
-
-lemma fetch_aft_erase_c_b:
- "fetch (mop_bef n @ tshift mp_up (2 * n)) (2 * n + 4) Bk
- = (R, Suc (2 * n + 1))"
-apply(subgoal_tac "length (mop_bef n) = 4 * n")
-apply(subgoal_tac "2*n + 4 = Suc (2*n + 3)", simp only: fetch.simps)
-apply(auto simp: nth_of.simps mp_up_def nth_append tshift.simps)
-done
-
-lemma fetch_left_moving_o:
- "(fetch (mop_bef n @ tshift mp_up (2 * n)) (2 * n + 5) Oc)
- = (L, 2*n + 6)"
-apply(subgoal_tac "length (mop_bef n) = 4 * n")
-apply(subgoal_tac "2*n + 5 = Suc (2*n + 4)", simp only: fetch.simps)
-apply(auto simp: nth_of.simps mp_up_def nth_append tshift.simps)
-done
-
-lemma fetch_left_moving_b:
- "(fetch (mop_bef n @ tshift mp_up (2 * n)) (2 * n + 5) Bk)
- = (L, 2*n + 5)"
-apply(subgoal_tac "length (mop_bef n) = 4 * n")
-apply(subgoal_tac "2*n + 5 = Suc (2*n + 4)", simp only: fetch.simps)
-apply(auto simp: nth_of.simps mp_up_def nth_append tshift.simps)
-done
-
-lemma fetch_jump_over2_b:
- "(fetch (mop_bef n @ tshift mp_up (2 * n)) (2 * n + 6) Bk)
- = (R, 0)"
-apply(subgoal_tac "length (mop_bef n) = 4 * n")
-apply(subgoal_tac "2*n + 6 = Suc (2*n + 5)", simp only: fetch.simps)
-apply(auto simp: nth_of.simps mp_up_def nth_append tshift.simps)
-done
-
-lemma fetch_jump_over2_o:
-"(fetch (mop_bef n @ tshift mp_up (2 * n)) (2 * n + 6) Oc)
- = (L, 2*n + 6)"
-apply(subgoal_tac "length (mop_bef n) = 4 * n")
-apply(subgoal_tac "2*n + 6 = Suc (2*n + 5)", simp only: fetch.simps)
-apply(auto simp: nth_of.simps mp_up_def nth_append tshift.simps)
-done
-
-lemmas mopupfetchs =
-fetch_bef_erase_a_o fetch_bef_erase_a_b fetch_bef_erase_b_b
-fetch_jump_over1_o fetch_jump_over1_b fetch_aft_erase_a_o
-fetch_aft_erase_a_b fetch_aft_erase_b_b fetch_aft_erase_c_o
-fetch_aft_erase_c_b fetch_left_moving_o fetch_left_moving_b
-fetch_jump_over2_b fetch_jump_over2_o
-
-lemma [simp]:
-"\<lbrakk>n < length lm; 0 < s; s mod 2 = Suc 0;
- mopup_bef_erase_a (s, l, Oc # xs) lm n ires;
- Suc s \<le> 2 * n\<rbrakk> \<Longrightarrow>
- mopup_bef_erase_b (Suc s, l, Bk # xs) lm n ires"
-apply(auto simp: mopup_bef_erase_a.simps mopup_bef_erase_b.simps )
-apply(rule_tac x = "m - 1" in exI, rule_tac x = rn in exI)
-apply(case_tac m, simp, simp)
-done
-
-lemma mopup_false1:
- "\<lbrakk>0 < s; s \<le> 2 * n; s mod 2 = Suc 0; \<not> Suc s \<le> 2 * n\<rbrakk>
- \<Longrightarrow> RR"
-apply(arith)
-done
-
-lemma [simp]:
- "\<lbrakk>n < length lm; 0 < s; s \<le> 2 * n; s mod 2 = Suc 0;
- mopup_bef_erase_a (s, l, Oc # xs) lm n ires; r = Oc # xs\<rbrakk>
- \<Longrightarrow> (Suc s \<le> 2 * n \<longrightarrow> mopup_bef_erase_b (Suc s, l, Bk # xs) lm n ires) \<and>
- (\<not> Suc s \<le> 2 * n \<longrightarrow> mopup_jump_over1 (Suc s, l, Bk # xs) lm n ires) "
-apply(auto elim: mopup_false1)
-done
-
-lemma drop_abc_lm_v_simp[simp]:
- "n < length lm \<Longrightarrow> drop n lm = abc_lm_v lm n # drop (Suc n) lm"
-apply(auto simp: abc_lm_v.simps)
-apply(drule drop_Suc_conv_tl, simp)
-done
-lemma [simp]: "(\<exists>rna. Bk\<^bsup>rn\<^esup> = Bk # Bk\<^bsup>rna\<^esup>) \<or> Bk\<^bsup>rn\<^esup> = []"
-apply(case_tac rn, simp, auto)
-done
-
-lemma [simp]: "\<exists>lna. Bk # Bk\<^bsup>ln\<^esup> = Bk\<^bsup>lna\<^esup>"
-apply(rule_tac x = "Suc ln" in exI, auto)
-done
-
-lemma mopup_bef_erase_a_2_jump_over[simp]:
- "\<lbrakk>n < length lm; 0 < s; s mod 2 = Suc 0;
- mopup_bef_erase_a (s, l, Bk # xs) lm n ires; Suc s = 2 * n\<rbrakk>
-\<Longrightarrow> mopup_jump_over1 (Suc (2 * n), Bk # l, xs) lm n ires"
-apply(auto simp: mopup_bef_erase_a.simps mopup_jump_over1.simps)
-apply(case_tac m, simp)
-apply(rule_tac x = "Suc ln" in exI, rule_tac x = 0 in exI,
- simp add: tape_of_nl_abv)
-apply(case_tac "drop (Suc n) lm", auto simp: tape_of_nat_list.simps )
-done
-
-lemma Suc_Suc_div: "\<lbrakk>0 < s; s mod 2 = Suc 0; Suc (Suc s) \<le> 2 * n\<rbrakk>
- \<Longrightarrow> (Suc (Suc (s div 2))) \<le> n"
-apply(arith)
-done
-
-lemma mopup_bef_erase_a_2_a[simp]:
- "\<lbrakk>n < length lm; 0 < s; s mod 2 = Suc 0;
- mopup_bef_erase_a (s, l, Bk # xs) lm n ires;
- Suc (Suc s) \<le> 2 * n\<rbrakk> \<Longrightarrow>
- mopup_bef_erase_a (Suc (Suc s), Bk # l, xs) lm n ires"
-apply(auto simp: mopup_bef_erase_a.simps )
-apply(subgoal_tac "drop (Suc (Suc (s div 2))) lm \<noteq> []")
-apply(case_tac m, simp)
-apply(rule_tac x = "Suc (abc_lm_v lm (Suc (s div 2)))" in exI,
- rule_tac x = rn in exI, simp, simp)
-apply(subgoal_tac "(Suc (Suc (s div 2))) \<le> n", simp,
- rule_tac Suc_Suc_div, auto)
-done
-
-lemma mopup_false2:
- "\<lbrakk>n < length lm; 0 < s; s \<le> 2 * n;
- s mod 2 = Suc 0; Suc s \<noteq> 2 * n;
- \<not> Suc (Suc s) \<le> 2 * n\<rbrakk> \<Longrightarrow> RR"
-apply(arith)
-done
-
-lemma [simp]:
- "\<lbrakk>n < length lm; 0 < s; s \<le> 2 * n;
- s mod 2 = Suc 0;
- mopup_bef_erase_a (s, l, Bk # xs) lm n ires;
- r = Bk # xs\<rbrakk>
- \<Longrightarrow> (Suc s = 2 * n \<longrightarrow>
- mopup_jump_over1 (Suc (2 * n), Bk # l, xs) lm n ires) \<and>
- (Suc s \<noteq> 2 * n \<longrightarrow>
- (Suc (Suc s) \<le> 2 * n \<longrightarrow>
- mopup_bef_erase_a (Suc (Suc s), Bk # l, xs) lm n ires) \<and>
- (\<not> Suc (Suc s) \<le> 2 * n \<longrightarrow>
- mopup_aft_erase_a (Suc (Suc s), Bk # l, xs) lm n ires))"
-apply(auto elim: mopup_false2)
-done
-
-lemma [simp]: "mopup_bef_erase_a (s, l, []) lm n ires \<Longrightarrow>
- mopup_bef_erase_a (s, l, [Bk]) lm n ires"
-apply(auto simp: mopup_bef_erase_a.simps)
-done
-
-lemma [simp]:
- "\<lbrakk>n < length lm; 0 < s; s \<le> 2 * n; s mod 2 = Suc 0;
- mopup_bef_erase_a (s, l, []) lm n ires; r = []\<rbrakk>
- \<Longrightarrow> (Suc s = 2 * n \<longrightarrow>
- mopup_jump_over1 (Suc (2 * n), Bk # l, []) lm n ires) \<and>
- (Suc s \<noteq> 2 * n \<longrightarrow>
- (Suc (Suc s) \<le> 2 * n \<longrightarrow>
- mopup_bef_erase_a (Suc (Suc s), Bk # l, []) lm n ires) \<and>
- (\<not> Suc (Suc s) \<le> 2 * n \<longrightarrow>
- mopup_aft_erase_a (Suc (Suc s), Bk # l, []) lm n ires))"
-apply(auto)
-done
-
-lemma "mopup_bef_erase_b (s, l, Oc # xs) lm n ires \<Longrightarrow> l \<noteq> []"
-apply(auto simp: mopup_bef_erase_b.simps)
-done
-
-lemma [simp]: "mopup_bef_erase_b (s, l, Oc # xs) lm n ires = False"
-apply(auto simp: mopup_bef_erase_b.simps )
-done
-
-lemma [simp]: "\<lbrakk>0 < s; s \<le> 2 *n; s mod 2 \<noteq> Suc 0\<rbrakk> \<Longrightarrow>
- (s - Suc 0) mod 2 = Suc 0"
-apply(arith)
-done
-
-lemma [simp]: "\<lbrakk>0 < s; s \<le> 2 *n; s mod 2 \<noteq> Suc 0\<rbrakk> \<Longrightarrow>
- s - Suc 0 \<le> 2 * n"
-apply(simp)
-done
-
-lemma [simp]: "\<lbrakk>0 < s; s \<le> 2 *n; s mod 2 \<noteq> Suc 0\<rbrakk> \<Longrightarrow> \<not> s \<le> Suc 0"
-apply(arith)
-done
-
-lemma [simp]: "\<lbrakk>n < length lm; 0 < s; s \<le> 2 * n;
- s mod 2 \<noteq> Suc 0;
- mopup_bef_erase_b (s, l, Bk # xs) lm n ires; r = Bk # xs\<rbrakk>
- \<Longrightarrow> mopup_bef_erase_a (s - Suc 0, Bk # l, xs) lm n ires"
-apply(auto simp: mopup_bef_erase_b.simps mopup_bef_erase_a.simps)
-done
-
-lemma [simp]: "\<lbrakk>mopup_bef_erase_b (s, l, []) lm n ires\<rbrakk> \<Longrightarrow>
- mopup_bef_erase_a (s - Suc 0, Bk # l, []) lm n ires"
-apply(auto simp: mopup_bef_erase_b.simps mopup_bef_erase_a.simps)
-done
-
-lemma [simp]:
- "\<lbrakk>n < length lm;
- mopup_jump_over1 (Suc (2 * n), l, Oc # xs) lm n ires;
- r = Oc # xs\<rbrakk>
- \<Longrightarrow> mopup_jump_over1 (Suc (2 * n), Oc # l, xs) lm n ires"
-apply(auto simp: mopup_jump_over1.simps)
-apply(rule_tac x = ln in exI, rule_tac x = "Suc m1" in exI,
- rule_tac x = "m2 - 1" in exI)
-apply(case_tac "m2", simp, simp, rule_tac x = rn in exI, simp)
-apply(rule_tac x = ln in exI, rule_tac x = "Suc m1" in exI,
- rule_tac x = "m2 - 1" in exI)
-apply(case_tac m2, simp, simp)
-done
-
-lemma mopup_jump_over1_2_aft_erase_a[simp]:
- "\<lbrakk>n < length lm; mopup_jump_over1 (Suc (2 * n), l, Bk # xs) lm n ires\<rbrakk>
- \<Longrightarrow> mopup_aft_erase_a (Suc (Suc (2 * n)), Bk # l, xs) lm n ires"
-apply(simp only: mopup_jump_over1.simps mopup_aft_erase_a.simps)
-apply(erule_tac exE)+
-apply(rule_tac x = ln in exI, rule_tac x = "Suc 0" in exI)
-apply(case_tac m2, simp)
-apply(rule_tac x = rn in exI, rule_tac x = "drop (Suc n) lm" in exI,
- simp)
-apply(simp)
-done
-
-lemma [simp]:
- "\<lbrakk>n < length lm; mopup_jump_over1 (Suc (2 * n), l, []) lm n ires\<rbrakk> \<Longrightarrow>
- mopup_aft_erase_a (Suc (Suc (2 * n)), Bk # l, []) lm n ires"
-apply(rule mopup_jump_over1_2_aft_erase_a, simp)
-apply(auto simp: mopup_jump_over1.simps)
-apply(rule_tac x = ln in exI, rule_tac x = m1 in exI,
- rule_tac x = m2 in exI, simp add: )
-apply(rule_tac x = 0 in exI, auto)
-done
-
-lemma [simp]:
- "\<lbrakk>n < length lm;
- mopup_aft_erase_a (Suc (Suc (2 * n)), l, Oc # xs) lm n ires\<rbrakk>
- \<Longrightarrow> mopup_aft_erase_b (Suc (Suc (Suc (2 * n))), l, Bk # xs) lm n ires"
-apply(auto simp: mopup_aft_erase_a.simps mopup_aft_erase_b.simps )
-apply(case_tac ml, simp, case_tac rn, simp, simp)
-apply(case_tac list, auto simp: tape_of_nl_abv
- tape_of_nat_list.simps )
-apply(case_tac a, simp, rule_tac x = rn in exI,
- rule_tac x = "[]" in exI,
- simp add: tape_of_nat_list.simps, simp)
-apply(rule_tac x = rn in exI, rule_tac x = "[nat]" in exI,
- simp add: tape_of_nat_list.simps )
-apply(case_tac a, simp, rule_tac x = rn in exI,
- rule_tac x = "aa # lista" in exI, simp, simp)
-apply(rule_tac x = rn in exI, rule_tac x = "nat # aa # lista" in exI,
- simp add: tape_of_nat_list.simps )
-done
-
-lemma [simp]:
- "mopup_aft_erase_a (Suc (Suc (2 * n)), l, Bk # xs) lm n ires \<Longrightarrow> l \<noteq> []"
-apply(auto simp: mopup_aft_erase_a.simps)
-done
-
-lemma [simp]:
- "\<lbrakk>n < length lm;
- mopup_aft_erase_a (Suc (Suc (2 * n)), l, Bk # xs) lm n ires\<rbrakk>
- \<Longrightarrow> mopup_left_moving (5 + 2 * n, tl l, hd l # Bk # xs) lm n ires"
-apply(simp only: mopup_aft_erase_a.simps mopup_left_moving.simps)
-apply(erule exE)+
-apply(case_tac lnr, simp)
-apply(rule_tac x = lnl in exI, simp, rule_tac x = rn in exI, simp)
-apply(subgoal_tac "ml = []", simp)
-apply(rule_tac xs = xs and rn = rn in BkCons_nil, simp, auto)
-apply(subgoal_tac "ml = []", auto)
-apply(rule_tac xs = xs and rn = rn in BkCons_nil, simp)
-done
-
-lemma [simp]:
- "mopup_aft_erase_a (Suc (Suc (2 * n)), l, []) lm n ires \<Longrightarrow> l \<noteq> []"
-apply(simp only: mopup_aft_erase_a.simps)
-apply(erule exE)+
-apply(auto)
-done
-
-lemma [simp]:
- "\<lbrakk>n < length lm; mopup_aft_erase_a (Suc (Suc (2 * n)), l, []) lm n ires\<rbrakk>
- \<Longrightarrow> mopup_left_moving (5 + 2 * n, tl l, [hd l]) lm n ires"
-apply(simp only: mopup_aft_erase_a.simps mopup_left_moving.simps)
-apply(erule exE)+
-apply(subgoal_tac "ml = [] \<and> rn = 0", erule conjE, erule conjE, simp)
-apply(case_tac lnr, simp, rule_tac x = lnl in exI, simp,
- rule_tac x = 0 in exI, simp)
-apply(rule_tac x = lnl in exI, rule_tac x = nat in exI,
- rule_tac x = "Suc 0" in exI, simp)
-apply(case_tac ml, simp, case_tac rn, simp, simp)
-apply(case_tac list, auto simp: tape_of_nl_abv tape_of_nat_list.simps)
-done
-
-lemma [simp]: "mopup_aft_erase_b (2 * n + 3, l, Oc # xs) lm n ires = False"
-apply(auto simp: mopup_aft_erase_b.simps )
-done
-
-lemma [simp]:
- "\<lbrakk>n < length lm;
- mopup_aft_erase_c (2 * n + 4, l, Oc # xs) lm n ires\<rbrakk>
- \<Longrightarrow> mopup_aft_erase_b (Suc (Suc (Suc (2 * n))), l, Bk # xs) lm n ires"
-apply(auto simp: mopup_aft_erase_c.simps mopup_aft_erase_b.simps )
-apply(case_tac ml, simp, case_tac rn, simp, simp, simp)
-apply(case_tac list, auto simp: tape_of_nl_abv
- tape_of_nat_list.simps tape_of_nat_abv )
-apply(case_tac a, rule_tac x = rn in exI,
- rule_tac x = "[]" in exI, simp add: tape_of_nat_list.simps)
-apply(rule_tac x = rn in exI, rule_tac x = "[nat]" in exI,
- simp add: tape_of_nat_list.simps )
-apply(case_tac a, simp, rule_tac x = rn in exI,
- rule_tac x = "aa # lista" in exI, simp)
-apply(rule_tac x = rn in exI, rule_tac x = "nat # aa # lista" in exI,
- simp add: tape_of_nat_list.simps )
-done
-
-lemma mopup_aft_erase_c_aft_erase_a[simp]:
- "\<lbrakk>n < length lm; mopup_aft_erase_c (2 * n + 4, l, Bk # xs) lm n ires\<rbrakk>
- \<Longrightarrow> mopup_aft_erase_a (Suc (Suc (2 * n)), Bk # l, xs) lm n ires"
-apply(simp only: mopup_aft_erase_c.simps mopup_aft_erase_a.simps )
-apply(erule_tac exE)+
-apply(erule conjE, erule conjE, erule disjE)
-apply(subgoal_tac "ml = []", simp, case_tac rn,
- simp, simp, rule conjI)
-apply(rule_tac x = lnl in exI, rule_tac x = "Suc lnr" in exI, simp)
-apply(rule_tac x = nat in exI, rule_tac x = "[]" in exI, simp)
-apply(rule_tac xs = xs and rn = rn in BkCons_nil, simp, simp,
- rule conjI)
-apply(rule_tac x = lnl in exI, rule_tac x = "Suc lnr" in exI, simp)
-apply(rule_tac x = rn in exI, rule_tac x = "ml" in exI, simp)
-done
-
-lemma [simp]:
- "\<lbrakk>n < length lm; mopup_aft_erase_c (2 * n + 4, l, []) lm n ires\<rbrakk>
- \<Longrightarrow> mopup_aft_erase_a (Suc (Suc (2 * n)), Bk # l, []) lm n ires"
-apply(rule mopup_aft_erase_c_aft_erase_a, simp)
-apply(simp only: mopup_aft_erase_c.simps)
-apply(erule exE)+
-apply(rule_tac x = lnl in exI, rule_tac x = lnr in exI, simp add: )
-apply(rule_tac x = 0 in exI, rule_tac x = "[]" in exI, simp)
-done
-
-lemma mopup_aft_erase_b_2_aft_erase_c[simp]:
- "\<lbrakk>n < length lm; mopup_aft_erase_b (2 * n + 3, l, Bk # xs) lm n ires\<rbrakk>
- \<Longrightarrow> mopup_aft_erase_c (4 + 2 * n, Bk # l, xs) lm n ires"
-apply(auto simp: mopup_aft_erase_b.simps mopup_aft_erase_c.simps)
-apply(rule_tac x = "lnl" in exI, rule_tac x = "Suc lnr" in exI, simp)
-apply(rule_tac x = "lnl" in exI, rule_tac x = "Suc lnr" in exI, simp)
-done
-
-lemma [simp]:
- "\<lbrakk>n < length lm; mopup_aft_erase_b (2 * n + 3, l, []) lm n ires\<rbrakk>
- \<Longrightarrow> mopup_aft_erase_c (4 + 2 * n, Bk # l, []) lm n ires"
-apply(rule_tac mopup_aft_erase_b_2_aft_erase_c, simp)
-apply(simp add: mopup_aft_erase_b.simps)
-done
-
-lemma [simp]:
- "mopup_left_moving (2 * n + 5, l, Oc # xs) lm n ires \<Longrightarrow> l \<noteq> []"
-apply(auto simp: mopup_left_moving.simps)
-done
-
-lemma [simp]:
- "\<lbrakk>n < length lm; mopup_left_moving (2 * n + 5, l, Oc # xs) lm n ires\<rbrakk>
- \<Longrightarrow> mopup_jump_over2 (2 * n + 6, tl l, hd l # Oc # xs) lm n ires"
-apply(simp only: mopup_left_moving.simps mopup_jump_over2.simps)
-apply(erule_tac exE)+
-apply(erule conjE, erule disjE, erule conjE)
-apply(case_tac rn, simp, simp add: )
-apply(case_tac "hd l", simp add: )
-apply(case_tac "abc_lm_v lm n", simp)
-apply(rule_tac x = "lnl" in exI, rule_tac x = rn in exI,
- rule_tac x = "Suc 0" in exI, rule_tac x = 0 in exI)
-apply(case_tac lnl, simp, simp, simp add: exp_ind[THEN sym], simp)
-apply(case_tac "abc_lm_v lm n", simp)
-apply(case_tac lnl, simp, simp)
-apply(rule_tac x = lnl in exI, rule_tac x = rn in exI)
-apply(rule_tac x = nat in exI, rule_tac x = "Suc (Suc 0)" in exI, simp)
-done
-
-lemma [simp]: "mopup_left_moving (2 * n + 5, l, xs) lm n ires \<Longrightarrow> l \<noteq> []"
-apply(auto simp: mopup_left_moving.simps)
-done
-
-lemma [simp]:
- "\<lbrakk>n < length lm; mopup_left_moving (2 * n + 5, l, Bk # xs) lm n ires\<rbrakk>
- \<Longrightarrow> mopup_left_moving (2 * n + 5, tl l, hd l # Bk # xs) lm n ires"
-apply(simp only: mopup_left_moving.simps)
-apply(erule exE)+
-apply(case_tac lnr, simp)
-apply(rule_tac x = lnl in exI, rule_tac x = 0 in exI,
- rule_tac x = rn in exI, simp, simp)
-apply(rule_tac x = lnl in exI, rule_tac x = nat in exI, simp)
-done
-
-lemma [simp]:
-"\<lbrakk>n < length lm; mopup_left_moving (2 * n + 5, l, []) lm n ires\<rbrakk>
- \<Longrightarrow> mopup_left_moving (2 * n + 5, tl l, [hd l]) lm n ires"
-apply(simp only: mopup_left_moving.simps)
-apply(erule exE)+
-apply(case_tac lnr, simp)
-apply(rule_tac x = lnl in exI, rule_tac x = 0 in exI,
- rule_tac x = 0 in exI, simp, auto)
-done
-
-lemma [simp]:
- "mopup_jump_over2 (2 * n + 6, l, Oc # xs) lm n ires \<Longrightarrow> l \<noteq> []"
-apply(auto simp: mopup_jump_over2.simps )
-done
-
-lemma [intro]: "\<exists>lna. Bk # Bk\<^bsup>ln\<^esup> = Bk\<^bsup>lna\<^esup> @ [Bk]"
-apply(simp only: exp_ind[THEN sym], auto)
-done
-
-lemma [simp]:
-"\<lbrakk>n < length lm; mopup_jump_over2 (2 * n + 6, l, Oc # xs) lm n ires\<rbrakk>
- \<Longrightarrow> mopup_jump_over2 (2 * n + 6, tl l, hd l # Oc # xs) lm n ires"
-apply(simp only: mopup_jump_over2.simps)
-apply(erule_tac exE)+
-apply(simp add: , erule conjE, erule_tac conjE)
-apply(case_tac m1, simp)
-apply(rule_tac x = ln in exI, rule_tac x = rn in exI,
- rule_tac x = 0 in exI, simp)
-apply(case_tac ln, simp, simp, simp only: exp_ind[THEN sym], simp)
-apply(rule_tac x = ln in exI, rule_tac x = rn in exI,
- rule_tac x = nat in exI, rule_tac x = "Suc m2" in exI, simp)
-done
-
-lemma [simp]: "\<exists>rna. Oc # Oc\<^bsup>a\<^esup> @ Bk\<^bsup>rn\<^esup> = <a> @ Bk\<^bsup>rna\<^esup>"
-apply(case_tac a, auto simp: tape_of_nat_abv )
-done
-
-lemma [simp]:
- "\<lbrakk>n < length lm; mopup_jump_over2 (2 * n + 6, l, Bk # xs) lm n ires\<rbrakk>
- \<Longrightarrow> mopup_stop (0, Bk # l, xs) lm n ires"
-apply(auto simp: mopup_jump_over2.simps mopup_stop.simps)
-done
-
-lemma [simp]: "mopup_jump_over2 (2 * n + 6, l, []) lm n ires = False"
-apply(simp only: mopup_jump_over2.simps, simp)
-done
-
-lemma mopup_inv_step:
- "\<lbrakk>n < length lm; mopup_inv (s, l, r) lm n ires\<rbrakk>
- \<Longrightarrow> mopup_inv (t_step (s, l, r)
- ((mop_bef n @ tshift mp_up (2 * n)), 0)) lm n ires"
-apply(auto split:if_splits simp add:t_step.simps,
- tactic {* ALLGOALS (resolve_tac [@{thm "fetch_intro"}]) *})
-apply(simp_all add: mopupfetchs new_tape.simps)
-done
-
-declare mopup_inv.simps[simp del]
-
-lemma mopup_inv_steps:
-"\<lbrakk>n < length lm; mopup_inv (s, l, r) lm n ires\<rbrakk> \<Longrightarrow>
- mopup_inv (t_steps (s, l, r)
- ((mop_bef n @ tshift mp_up (2 * n)), 0) stp) lm n ires"
-apply(induct stp, simp add: t_steps.simps)
-apply(simp add: t_steps_ind)
-apply(case_tac "(t_steps (s, l, r)
- (mop_bef n @ tshift mp_up (2 * n), 0) stp)", simp)
-apply(rule_tac mopup_inv_step, simp, simp)
-done
-
-lemma [simp]:
- "\<lbrakk>n < length lm; Suc 0 \<le> n\<rbrakk> \<Longrightarrow>
- mopup_bef_erase_a (Suc 0, Bk\<^bsup>ln\<^esup> @ Bk # Bk # ires, <lm> @ Bk\<^bsup>rn\<^esup>) lm n ires"
-apply(auto simp: mopup_bef_erase_a.simps abc_lm_v.simps)
-apply(case_tac lm, simp, case_tac list, simp, simp)
-apply(rule_tac x = "Suc a" in exI, rule_tac x = rn in exI, simp)
-done
-
-lemma [simp]:
- "lm \<noteq> [] \<Longrightarrow> mopup_jump_over1 (Suc 0, Bk\<^bsup>ln\<^esup> @ Bk # Bk # ires, <lm> @ Bk\<^bsup>rn\<^esup>) lm 0 ires"
-apply(auto simp: mopup_jump_over1.simps)
-apply(rule_tac x = ln in exI, rule_tac x = 0 in exI, simp add: )
-apply(case_tac lm, simp, simp add: abc_lm_v.simps)
-apply(case_tac rn, simp)
-apply(case_tac list, rule_tac disjI2,
- simp add: tape_of_nl_abv tape_of_nat_list.simps)
-apply(rule_tac disjI1,
- simp add: tape_of_nl_abv tape_of_nat_list.simps )
-apply(rule_tac disjI1, case_tac list,
- simp add: tape_of_nl_abv tape_of_nat_list.simps,
- rule_tac x = nat in exI, simp)
-apply(simp add: tape_of_nl_abv tape_of_nat_list.simps )
-done
-
-lemma mopup_init:
- "\<lbrakk>n < length lm; crsp_l ly (as, lm) (ac, l, r) ires\<rbrakk> \<Longrightarrow>
- mopup_inv (Suc 0, l, r) lm n ires"
-apply(auto simp: crsp_l.simps mopup_inv.simps)
-apply(case_tac n, simp, auto simp: mopup_bef_erase_a.simps )
-apply(rule_tac x = "Suc (hd lm)" in exI, rule_tac x = rn in exI, simp)
-apply(case_tac lm, simp, case_tac list, simp, case_tac lista, simp add: abc_lm_v.simps)
-apply(simp add: tape_of_nl_abv tape_of_nat_list.simps abc_lm_v.simps)
-apply(simp add: mopup_jump_over1.simps)
-apply(rule_tac x = 0 in exI, rule_tac x = 0 in exI, auto)
-apply(case_tac [!] n, simp_all)
-apply(case_tac [!] lm, simp, case_tac list, simp)
-apply(case_tac rn, simp add: tape_of_nl_abv tape_of_nat_list.simps abc_lm_v.simps)
-apply(erule_tac x = nat in allE, simp add: tape_of_nl_abv tape_of_nat_list.simps abc_lm_v.simps)
-apply(simp add: abc_lm_v.simps, auto)
-apply(case_tac list, simp_all add: tape_of_nl_abv tape_of_nat_list.simps abc_lm_v.simps)
-apply(erule_tac x = rn in allE, simp_all)
-done
-
-fun abc_mopup_stage1 :: "t_conf \<Rightarrow> nat \<Rightarrow> nat"
- where
- "abc_mopup_stage1 (s, l, r) n =
- (if s > 0 \<and> s \<le> 2*n then 6
- else if s = 2*n + 1 then 4
- else if s \<ge> 2*n + 2 \<and> s \<le> 2*n + 4 then 3
- else if s = 2*n + 5 then 2
- else if s = 2*n + 6 then 1
- else 0)"
-
-fun abc_mopup_stage2 :: "t_conf \<Rightarrow> nat \<Rightarrow> nat"
- where
- "abc_mopup_stage2 (s, l, r) n =
- (if s > 0 \<and> s \<le> 2*n then length r
- else if s = 2*n + 1 then length r
- else if s = 2*n + 5 then length l
- else if s = 2*n + 6 then length l
- else if s \<ge> 2*n + 2 \<and> s \<le> 2*n + 4 then length r
- else 0)"
-
-fun abc_mopup_stage3 :: "t_conf \<Rightarrow> nat \<Rightarrow> nat"
- where
- "abc_mopup_stage3 (s, l, r) n =
- (if s > 0 \<and> s \<le> 2*n then
- if hd r = Bk then 0
- else 1
- else if s = 2*n + 2 then 1
- else if s = 2*n + 3 then 0
- else if s = 2*n + 4 then 2
- else 0)"
-
-fun abc_mopup_measure :: "(t_conf \<times> nat) \<Rightarrow> (nat \<times> nat \<times> nat)"
- where
- "abc_mopup_measure (c, n) =
- (abc_mopup_stage1 c n, abc_mopup_stage2 c n,
- abc_mopup_stage3 c n)"
-
-definition abc_mopup_LE ::
- "(((nat \<times> block list \<times> block list) \<times> nat) \<times>
- ((nat \<times> block list \<times> block list) \<times> nat)) set"
- where
- "abc_mopup_LE \<equiv> (inv_image lex_triple abc_mopup_measure)"
-
-lemma wf_abc_mopup_le[intro]: "wf abc_mopup_LE"
-by(auto intro:wf_inv_image wf_lex_triple simp:abc_mopup_LE_def)
-
-lemma [simp]: "mopup_bef_erase_a (a, aa, []) lm n ires = False"
-apply(auto simp: mopup_bef_erase_a.simps)
-done
-
-lemma [simp]: "mopup_bef_erase_b (a, aa, []) lm n ires = False"
-apply(auto simp: mopup_bef_erase_b.simps)
-done
-
-lemma [simp]: "mopup_aft_erase_b (2 * n + 3, aa, []) lm n ires = False"
-apply(auto simp: mopup_aft_erase_b.simps)
-done
-
-lemma mopup_halt_pre:
- "\<lbrakk>n < length lm; mopup_inv (Suc 0, l, r) lm n ires; wf abc_mopup_LE\<rbrakk>
- \<Longrightarrow> \<forall>na. \<not> (\<lambda>(s, l, r) n. s = 0) (t_steps (Suc 0, l, r)
- (mop_bef n @ tshift mp_up (2 * n), 0) na) n \<longrightarrow>
- ((t_steps (Suc 0, l, r) (mop_bef n @ tshift mp_up (2 * n), 0)
- (Suc na), n),
- t_steps (Suc 0, l, r) (mop_bef n @ tshift mp_up (2 * n), 0)
- na, n) \<in> abc_mopup_LE"
-apply(rule allI, rule impI, simp add: t_steps_ind)
-apply(subgoal_tac "mopup_inv (t_steps (Suc 0, l, r)
- (mop_bef n @ tshift mp_up (2 * n), 0) na) lm n ires")
-apply(case_tac "(t_steps (Suc 0, l, r)
- (mop_bef n @ tshift mp_up (2 * n), 0) na)", simp)
-proof -
- fix na a b c
- assume "n < length lm" "mopup_inv (a, b, c) lm n ires" "0 < a"
- thus "((t_step (a, b, c) (mop_bef n @ tshift mp_up (2 * n), 0), n),
- (a, b, c), n) \<in> abc_mopup_LE"
- apply(auto split:if_splits simp add:t_step.simps mopup_inv.simps,
- tactic {* ALLGOALS (resolve_tac [@{thm "fetch_intro"}]) *})
- apply(simp_all add: mopupfetchs new_tape.simps abc_mopup_LE_def
- lex_triple_def lex_pair_def )
- done
-next
- fix na
- assume "n < length lm" "mopup_inv (Suc 0, l, r) lm n ires"
- thus "mopup_inv (t_steps (Suc 0, l, r)
- (mop_bef n @ tshift mp_up (2 * n), 0) na) lm n ires"
- apply(rule mopup_inv_steps)
- done
-qed
-
-lemma mopup_halt: "\<lbrakk>n < length lm; crsp_l ly (as, lm) (s, l, r) ires\<rbrakk> \<Longrightarrow>
- \<exists> stp. (\<lambda> (s, l, r). s = 0) (t_steps (Suc 0, l, r)
- ((mop_bef n @ tshift mp_up (2 * n)), 0) stp)"
-apply(subgoal_tac "mopup_inv (Suc 0, l, r) lm n ires")
-apply(insert wf_abc_mopup_le)
-apply(insert halt_lemma[of abc_mopup_LE
- "\<lambda> ((s, l, r), n). s = 0"
- "\<lambda> stp. (t_steps (Suc 0, l, r) ((mop_bef n @ tshift mp_up (2 * n))
- , 0) stp, n)"], auto)
-apply(insert mopup_halt_pre[of n lm l r], simp, erule exE)
-apply(rule_tac x = na in exI, case_tac "(t_steps (Suc 0, l, r)
- (mop_bef n @ tshift mp_up (2 * n), 0) na)", simp)
-apply(rule_tac mopup_init, auto)
-done
-(***End: mopup stop****)
-
-lemma mopup_halt_conf_pre:
- "\<lbrakk>n < length lm; crsp_l ly (as, lm) (s, l, r) ires\<rbrakk>
- \<Longrightarrow> \<exists> na. (\<lambda> (s', l', r'). s' = 0 \<and> mopup_stop (s', l', r') lm n ires)
- (t_steps (Suc 0, l, r)
- ((mop_bef n @ tshift mp_up (2 * n)), 0) na)"
-apply(subgoal_tac "\<exists> stp. (\<lambda> (s, l, r). s = 0)
- (t_steps (Suc 0, l, r) ((mop_bef n @ tshift mp_up (2 * n)), 0) stp)",
- erule exE)
-apply(rule_tac x = stp in exI,
- case_tac "(t_steps (Suc 0, l, r)
- (mop_bef n @ tshift mp_up (2 * n), 0) stp)", simp)
-apply(subgoal_tac " mopup_inv (Suc 0, l, r) lm n ires")
-apply(subgoal_tac "mopup_inv (t_steps (Suc 0, l, r)
- (mop_bef n @ tshift mp_up (2 * n), 0) stp) lm n ires", simp)
-apply(simp only: mopup_inv.simps)
-apply(rule_tac mopup_inv_steps, simp, simp)
-apply(rule_tac mopup_init, simp, simp)
-apply(rule_tac mopup_halt, simp, simp)
-done
-
-lemma mopup_halt_conf:
- assumes len: "n < length lm"
- and correspond: "crsp_l ly (as, lm) (s, l, r) ires"
- shows
- "\<exists> na. (\<lambda> (s', l', r'). ((\<exists>ln rn. s' = 0 \<and> l' = Bk\<^bsup>ln\<^esup> @ Bk # Bk # ires
- \<and> r' = Oc\<^bsup>Suc (abc_lm_v lm n)\<^esup> @ Bk\<^bsup>rn\<^esup>)))
- (t_steps (Suc 0, l, r)
- ((mop_bef n @ tshift mp_up (2 * n)), 0) na)"
-using len correspond mopup_halt_conf_pre[of n lm ly as s l r ires]
-apply(simp add: mopup_stop.simps tape_of_nat_abv tape_of_nat_list.simps)
-done
-
-subsection {* Final results about Abacus machine *}
-
-lemma mopup_halt_bef: "\<lbrakk>n < length lm; crsp_l ly (as, lm) (s, l, r) ires\<rbrakk>
- \<Longrightarrow> \<exists>stp. (\<lambda>(s, l, r). s \<noteq> 0 \<and> ((\<lambda> (s', l', r'). s' = 0)
- (t_step (s, l, r) (mop_bef n @ tshift mp_up (2 * n), 0))))
- (t_steps (Suc 0, l, r) (mop_bef n @ tshift mp_up (2 * n), 0) stp)"
-apply(insert mopup_halt[of n lm ly as s l r ires], simp, erule_tac exE)
-proof -
- fix stp
- assume "n < length lm"
- "crsp_l ly (as, lm) (s, l, r) ires"
- "(\<lambda>(s, l, r). s = 0)
- (t_steps (Suc 0, l, r)
- (mop_bef n @ tshift mp_up (2 * n), 0) stp)"
- thus "\<exists>stp. (\<lambda>(s, ab). 0 < s \<and> (\<lambda>(s', l', r'). s' = 0)
- (t_step (s, ab) (mop_bef n @ tshift mp_up (2 * n), 0)))
- (t_steps (Suc 0, l, r) (mop_bef n @ tshift mp_up (2 * n), 0) stp)"
- proof(induct stp, simp add: t_steps.simps, simp)
- fix stpa
- assume h1:
- "(\<lambda>(s, l, r). s = 0) (t_steps (Suc 0, l, r)
- (mop_bef n @ tshift mp_up (2 * n), 0) stpa) \<Longrightarrow>
- \<exists>stp. (\<lambda>(s, ab). 0 < s \<and> (\<lambda>(s', l', r'). s' = 0)
- (t_step (s, ab) (mop_bef n @ tshift mp_up (2 * n), 0)))
- (t_steps (Suc 0, l, r)
- (mop_bef n @ tshift mp_up (2 * n), 0) stp)"
- and h2:
- "(\<lambda>(s, l, r). s = 0) (t_steps (Suc 0, l, r)
- (mop_bef n @ tshift mp_up (2 * n), 0) (Suc stpa))"
- "n < length lm"
- "crsp_l ly (as, lm) (s, l, r) ires"
- thus "\<exists>stp. (\<lambda>(s, ab). 0 < s \<and> (\<lambda>(s', l', r'). s' = 0)
- (t_step (s, ab) (mop_bef n @ tshift mp_up (2 * n), 0))) (
- t_steps (Suc 0, l, r)
- (mop_bef n @ tshift mp_up (2 * n), 0) stp)"
- apply(case_tac "(\<lambda>(s, l, r). s = 0) (t_steps (Suc 0, l, r)
- (mop_bef n @ tshift mp_up (2 * n), 0) stpa)",
- simp)
- apply(rule_tac x = "stpa" in exI)
- apply(case_tac "(t_steps (Suc 0, l, r)
- (mop_bef n @ tshift mp_up (2 * n), 0) stpa)",
- simp add: t_steps_ind)
- done
- qed
-qed
-
-lemma tshift_nth_state0: "\<lbrakk>n < length tp; tp ! n = (a, 0)\<rbrakk>
- \<Longrightarrow> tshift tp off ! n = (a, 0)"
-apply(induct n, case_tac tp, simp)
-apply(auto simp: tshift.simps)
-done
-
-lemma shift_length: "length (tshift tp n) = length tp"
-apply(auto simp: tshift.simps)
-done
-
-lemma even_Suc_le: "\<lbrakk>y mod 2 = 0; 2 * x < y\<rbrakk> \<Longrightarrow> Suc (2 * x) < y"
-by arith
-
-lemma [simp]: "(4::nat) * n mod 2 = 0"
-by arith
-
-lemma tm_append_fetch_equal:
- "\<lbrakk>t_ncorrect (tm_of aprog); s'> 0;
- fetch (mop_bef n @ tshift mp_up (2 * n)) s' b = (a, 0)\<rbrakk>
-\<Longrightarrow> fetch (tm_of aprog @ tshift (mop_bef n @ tshift mp_up (2 * n))
- (length (tm_of aprog) div 2)) (s' + length (tm_of aprog) div 2) b
- = (a, 0)"
-apply(case_tac s', simp)
-apply(auto simp: fetch.simps nth_of.simps t_ncorrect.simps shift_length nth_append
- tshift.simps split: list.splits block.splits split: if_splits)
-done
-
-lemma [simp]:
- "\<lbrakk>t_ncorrect (tm_of aprog);
- t_step (s', l', r') (mop_bef n @ tshift mp_up (2 * n), 0) =
- (0, l'', r''); s' > 0\<rbrakk>
- \<Longrightarrow> t_step (s' + length (tm_of aprog) div 2, l', r')
- (tm_of aprog @ tshift (mop_bef n @ tshift mp_up (2 * n))
- (length (tm_of aprog) div 2), 0) = (0, l'', r'')"
-apply(simp add: t_step.simps)
-apply(subgoal_tac
- "(fetch (mop_bef n @ tshift mp_up (2 * n)) s'
- (case r' of [] \<Rightarrow> Bk | Bk # xs \<Rightarrow> Bk | Oc # xs \<Rightarrow> Oc))
- = (fetch (tm_of aprog @ tshift (mop_bef n @ tshift mp_up (2 * n))
- (length (tm_of aprog) div 2)) (s' + length (tm_of aprog) div 2)
- (case r' of [] \<Rightarrow> Bk | Bk # xs \<Rightarrow> Bk | Oc # xs \<Rightarrow> Oc))", simp)
-apply(case_tac "(fetch (mop_bef n @ tshift mp_up (2 * n)) s'
- (case r' of [] \<Rightarrow> Bk | Bk # xs \<Rightarrow> Bk | Oc # xs \<Rightarrow> Oc))", simp)
-apply(drule_tac tm_append_fetch_equal, auto)
-done
-
-lemma [intro]:
- "start_of (layout_of aprog) (length aprog) - Suc 0 =
- length (tm_of aprog) div 2"
-apply(subgoal_tac "abc2t_correct aprog")
-apply(insert pre_lheq[of "concat (take (length aprog)
- (tms_of aprog))" "length aprog" aprog], simp add: tm_of.simps)
-by auto
-
-lemma tm_append_stop_step:
- "\<lbrakk>t_ncorrect (tm_of aprog);
- t_ncorrect (mop_bef n @ tshift mp_up (2 * n)); n < length lm;
- (t_steps (Suc 0, l, r) (mop_bef n @ tshift mp_up (2 * n), 0) stp) =
- (s', l', r');
- s' \<noteq> 0;
- t_step (s', l', r') (mop_bef n @ tshift mp_up (2 * n), 0)
- = (0, l'', r'')\<rbrakk>
- \<Longrightarrow>
-(t_steps ((start_of (layout_of aprog) (length aprog), l, r))
- (tm_of aprog @ tshift (mop_bef n @ tshift mp_up (2 * n))
- (start_of (layout_of aprog) (length aprog) - Suc 0), 0) (Suc stp))
- = (0, l'', r'')"
-apply(insert tm_append_steps_equal[of "tm_of aprog"
- "(mop_bef n @ tshift mp_up (2 * n))"
- "(start_of (layout_of aprog) (length aprog) - Suc 0)"
- "Suc 0" l r stp], simp)
-apply(subgoal_tac "(start_of (layout_of aprog) (length aprog) - Suc 0)
- = (length (tm_of aprog) div 2)", simp add: t_steps_ind)
-apply(case_tac
- "(t_steps (start_of (layout_of aprog) (length aprog), l, r)
- (tm_of aprog @ tshift (mop_bef n @ tshift mp_up (2 * n))
- (length (tm_of aprog) div 2), 0) stp)", simp)
-apply(subgoal_tac "start_of (layout_of aprog) (length aprog) > 0",
- case_tac "start_of (layout_of aprog) (length aprog)",
- simp, simp)
-apply(rule startof_not0, auto)
-done
-
-lemma start_of_out_range:
-"as \<ge> length aprog \<Longrightarrow>
- start_of (layout_of aprog) as =
- start_of (layout_of aprog) (length aprog)"
-apply(induct as, simp)
-apply(case_tac "length aprog = Suc as", simp)
-apply(simp add: start_of.simps)
-done
-
-lemma [intro]: "t_ncorrect (tm_of aprog)"
-apply(simp add: tm_of.simps)
-apply(insert tms_mod2[of "length aprog" aprog],
- simp add: t_ncorrect.simps)
-done
-
-lemma abacus_turing_eq_halt_case_pre:
- "\<lbrakk>ly = layout_of aprog;
- tprog = tm_of aprog;
- crsp_l ly ac tc ires;
- n < length am;
- abc_steps_l ac aprog stp = (as, am);
- mop_ss = start_of ly (length aprog);
- as \<ge> length aprog\<rbrakk>
- \<Longrightarrow> \<exists> stp. (\<lambda> (s, l, r). s = 0 \<and> mopup_inv (0, l, r) am n ires)
- (t_steps tc (tprog @ (tMp n (mop_ss - 1)), 0) stp)"
-apply(insert steps_crsp[of ly aprog tprog ac tc ires "(as, am)" stp], auto)
-apply(case_tac "(t_steps tc (tm_of aprog, 0) n')",
- simp add: tMp.simps)
-apply(subgoal_tac "t_ncorrect (mop_bef n @ tshift mp_up (2 * n))")
-proof -
- fix n' a b c
- assume h1:
- "crsp_l (layout_of aprog) ac tc ires"
- "abc_steps_l ac aprog stp = (as, am)"
- "length aprog \<le> as"
- "crsp_l (layout_of aprog) (as, am) (a, b, c) ires"
- "t_steps tc (tm_of aprog, 0) n' = (a, b, c)"
- "n < length am"
- "t_ncorrect (mop_bef n @ tshift mp_up (2 * n))"
- hence h2:
- "t_steps tc (tm_of aprog @ tshift (mop_bef n @ tshift mp_up (2 * n))
- (start_of (layout_of aprog) (length aprog) - Suc 0), 0) n'
- = (a, b, c)"
- apply(rule_tac tm_append_steps, simp)
- apply(simp add: crsp_l.simps, auto)
- apply(simp add: crsp_l.simps)
- apply(rule startof_not0)
- done
- from h1 have h3:
- "\<exists>stp. (\<lambda>(s, l, r). s \<noteq> 0 \<and> ((\<lambda> (s', l', r'). s' = 0)
- (t_step (s, l, r) (mop_bef n @ tshift mp_up (2 * n), 0))))
- (t_steps (Suc 0, b, c)
- (mop_bef n @ tshift mp_up (2 * n), 0) stp)"
- apply(rule_tac mopup_halt_bef, auto)
- done
- from h1 and h2 and h3 show
- "\<exists>stp. case t_steps tc (tm_of aprog @ tshift (mop_bef n @ tshift mp_up (2 * n))
- (start_of (layout_of aprog) (length aprog) - Suc 0), 0) stp of (s, ab)
- \<Rightarrow> s = 0 \<and> mopup_inv (0, ab) am n ires"
- proof(erule_tac exE,
- case_tac "(t_steps (Suc 0, b, c)
- (mop_bef n @ tshift mp_up (2 * n), 0) stpa)", simp,
- case_tac "(t_step (aa, ba, ca)
- (mop_bef n @ tshift mp_up (2 * n), 0))", simp)
- fix stpa aa ba ca aaa baa caa
- assume g1: "0 < aa \<and> aaa = 0"
- "t_steps (Suc 0, b, c)
- (mop_bef n @ tshift mp_up (2 * n), 0) stpa = (aa, ba,ca)"
- "t_step (aa, ba, ca) (mop_bef n @ tshift mp_up (2 * n), 0)
- = (0, baa, caa)"
- from h1 and this have g2:
- "t_steps (start_of (layout_of aprog) (length aprog), b, c)
- (tm_of aprog @ tshift (mop_bef n @ tshift mp_up (2 * n))
- (start_of (layout_of aprog) (length aprog) - Suc 0), 0)
- (Suc stpa) = (0, baa, caa)"
- apply(rule_tac tm_append_stop_step, auto)
- done
- from h1 and h2 and g1 and this show "?thesis"
- apply(rule_tac x = "n' + Suc stpa" in exI)
- apply(simp add: t_steps_ind del: t_steps.simps)
- apply(subgoal_tac "a = start_of (layout_of aprog)
- (length aprog)", simp)
- apply(insert mopup_inv_steps[of n am "Suc 0" b c ires "Suc stpa"],
- simp add: t_steps_ind)
- apply(subgoal_tac "mopup_inv (Suc 0, b, c) am n ires", simp)
- apply(rule_tac mopup_init, simp, simp)
- apply(simp add: crsp_l.simps)
- apply(erule_tac start_of_out_range)
- done
- qed
-next
- show " t_ncorrect (mop_bef n @ tshift mp_up (2 * n))"
- apply(auto simp: t_ncorrect.simps tshift.simps mp_up_def)
- done
-qed
-
-text {*
- One of the main theorems about Abacus compilation.
-*}
-
-lemma abacus_turing_eq_halt_case:
- assumes layout:
- -- {* There is an Abacus program @{text "aprog"} with layout @{text "ly"}: *}
- "ly = layout_of aprog"
- and complied:
- -- {* The TM compiled from @{text "aprog"} is @{text "tprog"}: *}
- "tprog = tm_of aprog"
- and correspond:
- -- {* TM configuration @{text "tc"} and Abacus configuration @{text "ac"}
- are in correspondence: *}
- "crsp_l ly ac tc ires"
- and halt_state:
- -- {* @{text "as"} is a program label outside the range of @{text "aprog"}. So
- if Abacus is in such a state, it is in halt state: *}
- "as \<ge> length aprog"
- and abc_exec:
- -- {* Supposing after @{text "stp"} step of execution, Abacus program @{text "aprog"}
- reaches such a halt state: *}
- "abc_steps_l ac aprog stp = (as, am)"
- and rs_len:
- -- {* @{text "n"} is a memory address in the range of Abacus memory @{text "am"}: *}
- "n < length am"
- and mopup_start:
- -- {* The startling label for mopup mahines, according to the layout and Abacus program
- should be @{text "mop_ss"}: *}
- "mop_ss = start_of ly (length aprog)"
- shows
- -- {*
- After @{text "stp"} steps of execution of the TM composed of @{text "tprog"} and the mopup
- TM @{text "(tMp n (mop_ss - 1))"} will halt and gives rise to a configuration which
- only hold the content of memory cell @{text "n"}:
- *}
- "\<exists> stp. (\<lambda> (s, l, r). \<exists> ln rn. s = 0 \<and> l = Bk\<^bsup>ln\<^esup> @ Bk # Bk # ires
- \<and> r = Oc\<^bsup>Suc (abc_lm_v am n)\<^esup> @ Bk\<^bsup>rn\<^esup>)
- (t_steps tc (tprog @ (tMp n (mop_ss - 1)), 0) stp)"
-proof -
- from layout complied correspond halt_state abc_exec rs_len mopup_start
- and abacus_turing_eq_halt_case_pre [of ly aprog tprog ac tc ires n am stp as mop_ss]
- show "?thesis"
- apply(simp add: mopup_inv.simps mopup_stop.simps tape_of_nat_abv)
- done
-qed
-
-lemma abc_unhalt_case_zero:
-"\<lbrakk>crsp_l (layout_of aprog) ac tc ires;
- n < length am;
- \<forall>stp. (\<lambda>(as, am). as < length aprog) (abc_steps_l ac aprog stp)\<rbrakk>
- \<Longrightarrow> \<exists>astp bstp. 0 \<le> bstp \<and>
- crsp_l (layout_of aprog) (abc_steps_l ac aprog astp)
- (t_steps tc (tm_of aprog, 0) bstp) ires"
-apply(rule_tac x = "Suc 0" in exI)
-apply(case_tac " abc_steps_l ac aprog (Suc 0)", simp)
-proof -
- fix a b
- assume "crsp_l (layout_of aprog) ac tc ires"
- "abc_steps_l ac aprog (Suc 0) = (a, b)"
- thus "\<exists>bstp. crsp_l (layout_of aprog) (a, b)
- (t_steps tc (tm_of aprog, 0) bstp) ires"
- apply(insert steps_crsp[of "layout_of aprog" aprog
- "tm_of aprog" ac tc ires "(a, b)" "Suc 0"], auto)
- done
-qed
-
-declare abc_steps_l.simps[simp del]
-
-lemma abc_steps_ind:
- "let (as, am) = abc_steps_l ac aprog stp in
- abc_steps_l ac aprog (Suc stp) =
- abc_step_l (as, am) (abc_fetch as aprog) "
-proof(simp)
- show "(\<lambda>(as, am). abc_steps_l ac aprog (Suc stp) =
- abc_step_l (as, am) (abc_fetch as aprog))
- (abc_steps_l ac aprog stp)"
- proof(induct stp arbitrary: ac)
- fix ac
- show "(\<lambda>(as, am). abc_steps_l ac aprog (Suc 0) =
- abc_step_l (as, am) (abc_fetch as aprog))
- (abc_steps_l ac aprog 0)"
- apply(case_tac "ac:: nat \<times> nat list",
- simp add: abc_steps_l.simps)
- apply(case_tac "(abc_step_l (a, b) (abc_fetch a aprog))",
- simp add: abc_steps_l.simps)
- done
- next
- fix stp ac
- assume h1:
- "(\<And>ac. (\<lambda>(as, am). abc_steps_l ac aprog (Suc stp) =
- abc_step_l (as, am) (abc_fetch as aprog))
- (abc_steps_l ac aprog stp))"
- thus
- "(\<lambda>(as, am). abc_steps_l ac aprog (Suc (Suc stp)) =
- abc_step_l (as, am) (abc_fetch as aprog))
- (abc_steps_l ac aprog (Suc stp))"
- apply(case_tac "ac::nat \<times> nat list", simp)
- apply(subgoal_tac
- "abc_steps_l (a, b) aprog (Suc (Suc stp)) =
- abc_steps_l (abc_step_l (a, b) (abc_fetch a aprog))
- aprog (Suc stp)", simp)
- apply(case_tac "(abc_step_l (a, b) (abc_fetch a aprog))", simp)
- proof -
- fix a b aa ba
- assume h2: "abc_step_l (a, b) (abc_fetch a aprog) = (aa, ba)"
- from h1 and h2 show
- "(\<lambda>(as, am). abc_steps_l (aa, ba) aprog (Suc stp) =
- abc_step_l (as, am) (abc_fetch as aprog))
- (abc_steps_l (a, b) aprog (Suc stp))"
- apply(insert h1[of "(aa, ba)"])
- apply(simp add: abc_steps_l.simps)
- apply(insert h2, simp)
- done
- next
- fix a b
- show
- "abc_steps_l (a, b) aprog (Suc (Suc stp)) =
- abc_steps_l (abc_step_l (a, b) (abc_fetch a aprog))
- aprog (Suc stp)"
- apply(simp only: abc_steps_l.simps)
- done
- qed
- qed
-qed
-
-lemma abc_unhalt_case_induct:
- "\<lbrakk>crsp_l (layout_of aprog) ac tc ires;
- n < length am;
- \<forall>stp. (\<lambda>(as, am). as < length aprog) (abc_steps_l ac aprog stp);
- stp \<le> bstp;
- crsp_l (layout_of aprog) (abc_steps_l ac aprog astp)
- (t_steps tc (tm_of aprog, 0) bstp) ires\<rbrakk>
- \<Longrightarrow> \<exists>astp bstp. Suc stp \<le> bstp \<and> crsp_l (layout_of aprog)
- (abc_steps_l ac aprog astp) (t_steps tc (tm_of aprog, 0) bstp) ires"
-apply(rule_tac x = "Suc astp" in exI)
-apply(case_tac "abc_steps_l ac aprog astp")
-proof -
- fix a b
- assume
- "\<forall>stp. (\<lambda>(as, am). as < length aprog)
- (abc_steps_l ac aprog stp)"
- "stp \<le> bstp"
- "crsp_l (layout_of aprog) (abc_steps_l ac aprog astp)
- (t_steps tc (tm_of aprog, 0) bstp) ires"
- "abc_steps_l ac aprog astp = (a, b)"
- thus
- "\<exists>bstp\<ge>Suc stp. crsp_l (layout_of aprog)
- (abc_steps_l ac aprog (Suc astp))
- (t_steps tc (tm_of aprog, 0) bstp) ires"
- apply(insert crsp_inside[of "layout_of aprog" aprog
- "tm_of aprog" a b "(t_steps tc (tm_of aprog, 0) bstp)" "ires"], auto)
- apply(erule_tac x = astp in allE, auto)
- apply(rule_tac x = "bstp + stpa" in exI, simp)
- apply(insert abc_steps_ind[of ac aprog "astp"], simp)
- done
-qed
-
-lemma abc_unhalt_case:
- "\<lbrakk>crsp_l (layout_of aprog) ac tc ires;
- \<forall>stp. (\<lambda>(as, am). as < length aprog) (abc_steps_l ac aprog stp)\<rbrakk>
- \<Longrightarrow> (\<exists> astp bstp. bstp \<ge> stp \<and>
- crsp_l (layout_of aprog) (abc_steps_l ac aprog astp)
- (t_steps tc (tm_of aprog, 0) bstp) ires)"
-apply(induct stp)
-apply(rule_tac abc_unhalt_case_zero, auto)
-apply(rule_tac abc_unhalt_case_induct, auto)
-done
-
-lemma abacus_turing_eq_unhalt_case_pre:
- "\<lbrakk>ly = layout_of aprog;
- tprog = tm_of aprog;
- crsp_l ly ac tc ires;
- \<forall> stp. ((\<lambda> (as, am). as < length aprog)
- (abc_steps_l ac aprog stp));
- mop_ss = start_of ly (length aprog)\<rbrakk>
- \<Longrightarrow> (\<not> (\<exists> stp. (\<lambda> (s, l, r). s = 0)
- (t_steps tc (tprog @ (tMp n (mop_ss - 1)), 0) stp)))"
- apply(auto)
-proof -
- fix stp a b
- assume h1:
- "crsp_l (layout_of aprog) ac tc ires"
- "\<forall>stp. (\<lambda>(as, am). as < length aprog) (abc_steps_l ac aprog stp)"
- "t_steps tc (tm_of aprog @ tMp n (start_of (layout_of aprog)
- (length aprog) - Suc 0), 0) stp = (0, a, b)"
- thus "False"
- proof(insert abc_unhalt_case[of aprog ac tc ires stp], auto,
- case_tac "(abc_steps_l ac aprog astp)",
- case_tac "(t_steps tc (tm_of aprog, 0) bstp)", simp)
- fix astp bstp aa ba aaa baa c
- assume h2:
- "abc_steps_l ac aprog astp = (aa, ba)" "stp \<le> bstp"
- "t_steps tc (tm_of aprog, 0) bstp = (aaa, baa, c)"
- "crsp_l (layout_of aprog) (aa, ba) (aaa, baa, c) ires"
- hence h3:
- "t_steps tc (tm_of aprog @ tMp n
- (start_of (layout_of aprog) (length aprog) - Suc 0), 0) bstp
- = (aaa, baa, c)"
- apply(intro tm_append_steps, auto)
- apply(simp add: crsp_l.simps, rule startof_not0)
- done
- from h2 have h4: "\<exists> diff. bstp = stp + diff"
- apply(rule_tac x = "bstp - stp" in exI, simp)
- done
- from h4 and h3 and h2 and h1 show "?thesis"
- apply(auto)
- apply(simp add: state0_ind crsp_l.simps)
- apply(subgoal_tac "start_of (layout_of aprog) aa > 0", simp)
- apply(rule startof_not0)
- done
- qed
-qed
-
-lemma abacus_turing_eq_unhalt_case:
- assumes layout:
- -- {* There is an Abacus program @{text "aprog"} with layout @{text "ly"}: *}
- "ly = layout_of aprog"
- and compiled:
- -- {* The TM compiled from @{text "aprog"} is @{text "tprog"}: *}
- "tprog = tm_of aprog"
- and correspond:
- -- {*
- TM configuration @{text "tc"} and Abacus configuration @{text "ac"}
- are in correspondence:
- *}
- "crsp_l ly ac tc ires"
- and abc_unhalt:
- -- {*
- If, no matter how many steps the Abacus program @{text "aprog"} executes, it
- may never reach a halt state.
- *}
- "\<forall> stp. ((\<lambda> (as, am). as < length aprog)
- (abc_steps_l ac aprog stp))"
- and mopup_start: "mop_ss = start_of ly (length aprog)"
- shows
- -- {*
- The the TM composed of TM @{text "tprog"} and the moupup TM may never reach a halt state as well.
- *}
- "\<not> (\<exists> stp. (\<lambda> (s, l, r). s = 0)
- (t_steps tc (tprog @ (tMp n (mop_ss - 1)), 0) stp))"
- using layout compiled correspond abc_unhalt mopup_start
- apply(rule_tac abacus_turing_eq_unhalt_case_pre, auto)
- done
-
-
-definition abc_list_crsp:: "nat list \<Rightarrow> nat list \<Rightarrow> bool"
- where
- "abc_list_crsp xs ys = (\<exists> n. xs = ys @ 0\<^bsup>n\<^esup> \<or> ys = xs @ 0\<^bsup>n\<^esup>)"
-lemma [intro]: "abc_list_crsp (lm @ 0\<^bsup>m\<^esup>) lm"
-apply(auto simp: abc_list_crsp_def)
-done
-
-lemma abc_list_crsp_lm_v:
- "abc_list_crsp lma lmb \<Longrightarrow> abc_lm_v lma n = abc_lm_v lmb n"
-apply(auto simp: abc_list_crsp_def abc_lm_v.simps
- nth_append exponent_def)
-done
-
-lemma rep_app_cons_iff:
- "k < n \<Longrightarrow> replicate n a[k:=b] =
- replicate k a @ b # replicate (n - k - 1) a"
-apply(induct n arbitrary: k, simp)
-apply(simp split:nat.splits)
-done
-
-lemma abc_list_crsp_lm_s:
- "abc_list_crsp lma lmb \<Longrightarrow>
- abc_list_crsp (abc_lm_s lma m n) (abc_lm_s lmb m n)"
-apply(auto simp: abc_list_crsp_def abc_lm_v.simps abc_lm_s.simps)
-apply(simp_all add: list_update_append, auto simp: exponent_def)
-proof -
- fix na
- assume h: "m < length lmb + na" " \<not> m < length lmb"
- hence "m - length lmb < na" by simp
- hence "replicate na 0[(m- length lmb):= n] =
- replicate (m - length lmb) 0 @ n #
- replicate (na - (m - length lmb) - 1) 0"
- apply(erule_tac rep_app_cons_iff)
- done
- thus "\<exists>nb. replicate na 0[m - length lmb := n] =
- replicate (m - length lmb) 0 @ n # replicate nb 0 \<or>
- replicate (m - length lmb) 0 @ [n] =
- replicate na 0[m - length lmb := n] @ replicate nb 0"
- apply(auto)
- done
-next
- fix na
- assume h: "\<not> m < length lmb + na"
- show
- "\<exists>nb. replicate na 0 @ replicate (m - (length lmb + na)) 0 @ [n] =
- replicate (m - length lmb) 0 @ n # replicate nb 0 \<or>
- replicate (m - length lmb) 0 @ [n] =
- replicate na 0 @
- replicate (m - (length lmb + na)) 0 @ n # replicate nb 0"
- apply(rule_tac x = 0 in exI, simp, auto)
- using h
- apply(simp add: replicate_add[THEN sym])
- done
-next
- fix na
- assume h: "\<not> m < length lma" "m < length lma + na"
- hence "m - length lma < na" by simp
- hence
- "replicate na 0[(m- length lma):= n] = replicate (m - length lma)
- 0 @ n # replicate (na - (m - length lma) - 1) 0"
- apply(erule_tac rep_app_cons_iff)
- done
- thus "\<exists>nb. replicate (m - length lma) 0 @ [n] =
- replicate na 0[m - length lma := n] @ replicate nb 0
- \<or> replicate na 0[m - length lma := n] =
- replicate (m - length lma) 0 @ n # replicate nb 0"
- apply(auto)
- done
-next
- fix na
- assume "\<not> m < length lma + na"
- thus " \<exists>nb. replicate (m - length lma) 0 @ [n] =
- replicate na 0 @
- replicate (m - (length lma + na)) 0 @ n # replicate nb 0
- \<or> replicate na 0 @
- replicate (m - (length lma + na)) 0 @ [n] =
- replicate (m - length lma) 0 @ n # replicate nb 0"
- apply(rule_tac x = 0 in exI, simp, auto)
- apply(simp add: replicate_add[THEN sym])
- done
-qed
-
-lemma abc_list_crsp_step:
- "\<lbrakk>abc_list_crsp lma lmb; abc_step_l (aa, lma) i = (a, lma');
- abc_step_l (aa, lmb) i = (a', lmb')\<rbrakk>
- \<Longrightarrow> a' = a \<and> abc_list_crsp lma' lmb'"
-apply(case_tac i, auto simp: abc_step_l.simps
- abc_list_crsp_lm_s abc_list_crsp_lm_v Let_def
- split: abc_inst.splits if_splits)
-done
-
-lemma abc_steps_red:
- "abc_steps_l ac aprog stp = (as, am) \<Longrightarrow>
- abc_steps_l ac aprog (Suc stp) =
- abc_step_l (as, am) (abc_fetch as aprog)"
-using abc_steps_ind[of ac aprog stp]
-apply(simp)
-done
-
-lemma abc_list_crsp_steps:
- "\<lbrakk>abc_steps_l (0, lm @ 0\<^bsup>m\<^esup>) aprog stp = (a, lm'); aprog \<noteq> []\<rbrakk>
- \<Longrightarrow> \<exists> lma. abc_steps_l (0, lm) aprog stp = (a, lma) \<and>
- abc_list_crsp lm' lma"
-apply(induct stp arbitrary: a lm', simp add: abc_steps_l.simps, auto)
-apply(case_tac "abc_steps_l (0, lm @ 0\<^bsup>m\<^esup>) aprog stp",
- simp add: abc_steps_ind)
-proof -
- fix stp a lm' aa b
- assume ind:
- "\<And>a lm'. aa = a \<and> b = lm' \<Longrightarrow>
- \<exists>lma. abc_steps_l (0, lm) aprog stp = (a, lma) \<and>
- abc_list_crsp lm' lma"
- and h: "abc_steps_l (0, lm @ 0\<^bsup>m\<^esup>) aprog (Suc stp) = (a, lm')"
- "abc_steps_l (0, lm @ 0\<^bsup>m\<^esup>) aprog stp = (aa, b)"
- "aprog \<noteq> []"
- hence g1: "abc_steps_l (0, lm @ 0\<^bsup>m\<^esup>) aprog (Suc stp)
- = abc_step_l (aa, b) (abc_fetch aa aprog)"
- apply(rule_tac abc_steps_red, simp)
- done
- have "\<exists>lma. abc_steps_l (0, lm) aprog stp = (aa, lma) \<and>
- abc_list_crsp b lma"
- apply(rule_tac ind, simp)
- done
- from this obtain lma where g2:
- "abc_steps_l (0, lm) aprog stp = (aa, lma) \<and>
- abc_list_crsp b lma" ..
- hence g3: "abc_steps_l (0, lm) aprog (Suc stp)
- = abc_step_l (aa, lma) (abc_fetch aa aprog)"
- apply(rule_tac abc_steps_red, simp)
- done
- show "\<exists>lma. abc_steps_l (0, lm) aprog (Suc stp) = (a, lma) \<and>
- abc_list_crsp lm' lma"
- using g1 g2 g3 h
- apply(auto)
- apply(case_tac "abc_step_l (aa, b) (abc_fetch aa aprog)",
- case_tac "abc_step_l (aa, lma) (abc_fetch aa aprog)", simp)
- apply(rule_tac abc_list_crsp_step, auto)
- done
-qed
-
-lemma [simp]: "(case ca of [] \<Rightarrow> Bk | Bk # xs \<Rightarrow> Bk | Oc # xs \<Rightarrow> Oc) =
- (case ca of [] \<Rightarrow> Bk | x # xs \<Rightarrow> x)"
-by(case_tac ca, simp_all, case_tac a, simp, simp)
-
-lemma steps_eq: "length t mod 2 = 0 \<Longrightarrow>
- t_steps c (t, 0) stp = steps c t stp"
-apply(induct stp)
-apply(simp add: steps.simps t_steps.simps)
-apply(simp add:tstep_red t_steps_ind)
-apply(case_tac "steps c t stp", simp)
-apply(auto simp: t_step.simps tstep.simps)
-done
-
-lemma crsp_l_start: "crsp_l ly (0, lm) (Suc 0, Bk # Bk # ires, <lm> @ Bk\<^bsup>rn\<^esup>) ires"
-apply(simp add: crsp_l.simps, auto simp: start_of.simps)
-done
-
-lemma t_ncorrect_app: "\<lbrakk>t_ncorrect t1; t_ncorrect t2\<rbrakk> \<Longrightarrow>
- t_ncorrect (t1 @ t2)"
-apply(simp add: t_ncorrect.simps, auto)
-done
-
-lemma [simp]:
- "(length (tm_of aprog) +
- length (tMp n (start_of ly (length aprog) - Suc 0))) mod 2 = 0"
-apply(subgoal_tac
- "t_ncorrect (tm_of aprog @ tMp n
- (start_of ly (length aprog) - Suc 0))")
-apply(simp add: t_ncorrect.simps)
-apply(rule_tac t_ncorrect_app,
- auto simp: tMp.simps t_ncorrect.simps tshift.simps mp_up_def)
-apply(subgoal_tac
- "t_ncorrect (tm_of aprog)", simp add: t_ncorrect.simps)
-apply(auto)
-done
-
-lemma [simp]: "takeWhile (\<lambda>a. a = Oc)
- (replicate rs Oc @ replicate rn Bk) = replicate rs Oc"
-apply(induct rs, auto)
-apply(induct rn, auto)
-done
-
-lemma abacus_turing_eq_halt':
- "\<lbrakk>ly = layout_of aprog;
- tprog = tm_of aprog;
- n < length am;
- abc_steps_l (0, lm) aprog stp = (as, am);
- mop_ss = start_of ly (length aprog);
- as \<ge> length aprog\<rbrakk>
- \<Longrightarrow> \<exists> stp m l. steps (Suc 0, Bk # Bk # ires, <lm> @ Bk\<^bsup>rn\<^esup>)
- (tprog @ (tMp n (mop_ss - 1))) stp
- = (0, Bk\<^bsup>m\<^esup> @ Bk # Bk # ires, Oc\<^bsup>Suc (abc_lm_v am n)\<^esup> @ Bk\<^bsup>l\<^esup>)"
-apply(drule_tac tc = "(Suc 0, Bk # Bk # ires, <lm> @ Bk\<^bsup>rn\<^esup>)" in
- abacus_turing_eq_halt_case, auto intro: crsp_l_start)
-apply(subgoal_tac
- "length (tm_of aprog @ tMp n
- (start_of ly (length aprog) - Suc 0)) mod 2 = 0")
-apply(simp add: steps_eq)
-apply(rule_tac x = stpa in exI,
- simp add: exponent_def, auto)
-done
-
-
-lemma list_length: "xs = ys \<Longrightarrow> length xs = length ys"
-by simp
-lemma [elim]: "tinres (Bk\<^bsup>m\<^esup>) b \<Longrightarrow> \<exists>m. b = Bk\<^bsup>m\<^esup>"
-apply(auto simp: tinres_def)
-apply(rule_tac x = "m-n" in exI,
- auto simp: exponent_def replicate_add[THEN sym])
-apply(case_tac "m < n", auto)
-apply(drule_tac list_length, auto)
-apply(subgoal_tac "\<exists> d. m = d + n", auto simp: replicate_add)
-apply(rule_tac x = "m - n" in exI, simp)
-done
-lemma [intro]: "tinres [Bk] (Bk\<^bsup>k\<^esup>) "
-apply(auto simp: tinres_def exponent_def)
-apply(case_tac k, auto)
-apply(rule_tac x = "Suc 0" in exI, simp)
-done
-
-lemma abacus_turing_eq_halt_pre:
- "\<lbrakk>ly = layout_of aprog;
- tprog = tm_of aprog;
- n < length am;
- abc_steps_l (0, lm) aprog stp = (as, am);
- mop_ss = start_of ly (length aprog);
- as \<ge> length aprog\<rbrakk>
- \<Longrightarrow> \<exists> stp m l. steps (Suc 0, Bk # Bk # ires, <lm> @ Bk\<^bsup>rn\<^esup>)
- (tprog @ (tMp n (mop_ss - 1))) stp
- = (0, Bk\<^bsup>m\<^esup> @ Bk # Bk # ires, Oc\<^bsup>Suc (abc_lm_v am n)\<^esup> @ Bk\<^bsup>l\<^esup>)"
-using abacus_turing_eq_halt'
-apply(simp)
-done
-
-
-text {*
- Main theorem for the case when the original Abacus program does halt.
-*}
-
-lemma abacus_turing_eq_halt:
- assumes layout:
- "ly = layout_of aprog"
- -- {* There is an Abacus program @{text "aprog"} with layout @{text "ly"}: *}
- and compiled: "tprog = tm_of aprog"
- -- {* The TM compiled from @{text "aprog"} is @{text "tprog"}: *}
- and halt_state:
- -- {* @{text "as"} is a program label outside the range of @{text "aprog"}. So
- if Abacus is in such a state, it is in halt state: *}
- "as \<ge> length aprog"
- and abc_exec:
- -- {* Supposing after @{text "stp"} step of execution, Abacus program @{text "aprog"}
- reaches such a halt state: *}
- "abc_steps_l (0, lm) aprog stp = (as, am)"
- and rs_locate:
- -- {* @{text "n"} is a memory address in the range of Abacus memory @{text "am"}: *}
- "n < length am"
- and mopup_start:
- -- {* The startling label for mopup mahines, according to the layout and Abacus program
- should be @{text "mop_ss"}: *}
- "mop_ss = start_of ly (length aprog)"
- shows
- -- {*
- After @{text "stp"} steps of execution of the TM composed of @{text "tprog"} and the mopup
- TM @{text "(tMp n (mop_ss - 1))"} will halt and gives rise to a configuration which
- only hold the content of memory cell @{text "n"}:
- *}
- "\<exists> stp m l. steps (Suc 0, Bk # Bk # ires, <lm> @ Bk\<^bsup>rn\<^esup>) (tprog @ (tMp n (mop_ss - 1))) stp
- = (0, Bk\<^bsup>m\<^esup> @ Bk # Bk # ires, Oc\<^bsup>Suc (abc_lm_v am n)\<^esup> @ Bk\<^bsup>l\<^esup>)"
- using layout compiled halt_state abc_exec rs_locate mopup_start
- by(rule_tac abacus_turing_eq_halt_pre, auto)
-
-lemma abacus_turing_eq_uhalt':
- "\<lbrakk>ly = layout_of aprog;
- tprog = tm_of aprog;
- \<forall> stp. ((\<lambda> (as, am). as < length aprog)
- (abc_steps_l (0, lm) aprog stp));
- mop_ss = start_of ly (length aprog)\<rbrakk>
- \<Longrightarrow> (\<not> (\<exists> stp. isS0 (steps (Suc 0, [Bk, Bk], <lm>)
- (tprog @ (tMp n (mop_ss - 1))) stp)))"
-apply(drule_tac tc = "(Suc 0, [Bk, Bk], <lm>)" and n = n and ires = "[]" in
- abacus_turing_eq_unhalt_case, auto intro: crsp_l_start)
-apply(simp add: crsp_l.simps start_of.simps)
-apply(erule_tac x = stp in allE, erule_tac x = stp in allE)
-apply(subgoal_tac
- "length (tm_of aprog @ tMp n
- (start_of ly (length aprog) - Suc 0)) mod 2 = 0")
-apply(simp add: steps_eq, auto simp: isS0_def)
-done
-
-text {*
- Main theorem for the case when the original Abacus program does not halt.
- *}
-lemma abacus_turing_eq_uhalt:
- assumes layout:
- -- {* There is an Abacus program @{text "aprog"} with layout @{text "ly"}: *}
- "ly = layout_of aprog"
- and compiled:
- -- {* The TM compiled from @{text "aprog"} is @{text "tprog"}: *}
- "tprog = tm_of aprog"
- and abc_unhalt:
- -- {*
- If, no matter how many steps the Abacus program @{text "aprog"} executes, it
- may never reach a halt state.
- *}
- "\<forall> stp. ((\<lambda> (as, am). as < length aprog)
- (abc_steps_l (0, lm) aprog stp))"
- and mop_start: "mop_ss = start_of ly (length aprog)"
- shows
- -- {*
- The the TM composed of TM @{text "tprog"} and the moupup TM may never reach a halt state as well.
- *}
- "\<not> (\<exists> stp. isS0 (steps (Suc 0, [Bk, Bk], <lm>)
- (tprog @ (tMp n (mop_ss - 1))) stp))"
- using abacus_turing_eq_uhalt'
- layout compiled abc_unhalt mop_start
- by(auto)
-
-end
-