hcju/isabelleps huggingface dataset:
not_High_Low: 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 Def :: "'b \<Rightarrow> 'd set"   and Use :: "'b \<Rightarrow> 'd set"   and state_val :: "'c \<Rightarrow> 'd \<Rightarrow> 'e"   and backward_slice :: "'b set \<Rightarrow> 'b set"   and Exit :: "'b"   and H :: "'d set"   and L :: "'d set"   and High :: "'b"   and Low :: "'b"   and V :: "'d" assumes "NonInterferenceIntraGraph sourcenode targetnode kind valid_edge Entry Def Use state_val backward_slice Exit H L High Low"   and "V \<notin> H" shows "V \<in> L"
simps: fixes \<sigma> :: "'a \<Rightarrow> ('b, 'a, 'c) val option"   and \<tau> :: "'a \<Rightarrow> ('b, 'a, 'c) val option"   and e :: "('b, 'a, 'c) expr" shows "doms (\<sigma>, \<tau>, e) = dom \<sigma> \<union> dom \<tau>"
assms_2: shows "\<langle>v|w\<rangle> = 1"
separating_of_bipartite: fixes S :: "('v + 'v) set" assumes "separating (bipartite_web_of \<Gamma>) S" shows "separating \<Gamma> (separating_of_bipartite S)"
sup_red_f_in_red_liminf: fixes Ns :: "'a set llist" assumes "chain (\<rhd>) Ns" shows "Sup_llist (lmap Red_F Ns) \<subseteq> Red_F (Liminf_llist Ns)"
disjnt_Union1: fixes \<A> :: "'a set set"   and B :: "'a set" shows "disjnt (\<Union> \<A>) B = (\<forall>A\<in>\<A>. disjnt A B)"
cr_scalar_def: fixes R :: "V \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "V"   and y :: "'b" shows "cr_scalar R x y = (\<exists>a. x = [a]\<^sub>\<circ> \<and> R a y)"
nomatchD: fixes \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and r :: "'c rule list"   and s :: "state"   and t :: "state"   and m :: "'c match_expr"   and a :: "action" assumes "\<Gamma>,\<gamma>,p\<turnstile> \<langle>r, s\<rangle> \<Rightarrow> t"   and "r = [Rule m a]"   and "s = Undecided"   and "\<not> matches \<gamma> m p" shows "t = Undecided"
nofail_SPEC: fixes m :: "'b nres" assumes "nofail m" shows "m \<le> SPEC (\<lambda>_. True)"
eventually_at_rightI: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and "open" :: "'c set \<Rightarrow> bool"   and a :: "'c"   and b :: "'c"   and P :: "'c \<Rightarrow> bool" assumes "class.order_topology less_eq less open"   and "\<And>x. x \<in> ord.greaterThanLessThan less a b \<Longrightarrow> P x"   and "less a b" shows "eventually P (topological_space.at_within open a (ord.greaterThan less a))"
AKcryptSK_not_AKcryptSK: fixes authK :: "nat"   and servK :: "nat"   and evs :: "event list"   and K :: "nat" assumes "AKcryptSK authK servK evs"   and "evs \<in> kerbIV_gets" shows "\<not> AKcryptSK servK K evs"
UP_def: fixes V\<^sub>B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H\<^sub>B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a>\<^sub>B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<i>\<^sub>B :: "'b \<Rightarrow> 'b"   and src\<^sub>B :: "'b \<Rightarrow> 'b"   and trg\<^sub>B :: "'b \<Rightarrow> 'b"   and \<mu> :: "'b" assumes "strictified_bicategory V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B" shows "strictified_bicategory.UP V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B \<mu> \<equiv> if partial_magma.arr V\<^sub>B \<mu> then concrete_category.MkArr (bicategorical_language.Prim (partial_magma.dom V\<^sub>B \<mu>)) (bicategorical_language.Prim (partial_magma.cod V\<^sub>B \<mu>)) \<mu> else partial_magma.null (concrete_category.COMP {A. bicategorical_language.Nml V\<^sub>B src\<^sub>B trg\<^sub>B A \<and> bicategorical_language.Ide V\<^sub>B src\<^sub>B trg\<^sub>B A} (\<lambda>A B. {\<mu>. bicategorical_language.Src src\<^sub>B A = bicategorical_language.Src src\<^sub>B B \<and> bicategorical_language.Trg trg\<^sub>B A = bicategorical_language.Trg trg\<^sub>B B \<and> partial_magma.in_hom V\<^sub>B \<mu> (Coherence.evaluation_map.eval V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B (\<lambda>\<mu>. if partial_magma.arr V\<^sub>B \<mu> then \<mu> else partial_magma.null V\<^sub>B) A) (Coherence.evaluation_map.eval V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B (\<lambda>\<mu>. if partial_magma.arr V\<^sub>B \<mu> then \<mu> else partial_magma.null V\<^sub>B) B)}) (\<lambda>_ _ _. V\<^sub>B))"
cases: fixes a :: "'node SDG_node"   and S :: "'node SDG_node set" assumes "a \<in> combine_SDG_slices S" obtains n :: "'node SDG_node" where "a = na__" and "na__ \<in> S"   | n' :: "'node SDG_node" and n'' :: "'node SDG_node" and p :: "'pname" and n :: "'node SDG_node" where "a = n" and "n' \<in> S" and "n'' s-p\<rightarrow>\<^bsub>ret\<^esub> CFG_node (parent_node n')" and "n \<in> sum_SDG_slice2 n'"
assms: fixes c :: "'a" assumes "c \<in> set cs" shows "P c w"
functional_relation: fixes R :: "'d \<Rightarrow> 'e \<Rightarrow> bool" assumes "right_unique R"   and "left_total R" shows "\<forall>x. \<exists>!y. R x y"
SUP_empty: fixes Inf :: "'d set \<Rightarrow> 'd"   and Sup :: "'d set \<Rightarrow> 'd"   and inf :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and sup :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and bot :: "'d"   and top :: "'d"   and f :: "'e \<Rightarrow> 'd" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "Sup (f ` {}) = bot"
assm: shows "v=u--w \<and> restrict w (clm ts) c \<noteq> bot \<and> consec (lan u) (lan w)"
type_less_x_Collection_intro_1: fixes \<tau> :: "'a OCL_Types.type"   and \<rho> :: "'a OCL_Types.type"   and \<sigma> :: "'a OCL_Types.type" assumes "\<tau> = Collection \<rho>"   and "\<rho> < \<sigma>" shows "\<tau> < Collection \<sigma>"
set_disconnected_nodes_ok: fixes h :: "(_) heap"   and document_ptr :: "(_) document_ptr"   and node_ptrs :: "(_) node_ptr list" assumes "ShadowRootClass.type_wf h"   and "document_ptr |\<in>| document_ptr_kinds h" shows "h \<turnstile> ok set_disconnected_nodes document_ptr node_ptrs"
objI_src: fixes a :: "'c" assumes "C.arr a"   and "src\<^sub>C a = a" shows "C.obj a"
msubmodule_def: fixes R :: "('e, 'f) Ring_scheme"   and M :: "('g, 'e, 'h) Module_scheme"   and M1 :: "('g, 'e, 'h) Module_scheme" shows "msubmodule R M M1 = (miota R M1 M \<in> mHom R M1 M \<and> carrier M1 \<subseteq> carrier M)"
inf_sup_distrib2: fixes y :: "'a multiset"   and z :: "'a multiset"   and x :: "'a multiset" shows "y \<union># z \<inter># x = y \<inter># x \<union># (z \<inter># x)"
CFG_axioms: shows "CFG sourcenode targetnode valid_edge (_Entry_)"
real_divl_lower_bound: fixes x :: "real"   and y :: "real"   and prec :: "nat" assumes "0 \<le> x"   and "0 \<le> y" shows "0 \<le> real_divl prec x y"
Await: fixes s :: "'c"   and b :: "'c set"   and P :: "'c com"   and t :: "'c" assumes "s \<in> b"   and "(Some P, s) -c*\<rightarrow> (None, t)" shows "(Some (Await b P), s) -c\<rightarrow> (None, t)"
inext_induct: fixes P :: "nat \<Rightarrow> bool"   and I :: "nat set"   and n :: "nat" assumes "P (iMin I)"   and "\<And>n. \<lbrakk>n \<in> I; P n\<rbrakk> \<Longrightarrow> P (inext n I)"   and "n \<in> I" shows "P n"
self_append_conv: fixes xs :: "'d list"   and ys :: "'d list" shows "(xs = xs @ ys) = (ys = [])"
tendsF: fixes x :: "'a" assumes "x \<notin> NF" shows "(\<lambda>n. F n x) \<longlonglongrightarrow> (if x \<in> T then f x else (0::'b))"
J_csp_def: shows "J_csp \<equiv> \<lambda>(A, A'). (ok A \<longrightarrow> ok A') \<and> tr A = tr A' \<and> wait A = wait A' \<and> ref A = ref A' \<and> alpha_rp.more A = alpha_rp.more A'"
prime_factorization_prime_power: 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 unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and p :: "'b"   and n :: "nat" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize"   and "normalization_semidom.prime zero times one normalize p" shows "factorial_semiring.prime_factorization zero times one normalize (power.power one times p n) = replicate_mset n p"
image_subset_iff: fixes f :: "'c \<Rightarrow> 'b"   and A :: "'c set"   and B :: "'b set" shows "order.greater_eq B (f ` A) = (\<forall>x\<in>A. f x \<in> B)"
simps_4: fixes h :: "'b \<Rightarrow> 'a fm"   and \<phi>\<^sub>1 :: "'b fm"   and \<phi>\<^sub>2 :: "'b fm" shows "amap\<^sub>f\<^sub>m h (And \<phi>\<^sub>1 \<phi>\<^sub>2) = and (amap\<^sub>f\<^sub>m h \<phi>\<^sub>1) (amap\<^sub>f\<^sub>m h \<phi>\<^sub>2)"
D_def: fixes fs :: "int vec list" shows "D fs = nat (prod (d fs) {..<length fs})"
funpow_Suc_right: fixes n :: "nat"   and f :: "'a \<Rightarrow> 'a" shows "f ^^ Suc n = f ^^ n \<circ> f"
between_mem_convex_hull: fixes a :: "'a"   and b :: "'a"   and x :: "'a" shows "between (a, b) x = (x \<in> convex hull {a, b})"
a_set_tag_name_def: fixes ptr :: "(_) element_ptr"   and tag :: "char list" shows "CD.a_set_tag_name ptr tag = Heap_Error_Monad.bind (get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t ptr attrs) (\<lambda>m. put_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t ptr tag_name_update tag)"
diagonalize_2x2_aux_works: fixes A :: "(('b, 2) Finite_Cartesian_Product.vec, 2) Finite_Cartesian_Product.vec"   and P :: "(('b, 2) Finite_Cartesian_Product.vec, 2) Finite_Cartesian_Product.vec"   and A_input :: "(('b, 2) Finite_Cartesian_Product.vec, 2) Finite_Cartesian_Product.vec"   and Q :: "(('b, 2) Finite_Cartesian_Product.vec, 2) Finite_Cartesian_Product.vec"   and P' :: "(('b, 2) Finite_Cartesian_Product.vec, 2) Finite_Cartesian_Product.vec"   and D :: "(('b, 2) Finite_Cartesian_Product.vec, 2) Finite_Cartesian_Product.vec"   and Q' :: "(('b, 2) Finite_Cartesian_Product.vec, 2) Finite_Cartesian_Product.vec" assumes "A = P ** A_input ** Q"   and "invertible P"   and "invertible Q"   and "(P', D, Q') = diagonalize_2x2_aux (P, A, Q)"   and "A $h 0 $h 0 \<noteq> (0::'b)" shows "D = P' ** A_input ** Q' \<and> invertible P' \<and> invertible Q' \<and> isDiagonal D"
dom_src: fixes \<mu> :: "'a" assumes "C.arr \<mu>" shows "C.dom (src\<^sub>C \<mu>) = src\<^sub>C \<mu>"
fastprod_def: fixes return :: "int \<Rightarrow> 'a"   and bind :: "'a \<Rightarrow> (int \<Rightarrow> 'a) \<Rightarrow> 'a"   and fail :: "'a"   and catch :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and xs :: "int list" shows "fast_product_base.fastprod return bind fail catch xs = catch (fast_product_base.work return bind fail xs) (return 0)"
Pi_coprime_pre: fixes ps :: "nat list"   and i :: "nat" assumes "length ps \<le> i" shows "coprime (Pi (Suc i)) (godel_code' ps (Suc 0))"
normalize_boundary_iff: fixes p :: "(real, 2) vec" shows "(\<bar>normalize p $ 1\<bar> = 1) = (p $ 2 = 0)"
arc_pos_mult: fixes x :: "'a"   and y :: "'a" assumes "arc_pos x"   and "arc_pos y" shows "arc_pos (x * y)"
af\<^sub>G_implies_af_eval\<^sub>G: fixes S :: "'b ltl set"   and \<G> :: "'b ltl set"   and \<phi> :: "'b ltl"   and w :: "nat \<Rightarrow> 'b set"   and j :: "nat" assumes "S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> (prefix j w))"   and "\<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>"   and "\<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i \<le> j\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (w [i \<rightarrow> j]))" shows "S \<Turnstile>\<^sub>P af \<phi> (prefix j w)"
tensor_one_def: shows "1 = tensor_from_vec [] [1::'a]"
adj_msog_single: fixes xs :: "(accent \<times> 'a) list"   and zs :: "(accent \<times> 'a) list"   and x :: "accent \<times> 'a" shows "adj_msog xs zs (x, []) = (x, case fst x of Acute \<Rightarrow> zs | Grave \<Rightarrow> xs | Macron \<Rightarrow> [])"
enum_all_fin_string: shows "enum_class.enum_all \<equiv> fin_string_enum_all"
assms_5: shows "\<forall>p\<in>set (p\<^sub>0 # ps). real_of_int l - \<delta> \<le> real_of_int (fst p) \<and> real_of_int (fst p) \<le> real_of_int l + \<delta>"
bot_eq_sup_iff: fixes x :: "'e multiset"   and y :: "'e multiset" shows "({#} = x \<union># y) = (x = {#} \<and> y = {#})"
de_morgan2: fixes X :: "'a set" shows "- \<Sqinter> X = \<Squnion> (uminus ` X)"
wordinterval_subset_set_eq: fixes r1 :: "'b wordinterval"   and r2 :: "'b wordinterval" shows "wordinterval_subset r1 r2 = (wordinterval_to_set r1 \<subseteq> wordinterval_to_set r2)"
log_le_iff: fixes b :: "real"   and x :: "real"   and y :: "real" assumes "1 < b"   and "0 < x" shows "(log b x \<le> y) = (x \<le> b powr y)"
assms_2: fixes as :: "'a stream"   and tr :: "'a treeFI" assumes "phi as tr" shows "shd as = lab tr"
exhaust: fixes y :: "'b extT" obtains x1 :: "'b" where "y = extT.Val x1"   | "y = extT.Top"
word_unat_less_le: fixes a :: "'b word"   and b :: "nat" assumes "a \<le> word_of_nat b" shows "unat a \<le> b"
p'_irred: shows "\<not> is_sig_red (\<prec>\<^sub>t) (\<preceq>) (set bs) p'"
WN_onE: fixes r :: "('d \<times> 'd) set"   and A :: "'d set"   and a :: "'d" assumes "WN_on r A"   and "a \<in> A" obtains b :: "'d" where "(a, b) \<in> r\<^sup>!"
indRelR_reflects_pred_iff_indRelL_preserves_pred: fixes Pred :: "('procS, 'procT) Proc \<Rightarrow> bool" shows "rel_reflects_pred indRelR Pred = rel_preserves_pred indRelL Pred"
collapse_1: fixes llist :: "'b llist" assumes "lnull llist" shows "llist = LNil"
subsequence_drop: fixes i :: "nat"   and w :: "nat \<Rightarrow> 'b"   and j :: "nat"   and k :: "nat" shows "drop i (w [j \<rightarrow> k]) = w [j + i \<rightarrow> k]"
inverse_distrib_swap: fixes a :: "'a"   and b :: "'a" shows "inverse (a \<cdot> b) = inverse b \<cdot> inverse a"
Push_inject: fixes i :: "'a + nat"   and f :: "nat \<Rightarrow> 'a + nat"   and j :: "'a + nat"   and g :: "nat \<Rightarrow> 'a + nat" assumes "Push i f = Push j g" obtains "i = j" and "f = g"
G_nested_propos_alt_def: fixes \<phi> :: "'b ltl" shows "\<^bold>G \<phi> = nested_propos \<phi> \<inter> {\<psi>. \<exists>x. \<psi> = G x}"
cbnd_mono: fixes b :: "nat"   and d :: "nat"   and a :: "nat" assumes "b \<le> d" shows "cbnd a b \<le> cbnd a d"
c_max_def: fixes S :: "('d, 'e) Ring_scheme"   and c :: "nat \<times> (nat \<Rightarrow> 'd)" shows "c_max S c = (if {j. j \<le> fst c \<and> snd c j \<noteq> \<zero>\<^bsub>S\<^esub>} = {} then 0 else n_max {j. j \<le> fst c \<and> snd c j \<noteq> \<zero>\<^bsub>S\<^esub>})"
simps_2: fixes A :: "'b mat"   and k :: "nat" shows "A ^\<^sub>m Suc k = A ^\<^sub>m k * A"
matrix_to_iarray_matrix_matrix_mult: fixes A :: "(('a, 'd) vec, 'c) vec"   and B :: "(('a, 'b) vec, 'd) vec" shows "matrix_to_iarray (A ** B) = matrix_to_iarray A **i matrix_to_iarray B"
LSC_Top_unused: fixes \<Gamma> :: "'b formula multiset" assumes "\<Gamma> \<Rightarrow>\<^sub>n"   and "is_nnf_mset \<Gamma>" shows "\<Gamma> - (\<^bold>\<not> \<bottom>, {#}) \<Rightarrow>\<^sub>n"
Nlt0_iff: fixes x :: "int \<times> int" assumes "isnormNum x" shows "(INum x < (0::'a)) = 0>\<^sub>N x"
finite_cyclic_subgroup_order: fixes x :: "'a" assumes "x \<in> carrier G" shows "finite (carrier (subgroup_generated G {x})) = (ord x \<noteq> 0)"
axioms_1: fixes G :: "('e, 'f) monoid_scheme" assumes "Group.group G" shows "Group.monoid G"
rts0d: fixes p :: "nat"   and i :: "nat" assumes "correct p (te p (i + 1))" shows "te p (i + 1) - te p i \<le> rmax"
oreachable_weakenE: fixes s :: "'e \<times> 'f"   and A :: "('e \<times> 'f, 'g) automaton"   and PS :: "'e \<Rightarrow> 'e \<Rightarrow> 'g \<Rightarrow> bool"   and PU :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and QS :: "'e \<Rightarrow> 'e \<Rightarrow> 'g \<Rightarrow> bool"   and QU :: "'e \<Rightarrow> 'e \<Rightarrow> bool" assumes "s \<in> oreachable A PS PU"   and "\<And>s s' a. PS s s' a \<Longrightarrow> QS s s' a"   and "\<And>s s'. PU s s' \<Longrightarrow> QU s s'" shows "s \<in> oreachable A QS QU"
simps_8: fixes c :: "int"   and e :: "Ferrack.num" shows "isrlfm (Ge (CN 0 c e)) = (0 < c \<and> numbound0 e)"
disjointI: fixes a :: "'e set"   and b :: "'e set" assumes "\<And>x. \<lbrakk>x \<in> a; x \<in> b\<rbrakk> \<Longrightarrow> False" shows "a \<inter> b = {}"
fac_dummy: fixes x :: "nat"   and y :: "nat" shows "rec_exec rec_dummyfac [x, y] = y!"
UN_extend_simps_4: fixes A :: "'c \<Rightarrow> 'b set"   and C :: "'c set"   and B :: "'b set" shows "\<Union> (A ` C) \<inter> B = (\<Union>x\<in>C. A x \<inter> B)"
euclidean_size_greater_0_iff: 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" assumes "class.euclidean_semiring divide plus minus zero times one modulo euclidean_size" shows "(0 < euclidean_size b) = (b \<noteq> zero)"
carrier_many_calls: fixes x :: "'b" shows "carrier (many_calls x) = {x}"
rStatImp: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi" assumes "(\<Psi>, P, Q) \<in> X" shows "\<Psi> \<rhd> P \<lessapprox>\<^sub>w<(X \<union> weakenBisim)> Q"
bounded_cbilinear_axioms: shows "bounded_cbilinear (**)"
bnFreshResidual: fixes \<alpha> :: "'d action"   and P :: "('d, 'e, 'f) psi" shows "bn \<alpha> \<sharp>* (\<alpha> \<prec> P) = bn \<alpha> \<sharp>* subject \<alpha>"
Int_space_eq2: fixes x :: "'a set" assumes "x \<in> M" shows "x \<inter> \<Omega> = x"
induceES_def: fixes SES :: "('b, 'a) SES_rec" shows "induceES SES \<equiv> \<lparr>E_ES = E\<^bsub>SES\<^esub>, I_ES = I\<^bsub>SES\<^esub>, O_ES = O\<^bsub>SES\<^esub>, Tr_ES = possible_traces SES\<rparr>"
ag_inv_inj: fixes x :: "'a"   and y :: "'a" assumes "x \<in> carrier M"   and "y \<in> carrier M"   and "x \<noteq> y" shows "-\<^sub>a x \<noteq> -\<^sub>a y"
B: fixes h :: "complex \<Rightarrow> complex"   and z :: "complex"   and i :: "nat" assumes "h \<in> \<H>"   and "z \<in> K i" shows "cmod (h z) \<le> B i"
preserves_dom: fixes A :: "'ag \<Rightarrow> 'ag \<Rightarrow> 'ag"   and B :: "'ah \<Rightarrow> 'ah \<Rightarrow> 'ah"   and F :: "'ag \<Rightarrow> 'ah"   and G :: "'ag \<Rightarrow> 'ah"   and \<tau> :: "'ag \<Rightarrow> 'ah"   and f :: "'ag" assumes "natural_transformation A B F G \<tau>"   and "partial_magma.arr A f" shows "partial_magma.dom B (\<tau> f) = F (partial_magma.dom A f)"
disc_eq_case_1: fixes list :: "'b list" shows "(list = []) = (case list of [] \<Rightarrow> True | uu_ # uua_ \<Rightarrow> False)"
simps: fixes Gcd :: "'c poly \<Rightarrow> 'c poly \<Rightarrow> 'c poly"   and bn :: "'c poly"   and cn :: "'c poly"   and i :: "nat"   and sqr :: "('c poly \<times> nat) list" shows "yun_gcd.yun_factorization_main Gcd bn cn i sqr = (if bn = 1 then sqr else let dn = cn - pderiv bn; an = Gcd bn dn in yun_gcd.yun_factorization_main Gcd (bn div an) (dn div an) (Suc i) ((an, i) # sqr))"
a: fixes n :: "nat" assumes "steps0 (1, tp) (A |+| B) n = (Suc (length A div 2), tp')" shows "thesis"
lrb_def: fixes ur :: "('s \<times> 's) set" shows "lrb ur \<equiv> \<forall>s\<in>S\<^bsub>SES\<^esub>. \<forall>c\<in>C\<^bsub>\<V>\<^esub>. reachable SES s \<longrightarrow> (\<exists>s'\<in>S\<^bsub>SES\<^esub>. s c\<longrightarrow>\<^bsub>SES\<^esub> s' \<and> (s, s') \<in> ur)"
node_castTE': fixes i :: "nat"   and s :: "'c"   and R :: "nat set"   and RM :: "nat set"   and m :: "'d"   and s' :: "'c net_state"   and S :: "('c \<times> 'd seq_action \<times> 'c) set" assumes "(NodeS i s R, RM:*cast(m), s') \<in> node_sos S" obtains s'a :: "'c" where "RM = R" and "s' = NodeS i s'a__ R" and "(s, broadcast m, s'a__) \<in> S"   | D :: "nat set" and s'a :: "'c" where "RM = R \<inter> D" and "s' = NodeS i s'aa__ R" and "(s, groupcast D m, s'aa__) \<in> S"   | d :: "nat" and s'a :: "'c" where "RM = {d}" and "s' = NodeS i s'a R" and "(s, unicast d m, s'a) \<in> S" and "d \<in> R"
functional_def: fixes g :: "('c \<times> 'd) set" shows "functional g = inj_on fst g"
extend_eval: 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 :: "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 E :: "char list \<Rightarrow> ty 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)" assumes "wwf_prog P"   and "P,E \<turnstile> \<langle>e,s\<rangle> \<rightarrow>* \<langle>e'',s''\<rangle>"   and "P,E \<turnstile> \<langle>e'',s''\<rangle> \<Rightarrow> \<langle>e',s'\<rangle>" shows "P,E \<turnstile> \<langle>e,s\<rangle> \<Rightarrow> \<langle>e',s'\<rangle>"
sum_pmf_eq_1: fixes A :: "'b set"   and p :: "'b pmf" assumes "finite A"   and "set_pmf p \<subseteq> A" shows "sum (pmf p) A = 1"
simCasesCont: fixes Rel :: "(pi \<times> pi) set"   and Q :: "pi"   and P :: "pi"   and C :: "'a" assumes "eqvt Rel"   and "\<And>a x Q'. \<lbrakk>Q \<longmapsto> a<\<nu>x> \<prec> Q'; x \<sharp> P; x \<sharp> Q; x \<noteq> a; x \<sharp> C\<rbrakk> \<Longrightarrow> \<exists>P'. P \<Longrightarrow>a<\<nu>x> \<prec> P' \<and> (P', Q') \<in> Rel"   and "\<And>\<alpha> Q'. Q \<longmapsto> \<alpha> \<prec> Q' \<Longrightarrow> \<exists>P'. P \<Longrightarrow>\<^sup>^\<alpha> \<prec> P' \<and> (P', Q') \<in> Rel" shows "P \<leadsto><Rel> Q"
le_imp_sub_mod_le: fixes x :: "nat"   and y :: "nat"   and m :: "nat" assumes "x \<le> y" shows "x \<le> y - (y - x) mod m"
elims_2: fixes x :: "'b extB"   and xa :: "'b extB" assumes "x < xa" obtains x :: "'b extB" and y :: "'b extB" where "xa__ = x" and "xa = y" and "x \<le> y \<and> \<not> y \<le> x"
arrI: fixes C :: "'ag \<Rightarrow> 'ag \<Rightarrow> 'ag"   and f :: "'ag"   and a :: "'ag"   and b :: "'ag" assumes "partial_magma C"   and "partial_magma.in_hom C f a b" shows "partial_magma.arr C f"
divide: fixes e :: "'a \<times> 'a"   and H :: "'a pair_pre_digraph"   and w :: "'a"   and G :: "'a pair_pre_digraph" assumes "e \<in> parcs H"   and "w \<notin> pverts H"   and "pair_sd G H" shows "pair_sd G (subdivide H e w)"
options_def: fixes ps :: "(char list \<times> (xml \<Rightarrow> char list +\<^sub>\<bottom> 'a)) list"   and x :: "xml" shows "Xmlt.options ps x = (case map_of ps (Xml.tag x) of None \<Rightarrow> Strict_Sum.error (concat [''expected one of: '', concat (map (\<lambda>p. fst p @ '' '') ps), ''\<newline>'', ''but found'', ''\<newline>'', show x]) | Some p \<Rightarrow> p x)"
emb_lset_mono: fixes xs :: "'c llist"   and ys :: "'c llist"   and x :: "'c" assumes "emb xs ys"   and "x \<in> lset xs" shows "x \<in> lset ys"
conditional_iff: fixes P :: "'a upred"   and Q :: "'a upred"   and R :: "'a upred" shows "((P \<Rightarrow> Q) = (P \<Rightarrow> R)) = `P \<Rightarrow> Q \<Leftrightarrow> R`"
indexed_pmult_def: fixes P :: "'d multiset \<Rightarrow> 'a"   and i :: "'d" shows "P \<Otimes> i = (\<lambda>m. if i \<in># m then P (m - {#i#}) else \<zero>)"
weak_unit_in_vhom: fixes a :: "'a" assumes "weak_unit a" shows "\<guillemotleft>a : a \<Rightarrow> a\<guillemotright>"
rmlab_same_label1: fixes l :: "'a"   and l' :: "'a"   and A :: "'a \<Rightarrow> 'b set" assumes "l = l'" shows "rmlab l A l' = UNIV"
L1_I_2: fixes M :: "'b measure"   and f :: "'b \<Rightarrow> real" assumes "integrable M f" shows "Norm (\<LL> 1 M) f = LINT x|M. \<bar>f x\<bar>"
ereal_inverse_antimono_strict: fixes x :: "ereal"   and y :: "ereal" assumes "0 \<le> x"   and "x < y" shows "inverse y < inverse x"
S\<^sub>e_def: fixes K :: "'a \<Rightarrow> 'a pmf set"   and S :: "'a set"   and S1 :: "'a set"   and S2 :: "'a set" assumes "Reachability_Problem K S S1 S2" shows "Reachability_Problem.S\<^sub>e K S S1 S2 = {s \<in> S. Reachability_Problem.p K S S1 S2 s = 0}"