--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/Nominal/activities/tphols09/IDW/CU-Ex1.thy	Wed Mar 30 17:27:34 2016 +0100
@@ -0,0 +1,461 @@
+theory Ex
+imports Main
+begin
+
+text {*
+  An example of an apply-proof and the
+  corresponding ML-code.
+*}
+
+lemma disj_swap: 
+  shows "P \<or> Q \<Longrightarrow> Q \<or> P"
+apply(erule disjE)
+apply(rule disjI2)
+apply(assumption)
+apply(rule disjI1)
+apply(assumption)
+done
+
+ML {*
+let
+  val ctxt  = @{context}
+  val goal  = @{prop "P \<or> Q \<Longrightarrow> Q \<or> P"}
+  val facts = []
+  val schms  = ["P", "Q"]
+in
+  Goal.prove ctxt schms facts goal
+   (fn _ =>
+      etac @{thm disjE} 1
+      THEN rtac @{thm disjI2} 1
+      THEN atac 1
+      THEN rtac @{thm disjI1} 1
+      THEN atac 1)
+end
+*}
+
+text {*  
+  Tactics coded in ML can be applied / tried out in 
+  apply-scripts using "tactic". "THEN" is a tactic
+  combinator that just strings tactics together.
+  Tactic combinators are also called tacticals.
+*}
+
+ML {*
+val foo_tac =
+      (etac @{thm disjE} 1
+       THEN rtac @{thm disjI2} 1
+       THEN atac 1
+       THEN rtac @{thm disjI1} 1
+       THEN atac 1)
+*}
+
+lemma 
+  shows "P \<or> Q \<Longrightarrow> Q \<or> P"
+apply(tactic {* foo_tac *})
+done
+
+text {* 
+  Coding tactics with explicit subgoal addressing
+  is quite brittle and limits the usage of the tactics.
+  This can be avoided using the `primed' versions
+  of the tactic comibinators.
+*}
+
+ML {*
+val foo_tac' =
+      (etac @{thm disjE}
+       THEN' rtac @{thm disjI2}
+       THEN' atac
+       THEN' rtac @{thm disjI1}
+       THEN' atac)
+*}
+
+text {*
+  Now the tactic can be used on any subgoal.
+*}
+
+lemma "P1 \<or> Q1 \<Longrightarrow> Q1 \<or> P1"
+  and "P2 \<or> Q2 \<Longrightarrow> Q2 \<or> P2"
+apply(tactic {* foo_tac' 2 *})
+apply(tactic {* foo_tac' 1 *})
+done
+
+text {* 
+  The type of a tactic is from a theorem
+  to a lazy sequence of (successor) theorems.
+
+  type tactic = thm -> thm Seq.seq 
+
+  The simplest tactics are no_tac (always
+  fails) and all_tac (always succeds, but 
+  does not make any progress.
+*}
+
+
+ML {* fun no_tac thm = Seq.empty *}
+ML {* fun all_tac thm = Seq.single thm *}
+
+text {*
+  The lazy sequence can be explored using
+  "back".
+*}
+
+lemma "\<lbrakk>P \<or> Q; P \<or> Q\<rbrakk> \<Longrightarrow> Q \<or> P"
+apply(tactic {* foo_tac' 1 *})
+back
+done
+
+text {* 
+  It might be surprising that a goalstate of
+  an "unfinished" proof is a theorem. Below
+  we show the internals.
+
+  In general a goalstate is of the form
+
+  A1 \<dots> An \<Longrightarrow> #C
+
+  where the Ai are the open subgoals and C
+  is the theorem to be proved, protected by
+  the constant prop. This constant is usually
+  not visible.
+*}
+
+ML {*
+fun my_print_tac ctxt thm =
+let
+   val _ = tracing (Syntax.string_of_term ctxt (prop_of thm))
+in
+   Seq.single thm
+end
+*}
+
+notation (output) "prop" ("#_" [1000] 1000)
+
+lemma shows "\<lbrakk>A; B\<rbrakk> \<Longrightarrow> A \<and> B"
+apply(tactic {* my_print_tac @{context} *})
+
+apply(rule conjI)
+apply(tactic {* my_print_tac @{context} *})
+
+apply(assumption)
+apply(tactic {* my_print_tac @{context} *})
+
+apply(assumption)
+apply(tactic {* my_print_tac @{context} *})
+done
+
+notation (output) "prop" ("_" [1000] 1000)
+
+text {* 
+  There are some simple tactics corresponding
+  to rule, erule and drule (from apply scripts).
+  The examples should be self-explanatory.
+*}
+
+lemma shows "P \<Longrightarrow> P"
+apply(tactic {* atac 1 *})
+oops
+
+lemma shows "P \<and> Q"
+apply(tactic {* resolve_tac [@{thm conjI}] 1 *})
+oops
+
+lemma shows "P \<and> Q \<Longrightarrow> False"
+apply(tactic {* eresolve_tac [@{thm conjE}] 1 *})
+oops
+
+lemma shows "False \<and> True \<Longrightarrow> False"
+apply(tactic {* dresolve_tac [@{thm conjunct2}] 1 *})
+oops
+
+text {*
+  For all sorts of operations on the goalstate there
+  is a corresponding tactic. The following corresponds
+  to "insert".
+*}
+
+lemma shows "True = False"
+apply(tactic {* cut_facts_tac [@{thm True_def}, @{thm False_def}] 1 *})
+oops
+
+text {* 
+  Because of schematic variables, theorems need often
+  to be pre-instantiated.  
+*}
+
+lemma shows "\<forall>x \<in> A. P x \<Longrightarrow> Q x"
+apply(tactic {* dresolve_tac [@{thm bspec}] 1 *})
+oops
+
+ML {* @{thm disjI1} RS @{thm conjI} *}
+
+text {*
+  Tactic combinators: every tactic (THEN),
+  first applicable tactic (ORELSE).
+*}
+
+ML {*
+val foo_tac' = 
+  EVERY' [etac @{thm disjE}, 
+          rtac @{thm disjI2},
+          atac, 
+          rtac @{thm disjI1}, 
+          atac]
+*}
+
+
+ML {*
+val select_tac = 
+  FIRST' [rtac @{thm conjI}, 
+          rtac @{thm impI},
+          rtac @{thm notI}, 
+          rtac @{thm allI}, K all_tac]
+*}
+
+text {*
+  The most convenient order to apply tactics to subgoal
+  is in reverse order. 
+*}
+
+lemma 
+  shows "A \<and> B" and "A \<longrightarrow> B \<longrightarrow> C" and " \<forall> x. D x" and "E \<Longrightarrow> F"
+apply(tactic {* select_tac 4 *})
+apply(tactic {* select_tac 3 *})
+apply(tactic {* select_tac 2 *})
+apply(tactic {* select_tac 1 *})
+oops
+
+text {*
+  The tactic select_tac can also be written using the 
+  combinator TRY.
+*}
+
+ML {*
+val select_tac' = 
+  TRY o FIRST' [rtac @{thm conjI}, 
+                rtac @{thm impI},
+                rtac @{thm notI}, 
+                rtac @{thm allI}]
+*}
+
+text {*
+  Repeated application of tactics. 
+*}
+
+ML {*
+val repeat_sel_tac = 
+  REPEAT o CHANGED o select_tac'
+*}
+
+lemma 
+  shows "A \<and> (B \<and> C)" and "A \<longrightarrow> B \<longrightarrow> C" and " \<forall> x. D x" and "E \<Longrightarrow> F"
+apply(tactic {* repeat_sel_tac 4 *})
+apply(tactic {* repeat_sel_tac 3 *})
+apply(tactic {* repeat_sel_tac 2 *})
+apply(tactic {* repeat_sel_tac 1 *})
+oops
+
+text {*  
+  Application of a tactics to all new subgoals.
+*}
+
+ML {*
+val repeat_all_new_sel_tac = 
+  TRY o REPEAT_ALL_NEW (CHANGED o select_tac')
+*}
+
+lemma 
+  shows "A \<and> (B \<and> C)" and "A \<longrightarrow> B \<longrightarrow> C" and " \<forall> x. D x" and "E \<Longrightarrow> F"
+apply(tactic {* repeat_all_new_sel_tac 4 *})
+apply(tactic {* repeat_all_new_sel_tac 3 *})
+apply(tactic {* repeat_all_new_sel_tac 2 *})
+apply(tactic {* repeat_all_new_sel_tac 1 *})
+oops
+
+text {* 
+  The tactical DEPTH_SOLVED applies tactics in depth first
+  search including backtracking.
+
+  This can be used to implement a decision procedure for 
+  propositional intuitionistic logic inspired by work of 
+  Roy Dyckhoff.
+*}
+
+
+lemma impE1:
+  shows "\<lbrakk>A \<longrightarrow> B; A; \<lbrakk>A; B\<rbrakk> \<Longrightarrow> R\<rbrakk> \<Longrightarrow> R"
+by iprover
+
+lemma impE2:
+  shows "\<lbrakk>(C \<and> D) \<longrightarrow> B; C \<longrightarrow> (D \<longrightarrow>B) \<Longrightarrow> R\<rbrakk> \<Longrightarrow> R"
+by iprover
+
+lemma impE3:
+  shows "\<lbrakk>(C \<or> D) \<longrightarrow> B; \<lbrakk>C \<longrightarrow> B; D \<longrightarrow> B\<rbrakk> \<Longrightarrow> R\<rbrakk> \<Longrightarrow> R"
+by iprover
+
+lemma impE4:
+  shows "\<lbrakk>(C \<longrightarrow> D) \<longrightarrow> B; D \<longrightarrow> B \<Longrightarrow> C \<longrightarrow> D; B \<Longrightarrow> R\<rbrakk> \<Longrightarrow> R"
+by iprover
+
+lemma impE5:
+  shows "\<lbrakk>(C = D) \<longrightarrow> B; (C \<longrightarrow> D) \<longrightarrow> ((D \<longrightarrow> C) \<longrightarrow> B) \<Longrightarrow> R\<rbrakk> \<Longrightarrow> R"
+by iprover
+
+ML {*
+val apply_tac =
+let
+  val intros = [@{thm conjI}, @{thm disjI1}, @{thm disjI2}, @{thm impI}, @{thm iffI}]
+  val elims  = [@{thm FalseE}, @{thm conjE}, @{thm disjE}, @{thm iffE},
+                @{thm impE2}, @{thm impE3}, @{thm impE4}, @{thm impE5}, @{thm impE1}]
+in
+  atac 
+  ORELSE' resolve_tac intros
+  ORELSE' eresolve_tac elims
+end
+*}
+
+lemma
+  shows "((((P \<longrightarrow> Q) \<longrightarrow> P) \<longrightarrow> P) \<longrightarrow> Q) \<longrightarrow> Q"
+apply(tactic {* (DEPTH_SOLVE o apply_tac) 1 *})
+done
+
+lemma 
+  shows "((A \<longrightarrow> B) \<longrightarrow> A) \<longrightarrow> A"
+apply(tactic {* (TRY o DEPTH_SOLVE o apply_tac) 1 *})
+oops
+
+lemma
+  shows "(A \<and> B \<or> ((((A \<longrightarrow> False) \<longrightarrow> False) \<longrightarrow> Q) \<or> (B \<longrightarrow> Q)) \<longrightarrow> Q) \<longrightarrow> Q"
+apply(tactic {* (TRY o DEPTH_SOLVE o apply_tac) 1 *})
+oops
+
+text {* 
+  The SUBPROOF tactic gives you full access to
+  all components of a goal state.
+*}
+
+ML {*
+fun subproof_tac {asms, concl, prems, params, context, schematics} =
+let
+  fun out f xs = commas (map (Syntax.string_of_term context o f) xs)
+  val str_of_asms   = out term_of asms
+  val str_of_concl  = out term_of [concl]
+  val str_of_prems  = out prop_of prems
+  val str_of_params = out term_of params
+  val str_of_schms  = out term_of (snd schematics)
+  val s = ["params: "      ^ str_of_params,
+           "schematics: "  ^ str_of_schms,
+           "premises: "    ^ str_of_prems,
+           "assumptions: " ^ str_of_asms,
+           "conclusion: "  ^ str_of_concl]
+in
+   tracing (cat_lines s); no_tac
+end
+*}
+
+lemma shows "\<And>x y. A x y \<Longrightarrow> B y x \<longrightarrow> C (?z y) x"
+apply(tactic {* SUBPROOF subproof_tac @{context} 1 *})?
+apply(rule impI)
+apply(tactic {* SUBPROOF subproof_tac @{context} 1 *})?
+oops
+
+ML {*
+val my_atac = SUBPROOF (fn {prems, ...} => resolve_tac prems 1)
+*}
+
+lemma shows "\<lbrakk>B x y; A x y; C x y \<rbrakk> \<Longrightarrow> A x y"
+apply(tactic {* my_atac @{context} 1 *})
+done
+
+
+text {*
+  Setting up the goal state using the function 
+  Goal.prove. It expects the goal as a term, we
+  have to construct manually.
+*}
+
+ML {* 
+open HOLogic
+
+fun P n = @{term "P::nat \<Rightarrow> bool"} $ (mk_number @{typ "nat"} n) 
+
+fun rhs 1 = P 1
+  | rhs n = mk_conj (P n, rhs (n - 1))
+
+fun lhs 1 n = mk_imp (mk_eq (P 1, P n), rhs n)
+  | lhs m n =  mk_conj (mk_imp 
+                 (mk_eq (P (m - 1), P m), rhs n), lhs (m - 1) n)
+*}
+
+ML {*
+fun de_bruijn_test ctxt n =
+let 
+  val i = 2*n + 1
+  val goal = mk_Trueprop (mk_imp (lhs i i, rhs i))
+in
+  Syntax.string_of_term ctxt goal
+  |> writeln
+end;
+
+de_bruijn_test @{context} 1   
+*}
+
+ML {*
+fun de_bruijn ctxt n =
+let 
+  val i = 2*n+1
+  val goal = mk_Trueprop (mk_imp (lhs i i, rhs i))
+in
+  Goal.prove ctxt ["P"] [] goal
+   (fn _ => (DEPTH_SOLVE o apply_tac) 1)
+end 
+*}
+
+ML{* de_bruijn @{context} 1 *}
+
+text {*
+  Below is the version which constructs the terms
+  using fixed variables.
+*}
+
+ML {*
+fun de_bruijn ctxt n = 
+let 
+  val i = 2*n+1
+  val bs = replicate (i+1) "b"
+  val (nbs, ctxt') = Variable.variant_fixes bs ctxt
+  val fbs = map (fn z => Free (z, @{typ "bool"})) nbs
+
+  fun P n = nth fbs n 
+  
+  fun rhs 0 = @{term "True"}
+    | rhs 1 = P 1
+    | rhs n = mk_conj (P n, rhs (n - 1))
+
+  fun lhs 1 n = mk_imp (mk_eq (P 1, P n), rhs n)
+    | lhs m n =  mk_conj (mk_imp 
+                   (mk_eq (P (m - 1), P m), rhs n), lhs (m - 1) n)
+
+  val goal = mk_Trueprop (mk_imp (lhs i i, rhs i))
+in
+  Goal.prove ctxt' [] [] goal
+   (fn _ => (DEPTH_SOLVE o apply_tac) 1)
+  (*|> singleton (ProofContext.export ctxt' ctxt)*)
+end
+*}
+
+ML {* de_bruijn @{context} 1 *}
+
+
+end
+
+ 
+
+
+
+
+
+
+
+