A Proof Calculus with Case Analysis as its Only Rule

A **Proof** **Calculus****with** **C ase**

Dans un effort louable pour ne p**as** «pensernégativement», certaines personnes répètent desaffirmations positives comme des sortes de mantr**as**qui auraient le pouvoir magique de contrer lesmessages du Critique ou du Juge. Cela ne fait,malheureusement, que recouvrir leur voix.Une des manières de se libérer d’eux est de lesentendre clairement et de comprendre la réelle raisond’être de ces jugements et commentaires négatifs.Page 2Le premier point à comprendre est que ce Critiqueest lié à nos besoins. Prenons Marie. Marie, commetout être humain a des besoins. Elle a besoin desécurité financière, d’être fiable, de se sentir utile etappréciée. Pour satisfaire ces besoins, elle adéveloppé une personnalité de femme responsable,travaillant dur, aimable et sympathique.En même temps, Marie a besoin de se maintenir enforme, d’être en bonne santé, elle a besoin de plaisir,de sensualité, de repos. Elle devient aussiconsciente d’un besoin de vie spirituelle et artistique.C’est-à-dire que certaines parties d’elle ont vraimentbesoin de faire la sieste, d’aller marcher sur la plage,de prendre des bains moussants, de méditer et deprendre des cours de peinture.Marie, comme toutes les personnes normales et enbonne santé est constituée, au niveau psychique,d’une mosaïque de personnes différentes. Chacuned’elles a des besoins particuliers.Lorsque Marie est identifiée à la femme qui aimetravailler dur et atteindre ses buts, elle ne peut p**as**prendre de pause, elle n’en ressent p**as** le besoin.Elle ne désire jamais aller faire de sieste ou marchersur la plage car «ce n’est p**as** de cette façon que jepourrai payer mes factures ou meconstituer une retraite». Lorsquecette part d’elle domine, Marie estcomplètement focalisée sur lebesoin de travailler et de gagner savie.Lorsqu’elle est identifiée à lapersonne responsable, aimable etgentille, prendre du temps pour elleest également impossible car«comment les gens pourront-ils mefaire confiance si je ne suis p**as** làlorsqu’ils ont besoin de moi ?»Tant que Marie est identifiée à cesvoix, elle ne peut trouver du tempspour les parties qui aiment sereposer, se cultiver, ou donner dutemps à la sexualité, à la sensualité,à la partie fragile et vulnérable d’ellemêmeou à celle qui a besoin d’une vie spirituelle.Ces parts d’elles finissent par n’avoir plus aucundroit, aucune place, aucun territoire.Les parties dominantes qui se sont développéespour répondre aux besoin d’amour et de sécuritéfinancière sont extrêmement valables, mais elles nereprésentent p**as** la totalité de son potentiel.Lorsqu’elles envahissent Marie, elle perd le contactavec les besoins -et les ressources- des autres partsd’elle-même.L’entrée du Critique intérieur.Le Critique intérieurarrive lorsqu’il existeune ou plusieursparties dominantesdans la personnalitéd’une personne quiempêchent leursopposés de vivre. Ilsignale un déséquilibre,une identification, uneabsence de consciencede la valeur des partiesopposées.Viens voir par là,j’ai un truc à te dire...Chez Marie, les besoins de détente, de repos, devacances, de temps libre, sont très concrets et réels.Lorsque les personnalités qui peuvent satisfaire cesbesoins prennent le dessus, le Critique estinstantanément là : Marie ne suit plus les règles decomportement des parties dominantes, toutes lesalarmes sonnent. Le Critique intérieur est celui qui semet en alerte quand les besoins des partiesdominantes ne sont plus satisfa**its**.Marie doit travailler, gagner sa vie,être disponible aux autres, elle nedoit p**as** g**as**piller son temps, ni êtreégoïste.Mais le Critique est aussi sensible aumanque d’équilibre. Lorsque Marie nefait que travailler, gagner sa vie,répondre aux besoins des autres, iln’est p**as** satisfait non plus. Il saitqu’elle est en déséquilibre et ildégaine : «Tout le monde a une vieprivée sauf toi, quelque chose ne vap**as** chez toi !» «Tu v**as** devenirexactement comme ton père, uneaccro au travail.» Finalement quoi queMarie f**as**se, son Critique la fustige.Lorsque l’identification à une partie esttotale, le Critique pèse un tonne.Association Voice Dialogue Sud

A **Proof** **Calculus** **with** **C ase**

4 Ján Kľuka and Paul J. VodaDefinition: ∀x ∀y (f (x) = y ↔ · · · ).Lemma n: ∀x (· · · f · · · ) — proof· · · q.e.d..Theorem m: ∀x ∀y (· · · f · · · ) — proof· · ·Definition: ∀x ∀y (g(x) = y ↔ · · · )Lemma i: ∀x (· · · g · · · f · · · )— proof· · · q.e.d.c**as**e analysis on · · · ∨ · · · :1. · · ·use of Lemma n· · ·2. use of Lemma i· · · q.e.d.Fig. 1. A schema of development of proofs by a proof **as**sistant in extensions of theories.3 Formul**as**3.1 Formul**as** and sentences. We use the standard notion of countable languagesfor first-order logic (denoted by L) **with** predicate symbols (R, . . . ) andfunction symbols (f , g, . . . ). We denote symbols of any kind by F , G, and setsof symbols by F. We denote by L[F] the language L extended **with** the functionand predicate symbols F. We will abbreviate e 1 , . . . , e n to ⃗e if n is known fromthe context of irrelevant.Terms (t, s, . . . ) are built up from variables (x, y, z, . . . ) and applications offunction symbols (including constants). Atomic formul**as** are applications R(⃗s)or equalities s = t. Formul**as** are the le**as**t cl**as**s containing atomic formul**as**,⊤, ⊥, and closed under existential quantification ∃x A and if-then-else constructIf(A, B, C) **with** A, B, C formul**as**. Prime formul**as** (designated by P ) are atomicor existentially quantified formul**as**.We will usually display If(A, B, C), whose intended interpretation is (A ∧B) ∨ (¬A ∧ C), **as**AB C .We call A the guard.We use the standard notions of free and bound variables. We use A[x/t]to designate the substitution of a closed term t for all free occurrences of thevariable x in A. A sentence is a closed formula, i.e., a formula **with** no freevariables.

6 Ján Kľuka and Paul J. Voda3.4 Finite sequences of sentences. We will use a computer science list notationfor finite sequences of sentences. Empty sequence is 〈〉 and the prefixing of **as**equence Π **with** a sentence A will be designated by the pairing function 〈A|Π〉.Pairing is such that 〈〉 ≠ 〈A|Π〉. We use abbreviations 〈A, B |Π〉 := 〈A|〈B | Π〉〉and 〈A 1 , . . . , A n 〉 := 〈A 1 , . . . , A n | 〈〉〉. Concatenation ⊕ of sequences is defined**as** 〈〉 ⊕ Φ := Φ and 〈A | Π〉 ⊕ Φ := 〈A | Π ⊕ Φ〉. All declarative programminglanguages use lists in such a form. The presented notation is PROLOG’s but wehave replaced **its** square brackets **with** the angle ones.The sequence Π ′ extends Π if Π ′ = Π ⊕ Φ for some Φ. We defineSet(Π) = {A | Π = Π 1 ⊕ 〈A | Π 2 〉 for some Π 1 , Π 2 }.We adopt a convention that an occurrence of a sequence Π in a position wherea set is expected stands for Set(Π). Thus, for instance A ∈ Π stands for A ∈Set(Π) and T, Π Π ′ for T, Set(Π) Set(Π ′ ).Note that Set(Π ⊕Π ′ ) = Set(Π)∪Set(Π ′ ) and if Π ′ extends Π then Π ⊆ Π ′ .3.5 Paths through sentences. We define a partial function (A) Π selectinga subsentence of A at Π to be defined only if it satisfies:( P)(A) 〈〉 :≡ A (P ) 〈A〉 :≡⊤ ⊥( A)( A):≡ (B) Π :≡ (C) Π .B C 〈A|Π〉B C 〈¬A|Π〉Note that we have (A) Π1⊕Π 2≡ ((A) Π1 ) Π2 if defined.The set of all paths in A is defined **as** Ps(A) := {Π | (A) Π is defined}. Wesay that Π is a path through A if Π selects ⊤ or ⊥ in A. We say that A h**as** ahole at Π if Π selects ⊥ in A. A sentence A is syntactically valid if it h**as** noholes, i.e., if all paths through A select ⊤.We define a ternary partial replacement function A[Π := D] to yield sentenceslike A but **with** the subsentence selected by Π (if any) replaced by D.The function is defined only when it satisfies:3.6 Lemma.( PA[〈〉 := D] :≡ D P [〈A〉 := D] :≡⊤ ⊥( A)A[〈A | Π〉 := D] :≡B CB[Π := D]( A)[〈¬A | Π〉 := D] :≡BC)[〈A〉 := D]CAB C[Π := D] .(i) For every sentence A ∈ L and a structure M for L, there is a uniquepath Π through A such that M Π.(ii) Π A ↔ (A) Π for every Π ∈ Ps(A).〈A〉

A **Proof** **Calculus** **with** **C ase**

10 Ján Kľuka and Paul J. Vodaof Henkin axioms is regular for C.Construction of the witnessing expansion is given, e.g., in [Bar77].5.2 Hintikka sets. A theory H in L simply consistent if for each sentenceA ∈ H we have (¬A) /∈ H. A simply consistent theory H is maximal if it h**as** noproper simply consistent extension in L, i.e., if for all A ∈ L either A or ¬A isin H.Let L[C] be the witnessing expansion of L. A theory H in L[C] is called aHintikka set if the following conditions are satisfied:(i) ⊤ ∈ H, and (t = t) ∈ H for all terms t in L[C].(ii) If (s = t) ∈ H and P [x/s] ∈ H for atomic P , then P [x/t] ∈ H.(iii) IfA∈ H, then A ∈ H and B ∈ H, or (¬A) ∈ H and C ∈ H.B C(iv) If ∃x A ∈ H, then A[x/c ∃xA ] ∈ H.(v) If ∀x A ∈ H, then A[x/t] ∈ H for all terms t in L[C].5.3 Lemma. Every maximal simply consistent Hintikka set h**as** a model.**Proof**. For a similar construction see, e.g., [Bar77,Sho67].⊓⊔5.4 Brute-force proof construction. Let A be a sentence, T a theory, bothin L, and He be **as** in Par. 5.1. To any pair of sentences D, B in L[C] we define **as**entence BF(D, B) which extends D **with** the sentenceBat **its** syntactically⊥ ⊥consistent holes:BF(D, B) :≡ D[Π 1 := E 1 ] · · · [Π n := E n ]where Π 1 , . . . , Π n (n ≥ 0) are paths to all holes in D, and for j = 1, . . . , n{⊤ if Π i is syntactically inconsistent **with** T and He,E j :≡Botherwise.⊥ ⊥For an enumeration {B i } ∞ i=0 of all sentences in L[C] we define a sequence ofsentences {D i } ∞ i=0 by D 0 :≡ A and D i+1 :≡ BF(D i , B i ).5.5 Theorem (Completeness). If T L A, then T ⊢ L A.**Proof**. Assume T L A, construct the sequence {D i } ∞ i=0 **as** in Par. 5.4, and letΠ 1 , . . . , Π n be paths leading to all holes in A. A straightforward induction on iproves:For each i > 0 and each path Π through D i extending some Π j , we have(D i ) Π ≡ ⊤ iff Π is syntactically inconsistent **with** T and He.We consider two c**as**es. If D i ≡ D i+1 for some i, then D i h**as** no holes, and sothe sentences (D i ) Π1 , . . . , (D i ) Πn are syntactically valid. Since by the aboveproperty, each (D i ) Πj plugs the hole in A at Π j w.r.t. T and He, we haveD i ⊢⃗ F T L A.

A **Proof** **Calculus** **with** **C ase**

12 Ján Kľuka and Paul J. Vodaof ¬C. By (†), we must have (¬C) /∈ H. Hence C ∈ H by the maximality of H.The c**as**e when (¬B) ∈ H, which implies D ∈ H, is similar.(iv) If ∃x B ∈ H then, since 〈∃x B, ¬B[x/c ∃xB ]〉 ⊆ H would violate (†)(4),we must have B[x/c ∃xB ] ∈ H by the maximality of H.(v) For any t, if ∀x B ≡ ¬∃x ¬B ∈ H, then, since 〈¬B[x/t], ¬∃x ¬B〉 ⊆ Hwould violate (†)(1), we must have B[x/t] ∈ H by the maximality of H. ⊓⊔6 Methods of **Proof** ConstructionWe have introduced a new proof calculus and demonstrated that **its** conceptof provability coincides, **as** it should, **with** logical consequence. It remains toconvince the reader that the calculus offers advantages over the standard proofsystems. In this section we will show that• we can construct proofs incrementally by applying local proof rules,• the rules are flexible enough to emulate Hilbert, **as** well **as** Gentzen style ofproofs,• we can, because of definition hiding, transform proofs **with** flexibility notafforded by the l**as**t two types of calculi.6.1 Derivation rules. For the duration of this section we fix a (possibly empty)theory T in some L. A derivation rule (R) is of the formΓ(R) Dwhere Γ is a finite set of sentences (possibly empty) in L, called premises, andthe sentence D in L[ ⃗ F ] is the conclusion. The rule is sound if the following isconstructively proved:for all paths Π such that Γ ⊆ Π, and every A in L[ ⃗ F ] **with** a hole at Πand such that T ⊢ L[ ⃗ F ] A[Π := D], we have T ⊢ L A.For the soundness of the rule (R) it suffices that D is such that every path Π ′selecting ⊤ in D (if any) leads to inconsistency: T, S, Γ, Π ′ ⊢ L[ ⃗ F ] ⊥ for some S.This is an e**as**y consequence of the definition of proofs. The simplest proof of thel**as**t condition is ⊤ when Π ⊕ Π ′ is syntactically inconsistent **with** T and S.6.2 The c**as**e analysis (cut) method. The c**as**e analysis (cut) method ischaracterized by two rules:(Close)(Cut)Γ⊤A⊥ ⊥if Γ is synt. inconsistent **with** T and some SSoundness of the two rules follows from the sufficient condition in Par. 6.1.

A **Proof** **Calculus** **with** **C ase**

14 Ján Kľuka and Paul J. Vodato (Close). However, one can e**as**ily see thatB ⊤` A ´B C¬B ⊤ ⊢A ⊤⊤A , A, ¬B B C L ⊥where B ⊤ is just like B **with** all holes replaced by ⊤. There is an similar witnessofA, ¬A, ¬C ⊢ B C L ⊥. This is by Par. 6.1 sufficient for soundness of the rule.The sequent rules are (Ax), (If), (Th), (L∀), (L∃), (Id), (Subst).We now present a translation of a proof D of T ⊢ A **with** all inferencesin D by sequent rules to a one-sided sequent calculus **with** sequents Γ ⇒ . Suchsequents are dual to the well-known Schütte-Tait one-sided sequents ⇒ ∆ in thecalculus called GS in [TS00]. Assume that a sequent rule Γ h**as** been appliedCat a path Π ∈ Ps(D) leading through a hole of A. We translate it to the sequentcalculus inferenceΠ 1 , Γ, Π ⇒ . . . Π n , Γ, Π ⇒Γ, Π ⇒where Π 1 , . . . , Π n are all paths to holes in C. For instance, sequent calculusinferences corresponding to (Ax), (If), and (L∀) are respectively:A, B, Π ⇒ ¬A, C, Π ⇒ A[x/t], Π ⇒(Ax) (If)P, ¬P, Π ⇒ A, Π ⇒ (L∀)∀x A, Π ⇒ .B C6.5 Considerations of completeness. In this section we have concentratedon the soundness of derivation rules. We know that the cut method is alsocomplete, because we have proved the completeness theorem 5.5 by **its** two rules.Incidentally, we have proved more than completeness, we have also semanticallydemonstrated the conservativity of extensions by definitions because we neverhad to use other extension axioms than Henkin’s. It can be shown that also theunboxing and sequent methods are complete in this sense.When it comes to proving sentencess A in L[ ⃗ F ] by definition hiding, wherethe extension axioms defining the symbols ⃗ F are local to the proofs, then wecannot hope for the completeness. This is because A can express a property of ⃗ Fnot satisfiable by any extensions by definitions.6.6 Inversion. The rule (L∀) in the one-sided sequent calculus is not invertible.This means that given a proof D of the conclusion we cannot in generaleliminate all **its** uses **with** terms t 1 , . . . , t n in D and obtain a proof ofA[x/t 1 ], . . . , A[x/t n ], Γ ⇒. This is because the terms t i may contain eigenvariablesintroduced by (L∃) in D.In our calculus we have weakened the eigen-variable conditions to the requirementof regularity of extension axioms. This allows arbitrary permutationswhere we can, for instance, replaceAD 1 BBA A .D 1 D 2 D 1 D 3anywhere in D byD 2 D 3Thus, given a proof of ∃x A, we can permute all applications of (L∀) to ∀x ¬Ain the proof to obtain a proof of ∨ i A[x/t i]. Note that this is an example of

A **Proof** **Calculus** **with** **C ase**