ran isabelle update_op on all sources
authornipkow
Wed Jan 10 15:25:09 2018 +0100 (16 months ago)
changeset 67399eab6ce8368fa
parent 67398 5eb932e604a2
child 67400 bbed46f40cf5
ran isabelle update_op on all sources
src/Benchmarks/Quickcheck_Benchmark/Needham_Schroeder_Base.thy
src/Benchmarks/Quickcheck_Benchmark/Needham_Schroeder_Guided_Attacker_Example.thy
src/Benchmarks/Quickcheck_Benchmark/Needham_Schroeder_Unguided_Attacker_Example.thy
src/Benchmarks/Record_Benchmark/Record_Benchmark.thy
src/CTT/CTT.thy
src/Doc/Corec/Corec.thy
src/Doc/Datatypes/Datatypes.thy
src/Doc/Eisbach/Preface.thy
src/Doc/How_to_Prove_it/How_to_Prove_it.thy
src/Doc/Isar_Ref/Framework.thy
src/Doc/Isar_Ref/Generic.thy
src/Doc/Isar_Ref/HOL_Specific.thy
src/Doc/Isar_Ref/Proof_Script.thy
src/Doc/Isar_Ref/Symbols.thy
src/Doc/JEdit/JEdit.thy
src/Doc/Locales/Examples1.thy
src/Doc/Locales/Examples2.thy
src/Doc/Main/Main_Doc.thy
src/Doc/Prog_Prove/Basics.thy
src/Doc/Prog_Prove/LaTeXsugar.thy
src/Doc/System/Environment.thy
src/Doc/System/Misc.thy
src/Doc/System/Presentation.thy
src/Doc/Tutorial/Types/Pairs.thy
src/FOL/ex/Locale_Test/Locale_Test1.thy
src/FOLP/IFOLP.thy
src/HOL/Algebra/Complete_Lattice.thy
src/HOL/Algebra/Divisibility.thy
src/HOL/Algebra/Group.thy
src/HOL/Algebra/IntRing.thy
src/HOL/Algebra/Multiplicative_Group.thy
src/HOL/Algebra/Order.thy
src/HOL/Algebra/QuotRing.thy
src/HOL/Algebra/Sylow.thy
src/HOL/Algebra/UnivPoly.thy
src/HOL/Analysis/Ball_Volume.thy
src/HOL/Analysis/Binary_Product_Measure.thy
src/HOL/Analysis/Bochner_Integration.thy
src/HOL/Analysis/Borel_Space.thy
src/HOL/Analysis/Bounded_Linear_Function.thy
src/HOL/Analysis/Brouwer_Fixpoint.thy
src/HOL/Analysis/Cartesian_Euclidean_Space.thy
src/HOL/Analysis/Cauchy_Integral_Theorem.thy
src/HOL/Analysis/Complex_Analysis_Basics.thy
src/HOL/Analysis/Conformal_Mappings.thy
src/HOL/Analysis/Connected.thy
src/HOL/Analysis/Convex_Euclidean_Space.thy
src/HOL/Analysis/Derivative.thy
src/HOL/Analysis/Determinants.thy
src/HOL/Analysis/Embed_Measure.thy
src/HOL/Analysis/Equivalence_Lebesgue_Henstock_Integration.thy
src/HOL/Analysis/Euclidean_Space.thy
src/HOL/Analysis/Finite_Cartesian_Product.thy
src/HOL/Analysis/Finite_Product_Measure.thy
src/HOL/Analysis/Further_Topology.thy
src/HOL/Analysis/Gamma_Function.thy
src/HOL/Analysis/Great_Picard.thy
src/HOL/Analysis/Harmonic_Numbers.thy
src/HOL/Analysis/Henstock_Kurzweil_Integration.thy
src/HOL/Analysis/Homeomorphism.thy
src/HOL/Analysis/Improper_Integral.thy
src/HOL/Analysis/Inner_Product.thy
src/HOL/Analysis/Lebesgue_Measure.thy
src/HOL/Analysis/Linear_Algebra.thy
src/HOL/Analysis/Measure_Space.thy
src/HOL/Analysis/Nonnegative_Lebesgue_Integration.thy
src/HOL/Analysis/Path_Connected.thy
src/HOL/Analysis/Polytope.thy
src/HOL/Analysis/Riemann_Mapping.thy
src/HOL/Analysis/Set_Integral.thy
src/HOL/Analysis/Sigma_Algebra.thy
src/HOL/Analysis/Starlike.thy
src/HOL/Analysis/Tagged_Division.thy
src/HOL/Analysis/Topology_Euclidean_Space.thy
src/HOL/Analysis/ex/Approximations.thy
src/HOL/Analysis/normarith.ML
src/HOL/BNF_Composition.thy
src/HOL/BNF_Def.thy
src/HOL/BNF_Fixpoint_Base.thy
src/HOL/Bali/AxCompl.thy
src/HOL/Bali/AxExample.thy
src/HOL/Bali/AxSem.thy
src/HOL/Bali/State.thy
src/HOL/Basic_BNF_LFPs.thy
src/HOL/Basic_BNFs.thy
src/HOL/Binomial.thy
src/HOL/Cardinals/Bounded_Set.thy
src/HOL/Code_Numeral.thy
src/HOL/Complete_Lattices.thy
src/HOL/Complete_Partial_Order.thy
src/HOL/Computational_Algebra/Euclidean_Algorithm.thy
src/HOL/Computational_Algebra/Formal_Power_Series.thy
src/HOL/Computational_Algebra/Nth_Powers.thy
src/HOL/Computational_Algebra/Polynomial.thy
src/HOL/Computational_Algebra/Polynomial_FPS.thy
src/HOL/Computational_Algebra/Squarefree.thy
src/HOL/Corec_Examples/Tests/GPV_Bare_Bones.thy
src/HOL/Corec_Examples/Tests/Iterate_GPV.thy
src/HOL/Corec_Examples/Tests/Misc_Mono.thy
src/HOL/Data_Structures/Base_FDS.thy
src/HOL/Data_Structures/Binomial_Heap.thy
src/HOL/Data_Structures/Sorted_Less.thy
src/HOL/Datatype_Examples/Derivation_Trees/DTree.thy
src/HOL/Datatype_Examples/Derivation_Trees/Parallel_Composition.thy
src/HOL/Datatype_Examples/Stream_Processor.thy
src/HOL/Decision_Procs/Algebra_Aux.thy
src/HOL/Decision_Procs/Cooper.thy
src/HOL/Decision_Procs/Ferrack.thy
src/HOL/Decision_Procs/MIR.thy
src/HOL/Decision_Procs/Parametric_Ferrante_Rackoff.thy
src/HOL/Decision_Procs/Polynomial_List.thy
src/HOL/Decision_Procs/approximation.ML
src/HOL/Decision_Procs/langford.ML
src/HOL/Deriv.thy
src/HOL/Enum.thy
src/HOL/Equiv_Relations.thy
src/HOL/Factorial.thy
src/HOL/Filter.thy
src/HOL/Fun.thy
src/HOL/GCD.thy
src/HOL/Groups_List.thy
src/HOL/HOL.thy
src/HOL/HOLCF/Cpodef.thy
src/HOL/HOLCF/Discrete.thy
src/HOL/HOLCF/Domain.thy
src/HOL/HOLCF/Fun_Cpo.thy
src/HOL/HOLCF/Library/List_Cpo.thy
src/HOL/HOLCF/Product_Cpo.thy
src/HOL/HOLCF/Universal.thy
src/HOL/HOLCF/Up.thy
src/HOL/IMP/Abs_Int2_ivl.thy
src/HOL/IMP/Abs_Int3.thy
src/HOL/IMP/Abs_State.thy
src/HOL/IMP/Collecting.thy
src/HOL/IMP/Compiler2.thy
src/HOL/Imperative_HOL/Heap_Monad.thy
src/HOL/Import/HOL_Light_Maps.thy
src/HOL/Int.thy
src/HOL/Lattices.thy
src/HOL/Library/BNF_Corec.thy
src/HOL/Library/Cancellation.thy
src/HOL/Library/Cardinality.thy
src/HOL/Library/Char_ord.thy
src/HOL/Library/Code_Real_Approx_By_Float.thy
src/HOL/Library/Complete_Partial_Order2.thy
src/HOL/Library/Conditional_Parametricity.thy
src/HOL/Library/Countable.thy
src/HOL/Library/Countable_Set.thy
src/HOL/Library/Countable_Set_Type.thy
src/HOL/Library/DAList_Multiset.thy
src/HOL/Library/Diagonal_Subsequence.thy
src/HOL/Library/Discrete.thy
src/HOL/Library/Disjoint_Sets.thy
src/HOL/Library/Dlist.thy
src/HOL/Library/Extended_Nonnegative_Real.thy
src/HOL/Library/FSet.thy
src/HOL/Library/Finite_Map.thy
src/HOL/Library/Float.thy
src/HOL/Library/LaTeXsugar.thy
src/HOL/Library/Linear_Temporal_Logic_on_Streams.thy
src/HOL/Library/ListVector.thy
src/HOL/Library/Mapping.thy
src/HOL/Library/More_List.thy
src/HOL/Library/Multiset_Permutations.thy
src/HOL/Library/Nat_Bijection.thy
src/HOL/Library/Numeral_Type.thy
src/HOL/Library/Open_State_Syntax.thy
src/HOL/Library/Pattern_Aliases.thy
src/HOL/Library/Perm.thy
src/HOL/Library/Permutations.thy
src/HOL/Library/Predicate_Compile_Alternative_Defs.thy
src/HOL/Library/Quotient_List.thy
src/HOL/Library/Quotient_Product.thy
src/HOL/Library/Quotient_Set.thy
src/HOL/Library/Quotient_Sum.thy
src/HOL/Library/RBT_Set.thy
src/HOL/Library/State_Monad.thy
src/HOL/Library/Stream.thy
src/HOL/Library/Sublist.thy
src/HOL/Library/Subseq_Order.thy
src/HOL/Library/Sum_of_Squares/sum_of_squares.ML
src/HOL/Library/Tree.thy
src/HOL/Library/Tree_Real.thy
src/HOL/Library/Uprod.thy
src/HOL/Library/conditional_parametricity.ML
src/HOL/Library/positivstellensatz.ML
src/HOL/Library/refute.ML
src/HOL/Lifting.thy
src/HOL/Lifting_Set.thy
src/HOL/Limits.thy
src/HOL/List.thy
src/HOL/Matrix_LP/Cplex_tools.ML
src/HOL/Matrix_LP/Matrix.thy
src/HOL/Metis_Examples/Abstraction.thy
src/HOL/Metis_Examples/Proxies.thy
src/HOL/MicroJava/BV/BVExample.thy
src/HOL/MicroJava/BV/Typing_Framework_JVM.thy
src/HOL/MicroJava/DFA/Listn.thy
src/HOL/Mirabelle/Tools/mirabelle_quickcheck.ML
src/HOL/Mirabelle/Tools/mirabelle_sledgehammer.ML
src/HOL/NanoJava/Equivalence.thy
src/HOL/Nat.thy
src/HOL/Nitpick.thy
src/HOL/Nitpick_Examples/Core_Nits.thy
src/HOL/Nitpick_Examples/Manual_Nits.thy
src/HOL/Nitpick_Examples/Mini_Nits.thy
src/HOL/Nitpick_Examples/Mono_Nits.thy
src/HOL/Nitpick_Examples/Refute_Nits.thy
src/HOL/Nominal/Examples/Standardization.thy
src/HOL/Nominal/nominal_datatype.ML
src/HOL/Nominal/nominal_inductive.ML
src/HOL/Nominal/nominal_inductive2.ML
src/HOL/Nonstandard_Analysis/Free_Ultrafilter.thy
src/HOL/Nonstandard_Analysis/HDeriv.thy
src/HOL/Nonstandard_Analysis/HLog.thy
src/HOL/Nonstandard_Analysis/HyperDef.thy
src/HOL/Nonstandard_Analysis/NatStar.thy
src/HOL/Nonstandard_Analysis/StarDef.thy
src/HOL/Num.thy
src/HOL/Number_Theory/Gauss.thy
src/HOL/Number_Theory/Pocklington.thy
src/HOL/Number_Theory/Totient.thy
src/HOL/Option.thy
src/HOL/Partial_Function.thy
src/HOL/Predicate.thy
src/HOL/Predicate_Compile_Examples/Hotel_Example_Prolog.thy
src/HOL/Predicate_Compile_Examples/Hotel_Example_Small_Generator.thy
src/HOL/Predicate_Compile_Examples/Predicate_Compile_Tests.thy
src/HOL/Probability/Distributions.thy
src/HOL/Probability/Fin_Map.thy
src/HOL/Probability/Giry_Monad.thy
src/HOL/Probability/Helly_Selection.thy
src/HOL/Probability/Independent_Family.thy
src/HOL/Probability/Infinite_Product_Measure.thy
src/HOL/Probability/PMF_Impl.thy
src/HOL/Probability/Probability_Mass_Function.thy
src/HOL/Probability/Probability_Measure.thy
src/HOL/Probability/Projective_Limit.thy
src/HOL/Probability/Random_Permutations.thy
src/HOL/Probability/SPMF.thy
src/HOL/Probability/Stream_Space.thy
src/HOL/Probability/ex/Dining_Cryptographers.thy
src/HOL/Probability/ex/Koepf_Duermuth_Countermeasure.thy
src/HOL/Probability/ex/Measure_Not_CCC.thy
src/HOL/Prolog/Test.thy
src/HOL/Proofs/Extraction/Higman_Extraction.thy
src/HOL/Proofs/Lambda/ListApplication.thy
src/HOL/Proofs/Lambda/Standardization.thy
src/HOL/Proofs/ex/Proof_Terms.thy
src/HOL/Quickcheck_Examples/Quickcheck_Examples.thy
src/HOL/Quickcheck_Narrowing.thy
src/HOL/Quickcheck_Random.thy
src/HOL/Quotient.thy
src/HOL/Quotient_Examples/Lift_FSet.thy
src/HOL/Quotient_Examples/Lift_Fun.thy
src/HOL/Quotient_Examples/Quotient_FSet.thy
src/HOL/Quotient_Examples/Quotient_Int.thy
src/HOL/Quotient_Examples/Quotient_Rat.thy
src/HOL/Real.thy
src/HOL/Real_Vector_Spaces.thy
src/HOL/Relation.thy
src/HOL/SMT.thy
src/HOL/SMT_Examples/SMT_Examples.thy
src/HOL/SMT_Examples/boogie.ML
src/HOL/SPARK/SPARK.thy
src/HOL/Set_Interval.thy
src/HOL/String.thy
src/HOL/TLA/Intensional.thy
src/HOL/TPTP/TPTP_Parser/tptp_reconstruct_library.ML
src/HOL/TPTP/TPTP_Proof_Reconstruction.thy
src/HOL/TPTP/TPTP_Proof_Reconstruction_Test.thy
src/HOL/TPTP/TPTP_Proof_Reconstruction_Test_Units.thy
src/HOL/Tools/ATP/atp_problem.ML
src/HOL/Tools/ATP/atp_problem_generate.ML
src/HOL/Tools/BNF/bnf_def_tactics.ML
src/HOL/Tools/BNF/bnf_fp_def_sugar_tactics.ML
src/HOL/Tools/BNF/bnf_fp_n2m_tactics.ML
src/HOL/Tools/BNF/bnf_fp_util_tactics.ML
src/HOL/Tools/BNF/bnf_gfp.ML
src/HOL/Tools/BNF/bnf_gfp_tactics.ML
src/HOL/Tools/BNF/bnf_lfp.ML
src/HOL/Tools/BNF/bnf_lfp_tactics.ML
src/HOL/Tools/Ctr_Sugar/ctr_sugar.ML
src/HOL/Tools/Lifting/lifting_info.ML
src/HOL/Tools/Nitpick/nitpick_commands.ML
src/HOL/Tools/Nitpick/nitpick_mono.ML
src/HOL/Tools/Nunchaku/nunchaku_collect.ML
src/HOL/Tools/Nunchaku/nunchaku_commands.ML
src/HOL/Tools/Nunchaku/nunchaku_reconstruct.ML
src/HOL/Tools/Predicate_Compile/code_prolog.ML
src/HOL/Tools/Predicate_Compile/core_data.ML
src/HOL/Tools/Predicate_Compile/predicate_compile_aux.ML
src/HOL/Tools/Qelim/cooper.ML
src/HOL/Tools/SMT/smt_systems.ML
src/HOL/Tools/SMT/smtlib.ML
src/HOL/Tools/SMT/smtlib_interface.ML
src/HOL/Tools/Sledgehammer/sledgehammer_commands.ML
src/HOL/Tools/Sledgehammer/sledgehammer_isar_proof.ML
src/HOL/Tools/Transfer/transfer.ML
src/HOL/Tools/datatype_realizer.ML
src/HOL/Tools/functor.ML
src/HOL/Tools/groebner.ML
src/HOL/Tools/rewrite_hol_proof.ML
src/HOL/Tools/sat_solver.ML
src/HOL/Tools/semiring_normalizer.ML
src/HOL/Tools/set_comprehension_pointfree.ML
src/HOL/Transcendental.thy
src/HOL/Transfer.thy
src/HOL/Transitive_Closure.thy
src/HOL/Types_To_Sets/Examples/Finite.thy
src/HOL/Types_To_Sets/Examples/T2_Spaces.thy
src/HOL/Wellfounded.thy
src/HOL/Word/Bool_List_Representation.thy
src/HOL/Word/Misc_Typedef.thy
src/HOL/Word/Word.thy
src/HOL/Word/WordBitwise.thy
src/HOL/Zorn.thy
src/HOL/ex/Ballot.thy
src/HOL/ex/Coercion_Examples.thy
src/HOL/ex/Conditional_Parametricity_Examples.thy
src/HOL/ex/Erdoes_Szekeres.thy
src/HOL/ex/LocaleTest2.thy
src/HOL/ex/Refute_Examples.thy
src/HOL/ex/Transfer_Debug.thy
src/HOL/ex/Transfer_Int_Nat.thy
src/HOL/ex/While_Combinator_Example.thy
src/ZF/Constructible/Rank.thy
src/ZF/Constructible/Rank_Separation.thy
src/ZF/Induct/FoldSet.thy
src/ZF/Order.thy
src/ZF/Perm.thy
src/ZF/UNITY/Follows.thy
src/ZF/UNITY/Monotonicity.thy
src/ZF/UNITY/MultisetSum.thy
src/ZF/UNITY/Mutex.thy
src/ZF/ex/Limit.thy
     1.1 --- a/src/Benchmarks/Quickcheck_Benchmark/Needham_Schroeder_Base.thy	Wed Jan 10 15:21:49 2018 +0100
     1.2 +++ b/src/Benchmarks/Quickcheck_Benchmark/Needham_Schroeder_Base.thy	Wed Jan 10 15:25:09 2018 +0100
     1.3 @@ -193,4 +193,4 @@
     1.4  setup Exhaustive_Generators.setup_exhaustive_datatype_interpretation
     1.5  declare [[quickcheck_full_support = false]]
     1.6  
     1.7 -end
     1.8 \ No newline at end of file
     1.9 +end
     2.1 --- a/src/Benchmarks/Quickcheck_Benchmark/Needham_Schroeder_Guided_Attacker_Example.thy	Wed Jan 10 15:21:49 2018 +0100
     2.2 +++ b/src/Benchmarks/Quickcheck_Benchmark/Needham_Schroeder_Guided_Attacker_Example.thy	Wed Jan 10 15:25:09 2018 +0100
     2.3 @@ -97,4 +97,4 @@
     2.4      done
     2.5  qed
     2.6  
     2.7 -end
     2.8 \ No newline at end of file
     2.9 +end
     3.1 --- a/src/Benchmarks/Quickcheck_Benchmark/Needham_Schroeder_Unguided_Attacker_Example.thy	Wed Jan 10 15:21:49 2018 +0100
     3.2 +++ b/src/Benchmarks/Quickcheck_Benchmark/Needham_Schroeder_Unguided_Attacker_Example.thy	Wed Jan 10 15:25:09 2018 +0100
     3.3 @@ -93,4 +93,4 @@
     3.4      done
     3.5  qed
     3.6  
     3.7 -end
     3.8 \ No newline at end of file
     3.9 +end
     4.1 --- a/src/Benchmarks/Record_Benchmark/Record_Benchmark.thy	Wed Jan 10 15:21:49 2018 +0100
     4.2 +++ b/src/Benchmarks/Record_Benchmark/Record_Benchmark.thy	Wed Jan 10 15:25:09 2018 +0100
     4.3 @@ -424,4 +424,4 @@
     4.4  
     4.5  print_record many_B
     4.6  
     4.7 -end
     4.8 \ No newline at end of file
     4.9 +end
     5.1 --- a/src/CTT/CTT.thy	Wed Jan 10 15:21:49 2018 +0100
     5.2 +++ b/src/CTT/CTT.thy	Wed Jan 10 15:25:09 2018 +0100
     5.3 @@ -434,7 +434,7 @@
     5.4  
     5.5  (*0 subgoals vs 1 or more*)
     5.6  val (safe0_brls, safep_brls) =
     5.7 -    List.partition (curry (op =) 0 o subgoals_of_brl) safe_brls
     5.8 +    List.partition (curry (=) 0 o subgoals_of_brl) safe_brls
     5.9  
    5.10  fun safestep_tac ctxt thms i =
    5.11      form_tac ctxt ORELSE
     6.1 --- a/src/Doc/Corec/Corec.thy	Wed Jan 10 15:21:49 2018 +0100
     6.2 +++ b/src/Doc/Corec/Corec.thy	Wed Jan 10 15:25:09 2018 +0100
     6.3 @@ -280,7 +280,7 @@
     6.4  @{command corec}, @{command corecursive}, and @{command friend_of_corec}. In
     6.5  particular, nesting through the function type can be expressed using
     6.6  @{text \<lambda>}-abstractions and function applications rather than through composition
     6.7 -(@{term "op \<circ>"}, the map function for @{text \<Rightarrow>}). For example:
     6.8 +(@{term "(\<circ>)"}, the map function for @{text \<Rightarrow>}). For example:
     6.9  \<close>
    6.10  
    6.11      codatatype 'a language =
    6.12 @@ -892,7 +892,7 @@
    6.13  
    6.14  text \<open>
    6.15  The @{method transfer_prover_eq} proof method replaces the equality relation
    6.16 -@{term "op ="} with compound relator expressions according to
    6.17 +@{term "(=)"} with compound relator expressions according to
    6.18  @{thm [source] relator_eq} before calling @{method transfer_prover} on the
    6.19  current subgoal. It tends to work better than plain @{method transfer_prover} on
    6.20  the parametricity proof obligations of @{command corecursive} and
     7.1 --- a/src/Doc/Datatypes/Datatypes.thy	Wed Jan 10 15:21:49 2018 +0100
     7.2 +++ b/src/Doc/Datatypes/Datatypes.thy	Wed Jan 10 15:25:09 2018 +0100
     7.3 @@ -1410,7 +1410,7 @@
     7.4  \noindent
     7.5  The same principle applies for arbitrary type constructors through which
     7.6  recursion is possible. Notably, the map function for the function type
     7.7 -(@{text \<Rightarrow>}) is simply composition (@{text "op \<circ>"}):
     7.8 +(@{text \<Rightarrow>}) is simply composition (@{text "(\<circ>)"}):
     7.9  \<close>
    7.10  
    7.11      primrec (*<*)(in early) (*>*)relabel_ft :: "('a \<Rightarrow> 'a) \<Rightarrow> 'a ftree \<Rightarrow> 'a ftree" where
    7.12 @@ -1436,7 +1436,7 @@
    7.13  text \<open>
    7.14  \noindent
    7.15  For recursion through curried $n$-ary functions, $n$ applications of
    7.16 -@{term "op \<circ>"} are necessary. The examples below illustrate the case where
    7.17 +@{term "(\<circ>)"} are necessary. The examples below illustrate the case where
    7.18  $n = 2$:
    7.19  \<close>
    7.20  
    7.21 @@ -1446,7 +1446,7 @@
    7.22  
    7.23      primrec (*<*)(in early) (*>*)relabel_ft2 :: "('a \<Rightarrow> 'a) \<Rightarrow> 'a ftree2 \<Rightarrow> 'a ftree2" where
    7.24        "relabel_ft2 f (FTLeaf2 x) = FTLeaf2 (f x)"
    7.25 -    | "relabel_ft2 f (FTNode2 g) = FTNode2 (op \<circ> (op \<circ> (relabel_ft2 f)) g)"
    7.26 +    | "relabel_ft2 f (FTNode2 g) = FTNode2 ((\<circ>) ((\<circ>) (relabel_ft2 f)) g)"
    7.27  
    7.28  text \<open>\blankline\<close>
    7.29  
    7.30 @@ -2264,7 +2264,7 @@
    7.31  text \<open>
    7.32  \noindent
    7.33  The map function for the function type (@{text \<Rightarrow>}) is composition
    7.34 -(@{text "op \<circ>"}). For convenience, corecursion through functions can
    7.35 +(@{text "(\<circ>)"}). For convenience, corecursion through functions can
    7.36  also be expressed using $\lambda$-abstractions and function application rather
    7.37  than through composition. For example:
    7.38  \<close>
    7.39 @@ -2291,7 +2291,7 @@
    7.40  text \<open>
    7.41  \noindent
    7.42  For recursion through curried $n$-ary functions, $n$ applications of
    7.43 -@{term "op \<circ>"} are necessary. The examples below illustrate the case where
    7.44 +@{term "(\<circ>)"} are necessary. The examples below illustrate the case where
    7.45  $n = 2$:
    7.46  \<close>
    7.47  
    7.48 @@ -2303,7 +2303,7 @@
    7.49      primcorec
    7.50        (*<*)(in early) (*>*)sm2_of_dfa :: "('q \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'q) \<Rightarrow> 'q set \<Rightarrow> 'q \<Rightarrow> ('a, 'b) sm2"
    7.51      where
    7.52 -      "sm2_of_dfa \<delta> F q = SM2 (q \<in> F) (op \<circ> (op \<circ> (sm2_of_dfa \<delta> F)) (\<delta> q))"
    7.53 +      "sm2_of_dfa \<delta> F q = SM2 (q \<in> F) ((\<circ>) ((\<circ>) (sm2_of_dfa \<delta> F)) (\<delta> q))"
    7.54  
    7.55  text \<open>\blankline\<close>
    7.56  
    7.57 @@ -2782,7 +2782,7 @@
    7.58  
    7.59  text \<open>\blankline\<close>
    7.60  
    7.61 -    lift_definition map_fn :: "('a \<Rightarrow> 'b) \<Rightarrow> ('d, 'a) fn \<Rightarrow> ('d, 'b) fn" is "op \<circ>" .
    7.62 +    lift_definition map_fn :: "('a \<Rightarrow> 'b) \<Rightarrow> ('d, 'a) fn \<Rightarrow> ('d, 'b) fn" is "(\<circ>)" .
    7.63      lift_definition set_fn :: "('d, 'a) fn \<Rightarrow> 'a set" is range .
    7.64  
    7.65  text \<open>\blankline\<close>
    7.66 @@ -2795,7 +2795,7 @@
    7.67      lift_definition
    7.68        rel_fn :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> ('d, 'a) fn \<Rightarrow> ('d, 'b) fn \<Rightarrow> bool"
    7.69      is
    7.70 -      "rel_fun (op =)" .
    7.71 +      "rel_fun (=)" .
    7.72  
    7.73  text \<open>\blankline\<close>
    7.74  
    7.75 @@ -2819,7 +2819,7 @@
    7.76          by transfer auto
    7.77      next
    7.78        fix f :: "'a \<Rightarrow> 'b"
    7.79 -      show "set_fn \<circ> map_fn f = op ` f \<circ> set_fn"
    7.80 +      show "set_fn \<circ> map_fn f = (`) f \<circ> set_fn"
    7.81          by transfer (auto simp add: comp_def)
    7.82      next
    7.83        show "card_order (natLeq +c |UNIV :: 'd set| )"
     8.1 --- a/src/Doc/Eisbach/Preface.thy	Wed Jan 10 15:21:49 2018 +0100
     8.2 +++ b/src/Doc/Eisbach/Preface.thy	Wed Jan 10 15:25:09 2018 +0100
     8.3 @@ -33,4 +33,4 @@
     8.4    with existing Isar concepts such as @{command named_theorems}.
     8.5  \<close>
     8.6  
     8.7 -end
     8.8 \ No newline at end of file
     8.9 +end
     9.1 --- a/src/Doc/How_to_Prove_it/How_to_Prove_it.thy	Wed Jan 10 15:21:49 2018 +0100
     9.2 +++ b/src/Doc/How_to_Prove_it/How_to_Prove_it.thy	Wed Jan 10 15:25:09 2018 +0100
     9.3 @@ -131,4 +131,4 @@
     9.4  
     9.5  (*<*)
     9.6  end
     9.7 -(*>*)
     9.8 \ No newline at end of file
     9.9 +(*>*)
    10.1 --- a/src/Doc/Isar_Ref/Framework.thy	Wed Jan 10 15:21:49 2018 +0100
    10.2 +++ b/src/Doc/Isar_Ref/Framework.thy	Wed Jan 10 15:25:09 2018 +0100
    10.3 @@ -1028,4 +1028,4 @@
    10.4    ..\<close>''.
    10.5  \<close>
    10.6  
    10.7 -end
    10.8 \ No newline at end of file
    10.9 +end
    11.1 --- a/src/Doc/Isar_Ref/Generic.thy	Wed Jan 10 15:21:49 2018 +0100
    11.2 +++ b/src/Doc/Isar_Ref/Generic.thy	Wed Jan 10 15:25:09 2018 +0100
    11.3 @@ -387,7 +387,7 @@
    11.4  lemma "0 + (x + 0) = x + 0 + 0" apply simp? oops
    11.5  
    11.6  text \<open>
    11.7 -  The above attempt \<^emph>\<open>fails\<close>, because @{term "0"} and @{term "op +"} in the
    11.8 +  The above attempt \<^emph>\<open>fails\<close>, because @{term "0"} and @{term "(+)"} in the
    11.9    HOL library are declared as generic type class operations, without stating
   11.10    any algebraic laws yet. More specific types are required to get access to
   11.11    certain standard simplifications of the theory context, e.g.\ like this:\<close>
    12.1 --- a/src/Doc/Isar_Ref/HOL_Specific.thy	Wed Jan 10 15:21:49 2018 +0100
    12.2 +++ b/src/Doc/Isar_Ref/HOL_Specific.thy	Wed Jan 10 15:25:09 2018 +0100
    12.3 @@ -1452,7 +1452,7 @@
    12.4    \<open>Quotient_thm\<close>). Optional theorems \<open>pcr_def\<close> and \<open>pcr_cr_eq_thm\<close> can be
    12.5    specified to register the parametrized correspondence relation for \<open>\<tau>\<close>.
    12.6    E.g.\ for @{typ "'a dlist"}, \<open>pcr_def\<close> is \<open>pcr_dlist A \<equiv> list_all2 A \<circ>\<circ>
    12.7 -  cr_dlist\<close> and \<open>pcr_cr_eq_thm\<close> is \<open>pcr_dlist (op =) = (op =)\<close>. This attribute
    12.8 +  cr_dlist\<close> and \<open>pcr_cr_eq_thm\<close> is \<open>pcr_dlist (=) = (=)\<close>. This attribute
    12.9    is rather used for low-level manipulation with set-up of the Lifting package
   12.10    because using of the bundle \<open>\<tau>.lifting\<close> together with the commands @{command
   12.11    (HOL) lifting_forget} and @{command (HOL) lifting_update} is preferred for
   12.12 @@ -1539,7 +1539,7 @@
   12.13    Lemmas involving predicates on relations can also be registered using the
   12.14    same attribute. For example:
   12.15  
   12.16 -    \<open>bi_unique A \<Longrightarrow> (list_all2 A ===> op =) distinct distinct\<close> \\
   12.17 +    \<open>bi_unique A \<Longrightarrow> (list_all2 A ===> (=)) distinct distinct\<close> \\
   12.18      \<open>\<lbrakk>bi_unique A; bi_unique B\<rbrakk> \<Longrightarrow> bi_unique (rel_prod A B)\<close>
   12.19  
   12.20    Preservation of predicates on relations (\<open>bi_unique, bi_total, right_unique,
   12.21 @@ -1555,7 +1555,7 @@
   12.22    quantifiers are transferred.
   12.23  
   12.24    \<^descr> @{attribute (HOL) relator_eq} attribute collects identity laws for
   12.25 -  relators of various type constructors, e.g. @{term "rel_set (op =) = (op =)"}.
   12.26 +  relators of various type constructors, e.g. @{term "rel_set (=) = (=)"}.
   12.27    The @{method (HOL) transfer} method uses these lemmas to infer
   12.28    transfer rules for non-polymorphic constants on the fly. For examples see
   12.29    \<^file>\<open>~~/src/HOL/Lifting_Set.thy\<close> or \<^file>\<open>~~/src/HOL/Lifting.thy\<close>. This property
   12.30 @@ -1894,7 +1894,7 @@
   12.31      @{term "Quickcheck_Narrowing.sum (Quickcheck_Narrowing.cons [])
   12.32                (Quickcheck_Narrowing.apply
   12.33                  (Quickcheck_Narrowing.apply
   12.34 -                  (Quickcheck_Narrowing.cons (op #))
   12.35 +                  (Quickcheck_Narrowing.cons (#))
   12.36                    narrowing)
   12.37                  narrowing)"}.
   12.38      If a symbolic variable of type @{typ "_ list"} is evaluated, it is
   12.39 @@ -1909,7 +1909,7 @@
   12.40      "Quickcheck_Narrowing.Narrowing_constructor i args"}, where @{term "i ::
   12.41      integer"} denotes the index in the sum of refinements. In the above
   12.42      example for lists, @{term "0"} corresponds to @{term "[]"} and @{term "1"}
   12.43 -    to @{term "op #"}.
   12.44 +    to @{term "(#)"}.
   12.45  
   12.46      The command @{command (HOL) "code_datatype"} sets up @{const
   12.47      partial_term_of} such that the @{term "i"}-th refinement is interpreted as
    13.1 --- a/src/Doc/Isar_Ref/Proof_Script.thy	Wed Jan 10 15:21:49 2018 +0100
    13.2 +++ b/src/Doc/Isar_Ref/Proof_Script.thy	Wed Jan 10 15:25:09 2018 +0100
    13.3 @@ -278,4 +278,4 @@
    13.4    definitions, it should not never appear in production theories.
    13.5  \<close>
    13.6  
    13.7 -end
    13.8 \ No newline at end of file
    13.9 +end
    14.1 --- a/src/Doc/Isar_Ref/Symbols.thy	Wed Jan 10 15:21:49 2018 +0100
    14.2 +++ b/src/Doc/Isar_Ref/Symbols.thy	Wed Jan 10 15:25:09 2018 +0100
    14.3 @@ -37,4 +37,4 @@
    14.4    \end{center}
    14.5  \<close>
    14.6  
    14.7 -end
    14.8 \ No newline at end of file
    14.9 +end
    15.1 --- a/src/Doc/JEdit/JEdit.thy	Wed Jan 10 15:21:49 2018 +0100
    15.2 +++ b/src/Doc/JEdit/JEdit.thy	Wed Jan 10 15:25:09 2018 +0100
    15.3 @@ -2171,4 +2171,4 @@
    15.4    which sometimes helps in low-memory situations.
    15.5  \<close>
    15.6  
    15.7 -end
    15.8 \ No newline at end of file
    15.9 +end
    16.1 --- a/src/Doc/Locales/Examples1.thy	Wed Jan 10 15:21:49 2018 +0100
    16.2 +++ b/src/Doc/Locales/Examples1.thy	Wed Jan 10 15:25:09 2018 +0100
    16.3 @@ -19,8 +19,8 @@
    16.4    Section~\ref{sec:local-interpretation}.
    16.5  
    16.6    As an example, consider the type of integers @{typ int}.  The
    16.7 -  relation @{term "op \<le>"} is a total order over @{typ int}.  We start
    16.8 -  with the interpretation that @{term "op \<le>"} is a partial order.  The
    16.9 +  relation @{term "(\<le>)"} is a total order over @{typ int}.  We start
   16.10 +  with the interpretation that @{term "(\<le>)"} is a partial order.  The
   16.11    facilities of the interpretation command are explored gradually in
   16.12    three versions.
   16.13  \<close>
   16.14 @@ -32,11 +32,11 @@
   16.15  text \<open>
   16.16    The command \isakeyword{interpretation} is for the interpretation of
   16.17    locale in theories.  In the following example, the parameter of locale
   16.18 -  @{text partial_order} is replaced by @{term "op \<le> :: int \<Rightarrow> int \<Rightarrow>
   16.19 +  @{text partial_order} is replaced by @{term "(\<le>) :: int \<Rightarrow> int \<Rightarrow>
   16.20    bool"} and the locale instance is interpreted in the current
   16.21    theory.\<close>
   16.22  
   16.23 -  interpretation %visible int: partial_order "op \<le> :: int \<Rightarrow> int \<Rightarrow> bool"
   16.24 +  interpretation %visible int: partial_order "(\<le>) :: int \<Rightarrow> int \<Rightarrow> bool"
   16.25  txt \<open>\normalsize
   16.26    The argument of the command is a simple \emph{locale expression}
   16.27    consisting of the name of the interpreted locale, which is
   16.28 @@ -70,17 +70,17 @@
   16.29    The prefix @{text int} is applied to all names introduced in locale
   16.30    conclusions including names introduced in definitions.  The
   16.31    qualified name @{text int.less} is short for
   16.32 -  the interpretation of the definition, which is @{text "partial_order.less op \<le>"}.
   16.33 +  the interpretation of the definition, which is @{text "partial_order.less (\<le>)"}.
   16.34    Qualified name and expanded form may be used almost
   16.35    interchangeably.%
   16.36 -\footnote{Since @{term "op \<le>"} is polymorphic, for @{text "partial_order.less op \<le>"} a
   16.37 +\footnote{Since @{term "(\<le>)"} is polymorphic, for @{text "partial_order.less (\<le>)"} a
   16.38    more general type will be inferred than for @{text int.less} which
   16.39    is over type @{typ int}.}
   16.40    The former is preferred on output, as for example in the theorem
   16.41    @{thm [source] int.less_le_trans}: @{thm [display, indent=2]
   16.42    int.less_le_trans}
   16.43    Both notations for the strict order are not satisfactory.  The
   16.44 -  constant @{term "op <"} is the strict order for @{typ int}.
   16.45 +  constant @{term "(<)"} is the strict order for @{typ int}.
   16.46    In order to allow for the desired replacement, interpretation
   16.47    accepts \emph{equations} in addition to the parameter instantiation.
   16.48    These follow the locale expression and are indicated with the
    17.1 --- a/src/Doc/Locales/Examples2.thy	Wed Jan 10 15:21:49 2018 +0100
    17.2 +++ b/src/Doc/Locales/Examples2.thy	Wed Jan 10 15:25:09 2018 +0100
    17.3 @@ -1,18 +1,18 @@
    17.4  theory Examples2
    17.5  imports Examples
    17.6  begin
    17.7 -  interpretation %visible int: partial_order "op \<le> :: [int, int] \<Rightarrow> bool"
    17.8 +  interpretation %visible int: partial_order "(\<le>) :: [int, int] \<Rightarrow> bool"
    17.9      rewrites "int.less x y = (x < y)"
   17.10    proof -
   17.11      txt \<open>\normalsize The goals are now:
   17.12        @{subgoals [display]}
   17.13        The proof that~@{text \<le>} is a partial order is as above.\<close>
   17.14 -    show "partial_order (op \<le> :: int \<Rightarrow> int \<Rightarrow> bool)"
   17.15 +    show "partial_order ((\<le>) :: int \<Rightarrow> int \<Rightarrow> bool)"
   17.16        by unfold_locales auto
   17.17      txt \<open>\normalsize The second goal is shown by unfolding the
   17.18        definition of @{term "partial_order.less"}.\<close>
   17.19 -    show "partial_order.less op \<le> x y = (x < y)"
   17.20 -      unfolding partial_order.less_def [OF \<open>partial_order (op \<le>)\<close>]
   17.21 +    show "partial_order.less (\<le>) x y = (x < y)"
   17.22 +      unfolding partial_order.less_def [OF \<open>partial_order (\<le>)\<close>]
   17.23        by auto
   17.24    qed
   17.25  
    18.1 --- a/src/Doc/Main/Main_Doc.thy	Wed Jan 10 15:21:49 2018 +0100
    18.2 +++ b/src/Doc/Main/Main_Doc.thy	Wed Jan 10 15:25:09 2018 +0100
    18.3 @@ -297,7 +297,7 @@
    18.4  @{const Transitive_Closure.trancl} & @{term_type_only Transitive_Closure.trancl "('a*'a)set\<Rightarrow>('a*'a)set"}\\
    18.5  @{const Transitive_Closure.reflcl} & @{term_type_only Transitive_Closure.reflcl "('a*'a)set\<Rightarrow>('a*'a)set"}\\
    18.6  @{const Transitive_Closure.acyclic} & @{term_type_only Transitive_Closure.acyclic "('a*'a)set\<Rightarrow>bool"}\\
    18.7 -@{const compower} & @{term_type_only "op ^^ :: ('a*'a)set\<Rightarrow>nat\<Rightarrow>('a*'a)set" "('a*'a)set\<Rightarrow>nat\<Rightarrow>('a*'a)set"}\\
    18.8 +@{const compower} & @{term_type_only "(^^) :: ('a*'a)set\<Rightarrow>nat\<Rightarrow>('a*'a)set" "('a*'a)set\<Rightarrow>nat\<Rightarrow>('a*'a)set"}\\
    18.9  \end{tabular}
   18.10  
   18.11  \subsubsection*{Syntax}
   18.12 @@ -345,15 +345,15 @@
   18.13  \<^bigskip>
   18.14  
   18.15  \begin{tabular}{@ {} lllllll @ {}}
   18.16 -@{term "op + :: nat \<Rightarrow> nat \<Rightarrow> nat"} &
   18.17 -@{term "op - :: nat \<Rightarrow> nat \<Rightarrow> nat"} &
   18.18 -@{term "op * :: nat \<Rightarrow> nat \<Rightarrow> nat"} &
   18.19 -@{term "op ^ :: nat \<Rightarrow> nat \<Rightarrow> nat"} &
   18.20 -@{term "op div :: nat \<Rightarrow> nat \<Rightarrow> nat"}&
   18.21 -@{term "op mod :: nat \<Rightarrow> nat \<Rightarrow> nat"}&
   18.22 -@{term "op dvd :: nat \<Rightarrow> nat \<Rightarrow> bool"}\\
   18.23 -@{term "op \<le> :: nat \<Rightarrow> nat \<Rightarrow> bool"} &
   18.24 -@{term "op < :: nat \<Rightarrow> nat \<Rightarrow> bool"} &
   18.25 +@{term "(+) :: nat \<Rightarrow> nat \<Rightarrow> nat"} &
   18.26 +@{term "(-) :: nat \<Rightarrow> nat \<Rightarrow> nat"} &
   18.27 +@{term "( * ) :: nat \<Rightarrow> nat \<Rightarrow> nat"} &
   18.28 +@{term "(^) :: nat \<Rightarrow> nat \<Rightarrow> nat"} &
   18.29 +@{term "(div) :: nat \<Rightarrow> nat \<Rightarrow> nat"}&
   18.30 +@{term "(mod) :: nat \<Rightarrow> nat \<Rightarrow> nat"}&
   18.31 +@{term "(dvd) :: nat \<Rightarrow> nat \<Rightarrow> bool"}\\
   18.32 +@{term "(\<le>) :: nat \<Rightarrow> nat \<Rightarrow> bool"} &
   18.33 +@{term "(<) :: nat \<Rightarrow> nat \<Rightarrow> bool"} &
   18.34  @{term "min :: nat \<Rightarrow> nat \<Rightarrow> nat"} &
   18.35  @{term "max :: nat \<Rightarrow> nat \<Rightarrow> nat"} &
   18.36  @{term "Min :: nat set \<Rightarrow> nat"} &
   18.37 @@ -362,8 +362,8 @@
   18.38  
   18.39  \begin{tabular}{@ {} l @ {~::~} l @ {}}
   18.40  @{const Nat.of_nat} & @{typeof Nat.of_nat}\\
   18.41 -@{term "op ^^ :: ('a \<Rightarrow> 'a) \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> 'a"} &
   18.42 -  @{term_type_only "op ^^ :: ('a \<Rightarrow> 'a) \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> 'a" "('a \<Rightarrow> 'a) \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> 'a"}
   18.43 +@{term "(^^) :: ('a \<Rightarrow> 'a) \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> 'a"} &
   18.44 +  @{term_type_only "(^^) :: ('a \<Rightarrow> 'a) \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> 'a" "('a \<Rightarrow> 'a) \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> 'a"}
   18.45  \end{tabular}
   18.46  
   18.47  \section*{Int}
   18.48 @@ -372,16 +372,16 @@
   18.49  \<^bigskip>
   18.50  
   18.51  \begin{tabular}{@ {} llllllll @ {}}
   18.52 -@{term "op + :: int \<Rightarrow> int \<Rightarrow> int"} &
   18.53 -@{term "op - :: int \<Rightarrow> int \<Rightarrow> int"} &
   18.54 +@{term "(+) :: int \<Rightarrow> int \<Rightarrow> int"} &
   18.55 +@{term "(-) :: int \<Rightarrow> int \<Rightarrow> int"} &
   18.56  @{term "uminus :: int \<Rightarrow> int"} &
   18.57 -@{term "op * :: int \<Rightarrow> int \<Rightarrow> int"} &
   18.58 -@{term "op ^ :: int \<Rightarrow> nat \<Rightarrow> int"} &
   18.59 -@{term "op div :: int \<Rightarrow> int \<Rightarrow> int"}&
   18.60 -@{term "op mod :: int \<Rightarrow> int \<Rightarrow> int"}&
   18.61 -@{term "op dvd :: int \<Rightarrow> int \<Rightarrow> bool"}\\
   18.62 -@{term "op \<le> :: int \<Rightarrow> int \<Rightarrow> bool"} &
   18.63 -@{term "op < :: int \<Rightarrow> int \<Rightarrow> bool"} &
   18.64 +@{term "( * ) :: int \<Rightarrow> int \<Rightarrow> int"} &
   18.65 +@{term "(^) :: int \<Rightarrow> nat \<Rightarrow> int"} &
   18.66 +@{term "(div) :: int \<Rightarrow> int \<Rightarrow> int"}&
   18.67 +@{term "(mod) :: int \<Rightarrow> int \<Rightarrow> int"}&
   18.68 +@{term "(dvd) :: int \<Rightarrow> int \<Rightarrow> bool"}\\
   18.69 +@{term "(\<le>) :: int \<Rightarrow> int \<Rightarrow> bool"} &
   18.70 +@{term "(<) :: int \<Rightarrow> int \<Rightarrow> bool"} &
   18.71  @{term "min :: int \<Rightarrow> int \<Rightarrow> int"} &
   18.72  @{term "max :: int \<Rightarrow> int \<Rightarrow> int"} &
   18.73  @{term "Min :: int set \<Rightarrow> int"} &
    19.1 --- a/src/Doc/Prog_Prove/Basics.thy	Wed Jan 10 15:21:49 2018 +0100
    19.2 +++ b/src/Doc/Prog_Prove/Basics.thy	Wed Jan 10 15:25:09 2018 +0100
    19.3 @@ -40,9 +40,9 @@
    19.4  
    19.5  \begin{warn}
    19.6  There are many predefined infix symbols like @{text "+"} and @{text"\<le>"}.
    19.7 -The name of the corresponding binary function is @{term"op +"},
    19.8 +The name of the corresponding binary function is @{term"(+)"},
    19.9  not just @{text"+"}. That is, @{term"x + y"} is nice surface syntax
   19.10 -(``syntactic sugar'') for \noquotes{@{term[source]"op + x y"}}.
   19.11 +(``syntactic sugar'') for \noquotes{@{term[source]"(+) x y"}}.
   19.12  \end{warn}
   19.13  
   19.14  HOL also supports some basic constructs from functional programming:
   19.15 @@ -152,4 +152,4 @@
   19.16  *}
   19.17  (*<*)
   19.18  end
   19.19 -(*>*)
   19.20 \ No newline at end of file
   19.21 +(*>*)
    20.1 --- a/src/Doc/Prog_Prove/LaTeXsugar.thy	Wed Jan 10 15:21:49 2018 +0100
    20.2 +++ b/src/Doc/Prog_Prove/LaTeXsugar.thy	Wed Jan 10 15:25:09 2018 +0100
    20.3 @@ -60,4 +60,4 @@
    20.4  *}
    20.5  
    20.6  end
    20.7 -(*>*)
    20.8 \ No newline at end of file
    20.9 +(*>*)
    21.1 --- a/src/Doc/System/Environment.thy	Wed Jan 10 15:21:49 2018 +0100
    21.2 +++ b/src/Doc/System/Environment.thy	Wed Jan 10 15:25:09 2018 +0100
    21.3 @@ -505,4 +505,4 @@
    21.4    characters are \<open>\<^bold>X\<^bold>Y\<close>.
    21.5  \<close>
    21.6  
    21.7 -end
    21.8 \ No newline at end of file
    21.9 +end
    22.1 --- a/src/Doc/System/Misc.thy	Wed Jan 10 15:21:49 2018 +0100
    22.2 +++ b/src/Doc/System/Misc.thy	Wed Jan 10 15:25:09 2018 +0100
    22.3 @@ -226,4 +226,4 @@
    22.4    id of the @{setting ISABELLE_HOME} directory.
    22.5  \<close>
    22.6  
    22.7 -end
    22.8 \ No newline at end of file
    22.9 +end
    23.1 --- a/src/Doc/System/Presentation.thy	Wed Jan 10 15:21:49 2018 +0100
    23.2 +++ b/src/Doc/System/Presentation.thy	Wed Jan 10 15:25:09 2018 +0100
    23.3 @@ -286,4 +286,4 @@
    23.4  isabelle latex -o pdf\<close>}
    23.5  \<close>
    23.6  
    23.7 -end
    23.8 \ No newline at end of file
    23.9 +end
    24.1 --- a/src/Doc/Tutorial/Types/Pairs.thy	Wed Jan 10 15:21:49 2018 +0100
    24.2 +++ b/src/Doc/Tutorial/Types/Pairs.thy	Wed Jan 10 15:25:09 2018 +0100
    24.3 @@ -110,7 +110,7 @@
    24.4  @{subgoals[display,indent=0]}
    24.5  Again, simplification produces a term suitable for @{thm[source]prod.split}
    24.6  as above. If you are worried about the strange form of the premise:
    24.7 -@{text"case_prod (op =)"} is short for @{term"\<lambda>(x,y). x=y"}.
    24.8 +@{text"case_prod (=)"} is short for @{term"\<lambda>(x,y). x=y"}.
    24.9  The same proof procedure works for
   24.10  *}
   24.11  
    25.1 --- a/src/FOL/ex/Locale_Test/Locale_Test1.thy	Wed Jan 10 15:21:49 2018 +0100
    25.2 +++ b/src/FOL/ex/Locale_Test/Locale_Test1.thy	Wed Jan 10 15:25:09 2018 +0100
    25.3 @@ -68,7 +68,7 @@
    25.4      and linv: "inv(x) ** x = one"
    25.5  print_locale! lgrp thm lgrp_def lgrp_axioms_def
    25.6  
    25.7 -locale add_lgrp = semi "op ++" for sum (infixl "++" 60) +
    25.8 +locale add_lgrp = semi "(++)" for sum (infixl "++" 60) +
    25.9    fixes zero and neg
   25.10    assumes lzero: "zero ++ x = x"
   25.11      and lneg: "neg(x) ++ x = zero"
   25.12 @@ -106,7 +106,7 @@
   25.13  end
   25.14  print_locale! logic
   25.15  
   25.16 -locale use_decl = l: logic + semi "op ||"
   25.17 +locale use_decl = l: logic + semi "(||)"
   25.18  print_locale! use_decl thm use_decl_def
   25.19  
   25.20  locale extra_type =
   25.21 @@ -376,7 +376,7 @@
   25.22  
   25.23  end
   25.24  
   25.25 -sublocale order_with_def < dual: order' "op >>"
   25.26 +sublocale order_with_def < dual: order' "(>>)"
   25.27    apply unfold_locales
   25.28    unfolding greater_def
   25.29    apply (rule refl) apply (blast intro: trans)
   25.30 @@ -438,13 +438,13 @@
   25.31  
   25.32  subsection \<open>Sublocale, then interpretation in theory\<close>
   25.33  
   25.34 -interpretation int?: lgrp "op +" "0" "minus"
   25.35 +interpretation int?: lgrp "(+)" "0" "minus"
   25.36  proof unfold_locales
   25.37  qed (rule int_assoc int_zero int_minus)+
   25.38  
   25.39  thm int.assoc int.semi_axioms
   25.40  
   25.41 -interpretation int2?: semi "op +"
   25.42 +interpretation int2?: semi "(+)"
   25.43    by unfold_locales  (* subsumed, thm int2.assoc not generated *)
   25.44  
   25.45  ML \<open>(Global_Theory.get_thms @{theory} "int2.assoc";
   25.46 @@ -457,7 +457,7 @@
   25.47  
   25.48  subsection \<open>Interpretation in theory, then sublocale\<close>
   25.49  
   25.50 -interpretation fol: logic "op +" "minus"
   25.51 +interpretation fol: logic "(+)" "minus"
   25.52    by unfold_locales (rule int_assoc int_minus2)+
   25.53  
   25.54  locale logic2 =
   25.55 @@ -503,11 +503,11 @@
   25.56  
   25.57  end
   25.58  
   25.59 -interpretation x: logic_o "op \<and>" "Not"
   25.60 +interpretation x: logic_o "(\<and>)" "Not"
   25.61    rewrites bool_logic_o: "x.lor_o(x, y) \<longleftrightarrow> x \<or> y"
   25.62  proof -
   25.63 -  show bool_logic_o: "PROP logic_o(op \<and>, Not)" by unfold_locales fast+
   25.64 -  show "logic_o.lor_o(op \<and>, Not, x, y) \<longleftrightarrow> x \<or> y"
   25.65 +  show bool_logic_o: "PROP logic_o((\<and>), Not)" by unfold_locales fast+
   25.66 +  show "logic_o.lor_o((\<and>), Not, x, y) \<longleftrightarrow> x \<or> y"
   25.67      by (unfold logic_o.lor_o_def [OF bool_logic_o]) fast
   25.68  qed
   25.69  
   25.70 @@ -759,8 +759,8 @@
   25.71  
   25.72  text \<open>Interpreted versions\<close>
   25.73  
   25.74 -lemma "0 = glob_one ((op +))" by (rule int.def.one_def)
   25.75 -lemma "- x = glob_inv(op +, x)" by (rule int.def.inv_def)
   25.76 +lemma "0 = glob_one ((+))" by (rule int.def.one_def)
   25.77 +lemma "- x = glob_inv((+), x)" by (rule int.def.inv_def)
   25.78  
   25.79  text \<open>Roundup applies rewrite morphisms at declaration level in DFS tree\<close>
   25.80  
   25.81 @@ -792,18 +792,18 @@
   25.82  
   25.83  lemma True
   25.84  proof
   25.85 -  interpret "local": lgrp "op +" "0" "minus"
   25.86 +  interpret "local": lgrp "(+)" "0" "minus"
   25.87      by unfold_locales  (* subsumed *)
   25.88    {
   25.89      fix zero :: int
   25.90      assume "!!x. zero + x = x" "!!x. (-x) + x = zero"
   25.91 -    then interpret local_fixed: lgrp "op +" zero "minus"
   25.92 +    then interpret local_fixed: lgrp "(+)" zero "minus"
   25.93        by unfold_locales
   25.94      thm local_fixed.lone
   25.95 -    print_dependencies! lgrp "op +" 0 minus + lgrp "op +" zero minus
   25.96 +    print_dependencies! lgrp "(+)" 0 minus + lgrp "(+)" zero minus
   25.97    }
   25.98    assume "!!x zero. zero + x = x" "!!x zero. (-x) + x = zero"
   25.99 -  then interpret local_free: lgrp "op +" zero "minus" for zero
  25.100 +  then interpret local_free: lgrp "(+)" zero "minus" for zero
  25.101      by unfold_locales
  25.102    thm local_free.lone [where ?zero = 0]
  25.103  qed
    26.1 --- a/src/FOLP/IFOLP.thy	Wed Jan 10 15:21:49 2018 +0100
    26.2 +++ b/src/FOLP/IFOLP.thy	Wed Jan 10 15:25:09 2018 +0100
    26.3 @@ -259,7 +259,7 @@
    26.4            and concl = discard_proof (Logic.strip_assums_concl prem)
    26.5        in
    26.6            if exists (fn hyp => hyp aconv concl) hyps
    26.7 -          then case distinct (op =) (filter (fn hyp => Term.could_unify (hyp, concl)) hyps) of
    26.8 +          then case distinct (=) (filter (fn hyp => Term.could_unify (hyp, concl)) hyps) of
    26.9                     [_] => assume_tac ctxt i
   26.10                   |  _  => no_tac
   26.11            else no_tac
   26.12 @@ -522,7 +522,7 @@
   26.13  lemmas pred_congs = pred1_cong pred2_cong pred3_cong
   26.14  
   26.15  (*special case for the equality predicate!*)
   26.16 -lemmas eq_cong = pred2_cong [where P = "op ="]
   26.17 +lemmas eq_cong = pred2_cong [where P = "(=)"]
   26.18  
   26.19  
   26.20  (*** Simplifications of assumed implications.
    27.1 --- a/src/HOL/Algebra/Complete_Lattice.thy	Wed Jan 10 15:21:49 2018 +0100
    27.2 +++ b/src/HOL/Algebra/Complete_Lattice.thy	Wed Jan 10 15:25:09 2018 +0100
    27.3 @@ -1171,7 +1171,7 @@
    27.4  subsubsection \<open>The Powerset of a Set is a Complete Lattice\<close>
    27.5  
    27.6  theorem powerset_is_complete_lattice:
    27.7 -  "complete_lattice \<lparr>carrier = Pow A, eq = op =, le = (op \<subseteq>)\<rparr>"
    27.8 +  "complete_lattice \<lparr>carrier = Pow A, eq = (=), le = (\<subseteq>)\<rparr>"
    27.9    (is "complete_lattice ?L")
   27.10  proof (rule partial_order.complete_latticeI)
   27.11    show "partial_order ?L"
    28.1 --- a/src/HOL/Algebra/Divisibility.thy	Wed Jan 10 15:21:49 2018 +0100
    28.2 +++ b/src/HOL/Algebra/Divisibility.thy	Wed Jan 10 15:25:09 2018 +0100
    28.3 @@ -145,7 +145,7 @@
    28.4  definition associated :: "[_, 'a, 'a] \<Rightarrow> bool" (infix "\<sim>\<index>" 55)
    28.5    where "a \<sim>\<^bsub>G\<^esub> b \<longleftrightarrow> a divides\<^bsub>G\<^esub> b \<and> b divides\<^bsub>G\<^esub> a"
    28.6  
    28.7 -abbreviation "division_rel G \<equiv> \<lparr>carrier = carrier G, eq = op \<sim>\<^bsub>G\<^esub>, le = (op divides\<^bsub>G\<^esub>)\<rparr>"
    28.8 +abbreviation "division_rel G \<equiv> \<lparr>carrier = carrier G, eq = (\<sim>\<^bsub>G\<^esub>), le = (divides\<^bsub>G\<^esub>)\<rparr>"
    28.9  
   28.10  definition properfactor :: "[_, 'a, 'a] \<Rightarrow> bool"
   28.11    where "properfactor G a b \<longleftrightarrow> a divides\<^bsub>G\<^esub> b \<and> \<not>(b divides\<^bsub>G\<^esub> a)"
   28.12 @@ -857,13 +857,13 @@
   28.13  subsubsection \<open>Function definitions\<close>
   28.14  
   28.15  definition factors :: "[_, 'a list, 'a] \<Rightarrow> bool"
   28.16 -  where "factors G fs a \<longleftrightarrow> (\<forall>x \<in> (set fs). irreducible G x) \<and> foldr (op \<otimes>\<^bsub>G\<^esub>) fs \<one>\<^bsub>G\<^esub> = a"
   28.17 +  where "factors G fs a \<longleftrightarrow> (\<forall>x \<in> (set fs). irreducible G x) \<and> foldr (\<otimes>\<^bsub>G\<^esub>) fs \<one>\<^bsub>G\<^esub> = a"
   28.18  
   28.19  definition wfactors ::"[_, 'a list, 'a] \<Rightarrow> bool"
   28.20 -  where "wfactors G fs a \<longleftrightarrow> (\<forall>x \<in> (set fs). irreducible G x) \<and> foldr (op \<otimes>\<^bsub>G\<^esub>) fs \<one>\<^bsub>G\<^esub> \<sim>\<^bsub>G\<^esub> a"
   28.21 +  where "wfactors G fs a \<longleftrightarrow> (\<forall>x \<in> (set fs). irreducible G x) \<and> foldr (\<otimes>\<^bsub>G\<^esub>) fs \<one>\<^bsub>G\<^esub> \<sim>\<^bsub>G\<^esub> a"
   28.22  
   28.23  abbreviation list_assoc :: "('a,_) monoid_scheme \<Rightarrow> 'a list \<Rightarrow> 'a list \<Rightarrow> bool" (infix "[\<sim>]\<index>" 44)
   28.24 -  where "list_assoc G \<equiv> list_all2 (op \<sim>\<^bsub>G\<^esub>)"
   28.25 +  where "list_assoc G \<equiv> list_all2 (\<sim>\<^bsub>G\<^esub>)"
   28.26  
   28.27  definition essentially_equal :: "[_, 'a list, 'a list] \<Rightarrow> bool"
   28.28    where "essentially_equal G fs1 fs2 \<longleftrightarrow> (\<exists>fs1'. fs1 <~~> fs1' \<and> fs1' [\<sim>]\<^bsub>G\<^esub> fs2)"
   28.29 @@ -1043,12 +1043,12 @@
   28.30  
   28.31  lemma (in monoid) multlist_closed [simp, intro]:
   28.32    assumes ascarr: "set fs \<subseteq> carrier G"
   28.33 -  shows "foldr (op \<otimes>) fs \<one> \<in> carrier G"
   28.34 +  shows "foldr (\<otimes>) fs \<one> \<in> carrier G"
   28.35    using ascarr by (induct fs) simp_all
   28.36  
   28.37  lemma  (in comm_monoid) multlist_dividesI (*[intro]*):
   28.38    assumes "f \<in> set fs" and "f \<in> carrier G" and "set fs \<subseteq> carrier G"
   28.39 -  shows "f divides (foldr (op \<otimes>) fs \<one>)"
   28.40 +  shows "f divides (foldr (\<otimes>) fs \<one>)"
   28.41    using assms
   28.42    apply (induct fs)
   28.43     apply simp
   28.44 @@ -1062,7 +1062,7 @@
   28.45  lemma (in comm_monoid_cancel) multlist_listassoc_cong:
   28.46    assumes "fs [\<sim>] fs'"
   28.47      and "set fs \<subseteq> carrier G" and "set fs' \<subseteq> carrier G"
   28.48 -  shows "foldr (op \<otimes>) fs \<one> \<sim> foldr (op \<otimes>) fs' \<one>"
   28.49 +  shows "foldr (\<otimes>) fs \<one> \<sim> foldr (\<otimes>) fs' \<one>"
   28.50    using assms
   28.51  proof (induct fs arbitrary: fs', simp)
   28.52    case (Cons a as fs')
   28.53 @@ -1073,16 +1073,16 @@
   28.54      assume "a \<sim> b"
   28.55        and acarr: "a \<in> carrier G" and bcarr: "b \<in> carrier G"
   28.56        and ascarr: "set as \<subseteq> carrier G"
   28.57 -    then have p: "a \<otimes> foldr op \<otimes> as \<one> \<sim> b \<otimes> foldr op \<otimes> as \<one>"
   28.58 +    then have p: "a \<otimes> foldr (\<otimes>) as \<one> \<sim> b \<otimes> foldr (\<otimes>) as \<one>"
   28.59        by (fast intro: mult_cong_l)
   28.60      also
   28.61      assume "as [\<sim>] bs"
   28.62        and bscarr: "set bs \<subseteq> carrier G"
   28.63 -      and "\<And>fs'. \<lbrakk>as [\<sim>] fs'; set fs' \<subseteq> carrier G\<rbrakk> \<Longrightarrow> foldr op \<otimes> as \<one> \<sim> foldr op \<otimes> fs' \<one>"
   28.64 -    then have "foldr op \<otimes> as \<one> \<sim> foldr op \<otimes> bs \<one>" by simp
   28.65 -    with ascarr bscarr bcarr have "b \<otimes> foldr op \<otimes> as \<one> \<sim> b \<otimes> foldr op \<otimes> bs \<one>"
   28.66 +      and "\<And>fs'. \<lbrakk>as [\<sim>] fs'; set fs' \<subseteq> carrier G\<rbrakk> \<Longrightarrow> foldr (\<otimes>) as \<one> \<sim> foldr (\<otimes>) fs' \<one>"
   28.67 +    then have "foldr (\<otimes>) as \<one> \<sim> foldr (\<otimes>) bs \<one>" by simp
   28.68 +    with ascarr bscarr bcarr have "b \<otimes> foldr (\<otimes>) as \<one> \<sim> b \<otimes> foldr (\<otimes>) bs \<one>"
   28.69        by (fast intro: mult_cong_r)
   28.70 -    finally show "a \<otimes> foldr op \<otimes> as \<one> \<sim> b \<otimes> foldr op \<otimes> bs \<one>"
   28.71 +    finally show "a \<otimes> foldr (\<otimes>) as \<one> \<sim> b \<otimes> foldr (\<otimes>) bs \<one>"
   28.72        by (simp add: ascarr bscarr acarr bcarr)
   28.73    qed
   28.74  qed
   28.75 @@ -1090,21 +1090,21 @@
   28.76  lemma (in comm_monoid) multlist_perm_cong:
   28.77    assumes prm: "as <~~> bs"
   28.78      and ascarr: "set as \<subseteq> carrier G"
   28.79 -  shows "foldr (op \<otimes>) as \<one> = foldr (op \<otimes>) bs \<one>"
   28.80 +  shows "foldr (\<otimes>) as \<one> = foldr (\<otimes>) bs \<one>"
   28.81    using prm ascarr
   28.82    apply (induct, simp, clarsimp simp add: m_ac, clarsimp)
   28.83  proof clarsimp
   28.84    fix xs ys zs
   28.85    assume "xs <~~> ys"  "set xs \<subseteq> carrier G"
   28.86    then have "set ys \<subseteq> carrier G" by (rule perm_closed)
   28.87 -  moreover assume "set ys \<subseteq> carrier G \<Longrightarrow> foldr op \<otimes> ys \<one> = foldr op \<otimes> zs \<one>"
   28.88 -  ultimately show "foldr op \<otimes> ys \<one> = foldr op \<otimes> zs \<one>" by simp
   28.89 +  moreover assume "set ys \<subseteq> carrier G \<Longrightarrow> foldr (\<otimes>) ys \<one> = foldr (\<otimes>) zs \<one>"
   28.90 +  ultimately show "foldr (\<otimes>) ys \<one> = foldr (\<otimes>) zs \<one>" by simp
   28.91  qed
   28.92  
   28.93  lemma (in comm_monoid_cancel) multlist_ee_cong:
   28.94    assumes "essentially_equal G fs fs'"
   28.95      and "set fs \<subseteq> carrier G" and "set fs' \<subseteq> carrier G"
   28.96 -  shows "foldr (op \<otimes>) fs \<one> \<sim> foldr (op \<otimes>) fs' \<one>"
   28.97 +  shows "foldr (\<otimes>) fs \<one> \<sim> foldr (\<otimes>) fs' \<one>"
   28.98    using assms
   28.99    apply (elim essentially_equalE)
  28.100    apply (simp add: multlist_perm_cong multlist_listassoc_cong perm_closed)
  28.101 @@ -1116,27 +1116,27 @@
  28.102  lemma wfactorsI:
  28.103    fixes G (structure)
  28.104    assumes "\<forall>f\<in>set fs. irreducible G f"
  28.105 -    and "foldr (op \<otimes>) fs \<one> \<sim> a"
  28.106 +    and "foldr (\<otimes>) fs \<one> \<sim> a"
  28.107    shows "wfactors G fs a"
  28.108    using assms unfolding wfactors_def by simp
  28.109  
  28.110  lemma wfactorsE:
  28.111    fixes G (structure)
  28.112    assumes wf: "wfactors G fs a"
  28.113 -    and e: "\<lbrakk>\<forall>f\<in>set fs. irreducible G f; foldr (op \<otimes>) fs \<one> \<sim> a\<rbrakk> \<Longrightarrow> P"
  28.114 +    and e: "\<lbrakk>\<forall>f\<in>set fs. irreducible G f; foldr (\<otimes>) fs \<one> \<sim> a\<rbrakk> \<Longrightarrow> P"
  28.115    shows "P"
  28.116    using wf unfolding wfactors_def by (fast dest: e)
  28.117  
  28.118  lemma (in monoid) factorsI:
  28.119    assumes "\<forall>f\<in>set fs. irreducible G f"
  28.120 -    and "foldr (op \<otimes>) fs \<one> = a"
  28.121 +    and "foldr (\<otimes>) fs \<one> = a"
  28.122    shows "factors G fs a"
  28.123    using assms unfolding factors_def by simp
  28.124  
  28.125  lemma factorsE:
  28.126    fixes G (structure)
  28.127    assumes f: "factors G fs a"
  28.128 -    and e: "\<lbrakk>\<forall>f\<in>set fs. irreducible G f; foldr (op \<otimes>) fs \<one> = a\<rbrakk> \<Longrightarrow> P"
  28.129 +    and e: "\<lbrakk>\<forall>f\<in>set fs. irreducible G f; foldr (\<otimes>) fs \<one> = a\<rbrakk> \<Longrightarrow> P"
  28.130    shows "P"
  28.131    using f unfolding factors_def by (simp add: e)
  28.132  
  28.133 @@ -1185,14 +1185,14 @@
  28.134    from fs wf have "irreducible G f" by (simp add: wfactors_def)
  28.135    then have fnunit: "f \<notin> Units G" by (fast elim: irreducibleE)
  28.136  
  28.137 -  from fs wf have a: "f \<otimes> foldr (op \<otimes>) fs' \<one> \<sim> a" by (simp add: wfactors_def)
  28.138 +  from fs wf have a: "f \<otimes> foldr (\<otimes>) fs' \<one> \<sim> a" by (simp add: wfactors_def)
  28.139  
  28.140    note aunit
  28.141    also from fs wf
  28.142 -  have a: "f \<otimes> foldr (op \<otimes>) fs' \<one> \<sim> a" by (simp add: wfactors_def)
  28.143 -  have "a \<sim> f \<otimes> foldr (op \<otimes>) fs' \<one>"
  28.144 +  have a: "f \<otimes> foldr (\<otimes>) fs' \<one> \<sim> a" by (simp add: wfactors_def)
  28.145 +  have "a \<sim> f \<otimes> foldr (\<otimes>) fs' \<one>"
  28.146      by (simp add: Units_closed[OF aunit] a[symmetric])
  28.147 -  finally have "f \<otimes> foldr (op \<otimes>) fs' \<one> \<in> Units G" by simp
  28.148 +  finally have "f \<otimes> foldr (\<otimes>) fs' \<one> \<in> Units G" by simp
  28.149    then have "f \<in> Units G" by (intro unit_factor[of f], simp+)
  28.150    with fnunit show ?thesis by contradiction
  28.151  qed
  28.152 @@ -1209,10 +1209,10 @@
  28.153    apply (elim wfactorsE, intro wfactorsI)
  28.154     apply (metis assms(2) assms(4) assms(5) irrlist_listassoc_cong)
  28.155  proof -
  28.156 -  from asc[symmetric] have "foldr op \<otimes> fs' \<one> \<sim> foldr op \<otimes> fs \<one>"
  28.157 +  from asc[symmetric] have "foldr (\<otimes>) fs' \<one> \<sim> foldr (\<otimes>) fs \<one>"
  28.158      by (simp add: multlist_listassoc_cong carr)
  28.159 -  also assume "foldr op \<otimes> fs \<one> \<sim> a"
  28.160 -  finally show "foldr op \<otimes> fs' \<one> \<sim> a" by (simp add: carr)
  28.161 +  also assume "foldr (\<otimes>) fs \<one> \<sim> a"
  28.162 +  finally show "foldr (\<otimes>) fs' \<one> \<sim> a" by (simp add: carr)
  28.163  qed
  28.164  
  28.165  lemma (in comm_monoid) wfactors_perm_cong_l:
  28.166 @@ -1251,8 +1251,8 @@
  28.167    shows "wfactors G fs a'"
  28.168    using fac
  28.169  proof (elim wfactorsE, intro wfactorsI)
  28.170 -  assume "foldr op \<otimes> fs \<one> \<sim> a" also note aa'
  28.171 -  finally show "foldr op \<otimes> fs \<one> \<sim> a'" by simp
  28.172 +  assume "foldr (\<otimes>) fs \<one> \<sim> a" also note aa'
  28.173 +  finally show "foldr (\<otimes>) fs \<one> \<sim> a'" by simp
  28.174  qed
  28.175  
  28.176  
  28.177 @@ -1296,11 +1296,11 @@
  28.178    using afs bfs
  28.179  proof (elim wfactorsE)
  28.180    from prm have [simp]: "set bs \<subseteq> carrier G" by (simp add: perm_closed)
  28.181 -  assume "foldr op \<otimes> as \<one> \<sim> a"
  28.182 -  then have "a \<sim> foldr op \<otimes> as \<one>" by (rule associated_sym, simp+)
  28.183 +  assume "foldr (\<otimes>) as \<one> \<sim> a"
  28.184 +  then have "a \<sim> foldr (\<otimes>) as \<one>" by (rule associated_sym, simp+)
  28.185    also from prm
  28.186 -  have "foldr op \<otimes> as \<one> = foldr op \<otimes> bs \<one>" by (rule multlist_perm_cong, simp)
  28.187 -  also assume "foldr op \<otimes> bs \<one> \<sim> b"
  28.188 +  have "foldr (\<otimes>) as \<one> = foldr (\<otimes>) bs \<one>" by (rule multlist_perm_cong, simp)
  28.189 +  also assume "foldr (\<otimes>) bs \<one> \<sim> b"
  28.190    finally show "a \<sim> b" by simp
  28.191  qed
  28.192  
  28.193 @@ -1313,11 +1313,11 @@
  28.194    shows "a \<sim> b"
  28.195    using afs bfs
  28.196  proof (elim wfactorsE)
  28.197 -  assume "foldr op \<otimes> as \<one> \<sim> a"
  28.198 -  then have "a \<sim> foldr op \<otimes> as \<one>" by (rule associated_sym, simp+)
  28.199 +  assume "foldr (\<otimes>) as \<one> \<sim> a"
  28.200 +  then have "a \<sim> foldr (\<otimes>) as \<one>" by (rule associated_sym, simp+)
  28.201    also from assoc
  28.202 -  have "foldr op \<otimes> as \<one> \<sim> foldr op \<otimes> bs \<one>" by (rule multlist_listassoc_cong, simp+)
  28.203 -  also assume "foldr op \<otimes> bs \<one> \<sim> b"
  28.204 +  have "foldr (\<otimes>) as \<one> \<sim> foldr (\<otimes>) bs \<one>" by (rule multlist_listassoc_cong, simp+)
  28.205 +  also assume "foldr (\<otimes>) bs \<one> \<sim> b"
  28.206    finally show "a \<sim> b" by simp
  28.207  qed
  28.208  
  28.209 @@ -2912,10 +2912,10 @@
  28.210  lemma (in comm_monoid_cancel) multlist_prime_pos:
  28.211    assumes carr: "a \<in> carrier G"  "set as \<subseteq> carrier G"
  28.212      and aprime: "prime G a"
  28.213 -    and "a divides (foldr (op \<otimes>) as \<one>)"
  28.214 +    and "a divides (foldr (\<otimes>) as \<one>)"
  28.215    shows "\<exists>i<length as. a divides (as!i)"
  28.216  proof -
  28.217 -  have r[rule_format]: "set as \<subseteq> carrier G \<and> a divides (foldr (op \<otimes>) as \<one>)
  28.218 +  have r[rule_format]: "set as \<subseteq> carrier G \<and> a divides (foldr (\<otimes>) as \<one>)
  28.219      \<longrightarrow> (\<exists>i. i < length as \<and> a divides (as!i))"
  28.220      apply (induct as)
  28.221       apply clarsimp defer 1
  28.222 @@ -2928,10 +2928,10 @@
  28.223        by (elim primeE, simp)
  28.224    next
  28.225      fix aa as
  28.226 -    assume ih[rule_format]: "a divides foldr op \<otimes> as \<one> \<longrightarrow> (\<exists>i<length as. a divides as ! i)"
  28.227 +    assume ih[rule_format]: "a divides foldr (\<otimes>) as \<one> \<longrightarrow> (\<exists>i<length as. a divides as ! i)"
  28.228        and carr': "aa \<in> carrier G"  "set as \<subseteq> carrier G"
  28.229 -      and "a divides aa \<otimes> foldr op \<otimes> as \<one>"
  28.230 -    with carr aprime have "a divides aa \<or> a divides foldr op \<otimes> as \<one>"
  28.231 +      and "a divides aa \<otimes> foldr (\<otimes>) as \<one>"
  28.232 +    with carr aprime have "a divides aa \<or> a divides foldr (\<otimes>) as \<one>"
  28.233        by (intro prime_divides) simp+
  28.234      then show "\<exists>i<Suc (length as). a divides (aa # as) ! i"
  28.235      proof
  28.236 @@ -2940,7 +2940,7 @@
  28.237        have "0 < Suc (length as)" by simp
  28.238        with p1 show ?thesis by fast
  28.239      next
  28.240 -      assume "a divides foldr op \<otimes> as \<one>"
  28.241 +      assume "a divides foldr (\<otimes>) as \<one>"
  28.242        from ih [OF this] obtain i where "a divides as ! i" and len: "i < length as" by auto
  28.243        then have p1: "a divides (aa#as) ! (Suc i)" by simp
  28.244        from len have "Suc i < Suc (length as)" by simp
  28.245 @@ -2998,9 +2998,9 @@
  28.246      note carr [simp] = acarr ahcarr ascarr as'carr a'carr
  28.247  
  28.248      note ahdvda
  28.249 -    also from afs' have "a divides (foldr (op \<otimes>) as' \<one>)"
  28.250 +    also from afs' have "a divides (foldr (\<otimes>) as' \<one>)"
  28.251        by (elim wfactorsE associatedE, simp)
  28.252 -    finally have "ah divides (foldr (op \<otimes>) as' \<one>)"
  28.253 +    finally have "ah divides (foldr (\<otimes>) as' \<one>)"
  28.254        by simp
  28.255      with ahprime have "\<exists>i<length as'. ah divides as'!i"
  28.256        by (intro multlist_prime_pos) simp_all
    29.1 --- a/src/HOL/Algebra/Group.thy	Wed Jan 10 15:21:49 2018 +0100
    29.2 +++ b/src/HOL/Algebra/Group.thy	Wed Jan 10 15:25:09 2018 +0100
    29.3 @@ -676,7 +676,7 @@
    29.4  
    29.5  (* Contributed by Joachim Breitner *)
    29.6  lemma (in group) int_pow_is_hom:
    29.7 -  "x \<in> carrier G \<Longrightarrow> (op[^] x) \<in> hom \<lparr> carrier = UNIV, mult = op +, one = 0::int \<rparr> G "
    29.8 +  "x \<in> carrier G \<Longrightarrow> (([^]) x) \<in> hom \<lparr> carrier = UNIV, mult = (+), one = 0::int \<rparr> G "
    29.9    unfolding hom_def by (simp add: int_pow_mult)
   29.10  
   29.11  
   29.12 @@ -773,7 +773,7 @@
   29.13  text_raw \<open>\label{sec:subgroup-lattice}\<close>
   29.14  
   29.15  theorem (in group) subgroups_partial_order:
   29.16 -  "partial_order \<lparr>carrier = {H. subgroup H G}, eq = op =, le = (op \<subseteq>)\<rparr>"
   29.17 +  "partial_order \<lparr>carrier = {H. subgroup H G}, eq = (=), le = (\<subseteq>)\<rparr>"
   29.18    by standard simp_all
   29.19  
   29.20  lemma (in group) subgroup_self:
   29.21 @@ -818,7 +818,7 @@
   29.22  qed
   29.23  
   29.24  theorem (in group) subgroups_complete_lattice:
   29.25 -  "complete_lattice \<lparr>carrier = {H. subgroup H G}, eq = op =, le = (op \<subseteq>)\<rparr>"
   29.26 +  "complete_lattice \<lparr>carrier = {H. subgroup H G}, eq = (=), le = (\<subseteq>)\<rparr>"
   29.27      (is "complete_lattice ?L")
   29.28  proof (rule partial_order.complete_lattice_criterion1)
   29.29    show "partial_order ?L" by (rule subgroups_partial_order)
    30.1 --- a/src/HOL/Algebra/IntRing.thy	Wed Jan 10 15:21:49 2018 +0100
    30.2 +++ b/src/HOL/Algebra/IntRing.thy	Wed Jan 10 15:25:09 2018 +0100
    30.3 @@ -23,7 +23,7 @@
    30.4  subsection \<open>\<open>\<Z>\<close>: The Set of Integers as Algebraic Structure\<close>
    30.5  
    30.6  abbreviation int_ring :: "int ring" ("\<Z>")
    30.7 -  where "int_ring \<equiv> \<lparr>carrier = UNIV, mult = op *, one = 1, zero = 0, add = (op +)\<rparr>"
    30.8 +  where "int_ring \<equiv> \<lparr>carrier = UNIV, mult = ( * ), one = 1, zero = 0, add = (+)\<rparr>"
    30.9  
   30.10  lemma int_Zcarr [intro!, simp]: "k \<in> carrier \<Z>"
   30.11    by simp
   30.12 @@ -172,27 +172,27 @@
   30.13    by simp_all
   30.14  
   30.15  interpretation int (* FIXME [unfolded UNIV] *) :
   30.16 -  partial_order "\<lparr>carrier = UNIV::int set, eq = op =, le = (op \<le>)\<rparr>"
   30.17 -  rewrites "carrier \<lparr>carrier = UNIV::int set, eq = op =, le = (op \<le>)\<rparr> = UNIV"
   30.18 -    and "le \<lparr>carrier = UNIV::int set, eq = op =, le = (op \<le>)\<rparr> x y = (x \<le> y)"
   30.19 -    and "lless \<lparr>carrier = UNIV::int set, eq = op =, le = (op \<le>)\<rparr> x y = (x < y)"
   30.20 +  partial_order "\<lparr>carrier = UNIV::int set, eq = (=), le = (\<le>)\<rparr>"
   30.21 +  rewrites "carrier \<lparr>carrier = UNIV::int set, eq = (=), le = (\<le>)\<rparr> = UNIV"
   30.22 +    and "le \<lparr>carrier = UNIV::int set, eq = (=), le = (\<le>)\<rparr> x y = (x \<le> y)"
   30.23 +    and "lless \<lparr>carrier = UNIV::int set, eq = (=), le = (\<le>)\<rparr> x y = (x < y)"
   30.24  proof -
   30.25 -  show "partial_order \<lparr>carrier = UNIV::int set, eq = op =, le = (op \<le>)\<rparr>"
   30.26 +  show "partial_order \<lparr>carrier = UNIV::int set, eq = (=), le = (\<le>)\<rparr>"
   30.27      by standard simp_all
   30.28 -  show "carrier \<lparr>carrier = UNIV::int set, eq = op =, le = (op \<le>)\<rparr> = UNIV"
   30.29 +  show "carrier \<lparr>carrier = UNIV::int set, eq = (=), le = (\<le>)\<rparr> = UNIV"
   30.30      by simp
   30.31 -  show "le \<lparr>carrier = UNIV::int set, eq = op =, le = (op \<le>)\<rparr> x y = (x \<le> y)"
   30.32 +  show "le \<lparr>carrier = UNIV::int set, eq = (=), le = (\<le>)\<rparr> x y = (x \<le> y)"
   30.33      by simp
   30.34 -  show "lless \<lparr>carrier = UNIV::int set, eq = op =, le = (op \<le>)\<rparr> x y = (x < y)"
   30.35 +  show "lless \<lparr>carrier = UNIV::int set, eq = (=), le = (\<le>)\<rparr> x y = (x < y)"
   30.36      by (simp add: lless_def) auto
   30.37  qed
   30.38  
   30.39  interpretation int (* FIXME [unfolded UNIV] *) :
   30.40 -  lattice "\<lparr>carrier = UNIV::int set, eq = op =, le = (op \<le>)\<rparr>"
   30.41 -  rewrites "join \<lparr>carrier = UNIV::int set, eq = op =, le = (op \<le>)\<rparr> x y = max x y"
   30.42 -    and "meet \<lparr>carrier = UNIV::int set, eq = op =, le = (op \<le>)\<rparr> x y = min x y"
   30.43 +  lattice "\<lparr>carrier = UNIV::int set, eq = (=), le = (\<le>)\<rparr>"
   30.44 +  rewrites "join \<lparr>carrier = UNIV::int set, eq = (=), le = (\<le>)\<rparr> x y = max x y"
   30.45 +    and "meet \<lparr>carrier = UNIV::int set, eq = (=), le = (\<le>)\<rparr> x y = min x y"
   30.46  proof -
   30.47 -  let ?Z = "\<lparr>carrier = UNIV::int set, eq = op =, le = (op \<le>)\<rparr>"
   30.48 +  let ?Z = "\<lparr>carrier = UNIV::int set, eq = (=), le = (\<le>)\<rparr>"
   30.49    show "lattice ?Z"
   30.50      apply unfold_locales
   30.51      apply (simp add: least_def Upper_def)
   30.52 @@ -214,7 +214,7 @@
   30.53  qed
   30.54  
   30.55  interpretation int (* [unfolded UNIV] *) :
   30.56 -  total_order "\<lparr>carrier = UNIV::int set, eq = op =, le = (op \<le>)\<rparr>"
   30.57 +  total_order "\<lparr>carrier = UNIV::int set, eq = (=), le = (\<le>)\<rparr>"
   30.58    by standard clarsimp
   30.59  
   30.60  
    31.1 --- a/src/HOL/Algebra/Multiplicative_Group.thy	Wed Jan 10 15:21:49 2018 +0100
    31.2 +++ b/src/HOL/Algebra/Multiplicative_Group.thy	Wed Jan 10 15:25:09 2018 +0100
    31.3 @@ -580,7 +580,7 @@
    31.4  lemma mult_mult_of: "mult (mult_of R) = mult R"
    31.5   by (simp add: mult_of_def)
    31.6  
    31.7 -lemma nat_pow_mult_of: "op [^]\<^bsub>mult_of R\<^esub> = (op [^]\<^bsub>R\<^esub> :: _ \<Rightarrow> nat \<Rightarrow> _)"
    31.8 +lemma nat_pow_mult_of: "([^]\<^bsub>mult_of R\<^esub>) = (([^]\<^bsub>R\<^esub>) :: _ \<Rightarrow> nat \<Rightarrow> _)"
    31.9    by (simp add: mult_of_def fun_eq_iff nat_pow_def)
   31.10  
   31.11  lemma one_mult_of: "\<one>\<^bsub>mult_of R\<^esub> = \<one>\<^bsub>R\<^esub>"
   31.12 @@ -762,7 +762,7 @@
   31.13    note mult_of_simps[simp]
   31.14    have finite': "finite (carrier (mult_of R))" using finite by (rule finite_mult_of)
   31.15  
   31.16 -  interpret G:group "mult_of R" rewrites "op [^]\<^bsub>mult_of R\<^esub> = (op [^] :: _ \<Rightarrow> nat \<Rightarrow> _)" and "\<one>\<^bsub>mult_of R\<^esub> = \<one>"
   31.17 +  interpret G:group "mult_of R" rewrites "([^]\<^bsub>mult_of R\<^esub>) = (([^]) :: _ \<Rightarrow> nat \<Rightarrow> _)" and "\<one>\<^bsub>mult_of R\<^esub> = \<one>"
   31.18      by (rule field_mult_group) simp_all
   31.19  
   31.20    from exists
   31.21 @@ -826,7 +826,7 @@
   31.22    have finite': "finite (carrier (mult_of R))" using finite by (rule finite_mult_of)
   31.23  
   31.24    interpret G: group "mult_of R" rewrites
   31.25 -      "op [^]\<^bsub>mult_of R\<^esub> = (op [^] :: _ \<Rightarrow> nat \<Rightarrow> _)" and "\<one>\<^bsub>mult_of R\<^esub> = \<one>"
   31.26 +      "([^]\<^bsub>mult_of R\<^esub>) = (([^]) :: _ \<Rightarrow> nat \<Rightarrow> _)" and "\<one>\<^bsub>mult_of R\<^esub> = \<one>"
   31.27      by (rule field_mult_group) (simp_all add: fun_eq_iff nat_pow_def)
   31.28  
   31.29    let ?N = "\<lambda> x . card {a \<in> carrier (mult_of R). group.ord (mult_of R) a  = x}"
    32.1 --- a/src/HOL/Algebra/Order.thy	Wed Jan 10 15:21:49 2018 +0100
    32.2 +++ b/src/HOL/Algebra/Order.thy	Wed Jan 10 15:25:09 2018 +0100
    32.3 @@ -22,7 +22,7 @@
    32.4  abbreviation inv_gorder :: "_ \<Rightarrow> 'a gorder" where
    32.5    "inv_gorder L \<equiv>
    32.6     \<lparr> carrier = carrier L,
    32.7 -     eq = op .=\<^bsub>L\<^esub>,
    32.8 +     eq = (.=\<^bsub>L\<^esub>),
    32.9       le = (\<lambda> x y. y \<sqsubseteq>\<^bsub>L \<^esub>x) \<rparr>"
   32.10  
   32.11  lemma inv_gorder_inv:
   32.12 @@ -591,7 +591,7 @@
   32.13  subsection \<open>Partial orders where \<open>eq\<close> is the Equality\<close>
   32.14  
   32.15  locale partial_order = weak_partial_order +
   32.16 -  assumes eq_is_equal: "op .= = op ="
   32.17 +  assumes eq_is_equal: "(.=) = (=)"
   32.18  begin
   32.19  
   32.20  declare weak_le_antisym [rule del]
    33.1 --- a/src/HOL/Algebra/QuotRing.thy	Wed Jan 10 15:21:49 2018 +0100
    33.2 +++ b/src/HOL/Algebra/QuotRing.thy	Wed Jan 10 15:25:09 2018 +0100
    33.3 @@ -115,7 +115,7 @@
    33.4  
    33.5  text \<open>This is a ring homomorphism\<close>
    33.6  
    33.7 -lemma (in ideal) rcos_ring_hom: "(op +> I) \<in> ring_hom R (R Quot I)"
    33.8 +lemma (in ideal) rcos_ring_hom: "((+>) I) \<in> ring_hom R (R Quot I)"
    33.9  apply (rule ring_hom_memI)
   33.10     apply (simp add: FactRing_def a_rcosetsI[OF a_subset])
   33.11    apply (simp add: FactRing_def rcoset_mult_add)
   33.12 @@ -123,7 +123,7 @@
   33.13  apply (simp add: FactRing_def)
   33.14  done
   33.15  
   33.16 -lemma (in ideal) rcos_ring_hom_ring: "ring_hom_ring R (R Quot I) (op +> I)"
   33.17 +lemma (in ideal) rcos_ring_hom_ring: "ring_hom_ring R (R Quot I) ((+>) I)"
   33.18  apply (rule ring_hom_ringI)
   33.19       apply (rule is_ring, rule quotient_is_ring)
   33.20     apply (simp add: FactRing_def a_rcosetsI[OF a_subset])
   33.21 @@ -151,7 +151,7 @@
   33.22  text \<open>Cosets as a ring homomorphism on crings\<close>
   33.23  lemma (in ideal) rcos_ring_hom_cring:
   33.24    assumes "cring R"
   33.25 -  shows "ring_hom_cring R (R Quot I) (op +> I)"
   33.26 +  shows "ring_hom_cring R (R Quot I) ((+>) I)"
   33.27  proof -
   33.28    interpret cring R by fact
   33.29    show ?thesis
   33.30 @@ -203,7 +203,7 @@
   33.31  
   33.32  text \<open>Generating right cosets of a prime ideal is a homomorphism
   33.33          on commutative rings\<close>
   33.34 -lemma (in primeideal) rcos_ring_hom_cring: "ring_hom_cring R (R Quot I) (op +> I)"
   33.35 +lemma (in primeideal) rcos_ring_hom_cring: "ring_hom_cring R (R Quot I) ((+>) I)"
   33.36    by (rule rcos_ring_hom_cring) (rule is_cring)
   33.37  
   33.38  
    34.1 --- a/src/HOL/Algebra/Sylow.thy	Wed Jan 10 15:21:49 2018 +0100
    34.2 +++ b/src/HOL/Algebra/Sylow.thy	Wed Jan 10 15:25:09 2018 +0100
    34.3 @@ -80,7 +80,7 @@
    34.4    proof
    34.5      show "inj_on (\<lambda>z\<in>H. m1 \<otimes> z) H"
    34.6        by (simp add: inj_on_def l_cancel [of m1 x y, THEN iffD1] H_def m1)
    34.7 -    show "restrict (op \<otimes> m1) H \<in> H \<rightarrow> M1"
    34.8 +    show "restrict ((\<otimes>) m1) H \<in> H \<rightarrow> M1"
    34.9      proof (rule restrictI)
   34.10        fix z
   34.11        assume zH: "z \<in> H"
    35.1 --- a/src/HOL/Algebra/UnivPoly.thy	Wed Jan 10 15:21:49 2018 +0100
    35.2 +++ b/src/HOL/Algebra/UnivPoly.thy	Wed Jan 10 15:25:09 2018 +0100
    35.3 @@ -1808,7 +1808,7 @@
    35.4  
    35.5  definition
    35.6    INTEG :: "int ring"
    35.7 -  where "INTEG = \<lparr>carrier = UNIV, mult = op *, one = 1, zero = 0, add = (op +)\<rparr>"
    35.8 +  where "INTEG = \<lparr>carrier = UNIV, mult = ( * ), one = 1, zero = 0, add = (+)\<rparr>"
    35.9  
   35.10  lemma INTEG_cring: "cring INTEG"
   35.11    by (unfold INTEG_def) (auto intro!: cringI abelian_groupI comm_monoidI
    36.1 --- a/src/HOL/Analysis/Ball_Volume.thy	Wed Jan 10 15:21:49 2018 +0100
    36.2 +++ b/src/HOL/Analysis/Ball_Volume.thy	Wed Jan 10 15:25:09 2018 +0100
    36.3 @@ -144,7 +144,7 @@
    36.4         (auto simp: mult_ac ennreal_mult' [symmetric] indicator_def intro!: nn_integral_cong)
    36.5    also have "(\<integral>\<^sup>+ (y::real). indicator {-r..r} y * (sqrt (r ^ 2 - y ^ 2)) ^ card A \<partial>lborel) =
    36.6                 (\<integral>\<^sup>+ (y::real). r ^ card A * indicator {-1..1} y * (sqrt (1 - y ^ 2)) ^ card A  
    36.7 -               \<partial>(distr lborel borel (op * (1/r))))" using \<open>r > 0\<close>
    36.8 +               \<partial>(distr lborel borel (( * ) (1/r))))" using \<open>r > 0\<close>
    36.9      by (subst nn_integral_distr)
   36.10         (auto simp: indicator_def field_simps real_sqrt_divide intro!: nn_integral_cong)
   36.11    also have "\<dots> = (\<integral>\<^sup>+ x. ennreal (r ^ Suc (card A)) * 
    37.1 --- a/src/HOL/Analysis/Binary_Product_Measure.thy	Wed Jan 10 15:21:49 2018 +0100
    37.2 +++ b/src/HOL/Analysis/Binary_Product_Measure.thy	Wed Jan 10 15:25:09 2018 +0100
    37.3 @@ -163,7 +163,7 @@
    37.4  
    37.5  lemma measurable_split_conv:
    37.6    "(\<lambda>(x, y). f x y) \<in> measurable A B \<longleftrightarrow> (\<lambda>x. f (fst x) (snd x)) \<in> measurable A B"
    37.7 -  by (intro arg_cong2[where f="op \<in>"]) auto
    37.8 +  by (intro arg_cong2[where f="(\<in>)"]) auto
    37.9  
   37.10  lemma measurable_pair_swap': "(\<lambda>(x,y). (y, x)) \<in> measurable (M1 \<Otimes>\<^sub>M M2) (M2 \<Otimes>\<^sub>M M1)"
   37.11    by (auto intro!: measurable_Pair simp: measurable_split_conv)
    38.1 --- a/src/HOL/Analysis/Bochner_Integration.thy	Wed Jan 10 15:21:49 2018 +0100
    38.2 +++ b/src/HOL/Analysis/Bochner_Integration.thy	Wed Jan 10 15:25:09 2018 +0100
    38.3 @@ -497,7 +497,7 @@
    38.4    have "ennreal (norm (?s - ?t)) = norm (simple_bochner_integral M (\<lambda>x. s x - t x))"
    38.5      using s t by (subst simple_bochner_integral_diff) auto
    38.6    also have "\<dots> \<le> simple_bochner_integral M (\<lambda>x. norm (s x - t x))"
    38.7 -    using simple_bochner_integrable_compose2[of "op -" M "s" "t"] s t
    38.8 +    using simple_bochner_integrable_compose2[of "(-)" M "s" "t"] s t
    38.9      by (auto intro!: simple_bochner_integral_norm_bound)
   38.10    also have "\<dots> = (\<integral>\<^sup>+x. norm (s x - t x) \<partial>M)"
   38.11      using simple_bochner_integrable_compose2[of "\<lambda>x y. norm (x - y)" M "s" "t"] s t
   38.12 @@ -1855,7 +1855,7 @@
   38.13  
   38.14    have sf[measurable]: "\<And>i. simple_function M (s' i)"
   38.15      unfolding s'_def using s(1)
   38.16 -    by (intro simple_function_compose2[where h="op *\<^sub>R"] simple_function_indicator) auto
   38.17 +    by (intro simple_function_compose2[where h="( *\<^sub>R)"] simple_function_indicator) auto
   38.18  
   38.19    { fix i
   38.20      have "\<And>z. {y. s' i z = y \<and> y \<in> s' i ` space M \<and> y \<noteq> 0 \<and> z \<in> space M} =
   38.21 @@ -2282,7 +2282,7 @@
   38.22    shows "integrable (density M g) f \<longleftrightarrow> integrable M (\<lambda>x. g x *\<^sub>R f x)"
   38.23    unfolding integrable_iff_bounded using nn
   38.24    apply (simp add: nn_integral_density less_top[symmetric])
   38.25 -  apply (intro arg_cong2[where f="op ="] refl nn_integral_cong_AE)
   38.26 +  apply (intro arg_cong2[where f="(=)"] refl nn_integral_cong_AE)
   38.27    apply (auto simp: ennreal_mult)
   38.28    done
   38.29  
    39.1 --- a/src/HOL/Analysis/Borel_Space.thy	Wed Jan 10 15:21:49 2018 +0100
    39.2 +++ b/src/HOL/Analysis/Borel_Space.thy	Wed Jan 10 15:25:09 2018 +0100
    39.3 @@ -110,8 +110,8 @@
    39.4    show "eventually (\<lambda>h. (f (x + h) - f x) / h \<ge> 0) (at 0)"
    39.5    proof (subst eventually_at_topological, intro exI conjI ballI impI)
    39.6      have "open (interior A)" by simp
    39.7 -    hence "open (op + (-x) ` interior A)" by (rule open_translation)
    39.8 -    also have "(op + (-x) ` interior A) = ?A'" by auto
    39.9 +    hence "open ((+) (-x) ` interior A)" by (rule open_translation)
   39.10 +    also have "((+) (-x) ` interior A) = ?A'" by auto
   39.11      finally show "open ?A'" .
   39.12    next
   39.13      from \<open>x \<in> interior A\<close> show "0 \<in> ?A'" by auto
   39.14 @@ -1701,7 +1701,7 @@
   39.15  
   39.16  definition [simp]: "is_borel f M \<longleftrightarrow> f \<in> borel_measurable M"
   39.17  
   39.18 -lemma is_borel_transfer[transfer_rule]: "rel_fun (rel_fun op = pcr_ennreal) op = is_borel is_borel"
   39.19 +lemma is_borel_transfer[transfer_rule]: "rel_fun (rel_fun (=) pcr_ennreal) (=) is_borel is_borel"
   39.20    unfolding is_borel_def[abs_def]
   39.21  proof (safe intro!: rel_funI ext dest!: rel_fun_eq_pcr_ennreal[THEN iffD1])
   39.22    fix f and M :: "'a measure"
    40.1 --- a/src/HOL/Analysis/Bounded_Linear_Function.thy	Wed Jan 10 15:21:49 2018 +0100
    40.2 +++ b/src/HOL/Analysis/Bounded_Linear_Function.thy	Wed Jan 10 15:25:09 2018 +0100
    40.3 @@ -569,7 +569,7 @@
    40.4  qed (auto simp: assms intro!: blinfun.comp)
    40.5  
    40.6  lemma transfer_bounded_bilinear_bounded_linear[transfer_rule]:
    40.7 -  "(rel_fun (rel_fun op = (pcr_blinfun op = op =)) op =) bounded_bilinear bounded_linear"
    40.8 +  "(rel_fun (rel_fun (=) (pcr_blinfun (=) (=))) (=)) bounded_bilinear bounded_linear"
    40.9    by (auto simp: pcr_blinfun_def cr_blinfun_def rel_fun_def OO_def
   40.10      intro!: transfer_bounded_bilinear_bounded_linearI)
   40.11  
   40.12 @@ -623,7 +623,7 @@
   40.13  lift_definition blinfun_compose::
   40.14    "'a::real_normed_vector \<Rightarrow>\<^sub>L 'b::real_normed_vector \<Rightarrow>
   40.15      'c::real_normed_vector \<Rightarrow>\<^sub>L 'a \<Rightarrow>
   40.16 -    'c \<Rightarrow>\<^sub>L 'b" (infixl "o\<^sub>L" 55) is "op o"
   40.17 +    'c \<Rightarrow>\<^sub>L 'b" (infixl "o\<^sub>L" 55) is "(o)"
   40.18    parametric comp_transfer
   40.19    unfolding o_def
   40.20    by (rule bounded_linear_compose)
   40.21 @@ -635,7 +635,7 @@
   40.22    "norm (f o\<^sub>L g) \<le> norm f * norm g"
   40.23    by transfer (rule onorm_compose)
   40.24  
   40.25 -lemma bounded_bilinear_blinfun_compose[bounded_bilinear]: "bounded_bilinear op o\<^sub>L"
   40.26 +lemma bounded_bilinear_blinfun_compose[bounded_bilinear]: "bounded_bilinear (o\<^sub>L)"
   40.27    by unfold_locales
   40.28      (auto intro!: blinfun_eqI exI[where x=1] simp: blinfun.bilinear_simps norm_blinfun_compose)
   40.29  
   40.30 @@ -645,7 +645,7 @@
   40.31    by (auto simp: blinfun.bilinear_simps intro!: blinfun_eqI)
   40.32  
   40.33  
   40.34 -lift_definition blinfun_inner_right::"'a::real_inner \<Rightarrow> 'a \<Rightarrow>\<^sub>L real" is "op \<bullet>"
   40.35 +lift_definition blinfun_inner_right::"'a::real_inner \<Rightarrow> 'a \<Rightarrow>\<^sub>L real" is "(\<bullet>)"
   40.36    by (rule bounded_linear_inner_right)
   40.37  declare blinfun_inner_right.rep_eq[simp]
   40.38  
   40.39 @@ -661,7 +661,7 @@
   40.40    by transfer (rule bounded_bilinear.flip[OF bounded_bilinear_inner])
   40.41  
   40.42  
   40.43 -lift_definition blinfun_scaleR_right::"real \<Rightarrow> 'a \<Rightarrow>\<^sub>L 'a::real_normed_vector" is "op *\<^sub>R"
   40.44 +lift_definition blinfun_scaleR_right::"real \<Rightarrow> 'a \<Rightarrow>\<^sub>L 'a::real_normed_vector" is "( *\<^sub>R)"
   40.45    by (rule bounded_linear_scaleR_right)
   40.46  declare blinfun_scaleR_right.rep_eq[simp]
   40.47  
   40.48 @@ -677,7 +677,7 @@
   40.49    by transfer (rule bounded_bilinear.flip[OF bounded_bilinear_scaleR])
   40.50  
   40.51  
   40.52 -lift_definition blinfun_mult_right::"'a \<Rightarrow> 'a \<Rightarrow>\<^sub>L 'a::real_normed_algebra" is "op *"
   40.53 +lift_definition blinfun_mult_right::"'a \<Rightarrow> 'a \<Rightarrow>\<^sub>L 'a::real_normed_algebra" is "( * )"
   40.54    by (rule bounded_linear_mult_right)
   40.55  declare blinfun_mult_right.rep_eq[simp]
   40.56  
    41.1 --- a/src/HOL/Analysis/Brouwer_Fixpoint.thy	Wed Jan 10 15:21:49 2018 +0100
    41.2 +++ b/src/HOL/Analysis/Brouwer_Fixpoint.thy	Wed Jan 10 15:25:09 2018 +0100
    41.3 @@ -146,7 +146,7 @@
    41.4      by (subst card_Un_disjoint[symmetric])
    41.5         (auto simp: nF_def intro!: sum.cong arg_cong[where f=card])
    41.6    also have "\<dots> = 1 * card {f\<in>F. compo' f \<and> bnd f} + 2 * card {f\<in>F. compo' f \<and> \<not> bnd f}"
    41.7 -    using assms(4,5) by (fastforce intro!: arg_cong2[where f="op +"] sum_multicount)
    41.8 +    using assms(4,5) by (fastforce intro!: arg_cong2[where f="(+)"] sum_multicount)
    41.9    finally have "odd ((\<Sum>s | s \<in> S \<and> \<not> compo s. nF s) + card {s\<in>S. compo s})"
   41.10      using assms(6,8) by simp
   41.11    moreover have "(\<Sum>s | s \<in> S \<and> \<not> compo s. nF s) =
   41.12 @@ -4221,13 +4221,13 @@
   41.13          show "ANR (C \<inter> \<Union>\<U>)"
   41.14            unfolding Int_Union
   41.15          proof (rule Suc)
   41.16 -          show "finite (op \<inter> C ` \<U>)"
   41.17 +          show "finite ((\<inter>) C ` \<U>)"
   41.18              by (simp add: insert.hyps(1))
   41.19 -          show "\<And>Ca. Ca \<in> op \<inter> C ` \<U> \<Longrightarrow> closed Ca"
   41.20 +          show "\<And>Ca. Ca \<in> (\<inter>) C ` \<U> \<Longrightarrow> closed Ca"
   41.21              by (metis (no_types, hide_lams) Suc.prems(2) closed_Int subsetD imageE insert.prems insertI1 insertI2)
   41.22 -          show "\<And>Ca. Ca \<in> op \<inter> C ` \<U> \<Longrightarrow> convex Ca"
   41.23 +          show "\<And>Ca. Ca \<in> (\<inter>) C ` \<U> \<Longrightarrow> convex Ca"
   41.24              by (metis (mono_tags, lifting) Suc.prems(3) convex_Int imageE insert.prems insert_subset subsetCE)
   41.25 -          show "card (op \<inter> C ` \<U>) < n"
   41.26 +          show "card ((\<inter>) C ` \<U>) < n"
   41.27            proof -
   41.28              have "card \<T> \<le> n"
   41.29                by (meson Suc.prems(4) not_less not_less_eq)
    42.1 --- a/src/HOL/Analysis/Cartesian_Euclidean_Space.thy	Wed Jan 10 15:21:49 2018 +0100
    42.2 +++ b/src/HOL/Analysis/Cartesian_Euclidean_Space.thy	Wed Jan 10 15:25:09 2018 +0100
    42.3 @@ -39,7 +39,7 @@
    42.4  instantiation vec :: (times, finite) times
    42.5  begin
    42.6  
    42.7 -definition "op * \<equiv> (\<lambda> x y.  (\<chi> i. (x$i) * (y$i)))"
    42.8 +definition "( * ) \<equiv> (\<lambda> x y.  (\<chi> i. (x$i) * (y$i)))"
    42.9  instance ..
   42.10  
   42.11  end
   42.12 @@ -714,9 +714,9 @@
   42.13        unfolding matrix_vector_mul_assoc B matrix_vector_mul_lid . }
   42.14    moreover
   42.15    { assume A: "\<forall>x y. A *v x = A *v y \<longrightarrow> x = y"
   42.16 -    hence i: "inj (op *v A)" unfolding inj_on_def by auto
   42.17 +    hence i: "inj (( *v) A)" unfolding inj_on_def by auto
   42.18      from linear_injective_left_inverse[OF matrix_vector_mul_linear i]
   42.19 -    obtain g where g: "linear g" "g \<circ> op *v A = id" by blast
   42.20 +    obtain g where g: "linear g" "g \<circ> ( *v) A = id" by blast
   42.21      have "matrix g ** A = mat 1"
   42.22        unfolding matrix_eq matrix_vector_mul_lid matrix_vector_mul_assoc[symmetric] matrix_works[OF g(1)]
   42.23        using g(2) by (simp add: fun_eq_iff)
   42.24 @@ -738,11 +738,11 @@
   42.25      { fix x :: "real ^ 'm"
   42.26        have "A *v (B *v x) = x"
   42.27          by (simp add: matrix_vector_mul_lid matrix_vector_mul_assoc AB) }
   42.28 -    hence "surj (op *v A)" unfolding surj_def by metis }
   42.29 +    hence "surj (( *v) A)" unfolding surj_def by metis }
   42.30    moreover
   42.31 -  { assume sf: "surj (op *v A)"
   42.32 +  { assume sf: "surj (( *v) A)"
   42.33      from linear_surjective_right_inverse[OF matrix_vector_mul_linear sf]
   42.34 -    obtain g:: "real ^'m \<Rightarrow> real ^'n" where g: "linear g" "op *v A \<circ> g = id"
   42.35 +    obtain g:: "real ^'m \<Rightarrow> real ^'n" where g: "linear g" "( *v) A \<circ> g = id"
   42.36        by blast
   42.37  
   42.38      have "A ** (matrix g) = mat 1"
   42.39 @@ -878,7 +878,7 @@
   42.40  proof -
   42.41    { fix A A' :: "real ^'n^'n"
   42.42      assume AA': "A ** A' = mat 1"
   42.43 -    have sA: "surj (op *v A)"
   42.44 +    have sA: "surj (( *v) A)"
   42.45        unfolding surj_def
   42.46        apply clarify
   42.47        apply (rule_tac x="(A' *v y)" in exI)
   42.48 @@ -1235,7 +1235,7 @@
   42.49      where "finite s" "cbox (x - (\<chi> i. d)) (x + (\<chi> i. d)) = convex hull s"
   42.50  proof -
   42.51    from assms obtain s where "finite s"
   42.52 -    and "cbox (x - sum (op *\<^sub>R d) Basis) (x + sum (op *\<^sub>R d) Basis) = convex hull s"
   42.53 +    and "cbox (x - sum (( *\<^sub>R) d) Basis) (x + sum (( *\<^sub>R) d) Basis) = convex hull s"
   42.54      by (rule cube_convex_hull)
   42.55    with that[of s] show thesis
   42.56      by (simp add: const_vector_cart)
    43.1 --- a/src/HOL/Analysis/Cauchy_Integral_Theorem.thy	Wed Jan 10 15:21:49 2018 +0100
    43.2 +++ b/src/HOL/Analysis/Cauchy_Integral_Theorem.thy	Wed Jan 10 15:25:09 2018 +0100
    43.3 @@ -236,7 +236,7 @@
    43.4  
    43.5  lemma continuous_on_joinpaths_D1:
    43.6      "continuous_on {0..1} (g1 +++ g2) \<Longrightarrow> continuous_on {0..1} g1"
    43.7 -  apply (rule continuous_on_eq [of _ "(g1 +++ g2) o (op*(inverse 2))"])
    43.8 +  apply (rule continuous_on_eq [of _ "(g1 +++ g2) o (( * )(inverse 2))"])
    43.9    apply (rule continuous_intros | simp)+
   43.10    apply (auto elim!: continuous_on_subset simp: joinpaths_def)
   43.11    done
   43.12 @@ -251,10 +251,10 @@
   43.13  lemma piecewise_differentiable_D1:
   43.14      "(g1 +++ g2) piecewise_differentiable_on {0..1} \<Longrightarrow> g1 piecewise_differentiable_on {0..1}"
   43.15    apply (clarsimp simp add: piecewise_differentiable_on_def dest!: continuous_on_joinpaths_D1)
   43.16 -  apply (rule_tac x="insert 1 ((op*2)`s)" in exI)
   43.17 +  apply (rule_tac x="insert 1 ((( * )2)`s)" in exI)
   43.18    apply simp
   43.19    apply (intro ballI)
   43.20 -  apply (rule_tac d="dist (x/2) (1/2)" and f = "(g1 +++ g2) o (op*(inverse 2))"
   43.21 +  apply (rule_tac d="dist (x/2) (1/2)" and f = "(g1 +++ g2) o (( * )(inverse 2))"
   43.22         in differentiable_transform_within)
   43.23    apply (auto simp: dist_real_def joinpaths_def)
   43.24    apply (rule differentiable_chain_within derivative_intros | simp)+
   43.25 @@ -537,40 +537,40 @@
   43.26               and co12: "continuous_on ({0..1} - s) (\<lambda>x. vector_derivative (g1 +++ g2) (at x))"
   43.27               and g12D: "\<forall>x\<in>{0..1} - s. g1 +++ g2 differentiable at x"
   43.28      using assms  by (auto simp: piecewise_C1_differentiable_on_def C1_differentiable_on_eq)
   43.29 -  then have g1D: "g1 differentiable at x" if "x \<in> {0..1} - insert 1 (op * 2 ` s)" for x
   43.30 -    apply (rule_tac d="dist (x/2) (1/2)" and f = "(g1 +++ g2) o (op*(inverse 2))" in differentiable_transform_within)
   43.31 +  then have g1D: "g1 differentiable at x" if "x \<in> {0..1} - insert 1 (( * ) 2 ` s)" for x
   43.32 +    apply (rule_tac d="dist (x/2) (1/2)" and f = "(g1 +++ g2) o (( * )(inverse 2))" in differentiable_transform_within)
   43.33      using that
   43.34      apply (simp_all add: dist_real_def joinpaths_def)
   43.35      apply (rule differentiable_chain_at derivative_intros | force)+
   43.36      done
   43.37 -  have [simp]: "vector_derivative (g1 \<circ> op * 2) (at (x/2)) = 2 *\<^sub>R vector_derivative g1 (at x)"
   43.38 -               if "x \<in> {0..1} - insert 1 (op * 2 ` s)" for x
   43.39 +  have [simp]: "vector_derivative (g1 \<circ> ( * ) 2) (at (x/2)) = 2 *\<^sub>R vector_derivative g1 (at x)"
   43.40 +               if "x \<in> {0..1} - insert 1 (( * ) 2 ` s)" for x
   43.41      apply (subst vector_derivative_chain_at)
   43.42      using that
   43.43      apply (rule derivative_eq_intros g1D | simp)+
   43.44      done
   43.45    have "continuous_on ({0..1/2} - insert (1/2) s) (\<lambda>x. vector_derivative (g1 +++ g2) (at x))"
   43.46      using co12 by (rule continuous_on_subset) force
   43.47 -  then have coDhalf: "continuous_on ({0..1/2} - insert (1/2) s) (\<lambda>x. vector_derivative (g1 o op*2) (at x))"
   43.48 +  then have coDhalf: "continuous_on ({0..1/2} - insert (1/2) s) (\<lambda>x. vector_derivative (g1 o ( * )2) (at x))"
   43.49      apply (rule continuous_on_eq [OF _ vector_derivative_at])
   43.50 -    apply (rule_tac f="g1 o op*2" and d="dist x (1/2)" in has_vector_derivative_transform_within)
   43.51 +    apply (rule_tac f="g1 o ( * )2" and d="dist x (1/2)" in has_vector_derivative_transform_within)
   43.52      apply (simp_all add: dist_norm joinpaths_def vector_derivative_works [symmetric])
   43.53      apply (force intro: g1D differentiable_chain_at)
   43.54      apply auto
   43.55      done
   43.56 -  have "continuous_on ({0..1} - insert 1 (op * 2 ` s))
   43.57 -                      ((\<lambda>x. 1/2 * vector_derivative (g1 o op*2) (at x)) o op*(1/2))"
   43.58 +  have "continuous_on ({0..1} - insert 1 (( * ) 2 ` s))
   43.59 +                      ((\<lambda>x. 1/2 * vector_derivative (g1 o ( * )2) (at x)) o ( * )(1/2))"
   43.60      apply (rule continuous_intros)+
   43.61      using coDhalf
   43.62      apply (simp add: scaleR_conv_of_real image_set_diff image_image)
   43.63      done
   43.64 -  then have con_g1: "continuous_on ({0..1} - insert 1 (op * 2 ` s)) (\<lambda>x. vector_derivative g1 (at x))"
   43.65 +  then have con_g1: "continuous_on ({0..1} - insert 1 (( * ) 2 ` s)) (\<lambda>x. vector_derivative g1 (at x))"
   43.66      by (rule continuous_on_eq) (simp add: scaleR_conv_of_real)
   43.67    have "continuous_on {0..1} g1"
   43.68      using continuous_on_joinpaths_D1 assms piecewise_C1_differentiable_on_def by blast
   43.69    with \<open>finite s\<close> show ?thesis
   43.70      apply (clarsimp simp add: piecewise_C1_differentiable_on_def C1_differentiable_on_eq)
   43.71 -    apply (rule_tac x="insert 1 ((op*2)`s)" in exI)
   43.72 +    apply (rule_tac x="insert 1 ((( * )2)`s)" in exI)
   43.73      apply (simp add: g1D con_g1)
   43.74    done
   43.75  qed
   43.76 @@ -785,7 +785,7 @@
   43.77  proof -
   43.78    obtain s where "finite s" "g C1_differentiable_on ({0..1} - s)"
   43.79      using assms by (auto simp: valid_path_def piecewise_C1_differentiable_on_def)
   43.80 -  then have "finite (op - 1 ` s)" "(reversepath g C1_differentiable_on ({0..1} - op - 1 ` s))"
   43.81 +  then have "finite ((-) 1 ` s)" "(reversepath g C1_differentiable_on ({0..1} - (-) 1 ` s))"
   43.82      apply (auto simp: reversepath_def)
   43.83      apply (rule C1_differentiable_compose [of "\<lambda>x::real. 1-x" _ g, unfolded o_def])
   43.84      apply (auto simp: C1_differentiable_on_eq)
   43.85 @@ -805,7 +805,7 @@
   43.86      shows "(f has_contour_integral (-i)) (reversepath g)"
   43.87  proof -
   43.88    { fix s x
   43.89 -    assume xs: "g C1_differentiable_on ({0..1} - s)" "x \<notin> op - 1 ` s" "0 \<le> x" "x \<le> 1"
   43.90 +    assume xs: "g C1_differentiable_on ({0..1} - s)" "x \<notin> (-) 1 ` s" "0 \<le> x" "x \<le> 1"
   43.91        have "vector_derivative (\<lambda>x. g (1 - x)) (at x within {0..1}) =
   43.92              - vector_derivative g (at (1 - x) within {0..1})"
   43.93        proof -
   43.94 @@ -870,7 +870,7 @@
   43.95      using assms
   43.96      apply (auto simp: valid_path_def piecewise_C1_differentiable_on_def continuous_on_joinpaths)
   43.97      apply (rule continuous_intros | simp)+
   43.98 -    apply (force intro: finite_vimageI [where h = "op*2"] inj_onI)
   43.99 +    apply (force intro: finite_vimageI [where h = "( * )2"] inj_onI)
  43.100      done
  43.101    moreover have "(g2 o (\<lambda>x. 2*x-1)) piecewise_C1_differentiable_on {1/2..1}"
  43.102      apply (rule piecewise_C1_differentiable_compose)
  43.103 @@ -941,9 +941,9 @@
  43.104      apply (auto simp: algebra_simps vector_derivative_works)
  43.105      done
  43.106    have "((\<lambda>x. f ((g1 +++ g2) x) * vector_derivative (g1 +++ g2) (at x)) has_integral i1) {0..1/2}"
  43.107 -    apply (rule has_integral_spike_finite [OF _ _ i1, of "insert (1/2) (op*2 -` s1)"])
  43.108 +    apply (rule has_integral_spike_finite [OF _ _ i1, of "insert (1/2) (( * )2 -` s1)"])
  43.109      using s1
  43.110 -    apply (force intro: finite_vimageI [where h = "op*2"] inj_onI)
  43.111 +    apply (force intro: finite_vimageI [where h = "( * )2"] inj_onI)
  43.112      apply (clarsimp simp add: joinpaths_def scaleR_conv_of_real mult_ac g1)
  43.113      done
  43.114    moreover have "((\<lambda>x. f ((g1 +++ g2) x) * vector_derivative (g1 +++ g2) (at x)) has_integral i2) {1/2..1}"
  43.115 @@ -1437,7 +1437,7 @@
  43.116        by (simp add: has_vector_derivative_def scaleR_conv_of_real)
  43.117      have "(f has_field_derivative (f' (g x))) (at (g x) within g ` {a..b})"
  43.118        using assms by (metis a atLeastAtMost_iff b DERIV_subset image_subset_iff less_eq_real_def)
  43.119 -    then have fdiff: "(f has_derivative op * (f' (g x))) (at (g x) within g ` {a..b})"
  43.120 +    then have fdiff: "(f has_derivative ( * ) (f' (g x))) (at (g x) within g ` {a..b})"
  43.121        by (simp add: has_field_derivative_def)
  43.122      have "((\<lambda>x. f (g x)) has_vector_derivative f' (g x) * vector_derivative g (at x within {a..b})) (at x within {a..b})"
  43.123        using diff_chain_within [OF gdiff fdiff]
  43.124 @@ -3589,11 +3589,11 @@
  43.125    assumes \<gamma>: "valid_path \<gamma>" and 0: "0 \<notin> path_image \<gamma>"
  43.126    shows "winding_number(uminus \<circ> \<gamma>) 0 = winding_number \<gamma> 0"
  43.127  proof -
  43.128 -  have "op / 1 contour_integrable_on \<gamma>"
  43.129 +  have "(/) 1 contour_integrable_on \<gamma>"
  43.130      using "0" \<gamma> contour_integrable_inversediff by fastforce
  43.131 -  then have "((\<lambda>z. 1/z) has_contour_integral contour_integral \<gamma> (op / 1)) \<gamma>"
  43.132 +  then have "((\<lambda>z. 1/z) has_contour_integral contour_integral \<gamma> ((/) 1)) \<gamma>"
  43.133      by (rule has_contour_integral_integral)
  43.134 -  then have "((\<lambda>z. 1 / - z) has_contour_integral - contour_integral \<gamma> (op / 1)) \<gamma>"
  43.135 +  then have "((\<lambda>z. 1 / - z) has_contour_integral - contour_integral \<gamma> ((/) 1)) \<gamma>"
  43.136      using has_contour_integral_neg by auto
  43.137    then show ?thesis
  43.138      using assms
  43.139 @@ -5914,7 +5914,7 @@
  43.140    case 0 then show ?case by simp
  43.141  next
  43.142    case (Suc n z)
  43.143 -  have holo0: "f holomorphic_on op * u ` s"
  43.144 +  have holo0: "f holomorphic_on ( * ) u ` s"
  43.145      by (meson fg f holomorphic_on_subset image_subset_iff)
  43.146    have holo1: "(\<lambda>w. f (u * w)) holomorphic_on s"
  43.147      apply (rule holomorphic_on_compose [where g=f, unfolded o_def])
  43.148 @@ -5942,7 +5942,7 @@
  43.149      apply (blast intro: fg)
  43.150      done
  43.151    also have "... = u * u ^ n * deriv ((deriv ^^ n) f) (u * z)"
  43.152 -      apply (subst complex_derivative_chain [where g = "(deriv ^^ n) f" and f = "op* u", unfolded o_def])
  43.153 +      apply (subst complex_derivative_chain [where g = "(deriv ^^ n) f" and f = "( * ) u", unfolded o_def])
  43.154        apply (rule derivative_intros)
  43.155        using Suc.prems field_differentiable_def f fg has_field_derivative_higher_deriv t apply blast
  43.156        apply (simp add: deriv_linear)
    44.1 --- a/src/HOL/Analysis/Complex_Analysis_Basics.thy	Wed Jan 10 15:21:49 2018 +0100
    44.2 +++ b/src/HOL/Analysis/Complex_Analysis_Basics.thy	Wed Jan 10 15:25:09 2018 +0100
    44.3 @@ -16,7 +16,7 @@
    44.4  
    44.5  lemma has_derivative_mult_right:
    44.6    fixes c:: "'a :: real_normed_algebra"
    44.7 -  shows "((op * c) has_derivative (op * c)) F"
    44.8 +  shows "((( * ) c) has_derivative (( * ) c)) F"
    44.9  by (rule has_derivative_mult_right [OF has_derivative_id])
   44.10  
   44.11  lemma has_derivative_of_real[derivative_intros, simp]:
   44.12 @@ -25,7 +25,7 @@
   44.13  
   44.14  lemma has_vector_derivative_real_field:
   44.15    "DERIV f (of_real a) :> f' \<Longrightarrow> ((\<lambda>x. f (of_real x)) has_vector_derivative f') (at a within s)"
   44.16 -  using has_derivative_compose[of of_real of_real a _ f "op * f'"]
   44.17 +  using has_derivative_compose[of of_real of_real a _ f "( * ) f'"]
   44.18    by (simp add: scaleR_conv_of_real ac_simps has_vector_derivative_def has_field_derivative_def)
   44.19  lemmas has_vector_derivative_real_complex = has_vector_derivative_real_field
   44.20  
   44.21 @@ -69,10 +69,10 @@
   44.22      shows  "b \<le> Im(l)"
   44.23    by (metis assms tendsto_le [OF _ _ tendsto_const]  tendsto_Im)
   44.24  
   44.25 -lemma lambda_zero: "(\<lambda>h::'a::mult_zero. 0) = op * 0"
   44.26 +lemma lambda_zero: "(\<lambda>h::'a::mult_zero. 0) = ( * ) 0"
   44.27    by auto
   44.28  
   44.29 -lemma lambda_one: "(\<lambda>x::'a::monoid_mult. x) = op * 1"
   44.30 +lemma lambda_one: "(\<lambda>x::'a::monoid_mult. x) = ( * ) 1"
   44.31    by auto
   44.32  
   44.33  lemma continuous_mult_left:
   44.34 @@ -315,7 +315,7 @@
   44.35  lemma holomorphic_cong: "s = t ==> (\<And>x. x \<in> s \<Longrightarrow> f x = g x) \<Longrightarrow> f holomorphic_on s \<longleftrightarrow> g holomorphic_on t"
   44.36    by (metis holomorphic_transform)
   44.37  
   44.38 -lemma holomorphic_on_linear [simp, holomorphic_intros]: "(op * c) holomorphic_on s"
   44.39 +lemma holomorphic_on_linear [simp, holomorphic_intros]: "(( * ) c) holomorphic_on s"
   44.40    unfolding holomorphic_on_def by (metis field_differentiable_linear)
   44.41  
   44.42  lemma holomorphic_on_const [simp, holomorphic_intros]: "(\<lambda>z. c) holomorphic_on s"
   44.43 @@ -570,7 +570,7 @@
   44.44    finally show ?thesis .
   44.45  qed
   44.46  
   44.47 -lemma analytic_on_linear [analytic_intros,simp]: "(op * c) analytic_on s"
   44.48 +lemma analytic_on_linear [analytic_intros,simp]: "(( * ) c) analytic_on s"
   44.49    by (auto simp add: analytic_on_holomorphic)
   44.50  
   44.51  lemma analytic_on_const [analytic_intros,simp]: "(\<lambda>z. c) analytic_on s"
   44.52 @@ -839,7 +839,7 @@
   44.53    show ?thesis
   44.54    unfolding has_field_derivative_def
   44.55    proof (rule has_derivative_sequence [OF cvs _ _ x])
   44.56 -    show "\<forall>n. \<forall>x\<in>s. (f n has_derivative (op * (f' n x))) (at x within s)"
   44.57 +    show "\<forall>n. \<forall>x\<in>s. (f n has_derivative (( * ) (f' n x))) (at x within s)"
   44.58        by (metis has_field_derivative_def df)
   44.59    next show "(\<lambda>n. f n x) \<longlonglongrightarrow> l"
   44.60      by (rule tf)
   44.61 @@ -906,9 +906,9 @@
   44.62    then obtain g where g: "\<And>x. x \<in> s \<Longrightarrow> (\<lambda>n. f n x) sums g x"
   44.63      "\<And>x. x \<in> s \<Longrightarrow> (g has_field_derivative g' x) (at x within s)" by blast
   44.64    from g[OF x] show "summable (\<lambda>n. f n x)" by (auto simp: summable_def)
   44.65 -  from g(2)[OF x] have g': "(g has_derivative op * (g' x)) (at x)"
   44.66 +  from g(2)[OF x] have g': "(g has_derivative ( * ) (g' x)) (at x)"
   44.67      by (simp add: has_field_derivative_def s)
   44.68 -  have "((\<lambda>x. \<Sum>n. f n x) has_derivative op * (g' x)) (at x)"
   44.69 +  have "((\<lambda>x. \<Sum>n. f n x) has_derivative ( * ) (g' x)) (at x)"
   44.70      by (rule has_derivative_transform_within_open[OF g' \<open>open s\<close> x])
   44.71         (insert g, auto simp: sums_iff)
   44.72    thus "(\<lambda>x. \<Sum>n. f n x) field_differentiable (at x)" unfolding differentiable_def
    45.1 --- a/src/HOL/Analysis/Conformal_Mappings.thy	Wed Jan 10 15:21:49 2018 +0100
    45.2 +++ b/src/HOL/Analysis/Conformal_Mappings.thy	Wed Jan 10 15:25:09 2018 +0100
    45.3 @@ -1141,10 +1141,10 @@
    45.4        done
    45.5      with \<open>e>0\<close> show ?thesis by force
    45.6    qed
    45.7 -  have "inj (op * (deriv f \<xi>))"
    45.8 +  have "inj (( * ) (deriv f \<xi>))"
    45.9      using dnz by simp
   45.10 -  then obtain g' where g': "linear g'" "g' \<circ> op * (deriv f \<xi>) = id"
   45.11 -    using linear_injective_left_inverse [of "op * (deriv f \<xi>)"]
   45.12 +  then obtain g' where g': "linear g'" "g' \<circ> ( * ) (deriv f \<xi>) = id"
   45.13 +    using linear_injective_left_inverse [of "( * ) (deriv f \<xi>)"]
   45.14      by (auto simp: linear_times)
   45.15    show ?thesis
   45.16      apply (rule has_derivative_locally_injective [OF S, where f=f and f' = "\<lambda>z h. deriv f z * h" and g' = g'])
   45.17 @@ -2093,10 +2093,10 @@
   45.18      apply (simp add: C_def False fo)
   45.19      apply (simp add: derivative_intros dfa complex_derivative_chain)
   45.20      done
   45.21 -  have sb1: "op * (C * r) ` (\<lambda>z. f (a + of_real r * z) / (C * r)) ` ball 0 1
   45.22 +  have sb1: "( * ) (C * r) ` (\<lambda>z. f (a + of_real r * z) / (C * r)) ` ball 0 1
   45.23               \<subseteq> f ` ball a r"
   45.24      using \<open>0 < r\<close> by (auto simp: dist_norm norm_mult C_def False)
   45.25 -  have sb2: "ball (C * r * b) r' \<subseteq> op * (C * r) ` ball b t"
   45.26 +  have sb2: "ball (C * r * b) r' \<subseteq> ( * ) (C * r) ` ball b t"
   45.27               if "1 / 12 < t" for b t
   45.28    proof -
   45.29      have *: "r * cmod (deriv f a) / 12 \<le> r * (t * cmod (deriv f a))"
   45.30 @@ -2602,7 +2602,7 @@
   45.31      by (metis DiffD2 Ints_cases insertI1 subset_eq valid_path_imp_path)
   45.32    define p_circ where "p_circ \<equiv> circlepath p (h p)"
   45.33    define p_circ_pt where "p_circ_pt \<equiv> linepath (p+h p) (p+h p)"
   45.34 -  define n_circ where "n_circ \<equiv> \<lambda>n. (op +++ p_circ ^^ n) p_circ_pt"
   45.35 +  define n_circ where "n_circ \<equiv> \<lambda>n. ((+++) p_circ ^^ n) p_circ_pt"
   45.36    define cp where "cp \<equiv> if n\<ge>0 then reversepath (n_circ (nat n)) else n_circ (nat (- n))"
   45.37    have n_circ:"valid_path (n_circ k)"
   45.38        "winding_number (n_circ k) p = k"
   45.39 @@ -3861,7 +3861,7 @@
   45.40            then show "h field_differentiable at (\<gamma> x)" 
   45.41              unfolding t_def field_differentiable_def by blast
   45.42          qed
   45.43 -      then have " (op / 1 has_contour_integral 0) (h \<circ> \<gamma>)
   45.44 +      then have " ((/) 1 has_contour_integral 0) (h \<circ> \<gamma>)
   45.45            = ((\<lambda>x. deriv h x / h x) has_contour_integral 0) \<gamma>"
   45.46          unfolding has_contour_integral
   45.47          apply (intro has_integral_spike_eq[OF negligible_finite, OF \<open>finite spikes\<close>])
    46.1 --- a/src/HOL/Analysis/Connected.thy	Wed Jan 10 15:21:49 2018 +0100
    46.2 +++ b/src/HOL/Analysis/Connected.thy	Wed Jan 10 15:25:09 2018 +0100
    46.3 @@ -2165,14 +2165,14 @@
    46.4          unfolding dist_norm
    46.5          using norm_scaleR[of c "(1 / c) *\<^sub>R y - x", unfolded scaleR_right_diff_distrib, unfolded scaleR_scaleR] assms(1)
    46.6            assms(1)[unfolded zero_less_abs_iff[symmetric]] by (simp del:zero_less_abs_iff)
    46.7 -      then have "y \<in> op *\<^sub>R c ` s"
    46.8 -        using rev_image_eqI[of "(1 / c) *\<^sub>R y" s y "op *\<^sub>R c"]
    46.9 +      then have "y \<in> ( *\<^sub>R) c ` s"
   46.10 +        using rev_image_eqI[of "(1 / c) *\<^sub>R y" s y "( *\<^sub>R) c"]
   46.11          using e[THEN spec[where x="(1 / c) *\<^sub>R y"]]
   46.12          using assms(1)
   46.13          unfolding dist_norm scaleR_scaleR
   46.14          by auto
   46.15      }
   46.16 -    ultimately have "\<exists>e>0. \<forall>x'. dist x' (c *\<^sub>R x) < e \<longrightarrow> x' \<in> op *\<^sub>R c ` s"
   46.17 +    ultimately have "\<exists>e>0. \<forall>x'. dist x' (c *\<^sub>R x) < e \<longrightarrow> x' \<in> ( *\<^sub>R) c ` s"
   46.18        apply (rule_tac x="e * \<bar>c\<bar>" in exI, auto)
   46.19        done
   46.20    }
   46.21 @@ -2199,7 +2199,7 @@
   46.22      have "continuous (at x) (\<lambda>x. x - a)"
   46.23        by (intro continuous_diff continuous_ident continuous_const)
   46.24    }
   46.25 -  moreover have "{x. x - a \<in> S} = op + a ` S"
   46.26 +  moreover have "{x. x - a \<in> S} = (+) a ` S"
   46.27      by force
   46.28    ultimately show ?thesis
   46.29      by (metis assms continuous_open_vimage vimage_def)
   46.30 @@ -2212,10 +2212,10 @@
   46.31  proof -
   46.32    have *: "(\<lambda>x. a + c *\<^sub>R x) = (\<lambda>x. a + x) \<circ> (\<lambda>x. c *\<^sub>R x)"
   46.33      unfolding o_def ..
   46.34 -  have "op + a ` op *\<^sub>R c ` S = (op + a \<circ> op *\<^sub>R c) ` S"
   46.35 +  have "(+) a ` ( *\<^sub>R) c ` S = ((+) a \<circ> ( *\<^sub>R) c) ` S"
   46.36      by auto
   46.37    then show ?thesis
   46.38 -    using assms open_translation[of "op *\<^sub>R c ` S" a]
   46.39 +    using assms open_translation[of "( *\<^sub>R) c ` S" a]
   46.40      unfolding *
   46.41      by auto
   46.42  qed
   46.43 @@ -2225,13 +2225,13 @@
   46.44    shows "interior ((\<lambda>x. a + x) ` S) = (\<lambda>x. a + x) ` (interior S)"
   46.45  proof (rule set_eqI, rule)
   46.46    fix x
   46.47 -  assume "x \<in> interior (op + a ` S)"
   46.48 -  then obtain e where "e > 0" and e: "ball x e \<subseteq> op + a ` S"
   46.49 +  assume "x \<in> interior ((+) a ` S)"
   46.50 +  then obtain e where "e > 0" and e: "ball x e \<subseteq> (+) a ` S"
   46.51      unfolding mem_interior by auto
   46.52    then have "ball (x - a) e \<subseteq> S"
   46.53      unfolding subset_eq Ball_def mem_ball dist_norm
   46.54      by (auto simp: diff_diff_eq)
   46.55 -  then show "x \<in> op + a ` interior S"
   46.56 +  then show "x \<in> (+) a ` interior S"
   46.57      unfolding image_iff
   46.58      apply (rule_tac x="x - a" in bexI)
   46.59      unfolding mem_interior
   46.60 @@ -2240,7 +2240,7 @@
   46.61      done
   46.62  next
   46.63    fix x
   46.64 -  assume "x \<in> op + a ` interior S"
   46.65 +  assume "x \<in> (+) a ` interior S"
   46.66    then obtain y e where "e > 0" and e: "ball y e \<subseteq> S" and y: "x = a + y"
   46.67      unfolding image_iff Bex_def mem_interior by auto
   46.68    {
   46.69 @@ -2251,12 +2251,12 @@
   46.70        using e[unfolded subset_eq, THEN bspec[where x="z - a"]]
   46.71        unfolding mem_ball dist_norm y group_add_class.diff_diff_eq2 *
   46.72        by auto
   46.73 -    then have "z \<in> op + a ` S"
   46.74 +    then have "z \<in> (+) a ` S"
   46.75        unfolding image_iff by (auto intro!: bexI[where x="z - a"])
   46.76    }
   46.77 -  then have "ball x e \<subseteq> op + a ` S"
   46.78 +  then have "ball x e \<subseteq> (+) a ` S"
   46.79      unfolding subset_eq by auto
   46.80 -  then show "x \<in> interior (op + a ` S)"
   46.81 +  then show "x \<in> interior ((+) a ` S)"
   46.82      unfolding mem_interior using \<open>e > 0\<close> by auto
   46.83  qed
   46.84  
   46.85 @@ -2574,7 +2574,7 @@
   46.86    assumes "compact s"
   46.87    shows "compact ((\<lambda>x. a + c *\<^sub>R x) ` s)"
   46.88  proof -
   46.89 -  have "op + a ` op *\<^sub>R c ` s = (\<lambda>x. a + c *\<^sub>R x) ` s"
   46.90 +  have "(+) a ` ( *\<^sub>R) c ` s = (\<lambda>x. a + c *\<^sub>R x) ` s"
   46.91      by auto
   46.92    then show ?thesis
   46.93      using compact_translation[OF compact_scaling[OF assms], of a c] by auto
   46.94 @@ -3007,7 +3007,7 @@
   46.95    assumes "closed S"
   46.96    shows "closed ((\<lambda>x. a + x) ` S)"
   46.97  proof -
   46.98 -  have "(\<Union>x\<in> {a}. \<Union>y \<in> S. {x + y}) = (op + a ` S)" by auto
   46.99 +  have "(\<Union>x\<in> {a}. \<Union>y \<in> S. {x + y}) = ((+) a ` S)" by auto
  46.100    then show ?thesis
  46.101      using compact_closed_sums[OF compact_sing[of a] assms] by auto
  46.102  qed
  46.103 @@ -3038,7 +3038,7 @@
  46.104    fixes a :: "'a::real_normed_vector"
  46.105    shows "closure ((\<lambda>x. a + x) ` s) = (\<lambda>x. a + x) ` (closure s)"
  46.106  proof -
  46.107 -  have *: "op + a ` (- s) = - op + a ` s"
  46.108 +  have *: "(+) a ` (- s) = - (+) a ` s"
  46.109      apply auto
  46.110      unfolding image_iff
  46.111      apply (rule_tac x="x - a" in bexI, auto)
  46.112 @@ -3058,7 +3058,7 @@
  46.113  
  46.114  lemma sphere_translation:
  46.115    fixes a :: "'n::euclidean_space"
  46.116 -  shows "sphere (a+c) r = op+ a ` sphere c r"
  46.117 +  shows "sphere (a+c) r = (+) a ` sphere c r"
  46.118  apply safe
  46.119  apply (rule_tac x="x-a" in image_eqI)
  46.120  apply (auto simp: dist_norm algebra_simps)
  46.121 @@ -3066,7 +3066,7 @@
  46.122  
  46.123  lemma cball_translation:
  46.124    fixes a :: "'n::euclidean_space"
  46.125 -  shows "cball (a+c) r = op+ a ` cball c r"
  46.126 +  shows "cball (a+c) r = (+) a ` cball c r"
  46.127  apply safe
  46.128  apply (rule_tac x="x-a" in image_eqI)
  46.129  apply (auto simp: dist_norm algebra_simps)
  46.130 @@ -3074,7 +3074,7 @@
  46.131  
  46.132  lemma ball_translation:
  46.133    fixes a :: "'n::euclidean_space"
  46.134 -  shows "ball (a+c) r = op+ a ` ball c r"
  46.135 +  shows "ball (a+c) r = (+) a ` ball c r"
  46.136  apply safe
  46.137  apply (rule_tac x="x-a" in image_eqI)
  46.138  apply (auto simp: dist_norm algebra_simps)
  46.139 @@ -3420,7 +3420,7 @@
  46.140  
  46.141  lemma homeomorphism_translation:
  46.142    fixes a :: "'a :: real_normed_vector"
  46.143 -  shows "homeomorphism (op + a ` S) S (op + (- a)) (op + a)"
  46.144 +  shows "homeomorphism ((+) a ` S) S ((+) (- a)) ((+) a)"
  46.145  unfolding homeomorphism_def by (auto simp: algebra_simps continuous_intros)
  46.146  
  46.147  lemma homeomorphism_ident: "homeomorphism T T (\<lambda>a. a) (\<lambda>a. a)"
  46.148 @@ -3639,7 +3639,7 @@
  46.149    assumes "c \<noteq> 0"
  46.150    shows "s homeomorphic ((\<lambda>x. a + c *\<^sub>R x) ` s)"
  46.151  proof -
  46.152 -  have *: "op + a ` op *\<^sub>R c ` s = (\<lambda>x. a + c *\<^sub>R x) ` s" by auto
  46.153 +  have *: "(+) a ` ( *\<^sub>R) c ` s = (\<lambda>x. a + c *\<^sub>R x) ` s" by auto
  46.154    show ?thesis
  46.155      using homeomorphic_trans
  46.156      using homeomorphic_scaling[OF assms, of s]
  46.157 @@ -4540,9 +4540,9 @@
  46.158      proof
  46.159        show "countable ((\<lambda>C. S \<inter> C) ` \<C>)"
  46.160          by (simp add: \<open>countable \<C>\<close>)
  46.161 -      show "\<And>C. C \<in> op \<inter> S ` \<C> \<Longrightarrow> openin (subtopology euclidean S) C"
  46.162 +      show "\<And>C. C \<in> (\<inter>) S ` \<C> \<Longrightarrow> openin (subtopology euclidean S) C"
  46.163          using ope by auto
  46.164 -      show "\<And>T. openin (subtopology euclidean S) T \<Longrightarrow> \<exists>\<U>\<subseteq>op \<inter> S ` \<C>. T = \<Union>\<U>"
  46.165 +      show "\<And>T. openin (subtopology euclidean S) T \<Longrightarrow> \<exists>\<U>\<subseteq>(\<inter>) S ` \<C>. T = \<Union>\<U>"
  46.166          by (metis \<C> image_mono inf_Sup openin_open)
  46.167      qed
  46.168    qed
    47.1 --- a/src/HOL/Analysis/Convex_Euclidean_Space.thy	Wed Jan 10 15:21:49 2018 +0100
    47.2 +++ b/src/HOL/Analysis/Convex_Euclidean_Space.thy	Wed Jan 10 15:25:09 2018 +0100
    47.3 @@ -124,7 +124,7 @@
    47.4    by (rule ccontr) auto
    47.5  
    47.6  lemma subset_translation_eq [simp]:
    47.7 -    fixes a :: "'a::real_vector" shows "op + a ` s \<subseteq> op + a ` t \<longleftrightarrow> s \<subseteq> t"
    47.8 +    fixes a :: "'a::real_vector" shows "(+) a ` s \<subseteq> (+) a ` t \<longleftrightarrow> s \<subseteq> t"
    47.9    by auto
   47.10  
   47.11  lemma translate_inj_on:
   47.12 @@ -714,7 +714,7 @@
   47.13    assumes "convex S"
   47.14    shows "convex ((\<lambda>x. a + c *\<^sub>R x) ` S)"
   47.15  proof -
   47.16 -  have "(\<lambda>x. a + c *\<^sub>R x) ` S = op + a ` op *\<^sub>R c ` S"
   47.17 +  have "(\<lambda>x. a + c *\<^sub>R x) ` S = (+) a ` ( *\<^sub>R) c ` S"
   47.18      by auto
   47.19    then show ?thesis
   47.20      using convex_translation[OF convex_scaling[OF assms], of a c] by auto
   47.21 @@ -1330,12 +1330,12 @@
   47.22  
   47.23  lemma closure_scaleR:
   47.24    fixes S :: "'a::real_normed_vector set"
   47.25 -  shows "(op *\<^sub>R c) ` (closure S) = closure ((op *\<^sub>R c) ` S)"
   47.26 +  shows "(( *\<^sub>R) c) ` (closure S) = closure ((( *\<^sub>R) c) ` S)"
   47.27  proof
   47.28 -  show "(op *\<^sub>R c) ` (closure S) \<subseteq> closure ((op *\<^sub>R c) ` S)"
   47.29 +  show "(( *\<^sub>R) c) ` (closure S) \<subseteq> closure ((( *\<^sub>R) c) ` S)"
   47.30      using bounded_linear_scaleR_right
   47.31      by (rule closure_bounded_linear_image_subset)
   47.32 -  show "closure ((op *\<^sub>R c) ` S) \<subseteq> (op *\<^sub>R c) ` (closure S)"
   47.33 +  show "closure ((( *\<^sub>R) c) ` S) \<subseteq> (( *\<^sub>R) c) ` (closure S)"
   47.34      by (intro closure_minimal image_mono closure_subset closed_scaling closed_closure)
   47.35  qed
   47.36  
   47.37 @@ -2154,7 +2154,7 @@
   47.38  
   47.39  lemma cone_iff:
   47.40    assumes "S \<noteq> {}"
   47.41 -  shows "cone S \<longleftrightarrow> 0 \<in> S \<and> (\<forall>c. c > 0 \<longrightarrow> (op *\<^sub>R c) ` S = S)"
   47.42 +  shows "cone S \<longleftrightarrow> 0 \<in> S \<and> (\<forall>c. c > 0 \<longrightarrow> (( *\<^sub>R) c) ` S = S)"
   47.43  proof -
   47.44    {
   47.45      assume "cone S"
   47.46 @@ -2164,7 +2164,7 @@
   47.47        {
   47.48          fix x
   47.49          assume "x \<in> S"
   47.50 -        then have "x \<in> (op *\<^sub>R c) ` S"
   47.51 +        then have "x \<in> (( *\<^sub>R) c) ` S"
   47.52            unfolding image_def
   47.53            using \<open>cone S\<close> \<open>c>0\<close> mem_cone[of S x "1/c"]
   47.54              exI[of "(\<lambda>t. t \<in> S \<and> x = c *\<^sub>R t)" "(1 / c) *\<^sub>R x"]
   47.55 @@ -2173,19 +2173,19 @@
   47.56        moreover
   47.57        {
   47.58          fix x
   47.59 -        assume "x \<in> (op *\<^sub>R c) ` S"
   47.60 +        assume "x \<in> (( *\<^sub>R) c) ` S"
   47.61          then have "x \<in> S"
   47.62            using \<open>cone S\<close> \<open>c > 0\<close>
   47.63            unfolding cone_def image_def \<open>c > 0\<close> by auto
   47.64        }
   47.65 -      ultimately have "(op *\<^sub>R c) ` S = S" by auto
   47.66 +      ultimately have "(( *\<^sub>R) c) ` S = S" by auto
   47.67      }
   47.68 -    then have "0 \<in> S \<and> (\<forall>c. c > 0 \<longrightarrow> (op *\<^sub>R c) ` S = S)"
   47.69 +    then have "0 \<in> S \<and> (\<forall>c. c > 0 \<longrightarrow> (( *\<^sub>R) c) ` S = S)"
   47.70        using \<open>cone S\<close> cone_contains_0[of S] assms by auto
   47.71    }
   47.72    moreover
   47.73    {
   47.74 -    assume a: "0 \<in> S \<and> (\<forall>c. c > 0 \<longrightarrow> (op *\<^sub>R c) ` S = S)"
   47.75 +    assume a: "0 \<in> S \<and> (\<forall>c. c > 0 \<longrightarrow> (( *\<^sub>R) c) ` S = S)"
   47.76      {
   47.77        fix x
   47.78        assume "x \<in> S"
   47.79 @@ -2271,9 +2271,9 @@
   47.80    then show ?thesis by auto
   47.81  next
   47.82    case False
   47.83 -  then have "0 \<in> S \<and> (\<forall>c. c > 0 \<longrightarrow> op *\<^sub>R c ` S = S)"
   47.84 +  then have "0 \<in> S \<and> (\<forall>c. c > 0 \<longrightarrow> ( *\<^sub>R) c ` S = S)"
   47.85      using cone_iff[of S] assms by auto
   47.86 -  then have "0 \<in> closure S \<and> (\<forall>c. c > 0 \<longrightarrow> op *\<^sub>R c ` closure S = closure S)"
   47.87 +  then have "0 \<in> closure S \<and> (\<forall>c. c > 0 \<longrightarrow> ( *\<^sub>R) c ` closure S = closure S)"
   47.88      using closure_subset by (auto simp add: closure_scaleR)
   47.89    then show ?thesis
   47.90      using False cone_iff[of "closure S"] by auto
   47.91 @@ -3344,7 +3344,7 @@
   47.92  proof -
   47.93    obtain x where x: "x \<in> S \<and> x \<in> affine hull (S - {x})"
   47.94      using assms affine_dependent_def by auto
   47.95 -  have "op + a ` (S - {x}) = op + a ` S - {a + x}"
   47.96 +  have "(+) a ` (S - {x}) = (+) a ` S - {a + x}"
   47.97      by auto
   47.98    then have "a + x \<in> affine hull ((\<lambda>x. a + x) ` S - {a + x})"
   47.99      using affine_hull_translation[of a "S - {x}"] x by auto
  47.100 @@ -3409,7 +3409,7 @@
  47.101    assumes "a \<notin> S"
  47.102    shows "affine_dependent (insert a S) \<longleftrightarrow> dependent ((\<lambda>x. -a + x) ` S)"
  47.103  proof -
  47.104 -  have "(op + (- a) ` S) = {x - a| x . x : S}" by auto
  47.105 +  have "((+) (- a) ` S) = {x - a| x . x : S}" by auto
  47.106    then show ?thesis
  47.107      using affine_dependent_translation_eq[of "(insert a S)" "-a"]
  47.108        affine_dependent_imp_dependent2 assms
  47.109 @@ -3448,7 +3448,7 @@
  47.110      then have "insert 0 ((\<lambda>x. -a+x) ` (s - {a})) = (\<lambda>x. -a+x) ` s"
  47.111        by auto
  47.112      then have "span ((\<lambda>x. -a+x) ` (s - {a}))=span ((\<lambda>x. -a+x) ` s)"
  47.113 -      using span_insert_0[of "op + (- a) ` (s - {a})"] by (auto simp del: uminus_add_conv_diff)
  47.114 +      using span_insert_0[of "(+) (- a) ` (s - {a})"] by (auto simp del: uminus_add_conv_diff)
  47.115      moreover have "{x - a |x. x \<in> (s - {a})} = ((\<lambda>x. -a+x) ` (s - {a}))"
  47.116        by auto
  47.117      moreover have "insert a (s - {a}) = insert a s"
  47.118 @@ -4024,14 +4024,14 @@
  47.119    case False
  47.120    then obtain c S' where "c \<notin> S'" "S = insert c S'"
  47.121      by (meson equals0I mk_disjoint_insert)
  47.122 -  have "dim (op + (-c) ` S) < DIM('a)"
  47.123 +  have "dim ((+) (-c) ` S) < DIM('a)"
  47.124      by (metis \<open>S = insert c S'\<close> aff_dim_eq_dim assms hull_inc insertI1 of_nat_less_imp_less)
  47.125 -  then obtain a where "a \<noteq> 0" "span (op + (-c) ` S) \<subseteq> {x. a \<bullet> x = 0}"
  47.126 +  then obtain a where "a \<noteq> 0" "span ((+) (-c) ` S) \<subseteq> {x. a \<bullet> x = 0}"
  47.127      using lowdim_subset_hyperplane by blast
  47.128    moreover
  47.129 -  have "a \<bullet> w = a \<bullet> c" if "span (op + (- c) ` S) \<subseteq> {x. a \<bullet> x = 0}" "w \<in> S" for w
  47.130 +  have "a \<bullet> w = a \<bullet> c" if "span ((+) (- c) ` S) \<subseteq> {x. a \<bullet> x = 0}" "w \<in> S" for w
  47.131    proof -
  47.132 -    have "w-c \<in> span (op + (- c) ` S)"
  47.133 +    have "w-c \<in> span ((+) (- c) ` S)"
  47.134        by (simp add: span_superset \<open>w \<in> S\<close>)
  47.135      with that have "w-c \<in> {x. a \<bullet> x = 0}"
  47.136        by blast
  47.137 @@ -4235,7 +4235,7 @@
  47.138      unfolding cball_def dist_norm by auto
  47.139    then have "aff_dim (cball (0 :: 'n::euclidean_space) e) \<le> aff_dim (cball a e)"
  47.140      using aff_dim_translation_eq[of a "cball 0 e"]
  47.141 -          aff_dim_subset[of "op + a ` cball 0 e" "cball a e"]
  47.142 +          aff_dim_subset[of "(+) a ` cball 0 e" "cball a e"]
  47.143      by auto
  47.144    moreover have "aff_dim (cball (0 :: 'n::euclidean_space) e) = int (DIM('n))"
  47.145      using hull_inc[of "(0 :: 'n::euclidean_space)" "cball 0 e"]
  47.146 @@ -4805,7 +4805,7 @@
  47.147      assume "S \<noteq> {}"
  47.148      then obtain L where L: "subspace L" "affine_parallel S L"
  47.149        using assms affine_parallel_subspace[of S] by auto
  47.150 -    then obtain a where a: "S = (op + a ` L)"
  47.151 +    then obtain a where a: "S = ((+) a ` L)"
  47.152        using affine_parallel_def[of L S] affine_parallel_commut by auto
  47.153      from L have "closed L" using closed_subspace by auto
  47.154      then have "closed S"
  47.155 @@ -5000,7 +5000,7 @@
  47.156        translation_assoc[of "-a" "a"]
  47.157      by auto
  47.158    then have "((\<lambda>x. a + x) ` rel_interior S) \<supseteq> rel_interior ((\<lambda>x. a + x) ` S)"
  47.159 -    using translation_inverse_subset[of a "rel_interior (op + a ` S)" "rel_interior S"]
  47.160 +    using translation_inverse_subset[of a "rel_interior ((+) a ` S)" "rel_interior S"]
  47.161      by auto
  47.162    then show ?thesis
  47.163      using rel_interior_translation_aux[of a S] by auto
  47.164 @@ -5141,7 +5141,7 @@
  47.165    shows "affine hull (insert 0 d) = {x::'a::euclidean_space. \<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x\<bullet>i = 0}"
  47.166    (is "affine hull (insert 0 ?A) = ?B")
  47.167  proof -
  47.168 -  have *: "\<And>A. op + (0::'a) ` A = A" "\<And>A. op + (- (0::'a)) ` A = A"
  47.169 +  have *: "\<And>A. (+) (0::'a) ` A = A" "\<And>A. (+) (- (0::'a)) ` A = A"
  47.170      by auto
  47.171    show ?thesis
  47.172      unfolding affine_hull_insert_span_gen span_substd_basis[OF assms,symmetric] * ..
  47.173 @@ -5982,7 +5982,7 @@
  47.174        unfolding k_def
  47.175      proof (subst less_cSUP_iff)
  47.176        show "T \<noteq> {}" by fact
  47.177 -      show "bdd_above (op \<bullet> a ` T)"
  47.178 +      show "bdd_above ((\<bullet>) a ` T)"
  47.179          using ab[rule_format, of y] \<open>y \<in> S\<close>
  47.180          by (intro bdd_aboveI2[where M="inner a y - b"]) (auto simp: field_simps intro: less_imp_le)
  47.181      qed (auto intro!: bexI[of _ x] \<open>0<b\<close>)
  47.182 @@ -6073,7 +6073,7 @@
  47.183      using assms(3-5) by fastforce
  47.184    then have *: "\<And>x y. x \<in> t \<Longrightarrow> y \<in> s \<Longrightarrow> inner a y \<le> inner a x"
  47.185      by (force simp add: inner_diff)
  47.186 -  then have bdd: "bdd_above ((op \<bullet> a)`s)"
  47.187 +  then have bdd: "bdd_above (((\<bullet>) a)`s)"
  47.188      using \<open>t \<noteq> {}\<close> by (auto intro: bdd_aboveI2[OF *])
  47.189    show ?thesis
  47.190      using \<open>a\<noteq>0\<close>
  47.191 @@ -6209,19 +6209,19 @@
  47.192    then show ?thesis by auto
  47.193  next
  47.194    case False
  47.195 -  then have *: "0 \<in> S \<and> (\<forall>c. c > 0 \<longrightarrow> op *\<^sub>R c ` S = S)"
  47.196 +  then have *: "0 \<in> S \<and> (\<forall>c. c > 0 \<longrightarrow> ( *\<^sub>R) c ` S = S)"
  47.197      using cone_iff[of S] assms by auto
  47.198    {
  47.199      fix c :: real
  47.200      assume "c > 0"
  47.201 -    then have "op *\<^sub>R c ` (convex hull S) = convex hull (op *\<^sub>R c ` S)"
  47.202 +    then have "( *\<^sub>R) c ` (convex hull S) = convex hull (( *\<^sub>R) c ` S)"
  47.203        using convex_hull_scaling[of _ S] by auto
  47.204      also have "\<dots> = convex hull S"
  47.205        using * \<open>c > 0\<close> by auto
  47.206 -    finally have "op *\<^sub>R c ` (convex hull S) = convex hull S"
  47.207 +    finally have "( *\<^sub>R) c ` (convex hull S) = convex hull S"
  47.208        by auto
  47.209    }
  47.210 -  then have "0 \<in> convex hull S" "\<And>c. c > 0 \<Longrightarrow> (op *\<^sub>R c ` (convex hull S)) = (convex hull S)"
  47.211 +  then have "0 \<in> convex hull S" "\<And>c. c > 0 \<Longrightarrow> (( *\<^sub>R) c ` (convex hull S)) = (convex hull S)"
  47.212      using * hull_subset[of S convex] by auto
  47.213    then show ?thesis
  47.214      using \<open>S \<noteq> {}\<close> cone_iff[of "convex hull S"] by auto
  47.215 @@ -7030,7 +7030,7 @@
  47.216    fixes a :: "'a::euclidean_space"
  47.217    assumes "cbox a b \<noteq> {}"
  47.218      shows "cbox a b =
  47.219 -           op + a ` (\<lambda>x. \<Sum>k\<in>Basis. ((b \<bullet> k - a \<bullet> k) * (x \<bullet> k)) *\<^sub>R k) ` cbox 0 One"
  47.220 +           (+) a ` (\<lambda>x. \<Sum>k\<in>Basis. ((b \<bullet> k - a \<bullet> k) * (x \<bullet> k)) *\<^sub>R k) ` cbox 0 One"
  47.221  using assms
  47.222  apply (simp add: box_ne_empty image_stretch_interval cbox_translation [symmetric])
  47.223  apply (simp add: min_def max_def algebra_simps sum_subtractf euclidean_representation)
  47.224 @@ -7048,7 +7048,7 @@
  47.225      by (blast intro: unit_cube_convex_hull)
  47.226    have lin: "linear (\<lambda>x. \<Sum>k\<in>Basis. ((b \<bullet> k - a \<bullet> k) * (x \<bullet> k)) *\<^sub>R k)"
  47.227      by (rule linear_compose_sum) (auto simp: algebra_simps linearI)
  47.228 -  have "finite (op + a ` (\<lambda>x. \<Sum>k\<in>Basis. ((b \<bullet> k - a \<bullet> k) * (x \<bullet> k)) *\<^sub>R k) ` s)"
  47.229 +  have "finite ((+) a ` (\<lambda>x. \<Sum>k\<in>Basis. ((b \<bullet> k - a \<bullet> k) * (x \<bullet> k)) *\<^sub>R k) ` s)"
  47.230      by (rule finite_imageI \<open>finite s\<close>)+
  47.231    then show ?thesis
  47.232      apply (rule that)
    48.1 --- a/src/HOL/Analysis/Derivative.thy	Wed Jan 10 15:21:49 2018 +0100
    48.2 +++ b/src/HOL/Analysis/Derivative.thy	Wed Jan 10 15:25:09 2018 +0100
    48.3 @@ -91,7 +91,7 @@
    48.4  lemma has_derivative_right:
    48.5    fixes f :: "real \<Rightarrow> real"
    48.6      and y :: "real"
    48.7 -  shows "(f has_derivative (op * y)) (at x within ({x <..} \<inter> I)) \<longleftrightarrow>
    48.8 +  shows "(f has_derivative (( * ) y)) (at x within ({x <..} \<inter> I)) \<longleftrightarrow>
    48.9      ((\<lambda>t. (f x - f t) / (x - t)) \<longlongrightarrow> y) (at x within ({x <..} \<inter> I))"
   48.10  proof -
   48.11    have "((\<lambda>t. (f t - (f x + y * (t - x))) / \<bar>t - x\<bar>) \<longlongrightarrow> 0) (at x within ({x<..} \<inter> I)) \<longleftrightarrow>
   48.12 @@ -245,7 +245,7 @@
   48.13  
   48.14  lemma has_derivative_sqnorm_at [derivative_intros, simp]:
   48.15     "((\<lambda>x. (norm x)\<^sup>2) has_derivative (\<lambda>x. 2 *\<^sub>R (a \<bullet> x))) (at a)"
   48.16 -using has_derivative_bilinear_at [of id id a id id "op  \<bullet>"]
   48.17 +using has_derivative_bilinear_at [of id id a id id "(\<bullet>)"]
   48.18  by (auto simp: inner_commute dot_square_norm bounded_bilinear_inner)
   48.19  
   48.20  lemma differentiable_sqnorm_at [derivative_intros, simp]:
   48.21 @@ -1112,7 +1112,7 @@
   48.22    shows "norm (f (x0 + a) - f x0) \<le> norm a * B"
   48.23  proof -
   48.24    let ?G = "(\<lambda>x. x0 + x *\<^sub>R a) ` {0..1}"
   48.25 -  have "?G = op + x0 ` (\<lambda>x. x *\<^sub>R a) ` {0..1}" by auto
   48.26 +  have "?G = (+) x0 ` (\<lambda>x. x *\<^sub>R a) ` {0..1}" by auto
   48.27    also have "convex \<dots>"
   48.28      by (intro convex_translation convex_scaled convex_real_interval)
   48.29    finally have "convex ?G" .
   48.30 @@ -1164,7 +1164,7 @@
   48.31    assumes "convex s" "\<And>x. x \<in> s \<Longrightarrow> (f has_field_derivative 0) (at x within s)"
   48.32    shows   "\<exists>c. \<forall>x\<in>s. f (x) = (c :: 'a :: real_normed_field)"
   48.33  proof (rule has_derivative_zero_constant)
   48.34 -  have A: "op * 0 = (\<lambda>_. 0 :: 'a)" by (intro ext) simp
   48.35 +  have A: "( * ) 0 = (\<lambda>_. 0 :: 'a)" by (intro ext) simp
   48.36    fix x assume "x \<in> s" thus "(f has_derivative (\<lambda>h. 0)) (at x within s)"
   48.37      using assms(2)[of x] by (simp add: has_field_derivative_def A)
   48.38  qed fact
   48.39 @@ -2292,9 +2292,9 @@
   48.40    then obtain g where g: "\<And>x. x \<in> s \<Longrightarrow> (\<lambda>n. f n x) sums g x"
   48.41      "\<And>x. x \<in> s \<Longrightarrow> (g has_field_derivative g' x) (at x within s)" by blast
   48.42    from g[OF x] show "summable (\<lambda>n. f n x)" by (auto simp: summable_def)
   48.43 -  from g(2)[OF x] have g': "(g has_derivative op * (g' x)) (at x)"
   48.44 +  from g(2)[OF x] have g': "(g has_derivative ( * ) (g' x)) (at x)"
   48.45      by (simp add: has_field_derivative_def s)
   48.46 -  have "((\<lambda>x. \<Sum>n. f n x) has_derivative op * (g' x)) (at x)"
   48.47 +  have "((\<lambda>x. \<Sum>n. f n x) has_derivative ( * ) (g' x)) (at x)"
   48.48      by (rule has_derivative_transform_within_open[OF g' \<open>open s\<close> x])
   48.49         (insert g, auto simp: sums_iff)
   48.50    thus "(\<lambda>x. \<Sum>n. f n x) differentiable (at x)" unfolding differentiable_def
   48.51 @@ -2645,7 +2645,7 @@
   48.52    unfolding field_differentiable_def
   48.53    by (metis DERIV_subset top_greatest)
   48.54  
   48.55 -lemma field_differentiable_linear [simp,derivative_intros]: "(op * c) field_differentiable F"
   48.56 +lemma field_differentiable_linear [simp,derivative_intros]: "(( * ) c) field_differentiable F"
   48.57  proof -
   48.58    show ?thesis
   48.59      unfolding field_differentiable_def has_field_derivative_def mult_commute_abs
   48.60 @@ -2675,7 +2675,7 @@
   48.61    by (metis field_differentiable_add)
   48.62  
   48.63  lemma field_differentiable_add_const [simp,derivative_intros]:
   48.64 -     "op + c field_differentiable F"
   48.65 +     "(+) c field_differentiable F"
   48.66    by (simp add: field_differentiable_add)
   48.67  
   48.68  lemma field_differentiable_sum [derivative_intros]:
    49.1 --- a/src/HOL/Analysis/Determinants.thy	Wed Jan 10 15:21:49 2018 +0100
    49.2 +++ b/src/HOL/Analysis/Determinants.thy	Wed Jan 10 15:25:09 2018 +0100
    49.3 @@ -819,7 +819,7 @@
    49.4        unfolding matrix_right_invertible_independent_rows
    49.5        by blast
    49.6      have *: "\<And>(a::real^'n) b. a + b = 0 \<Longrightarrow> -a = b"
    49.7 -      apply (drule_tac f="op + (- a)" in cong[OF refl])
    49.8 +      apply (drule_tac f="(+) (- a)" in cong[OF refl])
    49.9        apply (simp only: ab_left_minus add.assoc[symmetric])
   49.10        apply simp
   49.11        done
    50.1 --- a/src/HOL/Analysis/Embed_Measure.thy	Wed Jan 10 15:21:49 2018 +0100
    50.2 +++ b/src/HOL/Analysis/Embed_Measure.thy	Wed Jan 10 15:25:09 2018 +0100
    50.3 @@ -159,15 +159,15 @@
    50.4    from assms(1) interpret sigma_finite_measure M .
    50.5    from sigma_finite_countable obtain A where
    50.6        A_props: "countable A" "A \<subseteq> sets M" "\<Union>A = space M" "\<And>X. X\<in>A \<Longrightarrow> emeasure M X \<noteq> \<infinity>" by blast
    50.7 -  from A_props have "countable (op ` f`A)" by auto
    50.8 +  from A_props have "countable ((`) f`A)" by auto
    50.9    moreover
   50.10 -  from inj and A_props have "op ` f`A \<subseteq> sets (embed_measure M f)"
   50.11 +  from inj and A_props have "(`) f`A \<subseteq> sets (embed_measure M f)"
   50.12      by (auto simp: sets_embed_measure)
   50.13    moreover
   50.14 -  from A_props and inj have "\<Union>(op ` f`A) = space (embed_measure M f)"
   50.15 +  from A_props and inj have "\<Union>((`) f`A) = space (embed_measure M f)"
   50.16      by (auto simp: space_embed_measure intro!: imageI)
   50.17    moreover
   50.18 -  from A_props and inj have "\<forall>a\<in>op ` f ` A. emeasure (embed_measure M f) a \<noteq> \<infinity>"
   50.19 +  from A_props and inj have "\<forall>a\<in>(`) f ` A. emeasure (embed_measure M f) a \<noteq> \<infinity>"
   50.20      by (intro ballI, subst emeasure_embed_measure)
   50.21         (auto simp: inj_vimage_image_eq intro: in_sets_embed_measure)
   50.22    ultimately show ?thesis by - (standard, blast)
   50.23 @@ -189,7 +189,7 @@
   50.24    by(rule embed_measure_count_space')(erule subset_inj_on, simp)
   50.25  
   50.26  lemma sets_embed_measure_alt:
   50.27 -    "inj f \<Longrightarrow> sets (embed_measure M f) = (op` f) ` sets M"
   50.28 +    "inj f \<Longrightarrow> sets (embed_measure M f) = ((`) f) ` sets M"
   50.29    by (auto simp: sets_embed_measure)
   50.30  
   50.31  lemma emeasure_embed_measure_image':
   50.32 @@ -210,7 +210,7 @@
   50.33    assumes "inj f"
   50.34    shows "embed_measure A f = embed_measure B f \<longleftrightarrow> A = B" (is "?M = ?N \<longleftrightarrow> _")
   50.35  proof
   50.36 -  from assms have I: "inj (op` f)" by (auto intro: injI dest: injD)
   50.37 +  from assms have I: "inj ((`) f)" by (auto intro: injI dest: injD)
   50.38    assume asm: "?M = ?N"
   50.39    hence "sets (embed_measure A f) = sets (embed_measure B f)" by simp
   50.40    with assms have "sets A = sets B" by (simp only: I inj_image_eq_iff sets_embed_measure_alt)
   50.41 @@ -369,7 +369,7 @@
   50.42  lemma nn_integral_monotone_convergence_SUP_countable:
   50.43    fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> ennreal"
   50.44    assumes nonempty: "Y \<noteq> {}"
   50.45 -  and chain: "Complete_Partial_Order.chain op \<le> (f ` Y)"
   50.46 +  and chain: "Complete_Partial_Order.chain (\<le>) (f ` Y)"
   50.47    and countable: "countable B"
   50.48    shows "(\<integral>\<^sup>+ x. (SUP i:Y. f i x) \<partial>count_space B) = (SUP i:Y. (\<integral>\<^sup>+ x. f i x \<partial>count_space B))"
   50.49    (is "?lhs = ?rhs")
   50.50 @@ -383,7 +383,7 @@
   50.51      by(simp add: nn_integral_count_space_indicator ennreal_indicator[symmetric] SUP_mult_right_ennreal nonempty)
   50.52    also have "\<dots> = (SUP i:Y. \<integral>\<^sup>+ x. ?f i x \<partial>count_space UNIV)"
   50.53    proof(rule nn_integral_monotone_convergence_SUP_nat)
   50.54 -    show "Complete_Partial_Order.chain op \<le> (?f ` Y)"
   50.55 +    show "Complete_Partial_Order.chain (\<le>) (?f ` Y)"
   50.56        by(rule chain_imageI[OF chain, unfolded image_image])(auto intro!: le_funI split: split_indicator dest: le_funD)
   50.57    qed fact
   50.58    also have "\<dots> = (SUP i:Y. \<integral>\<^sup>+ x. f i (from_nat_into B x) \<partial>count_space (to_nat_on B ` B))"
    51.1 --- a/src/HOL/Analysis/Equivalence_Lebesgue_Henstock_Integration.thy	Wed Jan 10 15:21:49 2018 +0100
    51.2 +++ b/src/HOL/Analysis/Equivalence_Lebesgue_Henstock_Integration.thy	Wed Jan 10 15:25:09 2018 +0100
    51.3 @@ -1041,10 +1041,10 @@
    51.4        and eq1: "\<And>c x. \<lbrakk>(a + c *\<^sub>R x) \<in> S; 0 \<le> c; a + x \<in> S\<rbrakk> \<Longrightarrow> c = 1"
    51.5      shows "negligible S"
    51.6  proof -
    51.7 -  have "negligible (op + (-a) ` S)"
    51.8 +  have "negligible ((+) (-a) ` S)"
    51.9    proof (subst negligible_on_intervals, intro allI)
   51.10      fix u v
   51.11 -    show "negligible (op + (- a) ` S \<inter> cbox u v)"
   51.12 +    show "negligible ((+) (- a) ` S \<inter> cbox u v)"
   51.13        unfolding negligible_iff_null_sets
   51.14        apply (rule starlike_negligible_compact)
   51.15         apply (simp add: assms closed_translation closed_Int_compact, clarify)
   51.16 @@ -1831,7 +1831,7 @@
   51.17            qed
   51.18            also have "\<dots> = (\<Sum>(i,l) \<in> d \<times> snd ` p. norm (integral (i\<inter>l) f))"
   51.19              by (simp add: sum.cartesian_product)
   51.20 -          also have "\<dots> = (\<Sum>x \<in> d \<times> snd ` p. norm (integral (case_prod op \<inter> x) f))"
   51.21 +          also have "\<dots> = (\<Sum>x \<in> d \<times> snd ` p. norm (integral (case_prod (\<inter>) x) f))"
   51.22              by (force simp: split_def intro!: sum.cong)
   51.23            also have "\<dots> = (\<Sum>k\<in>{i \<inter> l |i l. i \<in> d \<and> l \<in> snd ` p}. norm (integral k f))"
   51.24            proof -
    52.1 --- a/src/HOL/Analysis/Euclidean_Space.thy	Wed Jan 10 15:21:49 2018 +0100
    52.2 +++ b/src/HOL/Analysis/Euclidean_Space.thy	Wed Jan 10 15:25:09 2018 +0100
    52.3 @@ -242,7 +242,7 @@
    52.4  
    52.5  lemma DIM_prod[simp]: "DIM('a \<times> 'b) = DIM('a) + DIM('b)"
    52.6    unfolding Basis_prod_def
    52.7 -  by (subst card_Un_disjoint) (auto intro!: card_image arg_cong2[where f="op +"] inj_onI)
    52.8 +  by (subst card_Un_disjoint) (auto intro!: card_image arg_cong2[where f="(+)"] inj_onI)
    52.9  
   52.10  end
   52.11  
    53.1 --- a/src/HOL/Analysis/Finite_Cartesian_Product.thy	Wed Jan 10 15:21:49 2018 +0100
    53.2 +++ b/src/HOL/Analysis/Finite_Cartesian_Product.thy	Wed Jan 10 15:25:09 2018 +0100
    53.3 @@ -142,13 +142,13 @@
    53.4  
    53.5  instantiation vec :: (plus, finite) plus
    53.6  begin
    53.7 -  definition "op + \<equiv> (\<lambda> x y. (\<chi> i. x$i + y$i))"
    53.8 +  definition "(+) \<equiv> (\<lambda> x y. (\<chi> i. x$i + y$i))"
    53.9    instance ..
   53.10  end
   53.11  
   53.12  instantiation vec :: (minus, finite) minus
   53.13  begin
   53.14 -  definition "op - \<equiv> (\<lambda> x y. (\<chi> i. x$i - y$i))"
   53.15 +  definition "(-) \<equiv> (\<lambda> x y. (\<chi> i. x$i - y$i))"
   53.16    instance ..
   53.17  end
   53.18  
    54.1 --- a/src/HOL/Analysis/Finite_Product_Measure.thy	Wed Jan 10 15:21:49 2018 +0100
    54.2 +++ b/src/HOL/Analysis/Finite_Product_Measure.thy	Wed Jan 10 15:25:09 2018 +0100
    54.3 @@ -929,7 +929,7 @@
    54.4        using E by (subst insert) (auto intro!: prod.cong)
    54.5      also have "(\<Prod>j\<in>I. if j \<in> J - {i} then emeasure (M j) (E j) else emeasure (M j) (space (M j))) *
    54.6         emeasure (M i) (if i \<in> J then E i else space (M i)) = (\<Prod>j\<in>insert i I. ?f J E j)"
    54.7 -      using insert by (auto simp: mult.commute intro!: arg_cong2[where f="op *"] prod.cong)
    54.8 +      using insert by (auto simp: mult.commute intro!: arg_cong2[where f="( * )"] prod.cong)
    54.9      also have "\<dots> = (\<Prod>j\<in>J \<union> ?I. ?f J E j)"
   54.10        using insert(1,2) J E by (intro prod.mono_neutral_right) auto
   54.11      finally show "?\<mu> ?p = \<dots>" .
    55.1 --- a/src/HOL/Analysis/Further_Topology.thy	Wed Jan 10 15:21:49 2018 +0100
    55.2 +++ b/src/HOL/Analysis/Further_Topology.thy	Wed Jan 10 15:25:09 2018 +0100
    55.3 @@ -527,7 +527,7 @@
    55.4                          and gh: "\<And>x. x \<in> rel_frontier D \<Longrightarrow> g x = h x"
    55.5            by (metis *)
    55.6          have "D \<inter> E \<subseteq> rel_frontier D"
    55.7 -             if "E \<in> \<G> \<union> {D. Bex \<F> (op face_of D) \<and> aff_dim D < int p}" for E
    55.8 +             if "E \<in> \<G> \<union> {D. Bex \<F> ((face_of) D) \<and> aff_dim D < int p}" for E
    55.9          proof (rule face_of_subset_rel_frontier)
   55.10            show "D \<inter> E face_of D"
   55.11              using that \<open>C \<in> \<F>\<close> \<open>D face_of C\<close> face
   55.12 @@ -2173,21 +2173,21 @@
   55.13    case False
   55.14    obtain a b where "a \<in> S" "a \<in> U" "b \<in> V"
   55.15      using False fim ope openin_contains_cball by fastforce
   55.16 -  have "openin (subtopology euclidean (op + (- b) ` V)) ((op + (- b) \<circ> f \<circ> op + a) ` op + (- a) ` S)"
   55.17 +  have "openin (subtopology euclidean ((+) (- b) ` V)) (((+) (- b) \<circ> f \<circ> (+) a) ` (+) (- a) ` S)"
   55.18    proof (rule invariance_of_domain_subspaces)
   55.19 -    show "openin (subtopology euclidean (op + (- a) ` U)) (op + (- a) ` S)"
   55.20 +    show "openin (subtopology euclidean ((+) (- a) ` U)) ((+) (- a) ` S)"
   55.21        by (metis ope homeomorphism_imp_open_map homeomorphism_translation translation_galois)
   55.22 -    show "subspace (op + (- a) ` U)"
   55.23 +    show "subspace ((+) (- a) ` U)"
   55.24        by (simp add: \<open>a \<in> U\<close> affine_diffs_subspace \<open>affine U\<close>)
   55.25 -    show "subspace (op + (- b) ` V)"
   55.26 +    show "subspace ((+) (- b) ` V)"
   55.27        by (simp add: \<open>b \<in> V\<close> affine_diffs_subspace \<open>affine V\<close>)
   55.28 -    show "dim (op + (- b) ` V) \<le> dim (op + (- a) ` U)"
   55.29 +    show "dim ((+) (- b) ` V) \<le> dim ((+) (- a) ` U)"
   55.30        by (metis \<open>a \<in> U\<close> \<open>b \<in> V\<close> aff_dim_eq_dim affine_hull_eq aff of_nat_le_iff)
   55.31 -    show "continuous_on (op + (- a) ` S) (op + (- b) \<circ> f \<circ> op + a)"
   55.32 +    show "continuous_on ((+) (- a) ` S) ((+) (- b) \<circ> f \<circ> (+) a)"
   55.33        by (metis contf continuous_on_compose homeomorphism_cont2 homeomorphism_translation translation_galois)
   55.34 -    show "(op + (- b) \<circ> f \<circ> op + a) ` op + (- a) ` S \<subseteq> op + (- b) ` V"
   55.35 +    show "((+) (- b) \<circ> f \<circ> (+) a) ` (+) (- a) ` S \<subseteq> (+) (- b) ` V"
   55.36        using fim by auto
   55.37 -    show "inj_on (op + (- b) \<circ> f \<circ> op + a) (op + (- a) ` S)"
   55.38 +    show "inj_on ((+) (- b) \<circ> f \<circ> (+) a) ((+) (- a) ` S)"
   55.39        by (auto simp: inj_on_def) (meson inj_onD injf)
   55.40    qed
   55.41    then show ?thesis
   55.42 @@ -2204,19 +2204,19 @@
   55.43  proof -
   55.44    obtain a b where "a \<in> S" "a \<in> U" "b \<in> V"
   55.45      using \<open>S \<noteq> {}\<close> fim ope openin_contains_cball by fastforce
   55.46 -  have "dim (op + (- a) ` U) \<le> dim (op + (- b) ` V)"
   55.47 +  have "dim ((+) (- a) ` U) \<le> dim ((+) (- b) ` V)"
   55.48    proof (rule invariance_of_dimension_subspaces)
   55.49 -    show "openin (subtopology euclidean (op + (- a) ` U)) (op + (- a) ` S)"
   55.50 +    show "openin (subtopology euclidean ((+) (- a) ` U)) ((+) (- a) ` S)"
   55.51        by (metis ope homeomorphism_imp_open_map homeomorphism_translation translation_galois)
   55.52 -    show "subspace (op + (- a) ` U)"
   55.53 +    show "subspace ((+) (- a) ` U)"
   55.54        by (simp add: \<open>a \<in> U\<close> affine_diffs_subspace \<open>affine U\<close>)
   55.55 -    show "subspace (op + (- b) ` V)"
   55.56 +    show "subspace ((+) (- b) ` V)"
   55.57        by (simp add: \<open>b \<in> V\<close> affine_diffs_subspace \<open>affine V\<close>)
   55.58 -    show "continuous_on (op + (- a) ` S) (op + (- b) \<circ> f \<circ> op + a)"
   55.59 +    show "continuous_on ((+) (- a) ` S) ((+) (- b) \<circ> f \<circ> (+) a)"
   55.60        by (metis contf continuous_on_compose homeomorphism_cont2 homeomorphism_translation translation_galois)
   55.61 -    show "(op + (- b) \<circ> f \<circ> op + a) ` op + (- a) ` S \<subseteq> op + (- b) ` V"
   55.62 +    show "((+) (- b) \<circ> f \<circ> (+) a) ` (+) (- a) ` S \<subseteq> (+) (- b) ` V"
   55.63        using fim by auto
   55.64 -    show "inj_on (op + (- b) \<circ> f \<circ> op + a) (op + (- a) ` S)"
   55.65 +    show "inj_on ((+) (- b) \<circ> f \<circ> (+) a) ((+) (- a) ` S)"
   55.66        by (auto simp: inj_on_def) (meson inj_onD injf)
   55.67    qed (use \<open>S \<noteq> {}\<close> in auto)
   55.68    then show ?thesis
   55.69 @@ -2762,7 +2762,7 @@
   55.70    case False
   55.71    then obtain a b where "a \<in> S" "b \<in> T"
   55.72      by blast
   55.73 -  then have "subspace (op + (- a) ` S)" "subspace (op + (- b) ` T)"
   55.74 +  then have "subspace ((+) (- a) ` S)" "subspace ((+) (- b) ` T)"
   55.75      using affine_diffs_subspace assms by blast+
   55.76    then show ?thesis
   55.77      by (metis affine_imp_convex assms homeomorphic_affine_sets homeomorphic_convex_sets)
   55.78 @@ -3539,12 +3539,12 @@
   55.79        by (intro conjI contg continuous_intros)
   55.80      show "(complex_of_real \<circ> g) ` S \<subseteq> \<real>"
   55.81        by auto
   55.82 -    show "continuous_on \<real> (exp \<circ> op*\<i>)"
   55.83 +    show "continuous_on \<real> (exp \<circ> ( * )\<i>)"
   55.84        by (intro continuous_intros)
   55.85 -    show "(exp \<circ> op*\<i>) ` \<real> \<subseteq> sphere 0 1"
   55.86 +    show "(exp \<circ> ( * )\<i>) ` \<real> \<subseteq> sphere 0 1"
   55.87        by (auto simp: complex_is_Real_iff)
   55.88    qed (auto simp: convex_Reals convex_imp_contractible)
   55.89 -  moreover have "\<And>x. x \<in> S \<Longrightarrow> (exp \<circ> op*\<i> \<circ> (complex_of_real \<circ> g)) x = f x"
   55.90 +  moreover have "\<And>x. x \<in> S \<Longrightarrow> (exp \<circ> ( * )\<i> \<circ> (complex_of_real \<circ> g)) x = f x"
   55.91      by (simp add: g)
   55.92    ultimately show ?lhs
   55.93      apply (rule_tac x=a in exI)
    56.1 --- a/src/HOL/Analysis/Gamma_Function.thy	Wed Jan 10 15:21:49 2018 +0100
    56.2 +++ b/src/HOL/Analysis/Gamma_Function.thy	Wed Jan 10 15:25:09 2018 +0100
    56.3 @@ -2038,7 +2038,7 @@
    56.4  
    56.5      moreover from z double_in_nonpos_Ints_imp[of z] have "2 * z \<notin> \<int>\<^sub>\<le>\<^sub>0" by auto
    56.6      hence "?g \<longlonglongrightarrow> ?powr 2 (2*z) * Gamma z * Gamma (z+1/2) / Gamma (2*z)"
    56.7 -      using LIMSEQ_subseq_LIMSEQ[OF Gamma_series'_LIMSEQ, of "op*2" "2*z"]
    56.8 +      using LIMSEQ_subseq_LIMSEQ[OF Gamma_series'_LIMSEQ, of "( * )2" "2*z"]
    56.9        by (intro tendsto_intros Gamma_series'_LIMSEQ)
   56.10           (simp_all add: o_def strict_mono_def Gamma_eq_zero_iff)
   56.11      ultimately have "?h \<longlonglongrightarrow> ?powr 2 (2*z) * Gamma z * Gamma (z+1/2) / Gamma (2*z)"
   56.12 @@ -3088,7 +3088,7 @@
   56.13    proof (rule nn_integral_cong, goal_cases)
   56.14      case (1 t)
   56.15      have "(\<integral>\<^sup>+u. ennreal (indicator ({0..}\<times>{0..}) (t,u) * t powr (a - 1) * 
   56.16 -                              u powr (b - 1) / exp (t + u)) \<partial>distr lborel borel (op + (-t))) =
   56.17 +                              u powr (b - 1) / exp (t + u)) \<partial>distr lborel borel ((+) (-t))) =
   56.18                 (\<integral>\<^sup>+u. ennreal (indicator ({0..}\<times>{t..}) (t,u) * t powr (a - 1) * 
   56.19                                (u - t) powr (b - 1) / exp u) \<partial>lborel)"
   56.20        by (subst nn_integral_distr) (auto intro!: nn_integral_cong simp: indicator_def)
   56.21 @@ -3116,10 +3116,10 @@
   56.22        case True
   56.23        have "(\<integral>\<^sup>+t. ennreal (indicator {0..u} t * t powr (a - 1) * (u - t) powr (b - 1)) \<partial>lborel) = 
   56.24                (\<integral>\<^sup>+t. ennreal (indicator {0..1} t * (u * t) powr (a - 1) * (u - u * t) powr (b - 1)) 
   56.25 -                \<partial>distr lborel borel (op * (1 / u)))" (is "_ = nn_integral _ ?f")
   56.26 +                \<partial>distr lborel borel (( * ) (1 / u)))" (is "_ = nn_integral _ ?f")
   56.27          using True
   56.28          by (subst nn_integral_distr) (auto simp: indicator_def field_simps intro!: nn_integral_cong)
   56.29 -      also have "distr lborel borel (op * (1 / u)) = density lborel (\<lambda>_. u)"
   56.30 +      also have "distr lborel borel (( * ) (1 / u)) = density lborel (\<lambda>_. u)"
   56.31          using \<open>u > 0\<close> by (subst lborel_distr_mult) auto
   56.32        also have "nn_integral \<dots> ?f = (\<integral>\<^sup>+x. ennreal (indicator {0..1} x * (u * (u * x) powr (a - 1) *
   56.33                                                (u * (1 - x)) powr (b - 1))) \<partial>lborel)" using \<open>u > 0\<close>
    57.1 --- a/src/HOL/Analysis/Great_Picard.thy	Wed Jan 10 15:21:49 2018 +0100
    57.2 +++ b/src/HOL/Analysis/Great_Picard.thy	Wed Jan 10 15:25:09 2018 +0100
    57.3 @@ -596,7 +596,7 @@
    57.4        proof (cases "k=1")
    57.5          case True
    57.6          then have "\<exists>x. k * x + l \<noteq> a + x" for a
    57.7 -          using l non [of a] ext [of f "op + a"]
    57.8 +          using l non [of a] ext [of f "(+) a"]
    57.9            by (metis add.commute diff_eq_eq)
   57.10          with True show ?thesis by auto
   57.11        next
    58.1 --- a/src/HOL/Analysis/Harmonic_Numbers.thy	Wed Jan 10 15:21:49 2018 +0100
    58.2 +++ b/src/HOL/Analysis/Harmonic_Numbers.thy	Wed Jan 10 15:25:09 2018 +0100
    58.3 @@ -225,7 +225,7 @@
    58.4      by (intro summable_Leibniz(1) decseq_imp_monoseq decseq_SucI) simp_all
    58.5    hence A: "(\<lambda>n. \<Sum>k<n. (-1)^k / real_of_nat (Suc k)) \<longlonglongrightarrow> (\<Sum>k. (-1)^k / real_of_nat (Suc k))"
    58.6      by (simp add: summable_sums_iff divide_inverse sums_def)
    58.7 -  from filterlim_compose[OF this filterlim_subseq[of "op * (2::nat)"]]
    58.8 +  from filterlim_compose[OF this filterlim_subseq[of "( * ) (2::nat)"]]
    58.9      have "(\<lambda>n. \<Sum>k<2*n. (-1)^k / real_of_nat (Suc k)) \<longlonglongrightarrow> (\<Sum>k. (-1)^k / real_of_nat (Suc k))"
   58.10      by (simp add: strict_mono_def)
   58.11    ultimately have "(\<Sum>k. (- 1) ^ k / real_of_nat (Suc k)) = ln 2" by (intro LIMSEQ_unique)
    59.1 --- a/src/HOL/Analysis/Henstock_Kurzweil_Integration.thy	Wed Jan 10 15:21:49 2018 +0100
    59.2 +++ b/src/HOL/Analysis/Henstock_Kurzweil_Integration.thy	Wed Jan 10 15:25:09 2018 +0100
    59.3 @@ -1396,7 +1396,7 @@
    59.4  
    59.5  lemma operative_integralI:
    59.6    fixes f :: "'a::euclidean_space \<Rightarrow> 'b::banach"
    59.7 -  shows "operative (lift_option op +) (Some 0)
    59.8 +  shows "operative (lift_option (+)) (Some 0)
    59.9      (\<lambda>i. if f integrable_on i then Some (integral i f) else None)"
   59.10  proof -
   59.11    interpret comm_monoid "lift_option plus" "Some (0::'b)"
   59.12 @@ -1408,7 +1408,7 @@
   59.13      fix k :: 'a
   59.14      assume k: "k \<in> Basis"
   59.15      show "(if f integrable_on cbox a b then Some (integral (cbox a b) f) else None) =
   59.16 -          lift_option op + (if f integrable_on cbox a b \<inter> {x. x \<bullet> k \<le> c} then Some (integral (cbox a b \<inter> {x. x \<bullet> k \<le> c}) f) else None)
   59.17 +          lift_option (+) (if f integrable_on cbox a b \<inter> {x. x \<bullet> k \<le> c} then Some (integral (cbox a b \<inter> {x. x \<bullet> k \<le> c}) f) else None)
   59.18            (if f integrable_on cbox a b \<inter> {x. c \<le> x \<bullet> k} then Some (integral (cbox a b \<inter> {x. c \<le> x \<bullet> k}) f) else None)"
   59.19      proof (cases "f integrable_on cbox a b")
   59.20        case True
   59.21 @@ -2176,11 +2176,11 @@
   59.22          unfolding sum_distrib_left by (metis divide_inverse inverse_eq_divide power_one_over)
   59.23        also have "\<dots> < e/2 * 2"
   59.24        proof (rule mult_strict_left_mono)
   59.25 -        have "sum (op ^ (1/2)) {..N + 1} = sum (op ^ (1/2::real)) {..<N + 2}"
   59.26 +        have "sum ((^) (1/2)) {..N + 1} = sum ((^) (1/2::real)) {..<N + 2}"
   59.27            using lessThan_Suc_atMost by auto
   59.28          also have "... < 2"
   59.29            by (auto simp: geometric_sum)
   59.30 -        finally show "sum (op ^ (1/2::real)) {..N + 1} < 2" .
   59.31 +        finally show "sum ((^) (1/2::real)) {..N + 1} < 2" .
   59.32        qed (use \<open>0 < e\<close> in auto)
   59.33        finally  show ?thesis by auto
   59.34      qed
   59.35 @@ -2479,9 +2479,9 @@
   59.36    qed
   59.37  qed
   59.38  
   59.39 -lemma comm_monoid_set_F_and: "comm_monoid_set.F op \<and> True f s \<longleftrightarrow> (finite s \<longrightarrow> (\<forall>x\<in>s. f x))"
   59.40 +lemma comm_monoid_set_F_and: "comm_monoid_set.F (\<and>) True f s \<longleftrightarrow> (finite s \<longrightarrow> (\<forall>x\<in>s. f x))"
   59.41  proof -
   59.42 -  interpret bool: comm_monoid_set "op \<and>" True
   59.43 +  interpret bool: comm_monoid_set "(\<and>)" True
   59.44      proof qed auto
   59.45    show ?thesis
   59.46      by (induction s rule: infinite_finite_induct) auto
   59.47 @@ -2587,8 +2587,8 @@
   59.48      fix e :: real
   59.49      assume e: "e > 0"
   59.50      {
   59.51 -      assume "\<forall>e>0. ?P e op <"
   59.52 -      then show "?P (e * content (cbox a b)) op \<le>"
   59.53 +      assume "\<forall>e>0. ?P e (<)"
   59.54 +      then show "?P (e * content (cbox a b)) (\<le>)"
   59.55          apply (erule_tac x="e * content (cbox a b)" in allE)
   59.56          apply (erule impE)
   59.57          defer
   59.58 @@ -2598,8 +2598,8 @@
   59.59          done
   59.60      }
   59.61      {
   59.62 -      assume "\<forall>e>0. ?P (e * content (cbox a b)) op \<le>"
   59.63 -      then show "?P e op <"
   59.64 +      assume "\<forall>e>0. ?P (e * content (cbox a b)) (\<le>)"
   59.65 +      then show "?P e (<)"
   59.66          apply (erule_tac x="e/2 / content (cbox a b)" in allE)
   59.67          apply (erule impE)
   59.68          defer
   59.69 @@ -2983,7 +2983,7 @@
   59.70      "\<lambda>i. if f integrable_on i then Some (integral i f) else None"
   59.71      using operative_integralI by (rule operative_realI)
   59.72    from \<open>a \<le> c\<close> \<open>c \<le> b\<close> ac cb coalesce_less_eq
   59.73 -  have *: "lift_option op +
   59.74 +  have *: "lift_option (+)
   59.75               (if f integrable_on {a..c} then Some (integral {a..c} f) else None)
   59.76               (if f integrable_on {c..b} then Some (integral {c..b} f) else None) =
   59.77              (if f integrable_on {a..b} then Some (integral {a..b} f) else None)"
   59.78 @@ -4609,9 +4609,9 @@
   59.79  
   59.80  lemma negligible_translation:
   59.81    assumes "negligible S"
   59.82 -    shows "negligible (op + c ` S)"
   59.83 +    shows "negligible ((+) c ` S)"
   59.84  proof -
   59.85 -  have inj: "inj (op + c)"
   59.86 +  have inj: "inj ((+) c)"
   59.87      by simp
   59.88    show ?thesis
   59.89    using assms
   59.90 @@ -4620,9 +4620,9 @@
   59.91      assume "\<forall>x y. (indicator S has_integral 0) (cbox x y)"
   59.92      then have *: "(indicator S has_integral 0) (cbox (a-c) (b-c))"
   59.93        by (meson Diff_iff assms has_integral_negligible indicator_simps(2))
   59.94 -    have eq: "indicator (op + c ` S) = (\<lambda>x. indicator S (x - c))"
   59.95 +    have eq: "indicator ((+) c ` S) = (\<lambda>x. indicator S (x - c))"
   59.96        by (force simp add: indicator_def)
   59.97 -    show "(indicator (op + c ` S) has_integral 0) (cbox a b)"
   59.98 +    show "(indicator ((+) c ` S) has_integral 0) (cbox a b)"
   59.99        using has_integral_affinity [OF *, of 1 "-c"]
  59.100              cbox_translation [of "c" "-c+a" "-c+b"]
  59.101        by (simp add: eq add.commute)
  59.102 @@ -4630,7 +4630,7 @@
  59.103  qed
  59.104  
  59.105  lemma negligible_translation_rev:
  59.106 -  assumes "negligible (op + c ` S)"
  59.107 +  assumes "negligible ((+) c ` S)"
  59.108      shows "negligible S"
  59.109  by (metis negligible_translation [OF assms, of "-c"] translation_galois)
  59.110  
  59.111 @@ -4838,8 +4838,8 @@
  59.112        using real_arch_simple by blast
  59.113      have "ball 0 B \<subseteq> ?cube n" if n: "n \<ge> N" for n
  59.114      proof -
  59.115 -      have "sum (op *\<^sub>R (- real n)) Basis \<bullet> i \<le> x \<bullet> i \<and>
  59.116 -            x \<bullet> i \<le> sum (op *\<^sub>R (real n)) Basis \<bullet> i"
  59.117 +      have "sum (( *\<^sub>R) (- real n)) Basis \<bullet> i \<le> x \<bullet> i \<and>
  59.118 +            x \<bullet> i \<le> sum (( *\<^sub>R) (real n)) Basis \<bullet> i"
  59.119          if "norm x < B" "i \<in> Basis" for x i::'n
  59.120            using Basis_le_norm[of i x] n N that by (auto simp add: field_simps sum_negf)
  59.121        then show ?thesis
  59.122 @@ -4874,7 +4874,7 @@
  59.123          fix x :: 'n
  59.124          assume x: "x \<in> ball 0 B"
  59.125          have "\<lbrakk>norm (0 - x) < B; i \<in> Basis\<rbrakk>
  59.126 -              \<Longrightarrow> sum (op *\<^sub>R (-n)) Basis \<bullet> i\<le> x \<bullet> i \<and> x \<bullet> i \<le> sum (op *\<^sub>R n) Basis \<bullet> i" for i
  59.127 +              \<Longrightarrow> sum (( *\<^sub>R) (-n)) Basis \<bullet> i\<le> x \<bullet> i \<and> x \<bullet> i \<le> sum (( *\<^sub>R) n) Basis \<bullet> i" for i
  59.128            using Basis_le_norm[of i x] n by (auto simp add: field_simps sum_negf)
  59.129          then show "x \<in> ?cube n"
  59.130            using x by (auto simp: mem_box dist_norm)
  59.131 @@ -5925,7 +5925,7 @@
  59.132      and bou: "bounded (range(\<lambda>k. integral S (f k)))"
  59.133    shows "g integrable_on S \<and> (\<lambda>k. integral S (f k)) \<longlonglongrightarrow> integral S g"
  59.134  proof -
  59.135 -  have *: "range(\<lambda>k. integral S (\<lambda>x. - f k x)) = op *\<^sub>R (- 1) ` (range(\<lambda>k. integral S (f k)))"
  59.136 +  have *: "range(\<lambda>k. integral S (\<lambda>x. - f k x)) = ( *\<^sub>R) (- 1) ` (range(\<lambda>k. integral S (f k)))"
  59.137      by force
  59.138    have "(\<lambda>x. - g x) integrable_on S \<and> (\<lambda>k. integral S (\<lambda>x. - f k x)) \<longlonglongrightarrow> integral S (\<lambda>x. - g x)"
  59.139    proof (rule monotone_convergence_increasing)
    60.1 --- a/src/HOL/Analysis/Homeomorphism.thy	Wed Jan 10 15:21:49 2018 +0100
    60.2 +++ b/src/HOL/Analysis/Homeomorphism.thy	Wed Jan 10 15:25:09 2018 +0100
    60.3 @@ -549,31 +549,31 @@
    60.4        and starlike_compact_projective2:
    60.5              "S homeomorphic cball a 1 \<inter> affine hull S"
    60.6  proof -
    60.7 -  have 1: "compact (op+ (-a) ` S)" by (meson assms compact_translation)
    60.8 -  have 2: "0 \<in> rel_interior (op+ (-a) ` S)"
    60.9 +  have 1: "compact ((+) (-a) ` S)" by (meson assms compact_translation)
   60.10 +  have 2: "0 \<in> rel_interior ((+) (-a) ` S)"
   60.11      by (simp add: a rel_interior_translation)
   60.12 -  have 3: "open_segment 0 x \<subseteq> rel_interior (op+ (-a) ` S)" if "x \<in> (op+ (-a) ` S)" for x
   60.13 +  have 3: "open_segment 0 x \<subseteq> rel_interior ((+) (-a) ` S)" if "x \<in> ((+) (-a) ` S)" for x
   60.14    proof -
   60.15      have "x+a \<in> S" using that by auto
   60.16      then have "open_segment a (x+a) \<subseteq> rel_interior S" by (metis star)
   60.17      then show ?thesis using open_segment_translation
   60.18        using rel_interior_translation by fastforce
   60.19    qed
   60.20 -  have "S - rel_interior S homeomorphic (op+ (-a) ` S) - rel_interior (op+ (-a) ` S)"
   60.21 +  have "S - rel_interior S homeomorphic ((+) (-a) ` S) - rel_interior ((+) (-a) ` S)"
   60.22      by (metis rel_interior_translation translation_diff homeomorphic_translation)
   60.23 -  also have "... homeomorphic sphere 0 1 \<inter> affine hull (op+ (-a) ` S)"
   60.24 +  also have "... homeomorphic sphere 0 1 \<inter> affine hull ((+) (-a) ` S)"
   60.25      by (rule starlike_compact_projective1_0 [OF 1 2 3])
   60.26 -  also have "... = op+ (-a) ` (sphere a 1 \<inter> affine hull S)"
   60.27 +  also have "... = (+) (-a) ` (sphere a 1 \<inter> affine hull S)"
   60.28      by (metis affine_hull_translation left_minus sphere_translation translation_Int)
   60.29    also have "... homeomorphic sphere a 1 \<inter> affine hull S"
   60.30      using homeomorphic_translation homeomorphic_sym by blast
   60.31    finally show "S - rel_interior S homeomorphic sphere a 1 \<inter> affine hull S" .
   60.32  
   60.33 -  have "S homeomorphic (op+ (-a) ` S)"
   60.34 +  have "S homeomorphic ((+) (-a) ` S)"
   60.35      by (metis homeomorphic_translation)
   60.36 -  also have "... homeomorphic cball 0 1 \<inter> affine hull (op+ (-a) ` S)"
   60.37 +  also have "... homeomorphic cball 0 1 \<inter> affine hull ((+) (-a) ` S)"
   60.38      by (rule starlike_compact_projective2_0 [OF 1 2 3])
   60.39 -  also have "... = op+ (-a) ` (cball a 1 \<inter> affine hull S)"
   60.40 +  also have "... = (+) (-a) ` (cball a 1 \<inter> affine hull S)"
   60.41      by (metis affine_hull_translation left_minus cball_translation translation_Int)
   60.42    also have "... homeomorphic cball a 1 \<inter> affine hull S"
   60.43      using homeomorphic_translation homeomorphic_sym by blast
   60.44 @@ -623,13 +623,13 @@
   60.45    have starT: "\<And>x. x \<in> T \<Longrightarrow> open_segment b x \<subseteq> rel_interior T"
   60.46      using rel_interior_closure_convex_segment
   60.47            b \<open>convex T\<close> closure_subset subsetCE by blast
   60.48 -  let ?aS = "op+ (-a) ` S" and ?bT = "op+ (-b) ` T"
   60.49 +  let ?aS = "(+) (-a) ` S" and ?bT = "(+) (-b) ` T"
   60.50    have 0: "0 \<in> affine hull ?aS" "0 \<in> affine hull ?bT"
   60.51      by (metis a b subsetD hull_inc image_eqI left_minus rel_interior_subset)+
   60.52    have subs: "subspace (span ?aS)" "subspace (span ?bT)"
   60.53      by (rule subspace_span)+
   60.54    moreover
   60.55 -  have "dim (span (op + (- a) ` S)) = dim (span (op + (- b) ` T))"
   60.56 +  have "dim (span ((+) (- a) ` S)) = dim (span ((+) (- b) ` T))"
   60.57      by (metis 0 aff_dim_translation_eq aff_dim_zero affeq dim_span nat_int)
   60.58    ultimately obtain f g where "linear f" "linear g"
   60.59                  and fim: "f ` span ?aS = span ?bT"
   60.60 @@ -649,9 +649,9 @@
   60.61      by (metis b affine_hull_span_0 subsetD hull_inc image_eqI left_minus rel_interior_subset)
   60.62    have "S homeomorphic cball a 1 \<inter> affine hull S"
   60.63      by (rule starlike_compact_projective2 [OF \<open>compact S\<close> a starS])
   60.64 -  also have "... homeomorphic op+ (-a) ` (cball a 1 \<inter> affine hull S)"
   60.65 +  also have "... homeomorphic (+) (-a) ` (cball a 1 \<inter> affine hull S)"
   60.66      by (metis homeomorphic_translation)
   60.67 -  also have "... = cball 0 1 \<inter> op+ (-a) ` (affine hull S)"
   60.68 +  also have "... = cball 0 1 \<inter> (+) (-a) ` (affine hull S)"
   60.69      by (auto simp: dist_norm)
   60.70    also have "... = cball 0 1 \<inter> span ?aS"
   60.71      using eqspanS affine_hull_translation by blast
   60.72 @@ -666,9 +666,9 @@
   60.73           using gim gno apply (force simp:, clarify)
   60.74          by (metis IntI fim1 gf image_eqI)
   60.75      qed (auto simp: fg gf)
   60.76 -  also have "... = cball 0 1 \<inter> op+ (-b) ` (affine hull T)"
   60.77 +  also have "... = cball 0 1 \<inter> (+) (-b) ` (affine hull T)"
   60.78      using eqspanT affine_hull_translation by blast
   60.79 -  also have "... = op+ (-b) ` (cball b 1 \<inter> affine hull T)"
   60.80 +  also have "... = (+) (-b) ` (cball b 1 \<inter> affine hull T)"
   60.81      by (auto simp: dist_norm)
   60.82    also have "... homeomorphic (cball b 1 \<inter> affine hull T)"
   60.83      by (metis homeomorphic_translation homeomorphic_sym)
   60.84 @@ -678,9 +678,9 @@
   60.85  
   60.86    have "S - rel_interior S homeomorphic sphere a 1 \<inter> affine hull S"
   60.87      by (rule starlike_compact_projective1 [OF \<open>compact S\<close> a starS])
   60.88 -  also have "... homeomorphic op+ (-a) ` (sphere a 1 \<inter> affine hull S)"
   60.89 +  also have "... homeomorphic (+) (-a) ` (sphere a 1 \<inter> affine hull S)"
   60.90      by (metis homeomorphic_translation)
   60.91 -  also have "... = sphere 0 1 \<inter> op+ (-a) ` (affine hull S)"
   60.92 +  also have "... = sphere 0 1 \<inter> (+) (-a) ` (affine hull S)"
   60.93      by (auto simp: dist_norm)
   60.94    also have "... = sphere 0 1 \<inter> span ?aS"
   60.95      using eqspanS affine_hull_translation by blast
   60.96 @@ -695,9 +695,9 @@
   60.97          using gim gno apply (force simp:, clarify)
   60.98          by (metis IntI fim1 gf image_eqI)
   60.99      qed (auto simp: fg gf)
  60.100 -  also have "... = sphere 0 1 \<inter> op+ (-b) ` (affine hull T)"
  60.101 +  also have "... = sphere 0 1 \<inter> (+) (-b) ` (affine hull T)"
  60.102      using eqspanT affine_hull_translation by blast
  60.103 -  also have "... = op+ (-b) ` (sphere b 1 \<inter> affine hull T)"
  60.104 +  also have "... = (+) (-b) ` (sphere b 1 \<inter> affine hull T)"
  60.105      by (auto simp: dist_norm)
  60.106    also have "... homeomorphic (sphere b 1 \<inter> affine hull T)"
  60.107      by (metis homeomorphic_translation homeomorphic_sym)
  60.108 @@ -833,11 +833,11 @@
  60.109    then have inj: "inj (\<lambda>x::'a. x /\<^sub>R norm (a - b))"
  60.110      by (simp add: inj_on_def)
  60.111    have "((sphere a r \<inter> T) - {b}) homeomorphic
  60.112 -        op+ (-a) ` ((sphere a r \<inter> T) - {b})"
  60.113 +        (+) (-a) ` ((sphere a r \<inter> T) - {b})"
  60.114      by (rule homeomorphic_translation)
  60.115 -  also have "... homeomorphic op *\<^sub>R (inverse r) ` op + (- a) ` (sphere a r \<inter> T - {b})"
  60.116 +  also have "... homeomorphic ( *\<^sub>R) (inverse r) ` (+) (- a) ` (sphere a r \<inter> T - {b})"
  60.117      by (metis \<open>0 < r\<close> homeomorphic_scaling inverse_inverse_eq inverse_zero less_irrefl)
  60.118 -  also have "... = sphere 0 1 \<inter> (op *\<^sub>R (inverse r) ` op + (- a) ` T) - {(b - a) /\<^sub>R r}"
  60.119 +  also have "... = sphere 0 1 \<inter> (( *\<^sub>R) (inverse r) ` (+) (- a) ` T) - {(b - a) /\<^sub>R r}"
  60.120      using assms by (auto simp: dist_norm norm_minus_commute divide_simps)
  60.121    also have "... homeomorphic p"
  60.122      apply (rule homeomorphic_punctured_affine_sphere_affine_01)
  60.123 @@ -928,27 +928,27 @@
  60.124    then obtain a where "a \<in> S" by auto
  60.125    obtain i::'n where i: "i \<in> Basis" "i \<noteq> 0"
  60.126      using SOME_Basis Basis_zero by force
  60.127 -  have "0 \<in> affine hull (op + (- a) ` S)"
  60.128 +  have "0 \<in> affine hull ((+) (- a) ` S)"
  60.129      by (simp add: \<open>a \<in> S\<close> hull_inc)
  60.130 -  then have "dim (op + (- a) ` S) = aff_dim (op + (- a) ` S)"
  60.131 +  then have "dim ((+) (- a) ` S) = aff_dim ((+) (- a) ` S)"
  60.132      by (simp add: aff_dim_zero)
  60.133    also have "... < DIM('n)"
  60.134      by (simp add: aff_dim_translation_eq assms)
  60.135 -  finally have dd: "dim (op + (- a) ` S) < DIM('n)"
  60.136 +  finally have dd: "dim ((+) (- a) ` S) < DIM('n)"
  60.137      by linarith
  60.138    obtain T where "subspace T" and Tsub: "T \<subseteq> {x. i \<bullet> x = 0}"
  60.139 -             and dimT: "dim T = dim (op + (- a) ` S)"
  60.140 -    apply (rule choose_subspace_of_subspace [of "dim (op + (- a) ` S)" "{x::'n. i \<bullet> x = 0}"])
  60.141 +             and dimT: "dim T = dim ((+) (- a) ` S)"
  60.142 +    apply (rule choose_subspace_of_subspace [of "dim ((+) (- a) ` S)" "{x::'n. i \<bullet> x = 0}"])
  60.143       apply (simp add: dim_hyperplane [OF \<open>i \<noteq> 0\<close>])
  60.144       apply (metis DIM_positive Suc_pred dd not_le not_less_eq_eq)
  60.145      apply (metis span_eq subspace_hyperplane)
  60.146      done
  60.147 -  have "subspace (span (op + (- a) ` S))"
  60.148 +  have "subspace (span ((+) (- a) ` S))"
  60.149      using subspace_span by blast
  60.150    then obtain h k where "linear h" "linear k"
  60.151 -               and heq: "h ` span (op + (- a) ` S) = T"
  60.152 -               and keq:"k ` T = span (op + (- a) ` S)"
  60.153 -               and hinv [simp]:  "\<And>x. x \<in> span (op + (- a) ` S) \<Longrightarrow> k(h x) = x"
  60.154 +               and heq: "h ` span ((+) (- a) ` S) = T"
  60.155 +               and keq:"k ` T = span ((+) (- a) ` S)"
  60.156 +               and hinv [simp]:  "\<And>x. x \<in> span ((+) (- a) ` S) \<Longrightarrow> k(h x) = x"
  60.157                 and kinv [simp]:  "\<And>x. x \<in> T \<Longrightarrow> h(k x) = x"
  60.158      apply (rule isometries_subspaces [OF _ \<open>subspace T\<close>])
  60.159      apply (auto simp: dimT)
  60.160 @@ -964,38 +964,38 @@
  60.161      by (metis DIM_positive Suc_eq_plus1 add.left_neutral diff_add_cancel not_le not_less_eq_eq of_nat_1 of_nat_diff)
  60.162    then obtain f g where fg: "homeomorphism (sphere 0 1 - {i}) {x. i \<bullet> x = 0} f g"
  60.163      by (force simp: homeomorphic_def)
  60.164 -  have "h ` op + (- a) ` S \<subseteq> T"
  60.165 +  have "h ` (+) (- a) ` S \<subseteq> T"
  60.166      using heq span_clauses(1) span_linear_image by blast
  60.167 -  then have "g ` h ` op + (- a) ` S \<subseteq> g ` {x. i \<bullet> x = 0}"
  60.168 +  then have "g ` h ` (+) (- a) ` S \<subseteq> g ` {x. i \<bullet> x = 0}"
  60.169      using Tsub by (simp add: image_mono)
  60.170    also have "... \<subseteq> sphere 0 1 - {i}"
  60.171      by (simp add: fg [unfolded homeomorphism_def])
  60.172 -  finally have gh_sub_sph: "(g \<circ> h) ` op + (- a) ` S \<subseteq> sphere 0 1 - {i}"
  60.173 +  finally have gh_sub_sph: "(g \<circ> h) ` (+) (- a) ` S \<subseteq> sphere 0 1 - {i}"
  60.174      by (metis image_comp)
  60.175 -  then have gh_sub_cb: "(g \<circ> h) ` op + (- a) ` S \<subseteq> cball 0 1"
  60.176 +  then have gh_sub_cb: "(g \<circ> h) ` (+) (- a) ` S \<subseteq> cball 0 1"
  60.177      by (metis Diff_subset order_trans sphere_cball)
  60.178    have [simp]: "\<And>u. u \<in> S \<Longrightarrow> norm (g (h (u - a))) = 1"
  60.179      using gh_sub_sph [THEN subsetD] by (auto simp: o_def)
  60.180 -  have ghcont: "continuous_on (op + (- a) ` S) (\<lambda>x. g (h x))"
  60.181 +  have ghcont: "continuous_on ((+) (- a) ` S) (\<lambda>x. g (h x))"
  60.182      apply (rule continuous_on_compose2 [OF homeomorphism_cont2 [OF fg] hcont], force)
  60.183      done
  60.184 -  have kfcont: "continuous_on ((g \<circ> h \<circ> op + (- a)) ` S) (\<lambda>x. k (f x))"
  60.185 +  have kfcont: "continuous_on ((g \<circ> h \<circ> (+) (- a)) ` S) (\<lambda>x. k (f x))"
  60.186      apply (rule continuous_on_compose2 [OF kcont])
  60.187      using homeomorphism_cont1 [OF fg] gh_sub_sph apply (force intro: continuous_on_subset, blast)
  60.188      done
  60.189 -  have "S homeomorphic op + (- a) ` S"
  60.190 +  have "S homeomorphic (+) (- a) ` S"
  60.191      by (simp add: homeomorphic_translation)
  60.192 -  also have Shom: "\<dots> homeomorphic (g \<circ> h) ` op + (- a) ` S"
  60.193 +  also have Shom: "\<dots> homeomorphic (g \<circ> h) ` (+) (- a) ` S"
  60.194      apply (simp add: homeomorphic_def homeomorphism_def)
  60.195      apply (rule_tac x="g \<circ> h" in exI)
  60.196      apply (rule_tac x="k \<circ> f" in exI)
  60.197      apply (auto simp: ghcont kfcont span_clauses(1) homeomorphism_apply2 [OF fg] image_comp)
  60.198      apply (force simp: o_def homeomorphism_apply2 [OF fg] span_clauses(1))
  60.199      done
  60.200 -  finally have Shom: "S homeomorphic (g \<circ> h) ` op + (- a) ` S" .
  60.201 +  finally have Shom: "S homeomorphic (g \<circ> h) ` (+) (- a) ` S" .
  60.202    show ?thesis
  60.203 -    apply (rule_tac U = "ball 0 1 \<union> image (g o h) (op + (- a) ` S)"
  60.204 -                and T = "image (g o h) (op + (- a) ` S)"
  60.205 +    apply (rule_tac U = "ball 0 1 \<union> image (g o h) ((+) (- a) ` S)"
  60.206 +                and T = "image (g o h) ((+) (- a) ` S)"
  60.207                      in that)
  60.208      apply (rule convex_intermediate_ball [of 0 1], force)
  60.209      using gh_sub_cb apply force
  60.210 @@ -2210,26 +2210,26 @@
  60.211                  and pq'_eq: "\<And>t. t \<in> {0..1} \<Longrightarrow> p (q' t) = (f \<circ> g +++ reversepath g') t"
  60.212          using covering_space_lift_homotopic_path [OF cov homotopic_paths_sym [OF homS] \<open>path q\<close> piq refl refl]
  60.213          by auto
  60.214 -      have "q' t = (h \<circ> op *\<^sub>R 2) t" if "0 \<le> t" "t \<le> 1/2" for t
  60.215 -      proof (rule covering_space_lift_unique [OF cov, of q' 0 "h \<circ> op *\<^sub>R 2" "{0..1/2}" "f \<circ> g \<circ> op *\<^sub>R 2" t])
  60.216 -        show "q' 0 = (h \<circ> op *\<^sub>R 2) 0"
  60.217 +      have "q' t = (h \<circ> ( *\<^sub>R) 2) t" if "0 \<le> t" "t \<le> 1/2" for t
  60.218 +      proof (rule covering_space_lift_unique [OF cov, of q' 0 "h \<circ> ( *\<^sub>R) 2" "{0..1/2}" "f \<circ> g \<circ> ( *\<^sub>R) 2" t])
  60.219 +        show "q' 0 = (h \<circ> ( *\<^sub>R) 2) 0"
  60.220            by (metis \<open>pathstart q' = pathstart q\<close> comp_def g h pastq pathstart_def pth_4(2))
  60.221 -        show "continuous_on {0..1/2} (f \<circ> g \<circ> op *\<^sub>R 2)"
  60.222 +        show "continuous_on {0..1/2} (f \<circ> g \<circ> ( *\<^sub>R) 2)"
  60.223            apply (intro continuous_intros continuous_on_compose continuous_on_path [OF \<open>path g\<close>] continuous_on_subset [OF contf])
  60.224            using g(2) path_image_def by fastforce+
  60.225 -        show "(f \<circ> g \<circ> op *\<^sub>R 2) ` {0..1/2} \<subseteq> S"
  60.226 +        show "(f \<circ> g \<circ> ( *\<^sub>R) 2) ` {0..1/2} \<subseteq> S"
  60.227            using g(2) path_image_def fim by fastforce
  60.228 -        show "(h \<circ> op *\<^sub>R 2) ` {0..1/2} \<subseteq> C"
  60.229 +        show "(h \<circ> ( *\<^sub>R) 2) ` {0..1/2} \<subseteq> C"
  60.230            using h path_image_def by fastforce
  60.231          show "q' ` {0..1/2} \<subseteq> C"
  60.232            using \<open>path_image q' \<subseteq> C\<close> path_image_def by fastforce
  60.233 -        show "\<And>x. x \<in> {0..1/2} \<Longrightarrow> (f \<circ> g \<circ> op *\<^sub>R 2) x = p (q' x)"
  60.234 +        show "\<And>x. x \<in> {0..1/2} \<Longrightarrow> (f \<circ> g \<circ> ( *\<^sub>R) 2) x = p (q' x)"
  60.235            by (auto simp: joinpaths_def pq'_eq)
  60.236 -        show "\<And>x. x \<in> {0..1/2} \<Longrightarrow> (f \<circ> g \<circ> op *\<^sub>R 2) x = p ((h \<circ> op *\<^sub>R 2) x)"
  60.237 +        show "\<And>x. x \<in> {0..1/2} \<Longrightarrow> (f \<circ> g \<circ> ( *\<^sub>R) 2) x = p ((h \<circ> ( *\<^sub>R) 2) x)"
  60.238            by (simp add: phg)
  60.239          show "continuous_on {0..1/2} q'"
  60.240            by (simp add: continuous_on_path \<open>path q'\<close>)
  60.241 -        show "continuous_on {0..1/2} (h \<circ> op *\<^sub>R 2)"
  60.242 +        show "continuous_on {0..1/2} (h \<circ> ( *\<^sub>R) 2)"
  60.243            apply (intro continuous_intros continuous_on_compose continuous_on_path [OF \<open>path h\<close>], force)
  60.244            done
  60.245        qed (use that in auto)
    61.1 --- a/src/HOL/Analysis/Improper_Integral.thy	Wed Jan 10 15:21:49 2018 +0100
    61.2 +++ b/src/HOL/Analysis/Improper_Integral.thy	Wed Jan 10 15:25:09 2018 +0100
    61.3 @@ -967,7 +967,7 @@
    61.4                  by (simp add: sum.union_disjoint T''_eq disj \<open>finite A\<close> \<open>finite B\<close>)
    61.5                also have "... = (\<Sum>(x,K) \<in> A. norm (integral K h - integral K f)) +
    61.6                                 (\<Sum>(x,K) \<in> B. norm (?CI K h x + integral K f))"
    61.7 -                by (auto simp: A_def B_def f norm_minus_commute intro!: sum.cong arg_cong2 [where f= "op+"])
    61.8 +                by (auto simp: A_def B_def f norm_minus_commute intro!: sum.cong arg_cong2 [where f= "(+)"])
    61.9                also have "... \<le> (\<Sum>(x,K)\<in>A. norm (integral K h)) +
   61.10                                   (\<Sum>(x,K)\<in>(\<lambda>(x,K). (x,K \<inter> {x. x \<bullet> i \<le> c})) ` A. norm (integral K h))
   61.11                               + ((\<Sum>(x,K)\<in>B. norm (?CI K h x)) +
   61.12 @@ -1663,4 +1663,4 @@
   61.13  qed
   61.14  
   61.15  end
   61.16 -  
   61.17 \ No newline at end of file
   61.18 +  
    62.1 --- a/src/HOL/Analysis/Inner_Product.thy	Wed Jan 10 15:21:49 2018 +0100
    62.2 +++ b/src/HOL/Analysis/Inner_Product.thy	Wed Jan 10 15:25:09 2018 +0100
    62.3 @@ -239,7 +239,7 @@
    62.4  instantiation real :: real_inner
    62.5  begin
    62.6  
    62.7 -definition inner_real_def [simp]: "inner = op *"
    62.8 +definition inner_real_def [simp]: "inner = ( * )"
    62.9  
   62.10  instance
   62.11  proof
    63.1 --- a/src/HOL/Analysis/Lebesgue_Measure.thy	Wed Jan 10 15:21:49 2018 +0100
    63.2 +++ b/src/HOL/Analysis/Lebesgue_Measure.thy	Wed Jan 10 15:25:09 2018 +0100
    63.3 @@ -750,10 +750,10 @@
    63.4    finally show "lebesgue = density (distr lebesgue lebesgue T) (\<lambda>_. (\<Prod>j\<in>Basis. \<bar>c j\<bar>))" .
    63.5  qed
    63.6  
    63.7 -lemma lebesgue_measurable_scaling[measurable]: "op *\<^sub>R x \<in> lebesgue \<rightarrow>\<^sub>M lebesgue"
    63.8 +lemma lebesgue_measurable_scaling[measurable]: "( *\<^sub>R) x \<in> lebesgue \<rightarrow>\<^sub>M lebesgue"
    63.9  proof cases
   63.10    assume "x = 0"
   63.11 -  then have "op *\<^sub>R x = (\<lambda>x. 0::'a)"
   63.12 +  then have "( *\<^sub>R) x = (\<lambda>x. 0::'a)"
   63.13      by (auto simp: fun_eq_iff)
   63.14    then show ?thesis by auto
   63.15  next
   63.16 @@ -843,9 +843,9 @@
   63.17  
   63.18  lemma lborel_distr_mult:
   63.19    assumes "(c::real) \<noteq> 0"
   63.20 -  shows "distr lborel borel (op * c) = density lborel (\<lambda>_. inverse \<bar>c\<bar>)"
   63.21 +  shows "distr lborel borel (( * ) c) = density lborel (\<lambda>_. inverse \<bar>c\<bar>)"
   63.22  proof-
   63.23 -  have "distr lborel borel (op * c) = distr lborel lborel (op * c)" by (simp cong: distr_cong)
   63.24 +  have "distr lborel borel (( * ) c) = distr lborel lborel (( * ) c)" by (simp cong: distr_cong)
   63.25    also from assms have "... = density lborel (\<lambda>_. inverse \<bar>c\<bar>)"
   63.26      by (subst lborel_real_affine[of "inverse c" 0]) (auto simp: o_def distr_density_distr)
   63.27    finally show ?thesis .
   63.28 @@ -853,18 +853,18 @@
   63.29  
   63.30  lemma lborel_distr_mult':
   63.31    assumes "(c::real) \<noteq> 0"
   63.32 -  shows "lborel = density (distr lborel borel (op * c)) (\<lambda>_. \<bar>c\<bar>)"
   63.33 +  shows "lborel = density (distr lborel borel (( * ) c)) (\<lambda>_. \<bar>c\<bar>)"
   63.34  proof-
   63.35    have "lborel = density lborel (\<lambda>_. 1)" by (rule density_1[symmetric])
   63.36    also from assms have "(\<lambda>_. 1 :: ennreal) = (\<lambda>_. inverse \<bar>c\<bar> * \<bar>c\<bar>)" by (intro ext) simp
   63.37    also have "density lborel ... = density (density lborel (\<lambda>_. inverse \<bar>c\<bar>)) (\<lambda>_. \<bar>c\<bar>)"
   63.38      by (subst density_density_eq) (auto simp: ennreal_mult)
   63.39 -  also from assms have "density lborel (\<lambda>_. inverse \<bar>c\<bar>) = distr lborel borel (op * c)"
   63.40 +  also from assms have "density lborel (\<lambda>_. inverse \<bar>c\<bar>) = distr lborel borel (( * ) c)"
   63.41      by (rule lborel_distr_mult[symmetric])
   63.42    finally show ?thesis .
   63.43  qed
   63.44  
   63.45 -lemma lborel_distr_plus: "distr lborel borel (op + c) = (lborel :: real measure)"
   63.46 +lemma lborel_distr_plus: "distr lborel borel ((+) c) = (lborel :: real measure)"
   63.47    by (subst lborel_real_affine[of 1 c]) (auto simp: density_1 one_ennreal_def[symmetric])
   63.48  
   63.49  interpretation lborel: sigma_finite_measure lborel
   63.50 @@ -885,7 +885,7 @@
   63.51      "box (la, lb) (ua, ub) = box la ua \<times> box lb ub"
   63.52      using lu[of _ 0] lu[of 0] by (auto intro!: inj_onI simp add: Basis_prod_def ball_Un box_def)
   63.53    show "emeasure (lborel \<Otimes>\<^sub>M lborel) (box (la, lb) (ua, ub)) =
   63.54 -      ennreal (prod (op \<bullet> ((ua, ub) - (la, lb))) Basis)"
   63.55 +      ennreal (prod ((\<bullet>) ((ua, ub) - (la, lb))) Basis)"
   63.56      by (simp add: lborel.emeasure_pair_measure_Times Basis_prod_def prod.union_disjoint
   63.57                    prod.reindex ennreal_mult inner_diff_left prod_nonneg)
   63.58  qed (simp add: borel_prod[symmetric])
   63.59 @@ -994,7 +994,7 @@
   63.60  
   63.61    let ?f = "\<lambda>n. root DIM('a) (Suc n)"
   63.62  
   63.63 -  have vimage_eq_image: "op *\<^sub>R (?f n) -` S = op *\<^sub>R (1 / ?f n) ` S" for n
   63.64 +  have vimage_eq_image: "( *\<^sub>R) (?f n) -` S = ( *\<^sub>R) (1 / ?f n) ` S" for n
   63.65      apply safe
   63.66      subgoal for x by (rule image_eqI[of _ _ "?f n *\<^sub>R x"]) auto
   63.67      subgoal by auto
   63.68 @@ -1016,20 +1016,20 @@
   63.69      by (intro summable_iff_suminf_neq_top) (auto simp add: inverse_eq_divide)
   63.70    then have "top * emeasure lebesgue S = (\<Sum>n. (1 / ?f n)^DIM('a) * emeasure lebesgue S)"
   63.71      unfolding ennreal_suminf_multc eq by simp
   63.72 -  also have "\<dots> = (\<Sum>n. emeasure lebesgue (op *\<^sub>R (?f n) -` S))"
   63.73 +  also have "\<dots> = (\<Sum>n. emeasure lebesgue (( *\<^sub>R) (?f n) -` S))"
   63.74      unfolding vimage_eq_image using emeasure_lebesgue_affine[of "1 / ?f n" 0 S for n] by simp
   63.75 -  also have "\<dots> = emeasure lebesgue (\<Union>n. op *\<^sub>R (?f n) -` S)"
   63.76 +  also have "\<dots> = emeasure lebesgue (\<Union>n. ( *\<^sub>R) (?f n) -` S)"
   63.77    proof (intro suminf_emeasure)
   63.78 -    show "disjoint_family (\<lambda>n. op *\<^sub>R (?f n) -` S)"
   63.79 +    show "disjoint_family (\<lambda>n. ( *\<^sub>R) (?f n) -` S)"
   63.80        unfolding disjoint_family_on_def
   63.81      proof safe
   63.82        fix m n :: nat and x assume "m \<noteq> n" "?f m *\<^sub>R x \<in> S" "?f n *\<^sub>R x \<in> S"
   63.83        with eq1[of "?f m / ?f n" "?f n *\<^sub>R x"] show "x \<in> {}"
   63.84          by auto
   63.85      qed
   63.86 -    have "op *\<^sub>R (?f i) -` S \<in> sets lebesgue" for i
   63.87 +    have "( *\<^sub>R) (?f i) -` S \<in> sets lebesgue" for i
   63.88        using measurable_sets[OF lebesgue_measurable_scaling[of "?f i"] S] by auto
   63.89 -    then show "range (\<lambda>i. op *\<^sub>R (?f i) -` S) \<subseteq> sets lebesgue"
   63.90 +    then show "range (\<lambda>i. ( *\<^sub>R) (?f i) -` S) \<subseteq> sets lebesgue"
   63.91        by auto
   63.92    qed
   63.93    also have "\<dots> \<le> emeasure lebesgue (ball 0 M :: 'a set)"
    64.1 --- a/src/HOL/Analysis/Linear_Algebra.thy	Wed Jan 10 15:21:49 2018 +0100
    64.2 +++ b/src/HOL/Analysis/Linear_Algebra.thy	Wed Jan 10 15:25:09 2018 +0100
    64.3 @@ -980,7 +980,7 @@
    64.4        by (intro sum.mono_neutral_cong_left) (auto intro: X)
    64.5      also have "\<dots> = (\<Sum>z\<in>{z. X x z \<noteq> 0}. X x z *\<^sub>R z) + (\<Sum>z\<in>{z. X y z \<noteq> 0}. X y z *\<^sub>R z)"
    64.6        by (auto simp add: scaleR_add_left sum.distrib
    64.7 -               intro!: arg_cong2[where f="op +"]  sum.mono_neutral_cong_right X)
    64.8 +               intro!: arg_cong2[where f="(+)"]  sum.mono_neutral_cong_right X)
    64.9      also have "\<dots> = x + y"
   64.10        by (simp add: X(3)[symmetric])
   64.11      also have "\<dots> = (\<Sum>z | X (x + y) z \<noteq> 0. X (x + y) z *\<^sub>R z)"
   64.12 @@ -1024,7 +1024,7 @@
   64.13      show "g (x + y) = g x + g y"
   64.14        unfolding g_def X_add *
   64.15        by (auto simp add: scaleR_add_left sum.distrib
   64.16 -               intro!: arg_cong2[where f="op +"]  sum.mono_neutral_cong_right X)
   64.17 +               intro!: arg_cong2[where f="(+)"]  sum.mono_neutral_cong_right X)
   64.18    next
   64.19      show "g (r *\<^sub>R x) = r *\<^sub>R g x" for r x
   64.20        by (auto simp add: g_def X_cmult scaleR_sum_right intro!: sum.mono_neutral_cong_left X)
   64.21 @@ -1398,7 +1398,7 @@
   64.22    by (simp add: norm_eq_sqrt_inner)
   64.23  
   64.24  
   64.25 -text\<open>Equality of vectors in terms of @{term "op \<bullet>"} products.\<close>
   64.26 +text\<open>Equality of vectors in terms of @{term "(\<bullet>)"} products.\<close>
   64.27  
   64.28  lemma linear_componentwise:
   64.29    fixes f:: "'a::euclidean_space \<Rightarrow> 'b::real_inner"
    65.1 --- a/src/HOL/Analysis/Measure_Space.thy	Wed Jan 10 15:21:49 2018 +0100
    65.2 +++ b/src/HOL/Analysis/Measure_Space.thy	Wed Jan 10 15:25:09 2018 +0100
    65.3 @@ -2348,7 +2348,7 @@
    65.4    case True
    65.5    show ?thesis
    65.6    proof (rule emeasure_measure_of[OF restrict_space_def])
    65.7 -    show "op \<inter> \<Omega> ` sets M \<subseteq> Pow (\<Omega> \<inter> space M)" "A \<in> sets (restrict_space M \<Omega>)"
    65.8 +    show "(\<inter>) \<Omega> ` sets M \<subseteq> Pow (\<Omega> \<inter> space M)" "A \<in> sets (restrict_space M \<Omega>)"
    65.9        using \<open>A \<subseteq> \<Omega>\<close> \<open>A \<in> sets M\<close> sets.space_closed by (auto simp: sets_restrict_space)
   65.10      show "positive (sets (restrict_space M \<Omega>)) (emeasure M)"
   65.11        by (auto simp: positive_def)
   65.12 @@ -2427,7 +2427,7 @@
   65.13    from sigma_finite_countable obtain C
   65.14      where C: "countable C" "C \<subseteq> sets M" "(\<Union>C) = space M" "\<forall>a\<in>C. emeasure M a \<noteq> \<infinity>"
   65.15      by blast
   65.16 -  let ?C = "op \<inter> A ` C"
   65.17 +  let ?C = "(\<inter>) A ` C"
   65.18    from C have "countable ?C" "?C \<subseteq> sets (restrict_space M A)" "(\<Union>?C) = space (restrict_space M A)"
   65.19      by(auto simp add: sets_restrict_space space_restrict_space)
   65.20    moreover {
   65.21 @@ -2861,7 +2861,7 @@
   65.22        also have "\<dots> = ?S (\<Union>i. X i)"
   65.23          unfolding UN_extend_simps(4)
   65.24          by (auto simp add: suminf_add[symmetric] Diff_eq[symmetric] simp del: UN_simps
   65.25 -                 intro!: SUP_cong arg_cong2[where f="op +"] suminf_emeasure
   65.26 +                 intro!: SUP_cong arg_cong2[where f="(+)"] suminf_emeasure
   65.27                           disjoint_family_on_bisimulation[OF \<open>disjoint_family X\<close>])
   65.28        finally show "(\<Sum>i. ?S (X i)) = ?S (\<Union>i. X i)" .
   65.29      qed
   65.30 @@ -3395,13 +3395,13 @@
   65.31  
   65.32  lemma emeasure_SUP_chain:
   65.33    assumes sets: "\<And>i. i \<in> A \<Longrightarrow> sets (M i) = sets N" "X \<in> sets N"
   65.34 -  assumes ch: "Complete_Partial_Order.chain op \<le> (M ` A)" and "A \<noteq> {}"
   65.35 +  assumes ch: "Complete_Partial_Order.chain (\<le>) (M ` A)" and "A \<noteq> {}"
   65.36    shows "emeasure (SUP i:A. M i) X = (SUP i:A. emeasure (M i) X)"
   65.37  proof (subst emeasure_SUP[OF sets \<open>A \<noteq> {}\<close>])
   65.38    show "(SUP J:{J. J \<noteq> {} \<and> finite J \<and> J \<subseteq> A}. emeasure (SUPREMUM J M) X) = (SUP i:A. emeasure (M i) X)"
   65.39    proof (rule SUP_eq)
   65.40      fix J assume "J \<in> {J. J \<noteq> {} \<and> finite J \<and> J \<subseteq> A}"
   65.41 -    then have J: "Complete_Partial_Order.chain op \<le> (M ` J)" "finite J" "J \<noteq> {}" and "J \<subseteq> A"
   65.42 +    then have J: "Complete_Partial_Order.chain (\<le>) (M ` J)" "finite J" "J \<noteq> {}" and "J \<subseteq> A"
   65.43        using ch[THEN chain_subset, of "M`J"] by auto
   65.44      with in_chain_finite[OF J(1)] obtain j where "j \<in> J" "(SUP j:J. M j) = M j"
   65.45        by auto
    66.1 --- a/src/HOL/Analysis/Nonnegative_Lebesgue_Integration.thy	Wed Jan 10 15:21:49 2018 +0100
    66.2 +++ b/src/HOL/Analysis/Nonnegative_Lebesgue_Integration.thy	Wed Jan 10 15:25:09 2018 +0100
    66.3 @@ -276,11 +276,11 @@
    66.4    thus ?thesis by (simp_all add: comp_def)
    66.5  qed
    66.6  
    66.7 -lemmas simple_function_add[intro, simp] = simple_function_compose2[where h="op +"]
    66.8 -  and simple_function_diff[intro, simp] = simple_function_compose2[where h="op -"]
    66.9 +lemmas simple_function_add[intro, simp] = simple_function_compose2[where h="(+)"]
   66.10 +  and simple_function_diff[intro, simp] = simple_function_compose2[where h="(-)"]
   66.11    and simple_function_uminus[intro, simp] = simple_function_compose[where g="uminus"]
   66.12 -  and simple_function_mult[intro, simp] = simple_function_compose2[where h="op *"]
   66.13 -  and simple_function_div[intro, simp] = simple_function_compose2[where h="op /"]
   66.14 +  and simple_function_mult[intro, simp] = simple_function_compose2[where h="( * )"]
   66.15 +  and simple_function_div[intro, simp] = simple_function_compose2[where h="(/)"]
   66.16    and simple_function_inverse[intro, simp] = simple_function_compose[where g="inverse"]
   66.17    and simple_function_max[intro, simp] = simple_function_compose2[where h=max]
   66.18  
   66.19 @@ -760,7 +760,7 @@
   66.20      using assms by (intro simple_function_partition) auto
   66.21    also have "\<dots> = (\<Sum>y\<in>(\<lambda>x. (f x, indicator A x::ennreal))`space M.
   66.22      if snd y = 1 then fst y * emeasure M (f -` {fst y} \<inter> space M \<inter> A) else 0)"
   66.23 -    by (auto simp: indicator_def split: if_split_asm intro!: arg_cong2[where f="op *"] arg_cong2[where f=emeasure] sum.cong)
   66.24 +    by (auto simp: indicator_def split: if_split_asm intro!: arg_cong2[where f="( * )"] arg_cong2[where f=emeasure] sum.cong)
   66.25    also have "\<dots> = (\<Sum>y\<in>(\<lambda>x. (f x, 1::ennreal))`A. fst y * emeasure M (f -` {fst y} \<inter> space M \<inter> A))"
   66.26      using assms by (subst sum.If_cases) (auto intro!: simple_functionD(1) simp: eq)
   66.27    also have "\<dots> = (\<Sum>y\<in>fst`(\<lambda>x. (f x, 1::ennreal))`A. y * emeasure M (f -` {y} \<inter> space M \<inter> A))"
   66.28 @@ -1873,7 +1873,7 @@
   66.29  
   66.30  lemma nn_integral_monotone_convergence_SUP_nat:
   66.31    fixes f :: "'a \<Rightarrow> nat \<Rightarrow> ennreal"
   66.32 -  assumes chain: "Complete_Partial_Order.chain op \<le> (f ` Y)"
   66.33 +  assumes chain: "Complete_Partial_Order.chain (\<le>) (f ` Y)"
   66.34    and nonempty: "Y \<noteq> {}"
   66.35    shows "(\<integral>\<^sup>+ x. (SUP i:Y. f i x) \<partial>count_space UNIV) = (SUP i:Y. (\<integral>\<^sup>+ x. f i x \<partial>count_space UNIV))"
   66.36    (is "?lhs = ?rhs" is "integral\<^sup>N ?M _ = _")
   66.37 @@ -1913,7 +1913,7 @@
   66.38          case True
   66.39          let ?Y = "I ` {..<m}"
   66.40          have "f ` ?Y \<subseteq> f ` Y" using I by auto
   66.41 -        with chain have chain': "Complete_Partial_Order.chain op \<le> (f ` ?Y)" by(rule chain_subset)
   66.42 +        with chain have chain': "Complete_Partial_Order.chain (\<le>) (f ` ?Y)" by(rule chain_subset)
   66.43          hence "Sup (f ` ?Y) \<in> f ` ?Y"
   66.44            by(rule ccpo_class.in_chain_finite)(auto simp add: True lessThan_empty_iff)
   66.45          then obtain m' where "m' < m" and m': "(SUP i:?Y. f i) = f (I m')" by auto
    67.1 --- a/src/HOL/Analysis/Path_Connected.thy	Wed Jan 10 15:21:49 2018 +0100
    67.2 +++ b/src/HOL/Analysis/Path_Connected.thy	Wed Jan 10 15:25:09 2018 +0100
    67.3 @@ -1659,7 +1659,7 @@
    67.4    fixes a :: "'a :: topological_group_add"
    67.5    shows "path_connected ((\<lambda>x. a + x) ` S) = path_connected S"
    67.6  proof -
    67.7 -  have "\<forall>x y. op + (x::'a) ` op + (0 - x) ` y = y"
    67.8 +  have "\<forall>x y. (+) (x::'a) ` (+) (0 - x) ` y = y"
    67.9      by (simp add: image_image)
   67.10    then show ?thesis
   67.11      by (metis (no_types) path_connected_translationI)
   67.12 @@ -1738,7 +1738,7 @@
   67.13      case True then show ?thesis
   67.14        by (simp add: path_component_refl_eq pathstart_def)
   67.15    next
   67.16 -    case False have "continuous_on {0..1} (p o (op* y))"
   67.17 +    case False have "continuous_on {0..1} (p o (( * ) y))"
   67.18        apply (rule continuous_intros)+
   67.19        using p [unfolded path_def] y
   67.20        apply (auto simp: mult_le_one intro: continuous_on_subset [of _ p])
   67.21 @@ -2012,7 +2012,7 @@
   67.22      by (intro path_connected_continuous_image path_connected_punctured_universe assms)
   67.23    with eq have "path_connected (sphere (0::'a) r)"
   67.24      by auto
   67.25 -  then have "path_connected(op + a ` (sphere (0::'a) r))"
   67.26 +  then have "path_connected((+) a ` (sphere (0::'a) r))"
   67.27      by (simp add: path_connected_translation)
   67.28    then show ?thesis
   67.29      by (metis add.right_neutral sphere_translation)
   67.30 @@ -2241,7 +2241,7 @@
   67.31    assumes "DIM('a) = 1" and "r > 0"
   67.32    obtains x y where "sphere a r = {x,y} \<and> dist x y = 2*r"
   67.33  proof -
   67.34 -  have "sphere a r = op + a ` sphere 0 r"
   67.35 +  have "sphere a r = (+) a ` sphere 0 r"
   67.36      by (metis add.right_neutral sphere_translation)
   67.37    then show ?thesis
   67.38      using sphere_1D_doubleton_zero [OF assms]
   67.39 @@ -2282,7 +2282,7 @@
   67.40    assumes 2: "2 \<le> DIM('N)" and pc: "path_connected {r. 0 \<le> r \<and> P r}"
   67.41    shows "path_connected {x. P(norm(x - a))}"
   67.42  proof -
   67.43 -  have "{x. P(norm(x - a))} = op+ a ` {x. P(norm x)}"
   67.44 +  have "{x. P(norm(x - a))} = (+) a ` {x. P(norm x)}"
   67.45      by force
   67.46    moreover have "path_connected {x::'N. P(norm x)}"
   67.47    proof -
   67.48 @@ -6534,13 +6534,13 @@
   67.49  next
   67.50    case False
   67.51    then obtain a b where ab: "a \<in> S" "b \<in> T" by auto
   67.52 -  then have ss: "subspace (op + (- a) ` S)" "subspace (op + (- b) ` T)"
   67.53 +  then have ss: "subspace ((+) (- a) ` S)" "subspace ((+) (- b) ` T)"
   67.54      using affine_diffs_subspace assms by blast+
   67.55 -  have dd: "dim (op + (- a) ` S) = dim (op + (- b) ` T)"
   67.56 +  have dd: "dim ((+) (- a) ` S) = dim ((+) (- b) ` T)"
   67.57      using assms ab  by (simp add: aff_dim_eq_dim  [OF hull_inc] image_def)
   67.58 -  have "S homeomorphic (op + (- a) ` S)"
   67.59 +  have "S homeomorphic ((+) (- a) ` S)"
   67.60      by (simp add: homeomorphic_translation)
   67.61 -  also have "... homeomorphic (op + (- b) ` T)"
   67.62 +  also have "... homeomorphic ((+) (- b) ` T)"
   67.63      by (rule homeomorphic_subspaces [OF ss dd])
   67.64    also have "... homeomorphic T"
   67.65      using homeomorphic_sym homeomorphic_translation by auto
   67.66 @@ -6792,7 +6792,7 @@
   67.67  
   67.68  lemma homotopy_eqv_translation:
   67.69      fixes S :: "'a::real_normed_vector set"
   67.70 -    shows "op + a ` S homotopy_eqv S"
   67.71 +    shows "(+) a ` S homotopy_eqv S"
   67.72    apply (rule homeomorphic_imp_homotopy_eqv)
   67.73    using homeomorphic_translation homeomorphic_sym by blast
   67.74  
    68.1 --- a/src/HOL/Analysis/Polytope.thy	Wed Jan 10 15:21:49 2018 +0100
    68.2 +++ b/src/HOL/Analysis/Polytope.thy	Wed Jan 10 15:25:09 2018 +0100
    68.3 @@ -18,9 +18,9 @@
    68.4    unfolding face_of_def by blast
    68.5  
    68.6  lemma face_of_translation_eq [simp]:
    68.7 -    "(op + a ` T face_of op + a ` S) \<longleftrightarrow> T face_of S"
    68.8 +    "((+) a ` T face_of (+) a ` S) \<longleftrightarrow> T face_of S"
    68.9  proof -
   68.10 -  have *: "\<And>a T S. T face_of S \<Longrightarrow> (op + a ` T face_of op + a ` S)"
   68.11 +  have *: "\<And>a T S. T face_of S \<Longrightarrow> ((+) a ` T face_of (+) a ` S)"
   68.12      apply (simp add: face_of_def Ball_def, clarify)
   68.13      apply (drule open_segment_translation_eq [THEN iffD1])
   68.14      using inj_image_mem_iff inj_add_left apply metis
   68.15 @@ -1125,7 +1125,7 @@
   68.16            and ab: "\<And>x. x \<in> closure(convex hull {x. x extreme_point_of S}) \<Longrightarrow> b < a \<bullet> x"
   68.17        using separating_hyperplane_closed_point [of "closure(convex hull {x. x extreme_point_of S})"]
   68.18        by blast
   68.19 -    have "continuous_on S (op \<bullet> a)"
   68.20 +    have "continuous_on S ((\<bullet>) a)"
   68.21        by (rule continuous_intros)+
   68.22      then obtain m where "m \<in> S" and m: "\<And>y. y \<in> S \<Longrightarrow> a \<bullet> m \<le> a \<bullet> y"
   68.23        using continuous_attains_inf [of S "\<lambda>x. a \<bullet> x"] \<open>compact S\<close> \<open>u \<in> S\<close>
   68.24 @@ -1206,9 +1206,9 @@
   68.25    show "S \<subseteq> convex hull {x. x extreme_point_of S}"
   68.26    proof
   68.27      fix a assume [simp]: "a \<in> S"
   68.28 -    have 1: "compact (op + (- a) ` S)"
   68.29 +    have 1: "compact ((+) (- a) ` S)"
   68.30        by (simp add: \<open>compact S\<close> compact_translation)
   68.31 -    have 2: "convex (op + (- a) ` S)"
   68.32 +    have 2: "convex ((+) (- a) ` S)"
   68.33        by (simp add: \<open>convex S\<close> convex_translation)
   68.34      show a_invex: "a \<in> convex hull {x. x extreme_point_of S}"
   68.35        using Krein_Milman_Minkowski_aux [OF refl 1 2]
   68.36 @@ -1630,9 +1630,9 @@
   68.37  proof -
   68.38    obtain v where "finite v" "S = convex hull v"
   68.39      using assms polytope_def by auto
   68.40 -  have "finite (op hull convex ` {T. T \<subseteq> v})"
   68.41 +  have "finite ((hull) convex ` {T. T \<subseteq> v})"
   68.42      by (simp add: \<open>finite v\<close>)
   68.43 -  moreover have "{F. F face_of S} \<subseteq> (op hull convex ` {T. T \<subseteq> v})"
   68.44 +  moreover have "{F. F face_of S} \<subseteq> ((hull) convex ` {T. T \<subseteq> v})"
   68.45      by (metis (no_types, lifting) \<open>finite v\<close> \<open>S = convex hull v\<close> face_of_convex_hull_subset finite_imp_compact image_eqI mem_Collect_eq subsetI)
   68.46    ultimately show ?thesis
   68.47      by (blast intro: finite_subset)
   68.48 @@ -2503,9 +2503,9 @@
   68.49        case 1 then show ?thesis .
   68.50      next
   68.51        case 2
   68.52 -      have "Collect (op \<in> x) \<notin> Collect (op \<in> (\<Union>{A. A facet_of S}))"
   68.53 +      have "Collect ((\<in>) x) \<notin> Collect ((\<in>) (\<Union>{A. A facet_of S}))"
   68.54          using xnot by fastforce
   68.55 -      then have "F \<notin> Collect (op \<in> h)"
   68.56 +      then have "F \<notin> Collect ((\<in>) h)"
   68.57          using 2 \<open>x \<in> S\<close> facet by blast
   68.58        with \<open>h \<in> F\<close> have "\<Inter>F \<subseteq> S \<inter> {x. a h \<bullet> x = b h}" by blast
   68.59        with 2 that \<open>x \<in> \<Inter>F\<close> show ?thesis
   68.60 @@ -2661,7 +2661,7 @@
   68.61      apply (auto simp: \<open>bij h\<close> bij_is_surj image_f_inv_f)
   68.62      done
   68.63    have "inj h" using bij_is_inj assms by blast
   68.64 -  then have injim: "inj_on (op ` h) A" for A
   68.65 +  then have injim: "inj_on ((`) h) A" for A
   68.66      by (simp add: inj_on_def inj_image_eq_iff)
   68.67    show ?thesis
   68.68      using \<open>linear h\<close> \<open>inj h\<close>
    69.1 --- a/src/HOL/Analysis/Riemann_Mapping.thy	Wed Jan 10 15:21:49 2018 +0100
    69.2 +++ b/src/HOL/Analysis/Riemann_Mapping.thy	Wed Jan 10 15:25:09 2018 +0100
    69.3 @@ -138,17 +138,17 @@
    69.4      proof (intro bdd_aboveI exI ballI, clarify)
    69.5        show "norm (deriv f 0) \<le> 1 / r" if "f \<in> F" for f
    69.6        proof -
    69.7 -        have r01: "op * (complex_of_real r) ` ball 0 1 \<subseteq> S"
    69.8 +        have r01: "( * ) (complex_of_real r) ` ball 0 1 \<subseteq> S"
    69.9            using that \<open>r > 0\<close> by (auto simp: norm_mult r [THEN subsetD])
   69.10 -        then have "f holomorphic_on op * (complex_of_real r) ` ball 0 1"
   69.11 +        then have "f holomorphic_on ( * ) (complex_of_real r) ` ball 0 1"
   69.12            using holomorphic_on_subset [OF holF] by (simp add: that)
   69.13          then have holf: "f \<circ> (\<lambda>z. (r * z)) holomorphic_on (ball 0 1)"
   69.14            by (intro holomorphic_intros holomorphic_on_compose)
   69.15 -        have f0: "(f \<circ> op * (complex_of_real r)) 0 = 0"
   69.16 +        have f0: "(f \<circ> ( * ) (complex_of_real r)) 0 = 0"
   69.17            using F_def that by auto
   69.18          have "f ` S \<subseteq> ball 0 1"
   69.19            using F_def that by blast
   69.20 -        with r01 have fr1: "\<And>z. norm z < 1 \<Longrightarrow> norm ((f \<circ> op*(of_real r))z) < 1"
   69.21 +        with r01 have fr1: "\<And>z. norm z < 1 \<Longrightarrow> norm ((f \<circ> ( * )(of_real r))z) < 1"
   69.22            by force
   69.23          have *: "((\<lambda>w. f (r * w)) has_field_derivative deriv f (r * z) * r) (at z)"
   69.24            if "z \<in> ball 0 1" for z::complex
   69.25 @@ -162,7 +162,7 @@
   69.26            using * [of 0] by simp
   69.27          have deq: "deriv (\<lambda>x. f (complex_of_real r * x)) 0 = deriv f 0 * complex_of_real r"
   69.28            using DERIV_imp_deriv df0 by blast
   69.29 -        have "norm (deriv (f \<circ> op * (complex_of_real r)) 0) \<le> 1"
   69.30 +        have "norm (deriv (f \<circ> ( * ) (complex_of_real r)) 0) \<le> 1"
   69.31            by (auto intro: Schwarz_Lemma [OF holf f0 fr1, of 0])
   69.32          with \<open>r > 0\<close> show ?thesis
   69.33            by (simp add: deq norm_mult divide_simps o_def)
    70.1 --- a/src/HOL/Analysis/Set_Integral.thy	Wed Jan 10 15:21:49 2018 +0100
    70.2 +++ b/src/HOL/Analysis/Set_Integral.thy	Wed Jan 10 15:25:09 2018 +0100
    70.3 @@ -310,7 +310,7 @@
    70.4      by (auto intro!: set_integral_Un set_integrable_subset[OF f])
    70.5    also have "\<dots> = (LINT x:A|M. f x) + (LINT x:B|M. f x)"
    70.6      using ae
    70.7 -    by (intro arg_cong2[where f="op+"] set_integral_cong_set)
    70.8 +    by (intro arg_cong2[where f="(+)"] set_integral_cong_set)
    70.9         (auto intro!: set_borel_measurable_subset[OF f'])
   70.10    finally show ?thesis .
   70.11  qed
    71.1 --- a/src/HOL/Analysis/Sigma_Algebra.thy	Wed Jan 10 15:21:49 2018 +0100
    71.2 +++ b/src/HOL/Analysis/Sigma_Algebra.thy	Wed Jan 10 15:25:09 2018 +0100
    71.3 @@ -217,7 +217,7 @@
    71.4  subsubsection \<open>Restricted algebras\<close>
    71.5  
    71.6  abbreviation (in algebra)
    71.7 -  "restricted_space A \<equiv> (op \<inter> A) ` M"
    71.8 +  "restricted_space A \<equiv> ((\<inter>) A) ` M"
    71.9  
   71.10  lemma (in algebra) restricted_algebra:
   71.11    assumes "A \<in> M" shows "algebra A (restricted_space A)"
   71.12 @@ -617,11 +617,11 @@
   71.13  
   71.14  lemma sigma_sets_Int:
   71.15    assumes "A \<in> sigma_sets sp st" "A \<subseteq> sp"
   71.16 -  shows "op \<inter> A ` sigma_sets sp st = sigma_sets A (op \<inter> A ` st)"
   71.17 +  shows "(\<inter>) A ` sigma_sets sp st = sigma_sets A ((\<inter>) A ` st)"
   71.18  proof (intro equalityI subsetI)
   71.19 -  fix x assume "x \<in> op \<inter> A ` sigma_sets sp st"
   71.20 +  fix x assume "x \<in> (\<inter>) A ` sigma_sets sp st"
   71.21    then obtain y where "y \<in> sigma_sets sp st" "x = y \<inter> A" by auto
   71.22 -  then have "x \<in> sigma_sets (A \<inter> sp) (op \<inter> A ` st)"
   71.23 +  then have "x \<in> sigma_sets (A \<inter> sp) ((\<inter>) A ` st)"
   71.24    proof (induct arbitrary: x)
   71.25      case (Compl a)
   71.26      then show ?case
   71.27 @@ -632,11 +632,11 @@
   71.28        by (auto intro!: sigma_sets.Union
   71.29                 simp add: UN_extend_simps simp del: UN_simps)
   71.30    qed (auto intro!: sigma_sets.intros(2-))
   71.31 -  then show "x \<in> sigma_sets A (op \<inter> A ` st)"
   71.32 +  then show "x \<in> sigma_sets A ((\<inter>) A ` st)"
   71.33      using \<open>A \<subseteq> sp\<close> by (simp add: Int_absorb2)
   71.34  next
   71.35 -  fix x assume "x \<in> sigma_sets A (op \<inter> A ` st)"
   71.36 -  then show "x \<in> op \<inter> A ` sigma_sets sp st"
   71.37 +  fix x assume "x \<in> sigma_sets A ((\<inter>) A ` st)"
   71.38 +  then show "x \<in> (\<inter>) A ` sigma_sets sp st"
   71.39    proof induct
   71.40      case (Compl a)
   71.41      then obtain x where "a = A \<inter> x" "x \<in> sigma_sets sp st" by auto
   71.42 @@ -2087,7 +2087,7 @@
   71.43  subsubsection \<open>Restricted Space Sigma Algebra\<close>
   71.44  
   71.45  definition restrict_space where
   71.46 -  "restrict_space M \<Omega> = measure_of (\<Omega> \<inter> space M) ((op \<inter> \<Omega>) ` sets M) (emeasure M)"
   71.47 +  "restrict_space M \<Omega> = measure_of (\<Omega> \<inter> space M) (((\<inter>) \<Omega>) ` sets M) (emeasure M)"
   71.48  
   71.49  lemma space_restrict_space: "space (restrict_space M \<Omega>) = \<Omega> \<inter> space M"
   71.50    using sets.sets_into_space unfolding restrict_space_def by (subst space_measure_of) auto
   71.51 @@ -2095,10 +2095,10 @@
   71.52  lemma space_restrict_space2: "\<Omega> \<in> sets M \<Longrightarrow> space (restrict_space M \<Omega>) = \<Omega>"
   71.53    by (simp add: space_restrict_space sets.sets_into_space)
   71.54  
   71.55 -lemma sets_restrict_space: "sets (restrict_space M \<Omega>) = (op \<inter> \<Omega>) ` sets M"
   71.56 +lemma sets_restrict_space: "sets (restrict_space M \<Omega>) = ((\<inter>) \<Omega>) ` sets M"
   71.57    unfolding restrict_space_def
   71.58  proof (subst sets_measure_of)
   71.59 -  show "op \<inter> \<Omega> ` sets M \<subseteq> Pow (\<Omega> \<inter> space M)"
   71.60 +  show "(\<inter>) \<Omega> ` sets M \<subseteq> Pow (\<Omega> \<inter> space M)"
   71.61      by (auto dest: sets.sets_into_space)
   71.62    have "sigma_sets (\<Omega> \<inter> space M) {((\<lambda>x. x) -` X) \<inter> (\<Omega> \<inter> space M) | X. X \<in> sets M} =
   71.63      (\<lambda>X. X \<inter> (\<Omega> \<inter> space M)) ` sets M"
   71.64 @@ -2106,9 +2106,9 @@
   71.65         (auto simp add: sets.sigma_sets_eq)
   71.66    moreover have "{((\<lambda>x. x) -` X) \<inter> (\<Omega> \<inter> space M) | X. X \<in> sets M} = (\<lambda>X. X \<inter> (\<Omega> \<inter> space M)) `  sets M"
   71.67      by auto
   71.68 -  moreover have "(\<lambda>X. X \<inter> (\<Omega> \<inter> space M)) `  sets M = (op \<inter> \<Omega>) ` sets M"
   71.69 +  moreover have "(\<lambda>X. X \<inter> (\<Omega> \<inter> space M)) `  sets M = ((\<inter>) \<Omega>) ` sets M"
   71.70      by (intro image_cong) (auto dest: sets.sets_into_space)
   71.71 -  ultimately show "sigma_sets (\<Omega> \<inter> space M) (op \<inter> \<Omega> ` sets M) = op \<inter> \<Omega> ` sets M"
   71.72 +  ultimately show "sigma_sets (\<Omega> \<inter> space M) ((\<inter>) \<Omega> ` sets M) = (\<inter>) \<Omega> ` sets M"
   71.73      by simp
   71.74  qed
   71.75  
    72.1 --- a/src/HOL/Analysis/Starlike.thy	Wed Jan 10 15:21:49 2018 +0100
    72.2 +++ b/src/HOL/Analysis/Starlike.thy	Wed Jan 10 15:25:09 2018 +0100
    72.3 @@ -1277,13 +1277,13 @@
    72.4        using as(3)
    72.5        unfolding substdbasis_expansion_unique[OF assms]
    72.6        by auto
    72.7 -    then have **: "sum u ?D = sum (op \<bullet> x) ?D"
    72.8 +    then have **: "sum u ?D = sum ((\<bullet>) x) ?D"
    72.9        apply -
   72.10        apply (rule sum.cong)
   72.11        using assms
   72.12        apply auto
   72.13        done
   72.14 -    have "(\<forall>i\<in>Basis. 0 \<le> x\<bullet>i) \<and> sum (op \<bullet> x) ?D \<le> 1"
   72.15 +    have "(\<forall>i\<in>Basis. 0 \<le> x\<bullet>i) \<and> sum ((\<bullet>) x) ?D \<le> 1"
   72.16      proof (rule,rule)
   72.17        fix i :: 'a
   72.18        assume i: "i \<in> Basis"
   72.19 @@ -1296,11 +1296,11 @@
   72.20          using \<open>(\<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x \<bullet> i = 0)\<close>[rule_format, OF i] by auto
   72.21        ultimately show "0 \<le> x\<bullet>i" by auto
   72.22      qed (insert as(2)[unfolded **], auto)
   72.23 -    then show "(\<forall>i\<in>Basis. 0 \<le> x\<bullet>i) \<and> sum (op \<bullet> x) ?D \<le> 1 \<and> (\<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x \<bullet> i = 0)"
   72.24 +    then show "(\<forall>i\<in>Basis. 0 \<le> x\<bullet>i) \<and> sum ((\<bullet>) x) ?D \<le> 1 \<and> (\<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x \<bullet> i = 0)"
   72.25        using \<open>(\<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x \<bullet> i = 0)\<close> by auto
   72.26    next
   72.27      fix x :: "'a::euclidean_space"
   72.28 -    assume as: "\<forall>i\<in>Basis. 0 \<le> x \<bullet> i" "sum (op \<bullet> x) ?D \<le> 1" "(\<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x \<bullet> i = 0)"
   72.29 +    assume as: "\<forall>i\<in>Basis. 0 \<le> x \<bullet> i" "sum ((\<bullet>) x) ?D \<le> 1" "(\<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x \<bullet> i = 0)"
   72.30      show "\<exists>u. (\<forall>x\<in>?D. 0 \<le> u x) \<and> sum u ?D \<le> 1 \<and> (\<Sum>x\<in>?D. u x *\<^sub>R x) = x"
   72.31        using as d
   72.32        unfolding substdbasis_expansion_unique[OF assms]
   72.33 @@ -1329,8 +1329,8 @@
   72.34  proof -
   72.35    fix x :: 'a
   72.36    fix e
   72.37 -  assume "e > 0" and as: "\<forall>xa. dist x xa < e \<longrightarrow> (\<forall>x\<in>Basis. 0 \<le> xa \<bullet> x) \<and> sum (op \<bullet> xa) Basis \<le> 1"
   72.38 -  show "(\<forall>xa\<in>Basis. 0 < x \<bullet> xa) \<and> sum (op \<bullet> x) Basis < 1"
   72.39 +  assume "e > 0" and as: "\<forall>xa. dist x xa < e \<longrightarrow> (\<forall>x\<in>Basis. 0 \<le> xa \<bullet> x) \<and> sum ((\<bullet>) xa) Basis \<le> 1"
   72.40 +  show "(\<forall>xa\<in>Basis. 0 < x \<bullet> xa) \<and> sum ((\<bullet>) x) Basis < 1"
   72.41      apply safe
   72.42    proof -
   72.43      fix i :: 'a
   72.44 @@ -1346,12 +1346,12 @@
   72.45      have "\<And>i. i \<in> Basis \<Longrightarrow> (x + (e / 2) *\<^sub>R (SOME i. i\<in>Basis)) \<bullet> i =
   72.46        x\<bullet>i + (if i = (SOME i. i\<in>Basis) then e/2 else 0)"
   72.47        by (auto simp: SOME_Basis inner_Basis inner_simps)
   72.48 -    then have *: "sum (op \<bullet> (x + (e / 2) *\<^sub>R (SOME i. i\<in>Basis))) Basis =
   72.49 +    then have *: "sum ((\<bullet>) (x + (e / 2) *\<^sub>R (SOME i. i\<in>Basis))) Basis =
   72.50        sum (\<lambda>i. x\<bullet>i + (if (SOME i. i\<in>Basis) = i then e/2 else 0)) Basis"
   72.51        apply (rule_tac sum.cong)
   72.52        apply auto
   72.53        done
   72.54 -    have "sum (op \<bullet> x) Basis < sum (op \<bullet> (x + (e / 2) *\<^sub>R (SOME i. i\<in>Basis))) Basis"
   72.55 +    have "sum ((\<bullet>) x) Basis < sum ((\<bullet>) (x + (e / 2) *\<^sub>R (SOME i. i\<in>Basis))) Basis"
   72.56        unfolding * sum.distrib
   72.57        using \<open>e > 0\<close> DIM_positive[where 'a='a]
   72.58        apply (subst sum.delta')
   72.59 @@ -1362,18 +1362,18 @@
   72.60        apply (drule_tac as[rule_format])
   72.61        apply auto
   72.62        done
   72.63 -    finally show "sum (op \<bullet> x) Basis < 1" by auto
   72.64 +    finally show "sum ((\<bullet>) x) Basis < 1" by auto
   72.65    qed
   72.66  next
   72.67    fix x :: 'a
   72.68 -  assume as: "\<forall>i\<in>Basis. 0 < x \<bullet> i" "sum (op \<bullet> x) Basis < 1"
   72.69 +  assume as: "\<forall>i\<in>Basis. 0 < x \<bullet> i" "sum ((\<bullet>) x) Basis < 1"
   72.70    obtain a :: 'b where "a \<in> UNIV" using UNIV_witness ..
   72.71 -  let ?d = "(1 - sum (op \<bullet> x) Basis) / real (DIM('a))"
   72.72 -  show "\<exists>e>0. \<forall>y. dist x y < e \<longrightarrow> (\<forall>i\<in>Basis. 0 \<le> y \<bullet> i) \<and> sum (op \<bullet> y) Basis \<le> 1"
   72.73 -  proof (rule_tac x="min (Min ((op \<bullet> x) ` Basis)) D" for D in exI, intro conjI impI allI)
   72.74 +  let ?d = "(1 - sum ((\<bullet>) x) Basis) / real (DIM('a))"
   72.75 +  show "\<exists>e>0. \<forall>y. dist x y < e \<longrightarrow> (\<forall>i\<in>Basis. 0 \<le> y \<bullet> i) \<and> sum ((\<bullet>) y) Basis \<le> 1"
   72.76 +  proof (rule_tac x="min (Min (((\<bullet>) x) ` Basis)) D" for D in exI, intro conjI impI allI)
   72.77      fix y
   72.78 -    assume y: "dist x y < min (Min (op \<bullet> x ` Basis)) ?d"
   72.79 -    have "sum (op \<bullet> y) Basis \<le> sum (\<lambda>i. x\<bullet>i + ?d) Basis"
   72.80 +    assume y: "dist x y < min (Min ((\<bullet>) x ` Basis)) ?d"
   72.81 +    have "sum ((\<bullet>) y) Basis \<le> sum (\<lambda>i. x\<bullet>i + ?d) Basis"
   72.82      proof (rule sum_mono)
   72.83        fix i :: 'a
   72.84        assume i: "i \<in> Basis"
   72.85 @@ -1389,7 +1389,7 @@
   72.86      also have "\<dots> \<le> 1"
   72.87        unfolding sum.distrib sum_constant
   72.88        by (auto simp add: Suc_le_eq)
   72.89 -    finally show "sum (op \<bullet> y) Basis \<le> 1" .
   72.90 +    finally show "sum ((\<bullet>) y) Basis \<le> 1" .
   72.91      show "(\<forall>i\<in>Basis. 0 \<le> y \<bullet> i)"
   72.92      proof safe
   72.93        fix i :: 'a
   72.94 @@ -1405,11 +1405,11 @@
   72.95          by (auto simp: inner_simps)
   72.96      qed
   72.97    next
   72.98 -    have "Min ((op \<bullet> x) ` Basis) > 0"
   72.99 +    have "Min (((\<bullet>) x) ` Basis) > 0"
  72.100        using as by simp
  72.101      moreover have "?d > 0"
  72.102        using as by (auto simp: Suc_le_eq)
  72.103 -    ultimately show "0 < min (Min (op \<bullet> x ` Basis)) ((1 - sum (op \<bullet> x) Basis) / real DIM('a))"
  72.104 +    ultimately show "0 < min (Min ((\<bullet>) x ` Basis)) ((1 - sum ((\<bullet>) x) Basis) / real DIM('a))"
  72.105        by linarith
  72.106    qed 
  72.107  qed
  72.108 @@ -1436,7 +1436,7 @@
  72.109      show "0 < ?a \<bullet> i"
  72.110        unfolding **[OF i] by (auto simp add: Suc_le_eq DIM_positive)
  72.111    next
  72.112 -    have "sum (op \<bullet> ?a) ?D = sum (\<lambda>i. inverse (2 * real DIM('a))) ?D"
  72.113 +    have "sum ((\<bullet>) ?a) ?D = sum (\<lambda>i. inverse (2 * real DIM('a))) ?D"
  72.114        apply (rule sum.cong)
  72.115        apply rule
  72.116        apply auto
  72.117 @@ -1444,7 +1444,7 @@
  72.118      also have "\<dots> < 1"
  72.119        unfolding sum_constant divide_inverse[symmetric]
  72.120        by (auto simp add: field_simps)
  72.121 -    finally show "sum (op \<bullet> ?a) ?D < 1" by auto
  72.122 +    finally show "sum ((\<bullet>) ?a) ?D < 1" by auto
  72.123    qed
  72.124  qed
  72.125  
  72.126 @@ -1478,11 +1478,11 @@
  72.127          "ball x e \<inter> {xa. (\<forall>i\<in>Basis. i \<notin> d \<longrightarrow> xa\<bullet>i = 0)} \<subseteq> convex hull (insert 0 ?p)"
  72.128          using mem_rel_interior_ball[of x "convex hull (insert 0 ?p)"] h0 by auto
  72.129        then have as: "\<forall>xa. dist x xa < e \<and> (\<forall>i\<in>Basis. i \<notin> d \<longrightarrow> xa\<bullet>i = 0) \<longrightarrow>
  72.130 -        (\<forall>i\<in>d. 0 \<le> xa \<bullet> i) \<and> sum (op \<bullet> xa) d \<le> 1"
  72.131 +        (\<forall>i\<in>d. 0 \<le> xa \<bullet> i) \<and> sum ((\<bullet>) xa) d \<le> 1"
  72.132          unfolding ball_def unfolding substd_simplex[OF assms] using assms by auto
  72.133        have x0: "(\<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x\<bullet>i = 0)"
  72.134          using x rel_interior_subset  substd_simplex[OF assms] by auto
  72.135 -      have "(\<forall>i\<in>d. 0 < x \<bullet> i) \<and> sum (op \<bullet> x) d < 1 \<and> (\<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x\<bullet>i = 0)"
  72.136 +      have "(\<forall>i\<in>d. 0 < x \<bullet> i) \<and> sum ((\<bullet>) x) d < 1 \<and> (\<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x\<bullet>i = 0)"
  72.137          apply rule
  72.138          apply rule
  72.139        proof -
  72.140 @@ -1509,14 +1509,14 @@
  72.141            by auto
  72.142          have "\<And>i. i \<in> Basis \<Longrightarrow> (x + (e / 2) *\<^sub>R a) \<bullet> i = x\<bullet>i + (if i = a then e/2 else 0)"
  72.143            using a d by (auto simp: inner_simps inner_Basis)
  72.144 -        then have *: "sum (op \<bullet> (x + (e / 2) *\<^sub>R a)) d =
  72.145 +        then have *: "sum ((\<bullet>) (x + (e / 2) *\<^sub>R a)) d =
  72.146            sum (\<lambda>i. x\<bullet>i + (if a = i then e/2 else 0)) d"
  72.147            using d by (intro sum.cong) auto
  72.148          have "a \<in> Basis"
  72.149            using \<open>a \<in> d\<close> d by auto
  72.150          then have h1: "(\<forall>i\<in>Basis. i \<notin> d \<longrightarrow> (x + (e / 2) *\<^sub>R a) \<bullet> i = 0)"
  72.151            using x0 d \<open>a\<in>d\<close> by (auto simp add: inner_add_left inner_Basis)
  72.152 -        have "sum (op \<bullet> x) d < sum (op \<bullet> (x + (e / 2) *\<^sub>R a)) d"
  72.153 +        have "sum ((\<bullet>) x) d < sum ((\<bullet>) (x + (e / 2) *\<^sub>R a)) d"
  72.154            unfolding * sum.distrib
  72.155            using \<open>e > 0\<close> \<open>a \<in> d\<close>
  72.156            using \<open>finite d\<close>
  72.157 @@ -1524,7 +1524,7 @@
  72.158          also have "\<dots> \<le> 1"
  72.159            using ** h1 as[rule_format, of "x + (e / 2) *\<^sub>R a"]
  72.160            by auto
  72.161 -        finally show "sum (op \<bullet> x) d < 1 \<and> (\<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x\<bullet>i = 0)"
  72.162 +        finally show "sum ((\<bullet>) x) d < 1 \<and> (\<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x\<bullet>i = 0)"
  72.163            using x0 by auto
  72.164        qed
  72.165      }
  72.166 @@ -1543,28 +1543,28 @@
  72.167          unfolding substd_simplex[OF assms] by fastforce
  72.168        obtain a where a: "a \<in> d"
  72.169          using \<open>d \<noteq> {}\<close> by auto
  72.170 -      let ?d = "(1 - sum (op \<bullet> x) d) / real (card d)"
  72.171 +      let ?d = "(1 - sum ((\<bullet>) x) d) / real (card d)"
  72.172        have "0 < card d" using \<open>d \<noteq> {}\<close> \<open>finite d\<close>
  72.173          by (simp add: card_gt_0_iff)
  72.174 -      have "Min ((op \<bullet> x) ` d) > 0"
  72.175 +      have "Min (((\<bullet>) x) ` d) > 0"
  72.176          using as \<open>d \<noteq> {}\<close> \<open>finite d\<close> by (simp add: Min_gr_iff)
  72.177        moreover have "?d > 0" using as using \<open>0 < card d\<close> by auto
  72.178 -      ultimately have h3: "min (Min ((op \<bullet> x) ` d)) ?d > 0"
  72.179 +      ultimately have h3: "min (Min (((\<bullet>) x) ` d)) ?d > 0"
  72.180          by auto
  72.181  
  72.182        have "x \<in> rel_interior (convex hull (insert 0 ?p))"
  72.183          unfolding rel_interior_ball mem_Collect_eq h0
  72.184          apply (rule,rule h2)
  72.185          unfolding substd_simplex[OF assms]
  72.186 -        apply (rule_tac x="min (Min ((op \<bullet> x) ` d)) ?d" in exI)
  72.187 +        apply (rule_tac x="min (Min (((\<bullet>) x) ` d)) ?d" in exI)
  72.188          apply (rule, rule h3)
  72.189          apply safe
  72.190          unfolding mem_ball
  72.191        proof -
  72.192          fix y :: 'a
  72.193 -        assume y: "dist x y < min (Min (op \<bullet> x ` d)) ?d"
  72.194 +        assume y: "dist x y < min (Min ((\<bullet>) x ` d)) ?d"
  72.195          assume y2: "\<forall>i\<in>Basis. i \<notin> d \<longrightarrow> y\<bullet>i = 0"
  72.196 -        have "sum (op \<bullet> y) d \<le> sum (\<lambda>i. x\<bullet>i + ?d) d"
  72.197 +        have "sum ((\<bullet>) y) d \<le> sum (\<lambda>i. x\<bullet>i + ?d) d"
  72.198          proof (rule sum_mono)
  72.199            fix i
  72.200            assume "i \<in> d"
  72.201 @@ -1581,7 +1581,7 @@
  72.202          also have "\<dots> \<le> 1"
  72.203            unfolding sum.distrib sum_constant  using \<open>0 < card d\<close>
  72.204            by auto
  72.205 -        finally show "sum (op \<bullet> y) d \<le> 1" .
  72.206 +        finally show "sum ((\<bullet>) y) d \<le> 1" .
  72.207  
  72.208          fix i :: 'a
  72.209          assume i: "i \<in> Basis"
  72.210 @@ -1590,7 +1590,7 @@
  72.211            case True
  72.212            have "norm (x - y) < x\<bullet>i"
  72.213              using y[unfolded min_less_iff_conj dist_norm, THEN conjunct1]
  72.214 -            using Min_gr_iff[of "op \<bullet> x ` d" "norm (x - y)"] \<open>0 < card d\<close> \<open>i:d\<close>
  72.215 +            using Min_gr_iff[of "(\<bullet>) x ` d" "norm (x - y)"] \<open>0 < card d\<close> \<open>i:d\<close>
  72.216              by (simp add: card_gt_0_iff)
  72.217            then show "0 \<le> y\<bullet>i"
  72.218              using Basis_le_norm[OF i, of "x - y"] and as(1)[rule_format]
  72.219 @@ -1600,7 +1600,7 @@
  72.220      }
  72.221      ultimately have
  72.222        "\<And>x. x \<in> rel_interior (convex hull insert 0 d) \<longleftrightarrow>
  72.223 -        x \<in> {x. (\<forall>i\<in>d. 0 < x \<bullet> i) \<and> sum (op \<bullet> x) d < 1 \<and> (\<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x \<bullet> i = 0)}"
  72.224 +        x \<in> {x. (\<forall>i\<in>d. 0 < x \<bullet> i) \<and> sum ((\<bullet>) x) d < 1 \<and> (\<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x \<bullet> i = 0)}"
  72.225        by blast
  72.226      then show ?thesis by (rule set_eqI)
  72.227    qed
  72.228 @@ -1645,12 +1645,12 @@
  72.229        by auto
  72.230      finally show "0 < ?a \<bullet> i" by auto
  72.231    next
  72.232 -    have "sum (op \<bullet> ?a) ?D = sum (\<lambda>i. inverse (2 * real (card d))) ?D"
  72.233 +    have "sum ((\<bullet>) ?a) ?D = sum (\<lambda>i. inverse (2 * real (card d))) ?D"
  72.234        by (rule sum.cong) (rule refl, rule **)
  72.235      also have "\<dots> < 1"
  72.236        unfolding sum_constant divide_real_def[symmetric]
  72.237        by (auto simp add: field_simps)
  72.238 -    finally show "sum (op \<bullet> ?a) ?D < 1" by auto
  72.239 +    finally show "sum ((\<bullet>) ?a) ?D < 1" by auto
  72.240    next
  72.241      fix i
  72.242      assume "i \<in> Basis" and "i \<notin> d"
  72.243 @@ -1748,10 +1748,10 @@
  72.244    {
  72.245      assume "S \<noteq> {}"
  72.246      then obtain a where "a \<in> S" by auto
  72.247 -    then have "0 \<in> op + (-a) ` S"
  72.248 +    then have "0 \<in> (+) (-a) ` S"
  72.249        using assms exI[of "(\<lambda>x. x \<in> S \<and> - a + x = 0)" a] by auto
  72.250 -    then have "rel_interior (op + (-a) ` S) \<noteq> {}"
  72.251 -      using rel_interior_convex_nonempty_aux[of "op + (-a) ` S"]
  72.252 +    then have "rel_interior ((+) (-a) ` S) \<noteq> {}"
  72.253 +      using rel_interior_convex_nonempty_aux[of "(+) (-a) ` S"]
  72.254          convex_translation[of S "-a"] assms
  72.255        by auto
  72.256      then have "rel_interior S \<noteq> {}"
  72.257 @@ -2958,22 +2958,22 @@
  72.258  lemma rel_interior_scaleR:
  72.259    fixes S :: "'n::euclidean_space set"
  72.260    assumes "c \<noteq> 0"
  72.261 -  shows "(op *\<^sub>R c) ` (rel_interior S) = rel_interior ((op *\<^sub>R c) ` S)"
  72.262 -  using rel_interior_injective_linear_image[of "(op *\<^sub>R c)" S]
  72.263 -    linear_conv_bounded_linear[of "op *\<^sub>R c"] linear_scaleR injective_scaleR[of c] assms
  72.264 +  shows "(( *\<^sub>R) c) ` (rel_interior S) = rel_interior ((( *\<^sub>R) c) ` S)"
  72.265 +  using rel_interior_injective_linear_image[of "(( *\<^sub>R) c)" S]
  72.266 +    linear_conv_bounded_linear[of "( *\<^sub>R) c"] linear_scaleR injective_scaleR[of c] assms
  72.267    by auto
  72.268  
  72.269  lemma rel_interior_convex_scaleR:
  72.270    fixes S :: "'n::euclidean_space set"
  72.271    assumes "convex S"
  72.272 -  shows "(op *\<^sub>R c) ` (rel_interior S) = rel_interior ((op *\<^sub>R c) ` S)"
  72.273 +  shows "(( *\<^sub>R) c) ` (rel_interior S) = rel_interior ((( *\<^sub>R) c) ` S)"
  72.274    by (metis assms linear_scaleR rel_interior_convex_linear_image)
  72.275  
  72.276  lemma convex_rel_open_scaleR:
  72.277    fixes S :: "'n::euclidean_space set"
  72.278    assumes "convex S"
  72.279      and "rel_open S"
  72.280 -  shows "convex ((op *\<^sub>R c) ` S) \<and> rel_open ((op *\<^sub>R c) ` S)"
  72.281 +  shows "convex ((( *\<^sub>R) c) ` S) \<and> rel_open ((( *\<^sub>R) c) ` S)"
  72.282    by (metis assms convex_scaling rel_interior_convex_scaleR rel_open_def)
  72.283  
  72.284  lemma convex_rel_open_finite_inter:
  72.285 @@ -3142,10 +3142,10 @@
  72.286      by (simp add: rel_interior_empty cone_0)
  72.287  next
  72.288    case False
  72.289 -  then have *: "0 \<in> S \<and> (\<forall>c. c > 0 \<longrightarrow> op *\<^sub>R c ` S = S)"
  72.290 +  then have *: "0 \<in> S \<and> (\<forall>c. c > 0 \<longrightarrow> ( *\<^sub>R) c ` S = S)"
  72.291      using cone_iff[of S] assms by auto
  72.292    then have *: "0 \<in> ({0} \<union> rel_interior S)"
  72.293 -    and "\<forall>c. c > 0 \<longrightarrow> op *\<^sub>R c ` ({0} \<union> rel_interior S) = ({0} \<union> rel_interior S)"
  72.294 +    and "\<forall>c. c > 0 \<longrightarrow> ( *\<^sub>R) c ` ({0} \<union> rel_interior S) = ({0} \<union> rel_interior S)"
  72.295      by (auto simp add: rel_interior_scaleR)
  72.296    then show ?thesis
  72.297      using cone_iff[of "{0} \<union> rel_interior S"] by auto
  72.298 @@ -3155,7 +3155,7 @@
  72.299    fixes S :: "'m::euclidean_space set"
  72.300    assumes "convex S"
  72.301    shows "(c, x) \<in> rel_interior (cone hull ({(1 :: real)} \<times> S)) \<longleftrightarrow>
  72.302 -    c > 0 \<and> x \<in> ((op *\<^sub>R c) ` (rel_interior S))"
  72.303 +    c > 0 \<and> x \<in> ((( *\<^sub>R) c) ` (rel_interior S))"
  72.304  proof (cases "S = {}")
  72.305    case True
  72.306    then show ?thesis
  72.307 @@ -3188,9 +3188,9 @@
  72.308    {
  72.309      fix c :: real
  72.310      assume "c > 0"
  72.311 -    then have "f c = (op *\<^sub>R c ` S)"
  72.312 +    then have "f c = (( *\<^sub>R) c ` S)"
  72.313        using f_def cone_hull_expl[of "{1 :: real} \<times> S"] by auto
  72.314 -    then have "rel_interior (f c) = op *\<^sub>R c ` rel_interior S"
  72.315 +    then have "rel_interior (f c) = ( *\<^sub>R) c ` rel_interior S"
  72.316        using rel_interior_convex_scaleR[of S c] assms by auto
  72.317    }
  72.318    then show ?thesis using * ** by auto
  72.319 @@ -5154,9 +5154,9 @@
  72.320      let ?\<C> = "insert (U \<union> V) ((\<lambda>S. N - S) ` \<F>)"
  72.321      obtain \<D> where "\<D> \<subseteq> ?\<C>" "finite \<D>" "K \<subseteq> \<Union>\<D>"
  72.322      proof (rule compactE [OF \<open>compact K\<close>])
  72.323 -      show "K \<subseteq> \<Union>insert (U \<union> V) (op - N ` \<F>)"
  72.324 +      show "K \<subseteq> \<Union>insert (U \<union> V) ((-) N ` \<F>)"
  72.325          using \<open>K \<subseteq> N\<close> ABeq \<open>A \<subseteq> U\<close> \<open>B \<subseteq> V\<close> by auto
  72.326 -      show "\<And>B. B \<in> insert (U \<union> V) (op - N ` \<F>) \<Longrightarrow> open B"
  72.327 +      show "\<And>B. B \<in> insert (U \<union> V) ((-) N ` \<F>) \<Longrightarrow> open B"
  72.328          by (auto simp:  \<open>open U\<close> \<open>open V\<close> open_Un \<open>open N\<close> cc compact_imp_closed open_Diff)
  72.329      qed
  72.330      then have "finite(\<D> - {U \<union> V})"
  72.331 @@ -5195,7 +5195,7 @@
  72.332                by auto
  72.333            next
  72.334              assume "N - X \<subseteq> N - J"
  72.335 -            with J have "N - X \<union> UNION \<H> (op - N) \<subseteq> N - J"
  72.336 +            with J have "N - X \<union> UNION \<H> ((-) N) \<subseteq> N - J"
  72.337                by auto
  72.338              with \<open>J \<in> \<H>\<close> show ?thesis
  72.339                by blast
  72.340 @@ -5234,9 +5234,9 @@
  72.341      using X by blast
  72.342    moreover have "connected (\<Inter>T\<in>\<F>. X \<inter> T)"
  72.343    proof (rule connected_chain)
  72.344 -    show "\<And>T. T \<in> op \<inter> X ` \<F> \<Longrightarrow> compact T \<and> connected T"
  72.345 +    show "\<And>T. T \<in> (\<inter>) X ` \<F> \<Longrightarrow> compact T \<and> connected T"
  72.346        using cc X by auto (metis inf.absorb2 inf.orderE local.linear)
  72.347 -    show "\<And>S T. S \<in> op \<inter> X ` \<F> \<and> T \<in> op \<inter> X ` \<F> \<Longrightarrow> S \<subseteq> T \<or> T \<subseteq> S"
  72.348 +    show "\<And>S T. S \<in> (\<inter>) X ` \<F> \<and> T \<in> (\<inter>) X ` \<F> \<Longrightarrow> S \<subseteq> T \<or> T \<subseteq> S"
  72.349        using local.linear by blast
  72.350    qed
  72.351    ultimately show ?thesis
  72.352 @@ -5599,7 +5599,7 @@
  72.353      using assms interior_subset by blast
  72.354    then obtain e where "e > 0" and e: "cball a e \<subseteq> T"
  72.355      using mem_interior_cball by blast
  72.356 -  have *: "x \<in> op + a ` span ((\<lambda>x. x - a) ` (S \<inter> T))" if "x \<in> S" for x
  72.357 +  have *: "x \<in> (+) a ` span ((\<lambda>x. x - a) ` (S \<inter> T))" if "x \<in> S" for x
  72.358    proof (cases "x = a")
  72.359      case True with that span_0 eq_add_iff image_def mem_Collect_eq show ?thesis
  72.360        by blast
  72.361 @@ -5870,19 +5870,19 @@
  72.362      done
  72.363    next
  72.364      case False
  72.365 -    have xc_im: "x \<in> op + c ` {y. a \<bullet> y = 0}" if "a \<bullet> x = a \<bullet> c" for a x
  72.366 +    have xc_im: "x \<in> (+) c ` {y. a \<bullet> y = 0}" if "a \<bullet> x = a \<bullet> c" for a x
  72.367      proof -
  72.368        have "\<exists>y. a \<bullet> y = 0 \<and> c + y = x"
  72.369          by (metis that add.commute diff_add_cancel inner_commute inner_diff_left right_minus_eq)
  72.370 -      then show "x \<in> op + c ` {y. a \<bullet> y = 0}"
  72.371 +      then show "x \<in> (+) c ` {y. a \<bullet> y = 0}"
  72.372          by blast
  72.373      qed
  72.374      have 2: "span ((\<lambda>x. x - c) ` S) = {x. a \<bullet> x = 0}"
  72.375 -         if "op + c ` span ((\<lambda>x. x - c) ` S) = {x. a \<bullet> x = b}" for a b
  72.376 +         if "(+) c ` span ((\<lambda>x. x - c) ` S) = {x. a \<bullet> x = b}" for a b
  72.377      proof -
  72.378        have "b = a \<bullet> c"
  72.379          using span_0 that by fastforce
  72.380 -      with that have "op + c ` span ((\<lambda>x. x - c) ` S) = {x. a \<bullet> x = a \<bullet> c}"
  72.381 +      with that have "(+) c ` span ((\<lambda>x. x - c) ` S) = {x. a \<bullet> x = a \<bullet> c}"
  72.382          by simp
  72.383        then have "span ((\<lambda>x. x - c) ` S) = (\<lambda>x. x - c) ` {x. a \<bullet> x = a \<bullet> c}"
  72.384          by (metis (no_types) image_cong translation_galois uminus_add_conv_diff)
  72.385 @@ -6183,21 +6183,21 @@
  72.386                  and "\<And>x. x \<in> S \<Longrightarrow> a \<bullet> x \<ge> b"
  72.387  proof -
  72.388  from separating_hyperplane_set_0_inspan [of "image (\<lambda>x. -z + x) S"]
  72.389 -  have "convex (op + (- z) ` S)"
  72.390 +  have "convex ((+) (- z) ` S)"
  72.391      by (simp add: \<open>convex S\<close>)
  72.392 -  moreover have "op + (- z) ` S \<noteq> {}"
  72.393 +  moreover have "(+) (- z) ` S \<noteq> {}"
  72.394      by (simp add: \<open>S \<noteq> {}\<close>)
  72.395 -  moreover have "0 \<notin> op + (- z) ` S"
  72.396 +  moreover have "0 \<notin> (+) (- z) ` S"
  72.397      using zno by auto
  72.398 -  ultimately obtain a where "a \<in> span (op + (- z) ` S)" "a \<noteq> 0"
  72.399 -                  and a:  "\<And>x. x \<in> (op + (- z) ` S) \<Longrightarrow> 0 \<le> a \<bullet> x"
  72.400 +  ultimately obtain a where "a \<in> span ((+) (- z) ` S)" "a \<noteq> 0"
  72.401 +                  and a:  "\<And>x. x \<in> ((+) (- z) ` S) \<Longrightarrow> 0 \<le> a \<bullet> x"
  72.402      using separating_hyperplane_set_0_inspan [of "image (\<lambda>x. -z + x) S"]
  72.403      by blast
  72.404    then have szx: "\<And>x. x \<in> S \<Longrightarrow> a \<bullet> z \<le> a \<bullet> x"
  72.405      by (metis (no_types, lifting) imageI inner_minus_right inner_right_distrib minus_add neg_le_0_iff_le neg_le_iff_le real_add_le_0_iff)
  72.406    show ?thesis
  72.407      apply (rule_tac a=a and b = "a  \<bullet> z" in that, simp_all)
  72.408 -    using \<open>a \<in> span (op + (- z) ` S)\<close> affine_hull_insert_span_gen apply blast
  72.409 +    using \<open>a \<in> span ((+) (- z) ` S)\<close> affine_hull_insert_span_gen apply blast
  72.410      apply (simp_all add: \<open>a \<noteq> 0\<close> szx)
  72.411      done
  72.412  qed
  72.413 @@ -6216,7 +6216,7 @@
  72.414      by (auto simp: rel_interior_eq_empty convex_rel_interior)
  72.415    have le_ay: "a \<bullet> x \<le> a \<bullet> y" if "y \<in> S" for y
  72.416    proof -
  72.417 -    have con: "continuous_on (closure (rel_interior S)) (op \<bullet> a)"
  72.418 +    have con: "continuous_on (closure (rel_interior S)) ((\<bullet>) a)"
  72.419        by (rule continuous_intros continuous_on_subset | blast)+
  72.420      have y: "y \<in> closure (rel_interior S)"
  72.421        using \<open>convex S\<close> closure_def convex_closure_rel_interior \<open>y \<in> S\<close>
  72.422 @@ -6701,17 +6701,17 @@
  72.423    obtain c where "c \<in> S" and c: "a \<bullet> c = b"
  72.424      using assms by force
  72.425    with affine_diffs_subspace [OF \<open>affine S\<close>]
  72.426 -  have "subspace (op + (- c) ` S)" by blast
  72.427 -  then have aff: "affine (op + (- c) ` S)"
  72.428 +  have "subspace ((+) (- c) ` S)" by blast
  72.429 +  then have aff: "affine ((+) (- c) ` S)"
  72.430      by (simp add: subspace_imp_affine)
  72.431 -  have 0: "0 \<in> op + (- c) ` S"
  72.432 +  have 0: "0 \<in> (+) (- c) ` S"
  72.433      by (simp add: \<open>c \<in> S\<close>)
  72.434 -  obtain d where "d \<in> S" and "a \<bullet> d \<noteq> b" and dc: "d-c \<in> op + (- c) ` S"
  72.435 +  obtain d where "d \<in> S" and "a \<bullet> d \<noteq> b" and dc: "d-c \<in> (+) (- c) ` S"
  72.436      using assms by auto
  72.437    then have adc: "a \<bullet> (d - c) \<noteq> 0"
  72.438      by (simp add: c inner_diff_right)
  72.439 -  let ?U = "op + (c+c) ` {x + y |x y. x \<in> op + (- c) ` S \<and> a \<bullet> y = 0}"
  72.440 -  have "u + v \<in> op + (c + c) ` {x + v |x v. x \<in> op + (- c) ` S \<and> a \<bullet> v = 0}"
  72.441 +  let ?U = "(+) (c+c) ` {x + y |x y. x \<in> (+) (- c) ` S \<and> a \<bullet> y = 0}"
  72.442 +  have "u + v \<in> (+) (c + c) ` {x + v |x v. x \<in> (+) (- c) ` S \<and> a \<bullet> v = 0}"
  72.443                if "u \<in> S" "b = a \<bullet> v" for u v
  72.444      apply (rule_tac x="u+v-c-c" in image_eqI)
  72.445      apply (simp_all add: algebra_simps)
  72.446 @@ -6724,7 +6724,7 @@
  72.447      by (metis add.left_commute c inner_right_distrib pth_d)
  72.448    ultimately have "{x + y |x y. x \<in> S \<and> a \<bullet> y = b} = ?U"
  72.449      by (fastforce simp: algebra_simps)
  72.450 -  also have "... = op + (c+c) ` UNIV"
  72.451 +  also have "... = (+) (c+c) ` UNIV"
  72.452      by (simp add: affine_hyperplane_sums_eq_UNIV_0 [OF aff 0 dc adc])
  72.453    also have "... = UNIV"
  72.454      by (simp add: translation_UNIV)
  72.455 @@ -6856,14 +6856,14 @@
  72.456      shows "aff_dim {x + y| x y. x \<in> S \<and> y \<in> T} = (aff_dim S + aff_dim T) - aff_dim(S \<inter> T)"
  72.457  proof -
  72.458    obtain a where a: "a \<in> S" "a \<in> T" using assms by force
  72.459 -  have aff: "affine (op+ (-a) ` S)"  "affine (op+ (-a) ` T)"
  72.460 +  have aff: "affine ((+) (-a) ` S)"  "affine ((+) (-a) ` T)"
  72.461      using assms by (auto simp: affine_translation [symmetric])
  72.462 -  have zero: "0 \<in> (op+ (-a) ` S)"  "0 \<in> (op+ (-a) ` T)"
  72.463 +  have zero: "0 \<in> ((+) (-a) ` S)"  "0 \<in> ((+) (-a) ` T)"
  72.464      using a assms by auto
  72.465 -  have [simp]: "{x + y |x y. x \<in> op + (- a) ` S \<and> y \<in> op + (- a) ` T} =
  72.466 -        op + (- 2 *\<^sub>R a) ` {x + y| x y. x \<in> S \<and> y \<in> T}"
  72.467 +  have [simp]: "{x + y |x y. x \<in> (+) (- a) ` S \<and> y \<in> (+) (- a) ` T} =
  72.468 +        (+) (- 2 *\<^sub>R a) ` {x + y| x y. x \<in> S \<and> y \<in> T}"
  72.469      by (force simp: algebra_simps scaleR_2)
  72.470 -  have [simp]: "op + (- a) ` S \<inter> op + (- a) ` T = op + (- a) ` (S \<inter> T)"
  72.471 +  have [simp]: "(+) (- a) ` S \<inter> (+) (- a) ` T = (+) (- a) ` (S \<inter> T)"
  72.472      by auto
  72.473    show ?thesis
  72.474      using aff_dim_sums_Int_0 [OF aff zero]
  72.475 @@ -7457,11 +7457,11 @@
  72.476  next
  72.477    case False
  72.478    then obtain z where z: "z \<in> S \<inter> T" by blast
  72.479 -  then have "subspace (op + (- z) ` S)"
  72.480 +  then have "subspace ((+) (- z) ` S)"
  72.481      by (meson IntD1 affine_diffs_subspace \<open>affine S\<close>)
  72.482 -  moreover have "int (dim (op + (- z) ` T)) < int (dim (op + (- z) ` S))"
  72.483 +  moreover have "int (dim ((+) (- z) ` T)) < int (dim ((+) (- z) ` S))"
  72.484      using z less by (simp add: aff_dim_eq_dim [symmetric] hull_inc)
  72.485 -  ultimately have "closure((op + (- z) ` S) - (op + (- z) ` T)) = (op + (- z) ` S)"
  72.486 +  ultimately have "closure(((+) (- z) ` S) - ((+) (- z) ` T)) = ((+) (- z) ` S)"
  72.487      by (simp add: dense_complement_subspace)
  72.488    then show ?thesis
  72.489      by (metis closure_translation translation_diff translation_invert)
  72.490 @@ -7535,19 +7535,19 @@
  72.491    then obtain z where "z \<in> S" and z: "a \<bullet> z = b"
  72.492      using assms by auto
  72.493    with affine_diffs_subspace [OF \<open>affine S\<close>]
  72.494 -  have sub: "subspace (op + (- z) ` S)" by blast
  72.495 -  then have aff: "affine (op + (- z) ` S)" and span: "span (op + (- z) ` S) = (op + (- z) ` S)"
  72.496 +  have sub: "subspace ((+) (- z) ` S)" by blast
  72.497 +  then have aff: "affine ((+) (- z) ` S)" and span: "span ((+) (- z) ` S) = ((+) (- z) ` S)"
  72.498      by (auto simp: subspace_imp_affine)
  72.499 -  obtain a' a'' where a': "a' \<in> span (op + (- z) ` S)" and a: "a = a' + a''"
  72.500 -                  and "\<And>w. w \<in> span (op + (- z) ` S) \<Longrightarrow> orthogonal a'' w"
  72.501 -      using orthogonal_subspace_decomp_exists [of "op + (- z) ` S" "a"] by metis
  72.502 +  obtain a' a'' where a': "a' \<in> span ((+) (- z) ` S)" and a: "a = a' + a''"
  72.503 +                  and "\<And>w. w \<in> span ((+) (- z) ` S) \<Longrightarrow> orthogonal a'' w"
  72.504 +      using orthogonal_subspace_decomp_exists [of "(+) (- z) ` S" "a"] by metis
  72.505    then have "\<And>w. w \<in> S \<Longrightarrow> a'' \<bullet> (w-z) = 0"
  72.506      by (simp add: imageI orthogonal_def span)
  72.507    then have a'': "\<And>w. w \<in> S \<Longrightarrow> a'' \<bullet> w = (a - a') \<bullet> z"
  72.508      by (simp add: a inner_diff_right)
  72.509    then have ba'': "\<And>w. w \<in> S \<Longrightarrow> a'' \<bullet> w = b - a' \<bullet> z"
  72.510      by (simp add: inner_diff_left z)
  72.511 -  have "\<And>w. w \<in> op + (- z) ` S \<Longrightarrow> (w + a') \<in> op + (- z) ` S"
  72.512 +  have "\<And>w. w \<in> (+) (- z) ` S \<Longrightarrow> (w + a') \<in> (+) (- z) ` S"
  72.513      by (metis subspace_add a' span_eq sub)
  72.514    then have Sclo: "\<And>w. w \<in> S \<Longrightarrow> (w + a') \<in> S"
  72.515      by fastforce
  72.516 @@ -7705,25 +7705,25 @@
  72.517  next
  72.518    case False
  72.519    with assms obtain a where "a \<in> S" "0 \<le> d" by auto
  72.520 -  with assms have ss: "subspace (op + (- a) ` S)"
  72.521 +  with assms have ss: "subspace ((+) (- a) ` S)"
  72.522      by (simp add: affine_diffs_subspace)
  72.523 -  have "nat d \<le> dim (op + (- a) ` S)"
  72.524 +  have "nat d \<le> dim ((+) (- a) ` S)"
  72.525      by (metis aff_dim_subspace aff_dim_translation_eq dle nat_int nat_mono ss)
  72.526 -  then obtain T where "subspace T" and Tsb: "T \<subseteq> span (op + (- a) ` S)"
  72.527 +  then obtain T where "subspace T" and Tsb: "T \<subseteq> span ((+) (- a) ` S)"
  72.528                    and Tdim: "dim T = nat d"
  72.529 -    using choose_subspace_of_subspace [of "nat d" "op + (- a) ` S"] by blast
  72.530 +    using choose_subspace_of_subspace [of "nat d" "(+) (- a) ` S"] by blast
  72.531    then have "affine T"
  72.532      using subspace_affine by blast
  72.533 -  then have "affine (op + a ` T)"
  72.534 +  then have "affine ((+) a ` T)"
  72.535      by (metis affine_hull_eq affine_hull_translation)
  72.536 -  moreover have "op + a ` T \<subseteq> S"
  72.537 +  moreover have "(+) a ` T \<subseteq> S"
  72.538    proof -
  72.539 -    have "T \<subseteq> op + (- a) ` S"
  72.540 +    have "T \<subseteq> (+) (- a) ` S"
  72.541        by (metis (no_types) span_eq Tsb ss)
  72.542 -    then show "op + a ` T \<subseteq> S"
  72.543 +    then show "(+) a ` T \<subseteq> S"
  72.544        using add_ac by auto
  72.545    qed
  72.546 -  moreover have "aff_dim (op + a ` T) = d"
  72.547 +  moreover have "aff_dim ((+) a ` T) = d"
  72.548      by (simp add: aff_dim_subspace Tdim \<open>0 \<le> d\<close> \<open>subspace T\<close> aff_dim_translation_eq)
  72.549    ultimately show ?thesis
  72.550      by (rule that)
  72.551 @@ -8223,4 +8223,4 @@
  72.552  qed
  72.553  
  72.554  end
  72.555 -  
  72.556 \ No newline at end of file
  72.557 +  
    73.1 --- a/src/HOL/Analysis/Tagged_Division.thy	Wed Jan 10 15:21:49 2018 +0100
    73.2 +++ b/src/HOL/Analysis/Tagged_Division.thy	Wed Jan 10 15:25:09 2018 +0100
    73.3 @@ -1838,7 +1838,7 @@
    73.4  proof -
    73.5    let ?f = "(\<lambda>k::(real) set. if k = {} then 0 else f(interval_upperbound k) - f(interval_lowerbound k))"
    73.6    interpret operative_real plus 0 ?f
    73.7 -    rewrites "comm_monoid_set.F op + 0 = sum"
    73.8 +    rewrites "comm_monoid_set.F (+) 0 = sum"
    73.9      by standard[1] (auto simp add: sum_def)
   73.10    have p_td: "p tagged_division_of cbox a b"
   73.11      using assms(2) box_real(2) by presburger
    74.1 --- a/src/HOL/Analysis/Topology_Euclidean_Space.thy	Wed Jan 10 15:21:49 2018 +0100
    74.2 +++ b/src/HOL/Analysis/Topology_Euclidean_Space.thy	Wed Jan 10 15:25:09 2018 +0100
    74.3 @@ -1141,7 +1141,7 @@
    74.4  
    74.5  lemma image_add_ball [simp]:
    74.6    fixes a :: "'a::real_normed_vector"
    74.7 -  shows "op + b ` ball a r = ball (a+b) r"
    74.8 +  shows "(+) b ` ball a r = ball (a+b) r"
    74.9  apply (intro equalityI subsetI)
   74.10  apply (force simp: dist_norm)
   74.11  apply (rule_tac x="x-b" in image_eqI)
   74.12 @@ -1150,7 +1150,7 @@
   74.13  
   74.14  lemma image_add_cball [simp]:
   74.15    fixes a :: "'a::real_normed_vector"
   74.16 -  shows "op + b ` cball a r = cball (a+b) r"
   74.17 +  shows "(+) b ` cball a r = cball (a+b) r"
   74.18  apply (intro equalityI subsetI)
   74.19  apply (force simp: dist_norm)
   74.20  apply (rule_tac x="x-b" in image_eqI)
   74.21 @@ -5565,9 +5565,9 @@
   74.22  
   74.23  lemma open_box[intro]: "open (box a b)"
   74.24  proof -
   74.25 -  have "open (\<Inter>i\<in>Basis. (op \<bullet> i) -` {a \<bullet> i <..< b \<bullet> i})"
   74.26 +  have "open (\<Inter>i\<in>Basis. ((\<bullet>) i) -` {a \<bullet> i <..< b \<bullet> i})"
   74.27      by (auto intro!: continuous_open_vimage continuous_inner continuous_ident continuous_const)
   74.28 -  also have "(\<Inter>i\<in>Basis. (op \<bullet> i) -` {a \<bullet> i <..< b \<bullet> i}) = box a b"
   74.29 +  also have "(\<Inter>i\<in>Basis. ((\<bullet>) i) -` {a \<bullet> i <..< b \<bullet> i}) = box a b"
   74.30      by (auto simp: box_def inner_commute)
   74.31    finally show ?thesis .
   74.32  qed
    75.1 --- a/src/HOL/Analysis/ex/Approximations.thy	Wed Jan 10 15:21:49 2018 +0100
    75.2 +++ b/src/HOL/Analysis/ex/Approximations.thy	Wed Jan 10 15:25:09 2018 +0100
    75.3 @@ -567,7 +567,7 @@
    75.4      where "b = 8428294561696506782041394632 / 503593538783547230635598424135"
    75.5    \<comment> \<open>The introduction of this constant prevents the simplifier from applying solvers that
    75.6        we don't want. We want it to simply evaluate the terms to rational constants.}\<close>
    75.7 -  define eq :: "real \<Rightarrow> real \<Rightarrow> bool" where "eq = op ="
    75.8 +  define eq :: "real \<Rightarrow> real \<Rightarrow> bool" where "eq = (=)"
    75.9  
   75.10    \<comment> \<open>Splitting the computation into several steps has the advantage that simplification can
   75.11        be done in parallel\<close>
   75.12 @@ -679,7 +679,7 @@
   75.13             337877029279505250241149903214554249587517250716358486542628059"
   75.14    let ?pi'' = "3882327391761098513316067116522233897127356523627918964967729040413954225768920394233198626889767468122598417405434625348404038165437924058179155035564590497837027530349 /
   75.15                 1235783190199688165469648572769847552336447197542738425378629633275352407743112409829873464564018488572820294102599160968781449606552922108667790799771278860366957772800"
   75.16 -  define eq :: "real \<Rightarrow> real \<Rightarrow> bool" where "eq = op ="
   75.17 +  define eq :: "real \<Rightarrow> real \<Rightarrow> bool" where "eq = (=)"
   75.18  
   75.19    have "abs (pi - pi_approx2 4) \<le> inverse (2^183)" by (rule pi_approx2') simp_all
   75.20    also have "pi_approx2 4 = 48 * arctan_approx 24 (1 / 18) +
    76.1 --- a/src/HOL/Analysis/normarith.ML	Wed Jan 10 15:21:49 2018 +0100
    76.2 +++ b/src/HOL/Analysis/normarith.ML	Wed Jan 10 15:25:09 2018 +0100
    76.3 @@ -24,9 +24,9 @@
    76.4       Const(@{const_name norm}, _) $ _ => insert (eq_pair bool_eq (op aconvc)) (b,Thm.dest_arg t) acc
    76.5     | _ => acc
    76.6   fun find_normedterms t acc = case Thm.term_of t of
    76.7 -    @{term "op + :: real => _"}$_$_ =>
    76.8 +    @{term "(+) :: real => _"}$_$_ =>
    76.9              find_normedterms (Thm.dest_arg1 t) (find_normedterms (Thm.dest_arg t) acc)
   76.10 -      | @{term "op * :: real => _"}$_$_ =>
   76.11 +      | @{term "( * ) :: real => _"}$_$_ =>
   76.12              if not (is_ratconst (Thm.dest_arg1 t)) then acc else
   76.13              augment_norm (dest_ratconst (Thm.dest_arg1 t) >= @0)
   76.14                        (Thm.dest_arg t) acc
   76.15 @@ -82,8 +82,8 @@
   76.16     | SOME _ => fns) ts []
   76.17  
   76.18  fun replacenegnorms cv t = case Thm.term_of t of
   76.19 -  @{term "op + :: real => _"}$_$_ => binop_conv (replacenegnorms cv) t
   76.20 -| @{term "op * :: real => _"}$_$_ =>
   76.21 +  @{term "(+) :: real => _"}$_$_ => binop_conv (replacenegnorms cv) t
   76.22 +| @{term "( * ) :: real => _"}$_$_ =>
   76.23      if dest_ratconst (Thm.dest_arg1 t) < @0 then arg_conv cv t else Thm.reflexive t
   76.24  | _ => Thm.reflexive t
   76.25  (*
   76.26 @@ -149,7 +149,7 @@
   76.27  let val (a, b) = Rat.dest x
   76.28  in
   76.29   if b = 1 then Numeral.mk_cnumber @{ctyp "real"} a
   76.30 -  else Thm.apply (Thm.apply @{cterm "op / :: real => _"}
   76.31 +  else Thm.apply (Thm.apply @{cterm "(/) :: real => _"}
   76.32                     (Numeral.mk_cnumber @{ctyp "real"} a))
   76.33          (Numeral.mk_cnumber @{ctyp "real"} b)
   76.34  end;
    77.1 --- a/src/HOL/BNF_Composition.thy	Wed Jan 10 15:21:49 2018 +0100
    77.2 +++ b/src/HOL/BNF_Composition.thy	Wed Jan 10 15:25:09 2018 +0100
    77.3 @@ -154,7 +154,7 @@
    77.4  bnf DEADID: 'a
    77.5    map: "id :: 'a \<Rightarrow> 'a"
    77.6    bd: natLeq
    77.7 -  rel: "op = :: 'a \<Rightarrow> 'a \<Rightarrow> bool"
    77.8 +  rel: "(=) :: 'a \<Rightarrow> 'a \<Rightarrow> bool"
    77.9    by (auto simp add: natLeq_card_order natLeq_cinfinite)
   77.10  
   77.11  definition id_bnf :: "'a \<Rightarrow> 'a" where
    78.1 --- a/src/HOL/BNF_Def.thy	Wed Jan 10 15:21:49 2018 +0100
    78.2 +++ b/src/HOL/BNF_Def.thy	Wed Jan 10 15:25:09 2018 +0100
    78.3 @@ -57,7 +57,7 @@
    78.4    using assms unfolding rel_set_def by simp
    78.5  
    78.6  lemma predicate2_transferD:
    78.7 -   "\<lbrakk>rel_fun R1 (rel_fun R2 (op =)) P Q; a \<in> A; b \<in> B; A \<subseteq> {(x, y). R1 x y}; B \<subseteq> {(x, y). R2 x y}\<rbrakk> \<Longrightarrow>
    78.8 +   "\<lbrakk>rel_fun R1 (rel_fun R2 (=)) P Q; a \<in> A; b \<in> B; A \<subseteq> {(x, y). R1 x y}; B \<subseteq> {(x, y). R2 x y}\<rbrakk> \<Longrightarrow>
    78.9     P (fst a) (fst b) \<longleftrightarrow> Q (snd a) (snd b)"
   78.10    unfolding rel_fun_def by (blast dest!: Collect_case_prodD)
   78.11  
   78.12 @@ -102,13 +102,13 @@
   78.13  definition csquare where
   78.14    "csquare A f1 f2 p1 p2 \<longleftrightarrow> (\<forall> a \<in> A. f1 (p1 a) = f2 (p2 a))"
   78.15  
   78.16 -lemma eq_alt: "op = = Grp UNIV id"
   78.17 +lemma eq_alt: "(=) = Grp UNIV id"
   78.18    unfolding Grp_def by auto
   78.19  
   78.20 -lemma leq_conversepI: "R = op = \<Longrightarrow> R \<le> R^--1"
   78.21 +lemma leq_conversepI: "R = (=) \<Longrightarrow> R \<le> R^--1"
   78.22    by auto
   78.23  
   78.24 -lemma leq_OOI: "R = op = \<Longrightarrow> R \<le> R OO R"
   78.25 +lemma leq_OOI: "R = (=) \<Longrightarrow> R \<le> R OO R"
   78.26    by auto
   78.27  
   78.28  lemma OO_Grp_alt: "(Grp A f)^--1 OO Grp A g = (\<lambda>x y. \<exists>z. z \<in> A \<and> f z = x \<and> g z = y)"
   78.29 @@ -217,13 +217,13 @@
   78.30  lemma comp_apply_eq: "f (g x) = h (k x) \<Longrightarrow> (f \<circ> g) x = (h \<circ> k) x"
   78.31    unfolding comp_apply by assumption
   78.32  
   78.33 -lemma refl_ge_eq: "(\<And>x. R x x) \<Longrightarrow> op = \<le> R"
   78.34 +lemma refl_ge_eq: "(\<And>x. R x x) \<Longrightarrow> (=) \<le> R"
   78.35    by auto
   78.36  
   78.37 -lemma ge_eq_refl: "op = \<le> R \<Longrightarrow> R x x"
   78.38 +lemma ge_eq_refl: "(=) \<le> R \<Longrightarrow> R x x"
   78.39    by auto
   78.40  
   78.41 -lemma reflp_eq: "reflp R = (op = \<le> R)"
   78.42 +lemma reflp_eq: "reflp R = ((=) \<le> R)"
   78.43    by (auto simp: reflp_def fun_eq_iff)
   78.44  
   78.45  lemma transp_relcompp: "transp r \<longleftrightarrow> r OO r \<le> r"
   78.46 @@ -244,7 +244,7 @@
   78.47  lemma eq_onp_to_eq: "eq_onp P x y \<Longrightarrow> x = y"
   78.48    by (simp add: eq_onp_def)
   78.49  
   78.50 -lemma eq_onp_top_eq_eq: "eq_onp top = op ="
   78.51 +lemma eq_onp_top_eq_eq: "eq_onp top = (=)"
   78.52    by (simp add: eq_onp_def)
   78.53  
   78.54  lemma eq_onp_same_args: "eq_onp P x x = P x"
   78.55 @@ -259,7 +259,7 @@
   78.56  lemma eq_onp_mono0: "\<forall>x\<in>A. P x \<longrightarrow> Q x \<Longrightarrow> \<forall>x\<in>A. \<forall>y\<in>A. eq_onp P x y \<longrightarrow> eq_onp Q x y"
   78.57    unfolding eq_onp_def by auto
   78.58  
   78.59 -lemma eq_onp_True: "eq_onp (\<lambda>_. True) = (op =)"
   78.60 +lemma eq_onp_True: "eq_onp (\<lambda>_. True) = (=)"
   78.61    unfolding eq_onp_def by simp
   78.62  
   78.63  lemma Ball_image_comp: "Ball (f ` A) g = Ball A (g \<circ> f)"
    79.1 --- a/src/HOL/BNF_Fixpoint_Base.thy	Wed Jan 10 15:21:49 2018 +0100
    79.2 +++ b/src/HOL/BNF_Fixpoint_Base.thy	Wed Jan 10 15:25:09 2018 +0100
    79.3 @@ -131,10 +131,10 @@
    79.4  lemma subst_eq_imp: "(\<forall>a b. a = b \<longrightarrow> P a b) \<equiv> (\<forall>a. P a a)"
    79.5    by auto
    79.6  
    79.7 -lemma eq_subset: "op = \<le> (\<lambda>a b. P a b \<or> a = b)"
    79.8 +lemma eq_subset: "(=) \<le> (\<lambda>a b. P a b \<or> a = b)"
    79.9    by auto
   79.10  
   79.11 -lemma eq_le_Grp_id_iff: "(op = \<le> Grp (Collect R) id) = (All R)"
   79.12 +lemma eq_le_Grp_id_iff: "((=) \<le> Grp (Collect R) id) = (All R)"
   79.13    unfolding Grp_def id_apply by blast
   79.14  
   79.15  lemma Grp_id_mono_subst: "(\<And>x y. Grp P id x y \<Longrightarrow> Grp Q id (f x) (f y)) \<equiv>
   79.16 @@ -211,10 +211,10 @@
   79.17    by simp
   79.18  
   79.19  lemma comp_transfer:
   79.20 -  "rel_fun (rel_fun B C) (rel_fun (rel_fun A B) (rel_fun A C)) (op \<circ>) (op \<circ>)"
   79.21 +  "rel_fun (rel_fun B C) (rel_fun (rel_fun A B) (rel_fun A C)) (\<circ>) (\<circ>)"
   79.22    unfolding rel_fun_def by simp
   79.23  
   79.24 -lemma If_transfer: "rel_fun (op =) (rel_fun A (rel_fun A A)) If If"
   79.25 +lemma If_transfer: "rel_fun (=) (rel_fun A (rel_fun A A)) If If"
   79.26    unfolding rel_fun_def by simp
   79.27  
   79.28  lemma Abs_transfer:
    80.1 --- a/src/HOL/Bali/AxCompl.thy	Wed Jan 10 15:21:49 2018 +0100
    80.2 +++ b/src/HOL/Bali/AxCompl.thy	Wed Jan 10 15:25:09 2018 +0100
    80.3 @@ -119,7 +119,7 @@
    80.4    remember_init_state :: "state assn" ("\<doteq>")
    80.5    where "\<doteq> \<equiv> \<lambda>Y s Z. s = Z"
    80.6  
    80.7 -lemma remember_init_state_def2 [simp]: "\<doteq> Y = op ="
    80.8 +lemma remember_init_state_def2 [simp]: "\<doteq> Y = (=)"
    80.9  apply (unfold remember_init_state_def)
   80.10  apply (simp (no_asm))
   80.11  done
    81.1 --- a/src/HOL/Bali/AxExample.thy	Wed Jan 10 15:21:49 2018 +0100
    81.2 +++ b/src/HOL/Bali/AxExample.thy	Wed Jan 10 15:25:09 2018 +0100
    81.3 @@ -42,7 +42,7 @@
    81.4  
    81.5  ML \<open>
    81.6  fun inst1_tac ctxt s t xs st =
    81.7 -  (case AList.lookup (op =) (rev (Term.add_var_names (Thm.prop_of st) [])) s of
    81.8 +  (case AList.lookup (=) (rev (Term.add_var_names (Thm.prop_of st) [])) s of
    81.9      SOME i => PRIMITIVE (Rule_Insts.read_instantiate ctxt [(((s, i), Position.none), t)] xs) st
   81.10    | NONE => Seq.empty);
   81.11  
    82.1 --- a/src/HOL/Bali/AxSem.thy	Wed Jan 10 15:21:49 2018 +0100
    82.2 +++ b/src/HOL/Bali/AxSem.thy	Wed Jan 10 15:25:09 2018 +0100
    82.3 @@ -799,7 +799,7 @@
    82.4  done
    82.5  
    82.6  (* unused *)
    82.7 -lemma ax_nochange_lemma: "\<lbrakk>P Y s; All (op = w)\<rbrakk> \<Longrightarrow> P w s"
    82.8 +lemma ax_nochange_lemma: "\<lbrakk>P Y s; All ((=) w)\<rbrakk> \<Longrightarrow> P w s"
    82.9  apply auto
   82.10  done
   82.11  
    83.1 --- a/src/HOL/Bali/State.thy	Wed Jan 10 15:21:49 2018 +0100
    83.2 +++ b/src/HOL/Bali/State.thy	Wed Jan 10 15:25:09 2018 +0100
    83.3 @@ -604,7 +604,7 @@
    83.4  apply clarify
    83.5  done
    83.6  
    83.7 -lemma state_not_single: "All (op = (x::state)) \<Longrightarrow> R"
    83.8 +lemma state_not_single: "All ((=) (x::state)) \<Longrightarrow> R"
    83.9  apply (drule_tac x = "(if abrupt x = None then Some x' else None, y)" for x' y in spec)
   83.10  apply clarsimp
   83.11  done
    84.1 --- a/src/HOL/Basic_BNF_LFPs.thy	Wed Jan 10 15:21:49 2018 +0100
    84.2 +++ b/src/HOL/Basic_BNF_LFPs.thy	Wed Jan 10 15:25:09 2018 +0100
    84.3 @@ -30,7 +30,7 @@
    84.4  lemma xtor_xtor: "xtor (xtor x) = x"
    84.5    unfolding xtor_def by (rule refl)
    84.6  
    84.7 -lemmas xtor_inject = xtor_rel[of "op ="]
    84.8 +lemmas xtor_inject = xtor_rel[of "(=)"]
    84.9  
   84.10  lemma xtor_rel_induct: "(\<And>x y. vimage2p id_bnf id_bnf R x y \<Longrightarrow> IR (xtor x) (xtor y)) \<Longrightarrow> R \<le> IR"
   84.11    unfolding xtor_def vimage2p_def id_bnf_def ..
   84.12 @@ -96,7 +96,7 @@
   84.13    (\<not> isl a \<longrightarrow> \<not> isl b \<longrightarrow> R2 (projr a) (projr b)))"
   84.14    by (cases a b rule: sum.exhaust[case_product sum.exhaust]) simp_all
   84.15  
   84.16 -lemma isl_transfer: "rel_fun (rel_sum A B) (op =) isl isl"
   84.17 +lemma isl_transfer: "rel_fun (rel_sum A B) (=) isl isl"
   84.18    unfolding rel_fun_def rel_sum_sel by simp
   84.19  
   84.20  lemma rel_prod_sel: "rel_prod R1 R2 p q = (R1 (fst p) (fst q) \<and> R2 (snd p) (snd q))"
    85.1 --- a/src/HOL/Basic_BNFs.thy	Wed Jan 10 15:21:49 2018 +0100
    85.2 +++ b/src/HOL/Basic_BNFs.thy	Wed Jan 10 15:25:09 2018 +0100
    85.3 @@ -188,22 +188,22 @@
    85.4  qed auto
    85.5  
    85.6  bnf "'a \<Rightarrow> 'b"
    85.7 -  map: "op \<circ>"
    85.8 +  map: "(\<circ>)"
    85.9    sets: range
   85.10    bd: "natLeq +c |UNIV :: 'a set|"
   85.11 -  rel: "rel_fun op ="
   85.12 +  rel: "rel_fun (=)"
   85.13    pred: "pred_fun (\<lambda>_. True)"
   85.14  proof
   85.15    fix f show "id \<circ> f = id f" by simp
   85.16  next
   85.17 -  fix f g show "op \<circ> (g \<circ> f) = op \<circ> g \<circ> op \<circ> f"
   85.18 +  fix f g show "(\<circ>) (g \<circ> f) = (\<circ>) g \<circ> (\<circ>) f"
   85.19    unfolding comp_def[abs_def] ..
   85.20  next
   85.21    fix x f g
   85.22    assume "\<And>z. z \<in> range x \<Longrightarrow> f z = g z"
   85.23    thus "f \<circ> x = g \<circ> x" by auto
   85.24  next
   85.25 -  fix f show "range \<circ> op \<circ> f = op ` f \<circ> range"
   85.26 +  fix f show "range \<circ> (\<circ>) f = (`) f \<circ> range"
   85.27      by (auto simp add: fun_eq_iff)
   85.28  next
   85.29    show "card_order (natLeq +c |UNIV| )" (is "_ (_ +c ?U)")
   85.30 @@ -222,10 +222,10 @@
   85.31    finally show "|range f| \<le>o natLeq +c ?U" .
   85.32  next
   85.33    fix R S
   85.34 -  show "rel_fun op = R OO rel_fun op = S \<le> rel_fun op = (R OO S)" by (auto simp: rel_fun_def)
   85.35 +  show "rel_fun (=) R OO rel_fun (=) S \<le> rel_fun (=) (R OO S)" by (auto simp: rel_fun_def)
   85.36  next
   85.37    fix R
   85.38 -  show "rel_fun op = R = (\<lambda>x y.
   85.39 +  show "rel_fun (=) R = (\<lambda>x y.
   85.40      \<exists>z. range z \<subseteq> {(x, y). R x y} \<and> fst \<circ> z = x \<and> snd \<circ> z = y)"
   85.41    unfolding rel_fun_def subset_iff by (force simp: fun_eq_iff[symmetric])
   85.42  qed (auto simp: pred_fun_def)
    86.1 --- a/src/HOL/Binomial.thy	Wed Jan 10 15:21:49 2018 +0100
    86.2 +++ b/src/HOL/Binomial.thy	Wed Jan 10 15:25:09 2018 +0100
    86.3 @@ -377,15 +377,15 @@
    86.4    case False
    86.5    then have "n < k"
    86.6      by (simp add: not_le)
    86.7 -  then have "0 \<in> (op - n) ` {0..<k}"
    86.8 +  then have "0 \<in> ((-) n) ` {0..<k}"
    86.9      by auto
   86.10 -  then have "prod (op - n) {0..<k} = 0"
   86.11 +  then have "prod ((-) n) {0..<k} = 0"
   86.12      by (auto intro: prod_zero)
   86.13    with \<open>n < k\<close> show ?thesis
   86.14      by (simp add: binomial_eq_0 gbinomial_prod_rev prod_zero)
   86.15  next
   86.16    case True
   86.17 -  from True have *: "prod (op - n) {0..<k} = \<Prod>{Suc (n - k)..n}"
   86.18 +  from True have *: "prod ((-) n) {0..<k} = \<Prod>{Suc (n - k)..n}"
   86.19      by (intro prod.reindex_bij_witness[of _ "\<lambda>i. n - i" "\<lambda>i. n - i"]) auto
   86.20    from True have "n choose k = fact n div (fact k * fact (n - k))"
   86.21      by (rule binomial_fact')
   86.22 @@ -1252,15 +1252,15 @@
   86.23  using assms
   86.24  proof (induction xs ys rule: shuffle.induct)
   86.25    case (3 x xs y ys)
   86.26 -  have "shuffle (x # xs) (y # ys) = op # x ` shuffle xs (y # ys) \<union> op # y ` shuffle (x # xs) ys"
   86.27 +  have "shuffle (x # xs) (y # ys) = (#) x ` shuffle xs (y # ys) \<union> (#) y ` shuffle (x # xs) ys"
   86.28      by (rule shuffle.simps)
   86.29 -  also have "card \<dots> = card (op # x ` shuffle xs (y # ys)) + card (op # y ` shuffle (x # xs) ys)"
   86.30 +  also have "card \<dots> = card ((#) x ` shuffle xs (y # ys)) + card ((#) y ` shuffle (x # xs) ys)"
   86.31      by (rule card_Un_disjoint) (insert "3.prems", auto)
   86.32 -  also have "card (op # x ` shuffle xs (y # ys)) = card (shuffle xs (y # ys))"
   86.33 +  also have "card ((#) x ` shuffle xs (y # ys)) = card (shuffle xs (y # ys))"
   86.34      by (rule card_image) auto
   86.35    also have "\<dots> = (length xs + length (y # ys)) choose length xs"
   86.36      using "3.prems" by (intro "3.IH") auto
   86.37 -  also have "card (op # y ` shuffle (x # xs) ys) = card (shuffle (x # xs) ys)"
   86.38 +  also have "card ((#) y ` shuffle (x # xs) ys) = card (shuffle (x # xs) ys)"
   86.39      by (rule card_image) auto
   86.40    also have "\<dots> = (length (x # xs) + length ys) choose length (x # xs)"
   86.41      using "3.prems" by (intro "3.IH") auto
   86.42 @@ -1304,7 +1304,7 @@
   86.43    "(n choose k) =
   86.44        (if k > n then 0
   86.45         else if 2 * k > n then (n choose (n - k))
   86.46 -       else (fold_atLeastAtMost_nat (op * ) (n-k+1) n 1 div fact k))"
   86.47 +       else (fold_atLeastAtMost_nat (( * ) ) (n-k+1) n 1 div fact k))"
   86.48  proof -
   86.49    {
   86.50      assume "k \<le> n"
    87.1 --- a/src/HOL/Cardinals/Bounded_Set.thy	Wed Jan 10 15:21:49 2018 +0100
    87.2 +++ b/src/HOL/Cardinals/Bounded_Set.thy	Wed Jan 10 15:25:09 2018 +0100
    87.3 @@ -88,7 +88,7 @@
    87.4    then show "map_bset f X = map_bset g X" by transfer force
    87.5  next
    87.6    fix f
    87.7 -  show "set_bset \<circ> map_bset f = op ` f \<circ> set_bset" by (rule ext, transfer) auto
    87.8 +  show "set_bset \<circ> map_bset f = (`) f \<circ> set_bset" by (rule ext, transfer) auto
    87.9  next
   87.10    fix X :: "'a set['k]"
   87.11    show "|set_bset X| \<le>o natLeq +c |UNIV :: 'k set|"
   87.12 @@ -169,7 +169,7 @@
   87.13    by transfer (auto simp: rel_set_def split: option.splits)
   87.14  
   87.15  lemma rel_bgraph[simp]:
   87.16 -  "rel_bset (rel_prod (op =) R) (bgraph f1) (bgraph f2) = rel_fun (op =) (rel_option R) f1 f2"
   87.17 +  "rel_bset (rel_prod (=) R) (bgraph f1) (bgraph f2) = rel_fun (=) (rel_option R) f1 f2"
   87.18    apply transfer
   87.19    apply (auto simp: rel_fun_def rel_option_iff rel_set_def split: option.splits)
   87.20    using option.collapse apply fastforce+
   87.21 @@ -195,7 +195,7 @@
   87.22    apply (rule ordLeq_csum2[OF card_of_Card_order])
   87.23    done
   87.24  
   87.25 -lift_definition bmember :: "'a \<Rightarrow> 'a set['k] \<Rightarrow> bool" is "op \<in>" .
   87.26 +lift_definition bmember :: "'a \<Rightarrow> 'a set['k] \<Rightarrow> bool" is "(\<in>)" .
   87.27  
   87.28  lemma bmember_bCollect[simp]: "bmember a (bCollect P) = P a"
   87.29    by transfer simp
    88.1 --- a/src/HOL/Code_Numeral.thy	Wed Jan 10 15:21:49 2018 +0100
    88.2 +++ b/src/HOL/Code_Numeral.thy	Wed Jan 10 15:25:09 2018 +0100
    88.3 @@ -634,7 +634,7 @@
    88.4  setup \<open>
    88.5    fold (fn target =>
    88.6      Numeral.add_code @{const_name Code_Numeral.Pos} I Code_Printer.literal_numeral target
    88.7 -    #> Numeral.add_code @{const_name Code_Numeral.Neg} (op ~) Code_Printer.literal_numeral target)
    88.8 +    #> Numeral.add_code @{const_name Code_Numeral.Neg} (~) Code_Printer.literal_numeral target)
    88.9      ["SML", "OCaml", "Haskell", "Scala"]
   88.10  \<close>
   88.11  
    89.1 --- a/src/HOL/Complete_Lattices.thy	Wed Jan 10 15:21:49 2018 +0100
    89.2 +++ b/src/HOL/Complete_Lattices.thy	Wed Jan 10 15:25:09 2018 +0100
    89.3 @@ -128,7 +128,7 @@
    89.4      by (auto intro: Inf_greatest simp only: Inf_empty [symmetric])
    89.5  qed
    89.6  
    89.7 -lemma dual_complete_lattice: "class.complete_lattice Sup Inf sup (op \<ge>) (op >) inf \<top> \<bottom>"
    89.8 +lemma dual_complete_lattice: "class.complete_lattice Sup Inf sup (\<ge>) (>) inf \<top> \<bottom>"
    89.9    by (auto intro!: class.complete_lattice.intro dual_lattice)
   89.10      (unfold_locales, (fact Inf_empty Sup_empty Sup_upper Sup_least Inf_lower Inf_greatest)+)
   89.11  
   89.12 @@ -465,7 +465,7 @@
   89.13    by (simp add: inf_Sup)
   89.14  
   89.15  lemma dual_complete_distrib_lattice:
   89.16 -  "class.complete_distrib_lattice Sup Inf sup (op \<ge>) (op >) inf \<top> \<bottom>"
   89.17 +  "class.complete_distrib_lattice Sup Inf sup (\<ge>) (>) inf \<top> \<bottom>"
   89.18    apply (rule class.complete_distrib_lattice.intro)
   89.19     apply (fact dual_complete_lattice)
   89.20    apply (rule class.complete_distrib_lattice_axioms.intro)
   89.21 @@ -528,7 +528,7 @@
   89.22  begin
   89.23  
   89.24  lemma dual_complete_boolean_algebra:
   89.25 -  "class.complete_boolean_algebra Sup Inf sup (op \<ge>) (op >) inf \<top> \<bottom> (\<lambda>x y. x \<squnion> - y) uminus"
   89.26 +  "class.complete_boolean_algebra Sup Inf sup (\<ge>) (>) inf \<top> \<bottom> (\<lambda>x y. x \<squnion> - y) uminus"
   89.27    by (rule class.complete_boolean_algebra.intro,
   89.28        rule dual_complete_distrib_lattice,
   89.29        rule dual_boolean_algebra)
   89.30 @@ -560,7 +560,7 @@
   89.31  begin
   89.32  
   89.33  lemma dual_complete_linorder:
   89.34 -  "class.complete_linorder Sup Inf sup (op \<ge>) (op >) inf \<top> \<bottom>"
   89.35 +  "class.complete_linorder Sup Inf sup (\<ge>) (>) inf \<top> \<bottom>"
   89.36    by (rule class.complete_linorder.intro, rule dual_complete_lattice, rule dual_linorder)
   89.37  
   89.38  lemma complete_linorder_inf_min: "inf = min"
   89.39 @@ -1205,7 +1205,7 @@
   89.40  lemma UN_singleton [simp]: "(\<Union>x\<in>A. {x}) = A"
   89.41    by blast
   89.42  
   89.43 -lemma inj_on_image: "inj_on f (\<Union>A) \<Longrightarrow> inj_on (op ` f) A"
   89.44 +lemma inj_on_image: "inj_on f (\<Union>A) \<Longrightarrow> inj_on ((`) f) A"
   89.45    unfolding inj_on_def by blast
   89.46  
   89.47  
    90.1 --- a/src/HOL/Complete_Partial_Order.thy	Wed Jan 10 15:21:49 2018 +0100
    90.2 +++ b/src/HOL/Complete_Partial_Order.thy	Wed Jan 10 15:25:09 2018 +0100
    90.3 @@ -51,7 +51,7 @@
    90.4  lemma chain_empty: "chain ord {}"
    90.5    by (simp add: chain_def)
    90.6  
    90.7 -lemma chain_equality: "chain op = A \<longleftrightarrow> (\<forall>x\<in>A. \<forall>y\<in>A. x = y)"
    90.8 +lemma chain_equality: "chain (=) A \<longleftrightarrow> (\<forall>x\<in>A. \<forall>y\<in>A. x = y)"
    90.9    by (auto simp add: chain_def)
   90.10  
   90.11  lemma chain_subset: "chain ord A \<Longrightarrow> B \<subseteq> A \<Longrightarrow> chain ord B"
   90.12 @@ -72,11 +72,11 @@
   90.13  \<close>
   90.14  
   90.15  class ccpo = order + Sup +
   90.16 -  assumes ccpo_Sup_upper: "chain (op \<le>) A \<Longrightarrow> x \<in> A \<Longrightarrow> x \<le> Sup A"
   90.17 -  assumes ccpo_Sup_least: "chain (op \<le>) A \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> x \<le> z) \<Longrightarrow> Sup A \<le> z"
   90.18 +  assumes ccpo_Sup_upper: "chain (\<le>) A \<Longrightarrow> x \<in> A \<Longrightarrow> x \<le> Sup A"
   90.19 +  assumes ccpo_Sup_least: "chain (\<le>) A \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> x \<le> z) \<Longrightarrow> Sup A \<le> z"
   90.20  begin
   90.21  
   90.22 -lemma chain_singleton: "Complete_Partial_Order.chain op \<le> {x}"
   90.23 +lemma chain_singleton: "Complete_Partial_Order.chain (\<le>) {x}"
   90.24    by (rule chainI) simp
   90.25  
   90.26  lemma ccpo_Sup_singleton [simp]: "\<Squnion>{x} = x"
   90.27 @@ -92,17 +92,17 @@
   90.28    for f :: "'a \<Rightarrow> 'a"
   90.29    where
   90.30      step: "x \<in> iterates f \<Longrightarrow> f x \<in> iterates f"
   90.31 -  | Sup: "chain (op \<le>) M \<Longrightarrow> \<forall>x\<in>M. x \<in> iterates f \<Longrightarrow> Sup M \<in> iterates f"
   90.32 +  | Sup: "chain (\<le>) M \<Longrightarrow> \<forall>x\<in>M. x \<in> iterates f \<Longrightarrow> Sup M \<in> iterates f"
   90.33  
   90.34  end
   90.35  
   90.36 -lemma iterates_le_f: "x \<in> iterates f \<Longrightarrow> monotone (op \<le>) (op \<le>) f \<Longrightarrow> x \<le> f x"
   90.37 +lemma iterates_le_f: "x \<in> iterates f \<Longrightarrow> monotone (\<le>) (\<le>) f \<Longrightarrow> x \<le> f x"
   90.38    by (induct x rule: iterates.induct)
   90.39      (force dest: monotoneD intro!: ccpo_Sup_upper ccpo_Sup_least)+
   90.40  
   90.41  lemma chain_iterates:
   90.42 -  assumes f: "monotone (op \<le>) (op \<le>) f"
   90.43 -  shows "chain (op \<le>) (iterates f)" (is "chain _ ?C")
   90.44 +  assumes f: "monotone (\<le>) (\<le>) f"
   90.45 +  shows "chain (\<le>) (iterates f)" (is "chain _ ?C")
   90.46  proof (rule chainI)
   90.47    fix x y
   90.48    assume "x \<in> ?C" "y \<in> ?C"
   90.49 @@ -117,7 +117,7 @@
   90.50        with IH f show ?case by (auto dest: monotoneD)
   90.51      next
   90.52        case (Sup M)
   90.53 -      then have chM: "chain (op \<le>) M"
   90.54 +      then have chM: "chain (\<le>) M"
   90.55          and IH': "\<And>z. z \<in> M \<Longrightarrow> f x \<le> z \<or> z \<le> f x" by auto
   90.56        show "f x \<le> Sup M \<or> Sup M \<le> f x"
   90.57        proof (cases "\<exists>z\<in>M. f x \<le> z")
   90.58 @@ -164,13 +164,13 @@
   90.59    where "fixp f = Sup (iterates f)"
   90.60  
   90.61  lemma iterates_fixp:
   90.62 -  assumes f: "monotone (op \<le>) (op \<le>) f"
   90.63 +  assumes f: "monotone (\<le>) (\<le>) f"
   90.64    shows "fixp f \<in> iterates f"
   90.65    unfolding fixp_def
   90.66    by (simp add: iterates.Sup chain_iterates f)
   90.67  
   90.68  lemma fixp_unfold:
   90.69 -  assumes f: "monotone (op \<le>) (op \<le>) f"
   90.70 +  assumes f: "monotone (\<le>) (\<le>) f"
   90.71    shows "fixp f = f (fixp f)"
   90.72  proof (rule antisym)
   90.73    show "fixp f \<le> f (fixp f)"
   90.74 @@ -182,7 +182,7 @@
   90.75  qed
   90.76  
   90.77  lemma fixp_lowerbound:
   90.78 -  assumes f: "monotone (op \<le>) (op \<le>) f"
   90.79 +  assumes f: "monotone (\<le>) (\<le>) f"
   90.80      and z: "f z \<le> z"
   90.81    shows "fixp f \<le> z"
   90.82    unfolding fixp_def
   90.83 @@ -228,8 +228,8 @@
   90.84  setup \<open>Sign.map_naming Name_Space.parent_path\<close>
   90.85  
   90.86  lemma (in ccpo) fixp_induct:
   90.87 -  assumes adm: "ccpo.admissible Sup (op \<le>) P"
   90.88 -  assumes mono: "monotone (op \<le>) (op \<le>) f"
   90.89 +  assumes adm: "ccpo.admissible Sup (\<le>) P"
   90.90 +  assumes mono: "monotone (\<le>) (\<le>) f"
   90.91    assumes bot: "P (Sup {})"
   90.92    assumes step: "\<And>x. P x \<Longrightarrow> P (f x)"
   90.93    shows "P (fixp f)"
   90.94 @@ -284,12 +284,12 @@
   90.95  
   90.96  lemma admissible_disj:
   90.97    fixes P Q :: "'a \<Rightarrow> bool"
   90.98 -  assumes P: "ccpo.admissible Sup (op \<le>) (\<lambda>x. P x)"
   90.99 -  assumes Q: "ccpo.admissible Sup (op \<le>) (\<lambda>x. Q x)"
  90.100 -  shows "ccpo.admissible Sup (op \<le>) (\<lambda>x. P x \<or> Q x)"
  90.101 +  assumes P: "ccpo.admissible Sup (\<le>) (\<lambda>x. P x)"
  90.102 +  assumes Q: "ccpo.admissible Sup (\<le>) (\<lambda>x. Q x)"
  90.103 +  shows "ccpo.admissible Sup (\<le>) (\<lambda>x. P x \<or> Q x)"
  90.104  proof (rule ccpo.admissibleI)
  90.105    fix A :: "'a set"
  90.106 -  assume chain: "chain (op \<le>) A"
  90.107 +  assume chain: "chain (\<le>) A"
  90.108    assume A: "A \<noteq> {}" and P_Q: "\<forall>x\<in>A. P x \<or> Q x"
  90.109    have "(\<exists>x\<in>A. P x) \<and> (\<forall>x\<in>A. \<exists>y\<in>A. x \<le> y \<and> P y) \<or> (\<exists>x\<in>A. Q x) \<and> (\<forall>x\<in>A. \<exists>y\<in>A. x \<le> y \<and> Q y)"
  90.110      (is "?P \<or> ?Q" is "?P1 \<and> ?P2 \<or> _")
  90.111 @@ -330,7 +330,7 @@
  90.112    moreover
  90.113    have "Sup A = Sup {x \<in> A. P x}" if "\<forall>x\<in>A. \<exists>y\<in>A. x \<le> y \<and> P y" for P
  90.114    proof (rule antisym)
  90.115 -    have chain_P: "chain (op \<le>) {x \<in> A. P x}"
  90.116 +    have chain_P: "chain (\<le>) {x \<in> A. P x}"
  90.117        by (rule chain_compr [OF chain])
  90.118      show "Sup A \<le> Sup {x \<in> A. P x}"
  90.119        apply (rule ccpo_Sup_least [OF chain])
  90.120 @@ -368,7 +368,7 @@
  90.121    assumes mono: "mono f"
  90.122    shows "lfp f = fixp f"
  90.123  proof (rule antisym)
  90.124 -  from mono have f': "monotone (op \<le>) (op \<le>) f"
  90.125 +  from mono have f': "monotone (\<le>) (\<le>) f"
  90.126      unfolding mono_def monotone_def .
  90.127    show "lfp f \<le> fixp f"
  90.128      by (rule lfp_lowerbound, subst fixp_unfold [OF f'], rule order_refl)
    91.1 --- a/src/HOL/Computational_Algebra/Euclidean_Algorithm.thy	Wed Jan 10 15:21:49 2018 +0100
    91.2 +++ b/src/HOL/Computational_Algebra/Euclidean_Algorithm.thy	Wed Jan 10 15:25:09 2018 +0100
    91.3 @@ -422,7 +422,7 @@
    91.4      Euclidean_Algorithm.gcd Euclidean_Algorithm.lcm
    91.5      Euclidean_Algorithm.Gcd Euclidean_Algorithm.Lcm
    91.6      divide plus minus unit_factor normalize
    91.7 -    rewrites "dvd.dvd op * = Rings.dvd"
    91.8 +    rewrites "dvd.dvd ( * ) = Rings.dvd"
    91.9      by (fact semiring_Gcd) (simp add: dvd.dvd_def dvd_def fun_eq_iff)
   91.10    show [simp]: "Euclidean_Algorithm.gcd = (gcd :: 'a \<Rightarrow> _)"
   91.11    proof (rule ext)+
   91.12 @@ -576,7 +576,7 @@
   91.13      "Euclidean_Algorithm.gcd" "Euclidean_Algorithm.lcm"
   91.14      "Euclidean_Algorithm.Gcd" "Euclidean_Algorithm.Lcm"
   91.15      divide plus minus unit_factor normalize
   91.16 -    rewrites "dvd.dvd op * = Rings.dvd"
   91.17 +    rewrites "dvd.dvd ( * ) = Rings.dvd"
   91.18      by (fact semiring_Gcd) (simp add: dvd.dvd_def dvd_def fun_eq_iff)
   91.19    show [simp]: "(Euclidean_Algorithm.gcd :: nat \<Rightarrow> _) = gcd"
   91.20    proof (rule ext)+
   91.21 @@ -613,7 +613,7 @@
   91.22      "Euclidean_Algorithm.gcd" "Euclidean_Algorithm.lcm"
   91.23      "Euclidean_Algorithm.Gcd" "Euclidean_Algorithm.Lcm"
   91.24      divide plus minus unit_factor normalize
   91.25 -    rewrites "dvd.dvd op * = Rings.dvd"
   91.26 +    rewrites "dvd.dvd ( * ) = Rings.dvd"
   91.27      by (fact semiring_Gcd) (simp add: dvd.dvd_def dvd_def fun_eq_iff)
   91.28    show [simp]: "(Euclidean_Algorithm.gcd :: int \<Rightarrow> _) = gcd"
   91.29    proof (rule ext)+
    92.1 --- a/src/HOL/Computational_Algebra/Formal_Power_Series.thy	Wed Jan 10 15:21:49 2018 +0100
    92.2 +++ b/src/HOL/Computational_Algebra/Formal_Power_Series.thy	Wed Jan 10 15:25:09 2018 +0100
    92.3 @@ -51,7 +51,7 @@
    92.4  
    92.5  instantiation fps :: (plus) plus
    92.6  begin
    92.7 -  definition fps_plus_def: "op + = (\<lambda>f g. Abs_fps (\<lambda>n. f $ n + g $ n))"
    92.8 +  definition fps_plus_def: "(+) = (\<lambda>f g. Abs_fps (\<lambda>n. f $ n + g $ n))"
    92.9    instance ..
   92.10  end
   92.11  
   92.12 @@ -60,7 +60,7 @@
   92.13  
   92.14  instantiation fps :: (minus) minus
   92.15  begin
   92.16 -  definition fps_minus_def: "op - = (\<lambda>f g. Abs_fps (\<lambda>n. f $ n - g $ n))"
   92.17 +  definition fps_minus_def: "(-) = (\<lambda>f g. Abs_fps (\<lambda>n. f $ n - g $ n))"
   92.18    instance ..
   92.19  end
   92.20  
   92.21 @@ -78,7 +78,7 @@
   92.22  
   92.23  instantiation fps :: ("{comm_monoid_add, times}") times
   92.24  begin
   92.25 -  definition fps_times_def: "op * = (\<lambda>f g. Abs_fps (\<lambda>n. \<Sum>i=0..n. f $ i * g $ (n - i)))"
   92.26 +  definition fps_times_def: "( * ) = (\<lambda>f g. Abs_fps (\<lambda>n. \<Sum>i=0..n. f $ i * g $ (n - i)))"
   92.27    instance ..
   92.28  end
   92.29  
   92.30 @@ -151,7 +151,7 @@
   92.31    fixes n :: nat
   92.32      and f :: "nat \<Rightarrow> nat \<Rightarrow> 'a::comm_monoid_add"
   92.33    shows "(\<Sum>i=0..n. f i (n - i)) = (\<Sum>i=0..n. f (n - i) i)"
   92.34 -  by (rule sum.reindex_bij_witness[where i="op - n" and j="op - n"]) auto
   92.35 +  by (rule sum.reindex_bij_witness[where i="(-) n" and j="(-) n"]) auto
   92.36  
   92.37  instance fps :: (comm_semiring_0) ab_semigroup_mult
   92.38  proof
   92.39 @@ -1563,10 +1563,10 @@
   92.40          of_nat i* f $ i * g $ ((n + 1) - i)"
   92.41      have s0: "sum (\<lambda>i. of_nat i * f $ i * g $ (n + 1 - i)) ?Zn1 =
   92.42        sum (\<lambda>i. of_nat (n + 1 - i) * f $ (n + 1 - i) * g $ i) ?Zn1"
   92.43 -       by (rule sum.reindex_bij_witness[where i="op - (n + 1)" and j="op - (n + 1)"]) auto
   92.44 +       by (rule sum.reindex_bij_witness[where i="(-) (n + 1)" and j="(-) (n + 1)"]) auto
   92.45      have s1: "sum (\<lambda>i. f $ i * g $ (n + 1 - i)) ?Zn1 =
   92.46        sum (\<lambda>i. f $ (n + 1 - i) * g $ i) ?Zn1"
   92.47 -       by (rule sum.reindex_bij_witness[where i="op - (n + 1)" and j="op - (n + 1)"]) auto
   92.48 +       by (rule sum.reindex_bij_witness[where i="(-) (n + 1)" and j="(-) (n + 1)"]) auto
   92.49      have "(f * ?D g + ?D f * g)$n = (?D g * f + ?D f * g)$n"
   92.50        by (simp only: mult.commute)
   92.51      also have "\<dots> = (\<Sum>i = 0..n. ?g i)"
   92.52 @@ -2040,7 +2040,7 @@
   92.53    (* If f reprents {a_n} and P is a polynomial, then
   92.54          P(xD) f represents {P(n) a_n}*)
   92.55  
   92.56 -definition "fps_XD = op * fps_X \<circ> fps_deriv"
   92.57 +definition "fps_XD = ( * ) fps_X \<circ> fps_deriv"
   92.58  
   92.59  lemma fps_XD_add[simp]:"fps_XD (a + b) = fps_XD a + fps_XD (b :: 'a::comm_ring_1 fps)"
   92.60    by (simp add: fps_XD_def field_simps)
   92.61 @@ -2115,10 +2115,10 @@
   92.62    let ?fps_X = "1 - (fps_X::'a fps)"
   92.63    have th0: "?fps_X $ 0 \<noteq> 0"
   92.64      by simp
   92.65 -  have "a /?fps_X = ?fps_X *  Abs_fps (\<lambda>n::nat. sum (op $ a) {0..n}) * inverse ?fps_X"
   92.66 +  have "a /?fps_X = ?fps_X *  Abs_fps (\<lambda>n::nat. sum (($) a) {0..n}) * inverse ?fps_X"
   92.67      using fps_divide_fps_X_minus1_sum_lemma[of a, symmetric] th0
   92.68      by (simp add: fps_divide_def mult.assoc)
   92.69 -  also have "\<dots> = (inverse ?fps_X * ?fps_X) * Abs_fps (\<lambda>n::nat. sum (op $ a) {0..n}) "
   92.70 +  also have "\<dots> = (inverse ?fps_X * ?fps_X) * Abs_fps (\<lambda>n::nat. sum (($) a) {0..n}) "
   92.71      by (simp add: ac_simps)
   92.72    finally show ?thesis
   92.73      by (simp add: inverse_mult_eq_1[OF th0])
    93.1 --- a/src/HOL/Computational_Algebra/Nth_Powers.thy	Wed Jan 10 15:21:49 2018 +0100
    93.2 +++ b/src/HOL/Computational_Algebra/Nth_Powers.thy	Wed Jan 10 15:25:09 2018 +0100
    93.3 @@ -305,4 +305,4 @@
    93.4      by (auto intro!: Max_mono)
    93.5  qed auto
    93.6  
    93.7 -end
    93.8 \ No newline at end of file
    93.9 +end
    94.1 --- a/src/HOL/Computational_Algebra/Polynomial.thy	Wed Jan 10 15:21:49 2018 +0100
    94.2 +++ b/src/HOL/Computational_Algebra/Polynomial.thy	Wed Jan 10 15:25:09 2018 +0100
    94.3 @@ -891,7 +891,7 @@
    94.4  lemma smult_monom: "smult a (monom b n) = monom (a * b) n"
    94.5    by (induct n) (simp_all add: monom_0 monom_Suc)
    94.6  
    94.7 -lemma smult_Poly: "smult c (Poly xs) = Poly (map (op * c) xs)"
    94.8 +lemma smult_Poly: "smult c (Poly xs) = Poly (map (( * ) c) xs)"
    94.9    by (auto simp: poly_eq_iff nth_default_def)
   94.10  
   94.11  lemma degree_smult_eq [simp]: "degree (smult a p) = (if a = 0 then 0 else degree p)"
   94.12 @@ -1161,7 +1161,7 @@
   94.13        simp del: upt_Suc)
   94.14  
   94.15  lemma coeffs_map_poly [code abstract]:
   94.16 -  "coeffs (map_poly f p) = strip_while (op = 0) (map f (coeffs p))"
   94.17 +  "coeffs (map_poly f p) = strip_while ((=) 0) (map f (coeffs p))"
   94.18    by (simp add: map_poly_def)
   94.19  
   94.20  lemma coeffs_map_poly':
   94.21 @@ -1949,7 +1949,7 @@
   94.22  qed
   94.23  
   94.24  lemma dropWhile_replicate_append:
   94.25 -  "dropWhile (op = a) (replicate n a @ ys) = dropWhile (op = a) ys"
   94.26 +  "dropWhile ((=) a) (replicate n a @ ys) = dropWhile ((=) a) ys"
   94.27    by (induct n) simp_all
   94.28  
   94.29  lemma Poly_append_replicate_0: "Poly (xs @ replicate n 0) = Poly xs"
   94.30 @@ -2233,8 +2233,8 @@
   94.31    by (simp add: poly_eq_iff coeff_poly_cutoff)
   94.32  
   94.33  lemma coeffs_poly_cutoff [code abstract]:
   94.34 -  "coeffs (poly_cutoff n p) = strip_while (op = 0) (take n (coeffs p))"
   94.35 -proof (cases "strip_while (op = 0) (take n (coeffs p)) = []")
   94.36 +  "coeffs (poly_cutoff n p) = strip_while ((=) 0) (take n (coeffs p))"
   94.37 +proof (cases "strip_while ((=) 0) (take n (coeffs p)) = []")
   94.38    case True
   94.39    then have "coeff (poly_cutoff n p) k = 0" for k
   94.40      unfolding coeff_poly_cutoff
   94.41 @@ -2245,9 +2245,9 @@
   94.42      by (subst True) simp_all
   94.43  next
   94.44    case False
   94.45 -  have "no_trailing (op = 0) (strip_while (op = 0) (take n (coeffs p)))"
   94.46 +  have "no_trailing ((=) 0) (strip_while ((=) 0) (take n (coeffs p)))"
   94.47      by simp
   94.48 -  with False have "last (strip_while (op = 0) (take n (coeffs p))) \<noteq> 0"
   94.49 +  with False have "last (strip_while ((=) 0) (take n (coeffs p))) \<noteq> 0"
   94.50      unfolding no_trailing_unfold by auto
   94.51    then show ?thesis
   94.52      by (intro coeffs_eqI)
   94.53 @@ -2261,7 +2261,7 @@
   94.54    where "reflect_poly p = Poly (rev (coeffs p))"
   94.55  
   94.56  lemma coeffs_reflect_poly [code abstract]:
   94.57 -  "coeffs (reflect_poly p) = rev (dropWhile (op = 0) (coeffs p))"
   94.58 +  "coeffs (reflect_poly p) = rev (dropWhile ((=) 0) (coeffs p))"
   94.59    by (simp add: reflect_poly_def)
   94.60  
   94.61  lemma reflect_poly_0 [simp]: "reflect_poly 0 = 0"
   94.62 @@ -3803,10 +3803,10 @@
   94.63    where
   94.64      "pseudo_divmod_main_list lc q r d (Suc n) =
   94.65        (let
   94.66 -        rr = map (op * lc) r;
   94.67 +        rr = map (( * ) lc) r;
   94.68          a = hd r;
   94.69 -        qqq = cCons a (map (op * lc) q);
   94.70 -        rrr = tl (if a = 0 then rr else minus_poly_rev_list rr (map (op * a) d))
   94.71 +        qqq = cCons a (map (( * ) lc) q);
   94.72 +        rrr = tl (if a = 0 then rr else minus_poly_rev_list rr (map (( * ) a) d))
   94.73         in pseudo_divmod_main_list lc qqq rrr d n)"
   94.74    | "pseudo_divmod_main_list lc q r d 0 = (q, r)"
   94.75  
   94.76 @@ -3814,9 +3814,9 @@
   94.77    where
   94.78      "pseudo_mod_main_list lc r d (Suc n) =
   94.79        (let
   94.80 -        rr = map (op * lc) r;
   94.81 +        rr = map (( * ) lc) r;
   94.82          a = hd r;
   94.83 -        rrr = tl (if a = 0 then rr else minus_poly_rev_list rr (map (op * a) d))
   94.84 +        rrr = tl (if a = 0 then rr else minus_poly_rev_list rr (map (( * ) a) d))
   94.85         in pseudo_mod_main_list lc rrr d n)"
   94.86    | "pseudo_mod_main_list lc r d 0 = r"
   94.87  
   94.88 @@ -3828,7 +3828,7 @@
   94.89        (let
   94.90          a = hd r;
   94.91          qqq = cCons a q;
   94.92 -        rr = tl (if a = 0 then r else minus_poly_rev_list r (map (op * a) d))
   94.93 +        rr = tl (if a = 0 then r else minus_poly_rev_list r (map (( * ) a) d))
   94.94         in divmod_poly_one_main_list qqq rr d n)"
   94.95    | "divmod_poly_one_main_list q r d 0 = (q, r)"
   94.96  
   94.97 @@ -3837,7 +3837,7 @@
   94.98      "mod_poly_one_main_list r d (Suc n) =
   94.99        (let
  94.100          a = hd r;
  94.101 -        rr = tl (if a = 0 then r else minus_poly_rev_list r (map (op * a) d))
  94.102 +        rr = tl (if a = 0 then r else minus_poly_rev_list r (map (( * ) a) d))
  94.103         in mod_poly_one_main_list rr d n)"
  94.104    | "mod_poly_one_main_list r d 0 = r"
  94.105  
  94.106 @@ -3858,7 +3858,7 @@
  94.107          re = pseudo_mod_main_list (hd rq) (rev p) rq (1 + length p - length q)
  94.108         in rev re))"
  94.109  
  94.110 -lemma minus_zero_does_nothing: "minus_poly_rev_list x (map (op * 0) y) = x"
  94.111 +lemma minus_zero_does_nothing: "minus_poly_rev_list x (map (( * ) 0) y) = x"
  94.112    for x :: "'a::ring list"
  94.113    by (induct x y rule: minus_poly_rev_list.induct) auto
  94.114  
  94.115 @@ -3866,8 +3866,8 @@
  94.116    by (induct xs ys rule: minus_poly_rev_list.induct) auto
  94.117  
  94.118  lemma if_0_minus_poly_rev_list:
  94.119 -  "(if a = 0 then x else minus_poly_rev_list x (map (op * a) y)) =
  94.120 -    minus_poly_rev_list x (map (op * a) y)"
  94.121 +  "(if a = 0 then x else minus_poly_rev_list x (map (( * ) a) y)) =
  94.122 +    minus_poly_rev_list x (map (( * ) a) y)"
  94.123    for a :: "'a::ring"
  94.124    by(cases "a = 0") (simp_all add: minus_zero_does_nothing)
  94.125  
  94.126 @@ -3909,7 +3909,7 @@
  94.127  
  94.128  lemma head_minus_poly_rev_list:
  94.129    "length d \<le> length r \<Longrightarrow> d \<noteq> [] \<Longrightarrow>
  94.130 -    hd (minus_poly_rev_list (map (op * (last d)) r) (map (op * (hd r)) (rev d))) = 0"
  94.131 +    hd (minus_poly_rev_list (map (( * ) (last d)) r) (map (( * ) (hd r)) (rev d))) = 0"
  94.132    for d r :: "'a::comm_ring list"
  94.133  proof (induct r)
  94.134    case Nil
  94.135 @@ -3919,7 +3919,7 @@
  94.136    then show ?case by (cases "rev d") (simp_all add: ac_simps)
  94.137  qed
  94.138  
  94.139 -lemma Poly_map: "Poly (map (op * a) p) = smult a (Poly p)"
  94.140 +lemma Poly_map: "Poly (map (( * ) a) p) = smult a (Poly p)"
  94.141  proof (induct p)
  94.142    case Nil
  94.143    then show ?case by simp
  94.144 @@ -3947,9 +3947,9 @@
  94.145    with \<open>d \<noteq> []\<close> have "r \<noteq> []"
  94.146      using Suc_leI length_greater_0_conv list.size(3) by fastforce
  94.147    let ?a = "(hd (rev r))"
  94.148 -  let ?rr = "map (op * lc) (rev r)"
  94.149 -  let ?rrr = "rev (tl (minus_poly_rev_list ?rr (map (op * ?a) (rev d))))"
  94.150 -  let ?qq = "cCons ?a (map (op * lc) q)"
  94.151 +  let ?rr = "map (( * ) lc) (rev r)"
  94.152 +  let ?rrr = "rev (tl (minus_poly_rev_list ?rr (map (( * ) ?a) (rev d))))"
  94.153 +  let ?qq = "cCons ?a (map (( * ) lc) q)"
  94.154    from * Suc(3) have n: "n = (1 + length r - length d - 1)"
  94.155      by simp
  94.156    from * have rr_val:"(length ?rrr) = (length r - 1)"
  94.157 @@ -3982,12 +3982,12 @@
  94.158      case 2
  94.159      show ?case
  94.160      proof (subst Poly_on_rev_starting_with_0, goal_cases)
  94.161 -      show "hd (minus_poly_rev_list (map (op * lc) (rev r)) (map (op * (hd (rev r))) (rev d))) = 0"
  94.162 +      show "hd (minus_poly_rev_list (map (( * ) lc) (rev r)) (map (( * ) (hd (rev r))) (rev d))) = 0"
  94.163          by (fold lc, subst head_minus_poly_rev_list, insert * \<open>d \<noteq> []\<close>, auto)
  94.164        from * have "length d \<le> length r"
  94.165          by simp
  94.166        then show "smult lc (Poly r) - monom (coeff (Poly r) (length r - 1)) n * Poly d =
  94.167 -          Poly (rev (minus_poly_rev_list (map (op * lc) (rev r)) (map (op * (hd (rev r))) (rev d))))"
  94.168 +          Poly (rev (minus_poly_rev_list (map (( * ) lc) (rev r)) (map (( * ) (hd (rev r))) (rev d))))"
  94.169          by (fold rev_map) (auto simp add: n smult_monom_mult Poly_map hd_rev [symmetric]
  94.170              minus_poly_rev_list)
  94.171      qed
  94.172 @@ -4092,9 +4092,9 @@
  94.173          let
  94.174            cf = coeffs f;
  94.175            ilc = inverse (last cg);
  94.176 -          ch = map (op * ilc) cg;
  94.177 +          ch = map (( * ) ilc) cg;
  94.178            (q, r) = pseudo_divmod_main_list 1 [] (rev cf) (rev ch) (1 + length cf - length cg)
  94.179 -        in (poly_of_list (map (op * ilc) q), poly_of_list (rev r)))"
  94.180 +        in (poly_of_list (map (( * ) ilc) q), poly_of_list (rev r)))"
  94.181  proof -
  94.182    note d = pdivmod_via_pseudo_divmod pseudo_divmod_impl pseudo_divmod_list_def
  94.183    show ?thesis
  94.184 @@ -4113,10 +4113,10 @@
  94.185      have id2: "hd (rev (coeffs (smult ilc g))) = 1"
  94.186        by (subst hd_rev, insert id ilc, auto simp: coeffs_smult, subst last_map, auto simp: id ilc_def)
  94.187      have id3: "length (coeffs (smult ilc g)) = length (coeffs g)"
  94.188 -      "rev (coeffs (smult ilc g)) = rev (map (op * ilc) (coeffs g))"
  94.189 +      "rev (coeffs (smult ilc g)) = rev (map (( * ) ilc) (coeffs g))"
  94.190        unfolding coeffs_smult using ilc by auto
  94.191      obtain q r where pair:
  94.192 -      "pseudo_divmod_main_list 1 [] (rev (coeffs f)) (rev (map (op * ilc) (coeffs g)))
  94.193 +      "pseudo_divmod_main_list 1 [] (rev (coeffs f)) (rev (map (( * ) ilc) (coeffs g)))
  94.194          (1 + length (coeffs f) - length (coeffs g)) = (q, r)"
  94.195        by force
  94.196      show ?thesis
  94.197 @@ -4129,7 +4129,7 @@
  94.198  lemma pseudo_divmod_main_list_1: "pseudo_divmod_main_list 1 = divmod_poly_one_main_list"
  94.199  proof (intro ext, goal_cases)
  94.200    case (1 q r d n)
  94.201 -  have *: "map (op * 1) xs = xs" for xs :: "'a list"
  94.202 +  have *: "map (( * ) 1) xs = xs" for xs :: "'a list"
  94.203      by (induct xs) auto
  94.204    show ?case
  94.205      by (induct n arbitrary: q r d) (auto simp: * Let_def)
  94.206 @@ -4143,7 +4143,7 @@
  94.207          in if cr = 0 then divide_poly_main_list lc (cCons cr q) (tl r) d n else let
  94.208          a = cr div lc;
  94.209          qq = cCons a q;
  94.210 -        rr = minus_poly_rev_list r (map (op * a) d)
  94.211 +        rr = minus_poly_rev_list r (map (( * ) a) d)
  94.212         in if hd rr = 0 then divide_poly_main_list lc qq (tl rr) d n else [])"
  94.213    | "divide_poly_main_list lc q r d 0 = q"
  94.214  
  94.215 @@ -4153,7 +4153,7 @@
  94.216        cr = hd r;
  94.217        a = cr div lc;
  94.218        qq = cCons a q;
  94.219 -      rr = minus_poly_rev_list r (map (op * a) d)
  94.220 +      rr = minus_poly_rev_list r (map (( * ) a) d)
  94.221       in if hd rr = 0 then divide_poly_main_list lc qq (tl rr) d n else [])"
  94.222    by (simp add: Let_def minus_zero_does_nothing)
  94.223  
  94.224 @@ -4182,7 +4182,7 @@
  94.225          let
  94.226            cf = coeffs f;
  94.227            ilc = inverse (last cg);
  94.228 -          ch = map (op * ilc) cg;
  94.229 +          ch = map (( * ) ilc) cg;
  94.230            r = mod_poly_one_main_list (rev cf) (rev ch) (1 + length cf - length cg)
  94.231          in poly_of_list (rev r))"
  94.232    (is "_ = ?rhs")
  94.233 @@ -4201,14 +4201,14 @@
  94.234          let
  94.235            cf = coeffs f;
  94.236            ilc = inverse (last cg);
  94.237 -          ch = map (op * ilc) cg;
  94.238 +          ch = map (( * ) ilc) cg;
  94.239            q = fst (divmod_poly_one_main_list [] (rev cf) (rev ch) (1 + length cf - length cg))
  94.240 -        in poly_of_list ((map (op * ilc) q)))"
  94.241 +        in poly_of_list ((map (( * ) ilc) q)))"
  94.242  
  94.243  text \<open>We do not declare the following lemma as code equation, since then polynomial division
  94.244    on non-fields will no longer be executable. However, a code-unfold is possible, since
  94.245    \<open>div_field_poly_impl\<close> is a bit more efficient than the generic polynomial division.\<close>
  94.246 -lemma div_field_poly_impl[code_unfold]: "op div = div_field_poly_impl"
  94.247 +lemma div_field_poly_impl[code_unfold]: "(div) = div_field_poly_impl"
  94.248  proof (intro ext)
  94.249    fix f g :: "'a poly"
  94.250    have "fst (f div g, f mod g) = div_field_poly_impl f g"
  94.251 @@ -4251,7 +4251,7 @@
  94.252      with r d have id:
  94.253        "?thesis \<longleftrightarrow>
  94.254          Poly (divide_poly_main_list lc (cCons (lcr div lc) q)
  94.255 -          (rev (rev (minus_poly_rev_list (rev rr) (rev (map (op * (lcr div lc)) dd))))) (rev d) n) =
  94.256 +          (rev (rev (minus_poly_rev_list (rev rr) (rev (map (( * ) (lcr div lc)) dd))))) (rev d) n) =
  94.257            divide_poly_main lc
  94.258              (monom 1 (Suc n) * Poly q + monom (lcr div lc) n)
  94.259              (Poly r - monom (lcr div lc) n * Poly d)
    95.1 --- a/src/HOL/Computational_Algebra/Polynomial_FPS.thy	Wed Jan 10 15:21:49 2018 +0100
    95.2 +++ b/src/HOL/Computational_Algebra/Polynomial_FPS.thy	Wed Jan 10 15:25:09 2018 +0100
    95.3 @@ -287,11 +287,11 @@
    95.4       (auto simp: prefix_length_def nth_Cons split: if_splits nat.splits)
    95.5  (* END TODO *)
    95.6    
    95.7 -lemma poly_subdegree_code [code]: "poly_subdegree p = prefix_length (op = 0) (coeffs p)"
    95.8 +lemma poly_subdegree_code [code]: "poly_subdegree p = prefix_length ((=) 0) (coeffs p)"
    95.9  proof (cases "p = 0")
   95.10    case False
   95.11    note [simp] = this
   95.12 -  define n where "n = prefix_length (op = 0) (coeffs p)"
   95.13 +  define n where "n = prefix_length ((=) 0) (coeffs p)"
   95.14    from False have "\<exists>k. coeff p k \<noteq> 0" by (auto simp: poly_eq_iff)
   95.15    hence ex: "\<exists>x\<in>set (coeffs p). x \<noteq> 0" by (auto simp: coeffs_def)
   95.16    hence n_less: "n < length (coeffs p)" and nonzero: "coeffs p ! n \<noteq> 0" 
   95.17 @@ -300,10 +300,10 @@
   95.18    proof (intro subdegreeI)
   95.19      from n_less have "fps_of_poly p $ n = coeffs p ! n"
   95.20        by (subst coeffs_nth) (simp_all add: degree_eq_length_coeffs)
   95.21 -    with nonzero show "fps_of_poly p $ prefix_length (op = 0) (coeffs p) \<noteq> 0"
   95.22 +    with nonzero show "fps_of_poly p $ prefix_length ((=) 0) (coeffs p) \<noteq> 0"
   95.23        unfolding n_def by simp
   95.24    next
   95.25 -    fix k assume A: "k < prefix_length (op = 0) (coeffs p)"
   95.26 +    fix k assume A: "k < prefix_length ((=) 0) (coeffs p)"
   95.27      also have "\<dots> \<le> length (coeffs p)" by (rule prefix_length_le_length)
   95.28      finally show "fps_of_poly p $ k = 0"
   95.29        using nth_less_prefix_length[OF A]
    96.1 --- a/src/HOL/Computational_Algebra/Squarefree.thy	Wed Jan 10 15:21:49 2018 +0100
    96.2 +++ b/src/HOL/Computational_Algebra/Squarefree.thy	Wed Jan 10 15:25:09 2018 +0100
    96.3 @@ -360,4 +360,4 @@
    96.4    "odd n \<Longrightarrow> square_part (x ^ n) = normalize (x ^ (n div 2) * square_part x)"
    96.5    by (subst square_part_odd_power' [symmetric]) auto
    96.6  
    96.7 -end
    96.8 \ No newline at end of file
    96.9 +end
    97.1 --- a/src/HOL/Corec_Examples/Tests/GPV_Bare_Bones.thy	Wed Jan 10 15:21:49 2018 +0100
    97.2 +++ b/src/HOL/Corec_Examples/Tests/GPV_Bare_Bones.thy	Wed Jan 10 15:25:09 2018 +0100
    97.3 @@ -39,9 +39,9 @@
    97.4  
    97.5  friend_of_corec bind_gpv'
    97.6  where "bind_gpv' r f =
    97.7 -GPV' (map_spmf (map_generat id id (op \<circ> (case_sum id (\<lambda>r. bind_gpv' r f))))
    97.8 +GPV' (map_spmf (map_generat id id ((\<circ>) (case_sum id (\<lambda>r. bind_gpv' r f))))
    97.9        (bind_spmf (the_gpv r)
   97.10 -        (case_generat (\<lambda>x. map_spmf (map_generat id id (op \<circ> Inl)) (the_gpv' (f x)))
   97.11 +        (case_generat (\<lambda>x. map_spmf (map_generat id id ((\<circ>) Inl)) (the_gpv' (f x)))
   97.12            (\<lambda>out c. return_spmf (IO out (\<lambda>input. Inr (c input)))))))"
   97.13    sorry
   97.14  
    98.1 --- a/src/HOL/Corec_Examples/Tests/Iterate_GPV.thy	Wed Jan 10 15:21:49 2018 +0100
    98.2 +++ b/src/HOL/Corec_Examples/Tests/Iterate_GPV.thy	Wed Jan 10 15:25:09 2018 +0100
    98.3 @@ -21,10 +21,10 @@
    98.4  
    98.5  primcorec bind_gpv :: "('a, 'out, 'in) gpv \<Rightarrow> ('a \<Rightarrow> ('b, 'out, 'in) gpv) \<Rightarrow> ('b, 'out, 'in) gpv"
    98.6  where
    98.7 -  "bind_gpv r f = GPV (map_spmf (map_generat id id (op \<circ> (case_sum id (\<lambda>r. bind_gpv r f))))
    98.8 +  "bind_gpv r f = GPV (map_spmf (map_generat id id ((\<circ>) (case_sum id (\<lambda>r. bind_gpv r f))))
    98.9       (bind_spmf (the_gpv r)
   98.10        (case_generat
   98.11 -        (\<lambda>x. map_spmf (map_generat id id (op \<circ> Inl)) (the_gpv (f x)))
   98.12 +        (\<lambda>x. map_spmf (map_generat id id ((\<circ>) Inl)) (the_gpv (f x)))
   98.13          (\<lambda>out c. return_spmf (IO out (\<lambda>input. Inr (c input)))))))"
   98.14  
   98.15  context includes lifting_syntax begin
   98.16 @@ -44,14 +44,14 @@
   98.17    by (cases x) auto
   98.18  
   98.19  friend_of_corec bind_gpv :: "('a, 'out, 'in) gpv \<Rightarrow> ('a \<Rightarrow> ('a, 'out, 'in) gpv) \<Rightarrow> ('a, 'out, 'in) gpv"
   98.20 -where "bind_gpv r f = GPV (map_spmf (map_generat id id (op \<circ> (case_sum id (\<lambda>r. bind_gpv r f))))
   98.21 +where "bind_gpv r f = GPV (map_spmf (map_generat id id ((\<circ>) (case_sum id (\<lambda>r. bind_gpv r f))))
   98.22       (bind_spmf (the_gpv r)
   98.23        (case_generat
   98.24 -        (\<lambda>x. map_spmf (map_generat id id (op \<circ> Inl)) (the_gpv (f x)))
   98.25 +        (\<lambda>x. map_spmf (map_generat id id ((\<circ>) Inl)) (the_gpv (f x)))
   98.26          (\<lambda>out c. return_spmf (IO out (\<lambda>input. Inr (c input)))))))"
   98.27  apply(rule bind_gpv.ctr)
   98.28  apply transfer_prover
   98.29  apply transfer_prover
   98.30  done
   98.31  
   98.32 -end
   98.33 \ No newline at end of file
   98.34 +end
    99.1 --- a/src/HOL/Corec_Examples/Tests/Misc_Mono.thy	Wed Jan 10 15:21:49 2018 +0100
    99.2 +++ b/src/HOL/Corec_Examples/Tests/Misc_Mono.thy	Wed Jan 10 15:25:09 2018 +0100
    99.3 @@ -149,7 +149,7 @@
    99.4  consts integerize_tree_list :: "'a list \<Rightarrow> int"
    99.5  
    99.6  lemma integerize_tree_list_transfer[transfer_rule]:
    99.7 -  "rel_fun (list_all2 R) op = integerize_tree_list integerize_tree_list"
    99.8 +  "rel_fun (list_all2 R) (=) integerize_tree_list integerize_tree_list"
    99.9    sorry
   99.10  
   99.11  corec (friend) f10a where
   100.1 --- a/src/HOL/Data_Structures/Base_FDS.thy	Wed Jan 10 15:21:49 2018 +0100
   100.2 +++ b/src/HOL/Data_Structures/Base_FDS.thy	Wed Jan 10 15:25:09 2018 +0100
   100.3 @@ -18,4 +18,4 @@
   100.4    "is_measure f \<Longrightarrow> is_measure g \<Longrightarrow> is_measure (size_prod f g)"
   100.5  by (rule is_measure_trivial)
   100.6  
   100.7 -end
   100.8 \ No newline at end of file
   100.9 +end
   101.1 --- a/src/HOL/Data_Structures/Binomial_Heap.thy	Wed Jan 10 15:21:49 2018 +0100
   101.2 +++ b/src/HOL/Data_Structures/Binomial_Heap.thy	Wed Jan 10 15:25:09 2018 +0100
   101.3 @@ -66,7 +66,7 @@
   101.4  
   101.5  definition invar_bheap :: "'a::linorder heap \<Rightarrow> bool" where
   101.6  "invar_bheap ts
   101.7 -  \<longleftrightarrow> (\<forall>t\<in>set ts. invar_btree t) \<and> (sorted_wrt (op <) (map rank ts))"
   101.8 +  \<longleftrightarrow> (\<forall>t\<in>set ts. invar_btree t) \<and> (sorted_wrt (<) (map rank ts))"
   101.9  
  101.10  text \<open>Ordering (heap) invariant\<close>
  101.11  fun invar_otree :: "'a::linorder tree \<Rightarrow> bool" where
  101.12 @@ -390,7 +390,7 @@
  101.13  to show that binomial heaps satisfy the specification of priority queues with merge.\<close>
  101.14  
  101.15  interpretation binheap: Priority_Queue_Merge
  101.16 -  where empty = "[]" and is_empty = "op = []" and insert = insert
  101.17 +  where empty = "[]" and is_empty = "(=) []" and insert = insert
  101.18    and get_min = get_min and del_min = del_min and merge = merge
  101.19    and invar = invar and mset = mset_heap
  101.20  proof (unfold_locales, goal_cases)
  101.21 @@ -453,14 +453,14 @@
  101.22    shows "2^length ts \<le> size (mset_heap ts) + 1"
  101.23  proof -
  101.24    from \<open>invar_bheap ts\<close> have 
  101.25 -    ASC: "sorted_wrt (op <) (map rank ts)" and
  101.26 +    ASC: "sorted_wrt (<) (map rank ts)" and
  101.27      TINV: "\<forall>t\<in>set ts. invar_btree t"
  101.28      unfolding invar_bheap_def by auto
  101.29        
  101.30    have "(2::nat)^length ts = (\<Sum>i\<in>{0..<length ts}. 2^i) + 1" 
  101.31      by (simp add: sum_power2)
  101.32    also have "\<dots> \<le> (\<Sum>t\<leftarrow>ts. 2^rank t) + 1"
  101.33 -    using sorted_wrt_less_sum_mono_lowerbound[OF _ ASC, of "op ^ (2::nat)"]
  101.34 +    using sorted_wrt_less_sum_mono_lowerbound[OF _ ASC, of "(^) (2::nat)"]
  101.35      using power_increasing[where a="2::nat"]  
  101.36      by (auto simp: o_def)
  101.37    also have "\<dots> = (\<Sum>t\<leftarrow>ts. size (mset_tree t)) + 1" using TINV   
   102.1 --- a/src/HOL/Data_Structures/Sorted_Less.thy	Wed Jan 10 15:21:49 2018 +0100
   102.2 +++ b/src/HOL/Data_Structures/Sorted_Less.thy	Wed Jan 10 15:25:09 2018 +0100
   102.3 @@ -12,7 +12,7 @@
   102.4  Could go into theory List under a name like @{term sorted_less}.\<close>
   102.5  
   102.6  abbreviation sorted :: "'a::linorder list \<Rightarrow> bool" where
   102.7 -"sorted \<equiv> sorted_wrt (op <)"
   102.8 +"sorted \<equiv> sorted_wrt (<)"
   102.9  
  102.10  lemma sorted_cons: "sorted (x#xs) \<Longrightarrow> sorted xs"
  102.11  by(simp add: sorted_wrt_Cons)
   103.1 --- a/src/HOL/Datatype_Examples/Derivation_Trees/DTree.thy	Wed Jan 10 15:21:49 2018 +0100
   103.2 +++ b/src/HOL/Datatype_Examples/Derivation_Trees/DTree.thy	Wed Jan 10 15:25:09 2018 +0100
   103.3 @@ -67,7 +67,7 @@
   103.4  lemma dtree_coinduct[elim, consumes 1, case_names Lift, induct pred: "HOL.eq"]:
   103.5  assumes phi: "\<phi> tr1 tr2" and
   103.6  Lift: "\<And> tr1 tr2. \<phi> tr1 tr2 \<Longrightarrow>
   103.7 -                  root tr1 = root tr2 \<and> rel_set (rel_sum op = \<phi>) (cont tr1) (cont tr2)"
   103.8 +                  root tr1 = root tr2 \<and> rel_set (rel_sum (=) \<phi>) (cont tr1) (cont tr2)"
   103.9  shows "tr1 = tr2"
  103.10  using phi apply(elim dtree.coinduct)
  103.11  apply(rule Lift[unfolded rel_set_cont]) .
   104.1 --- a/src/HOL/Datatype_Examples/Derivation_Trees/Parallel_Composition.thy	Wed Jan 10 15:21:49 2018 +0100
   104.2 +++ b/src/HOL/Datatype_Examples/Derivation_Trees/Parallel_Composition.thy	Wed Jan 10 15:25:09 2018 +0100
   104.3 @@ -68,7 +68,7 @@
   104.4  subsection\<open>Structural Coinduction Proofs\<close>
   104.5  
   104.6  lemma rel_set_rel_sum_eq[simp]:
   104.7 -"rel_set (rel_sum (op =) \<phi>) A1 A2 \<longleftrightarrow>
   104.8 +"rel_set (rel_sum (=) \<phi>) A1 A2 \<longleftrightarrow>
   104.9   Inl -` A1 = Inl -` A2 \<and> rel_set \<phi> (Inr -` A1) (Inr -` A2)"
  104.10  unfolding rel_set_rel_sum rel_set_eq ..
  104.11  
   105.1 --- a/src/HOL/Datatype_Examples/Stream_Processor.thy	Wed Jan 10 15:21:49 2018 +0100
   105.2 +++ b/src/HOL/Datatype_Examples/Stream_Processor.thy	Wed Jan 10 15:25:09 2018 +0100
   105.3 @@ -96,7 +96,7 @@
   105.4  
   105.5  primrec sp\<^sub>\<mu>_comp2 (infixl "o\<^sup>*\<^sub>\<mu>" 65) where
   105.6    "Put b sp o\<^sup>*\<^sub>\<mu> fsp = Put b (sp, In fsp)"
   105.7 -| "Get f o\<^sup>*\<^sub>\<mu> fsp = sp\<^sub>\<mu>_comp2R (op o\<^sup>*\<^sub>\<mu> o f) fsp"
   105.8 +| "Get f o\<^sup>*\<^sub>\<mu> fsp = sp\<^sub>\<mu>_comp2R ((o\<^sup>*\<^sub>\<mu>) o f) fsp"
   105.9  
  105.10  primcorec sp\<^sub>\<nu>_comp2 (infixl "o\<^sup>*\<^sub>\<nu>" 65) where
  105.11    "out (sp o\<^sup>*\<^sub>\<nu> sp') = map_sp\<^sub>\<mu> id (\<lambda>(sp, sp'). sp o\<^sup>*\<^sub>\<nu> sp') (out sp o\<^sup>*\<^sub>\<mu> out sp')"
  105.12 @@ -159,7 +159,7 @@
  105.13    end
  105.14  \<close>)
  105.15  
  105.16 -lemma sp\<^sub>\<mu>_rel_self: "\<lbrakk>op = \<le> R1; op = \<le> R2\<rbrakk> \<Longrightarrow> rel_sp\<^sub>\<mu> R1 R2 x x"
  105.17 +lemma sp\<^sub>\<mu>_rel_self: "\<lbrakk>(=) \<le> R1; (=) \<le> R2\<rbrakk> \<Longrightarrow> rel_sp\<^sub>\<mu> R1 R2 x x"
  105.18    by (erule (1) predicate2D[OF sp\<^sub>\<mu>.rel_mono]) (simp only: sp\<^sub>\<mu>.rel_eq)
  105.19  
  105.20  lemma sp\<^sub>\<nu>_comp_sp\<^sub>\<nu>_comp2: "sp o\<^sub>\<nu> sp' = sp o\<^sup>*\<^sub>\<nu> sp'"
   106.1 --- a/src/HOL/Decision_Procs/Algebra_Aux.thy	Wed Jan 10 15:21:49 2018 +0100
   106.2 +++ b/src/HOL/Decision_Procs/Algebra_Aux.thy	Wed Jan 10 15:25:09 2018 +0100
   106.3 @@ -9,7 +9,7 @@
   106.4  begin
   106.5  
   106.6  definition of_natural :: "('a, 'm) ring_scheme \<Rightarrow> nat \<Rightarrow> 'a" ("\<guillemotleft>_\<guillemotright>\<^sub>\<nat>\<index>")
   106.7 -  where "\<guillemotleft>n\<guillemotright>\<^sub>\<nat>\<^bsub>R\<^esub> = (op \<oplus>\<^bsub>R\<^esub> \<one>\<^bsub>R\<^esub> ^^ n) \<zero>\<^bsub>R\<^esub>"
   106.8 +  where "\<guillemotleft>n\<guillemotright>\<^sub>\<nat>\<^bsub>R\<^esub> = ((\<oplus>\<^bsub>R\<^esub>) \<one>\<^bsub>R\<^esub> ^^ n) \<zero>\<^bsub>R\<^esub>"
   106.9  
  106.10  definition of_integer :: "('a, 'm) ring_scheme \<Rightarrow> int \<Rightarrow> 'a" ("\<guillemotleft>_\<guillemotright>\<index>")
  106.11    where "\<guillemotleft>i\<guillemotright>\<^bsub>R\<^esub> = (if 0 \<le> i then \<guillemotleft>nat i\<guillemotright>\<^sub>\<nat>\<^bsub>R\<^esub> else \<ominus>\<^bsub>R\<^esub> \<guillemotleft>nat (- i)\<guillemotright>\<^sub>\<nat>\<^bsub>R\<^esub>)"
  106.12 @@ -234,7 +234,7 @@
  106.13    by (induct n) (simp_all add: m_ac)
  106.14  
  106.15  definition cring_class_ops :: "'a::comm_ring_1 ring"
  106.16 -  where "cring_class_ops \<equiv> \<lparr>carrier = UNIV, mult = op *, one = 1, zero = 0, add = (op +)\<rparr>"
  106.17 +  where "cring_class_ops \<equiv> \<lparr>carrier = UNIV, mult = ( * ), one = 1, zero = 0, add = (+)\<rparr>"
  106.18  
  106.19  lemma cring_class: "cring cring_class_ops"
  106.20    apply unfold_locales
   107.1 --- a/src/HOL/Decision_Procs/Cooper.thy	Wed Jan 10 15:21:49 2018 +0100
   107.2 +++ b/src/HOL/Decision_Procs/Cooper.thy	Wed Jan 10 15:25:09 2018 +0100
   107.3 @@ -2386,7 +2386,7 @@
   107.4  let
   107.5  
   107.6  fun num_of_term vs (t as Free (xn, xT)) =
   107.7 -      (case AList.lookup (op =) vs t of
   107.8 +      (case AList.lookup (=) vs t of
   107.9          NONE => error "Variable not found in the list!"
  107.10        | SOME n => @{code Bound} (@{code nat_of_integer} n))
  107.11    | num_of_term vs @{term "0::int"} = @{code C} (@{code int_of_integer} 0)
  107.12 @@ -2398,11 +2398,11 @@
  107.13        @{code C} (@{code int_of_integer} (~(HOLogic.dest_numeral t)))
  107.14    | num_of_term vs (Bound i) = @{code Bound} (@{code nat_of_integer} i)
  107.15    | num_of_term vs (@{term "uminus :: int \<Rightarrow> int"} $ t') = @{code Neg} (num_of_term vs t')
  107.16 -  | num_of_term vs (@{term "op + :: int \<Rightarrow> int \<Rightarrow> int"} $ t1 $ t2) =
  107.17 +  | num_of_term vs (@{term "(+) :: int \<Rightarrow> int \<Rightarrow> int"} $ t1 $ t2) =
  107.18        @{code Add} (num_of_term vs t1, num_of_term vs t2)
  107.19 -  | num_of_term vs (@{term "op - :: int \<Rightarrow> int \<Rightarrow> int"} $ t1 $ t2) =
  107.20 +  | num_of_term vs (@{term "(-) :: int \<Rightarrow> int \<Rightarrow> int"} $ t1 $ t2) =
  107.21        @{code Sub} (num_of_term vs t1, num_of_term vs t2)
  107.22 -  | num_of_term vs (@{term "op * :: int \<Rightarrow> int \<Rightarrow> int"} $ t1 $ t2) =
  107.23 +  | num_of_term vs (@{term "( * ) :: int \<Rightarrow> int \<Rightarrow> int"} $ t1 $ t2) =
  107.24        (case try HOLogic.dest_number t1 of
  107.25          SOME (_, i) => @{code Mul} (@{code int_of_integer} i, num_of_term vs t2)
  107.26        | NONE =>
  107.27 @@ -2413,17 +2413,17 @@
  107.28  
  107.29  fun fm_of_term ps vs @{term True} = @{code T}
  107.30    | fm_of_term ps vs @{term False} = @{code F}
  107.31 -  | fm_of_term ps vs (@{term "op < :: int \<Rightarrow> int \<Rightarrow> bool"} $ t1 $ t2) =
  107.32 +  | fm_of_term ps vs (@{term "(<) :: int \<Rightarrow> int \<Rightarrow> bool"} $ t1 $ t2) =
  107.33        @{code Lt} (@{code Sub} (num_of_term vs t1, num_of_term vs t2))
  107.34 -  | fm_of_term ps vs (@{term "op \<le> :: int \<Rightarrow> int \<Rightarrow> bool"} $ t1 $ t2) =
  107.35 +  | fm_of_term ps vs (@{term "(\<le>) :: int \<Rightarrow> int \<Rightarrow> bool"} $ t1 $ t2) =
  107.36        @{code Le} (@{code Sub} (num_of_term vs t1, num_of_term vs t2))
  107.37 -  | fm_of_term ps vs (@{term "op = :: int \<Rightarrow> int \<Rightarrow> bool"} $ t1 $ t2) =
  107.38 +  | fm_of_term ps vs (@{term "(=) :: int \<Rightarrow> int \<Rightarrow> bool"} $ t1 $ t2) =
  107.39        @{code Eq} (@{code Sub} (num_of_term vs t1, num_of_term vs t2))
  107.40 -  | fm_of_term ps vs (@{term "op dvd :: int \<Rightarrow> int \<Rightarrow> bool"} $ t1 $ t2) =
  107.41 +  | fm_of_term ps vs (@{term "(dvd) :: int \<Rightarrow> int \<Rightarrow> bool"} $ t1 $ t2) =
  107.42        (case try HOLogic.dest_number t1 of
  107.43          SOME (_, i) => @{code Dvd} (@{code int_of_integer} i, num_of_term vs t2)
  107.44        | NONE => error "num_of_term: unsupported dvd")
  107.45 -  | fm_of_term ps vs (@{term "op = :: bool \<Rightarrow> bool \<Rightarrow> bool"} $ t1 $ t2) =
  107.46 +  | fm_of_term ps vs (@{term "(=) :: bool \<Rightarrow> bool \<Rightarrow> bool"} $ t1 $ t2) =
  107.47        @{code Iff} (fm_of_term ps vs t1, fm_of_term ps vs t2)
  107.48    | fm_of_term ps vs (@{term HOL.conj} $ t1 $ t2) =
  107.49        @{code And} (fm_of_term ps vs t1, fm_of_term ps vs t2)
  107.50 @@ -2451,11 +2451,11 @@
  107.51          val q = @{code integer_of_nat} n
  107.52        in fst (the (find_first (fn (_, m) => q = m) vs)) end
  107.53    | term_of_num vs (@{code Neg} t') = @{term "uminus :: int \<Rightarrow> int"} $ term_of_num vs t'
  107.54 -  | term_of_num vs (@{code Add} (t1, t2)) = @{term "op + :: int \<Rightarrow> int \<Rightarrow> int"} $
  107.55 +  | term_of_num vs (@{code Add} (t1, t2)) = @{term "(+) :: int \<Rightarrow> int \<Rightarrow> int"} $
  107.56        term_of_num vs t1 $ term_of_num vs t2
  107.57 -  | term_of_num vs (@{code Sub} (t1, t2)) = @{term "op - :: int \<Rightarrow> int \<Rightarrow> int"} $
  107.58 +  | term_of_num vs (@{code Sub} (t1, t2)) = @{term "(-) :: int \<Rightarrow> int \<Rightarrow> int"} $
  107.59        term_of_num vs t1 $ term_of_num vs t2
  107.60 -  | term_of_num vs (@{code Mul} (i, t2)) = @{term "op * :: int \<Rightarrow> int \<Rightarrow> int"} $
  107.61 +  | term_of_num vs (@{code Mul} (i, t2)) = @{term "( * ) :: int \<Rightarrow> int \<Rightarrow> int"} $
  107.62        term_of_num vs (@{code C} i) $ term_of_num vs t2
  107.63    | term_of_num vs (@{code CN} (n, i, t)) =
  107.64        term_of_num vs (@{code Add} (@{code Mul} (i, @{code Bound} n), t));
  107.65 @@ -2463,19 +2463,19 @@
  107.66  fun term_of_fm ps vs @{code T} = @{term True}
  107.67    | term_of_fm ps vs @{code F} = @{term False}
  107.68    | term_of_fm ps vs (@{code Lt} t) =
  107.69 -      @{term "op < :: int \<Rightarrow> int \<Rightarrow> bool"} $ term_of_num vs t $ @{term "0::int"}
  107.70 +      @{term "(<) :: int \<Rightarrow> int \<Rightarrow> bool"} $ term_of_num vs t $ @{term "0::int"}
  107.71    | term_of_fm ps vs (@{code Le} t) =
  107.72 -      @{term "op \<le> :: int \<Rightarrow> int \<Rightarrow> bool"} $ term_of_num vs t $ @{term "0::int"}
  107.73 +      @{term "(\<le>) :: int \<Rightarrow> int \<Rightarrow> bool"} $ term_of_num vs t $ @{term "0::int"}
  107.74    | term_of_fm ps vs (@{code Gt} t) =
  107.75 -      @{term "op < :: int \<Rightarrow> int \<Rightarrow> bool"} $ @{term "0::int"} $ term_of_num vs t
  107.76 +      @{term "(<) :: int \<Rightarrow> int \<Rightarrow> bool"} $ @{term "0::int"} $ term_of_num vs t
  107.77    | term_of_fm ps vs (@{code Ge} t) =
  107.78 -      @{term "op \<le> :: int \<Rightarrow> int \<Rightarrow> bool"} $ @{term "0::int"} $ term_of_num vs t
  107.79 +      @{term "(\<le>) :: int \<Rightarrow> int \<Rightarrow> bool"} $ @{term "0::int"} $ term_of_num vs t
  107.80    | term_of_fm ps vs (@{code Eq} t) =
  107.81 -      @{term "op = :: int \<Rightarrow> int \<Rightarrow> bool"} $ term_of_num vs t $ @{term "0::int"}
  107.82 +      @{term "(=) :: int \<Rightarrow> int \<Rightarrow> bool"} $ term_of_num vs t $ @{term "0::int"}
  107.83    | term_of_fm ps vs (@{code NEq} t) =
  107.84        term_of_fm ps vs (@{code NOT} (@{code Eq} t))
  107.85    | term_of_fm ps vs (@{code Dvd} (i, t)) =
  107.86 -      @{term "op dvd :: int \<Rightarrow> int \<Rightarrow> bool"} $ term_of_num vs (@{code C} i) $ term_of_num vs t
  107.87 +      @{term "(dvd) :: int \<Rightarrow> int \<Rightarrow> bool"} $ term_of_num vs (@{code C} i) $ term_of_num vs t
  107.88    | term_of_fm ps vs (@{code NDvd} (i, t)) =
  107.89        term_of_fm ps vs (@{code NOT} (@{code Dvd} (i, t)))
  107.90    | term_of_fm ps vs (@{code NOT} t') =
  107.91 @@ -2487,7 +2487,7 @@
  107.92    | term_of_fm ps vs (@{code Imp} (t1, t2)) =
  107.93        HOLogic.imp $ term_of_fm ps vs t1 $ term_of_fm ps vs t2
  107.94    | term_of_fm ps vs (@{code Iff} (t1, t2)) =
  107.95 -      @{term "op = :: bool \<Rightarrow> bool \<Rightarrow> bool"} $ term_of_fm ps vs t1 $ term_of_fm ps vs t2
  107.96 +      @{term "(=) :: bool \<Rightarrow> bool \<Rightarrow> bool"} $ term_of_fm ps vs t1 $ term_of_fm ps vs t2
  107.97    | term_of_fm ps vs (@{code Closed} n) =
  107.98        let
  107.99          val q = @{code integer_of_nat} n
 107.100 @@ -2497,22 +2497,22 @@
 107.101  fun term_bools acc t =
 107.102    let
 107.103      val is_op =
 107.104 -      member (op =) [@{term HOL.conj}, @{term HOL.disj}, @{term HOL.implies},
 107.105 -      @{term "op = :: bool \<Rightarrow> _"},
 107.106 -      @{term "op = :: int \<Rightarrow> _"}, @{term "op < :: int \<Rightarrow> _"},
 107.107 -      @{term "op \<le> :: int \<Rightarrow> _"}, @{term "Not"}, @{term "All :: (int \<Rightarrow> _) \<Rightarrow> _"},
 107.108 +      member (=) [@{term HOL.conj}, @{term HOL.disj}, @{term HOL.implies},
 107.109 +      @{term "(=) :: bool \<Rightarrow> _"},
 107.110 +      @{term "(=) :: int \<Rightarrow> _"}, @{term "(<) :: int \<Rightarrow> _"},
 107.111 +      @{term "(\<le>) :: int \<Rightarrow> _"}, @{term "Not"}, @{term "All :: (int \<Rightarrow> _) \<Rightarrow> _"},
 107.112        @{term "Ex :: (int \<Rightarrow> _) \<Rightarrow> _"}, @{term "True"}, @{term "False"}]
 107.113      fun is_ty t = not (fastype_of t = HOLogic.boolT)
 107.114    in
 107.115      (case t of
 107.116        (l as f $ a) $ b =>
 107.117          if is_ty t orelse is_op t then term_bools (term_bools acc l) b
 107.118 -        else insert (op aconv) t acc
 107.119 +        else insert (aconv) t acc
 107.120      | f $ a =>
 107.121          if is_ty t orelse is_op t then term_bools (term_bools acc f) a
 107.122 -        else insert (op aconv) t acc
 107.123 +        else insert (aconv) t acc
 107.124      | Abs p => term_bools acc (snd (Syntax_Trans.variant_abs p))  (* FIXME !? *)
 107.125 -    | _ => if is_ty t orelse is_op t then acc else insert (op aconv) t acc)
 107.126 +    | _ => if is_ty t orelse is_op t then acc else insert (aconv) t acc)
 107.127    end;
 107.128  
 107.129  in
   108.1 --- a/src/HOL/Decision_Procs/Ferrack.thy	Wed Jan 10 15:21:49 2018 +0100
   108.2 +++ b/src/HOL/Decision_Procs/Ferrack.thy	Wed Jan 10 15:25:09 2018 +0100
   108.3 @@ -2472,11 +2472,11 @@
   108.4    | num_of_term vs @{term "1::real"} = mk_C 1
   108.5    | num_of_term vs (Bound i) = mk_Bound i
   108.6    | num_of_term vs (@{term "uminus :: real \<Rightarrow> real"} $ t') = @{code Neg} (num_of_term vs t')
   108.7 -  | num_of_term vs (@{term "op + :: real \<Rightarrow> real \<Rightarrow> real"} $ t1 $ t2) =
   108.8 +  | num_of_term vs (@{term "(+) :: real \<Rightarrow> real \<Rightarrow> real"} $ t1 $ t2) =
   108.9       @{code Add} (num_of_term vs t1, num_of_term vs t2)
  108.10 -  | num_of_term vs (@{term "op - :: real \<Rightarrow> real \<Rightarrow> real"} $ t1 $ t2) =
  108.11 +  | num_of_term vs (@{term "(-) :: real \<Rightarrow> real \<Rightarrow> real"} $ t1 $ t2) =
  108.12       @{code Sub} (num_of_term vs t1, num_of_term vs t2)
  108.13 -  | num_of_term vs (@{term "op * :: real \<Rightarrow> real \<Rightarrow> real"} $ t1 $ t2) = (case num_of_term vs t1
  108.14 +  | num_of_term vs (@{term "( * ) :: real \<Rightarrow> real \<Rightarrow> real"} $ t1 $ t2) = (case num_of_term vs t1
  108.15       of @{code C} i => @{code Mul} (i, num_of_term vs t2)
  108.16        | _ => error "num_of_term: unsupported multiplication")
  108.17    | num_of_term vs (@{term "real_of_int :: int \<Rightarrow> real"} $ t') =
  108.18 @@ -2488,13 +2488,13 @@
  108.19  
  108.20  fun fm_of_term vs @{term True} = @{code T}
  108.21    | fm_of_term vs @{term False} = @{code F}
  108.22 -  | fm_of_term vs (@{term "op < :: real \<Rightarrow> real \<Rightarrow> bool"} $ t1 $ t2) =
  108.23 +  | fm_of_term vs (@{term "(<) :: real \<Rightarrow> real \<Rightarrow> bool"} $ t1 $ t2) =
  108.24        @{code Lt} (@{code Sub} (num_of_term vs t1, num_of_term vs t2))
  108.25 -  | fm_of_term vs (@{term "op \<le> :: real \<Rightarrow> real \<Rightarrow> bool"} $ t1 $ t2) =
  108.26 +  | fm_of_term vs (@{term "(\<le>) :: real \<Rightarrow> real \<Rightarrow> bool"} $ t1 $ t2) =
  108.27        @{code Le} (@{code Sub} (num_of_term vs t1, num_of_term vs t2))
  108.28 -  | fm_of_term vs (@{term "op = :: real \<Rightarrow> real \<Rightarrow> bool"} $ t1 $ t2) =
  108.29 +  | fm_of_term vs (@{term "(=) :: real \<Rightarrow> real \<Rightarrow> bool"} $ t1 $ t2) =
  108.30        @{code Eq} (@{code Sub} (num_of_term vs t1, num_of_term vs t2))
  108.31 -  | fm_of_term vs (@{term "op \<longleftrightarrow> :: bool \<Rightarrow> bool \<Rightarrow> bool"} $ t1 $ t2) =
  108.32 +  | fm_of_term vs (@{term "(\<longleftrightarrow>) :: bool \<Rightarrow> bool \<Rightarrow> bool"} $ t1 $ t2) =
  108.33        @{code Iff} (fm_of_term vs t1, fm_of_term vs t2)
  108.34    | fm_of_term vs (@{term HOL.conj} $ t1 $ t2) = @{code And} (fm_of_term vs t1, fm_of_term vs t2)
  108.35    | fm_of_term vs (@{term HOL.disj} $ t1 $ t2) = @{code Or} (fm_of_term vs t1, fm_of_term vs t2)
  108.36 @@ -2510,32 +2510,32 @@
  108.37        HOLogic.mk_number HOLogic.intT (@{code integer_of_int} i)
  108.38    | term_of_num vs (@{code Bound} n) = Free (nth vs (@{code integer_of_nat} n))
  108.39    | term_of_num vs (@{code Neg} t') = @{term "uminus :: real \<Rightarrow> real"} $ term_of_num vs t'
  108.40 -  | term_of_num vs (@{code Add} (t1, t2)) = @{term "op + :: real \<Rightarrow> real \<Rightarrow> real"} $
  108.41 +  | term_of_num vs (@{code Add} (t1, t2)) = @{term "(+) :: real \<Rightarrow> real \<Rightarrow> real"} $
  108.42        term_of_num vs t1 $ term_of_num vs t2
  108.43 -  | term_of_num vs (@{code Sub} (t1, t2)) = @{term "op - :: real \<Rightarrow> real \<Rightarrow> real"} $
  108.44 +  | term_of_num vs (@{code Sub} (t1, t2)) = @{term "(-) :: real \<Rightarrow> real \<Rightarrow> real"} $
  108.45        term_of_num vs t1 $ term_of_num vs t2
  108.46 -  | term_of_num vs (@{code Mul} (i, t2)) = @{term "op * :: real \<Rightarrow> real \<Rightarrow> real"} $
  108.47 +  | term_of_num vs (@{code Mul} (i, t2)) = @{term "( * ) :: real \<Rightarrow> real \<Rightarrow> real"} $
  108.48        term_of_num vs (@{code C} i) $ term_of_num vs t2
  108.49    | term_of_num vs (@{code CN} (n, i, t)) = term_of_num vs (@{code Add} (@{code Mul} (i, @{code Bound} n), t));
  108.50  
  108.51  fun term_of_fm vs @{code T} = @{term True}
  108.52    | term_of_fm vs @{code F} = @{term False}
  108.53 -  | term_of_fm vs (@{code Lt} t) = @{term "op < :: real \<Rightarrow> real \<Rightarrow> bool"} $
  108.54 +  | term_of_fm vs (@{code Lt} t) = @{term "(<) :: real \<Rightarrow> real \<Rightarrow> bool"} $
  108.55        term_of_num vs t $ @{term "0::real"}
  108.56 -  | term_of_fm vs (@{code Le} t) = @{term "op \<le> :: real \<Rightarrow> real \<Rightarrow> bool"} $
  108.57 +  | term_of_fm vs (@{code Le} t) = @{term "(\<le>) :: real \<Rightarrow> real \<Rightarrow> bool"} $
  108.58        term_of_num vs t $ @{term "0::real"}
  108.59 -  | term_of_fm vs (@{code Gt} t) = @{term "op < :: real \<Rightarrow> real \<Rightarrow> bool"} $
  108.60 +  | term_of_fm vs (@{code Gt} t) = @{term "(<) :: real \<Rightarrow> real \<Rightarrow> bool"} $
  108.61        @{term "0::real"} $ term_of_num vs t
  108.62 -  | term_of_fm vs (@{code Ge} t) = @{term "op \<le> :: real \<Rightarrow> real \<Rightarrow> bool"} $
  108.63 +  | term_of_fm vs (@{code Ge} t) = @{term "(\<le>) :: real \<Rightarrow> real \<Rightarrow> bool"} $
  108.64        @{term "0::real"} $ term_of_num vs t
  108.65 -  | term_of_fm vs (@{code Eq} t) = @{term "op = :: real \<Rightarrow> real \<Rightarrow> bool"} $
  108.66 +  | term_of_fm vs (@{code Eq} t) = @{term "(=) :: real \<Rightarrow> real \<Rightarrow> bool"} $
  108.67        term_of_num vs t $ @{term "0::real"}
  108.68    | term_of_fm vs (@{code NEq} t) = term_of_fm vs (@{code NOT} (@{code Eq} t))
  108.69    | term_of_fm vs (@{code NOT} t') = HOLogic.Not $ term_of_fm vs t'
  108.70    | term_of_fm vs (@{code And} (t1, t2)) = HOLogic.conj $ term_of_fm vs t1 $ term_of_fm vs t2
  108.71    | term_of_fm vs (@{code Or} (t1, t2)) = HOLogic.disj $ term_of_fm vs t1 $ term_of_fm vs t2
  108.72    | term_of_fm vs (@{code Imp}  (t1, t2)) = HOLogic.imp $ term_of_fm vs t1 $ term_of_fm vs t2
  108.73 -  | term_of_fm vs (@{code Iff} (t1, t2)) = @{term "op \<longleftrightarrow> :: bool \<Rightarrow> bool \<Rightarrow> bool"} $
  108.74 +  | term_of_fm vs (@{code Iff} (t1, t2)) = @{term "(\<longleftrightarrow>) :: bool \<Rightarrow> bool \<Rightarrow> bool"} $
  108.75        term_of_fm vs t1 $ term_of_fm vs t2;
  108.76  
  108.77  in fn (ctxt, t) =>
   109.1 --- a/src/HOL/Decision_Procs/MIR.thy	Wed Jan 10 15:21:49 2018 +0100
   109.2 +++ b/src/HOL/Decision_Procs/MIR.thy	Wed Jan 10 15:25:09 2018 +0100
   109.3 @@ -5563,7 +5563,7 @@
   109.4  val mk_Dvd = @{code Dvd} o apfst @{code int_of_integer};
   109.5  val mk_Bound = @{code Bound} o @{code nat_of_integer};
   109.6  
   109.7 -fun num_of_term vs (t as Free (xn, xT)) = (case AList.lookup (op =) vs t
   109.8 +fun num_of_term vs (t as Free (xn, xT)) = (case AList.lookup (=) vs t
   109.9       of NONE => error "Variable not found in the list!"
  109.10        | SOME n => mk_Bound n)
  109.11    | num_of_term vs @{term "of_int (0::int)"} = mk_C 0
  109.12 @@ -5573,11 +5573,11 @@
  109.13    | num_of_term vs @{term "- 1::real"} = mk_C (~ 1)
  109.14    | num_of_term vs (Bound i) = mk_Bound i
  109.15    | num_of_term vs (@{term "uminus :: real \<Rightarrow> real"} $ t') = @{code Neg} (num_of_term vs t')
  109.16 -  | num_of_term vs (@{term "op + :: real \<Rightarrow> real \<Rightarrow> real"} $ t1 $ t2) =
  109.17 +  | num_of_term vs (@{term "(+) :: real \<Rightarrow> real \<Rightarrow> real"} $ t1 $ t2) =
  109.18        @{code Add} (num_of_term vs t1, num_of_term vs t2)
  109.19 -  | num_of_term vs (@{term "op - :: real \<Rightarrow> real \<Rightarrow> real"} $ t1 $ t2) =
  109.20 +  | num_of_term vs (@{term "(-) :: real \<Rightarrow> real \<Rightarrow> real"} $ t1 $ t2) =
  109.21        @{code Sub} (num_of_term vs t1, num_of_term vs t2)
  109.22 -  | num_of_term vs (@{term "op * :: real \<Rightarrow> real \<Rightarrow> real"} $ t1 $ t2) =
  109.23 +  | num_of_term vs (@{term "( * ) :: real \<Rightarrow> real \<Rightarrow> real"} $ t1 $ t2) =
  109.24        (case (num_of_term vs t1)
  109.25         of @{code C} i => @{code Mul} (i, num_of_term vs t2)
  109.26          | _ => error "num_of_term: unsupported Multiplication")
  109.27 @@ -5597,17 +5597,17 @@
  109.28  
  109.29  fun fm_of_term vs @{term True} = @{code T}
  109.30    | fm_of_term vs @{term False} = @{code F}
  109.31 -  | fm_of_term vs (@{term "op < :: real \<Rightarrow> real \<Rightarrow> bool"} $ t1 $ t2) =
  109.32 +  | fm_of_term vs (@{term "(<) :: real \<Rightarrow> real \<Rightarrow> bool"} $ t1 $ t2) =
  109.33        @{code Lt} (@{code Sub} (num_of_term vs t1, num_of_term vs t2))
  109.34 -  | fm_of_term vs (@{term "op \<le> :: real \<Rightarrow> real \<Rightarrow> bool"} $ t1 $ t2) =
  109.35 +  | fm_of_term vs (@{term "(\<le>) :: real \<Rightarrow> real \<Rightarrow> bool"} $ t1 $ t2) =
  109.36        @{code Le} (@{code Sub} (num_of_term vs t1, num_of_term vs t2))
  109.37 -  | fm_of_term vs (@{term "op = :: real \<Rightarrow> real \<Rightarrow> bool"} $ t1 $ t2) =
  109.38 +  | fm_of_term vs (@{term "(=) :: real \<Rightarrow> real \<Rightarrow> bool"} $ t1 $ t2) =
  109.39        @{code Eq} (@{code Sub} (num_of_term vs t1, num_of_term vs t2))
  109.40 -  | fm_of_term vs (@{term "op rdvd"} $ (@{term "of_int :: int \<Rightarrow> real"} $ (@{term "numeral :: _ \<Rightarrow> int"} $ t1)) $ t2) =
  109.41 +  | fm_of_term vs (@{term "(rdvd)"} $ (@{term "of_int :: int \<Rightarrow> real"} $ (@{term "numeral :: _ \<Rightarrow> int"} $ t1)) $ t2) =
  109.42        mk_Dvd (HOLogic.dest_numeral t1, num_of_term vs t2)
  109.43 -  | fm_of_term vs (@{term "op rdvd"} $ (@{term "of_int :: int \<Rightarrow> real"} $ (@{term "- numeral :: _ \<Rightarrow> int"} $ t1)) $ t2) =
  109.44 +  | fm_of_term vs (@{term "(rdvd)"} $ (@{term "of_int :: int \<Rightarrow> real"} $ (@{term "- numeral :: _ \<Rightarrow> int"} $ t1)) $ t2) =
  109.45        mk_Dvd (~ (HOLogic.dest_numeral t1), num_of_term vs t2)
  109.46 -  | fm_of_term vs (@{term "op = :: bool \<Rightarrow> bool \<Rightarrow> bool"} $ t1 $ t2) =
  109.47 +  | fm_of_term vs (@{term "(=) :: bool \<Rightarrow> bool \<Rightarrow> bool"} $ t1 $ t2) =
  109.48        @{code Iff} (fm_of_term vs t1, fm_of_term vs t2)
  109.49    | fm_of_term vs (@{term HOL.conj} $ t1 $ t2) =
  109.50        @{code And} (fm_of_term vs t1, fm_of_term vs t2)
  109.51 @@ -5632,11 +5632,11 @@
  109.52    | term_of_num vs (@{code Neg} (@{code Floor} (@{code Neg} t'))) =
  109.53        @{term "of_int :: int \<Rightarrow> real"} $ (@{term "ceiling :: real \<Rightarrow> int"} $ term_of_num vs t')
  109.54    | term_of_num vs (@{code Neg} t') = @{term "uminus :: real \<Rightarrow> real"} $ term_of_num vs t'
  109.55 -  | term_of_num vs (@{code Add} (t1, t2)) = @{term "op + :: real \<Rightarrow> real \<Rightarrow> real"} $
  109.56 +  | term_of_num vs (@{code Add} (t1, t2)) = @{term "(+) :: real \<Rightarrow> real \<Rightarrow> real"} $
  109.57        term_of_num vs t1 $ term_of_num vs t2
  109.58 -  | term_of_num vs (@{code Sub} (t1, t2)) = @{term "op - :: real \<Rightarrow> real \<Rightarrow> real"} $
  109.59 +  | term_of_num vs (@{code Sub} (t1, t2)) = @{term "(-) :: real \<Rightarrow> real \<Rightarrow> real"} $
  109.60        term_of_num vs t1 $ term_of_num vs t2
  109.61 -  | term_of_num vs (@{code Mul} (i, t2)) = @{term "op * :: real \<Rightarrow> real \<Rightarrow> real"} $
  109.62 +  | term_of_num vs (@{code Mul} (i, t2)) = @{term "( * ) :: real \<Rightarrow> real \<Rightarrow> real"} $
  109.63        term_of_num vs (@{code C} i) $ term_of_num vs t2
  109.64    | term_of_num vs (@{code Floor} t) = @{term "of_int :: int \<Rightarrow> real"} $ (@{term "floor :: real \<Rightarrow> int"} $ term_of_num vs t)
  109.65    | term_of_num vs (@{code CN} (n, i, t)) = term_of_num vs (@{code Add} (@{code Mul} (i, @{code Bound} n), t))
  109.66 @@ -5645,19 +5645,19 @@
  109.67  fun term_of_fm vs @{code T} = @{term True}
  109.68    | term_of_fm vs @{code F} = @{term False}
  109.69    | term_of_fm vs (@{code Lt} t) =
  109.70 -      @{term "op < :: real \<Rightarrow> real \<Rightarrow> bool"} $ term_of_num vs t $ @{term "0::real"}
  109.71 +      @{term "(<) :: real \<Rightarrow> real \<Rightarrow> bool"} $ term_of_num vs t $ @{term "0::real"}
  109.72    | term_of_fm vs (@{code Le} t) =
  109.73 -      @{term "op \<le> :: real \<Rightarrow> real \<Rightarrow> bool"} $ term_of_num vs t $ @{term "0::real"}
  109.74 +      @{term "(\<le>) :: real \<Rightarrow> real \<Rightarrow> bool"} $ term_of_num vs t $ @{term "0::real"}
  109.75    | term_of_fm vs (@{code Gt} t) =
  109.76 -      @{term "op < :: real \<Rightarrow> real \<Rightarrow> bool"} $ @{term "0::real"} $ term_of_num vs t
  109.77 +      @{term "(<) :: real \<Rightarrow> real \<Rightarrow> bool"} $ @{term "0::real"} $ term_of_num vs t
  109.78    | term_of_fm vs (@{code Ge} t) =
  109.79 -      @{term "op \<le> :: real \<Rightarrow> real \<Rightarrow> bool"} $ @{term "0::real"} $ term_of_num vs t
  109.80 +      @{term "(\<le>) :: real \<Rightarrow> real \<Rightarrow> bool"} $ @{term "0::real"} $ term_of_num vs t
  109.81    | term_of_fm vs (@{code Eq} t) =
  109.82 -      @{term "op = :: real \<Rightarrow> real \<Rightarrow> bool"} $ term_of_num vs t $ @{term "0::real"}
  109.83 +      @{term "(=) :: real \<Rightarrow> real \<Rightarrow> bool"} $ term_of_num vs t $ @{term "0::real"}
  109.84    | term_of_fm vs (@{code NEq} t) =
  109.85        term_of_fm vs (@{code NOT} (@{code Eq} t))
  109.86    | term_of_fm vs (@{code Dvd} (i, t)) =
  109.87 -      @{term "op rdvd"} $ term_of_num vs (@{code C} i) $ term_of_num vs t
  109.88 +      @{term "(rdvd)"} $ term_of_num vs (@{code C} i) $ term_of_num vs t
  109.89    | term_of_fm vs (@{code NDvd} (i, t)) =
  109.90        term_of_fm vs (@{code NOT} (@{code Dvd} (i, t)))
  109.91    | term_of_fm vs (@{code NOT} t') =
  109.92 @@ -5669,7 +5669,7 @@
  109.93    | term_of_fm vs (@{code Imp}  (t1, t2)) =
  109.94        HOLogic.imp $ term_of_fm vs t1 $ term_of_fm vs t2
  109.95    | term_of_fm vs (@{code Iff} (t1, t2)) =
  109.96 -      @{term "op = :: bool \<Rightarrow> bool \<Rightarrow> bool"} $ term_of_fm vs t1 $ term_of_fm vs t2;
  109.97 +      @{term "(=) :: bool \<Rightarrow> bool \<Rightarrow> bool"} $ term_of_fm vs t1 $ term_of_fm vs t2;
  109.98  
  109.99  in
 109.100    fn (ctxt, t) =>
   110.1 --- a/src/HOL/Decision_Procs/Parametric_Ferrante_Rackoff.thy	Wed Jan 10 15:21:49 2018 +0100
   110.2 +++ b/src/HOL/Decision_Procs/Parametric_Ferrante_Rackoff.thy	Wed Jan 10 15:25:09 2018 +0100
   110.3 @@ -4066,7 +4066,7 @@
   110.4  fun frpar_procedure alternative T ps fm =
   110.5    let
   110.6      val frpar = if alternative then @{code frpar2} else @{code frpar};
   110.7 -    val fs = subtract (op aconv) (map Free (Term.add_frees fm [])) ps;
   110.8 +    val fs = subtract (aconv) (map Free (Term.add_frees fm [])) ps;
   110.9      val eval = term_of_fm T fs ps o frpar o fm_of_term fs ps;
  110.10      val t = HOLogic.dest_Trueprop fm;
  110.11    in HOLogic.mk_Trueprop (HOLogic.mk_eq (t, eval t)) end;
   111.1 --- a/src/HOL/Decision_Procs/Polynomial_List.thy	Wed Jan 10 15:21:49 2018 +0100
   111.2 +++ b/src/HOL/Decision_Procs/Polynomial_List.thy	Wed Jan 10 15:25:09 2018 +0100
   111.3 @@ -186,7 +186,7 @@
   111.4      by (cases "x = zero") (auto simp add: distrib_left ac_simps)
   111.5  qed
   111.6  
   111.7 -lemma (in comm_semiring_0) poly_cmult_map: "poly (map (op * c) p) x = c * poly p x"
   111.8 +lemma (in comm_semiring_0) poly_cmult_map: "poly (map (( * ) c) p) x = c * poly p x"
   111.9    by (induct p) (auto simp add: distrib_left ac_simps)
  111.10  
  111.11  lemma (in comm_ring_1) poly_minus: "poly (-- p) x = - (poly p x)"
   112.1 --- a/src/HOL/Decision_Procs/approximation.ML	Wed Jan 10 15:21:49 2018 +0100
   112.2 +++ b/src/HOL/Decision_Procs/approximation.ML	Wed Jan 10 15:25:09 2018 +0100
   112.3 @@ -105,9 +105,9 @@
   112.4  
   112.5  fun calculated_subterms (@{const Trueprop} $ t) = calculated_subterms t
   112.6    | calculated_subterms (@{const HOL.implies} $ _ $ t) = calculated_subterms t
   112.7 -  | calculated_subterms (@{term "op <= :: real \<Rightarrow> real \<Rightarrow> bool"} $ t1 $ t2) = [t1, t2]
   112.8 -  | calculated_subterms (@{term "op < :: real \<Rightarrow> real \<Rightarrow> bool"} $ t1 $ t2) = [t1, t2]
   112.9 -  | calculated_subterms (@{term "op : :: real \<Rightarrow> real set \<Rightarrow> bool"} $ t1 $
  112.10 +  | calculated_subterms (@{term "(<=) :: real \<Rightarrow> real \<Rightarrow> bool"} $ t1 $ t2) = [t1, t2]
  112.11 +  | calculated_subterms (@{term "(<) :: real \<Rightarrow> real \<Rightarrow> bool"} $ t1 $ t2) = [t1, t2]
  112.12 +  | calculated_subterms (@{term "(:) :: real \<Rightarrow> real set \<Rightarrow> bool"} $ t1 $
  112.13                           (@{term "atLeastAtMost :: real \<Rightarrow> real \<Rightarrow> real set"} $ t2 $ t3)) = [t1, t2, t3]
  112.14    | calculated_subterms t = raise TERM ("calculated_subterms", [t])
  112.15  
  112.16 @@ -246,7 +246,7 @@
  112.17           |> approximate ctxt
  112.18           |> HOLogic.dest_list
  112.19           |> curry ListPair.zip (HOLogic.dest_list xs @ calculated_subterms arith_term)
  112.20 -         |> map (fn (elem, s) => @{term "op : :: real \<Rightarrow> real set \<Rightarrow> bool"} $ elem $ mk_result prec (dest_ivl s))
  112.21 +         |> map (fn (elem, s) => @{term "(:) :: real \<Rightarrow> real set \<Rightarrow> bool"} $ elem $ mk_result prec (dest_ivl s))
  112.22           |> foldr1 HOLogic.mk_conj))
  112.23  
  112.24  fun approx_arith prec ctxt t = realify t
   113.1 --- a/src/HOL/Decision_Procs/langford.ML	Wed Jan 10 15:21:49 2018 +0100
   113.2 +++ b/src/HOL/Decision_Procs/langford.ML	Wed Jan 10 15:25:09 2018 +0100
   113.3 @@ -145,7 +145,7 @@
   113.4                case ndx of
   113.5                  [] => NONE
   113.6                | _ =>
   113.7 -                conj_aci_rule (Thm.mk_binop @{cterm "op \<equiv> :: prop => _"} Pp
   113.8 +                conj_aci_rule (Thm.mk_binop @{cterm "(\<equiv>) :: prop => _"} Pp
   113.9                    (Thm.apply @{cterm Trueprop} (list_conj (ndx @ dx))))
  113.10                  |> Thm.abstract_rule xn x
  113.11                  |> Drule.arg_cong_rule e
  113.12 @@ -156,7 +156,7 @@
  113.13                  |> SOME
  113.14              end
  113.15          | _ =>
  113.16 -            conj_aci_rule (Thm.mk_binop @{cterm "op \<equiv> :: prop => _"} Pp
  113.17 +            conj_aci_rule (Thm.mk_binop @{cterm "(\<equiv>) :: prop => _"} Pp
  113.18                (Thm.apply @{cterm Trueprop} (list_conj (eqs @ neqs))))
  113.19              |> Thm.abstract_rule xn x |> Drule.arg_cong_rule e
  113.20              |> Conv.fconv_rule
   114.1 --- a/src/HOL/Deriv.thy	Wed Jan 10 15:21:49 2018 +0100
   114.2 +++ b/src/HOL/Deriv.thy	Wed Jan 10 15:25:09 2018 +0100
   114.3 @@ -31,7 +31,7 @@
   114.4  
   114.5  definition has_field_derivative :: "('a::real_normed_field \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a filter \<Rightarrow> bool"
   114.6      (infix "(has'_field'_derivative)" 50)
   114.7 -  where "(f has_field_derivative D) F \<longleftrightarrow> (f has_derivative op * D) F"
   114.8 +  where "(f has_field_derivative D) F \<longleftrightarrow> (f has_derivative ( * ) D) F"
   114.9  
  114.10  lemma DERIV_cong: "(f has_field_derivative X) F \<Longrightarrow> X = Y \<Longrightarrow> (f has_field_derivative Y) F"
  114.11    by simp
  114.12 @@ -150,7 +150,7 @@
  114.13  
  114.14  lemma field_has_derivative_at:
  114.15    fixes x :: "'a::real_normed_field"
  114.16 -  shows "(f has_derivative op * D) (at x) \<longleftrightarrow> (\<lambda>h. (f (x + h) - f x) / h) \<midarrow>0\<rightarrow> D"
  114.17 +  shows "(f has_derivative ( * ) D) (at x) \<longleftrightarrow> (\<lambda>h. (f (x + h) - f x) / h) \<midarrow>0\<rightarrow> D"
  114.18    apply (unfold has_derivative_at)
  114.19    apply (simp add: bounded_linear_mult_right)
  114.20    apply (simp cong: LIM_cong add: nonzero_norm_divide [symmetric])
  114.21 @@ -588,7 +588,7 @@
  114.22    by (rule has_derivative_eq_rhs[of f D]) (simp_all add: fun_eq_iff mult.commute)
  114.23  
  114.24  lemma has_field_derivative_imp_has_derivative:
  114.25 -  "(f has_field_derivative D) F \<Longrightarrow> (f has_derivative op * D) F"
  114.26 +  "(f has_field_derivative D) F \<Longrightarrow> (f has_derivative ( * ) D) F"
  114.27    by (simp add: has_field_derivative_def)
  114.28  
  114.29  lemma DERIV_subset:
  114.30 @@ -615,7 +615,7 @@
  114.31    assume "f differentiable at x within s"
  114.32    then obtain f' where *: "(f has_derivative f') (at x within s)"
  114.33      unfolding differentiable_def by auto
  114.34 -  then obtain c where "f' = (op * c)"
  114.35 +  then obtain c where "f' = (( * ) c)"
  114.36      by (metis real_bounded_linear has_derivative_bounded_linear mult.commute fun_eq_iff)
  114.37    with * show "\<exists>D. (f has_real_derivative D) (at x within s)"
  114.38      unfolding has_field_derivative_def by auto
  114.39 @@ -647,7 +647,7 @@
  114.40  lemma DERIV_def: "DERIV f x :> D \<longleftrightarrow> (\<lambda>h. (f (x + h) - f x) / h) \<midarrow>0\<rightarrow> D"
  114.41    unfolding field_has_derivative_at has_field_derivative_def has_field_derivative_iff ..
  114.42  
  114.43 -lemma mult_commute_abs: "(\<lambda>x. x * c) = op * c"
  114.44 +lemma mult_commute_abs: "(\<lambda>x. x * c) = ( * ) c"
  114.45    for c :: "'a::ab_semigroup_mult"
  114.46    by (simp add: fun_eq_iff mult.commute)
  114.47  
  114.48 @@ -830,7 +830,7 @@
  114.49      ((\<lambda>x. f x * c) has_field_derivative D * c) (at x within s)"
  114.50    using DERIV_cmult by (auto simp add: ac_simps)
  114.51  
  114.52 -lemma DERIV_cmult_Id [simp]: "(op * c has_field_derivative c) (at x within s)"
  114.53 +lemma DERIV_cmult_Id [simp]: "(( * ) c has_field_derivative c) (at x within s)"
  114.54    using DERIV_ident [THEN DERIV_cmult, where c = c and x = x] by simp
  114.55  
  114.56  lemma DERIV_cdivide:
  114.57 @@ -853,7 +853,7 @@
  114.58    shows "((\<lambda>x. inverse (f x)) has_field_derivative - (inverse (f x) * D * inverse (f x)))
  114.59      (at x within s)"
  114.60  proof -
  114.61 -  have "(f has_derivative (\<lambda>x. x * D)) = (f has_derivative op * D)"
  114.62 +  have "(f has_derivative (\<lambda>x. x * D)) = (f has_derivative ( * ) D)"
  114.63      by (rule arg_cong [of "\<lambda>x. x * D"]) (simp add: fun_eq_iff)
  114.64    with assms have "(f has_derivative (\<lambda>x. x * D)) (at x within s)"
  114.65      by (auto dest!: has_field_derivative_imp_has_derivative)
  114.66 @@ -906,7 +906,7 @@
  114.67  
  114.68  lemma DERIV_chain': "(f has_field_derivative D) (at x within s) \<Longrightarrow> DERIV g (f x) :> E \<Longrightarrow>
  114.69    ((\<lambda>x. g (f x)) has_field_derivative E * D) (at x within s)"
  114.70 -  using has_derivative_compose[of f "op * D" x s g "op * E"]
  114.71 +  using has_derivative_compose[of f "( * ) D" x s g "( * ) E"]
  114.72    by (simp only: has_field_derivative_def mult_commute_abs ac_simps)
  114.73  
  114.74  corollary DERIV_chain2: "DERIV f (g x) :> Da \<Longrightarrow> (g has_field_derivative Db) (at x within s) \<Longrightarrow>
  114.75 @@ -924,7 +924,7 @@
  114.76    "(f has_field_derivative Da) (at (g x) within (g ` s)) \<Longrightarrow>
  114.77      (g has_field_derivative Db) (at x within s) \<Longrightarrow>
  114.78      (f \<circ> g has_field_derivative Da * Db) (at x within s)"
  114.79 -  using has_derivative_in_compose [of g "op * Db" x s f "op * Da "]
  114.80 +  using has_derivative_in_compose [of g "( * ) Db" x s f "( * ) Da "]
  114.81    by (simp add: has_field_derivative_def o_def mult_commute_abs ac_simps)
  114.82  
  114.83  (*These two are from HOL Light: HAS_COMPLEX_DERIVATIVE_CHAIN*)
   115.1 --- a/src/HOL/Enum.thy	Wed Jan 10 15:21:49 2018 +0100
   115.2 +++ b/src/HOL/Enum.thy	Wed Jan 10 15:25:09 2018 +0100
   115.3 @@ -572,7 +572,7 @@
   115.4  
   115.5  instantiation finite_1 :: complete_boolean_algebra
   115.6  begin
   115.7 -definition [simp]: "op - = (\<lambda>_ _. a\<^sub>1)"
   115.8 +definition [simp]: "(-) = (\<lambda>_ _. a\<^sub>1)"
   115.9  definition [simp]: "uminus = (\<lambda>_. a\<^sub>1)"
  115.10  instance by intro_classes simp_all
  115.11  end
  115.12 @@ -584,9 +584,9 @@
  115.13  begin
  115.14  definition [simp]: "Groups.zero = a\<^sub>1"
  115.15  definition [simp]: "Groups.one = a\<^sub>1"
  115.16 -definition [simp]: "op + = (\<lambda>_ _. a\<^sub>1)"
  115.17 -definition [simp]: "op * = (\<lambda>_ _. a\<^sub>1)"
  115.18 -definition [simp]: "op mod = (\<lambda>_ _. a\<^sub>1)" 
  115.19 +definition [simp]: "(+) = (\<lambda>_ _. a\<^sub>1)"
  115.20 +definition [simp]: "( * ) = (\<lambda>_ _. a\<^sub>1)"
  115.21 +definition [simp]: "(mod) = (\<lambda>_ _. a\<^sub>1)" 
  115.22  definition [simp]: "abs = (\<lambda>_. a\<^sub>1)"
  115.23  definition [simp]: "sgn = (\<lambda>_. a\<^sub>1)"
  115.24  definition [simp]: "inverse = (\<lambda>_. a\<^sub>1)"
  115.25 @@ -688,10 +688,10 @@
  115.26  definition [simp]: "1 = a\<^sub>2"
  115.27  definition "x + y = (case (x, y) of (a\<^sub>1, a\<^sub>1) \<Rightarrow> a\<^sub>1 | (a\<^sub>2, a\<^sub>2) \<Rightarrow> a\<^sub>1 | _ \<Rightarrow> a\<^sub>2)"
  115.28  definition "uminus = (\<lambda>x :: finite_2. x)"
  115.29 -definition "op - = (op + :: finite_2 \<Rightarrow> _)"
  115.30 +definition "(-) = ((+) :: finite_2 \<Rightarrow> _)"
  115.31  definition "x * y = (case (x, y) of (a\<^sub>2, a\<^sub>2) \<Rightarrow> a\<^sub>2 | _ \<Rightarrow> a\<^sub>1)"
  115.32  definition "inverse = (\<lambda>x :: finite_2. x)"
  115.33 -definition "divide = (op * :: finite_2 \<Rightarrow> _)"
  115.34 +definition "divide = (( * ) :: finite_2 \<Rightarrow> _)"
  115.35  definition "x mod y = (case (x, y) of (a\<^sub>2, a\<^sub>1) \<Rightarrow> a\<^sub>2 | _ \<Rightarrow> a\<^sub>1)"
  115.36  definition "abs = (\<lambda>x :: finite_2. x)"
  115.37  definition "sgn = (\<lambda>x :: finite_2. x)"
   116.1 --- a/src/HOL/Equiv_Relations.thy	Wed Jan 10 15:21:49 2018 +0100
   116.2 +++ b/src/HOL/Equiv_Relations.thy	Wed Jan 10 15:25:09 2018 +0100
   116.3 @@ -472,7 +472,7 @@
   116.4  lemma equivp_reflp_symp_transp: "equivp R \<longleftrightarrow> reflp R \<and> symp R \<and> transp R"
   116.5    by (auto intro: equivpI elim: equivpE)
   116.6  
   116.7 -lemma identity_equivp: "equivp (op =)"
   116.8 +lemma identity_equivp: "equivp (=)"
   116.9    by (auto intro: equivpI reflpI sympI transpI)
  116.10  
  116.11  lemma equivp_reflp: "equivp R \<Longrightarrow> R x x"
   117.1 --- a/src/HOL/Factorial.thy	Wed Jan 10 15:21:49 2018 +0100
   117.2 +++ b/src/HOL/Factorial.thy	Wed Jan 10 15:25:09 2018 +0100
   117.3 @@ -389,13 +389,13 @@
   117.4  subsection \<open>Misc\<close>
   117.5  
   117.6  lemma fact_code [code]:
   117.7 -  "fact n = (of_nat (fold_atLeastAtMost_nat (op *) 2 n 1) :: 'a::semiring_char_0)"
   117.8 +  "fact n = (of_nat (fold_atLeastAtMost_nat (( * )) 2 n 1) :: 'a::semiring_char_0)"
   117.9  proof -
  117.10    have "fact n = (of_nat (\<Prod>{1..n}) :: 'a)"
  117.11      by (simp add: fact_prod)
  117.12    also have "\<Prod>{1..n} = \<Prod>{2..n}"
  117.13      by (intro prod.mono_neutral_right) auto
  117.14 -  also have "\<dots> = fold_atLeastAtMost_nat (op *) 2 n 1"
  117.15 +  also have "\<dots> = fold_atLeastAtMost_nat (( * )) 2 n 1"
  117.16      by (simp add: prod_atLeastAtMost_code)
  117.17    finally show ?thesis .
  117.18  qed
   118.1 --- a/src/HOL/Filter.thy	Wed Jan 10 15:21:49 2018 +0100
   118.2 +++ b/src/HOL/Filter.thy	Wed Jan 10 15:25:09 2018 +0100
   118.3 @@ -1372,11 +1372,11 @@
   118.4  begin
   118.5  
   118.6  definition rel_filter :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'a filter \<Rightarrow> 'b filter \<Rightarrow> bool"
   118.7 -where "rel_filter R F G = ((R ===> op =) ===> op =) (Rep_filter F) (Rep_filter G)"
   118.8 +where "rel_filter R F G = ((R ===> (=)) ===> (=)) (Rep_filter F) (Rep_filter G)"
   118.9  
  118.10  lemma rel_filter_eventually:
  118.11    "rel_filter R F G \<longleftrightarrow>
  118.12 -  ((R ===> op =) ===> op =) (\<lambda>P. eventually P F) (\<lambda>P. eventually P G)"
  118.13 +  ((R ===> (=)) ===> (=)) (\<lambda>P. eventually P F) (\<lambda>P. eventually P G)"
  118.14  by(simp add: rel_filter_def eventually_def)
  118.15  
  118.16  lemma filtermap_id [simp, id_simps]: "filtermap id = id"
  118.17 @@ -1408,14 +1408,14 @@
  118.18           fun_quotient[OF fun_quotient[OF Q identity_quotient] identity_quotient, unfolded Quotient_alt_def])
  118.19  
  118.20  lemma eventually_parametric [transfer_rule]:
  118.21 -  "((A ===> op =) ===> rel_filter A ===> op =) eventually eventually"
  118.22 +  "((A ===> (=)) ===> rel_filter A ===> (=)) eventually eventually"
  118.23  by(simp add: rel_fun_def rel_filter_eventually)
  118.24  
  118.25  lemma frequently_parametric [transfer_rule]:
  118.26 -  "((A ===> op =) ===> rel_filter A ===> op =) frequently frequently"
  118.27 +  "((A ===> (=)) ===> rel_filter A ===> (=)) frequently frequently"
  118.28    unfolding frequently_def[abs_def] by transfer_prover
  118.29  
  118.30 -lemma rel_filter_eq [relator_eq]: "rel_filter op = = op ="
  118.31 +lemma rel_filter_eq [relator_eq]: "rel_filter (=) = (=)"
  118.32  by(auto simp add: rel_filter_eventually rel_fun_eq fun_eq_iff filter_eq_iff)
  118.33  
  118.34  lemma rel_filter_mono [relator_mono]:
  118.35 @@ -1431,7 +1431,7 @@
  118.36  lemma is_filter_parametric_aux:
  118.37    assumes "is_filter F"
  118.38    assumes [transfer_rule]: "bi_total A" "bi_unique A"
  118.39 -  and [transfer_rule]: "((A ===> op =) ===> op =) F G"
  118.40 +  and [transfer_rule]: "((A ===> (=)) ===> (=)) F G"
  118.41    shows "is_filter G"
  118.42  proof -
  118.43    interpret is_filter F by fact
  118.44 @@ -1444,7 +1444,7 @@
  118.45      assume "G P'" "G Q'"
  118.46      moreover
  118.47      from bi_total_fun[OF \<open>bi_unique A\<close> bi_total_eq, unfolded bi_total_def]
  118.48 -    obtain P Q where [transfer_rule]: "(A ===> op =) P P'" "(A ===> op =) Q Q'" by blast
  118.49 +    obtain P Q where [transfer_rule]: "(A ===> (=)) P P'" "(A ===> (=)) Q Q'" by blast
  118.50      have "F P = G P'" "F Q = G Q'" by transfer_prover+
  118.51      ultimately have "F (\<lambda>x. P x \<and> Q x)" by(simp add: conj)
  118.52      moreover have "F (\<lambda>x. P x \<and> Q x) = G (\<lambda>x. P' x \<and> Q' x)" by transfer_prover
  118.53 @@ -1454,7 +1454,7 @@
  118.54      assume "\<forall>x. P' x \<longrightarrow> Q' x" "G P'"
  118.55      moreover
  118.56      from bi_total_fun[OF \<open>bi_unique A\<close> bi_total_eq, unfolded bi_total_def]
  118.57 -    obtain P Q where [transfer_rule]: "(A ===> op =) P P'" "(A ===> op =) Q Q'" by blast
  118.58 +    obtain P Q where [transfer_rule]: "(A ===> (=)) P P'" "(A ===> (=)) Q Q'" by blast
  118.59      have "F P = G P'" by transfer_prover
  118.60      moreover have "(\<forall>x. P x \<longrightarrow> Q x) \<longleftrightarrow> (\<forall>x. P' x \<longrightarrow> Q' x)" by transfer_prover
  118.61      ultimately have "F Q" by(simp add: mono)
  118.62 @@ -1465,7 +1465,7 @@
  118.63  
  118.64  lemma is_filter_parametric [transfer_rule]:
  118.65    "\<lbrakk> bi_total A; bi_unique A \<rbrakk>
  118.66 -  \<Longrightarrow> (((A ===> op =) ===> op =) ===> op =) is_filter is_filter"
  118.67 +  \<Longrightarrow> (((A ===> (=)) ===> (=)) ===> (=)) is_filter is_filter"
  118.68  apply(rule rel_funI)
  118.69  apply(rule iffI)
  118.70   apply(erule (3) is_filter_parametric_aux)
  118.71 @@ -1480,7 +1480,7 @@
  118.72  proof(rule left_totalI)
  118.73    fix F :: "'a filter"
  118.74    from bi_total_fun[OF bi_unique_fun[OF \<open>bi_total A\<close> bi_unique_eq] bi_total_eq]
  118.75 -  obtain G where [transfer_rule]: "((A ===> op =) ===> op =) (\<lambda>P. eventually P F) G"
  118.76 +  obtain G where [transfer_rule]: "((A ===> (=)) ===> (=)) (\<lambda>P. eventually P F) G"
  118.77      unfolding  bi_total_def by blast
  118.78    moreover have "is_filter (\<lambda>P. eventually P F) \<longleftrightarrow> is_filter G" by transfer_prover
  118.79    hence "is_filter G" by(simp add: eventually_def is_filter_Rep_filter)
  118.80 @@ -1509,7 +1509,7 @@
  118.81      unfolding filter_eq_iff
  118.82    proof
  118.83      fix P :: "'a \<Rightarrow> bool"
  118.84 -    obtain P' where [transfer_rule]: "(A ===> op =) P P'"
  118.85 +    obtain P' where [transfer_rule]: "(A ===> (=)) P P'"
  118.86        using left_total_fun[OF assms left_total_eq] unfolding left_total_def by blast
  118.87      have "eventually P F = eventually P' G"
  118.88        and "eventually P F' = eventually P' G" by transfer_prover+
  118.89 @@ -1579,11 +1579,11 @@
  118.90  begin
  118.91  
  118.92  lemma le_filter_parametric [transfer_rule]:
  118.93 -  "(rel_filter A ===> rel_filter A ===> op =) op \<le> op \<le>"
  118.94 +  "(rel_filter A ===> rel_filter A ===> (=)) (\<le>) (\<le>)"
  118.95  unfolding le_filter_def[abs_def] by transfer_prover
  118.96  
  118.97  lemma less_filter_parametric [transfer_rule]:
  118.98 -  "(rel_filter A ===> rel_filter A ===> op =) op < op <"
  118.99 +  "(rel_filter A ===> rel_filter A ===> (=)) (<) (<)"
 118.100  unfolding less_filter_def[abs_def] by transfer_prover
 118.101  
 118.102  context
   119.1 --- a/src/HOL/Fun.thy	Wed Jan 10 15:21:49 2018 +0100
   119.2 +++ b/src/HOL/Fun.thy	Wed Jan 10 15:25:09 2018 +0100
   119.3 @@ -84,7 +84,7 @@
   119.4  lemma image_eq_imp_comp: "f ` A = g ` B \<Longrightarrow> (h \<circ> f) ` A = (h \<circ> g) ` B"
   119.5    by (auto simp: comp_def elim!: equalityE)
   119.6  
   119.7 -lemma image_bind: "f ` (Set.bind A g) = Set.bind A (op ` f \<circ> g)"
   119.8 +lemma image_bind: "f ` (Set.bind A g) = Set.bind A ((`) f \<circ> g)"
   119.9    by (auto simp add: Set.bind_def)
  119.10  
  119.11  lemma bind_image: "Set.bind (f ` A) g = Set.bind A (g \<circ> f)"
   120.1 --- a/src/HOL/GCD.thy	Wed Jan 10 15:21:49 2018 +0100
   120.2 +++ b/src/HOL/GCD.thy	Wed Jan 10 15:25:09 2018 +0100
   120.3 @@ -996,25 +996,25 @@
   120.4  lemma dvd_Gcd_iff: "x dvd Gcd A \<longleftrightarrow> (\<forall>y\<in>A. x dvd y)"
   120.5    by (blast dest: dvd_GcdD intro: Gcd_greatest)
   120.6  
   120.7 -lemma Gcd_mult: "Gcd (op * c ` A) = normalize c * Gcd A"
   120.8 +lemma Gcd_mult: "Gcd (( * ) c ` A) = normalize c * Gcd A"
   120.9  proof (cases "c = 0")
  120.10    case True
  120.11    then show ?thesis by auto
  120.12  next
  120.13    case [simp]: False
  120.14 -  have "Gcd (op * c ` A) div c dvd Gcd A"
  120.15 +  have "Gcd (( * ) c ` A) div c dvd Gcd A"
  120.16      by (intro Gcd_greatest, subst div_dvd_iff_mult)
  120.17         (auto intro!: Gcd_greatest Gcd_dvd simp: mult.commute[of _ c])
  120.18 -  then have "Gcd (op * c ` A) dvd c * Gcd A"
  120.19 +  then have "Gcd (( * ) c ` A) dvd c * Gcd A"
  120.20      by (subst (asm) div_dvd_iff_mult) (auto intro: Gcd_greatest simp: mult_ac)
  120.21    also have "c * Gcd A = (normalize c * Gcd A) * unit_factor c"
  120.22      by (subst unit_factor_mult_normalize [symmetric]) (simp only: mult_ac)
  120.23 -  also have "Gcd (op * c ` A) dvd \<dots> \<longleftrightarrow> Gcd (op * c ` A) dvd normalize c * Gcd A"
  120.24 +  also have "Gcd (( * ) c ` A) dvd \<dots> \<longleftrightarrow> Gcd (( * ) c ` A) dvd normalize c * Gcd A"
  120.25      by (simp add: dvd_mult_unit_iff)
  120.26 -  finally have "Gcd (op * c ` A) dvd normalize c * Gcd A" .
  120.27 -  moreover have "normalize c * Gcd A dvd Gcd (op * c ` A)"
  120.28 +  finally have "Gcd (( * ) c ` A) dvd normalize c * Gcd A" .
  120.29 +  moreover have "normalize c * Gcd A dvd Gcd (( * ) c ` A)"
  120.30      by (intro Gcd_greatest) (auto intro: mult_dvd_mono Gcd_dvd)
  120.31 -  ultimately have "normalize (Gcd (op * c ` A)) = normalize (normalize c * Gcd A)"
  120.32 +  ultimately have "normalize (Gcd (( * ) c ` A)) = normalize (normalize c * Gcd A)"
  120.33      by (rule associatedI)
  120.34    then show ?thesis
  120.35      by (simp add: normalize_mult)
  120.36 @@ -1035,10 +1035,10 @@
  120.37  
  120.38  lemma Lcm_mult:
  120.39    assumes "A \<noteq> {}"
  120.40 -  shows "Lcm (op * c ` A) = normalize c * Lcm A"
  120.41 +  shows "Lcm (( * ) c ` A) = normalize c * Lcm A"
  120.42  proof (cases "c = 0")
  120.43    case True
  120.44 -  with assms have "op * c ` A = {0}"
  120.45 +  with assms have "( * ) c ` A = {0}"
  120.46      by auto
  120.47    with True show ?thesis by auto
  120.48  next
  120.49 @@ -1047,23 +1047,23 @@
  120.50      by blast
  120.51    have "c dvd c * x"
  120.52      by simp
  120.53 -  also from x have "c * x dvd Lcm (op * c ` A)"
  120.54 +  also from x have "c * x dvd Lcm (( * ) c ` A)"
  120.55      by (intro dvd_Lcm) auto
  120.56 -  finally have dvd: "c dvd Lcm (op * c ` A)" .
  120.57 -
  120.58 -  have "Lcm A dvd Lcm (op * c ` A) div c"
  120.59 +  finally have dvd: "c dvd Lcm (( * ) c ` A)" .
  120.60 +
  120.61 +  have "Lcm A dvd Lcm (( * ) c ` A) div c"
  120.62      by (intro Lcm_least dvd_mult_imp_div)
  120.63        (auto intro!: Lcm_least dvd_Lcm simp: mult.commute[of _ c])
  120.64 -  then have "c * Lcm A dvd Lcm (op * c ` A)"
  120.65 +  then have "c * Lcm A dvd Lcm (( * ) c ` A)"
  120.66      by (subst (asm) dvd_div_iff_mult) (auto intro!: Lcm_least simp: mult_ac dvd)
  120.67    also have "c * Lcm A = (normalize c * Lcm A) * unit_factor c"
  120.68      by (subst unit_factor_mult_normalize [symmetric]) (simp only: mult_ac)
  120.69 -  also have "\<dots> dvd Lcm (op * c ` A) \<longleftrightarrow> normalize c * Lcm A dvd Lcm (op * c ` A)"
  120.70 +  also have "\<dots> dvd Lcm (( * ) c ` A) \<longleftrightarrow> normalize c * Lcm A dvd Lcm (( * ) c ` A)"
  120.71      by (simp add: mult_unit_dvd_iff)
  120.72 -  finally have "normalize c * Lcm A dvd Lcm (op * c ` A)" .
  120.73 -  moreover have "Lcm (op * c ` A) dvd normalize c * Lcm A"
  120.74 +  finally have "normalize c * Lcm A dvd Lcm (( * ) c ` A)" .
  120.75 +  moreover have "Lcm (( * ) c ` A) dvd normalize c * Lcm A"
  120.76      by (intro Lcm_least) (auto intro: mult_dvd_mono dvd_Lcm)
  120.77 -  ultimately have "normalize (normalize c * Lcm A) = normalize (Lcm (op * c ` A))"
  120.78 +  ultimately have "normalize (normalize c * Lcm A) = normalize (Lcm (( * ) c ` A))"
  120.79      by (rule associatedI)
  120.80    then show ?thesis
  120.81      by (simp add: normalize_mult)
   121.1 --- a/src/HOL/Groups_List.thy	Wed Jan 10 15:21:49 2018 +0100
   121.2 +++ b/src/HOL/Groups_List.thy	Wed Jan 10 15:25:09 2018 +0100
   121.3 @@ -124,7 +124,7 @@
   121.4    by (induct xss) simp_all
   121.5  
   121.6  lemma (in monoid_add) length_product_lists:
   121.7 -  "length (product_lists xss) = foldr op * (map length xss) 1"
   121.8 +  "length (product_lists xss) = foldr ( * ) (map length xss) 1"
   121.9  proof (induct xss)
  121.10    case (Cons xs xss) then show ?case by (induct xs) (auto simp: length_concat o_def)
  121.11  qed simp
  121.12 @@ -234,7 +234,7 @@
  121.13  text \<open>General equivalence between @{const sum_list} and @{const sum}\<close>
  121.14  lemma (in monoid_add) sum_list_sum_nth:
  121.15    "sum_list xs = (\<Sum> i = 0 ..< length xs. xs ! i)"
  121.16 -  using interv_sum_list_conv_sum_set_nat [of "op ! xs" 0 "length xs"] by (simp add: map_nth)
  121.17 +  using interv_sum_list_conv_sum_set_nat [of "(!) xs" 0 "length xs"] by (simp add: map_nth)
  121.18  
  121.19  lemma sum_list_map_eq_sum_count:
  121.20    "sum_list (map f xs) = sum (\<lambda>x. count_list xs x * f x) (set xs)"
  121.21 @@ -292,7 +292,7 @@
  121.22  lemma sorted_wrt_less_sum_mono_lowerbound:
  121.23    fixes f :: "nat \<Rightarrow> ('b::ordered_comm_monoid_add)"
  121.24    assumes mono: "\<And>x y. x\<le>y \<Longrightarrow> f x \<le> f y"
  121.25 -  shows "sorted_wrt (op <) ns \<Longrightarrow>
  121.26 +  shows "sorted_wrt (<) ns \<Longrightarrow>
  121.27      (\<Sum>i\<in>{0..<length ns}. f i) \<le> (\<Sum>i\<leftarrow>ns. f i)"
  121.28  proof (induction ns rule: rev_induct)
  121.29    case Nil
  121.30 @@ -355,7 +355,7 @@
  121.31  lemma sum_list_transfer[transfer_rule]:
  121.32    includes lifting_syntax
  121.33    assumes [transfer_rule]: "A 0 0"
  121.34 -  assumes [transfer_rule]: "(A ===> A ===> A) op + op +"
  121.35 +  assumes [transfer_rule]: "(A ===> A ===> A) (+) (+)"
  121.36    shows "(list_all2 A ===> A) sum_list sum_list"
  121.37    unfolding sum_list.eq_foldr [abs_def]
  121.38    by transfer_prover
   122.1 --- a/src/HOL/HOL.thy	Wed Jan 10 15:21:49 2018 +0100
   122.2 +++ b/src/HOL/HOL.thy	Wed Jan 10 15:25:09 2018 +0100
   122.3 @@ -1140,7 +1140,7 @@
   122.4  \<close>
   122.5  
   122.6  definition simp_implies :: "prop \<Rightarrow> prop \<Rightarrow> prop"  (infixr "=simp=>" 1)
   122.7 -  where "simp_implies \<equiv> op \<Longrightarrow>"
   122.8 +  where "simp_implies \<equiv> (\<Longrightarrow>)"
   122.9  
  122.10  lemma simp_impliesI:
  122.11    assumes PQ: "(PROP P \<Longrightarrow> PROP Q)"
  122.12 @@ -1607,7 +1607,7 @@
  122.13      type T = ((term -> bool) * stamp) list;
  122.14      val empty = [];
  122.15      val extend = I;
  122.16 -    fun merge data : T = Library.merge (eq_snd op =) data;
  122.17 +    fun merge data : T = Library.merge (eq_snd (=)) data;
  122.18    );
  122.19    fun add m = Data.map (cons (m, stamp ()));
  122.20    fun matches thy t = exists (fn (m, _) => m t) (Data.get thy);
  122.21 @@ -1804,13 +1804,13 @@
  122.22    assumes equal_eq: "equal x y \<longleftrightarrow> x = y"
  122.23  begin
  122.24  
  122.25 -lemma equal: "equal = (op =)"
  122.26 +lemma equal: "equal = (=)"
  122.27    by (rule ext equal_eq)+
  122.28  
  122.29  lemma equal_refl: "equal x x \<longleftrightarrow> True"
  122.30    unfolding equal by rule+
  122.31  
  122.32 -lemma eq_equal: "(op =) \<equiv> equal"
  122.33 +lemma eq_equal: "(=) \<equiv> equal"
  122.34    by (rule eq_reflection) (rule ext, rule ext, rule sym, rule equal_eq)
  122.35  
  122.36  end
  122.37 @@ -1890,7 +1890,7 @@
  122.38  
  122.39  setup \<open>Sign.add_const_constraint (@{const_name equal}, SOME @{typ "'a::type \<Rightarrow> 'a \<Rightarrow> bool"})\<close>
  122.40  
  122.41 -lemma equal_alias_cert: "OFCLASS('a, equal_class) \<equiv> ((op = :: 'a \<Rightarrow> 'a \<Rightarrow> bool) \<equiv> equal)"
  122.42 +lemma equal_alias_cert: "OFCLASS('a, equal_class) \<equiv> (((=) :: 'a \<Rightarrow> 'a \<Rightarrow> bool) \<equiv> equal)"
  122.43    (is "?ofclass \<equiv> ?equal")
  122.44  proof
  122.45    assume "PROP ?ofclass"
   123.1 --- a/src/HOL/HOLCF/Cpodef.thy	Wed Jan 10 15:21:49 2018 +0100
   123.2 +++ b/src/HOL/HOLCF/Cpodef.thy	Wed Jan 10 15:25:09 2018 +0100
   123.3 @@ -21,7 +21,7 @@
   123.4  theorem typedef_po:
   123.5    fixes Abs :: "'a::po \<Rightarrow> 'b::type"
   123.6    assumes type: "type_definition Rep Abs A"
   123.7 -    and below: "op \<sqsubseteq> \<equiv> \<lambda>x y. Rep x \<sqsubseteq> Rep y"
   123.8 +    and below: "(\<sqsubseteq>) \<equiv> \<lambda>x y. Rep x \<sqsubseteq> Rep y"
   123.9    shows "OFCLASS('b, po_class)"
  123.10    apply (intro_classes, unfold below)
  123.11      apply (rule below_refl)
  123.12 @@ -51,14 +51,14 @@
  123.13  subsection \<open>Proving a subtype is chain-finite\<close>
  123.14  
  123.15  lemma ch2ch_Rep:
  123.16 -  assumes below: "op \<sqsubseteq> \<equiv> \<lambda>x y. Rep x \<sqsubseteq> Rep y"
  123.17 +  assumes below: "(\<sqsubseteq>) \<equiv> \<lambda>x y. Rep x \<sqsubseteq> Rep y"
  123.18    shows "chain S \<Longrightarrow> chain (\<lambda>i. Rep (S i))"
  123.19    unfolding chain_def below .
  123.20  
  123.21  theorem typedef_chfin:
  123.22    fixes Abs :: "'a::chfin \<Rightarrow> 'b::po"
  123.23    assumes type: "type_definition Rep Abs A"
  123.24 -    and below: "op \<sqsubseteq> \<equiv> \<lambda>x y. Rep x \<sqsubseteq> Rep y"
  123.25 +    and below: "(\<sqsubseteq>) \<equiv> \<lambda>x y. Rep x \<sqsubseteq> Rep y"
  123.26    shows "OFCLASS('b, chfin_class)"
  123.27    apply intro_classes
  123.28    apply (drule ch2ch_Rep [OF below])
  123.29 @@ -79,14 +79,14 @@
  123.30  \<close>
  123.31  
  123.32  lemma typedef_is_lubI:
  123.33 -  assumes below: "op \<sqsubseteq> \<equiv> \<lambda>x y. Rep x \<sqsubseteq> Rep y"
  123.34 +  assumes below: "(\<sqsubseteq>) \<equiv> \<lambda>x y. Rep x \<sqsubseteq> Rep y"
  123.35    shows "range (\<lambda>i. Rep (S i)) <<| Rep x \<Longrightarrow> range S <<| x"
  123.36    by (simp add: is_lub_def is_ub_def below)
  123.37  
  123.38  lemma Abs_inverse_lub_Rep:
  123.39    fixes Abs :: "'a::cpo \<Rightarrow> 'b::po"
  123.40    assumes type: "type_definition Rep Abs A"
  123.41 -    and below: "op \<sqsubseteq> \<equiv> \<lambda>x y. Rep x \<sqsubseteq> Rep y"
  123.42 +    and below: "(\<sqsubseteq>) \<equiv> \<lambda>x y. Rep x \<sqsubseteq> Rep y"
  123.43      and adm:  "adm (\<lambda>x. x \<in> A)"
  123.44    shows "chain S \<Longrightarrow> Rep (Abs (\<Squnion>i. Rep (S i))) = (\<Squnion>i. Rep (S i))"
  123.45    apply (rule type_definition.Abs_inverse [OF type])
  123.46 @@ -97,7 +97,7 @@
  123.47  theorem typedef_is_lub:
  123.48    fixes Abs :: "'a::cpo \<Rightarrow> 'b::po"
  123.49    assumes type: "type_definition Rep Abs A"
  123.50 -    and below: "op \<sqsubseteq> \<equiv> \<lambda>x y. Rep x \<sqsubseteq> Rep y"
  123.51 +    and below: "(\<sqsubseteq>) \<equiv> \<lambda>x y. Rep x \<sqsubseteq> Rep y"
  123.52      and adm: "adm (\<lambda>x. x \<in> A)"
  123.53    assumes S: "chain S"
  123.54    shows "range S <<| Abs (\<Squnion>i. Rep (S i))"
  123.55 @@ -117,7 +117,7 @@
  123.56  theorem typedef_cpo:
  123.57    fixes Abs :: "'a::cpo \<Rightarrow> 'b::po"
  123.58    assumes type: "type_definition Rep Abs A"
  123.59 -    and below: "op \<sqsubseteq> \<equiv> \<lambda>x y. Rep x \<sqsubseteq> Rep y"
  123.60 +    and below: "(\<sqsubseteq>) \<equiv> \<lambda>x y. Rep x \<sqsubseteq> Rep y"
  123.61      and adm: "adm (\<lambda>x. x \<in> A)"
  123.62    shows "OFCLASS('b, cpo_class)"
  123.63  proof
  123.64 @@ -136,7 +136,7 @@
  123.65  theorem typedef_cont_Rep:
  123.66    fixes Abs :: "'a::cpo \<Rightarrow> 'b::cpo"
  123.67    assumes type: "type_definition Rep Abs A"
  123.68 -    and below: "op \<sqsubseteq> \<equiv> \<lambda>x y. Rep x \<sqsubseteq> Rep y"
  123.69 +    and below: "(\<sqsubseteq>) \<equiv> \<lambda>x y. Rep x \<sqsubseteq> Rep y"
  123.70      and adm: "adm (\<lambda>x. x \<in> A)"
  123.71    shows "cont (\<lambda>x. f x) \<Longrightarrow> cont (\<lambda>x. Rep (f x))"
  123.72    apply (erule cont_apply [OF _ _ cont_const])
  123.73 @@ -157,7 +157,7 @@
  123.74    fixes Abs :: "'a::cpo \<Rightarrow> 'b::cpo"
  123.75    fixes f :: "'c::cpo \<Rightarrow> 'a::cpo"
  123.76    assumes type: "type_definition Rep Abs A"
  123.77 -    and below: "op \<sqsubseteq> \<equiv> \<lambda>x y. Rep x \<sqsubseteq> Rep y"
  123.78 +    and below: "(\<sqsubseteq>) \<equiv> \<lambda>x y. Rep x \<sqsubseteq> Rep y"
  123.79      and adm: "adm (\<lambda>x. x \<in> A)" (* not used *)
  123.80      and f_in_A: "\<And>x. f x \<in> A"
  123.81    shows "cont f \<Longrightarrow> cont (\<lambda>x. Abs (f x))"
  123.82 @@ -170,7 +170,7 @@
  123.83  theorem typedef_compact:
  123.84    fixes Abs :: "'a::cpo \<Rightarrow> 'b::cpo"
  123.85    assumes type: "type_definition Rep Abs A"
  123.86 -    and below: "op \<sqsubseteq> \<equiv> \<lambda>x y. Rep x \<sqsubseteq> Rep y"
  123.87 +    and below: "(\<sqsubseteq>) \<equiv> \<lambda>x y. Rep x \<sqsubseteq> Rep y"
  123.88      and adm: "adm (\<lambda>x. x \<in> A)"
  123.89    shows "compact (Rep k) \<Longrightarrow> compact k"
  123.90  proof (unfold compact_def)
  123.91 @@ -192,7 +192,7 @@
  123.92  theorem typedef_pcpo_generic:
  123.93    fixes Abs :: "'a::cpo \<Rightarrow> 'b::cpo"
  123.94    assumes type: "type_definition Rep Abs A"
  123.95 -    and below: "op \<sqsubseteq> \<equiv> \<lambda>x y. Rep x \<sqsubseteq> Rep y"
  123.96 +    and below: "(\<sqsubseteq>) \<equiv> \<lambda>x y. Rep x \<sqsubseteq> Rep y"
  123.97      and z_in_A: "z \<in> A"
  123.98      and z_least: "\<And>x. x \<in> A \<Longrightarrow> z \<sqsubseteq> x"
  123.99    shows "OFCLASS('b, pcpo_class)"
 123.100 @@ -211,7 +211,7 @@
 123.101  theorem typedef_pcpo:
 123.102    fixes Abs :: "'a::pcpo \<Rightarrow> 'b::cpo"
 123.103    assumes type: "type_definition Rep Abs A"
 123.104 -    and below: "op \<sqsubseteq> \<equiv> \<lambda>x y. Rep x \<sqsubseteq> Rep y"
 123.105 +    and below: "(\<sqsubseteq>) \<equiv> \<lambda>x y. Rep x \<sqsubseteq> Rep y"
 123.106      and bottom_in_A: "\<bottom> \<in> A"
 123.107    shows "OFCLASS('b, pcpo_class)"
 123.108    by (rule typedef_pcpo_generic [OF type below bottom_in_A], rule minimal)
 123.109 @@ -226,7 +226,7 @@
 123.110  
 123.111  theorem typedef_Abs_strict:
 123.112    assumes type: "type_definition Rep Abs A"
 123.113 -    and below: "op \<sqsubseteq> \<equiv> \<lambda>x y. Rep x \<sqsubseteq> Rep y"
 123.114 +    and below: "(\<sqsubseteq>) \<equiv> \<lambda>x y. Rep x \<sqsubseteq> Rep y"
 123.115      and bottom_in_A: "\<bottom> \<in> A"
 123.116    shows "Abs \<bottom> = \<bottom>"
 123.117    apply (rule bottomI, unfold below)
 123.118 @@ -235,7 +235,7 @@
 123.119  
 123.120  theorem typedef_Rep_strict:
 123.121    assumes type: "type_definition Rep Abs A"
 123.122 -    and below: "op \<sqsubseteq> \<equiv> \<lambda>x y. Rep x \<sqsubseteq> Rep y"
 123.123 +    and below: "(\<sqsubseteq>) \<equiv> \<lambda>x y. Rep x \<sqsubseteq> Rep y"
 123.124      and bottom_in_A: "\<bottom> \<in> A"
 123.125    shows "Rep \<bottom> = \<bottom>"
 123.126    apply (rule typedef_Abs_strict [OF type below bottom_in_A, THEN subst])
 123.127 @@ -244,7 +244,7 @@
 123.128  
 123.129  theorem typedef_Abs_bottom_iff:
 123.130    assumes type: "type_definition Rep Abs A"
 123.131 -    and below: "op \<sqsubseteq> \<equiv> \<lambda>x y. Rep x \<sqsubseteq> Rep y"
 123.132 +    and below: "(\<sqsubseteq>) \<equiv> \<lambda>x y. Rep x \<sqsubseteq> Rep y"
 123.133      and bottom_in_A: "\<bottom> \<in> A"
 123.134    shows "x \<in> A \<Longrightarrow> (Abs x = \<bottom>) = (x = \<bottom>)"
 123.135    apply (rule typedef_Abs_strict [OF type below bottom_in_A, THEN subst])
 123.136 @@ -253,7 +253,7 @@
 123.137  
 123.138  theorem typedef_Rep_bottom_iff:
 123.139    assumes type: "type_definition Rep Abs A"
 123.140 -    and below: "op \<sqsubseteq> \<equiv> \<lambda>x y. Rep x \<sqsubseteq> Rep y"
 123.141 +    and below: "(\<sqsubseteq>) \<equiv> \<lambda>x y. Rep x \<sqsubseteq> Rep y"
 123.142      and bottom_in_A: "\<bottom> \<in> A"
 123.143    shows "(Rep x = \<bottom>) = (x = \<bottom>)"
 123.144    apply (rule typedef_Rep_strict [OF type below bottom_in_A, THEN subst])
 123.145 @@ -266,7 +266,7 @@
 123.146  theorem typedef_flat:
 123.147    fixes Abs :: "'a::flat \<Rightarrow> 'b::pcpo"
 123.148    assumes type: "type_definition Rep Abs A"
 123.149 -    and below: "op \<sqsubseteq> \<equiv> \<lambda>x y. Rep x \<sqsubseteq> Rep y"
 123.150 +    and below: "(\<sqsubseteq>) \<equiv> \<lambda>x y. Rep x \<sqsubseteq> Rep y"
 123.151      and bottom_in_A: "\<bottom> \<in> A"
 123.152    shows "OFCLASS('b, flat_class)"
 123.153    apply (intro_classes)
   124.1 --- a/src/HOL/HOLCF/Discrete.thy	Wed Jan 10 15:21:49 2018 +0100
   124.2 +++ b/src/HOL/HOLCF/Discrete.thy	Wed Jan 10 15:25:09 2018 +0100
   124.3 @@ -15,7 +15,7 @@
   124.4  instantiation discr :: (type) discrete_cpo
   124.5  begin
   124.6  
   124.7 -definition "(op \<sqsubseteq> :: 'a discr \<Rightarrow> 'a discr \<Rightarrow> bool) = (op =)"
   124.8 +definition "((\<sqsubseteq>) :: 'a discr \<Rightarrow> 'a discr \<Rightarrow> bool) = (=)"
   124.9  
  124.10  instance
  124.11    by standard (simp add: below_discr_def)
   125.1 --- a/src/HOL/HOLCF/Domain.thy	Wed Jan 10 15:21:49 2018 +0100
   125.2 +++ b/src/HOL/HOLCF/Domain.thy	Wed Jan 10 15:25:09 2018 +0100
   125.3 @@ -97,7 +97,7 @@
   125.4    fixes Abs :: "udom \<Rightarrow> 'a::pcpo"
   125.5    fixes t :: "udom defl"
   125.6    assumes type: "type_definition Rep Abs (defl_set t)"
   125.7 -  assumes below: "op \<sqsubseteq> \<equiv> \<lambda>x y. Rep x \<sqsubseteq> Rep y"
   125.8 +  assumes below: "(\<sqsubseteq>) \<equiv> \<lambda>x y. Rep x \<sqsubseteq> Rep y"
   125.9    assumes emb: "emb \<equiv> (\<Lambda> x. Rep x)"
  125.10    assumes prj: "prj \<equiv> (\<Lambda> x. Abs (cast\<cdot>t\<cdot>x))"
  125.11    assumes defl: "defl \<equiv> (\<lambda> a::'a itself. t)"
   126.1 --- a/src/HOL/HOLCF/Fun_Cpo.thy	Wed Jan 10 15:21:49 2018 +0100
   126.2 +++ b/src/HOL/HOLCF/Fun_Cpo.thy	Wed Jan 10 15:25:09 2018 +0100
   126.3 @@ -14,7 +14,7 @@
   126.4  instantiation "fun"  :: (type, below) below
   126.5  begin
   126.6  
   126.7 -definition below_fun_def: "(op \<sqsubseteq>) \<equiv> (\<lambda>f g. \<forall>x. f x \<sqsubseteq> g x)"
   126.8 +definition below_fun_def: "(\<sqsubseteq>) \<equiv> (\<lambda>f g. \<forall>x. f x \<sqsubseteq> g x)"
   126.9  
  126.10  instance ..
  126.11  end
   127.1 --- a/src/HOL/HOLCF/Library/List_Cpo.thy	Wed Jan 10 15:21:49 2018 +0100
   127.2 +++ b/src/HOL/HOLCF/Library/List_Cpo.thy	Wed Jan 10 15:25:09 2018 +0100
   127.3 @@ -14,7 +14,7 @@
   127.4  begin
   127.5  
   127.6  definition
   127.7 -  "xs \<sqsubseteq> ys \<longleftrightarrow> list_all2 (op \<sqsubseteq>) xs ys"
   127.8 +  "xs \<sqsubseteq> ys \<longleftrightarrow> list_all2 (\<sqsubseteq>) xs ys"
   127.9  
  127.10  instance proof
  127.11    fix xs :: "'a list"
   128.1 --- a/src/HOL/HOLCF/Product_Cpo.thy	Wed Jan 10 15:21:49 2018 +0100
   128.2 +++ b/src/HOL/HOLCF/Product_Cpo.thy	Wed Jan 10 15:25:09 2018 +0100
   128.3 @@ -32,7 +32,7 @@
   128.4  instantiation prod :: (below, below) below
   128.5  begin
   128.6  
   128.7 -definition below_prod_def: "(op \<sqsubseteq>) \<equiv> \<lambda>p1 p2. (fst p1 \<sqsubseteq> fst p2 \<and> snd p1 \<sqsubseteq> snd p2)"
   128.8 +definition below_prod_def: "(\<sqsubseteq>) \<equiv> \<lambda>p1 p2. (fst p1 \<sqsubseteq> fst p2 \<and> snd p1 \<sqsubseteq> snd p2)"
   128.9  
  128.10  instance ..
  128.11  
   129.1 --- a/src/HOL/HOLCF/Universal.thy	Wed Jan 10 15:21:49 2018 +0100
   129.2 +++ b/src/HOL/HOLCF/Universal.thy	Wed Jan 10 15:25:09 2018 +0100
   129.3 @@ -47,7 +47,7 @@
   129.4  unfolding node_def less_Suc_eq_le set_encode_def
   129.5  apply (rule order_trans [OF _ le_prod_encode_2])
   129.6  apply (rule order_trans [OF _ le_prod_encode_2])
   129.7 -apply (rule order_trans [where y="sum (op ^ 2) {b}"])
   129.8 +apply (rule order_trans [where y="sum ((^) 2) {b}"])
   129.9  apply (simp add: nat_less_power2 [THEN order_less_imp_le])
  129.10  apply (erule sum_mono2, simp, simp)
  129.11  done
  129.12 @@ -263,7 +263,7 @@
  129.13  
  129.14  definition
  129.15    compact_le_def:
  129.16 -    "(op \<sqsubseteq>) \<equiv> (\<lambda>x y. Rep_compact_basis x \<sqsubseteq> Rep_compact_basis y)"
  129.17 +    "(\<sqsubseteq>) \<equiv> (\<lambda>x y. Rep_compact_basis x \<sqsubseteq> Rep_compact_basis y)"
  129.18  
  129.19  instance ..
  129.20  end
   130.1 --- a/src/HOL/HOLCF/Up.thy	Wed Jan 10 15:21:49 2018 +0100
   130.2 +++ b/src/HOL/HOLCF/Up.thy	Wed Jan 10 15:25:09 2018 +0100
   130.3 @@ -28,7 +28,7 @@
   130.4  begin
   130.5  
   130.6  definition below_up_def:
   130.7 -  "(op \<sqsubseteq>) \<equiv>
   130.8 +  "(\<sqsubseteq>) \<equiv>
   130.9      (\<lambda>x y.
  130.10        (case x of
  130.11          Ibottom \<Rightarrow> True
   131.1 --- a/src/HOL/IMP/Abs_Int2_ivl.thy	Wed Jan 10 15:21:49 2018 +0100
   131.2 +++ b/src/HOL/IMP/Abs_Int2_ivl.thy	Wed Jan 10 15:25:09 2018 +0100
   131.3 @@ -303,7 +303,7 @@
   131.4  by(drule (1) add_mono) simp
   131.5  
   131.6  global_interpretation Val_semilattice
   131.7 -where \<gamma> = \<gamma>_ivl and num' = num_ivl and plus' = "op +"
   131.8 +where \<gamma> = \<gamma>_ivl and num' = num_ivl and plus' = "(+)"
   131.9  proof (standard, goal_cases)
  131.10    case 1 thus ?case by transfer (simp add: le_iff_subset)
  131.11  next
  131.12 @@ -319,7 +319,7 @@
  131.13  
  131.14  
  131.15  global_interpretation Val_lattice_gamma
  131.16 -where \<gamma> = \<gamma>_ivl and num' = num_ivl and plus' = "op +"
  131.17 +where \<gamma> = \<gamma>_ivl and num' = num_ivl and plus' = "(+)"
  131.18  defines aval_ivl = aval'
  131.19  proof (standard, goal_cases)
  131.20    case 1 show ?case by(simp add: \<gamma>_inf)
  131.21 @@ -328,7 +328,7 @@
  131.22  qed
  131.23  
  131.24  global_interpretation Val_inv
  131.25 -where \<gamma> = \<gamma>_ivl and num' = num_ivl and plus' = "op +"
  131.26 +where \<gamma> = \<gamma>_ivl and num' = num_ivl and plus' = "(+)"
  131.27  and test_num' = in_ivl
  131.28  and inv_plus' = inv_plus_ivl and inv_less' = inv_less_ivl
  131.29  proof (standard, goal_cases)
  131.30 @@ -351,7 +351,7 @@
  131.31  qed
  131.32  
  131.33  global_interpretation Abs_Int_inv
  131.34 -where \<gamma> = \<gamma>_ivl and num' = num_ivl and plus' = "op +"
  131.35 +where \<gamma> = \<gamma>_ivl and num' = num_ivl and plus' = "(+)"
  131.36  and test_num' = in_ivl
  131.37  and inv_plus' = inv_plus_ivl and inv_less' = inv_less_ivl
  131.38  defines inv_aval_ivl = inv_aval'
  131.39 @@ -385,7 +385,7 @@
  131.40  by(auto simp: is_empty_rep_iff \<gamma>_rep_cases split: extended.splits)
  131.41  
  131.42  global_interpretation Abs_Int_inv_mono
  131.43 -where \<gamma> = \<gamma>_ivl and num' = num_ivl and plus' = "op +"
  131.44 +where \<gamma> = \<gamma>_ivl and num' = num_ivl and plus' = "(+)"
  131.45  and test_num' = in_ivl
  131.46  and inv_plus' = inv_plus_ivl and inv_less' = inv_less_ivl
  131.47  proof (standard, goal_cases)
   132.1 --- a/src/HOL/IMP/Abs_Int3.thy	Wed Jan 10 15:21:49 2018 +0100
   132.2 +++ b/src/HOL/IMP/Abs_Int3.thy	Wed Jan 10 15:25:09 2018 +0100
   132.3 @@ -56,10 +56,10 @@
   132.4  instantiation st :: ("{order_top,wn}")wn
   132.5  begin
   132.6  
   132.7 -lift_definition widen_st :: "'a st \<Rightarrow> 'a st \<Rightarrow> 'a st" is "map2_st_rep (op \<nabla>)"
   132.8 +lift_definition widen_st :: "'a st \<Rightarrow> 'a st \<Rightarrow> 'a st" is "map2_st_rep (\<nabla>)"
   132.9  by(auto simp: eq_st_def)
  132.10  
  132.11 -lift_definition narrow_st :: "'a st \<Rightarrow> 'a st \<Rightarrow> 'a st" is "map2_st_rep (op \<triangle>)"
  132.12 +lift_definition narrow_st :: "'a st \<Rightarrow> 'a st \<Rightarrow> 'a st" is "map2_st_rep (\<triangle>)"
  132.13  by(auto simp: eq_st_def)
  132.14  
  132.15  instance
  132.16 @@ -113,13 +113,13 @@
  132.17  
  132.18  instantiation acom :: (widen)widen