188 apply (tactic {* lift_tac_lam @{context} @{thm pi_lam_com} 1 *})

   188 apply (tactic {* lift_tac_lam @{context} @{thm pi_lam_com} 1 *})

   189 done

   189 done

   190 

   190 

   191 lemma fv_var: "fv (Var (a\<Colon>name)) = {a}"

   191 lemma fv_var: "fv (Var (a\<Colon>name)) = {a}"

   192 apply (tactic {* lift_tac_lam @{context} @{thm rfv_var} 1 *})

   192 apply (tactic {* lift_tac_lam @{context} @{thm rfv_var} 1 *})

   195 done

   193 done

   196 

   194 

   197 lemma fv_app: "fv (App (x\<Colon>lam) (xa\<Colon>lam)) = fv x \<union> fv xa"

   195 lemma fv_app: "fv (App (x\<Colon>lam) (xa\<Colon>lam)) = fv x \<union> fv xa"

   198 apply (tactic {* lift_tac_lam @{context} @{thm rfv_app} 1 *})

   196 apply (tactic {* lift_tac_lam @{context} @{thm rfv_app} 1 *})

   201 done

   197 done

   202 

   198 

   203 lemma fv_lam: "fv (Lam (a\<Colon>name) (x\<Colon>lam)) = fv x - {a}"

   199 lemma fv_lam: "fv (Lam (a\<Colon>name) (x\<Colon>lam)) = fv x - {a}"

   204 apply (tactic {* lift_tac_lam @{context} @{thm rfv_lam} 1 *})

   200 apply (tactic {* lift_tac_lam @{context} @{thm rfv_lam} 1 *})

   207 done

   201 done

   208 

   202 

   209 lemma a1: "(a\<Colon>name) = (b\<Colon>name) \<Longrightarrow> Var a = Var b"

   203 lemma a1: "(a\<Colon>name) = (b\<Colon>name) \<Longrightarrow> Var a = Var b"

   210 apply (tactic {* lift_tac_lam @{context} @{thm a1} 1 *})

   204 apply (tactic {* lift_tac_lam @{context} @{thm a1} 1 *})

   211 done

   205 done

   214 apply (tactic {* lift_tac_lam @{context} @{thm a2} 1 *})

   208 apply (tactic {* lift_tac_lam @{context} @{thm a2} 1 *})

   215 done

   209 done

   216 

   210 

   217 lemma a3: "\<lbrakk>(x\<Colon>lam) = [(a\<Colon>name, b\<Colon>name)] \<bullet> (xa\<Colon>lam); a \<notin> fv (Lam b x)\<rbrakk> \<Longrightarrow> Lam a x = Lam b xa"

   211 lemma a3: "\<lbrakk>(x\<Colon>lam) = [(a\<Colon>name, b\<Colon>name)] \<bullet> (xa\<Colon>lam); a \<notin> fv (Lam b x)\<rbrakk> \<Longrightarrow> Lam a x = Lam b xa"

   218 apply (tactic {* lift_tac_lam @{context} @{thm a3} 1 *})

   212 apply (tactic {* lift_tac_lam @{context} @{thm a3} 1 *})

   221 done

   213 done

   222 

   214 

   223 lemma alpha_cases: "\<lbrakk>a1 = a2; \<And>a b. \<lbrakk>a1 = Var a; a2 = Var b; a = b\<rbrakk> \<Longrightarrow> P;

   215 lemma alpha_cases: "\<lbrakk>a1 = a2; \<And>a b. \<lbrakk>a1 = Var a; a2 = Var b; a = b\<rbrakk> \<Longrightarrow> P;

   224      \<And>x xa xb xc. \<lbrakk>a1 = App x xb; a2 = App xa xc; x = xa; xb = xc\<rbrakk> \<Longrightarrow> P;

   216      \<And>x xa xb xc. \<lbrakk>a1 = App x xb; a2 = App xa xc; x = xa; xb = xc\<rbrakk> \<Longrightarrow> P;

   225      \<And>x a b xa. \<lbrakk>a1 = Lam a x; a2 = Lam b xa; x = [(a, b)] \<bullet> xa; a \<notin> fv (Lam b x)\<rbrakk> \<Longrightarrow> P\<rbrakk>

   217      \<And>x a b xa. \<lbrakk>a1 = Lam a x; a2 = Lam b xa; x = [(a, b)] \<bullet> xa; a \<notin> fv (Lam b x)\<rbrakk> \<Longrightarrow> P\<rbrakk>

   226     \<Longrightarrow> P"

   218     \<Longrightarrow> P"

   227 apply (tactic {* lift_tac_lam @{context} @{thm alpha.cases} 1 *})

   219 apply (tactic {* lift_tac_lam @{context} @{thm alpha.cases} 1 *})

   230 done

   220 done

   231 

   221 

   232 lemma alpha_induct: "\<lbrakk>(qx\<Colon>lam) = (qxa\<Colon>lam); \<And>(a\<Colon>name) b\<Colon>name. a = b \<Longrightarrow> (qxb\<Colon>lam \<Rightarrow> lam \<Rightarrow> bool) (Var a) (Var b);

   222 lemma alpha_induct: "\<lbrakk>(qx\<Colon>lam) = (qxa\<Colon>lam); \<And>(a\<Colon>name) b\<Colon>name. a = b \<Longrightarrow> (qxb\<Colon>lam \<Rightarrow> lam \<Rightarrow> bool) (Var a) (Var b);

   233      \<And>(x\<Colon>lam) (xa\<Colon>lam) (xb\<Colon>lam) xc\<Colon>lam. \<lbrakk>x = xa; qxb x xa; xb = xc; qxb xb xc\<rbrakk> \<Longrightarrow> qxb (App x xb) (App xa xc);

   223      \<And>(x\<Colon>lam) (xa\<Colon>lam) (xb\<Colon>lam) xc\<Colon>lam. \<lbrakk>x = xa; qxb x xa; xb = xc; qxb xb xc\<rbrakk> \<Longrightarrow> qxb (App x xb) (App xa xc);

   234      \<And>(x\<Colon>lam) (a\<Colon>name) (b\<Colon>name) xa\<Colon>lam.

   224      \<And>(x\<Colon>lam) (a\<Colon>name) (b\<Colon>name) xa\<Colon>lam.

   235         \<lbrakk>x = [(a, b)] \<bullet> xa; qxb x ([(a, b)] \<bullet> xa); a \<notin> fv (Lam b x)\<rbrakk> \<Longrightarrow> qxb (Lam a x) (Lam b xa)\<rbrakk>

   225         \<lbrakk>x = [(a, b)] \<bullet> xa; qxb x ([(a, b)] \<bullet> xa); a \<notin> fv (Lam b x)\<rbrakk> \<Longrightarrow> qxb (Lam a x) (Lam b xa)\<rbrakk>

   236     \<Longrightarrow> qxb qx qxa"

   226     \<Longrightarrow> qxb qx qxa"

   237 apply (tactic {* lift_tac_lam @{context} @{thm alpha.induct} 1 *})

   227 apply (tactic {* lift_tac_lam @{context} @{thm alpha.induct} 1 *})

   240 done

   228 done

   241 

   229 

   242 lemma var_inject: "(Var a = Var b) = (a = b)"

   230 lemma var_inject: "(Var a = Var b) = (a = b)"

   243 apply (tactic {* lift_tac_lam @{context} @{thm rvar_inject} 1 *})

   231 apply (tactic {* lift_tac_lam @{context} @{thm rvar_inject} 1 *})

   244 done

   232 done