163 apply (simp_all add: rlam.inject alpha_refl) |
163 apply (simp_all add: rlam.inject alpha_refl) |
164 done |
164 done |
165 |
165 |
166 |
166 |
167 lemma pi_var: "(pi\<Colon>('x \<times> 'x) list) \<bullet> Var a = Var (pi \<bullet> a)" |
167 lemma pi_var: "(pi\<Colon>('x \<times> 'x) list) \<bullet> Var a = Var (pi \<bullet> a)" |
168 apply (tactic {* lift_tac @{context} @{thm pi_var_com} 1 *}) |
168 apply (lifting pi_var_com) |
169 done |
169 done |
170 |
170 |
171 lemma pi_app: "(pi\<Colon>('x \<times> 'x) list) \<bullet> App (x\<Colon>lam) (xa\<Colon>lam) = App (pi \<bullet> x) (pi \<bullet> xa)" |
171 lemma pi_app: "(pi\<Colon>('x \<times> 'x) list) \<bullet> App (x\<Colon>lam) (xa\<Colon>lam) = App (pi \<bullet> x) (pi \<bullet> xa)" |
172 apply (tactic {* lift_tac @{context} @{thm pi_app_com} 1 *}) |
172 apply (lifting pi_app_com) |
173 done |
173 done |
174 |
174 |
175 lemma pi_lam: "(pi\<Colon>('x \<times> 'x) list) \<bullet> Lam (a\<Colon>name) (x\<Colon>lam) = Lam (pi \<bullet> a) (pi \<bullet> x)" |
175 lemma pi_lam: "(pi\<Colon>('x \<times> 'x) list) \<bullet> Lam (a\<Colon>name) (x\<Colon>lam) = Lam (pi \<bullet> a) (pi \<bullet> x)" |
176 apply (tactic {* lift_tac @{context} @{thm pi_lam_com} 1 *}) |
176 apply (lifting pi_lam_com) |
177 done |
177 done |
178 |
178 |
179 lemma fv_var: "fv (Var (a\<Colon>name)) = {a}" |
179 lemma fv_var: "fv (Var (a\<Colon>name)) = {a}" |
180 apply (tactic {* lift_tac @{context} @{thm rfv_var} 1 *}) |
180 apply (lifting rfv_var) |
181 done |
181 done |
182 |
182 |
183 lemma fv_app: "fv (App (x\<Colon>lam) (xa\<Colon>lam)) = fv x \<union> fv xa" |
183 lemma fv_app: "fv (App (x\<Colon>lam) (xa\<Colon>lam)) = fv x \<union> fv xa" |
184 apply (tactic {* lift_tac @{context} @{thm rfv_app} 1 *}) |
184 apply (lifting rfv_app) |
185 done |
185 done |
186 |
186 |
187 lemma fv_lam: "fv (Lam (a\<Colon>name) (x\<Colon>lam)) = fv x - {a}" |
187 lemma fv_lam: "fv (Lam (a\<Colon>name) (x\<Colon>lam)) = fv x - {a}" |
188 apply (tactic {* lift_tac @{context} @{thm rfv_lam} 1 *}) |
188 apply (lifting rfv_lam) |
189 done |
189 done |
190 |
190 |
191 lemma a1: "(a\<Colon>name) = (b\<Colon>name) \<Longrightarrow> Var a = Var b" |
191 lemma a1: "(a\<Colon>name) = (b\<Colon>name) \<Longrightarrow> Var a = Var b" |
192 apply (tactic {* lift_tac @{context} @{thm a1} 1 *}) |
192 apply (lifting a1) |
193 done |
193 done |
194 |
194 |
195 lemma a2: "\<lbrakk>(x\<Colon>lam) = (xa\<Colon>lam); (xb\<Colon>lam) = (xc\<Colon>lam)\<rbrakk> \<Longrightarrow> App x xb = App xa xc" |
195 lemma a2: "\<lbrakk>(x\<Colon>lam) = (xa\<Colon>lam); (xb\<Colon>lam) = (xc\<Colon>lam)\<rbrakk> \<Longrightarrow> App x xb = App xa xc" |
196 apply (tactic {* lift_tac @{context} @{thm a2} 1 *}) |
196 apply (lifting a2) |
197 done |
197 done |
198 |
198 |
199 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" |
199 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" |
200 apply (tactic {* lift_tac @{context} @{thm a3} 1 *}) |
200 apply (lifting a3) |
201 done |
201 done |
202 |
202 |
203 lemma alpha_cases: "\<lbrakk>a1 = a2; \<And>a b. \<lbrakk>a1 = Var a; a2 = Var b; a = b\<rbrakk> \<Longrightarrow> P; |
203 lemma alpha_cases: "\<lbrakk>a1 = a2; \<And>a b. \<lbrakk>a1 = Var a; a2 = Var b; a = b\<rbrakk> \<Longrightarrow> P; |
204 \<And>x xa xb xc. \<lbrakk>a1 = App x xb; a2 = App xa xc; x = xa; xb = xc\<rbrakk> \<Longrightarrow> P; |
204 \<And>x xa xb xc. \<lbrakk>a1 = App x xb; a2 = App xa xc; x = xa; xb = xc\<rbrakk> \<Longrightarrow> P; |
205 \<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> |
205 \<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> |
206 \<Longrightarrow> P" |
206 \<Longrightarrow> P" |
207 apply (tactic {* lift_tac @{context} @{thm alpha.cases} 1 *}) |
207 apply (lifting alpha.cases) |
208 done |
208 done |
209 |
209 |
210 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); |
210 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); |
211 \<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); |
211 \<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); |
212 \<And>(x\<Colon>lam) (a\<Colon>name) (b\<Colon>name) xa\<Colon>lam. |
212 \<And>(x\<Colon>lam) (a\<Colon>name) (b\<Colon>name) xa\<Colon>lam. |
213 \<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> |
213 \<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> |
214 \<Longrightarrow> qxb qx qxa" |
214 \<Longrightarrow> qxb qx qxa" |
215 apply (tactic {* lift_tac @{context} @{thm alpha.induct} 1 *}) |
215 apply (lifting alpha.induct) |
216 done |
216 done |
217 |
217 |
218 lemma var_inject: "(Var a = Var b) = (a = b)" |
218 lemma var_inject: "(Var a = Var b) = (a = b)" |
219 apply (tactic {* lift_tac @{context} @{thm rvar_inject} 1 *}) |
219 apply (lifting rvar_inject) |
220 done |
220 done |
221 |
221 |
222 lemma lam_induct:" \<lbrakk>\<And>name. P (Var name); \<And>lam1 lam2. \<lbrakk>P lam1; P lam2\<rbrakk> \<Longrightarrow> P (App lam1 lam2); |
222 lemma lam_induct:" \<lbrakk>\<And>name. P (Var name); \<And>lam1 lam2. \<lbrakk>P lam1; P lam2\<rbrakk> \<Longrightarrow> P (App lam1 lam2); |
223 \<And>name lam. P lam \<Longrightarrow> P (Lam name lam)\<rbrakk> \<Longrightarrow> P lam" |
223 \<And>name lam. P lam \<Longrightarrow> P (Lam name lam)\<rbrakk> \<Longrightarrow> P lam" |
224 apply (tactic {* lift_tac @{context} @{thm rlam.induct} 1 *}) |
224 apply (lifting rlam.induct) |
225 done |
225 done |
226 |
226 |
227 lemma var_supp: |
227 lemma var_supp: |
228 shows "supp (Var a) = ((supp a)::name set)" |
228 shows "supp (Var a) = ((supp a)::name set)" |
229 apply(simp add: supp_def) |
229 apply(simp add: supp_def) |