Quelle Example.thy Sprache: Isabelle

section \<open>Lambda Cube Examples\<close>

theory Example
imports Cube
begin

text \<open>Examples taken from:

  H. Barendregt. Introduction to Generalised Type Systems.
  J. Functional Programming.\<close>

method_setup depth_solve =
  \<open>Attrib.thms >> (fn thms => fn ctxt => METHOD (fn facts =>
    (DEPTH_SOLVE (HEADGOAL (assume_tac ctxt ORELSE' resolve_tac ctxt (facts @ thms))))))\

method_setup depth_solve1 =
  \<open>Attrib.thms >> (fn thms => fn ctxt => METHOD (fn facts =>
    (DEPTH_SOLVE_1 (HEADGOAL (assume_tac ctxt ORELSE' resolve_tac ctxt (facts @ thms))))))\

method_setup strip_asms =
  \<open>Attrib.thms >> (fn thms => fn ctxt => METHOD (fn facts =>
    REPEAT (resolve_tac ctxt @{thms strip_b strip_s} 1 THEN
    DEPTH_SOLVE_1 (assume_tac ctxt 1 ORELSE resolve_tac ctxt (facts @ thms) 1))))\<close>

subsection \<open>Simple types\<close>

schematic_goal "A:* \ A\A : ?T"
  by (depth_solve rules)

schematic_goal "A:* \ \<^bold>\a:A. a : ?T"
  by (depth_solve rules)

schematic_goal "A:* B:* b:B \ \<^bold>\x:A. b : ?T"
  by (depth_solve rules)

schematic_goal "A:* b:A \ (\<^bold>\a:A. a)\b: ?T"
  by (depth_solve rules)

schematic_goal "A:* B:* c:A b:B \ (\<^bold>\x:A. b)\ c: ?T"
  by (depth_solve rules)

schematic_goal "A:* B:* \ \<^bold>\a:A. \<^bold>\b:B. a : ?T"
  by (depth_solve rules)

subsection \<open>Second-order types\<close>

schematic_goal (in L2) "\ \<^bold>\A:*. \<^bold>\a:A. a : ?T"
  by (depth_solve rules)

schematic_goal (in L2) "A:* \ (\<^bold>\B:*. \<^bold>\b:B. b)\A : ?T"
  by (depth_solve rules)

schematic_goal (in L2) "A:* b:A \ (\<^bold>\B:*. \<^bold>\b:B. b) \ A \ b: ?T"
  by (depth_solve rules)

schematic_goal (in L2) "\ \<^bold>\B:*. \<^bold>\a:(\A:*.A).a \ ((\A:*.A)\B) \ a: ?T"
  by (depth_solve rules)

subsection \<open>Weakly higher-order propositional logic\<close>

schematic_goal (in Lomega) "\ \<^bold>\A:*.A\A : ?T"
  by (depth_solve rules)

schematic_goal (in Lomega) "B:* \ (\<^bold>\A:*.A\A) \ B : ?T"
  by (depth_solve rules)

schematic_goal (in Lomega) "B:* b:B \ (\<^bold>\y:B. b): ?T"
  by (depth_solve rules)

schematic_goal (in Lomega) "A:* F:*\* \ F\(F\A): ?T"
  by (depth_solve rules)

schematic_goal (in Lomega) "A:* \ \<^bold>\F:*\*.F\(F\A): ?T"
  by (depth_solve rules)

subsection \<open>LP\<close>

schematic_goal (in LP) "A:* \ A \ * : ?T"
  by (depth_solve rules)

schematic_goal (in LP) "A:* P:A\* a:A \ P\a: ?T"
  by (depth_solve rules)

schematic_goal (in LP) "A:* P:A\A\* a:A \ \a:A. P\a\a: ?T"
  by (depth_solve rules)

schematic_goal (in LP) "A:* P:A\* Q:A\* \ \a:A. P\a \ Q\a: ?T"
  by (depth_solve rules)

schematic_goal (in LP) "A:* P:A\* \ \a:A. P\a \ P\a: ?T"
  by (depth_solve rules)

schematic_goal (in LP) "A:* P:A\* \ \<^bold>\a:A. \<^bold>\x:P\a. x: ?T"
  by (depth_solve rules)

schematic_goal (in LP) "A:* P:A\* Q:* \ (\a:A. P\a\Q) \ (\a:A. P\a) \ Q : ?T"
  by (depth_solve rules)

schematic_goal (in LP) "A:* P:A\* Q:* a0:A \
        \<^bold>\<lambda>x:\<Prod>a:A. P\<cdot>a\<rightarrow>Q. \<^bold>\<lambda>y:\<Prod>a:A. P\<cdot>a. x\<cdot>a0\<cdot>(y\<cdot>a0): ?T"
  by (depth_solve rules)

subsection \<open>Omega-order types\<close>

schematic_goal (in L2) "A:* B:* \ \C:*.(A\B\C)\C : ?T"
  by (depth_solve rules)

schematic_goal (in Lomega2) "\ \<^bold>\A:*. \<^bold>\B:*.\C:*.(A\B\C)\C : ?T"
  by (depth_solve rules)

schematic_goal (in Lomega2) "\ \<^bold>\A:*. \<^bold>\B:*. \<^bold>\x:A. \<^bold>\y:B. x : ?T"
  by (depth_solve rules)

schematic_goal (in Lomega2) "A:* B:* \ ?p : (A\B) \ ((B\\P:*.P)\(A\\P:*.P))"
  apply (strip_asms rules)
  apply (rule lam_ss)
    apply (depth_solve1 rules)
   prefer 2
   apply (depth_solve1 rules)
  apply (rule lam_ss)
    apply (depth_solve1 rules)
   prefer 2
   apply (depth_solve1 rules)
  apply (rule lam_ss)
    apply assumption
   prefer 2
   apply (depth_solve1 rules)
  apply (erule pi_elim)
   apply assumption
  apply (erule pi_elim)
   apply assumption
  apply assumption
  done

subsection \<open>Second-order Predicate Logic\<close>

schematic_goal (in LP2) "A:* P:A\* \ \<^bold>\a:A. P\a\(\A:*.A) : ?T"
  by (depth_solve rules)

schematic_goal (in LP2) "A:* P:A\A\* \
    (\<Prod>a:A. \<Prod>b:A. P\<cdot>a\<cdot>b\<rightarrow>P\<cdot>b\<cdot>a\<rightarrow>\<Prod>P:*.P) \<rightarrow> \<Prod>a:A. P\<cdot>a\<cdot>a\<rightarrow>\<Prod>P:*.P : ?T"
  by (depth_solve rules)

schematic_goal (in LP2) "A:* P:A\A\* \
    ?p: (\<Prod>a:A. \<Prod>b:A. P\<cdot>a\<cdot>b\<rightarrow>P\<cdot>b\<cdot>a\<rightarrow>\<Prod>P:*.P) \<rightarrow> \<Prod>a:A. P\<cdot>a\<cdot>a\<rightarrow>\<Prod>P:*.P"
  \<comment> \<open>Antisymmetry implies irreflexivity:\<close>
  apply (strip_asms rules)
  apply (rule lam_ss)
    apply (depth_solve1 rules)
   prefer 2
   apply (depth_solve1 rules)
  apply (rule lam_ss)
    apply assumption
   prefer 2
   apply (depth_solve1 rules)
  apply (rule lam_ss)
    apply (depth_solve1 rules)
   prefer 2
   apply (depth_solve1 rules)
  apply (erule pi_elim, assumption, assumption?)+
  done

subsection \<open>LPomega\<close>

schematic_goal (in LPomega) "A:* \ \<^bold>\P:A\A\*. \<^bold>\a:A. P\a\a : ?T"
  by (depth_solve rules)

schematic_goal (in LPomega) "\ \<^bold>\A:*. \<^bold>\P:A\A\*. \<^bold>\a:A. P\a\a : ?T"
  by (depth_solve rules)

subsection \<open>Constructions\<close>

schematic_goal (in CC) "\ \<^bold>\A:*. \<^bold>\P:A\*. \<^bold>\a:A. P\a\\P:*.P: ?T"
  by (depth_solve rules)

schematic_goal (in CC) "\ \<^bold>\A:*. \<^bold>\P:A\*.\a:A. P\a: ?T"
  by (depth_solve rules)

schematic_goal (in CC) "A:* P:A\* a:A \ ?p : (\a:A. P\a)\P\a"
  apply (strip_asms rules)
  apply (rule lam_ss)
    apply (depth_solve1 rules)
   prefer 2
   apply (depth_solve1 rules)
  apply (erule pi_elim, assumption, assumption)
  done

subsection \<open>Some random examples\<close>

schematic_goal (in LP2) "A:* c:A f:A\A \
    \<^bold>\<lambda>a:A. \<Prod>P:A\<rightarrow>*.P\<cdot>c \<rightarrow> (\<Prod>x:A. P\<cdot>x\<rightarrow>P\<cdot>(f\<cdot>x)) \<rightarrow> P\<cdot>a : ?T"
  by (depth_solve rules)

schematic_goal (in CC) "\<^bold>\A:*. \<^bold>\c:A. \<^bold>\f:A\A.
    \<^bold>\<lambda>a:A. \<Prod>P:A\<rightarrow>*.P\<cdot>c \<rightarrow> (\<Prod>x:A. P\<cdot>x\<rightarrow>P\<cdot>(f\<cdot>x)) \<rightarrow> P\<cdot>a : ?T"
  by (depth_solve rules)

schematic_goal (in LP2)
  "A:* a:A b:A \ ?p: (\P:A\*.P\a\P\b) \ (\P:A\*.P\b\P\a)"
  \<comment> \<open>Symmetry of Leibnitz equality\<close>
  apply (strip_asms rules)
  apply (rule lam_ss)
    apply (depth_solve1 rules)
   prefer 2
   apply (depth_solve1 rules)
  apply (erule_tac a = "\<^bold>\x:A. \Q:A\*.Q\x\Q\a" in pi_elim)
   apply (depth_solve1 rules)
  apply (unfold beta)
  apply (erule imp_elim)
   apply (rule lam_bs)
     apply (depth_solve1 rules)
    prefer 2
    apply (depth_solve1 rules)
   apply (rule lam_ss)
     apply (depth_solve1 rules)
    prefer 2
    apply (depth_solve1 rules)
   apply assumption
  apply assumption
  done

end

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