theory PreEqu = Main + Swap + Terms + Disagreement + Fresh:

consts 

  equ :: "(fresh_envs \<times> trm \<times> trm) set"

syntax 

  "_equ_judge"   :: "fresh_envs \<Rightarrow> trm \<Rightarrow> trm \<Rightarrow> bool" (" _ \<turnstile> _ \<approx> _" [80,80,80] 80)

translations 

  "nabla \<turnstile> t1 \<approx> t2" \<rightleftharpoons> "(nabla,t1,t2) \<in> equ"

inductive equ 

intros

equ_abst_ab[intro!]: "\<lbrakk>a\<noteq>b;(nabla \<turnstile> a \<sharp> t2);(nabla \<turnstile> t1 \<approx> (swap [(a,b)] t2))\<rbrakk> 

                      \<Longrightarrow> (nabla \<turnstile> Abst a t1 \<approx> Abst b t2)"

equ_abst_aa[intro!]: "(nabla \<turnstile> t1 \<approx> t2) \<Longrightarrow> (nabla \<turnstile> Abst a t1 \<approx> Abst a t2)" 

equ_unit[intro!]:    "(nabla \<turnstile> Unit \<approx> Unit)"

equ_atom[intro!]:    "a=b\<Longrightarrow>nabla \<turnstile> Atom a \<approx> Atom b"

equ_susp[intro!]:    "(\<forall> c \<in> ds pi1 pi2. (c,X) \<in> nabla) \<Longrightarrow> (nabla \<turnstile> Susp pi1 X \<approx> Susp pi2 X)"

equ_paar[intro!]:    "\<lbrakk>(nabla \<turnstile> t1\<approx>t2);(nabla \<turnstile> s1\<approx>s2)\<rbrakk> \<Longrightarrow> (nabla \<turnstile> Paar t1 s1 \<approx> Paar t2 s2)"

equ_func[intro!]:    "(nabla \<turnstile> t1 \<approx> t2) \<Longrightarrow> (nabla \<turnstile> Func F t1 \<approx> Func F t2)" 

lemma equ_atom_elim[elim!]: "nabla\<turnstile>Atom a \<approx> Atom b \<Longrightarrow> a=b"

apply(ind_cases "nabla \<turnstile> Atom a \<approx> Atom b", auto)

done

lemma equ_susp_elim[elim!]: "(nabla \<turnstile> Susp pi1 X \<approx> Susp pi2 X) 

                             \<Longrightarrow> (\<forall> c \<in> ds pi1 pi2. (c,X)\<in> nabla)"

apply(ind_cases "nabla \<turnstile> Susp pi1 X \<approx> Susp pi2 X", auto)

done

lemma equ_paar_elim[elim!]: "(nabla \<turnstile> Paar s1 t1 \<approx> Paar s2 t2) \<Longrightarrow> 

                             (nabla \<turnstile> s1 \<approx> s2)\<and>(nabla \<turnstile> t1 \<approx> t2)"

apply(ind_cases "nabla \<turnstile> Paar s1 t1 \<approx> Paar s2 t2", auto)

done

lemma equ_func_elim[elim!]: "(nabla \<turnstile> Func F t1 \<approx> Func F t2) \<Longrightarrow> (nabla \<turnstile> t1 \<approx> t2)"

apply(ind_cases "nabla \<turnstile> Func F t1 \<approx> Func F t2", auto)

done

lemma equ_abst_aa_elim[elim!]: "(nabla \<turnstile> Abst a t1 \<approx> Abst a t2) \<Longrightarrow> (nabla \<turnstile> t1 \<approx> t2)"

apply(ind_cases "nabla \<turnstile> Abst a t1 \<approx> Abst a t2", auto)

done

lemma equ_abst_ab_elim[elim!]: "\<lbrakk>(nabla \<turnstile> Abst a t1 \<approx> Abst b t2);a\<noteq>b\<rbrakk> \<Longrightarrow> 

                                (nabla \<turnstile> t1 \<approx> (swap [(a,b)] t2))\<and>(nabla\<turnstile>a\<sharp>t2)"

apply(ind_cases "(nabla \<turnstile> Abst a t1 \<approx> Abst b t2)", auto)

done

lemma equ_depth: "nabla \<turnstile> t1 \<approx> t2 \<Longrightarrow> depth t1 = depth t2"

apply(erule equ.induct)

apply(simp_all)

done

lemma rev_pi_pi_equ: "(nabla\<turnstile>swap (rev pi) (swap pi t)\<approx>t)"

apply(induct_tac t)

apply(auto)

-- Susp

apply(drule_tac ds_cancel_pi_left[of _ "rev pi @ pi" _ "[]", THEN mp, simplified])

apply(simp only: ds_rev_pi_pi)

apply(simp only: ds_def)

apply(force)

done

lemma equ_pi_right: "\<forall>pi. (\<forall>a\<in>ds [] pi. nabla\<turnstile>a\<sharp>t) \<longrightarrow> (nabla\<turnstile>t\<approx>swap pi t)"

apply(induct_tac t)

apply(simp_all)

-- Abst

apply(rule allI)

apply(case_tac "(swapas pi list)=list") 

apply(simp)

apply(rule impI)

apply(rule equ_abst_aa)

apply(drule_tac x="pi" in spec)

apply(subgoal_tac "(\<forall>a\<in>ds [] pi.  nabla \<turnstile> a \<sharp> trm)")--A

apply(force)

--A

apply(rule ballI)

apply(drule_tac x=a in bspec)

apply(assumption)

apply(case_tac "list\<noteq>a")

apply(force dest!: fresh_abst_ab_elim)

apply(simp add: ds_def)

apply(rule impI)

apply(rule equ_abst_ab)

apply(force)

apply(drule_tac x="swapas (rev pi) list" in bspec)

apply(simp add: ds_def)

apply(rule conjI)

apply(subgoal_tac "swapas (rev pi) list \<in> atms (rev pi)") --B

apply(simp)

--B

apply(drule swapas_pi_ineq_a[THEN mp])

apply(rule swapas_pi_in_atms)

apply(simp)

apply(clarify)

apply(drule swapas_rev_pi_b)

apply(simp)

apply(force dest!: fresh_abst_ab_elim  swapas_rev_pi_b intro!: fresh_swap_right[THEN mp])

apply(drule_tac x="(list, swapas pi list)#pi" in spec)

apply(subgoal_tac "(\<forall>a\<in>ds [] ((list, swapas pi list) # pi).  nabla \<turnstile> a \<sharp> trm)")--C

apply(force simp add: swap_append[THEN sym])

--C

apply(rule ballI)

apply(drule_tac x="a" in bspec)

apply(rule_tac b="list" in ds_7)

apply(force)

apply(assumption)

apply(case_tac "list=a")

apply(simp)

apply(simp only: ds_def mem_Collect_eq)

apply(erule conjE)

apply(subgoal_tac "a\<noteq>swapas pi a")

apply(simp)

apply(force)

apply(force dest!: fresh_abst_ab_elim)

-- Susp

apply(rule allI)

apply(rule impI)

apply(rule equ_susp)

apply(rule ballI)

apply(subgoal_tac "swapas list1 c\<in>ds [] pi")--A

apply(force dest!: fresh_susp_elim)

--A

apply(rule ds_cancel_pi_left[THEN mp])

apply(simp)

-- Unit

apply(force)

-- Atom

apply(rule allI)

apply(rule impI)

apply(case_tac "(swapas pi list) = list")

apply(force)

apply(drule ds_elem)

apply(force dest!: fresh_atom_elim)

-- Paar

apply(force dest!: fresh_paar_elim)

-- Func

apply(force)

done

lemma pi_comm: "nabla\<turnstile>(swap (pi@[(a,b)]) t)\<approx>(swap ([(swapas pi a, swapas pi b)]@pi) t)"

apply(induct_tac t)

apply(simp_all)

-- Abst

apply(force simp add: swapas_comm)

-- Susp

apply(rule equ_susp)

apply(rule ballI)

apply(simp only: ds_def)

apply(simp only: mem_Collect_eq)

apply(erule conjE)

apply(subgoal_tac "swapas (pi@[(a,b)]) (swapas list1 c) =

                   swapas ([(swapas pi a,swapas pi b)]@pi) (swapas list1 c)")--A

apply(simp add: swapas_append[THEN sym])

--A

apply(simp only: swapas_comm)

-- Units

apply(rule equ_unit)

-- Atom

apply(force dest!: swapas_rev_pi_b swapas_rev_pi_a simp add: swapas_append)

--Paar

apply(force)

--Func

apply(force)

done

lemma l3_jud: "(nabla \<turnstile> t1\<approx>t2) \<Longrightarrow> (nabla \<turnstile> a\<sharp>t1) \<longrightarrow> (nabla \<turnstile> a\<sharp>t2)"

apply(erule equ.induct)

apply(simp_all)

--Abst.ab

apply(rule impI)

apply(case_tac "aa=a")

apply(force)

apply(case_tac "b=a")

apply(force)

apply(force dest!: fresh_abst_ab_elim fresh_swap_left[THEN mp])

-- Abst.aa

apply(case_tac "a=aa")

apply(force)

apply(force dest!: fresh_abst_ab_elim)

-- Susp

apply(rule impI)

apply(drule fresh_susp_elim, rule fresh_susp)

apply(case_tac "swapas (rev pi1) a = swapas (rev pi2) a") 

apply(simp)

apply(drule_tac x="swapas (rev pi2) a" in bspec)

apply(rule ds_cancel_pi_left[THEN mp])

apply(subgoal_tac "swapas (pi1@(rev pi2)) a \<noteq> a")--A

apply(drule ds_elem)

apply(force simp add: ds_def swapas_append)

--A

apply(clarify)

apply(simp only: swapas_append)

apply(drule swapas_rev_pi_a)

apply(force)

apply(assumption)

-- Paar

apply(force dest!: fresh_paar_elim)

-- Func

apply(force dest!: fresh_func_elim)

done

lemma big: "\<forall>t1 t2 t3. (n=depth t1) \<longrightarrow>

             (((nabla\<turnstile>t1\<approx>t2)\<longrightarrow>(nabla\<turnstile>t2\<approx>t1))\<and>  

              (\<forall>pi. (nabla\<turnstile>t1\<approx>t2)\<longrightarrow>(nabla\<turnstile>swap pi t1\<approx>swap pi t2))\<and> 

              ((nabla\<turnstile>t1\<approx>t2)\<and>(nabla\<turnstile>t2\<approx>t3)\<longrightarrow>(nabla\<turnstile>t1\<approx>t3)))" 

apply(induct_tac n rule: nat_less_induct)

apply(rule allI)+apply(rule impI)

apply(rule conjI)

-- SYMMETRY

apply(rule impI)

apply(ind_cases "nabla \<turnstile> t1 \<approx> t2")

apply(simp_all)

-- Abst.ab

apply(rule equ_abst_ab)

apply(force) --abst.ab.first.premise

apply(rule_tac "t1.1"="swap [(a,b)] t2a" in l3_jud[THEN mp])

apply(drule_tac x="depth t1a" in spec)

apply(simp)

apply(rule fresh_swap_right[THEN mp])

apply(simp) --abst.ab.second.premise

apply(subgoal_tac "nabla \<turnstile> swap [(b, a)] t1a \<approx> t2a")

apply(drule_tac x="depth t1a" in spec)

apply(simp)

apply(subgoal_tac "nabla \<turnstile> swap [(b,a)] t1a \<approx> swap ([(b,a)]@[(a,b)]) t2a") --A

apply(subgoal_tac "nabla \<turnstile> swap ([(b,a)]@[(a,b)]) t2a \<approx> t2a") --B

apply(drule_tac x="depth t1a" in spec)

apply(simp (no_asm_use))

apply(drule_tac x="swap [(b,a)] t1a" in spec)

apply(simp (no_asm_use))

apply(drule_tac x="swap [(b,a),(a,b)] t2a" in spec)

apply(force)

--B

apply(subgoal_tac "nabla\<turnstile>t2a \<approx> swap ([(b, a)] @ [(a, b)]) t2a")--C

apply(drule_tac x="depth t1a" in spec)

apply(simp)

apply(drule_tac x="t2a" in spec)

apply(drule mp)

apply(drule equ_depth)

apply(force)

apply(best)

--C

apply(rule equ_pi_right[THEN spec,THEN mp])

apply(subgoal_tac "ds [] ([(b, a)] @ [(a, b)])={}")

apply(simp)

apply(simp add: ds_baab)

--A

apply(force simp only: swap_append)

-- Abst.aa

apply(force)

-- Unit

apply(rule equ_unit)

-- Atom

apply(force)

-- Susp

apply(force simp only: ds_sym)

-- Paar

apply(rule equ_paar)

apply(drule_tac x="depth t1a" in spec)

apply(simp add: Suc_max_left)

apply(drule_tac x="depth s1" in spec)

apply(simp add: Suc_max_right)

-- Func

apply(best)

-- ADD.PI

apply(rule conjI)

apply(rule impI)

apply(ind_cases "nabla \<turnstile> t1 \<approx> t2")

apply(simp_all)

-- Abst.ab

apply(rule allI)

apply(rule equ_abst_ab)

-- abst.ab.first.premise

apply(clarify)

apply(drule swapas_rev_pi_a)

apply(simp)

-- abst.ab.second.premise

apply(rule fresh_swap_right[THEN mp])

apply(simp)

-- abst.ab.third.premise

apply(subgoal_tac "nabla \<turnstile> swap pi t1a \<approx> swap (pi@[(a,b)]) t2a") --A

apply(subgoal_tac "nabla \<turnstile> swap (pi@[(a,b)]) t2a \<approx> swap ([(swapas pi a,swapas pi b)]@pi) t2a") --B

apply(drule_tac x="depth t1a" in spec)

apply(simp (no_asm_use))

apply(drule_tac x="swap pi t1a" in spec)

apply(simp (no_asm_use)) 

apply(drule_tac x="swap (pi@[(a,b)]) t2a" in spec)

apply(drule conjunct2)+

apply(drule_tac x="swap ((swapas pi a, swapas pi b) # pi) t2a" in spec)

apply(simp add: swap_append[THEN sym])

--B

apply(rule pi_comm)

apply(force simp only: swap_append)

-- A

apply(force simp only: swap_append)

-- Unit

apply(rule equ_unit)

-- Atom

apply(force)

-- Susp

apply(force simp only: ds_cancel_pi_front)

-- Paar

apply(rule allI)

apply(rule equ_paar)

apply(drule_tac x="depth t1a" in spec)

apply(simp only: Suc_max_left)

apply(drule_tac x="depth s1" in spec)

apply(simp only: Suc_max_right)

-- Func

apply(best)

-- TRANSITIVITY

apply(rule impI)

apply(erule conjE)

apply(ind_cases "nabla \<turnstile> t1 \<approx> t2")

apply(simp_all)

-- Abst.ab

apply(ind_cases "nabla \<turnstile> Abst b t2a \<approx> t3")

apply(simp)

apply(case_tac "ba=a")

apply(simp)

apply(rule equ_abst_aa)

apply(subgoal_tac "nabla\<turnstile>swap [(a,b)] t2a \<approx> t2b") --A

apply(best)

--A

apply(subgoal_tac "nabla\<turnstile>swap [(a,b)] t2a\<approx> swap ([(a,b)]@[(b,a)]) t2b") --B

apply(subgoal_tac "nabla\<turnstile>swap ([(a,b)]@[(b,a)]) t2b \<approx> t2b") --C

apply(drule_tac x="depth t1a" in spec)

apply(simp)

apply(drule_tac x="swap [(a,b)] t2a" in spec)

apply(drule equ_depth)

apply(simp (no_asm_use))

apply(best)

--C

apply(subgoal_tac "nabla\<turnstile>t2b \<approx> swap ([(a,b)]@[(b,a)]) t2b")--D

apply(drule_tac x="depth t1a" in spec)

apply(simp)

apply(drule_tac x="t2b" in spec)

apply(drule mp)

apply(force dest!: equ_depth)

apply(best)

--D

apply(rule equ_pi_right[THEN spec, THEN mp])

apply(simp add: ds_baab)

--B

apply(drule_tac x="depth t1a" in spec)

apply(simp)

apply(drule_tac x="t2a" in spec)

apply(drule equ_depth)

apply(simp) 

apply(drule_tac x="swap [(b, a)] t2b" in spec)

apply(drule conjunct2)

apply(drule conjunct1)

apply(simp)

apply(drule_tac x="[(a,b)]" in spec)

apply(simp add: swap_append[THEN sym])

-- Abst.ab

apply(rule equ_abst_ab)

-- abst.ab.first.premise

apply(force)

-- abst.ab.second.premise

apply(rule_tac "t1.1"="swap [(b,ba)] t2a" in l3_jud[THEN mp])

apply(subgoal_tac "nabla \<turnstile> swap [(b,ba)] t2a \<approx> swap ([(b,ba)]@[(b, ba)]) t2b") --A

apply(subgoal_tac "nabla\<turnstile>swap ([(b,ba)]@[(b,ba)]) t2b \<approx> t2b") --B

apply(drule_tac x="depth t1a" in spec)

apply(simp)

apply(drule_tac x="swap [(b, ba)] t2a" in spec)

apply(drule mp)

apply(force dest!: equ_depth)

apply(best)

--B

apply(subgoal_tac "nabla\<turnstile>t2b \<approx> swap ([(b,ba)] @ [(b,ba)]) t2b")--C

apply(drule_tac x="depth t1a" in spec)

apply(simp)

apply(drule_tac x="t2b" in spec)

apply(drule mp)

apply(force dest!: equ_depth)

apply(best)

-- C

apply(rule equ_pi_right[THEN spec, THEN mp])

apply(simp add: ds_abab)

--A

apply(drule_tac x="depth t1a" in spec)

apply(simp)

apply(drule_tac x="t2a" in spec)

apply(drule mp)

apply(force dest!: equ_depth)

apply(drule_tac x="swap [(b,ba)] t2b" in spec)

apply(drule conjunct2)

apply(drule conjunct1)

apply(simp)

apply(drule_tac x="[(b,ba)]" in spec)

apply(simp add: swap_append[THEN sym])

-- abst.ab.third.premise

apply(force intro!: fresh_swap_right[THEN mp])

-- very.complex

apply(subgoal_tac "nabla\<turnstile>t1a \<approx> swap ([(a,b)]@[(b,ba)]) t2b") --A

apply(subgoal_tac "nabla\<turnstile>swap ([(a,b)]@[(b,ba)]) t2b \<approx> swap [(a,ba)] t2b") --B

apply(drule_tac x="depth t1a" in spec)

apply(simp (no_asm_use))

apply(best)

--B

apply(subgoal_tac "nabla\<turnstile>swap [(a, ba)] t2b \<approx> swap [(a,b),(b,ba)] t2b")--C

apply(drule_tac x="depth t1a" in spec)

apply(simp)

apply(drule_tac x="swap [(a, ba)] t2b" in spec)

apply(drule mp)

apply(force dest!: equ_depth)

apply(force)

apply(subgoal_tac "nabla\<turnstile>swap [(a,ba)] t2b\<approx> swap [(a,ba)] (swap [(a,ba),(a,b),(b,ba)] t2b)") --D

apply(subgoal_tac "nabla\<turnstile>swap (rev [(a,ba)]) (swap [(a,ba)] (swap [(a,b),(b,ba)] t2b)) 

                        \<approx>swap [(a,b),(b,ba)] t2b") --E

apply(simp add: swap_append[THEN sym])

apply(drule_tac x="depth t1a" in spec)

apply(simp)

apply(drule_tac x="swap [(a,ba)] t2b" in spec)

apply(drule mp)

apply(force dest!: equ_depth)

apply(drule_tac x="swap [(a, ba), (a, ba), (a, b), (b, ba)] t2b" in spec)

apply(drule conjunct2)+

apply(best)

-- D

apply(rule rev_pi_pi_equ)

-- E

apply(subgoal_tac "nabla\<turnstile>t2b\<approx>swap [(a, ba), (a, b), (b, ba)] t2b") --F

apply(drule_tac x="depth t1a" in spec)

apply(simp)

apply(drule_tac x="t2b" in spec)

apply(drule mp)

apply(force dest!: equ_depth)

apply(best)

--F

apply(rule equ_pi_right[THEN spec, THEN mp])

apply(subgoal_tac "ds [] [(a,ba),(a,b),(b,ba)]={a,b}") -- G

apply(simp)

apply(drule_tac "t1.1"="t2a" and "t2.1"="swap [(b, ba)] t2b" and a1="a" in l3_jud[THEN mp])

apply(assumption)

apply(subgoal_tac "nabla \<turnstile> swapas (rev [(b,ba)]) a \<sharp> t2b") --H

apply(simp)

apply(case_tac "b=a")

apply(force)

apply(force)

--H

apply(rule fresh_swap_left[THEN mp])

apply(assumption)

--G

apply(rule ds_acabbc)

apply(assumption)+

--A

apply(subgoal_tac "nabla\<turnstile>swap [(a,b)] t2a\<approx>swap [(a,b)] (swap [(b,ba)] t2b)")--I

apply(drule_tac x="depth t1a" in spec)

apply(simp (no_asm_use))

apply(drule_tac x="t1a" in spec)

apply(simp (no_asm_use))

apply(drule_tac x="swap [(a,b)] t2a" in spec)

apply(drule conjunct2)+

apply(drule_tac x="swap [(a, b)] (swap [(b, ba)] t2b)" in spec)

apply(force simp add: swap_append[THEN sym])

--I 

apply(drule_tac x="depth t1a" in spec)

apply(simp (no_asm_use))

apply(drule_tac x="t2a" in spec)

apply(drule mp)

apply(force dest!: equ_depth)

apply(drule_tac x="swap [(b,ba)] t2b" in spec)

apply(best)

-- Abst.ab

apply(simp)

apply(rule equ_abst_ab)

apply(assumption)

apply(drule_tac "t1.1"="t2a" and "t2.1"="t2b" and a1="a" in l3_jud[THEN mp])

apply(assumption)+

apply(subgoal_tac "nabla\<turnstile>swap [(a, b)] t2a\<approx>swap [(a, b)] t2b") --A

apply(best)

--A

apply(drule_tac x="depth t1a" in spec)

apply(simp (no_asm_use))

apply(drule_tac x="t2a" in spec)

apply(drule mp)

apply(force dest!: equ_depth)

apply(drule_tac x="t2b" in spec)

apply(best)

-- Abst

apply(ind_cases "nabla \<turnstile> Abst a t2a \<approx> t3")

apply(best)

apply(best)

-- Susp

apply(ind_cases "nabla \<turnstile> Susp pi2 X \<approx> t3")

apply(simp)

apply(rule equ_susp)

apply(rule ballI)

apply(drule_tac "pi2.1"="pi2" in ds_trans[THEN mp])

apply(force)

-- Paar

apply(ind_cases "nabla \<turnstile> Paar t2a s2 \<approx> t3")

apply(simp)

apply(rule equ_paar)

apply(drule_tac x="depth t1a" in spec)

apply(simp (no_asm_use) add: Suc_max_left)

apply(best)

apply(drule_tac x="depth s1" in spec)

apply(simp (no_asm_use) add: Suc_max_right)

apply(best)

-- Func

apply(ind_cases "nabla \<turnstile> Func F t2a \<approx> t3")

apply(best)

done

lemma pi_right_equ_help: 

      "\<forall>t. (n=depth t) \<longrightarrow> (\<forall>pi. nabla\<turnstile>t\<approx>swap pi t\<longrightarrow>(\<forall>a\<in> ds [] pi. nabla\<turnstile>a\<sharp>t))"

apply(induct_tac n rule: nat_less_induct)

apply(rule allI)+

apply(rule impI)

apply(rule allI)+

apply(rule impI)

apply(ind_cases "nabla \<turnstile> t \<approx> swap pi t")

apply(simp_all)

--Abst.ab

apply(rule ballI)

apply(case_tac "aa=a")

apply(force)

apply(rule fresh_abst_ab)

apply(case_tac "aa=swapas (rev pi) a")

apply(simp)

apply(drule fresh_swap_left[THEN mp])

apply(assumption)

apply(drule_tac x="depth t1" in spec)

apply(simp)

apply(drule_tac x="t1" in spec)

apply(simp add: swap_append[THEN sym])

apply(drule_tac x="[(a, swapas pi a)] @ pi" in spec)