Higher Order Termination: Automatable Techniques for Proving Termination of Higher-Order Term Rewriting Systems

ion, 13 Abstraction-simple, 182 Accessibility Relation, 100 AFS, 55 AFSM, 12, 16 Algebraic Functional System, 55 Algebraic Functional System with Metavariables, 12, 16 Alpha-conversion, 11 Application, 11, 13 Application-free, 45 Applicative Syntax, 18 Approximation, 226 Argument Filtering, 202 Argument Function, 135, 202 Argument Preserving Function, 135 Arity, 10, 11, 13 Arrow Decreasingness, 100 Arrow Preservation, 100 Base Type, 11 Beta Rule, 12 Beta-first, 17 Beta-normal Form, 12 Beta-redex, 12 Beta-reduced Sub-meta-term, 155 Beta-saturating, 153 Binding, 11 Bit, 274 BRSMT, 155 C , 232 Cand, 155 Candidate Term, 145, 155 Chain-free, 148, 164 Clause, 269 Closed, 11 CNF, 269 Collapsing Dependency Pair, 157 Collapsing Rule, 152 Collapsing Set, 165 Combinatory Reduction System, 41, 71 Combinatory Reduction Systems with Extensions, 64 Compatible, 30 Complete Processor, 207, 215 Computability Path Ordering, 100 Computable, 113 Conjunctive Normal Form, 269 Constructor Symbol, 153 Contains Beta, 30 Context, 10, 11, 14 Contraction Scheme, 41, 68 CPO, 100 CRS, 71 CRSX, 64 CS, 68 cur, 19 Currying, 19 Cycle, 225 Dangling Variable, 152 Default Requirements, 197 Defined Symbol, 145, 153 Dependency Chain, 146, 158 Dependency Graph, 209, 225 Dependency Graph Approximation, 226 Dependency Graph Processor, 210, 227 Dependency Pair, 146 Dependency Pair Problem, 207, 214 Dependency Pair Processor, 207, 215 Domain, 10 DP, 146, 156 DP Problem, 214 Dynamic Dependency Pair, 157 Empty Set Processor, 208, 216 Essentially First-order Dependency Pair, 237 Essentially First-order Meta-term, 237 Essentially First-order Rule, 237 Eta-expansion, 12, 17 Eta-long Beta-normal Form, 12 Eta-long Form, 12 exp, 60 expL, 62 Expression Reduction System, 41 Extended Monotonic Algebra, 85 Extended Meta-application Processor, 220 Finite Dependency Pair Problem, 214 Finitely Branching, 231 First-order Dependency Chain, 146 First-order Dependency Pair, 146 First-order Dependency Pair Problem, 207 First-order Rewrite Rule, 10 First-order Term Rewriting System, 10 flat, 27 FMV, 13 Formative Chain-free, 177 Formative Dependency Chain, 176 Formative Rules, 173 Formative Rules Processor, 217 Formative Symbols, 173 FR, 173 Free Of Abstractions, 180 Free Variable, 11 Function Symbol, 10 Functional Meta-term, 13 Functional Type, 11 FV, 11, 13 Graph Approximation, 226 Head, 13 Head Variable, 57 Headmost Step, 15 Headmost Variable, 56 Higher-order Dependency Pair Problem, 214 Higher-order Iterative Path Ordering, 104 Higher-order Recursive Path Ordering, 98 Higher-order Rewrite Systems, 48 HOIPO, 104 HORPO, 98 HRS, 48 HRS-term, 48 IDTS, 45 ILPO, 96 Inductive Data Type System, 45 Infinite Dependency Pair Problem, 214 Innermost, 16 Input Type, 11 Introducing Minimality Processor, 217 Inverse Compatible, 36 Irreflexive Relation, 29 Iterative Lexicographic Path Ordering, 96 Lambda-abstraction, 11 Lambda-calculus, 11 Leading Variable, 56 Left-linear, 16 Lexicographic Extension, 93 Lexicographic Path Ordering, 95 Limited Functional, 57 LPO, 95 Markable Term, 109 Meta-application, 13 Meta-context, 15, 106 Meta-rewriting, 106 Meta-stable, 30 Meta-substitution, 106 Meta-term, 13 Meta-variable, 13 Meta-variable Application, 13 Meta-variable Conditions, 155 Minimal Dependency Chain, 158 Minimal Non-terminating, 159 MNT, 159 Monotonic, 30 Monotonic Algebra Approach, 75 Multiset Extension, 92 Non-collapsing Set, 165 Non-overlapping, 74 Norm, 287 oa, 57 Order, 11, 16 Orthogonal, 74 Outermost, 16 Output Arity, 57 Output Type, 11 Overlay, 243 Parameter, 273 Pattern, 15 Pattern Higher-order Rewrite System, 48 Pattern HRS, 48 PHO, 242 Plain Function Passing, 244 Pol, 87 Polynomial Interpretation, 75 Potentially Higher-order Symbol, 242 Pre-term, 48 Precedence, 93 Processor, 215 Projection Function, 229 PRS, 48 Quasi-ordering, 29 Quasi-simplification Ordering, 141 Recursive Path Ordering, 94 Reduction Ordering, 30, 31 Reduction Pair, 32 Reduction Pair Processor, 208, 216 Reduction Pair Processor with Usable Rules, 209 Reduction Strategy, 16 Reduction Triple, 164 Reflexive Relation, 29 Respect, 225 Respect Arity, 33 Restricted η-expansion, 12, 17 Rewrite Relation, 10, 15 Rewrite Rule, 10, 15 Right Arrow Subterm, 100 RPO, 94 Rule, 10, 15 Rule Removal, 31 Rule Removal Processor, 208, 217 Rule Scheme, 96 SAT-solver, 269 SCC, 225 Shape, 172 Signature, 10, 13 Simple Meta-applications, 26 Simple Types, 11 Sound Processor, 207, 215 SPFP, 250 Stable, 30 Standard Reduction Pair For (. . . ), 195 StarHorpo, 116 stat, 94 Static Candidate Term, 245 Static Dependency Chain, 246 Static Dependency Pair, 245 Status, 94 Strict Ordering, 29 Strong Reduction Pair, 30 Strong Reduction Pair for TRSs, 30 Strongly Connected Component, 210, 225 Strongly Monotonic, 84 Strongly Plain Function Passing, 250 Sub-metaterm, 15 Substitute, 11 Substitution, 10, 11, 14 Subterm, 15 Subterm Criterion, 211, 229 Subterm Criterion Processor, 211, 229 Symb, 172 Syntactic Variable, 65 tag, 182 Tagged Dependency Chain, 187 Tagged Reduction Pair For (. . . ), 195 Term, 10, 11, 13 Terminating, 10, 16 Termination Problem Database, 8 TFO, 242 Todo List, 282 Topmost Step, 15 TPDB, 270 Transitive Relation, 29 TRS, 10 Truly First-order (Meta-)term, 242 Truly First-order Rule, 243 Truly First-order Symbol, 242 Type, 11 Type Declaration, 11 Type Ordering, 100 Type-changing Function, 28 Typed Symbol, 173 typeof, 60 uncur, 20 Uncurrying, 19 UR, 231 Usable Rules, 150, 231 Usable Rules Processor, 209 Usable Symbol, 233