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theory Tutorial3s
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imports Lambda
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begin
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section {* Formalising Barendregt's Proof of the Substitution Lemma *}
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text {*
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The substitution lemma is another theorem where the variable
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convention plays a crucial role.
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Barendregt's proof of this lemma needs in the variable case a
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case distinction. One way to do this in Isar is to use blocks.
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A block consist of some assumptions and reasoning steps
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enclosed in curly braces, like
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{ \<dots>
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have "statement"
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have "last_statement_in_the_block"
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}
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Such a block may contain local assumptions like
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{ assume "A"
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assume "B"
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\<dots>
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have "C" by \<dots>
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}
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Where "C" is the last have-statement in this block. The behaviour
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of such a block to the 'outside' is the implication
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A \<Longrightarrow> B \<Longrightarrow> C
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Now if we want to prove a property "smth" using the case-distinctions
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P1, P2 and P3 then we can use the following reasoning:
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{ assume "P1"
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\<dots>
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have "smth"
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}
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moreover
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{ assume "P2"
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\<dots>
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have "smth"
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}
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moreover
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{ assume "P3"
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\<dots>
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have "smth"
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}
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ultimately have "smth" by blast
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The blocks establish the implications
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P1 \<Longrightarrow> smth
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P2 \<Longrightarrow> smth
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P3 \<Longrightarrow> smth
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If we know that P1, P2 and P3 cover all the cases, that is P1 \<or> P2 \<or> P3
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holds, then we have 'ultimately' established the property "smth"
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*}
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subsection {* Two preliminary facts *}
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lemma forget:
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shows "atom x \<sharp> t \<Longrightarrow> t[x ::= s] = t"
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by (nominal_induct t avoiding: x s rule: lam.strong_induct)
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(auto simp add: lam.fresh fresh_at_base)
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lemma fresh_fact:
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assumes a: "atom z \<sharp> s"
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and b: "z = y \<or> atom z \<sharp> t"
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shows "atom z \<sharp> t[y ::= s]"
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using a b
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by (nominal_induct t avoiding: z y s rule: lam.strong_induct)
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(auto simp add: lam.fresh fresh_at_base)
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section {* EXERCISE 10 *}
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text {*
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Fill in the cases 1.2 and 1.3 and the equational reasoning
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in the lambda-case.
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*}
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lemma
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assumes a: "x \<noteq> y"
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and b: "atom x \<sharp> L"
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shows "M[x ::= N][y ::= L] = M[y ::= L][x ::= N[y ::= L]]"
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using a b
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proof (nominal_induct M avoiding: x y N L rule: lam.strong_induct)
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case (Var z)
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have a1: "x \<noteq> y" by fact
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have a2: "atom x \<sharp> L" by fact
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show "Var z[x::=N][y::=L] = Var z[y::=L][x::=N[y::=L]]" (is "?LHS = ?RHS")
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proof -
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{ -- {* Case 1.1 *}
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assume c1: "z = x"
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have "(1)": "?LHS = N[y::=L]" using c1 by simp
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have "(2)": "?RHS = N[y::=L]" using c1 a1 by simp
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have "?LHS = ?RHS" using "(1)" "(2)" by simp
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}
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moreover
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{ -- {* Case 1.2 *}
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assume c2: "z = y" "z \<noteq> x"
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have "(1)": "?LHS = L" using c2 by simp
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have "(2)": "?RHS = L[x::=N[y::=L]]" using c2 by simp
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have "(3)": "L[x::=N[y::=L]] = L" using a2 forget by simp
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have "?LHS = ?RHS" using "(1)" "(2)" "(3)" by simp
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}
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moreover
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{ -- {* Case 1.3 *}
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assume c3: "z \<noteq> x" "z \<noteq> y"
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have "(1)": "?LHS = Var z" using c3 by simp
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have "(2)": "?RHS = Var z" using c3 by simp
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have "?LHS = ?RHS" using "(1)" "(2)" by simp
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}
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ultimately show "?LHS = ?RHS" by blast
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qed
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next
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case (Lam z M1) -- {* case 2: lambdas *}
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have ih: "\<lbrakk>x \<noteq> y; atom x \<sharp> L\<rbrakk> \<Longrightarrow> M1[x ::= N][y ::= L] = M1[y ::= L][x ::= N[y ::= L]]" by fact
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have a1: "x \<noteq> y" by fact
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have a2: "atom x \<sharp> L" by fact
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have fs: "atom z \<sharp> x" "atom z \<sharp> y" "atom z \<sharp> N" "atom z \<sharp> L" by fact+ -- {* !! *}
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then have b: "atom z \<sharp> N[y::=L]" by (simp add: fresh_fact)
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show "(Lam [z].M1)[x ::= N][y ::= L] = (Lam [z].M1)[y ::= L][x ::= N[y ::= L]]" (is "?LHS=?RHS")
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proof -
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have "?LHS = Lam [z].(M1[x ::= N][y ::= L])" using fs by simp
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also have "\<dots> = Lam [z].(M1[y ::= L][x ::= N[y ::= L]])" using ih a1 a2 by simp
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also have "\<dots> = (Lam [z].(M1[y ::= L]))[x ::= N[y ::= L]]" using b fs by simp
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also have "\<dots> = ?RHS" using fs by simp
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finally show "?LHS = ?RHS" by simp
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qed
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next
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case (App M1 M2) -- {* case 3: applications *}
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then show "(App M1 M2)[x::=N][y::=L] = (App M1 M2)[y::=L][x::=N[y::=L]]" by simp
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qed
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text {*
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Again the strong induction principle enables Isabelle to find
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the proof of the substitution lemma completely automatically.
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*}
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lemma substitution_lemma_version:
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assumes asm: "x \<noteq> y" "atom x \<sharp> L"
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shows "M[x::=N][y::=L] = M[y::=L][x::=N[y::=L]]"
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using asm
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by (nominal_induct M avoiding: x y N L rule: lam.strong_induct)
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(auto simp add: fresh_fact forget)
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subsection {* MINI EXERCISE *}
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text {*
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Compare and contrast Barendregt's reasoning and the
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formalised proofs.
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*}
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end
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