src/CTT/ex/Elimination.thy
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(*  Title:      CTT/ex/Elimination.thy
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    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
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    Copyright   1991  University of Cambridge
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Some examples taken from P. Martin-L\"of, Intuitionistic type theory
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(Bibliopolis, 1984).
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*)
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header "Examples with elimination rules"
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theory Elimination
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imports CTT
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begin
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text "This finds the functions fst and snd!"
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schematic_lemma [folded basic_defs]: "A type ==> ?a : (A*A) --> A"
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apply (tactic {* pc_tac [] 1 *})
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done
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schematic_lemma [folded basic_defs]: "A type ==> ?a : (A*A) --> A"
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apply (tactic {* pc_tac [] 1 *})
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back
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done
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text "Double negation of the Excluded Middle"
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schematic_lemma "A type ==> ?a : ((A + (A-->F)) --> F) --> F"
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apply (tactic "intr_tac []")
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apply (rule ProdE)
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apply assumption
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apply (tactic "pc_tac [] 1")
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done
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schematic_lemma "[| A type;  B type |] ==> ?a : (A*B) --> (B*A)"
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apply (tactic "pc_tac [] 1")
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done
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(*The sequent version (ITT) could produce an interesting alternative
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  by backtracking.  No longer.*)
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text "Binary sums and products"
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schematic_lemma "[| A type; B type; C type |] ==> ?a : (A+B --> C) --> (A-->C) * (B-->C)"
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apply (tactic "pc_tac [] 1")
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done
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(*A distributive law*)
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schematic_lemma "[| A type;  B type;  C type |] ==> ?a : A * (B+C)  -->  (A*B + A*C)"
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apply (tactic "pc_tac [] 1")
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done
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(*more general version, same proof*)
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schematic_lemma
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  assumes "A type"
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    and "!!x. x:A ==> B(x) type"
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    and "!!x. x:A ==> C(x) type"
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  shows "?a : (SUM x:A. B(x) + C(x)) --> (SUM x:A. B(x)) + (SUM x:A. C(x))"
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apply (tactic {* pc_tac (thms "prems") 1 *})
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done
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text "Construction of the currying functional"
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schematic_lemma "[| A type;  B type;  C type |] ==> ?a : (A*B --> C) --> (A--> (B-->C))"
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apply (tactic "pc_tac [] 1")
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done
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(*more general goal with same proof*)
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schematic_lemma
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  assumes "A type"
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    and "!!x. x:A ==> B(x) type"
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    and "!!z. z: (SUM x:A. B(x)) ==> C(z) type"
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  shows "?a : PROD f: (PROD z : (SUM x:A . B(x)) . C(z)).
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                      (PROD x:A . PROD y:B(x) . C(<x,y>))"
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apply (tactic {* pc_tac (thms "prems") 1 *})
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done
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text "Martin-Lof (1984), page 48: axiom of sum-elimination (uncurry)"
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schematic_lemma "[| A type;  B type;  C type |] ==> ?a : (A --> (B-->C)) --> (A*B --> C)"
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apply (tactic "pc_tac [] 1")
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done
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(*more general goal with same proof*)
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schematic_lemma
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  assumes "A type"
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    and "!!x. x:A ==> B(x) type"
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    and "!!z. z: (SUM x:A . B(x)) ==> C(z) type"
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  shows "?a : (PROD x:A . PROD y:B(x) . C(<x,y>))
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        --> (PROD z : (SUM x:A . B(x)) . C(z))"
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apply (tactic {* pc_tac (thms "prems") 1 *})
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done
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text "Function application"
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schematic_lemma "[| A type;  B type |] ==> ?a : ((A --> B) * A) --> B"
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apply (tactic "pc_tac [] 1")
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done
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text "Basic test of quantifier reasoning"
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schematic_lemma
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  assumes "A type"
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    and "B type"
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    and "!!x y.[| x:A;  y:B |] ==> C(x,y) type"
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  shows
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    "?a :     (SUM y:B . PROD x:A . C(x,y))
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          --> (PROD x:A . SUM y:B . C(x,y))"
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apply (tactic {* pc_tac (thms "prems") 1 *})
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done
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text "Martin-Lof (1984) pages 36-7: the combinator S"
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schematic_lemma
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  assumes "A type"
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    and "!!x. x:A ==> B(x) type"
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    and "!!x y.[| x:A; y:B(x) |] ==> C(x,y) type"
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  shows "?a :    (PROD x:A. PROD y:B(x). C(x,y))
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             --> (PROD f: (PROD x:A. B(x)). PROD x:A. C(x, f`x))"
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apply (tactic {* pc_tac (thms "prems") 1 *})
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done
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text "Martin-Lof (1984) page 58: the axiom of disjunction elimination"
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schematic_lemma
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  assumes "A type"
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    and "B type"
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    and "!!z. z: A+B ==> C(z) type"
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  shows "?a : (PROD x:A. C(inl(x))) --> (PROD y:B. C(inr(y)))
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          --> (PROD z: A+B. C(z))"
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apply (tactic {* pc_tac (thms "prems") 1 *})
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done
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(*towards AXIOM OF CHOICE*)
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schematic_lemma [folded basic_defs]:
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  "[| A type; B type; C type |] ==> ?a : (A --> B*C) --> (A-->B) * (A-->C)"
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apply (tactic "pc_tac [] 1")
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done
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(*Martin-Lof (1984) page 50*)
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text "AXIOM OF CHOICE!  Delicate use of elimination rules"
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schematic_lemma
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  assumes "A type"
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    and "!!x. x:A ==> B(x) type"
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    and "!!x y.[| x:A;  y:B(x) |] ==> C(x,y) type"
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  shows "?a : PROD h: (PROD x:A. SUM y:B(x). C(x,y)).
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                         (SUM f: (PROD x:A. B(x)). PROD x:A. C(x, f`x))"
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apply (tactic {* intr_tac (thms "prems") *})
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apply (tactic "add_mp_tac 2")
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apply (tactic "add_mp_tac 1")
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apply (erule SumE_fst)
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apply (rule replace_type)
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apply (rule subst_eqtyparg)
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apply (rule comp_rls)
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apply (rule_tac [4] SumE_snd)
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apply (tactic {* typechk_tac (thm "SumE_fst" :: thms "prems") *})
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done
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text "Axiom of choice.  Proof without fst, snd.  Harder still!"
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schematic_lemma [folded basic_defs]:
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  assumes "A type"
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    and "!!x. x:A ==> B(x) type"
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    and "!!x y.[| x:A;  y:B(x) |] ==> C(x,y) type"
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  shows "?a : PROD h: (PROD x:A. SUM y:B(x). C(x,y)).
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                         (SUM f: (PROD x:A. B(x)). PROD x:A. C(x, f`x))"
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apply (tactic {* intr_tac (thms "prems") *})
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(*Must not use add_mp_tac as subst_prodE hides the construction.*)
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apply (rule ProdE [THEN SumE], assumption)
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apply (tactic "TRYALL assume_tac")
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apply (rule replace_type)
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apply (rule subst_eqtyparg)
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apply (rule comp_rls)
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apply (erule_tac [4] ProdE [THEN SumE])
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apply (tactic {* typechk_tac (thms "prems") *})
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apply (rule replace_type)
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apply (rule subst_eqtyparg)
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apply (rule comp_rls)
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apply (tactic {* typechk_tac (thms "prems") *})
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apply assumption
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done
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text "Example of sequent_style deduction"
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(*When splitting z:A*B, the assumption C(z) is affected;  ?a becomes
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    lam u. split(u,%v w.split(v,%x y.lam z. <x,<y,z>>) ` w)     *)
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8feb2c4bef1a mark schematic statements explicitly;
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schematic_lemma
19761
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  assumes "A type"
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    and "B type"
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    and "!!z. z:A*B ==> C(z) type"
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  shows "?a : (SUM z:A*B. C(z)) --> (SUM u:A. SUM v:B. C(<u,v>))"
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apply (rule intr_rls)
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apply (tactic {* biresolve_tac safe_brls 2 *})
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(*Now must convert assumption C(z) into antecedent C(<kd,ke>) *)
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apply (rule_tac [2] a = "y" in ProdE)
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apply (tactic {* typechk_tac (thms "prems") *})
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apply (rule SumE, assumption)
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apply (tactic "intr_tac []")
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apply (tactic "TRYALL assume_tac")
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apply (tactic {* typechk_tac (thms "prems") *})
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done
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end