hcju/isabelleps huggingface dataset:
P1_adv_real_ideal_model_def: fixes D :: "'d \<times> 'grp \<Rightarrow> bool spmf"   and M :: "'grp \<times> 'grp"   and \<sigma> :: "bool"   and \<A> :: "('grp \<times> 'grp \<Rightarrow> 'grp \<Rightarrow> 'grp \<Rightarrow> 'grp \<Rightarrow> 'grp \<Rightarrow> 'grp \<Rightarrow> 'e \<Rightarrow> (('grp \<times> 'grp \<times> 'grp) \<times> 'f) spmf) \<times> ('grp \<Rightarrow> 'grp \<Rightarrow> 'grp \<Rightarrow> 'grp \<Rightarrow> 'grp \<Rightarrow> 'grp \<times> 'grp \<Rightarrow> 'f \<Rightarrow> ((('grp \<times> 'grp) \<times> 'grp \<times> 'grp) \<times> 'f) spmf) \<times> ('f \<Rightarrow> 'd spmf)"   and z :: "'e" shows "P1_adv_real_ideal_model D M \<sigma> \<A> z = \<bar>spmf (P1_real_model M \<sigma> z \<A> \<bind> D) True - spmf (P1_ideal_model M \<sigma> z \<A> \<bind> D) True\<bar>"
Prim_Invar2_ctd_loc_def: fixes w :: "'a set \<Rightarrow> nat"   and g :: "'a ugraph"   and r :: "'a"   and Q :: "'a \<Rightarrow> enat"   and \<pi> :: "'a \<Rightarrow> 'a option" shows "Prim_Invar2_ctd_loc w g r Q \<pi> \<equiv> Prim2.prim_invar2_ctd w g r Q \<pi>"
A3: shows "A3 \<in> carrier_mat 1 m1"
mult_t_closed: fixes x :: "'a"   and y :: "'a" shows "t (n x \<cdot> n y) = n x \<cdot> n y"
postdominate_def: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> 'c edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and Exit :: "'b"   and n' :: "'b"   and n :: "'b" assumes "Postdomination sourcenode targetnode kind valid_edge Entry Exit" shows "Postdomination.postdominate sourcenode targetnode valid_edge Exit n' n \<equiv> CFG.valid_node sourcenode targetnode valid_edge n \<and> CFG.valid_node sourcenode targetnode valid_edge n' \<and> (\<forall>as. CFG.path sourcenode targetnode valid_edge n as Exit \<longrightarrow> n' \<in> set (CFG.sourcenodes sourcenode as))"
sees_methods_fun: fixes P :: "'a prog"   and C :: "String.literal"   and Mm :: "String.literal \<Rightarrow> ((ty list \<times> ty \<times> 'a option) \<times> String.literal) option"   and Mm' :: "String.literal \<Rightarrow> ((ty list \<times> ty \<times> 'a option) \<times> String.literal) option" assumes "P \<turnstile> C sees_methods Mm"   and "P \<turnstile> C sees_methods Mm'" shows "Mm' = Mm"
tdev'_def: fixes p :: "nat"   and x :: "'a pdevs" shows "tdev' p x = sum_list' p (map (abs \<circ> snd) (list_of_pdevs x))"
subgraph_slim: shows "subgraph (with_proj S.slim) (with_proj slim)"
plusU_stateT_def: shows "plusU = (\<Lambda> a b. StateT\<cdot>(\<Lambda> s. mplus\<cdot>(runStateT\<cdot>a\<cdot>s)\<cdot>(runStateT\<cdot>b\<cdot>s)))"
vars_iff_subtermeq_set: fixes x :: "'a"   and M :: "('b, 'a) Term.term set" shows "(x \<in> fv\<^sub>s\<^sub>e\<^sub>t M) = (Var x \<sqsubseteq>\<^sub>s\<^sub>e\<^sub>t M)"
Cauchy_theorem_primitive: fixes S :: "complex set"   and f :: "complex \<Rightarrow> complex"   and f' :: "complex \<Rightarrow> complex"   and g :: "real \<Rightarrow> complex" assumes "\<And>x. x \<in> S \<Longrightarrow> (f has_field_derivative f' x) (at x within S)"   and "valid_path g"   and "path_image g \<subseteq> S"   and "pathfinish g = pathstart g" shows "(f' has_contour_integral 0) g"
below_Trivial_Aexp: fixes ae :: "Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and e :: "exp"   and n :: "Arity" shows "(ae \<sqsubseteq> Aexp_syn n e) = (edom ae \<subseteq> fv e)"
par_subst_comm: fixes A :: "'e \<Longrightarrow> 'f"   and B :: "'g \<Longrightarrow> 'f"   and \<sigma> :: "'f \<Rightarrow> 'f"   and \<rho> :: "'f \<Rightarrow> 'f" assumes "A \<bowtie> B" shows "\<sigma> [A|B]\<^sub>s \<rho> = \<rho> [B|A]\<^sub>s \<sigma>"
assms: shows "y \<notin> \<int>\<^sub>\<le>\<^sub>0"
pathVertices_alt: fixes p :: "(nat \<times> nat) list"   and u :: "nat" assumes "p \<noteq> []" shows "pathVertices u p = map fst p @ [snd (last p)]"
RSD_pareto_eqclass_indiff_set: fixes agents :: "'a set"   and alts :: "'b set"   and R :: "'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set"   and i :: "'a"   and x :: "'b"   and y :: "'b" assumes "RSD_pareto_eqclass agents alts R A"   and "i \<in> agents"   and "x \<in> A"   and "y \<in> A" shows "R i x y"
PAC_checker_l_step'_def: fixes a :: "(char list list \<times> int) list"   and b :: "char list code_status"   and c :: "char list set"   and d :: "(nat, (char list list \<times> int) list) fmap" shows "PAC_checker_l_step' a b c d = PAC_checker_l_step a (b, c, d)"
lincomb_union: fixes A :: "'a vec set"   and B :: "'a vec set"   and a :: "'a vec \<Rightarrow> 'a" assumes "finite (A \<union> B)"   and "A \<union> B \<subseteq> carrier_vec n"   and "A \<inter> B = {}" shows "lincomb a (A \<union> B) = lincomb a A + lincomb a B"
getParts_same_fw_behaviour: fixes A :: "'b word set"   and rs :: "'b simple_rule list"   and s1 :: "'b word"   and s2 :: "'b word" assumes "A \<in> set (map wordinterval_to_set (getParts rs))"   and "s1 \<in> A"   and "s2 \<in> A" shows "same_fw_behaviour s1 s2 rs"
assms_3: fixes R :: "pi"   and S :: "pi" assumes "Prop R S" shows "F R \<leadsto>\<guillemotleft>weakBisim\<guillemotright> F S"
worse_in_no_better: fixes x :: "'a"   and y :: "'a" assumes "x \<succeq> y" shows "y \<in> no_better_than y carrier relation"
3: shows "- - v\<^sup>T * w\<^sup>T \<le> top * e * (w\<^sup>T \<sqinter> - v\<^sup>T)\<^sup>\<star> \<squnion> - - v\<^sup>T"
fproductE2: fixes a :: "V"   and b :: "V"   and A :: "V"   and B :: "V" assumes "[a, b]\<^sub>\<circ> \<in>\<^sub>\<circ> A \<times>\<^sub>\<bullet> B" obtains "a \<in>\<^sub>\<circ> A" and "b \<in>\<^sub>\<circ> B"
list_all2_all_nthI: fixes a :: "'c list"   and b :: "'d list"   and P :: "'c \<Rightarrow> 'd \<Rightarrow> bool" assumes "length a = length b"   and "\<And>n. n < length a \<Longrightarrow> P (a ! n) (b ! n)" shows "list_all2 P a b"
option_to_ftb_def: fixes b :: "'d option" shows "option_to_ftb b \<equiv> case b of None \<Rightarrow> NoAction | Some a \<Rightarrow> Action a"
map_graph_in: fixes G :: "('a, 'b) labeled_graph"   and a :: "'b"   and b :: "'b"   and e :: "'a allegorical_term"   and f :: "'b \<Rightarrow> 'c" assumes "graph G"   and "(a, b) \<in> :G:\<lbrakk>e\<rbrakk>" shows "(f a, f b) \<in> :map_graph_fn G f:\<lbrakk>e\<rbrakk>"
infsetsum_def: fixes f :: "'b \<Rightarrow> 'a"   and A :: "'b set" shows "infsetsum f A = integral\<^sup>L (count_space A) f"
assms_2: shows "Group.group (G\<lparr>carrier := H\<rparr>)"
cases: fixes g :: "'g"   and a :: "'node list" assumes "local.path g a" obtains n :: "'node" where "a = [n]" and "n \<in> set (\<alpha>n g)" and "invar g"   | ns :: "'node list" and n' :: "'node" where "a = n' # ns" and "local.path g ns" and "n' \<in> set (predecessors g (hd ns))"
False: shows "\<not> \<Gamma>\<Turnstile>\<^sub>t\<^bsub>/F\<^esub> P c Q,A"
poincare_line_inversion_2: fixes z :: "complex_homo"   and w :: "complex_homo" assumes "z \<noteq> w" shows "on_circline (poincare_line z w) (inversion w)"
lemma_R_for_TOTAL: fixes \<psi> :: "nat \<Rightarrow> nat \<Rightarrow> nat option" shows "\<exists>d. \<forall>U i. learn_total \<psi> U (\<phi> i) \<longrightarrow> learn_total \<psi> U (totalizer d i)"
app_vimageD: fixes x :: "V"   and r :: "V"   and A :: "V" assumes "x \<in>\<^sub>\<circ> r `\<^sub>\<circ> A" shows "x \<in>\<^sub>\<circ> \<R>\<^sub>\<circ> (r \<restriction>\<^sup>l\<^sub>\<circ> A)"
parallel_reduction_diamond: fixes A :: "'a trm"   and B :: "'a trm"   and C :: "'a trm" assumes "A >> B"   and "A >> C" shows "\<exists>D. B >> D \<and> C >> D"
div_mult_self2: 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 b :: "'b"   and a :: "'b"   and c :: "'b" assumes "class.euclidean_semiring_cancel divide plus minus zero times one modulo euclidean_size"   and "b \<noteq> zero" shows "divide (plus a (times b c)) b = plus c (divide a b)"
leg1_is_map: shows "is_left_adjoint (tab\<^sub>1 (Maps.REP S\<^sub>1))"
submodule_pOp_closed: fixes M :: "('e, 'f, 'g) Module_scheme"   and R :: "('f, 'h) Ring_scheme"   and H :: "'e set"   and h :: "'e"   and k :: "'e" assumes "R module M"   and "submodule R M H"   and "h \<in> H"   and "k \<in> H" shows "h \<plusminus>\<^bsub>M\<^esub> k \<in> H"
RETURN_RES_refine_iff: fixes x :: "'b"   and R :: "('b \<times> 'c) set"   and Y :: "'c set" shows "(RETURN x \<le> \<Down> R (RES Y)) = (\<exists>y\<in>Y. (x, y) \<in> R)"
lepoll_funcset_right: fixes B :: "'a set"   and B' :: "'b set"   and A :: "'c set" assumes "B \<lesssim> B'" shows "A \<rightarrow>\<^sub>E B \<lesssim> A \<rightarrow>\<^sub>E B'"
size_image_mset: fixes f :: "'d \<Rightarrow> 'c"   and M :: "'d multiset" shows "size (image_mset f M) = size M"
naturally_isomorphic_symmetric: fixes A :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and B :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and F :: "'b \<Rightarrow> 'c"   and G :: "'b \<Rightarrow> 'c" assumes "naturally_isomorphic A B F G" shows "naturally_isomorphic A B G F"
join_le_iff: fixes le :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and join :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and Top :: "'a"   and x :: "'a"   and y :: "'a"   and z :: "'a" assumes "class.SL_top le join Top" shows "le (join x y) z = (le x z \<and> le y z)"
gauss2complex_eq_iff: fixes z :: "gauss_int"   and u :: "gauss_int" shows "(gauss2complex z = gauss2complex u) = (z = u)"
idem: fixes a :: "'a \<times> 'b" shows "sup_prod a a = a"
fmrel_on_fset_alt_def: fixes S :: "'d fset"   and P :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and m :: "('d, 'e) fmap"   and n :: "('d, 'f) fmap" shows "fmrel_on_fset S P m n = fBall S (\<lambda>x. rel_option P (fmlookup m x) (fmlookup n x))"
well_ord_adjunction: fixes X :: "'a set"   and a :: "'a" assumes "X \<subseteq> carrier D"   and "a \<in> carrier D"   and "\<forall>x\<in>X. x \<preceq> a"   and "Worder (Iod D X)" shows "Worder (Iod D (X \<union> {a}))"
conjuncts: fixes p :: "fm"   and vs :: "'b list"   and bs :: "'b list" shows "(\<forall>q\<in>set (conjuncts p). Ifm vs bs q) = Ifm vs bs p"
cmp_imp_equal: fixes x :: "'a"   and y :: "'a" assumes "x \<in> D"   and "y \<in> D"   and "cmp x y = EQUAL" shows "x = y"
naturality: fixes f :: "'a" assumes "A.arr f" shows "\<tau> (A.cod f) \<cdot>\<^sub>B F f = G f \<cdot>\<^sub>B \<tau> (A.dom f)"
rotateL_def: fixes t :: "('c, 'd) tree" shows "rotateL t = (if t = Tip then t else if rsub t = Tip then t else let rt = rsub t in Node (Node (lsub t) (key t) (nval t) (lsub rt)) (key rt) (nval rt) (rsub rt))"
f4: fixes r :: "real"   and ra :: "real" shows "- r + (ra + r) = ra"
finite_measure_mono: fixes A :: "'a set"   and B :: "'a set"   and M :: "'a pmf" assumes "A \<subseteq> B"   and "B \<in> measure_pmf.events M" shows "measure_pmf.prob M A \<le> measure_pmf.prob M B"
lnths_empty: fixes xs :: "'b llist" shows "lselect {} xs = <>"
Worder_ord_equivTr: fixes S :: "('c, 'd) Order_scheme"   and T :: "('e, 'f) Order_scheme" assumes "Worder S"   and "Worder T" shows "ord_equiv S T = (\<exists>f. ord_isom S T f)"
Gcd_dvd: fixes one :: "'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and gcd :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and lcm :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and Gcd :: "'a set \<Rightarrow> 'a"   and Lcm :: "'a set \<Rightarrow> 'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and a :: "'a"   and A :: "'a set" assumes "class.semiring_Gcd one zero times gcd lcm Gcd Lcm divide plus minus unit_factor normalize"   and "a \<in> A" shows "dvd.dvd times (Gcd A) a"
innermost_intvl: fixes f :: "atom fm"   and xs :: "real list"   and l :: "real"   and x :: "real"   and y :: "real" assumes "nqfree f"   and "nolb f xs l x"   and "l < x"   and "x \<notin> EQ f xs"   and "R.I f (x # xs)"   and "l < y"   and "y \<le> x" shows "R.I f (y # xs)"
is_category: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" assumes "dual_category C" shows "category (dual_category.comp C)"
CIOI: fixes \<xi> :: "'b"   and p :: "'c" assumes "(\<xi>, p) \<in> CI" shows "\<exists>\<sigma>. \<xi> = \<sigma> i \<and> (\<sigma>, p) \<in> OI"
l_find_slot_def: fixes find_slot :: "bool \<Rightarrow> (_) node_ptr \<Rightarrow> ((_) heap, exception, (_) element_ptr option) prog" shows "l_find_slot find_slot \<equiv> (\<forall>h open_flag slotable slot_opt. h \<turnstile> find_slot open_flag slotable \<rightarrow>\<^sub>r slot_opt \<longrightarrow> slotable |\<in>| node_ptr_kinds h) \<and> (\<forall>h open_flag slotable slot. h \<turnstile> find_slot open_flag slotable \<rightarrow>\<^sub>r Some slot \<longrightarrow> slot |\<in>| element_ptr_kinds h) \<and> (\<forall>open_flag slotable h. pure (find_slot open_flag slotable) h)"
eigen_equation: fixes k :: "complex"   and H :: "complex \<times> complex \<times> complex \<times> complex" shows "eigenval k H = (k\<^sup>2 - mat_trace H * k + mat_det H = 0)"
is_top_sorted_remove_elem: fixes R :: "('a \<times> 'a) set"   and l1 :: "'a list"   and x :: "'a"   and l2 :: "'a list" assumes "is_top_sorted R (l1 @ x # l2)" shows "is_top_sorted R (l1 @ l2)"
ftype_f_def: fixes P :: "cld list"   and ty :: "ty"   and f :: "char list" shows "ftype_f P ty f = (case find_path_ty_f P ty of None \<Rightarrow> None | Some ctxclds \<Rightarrow> ftype_in_path_f P ctxclds f)"
regions_partition': fixes \<R> :: "('d \<Rightarrow> real) set set"   and X :: "'d set"   and k :: "'d \<Rightarrow> nat"   and v :: "'d \<Rightarrow> real"   and v' :: "'d \<Rightarrow> real" assumes "\<R> = {region X I r |I r. valid_region X k I r}"   and "\<forall>x\<in>X. 0 \<le> v x"   and "\<forall>x\<in>X. 0 \<le> v' x"   and "v' \<in> [v]\<^sub>\<R>" shows "[v']\<^sub>\<R> = [v]\<^sub>\<R>"
get_proc_intra: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> ('c, 'd, 'e, 'f) edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and get_proc :: "'b \<Rightarrow> 'f"   and get_return_edges :: "'a \<Rightarrow> 'a set"   and procs :: "('f \<times> 'c list \<times> 'c list) list"   and Main :: "'f"   and a :: "'a" assumes "CFG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main"   and "valid_edge a"   and "intra_kind (kind a)" shows "get_proc (sourcenode a) = get_proc (targetnode a)"
Deduct2_with_False_def: fixes var :: "'a set"   and trm :: "'b set"   and fmla :: "'c set"   and Var :: "'a \<Rightarrow> 'b"   and FvarsT :: "'b \<Rightarrow> 'a set"   and substT :: "'b \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b"   and Fvars :: "'c \<Rightarrow> 'a set"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'c"   and eql :: "'b \<Rightarrow> 'b \<Rightarrow> 'c"   and cnj :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and imp :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and all :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and exi :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and fls :: "'c"   and num :: "'b set"   and prv :: "'c \<Rightarrow> bool"   and bprv :: "'c \<Rightarrow> bool" shows "Deduct2_with_False var trm fmla Var FvarsT substT Fvars subst eql cnj imp all exi fls num prv bprv \<equiv> Deduct_with_False var trm fmla Var FvarsT substT Fvars subst eql cnj imp all exi fls num prv \<and> Deduct_with_False var trm fmla Var FvarsT substT Fvars subst eql cnj imp all exi fls num bprv \<and> Deduct2_with_False_axioms fmla Fvars prv bprv"
0: shows "(\<lambda>n. Norm (\<LL> (ennreal p) (lebesgue_on S)) (g n - h)) \<longlonglongrightarrow> 0"
triangle_paths_neq: fixes a :: "'a"   and b :: "'a"   and c :: "'a"   and ab :: "'a set"   and ac :: "'a set" assumes "\<triangle> a b c"   and "path ab a b"   and "path ac a c" shows "ab \<noteq> ac"
disjunct_not_in: fixes A :: "'a multiset"   and B :: "'a multiset" shows "(A \<inter># B = {#}) = (\<forall>a. a \<notin># A \<or> a \<notin># B)"
raw_match'_produces_matcher: fixes T :: "typ"   and U :: "typ"   and subs :: "variable \<times> String.literal set \<Rightarrow> typ option"   and subs' :: "variable \<times> String.literal set \<Rightarrow> typ option" assumes "raw_match' T U subs = Some subs'" shows "tsubstT T (convert_subs subs') = U"
adjoint_dim_col: fixes A :: "'c mat" shows "dim_col (adjoint A) = dim_row A"
igSwapInp_def: fixes MOD :: "('c, 'd, 'e, 'f, 'g, 'h, 'b, 'i, 'j) model_scheme"   and zs :: "'e"   and z1 :: "'h"   and z2 :: "'h"   and inp :: "'a \<Rightarrow> 'b option" shows "igSwapInp MOD zs z1 z2 inp \<equiv> lift (igSwap MOD zs z1 z2) inp"
\<phi>_simps_4: shows "local.dom r\<^sub>0s\<^sub>1.\<phi> = r\<^sub>0 \<star> p\<^sub>1"
less_imp_le: fixes x :: "'c DBMEntry"   and y :: "'c DBMEntry" assumes "x \<prec> y" shows "x \<preceq> y"
prems: fixes i :: "'a" assumes "i \<in> insert i__ I" shows "F i \<in> sets M"
stableI: fixes F :: "'b program"   and A :: "'b set" assumes "F \<in> A co A" shows "F \<in> stable A"
Ana_keys_wf': fixes t :: "('fun, ('fun, 'atom) Term.term \<times> nat) Term.term"   and K :: "('fun, ('fun, 'atom) Term.term \<times> nat) Term.term list"   and T :: "('fun, ('fun, 'atom) Term.term \<times> nat) Term.term list"   and k :: "('fun, ('fun, 'atom) Term.term \<times> nat) Term.term" assumes "Ana t = (K, T)"   and "wf\<^sub>t\<^sub>r\<^sub>m t"   and "k \<in> set K" shows "wf\<^sub>t\<^sub>r\<^sub>m k"
prefixIsSubset: fixes p :: "'a list"   and l :: "'a list" assumes "isPrefix p l" shows "set p \<subseteq> set l"
hash_F_def: fixes h :: "'c \<Rightarrow> 'e"   and rhL :: "'d \<Rightarrow> 'f" shows "hash_F h rhL = map_sum id (map_prod h rhL)"
0: shows "to_fun f (a \<ominus> b) = to_fun f a \<oplus> deriv f a \<otimes> \<ominus> b \<oplus> c \<otimes> \<ominus> b [^] 2"
termMOD_igFreshIGVar: shows "igFreshIGVar termMOD"
plus_oracle_split: fixes P :: "(('h + 'i) \<times> 'j) spmf \<Rightarrow> bool"   and left :: "'j \<Rightarrow> 'k \<Rightarrow> ('h \<times> 'j) spmf"   and right :: "'j \<Rightarrow> 'l \<Rightarrow> ('i \<times> 'j) spmf"   and s :: "'j"   and lr :: "'k + 'l" shows "P ((left \<oplus>\<^sub>O right) s lr) = ((\<forall>x. lr = Inl x \<longrightarrow> P (map_spmf (apfst Inl) (left s x))) \<and> (\<forall>y. lr = Inr y \<longrightarrow> P (map_spmf (apfst Inr) (right s y))))"
restrict_gpv_parametric': assumes "bi_unique C"   and "bi_unique (=)" shows "(rel_\<I> C (=) ===> rel_gpv'' A C (=) ===> rel_gpv'' A C (=)) restrict_gpv restrict_gpv"
seteq: fixes A :: "'b set"   and B :: "'b set" shows "(A = B) = ((\<forall>a\<in>A. a \<in> B) \<and> (\<forall>b\<in>B. b \<in> A))"
inv_f_eq: fixes f :: "'d \<Rightarrow> 'e"   and x :: "'d"   and y :: "'e" assumes "inj f"   and "f x = y" shows "inv f y = x"
fg: shows "src\<^sub>B f = trg\<^sub>B g"
pair_completeI: fixes A :: "'b set"   and r :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "\<And>x y. \<lbrakk>x \<in> A; y \<in> A\<rbrakk> \<Longrightarrow> Ex (extreme_bound A r {x, y})" shows "pair_complete A r"
gcd_le1_int: fixes a :: "int"   and b :: "int" assumes "0 < a" shows "gcd a b \<le> a"
fbox_arden_eq: fixes x :: "'a"   and q :: "'a"   and p :: "'a" assumes "\<nabla>x = (zero_class.zero::'a)"   and "d q \<cdot>  |x] p = d p" shows " |x\<^sup>\<star>] q = d p"
nsg_subset: fixes N :: "'a set" assumes "G \<triangleright> N" shows "N \<subseteq> carrier G"
cmax_def: fixes r :: "('c \<times> 'c) set"   and s :: "('d \<times> 'd) set" shows "cmax r s = (if cinfinite r \<or> cinfinite s then czero +c r +c s else natLeq_on (max (card (Field r)) (card (Field s))) +c czero)"
set_map_encode_operator_effect: fixes \<Pi> :: "'b strips_problem"   and t :: "nat" shows "set (map (\<lambda>(t, op). encode_operator_effect \<Pi> t op) (List.product [0..<t] (\<Pi>\<^sub>\<O>))) = {encode_operator_effect \<Pi> k op |k op. (k, op) \<in> {0..<t} \<times> set (\<Pi>\<^sub>\<O>)}"
simple_path_joinE: fixes g1 :: "real \<Rightarrow> 'b"   and g2 :: "real \<Rightarrow> 'b" assumes "simple_path (g1 +++ g2)"   and "pathfinish g1 = pathstart g2" obtains "arc g1" and "arc g2" and "path_image g1 \<inter> path_image g2 \<subseteq> {pathstart g1, pathstart g2}"
fvi_Suc_bound: fixes b :: "nat"   and \<phi> :: "Formula.formula"   and n :: "nat" assumes "\<forall>i\<in>Formula.fvi (Suc b) \<phi>. i < n" shows "\<forall>i\<in>Formula.fvi b \<phi>. i < Suc n"
trnl\<^sub>\<eta>_eq: fixes u :: "'a"   and v :: "'a"   and \<mu> :: "'a" assumes "ide u"   and "ide v"   and "src f = trg v"   and "src g' = trg u"   and "\<mu> \<in> hom (f \<star> v) u" shows "A'.trnl\<^sub>\<eta> v \<mu> = (g' \<star> \<mu>) \<cdot> (\<eta>' \<star> v)"
simps_1: fixes r :: "rat" shows "real_of_2 (Rational r) = real_of_rat r"
count_single: fixes b :: "'a"   and a :: "'a" shows "count {#b#} a = (if b = a then 1 else 0)"
nodes_productF: fixes M1 :: "('d, 'e, 'f) FSM"   and M2 :: "('d, 'e, 'g) FSM"   and FAIL :: "'g \<times> 'f"   and PM :: "('d, 'e, 'g \<times> 'f) FSM" assumes "well_formed M1"   and "well_formed M2"   and "productF M2 M1 FAIL PM" shows "nodes PM \<subseteq> insert FAIL (nodes M2 \<times> nodes M1)"
g_value_image_filter_loc_def: fixes ops1 :: "('a, 'b, 'c, 'd) map_ops_scheme"   and ops2 :: "('a, 'e, 'f, 'g) map_ops_scheme" shows "g_value_image_filter_loc ops1 ops2 \<equiv> StdMap ops1 \<and> StdMap ops2"
ord_eq_le_trans: fixes a :: "'b"   and b :: "'b"   and c :: "'b" assumes "a = b"   and "b \<sqsubseteq> c" shows "a \<sqsubseteq> c"
cmp_in_hom_2: fixes a :: "'a"   and b :: "'a" assumes "C.ide a"   and "C.ide b"   and "src\<^sub>C a = trg\<^sub>C b" shows "\<guillemotleft>\<Phi> (a, b) : F a \<star>\<^sub>D F b \<Rightarrow>\<^sub>D F (a \<star>\<^sub>C b)\<guillemotright>"
domain_invol: fixes x :: "'a" shows "domain_bdlo_monoid.d (1::'a) (\<cdot>) (\<sqinter>) \<top> (domain_bdlo_monoid.d (1::'a) (\<cdot>) (\<sqinter>) \<top> x) = domain_bdlo_monoid.d (1::'a) (\<cdot>) (\<sqinter>) \<top> x"
mset_sort: fixes k :: "'c \<Rightarrow> 'a"   and xs :: "'c list" shows "mset (sort_key k xs) = mset xs"
wf_eq_append''': fixes V :: "'c set"   and S :: "('d, 'c) strand_step list"   and t' :: "('d, 'c) Term.term"   and a :: "poscheckvariant"   and t :: "('d, 'c) Term.term" assumes "wf\<^sub>s\<^sub>t V S"   and "fv t' \<subseteq> wfrestrictedvars\<^sub>s\<^sub>t S \<union> V" shows "wf\<^sub>s\<^sub>t V (S @ [\<langle>a: t \<doteq> t'\<rangle>\<^sub>s\<^sub>t])"
Rep_upsets: fixes x :: "'b upsets" shows "Rep_upsets x \<in> range \<Up>"
completeness_game_def: fixes h :: "'g"   and w :: "'h"   and e :: "'e" shows "\<Sigma>0.completeness_game h w e = init0 h w \<bind> (\<lambda>(r, a). response0 r w e \<bind> (\<lambda>z. return_spmf (check0 h a e z)))"
pt_bij3: fixes x :: "'a"   and y :: "'a"   and pi :: "('b \<times> 'b) list" assumes "x = y" shows "pi \<bullet> x = pi \<bullet> y"
inverse_arrowsE: fixes f :: "'a \<times> 'a"   and g :: "'a \<times> 'a" assumes "B.VV.inverse_arrows f g" obtains "B.VV.ide (B.VV.comp g f)" and "B.VV.ide (B.VV.comp f g)"
insert_compr: fixes a :: "'f"   and B :: "'f set" shows "insert a B = {x. x = a \<or> x \<in> B}"
less_ceiling_iff: fixes z :: "int"   and x :: "'a" shows "(z < \<lceil>x\<rceil>) = (of_int z < x)"
rev_induct: fixes P :: "'a list \<Rightarrow> bool"   and xs :: "'a list" assumes "P []"   and "\<And>x xs. P xs \<Longrightarrow> P (xs @ [x])" shows "P xs"
simps_2: fixes a :: "'b"   and as :: "'b list"   and ns :: "'b list list" shows "listprepend (a # as) ns = map ((#) a) ns @ listprepend as ns"
soundI2: fixes b :: "real"   and P :: "'c \<Rightarrow> real" assumes "bounded_by b P"   and "nneg P" shows "sound P"
extendEmpty: fixes C :: "('c, 'd) sequent" shows "extend ( \<Empt> \<Rightarrow>* \<Empt>) C = C"
fdisjnt_subset_right: fixes N' :: "'b fset"   and N :: "'b fset"   and M :: "'b fset" assumes "N' |\<subseteq>| N"   and "fdisjnt M N" shows "fdisjnt M N'"
mono_ge_id: fixes f :: "nat \<Rightarrow> nat"   and i :: "nat" assumes "\<forall>i. f i < f (Suc i)" shows "i \<le> f i"
se_mono_for_states_eq: fixes c1 :: "('c, 'd) conf"   and c2 :: "('c, 'd) conf"   and l :: "('c, 'd) label"   and c1' :: "('c, 'd) conf"   and c2' :: "('c, 'd) conf" assumes "states c1 = states c2"   and "se c1 l c1'"   and "se c2 l c2'" shows "states c2' = states c1'"