hcju/isabelleps huggingface dataset:
lowner_le_add: fixes A :: "complex mat"   and n :: "nat"   and B :: "complex mat"   and C :: "complex mat"   and D :: "complex mat" assumes "A \<in> carrier_mat n n"   and "B \<in> carrier_mat n n"   and "C \<in> carrier_mat n n"   and "D \<in> carrier_mat n n"   and "A \<le>\<^sub>L B"   and "C \<le>\<^sub>L D" shows "A + C \<le>\<^sub>L B + D"
hyps_3: fixes b :: "name"   and ba :: "lam" assumes "b \<sharp> M" shows "M[b::=ba] = M"
push_down_configuration_token_run: fixes \<Sigma> :: "'c set"   and \<delta> :: "'d \<Rightarrow> 'c \<Rightarrow> 'd"   and q\<^sub>0 :: "'d"   and w :: "nat \<Rightarrow> 'c"   and x :: "nat"   and q :: "'d"   and n :: "nat"   and y :: "nat" assumes "semi_mojmir \<Sigma> \<delta> q\<^sub>0 w"   and "x \<in> semi_mojmir_def.configuration \<delta> q\<^sub>0 w q n"   and "y \<in> semi_mojmir_def.configuration \<delta> q\<^sub>0 w q n" shows "x \<le> n \<and> y \<le> n \<and> semi_mojmir_def.token_run \<delta> q\<^sub>0 w x n = semi_mojmir_def.token_run \<delta> q\<^sub>0 w y n"
doubleton_eq_iff: fixes a :: "'a"   and b :: "'a"   and c :: "'a"   and d :: "'a" shows "(insert a (\<eta> b) = insert c (\<eta> d)) = (a = c \<and> b = d \<or> a = d \<and> b = c)"
list_all2_Cons: fixes P :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and x :: "'d"   and xs :: "'d list"   and y :: "'e"   and ys :: "'e list" shows "list_all2 P (x # xs) (y # ys) = (P x y \<and> list_all2 P xs ys)"
genus_impl_def: fixes G :: "'b list \<times> ('b \<times> 'b) list"   and M :: "('b \<times> 'b) list list" shows "genus_impl G M \<equiv> case G of (V, A) \<Rightarrow> (int (2 * card (sccs_verts_impl G)) - int (length (isolated_verts_impl G)) - (int (length V) - int (length A) div 2 + int (length (orbits_list_impl (lists_fc_succ M) A)))) div 2"
br: fixes vs :: "e list"   and n :: "nat"   and i :: "nat"   and lholed :: "Lholed"   and LI :: "e list"   and es :: "e list" assumes "const_list vs"   and "length vs = n"   and "Lfilled i lholed (vs @ [$Br i]) LI" shows "\<lparr>[Label n es LI]\<rparr> \<leadsto> \<lparr>vs @ es\<rparr>"
l_set_disconnected_nodes_get_disconnected_nodes_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def: fixes known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and type_wf :: "(_) heap \<Rightarrow> bool"   and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and set_disconnected_nodes :: "(_) document_ptr \<Rightarrow> (_) node_ptr list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and heap_is_wellformed :: "(_) heap \<Rightarrow> bool"   and parent_child_rel :: "(_) heap \<Rightarrow> ((_) object_ptr \<times> (_) object_ptr) set"   and get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog" shows "l_set_disconnected_nodes_get_disconnected_nodes_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M known_ptr type_wf get_disconnected_nodes get_disconnected_nodes_locs set_disconnected_nodes set_disconnected_nodes_locs heap_is_wellformed parent_child_rel get_child_nodes \<equiv> l_set_disconnected_nodes_get_disconnected_nodes type_wf get_disconnected_nodes get_disconnected_nodes_locs set_disconnected_nodes set_disconnected_nodes_locs \<and> l_heap_is_wellformed type_wf known_ptr heap_is_wellformed parent_child_rel get_child_nodes get_disconnected_nodes"
rep_eq: fixes x :: "'a"   and xa :: "('a, 'b) alist" shows "alist.impl_of (delete x xa) = delete_aux x (alist.impl_of xa)"
left_adjoint_preserved_by_iso: fixes f :: "('a S.E.term, 'a) S.arr"   and \<phi> :: "('a S.E.term, 'a) S.arr"   and f' :: "('a S.E.term, 'a) S.arr" assumes "S.is_left_adjoint f"   and "\<guillemotleft>\<phi> : f \<Rightarrow>\<^sub>S f'\<guillemotright>"   and "S.iso \<phi>" shows "S.is_left_adjoint f'"
m\<tau>move_False: shows "\<tau>multithreaded.m\<tau>move (\<lambda>s ta s'. False) = (\<lambda>s ta s'. False)"
SPEC_rule_conj_leofI2: fixes m :: "'a nres"   and \<Phi> :: "'a \<Rightarrow> bool"   and \<Psi> :: "'a \<Rightarrow> bool" assumes "m \<le>\<^sub>n SPEC \<Phi>"   and "m \<le> SPEC \<Psi>" shows "m \<le> SPEC (\<lambda>s. \<Phi> s \<and> \<Psi> s)"
bis_diag: fixes B1 :: "'d set"   and B2 :: "'e set"   and s1 :: "'d \<Rightarrow> ('f, 'd, 'e) F1"   and s2 :: "'e \<Rightarrow> ('f, 'd, 'e) F2" assumes "coalg B1 B2 s1 s2" shows "sbis B1 B2 s1 s2 (Id_on B1) (Id_on B2)"
subgroup_img_is_subgroup: fixes I :: "nat set" assumes "subgroup I G" shows "subgroup (f ` I) H"
uniqAppend_1: fixes l1 :: "'b list"   and l2 :: "'b list" assumes "uniq (l1 @ l2)" shows "uniq l1"
assms_5: shows "sds R \<prec>[SD (R i)] q"
homeomorphism_on_bij: fixes S :: "'c set"   and f :: "'c \<Rightarrow> 'd" assumes "homeomorphism_on S f" shows "bij_betw f S (f ` S)"
tinfinite_def: fixes V :: "Preliminaries.var set" shows "tinfinite V \<equiv> \<forall>\<sigma>. infinite (V \<inter> tpOfV -` {\<sigma>})"
ground_resolvent_subset: fixes CAs :: "'a literal multiset list"   and DA :: "'a literal multiset"   and S :: "'a literal multiset \<Rightarrow> 'a literal multiset"   and AAs :: "'a multiset list"   and As :: "'a list"   and \<sigma> :: "'s"   and E :: "'a literal multiset" assumes "is_ground_cls_list CAs"   and "is_ground_cls DA"   and "ord_resolve S CAs DA AAs As \<sigma> E" shows "E \<subseteq># \<Sum>\<^sub># (mset CAs) + DA"
assms: fixes T :: "Unary_PCF.type" assumes "\<And>T1 T2. T = T1 \<rightarrow> T2 \<Longrightarrow> P T1"   and "\<And>T1 T2. T = T1 \<rightarrow> T2 \<Longrightarrow> P T2" shows "P T"
sums_integral: fixes M :: "'a measure"   and f :: "nat \<Rightarrow> 'a \<Rightarrow> 'b" assumes "\<And>i. integrable M (f i)"   and "AE x in M. summable (\<lambda>i. norm (f i x))"   and "summable (\<lambda>i. LINT x|M. norm (f i x))" shows "(\<lambda>i. integral\<^sup>L M (f i)) sums (LINT x|M. (\<Sum>i. f i x))"
nullable_iff: fixes r :: "'b rexp" shows "nullable r = ([] \<in> lang r)"
that: fixes a :: "V" assumes "[a, b]\<^sub>\<circ> \<in>\<^sub>\<circ> r" shows "thesis"
basis_def: fixes A :: "complex vec set" shows "k.basis A = (k.lin_indpt A \<and> k.span A = mat_kernel Ag \<and> A \<subseteq> mat_kernel Ag)"
splitAtRec_distinct_fst: fixes vs :: "'c list"   and s :: "'c list"   and ram1 :: "'c" assumes "distinct vs"   and "distinct s"   and "set s \<inter> set vs = {}" shows "distinct (fst (splitAtRec ram1 s vs))"
reduced_ifex: fixes ifex_of :: "'d \<Rightarrow> 'e ifex"   and val :: "'d \<Rightarrow> ('e \<Rightarrow> bool) \<Rightarrow> bool"   and b :: "'d" assumes "reduced_bdt_checkers ifex_of val" shows "reduced (ifex_of b) {}"
Init_def: fixes F :: "'a \<Rightarrow> bool"   and sigma :: "behavior" shows "(sigma \<Turnstile> Init F) = F (first_world sigma)"
leq_def: shows "leq = (\<lambda>x y. x = y \<or> less x y)"
unit_disc_fix_id_moebius: shows "unit_disc_fix id_moebius"
same_orbit_char: fixes G :: "('c, 'd) monoid_scheme"   and \<phi> :: "'c \<Rightarrow> 'e \<Rightarrow> 'e"   and M :: "'e set"   and m :: "'e"   and n :: "'e" assumes "group_action G \<phi> M"   and "m \<in> M"   and "n \<in> M" shows "((m, n) \<in> group_action.same_orbit_rel G \<phi> M) = (\<exists>g\<in>carrier G. \<phi> g n = m)"
less_neq: fixes n :: "nat"   and m :: "nat" assumes "n < m" shows "n \<noteq> m"
split_asm: fixes P :: "'a \<Rightarrow> bool"   and f1 :: "'b \<Rightarrow> 'a"   and f2 :: "'c \<Rightarrow> 'd \<Rightarrow> 'a"   and generat :: "('b, 'c, 'd) generat" shows "P (case generat of Pure x \<Rightarrow> f1 x | IO x xa \<Rightarrow> f2 x xa) = (\<not> ((\<exists>x1. generat = Pure x1 \<and> \<not> P (f1 x1)) \<or> (\<exists>x21 x22. generat = IO x21 x22 \<and> \<not> P (f2 x21 x22))))"
to_tree_order_ok: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr" assumes "Shadow_DOM.heap_is_wellformed h"   and "ptr |\<in>| object_ptr_kinds h"   and "ShadowRootClass.known_ptrs h"   and "ShadowRootClass.type_wf h" shows "h \<turnstile> ok Shadow_DOM.to_tree_order ptr"
span_base: fixes a :: "'b \<Rightarrow>\<^sub>0 'c"   and S :: "('b \<Rightarrow>\<^sub>0 'c) set" assumes "a \<in> S" shows "a \<in> phull S"
DEFL_app: shows "DEFL('a\<cdot>'b) = TC('b)\<cdot>DEFL('a)"
ring_simprules_5: fixes R :: "('c, 'd) ring_scheme"   and x :: "'c"   and y :: "'c" assumes "ring R"   and "x \<in> carrier R"   and "y \<in> carrier R" shows "x \<otimes>\<^bsub>R\<^esub> y \<in> carrier R"
intros_1: fixes m1 :: "'a"   and i :: "nat"   and m2 :: "'a" assumes "Broadcast m1 \<sqsubset>\<^sup>i Broadcast m2" shows "hb m1 m2"
simps_6: fixes nm_name :: "char list"   and nm_receiver_violation :: "bool"   and nm_default :: "'b"   and nm_sinvar :: "'a list_graph \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> bool"   and nm_offending_flows :: "'a list_graph \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> ('a \<times> 'a) list list"   and nm_node_props :: "('a, 'b) TopoS_Params \<Rightarrow> 'a \<Rightarrow> 'b"   and nm_eval :: "'a list_graph \<Rightarrow> ('a, 'b) TopoS_Params \<Rightarrow> bool"   and more :: "'c" shows "nm_node_props \<lparr>nm_name = nm_name, nm_receiver_violation = nm_receiver_violation, nm_default = nm_default, nm_sinvar = nm_sinvar, nm_offending_flows = nm_offending_flows, nm_node_props = nm_node_props, nm_eval = nm_eval, \<dots> = more\<rparr> = nm_node_props"
dag_def: fixes p :: "ref"   and l :: "ref \<Rightarrow> ref"   and r :: "ref \<Rightarrow> ref" shows "dag p l r = (THE t. Dag p l r t)"
fcall_def: fixes init :: "'d \<Rightarrow> 'd"   and p :: "'e"   and return :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and result :: "'d \<Rightarrow> 'g"   and c :: "'g \<Rightarrow> ('d, 'e, 'f) com" shows "fcall init p return result c = call init p return (\<lambda>s t. c (result t))"
lincomb_def: fixes R :: "('a, 'b) ring_scheme"   and M :: "('a, 'c, 'd) module_scheme"   and a :: "'c \<Rightarrow> 'a"   and A :: "'c set" assumes "Module.module R M" shows "module.lincomb M a A = (\<Oplus>\<^bsub>M\<^esub>v\<in>A. a v \<odot>\<^bsub>M\<^esub> v)"
Comm1: fixes P :: "pi"   and a :: "name"   and x :: "name"   and P' :: "pi"   and Q :: "pi"   and b :: "name"   and Q' :: "pi" assumes "P \<longmapsto> a<x> \<prec> P'"   and "Q \<longmapsto> a[b] \<prec> Q'" shows "P \<parallel> Q \<longmapsto> \<tau> \<prec> P'[x::=b] \<parallel> Q'"
fin_digraphI: fixes G :: "('c, 'd) pre_digraph" assumes "wf_digraph G"   and "finite (verts G)"   and "finite (arcs G)" shows "fin_digraph G"
fldname_simp: fixes n :: "vname"   and c :: "qtname" shows "fldname (n, c) = n"
assms_2: fixes C :: "'a set" assumes "C \<in> components S" shows "continuous_on C f"
P1_real_model_def: fixes M :: "'a \<times> 'a"   and \<sigma> :: "bool"   and z :: "'f"   and \<A> :: "('a \<times> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'f \<Rightarrow> (('a \<times> 'a \<times> 'a) \<times> 'g) spmf) \<times> ('a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<Rightarrow> 'g \<Rightarrow> ((('a \<times> 'a) \<times> 'a \<times> 'a) \<times> 'g) spmf) \<times> ('g \<Rightarrow> 'e spmf)" shows "P1_real_model M \<sigma> z \<A> = (let (\<A>1, \<A>2, \<A>3) = \<A> in sample_uniform (order \<G>) \<bind> (\<lambda>r. sample_uniform (order \<G>) \<bind> (\<lambda>\<alpha>0. sample_uniform (order \<G>) \<bind> (\<lambda>\<alpha>1. let h0 = \<^bold>g [^] \<alpha>0; h1 = \<^bold>g [^] \<alpha>1; a = \<^bold>g [^] r; b0 = h0 [^] r \<otimes> (if \<sigma> then \<^bold>g else \<one>); b1 = h1 [^] r \<otimes> (if \<sigma> then \<^bold>g else \<one>) in \<A>1 M h0 h1 a b0 b1 z \<bind> (\<lambda>((in1, in2, in3), s). let (h, a, b) = (h0 \<otimes> inv h1, a, b0 \<otimes> inv b1) in funct_DH_ZK (in1, in2, in3) ((h, a, b), r) \<bind> (\<lambda>(b, uu_). assert_spmf b \<bind> (\<lambda>_. \<A>2 h0 h1 a b0 b1 M s \<bind> (\<lambda>(((w0, z0), w1, z1), s'). \<A>3 s' \<bind> (\<lambda>adv_out. return_spmf (adv_out, if \<sigma> then z1 \<otimes> inv w1 [^] \<alpha>1 else z0 \<otimes> inv w0 [^] \<alpha>0))))))))))"
normal_form_length_upper_bound_def: fixes \<phi> :: "'b" shows "normal_form_length_upper_bound \<phi> \<equiv> 2 ^ size \<phi> * (2 ^ (size \<phi> + 1) + 2 * (size \<phi> + 2)\<^sup>2)"
fresh_1: fixes a :: "name"   and x2 :: "name"   and x1 :: "name" shows "a \<sharp> x2<x1> = (a \<sharp> x2 \<and> a \<sharp> x1)"
simps_26: fixes va :: "\<I>"   and vb :: "'b formula" shows "safe_formula (Neg (Next va vb)) = False"
MetaOperationCallN: fixes \<Gamma> :: "String.literal \<rightharpoonup>\<^sub>f 'a OCL_Types.type"   and \<tau> :: "'a OCL_Types.type"   and op :: "metaop" shows "\<Gamma> \<turnstile> MetaOperationCall \<tau> op \<Rrightarrow> MetaOperationCall \<tau> op"
less_minus_iff: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a" assumes "class.ordered_ab_group_add plus zero minus uminus less_eq less" shows "less a (uminus b) = less b (uminus a)"
f_def: shows "f = (\<lambda>r. r * a mod k)"
IH: fixes Y :: "nat"   and r :: "nat" shows "shared_weight_net (witness Y r rs)"
det_lower_triangular: fixes n :: "nat"   and A :: "'a mat" assumes "\<And>i j. \<lbrakk>i < j; j < n\<rbrakk> \<Longrightarrow> A $$ (i, j) = (0::'a)"   and "A \<in> carrier_mat n n" shows "det A = prod_list (diag_mat A)"
take_is_prefix: fixes n :: "nat"   and xs :: "'a list" shows "prefix (take n xs) xs"
Invoke2: fixes p :: "'b"   and c :: "'c"   and i :: "nat" assumes "Inv p c t t' \<and> s = s'"   and "id1 \<le> i \<and> i < id2"   and "a = Invoke i p c" shows "thesis"
ss423bij: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "bijective x" shows "(y ; x\<^sup>T \<le> z) = (y \<le> z ; x)"
indRelTEQ_impl_TRel_is_weak_barbed_simulation: fixes TRel :: "('procT \<times> 'procT) set" assumes "weak_barbed_simulation (indRelTEQ TRel) (STCalWB SWB TWB)" shows "weak_barbed_simulation (TRel\<^sup>*) TWB"
intro: fixes callee :: "'a \<Rightarrow> 'b \<Rightarrow> ('c \<times> 'a) spmf"   and I :: "'a \<Rightarrow> bool"   and \<I> :: "('b, 'c) \<I>" assumes "\<And>s x y s'. \<lbrakk>(y, s') \<in> set_spmf (callee s x); I s; x \<in> outs_\<I> \<I>\<rbrakk> \<Longrightarrow> I s'"   and "\<And>s. I s \<Longrightarrow> \<I> \<turnstile>c callee s \<surd>" shows "callee_invariant_on callee I \<I>"
ApproximationTr0: fixes aa :: "'a"   and N :: "nat" assumes "aa \<in> carrier K" shows "(1\<^sub>r \<plusminus> -\<^sub>a aa^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup> \<in> carrier K"
abs_pred_term_def: fixes P :: "term \<Rightarrow> bool"   and t :: "term" shows "abs_pred P t = ((\<forall>x. t = Bound x \<longrightarrow> P t) \<and> (\<forall>t'. t = \<Lambda> t' \<longrightarrow> P t' \<longrightarrow> P t))"
Collect_mono: fixes P :: "'k \<Rightarrow> bool"   and Q :: "'k \<Rightarrow> bool" assumes "\<And>x. P x \<longrightarrow> Q x" shows "Collect P \<subseteq> Collect Q"
left_neutral: fixes a :: "'e multiset" shows "{#} \<union># a = a"
deadlockD2: fixes s :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x1 \<times> 'l \<Rightarrow>f nat) option) \<times> 'm1) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and t :: "'t"   and x :: "'x1"   and ln :: "'l \<Rightarrow>f nat"   and l :: "'l" assumes "r1.deadlock s"   and "thr s t = \<lfloor>(x, ln)\<rfloor>"   and "0 < ln $ l"   and "\<not> waiting (wset s t)" obtains l' :: "'l" and t' :: "'t" where "0 < ln $ l'" and "t \<noteq> t'" and "thr s t' \<noteq> None" and "has_lock (locks s $ l') t'"
I: fixes t :: "real" assumes "a \<le> t" shows "(f has_integral I t) {a..t}"
that: shows "x \<notin> fst ` set_spmf q"
design_points_nempty: fixes point_set :: "'b set"   and block_collection :: "'b set multiset" assumes "proper_design point_set block_collection" shows "point_set \<noteq> {}"
LIMSEQ_le: fixes X :: "nat \<Rightarrow> 'b"   and x :: "'b"   and Y :: "nat \<Rightarrow> 'b"   and y :: "'b" assumes "X \<longlonglongrightarrow> x"   and "Y \<longlonglongrightarrow> y"   and "\<exists>N. \<forall>n\<ge>N. X n \<le> Y n" shows "x \<le> y"
diff_numeral_special_11: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and one :: "'b" assumes "class.neg_numeral minus plus zero uminus" shows "minus one (uminus one) = numeral.numeral one plus (num.Bit0 num.One)"
mk_poly2: fixes i :: "nat"   and A :: "'b mat" assumes "i < dim_row A"   and "0 < dim_col A" shows "mk_poly2 A $ i = (\<Sum>j'<dim_col A. monom (A $$ (i, j')) (dim_col A - Suc j'))"
JKBP_axioms: shows "JKBP jkbp"
Sigma_Suc_simp_rewrite: fixes f :: "nat list \<Rightarrow> nat"   and xs :: "nat list"   and x :: "nat" shows "UF.Sigma f (xs @ [Suc x]) = UF.Sigma f (xs @ [x]) + f (xs @ [Suc x])"
oDeriv_plus: fixes a :: "ordinal" shows "oDeriv ((+) a) = (+) (a * \<omega>)"
fps_divide_nth_below: fixes n :: "nat"   and f :: "'a fps"   and g :: "'a fps" assumes "n < subdegree f - subdegree g" shows "fps_nth (f / g) n = (0::'a)"
directed_towards_mono: fixes s :: "'a"   and A :: "'a set"   and F :: "('a \<times> 'a) set"   and G :: "('a \<times> 'a) set" assumes "s \<in> directed_towards A F"   and "F \<subseteq> G" shows "s \<in> directed_towards A G"
k_invariantTD: fixes a :: "'a"   and ec :: "'rep"   and A :: "('ma, 'mt) AlgState"   and obs :: "'obs" assumes "KBP.k_isNode a ec"   and "KBP.k_memb ec A"   and "obs \<in> simObs a ` set (KBP.k_succs a ec)"   and "KBP.k_invariant a A" shows "\<exists>ec'. lookup tOps (aTrans A) (ec, obs) = Some ec' \<and> simAbs ec' \<in> simAbs ` set (KBP.k_succs a ec) \<and> simObs a ec' = obs"
not_outside_1: fixes outcomes :: "'b set"   and relation :: "('b pmf \<times> 'b pmf) set"   and u :: "'b \<Rightarrow> real"   and p :: "'b pmf"   and q :: "'b pmf" assumes "vNM_utility outcomes relation u"   and "p \<succeq>[relation] q" shows "p \<in> \<P> outcomes"
prems: shows "x \<in> r \<down>m add_mset p P"
state_rank_oldest_token: fixes p :: "'b"   and n :: "nat"   and i :: "nat"   and q :: "'b"   and j :: "nat"   and x :: "nat"   and y :: "nat" assumes "state_rank p n = Some i"   and "state_rank q n = Some j"   and "oldest_token p n = Some x"   and "oldest_token q n = Some y" shows "(i < j) = (x < y)"
elims_2: fixes x :: "'a option"   and xa :: "'a option" assumes "x \<sqsubseteq> xa" obtains x :: "'a" and y :: "'a" where "xa__ = Some x" and "xa = Some ya__" and "x \<sqsubseteq> ya__"   | y :: "'a option" where "xa__ = None" and "xa = y"
RecvMarker_given_channel: fixes channel :: "nat \<Rightarrow> ('d \<times> 'd) option"   and ev :: "('d, 'e, 'f) event"   and cid :: "nat"   and p :: "'d"   and q :: "'d"   and trans :: "'d \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> bool"   and send :: "nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and recv :: "nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and c :: "('d, 'e, 'f) configuration" assumes "distributed_system channel"   and "isRecvMarker ev"   and "getId ev = cid"   and "channel cid = Some (p, q)"   and "distributed_system.can_occur channel trans send recv ev c" shows "ev = RecvMarker cid q p"
intro: fixes P :: "'a \<Rightarrow> bool" assumes "\<And>x. P x" shows "locale_poly P"
Un_Int_eq_1: fixes S :: "'e set"   and T :: "'e set" shows "(S \<union> T) \<inter> S = S"
set_intros_6: fixes yo :: "'a"   and x33 :: "('a, 'b, 'c) expr"   and x31 :: "('a, 'b, 'c) expr"   and x32 :: "('a, 'b, 'c) expr" assumes "yo \<in> RID\<^sub>E x33" shows "yo \<in> RID\<^sub>E (Ite x31 x32 x33)"
that: fixes x :: "'a" assumes "a \<bullet> x = b" shows "thesis"
inf_top_right: fixes inf :: "'a::type \<Rightarrow> 'a::type \<Rightarrow> 'a::type"   and less_eq :: "'a::type \<Rightarrow> 'a::type \<Rightarrow> bool"   and less :: "'a::type \<Rightarrow> 'a::type \<Rightarrow> bool"   and top :: "'a::type"   and x :: "'a::type" assumes "class.bounded_semilattice_inf_top (inf::'a::type \<Rightarrow> 'a::type \<Rightarrow> 'a::type) (less_eq::'a::type \<Rightarrow> 'a::type \<Rightarrow> bool) (less::'a::type \<Rightarrow> 'a::type \<Rightarrow> bool) (top::'a::type)" shows "(inf::'a::type \<Rightarrow> 'a::type \<Rightarrow> 'a::type) (x::'a::type) (top::'a::type) = x"
in_one_stalk: fixes V :: "'c set" assumes "V \<in> {U. is_open\<^sub>Y U \<and> f x \<in> U}" shows "(V, one_str\<^sub>Y V) \<in> stfx.one_stalk V"
Eq: fixes t1 :: "dbtm"   and t2 :: "dbtm" assumes "wf_dbtm t1"   and "wf_dbtm t2" shows "wf_dbfm (DBEq t1 t2)"
T_simp: fixes f :: "'a"   and g :: "'a" assumes "arr f"   and "arr g" shows "tensor f g = f \<otimes> g"
assms: fixes x :: "'a"   and y :: "'b" shows "monotone orda ordb (\<lambda>f. pair f x y)"
BinOp2Reds: fixes e :: "expr"   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 e' :: "expr"   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 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 v :: "val"   and bop :: "bop" assumes "P,E \<turnstile> \<langle>e,s\<rangle> \<rightarrow>* \<langle>e',s'\<rangle>" shows "P,E \<turnstile> \<langle>Val v \<guillemotleft>bop\<guillemotright> e,s\<rangle> \<rightarrow>* \<langle>Val v \<guillemotleft>bop\<guillemotright> e',s'\<rangle>"
mor2ZFInj: fixes x :: "'b"   and y :: "'b" assumes "x \<in> mor"   and "y \<in> mor"   and "m2zx = m2zy" shows "x = y"
rev_pow: fixes x :: "'b list"   and m :: "nat" shows "rev (x \<^sup>@ m) = rev x \<^sup>@ m"
fr_p_2: shows "atom (p \<bullet> a) \<sharp> (p \<bullet> h) (p \<bullet> a)"
det_2: fixes A :: "(('b, 2) Finite_Cartesian_Product.vec, 2) Finite_Cartesian_Product.vec" shows "Determinants.det A = A $h 1 $h 1 * A $h 2 $h 2 - A $h 1 $h 2 * A $h 2 $h 1"
LConsE: fixes x :: "'b"   and xs :: "'b llist"   and A :: "'b set" shows "(x ## xs \<in> A\<^sup>\<infinity>) = (x \<in> A \<and> xs \<in> A\<^sup>\<infinity>)"
xt1_9: fixes b :: "'a"   and a :: "'a"   and P :: "bool" assumes "b < a"   and "a < b" shows "P"
ACI_lang: fixes r :: "'a rexp"   and s :: "'a rexp" assumes "ACI r s" shows "lang r = lang s"
imult_ile_mono: fixes i :: "enat"   and j :: "enat"   and k :: "enat"   and l :: "enat" assumes "i \<le> j"   and "k \<le> l" shows "i * k \<le> j * l"
product_part_in_second_factor: fixes x :: "i"   and y :: "i"   and z :: "i" assumes "CM"   and "O x y"   and "P z (x \<otimes> y)" shows "P z y"
Diff_iff: fixes c :: "'g"   and A :: "'g set"   and B :: "'g set" shows "(c \<in> A - B) = (c \<in> A \<and> c \<notin> B)"
ttree_rest_substitute2: fixes f :: "'a \<Rightarrow> 'a ttree"   and S :: "'a set"   and T :: "'a set"   and t :: "'a ttree" assumes "\<And>x. carrier (f x) \<subseteq> S"   and "const_on f (- S) TTree.empty" shows "ttree_restr S (substitute f T t) = substitute f T (ttree_restr S t)"
inject_2: fixes x41 :: "'a rexp"   and x42 :: "'a rexp"   and y41 :: "'a rexp"   and y42 :: "'a rexp" shows "(x41 +\<^sub>r x42 = y41 +\<^sub>r y42) = (x41 = y41 \<and> x42 = y42)"
sub1_preserves_dimensions_1: fixes q :: "'a"   and k :: "nat"   and l :: "nat"   and r :: "'a"   and A :: "'a mat"   and r' :: "'a"   and A' :: "'a mat" assumes "local.sub1 q k l (r, A) = (r', A')" shows "dim_row A' = dim_row A"
is_processT8_S: fixes s :: "'c event list"   and P :: "'c process"   and X :: "'c event set" assumes "s \<in> D P" shows "(s, X) \<in> F P"
limit_inter_empty: fixes w :: "nat \<Rightarrow> 'b"   and S :: "'b set" assumes "finite (range w)"   and "limit w \<inter> S = {}" shows "\<forall>\<^sub>\<infinity>n. w n \<notin> S"
YFtorNTNatTrans': fixes C :: "('d, 'e, 'f) LSCategory_scheme"   and f :: "'e" assumes "LSCategory C"   and "f \<in> mor\<^bsub>C\<^esub>" shows "NatTransP (YFtorNT' C f)"