theory Lambda +
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  imports "Nominal" +
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begin+
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+
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atom_decl name+
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+
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section {* Alpha-Equated Lambda-Terms *}+
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+
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nominal_datatype lam =+
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  Var "name"+
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| App "lam" "lam" +
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| Lam "\<guillemotleft>name\<guillemotright>lam" ("Lam [_]._")+
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+
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section {* Capture-Avoiding Substitution *}+
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+
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consts subst :: "lam \<Rightarrow> name \<Rightarrow> lam \<Rightarrow> lam"  ("_[_::=_]")+
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+
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nominal_primrec+
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  "(Var x)[y::=s] = (if x=y then s else (Var x))"+
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  "(App t1 t2)[y::=s] = App (t1[y::=s]) (t2[y::=s])"+
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  "x\<sharp>(y,s) \<Longrightarrow> (Lam [x].t)[y::=s] = Lam [x].(t[y::=s])"+
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apply(finite_guess)++
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apply(rule TrueI)++
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apply(simp add: abs_fresh)++
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apply(fresh_guess)++
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done+
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+
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lemma  subst_eqvt[eqvt]:+
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  fixes pi::"name prm"+
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  shows "pi\<bullet>(t1[x::=t2]) = (pi\<bullet>t1)[(pi\<bullet>x)::=(pi\<bullet>t2)]"+
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by (nominal_induct t1 avoiding: x t2 rule: lam.strong_induct)+
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   (auto simp add: perm_bij fresh_atm fresh_bij)+
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+
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lemma forget: +
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  assumes a: "x\<sharp>L"+
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  shows "L[x::=P] = L"+
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  using a +
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by (nominal_induct L avoiding: x P rule: lam.strong_induct)+
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   (auto simp add: abs_fresh fresh_atm)+
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+
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lemma fresh_fact:+
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  fixes z::"name"+
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  shows "\<lbrakk>z\<sharp>s; (z=y \<or> z\<sharp>t)\<rbrakk> \<Longrightarrow> z\<sharp>t[y::=s]"+
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by (nominal_induct t avoiding: z y s rule: lam.strong_induct)+
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   (auto simp add: abs_fresh fresh_prod fresh_atm)+
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+
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lemma subst_rename: +
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  assumes a: "y\<sharp>t"+
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  shows "t[x::=s] = ([(y,x)]\<bullet>t)[y::=s]"+
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using a +
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by (nominal_induct t avoiding: x y s rule: lam.strong_induct)+
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     (auto simp add: calc_atm fresh_atm abs_fresh)+
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+
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text {* +
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  The purpose of the two lemmas below is to work+
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  around some quirks in Isabelle's handling of +
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  meta_quantifiers and meta_implications. +
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  *} +
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+
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lemma meta_impCE:+
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  assumes major: "P ==> PROP Q"+
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    and 1: "~ P ==> R"+
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    and 2: "PROP Q ==> R"+
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  shows R+
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proof (cases P)+
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  assume P+
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  then have "PROP Q" by (rule major)+
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  then show R by (rule 2)+
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next+
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  assume "~ P"+
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  then show R by (rule 1)+
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qed+
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+
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declare meta_allE [elim]+
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    and meta_impCE [elim!]+
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+
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end    +
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