src/CTT/ex/Elimination.thy

tuned;

(*  Title:      CTT/ex/Elimination.thy
    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
    Copyright   1991  University of Cambridge

Some examples taken from P. Martin-L\"of, Intuitionistic type theory
(Bibliopolis, 1984).
*)

section "Examples with elimination rules"

theory Elimination
imports "../CTT"
begin

text "This finds the functions fst and snd!"

schematic_goal [folded basic_defs]: "A type \<Longrightarrow> ?a : (A \<times> A) \<longrightarrow> A"
apply pc
done

schematic_goal [folded basic_defs]: "A type \<Longrightarrow> ?a : (A \<times> A) \<longrightarrow> A"
apply pc
back
done

text "Double negation of the Excluded Middle"
schematic_goal "A type \<Longrightarrow> ?a : ((A + (A\<longrightarrow>F)) \<longrightarrow> F) \<longrightarrow> F"
apply intr
apply (rule ProdE)
apply assumption
apply pc
done

schematic_goal "\<lbrakk>A type; B type\<rbrakk> \<Longrightarrow> ?a : (A \<times> B) \<longrightarrow> (B \<times> A)"
apply pc
done
(*The sequent version (ITT) could produce an interesting alternative
  by backtracking.  No longer.*)

text "Binary sums and products"
schematic_goal "\<lbrakk>A type; B type; C type\<rbrakk> \<Longrightarrow> ?a : (A + B \<longrightarrow> C) \<longrightarrow> (A \<longrightarrow> C) \<times> (B \<longrightarrow> C)"
apply pc
done

(*A distributive law*)
schematic_goal "\<lbrakk>A type; B type; C type\<rbrakk> \<Longrightarrow> ?a : A \<times> (B + C) \<longrightarrow> (A \<times> B + A \<times> C)"
apply pc
done

(*more general version, same proof*)
schematic_goal
  assumes "A type"
    and "\<And>x. x:A \<Longrightarrow> B(x) type"
    and "\<And>x. x:A \<Longrightarrow> C(x) type"
  shows "?a : (\<Sum>x:A. B(x) + C(x)) \<longrightarrow> (\<Sum>x:A. B(x)) + (\<Sum>x:A. C(x))"
apply (pc assms)
done

text "Construction of the currying functional"
schematic_goal "\<lbrakk>A type; B type; C type\<rbrakk> \<Longrightarrow> ?a : (A \<times> B \<longrightarrow> C) \<longrightarrow> (A \<longrightarrow> (B \<longrightarrow> C))"
apply pc
done

(*more general goal with same proof*)
schematic_goal
  assumes "A type"
    and "\<And>x. x:A \<Longrightarrow> B(x) type"
    and "\<And>z. z: (\<Sum>x:A. B(x)) \<Longrightarrow> C(z) type"
  shows "?a : \<Prod>f: (\<Prod>z : (\<Sum>x:A . B(x)) . C(z)).
                      (\<Prod>x:A . \<Prod>y:B(x) . C(<x,y>))"
apply (pc assms)
done

text "Martin-Löf (1984), page 48: axiom of sum-elimination (uncurry)"
schematic_goal "\<lbrakk>A type; B type; C type\<rbrakk> \<Longrightarrow> ?a : (A \<longrightarrow> (B \<longrightarrow> C)) \<longrightarrow> (A \<times> B \<longrightarrow> C)"
apply pc
done

(*more general goal with same proof*)
schematic_goal
  assumes "A type"
    and "\<And>x. x:A \<Longrightarrow> B(x) type"
    and "\<And>z. z: (\<Sum>x:A . B(x)) \<Longrightarrow> C(z) type"
  shows "?a : (\<Prod>x:A . \<Prod>y:B(x) . C(<x,y>))
        \<longrightarrow> (\<Prod>z : (\<Sum>x:A . B(x)) . C(z))"
apply (pc assms)
done

text "Function application"
schematic_goal "\<lbrakk>A type; B type\<rbrakk> \<Longrightarrow> ?a : ((A \<longrightarrow> B) \<times> A) \<longrightarrow> B"
apply pc
done

text "Basic test of quantifier reasoning"
schematic_goal
  assumes "A type"
    and "B type"
    and "\<And>x y. \<lbrakk>x:A; y:B\<rbrakk> \<Longrightarrow> C(x,y) type"
  shows
    "?a :     (\<Sum>y:B . \<Prod>x:A . C(x,y))
          \<longrightarrow> (\<Prod>x:A . \<Sum>y:B . C(x,y))"
apply (pc assms)
done

text "Martin-Löf (1984) pages 36-7: the combinator S"
schematic_goal
  assumes "A type"
    and "\<And>x. x:A \<Longrightarrow> B(x) type"
    and "\<And>x y. \<lbrakk>x:A; y:B(x)\<rbrakk> \<Longrightarrow> C(x,y) type"
  shows "?a :    (\<Prod>x:A. \<Prod>y:B(x). C(x,y))
             \<longrightarrow> (\<Prod>f: (\<Prod>x:A. B(x)). \<Prod>x:A. C(x, f`x))"
apply (pc assms)
done

text "Martin-Löf (1984) page 58: the axiom of disjunction elimination"
schematic_goal
  assumes "A type"
    and "B type"
    and "\<And>z. z: A+B \<Longrightarrow> C(z) type"
  shows "?a : (\<Prod>x:A. C(inl(x))) \<longrightarrow> (\<Prod>y:B. C(inr(y)))
          \<longrightarrow> (\<Prod>z: A+B. C(z))"
apply (pc assms)
done

(*towards AXIOM OF CHOICE*)
schematic_goal [folded basic_defs]:
  "\<lbrakk>A type; B type; C type\<rbrakk> \<Longrightarrow> ?a : (A \<longrightarrow> B \<times> C) \<longrightarrow> (A \<longrightarrow> B) \<times> (A \<longrightarrow> C)"
apply pc
done


(*Martin-Löf (1984) page 50*)
text "AXIOM OF CHOICE!  Delicate use of elimination rules"
schematic_goal
  assumes "A type"
    and "\<And>x. x:A \<Longrightarrow> B(x) type"
    and "\<And>x y. \<lbrakk>x:A; y:B(x)\<rbrakk> \<Longrightarrow> C(x,y) type"
  shows "?a : (\<Prod>x:A. \<Sum>y:B(x). C(x,y)) \<longrightarrow> (\<Sum>f: (\<Prod>x:A. B(x)). \<Prod>x:A. C(x, f`x))"
apply (intr assms)
prefer 2 apply add_mp
prefer 2 apply add_mp
apply (erule SumE_fst)
apply (rule replace_type)
apply (rule subst_eqtyparg)
apply (rule comp_rls)
apply (rule_tac [4] SumE_snd)
apply (typechk SumE_fst assms)
done

text "Axiom of choice.  Proof without fst, snd.  Harder still!"
schematic_goal [folded basic_defs]:
  assumes "A type"
    and "\<And>x. x:A \<Longrightarrow> B(x) type"
    and "\<And>x y. \<lbrakk>x:A; y:B(x)\<rbrakk> \<Longrightarrow> C(x,y) type"
  shows "?a : (\<Prod>x:A. \<Sum>y:B(x). C(x,y)) \<longrightarrow> (\<Sum>f: (\<Prod>x:A. B(x)). \<Prod>x:A. C(x, f`x))"
apply (intr assms)
(*Must not use add_mp as subst_prodE hides the construction.*)
apply (rule ProdE [THEN SumE])
apply assumption
apply assumption
apply assumption
apply (rule replace_type)
apply (rule subst_eqtyparg)
apply (rule comp_rls)
apply (erule_tac [4] ProdE [THEN SumE])
apply (typechk assms)
apply (rule replace_type)
apply (rule subst_eqtyparg)
apply (rule comp_rls)
apply (typechk assms)
apply assumption
done

text "Example of sequent-style deduction"
(*When splitting z:A \<times> B, the assumption C(z) is affected;  ?a becomes
    \<^bold>\<lambda>u. split(u,\<lambda>v w.split(v,\<lambda>x y.\<^bold> \<lambda>z. <x,<y,z>>) ` w)     *)
schematic_goal
  assumes "A type"
    and "B type"
    and "\<And>z. z:A \<times> B \<Longrightarrow> C(z) type"
  shows "?a : (\<Sum>z:A \<times> B. C(z)) \<longrightarrow> (\<Sum>u:A. \<Sum>v:B. C(<u,v>))"
apply (rule intr_rls)
apply (tactic \<open>biresolve_tac @{context} safe_brls 2\<close>)
(*Now must convert assumption C(z) into antecedent C(<kd,ke>) *)
apply (rule_tac [2] a = "y" in ProdE)
apply (typechk assms)
apply (rule SumE, assumption)
apply intr
defer 1
apply assumption+
apply (typechk assms)
done

end