--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+set "baseuri" "cic:/matita/demo/propositional_sequent_calculus/".
+
+include "nat/plus.ma".
+include "nat/compare.ma".
+include "list/sort.ma".
+include "datatypes/constructors.ma".
+
+inductive Formula : Type ≝
+ FTrue: Formula
+ | FFalse: Formula
+ | FAtom: nat → Formula
+ | FAnd: Formula → Formula → Formula
+ | FOr: Formula → Formula → Formula
+ | FNot: Formula → Formula.
+
+definition interp ≝ nat → bool.
+
+let rec eval (interp:interp) F on F : bool ≝
+ match F with
+ [ FTrue ⇒ true
+ | FFalse ⇒ false
+ | FAtom n ⇒ interp n
+ | FAnd f1 f2 ⇒ eval interp f1 ∧ eval interp f2
+ | FOr f1 f2 ⇒ eval interp f1 ∨ eval interp f2
+ | FNot f ⇒ ¬ eval interp f
+ ].
+
+inductive not_nf : Formula → Prop ≝
+ NTrue: not_nf FTrue
+ | NFalse: not_nf FFalse
+ | NAtom: ∀n. not_nf (FAtom n)
+ | NAnd: ∀f1,f2. not_nf f1 → not_nf f2 → not_nf (FAnd f1 f2)
+ | NOr: ∀f1,f2. not_nf f1 → not_nf f2 → not_nf (FOr f1 f2)
+ | NNot: ∀n.not_nf (FNot (FAtom n)).
+
+let rec negate F ≝
+ match F with
+ [ FTrue ⇒ FFalse
+ | FFalse ⇒ FTrue
+ | FAtom n ⇒ FNot (FAtom n)
+ | FAnd f1 f2 ⇒ FOr (negate f1) (negate f2)
+ | FOr f1 f2 ⇒ FAnd (negate f1) (negate f2)
+ | FNot f ⇒ elim_not f]
+and elim_not F ≝
+ match F with
+ [ FTrue ⇒ FTrue
+ | FFalse ⇒ FFalse
+ | FAtom n ⇒ FAtom n
+ | FAnd f1 f2 ⇒ FAnd (elim_not f1) (elim_not f2)
+ | FOr f1 f2 ⇒ FOr (elim_not f1) (elim_not f2)
+ | FNot f ⇒ negate f
+ ].
+
+theorem not_nf_elim_not: ∀F.not_nf (elim_not F) ∧ not_nf (negate F).
+ intros;
+ elim F;
+ [1,2,3: simplify; autobatch
+ |4,5:
+ simplify;
+ elim H; clear H;
+ elim H1; clear H1;
+ split;
+ autobatch
+ |elim H; clear H;
+ split;
+ [ assumption
+ | assumption
+ ]
+ ]
+qed.
+
+theorem demorgan1: ∀b1,b2:bool. (¬ (b1 ∧ b2)) = ¬ b1 ∨ ¬ b2.
+ intros;
+ elim b1;
+ simplify;
+ reflexivity.
+qed.
+
+theorem demorgan2: ∀b1,b2:bool. (¬ (b1 ∨ b2)) = ¬ b1 ∧ ¬ b2.
+ intros;
+ elim b1;
+ simplify;
+ reflexivity.
+qed.
+
+theorem eq_notb_notb_b_b: ∀b:bool. (¬ ¬ b) = b.
+ intro;
+ elim b;
+ reflexivity.
+qed.
+
+theorem eq_eval_elim_not_eval:
+ ∀i,F. eval i (elim_not F) = eval i F ∧ eval i (negate F) = eval i (FNot F).
+ intros;
+ elim F;
+ [1,2,3: split; reflexivity
+ |4,5:
+ simplify;
+ elim H; clear H;
+ elim H1; clear H1;
+ split;
+ [1,3: autobatch
+ |replace with ((eval i (FNot f) ∨ eval i (FNot f1)) = ¬ (eval i f ∧ eval i f1));
+ [ simplify;
+ autobatch
+ | autobatch
+ ]
+ |replace with ((eval i (FNot f) ∧ eval i (FNot f1)) = ¬ (eval i f ∨ eval i f1));
+ [ simplify;
+ autobatch
+ | autobatch
+ ]
+ ]
+ |elim H; clear H;
+ split;
+ [ assumption
+ | change with (eval i (elim_not f) = ¬ ¬ eval i f);
+ autobatch
+ ]
+ ]
+qed.
+
+definition sequent ≝ (list Formula) × (list Formula).
+
+inductive derive: sequent → Prop ≝
+ ExchangeL: ∀l,l1,l2,f. derive 〈f::l1@l2,l〉 → derive 〈l1 @ [f] @ l2,l〉
+ | ExchangeR: ∀l,l1,l2,f. derive 〈l,f::l1@l2〉 → derive 〈l,l1 @ [f] @ l2〉
+ | Axiom: ∀l1,l2,f. derive 〈f::l1, f::l2〉
+ | TrueR: ∀l1,l2. derive 〈l1,FTrue::l2〉
+ | FalseL: ∀l1,l2. derive 〈FFalse::l1,l2〉
+ | AndR: ∀l1,l2,f1,f2.
+ derive 〈l1,f1::l2〉 → derive 〈l1,f2::l2〉 →
+ derive 〈l1,FAnd f1 f2::l2〉
+ | AndL: ∀l1,l2,f1,f2.
+ derive 〈f1 :: f2 :: l1,l2〉 → derive 〈FAnd f1 f2 :: l1,l2〉
+ | OrL: ∀l1,l2,f1,f2.
+ derive 〈f1::l1,l2〉 → derive 〈f2::l1,l2〉 →
+ derive 〈FOr f1 f2 :: l1,l2〉
+ | OrR: ∀l1,l2,f1,f2.
+ derive 〈l1,f1 :: f2 :: l2〉 → derive 〈l1,FOr f1 f2 :: l2〉
+ | NotR: ∀l1,l2,f.
+ derive 〈f::l1,l2〉 → derive 〈l1,FNot f :: l2〉
+ | NotL: ∀l1,l2,f.
+ derive 〈l1,f::l2〉 → derive 〈FNot f :: l1,l2〉.
+
+alias id "Nil" = "cic:/matita/list/list.ind#xpointer(1/1/1)".
+let rec and_of_list l ≝
+ match l with
+ [ Nil ⇒ FTrue
+ | Cons F l' ⇒ FAnd F (and_of_list l')
+ ].
+
+let rec or_of_list l ≝
+ match l with
+ [ Nil ⇒ FFalse
+ | Cons F l' ⇒ FOr F (or_of_list l')
+ ].
+
+definition formula_of_sequent ≝
+ λs.match s with [pair l1 l2 ⇒ FOr (FNot (and_of_list l1)) (or_of_list l2)].
+
+definition is_tautology ≝
+ λF. ∀i. eval i F = true.
+
+axiom assoc_orb: associative ? orb.
+axiom symm_orb: symmetric ? orb.
+axiom orb_not_b_b: ∀b:bool. (¬b ∨ b) = true.
+axiom distributive_orb_andb: distributive ? orb andb.
+axiom symm_andb: symmetric ? andb.
+axiom associative_andb: associative ? andb.
+axiom distributive_andb_orb: distributive ? andb orb.
+
+lemma and_of_list_permut:
+ ∀i,f,l1,l2. eval i (and_of_list (l1 @ (f::l2))) = eval i (and_of_list (f :: l1 @ l2)).
+ intros;
+ elim l1;
+ [ simplify;
+ reflexivity
+ | simplify in H ⊢ %;
+ rewrite > H;
+ autobatch paramodulation
+ ]
+qed.
+
+lemma or_of_list_permut:
+ ∀i,f,l1,l2. eval i (or_of_list (l1 @ (f::l2))) = eval i (or_of_list (f :: l1 @ l2)).
+ intros;
+ elim l1;
+ [ simplify;
+ reflexivity
+ | simplify in H ⊢ %;
+ rewrite > H;
+ autobatch paramodulation
+ ]
+qed.
+
+theorem soundness: ∀F. derive F → is_tautology (formula_of_sequent F).
+ intros;
+ elim H;
+ [ simplify in H2 ⊢ %;
+ intros;
+ lapply (H2 i); clear H2;
+ rewrite > and_of_list_permut;
+ simplify;
+ autobatch
+ | simplify in H2 ⊢ %;
+ intros;
+ lapply (H2 i); clear H2;
+ rewrite > or_of_list_permut;
+ simplify;
+ autobatch
+ | simplify;
+ intro;
+ rewrite > demorgan1;
+ rewrite < assoc_orb;
+ rewrite > assoc_orb in ⊢ (? ? (? % ?) ?);
+ rewrite > symm_orb in ⊢ (? ? (? (? ? %) ?) ?);
+ rewrite < assoc_orb;
+ rewrite > orb_not_b_b;
+ reflexivity
+ | simplify;
+ intros;
+ rewrite > symm_orb;
+ reflexivity
+ | simplify;
+ intros;
+ reflexivity
+ | simplify in H2 H4 ⊢ %;
+ intros;
+ lapply (H2 i); clear H2;
+ lapply (H4 i); clear H4;
+ rewrite > symm_orb in ⊢ (? ? (? ? %) ?);
+ rewrite > distributive_orb_andb;
+ autobatch paramodulation
+ | simplify in H2 ⊢ %;
+ intros;
+ lapply (H2 i); clear H2;
+ autobatch
+ | simplify in H2 H4 ⊢ %;
+ intros;
+ lapply (H2 i); clear H2;
+ lapply (H4 i); clear H4;
+ rewrite > symm_andb;
+ rewrite > distributive_andb_orb;
+ rewrite > demorgan2;
+ rewrite > symm_orb;
+ rewrite > distributive_orb_andb;
+ autobatch paramodulation
+ | simplify in H2 ⊢ %;
+ intros;
+ lapply (H2 i); clear H2;
+ autobatch
+ | simplify in H2 ⊢ %;
+ intros;
+ lapply (H2 i); clear H2;
+ autobatch paramodulation
+ | simplify in H2 ⊢ %;
+ intros;
+ lapply (H2 i); clear H2;
+ autobatch paramodulation
+ ]
+qed.
+
+alias num (instance 0) = "natural number".
+alias symbol "plus" = "natural plus".
+let rec size F ≝
+ match F with
+ [ FTrue ⇒ 0
+ | FFalse ⇒ 0
+ | FAtom n ⇒ 0
+ | FAnd f1 f2 ⇒ S (size f1 + size f2)
+ | FOr f1 f2 ⇒ S (size f1 + size f2)
+ | FNot f ⇒ S (size f)
+ ].
+
+let rec sizel l ≝
+ match l with
+ [ Nil ⇒ 0
+ | Cons F l' ⇒ size F + sizel l'
+ ].
+
+definition size_of_sequent ≝
+ λS.match S with [ pair l r ⇒ sizel l + sizel r].
+
+axiom weakeningR:
+ ∀l1,l2,F. derive 〈l1,l2〉 → derive 〈l1,F::l2〉.
+
+definition same_atom : Formula → Formula → bool.
+ intros;
+ elim f;
+ [3: elim f1;
+ [3: apply (eqb n n1)
+ |*: apply false
+ ]
+ |*: apply false
+ ]
+qed.
+
+definition symmetricb ≝
+ λA:Type.λeq:A → A → bool.
+ ∀x,y. eq x y = eq y x.
+
+theorem symmetricb_eqb: symmetricb ? eqb.
+ intro;
+ elim x;
+ elim y;
+ [1,2,3: reflexivity
+ | simplify;
+ autobatch
+ ]
+qed.
+
+theorem symmetricb_same_atom: symmetricb ? same_atom.
+ intro;
+ elim x;
+ [3:
+ elim y;
+ [3:
+ simplify;
+ apply symmetricb_eqb
+ |*: reflexivity
+ ]
+ |*: elim y; reflexivity
+ ]
+qed.
+
+definition transitiveb ≝
+ λA:Type.λeq:A → A → bool.
+ ∀x,y,z. eq x y = true → eq y z = eq x z.
+
+theorem transitiveb_same_atom: transitiveb ? same_atom.
+ intro;
+ elim x 0;
+ [3:
+ intros 2;
+ elim y 0;
+ [3:
+ intros 3;
+ simplify in H;
+ rewrite > (eqb_true_to_eq ? ? H);
+ reflexivity
+ |1,2:
+ intros;
+ simplify in H;
+ destruct H
+ |4,5:
+ intros;
+ simplify in H2;
+ destruct H2
+ | intros;
+ simplify in H1;
+ destruct H1
+ ]
+ |1,2:
+ intros;
+ simplify in H;
+ destruct H
+ |4,5:
+ intros;
+ simplify in H2;
+ destruct H2
+ | intros;
+ simplify in H1;
+ destruct H1
+ ]
+qed.
+
+theorem eq_to_eq_mem:
+ ∀A.∀eq: A → A → bool.transitiveb ? eq →
+ ∀x,y,l.eq x y = true → mem ? eq x l = mem ? eq y l.
+ intros;
+ elim l;
+ [ reflexivity
+ | simplify;
+ rewrite > (H ? ? ? H1);
+ rewrite > H2;
+ reflexivity
+ ]
+qed.
+
+theorem mem_to_exists_l1_l2:
+ ∀A,eq,n,l. (∀x,y. eq x y = true → x = y) → mem A eq n l = true → ∃l1,l2. l = l1 @ (n :: l2).
+ intros 4;
+ elim l;
+ [ simplify in H1;
+ destruct H1
+ | simplify in H2;
+ apply (bool_elim ? (eq n t));
+ intro;
+ [ apply (ex_intro ? ? []);
+ apply (ex_intro ? ? l1);
+ simplify;
+ rewrite > (H1 ? ? H3);
+ reflexivity
+ | rewrite > H3 in H2;
+ simplify in H2;
+ elim (H H1 H2);
+ elim H4;
+ rewrite > H5;
+ apply (ex_intro ? ? (t::a));
+ apply (ex_intro ? ? a1);
+ simplify;
+ reflexivity
+ ]
+ ]
+qed.
+
+lemma same_atom_to_eq: ∀f1,f2. same_atom f1 f2 = true → f1=f2.
+ intro;
+ elim f1;
+ [1,2:
+ simplify in H;
+ destruct H
+ | generalize in match H; clear H;
+ elim f2;
+ [1,2:
+ simplify in H;
+ destruct H
+ | simplify in H;
+ rewrite > (eqb_true_to_eq ? ? H);
+ reflexivity
+ |4,5:
+ simplify in H2;
+ destruct H2
+ | simplify in H2;
+ destruct H1
+ ]
+ |4,5:
+ simplify in H2;
+ destruct H2
+ |6:
+ simplify in H1;
+ destruct H1
+ ]
+qed.
+
+lemma same_atom_to_exists: ∀f1,f2. same_atom f1 f2 = true → ∃n. f1 = FAtom n.
+ intro;
+ elim f1;
+ [1,2:
+ simplify in H;
+ destruct H
+ | autobatch
+ |4,5:
+ simplify in H2;
+ destruct H2
+ | simplify in H1;
+ destruct H1
+ ]
+qed.
+
+lemma mem_same_atom_to_exists:
+ ∀f,l. mem ? same_atom f l = true → ∃n. f = FAtom n.
+ intros 2;
+ elim l;
+ [ simplify in H;
+ destruct H
+ | simplify in H1;
+ apply (bool_elim ? (same_atom f t));
+ intros;
+ [ elim (same_atom_to_exists ? ? H2);
+ autobatch
+ | rewrite > H2 in H1;
+ simplify in H1;
+ elim (H H1);
+ autobatch
+ ]
+ ]
+qed.
+
+lemma look_for_axiom:
+ ∀l1,l2.
+ (∃n,ll1,ll2,lr1,lr2. l1 = ll1 @ (FAtom n :: ll2) ∧ l2 = lr1 @ (FAtom n :: lr2))
+ ∨ ∀n1. (mem ? same_atom (FAtom n1) l1 ∧ mem ? same_atom (FAtom n1) l2) = false.
+ intro;
+ elim l1 1; clear l1;
+ [ intros;
+ right;
+ intros;
+ simplify;
+ reflexivity
+ | intros;
+ generalize in match (refl_eq ? (mem ? same_atom t l2));
+ elim (mem ? same_atom t l2) in ⊢ (? ? ? %→?);
+ [ left;
+ elim (mem_to_exists_l1_l2 ? ? ? ? same_atom_to_eq H1);
+ elim H2; clear H2;
+ elim (mem_same_atom_to_exists ? ? H1);
+ rewrite > H2 in H3;
+ apply (ex_intro ? ? a2);
+ rewrite > H2;
+ apply (ex_intro ? ? []);
+ simplify;
+ autobatch depth=5
+ | elim (H l2);
+ [ left;
+ decompose;
+ apply (ex_intro ? ? a);
+ apply (ex_intro ? ? (t::a1));
+ simplify;
+ apply (ex_intro ? ? a2);
+ apply (ex_intro ? ? a3);
+ autobatch
+ | right;
+ intro;
+ apply (bool_elim ? (same_atom t (FAtom n1)));
+ [ intro;
+ rewrite > (eq_to_eq_mem ? ? transitiveb_same_atom ? ? ? H3) in H1;
+ rewrite > H1;
+ autobatch
+ | intro;
+ change in ⊢ (? ? (? % ?) ?) with
+ (match same_atom (FAtom n1) t with
+ [true ⇒ true
+ |false ⇒ mem ? same_atom (FAtom n1) l
+ ]);
+ rewrite > symmetricb_same_atom;
+ rewrite > H3;
+ simplify;
+ apply H2
+ ]
+ ]
+ ]
+ ]
+qed.
+
+lemma eq_plus_n_m_O_to_eq_m_O: ∀n,m.n+m=0 → m=0.
+ intros 2;
+ elim n;
+ [ assumption
+ | simplify in H1;
+ destruct H1
+ ]
+qed.
+
+lemma not_eq_nil_append_cons: ∀A.∀l1,l2.∀x:A.¬ [] = l1 @ (x :: l2).
+ intros;
+ elim l1;
+ simplify;
+ intro;
+ [ destruct H
+ | destruct H1
+ ]
+qed.
+
+(*lemma foo: ∀x,l.
+ (¬eval
+ (λn:nat
+ .match eqb n x with
+ [true⇒true|false⇒mem Formula same_atom (FAtom n) l]) (and_of_list l)) =
+ (¬eval
+ (λn:nat.mem Formula same_atom (FAtom n) l) (and_of_list l)).
+ intros;
+ elim l;
+ [ reflexivity
+ | simplify in ⊢ (? ? (? (? ? %)) ?);
+ change in ⊢ (? ? (? %) ?) with
+ (eval (λn:nat
+ .match eqb n x in bool return λb:bool.bool with
+ [true⇒true|false⇒mem Formula same_atom (FAtom n) (t::l1)]) t
+ ∧
+ eval (λn:nat
+ .match eqb n x in bool return λb:bool.bool with
+ [true⇒true|false⇒mem Formula same_atom (FAtom n) (t::l1)])
+ (and_of_list l1));
+
+
+ ]
+qed.*)
+
+axiom daemon: False.
+
+lemma sizel_0_no_axiom_is_tautology:
+ ∀l1,l2. size_of_sequent 〈l1,l2〉 = 0 → is_tautology (formula_of_sequent 〈l1,l2〉) →
+ (∀n. (mem ? same_atom (FAtom n) l1 ∧ mem ? same_atom (FAtom n) l2) = false) →
+ (∃ll1,ll2. l1 = ll1 @ (FFalse :: ll2)) ∨ (∃ll1,ll2. l2 = ll1 @ (FTrue :: ll2)).
+ intros;
+ lapply (H1 (λn.mem ? same_atom (FAtom n) l1)); clear H1;
+ simplify in Hletin;
+ generalize in match Hletin; clear Hletin;
+ generalize in match H2; clear H2;
+ generalize in match H; clear H;
+ elim l1 0;
+ [ intros;
+ simplify in H2;
+ generalize in match H2; clear H2;
+ generalize in match H1; clear H1;
+ generalize in match H; clear H;
+ elim l2 0;
+ [ intros;
+ simplify in H2;
+ destruct H2
+ | simplify;
+ intro;
+ elim t;
+ [ right;
+ apply (ex_intro ? ? []);
+ simplify;
+ autobatch
+ | simplify in H3;
+ simplify in H1;
+ elim H;
+ [ elim H4;
+ elim H5;
+ elim (not_eq_nil_append_cons ? ? ? ? H6)
+ | elim H4;
+ right;
+ apply (ex_intro ? ? (FFalse::a));
+ simplify;
+ elim H5;
+ apply (ex_intro ? ? a1);
+ autobatch
+ |3,4: autobatch
+ | assumption
+ ]
+ | simplify in H1 H3;
+ elim (H H1 H2 H3); clear H;
+ [ elim H4;
+ elim H;
+ elim (not_eq_nil_append_cons ? ? ? ? H5)
+ | right;
+ elim H4;
+ apply (ex_intro ? ? (FAtom n::a));
+ simplify;
+ elim H;
+ autobatch
+ ]
+ |4,5:
+ simplify in H3;
+ destruct H3
+ | simplify in H2;
+ destruct H2
+ ]
+ ]
+ | intro;
+ elim t;
+ [ elim H;
+ [ left;
+ elim H4;
+ apply (ex_intro ? ? (FTrue::a));
+ simplify;
+ elim H5;
+ autobatch
+ | right;
+ assumption
+ | assumption
+ | lapply (H2 n); clear H2;
+ simplify in Hletin;
+ assumption
+ | simplify in H3;
+ assumption
+ ]
+ | left;
+ apply (ex_intro ? ? []);
+ simplify;
+ autobatch
+ | elim H;
+ [ left;
+ elim H4;
+ apply (ex_intro ? ? (FAtom n::a));
+ simplify;
+ elim H5;
+ autobatch
+ | right;
+ assumption
+ | assumption
+ | lapply (H2 n1); clear H2;
+ simplify in Hletin;
+ generalize in match Hletin; clear Hletin;
+ elim (eqb n1 n);
+ [ simplify in H2;
+ rewrite > H2;
+ autobatch
+ | simplify in H2;
+ assumption
+ ]
+ | simplify in H2;
+ lapply (H2 n); clear H2;
+ rewrite > eqb_n_n in Hletin;
+ simplify in Hletin;
+ simplify in H3;
+ rewrite > eqb_n_n in H3;
+ simplify in H3;
+(*
+ generalize in match H3;
+ generalize in match H1; clear H1;
+ generalize in match H; clear H;
+ elim l 0;
+ [ elim l2 0;
+ [ intros;
+ simplify in H2;
+ destruct H2
+ | intros;
+ simplify in H4 ⊢ %;
+ simplify in H;
+ rewrite > H;
+ [ autobatch
+ | intros;
+ apply H1;
+ | simplify in H2;
+ apply (eq_plus_n_m_O_to_eq_m_O ? ? H2)
+ |
+ ]
+
+ [ autobatch
+ | generalize in match H4; clear H4;
+ generalize in match H2; clear H2;
+ elim t1;
+ [
+ |
+ |
+ |4,5:
+ simplify in H5;
+ destruct H5
+ | simplify in H4;
+ destruct H4
+ ]
+ ]
+ ]
+ |
+ ]
+*) elim daemon
+ ]
+ |4,5:
+ simplify in H3;
+ destruct H3
+ | simplify in H2;
+ destruct H2
+ ]
+ ]
+qed.
+
+lemma completeness_base:
+ ∀S. size_of_sequent S = 0 → is_tautology (formula_of_sequent S) → derive S.
+ intro;
+ elim S 1; clear S;
+ simplify in ⊢ (?→%→?);
+ intros;
+ elim (look_for_axiom t t1);
+ [ decompose;
+ rewrite > H2; clear H2;
+ rewrite > H4; clear H4;
+ apply (exchangeL ? a1 a2 (FAtom a));
+ apply (exchangeR ? a3 a4 (FAtom a));
+ apply axiom
+ | elim (sizel_0_no_axiom_is_tautology t t1 H H1 H2);
+ [ decompose;
+ rewrite > H3;
+ apply (exchangeL ? a a1 FFalse);
+ apply falseL
+ | decompose;
+ rewrite > H3;
+ apply (exchangeR ? a a1 FTrue);
+ apply trueR
+ ]
+ ]
+qed.
+
+(*
+lemma completeness_step:
+ ∀l1,l2,n. size_of_sequent 〈l1,l2〉 = S n →
+ (∃ll1,ll2,f. l1 = ll1 @ (f::ll2) ∧ size f > 0) ∨
+ (∃ll1,ll2,f. l2 = ll1 @ (f::ll2) ∧ size f > 0).
+ intros 3;
+ elim l1 0;
+ [ elim l2 0;
+ [ intros;
+ simplify in H;
+ destruct H
+ | intros 3;
+ elim t;
+ [ elim (H H1);
+ [ left;
+ assumption
+ | right;
+ decompose;
+ apply (ex_intro ? ? (FTrue::a));
+ simplify;
+ autobatch depth=5
+ ]
+ | elim (H H1);
+ [ left;
+ assumption
+ | right;
+ decompose;
+ apply (ex_intro ? ? (FFalse::a));
+ simplify;
+ autobatch depth=5
+ ]
+ | elim (H H1);
+ [ left;
+ assumption
+ | right;
+ decompose;
+ apply (ex_intro ? ? (FAtom n1::a));
+ simplify;
+ autobatch depth=5
+ ]
+ | right;
+ apply (ex_intro ? ? []);
+ simplify;
+ apply (ex_intro ? ? l);
+ apply (ex_intro ? ? (FAnd f f1));
+ simplify;
+ split;
+ [ reflexivity
+ | unfold gt;
+ autobatch
+ ]
+ | right;
+ apply (ex_intro ? ? []);
+ simplify;
+ apply (ex_intro ? ? l);
+ apply (ex_intro ? ? (FOr f f1));
+ simplify;
+ split;
+ [ reflexivity
+ | unfold gt;
+ autobatch
+ ]
+ | right;
+ apply (ex_intro ? ? []);
+ simplify;
+ apply (ex_intro ? ? l);
+ apply (ex_intro ? ? (FNot f));
+ simplify;
+ split;
+ [ reflexivity
+ | unfold gt;
+ autobatch
+ ]
+ ]
+ ]
+ | intros 2;
+ elim t;
+ [1,2:(*,2,3:*)
+ elim (H H1);
+ decompose;
+ [ left;
+ autobatch depth=5
+ | right;
+ autobatch depth=5
+ ]
+ | left;
+ apply (ex_intro ? ? []);
+ simplify;
+ apply (ex_intro ? ? l);
+ apply (ex_intro ? ? (FAnd f f1));
+ unfold gt;
+ simplify;
+ autobatch
+ | left;
+ apply (ex_intro ? ? []);
+ simplify;
+ apply (ex_intro ? ? l);
+ apply (ex_intro ? ? (FOr f f1));
+ unfold gt;
+ simplify;
+ autobatch
+ | left;
+ apply (ex_intro ? ? []);
+ simplify;
+ apply (ex_intro ? ? l);
+ apply (ex_intro ? ? (FNot f));
+ unfold gt;
+ simplify;
+ autobatch
+ ]
+ ]
+qed.
+
+theorem completeness: ∀S. is_tautology (formula_of_sequent S) → derive S.
+ intro;
+ generalize in match (refl_eq ? (size_of_sequent S));
+ elim (size_of_sequent S) in ⊢ (? ? ? %→?);
+ [ apply completeness_base;
+ assumption
+ |
+ ]
+qed.
+
+ elim F;
+ [ autobatch
+ | simplify in H;
+ lapply (H (λx.true));
+ destruct Hletin
+ | simplify in H;
+ lapply (H (λx.false));
+ destruct Hletin
+ | apply AndR;
+ [ apply H;
+ intro;
+ lapply (H2 i); clear H2;
+ simplify in Hletin;
+ autobatch
+ | apply H1;
+ intro;
+ lapply (H2 i); clear H2;
+ simplify in Hletin;
+ autobatch
+ ]
+ | apply OrR;
+ simplify in H2;
+ | apply NotR;
+ simplify in H1;
+*)
+*)
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