hcju/isabelleps huggingface dataset:
assms_1: fixes x :: "'a"   and y :: "'b" shows "h x y = g' x * g y"
P_mod_n_m: fixes t :: "'a"   and g :: "'a"   and m :: "nat"   and n :: "nat" assumes "Idomain S"   and "t \<in> carrier S"   and "g \<in> carrier R"   and "m \<le> n"   and "P_mod R S X (S \<diamondsuit>\<^sub>p (t^\<^bsup>S Suc n\<^esup>)) g" shows "P_mod R S X (S \<diamondsuit>\<^sub>p (t^\<^bsup>S Suc m\<^esup>)) g"
sum_var_monom_mult: fixes m :: "'c monom"   and n :: "'c monom"   and x :: "'c" shows "sum_var (m * n) x = sum_var m x + sum_var n x"
split: fixes P :: "'d \<Rightarrow> bool"   and f1 :: "nat set \<Rightarrow> 'e \<Rightarrow> 'd"   and f2 :: "nat \<Rightarrow> nat \<Rightarrow> 'd"   and f3 :: "nat set \<Rightarrow> nat set \<Rightarrow> 'e \<Rightarrow> 'd"   and f4 :: "nat \<Rightarrow> nat \<Rightarrow> 'd"   and f5 :: "nat \<Rightarrow> nat \<Rightarrow> 'd"   and f6 :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 'd"   and f7 :: "'d"   and node_action :: "'e node_action" shows "P (case node_action of x:*cast(xa) \<Rightarrow> f1 x xa | x:deliver(xa) \<Rightarrow> f2 x xa | x\<not>xa:arrive(xb) \<Rightarrow> f3 x xa xb | connect(x, xa) \<Rightarrow> f4 x xa | disconnect(x, xa) \<Rightarrow> f5 x xa | x:newpkt(xa, xb) \<Rightarrow> f6 x xa xb | \<tau>\<^sub>n \<Rightarrow> f7) = ((\<forall>x11 x12. node_action = x11:*cast(x12) \<longrightarrow> P (f1 x11 x12)) \<and> (\<forall>x21 x22. node_action = x21:deliver(x22) \<longrightarrow> P (f2 x21 x22)) \<and> (\<forall>x31 x32 x33. node_action = x31\<not>x32:arrive(x33) \<longrightarrow> P (f3 x31 x32 x33)) \<and> (\<forall>x41 x42. node_action = connect(x41, x42) \<longrightarrow> P (f4 x41 x42)) \<and> (\<forall>x51 x52. node_action = disconnect(x51, x52) \<longrightarrow> P (f5 x51 x52)) \<and> (\<forall>x61 x62 x63. node_action = x61:newpkt(x62, x63) \<longrightarrow> P (f6 x61 x62 x63)) \<and> (node_action = \<tau>\<^sub>n \<longrightarrow> P f7))"
distinct_1: fixes x1 :: "'a"   and x21 :: "'a Trace"   and x22 :: "'a" shows "tInit x1 \<noteq> x21 \<leadsto> x22"
thm_relation_negation_2_2: fixes v :: "i"   and F :: "\<Pi>\<^sub>2"   and x :: "\<nu>"   and y :: "\<nu>" shows "[\<^bold>\<not>\<lparr>F\<^sup>-,x\<^sup>P,y\<^sup>P\<rparr> \<^bold>\<equiv> \<lparr>F,x\<^sup>P,y\<^sup>P\<rparr> in v]"
e: fixes x :: "'a"   and y :: "'a" assumes "x \<in> path_image g"   and "y \<in> - S" shows "e \<le> dist x y"
div_add_self2: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and euclidean_size :: "'a \<Rightarrow> nat"   and b :: "'a"   and a :: "'a" assumes "class.euclidean_semiring_cancel divide plus minus zero times one modulo euclidean_size"   and "b \<noteq> zero" shows "divide (plus a b) b = plus (divide a b) one"
ttree_belowI: fixes t :: "'a ttree"   and t' :: "'a ttree" assumes "\<And>xs. xs \<in> paths t \<Longrightarrow> xs \<in> paths t'" shows "t \<sqsubseteq> t'"
pseudo_proj_False_def: fixes n :: "nat" shows "pseudo_proj_False n = (\<lambda>w. (stake n w @ [False]) @- sconst True)"
points_index_one_unique: fixes B :: "'b set multiset"   and ps :: "'b set"   and bl :: "'b set"   and bl' :: "'b set" assumes "B index ps = 1"   and "bl \<in># B"   and "ps \<subseteq> bl"   and "bl' \<in># B"   and "ps \<subseteq> bl'" shows "bl = bl'"
prod_zero: fixes times :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and one :: "'e"   and plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and A :: "'f set"   and f :: "'f \<Rightarrow> 'e" assumes "class.comm_semiring_1 times one plus zero"   and "finite A"   and "\<exists>a\<in>A. f a = zero" shows "comm_monoid_mult.prod times one f A = zero"
cut_RestrictedP: fixes s :: "name"   and f :: "tm"   and k :: "tm"   and A :: "fm"   and H :: "fm set" assumes "atom s \<sharp> (f, k, A)"   and "\<forall>C\<in>H. atom s \<sharp> C"   and "insert (RestrictedP f k (Var s)) H \<turnstile> A" shows "H \<turnstile> A"
insertion_eq_IpolyI: fixes p :: "'a poly"   and xs :: "nat \<Rightarrow> 'a"   and ys :: "'a list" assumes "\<And>i. i < num_params p \<Longrightarrow> xs i = ys ! i" shows "insertion xs p = Ipoly ys p"
a2: shows "\<forall>x. \<exists>y. u x = t x + y"
Class_is_Left_Coset: fixes g :: "'a" assumes "g \<in> Ker" shows "L.Class g = g \<cdot>| L"
islimpt_UNIV: fixes x :: "'b" shows "x islimpt UNIV"
bot_fresh: fixes a :: "atom" shows "a \<sharp> \<bottom>"
e2ennreal_ereal: fixes x :: "real" shows "e2ennreal (ereal x) = ennreal x"
rep_eq: fixes x :: "'c \<Rightarrow>\<^sub>C\<^sub>L 'b"   and xa :: "'a \<Rightarrow>\<^sub>C\<^sub>L 'c" shows "(*\<^sub>V) (x o\<^sub>C\<^sub>L xa) = (*\<^sub>V) x \<circ> (*\<^sub>V) xa"
AssignE: fixes x :: "char list"   and a :: "aexp"   and s :: "char list \<Rightarrow> int"   and t :: "char list \<Rightarrow> int" assumes "(x ::= a, s) \<Rightarrow> t" obtains "t = s[a/x]"
assms_2: fixes r :: "((char list \<Rightarrow> int) \<times> (char list \<Rightarrow> int)) set" assumes "single_valued r" shows "single_valued (f r)"
to_tree_order_pure: fixes ptr :: "(_) object_ptr"   and h :: "(_) heap" shows "pure (to_tree_order ptr) h"
finite_imp_card_of_natLeq_on: fixes A :: "'a set" assumes "finite A" shows "|A| =o natLeq_on (card A)"
sup_selective: fixes x :: "'a"   and y :: "'a" shows "x \<squnion> y = x \<or> x \<squnion> y = y"
trans_F: fixes P :: "'a process"   and Q :: "'a process"   and S :: "'a process" assumes "P \<sqsubseteq>\<^sub>F Q"   and "Q \<sqsubseteq>\<^sub>F S" shows "P \<sqsubseteq>\<^sub>F S"
rec_eint_def: shows "rec_eint \<equiv> \<lambda>f1 f2 x. THE y. rec_set_eint f1 f2 x y"
sum_le_zero: fixes s :: "'sum" shows "(s \<le> (0::'sum)) = (s = (0::'sum))"
intro: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<i> :: "'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b"   and f :: "'b"   and g :: "'b"   and \<eta> :: "'b"   and \<epsilon> :: "'b" assumes "equivalence_in_bicategory V H \<a> \<i> src trg f g \<eta> \<epsilon>"   and "adjunction_in_bicategory V H \<a> \<i> src trg f g \<eta> \<epsilon>" shows "adjoint_equivalence_in_bicategory V H \<a> \<i> src trg f g \<eta> \<epsilon>"
lr_def: fixes F'' :: "'a filter"   and g'' :: "'a \<Rightarrow> real" shows "o[F''](g'') = {f. \<forall>c>0. \<forall>\<^sub>F x in F''. norm (f x) \<le> c * norm (g'' x)}"
SMP_subterms_subset: fixes M :: "('fun, 'var) Term.term set" shows "subterms\<^sub>s\<^sub>e\<^sub>t M \<subseteq> SMP M"
contra_subsetD: fixes A :: "'e set"   and B :: "'e set"   and c :: "'e" assumes "A \<subseteq> B"   and "c \<notin> B" shows "c \<notin> A"
that: fixes v :: "'t" assumes "v \<in> set (Koszul_syz_sigs fs)"   and "v adds\<^sub>t lt (fs ! i \<odot> monomial (1::'b) (term_of_pair (0::'a, j)) - fs ! j \<odot> monomial (1::'b) (term_of_pair (0::'a, i)))" shows "thesis"
measurable_subprob_algebra: fixes M :: "'a measure"   and K :: "'a \<Rightarrow> 'b measure"   and N :: "'b measure" assumes "\<And>a. a \<in> space M \<Longrightarrow> subprob_space (K a)"   and "\<And>a. a \<in> space M \<Longrightarrow> sets (K a) = sets N"   and "\<And>A. A \<in> sets N \<Longrightarrow> (\<lambda>a. emeasure (K a) A) \<in> borel_measurable M" shows "K \<in> M \<rightarrow>\<^sub>M subprob_algebra N"
matchs_app: fixes xs\<^sub>2 :: "term list"   and ys\<^sub>2 :: "'a list"   and xs\<^sub>1 :: "term list"   and ys\<^sub>1 :: "'a list" assumes "length xs\<^sub>2 = length ys\<^sub>2" shows "matchs (xs\<^sub>1 @ xs\<^sub>2) (ys\<^sub>1 @ ys\<^sub>2) = matchs xs\<^sub>1 ys\<^sub>1 \<bind> (\<lambda>env\<^sub>1. matchs xs\<^sub>2 ys\<^sub>2 \<bind> (\<lambda>env\<^sub>2. Some (env\<^sub>1 ++\<^sub>f env\<^sub>2)))"
is_sublist_rev: fixes a :: "'a"   and b :: "'a"   and zs :: "'a list" shows "is_sublist [a, b] (rev zs) = is_sublist [b, a] zs"
opp_add: fixes a :: "'b"   and b :: "'b"   and p\<^sub>1 :: "'b point"   and p\<^sub>2 :: "'b point" assumes "on_curve a b p\<^sub>1"   and "on_curve a b p\<^sub>2" shows "opp (ell_field_class.add a p\<^sub>1 p\<^sub>2) = ell_field_class.add a (opp p\<^sub>1) (opp p\<^sub>2)"
complex_of_real_mult_Complex: fixes r :: "real"   and x :: "real"   and y :: "real" shows "complex_of_real r * Complex x y = Complex (r * x) (r * y)"
assms_2: fixes x :: "'a" assumes "x \<in> S" shows "(f has_derivative f' x) (at x within S)"
of_fun_nth: fixes i :: "nat"   and n :: "nat"   and f :: "nat \<Rightarrow> 'b" assumes "i < n" shows "IArray.of_fun f n !! i = f i"
IJ_def: shows "IJ \<equiv> \<lambda>k. Sigma {..k} lessThan"
two_chain_valid_valid_cubes: shows "\<forall>two_cube\<in>two_chain_typeII. valid_two_cube two_cube"
exhaust: fixes y :: "('b, 'c) Stream" obtains "y = \<bottom>"   | cfun :: "'c \<rightarrow> ('b, 'c) Step" and s :: "'c" where "y = Stream\<cdot>cfun\<cdot>s" and "s \<noteq> \<bottom>"
le_square_hmset: fixes m :: "hmultiset" shows "m \<le> m * m"
finite_if_finite_subsets_card_bdd: fixes F :: "'e set"   and C :: "nat" assumes "\<And>G. \<lbrakk>G \<subseteq> F; finite G\<rbrakk> \<Longrightarrow> card G \<le> C" shows "finite F \<and> card F \<le> C"
bigo_sum_main: fixes A :: "'c \<Rightarrow> 'd set"   and h :: "'c \<Rightarrow> 'd \<Rightarrow> 'e"   and f :: "'c \<Rightarrow> 'd \<Rightarrow> 'e" assumes "\<forall>x. \<forall>y\<in>A x. (0::'e) \<le> h x y"   and "\<exists>c. \<forall>x. \<forall>y\<in>A x. \<bar>f x y\<bar> \<le> c * h x y" shows "(\<lambda>x. sum (f x) (A x)) \<in> O(\<lambda>x. sum (h x) (A x))"
split: fixes P :: "'e \<Rightarrow> bool"   and f1 :: "'e"   and f2 :: "'f \<Rightarrow> 'f list \<Rightarrow> 'e"   and list :: "'f list" shows "P (case list of [] \<Rightarrow> f1 | x # xa \<Rightarrow> f2 x xa) = ((list = [] \<longrightarrow> P f1) \<and> (\<forall>x21 x22. list = x21 # x22 \<longrightarrow> P (f2 x21 x22)))"
lower_character_lift_character_1: fixes \<chi> :: "'a \<Rightarrow> complex"   and z :: "complex" assumes "\<chi> \<in> characters (G\<lparr>carrier := H\<rparr>)" shows "(\<lambda>x. if x \<in> H then lift_character (\<chi>, z) x else 0) = \<chi>"
disc_eq_case: fixes prod :: "'d \<times> 'e" shows "(prod = prod) = (case prod of (uu_, uua_) \<Rightarrow> True)"
openin_Times: fixes S :: "'b set"   and S' :: "'b set"   and T :: "'c set"   and T' :: "'c set" assumes "openin (top_of_set S) S'"   and "openin (top_of_set T) T'" shows "openin (top_of_set (S \<times> T)) (S' \<times> T')"
simps_2: fixes y :: "('a, 'b) Combinators list" shows "removeShadowRules2 (DenyAll # y) = DenyAll # removeShadowRules2 y"
gcd_mult_unit1: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "is_unit a" shows "gcd (b * a) c = gcd b c"
empty_subls: fixes L :: "fterm literal set" shows "L \<cdot>\<^sub>l\<^sub>s \<epsilon> = L"
tllist_all2_refl: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "('a, 'b) tllist" shows "tllist_all2 P Q xs xs = ((\<forall>x\<in>tset xs. P x x) \<and> (tfinite xs \<longrightarrow> Q (terminal xs) (terminal xs)))"
is_processT3: fixes P :: "'c process" shows "\<forall>s t. (s @ t, {}) \<in> F P \<longrightarrow> (s, {}) \<in> F P"
length_filter_less: fixes x :: "'g"   and xs :: "'g list"   and P :: "'g \<Rightarrow> bool" assumes "x \<in> set xs"   and "\<not> P x" shows "length (filter P xs) < length xs"
Sup_insert: fixes a :: "'c \<times> 'd"   and A :: "('c \<times> 'd) set" shows "Sup_prod (insert a A) = sup_prod a (Sup_prod A)"
ex_eval_fmla_iff_exists_num: fixes e :: "(name, nat) finfun"   and k :: "name"   and A :: "fmla" shows "eval_fmla e (exi k A) = (\<exists>t. eval_fmla e (A(k::=t)) \<and> t \<in> num)"
rtrancl_rbt_impl_def: shows "rtrancl_rbt_impl = rtrancl_impl rm_succ (\<lambda>as bs. rs.union bs (rs.from_list as)) rs.memb (rs.empty ())"
set_eq_def: fixes S :: "('d, 'e) eq_object_scheme"   and A :: "'d set"   and B :: "'d set" shows "(A {.=}\<^bsub>S\<^esub> B) = ((\<forall>x\<in>A. x .\<in>\<^bsub>S\<^esub> B) \<and> (\<forall>x\<in>B. x .\<in>\<^bsub>S\<^esub> A))"
harr_is_varr: fixes \<mu> :: "'a" assumes "H.arr \<mu>" shows "arr \<mu>"
L: assumes "List.member (fin_string_upto n) L" shows "length (Rep_fin_string L) \<le> Suc n"
transitively_orientable_def: fixes x :: "'a" shows "bounded_distrib_allegory.transitively_orientable (\<sqinter>) (\<sqinter>) (\<le>) (\<squnion>) bot id x \<equiv> \<exists>y. (y \<squnion> id y = x \<and> y \<sqinter> id y = bot) \<and> y \<sqinter> y \<le> y"
ev_at_unique: fixes P :: "'a stream \<Rightarrow> bool"   and n :: "nat"   and \<omega> :: "'a stream"   and m :: "nat" assumes "ev_at P n \<omega>"   and "ev_at P m \<omega>" shows "n = m"
vec_carrier: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'b" shows "vec n f \<in> carrier_vec n"
append_assoc: fixes xs :: "'g list"   and ys :: "'g list"   and zs :: "'g list" shows "(xs @ ys) @ zs = xs @ ys @ zs"
as_needed_asses_submap_exec_iv: fixes v :: "'a"   and a :: "('a, 'b) fmap \<times> 'c"   and s :: "('a, 'b) fmap" assumes "v \<in> fmdom' (action_needed_asses a s)" shows "fmlookup (action_needed_asses a s) v = fmlookup s v \<and> fmlookup (action_needed_asses a s) v = fmlookup (fst a) v \<and> fmlookup (fst a) v = fmlookup s v"
A: shows "Crypt (sesK (d * e)) (Passwd n) \<in> parts (A \<union> spies evsFC5) \<longrightarrow> m' = n"
can_select_set_list_ex1: fixes P :: "'a \<Rightarrow> bool"   and A :: "'a list" shows "can_select P (set A) = list_ex1 P A"
f_def: fixes n :: "nat" shows "f n = sum g {i. i < n}"
open_Int_closure_subset: fixes S :: "'c set"   and T :: "'c set" assumes "open S" shows "S \<inter> closure T \<subseteq> closure (S \<inter> T)"
elims_3: fixes x :: "'a \<times> nat" assumes "\<not> is_valid_event_id x" obtains i :: "'a" and j :: "nat" where "x = (i, j)" and "\<not> j < length (events i)"
derived_set_of_Un: fixes X :: "'c topology"   and S :: "'c set"   and T :: "'c set" shows "X derived_set_of (S \<union> T) = X derived_set_of S \<union> X derived_set_of T"
eball_mono: fixes r :: "ereal"   and r' :: "ereal"   and z :: "'a" assumes "r \<le> r'" shows "eball z r \<subseteq> eball z r'"
tfr_stp_all_split_2: fixes x :: "('fun, 'var) strand_step"   and S :: "('fun, 'var) strand_step list" assumes "list_all tfr\<^sub>s\<^sub>t\<^sub>p (x # S)" shows "list_all tfr\<^sub>s\<^sub>t\<^sub>p S"
has_density_def: fixes M :: "'c measure"   and N :: "'c measure"   and f :: "'c \<Rightarrow> ennreal" shows "has_density M N f = (f \<in> borel_measurable N \<and> space N \<noteq> {} \<and> M = density N f)"
equivpI: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "reflp R"   and "symp R"   and "transp R" shows "equivp R"
map2_elemE: fixes z :: "'a"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> 'a"   and xs :: "'b list"   and ys :: "'c list" assumes "z \<in> set (Term_Utils.map2 f xs ys)" obtains x :: "'b" and y :: "'c" where "x \<in> set xs" and "y \<in> set ys" and "z = f x y"
i: shows "\<not> is_zero_row i (Hermite_of A ass res bezout)"
set_of_bv_def: fixes bs :: "bool list" shows "set_of_bv bs = {i. i < length bs \<and> bs ! i}"
induce_subgraph_conn_1: shows "strongly_connected (G \<restriction> verts (Digraph_Component.union c d))"
leq_l: shows "\<forall>x\<le>l. length (fst (c' (n + x))) = length (fst (c (n + x)))"
antisym_conv2: fixes less_eq :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and less :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and x :: "'k"   and y :: "'k" assumes "class.order less_eq less"   and "less_eq x y" shows "(\<not> less x y) = (x = y)"
args_to_set_def: fixes x :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<times> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<times> ((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list" shows "args_to_set x = fst ` (set (fst x) \<union> set (fst (snd x)) \<union> fst ` set (snd (snd x)) \<union> snd ` set (snd (snd x)))"
fw_shortest_path_up_to: fixes m :: "nat \<Rightarrow> nat \<Rightarrow> 'a"   and k :: "nat"   and n :: "nat"   and i' :: "nat"   and i :: "nat"   and j' :: "nat"   and j :: "nat" assumes "cycle_free_up_to m k n"   and "i' \<le> i"   and "j' \<le> j"   and "i \<le> n"   and "j \<le> n"   and "k \<le> n" shows "D m i' j' k = fw m n k i j i' j'"
less_divide_eq_numeral1_1: fixes a :: "'c"   and b :: "'c"   and w :: "num" shows "(a < b / numeral w) = (a * numeral w < b)"
orderE: fixes a :: "'b word"   and b :: "'b word" assumes "a \<le>s b" obtains "a = signed.min a b"
dagger_of_transpose_is_cnj: fixes M :: "complex mat" shows "M\<^sup>t\<^sup>\<dagger> = M\<^sup>\<star>"
Amin_le_addTr: fixes v :: "'a \<Rightarrow> ant"   and n :: "nat"   and f :: "nat \<Rightarrow> 'a" assumes "valuation K v" shows "(\<forall>j\<le>n. f j \<in> carrier K) \<longrightarrow> Amin n (v \<circ> f) \<le> v (\<Sigma>\<^sub>e K f n)"
has_ignore_fresh_ESem_axioms: shows "has_ignore_fresh_ESem ESem"
subst_clause'_def: fixes \<Phi> :: "'b ltln list"   and m :: "'b ltln \<Rightarrow> 'b ltln option" shows "subst_clause' \<Phi> m = fold (\<lambda>\<phi> acc. acc \<otimes>\<^sub>l list_dnf (subst \<phi> m)) \<Phi> {[]}"
assms_3: shows "\<lbrace>p \<cdot> b\<rbrace>x\<lbrace>q\<rbrace>"
inverse_transformation_def: fixes A :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and B :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and F :: "'b \<Rightarrow> 'c"   and G :: "'b \<Rightarrow> 'c"   and \<tau> :: "'b \<Rightarrow> 'c" shows "inverse_transformation A B F G \<tau> \<equiv> (category A \<and> category B) \<and> functor A B F \<and> functor A B G \<and> natural_isomorphism A B F G \<tau>"
C1: fixes S :: "ccs"   and T :: "ccs"   and U :: "ccs" shows "(S \<parallel> T \<parallel> U, S \<parallel> (T \<parallel> U)) \<in> Rel"
f: fixes n :: "nat" shows "(\<forall>y\<in>set (f n). SN_on {(s, t). fst (g s t)} {y}) \<and> fst (lex_ext g m (f n) (f (Suc n)))"
induced_def: fixes A :: "'b set"   and r :: "('b \<times> 'b) set" shows "induced A r = {(a, b). a \<in> A \<and> b \<in> A \<and> (a, b) \<in> r}"
poly_has_sgnx_left_right: fixes p :: "real poly"   and c :: "real"   and a :: "real" assumes "p \<noteq> 0" shows "(poly p has_sgnx c) (at_left a) = (if even (order a p) then (poly p has_sgnx c) (at_right a) else (poly p has_sgnx - c) (at_right a))"
rangeI: fixes f :: "'c \<Rightarrow> 'a"   and x :: "'c" shows "f x \<in> range f"
bound_trans: fixes X :: "'a set"   and b :: "'a"   and c :: "'a" assumes "bound X (\<sim>) b"   and "b \<sim> c"   and "X \<subseteq> A"   and "b \<in> A"   and "c \<in> A" shows "bound X (\<sim>) c"
card_derangements: fixes S :: "nat set"   and n :: "nat" assumes "finite S"   and "card S = n" shows "card (derangements S) = count_derangements n"
axioms_1: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'c" assumes "comp_fun_idem f" shows "comp_fun_commute f"
RModuleHom_def: fixes R :: "'a set"   and smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and M :: "'b set"   and smult' :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and T :: "'b \<Rightarrow> 'c" shows "RModuleHom R smult M smult' T \<equiv> RModule R smult M \<and> GroupHom M T \<and> RModuleHom_axioms R smult M smult' T"
induct: fixes P :: "Reflection_Examples.num \<Rightarrow> bool"   and a0 :: "Reflection_Examples.num" assumes "\<And>b. P (C b)"   and "\<And>n. P (num.Var n)"   and "\<And>t s. \<lbrakk>P t; P s\<rbrakk> \<Longrightarrow> P (Add t s)"   and "\<And>c t. P t \<Longrightarrow> P (num.Mul c t)"   and "\<And>n c t. \<lbrakk>P (num.Mul c (num.Var n)); P t\<rbrakk> \<Longrightarrow> P (CN n c t)" shows "P a0"
sf: shows "(\<forall>y. F y \<longrightarrow> P y x) \<and> (\<forall>y. P y x \<longrightarrow> (\<exists>z. F z \<and> O y z))"
assms_2: shows "set (injectionsAlg xs Y) = injections (set xs) (set Y)"
a: shows "d x \<cdot> y \<le> y \<cdot> d z"
bind_merge_merge: fixes y :: "'s"   and f :: "'a \<Rightarrow> 'm"   and g :: "'a \<Rightarrow> 'm" shows "bind (merge y f) (\<lambda>A. merge A g) = merge y (\<lambda>x. bind (f x) (\<lambda>A. merge A g))"
constant_transformation_axioms: shows "constant_transformation (\<cdot>\<^sub>J) (\<cdot>) a"
closure_box: fixes a :: "'a"   and b :: "'a" assumes "box a b \<noteq> {}" shows "closure (box a b) = cbox a b"
Radj_cancel: fixes x :: "'sz"   and y :: "'sz"   and \<nu> :: "(real, 'sz) vec \<times> (real, 'sz) vec" shows "Radj x y (Radj x y \<nu>) = \<nu>"
cases: fixes \<I> :: "('i, 'j) \<I>"   and \<I>' :: "('k, 'l) \<I>"   and cnv :: "('i, 'j, 'k, 'l) converter"   and a :: "('i \<Rightarrow> 'k) \<times> ('l \<Rightarrow> 'j)" assumes "wiring \<I> \<I>' cnv a" obtains f :: "'i \<Rightarrow> 'k" and g :: "'l \<Rightarrow> 'j" where "a = (f, g)" and "\<I>,\<I>' \<turnstile>\<^sub>C cnv \<sim> map_converter id id f g 1\<^sub>C" and "\<I>, \<I>' \<turnstile>\<^sub>C cnv \<surd>"
AbGroup: fixes R :: "'a set"   and smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and M :: "'b set" assumes "RModule R smult M" shows "AbGroup M"
top_neq_ennreal: fixes r :: "real" shows "top \<noteq> ennreal r"
UinvUUinv: shows "Uinv o\<^sub>C\<^sub>L U o\<^sub>C\<^sub>L Uinv = Uinv"
finite_borel_measure_axioms_def: fixes M :: "real measure" shows "finite_borel_measure_axioms M \<equiv> sets M = sets borel"
injective_open_imp_embedding_map: fixes X :: "'a topology"   and Y :: "'b topology"   and f :: "'a \<Rightarrow> 'b" assumes "continuous_map X Y f"   and "open_map X Y f"   and "inj_on f (topspace X)" shows "embedding_map X Y f"
AbstrLevels_A3_A32: fixes i :: "AbstrLevelsID" assumes "sA3 \<in> AbstrLevel i" shows "sA32 \<notin> AbstrLevel i"
intro: fixes A :: "'c set"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "semiattractive A less_eq"   and "well_related_set A less_eq" shows "pre_well_ordered_set A less_eq"
\<tau>_ide: fixes b :: "'c" assumes "Cop.ide b" shows "\<tau>' b = S.mkArr (Hom.set (b, a)) (F.SET b) (\<lambda>x. F.FUN (\<psi> (b, a) x) (T'.\<tau>.FUN a (\<phi> (a, a) a)))"
iarray_to_vec_nth: fixes A :: "'b iarray"   and i :: "'c" shows "iarray_to_vec A $ i = A !! mod_type_class.to_nat i"
lock_thread_okI: fixes ls :: "('d, 'e) locks"   and ts :: "'e \<Rightarrow> ('f \<times> 'd \<Rightarrow>f nat) option" assumes "\<And>l t. has_lock (ls $ l) t \<Longrightarrow> \<exists>xw. ts t = \<lfloor>xw\<rfloor>" shows "lock_thread_ok ls ts"
assms_2: fixes \<phi> :: "'a fm" assumes "nqfree \<phi>"   and "normal \<phi>" shows "normal (qe \<phi>)"
fls_lr_inverse_const_2: fixes b :: "'a"   and y :: "'a" shows "fls_right_inverse (fls_const b) y = fls_const y"
af_letter_nested_prop_atoms: fixes \<phi> :: "'a ltln"   and \<nu> :: "'a set" shows "nested_prop_atoms (af_letter \<phi> \<nu>) \<subseteq> nested_prop_atoms \<phi>"
less_imp_not_eq2: fixes x :: "'a"   and y :: "'a" assumes "lt x y" shows "(y = x) = False"
max_def: fixes less_eq :: "'i \<Rightarrow> 'i \<Rightarrow> bool"   and a :: "'i"   and b :: "'i" shows "ord.max less_eq a b = (if less_eq a b then b else a)"
FE_INV_PRES: fixes blues0 :: "'v set"   and reds0 :: "'v set"   and onstack0 :: "'v set"   and u0 :: "'v"   and it :: "'v set"   and blues :: "'v set"   and reds :: "'v set"   and onstack :: "'v set"   and t :: "'v"   and blues' :: "'v set"   and reds' :: "'v set"   and onstack' :: "'v set"   and cyc :: "'v blue_witness" assumes "pre (blues0, reds0, onstack0, u0)"   and "fe_inv (insert u0 blues0) u0 (insert u0 onstack0) it (blues, reds, onstack, NO_CYC)"   and "t \<in> it"   and "it \<subseteq> E `` {u0}"   and "t \<notin> blues"   and "post (blues, reds, onstack, t) (blues', reds', onstack', cyc)" shows "fe_inv (insert u0 blues0) u0 (insert u0 onstack0) (it - {t}) (blues', reds', onstack', prep_wit_blue u0 cyc)"
env_restr_flip: fixes x :: "'c"   and S :: "'c set"   and x' :: "'c"   and \<rho> :: "'c \<Rightarrow> 'd" assumes "x \<notin> S"   and "x' \<notin> S" shows "((x' \<leftrightarrow> x) \<bullet> \<rho>) f|` S = \<rho> f|` S"
ma_sqrt_main: fixes r :: "mini_alg" assumes "0 \<le> ma_rat r"   and "ma_coeff r = 0" shows "sqrt (real_of r) = real_of (ma_sqrt r)"
vpra_Cons_intra: fixes a :: "'edge"   and cs :: "'edge list"   and as :: "'edge list" assumes "intra_kind (kind a)"   and "valid_path_rev_aux cs as" shows "valid_path_rev_aux cs (a # as)"