Theory PropLog

theory PropLog
imports Main
(*  Title:      ZF/Induct/PropLog.thy
    Author:     Tobias Nipkow & Lawrence C Paulson
    Copyright   1993  University of Cambridge
*)

header {* Meta-theory of propositional logic *}

theory PropLog imports Main begin

text {*
  Datatype definition of propositional logic formulae and inductive
  definition of the propositional tautologies.

  Inductive definition of propositional logic.  Soundness and
  completeness w.r.t.\ truth-tables.

  Prove: If @{text "H |= p"} then @{text "G |= p"} where @{text "G ∈
  Fin(H)"}
*}


subsection {* The datatype of propositions *}

consts
  propn :: i

datatype propn =
    Fls
  | Var ("n ∈ nat")    ("#_" [100] 100)
  | Imp ("p ∈ propn", "q ∈ propn")    (infixr "=>" 90)


subsection {* The proof system *}

consts thms     :: "i => i"

abbreviation
  thms_syntax :: "[i,i] => o"    (infixl "|-" 50)
  where "H |- p == p ∈ thms(H)"

inductive
  domains "thms(H)"  "propn"
  intros
    H:  "[| p ∈ H;  p ∈ propn |] ==> H |- p"
    K:  "[| p ∈ propn;  q ∈ propn |] ==> H |- p=>q=>p"
    S:  "[| p ∈ propn;  q ∈ propn;  r ∈ propn |]
         ==> H |- (p=>q=>r) => (p=>q) => p=>r"
    DN: "p ∈ propn ==> H |- ((p=>Fls) => Fls) => p"
    MP: "[| H |- p=>q;  H |- p;  p ∈ propn;  q ∈ propn |] ==> H |- q"
  type_intros "propn.intros"

declare propn.intros [simp]


subsection {* The semantics *}

subsubsection {* Semantics of propositional logic. *}

consts
  is_true_fun :: "[i,i] => i"
primrec
  "is_true_fun(Fls, t) = 0"
  "is_true_fun(Var(v), t) = (if v ∈ t then 1 else 0)"
  "is_true_fun(p=>q, t) = (if is_true_fun(p,t) = 1 then is_true_fun(q,t) else 1)"

definition
  is_true :: "[i,i] => o"  where
  "is_true(p,t) == is_true_fun(p,t) = 1"
  -- {* this definition is required since predicates can't be recursive *}

lemma is_true_Fls [simp]: "is_true(Fls,t) <-> False"
  by (simp add: is_true_def)

lemma is_true_Var [simp]: "is_true(#v,t) <-> v ∈ t"
  by (simp add: is_true_def)

lemma is_true_Imp [simp]: "is_true(p=>q,t) <-> (is_true(p,t)-->is_true(q,t))"
  by (simp add: is_true_def)


subsubsection {* Logical consequence *}

text {*
  For every valuation, if all elements of @{text H} are true then so
  is @{text p}.
*}

definition
  logcon :: "[i,i] => o"    (infixl "|=" 50)  where
  "H |= p == ∀t. (∀q ∈ H. is_true(q,t)) --> is_true(p,t)"


text {*
  A finite set of hypotheses from @{text t} and the @{text Var}s in
  @{text p}.
*}

consts
  hyps :: "[i,i] => i"
primrec
  "hyps(Fls, t) = 0"
  "hyps(Var(v), t) = (if v ∈ t then {#v} else {#v=>Fls})"
  "hyps(p=>q, t) = hyps(p,t) ∪ hyps(q,t)"



subsection {* Proof theory of propositional logic *}

lemma thms_mono: "G ⊆ H ==> thms(G) ⊆ thms(H)"
  apply (unfold thms.defs)
  apply (rule lfp_mono)
    apply (rule thms.bnd_mono)+
  apply (assumption | rule univ_mono basic_monos)+
  done

lemmas thms_in_pl = thms.dom_subset [THEN subsetD]

inductive_cases ImpE: "p=>q ∈ propn"

lemma thms_MP: "[| H |- p=>q;  H |- p |] ==> H |- q"
  -- {* Stronger Modus Ponens rule: no typechecking! *}
  apply (rule thms.MP)
     apply (erule asm_rl thms_in_pl thms_in_pl [THEN ImpE])+
  done

lemma thms_I: "p ∈ propn ==> H |- p=>p"
  -- {*Rule is called @{text I} for Identity Combinator, not for Introduction. *}
  apply (rule thms.S [THEN thms_MP, THEN thms_MP])
      apply (rule_tac [5] thms.K)
       apply (rule_tac [4] thms.K)
         apply simp_all
  done


subsubsection {* Weakening, left and right *}

lemma weaken_left: "[| G ⊆ H;  G|-p |] ==> H|-p"
  -- {* Order of premises is convenient with @{text THEN} *}
  by (erule thms_mono [THEN subsetD])

lemma weaken_left_cons: "H |- p ==> cons(a,H) |- p"
  by (erule subset_consI [THEN weaken_left])

lemmas weaken_left_Un1  = Un_upper1 [THEN weaken_left]
lemmas weaken_left_Un2  = Un_upper2 [THEN weaken_left]

lemma weaken_right: "[| H |- q;  p ∈ propn |] ==> H |- p=>q"
  by (simp_all add: thms.K [THEN thms_MP] thms_in_pl)


subsubsection {* The deduction theorem *}

theorem deduction: "[| cons(p,H) |- q;  p ∈ propn |] ==>  H |- p=>q"
  apply (erule thms.induct)
      apply (blast intro: thms_I thms.H [THEN weaken_right])
     apply (blast intro: thms.K [THEN weaken_right])
    apply (blast intro: thms.S [THEN weaken_right])
   apply (blast intro: thms.DN [THEN weaken_right])
  apply (blast intro: thms.S [THEN thms_MP [THEN thms_MP]])
  done


subsubsection {* The cut rule *}

lemma cut: "[| H|-p;  cons(p,H) |- q |] ==>  H |- q"
  apply (rule deduction [THEN thms_MP])
    apply (simp_all add: thms_in_pl)
  done

lemma thms_FlsE: "[| H |- Fls; p ∈ propn |] ==> H |- p"
  apply (rule thms.DN [THEN thms_MP])
   apply (rule_tac [2] weaken_right)
    apply (simp_all add: propn.intros)
  done

lemma thms_notE: "[| H |- p=>Fls;  H |- p;  q ∈ propn |] ==> H |- q"
  by (erule thms_MP [THEN thms_FlsE])


subsubsection {* Soundness of the rules wrt truth-table semantics *}

theorem soundness: "H |- p ==> H |= p"
  apply (unfold logcon_def)
  apply (induct set: thms)
      apply auto
  done


subsection {* Completeness *}

subsubsection {* Towards the completeness proof *}

lemma Fls_Imp: "[| H |- p=>Fls; q ∈ propn |] ==> H |- p=>q"
  apply (frule thms_in_pl)
  apply (rule deduction)
   apply (rule weaken_left_cons [THEN thms_notE])
     apply (blast intro: thms.H elim: ImpE)+
  done

lemma Imp_Fls: "[| H |- p;  H |- q=>Fls |] ==> H |- (p=>q)=>Fls"
  apply (frule thms_in_pl)
  apply (frule thms_in_pl [of concl: "q=>Fls"])
  apply (rule deduction)
   apply (erule weaken_left_cons [THEN thms_MP])
   apply (rule consI1 [THEN thms.H, THEN thms_MP])
    apply (blast intro: weaken_left_cons elim: ImpE)+
  done

lemma hyps_thms_if:
    "p ∈ propn ==> hyps(p,t) |- (if is_true(p,t) then p else p=>Fls)"
  -- {* Typical example of strengthening the induction statement. *}
  apply simp
  apply (induct_tac p)
    apply (simp_all add: thms_I thms.H)
  apply (safe elim!: Fls_Imp [THEN weaken_left_Un1] Fls_Imp [THEN weaken_left_Un2])
  apply (blast intro: weaken_left_Un1 weaken_left_Un2 weaken_right Imp_Fls)+
  done

lemma logcon_thms_p: "[| p ∈ propn;  0 |= p |] ==> hyps(p,t) |- p"
  -- {* Key lemma for completeness; yields a set of assumptions satisfying @{text p} *}
  apply (drule hyps_thms_if)
  apply (simp add: logcon_def)
  done

text {*
  For proving certain theorems in our new propositional logic.
*}

lemmas propn_SIs = propn.intros deduction
  and propn_Is = thms_in_pl thms.H thms.H [THEN thms_MP]

text {*
  The excluded middle in the form of an elimination rule.
*}

lemma thms_excluded_middle:
    "[| p ∈ propn;  q ∈ propn |] ==> H |- (p=>q) => ((p=>Fls)=>q) => q"
  apply (rule deduction [THEN deduction])
    apply (rule thms.DN [THEN thms_MP])
     apply (best intro!: propn_SIs intro: propn_Is)+
  done

lemma thms_excluded_middle_rule:
  "[| cons(p,H) |- q;  cons(p=>Fls,H) |- q;  p ∈ propn |] ==> H |- q"
  -- {* Hard to prove directly because it requires cuts *}
  apply (rule thms_excluded_middle [THEN thms_MP, THEN thms_MP])
     apply (blast intro!: propn_SIs intro: propn_Is)+
  done


subsubsection {* Completeness -- lemmas for reducing the set of assumptions *}

text {*
  For the case @{prop "hyps(p,t)-cons(#v,Y) |- p"} we also have @{prop
  "hyps(p,t)-{#v} ⊆ hyps(p, t-{v})"}.
*}

lemma hyps_Diff:
    "p ∈ propn ==> hyps(p, t-{v}) ⊆ cons(#v=>Fls, hyps(p,t)-{#v})"
  by (induct set: propn) auto

text {*
  For the case @{prop "hyps(p,t)-cons(#v => Fls,Y) |- p"} we also have
  @{prop "hyps(p,t)-{#v=>Fls} ⊆ hyps(p, cons(v,t))"}.
*}

lemma hyps_cons:
    "p ∈ propn ==> hyps(p, cons(v,t)) ⊆ cons(#v, hyps(p,t)-{#v=>Fls})"
  by (induct set: propn) auto

text {* Two lemmas for use with @{text weaken_left} *}

lemma cons_Diff_same: "B-C ⊆ cons(a, B-cons(a,C))"
  by blast

lemma cons_Diff_subset2: "cons(a, B-{c}) - D ⊆ cons(a, B-cons(c,D))"
  by blast

text {*
  The set @{term "hyps(p,t)"} is finite, and elements have the form
  @{term "#v"} or @{term "#v=>Fls"}; could probably prove the stronger
  @{prop "hyps(p,t) ∈ Fin(hyps(p,0) ∪ hyps(p,nat))"}.
*}

lemma hyps_finite: "p ∈ propn ==> hyps(p,t) ∈ Fin(\<Union>v ∈ nat. {#v, #v=>Fls})"
  by (induct set: propn) auto

lemmas Diff_weaken_left = Diff_mono [OF _ subset_refl, THEN weaken_left]

text {*
  Induction on the finite set of assumptions @{term "hyps(p,t0)"}.  We
  may repeatedly subtract assumptions until none are left!
*}

lemma completeness_0_lemma [rule_format]:
    "[| p ∈ propn;  0 |= p |] ==> ∀t. hyps(p,t) - hyps(p,t0) |- p"
  apply (frule hyps_finite)
  apply (erule Fin_induct)
   apply (simp add: logcon_thms_p Diff_0)
  txt {* inductive step *}
  apply safe
   txt {* Case @{prop "hyps(p,t)-cons(#v,Y) |- p"} *}
   apply (rule thms_excluded_middle_rule)
     apply (erule_tac [3] propn.intros)
    apply (blast intro: cons_Diff_same [THEN weaken_left])
   apply (blast intro: cons_Diff_subset2 [THEN weaken_left]
     hyps_Diff [THEN Diff_weaken_left])
  txt {* Case @{prop "hyps(p,t)-cons(#v => Fls,Y) |- p"} *}
  apply (rule thms_excluded_middle_rule)
    apply (erule_tac [3] propn.intros)
   apply (blast intro: cons_Diff_subset2 [THEN weaken_left]
     hyps_cons [THEN Diff_weaken_left])
  apply (blast intro: cons_Diff_same [THEN weaken_left])
  done


subsubsection {* Completeness theorem *}

lemma completeness_0: "[| p ∈ propn;  0 |= p |] ==> 0 |- p"
  -- {* The base case for completeness *}
  apply (rule Diff_cancel [THEN subst])
  apply (blast intro: completeness_0_lemma)
  done

lemma logcon_Imp: "[| cons(p,H) |= q |] ==> H |= p=>q"
  -- {* A semantic analogue of the Deduction Theorem *}
  by (simp add: logcon_def)

lemma completeness:
     "H ∈ Fin(propn) ==> p ∈ propn ==> H |= p ==> H |- p"
  apply (induct arbitrary: p set: Fin)
   apply (safe intro!: completeness_0)
  apply (rule weaken_left_cons [THEN thms_MP])
   apply (blast intro!: logcon_Imp propn.intros)
  apply (blast intro: propn_Is)
  done

theorem thms_iff: "H ∈ Fin(propn) ==> H |- p <-> H |= p ∧ p ∈ propn"
  by (blast intro: soundness completeness thms_in_pl)

end