# Theory Class3

```theory Class3
imports Class2
begin

text ‹3rd Main Lemma›

lemma Cut_a_redu_elim:
assumes a: "Cut <a>.M (x).N ⟶⇩a R"
shows "(∃M'. R = Cut <a>.M' (x).N ∧ M ⟶⇩a M') ∨
(∃N'. R = Cut <a>.M (x).N' ∧ N ⟶⇩a N') ∨
(Cut <a>.M (x).N ⟶⇩c R) ∨
(Cut <a>.M (x).N ⟶⇩l R)"
using a
apply(erule_tac a_redu.cases)
apply(simp_all)
apply(rule disjI1)
apply(rule_tac x="[(a,aa)]∙M'" in exI)
apply(perm_simp add: fresh_left calc_atm a_redu.eqvt fresh_a_redu)
apply(rule_tac x="[(a,aa)]∙M'" in exI)
apply(perm_simp add: fresh_left calc_atm a_redu.eqvt fresh_a_redu)
apply(rule disjI2)
apply(rule disjI1)
apply(rule_tac x="[(x,xa)]∙N'" in exI)
apply(perm_simp add: fresh_left calc_atm a_redu.eqvt fresh_a_redu)
apply(rule_tac x="[(x,xa)]∙N'" in exI)
apply(perm_simp add: fresh_left calc_atm a_redu.eqvt fresh_a_redu)
done

lemma Cut_c_redu_elim:
assumes a: "Cut <a>.M (x).N ⟶⇩c R"
shows "(R = M{a:=(x).N} ∧ ¬fic M a) ∨
(R = N{x:=<a>.M} ∧ ¬fin N x)"
using a
apply(erule_tac c_redu.cases)
apply(simp_all)
apply(rule disjI1)
apply(drule_tac pi="[(a,aa)]" in fic.eqvt(2))
apply(perm_simp)
apply(drule_tac pi="[(a,aa)]" in fic.eqvt(2))
apply(perm_simp)
apply(rule disjI2)
apply(drule_tac pi="[(x,xa)]" in fin.eqvt(1))
apply(perm_simp)
apply(drule_tac pi="[(x,xa)]" in fin.eqvt(1))
apply(perm_simp)
done

lemma not_fic_crename_aux:
assumes a: "fic M c" "c♯(a,b)"
shows "fic (M[a⊢c>b]) c"
using a
apply(nominal_induct M avoiding: c a b rule: trm.strong_induct)
apply(auto dest!: fic_elims intro!: fic.intros simp add: fresh_prod fresh_atm rename_fresh abs_fresh)
done

lemma not_fic_crename:
assumes a: "¬(fic (M[a⊢c>b]) c)" "c♯(a,b)"
shows "¬(fic M c)"
using a
apply(auto dest:  not_fic_crename_aux)
done

lemma not_fin_crename_aux:
assumes a: "fin M y"
shows "fin (M[a⊢c>b]) y"
using a
apply(nominal_induct M avoiding: a b rule: trm.strong_induct)
apply(auto dest!: fin_elims intro!: fin.intros simp add: fresh_prod fresh_atm rename_fresh abs_fresh)
done

lemma not_fin_crename:
assumes a: "¬(fin (M[a⊢c>b]) y)"
shows "¬(fin M y)"
using a
apply(auto dest:  not_fin_crename_aux)
done

lemma crename_fresh_interesting1:
fixes c::"coname"
assumes a: "c♯(M[a⊢c>b])" "c♯(a,b)"
shows "c♯M"
using a
apply(nominal_induct M avoiding: c a b rule: trm.strong_induct)
apply(auto split: if_splits simp add: abs_fresh)
done

lemma crename_fresh_interesting2:
fixes x::"name"
assumes a: "x♯(M[a⊢c>b])"
shows "x♯M"
using a
apply(nominal_induct M avoiding: x a b rule: trm.strong_induct)
apply(auto split: if_splits simp add: abs_fresh abs_supp fin_supp fresh_atm)
done

lemma fic_crename:
assumes a: "fic (M[a⊢c>b]) c" "c♯(a,b)"
shows "fic M c"
using a
apply(nominal_induct M avoiding: c a b rule: trm.strong_induct)
apply(auto dest!: fic_elims intro!: fic.intros simp add: fresh_prod fresh_atm rename_fresh abs_fresh
split: if_splits)
apply(auto dest: crename_fresh_interesting1 simp add: fresh_prod fresh_atm)
done

lemma fin_crename:
assumes a: "fin (M[a⊢c>b]) x"
shows "fin M x"
using a
apply(nominal_induct M avoiding: x a b rule: trm.strong_induct)
apply(auto dest!: fin_elims intro!: fin.intros simp add: fresh_prod fresh_atm rename_fresh abs_fresh
split: if_splits)
apply(auto dest: crename_fresh_interesting2 simp add: fresh_prod fresh_atm)
done

lemma crename_Cut:
assumes a: "R[a⊢c>b] = Cut <c>.M (x).N" "c♯(a,b,N,R)" "x♯(M,R)"
shows "∃M' N'. R = Cut <c>.M' (x).N' ∧ M'[a⊢c>b] = M ∧ N'[a⊢c>b] = N ∧ c♯N' ∧ x♯M'"
using a
apply(nominal_induct R avoiding: a b c x M N rule: trm.strong_induct)
apply(auto split: if_splits)
apply(rule_tac x="[(name,x)]∙trm2" in exI)
apply(perm_simp)
apply(simp add: abs_fresh fresh_left calc_atm fresh_prod)
apply(drule sym)
apply(drule pt_bij1[OF pt_name_inst,OF at_name_inst])
apply(rule_tac x="[(coname,c)]∙trm1" in exI)
apply(perm_simp)
apply(simp add: abs_fresh fresh_left calc_atm fresh_prod)
apply(drule sym)
apply(drule pt_bij1[OF pt_coname_inst,OF at_coname_inst])
apply(rule_tac x="[(coname,c)]∙trm1" in exI)
apply(perm_simp)
apply(simp add: abs_fresh fresh_left calc_atm fresh_prod)
apply(rule_tac x="[(name,x)]∙trm2" in exI)
apply(perm_simp)
apply(simp add: abs_fresh fresh_left calc_atm fresh_prod)
apply(drule sym)
apply(drule pt_bij1[OF pt_coname_inst,OF at_coname_inst])
apply(drule sym)
apply(drule pt_bij1[OF pt_name_inst,OF at_name_inst])
apply(drule sym)
apply(drule pt_bij1[OF pt_name_inst,OF at_name_inst])
done

lemma crename_NotR:
assumes a: "R[a⊢c>b] = NotR (x).N c" "x♯R" "c♯(a,b)"
shows "∃N'. (R = NotR (x).N' c) ∧ N'[a⊢c>b] = N"
using a
apply(nominal_induct R avoiding: a b c x N rule: trm.strong_induct)
apply(auto split: if_splits simp add: fresh_prod fresh_atm alpha abs_fresh trm.inject)
apply(rule_tac x="[(name,x)]∙trm" in exI)
apply(perm_simp)
apply(simp add: abs_fresh fresh_left calc_atm fresh_prod)
apply(drule sym)
apply(drule pt_bij1[OF pt_name_inst,OF at_name_inst])
done

lemma crename_NotR':
assumes a: "R[a⊢c>b] = NotR (x).N c" "x♯R" "c♯a"
shows "(∃N'. (R = NotR (x).N' c) ∧ N'[a⊢c>b] = N) ∨ (∃N'. (R = NotR (x).N' a) ∧ b=c ∧ N'[a⊢c>b] = N)"
using a
apply(nominal_induct R avoiding: a b c x N rule: trm.strong_induct)
apply(auto split: if_splits simp add: fresh_prod fresh_atm abs_fresh alpha trm.inject)
apply(rule_tac x="[(name,x)]∙trm" in exI)
apply(perm_simp)
apply(simp add: abs_fresh fresh_left calc_atm fresh_prod)
apply(drule sym)
apply(drule pt_bij1[OF pt_name_inst,OF at_name_inst])
apply(rule_tac x="[(name,x)]∙trm" in exI)
apply(perm_simp)
apply(simp add: abs_fresh fresh_left calc_atm fresh_prod)
apply(drule sym)
apply(drule pt_bij1[OF pt_name_inst,OF at_name_inst])
done

lemma crename_NotR_aux:
assumes a: "R[a⊢c>b] = NotR (x).N c"
shows "(a=c ∧ a=b) ∨ (a≠c)"
using a
apply(nominal_induct R avoiding: a b c x N rule: trm.strong_induct)
apply(auto split: if_splits simp add: fresh_prod fresh_atm alpha abs_fresh trm.inject)
done

lemma crename_NotL:
assumes a: "R[a⊢c>b] = NotL <c>.N y" "c♯(R,a,b)"
shows "∃N'. (R = NotL <c>.N' y) ∧ N'[a⊢c>b] = N"
using a
apply(nominal_induct R avoiding: a b c y N rule: trm.strong_induct)
apply(auto split: if_splits simp add: fresh_prod fresh_atm alpha abs_fresh trm.inject)
apply(rule_tac x="[(coname,c)]∙trm" in exI)
apply(perm_simp)
apply(simp add: abs_fresh fresh_left calc_atm fresh_prod)
apply(drule sym)
apply(drule pt_bij1[OF pt_coname_inst,OF at_coname_inst])
done

lemma crename_AndL1:
assumes a: "R[a⊢c>b] = AndL1 (x).N y" "x♯R"
shows "∃N'. (R = AndL1 (x).N' y) ∧ N'[a⊢c>b] = N"
using a
apply(nominal_induct R avoiding: a b x y N rule: trm.strong_induct)
apply(auto split: if_splits simp add: fresh_prod fresh_atm alpha abs_fresh trm.inject)
apply(rule_tac x="[(name1,x)]∙trm" in exI)
apply(perm_simp)
apply(simp add: abs_fresh fresh_left calc_atm fresh_prod)
apply(drule sym)
apply(drule pt_bij1[OF pt_name_inst,OF at_name_inst])
done

lemma crename_AndL2:
assumes a: "R[a⊢c>b] = AndL2 (x).N y" "x♯R"
shows "∃N'. (R = AndL2 (x).N' y) ∧ N'[a⊢c>b] = N"
using a
apply(nominal_induct R avoiding: a b x y N rule: trm.strong_induct)
apply(auto split: if_splits simp add: fresh_prod fresh_atm alpha abs_fresh trm.inject)
apply(rule_tac x="[(name1,x)]∙trm" in exI)
apply(perm_simp)
apply(simp add: abs_fresh fresh_left calc_atm fresh_prod)
apply(drule sym)
apply(drule pt_bij1[OF pt_name_inst,OF at_name_inst])
done

lemma crename_AndR_aux:
assumes a: "R[a⊢c>b] = AndR <c>.M <d>.N e"
shows "(a=e ∧ a=b) ∨ (a≠e)"
using a
apply(nominal_induct R avoiding: a b c d e M N rule: trm.strong_induct)
apply(auto split: if_splits simp add: fresh_prod fresh_atm alpha abs_fresh trm.inject)
done

lemma crename_AndR:
assumes a: "R[a⊢c>b] = AndR <c>.M <d>.N e" "c♯(a,b,d,e,N,R)" "d♯(a,b,c,e,M,R)" "e♯(a,b)"
shows "∃M' N'. R = AndR <c>.M' <d>.N' e ∧ M'[a⊢c>b] = M ∧ N'[a⊢c>b] = N ∧ c♯N' ∧ d♯M'"
using a
apply(nominal_induct R avoiding: a b c d e M N rule: trm.strong_induct)
apply(auto split: if_splits simp add: trm.inject alpha)
apply(rule_tac x="[(coname2,d)]∙trm2" in exI)
apply(perm_simp)
apply(auto simp add: abs_fresh fresh_left calc_atm fresh_prod fresh_atm)[1]
apply(rule_tac x="[(coname1,c)]∙trm1" in exI)
apply(perm_simp)
apply(auto simp add: abs_fresh fresh_left calc_atm fresh_prod fresh_atm)[1]
apply(rule_tac x="[(coname1,c)]∙trm1" in exI)
apply(perm_simp)
apply(auto simp add: abs_fresh fresh_left calc_atm fresh_prod fresh_atm)[1]
apply(rule_tac x="[(coname2,d)]∙trm2" in exI)
apply(perm_simp)
apply(auto simp add: abs_fresh fresh_left calc_atm fresh_prod fresh_atm)[1]
apply(rule_tac x="[(coname1,c)]∙trm1" in exI)
apply(perm_simp)
apply(auto simp add: abs_fresh fresh_left calc_atm fresh_prod fresh_atm)[1]
apply(rule_tac x="[(coname1,c)]∙trm1" in exI)
apply(perm_simp)
apply(auto simp add: abs_fresh fresh_left calc_atm fresh_prod fresh_atm)[1]
apply(rule_tac x="[(coname2,d)]∙trm2" in exI)
apply(perm_simp)
apply(auto simp add: abs_fresh fresh_left calc_atm fresh_prod fresh_atm)[1]
apply(drule sym)
apply(drule pt_bij1[OF pt_coname_inst,OF at_coname_inst])
apply(drule_tac s="trm2[a⊢c>b]" in  sym)
apply(drule pt_bij1[OF pt_coname_inst,OF at_coname_inst])
done

lemma crename_AndR':
assumes a: "R[a⊢c>b] = AndR <c>.M <d>.N e" "c♯(a,b,d,e,N,R)" "d♯(a,b,c,e,M,R)" "e♯a"
shows "(∃M' N'. R = AndR <c>.M' <d>.N' e ∧ M'[a⊢c>b] = M ∧ N'[a⊢c>b] = N ∧ c♯N' ∧ d♯M') ∨
(∃M' N'. R = AndR <c>.M' <d>.N' a ∧ b=e ∧ M'[a⊢c>b] = M ∧ N'[a⊢c>b] = N ∧ c♯N' ∧ d♯M')"
using a [[simproc del: defined_all]]
apply(nominal_induct R avoiding: a b c d e M N rule: trm.strong_induct)
apply(auto split: if_splits simp add: trm.inject alpha)[1]
apply(auto split: if_splits simp add: trm.inject alpha)[1]
apply(auto split: if_splits simp add: trm.inject alpha)[1]
apply(auto split: if_splits simp add: trm.inject alpha)[1]
apply(simp add: abs_fresh fresh_left calc_atm fresh_prod fresh_atm trm.inject alpha)[1]
apply(case_tac "coname3=a")
apply(simp)
apply(rule_tac x="[(coname1,c)]∙trm1" in exI)
apply(perm_simp)
apply(auto simp add: abs_fresh fresh_left calc_atm fresh_prod fresh_atm)[1]
apply(rule_tac x="[(coname2,d)]∙trm2" in exI)
apply(perm_simp)
apply(auto simp add: abs_fresh fresh_left calc_atm fresh_prod fresh_atm trm.inject alpha split: if_splits)[1]
apply(drule sym)
apply(drule pt_bij1[OF pt_coname_inst,OF at_coname_inst])
apply(drule_tac s="trm2[a⊢c>e]" in  sym)
apply(drule pt_bij1[OF pt_coname_inst,OF at_coname_inst])
apply(simp)
apply(rule_tac x="[(coname1,c)]∙trm1" in exI)
apply(perm_simp)
apply(simp add: abs_fresh fresh_left calc_atm fresh_prod)
apply(rule_tac x="[(coname2,d)]∙trm2" in exI)
apply(perm_simp)
apply(auto simp add: abs_fresh fresh_left calc_atm fresh_prod fresh_atm trm.inject alpha split: if_splits)[1]
apply(drule sym)
apply(drule pt_bij1[OF pt_coname_inst,OF at_coname_inst])
apply(drule_tac s="trm2[a⊢c>b]" in  sym)
apply(drule pt_bij1[OF pt_coname_inst,OF at_coname_inst])
apply(auto split: if_splits simp add: fresh_prod fresh_atm alpha abs_fresh trm.inject)
done

lemma crename_OrR1_aux:
assumes a: "R[a⊢c>b] = OrR1 <c>.M e"
shows "(a=e ∧ a=b) ∨ (a≠e)"
using a
apply(nominal_induct R avoiding: a b c e M rule: trm.strong_induct)
apply(auto split: if_splits simp add: fresh_prod fresh_atm alpha abs_fresh trm.inject)
done

lemma crename_OrR1:
assumes a: "R[a⊢c>b] = OrR1 <c>.N d" "c♯(R,a,b)" "d♯(a,b)"
shows "∃N'. (R = OrR1 <c>.N' d) ∧ N'[a⊢c>b] = N"
using a
apply(nominal_induct R avoiding: a b c d N rule: trm.strong_induct)
apply(auto split: if_splits simp add: fresh_prod fresh_atm alpha abs_fresh trm.inject)
apply(rule_tac x="[(coname1,c)]∙trm" in exI)
apply(perm_simp)
apply(simp add: abs_fresh fresh_left calc_atm fresh_prod)
apply(drule sym)
apply(drule pt_bij1[OF pt_coname_inst,OF at_coname_inst])
done

lemma crename_OrR1':
assumes a: "R[a⊢c>b] = OrR1 <c>.N d" "c♯(R,a,b)" "d♯a"
shows "(∃N'. (R = OrR1 <c>.N' d) ∧ N'[a⊢c>b] = N) ∨
(∃N'. (R = OrR1 <c>.N' a) ∧ b=d ∧ N'[a⊢c>b] = N)"
using a
apply(nominal_induct R avoiding: a b c d N rule: trm.strong_induct)
apply(auto split: if_splits simp add: fresh_prod fresh_atm alpha abs_fresh trm.inject)
apply(rule_tac x="[(coname1,c)]∙trm" in exI)
apply(perm_simp)
apply(simp add: abs_fresh fresh_left calc_atm fresh_prod)
apply(drule sym)
apply(drule pt_bij1[OF pt_coname_inst,OF at_coname_inst])
apply(rule_tac x="[(coname1,c)]∙trm" in exI)
apply(perm_simp)
apply(simp add: abs_fresh fresh_left calc_atm fresh_prod)
apply(drule sym)
apply(drule pt_bij1[OF pt_coname_inst,OF at_coname_inst])
done

lemma crename_OrR2_aux:
assumes a: "R[a⊢c>b] = OrR2 <c>.M e"
shows "(a=e ∧ a=b) ∨ (a≠e)"
using a
apply(nominal_induct R avoiding: a b c e M rule: trm.strong_induct)
apply(auto split: if_splits simp add: fresh_prod fresh_atm alpha abs_fresh trm.inject)
done

lemma crename_OrR2:
assumes a: "R[a⊢c>b] = OrR2 <c>.N d" "c♯(R,a,b)" "d♯(a,b)"
shows "∃N'. (R = OrR2 <c>.N' d) ∧ N'[a⊢c>b] = N"
using a
apply(nominal_induct R avoiding: a b c d N rule: trm.strong_induct)
apply(auto split: if_splits simp add: fresh_prod fresh_atm alpha abs_fresh trm.inject)
apply(rule_tac x="[(coname1,c)]∙trm" in exI)
apply(perm_simp)
apply(simp add: abs_fresh fresh_left calc_atm fresh_prod)
apply(drule sym)
apply(drule pt_bij1[OF pt_coname_inst,OF at_coname_inst])
done

lemma crename_OrR2':
assumes a: "R[a⊢c>b] = OrR2 <c>.N d" "c♯(R,a,b)" "d♯a"
shows "(∃N'. (R = OrR2 <c>.N' d) ∧ N'[a⊢c>b] = N) ∨
(∃N'. (R = OrR2 <c>.N' a) ∧ b=d ∧ N'[a⊢c>b] = N)"
using a
apply(nominal_induct R avoiding: a b c d N rule: trm.strong_induct)
apply(auto split: if_splits simp add: fresh_prod fresh_atm alpha abs_fresh trm.inject)
apply(rule_tac x="[(coname1,c)]∙trm" in exI)
apply(perm_simp)
apply(simp add: abs_fresh fresh_left calc_atm fresh_prod)
apply(drule sym)
apply(drule pt_bij1[OF pt_coname_inst,OF at_coname_inst])
apply(rule_tac x="[(coname1,c)]∙trm" in exI)
apply(perm_simp)
apply(simp add: abs_fresh fresh_left calc_atm fresh_prod)
apply(drule sym)
apply(drule pt_bij1[OF pt_coname_inst,OF at_coname_inst])
done

lemma crename_OrL:
assumes a: "R[a⊢c>b] = OrL (x).M (y).N z" "x♯(y,z,N,R)" "y♯(x,z,M,R)"
shows "∃M' N'. R = OrL (x).M' (y).N' z ∧ M'[a⊢c>b] = M ∧ N'[a⊢c>b] = N ∧ x♯N' ∧ y♯M'"
using a
apply(nominal_induct R avoiding: a b x y z M N rule: trm.strong_induct)
apply(auto split: if_splits simp add: trm.inject alpha)
apply(rule_tac x="[(name2,y)]∙trm2" in exI)
apply(perm_simp)
apply(auto simp add: abs_fresh fresh_left calc_atm fresh_prod fresh_atm)[1]
apply(rule_tac x="[(name1,x)]∙trm1" in exI)
apply(perm_simp)
apply(auto simp add: abs_fresh fresh_left calc_atm fresh_prod fresh_atm)[1]
apply(rule_tac x="[(name1,x)]∙trm1" in exI)
apply(perm_simp)
apply(auto simp add: abs_fresh fresh_left calc_atm fresh_prod fresh_atm)[1]
apply(rule_tac x="[(name2,y)]∙trm2" in exI)
apply(perm_simp)
apply(auto simp add: abs_fresh fresh_left calc_atm fresh_prod fresh_atm)[1]
apply(drule sym)
apply(drule pt_bij1[OF pt_name_inst,OF at_name_inst])
apply(drule_tac s="trm2[a⊢c>b]" in  sym)
apply(drule pt_bij1[OF pt_name_inst,OF at_name_inst])
done

lemma crename_ImpL:
assumes a: "R[a⊢c>b] = ImpL <c>.M (y).N z" "c♯(a,b,N,R)" "y♯(z,M,R)"
shows "∃M' N'. R = ImpL <c>.M' (y).N' z ∧ M'[a⊢c>b] = M ∧ N'[a⊢c>b] = N ∧ c♯N' ∧ y♯M'"
using a
apply(nominal_induct R avoiding: a b c y z M N rule: trm.strong_induct)
apply(auto split: if_splits simp add: trm.inject alpha)
apply(rule_tac x="[(name1,y)]∙trm2" in exI)
apply(perm_simp)
apply(auto simp add: abs_fresh fresh_left calc_atm fresh_prod fresh_atm)[1]
apply(rule_tac x="[(coname,c)]∙trm1" in exI)
apply(perm_simp)
apply(auto simp add: abs_fresh fresh_left calc_atm fresh_prod fresh_atm)[1]
apply(rule_tac x="[(coname,c)]∙trm1" in exI)
apply(perm_simp)
apply(auto simp add: abs_fresh fresh_left calc_atm fresh_prod fresh_atm)[1]
apply(rule_tac x="[(name1,y)]∙trm2" in exI)
apply(perm_simp)
apply(auto simp add: abs_fresh fresh_left calc_atm fresh_prod fresh_atm)[1]
apply(drule sym)
apply(drule pt_bij1[OF pt_coname_inst,OF at_coname_inst])
apply(drule_tac s="trm2[a⊢c>b]" in  sym)
apply(drule pt_bij1[OF pt_name_inst,OF at_name_inst])
done

lemma crename_ImpR_aux:
assumes a: "R[a⊢c>b] = ImpR (x).<c>.M e"
shows "(a=e ∧ a=b) ∨ (a≠e)"
using a
apply(nominal_induct R avoiding: x a b c e M rule: trm.strong_induct)
apply(auto split: if_splits simp add: fresh_prod fresh_atm alpha abs_fresh trm.inject)
done

lemma crename_ImpR:
assumes a: "R[a⊢c>b] = ImpR (x).<c>.N d" "c♯(R,a,b)" "d♯(a,b)" "x♯R"
shows "∃N'. (R = ImpR (x).<c>.N' d) ∧ N'[a⊢c>b] = N"
using a
apply(nominal_induct R avoiding: a b x c d N rule: trm.strong_induct)
apply(auto split: if_splits simp add: fresh_prod fresh_atm abs_perm alpha abs_fresh trm.inject)
apply(rule_tac x="[(name,x)]∙trm" in exI)
apply(perm_simp)
apply(simp add: abs_fresh fresh_left calc_atm fresh_prod)
apply(rule_tac x="[(name,x)]∙[(coname1, c)]∙trm" in exI)
apply(perm_simp)
apply(simp add: abs_supp fin_supp abs_fresh fresh_left calc_atm fresh_prod)
apply(drule sym)
apply(drule pt_bij1[OF pt_coname_inst,OF at_coname_inst])
apply(drule pt_bij1[OF pt_name_inst,OF at_name_inst])
done

lemma crename_ImpR':
assumes a: "R[a⊢c>b] = ImpR (x).<c>.N d" "c♯(R,a,b)" "x♯R" "d♯a"
shows "(∃N'. (R = ImpR (x).<c>.N' d) ∧ N'[a⊢c>b] = N) ∨
(∃N'. (R = ImpR (x).<c>.N' a) ∧ b=d ∧ N'[a⊢c>b] = N)"
using a
apply(nominal_induct R avoiding: x a b c d N rule: trm.strong_induct)
apply(auto split: if_splits simp add: fresh_prod fresh_atm alpha abs_fresh trm.inject abs_perm calc_atm)
apply(rule_tac x="[(name,x)]∙[(coname1,c)]∙trm" in exI)
apply(perm_simp)
apply(simp add: abs_fresh fresh_left calc_atm fresh_prod abs_supp fin_supp)
apply(drule sym)
apply(drule pt_bij1[OF pt_coname_inst,OF at_coname_inst])
apply(drule pt_bij1[OF pt_name_inst,OF at_name_inst])
apply(rule_tac x="[(name,x)]∙[(coname1,c)]∙trm" in exI)
apply(perm_simp)
apply(simp add: abs_fresh fresh_left calc_atm fresh_prod abs_supp fin_supp)
apply(drule sym)
apply(drule pt_bij1[OF pt_coname_inst,OF at_coname_inst])
apply(drule pt_bij1[OF pt_name_inst,OF at_name_inst])
done

lemma crename_ax2:
assumes a: "N[a⊢c>b] = Ax x c"
shows "∃d. N = Ax x d"
using a
apply(nominal_induct N avoiding: a b rule: trm.strong_induct)
apply(auto split: if_splits)
done

lemma crename_interesting1:
assumes a: "distinct [a,b,c]"
shows "M[a⊢c>c][c⊢c>b] = M[c⊢c>b][a⊢c>b]"
using a
apply(nominal_induct M avoiding: a c b rule: trm.strong_induct)
apply(blast)
apply(rotate_tac 12)
apply(drule_tac x="a" in meta_spec)
apply(rotate_tac 15)
apply(drule_tac x="c" in meta_spec)
apply(rotate_tac 15)
apply(drule_tac x="b" in meta_spec)
apply(blast)
apply(blast)
apply(blast)
done

lemma crename_interesting2:
assumes a: "a≠c" "a≠d" "a≠b" "c≠d" "b≠c"
shows "M[a⊢c>b][c⊢c>d] = M[c⊢c>d][a⊢c>b]"
using a
apply(nominal_induct M avoiding: a c b d rule: trm.strong_induct)
done

lemma crename_interesting3:
shows "M[a⊢c>c][x⊢n>y] = M[x⊢n>y][a⊢c>c]"
apply(nominal_induct M avoiding: a c x y rule: trm.strong_induct)
done

lemma crename_credu:
assumes a: "(M[a⊢c>b]) ⟶⇩c M'"
shows "∃M0. M0[a⊢c>b]=M' ∧ M ⟶⇩c M0"
using a
apply(nominal_induct M≡"M[a⊢c>b]" M' avoiding: M a b rule: c_redu.strong_induct)
apply(drule sym)
apply(drule crename_Cut)
apply(simp)
apply(simp)
apply(auto)
apply(rule_tac x="M'{a:=(x).N'}" in exI)
apply(rule conjI)
apply(simp add: fresh_atm abs_fresh subst_comm fresh_prod)
apply(rule c_redu.intros)
apply(auto dest: not_fic_crename)[1]
apply(drule sym)
apply(drule crename_Cut)
apply(simp)
apply(simp)
apply(auto)
apply(rule_tac x="N'{x:=<a>.M'}" in exI)
apply(rule conjI)
apply(simp add: fresh_atm abs_fresh subst_comm fresh_prod)
apply(rule c_redu.intros)
apply(auto dest: not_fin_crename)[1]
done

lemma crename_lredu:
assumes a: "(M[a⊢c>b]) ⟶⇩l M'"
shows "∃M0. M0[a⊢c>b]=M' ∧ M ⟶⇩l M0"
using a
apply(nominal_induct M≡"M[a⊢c>b]" M' avoiding: M a b rule: l_redu.strong_induct)
apply(drule sym)
apply(drule crename_Cut)
apply(simp)
apply(auto simp add: abs_fresh fresh_prod fresh_atm)[1]
apply(case_tac "aa=ba")
apply(rule l_redu.intros)
apply(simp)
apply(assumption)
apply(frule crename_ax2)
apply(auto)[1]
apply(case_tac "d=aa")
apply(rule_tac x="M'[a⊢c>aa]" in exI)
apply(rule conjI)
apply(rule crename_interesting1)
apply(simp)
apply(rule l_redu.intros)
apply(simp)
apply(auto dest: fic_crename simp add: fresh_prod fresh_atm)[1]
apply(rule_tac x="M'[a⊢c>b]" in exI)
apply(rule conjI)
apply(rule crename_interesting2)
apply(simp)
apply(simp)
apply(simp)
apply(simp)
apply(simp)
apply(rule l_redu.intros)
apply(simp)
apply(auto dest: fic_crename simp add: fresh_prod fresh_atm)[1]
apply(drule sym)
apply(drule crename_Cut)
apply(auto simp add: abs_fresh fresh_prod fresh_atm)[1]
apply(case_tac "aa=b")
apply(rule l_redu.intros)
apply(simp)
apply(assumption)
apply(frule crename_ax2)
apply(auto)[1]
apply(case_tac "d=aa")
apply(rule_tac x="N'[x⊢n>y]" in exI)
apply(rule conjI)
apply(rule sym)
apply(rule crename_interesting3)
apply(rule l_redu.intros)
apply(simp)
apply(auto dest: fin_crename simp add: fresh_prod fresh_atm)[1]
(* LNot *)
apply(drule sym)
apply(drule crename_Cut)
apply(auto simp add: abs_fresh fresh_prod fresh_atm)[1]
apply(drule crename_NotR)
apply(auto simp add: abs_fresh fresh_prod fresh_atm)[1]
apply(drule crename_NotL)
apply(auto simp add: abs_fresh fresh_prod fresh_atm)[1]
apply(rule_tac x="Cut <b>.N'b (x).N'a" in exI)
apply(rule l_redu.intros)
apply(auto simp add: fresh_prod intro: crename_fresh_interesting2)[1]
apply(auto simp add: fresh_atm fresh_prod intro: crename_fresh_interesting2)[1]
apply(auto simp add: fresh_atm fresh_prod intro: crename_fresh_interesting1)[1]
apply(auto simp add: fresh_atm fresh_prod intro: crename_fresh_interesting1)[1]
(* LAnd1 *)
apply(auto dest: fin_crename simp add: fresh_prod fresh_atm)[1]
apply(drule sym)
apply(drule crename_Cut)
apply(auto)[1]
apply(drule crename_AndR)
apply(auto simp add: abs_fresh fresh_prod fresh_atm)[1]
apply(drule crename_AndL1)
apply(auto simp add: abs_fresh fresh_prod fresh_atm)[1]
apply(rule_tac x="Cut <a1>.M'a (x).N'a" in exI)
apply(rule l_redu.intros)
apply(auto simp add: fresh_atm abs_fresh fresh_prod intro: crename_fresh_interesting1)[1]
apply(auto simp add: abs_fresh fresh_atm fresh_prod intro: crename_fresh_interesting2)[1]
apply(auto simp add: fresh_atm fresh_prod intro: crename_fresh_interesting1)[1]
apply(auto simp add: fresh_atm fresh_prod intro: crename_fresh_interesting1)[1]
(* LAnd2 *)
apply(auto dest: fin_crename simp add: fresh_prod fresh_atm)[1]
apply(drule sym)
apply(drule crename_Cut)
apply(auto)[1]
apply(drule crename_AndR)
apply(auto simp add: abs_fresh fresh_prod fresh_atm)[1]
apply(drule crename_AndL2)
apply(auto simp add: abs_fresh fresh_prod fresh_atm)[1]
apply(rule_tac x="Cut <a2>.N'b (x).N'a" in exI)
apply(rule l_redu.intros)
apply(auto simp add: fresh_atm abs_fresh fresh_prod intro: crename_fresh_interesting1)[1]
apply(auto simp add: abs_fresh fresh_atm fresh_prod intro: crename_fresh_interesting2)[1]
apply(auto simp add: fresh_atm fresh_prod intro: crename_fresh_interesting1)[1]
apply(auto simp add: fresh_atm fresh_prod intro: crename_fresh_interesting1)[1]
(* LOr1 *)
apply(auto dest: fin_crename simp add: fresh_prod fresh_atm)[1]
apply(drule sym)
apply(drule crename_Cut)
apply(auto)[1]
apply(drule crename_OrL)
apply(auto simp add: abs_fresh fresh_prod fresh_atm)[1]
apply(drule crename_OrR1)
apply(auto simp add: abs_fresh fresh_prod fresh_atm)[1]
apply(auto)
apply(rule_tac x="Cut <a>.N' (x1).M'a" in exI)
apply(rule conjI)
apply(rule l_redu.intros)
apply(auto simp add: fresh_atm abs_fresh fresh_prod intro: crename_fresh_interesting1)[1]
apply(auto simp add: abs_fresh fresh_atm fresh_prod intro: crename_fresh_interesting2)[1]
apply(auto simp add: abs_fresh fresh_atm fresh_prod intro: crename_fresh_interesting1)[1]
apply(auto simp add: abs_fresh fresh_atm fresh_prod intro: crename_fresh_interesting1)[1]
(* LOr2 *)
apply(auto dest: fin_crename simp add: fresh_prod fresh_atm)[1]
apply(drule sym)
apply(drule crename_Cut)
apply(auto)[1]
apply(drule crename_OrL)
apply(auto simp add: abs_fresh fresh_prod fresh_atm)[1]
apply(drule crename_OrR2)
apply(auto simp add: abs_fresh fresh_prod fresh_atm)[1]
apply(auto)
apply(rule_tac x="Cut <a>.N' (x2).N'a" in exI)
apply(rule conjI)
apply(rule l_redu.intros)
apply(auto simp add: fresh_atm abs_fresh fresh_prod intro: crename_fresh_interesting1)[1]
apply(auto simp add: abs_fresh fresh_atm fresh_prod intro: crename_fresh_interesting2)[1]
apply(auto simp add: abs_fresh fresh_atm fresh_prod intro: crename_fresh_interesting1)[1]
apply(auto simp add: abs_fresh fresh_atm fresh_prod intro: crename_fresh_interesting1)[1]
(* ImpL *)
apply(auto dest: fin_crename simp add: fresh_prod fresh_atm)[1]
apply(drule sym)
apply(drule crename_Cut)
apply(simp add: fresh_prod abs_fresh fresh_atm abs_supp fin_supp)
apply(auto)[1]
apply(drule crename_ImpL)
apply(auto simp add: abs_fresh fresh_prod fresh_atm)[1]
apply(drule crename_ImpR)
apply(simp)
apply(auto simp add: abs_fresh fresh_prod fresh_atm)[1]
apply(rule_tac x="Cut <a>.(Cut <c>.M'a (x).N') (y).N'a" in exI)
apply(rule conjI)
apply(simp add: abs_fresh abs_supp fin_supp fresh_atm)[1]
apply(rule l_redu.intros)
apply(auto simp add: fresh_atm abs_fresh abs_supp fin_supp fresh_prod intro: crename_fresh_interesting2)[1]
apply(auto simp add: abs_fresh fresh_atm abs_supp fin_supp fresh_prod intro: crename_fresh_interesting1)[1]
apply(auto simp add: abs_fresh fresh_atm abs_supp fin_supp fresh_prod intro: crename_fresh_interesting2)[1]
apply(auto simp add: abs_fresh fresh_atm abs_supp fin_supp fresh_prod intro: crename_fresh_interesting1)[1]
apply(auto simp add: abs_fresh fresh_atm abs_supp fin_supp fresh_prod intro: crename_fresh_interesting1)[1]
apply(auto simp add: abs_fresh fresh_atm abs_supp fin_supp fresh_prod intro: crename_fresh_interesting1)[1]
done

lemma crename_aredu:
assumes a: "(M[a⊢c>b]) ⟶⇩a M'" "a≠b"
shows "∃M0. M0[a⊢c>b]=M' ∧ M ⟶⇩a M0"
using a
apply(nominal_induct "M[a⊢c>b]" M' avoiding: M a b rule: a_redu.strong_induct)
apply(drule  crename_lredu)
apply(blast)
apply(drule  crename_credu)
apply(blast)
(* Cut *)
apply(drule sym)
apply(drule crename_Cut)
apply(simp)
apply(simp)
apply(auto)[1]
apply(drule_tac x="M'a" in meta_spec)
apply(drule_tac x="aa" in meta_spec)
apply(drule_tac x="b" in meta_spec)
apply(auto)[1]
apply(rule_tac x="Cut <a>.M0 (x).N'" in exI)
apply(simp add: abs_fresh abs_supp fin_supp fresh_atm)[1]
apply(rule conjI)
apply(rule trans)
apply(rule crename.simps)
apply(simp add: abs_fresh abs_supp fin_supp fresh_atm)[1]
apply(drule crename_fresh_interesting2)
apply(simp)
apply(auto)[1]
apply(drule sym)
apply(drule crename_Cut)
apply(simp)
apply(simp)
apply(auto)[1]
apply(drule_tac x="N'a" in meta_spec)
apply(drule_tac x="aa" in meta_spec)
apply(drule_tac x="b" in meta_spec)
apply(auto)[1]
apply(rule_tac x="Cut <a>.M' (x).M0" in exI)
apply(simp add: abs_fresh abs_supp fin_supp fresh_atm)[1]
apply(rule conjI)
apply(rule trans)
apply(rule crename.simps)
apply(simp add: abs_fresh abs_supp fin_supp fresh_atm fresh_prod)[1]
apply(drule crename_fresh_interesting1)
apply(simp add: abs_fresh abs_supp fin_supp fresh_atm)[1]
apply(simp)
apply(simp)
apply(auto)[1]
(* NotL *)
apply(drule sym)
apply(drule crename_NotL)
apply(simp)
apply(auto)[1]
apply(drule_tac x="N'" in meta_spec)
apply(drule_tac x="aa" in meta_spec)
apply(drule_tac x="b" in meta_spec)
apply(auto)[1]
apply(rule_tac x="NotL <a>.M0 x" in exI)
apply(simp add: abs_fresh abs_supp fin_supp fresh_atm)[1]
apply(auto)[1]
(* NotR *)
apply(drule sym)
apply(frule crename_NotR_aux)
apply(erule disjE)
apply(auto)[1]
apply(drule crename_NotR')
apply(simp)
apply(erule disjE)
apply(auto)[1]
apply(drule_tac x="N'" in meta_spec)
apply(drule_tac x="aa" in meta_spec)
apply(drule_tac x="b" in meta_spec)
apply(auto)[1]
apply(rule_tac x="NotR (x).M0 a" in exI)
apply(simp add: abs_fresh abs_supp fin_supp fresh_atm)[1]
apply(auto)[1]
apply(auto)[1]
apply(drule_tac x="N'" in meta_spec)
apply(drule_tac x="aa" in meta_spec)
apply(drule_tac x="a" in meta_spec)
apply(auto)[1]
apply(rule_tac x="NotR (x).M0 aa" in exI)
apply(simp add: abs_fresh abs_supp fin_supp fresh_atm)[1]
apply(auto)[1]
(* AndR *)
apply(drule sym)
apply(frule crename_AndR_aux)
apply(erule disjE)
apply(auto)[1]
apply(drule crename_AndR')
apply(erule disjE)
apply(auto)[1]
apply(drule_tac x="M'a" in meta_spec)
apply(drule_tac x="aa" in meta_spec)
apply(drule_tac x="ba" in meta_spec)
apply(auto)[1]
apply(rule_tac x="AndR <a>.M0 <b>.N' c" in exI)
apply(simp add: abs_fresh abs_supp fin_supp fresh_atm)[1]
apply(auto)[1]
apply(rule trans)
apply(rule crename.simps)
apply(simp add: abs_fresh abs_supp fin_supp fresh_atm)[1]
apply(simp add: fresh_prod abs_fresh abs_supp fin_supp fresh_atm)[1]
apply(auto intro: fresh_a_redu)[1]
apply(simp add: fresh_prod abs_fresh abs_supp fin_supp fresh_atm)[1]
apply(simp add: fresh_prod abs_fresh abs_supp fin_supp fresh_atm)[1]
apply(auto)[1]
apply(drule_tac x="M'a" in meta_spec)
apply(drule_tac x="aa" in meta_spec)
apply(drule_tac x="c" in meta_spec)
apply(auto)[1]
apply(rule_tac x="AndR <a>.M0 <b>.N' aa" in exI)
apply(simp add: abs_fresh abs_supp fin_supp fresh_atm)[1]
apply(auto)[1]
apply(rule trans)
apply(rule crename.simps)
apply(simp add: abs_fresh abs_supp fin_supp fresh_atm)[1]
apply(simp add: fresh_prod abs_fresh abs_supp fin_supp fresh_atm)[1]
apply(auto intro: fresh_a_redu)[1]
apply(simp add: fresh_prod abs_fresh abs_supp fin_supp fresh_atm)[1]
apply(simp add: fresh_prod abs_fresh abs_supp fin_supp fresh_atm)[1]
apply(drule sym)
apply(frule crename_AndR_aux)
apply(erule disjE)
apply(auto)[1]
apply(drule crename_AndR')
apply(erule disjE)
apply(auto)[1]
apply(drule_tac x="N'a" in meta_spec)
apply(drule_tac x="aa" in meta_spec)
apply(drule_tac x="ba" in meta_spec)
apply(auto)[1]
apply(rule_tac x="AndR <a>.M' <b>.M0 c" in exI)
apply(simp add: abs_fresh abs_supp fin_supp fresh_atm)[1]
apply(auto)[1]
apply(rule trans)
apply(rule crename.simps)
apply(simp add: abs_fresh abs_supp fin_supp fresh_atm fresh_prod)[1]
apply(auto intro: fresh_a_redu)[1]
apply(simp add: fresh_prod abs_fresh abs_supp fin_supp fresh_atm)[1]
apply(simp add: fresh_prod abs_fresh abs_supp fin_supp fresh_atm)[1]
apply(simp add: fresh_prod abs_fresh abs_supp fin_supp fresh_atm)[1]
apply(auto)[1]
apply(drule_tac x="N'a" in meta_spec)
apply(drule_tac x="aa" in meta_spec)
apply(drule_tac x="c" in meta_spec)
apply(auto)[1]
apply(rule_tac x="AndR <a>.M' <b>.M0 aa" in exI)
apply(simp add: abs_fresh abs_supp fin_supp fresh_atm)[1]
apply(auto)[1]
apply(rule trans)
apply(rule crename.simps)
apply(simp add: abs_fresh abs_supp fin_supp fresh_atm fresh_prod)[1]
apply(auto intro: fresh_a_redu)[1]
apply(simp add: fresh_prod abs_fresh abs_supp fin_supp fresh_atm)[1]
apply(simp add: fresh_prod abs_fresh abs_supp fin_supp fresh_atm)[1]
apply(simp)
(* AndL1 *)
apply(drule sym)
apply(drule crename_AndL1)
apply(simp)
apply(auto)[1]
apply(drule_tac x="N'" in meta_spec)
apply(drule_tac x="a" in meta_spec)
apply(drule_tac x="b" in meta_spec)
apply(auto)[1]
apply(rule_tac x="AndL1 (x).M0 y" in exI)
apply(simp add: abs_fresh abs_supp fin_supp fresh_atm)[1]
apply(auto)[1]
(* AndL2 *)
apply(drule sym)
apply(drule crename_AndL2)
apply(simp)
apply(auto)[1]
apply(drule_tac x="N'" in meta_spec)
apply(drule_tac x="a" in meta_spec)
apply(drule_tac x="b" in meta_spec)
apply(auto)[1]
apply(rule_tac x="AndL2 (x).M0 y" in exI)
apply(simp add: abs_fresh abs_supp fin_supp fresh_atm)[1]
apply(auto)[1]
(* OrL *)
apply(drule sym)
apply(drule crename_OrL)
apply(simp)
apply(auto)[1]
apply(drule_tac x="M'a" in meta_spec)
apply(drule_tac x="a" in meta_spec)
apply(drule_tac x="b" in meta_spec)
apply(auto)[1]
apply(rule_tac x="OrL (x).M0 (y).N' z" in exI)
apply(simp add: abs_fresh abs_supp fin_supp fresh_atm)[1]
apply(auto)[1]
apply(rule trans)
apply(rule crename.simps)
apply(simp add: abs_fresh abs_supp fin_supp fresh_atm fresh_prod)[1]
apply(simp add: fresh_prod abs_fresh abs_supp fin_supp fresh_atm)[1]
apply(auto intro: fresh_a_redu)[1]
apply(simp add: fresh_prod abs_fresh abs_supp fin_supp fresh_atm)[1]
apply(simp)
apply(drule sym)
apply(drule crename_OrL)
apply(simp)
apply(auto)[1]
apply(drule_tac x="N'a" in meta_spec)
apply(drule_tac x="a" in meta_spec)
apply(drule_tac x="b" in meta_spec)
apply(auto)[1]
apply(rule_tac x="OrL (x).M' (y).M0 z" in exI)
apply(simp add: abs_fresh abs_supp fin_supp fresh_atm)[1]
apply(auto)[1]
apply(rule trans)
apply(rule crename.simps)
apply(simp add: abs_fresh abs_supp fin_supp fresh_atm fresh_prod)[1]
apply(auto intro: fresh_a_redu)[1]
apply(simp add: fresh_prod abs_fresh abs_supp fin_supp fresh_atm)[1]
apply(simp)
apply(simp)
(* OrR1 *)
apply(drule sym)
apply(frule crename_OrR1_aux)
apply(erule disjE)
apply(auto)[1]
apply(drule crename_OrR1')
apply(simp)
apply(erule disjE)
apply(auto)[1]
apply(drule_tac x="N'" in meta_spec)
apply(drule_tac x="aa" in meta_spec)
apply(drule_tac x="ba" in meta_spec)
apply(auto)[1]
apply(rule_tac x="OrR1 <a>.M0 b" in exI)
apply(simp add: abs_fresh abs_supp fin_supp fresh_atm)[1]
apply(auto)[1]
apply(auto)[1]
apply(drule_tac x="N'" in meta_spec)
apply(drule_tac x="aa" in meta_spec)
apply(drule_tac x="b" in meta_spec)
apply(auto)[1]
apply(rule_tac x="OrR1 <a>.M0 aa" in exI)
apply(simp add: abs_fresh abs_supp fin_supp fresh_atm)[1]
apply(auto)[1]
(* OrR2 *)
apply(drule sym)
apply(frule crename_OrR2_aux)
apply(erule disjE)
apply(auto)[1]
apply(drule crename_OrR2')
apply(simp)
apply(erule disjE)
apply(auto)[1]
apply(drule_tac x="N'" in meta_spec)
apply(drule_tac x="aa" in meta_spec)
apply(drule_tac x="ba" in meta_spec)
apply(auto)[1]
apply(rule_tac x="OrR2 <a>.M0 b" in exI)
apply(simp add: abs_fresh abs_supp fin_supp fresh_atm)[1]
apply(auto)[1]
apply(auto)[1]
apply(drule_tac x="N'" in meta_spec)
apply(drule_tac x="aa" in meta_spec)
apply(drule_tac x="b" in meta_spec)
apply(auto)[1]
apply(rule_tac x="OrR2 <a>.M0 aa" in exI)
apply(simp add: abs_fresh abs_supp fin_supp fresh_atm)[1]
apply(auto)[1]
(* ImpL *)
apply(drule sym)
apply(drule crename_ImpL)
apply(simp)
apply(simp)
apply(auto)[1]
apply(drule_tac x="M'a" in meta_spec)
apply(drule_tac x="aa" in meta_spec)
apply(drule_tac x="b" in meta_spec)
apply(auto)[1]
apply(rule_tac x="ImpL <a>.M0 (x).N' y" in exI)
apply(simp add: abs_fresh abs_supp fin_supp fresh_atm)[1]
apply(auto)[1]
apply(rule trans)
apply(rule crename.simps)
apply(simp add: abs_fresh abs_supp fin_supp fresh_atm fresh_prod)[1]
apply(simp add: fresh_prod abs_fresh abs_supp fin_supp fresh_atm)[1]
apply(auto intro: fresh_a_redu)[1]
apply(simp add: fresh_prod abs_fresh abs_supp fin_supp fresh_atm)[1]
apply(drule sym)
apply(drule crename_ImpL)
apply(simp)
apply(simp)
apply(auto)[1]
apply(drule_tac x="N'a" in meta_spec)
apply(drule_tac x="aa" in meta_spec)
apply(drule_tac x="b" in meta_spec)
apply(auto)[1]
apply(rule_tac x="ImpL <a>.M' (x).M0 y" in exI)
apply(simp add: abs_fresh abs_supp fin_supp fresh_atm)[1]
apply(auto)[1]
apply(rule trans)
apply(rule crename.simps)
apply(simp add: abs_fresh abs_supp fin_supp fresh_atm fresh_prod)[1]
apply(auto intro: fresh_a_redu)[1]
apply(simp add: fresh_prod abs_fresh abs_supp fin_supp fresh_atm)[1]
apply(simp)
(* ImpR *)
apply(drule sym)
apply(frule crename_ImpR_aux)
apply(erule disjE)
apply(auto)[1]
apply(drule crename_ImpR')
apply(simp)
apply(erule disjE)
apply(auto)[1]
apply(drule_tac x="N'" in meta_spec)
apply(drule_tac x="aa" in meta_spec)
apply(drule_tac x="ba" in meta_spec)
apply(auto)[1]
apply(rule_tac x="ImpR (x).<a>.M0 b" in exI)
apply(simp add: abs_fresh abs_supp fin_supp fresh_atm)[1]
apply(auto)[1]
apply(auto)[1]
apply(drule_tac x="N'" in meta_spec)
apply(drule_tac x="aa" in meta_spec)
apply(drule_tac x="b" in meta_spec)
apply(auto)[1]
apply(rule_tac x="ImpR (x).<a>.M0 aa" in exI)
apply(simp add: abs_fresh abs_supp fin_supp fresh_atm)[1]
apply(auto)[1]
done

lemma SNa_preserved_renaming1:
assumes a: "SNa M"
shows "SNa (M[a⊢c>b])"
using a
apply(induct rule: SNa_induct)
apply(case_tac "a=b")
apply(rule SNaI)
apply(drule crename_aredu)
apply(blast)+
done

lemma nrename_interesting1:
assumes a: "distinct [x,y,z]"
shows "M[x⊢n>z][z⊢n>y] = M[z⊢n>y][x⊢n>y]"
using a
apply(nominal_induct M avoiding: x y z rule: trm.strong_induct)
apply(blast)
apply(blast)
apply(rotate_tac 12)
apply(drule_tac x="x" in meta_spec)
apply(rotate_tac 15)
apply(drule_tac x="y" in meta_spec)
apply(rotate_tac 15)
apply(drule_tac x="z" in meta_spec)
apply(blast)
apply(rotate_tac 11)
apply(drule_tac x="x" in meta_spec)
apply(rotate_tac 14)
apply(drule_tac x="y" in meta_spec)
apply(rotate_tac 14)
apply(drule_tac x="z" in meta_spec)
apply(blast)
done

lemma nrename_interesting2:
assumes a: "x≠z" "x≠u" "x≠y" "z≠u" "y≠z"
shows "M[x⊢n>y][z⊢n>u] = M[z⊢n>u][x⊢n>y]"
using a
apply(nominal_induct M avoiding: x y z u rule: trm.strong_induct)
done

lemma not_fic_nrename_aux:
assumes a: "fic M c"
shows "fic (M[x⊢n>y]) c"
using a
apply(nominal_induct M avoiding: c x y rule: trm.strong_induct)
apply(auto dest!: fic_elims intro!: fic.intros simp add: fresh_prod fresh_atm rename_fresh abs_fresh)
done

lemma not_fic_nrename:
assumes a: "¬(fic (M[x⊢n>y]) c)"
shows "¬(fic M c)"
using a
apply(auto dest:  not_fic_nrename_aux)
done

lemma fin_nrename:
assumes a: "fin M z" "z♯(x,y)"
shows "fin (M[x⊢n>y]) z"
using a
apply(nominal_induct M avoiding: x y z rule: trm.strong_induct)
apply(auto dest!: fin_elims intro!: fin.intros simp add: fresh_prod fresh_atm rename_fresh abs_fresh
split: if_splits)
done

lemma nrename_fresh_interesting1:
fixes z::"name"
assumes a: "z♯(M[x⊢n>y])" "z♯(x,y)"
shows "z♯M"
using a
apply(nominal_induct M avoiding: x y z rule: trm.strong_induct)
apply(auto split: if_splits simp add: abs_fresh abs_supp fin_supp)
done

lemma nrename_fresh_interesting2:
fixes c::"coname"
assumes a: "c♯(M[x⊢n>y])"
shows "c♯M"
using a
apply(nominal_induct M avoiding: x y c rule: trm.strong_induct)
apply(auto split: if_splits simp add: abs_fresh abs_supp fin_supp fresh_atm)
done

lemma fin_nrename2:
assumes a: "fin (M[x⊢n>y]) z" "z♯(x,y)"
shows "fin M z"
using a
apply(nominal_induct M avoiding: x y z rule: trm.strong_induct)
apply(auto dest!: fin_elims intro!: fin.intros simp add: fresh_prod fresh_atm rename_fresh abs_fresh
split: if_splits)
apply(auto dest: nrename_fresh_interesting1 simp add: fresh_atm fresh_prod)
done

lemma nrename_Cut:
assumes a: "R[x⊢n>y] = Cut <c>.M (z).N" "c♯(N,R)" "z♯(x,y,M,R)"
shows "∃M' N'. R = Cut <c>.M' (z).N' ∧ M'[x⊢n>y] = M ∧ N'[x⊢n>y] = N ∧ c♯N' ∧ z♯M'"
using a
apply(nominal_induct R avoiding: c y x z M N rule: trm.strong_induct)
apply(auto split: if_splits)
apply(rule_tac x="[(coname,c)]∙trm1" in exI)
apply(perm_simp)
apply(simp add: abs_fresh fresh_left calc_atm fresh_prod)
apply(rule_tac x="[(name,z)]∙trm2" in exI)
apply(perm_simp)
apply(simp add: abs_fresh fresh_left calc_atm fresh_prod)
apply(rule conjI)
apply(drule sym)
apply(drule pt_bij1[OF pt_coname_inst,OF at_coname_inst])
apply(drule sym)
apply(drule sym)
apply(drule pt_bij1[OF pt_name_inst,OF at_name_inst])
done

lemma nrename_NotR:
assumes a: "R[x⊢n>y] = NotR (z).N c" "z♯(R,x,y)"
shows "∃N'. (R = NotR (z).N' c) ∧ N'[x⊢n>y] = N"
using a
apply(nominal_induct R avoiding: x y c z N rule: trm.strong_induct)
apply(auto split: if_splits simp add: fresh_prod fresh_atm alpha abs_fresh trm.inject)
apply(rule_tac x="[(name,z)]∙trm" in exI)
apply(perm_simp)
apply(simp add: abs_fresh fresh_left calc_atm fresh_prod)
apply(drule sym)
apply(drule pt_bij1[OF pt_name_inst,OF at_name_inst])
done

lemma nrename_NotL:
assumes a: "R[x⊢n>y] = NotL <c>.N z" "c♯R" "z♯(x,y)"
shows "∃N'. (R = NotL <c>.N' z) ∧ N'[x⊢n>y] = N"
using a
apply(nominal_induct R avoiding: x y c z N rule: trm.strong_induct)
apply(auto split: if_splits simp add: fresh_prod fresh_atm alpha abs_fresh trm.inject)
apply(rule_tac x="[(coname,c)]∙trm" in exI)
apply(perm_simp)
apply(simp add: abs_fresh fresh_left calc_atm fresh_prod)
apply(drule sym)
apply(drule pt_bij1[OF pt_coname_inst,OF at_coname_inst])
done

lemma nrename_NotL':
assumes a: "R[x⊢n>y] = NotL <c>.N u" "c♯R" "x≠y"
shows "(∃N'. (R = NotL <c>.N' u) ∧ N'[x⊢n>y] = N) ∨ (∃N'. (R = NotL <c>.N' x) ∧ y=u ∧ N'[x⊢n>y] = N)"
using a
apply(nominal_induct R avoiding: y u c x N rule: trm.strong_induct)
apply(auto split: if_splits simp add: fresh_prod fresh_atm abs_fresh alpha trm.inject)
apply(rule_tac x="[(coname,c)]∙trm" in exI)
apply(perm_simp)
apply(simp add: abs_fresh fresh_left calc_atm fresh_prod)
apply(drule sym)
apply(drule pt_bij1[OF pt_coname_inst,OF at_coname_inst])
apply(rule_tac x="[(coname,c)]∙trm" in exI)
apply(perm_simp)
apply(simp add: abs_fresh fresh_left calc_atm fresh_prod)
apply(drule sym)
apply(drule pt_bij1[OF pt_coname_inst,OF at_coname_inst])
done

lemma nrename_NotL_aux:
assumes a: "R[x⊢n>y] = NotL <c>.N u"
shows "(x=u ∧ x=y) ∨ (x≠u)"
using a
apply(nominal_induct R avoiding: y u c x N rule: trm.strong_induct)
apply(auto split: if_splits simp add: fresh_prod fresh_atm alpha abs_fresh trm.inject)
done

lemma nrename_AndL1:
assumes a: "R[x⊢n>y] = AndL1 (z).N u" "z♯(R,x,y)" "u♯(x,y)"
shows "∃N'. (R = AndL1 (z).N' u) ∧ N'[x⊢n>y] = N"
using a
apply(nominal_induct R avoiding: z u x y N rule: trm.strong_induct)
apply(auto split: if_splits simp add: fresh_prod fresh_atm alpha abs_fresh trm.inject)
apply(rule_tac x="[(name1,z)]∙trm" in exI)
apply(perm_simp)
apply(simp add: abs_fresh fresh_left calc_atm fresh_prod)
apply(drule sym)
apply(drule pt_bij1[OF pt_name_inst,OF at_name_inst])
done

lemma nrename_AndL1':
assumes a: "R[x⊢n>y] = AndL1 (v).N u" "v♯(R,u,x,y)" "x≠y"
shows "(∃N'. (R = AndL1 (v).N' u) ∧ N'[x⊢n>y] = N) ∨ (∃N'. (R = AndL1 (v).N' x) ∧ y=u ∧ N'[x⊢n>y] = N)"
using a
apply(nominal_induct R avoiding: y u v x N rule: trm.strong_induct)
apply(auto split: if_splits simp add: fresh_prod fresh_atm abs_fresh alpha trm.inject)
apply(rule_tac x="[(name1,v)]∙trm" in exI)
apply(perm_simp)
apply(simp add: abs_fresh fresh_left calc_atm fresh_prod)
apply(drule sym)
apply(drule pt_bij1[OF pt_name_inst,OF at_name_inst])
apply(rule_tac x="[(name1,v)]∙trm" in exI)
apply(perm_simp)
apply(simp add: abs_fresh fresh_left calc_atm fresh_prod)
apply(drule sym)
apply(drule pt_bij1[OF pt_name_inst,OF at_name_inst])
done

lemma nrename_AndL1_aux:
assumes a: "R[x⊢n>y] = AndL1 (v).N u"
shows "(x=u ∧ x=y) ∨ (x≠u)"
using a
apply(nominal_induct R avoiding: y u v x N rule: trm.strong_induct)
apply(auto split: if_splits simp add: fresh_prod fresh_atm alpha abs_fresh trm.inject)
done

lemma nrename_AndL2:
assumes a: "R[x⊢n>y] = AndL2 (z).N u" "z♯(R,x,y)" "u♯(x,y)"
shows "∃N'. (R = AndL2 (z).N' u) ∧ N'[x⊢n>y] = N"
using a
apply(nominal_induct R avoiding: z u x y N rule: trm.strong_induct)
apply(auto split: if_splits simp add: fresh_prod fresh_atm alpha abs_fresh trm.inject)
apply(rule_tac x="[(name1,z)]∙trm" in exI)
apply(perm_simp)
apply(simp add: abs_fresh fresh_left calc_atm fresh_prod)
apply(drule sym)
apply(drule pt_bij1[OF pt_name_inst,OF at_name_inst])
done

lemma nrename_AndL2':
assumes a: "R[x⊢n>y] = AndL2 (v).N u" "v♯(R,u,x,y)" "x≠y"
shows "(∃N'. (R = AndL2 (v).N' u) ∧ N'[x⊢n>y] = N) ∨ (∃N'. (R = AndL2 (v).N' x) ∧ y=u ∧ N'[x⊢n>y] = N)"
using a
apply(nominal_induct R avoiding: y u v x N rule: trm.strong_induct)
apply(auto split: if_splits simp add: fresh_prod fresh_atm abs_fresh alpha trm.inject)
apply(rule_tac x="[(name1,v)]∙trm" in exI)
apply(perm_simp)
apply(simp add: abs_fresh fresh_left calc_atm fresh_prod)
apply(drule sym)
apply(drule pt_bij1[OF pt_name_inst,OF at_name_inst])
apply(rule_tac x="[(name1,v)]∙trm" in exI)
apply(perm_simp)
apply(simp add: abs_fresh fresh_left calc_atm fresh_prod)
apply(drule sym)
apply(drule pt_bij1[OF pt_name_inst,OF at_name_inst])