Free Interpretation, Quotient Interpretation and Substitution of a Letter with a Term for First Order Languages

Free Interpretation, Quotient Interpretation and Substitution of a Letter with a Term for First Order Languages Fourth of a series of articles laying down the bases for classical first order model theory. This paper supplies a toolkit of constructions to work with languages and interpretations, and results relating them. The free interpretation of a language, having as a universe the set of terms of the language itself, is defined. The quotient of an interpreteation with respect to an equivalence relation is built, and shown to remain an interpretation when the relation respects it. Both the concepts of quotient and of respecting relation are defined in broadest terms, with respect to objects as general as possible. Along with the trivial symbol substitution generally defined in [11], the more complex substitution of a letter with a term is defined, basing right on the free interpretation just introduced, which is a novel approach, to the author's knowledge. A first important result shown is that the quotient operation commute in some sense with term evaluation and reassignment functors, both introduced in [13] (theorem 3, theorem 15). A second result proved is substitution lemma (theorem 10, corresponding to III.8.3 of [15]). This will be vital for proving satisfiability theorem and correctness of a certain sequent derivation rule in [14]. A third result supplied is that if two given languages coincide on the letters of a given FinSequence, their evaluation of it will also coincide. This too will be instrumental in [14] for proving correctness of another rule. Also, the Depth functor is shown to be invariant with respect to term substitution in a formula.

Summary.Fourth of a series of articles laying down the bases for classical first order model theory.This paper supplies a toolkit of constructions to work with languages and interpretations, and results relating them.The free interpretation of a language, having as a universe the set of terms of the language itself, is defined.
The quotient of an interpreteation with respect to an equivalence relation is built, and shown to remain an interpretation when the relation respects it.Both the concepts of quotient and of respecting relation are defined in broadest terms, with respect to objects as general as possible.
Along with the trivial symbol substitution generally defined in [11], the more complex substitution of a letter with a term is defined, basing right on the free interpretation just introduced, which is a novel approach, to the author's knowledge.A first important result shown is that the quotient operation commute in some sense with term evaluation and reassignment functors, both introduced in [13] (theorem 3, theorem 15).A second result proved is substitution lemma (theorem 10, corresponding to III.8.3 of [15]).This will be vital for proving satisfiability theorem and correctness of a certain sequent derivation rule in [14].A third result supplied is that if two given languages coincide on the letters of a given FinSequence, their evaluation of it will also coincide.This too will be instrumental in [14] for proving correctness of another rule.Also, the Depth functor is shown to be invariant with respect to term substitution in a formula.
MML identifier: FOMODEL3, version: 7.11.074.160.1126 1 The author wrote this paper as part of his PhD thesis research.Let us consider S and let s be a termal element of S.
Observe that s-compound (AllTermsOf S) |ar s| is AllTermsOf S-valued.Let us consider S and let s be a relational element of S. Note that s-compound (AllTermsOf S) |ar s| is AtomicFormulasOf S-valued.Let us consider S, let s be an of-atomic-formula element of S, and let X be a set.The functor X-freeInterpreter s is defined as follows: ( χ X,AtomicFormulasOf S qua binary relation), otherwise.Let us consider S, let s be an of-atomic-formula element of S, and let X be a set.Then X-freeInterpreter s is an interpreter of s and AllTermsOf S.
Let us consider S, X.The functor (S, X)-freeInterpreter yields a function and is defined as follows: (Def. 4) dom((S, X)-freeInterpreter) = OwnSymbolsOf S and for every own element s of S holds (S, X)-freeInterpreter(s) = X-freeInterpreter s.
Let us consider S, X.Note that (S, X)-freeInterpreter is function yielding.Let us consider S, X.Then (S, X)-freeInterpreter is an interpreter of S and AllTermsOf S.
Let us consider S, X.Note that (S, X)-freeInterpreter is (S, AllTermsOf S)interpreter-like.
Then (S, X)-freeInterpreter is an element of AllTermsOf S-InterpretersOf S.
Let X, Y be non empty sets, let R be a relation between X and Y , and let n be a natural number.The functor n-placesOf R yielding a relation between X n and Y n is defined as follows: Let X, Y be non empty sets, let R be a total relation between X and Y , and let n be a non zero natural number.Observe that n-placesOf R is total.
Let X, Y be non empty sets, let R be a total relation between X and Y , and let n be a natural number.Observe that n-placesOf R is total.
Let X, Y be non empty sets, let R be a relation between X and Y , and let n be a zero natural number.One can check that n-placesOf R is function-like.
Let X be a non empty set, let R be a binary relation on X, and let n be a natural number.The functor n-placesOf R yielding a binary relation on X n is defined by: (Def.6) n-placesOf R = n-placesOf(R qua relation between X and X).
Let X be a non empty set, let R be a binary relation on X, and let n be a zero natural number.Then n-placesOf R is a binary relation on X n and it can be characterized by the condition: Let X be a non empty set, let R be a symmetric total binary relation on X, and let us consider n.One can check that n-placesOf R is total.
Let X be a non empty set, let R be a symmetric total binary relation on X, and let us consider n.Observe that n-placesOf R is symmetric.
Let X be a non empty set, let R be a symmetric total binary relation on X, and let us consider n.Observe that n-placesOf R is symmetric and total.
Let X be a non empty set, let R be a transitive total binary relation on X, and let us consider n.Observe that n-placesOf R is transitive and total.
Let X be a non empty set, let R be an equivalence relation of X, and let us consider n.Observe that n-placesOf R is total, symmetric, and transitive.
Let X, Y be non empty sets, let E be an equivalence relation of X, let F be an equivalence relation of Y , and let R be a binary relation.The functor R quotient(E, F ) is defined by: (Def.8) R quotient(E, F ) = { e, f ; e ranges over elements of Classes E, f ranges over elements of Classes F : x,y : set Let X, Y be non empty sets, let E be an equivalence relation of X, let F be an equivalence relation of Y , and let R be a binary relation.Then R quotient(E, F ) is a relation between Classes E and Classes F.
Let E be a binary relation, let F be a binary relation, and let f be a function.We say that f is (E, F )-respecting if and only if: (Def.9) For all sets Let us consider S, U , let s be an of-atomic-formula element of S, let E be a binary relation on U , and let f be an interpreter of s and U .We say that f is E-respecting if and only if: Let X, Y be non empty sets, let E be an equivalence relation of X, and let F be an equivalence relation of Y .Observe that there exists a function from X into Y which is (E, F )-respecting.
Let us consider S, U , let s be an of-atomic-formula element of S, and let E be an equivalence relation of U .Note that there exists an interpreter of s and U which is E-respecting.
Let X, Y be non empty sets, let E be an equivalence relation of X, and let F be an equivalence relation of Y .One can verify that there exists a function which is (E, F )-respecting.
Let X be a non empty set, let E be an equivalence relation of X, and let us consider n.Then n-placesOf E is an equivalence relation of X n .
Let X be a non empty set and let x be an element of SmallestPartition(X).The functor DeTrivial x yielding an element of X is defined as follows: Let X be a non empty set.The functor peeler X yielding a function from {{ * } : * ∈ X} into X is defined as follows: (Def.12) For every element x of {{ * } : * ∈ X} holds (peeler X)(x) = DeTrivial x.
Let X be a non empty set and let E 1 be an equivalence relation of X.Note that every element of Classes E 1 is non empty.
Let X, Y be non empty sets, let E be an equivalence relation of X, let F be an equivalence relation of Y , and let f be an (E, F )-respecting function.One can check that f quotient(E, F ) is function-like.
Let X, Y be non empty sets, let E be an equivalence relation of X, let F be an equivalence relation of Y , and let R be a total relation between X and Y .One can check that R quotient(E, F ) is total.
Let X, Y be non empty sets, let E be an equivalence relation of X, let F be an equivalence relation of Y , and let f be an (E, F )-respecting function from X into Y .Then f quotient(E, F ) is a function from Classes E into Classes F.
Let X be a non empty set and let E be an equivalence relation of X.The functor E-class yields a function from X into Classes E and is defined by: (Def.13) For every element x of X holds E-class(x) = EqClass(E, x).
Let X be a non empty set and let E be an equivalence relation of X. Observe that E-class is onto.
Let X, Y be non empty sets.Note that there exists a relation between X and Y which is onto.
Let Y be a non empty set.Observe that there exists a Y -valued binary relation which is onto.
Let Y be a non empty set and let R be a Y -valued binary relation.Note that R is Y -defined.
Let Y be a non empty set and let R be an onto Y -valued binary relation.Note that R is total.
Let X, Y be non empty sets and let R be an onto relation between X and Y .One can check that R is total.
Let Y be a non empty set and let R be an onto Y -valued binary relation.Note that R is total.
Let us consider U , n and let E be an equivalence relation of U .The functor n -tuple2Class E yields a relation between (Classes E) n and Classes(n-placesOf E) and is defined as follows: Let us consider U , n and let E be an equivalence relation of U .Note that n -tuple2Class E is total.
Let us consider U , n and let E be an equivalence relation of U .Then n -tuple2Class E is a function from (Classes E) n into Classes(n-placesOf E).
Let us consider S, U , let s be an of-atomic-formula element of S, let E be an equivalence relation of U , and let f be an interpreter of s and U .The functor f quotient E is defined by: peeler Boolean, otherwise.Let us consider S, U , let s be an of-atomic-formula element of S, let E be an equivalence relation of U , and let f be an E-respecting interpreter of s and U .Then f quotient E is an interpreter of s and Classes E.

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would like to thank Marco Pedicini for his encouragement and support.

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Def. 14) n -tuple2Class E = (n-placesOf(E-class qua relation between U and Classes E) ) • (n-placesOf E)-class .Let us consider U , n and let E be an equivalence relation of U .Observe that n -tuple2Class E is function-like.

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Suppose TheNorSymbOf S 1 = TheNorSymbOf S 2 and TheEqSymbOf S 1 = TheEqSymbOf S 2 and (the adicity of S 1 ) OwnSymbolsOf S 1 = (the adicity of S 2 ) OwnSymbolsOf S 1 .Let I 1 be an element of U -InterpretersOf S 1 , I 2 be an element of U -InterpretersOf S 2 , and p 4 be a w.f.f.string of S 1 .Suppose I 1 OwnSymbolsOf S 1 = I 2 OwnSymbolsOf S 1 .Then there exists a w.f.f.string p 3 of S 2 such that p 3 = p 4 and I 2 -TruthEval p 3 = I 1 -TruthEval p 4 .(13) For all elements I 1 , I 2 of U -InterpretersOf S such that I 1 (rng p 2 ∩ OwnSymbolsOf S) = I 2 (rng p 2 ∩OwnSymbolsOf S) holds I 1 -TruthEval p 2 = I 2 -TruthEval p 2 .(14) For every element I of U -InterpretersOf S such that l is X-absent and X is I-satisfied holds X is (l, u) ReassignIn I-satisfied.(15) For every equivalence relation E of U and for every E-respecting element i of U -InterpretersOf S holds (l, E-class(u)) ReassignIn(i quotient E) = ((l, u) ReassignIn i) quotient E.