SFT: Rename Lambda to LambdaTerms, rename constants to match Lambda, remove smt proofs.
header {* The main lemma about Num and the Second Fixed Point Theorem *}+ −
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theory Theorem imports Consts begin+ −
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lemmas [simp] = b3[OF bI] b1 b4 b5 supp_Num[unfolded Num_def supp_ltgt] Num_def lam.fresh[unfolded fresh_def] fresh_def b6+ −
lemmas app = Ltgt1_app+ −
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lemma Num:+ −
shows "Num \<cdot> \<lbrace>M\<rbrace> \<approx> \<lbrace>\<lbrace>M\<rbrace>\<rbrace>"+ −
proof (induct M rule: lam.induct)+ −
case (Var n)+ −
have "Num \<cdot> \<lbrace>Var n\<rbrace> = Num \<cdot> (VAR \<cdot> Var n)" by simp+ −
also have "... = \<guillemotleft>[\<guillemotleft>[A1,A2,A3]\<guillemotright>]\<guillemotright> \<cdot> (VAR \<cdot> Var n)" by simp+ −
also have "... \<approx> VAR \<cdot> Var n \<cdot> \<guillemotleft>[A1,A2,A3]\<guillemotright>" using app .+ −
also have "... \<approx> \<guillemotleft>[A1,A2,A3]\<guillemotright> \<cdot> Umn 2 2 \<cdot> Var n \<cdot> \<guillemotleft>[A1,A2,A3]\<guillemotright>" using VAR_app .+ −
also have "... \<approx> A1 \<cdot> Var n \<cdot> \<guillemotleft>[A1,A2,A3]\<guillemotright>" using U_app by simp+ −
also have "... \<approx> F1 \<cdot> Var n" using A_app(1) .+ −
also have "... \<approx> APP \<cdot> \<lbrace>VAR\<rbrace> \<cdot> (VAR \<cdot> Var n)" using F_app(1) .+ −
also have "... = \<lbrace>\<lbrace>Var n\<rbrace>\<rbrace>" by simp+ −
finally show "Num \<cdot> \<lbrace>Var n\<rbrace> \<approx> \<lbrace>\<lbrace>Var n\<rbrace>\<rbrace>".+ −
next+ −
case (App M N)+ −
assume IH: "Num \<cdot> \<lbrace>M\<rbrace> \<approx> \<lbrace>\<lbrace>M\<rbrace>\<rbrace>" "Num \<cdot> \<lbrace>N\<rbrace> \<approx> \<lbrace>\<lbrace>N\<rbrace>\<rbrace>"+ −
have "Num \<cdot> \<lbrace>M \<cdot> N\<rbrace> = Num \<cdot> (APP \<cdot> \<lbrace>M\<rbrace> \<cdot> \<lbrace>N\<rbrace>)" by simp+ −
also have "... = \<guillemotleft>[\<guillemotleft>[A1,A2,A3]\<guillemotright>]\<guillemotright> \<cdot> (APP \<cdot> \<lbrace>M\<rbrace> \<cdot> \<lbrace>N\<rbrace>)" by simp+ −
also have "... \<approx> APP \<cdot> \<lbrace>M\<rbrace> \<cdot> \<lbrace>N\<rbrace> \<cdot> \<guillemotleft>[A1,A2,A3]\<guillemotright>" using app .+ −
also have "... \<approx> \<guillemotleft>[A1,A2,A3]\<guillemotright> \<cdot> Umn 2 1 \<cdot> \<lbrace>M\<rbrace> \<cdot> \<lbrace>N\<rbrace> \<cdot> \<guillemotleft>[A1,A2,A3]\<guillemotright>" using APP_app .+ −
also have "... \<approx> A2 \<cdot> \<lbrace>M\<rbrace> \<cdot> \<lbrace>N\<rbrace> \<cdot> \<guillemotleft>[A1,A2,A3]\<guillemotright>" using U_app by simp+ −
also have "... \<approx> F2 \<cdot> \<lbrace>M\<rbrace> \<cdot> \<lbrace>N\<rbrace> \<cdot> Num" using A_app(2) by simp+ −
also have "... \<approx> APP \<cdot> (APP \<cdot> \<lbrace>APP\<rbrace> \<cdot> (Num \<cdot> \<lbrace>M\<rbrace>)) \<cdot> (Num \<cdot> \<lbrace>N\<rbrace>)" using F_app(2) .+ −
also have "... \<approx> APP \<cdot> (APP \<cdot> \<lbrace>APP\<rbrace> \<cdot> (\<lbrace>\<lbrace>M\<rbrace>\<rbrace>)) \<cdot> (Num \<cdot> \<lbrace>N\<rbrace>)" using IH by simp+ −
also have "... \<approx> \<lbrace>\<lbrace>M \<cdot> N\<rbrace>\<rbrace>" using IH by simp+ −
finally show "Num \<cdot> \<lbrace>M \<cdot> N\<rbrace> \<approx> \<lbrace>\<lbrace>M \<cdot> N\<rbrace>\<rbrace>".+ −
next+ −
case (Lam x P)+ −
assume IH: "Num \<cdot> \<lbrace>P\<rbrace> \<approx> \<lbrace>\<lbrace>P\<rbrace>\<rbrace>"+ −
have "Num \<cdot> \<lbrace>\<integral> x. P\<rbrace> = Num \<cdot> (Abs \<cdot> \<integral> x. \<lbrace>P\<rbrace>)" by simp+ −
also have "... = \<guillemotleft>[\<guillemotleft>[A1,A2,A3]\<guillemotright>]\<guillemotright> \<cdot> (Abs \<cdot> \<integral> x. \<lbrace>P\<rbrace>)" by simp+ −
also have "... \<approx> Abs \<cdot> (\<integral> x. \<lbrace>P\<rbrace>) \<cdot> \<guillemotleft>[A1,A2,A3]\<guillemotright>" using app .+ −
also have "... \<approx> \<guillemotleft>[A1,A2,A3]\<guillemotright> \<cdot> Umn 2 0 \<cdot> (\<integral> x. \<lbrace>P\<rbrace>) \<cdot> \<guillemotleft>[A1,A2,A3]\<guillemotright>" using Abs_app .+ −
also have "... \<approx> A3 \<cdot> (\<integral> x. \<lbrace>P\<rbrace>) \<cdot> \<guillemotleft>[A1,A2,A3]\<guillemotright>" using U_app by simp+ −
also have "... \<approx> F3 \<cdot> (\<integral> x. \<lbrace>P\<rbrace>) \<cdot> \<guillemotleft>[\<guillemotleft>[A1,A2,A3]\<guillemotright>]\<guillemotright>" using A_app(3) .+ −
also have "... = F3 \<cdot> (\<integral> x. \<lbrace>P\<rbrace>) \<cdot> Num" by simp+ −
also have "... \<approx> APP \<cdot> \<lbrace>Abs\<rbrace> \<cdot> (Abs \<cdot> \<integral> x. (Num \<cdot> ((\<integral> x. \<lbrace>P\<rbrace>) \<cdot> Var x)))" by (rule F3_app) simp_all+ −
also have "... \<approx> APP \<cdot> \<lbrace>Abs\<rbrace> \<cdot> (Abs \<cdot> \<integral> x. (Num \<cdot> \<lbrace>P\<rbrace>))" using beta_app by simp+ −
also have "... \<approx> APP \<cdot> \<lbrace>Abs\<rbrace> \<cdot> (Abs \<cdot> \<integral> x. \<lbrace>\<lbrace>P\<rbrace>\<rbrace>)" using IH by simp+ −
also have "... = \<lbrace>\<lbrace>\<integral> x. P\<rbrace>\<rbrace>" by simp+ −
finally show "Num \<cdot> \<lbrace>\<integral> x. P\<rbrace> \<approx> \<lbrace>\<lbrace>\<integral> x. P\<rbrace>\<rbrace>" .+ −
qed+ −
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lemmas [simp] = Ap Num+ −
lemmas [simp del] = fresh_def Num_def+ −
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theorem SFP:+ −
fixes F :: lam+ −
shows "\<exists>X. X \<approx> F \<cdot> \<lbrace>X\<rbrace>"+ −
proof -+ −
obtain x :: var where [simp]:"atom x \<sharp> F" using obtain_fresh by blast+ −
def W \<equiv> "\<integral>x. (F \<cdot> (APP \<cdot> Var x \<cdot> (Num \<cdot> Var x)))"+ −
def X \<equiv> "W \<cdot> \<lbrace>W\<rbrace>"+ −
have a: "X = W \<cdot> \<lbrace>W\<rbrace>" unfolding X_def ..+ −
also have "... = (\<integral>x. (F \<cdot> (APP \<cdot> Var x \<cdot> (Num \<cdot> Var x)))) \<cdot> \<lbrace>W\<rbrace>" unfolding W_def ..+ −
also have "... \<approx> F \<cdot> (APP \<cdot> \<lbrace>W\<rbrace> \<cdot> (Num \<cdot> \<lbrace>W\<rbrace>))" by simp+ −
also have "... \<approx> F \<cdot> (APP \<cdot> \<lbrace>W\<rbrace> \<cdot> \<lbrace>\<lbrace>W\<rbrace>\<rbrace>)" by simp+ −
also have "... \<approx> F \<cdot> \<lbrace>W \<cdot> \<lbrace>W\<rbrace>\<rbrace>" by simp+ −
also have "... = F \<cdot> \<lbrace>X\<rbrace>" unfolding X_def ..+ −
finally show ?thesis by blast+ −
qed+ −
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end+ −