basic lexer for Isabelle/HOL theorem prover

3 files changed, 969 insertions, 2 deletions

diff --git a/pygments/lexers/_mapping.py b/pygments/lexers/_mapping.py
index f8454357..adecad2f 100644
--- a/pygments/lexers/_mapping.py
+++ b/pygments/lexers/_mapping.py
@@ -165,6 +165,7 @@ LEXERS = {
     'IoLexer': ('pygments.lexers.agile', 'Io', ('io',), ('*.io',), ('text/x-iosrc',)),
     'IokeLexer': ('pygments.lexers.jvm', 'Ioke', ('ioke', 'ik'), ('*.ik',), ('text/x-iokesrc',)),
     'IrcLogsLexer': ('pygments.lexers.text', 'IRC logs', ('irc',), ('*.weechatlog',), ('text/x-irclog',)),
+    'IsabelleLexer': ('pygments.lexers.functional', 'Isabelle', ('isabelle',), ('*.thy',), ('text/x-isabelle',)),
     'JadeLexer': ('pygments.lexers.web', 'Jade', ('jade',), ('*.jade',), ('text/x-jade',)),
     'JagsLexer': ('pygments.lexers.math', 'JAGS', ('jags',), ('*.jag', '*.bug'), ()),
     'JasminLexer': ('pygments.lexers.jvm', 'Jasmin', ('jasmin', 'jasminxt'), ('*.j',), ()),
diff --git a/pygments/lexers/functional.py b/pygments/lexers/functional.py
index a22c4f55..d39dda2e 100644
--- a/pygments/lexers/functional.py
+++ b/pygments/lexers/functional.py
@@ -20,8 +20,8 @@ __all__ = ['RacketLexer', 'SchemeLexer', 'CommonLispLexer', 'CryptolLexer',
            'HaskellLexer', 'AgdaLexer', 'LiterateCryptolLexer',
            'LiterateHaskellLexer', 'LiterateAgdaLexer', 'SMLLexer',
            'OcamlLexer', 'ErlangLexer', 'ErlangShellLexer', 'OpaLexer',
-           'CoqLexer', 'NewLispLexer', 'NixLexer', 'ElixirLexer',
-           'ElixirConsoleLexer', 'KokaLexer', 'IdrisLexer',
+           'CoqLexer', 'IsabelleLexer', 'NewLispLexer', 'NixLexer',
+           'ElixirLexer', 'ElixirConsoleLexer', 'KokaLexer', 'IdrisLexer',
            'LiterateIdrisLexer']
 
 
@@ -2734,6 +2734,221 @@ class OpaLexer(RegexLexer):
         ],
     }
 
+class IsabelleLexer(RegexLexer):
+    """
+    For the `Isabelle <http://isabelle.in.tum.de/>`_ proof assistant.
+
+    .. versionadded:: 2.0
+    """
+
+    name = 'Isabelle'
+    aliases = ['isabelle']
+    filenames = ['*.thy']
+    mimetypes = ['text/x-isabelle']
+
+    keyword_minor = [
+        'and', 'assumes', 'attach', 'avoids', 'binder', 'checking',
+        'class_instance', 'class_relation', 'code_module', 'congs',
+        'constant', 'constrains', 'datatypes', 'defines', 'file', 'fixes',
+        'for', 'functions', 'hints', 'identifier', 'if', 'imports', 'in',
+        'includes', 'infix', 'infixl', 'infixr', 'is', 'keywords', 'lazy',
+        'module_name', 'monos', 'morphisms', 'no_discs_sels', 'notes',
+        'obtains', 'open', 'output', 'overloaded', 'parametric', 'permissive',
+        'pervasive', 'rep_compat', 'shows', 'structure', 'type_class',
+        'type_constructor', 'unchecked', 'unsafe', 'where',
+    ]
+
+    keyword_diag = [
+        'ML_command', 'ML_val', 'class_deps', 'code_deps', 'code_thms',
+        'display_drafts', 'find_consts', 'find_theorems', 'find_unused_assms',
+        'full_prf', 'help', 'locale_deps', 'nitpick', 'pr', 'prf',
+        'print_abbrevs', 'print_antiquotations', 'print_attributes',
+        'print_binds', 'print_bnfs', 'print_bundles',
+        'print_case_translations', 'print_cases', 'print_claset',
+        'print_classes', 'print_codeproc', 'print_codesetup',
+        'print_coercions', 'print_commands', 'print_context',
+        'print_defn_rules', 'print_dependencies', 'print_facts',
+        'print_induct_rules', 'print_inductives', 'print_interps',
+        'print_locale', 'print_locales', 'print_methods', 'print_options',
+        'print_orders', 'print_quot_maps', 'print_quotconsts',
+        'print_quotients', 'print_quotientsQ3', 'print_quotmapsQ3',
+        'print_rules', 'print_simpset', 'print_state', 'print_statement',
+        'print_syntax', 'print_theorems', 'print_theory', 'print_trans_rules',
+        'prop', 'pwd', 'quickcheck', 'refute', 'sledgehammer', 'smt_status',
+        'solve_direct', 'spark_status', 'term', 'thm', 'thm_deps', 'thy_deps',
+        'try', 'try0', 'typ', 'unused_thms', 'value', 'values', 'welcome',
+        'print_ML_antiquotations', 'print_term_bindings', 'values_prolog', 
+    ]
+
+    keyword_thy = [ 'theory', 'begin', 'end', ]
+
+    keyword_section = [ 'header', 'chapter', ]
+
+    keyword_subsection = [
+        'section', 'subsection', 'subsubsection', 'sect', 'subsect',
+        'subsubsect',
+    ]
+
+    keyword_theory_decl = [
+        'ML', 'ML_file', 'abbreviation', 'adhoc_overloading', 'arities',
+        'atom_decl', 'attribute_setup', 'axiomatization', 'bundle',
+        'case_of_simps', 'class', 'classes', 'classrel', 'codatatype',
+        'code_abort', 'code_class', 'code_const', 'code_datatype',
+        'code_identifier', 'code_include', 'code_instance', 'code_modulename',
+        'code_monad', 'code_printing', 'code_reflect', 'code_reserved',
+        'code_type', 'coinductive', 'coinductive_set', 'consts', 'context',
+        'datatype', 'datatype_new', 'datatype_new_compat', 'declaration',
+        'declare', 'default_sort', 'defer_recdef', 'definition', 'defs',
+        'domain', 'domain_isomorphism', 'domaindef', 'equivariance',
+        'export_code', 'extract', 'extract_type', 'fixrec', 'fun',
+        'fun_cases', 'hide_class', 'hide_const', 'hide_fact', 'hide_type',
+        'import_const_map', 'import_file', 'import_tptp', 'import_type_map',
+        'inductive', 'inductive_set', 'instantiation', 'judgment', 'lemmas',
+        'lifting_forget', 'lifting_update', 'local_setup', 'locale',
+        'method_setup', 'nitpick_params', 'no_adhoc_overloading',
+        'no_notation', 'no_syntax', 'no_translations', 'no_type_notation',
+        'nominal_datatype', 'nonterminal', 'notation', 'notepad', 'oracle',
+        'overloading', 'parse_ast_translation', 'parse_translation',
+        'partial_function', 'primcorec', 'primrec', 'primrec_new',
+        'print_ast_translation', 'print_translation', 'quickcheck_generator',
+        'quickcheck_params', 'realizability', 'realizers', 'recdef', 'record',
+        'refute_params', 'setup', 'setup_lifting', 'simproc_setup',
+        'simps_of_case', 'sledgehammer_params', 'spark_end', 'spark_open',
+        'spark_open_siv', 'spark_open_vcg', 'spark_proof_functions',
+        'spark_types', 'statespace', 'syntax', 'syntax_declaration', 'text',
+        'text_raw', 'theorems', 'translations', 'type_notation',
+        'type_synonym', 'typed_print_translation', 'typedecl', 'hoarestate',
+        'install_C_file', 'install_C_types', 'wpc_setup', 'c_defs', 'c_types',
+        'memsafe', 'SML_export', 'SML_file', 'SML_import', 'approximate',
+        'bnf_axiomatization', 'cartouche', 'datatype_compat',
+        'free_constructors', 'functor', 'nominal_function',
+        'nominal_termination', 'permanent_interpretation',
+        'binds', 'defining', 'smt2_status', 'term_cartouche',
+        'boogie_file', 'datatype_compat', 'text_cartouche',
+    ]
+
+    keyword_theory_script = ['inductive_cases', 'inductive_simps', ]
+
+    keyword_theory_goal = [
+        'ax_specification', 'bnf', 'code_pred', 'corollary', 'cpodef',
+        'crunch', 'crunch_ignore', 
+        'enriched_type', 'function', 'instance', 'interpretation', 'lemma',
+        'lift_definition', 'nominal_inductive', 'nominal_inductive2',
+        'nominal_primrec', 'pcpodef', 'primcorecursive',
+        'quotient_definition', 'quotient_type', 'recdef_tc', 'rep_datatype',
+        'schematic_corollary', 'schematic_lemma', 'schematic_theorem',
+        'spark_vc', 'specification', 'subclass', 'sublocale', 'termination',
+        'theorem', 'typedef', 'wrap_free_constructors',
+    ]
+    
+    keyword_qed = [ 'by', 'done', 'qed' ]
+    keyword_abandon_proof = ['sorry', 'oops']
+
+    keyword_proof_goal = ['have', 'hence', 'interpret', ]
+
+    keyword_proof_block = [ 'next', 'proof', ]
+
+    keyword_proof_chain = [
+        'finally', 'from', 'then', 'ultimately', 'with',
+    ]
+
+    keyword_proof_decl = [
+        'ML_prf', 'also', 'include', 'including', 'let', 'moreover', 'note',
+        'txt', 'txt_raw', 'unfolding', 'using', 'write',
+    ]
+
+    keyword_proof_asm = [ 'assume', 'case', 'def', 'fix', 'presume', ]
+
+    keyword_proof_asm_goal = [ 'guess', 'obtain', 'show', 'thus', ]
+
+    keyword_proof_script = [
+        'apply', 'apply_end', 'apply_trace', 'back', 'defer', 'prefer',
+    ]
+
+    operators = [
+        '\\:\\:', '\\:', '\\(', '\\)', '\\[', '\\]', '_', '=', ',', '\\|',
+        '\\+', '\\-', '\\!', '\\?',
+	]
+
+    proof_operators = [ '\\{', '\\}', '\\.', '\\.\\.', ]
+
+    tokens = {
+        'root': [
+            (r'\s+', Text),
+            (r'\(\*', Comment, 'comment'),
+            (r'\{\*', Comment, 'text'),
+
+			(r'(%s)' % '|'.join(operators), Operator),
+			(r'(%s)' % '|'.join(proof_operators), Operator.Word),
+
+            (r'\b(%s)\b' % '|'.join(keyword_minor), Keyword.Pseudo),
+
+            (r'\b(%s)\b' % '|'.join(keyword_diag), Keyword.Type),
+
+            (r'\b(%s)\b' % '|'.join(keyword_thy), Keyword),
+            (r'\b(%s)\b' % '|'.join(keyword_theory_decl), Keyword),
+
+            (r'\b(%s)\b' % '|'.join(keyword_section), Generic.Heading),
+            (r'\b(%s)\b' % '|'.join(keyword_subsection), Generic.Subheading),
+
+            (r'\b(%s)\b' % '|'.join(keyword_theory_goal), Keyword.Namespace),
+            (r'\b(%s)\b' % '|'.join(keyword_theory_script), Keyword.Namespace),
+
+            (r'\b(%s)\b' % '|'.join(keyword_abandon_proof), Generic.Error),
+
+            (r'\b(%s)\b' % '|'.join(keyword_qed), Keyword),
+            (r'\b(%s)\b' % '|'.join(keyword_proof_goal), Keyword),
+            (r'\b(%s)\b' % '|'.join(keyword_proof_block), Keyword),
+            (r'\b(%s)\b' % '|'.join(keyword_proof_decl), Keyword),
+
+            (r'\b(%s)\b' % '|'.join(keyword_proof_chain), Keyword),
+            (r'\b(%s)\b' % '|'.join(keyword_proof_asm), Keyword),
+            (r'\b(%s)\b' % '|'.join(keyword_proof_asm_goal), Keyword),
+
+            (r'\b(%s)\b' % '|'.join(keyword_proof_script), Keyword.Pseudo),
+
+            (r'\\<\w*>', Text.Symbol),
+
+            (r"[^\W\d][.\w']*", Name),
+            (r"\?[^\W\d][.\w']*", Name),
+            (r"'[^\W\d][.\w']*", Name.Type),
+
+            (r'\d[\d_]*', Name), # display numbers as name
+            (r'0[xX][\da-fA-F][\da-fA-F_]*', Number.Hex),
+            (r'0[oO][0-7][0-7_]*', Number.Oct),
+            (r'0[bB][01][01_]*', Number.Bin),
+
+            (r'"', String, 'string'),
+            (r'`', String.Other, 'fact'),
+        ],
+        'comment': [
+            (r'[^(*)]+', Comment),
+            (r'\(\*', Comment, '#push'),
+            (r'\*\)', Comment, '#pop'),
+            (r'[(*)]', Comment),
+        ],
+        'text': [
+            (r'[^\*\}]+', Comment),
+            (r'\*\}', Comment, '#pop'),
+            (r'\*', Comment),
+            (r'\}', Comment),
+        ],
+        'string': [
+            (r'[^"\\]+', String),
+			(r'\\<\w*>', String.Symbol),
+            (r'\\"', String),
+            (r'\\', String),
+            (r'"', String, '#pop'),
+        ],
+        'fact': [
+            (r'[^`\\]+', String.Other),
+			(r'\\<\w*>', String.Symbol),
+            (r'\\`', String.Other),
+            (r'\\', String.Other),
+            (r'`', String.Other, '#pop'),
+        ],
+    }
+
 
 class CoqLexer(RegexLexer):
     """
diff --git a/tests/examplefiles/example.thy b/tests/examplefiles/example.thy
new file mode 100644
index 00000000..abaa1af8
--- /dev/null
+++ b/tests/examplefiles/example.thy
@@ -0,0 +1,751 @@
+(* from Isabelle2013-2 src/HOL/Power.thy; BSD license *)
+
+(*  Title:      HOL/Power.thy
+    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
+    Copyright   1997  University of Cambridge
+*)
+
+header {* Exponentiation *}
+
+theory Power
+imports Num
+begin
+
+subsection {* Powers for Arbitrary Monoids *}
+
+class power = one + times
+begin
+
+primrec power :: "'a \<Rightarrow> nat \<Rightarrow> 'a" (infixr "^" 80) where
+    power_0: "a ^ 0 = 1"
+  | power_Suc: "a ^ Suc n = a * a ^ n"
+
+notation (latex output)
+  power ("(_\<^bsup>_\<^esup>)" [1000] 1000)
+
+notation (HTML output)
+  power ("(_\<^bsup>_\<^esup>)" [1000] 1000)
+
+text {* Special syntax for squares. *}
+
+abbreviation (xsymbols)
+  power2 :: "'a \<Rightarrow> 'a"  ("(_\<^sup>2)" [1000] 999) where
+  "x\<^sup>2 \<equiv> x ^ 2"
+
+notation (latex output)
+  power2  ("(_\<^sup>2)" [1000] 999)
+
+notation (HTML output)
+  power2  ("(_\<^sup>2)" [1000] 999)
+
+end
+
+context monoid_mult
+begin
+
+subclass power .
+
+lemma power_one [simp]:
+  "1 ^ n = 1"
+  by (induct n) simp_all
+
+lemma power_one_right [simp]:
+  "a ^ 1 = a"
+  by simp
+
+lemma power_commutes:
+  "a ^ n * a = a * a ^ n"
+  by (induct n) (simp_all add: mult_assoc)
+
+lemma power_Suc2:
+  "a ^ Suc n = a ^ n * a"
+  by (simp add: power_commutes)
+
+lemma power_add:
+  "a ^ (m + n) = a ^ m * a ^ n"
+  by (induct m) (simp_all add: algebra_simps)
+
+lemma power_mult:
+  "a ^ (m * n) = (a ^ m) ^ n"
+  by (induct n) (simp_all add: power_add)
+
+lemma power2_eq_square: "a\<^sup>2 = a * a"
+  by (simp add: numeral_2_eq_2)
+
+lemma power3_eq_cube: "a ^ 3 = a * a * a"
+  by (simp add: numeral_3_eq_3 mult_assoc)
+
+lemma power_even_eq:
+  "a ^ (2 * n) = (a ^ n)\<^sup>2"
+  by (subst mult_commute) (simp add: power_mult)
+
+lemma power_odd_eq:
+  "a ^ Suc (2*n) = a * (a ^ n)\<^sup>2"
+  by (simp add: power_even_eq)
+
+lemma power_numeral_even:
+  "z ^ numeral (Num.Bit0 w) = (let w = z ^ (numeral w) in w * w)"
+  unfolding numeral_Bit0 power_add Let_def ..
+
+lemma power_numeral_odd:
+  "z ^ numeral (Num.Bit1 w) = (let w = z ^ (numeral w) in z * w * w)"
+  unfolding numeral_Bit1 One_nat_def add_Suc_right add_0_right
+  unfolding power_Suc power_add Let_def mult_assoc ..
+
+lemma funpow_times_power:
+  "(times x ^^ f x) = times (x ^ f x)"
+proof (induct "f x" arbitrary: f)
+  case 0 then show ?case by (simp add: fun_eq_iff)
+next
+  case (Suc n)
+  def g \<equiv> "\<lambda>x. f x - 1"
+  with Suc have "n = g x" by simp
+  with Suc have "times x ^^ g x = times (x ^ g x)" by simp
+  moreover from Suc g_def have "f x = g x + 1" by simp
+  ultimately show ?case by (simp add: power_add funpow_add fun_eq_iff mult_assoc)
+qed
+
+end
+
+context comm_monoid_mult
+begin
+
+lemma power_mult_distrib:
+  "(a * b) ^ n = (a ^ n) * (b ^ n)"
+  by (induct n) (simp_all add: mult_ac)
+
+end
+
+context semiring_numeral
+begin
+
+lemma numeral_sqr: "numeral (Num.sqr k) = numeral k * numeral k"
+  by (simp only: sqr_conv_mult numeral_mult)
+
+lemma numeral_pow: "numeral (Num.pow k l) = numeral k ^ numeral l"
+  by (induct l, simp_all only: numeral_class.numeral.simps pow.simps
+    numeral_sqr numeral_mult power_add power_one_right)
+
+lemma power_numeral [simp]: "numeral k ^ numeral l = numeral (Num.pow k l)"
+  by (rule numeral_pow [symmetric])
+
+end
+
+context semiring_1
+begin
+
+lemma of_nat_power:
+  "of_nat (m ^ n) = of_nat m ^ n"
+  by (induct n) (simp_all add: of_nat_mult)
+
+lemma power_zero_numeral [simp]: "(0::'a) ^ numeral k = 0"
+  by (simp add: numeral_eq_Suc)
+
+lemma zero_power2: "0\<^sup>2 = 0" (* delete? *)
+  by (rule power_zero_numeral)
+
+lemma one_power2: "1\<^sup>2 = 1" (* delete? *)
+  by (rule power_one)
+
+end
+
+context comm_semiring_1
+begin
+
+text {* The divides relation *}
+
+lemma le_imp_power_dvd:
+  assumes "m \<le> n" shows "a ^ m dvd a ^ n"
+proof
+  have "a ^ n = a ^ (m + (n - m))"
+    using `m \<le> n` by simp
+  also have "\<dots> = a ^ m * a ^ (n - m)"
+    by (rule power_add)
+  finally show "a ^ n = a ^ m * a ^ (n - m)" .
+qed
+
+lemma power_le_dvd:
+  "a ^ n dvd b \<Longrightarrow> m \<le> n \<Longrightarrow> a ^ m dvd b"
+  by (rule dvd_trans [OF le_imp_power_dvd])
+
+lemma dvd_power_same:
+  "x dvd y \<Longrightarrow> x ^ n dvd y ^ n"
+  by (induct n) (auto simp add: mult_dvd_mono)
+
+lemma dvd_power_le:
+  "x dvd y \<Longrightarrow> m \<ge> n \<Longrightarrow> x ^ n dvd y ^ m"
+  by (rule power_le_dvd [OF dvd_power_same])
+
+lemma dvd_power [simp]:
+  assumes "n > (0::nat) \<or> x = 1"
+  shows "x dvd (x ^ n)"
+using assms proof
+  assume "0 < n"
+  then have "x ^ n = x ^ Suc (n - 1)" by simp
+  then show "x dvd (x ^ n)" by simp
+next
+  assume "x = 1"
+  then show "x dvd (x ^ n)" by simp
+qed
+
+end
+
+context ring_1
+begin
+
+lemma power_minus:
+  "(- a) ^ n = (- 1) ^ n * a ^ n"
+proof (induct n)
+  case 0 show ?case by simp
+next
+  case (Suc n) then show ?case
+    by (simp del: power_Suc add: power_Suc2 mult_assoc)
+qed
+
+lemma power_minus_Bit0:
+  "(- x) ^ numeral (Num.Bit0 k) = x ^ numeral (Num.Bit0 k)"
+  by (induct k, simp_all only: numeral_class.numeral.simps power_add
+    power_one_right mult_minus_left mult_minus_right minus_minus)
+
+lemma power_minus_Bit1:
+  "(- x) ^ numeral (Num.Bit1 k) = - (x ^ numeral (Num.Bit1 k))"
+  by (simp only: eval_nat_numeral(3) power_Suc power_minus_Bit0 mult_minus_left)
+
+lemma power_neg_numeral_Bit0 [simp]:
+  "neg_numeral k ^ numeral (Num.Bit0 l) = numeral (Num.pow k (Num.Bit0 l))"
+  by (simp only: neg_numeral_def power_minus_Bit0 power_numeral)
+
+lemma power_neg_numeral_Bit1 [simp]:
+  "neg_numeral k ^ numeral (Num.Bit1 l) = neg_numeral (Num.pow k (Num.Bit1 l))"
+  by (simp only: neg_numeral_def power_minus_Bit1 power_numeral pow.simps)
+
+lemma power2_minus [simp]:
+  "(- a)\<^sup>2 = a\<^sup>2"
+  by (rule power_minus_Bit0)
+
+lemma power_minus1_even [simp]:
+  "-1 ^ (2*n) = 1"
+proof (induct n)
+  case 0 show ?case by simp
+next
+  case (Suc n) then show ?case by (simp add: power_add power2_eq_square)
+qed
+
+lemma power_minus1_odd:
+  "-1 ^ Suc (2*n) = -1"
+  by simp
+
+lemma power_minus_even [simp]:
+  "(-a) ^ (2*n) = a ^ (2*n)"
+  by (simp add: power_minus [of a])
+
+end
+
+context ring_1_no_zero_divisors
+begin
+
+lemma field_power_not_zero:
+  "a \<noteq> 0 \<Longrightarrow> a ^ n \<noteq> 0"
+  by (induct n) auto
+
+lemma zero_eq_power2 [simp]:
+  "a\<^sup>2 = 0 \<longleftrightarrow> a = 0"
+  unfolding power2_eq_square by simp
+
+lemma power2_eq_1_iff:
+  "a\<^sup>2 = 1 \<longleftrightarrow> a = 1 \<or> a = - 1"
+  unfolding power2_eq_square by (rule square_eq_1_iff)
+
+end
+
+context idom
+begin
+
+lemma power2_eq_iff: "x\<^sup>2 = y\<^sup>2 \<longleftrightarrow> x = y \<or> x = - y"
+  unfolding power2_eq_square by (rule square_eq_iff)
+
+end
+
+context division_ring
+begin
+
+text {* FIXME reorient or rename to @{text nonzero_inverse_power} *}
+lemma nonzero_power_inverse:
+  "a \<noteq> 0 \<Longrightarrow> inverse (a ^ n) = (inverse a) ^ n"
+  by (induct n)
+    (simp_all add: nonzero_inverse_mult_distrib power_commutes field_power_not_zero)
+
+end
+
+context field
+begin
+
+lemma nonzero_power_divide:
+  "b \<noteq> 0 \<Longrightarrow> (a / b) ^ n = a ^ n / b ^ n"
+  by (simp add: divide_inverse power_mult_distrib nonzero_power_inverse)
+
+end
+
+
+subsection {* Exponentiation on ordered types *}
+
+context linordered_ring (* TODO: move *)
+begin
+
+lemma sum_squares_ge_zero:
+  "0 \<le> x * x + y * y"
+  by (intro add_nonneg_nonneg zero_le_square)
+
+lemma not_sum_squares_lt_zero:
+  "\<not> x * x + y * y < 0"
+  by (simp add: not_less sum_squares_ge_zero)
+
+end
+
+context linordered_semidom
+begin
+
+lemma zero_less_power [simp]:
+  "0 < a \<Longrightarrow> 0 < a ^ n"
+  by (induct n) (simp_all add: mult_pos_pos)
+
+lemma zero_le_power [simp]:
+  "0 \<le> a \<Longrightarrow> 0 \<le> a ^ n"
+  by (induct n) (simp_all add: mult_nonneg_nonneg)
+
+lemma power_mono:
+  "a \<le> b \<Longrightarrow> 0 \<le> a \<Longrightarrow> a ^ n \<le> b ^ n"
+  by (induct n) (auto intro: mult_mono order_trans [of 0 a b])
+
+lemma one_le_power [simp]: "1 \<le> a \<Longrightarrow> 1 \<le> a ^ n"
+  using power_mono [of 1 a n] by simp
+
+lemma power_le_one: "\<lbrakk>0 \<le> a; a \<le> 1\<rbrakk> \<Longrightarrow> a ^ n \<le> 1"
+  using power_mono [of a 1 n] by simp
+
+lemma power_gt1_lemma:
+  assumes gt1: "1 < a"
+  shows "1 < a * a ^ n"
+proof -
+  from gt1 have "0 \<le> a"
+    by (fact order_trans [OF zero_le_one less_imp_le])
+  have "1 * 1 < a * 1" using gt1 by simp
+  also have "\<dots> \<le> a * a ^ n" using gt1
+    by (simp only: mult_mono `0 \<le> a` one_le_power order_less_imp_le
+        zero_le_one order_refl)
+  finally show ?thesis by simp
+qed
+
+lemma power_gt1:
+  "1 < a \<Longrightarrow> 1 < a ^ Suc n"
+  by (simp add: power_gt1_lemma)
+
+lemma one_less_power [simp]:
+  "1 < a \<Longrightarrow> 0 < n \<Longrightarrow> 1 < a ^ n"
+  by (cases n) (simp_all add: power_gt1_lemma)
+
+lemma power_le_imp_le_exp:
+  assumes gt1: "1 < a"
+  shows "a ^ m \<le> a ^ n \<Longrightarrow> m \<le> n"
+proof (induct m arbitrary: n)
+  case 0
+  show ?case by simp
+next
+  case (Suc m)
+  show ?case
+  proof (cases n)
+    case 0
+    with Suc.prems Suc.hyps have "a * a ^ m \<le> 1" by simp
+    with gt1 show ?thesis
+      by (force simp only: power_gt1_lemma
+          not_less [symmetric])
+  next
+    case (Suc n)
+    with Suc.prems Suc.hyps show ?thesis
+      by (force dest: mult_left_le_imp_le
+          simp add: less_trans [OF zero_less_one gt1])
+  qed
+qed
+
+text{*Surely we can strengthen this? It holds for @{text "0<a<1"} too.*}
+lemma power_inject_exp [simp]:
+  "1 < a \<Longrightarrow> a ^ m = a ^ n \<longleftrightarrow> m = n"
+  by (force simp add: order_antisym power_le_imp_le_exp)
+
+text{*Can relax the first premise to @{term "0<a"} in the case of the
+natural numbers.*}
+lemma power_less_imp_less_exp:
+  "1 < a \<Longrightarrow> a ^ m < a ^ n \<Longrightarrow> m < n"
+  by (simp add: order_less_le [of m n] less_le [of "a^m" "a^n"]
+    power_le_imp_le_exp)
+
+lemma power_strict_mono [rule_format]:
+  "a < b \<Longrightarrow> 0 \<le> a \<Longrightarrow> 0 < n \<longrightarrow> a ^ n < b ^ n"
+  by (induct n)
+   (auto simp add: mult_strict_mono le_less_trans [of 0 a b])
+
+text{*Lemma for @{text power_strict_decreasing}*}
+lemma power_Suc_less:
+  "0 < a \<Longrightarrow> a < 1 \<Longrightarrow> a * a ^ n < a ^ n"
+  by (induct n)
+    (auto simp add: mult_strict_left_mono)
+
+lemma power_strict_decreasing [rule_format]:
+  "n < N \<Longrightarrow> 0 < a \<Longrightarrow> a < 1 \<longrightarrow> a ^ N < a ^ n"
+proof (induct N)
+  case 0 then show ?case by simp
+next
+  case (Suc N) then show ?case 
+  apply (auto simp add: power_Suc_less less_Suc_eq)
+  apply (subgoal_tac "a * a^N < 1 * a^n")
+  apply simp
+  apply (rule mult_strict_mono) apply auto
+  done
+qed
+
+text{*Proof resembles that of @{text power_strict_decreasing}*}
+lemma power_decreasing [rule_format]:
+  "n \<le> N \<Longrightarrow> 0 \<le> a \<Longrightarrow> a \<le> 1 \<longrightarrow> a ^ N \<le> a ^ n"
+proof (induct N)
+  case 0 then show ?case by simp
+next
+  case (Suc N) then show ?case 
+  apply (auto simp add: le_Suc_eq)
+  apply (subgoal_tac "a * a^N \<le> 1 * a^n", simp)
+  apply (rule mult_mono) apply auto
+  done
+qed
+
+lemma power_Suc_less_one:
+  "0 < a \<Longrightarrow> a < 1 \<Longrightarrow> a ^ Suc n < 1"
+  using power_strict_decreasing [of 0 "Suc n" a] by simp
+
+text{*Proof again resembles that of @{text power_strict_decreasing}*}
+lemma power_increasing [rule_format]:
+  "n \<le> N \<Longrightarrow> 1 \<le> a \<Longrightarrow> a ^ n \<le> a ^ N"
+proof (induct N)
+  case 0 then show ?case by simp
+next
+  case (Suc N) then show ?case 
+  apply (auto simp add: le_Suc_eq)
+  apply (subgoal_tac "1 * a^n \<le> a * a^N", simp)
+  apply (rule mult_mono) apply (auto simp add: order_trans [OF zero_le_one])
+  done
+qed
+
+text{*Lemma for @{text power_strict_increasing}*}
+lemma power_less_power_Suc:
+  "1 < a \<Longrightarrow> a ^ n < a * a ^ n"
+  by (induct n) (auto simp add: mult_strict_left_mono less_trans [OF zero_less_one])
+
+lemma power_strict_increasing [rule_format]:
+  "n < N \<Longrightarrow> 1 < a \<longrightarrow> a ^ n < a ^ N"
+proof (induct N)
+  case 0 then show ?case by simp
+next
+  case (Suc N) then show ?case 
+  apply (auto simp add: power_less_power_Suc less_Suc_eq)
+  apply (subgoal_tac "1 * a^n < a * a^N", simp)
+  apply (rule mult_strict_mono) apply (auto simp add: less_trans [OF zero_less_one] less_imp_le)
+  done
+qed
+
+lemma power_increasing_iff [simp]:
+  "1 < b \<Longrightarrow> b ^ x \<le> b ^ y \<longleftrightarrow> x \<le> y"
+  by (blast intro: power_le_imp_le_exp power_increasing less_imp_le)
+
+lemma power_strict_increasing_iff [simp]:
+  "1 < b \<Longrightarrow> b ^ x < b ^ y \<longleftrightarrow> x < y"
+by (blast intro: power_less_imp_less_exp power_strict_increasing) 
+
+lemma power_le_imp_le_base:
+  assumes le: "a ^ Suc n \<le> b ^ Suc n"
+    and ynonneg: "0 \<le> b"
+  shows "a \<le> b"
+proof (rule ccontr)
+  assume "~ a \<le> b"
+  then have "b < a" by (simp only: linorder_not_le)
+  then have "b ^ Suc n < a ^ Suc n"
+    by (simp only: assms power_strict_mono)
+  from le and this show False
+    by (simp add: linorder_not_less [symmetric])
+qed
+
+lemma power_less_imp_less_base:
+  assumes less: "a ^ n < b ^ n"
+  assumes nonneg: "0 \<le> b"
+  shows "a < b"
+proof (rule contrapos_pp [OF less])
+  assume "~ a < b"
+  hence "b \<le> a" by (simp only: linorder_not_less)
+  hence "b ^ n \<le> a ^ n" using nonneg by (rule power_mono)
+  thus "\<not> a ^ n < b ^ n" by (simp only: linorder_not_less)
+qed
+
+lemma power_inject_base:
+  "a ^ Suc n = b ^ Suc n \<Longrightarrow> 0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> a = b"
+by (blast intro: power_le_imp_le_base antisym eq_refl sym)
+
+lemma power_eq_imp_eq_base:
+  "a ^ n = b ^ n \<Longrightarrow> 0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 < n \<Longrightarrow> a = b"
+  by (cases n) (simp_all del: power_Suc, rule power_inject_base)
+
+lemma power2_le_imp_le:
+  "x\<^sup>2 \<le> y\<^sup>2 \<Longrightarrow> 0 \<le> y \<Longrightarrow> x \<le> y"
+  unfolding numeral_2_eq_2 by (rule power_le_imp_le_base)
+
+lemma power2_less_imp_less:
+  "x\<^sup>2 < y\<^sup>2 \<Longrightarrow> 0 \<le> y \<Longrightarrow> x < y"
+  by (rule power_less_imp_less_base)
+
+lemma power2_eq_imp_eq:
+  "x\<^sup>2 = y\<^sup>2 \<Longrightarrow> 0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> x = y"
+  unfolding numeral_2_eq_2 by (erule (2) power_eq_imp_eq_base) simp
+
+end
+
+context linordered_ring_strict
+begin
+
+lemma sum_squares_eq_zero_iff:
+  "x * x + y * y = 0 \<longleftrightarrow> x = 0 \<and> y = 0"
+  by (simp add: add_nonneg_eq_0_iff)
+
+lemma sum_squares_le_zero_iff:
+  "x * x + y * y \<le> 0 \<longleftrightarrow> x = 0 \<and> y = 0"
+  by (simp add: le_less not_sum_squares_lt_zero sum_squares_eq_zero_iff)
+
+lemma sum_squares_gt_zero_iff:
+  "0 < x * x + y * y \<longleftrightarrow> x \<noteq> 0 \<or> y \<noteq> 0"
+  by (simp add: not_le [symmetric] sum_squares_le_zero_iff)
+
+end
+
+context linordered_idom
+begin
+
+lemma power_abs:
+  "abs (a ^ n) = abs a ^ n"
+  by (induct n) (auto simp add: abs_mult)
+
+lemma abs_power_minus [simp]:
+  "abs ((-a) ^ n) = abs (a ^ n)"
+  by (simp add: power_abs)
+
+lemma zero_less_power_abs_iff [simp, no_atp]:
+  "0 < abs a ^ n \<longleftrightarrow> a \<noteq> 0 \<or> n = 0"
+proof (induct n)
+  case 0 show ?case by simp
+next
+  case (Suc n) show ?case by (auto simp add: Suc zero_less_mult_iff)
+qed
+
+lemma zero_le_power_abs [simp]:
+  "0 \<le> abs a ^ n"
+  by (rule zero_le_power [OF abs_ge_zero])
+
+lemma zero_le_power2 [simp]:
+  "0 \<le> a\<^sup>2"
+  by (simp add: power2_eq_square)
+
+lemma zero_less_power2 [simp]:
+  "0 < a\<^sup>2 \<longleftrightarrow> a \<noteq> 0"
+  by (force simp add: power2_eq_square zero_less_mult_iff linorder_neq_iff)
+
+lemma power2_less_0 [simp]:
+  "\<not> a\<^sup>2 < 0"
+  by (force simp add: power2_eq_square mult_less_0_iff)
+
+lemma abs_power2 [simp]:
+  "abs (a\<^sup>2) = a\<^sup>2"
+  by (simp add: power2_eq_square abs_mult abs_mult_self)
+
+lemma power2_abs [simp]:
+  "(abs a)\<^sup>2 = a\<^sup>2"
+  by (simp add: power2_eq_square abs_mult_self)
+
+lemma odd_power_less_zero:
+  "a < 0 \<Longrightarrow> a ^ Suc (2*n) < 0"
+proof (induct n)
+  case 0
+  then show ?case by simp
+next
+  case (Suc n)
+  have "a ^ Suc (2 * Suc n) = (a*a) * a ^ Suc(2*n)"
+    by (simp add: mult_ac power_add power2_eq_square)
+  thus ?case
+    by (simp del: power_Suc add: Suc mult_less_0_iff mult_neg_neg)
+qed
+
+lemma odd_0_le_power_imp_0_le:
+  "0 \<le> a ^ Suc (2*n) \<Longrightarrow> 0 \<le> a"
+  using odd_power_less_zero [of a n]
+    by (force simp add: linorder_not_less [symmetric]) 
+
+lemma zero_le_even_power'[simp]:
+  "0 \<le> a ^ (2*n)"
+proof (induct n)
+  case 0
+    show ?case by simp
+next
+  case (Suc n)
+    have "a ^ (2 * Suc n) = (a*a) * a ^ (2*n)" 
+      by (simp add: mult_ac power_add power2_eq_square)
+    thus ?case
+      by (simp add: Suc zero_le_mult_iff)
+qed
+
+lemma sum_power2_ge_zero:
+  "0 \<le> x\<^sup>2 + y\<^sup>2"
+  by (intro add_nonneg_nonneg zero_le_power2)
+
+lemma not_sum_power2_lt_zero:
+  "\<not> x\<^sup>2 + y\<^sup>2 < 0"
+  unfolding not_less by (rule sum_power2_ge_zero)
+
+lemma sum_power2_eq_zero_iff:
+  "x\<^sup>2 + y\<^sup>2 = 0 \<longleftrightarrow> x = 0 \<and> y = 0"
+  unfolding power2_eq_square by (simp add: add_nonneg_eq_0_iff)
+
+lemma sum_power2_le_zero_iff:
+  "x\<^sup>2 + y\<^sup>2 \<le> 0 \<longleftrightarrow> x = 0 \<and> y = 0"
+  by (simp add: le_less sum_power2_eq_zero_iff not_sum_power2_lt_zero)
+
+lemma sum_power2_gt_zero_iff:
+  "0 < x\<^sup>2 + y\<^sup>2 \<longleftrightarrow> x \<noteq> 0 \<or> y \<noteq> 0"
+  unfolding not_le [symmetric] by (simp add: sum_power2_le_zero_iff)
+
+end
+
+
+subsection {* Miscellaneous rules *}
+
+lemma power_eq_if: "p ^ m = (if m=0 then 1 else p * (p ^ (m - 1)))"
+  unfolding One_nat_def by (cases m) simp_all
+
+lemma power2_sum:
+  fixes x y :: "'a::comm_semiring_1"
+  shows "(x + y)\<^sup>2 = x\<^sup>2 + y\<^sup>2 + 2 * x * y"
+  by (simp add: algebra_simps power2_eq_square mult_2_right)
+
+lemma power2_diff:
+  fixes x y :: "'a::comm_ring_1"
+  shows "(x - y)\<^sup>2 = x\<^sup>2 + y\<^sup>2 - 2 * x * y"
+  by (simp add: ring_distribs power2_eq_square mult_2) (rule mult_commute)
+
+lemma power_0_Suc [simp]:
+  "(0::'a::{power, semiring_0}) ^ Suc n = 0"
+  by simp
+
+text{*It looks plausible as a simprule, but its effect can be strange.*}
+lemma power_0_left:
+  "0 ^ n = (if n = 0 then 1 else (0::'a::{power, semiring_0}))"
+  by (induct n) simp_all
+
+lemma power_eq_0_iff [simp]:
+  "a ^ n = 0 \<longleftrightarrow>
+     a = (0::'a::{mult_zero,zero_neq_one,no_zero_divisors,power}) \<and> n \<noteq> 0"
+  by (induct n)
+    (auto simp add: no_zero_divisors elim: contrapos_pp)
+
+lemma (in field) power_diff:
+  assumes nz: "a \<noteq> 0"
+  shows "n \<le> m \<Longrightarrow> a ^ (m - n) = a ^ m / a ^ n"
+  by (induct m n rule: diff_induct) (simp_all add: nz field_power_not_zero)
+
+text{*Perhaps these should be simprules.*}
+lemma power_inverse:
+  fixes a :: "'a::division_ring_inverse_zero"
+  shows "inverse (a ^ n) = inverse a ^ n"
+apply (cases "a = 0")
+apply (simp add: power_0_left)
+apply (simp add: nonzero_power_inverse)
+done (* TODO: reorient or rename to inverse_power *)
+
+lemma power_one_over:
+  "1 / (a::'a::{field_inverse_zero, power}) ^ n =  (1 / a) ^ n"
+  by (simp add: divide_inverse) (rule power_inverse)
+
+lemma power_divide:
+  "(a / b) ^ n = (a::'a::field_inverse_zero) ^ n / b ^ n"
+apply (cases "b = 0")
+apply (simp add: power_0_left)
+apply (rule nonzero_power_divide)
+apply assumption
+done
+
+text {* Simprules for comparisons where common factors can be cancelled. *}
+
+lemmas zero_compare_simps =
+    add_strict_increasing add_strict_increasing2 add_increasing
+    zero_le_mult_iff zero_le_divide_iff 
+    zero_less_mult_iff zero_less_divide_iff 
+    mult_le_0_iff divide_le_0_iff 
+    mult_less_0_iff divide_less_0_iff 
+    zero_le_power2 power2_less_0
+
+
+subsection {* Exponentiation for the Natural Numbers *}
+
+lemma nat_one_le_power [simp]:
+  "Suc 0 \<le> i \<Longrightarrow> Suc 0 \<le> i ^ n"
+  by (rule one_le_power [of i n, unfolded One_nat_def])
+
+lemma nat_zero_less_power_iff [simp]:
+  "x ^ n > 0 \<longleftrightarrow> x > (0::nat) \<or> n = 0"
+  by (induct n) auto
+
+lemma nat_power_eq_Suc_0_iff [simp]: 
+  "x ^ m = Suc 0 \<longleftrightarrow> m = 0 \<or> x = Suc 0"
+  by (induct m) auto
+
+lemma power_Suc_0 [simp]:
+  "Suc 0 ^ n = Suc 0"
+  by simp
+
+text{*Valid for the naturals, but what if @{text"0<i<1"}?
+Premises cannot be weakened: consider the case where @{term "i=0"},
+@{term "m=1"} and @{term "n=0"}.*}
+lemma nat_power_less_imp_less:
+  assumes nonneg: "0 < (i\<Colon>nat)"
+  assumes less: "i ^ m < i ^ n"
+  shows "m < n"
+proof (cases "i = 1")
+  case True with less power_one [where 'a = nat] show ?thesis by simp
+next
+  case False with nonneg have "1 < i" by auto
+  from power_strict_increasing_iff [OF this] less show ?thesis ..
+qed
+
+lemma power_dvd_imp_le:
+  "i ^ m dvd i ^ n \<Longrightarrow> (1::nat) < i \<Longrightarrow> m \<le> n"
+  apply (rule power_le_imp_le_exp, assumption)
+  apply (erule dvd_imp_le, simp)
+  done
+
+lemma power2_nat_le_eq_le:
+  fixes m n :: nat
+  shows "m\<^sup>2 \<le> n\<^sup>2 \<longleftrightarrow> m \<le> n"
+  by (auto intro: power2_le_imp_le power_mono)
+
+lemma power2_nat_le_imp_le:
+  fixes m n :: nat
+  assumes "m\<^sup>2 \<le> n"
+  shows "m \<le> n"
+  using assms by (cases m) (simp_all add: power2_eq_square)
+
+
+
+subsection {* Code generator tweak *}
+
+lemma power_power_power [code]:
+  "power = power.power (1::'a::{power}) (op *)"
+  unfolding power_def power.power_def ..
+
+declare power.power.simps [code]
+
+code_identifier
+  code_module Power \<rightharpoonup> (SML) Arith and (OCaml) Arith and (Haskell) Arith
+
+end
+