hcju/isabelleps huggingface dataset:
sorted_list_of_set_1: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a set" assumes "class.linorder less_eq less"   and "finite A" shows "list.set (linorder.sorted_list_of_set less_eq A) = A"
is_Prim_def: fixes "term" :: "'b local.term" shows "is_Prim term = (\<exists>x1. term = \<^bold>\<langle>x1\<^bold>\<rangle>)"
emeasure_eq_AE: fixes A :: "'a set"   and B :: "'a set"   and M :: "'a measure" assumes "AE x in M. (x \<in> A) = (x \<in> B)"   and "A \<in> sets M"   and "B \<in> sets M" shows "emeasure M A = emeasure M B"
axioms_2: fixes R :: "('e, 'f) ring_scheme" assumes "cring R" shows "Group.comm_monoid R"
pos_le_divideC_eq: fixes c :: "complex"   and a :: "'a"   and b :: "'a" assumes "0 < c" shows "(a \<le> b /\<^sub>C c) = (c *\<^sub>C a \<le> b)"
lenlex_conv: fixes r :: "('b \<times> 'b) set" shows "lenlex r = {(xs, ys). length xs < length ys \<or> length xs = length ys \<and> (xs, ys) \<in> lex r}"
module_iff_vector_space: fixes s :: "'d \<Rightarrow> 'e \<Rightarrow> 'e" shows "module s = vector_space s"
blocksOf_def: fixes s :: "state"   and p :: "Proc" shows "blocksOf s p = {dblock s p} \<union> {disk s d p |d. d \<in> UNIV} \<union> {block br |br. br \<in> (\<Union>q. \<Union> (range (rdBy s p q)))}"
intro: fixes R :: "('c, 'd) ring_scheme" assumes "ring R"   and "Group.comm_monoid R" shows "cring R"
word_ao_absorbs_1: fixes x :: "'b word"   and y :: "'b word" shows "x && (y || x) = x"
MkIde_Cod: fixes a :: "('a E.term, 'a) arr" assumes "arr a" shows "MkIde (Cod a) = cod a"
heterogeneous_secrets_anti_mono: fixes t :: "('fun, 'var) Term.term"   and \<A> :: "('b strand_label \<times> ('fun, 'var) strand_step) list"   and Sec :: "('fun, 'var) Term.term set"   and Sec' :: "('fun, 'var) Term.term set" assumes "heterogeneous\<^sub>l\<^sub>s\<^sub>t t \<A> Sec"   and "Sec' \<subseteq> Sec" shows "heterogeneous\<^sub>l\<^sub>s\<^sub>t t \<A> Sec'"
Meet_empty: shows "\<Sqinter>{} = \<Squnion>UNIV"
ibl: assumes "(\<lambda>x. \<Sum>i. f i x) \<in> borel_measurable M"   and "\<And>i. (\<lambda>x. \<Sum>i<i. f i x) \<in> borel_measurable M" shows "integrable M (\<lambda>x. \<Sum>i. f i x)"
p_A_def: fixes n :: "nat" shows "p_A n = int (length (sw_A ! n))"
strongly_confluent_on_def: fixes r :: "('b \<times> 'b) set"   and A :: "'b set" shows "strongly_confluent_on r A = (\<forall>x\<in>A. \<forall>y z. (x, y) \<in> r \<and> (x, z) \<in> r \<longrightarrow> (\<exists>u. (y, u) \<in> r\<^sup>* \<and> (z, u) \<in> r\<^sup>=))"
ffold_rec: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and x :: "'b"   and A :: "'b fset"   and z :: "'c" assumes "comp_fun_commute f"   and "x |\<in>| A" shows "ffold f z A = f x (ffold f z (A |-| {|x|}))"
c3: shows "(\<exists>l\<in>set leq. evalUni (elimVarUni_atom L' (LeqUni l)) x) = (\<exists>(a', b', c')\<in>set leq. evalUni (elimVarUni_atom L' (LeqUni (a', b', c'))) x)"
language_def: fixes M :: "'b dfa" assumes "dfa M" shows "dfa.language M \<equiv> {xs. dfa.nextl M (dfa.init M) xs \<in> dfa.final M}"
mult_less_le_imp_less: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b"   and c :: "'b"   and d :: "'b" assumes "class.linordered_semiring_strict plus zero minus less_eq less times"   and "less a b"   and "less_eq c d"   and "less_eq zero a"   and "less zero c" shows "less (times a c) (times b d)"
scale_zero_right: fixes a :: "complex" shows "a *\<^sub>C (0::'a) = (0::'a)"
continuous_on_eq: fixes s :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" assumes "continuous_on s f"   and "\<And>x. x \<in> s \<Longrightarrow> f x = g x" shows "continuous_on s g"
contour_integrable_holomorphic_simple: fixes f :: "complex \<Rightarrow> complex"   and S :: "complex set"   and g :: "real \<Rightarrow> complex" assumes "f holomorphic_on S"   and "open S"   and "valid_path g"   and "path_image g \<subseteq> S" shows "f contour_integrable_on g"
is_processT4: fixes P :: "'a process" shows "\<forall>s X Y. (s, Y) \<in> F P \<and> X \<subseteq> Y \<longrightarrow> (s, X) \<in> F P"
A: shows "finite {t. \<exists>t2. x = t @ t2}"
doubleton_eq_iff: fixes a :: "'c"   and b :: "'c"   and c :: "'c"   and d :: "'c" shows "({a, b} = {c, d}) = (a = c \<and> b = d \<or> a = d \<and> b = c)"
mod_mult_self1: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and euclidean_size :: "'b \<Rightarrow> nat"   and a :: "'b"   and c :: "'b"   and b :: "'b" assumes "class.euclidean_semiring_cancel divide plus minus zero times one modulo euclidean_size" shows "modulo (plus a (times c b)) b = modulo a b"
target_size: fixes c :: "'b \<Rightarrow> 'c word" shows "target_size c = LENGTH('c)"
delay_bisimulation_final_base_flip: shows "delay_bisimulation_final_base trsys2 trsys1 (flip bisim) \<tau>move2 \<tau>move1 final2 final1"
AbstrLevels_S11opt_A93: fixes i :: "AbstrLevelsID" assumes "sS11opt \<in> AbstrLevel i" shows "sA93 \<notin> AbstrLevel i"
of_nat_totient_aux1: fixes ps :: "nat list"   and n :: "nat" assumes "\<And>p. p \<in> set ps \<Longrightarrow> prime p"   and "\<And>p. p \<in> set ps \<Longrightarrow> p dvd n"   and "distinct ps" shows "real (totient_aux1 n ps) = real n * (\<Prod>p\<in>set ps. 1 - 1 / real p)"
induct: fixes x :: "fmla"   and P :: "fmla \<Rightarrow> bool" assumes "x \<in> boolean_axioms"   and "\<And>A. P (A IMP A)"   and "\<And>A B. P (A IMP A OR B)"   and "\<And>A. P (A OR A IMP A)"   and "\<And>A B C. P (A OR B OR C IMP (A OR B) OR C)"   and "\<And>C A B. P (C OR A IMP (C IMP B) IMP A OR B)" shows "P x"
fmrestrict_set_insert_notin_dom_m: fixes xa :: "'c"   and N :: "('c, 'd) fmap"   and l1 :: "'c set" assumes "xa \<notin># dom_m N" shows "fmrestrict_set (insert xa l1) N = fmrestrict_set l1 N"
list_all_iff_all_interval_nat: fixes P :: "nat \<Rightarrow> bool"   and i :: "nat"   and j :: "nat" shows "list_all P [i..<j] = List.all_interval_nat P i j"
connex_union: fixes X :: "'b set"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and Y :: "'b set" assumes "connex X less_eq"   and "connex Y less_eq"   and "\<forall>x\<in>X. \<forall>y\<in>Y. less_eq x y \<or> less_eq y x" shows "connex (X \<union> Y) less_eq"
ro_ifex_unique: fixes x :: "'b ifex"   and y :: "'b ifex" assumes "ro_ifex x"   and "ro_ifex y"   and "\<And>ass. val_ifex x ass = val_ifex y ass" shows "x = y"
is_subset_1: fixes c :: "'a" assumes "c \<in> ta'.Qi" shows "c \<in> ta'.Q"
divmod_integer_def: fixes k :: "integer"   and l :: "integer" shows "divmod_integer k l = (k div l, k mod l)"
Init_simps2_4: fixes P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool" shows "Init (P \<and> Q) \<equiv> Init P \<and> Init Q"
inj_onTr1: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and x :: "'c"   and y :: "'c" assumes "inj_on f A"   and "x \<in> A"   and "y \<in> A"   and "f x = f y" shows "x = y"
splus_zero: fixes v :: "'t" shows "(0::'a) \<oplus> v = v"
sw: fixes n :: "nat"   and I :: "nat set" assumes "n < length cl"   and "I \<in> P n" shows "sum (wt (cl ! n) s) I = sum (wt (dl ! n) t) (F n I)"
assms_2: shows "pseudo_proj_True m x = pseudo_proj_True m y"
even_not_iff: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bit :: "'c \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and drop_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and take_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and "and" :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and or :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and xor :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and mask :: "nat \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and not :: "'c \<Rightarrow> 'c"   and a :: "'c" assumes "class.ring_bit_operations plus minus zero times one divide modulo bit push_bit drop_bit take_bit and or xor mask uminus not" shows "dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) (not a) = (\<not> dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) a)"
assms_3: fixes k :: "nat" shows "\<forall>x\<in>S. norm (f k x) \<le> h x"
all_ns_ns_mul_ext: fixes as :: "'a list"   and bs :: "'a list"   and ns :: "('a \<times> 'a) set"   and s :: "('a \<times> 'a) set" assumes "length as = length bs"   and "\<forall>i<length bs. (as ! i, bs ! i) \<in> ns" shows "(mset as, mset bs) \<in> ns_mul_ext ns s"
alwD: fixes P :: "'d stream \<Rightarrow> bool"   and x :: "'d stream" assumes "alw P x" shows "P x"
simps_13: fixes R :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and y21 :: "nat"   and y22 :: "nat"   and y23 :: "'b option"   and y24 :: "'b tree"   and y25 :: "'b tree" shows "\<not> rel_tree R (Node y21 y22 y23 y24 y25) Leaf"
results_in_zero: fixes summary :: "'a \<Rightarrow> 'a \<Rightarrow> 'b antichain"   and results_in :: "'c \<Rightarrow> 'b \<Rightarrow> 'c"   and t :: "'c" assumes "dataflow_topology summary results_in" shows "results_in t (0::'b) = t"
leof_RES_rule_2: fixes X :: "'c set"   and \<Phi> :: "'c \<Rightarrow> bool" assumes "\<And>x. x \<in> X \<Longrightarrow> \<Phi> x" shows "RES X \<le>\<^sub>n SPEC \<Phi>"
op_times_ivl_def: fixes a :: "'c set"   and b :: "'d set" shows "op_times_ivl a b = a \<times> b"
diff_add_zero: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and a :: "'b"   and b :: "'b" assumes "class.comm_monoid_diff plus minus zero" shows "minus a (plus a b) = zero"
simps_1: fixes u :: "nat" shows "c_unfold 0 u = []"
assms_5: shows "e ; s\<^sup>T \<le> x"
mono_set_iff: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and S :: "'b set" assumes "class.complete_linorder Inf Sup inf less_eq less sup bot top" shows "order.mono_set less_eq S = (S = ord.greaterThan less (Inf S) \<or> S = ord.atLeast less_eq (Inf S))"
int_simps_6: fixes P :: "'b \<Rightarrow> bool" shows "\<turnstile> P = \<not> P = #False"
distinct_11: shows "ConcatOp \<noteq> string_binop.GreaterOp"
pivot_unsat_id: fixes s :: "('i, 'a) state"   and x\<^sub>i :: "nat"   and x\<^sub>j :: "nat" assumes "\<triangle> (\<T> s)"   and "x\<^sub>i \<in> lvars (\<T> s)"   and "x\<^sub>j \<in> rvars_eq (eq_for_lvar (\<T> s) x\<^sub>i)" shows "\<U> (pivot x\<^sub>i x\<^sub>j s) = \<U> s"
asm_1: shows "\<beta> \<in> elts (\<omega> \<up> \<delta>)"
simps: fixes a :: "'b list"   and G :: "'b list set" shows "(a \<in> \<langle>G\<rangle>\<^sub>F) = (a = \<epsilon> \<or> (\<exists>w. a = w \<and> w \<in> G) \<or> (\<exists>w1 w2. a = w1 \<cdot> w2 \<and> w1 \<in> \<langle>G\<rangle>\<^sub>F \<and> w2 \<in> \<langle>G\<rangle>\<^sub>F) \<or> (\<exists>p q w. a = w \<and> p \<in> \<langle>G\<rangle>\<^sub>F \<and> q \<in> \<langle>G\<rangle>\<^sub>F \<and> p \<cdot> w \<in> \<langle>G\<rangle>\<^sub>F \<and> w \<cdot> q \<in> \<langle>G\<rangle>\<^sub>F))"
k_invariant_step_new: fixes x :: "'rep"   and ec :: "'rep"   and ec' :: "'rep"   and A :: "('ma, 'mt) AlgState" assumes "k_isNode x"   and "k_isNode ec"   and "ec' \<in> set (k_succs ec)"   and "simAbs ec = simAbs x" shows "\<exists>ec''. lookup tOps (aTrans (k_ins x A)) (ec, simObs a ec') = Some ec'' \<and> simAbs ec'' \<in> simAbs ` set (k_succs ec) \<and> simObs a ec'' = simObs a ec'"
extreme_points_of_translation: fixes a :: "'b"   and S :: "'b set" shows "{x. x extreme_point_of (+) a ` S} = (+) a ` {x. x extreme_point_of S}"
nonpos_Reals_inverse_I: fixes a :: "'b" assumes "a \<in> \<real>\<^sub>\<le>\<^sub>0" shows "inverse a \<in> \<real>\<^sub>\<le>\<^sub>0"
eval_cases_4: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list"   and V :: "char list"   and e :: "char list exp"   and s :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and e' :: "char list exp"   and s' :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)" assumes "P \<turnstile> \<langle>V:=e,s\<rangle> \<Rightarrow> \<langle>e',s'\<rangle>" obtains v :: "val" and h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option" and l :: "char list \<Rightarrow> val option" where "e' = unit" and "s' = (h, l(V \<mapsto> v))" and "P \<turnstile> \<langle>e,s\<rangle> \<Rightarrow> \<langle>Val v,(h, l)\<rangle>"   | e'a :: "char list exp" where "e' = throw e'a" and "P \<turnstile> \<langle>e,s\<rangle> \<Rightarrow> \<langle>throw e'a,s'\<rangle>"
asum_multTr_pm: fixes a :: "'a"   and m :: "int"   and n :: "int" assumes "a \<in> carrier A"   and "0 \<le> m"   and "n < 0" shows "aSum A (nat m) (aSum A (nat (- n)) (-\<^sub>a a)) = aSum A (nat (m * - n)) (-\<^sub>a a)"
commutative_monoid_def: fixes M :: "'c set"   and composition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unit :: "'c" shows "commutative_monoid M composition unit \<equiv> Group_Theory.monoid M composition unit \<and> commutative_monoid_axioms M composition"
insert_eqpoll_insert_iff: fixes a :: "'b"   and A :: "'b set"   and b :: "'c"   and B :: "'c set" assumes "a \<notin> A"   and "b \<notin> B" shows "(insert a A \<approx> insert b B) = (A \<approx> B)"
B_eq: shows "B = bezout_iterate (interchange_rows A (mod_type_class.from_nat i) (LEAST n. A $ n $ mod_type_class.from_nat k \<noteq> (0::'a) \<and> mod_type_class.from_nat i \<le> n)) (nrows A - Suc 0) (mod_type_class.from_nat i) (mod_type_class.from_nat k) bezout"
dist_triangle3: fixes dist :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and uniformity :: "('c \<times> 'c) filter"   and "open" :: "'c set \<Rightarrow> bool"   and x :: "'c"   and y :: "'c"   and a :: "'c" assumes "class.metric_space dist uniformity open" shows "dist x y \<le> dist a x + dist a y"
bigs_by_smalls: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and E :: "char list \<Rightarrow> ty option"   and es :: "expr list"   and s :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and es' :: "expr list"   and s' :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)" assumes "wwf_prog P"   and "P,E \<turnstile> \<langle>es,s\<rangle> [\<Rightarrow>] \<langle>es',s'\<rangle>" shows "P,E \<turnstile> \<langle>es,s\<rangle> [\<rightarrow>]* \<langle>es',s'\<rangle>"
fact_prod: fixes n :: "nat" shows "fact n = of_nat (\<Prod>{1..n})"
G_relpow_matpow_pos: fixes i :: "'n"   and j :: "'n"   and n :: "nat" assumes "(i, j) \<in> G ^^ n" shows "(0::'a) < matpow A n $h i $h j"
ack_bounds_PRIMREC: fixes f :: "nat list \<Rightarrow> nat" assumes "PRIMREC f" shows "\<exists>k. \<forall>l. f l < ack k (sum_list l)"
check_def: fixes q :: "'b queue" shows "check q = (if lenr q \<le> lenf q then exec2 q else let newstate = Rev 0 (front q) [] (rear q) [] in exec2 (q\<lparr>lenf := lenf q + lenr q, status := newstate, rear := [], lenr := 0\<rparr>))"
eq_\<I>_restrict_converter: fixes \<I> :: "('i, 'j) \<I>"   and \<I>' :: "('k, 'l) \<I>"   and cnv :: "('i, 'j, 'k, 'l) converter"   and A :: "'i set" assumes "\<I>, \<I>' \<turnstile>\<^sub>C cnv \<surd>"   and "outs_\<I> \<I> \<subseteq> A" shows "\<I>,\<I>' \<turnstile>\<^sub>C restrict_converter A \<I>' cnv \<sim> cnv"
stable_on_def: fixes Xd :: "'d \<Rightarrow> 'e"   and Xh :: "'d \<Rightarrow> 'f"   and Pd :: "'e \<Rightarrow> ('d \<times> 'd) set"   and Ch :: "'f \<Rightarrow> 'd set \<Rightarrow> 'd set"   and ds :: "'e set"   and X :: "'d set" assumes "Contracts Xd Xh Pd Ch" shows "Contracts.stable_on Pd Ch ds X = (Contracts.individually_rational_on Pd Ch ds X \<and> Contracts.stable_no_blocking_on Pd Ch ds X)"
card_union: fixes A :: "'b set"   and B :: "'b set" assumes "finite A"   and "finite B" shows "card (A \<union> B) = card A + card B - card (A \<inter> B)"
cone_def: fixes J :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and D :: "'a \<Rightarrow> 'b"   and a :: "'b"   and \<chi> :: "'a \<Rightarrow> 'b" shows "Limit.cone J C D a \<chi> \<equiv> (category C \<and> category J) \<and> diagram J C D \<and> constant_functor J C a \<and> natural_transformation J C (constant_functor.map J C a) D \<chi>"
closed_no_lconsts: fixes freshenLC :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and renameLCs :: "('a \<Rightarrow> 'a) \<Rightarrow> 'b \<Rightarrow> 'b"   and lconsts :: "'b \<Rightarrow> 'a set"   and closed :: "'b \<Rightarrow> bool"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and subst_lconsts :: "'c \<Rightarrow> 'a set"   and subst_renameLCs :: "('a \<Rightarrow> 'a) \<Rightarrow> 'c \<Rightarrow> 'c"   and anyP :: "'b"   and f :: "'b" assumes "Abstract_Formulas freshenLC renameLCs lconsts closed subst subst_lconsts subst_renameLCs anyP"   and "closed f" shows "lconsts f = {}"
monotone2monotone: fixes ordb :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and ordc :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> 'd"   and orda :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and t :: "'b \<Rightarrow> 'c" assumes "\<And>x. monotone ordb ordc (f x)"   and "monotone orda ordb t"   and "\<And>y. monotone orda ordc (\<lambda>x. f x y)"   and "transp ordc" shows "monotone orda ordc (\<lambda>x. f x (t x))"
not_arr_null: shows "\<not> F.B.arr F.B.null"
ts_prop_2: fixes k :: "cnf" shows "\<parallel>the_pb\<parallel>\<^bsub>k\<^esub>"
simps: fixes p :: "linear_poly" shows "dim_poly p = (if vars p = {} then 0 else Max (vars p) + 1)"
HMA_elements_mat: shows "(HMA_M ===> (=)) elements_mat elements_mat_h"
reflection: shows "rel_weakly_reflects_barbs Rel (STCalWB SWB TWB)"
inv_correspondence_into: fixes G :: "'a set"   and binop :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and e :: "'a"   and x :: "'a permutation" assumes "BinOpSetGroup G binop e"   and "x \<in> BinOpSetGroup.Abs_G_perm G binop ` G" shows "the_inv_into G (BinOpSetGroup.Abs_G_perm G binop) x \<in> G"
inf_period_3: fixes d :: "'a"   and D :: "'a"   and t :: "'a" assumes "d dvd D" shows "\<forall>x k. (d dvd x + t) = (d dvd x - k * D + t)"
ALGxy_append: fixes A :: "('c list \<Rightarrow> 'd pmf) \<times> ('c list \<times> 'd \<Rightarrow> 'c \<Rightarrow> ((nat \<times> nat list) \<times> 'd) pmf)"   and rs :: "'c list"   and r :: "'c"   and init :: "'c list"   and x :: "'c"   and y :: "'c" shows "ALGxy A (rs @ [r]) init x y = ALGxy A rs init x y + (if r \<in> {y, x} then ALG' A (rs @ [r]) init (length rs) y + ALG' A (rs @ [r]) init (length rs) x else 0)"
bind_mergec_merge: fixes y :: "'c cset"   and f :: "'c \<Rightarrow> 'm"   and g :: "'a \<Rightarrow> 'm" shows "bind (mergec y f) (\<lambda>A. merge A g) = mergec y (\<lambda>x. bind (f x) (\<lambda>A. merge A g))"
le_ucast_ucast_le: fixes x :: "'b word"   and y :: "'c word" assumes "x \<le> UCAST('c \<rightarrow> 'b) y" shows "UCAST('b \<rightarrow> 'c) x \<le> y"
interchange_rows_def: fixes A :: "(('d, 'e) Finite_Cartesian_Product.vec, 'f) Finite_Cartesian_Product.vec"   and a :: "'f"   and b :: "'f" shows "interchange_rows A a b = (\<chi>i j. if i = a then A $h b $h j else if i = b then A $h a $h j else A $h i $h j)"
contk: shows "continuous_map (prod_topology (top_of_set {0..1}) X2) X3 k"
fr_refinement_FrameTransmission: fixes n :: "nat"   and nSend :: "nat \<Rightarrow> nat \<Rightarrow> 'a Frame list"   and recv :: "nat \<Rightarrow> 'a Frame list"   and nReturn :: "nat \<Rightarrow> nat \<Rightarrow> 'a Frame list"   and nC :: "nat \<Rightarrow> Config"   and nStore :: "nat \<Rightarrow> nat \<Rightarrow> 'a Frame list"   and nGet :: "nat \<Rightarrow> nat \<Rightarrow> nat list" assumes "Cable n nSend recv"   and "\<forall>i<n. FlexRayController (nReturn i) recv (nC i) (nStore i) (nSend i) (nGet i)"   and "DisjointSchedules n nC"   and "IdenticCycleLength n nC" shows "FrameTransmission n nStore nReturn nGet nC"
AS_prec_dom_independent: shows "\<forall>s d a a'. current s \<noteq> d \<and> AS_precondition s d a \<longrightarrow> AS_precondition (kstep s a') d a"
hyperbola_method_bigo: fixes f :: "nat \<Rightarrow> 'a"   and g :: "nat \<Rightarrow> 'a"   and R :: "real \<Rightarrow> 'a"   and b :: "real \<Rightarrow> 'a"   and S :: "real \<Rightarrow> 'a"   and T :: "real \<Rightarrow> 'a" assumes "(\<lambda>x. sum_upto (\<lambda>n. f n * sum_upto g (x / real n)) (sqrt x) - R x) \<in> O(b)"   and "(\<lambda>x. sum_upto (\<lambda>n. sum_upto f (x / real n) * g n) (sqrt x) - S x) \<in> O(b)"   and "(\<lambda>x. sum_upto f (sqrt x) * sum_upto g (sqrt x) - T x) \<in> O(b)" shows "(\<lambda>x. sum_upto (dirichlet_prod f g) x - (R x + S x - T x)) \<in> O(b)"
intensional_rews_1: fixes c :: "'a"   and w :: "'b" shows "(#c) w \<equiv> c"
is_natural_1: fixes A :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and B :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and F :: "'c \<Rightarrow> 'd"   and G :: "'c \<Rightarrow> 'd"   and \<tau> :: "'c \<Rightarrow> 'd"   and f :: "'c" assumes "natural_transformation A B F G \<tau>"   and "partial_magma.arr A f" shows "B (G f) (\<tau> (partial_magma.dom A f)) = \<tau> f"
r: shows "r = 0 \<or> degree r < degree_m g"
step_sim: fixes a :: "'b"   and a' :: "'b"   and b :: "'a" assumes "(a, a') \<in> b.E"   and "(a, b) \<in> R\<inverse>" shows "\<exists>b'. (b, b') \<in> a.E \<and> (a', b') \<in> R\<inverse>"
routing_ipassmt_def: fixes rt :: "'b routing_rule list" shows "routing_ipassmt rt \<equiv> map ((apfst Iface \<circ>\<circ> apsnd) cidr_split) (routing_ipassmt_wi rt)"
fSum_eq: fixes n :: "nat"   and m :: "nat"   and f :: "nat \<Rightarrow> 'a"   and g :: "nat \<Rightarrow> 'a" assumes "\<forall>j\<in>nset n (n + m). f j \<in> carrier A"   and "\<forall>j\<in>nset n (n + m). g j \<in> carrier A"   and "\<forall>j\<in>nset n (n + m). f j = g j" shows "\<Sigma>\<^sub>f A f n (n + m) = \<Sigma>\<^sub>f A g n (n + m)"
assms_3: fixes x :: "'a"   and x' :: "'b" assumes "v = Some x"   and "v' = Some x'"   and "(x, x') \<in> Ra" shows "f x \<le> \<Down> Rb (f' x')"
csin_double: fixes a :: "real" shows "2 * complex_of_real (sin a) * complex_of_real (cos a) = complex_of_real (sin (2 * a))"
rtc_implies_cs: fixes x :: "'a"   and y :: "'a" assumes "x \<rightarrow>\<^sup>* y" shows "x \<down>\<^sup>* y"
distinct_1: fixes a :: "'d"   and psi :: "('d, 'e, 'f) psi"   and name' :: "name"   and boundOutput' :: "('d, 'e, 'f) boundOutput" shows "a \<prec>' psi \<noteq> \<lparr>\<nu>name'\<rparr>boundOutput'"
that: fixes B :: "real set" assumes "t \<in> B"   and "open B"   and "B \<subseteq> T"   and "D (\<lambda>\<tau>. \<phi> \<tau> s) \<mapsto> (\<lambda>\<tau>. \<tau> *\<^sub>R f (\<phi> t s)) at t within B" shows "thesis"
Operator_works: fixes h :: "nat"   and n_ops :: "nat"   and t :: "nat"   and k :: "nat" assumes "cnf_to_dimacs.valid_state_var h n_ops (Operator t k)" shows "cnf_to_dimacs.dimacs_to_var h n_ops (cnf_to_dimacs.var_to_dimacs h n_ops (Operator t k)) = Operator t k"
assms_2: shows "STRIPS_Semantics.is_parallel_solution_for_problem (\<phi> \<Psi> ) \<pi>"