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(* ========================================================================= *)
(* DERIVED RULES FOR CREATING FIRST ORDER LOGIC THEOREMS *)
(* Copyright (c) 2001 Joe Leslie-Hurd, distributed under the BSD License *)
(* ========================================================================= *)
signature Rule =
sig
(* ------------------------------------------------------------------------- *)
(* An equation consists of two terms (t,u) plus a theorem (stronger than) *)
(* t = u \/ C. *)
(* ------------------------------------------------------------------------- *)
type equation = (Term.term * Term.term) * Thm.thm
val ppEquation : equation Print.pp
val equationToString : equation -> string
(* Returns t = u if the equation theorem contains this literal *)
val equationLiteral : equation -> Literal.literal option
val reflEqn : Term.term -> equation
val symEqn : equation -> equation
val transEqn : equation -> equation -> equation
(* ------------------------------------------------------------------------- *)
(* A conversion takes a term t and either: *)
(* 1. Returns a term u together with a theorem (stronger than) t = u \/ C. *)
(* 2. Raises an Error exception. *)
(* ------------------------------------------------------------------------- *)
type conv = Term.term -> Term.term * Thm.thm
val allConv : conv
val noConv : conv
val thenConv : conv -> conv -> conv
val orelseConv : conv -> conv -> conv
val tryConv : conv -> conv
val repeatConv : conv -> conv
val firstConv : conv list -> conv
val everyConv : conv list -> conv
val rewrConv : equation -> Term.path -> conv
val pathConv : conv -> Term.path -> conv
val subtermConv : conv -> int -> conv
val subtermsConv : conv -> conv (* All function arguments *)
(* ------------------------------------------------------------------------- *)
(* Applying a conversion to every subterm, with some traversal strategy. *)
(* ------------------------------------------------------------------------- *)
val bottomUpConv : conv -> conv
val topDownConv : conv -> conv
val repeatTopDownConv : conv -> conv (* useful for rewriting *)
(* ------------------------------------------------------------------------- *)
(* A literule (bad pun) takes a literal L and either: *)
(* 1. Returns a literal L' with a theorem (stronger than) ~L \/ L' \/ C. *)
(* 2. Raises an Error exception. *)
(* ------------------------------------------------------------------------- *)
type literule = Literal.literal -> Literal.literal * Thm.thm
val allLiterule : literule
val noLiterule : literule
val thenLiterule : literule -> literule -> literule
val orelseLiterule : literule -> literule -> literule
val tryLiterule : literule -> literule
val repeatLiterule : literule -> literule
val firstLiterule : literule list -> literule
val everyLiterule : literule list -> literule
val rewrLiterule : equation -> Term.path -> literule
val pathLiterule : conv -> Term.path -> literule
val argumentLiterule : conv -> int -> literule
val allArgumentsLiterule : conv -> literule
(* ------------------------------------------------------------------------- *)
(* A rule takes one theorem and either deduces another or raises an Error *)
(* exception. *)
(* ------------------------------------------------------------------------- *)
type rule = Thm.thm -> Thm.thm
val allRule : rule
val noRule : rule
val thenRule : rule -> rule -> rule
val orelseRule : rule -> rule -> rule
val tryRule : rule -> rule
val changedRule : rule -> rule
val repeatRule : rule -> rule
val firstRule : rule list -> rule
val everyRule : rule list -> rule
val literalRule : literule -> Literal.literal -> rule
val rewrRule : equation -> Literal.literal -> Term.path -> rule
val pathRule : conv -> Literal.literal -> Term.path -> rule
val literalsRule : literule -> LiteralSet.set -> rule
val allLiteralsRule : literule -> rule
val convRule : conv -> rule (* All arguments of all literals *)
(* ------------------------------------------------------------------------- *)
(* *)
(* --------- reflexivity *)
(* x = x *)
(* ------------------------------------------------------------------------- *)
val reflexivityRule : Term.term -> Thm.thm
val reflexivity : Thm.thm
(* ------------------------------------------------------------------------- *)
(* *)
(* --------------------- symmetry *)
(* ~(x = y) \/ y = x *)
(* ------------------------------------------------------------------------- *)
val symmetryRule : Term.term -> Term.term -> Thm.thm
val symmetry : Thm.thm
(* ------------------------------------------------------------------------- *)
(* *)
(* --------------------------------- transitivity *)
(* ~(x = y) \/ ~(y = z) \/ x = z *)
(* ------------------------------------------------------------------------- *)
val transitivity : Thm.thm
(* ------------------------------------------------------------------------- *)
(* *)
(* ---------------------------------------------- functionCongruence (f,n) *)
(* ~(x0 = y0) \/ ... \/ ~(x{n-1} = y{n-1}) \/ *)
(* f x0 ... x{n-1} = f y0 ... y{n-1} *)
(* ------------------------------------------------------------------------- *)
val functionCongruence : Term.function -> Thm.thm
(* ------------------------------------------------------------------------- *)
(* *)
(* ---------------------------------------------- relationCongruence (R,n) *)
(* ~(x0 = y0) \/ ... \/ ~(x{n-1} = y{n-1}) \/ *)
(* ~R x0 ... x{n-1} \/ R y0 ... y{n-1} *)
(* ------------------------------------------------------------------------- *)
val relationCongruence : Atom.relation -> Thm.thm
(* ------------------------------------------------------------------------- *)
(* x = y \/ C *)
(* -------------- symEq (x = y) *)
(* y = x \/ C *)
(* ------------------------------------------------------------------------- *)
val symEq : Literal.literal -> rule
(* ------------------------------------------------------------------------- *)
(* ~(x = y) \/ C *)
(* ----------------- symNeq ~(x = y) *)
(* ~(y = x) \/ C *)
(* ------------------------------------------------------------------------- *)
val symNeq : Literal.literal -> rule
(* ------------------------------------------------------------------------- *)
(* sym (x = y) = symEq (x = y) /\ sym ~(x = y) = symNeq ~(x = y) *)
(* ------------------------------------------------------------------------- *)
val sym : Literal.literal -> rule
(* ------------------------------------------------------------------------- *)
(* ~(x = x) \/ C *)
(* ----------------- removeIrrefl *)
(* C *)
(* *)
(* where all irreflexive equalities. *)
(* ------------------------------------------------------------------------- *)
val removeIrrefl : rule
(* ------------------------------------------------------------------------- *)
(* x = y \/ y = x \/ C *)
(* ----------------------- removeSym *)
(* x = y \/ C *)
(* *)
(* where all duplicate copies of equalities and disequalities are removed. *)
(* ------------------------------------------------------------------------- *)
val removeSym : rule
(* ------------------------------------------------------------------------- *)
(* ~(v = t) \/ C *)
(* ----------------- expandAbbrevs *)
(* C[t/v] *)
(* *)
(* where t must not contain any occurrence of the variable v. *)
(* ------------------------------------------------------------------------- *)
val expandAbbrevs : rule
(* ------------------------------------------------------------------------- *)
(* simplify = isTautology + expandAbbrevs + removeSym *)
(* ------------------------------------------------------------------------- *)
val simplify : Thm.thm -> Thm.thm option
(* ------------------------------------------------------------------------- *)
(* C *)
(* -------- freshVars *)
(* C[s] *)
(* *)
(* where s is a renaming substitution chosen so that all of the variables in *)
(* C are replaced by fresh variables. *)
(* ------------------------------------------------------------------------- *)
val freshVars : rule
(* ------------------------------------------------------------------------- *)
(* C *)
(* ---------------------------- factor *)
(* C_s_1, C_s_2, ..., C_s_n *)
(* *)
(* where each s_i is a substitution that factors C, meaning that the theorem *)
(* *)
(* C_s_i = (removeIrrefl o removeSym o Thm.subst s_i) C *)
(* *)
(* has fewer literals than C. *)
(* *)
(* Also, if s is any substitution that factors C, then one of the s_i will *)
(* result in a theorem C_s_i that strictly subsumes the theorem C_s. *)
(* ------------------------------------------------------------------------- *)
val factor' : Thm.clause -> Subst.subst list
val factor : Thm.thm -> Thm.thm list
end
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