hcju/isabelleps huggingface dataset:
index_together: fixes point_set :: "'b set"   and block_collection :: "'b set multiset"   and groups :: "'b set set"   and index :: "int"   and G :: "'b set"   and x :: "'b"   and y :: "'b" assumes "GDD point_set block_collection groups index"   and "G \<in> groups"   and "x \<in> G"   and "y \<in> G"   and "x \<noteq> y" shows "block_collection index {x, y} = 0"
in_out_arcs_conv: fixes e :: "'c"   and G :: "('d, 'c) pre_digraph"   and v :: "'d" shows "(e \<in> out_arcs G v) = (e \<in> arcs G \<and> tail G e = v)"
bind_return: fixes f :: "'c \<Rightarrow> 'd measure"   and M :: "'c measure"   and N :: "'d measure"   and x :: "'c" assumes "f \<in> M \<rightarrow>\<^sub>M subprob_algebra N"   and "x \<in> space M" shows "return M x \<bind> f = f x"
dvdI: fixes a :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and b :: "'b"   and k :: "'b" assumes "a = times b k" shows "dvd.dvd times b a"
resolution_def: fixes P :: "'a set multiset multiset" shows "resolution P = (partition_on_mset \<B> P \<and> (\<forall>S\<in>#P. distinct_mset S \<and> resolution_class (set_mset S)))"
onorm_Sup_on_ball: fixes r :: "real"   and f :: "'c \<Rightarrow>\<^sub>L 'd"   and x :: "'c" assumes "0 < r" shows "norm f \<le> (SUP x\<in>ball x r. norm (f *\<^sub>v x)) / r"
times_left_mono: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and z :: "'a"   and y :: "'a"   and x :: "'a" assumes "SN_Orders.class.ordered_semiring_0 plus zero times less_eq"   and "less_eq zero z"   and "less_eq y x" shows "less_eq (times y z) (times x z)"
rel_mono: fixes R1 :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and R1a :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and R2 :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and R2a :: "'d \<Rightarrow> 'e \<Rightarrow> bool" assumes "R1 \<le> R1a"   and "R2 \<le> R2a" shows "rel_prod R1 R2 \<le> rel_prod R1a R2a"
IK_subset_extr: fixes IK :: "msg set"   and bad :: "agent set"   and chan :: "chan set" shows "IK \<subseteq> extr bad IK chan"
elims_2: fixes x :: "'a Point"   and xa :: "'a Point"   and xb :: "'a Point" assumes "collinear x xa xb" obtains p :: "'a Point" and q :: "'a Point" and r :: "'a Point" where "x = p" and "xa = q" and "xb = r" and "\<exists>\<alpha> \<beta>. \<alpha> + \<beta> = (1::'a) \<and> positionVector p = \<alpha> ** positionVector q \<oplus> \<beta> ** positionVector r"
rec_inseparable_non_computable: fixes A :: "nat set"   and B :: "nat set" assumes "A \<inter> B = {}"   and "rec_inseparable A B" shows "\<not> computable A"
assms: shows "Limit.cone J (\<cdot>\<^sub>A) D a \<chi>"
pairs_def: shows "pairs = {(x, y). \<exists>z. y = phi (- x) \<squnion> up_filter z}"
simps_2: fixes id :: "'b"   and f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and x :: "'b list"   and xs :: "'b list list"   and ys :: "'b list list" assumes "Matrix_Tensor.mult id f" shows "mult.Tensor f (x # xs) ys = mult.vec_mat_Tensor f x ys @ mult.Tensor f xs ys"
length_drop: fixes n :: "nat"   and xs :: "'b list" shows "length (drop n xs) = length xs - n"
bconfs_nonVals: fixes P :: "'a prog"   and sh :: "sheap"   and es :: "'b exp list" assumes "P,sh \<turnstile>\<^sub>b (es,True) \<surd>" shows "map_vals_of es = None"
simps_2: fixes l :: "'a tree"   and uu :: "'a"   and r :: "'a tree" shows "min_height \<langle>l, uu, r\<rangle> = min (min_height l) (min_height r) + 1"
dom_object: fixes CC :: "('b, 'c, 'd) category_scheme"   and f :: "'c" assumes "category CC"   and "f \<in> Ar\<^bsub>CC\<^esub>" shows "Dom\<^bsub>CC\<^esub> f \<in> Ob\<^bsub>CC\<^esub>"
funpow_mono: fixes f :: "'g \<Rightarrow> 'g"   and A :: "'g"   and B :: "'g"   and n :: "nat" assumes "mono f"   and "A \<le> B" shows "(f ^^ n) A \<le> (f ^^ n) B"
enabled_disj2: fixes G :: "state \<times> state \<Rightarrow> bool"   and F :: "state \<times> state \<Rightarrow> bool" shows "\<turnstile> Enabled G \<longrightarrow> Enabled (F \<or> G)"
wlp_loop_step_mono: fixes body :: "bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and t :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and u :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and G :: "'a \<Rightarrow> bool" assumes "nearly_healthy (wlp body)"   and "le_utrans t u"   and "\<And>P. unitary P \<Longrightarrow> unitary (t P)"   and "\<And>P. unitary P \<Longrightarrow> unitary (u P)" shows "le_utrans (wlp (body ;; Embed t \<^bsub>\<guillemotleft> G \<guillemotright>\<^esub>\<oplus> Skip)) (wlp (body ;; Embed u \<^bsub>\<guillemotleft> G \<guillemotright>\<^esub>\<oplus> Skip))"
ipath_def: fixes E :: "('c \<times> 'c) set"   and r :: "nat \<Rightarrow> 'c" shows "ipath E r \<equiv> \<forall>i. (r i, r (Suc i)) \<in> E"
linorder_cases: fixes x :: "'b word"   and y :: "'b word" obtains "x <s y"   | "x = y"   | "y <s x"
hom_simps_1: fixes x :: "'a list" assumes "x \<in> carrier (K [X])" shows "local.eval x x \<in> carrier R"
wf_interp_for_formula_FExists: fixes \<Sigma> :: "'b list"   and I :: "(nat + nat set) list"   and \<phi> :: "'b formula"   and w :: "'b list" assumes "formula \<Sigma>"   and "formula.wf_formula \<Sigma> (length I) (FExists \<phi>)"   and "w \<noteq> []" shows "formula.wf_interp_for_formula \<Sigma> (w, I) (FExists \<phi>) = (\<forall>p<length w. formula.wf_interp_for_formula \<Sigma> (w, Inl p # I) \<phi>)"
\<Delta>_\<Sigma>: fixes f :: "int \<Rightarrow> 'c"   and j :: "int" shows "\<Delta> (\<Sigma> f j) = f"
that: fixes L :: "real" assumes "X \<longlonglongrightarrow> L"   and "\<forall>i. L \<le> X i" shows "thesis"
wellfounded: fixes cnt :: "'a \<Rightarrow> bool"   and nxt :: "'a \<Rightarrow> 'a"   and sizef :: "'a \<Rightarrow> nat" assumes "iterator cnt nxt sizef" shows "wf {(nxt x, x) |x. cnt x}"
total_on_VWF: fixes A :: "V set" shows "total_on A VWF"
prime_sum_upto_mono: fixes f :: "nat \<Rightarrow> real"   and x :: "real"   and y :: "real" assumes "\<And>n. 0 < n \<Longrightarrow> 0 \<le> f n"   and "x \<le> y" shows "prime_sum_upto f x \<le> prime_sum_upto f y"
hb_has_a_reason: fixes m1 :: "'a"   and m2 :: "'a"   and i :: "nat" assumes "hb m1 m2"   and "Broadcast m2 \<in> set (history i)" shows "Deliver m1 \<in> set (history i) \<or> Broadcast m1 \<in> set (history i)"
IH: fixes s1 :: "('a, bool) fmap"   and s2 :: "('a, bool) fmap" assumes "s1 \<subseteq>\<^sub>f s2"   and "\<forall>a. ListMem a as \<longrightarrow> fmdom' (fst a) \<subseteq> fmdom' s1" shows "exec_plan s1 as \<subseteq>\<^sub>f exec_plan s2 as"
axioms_2: fixes G :: "('a, 'b) graph" assumes "valid_unMultigraph G" shows "valid_unMultigraph_axioms G"
iso_is_arr: fixes f :: "('a S.E.term, 'a) S.arr \<times> ('a S.E.term, 'a) S.arr" assumes "S.VV.iso f" shows "S.VV.arr f"
UNIV_ipv4addrset: shows "UNIV = {0..max_ipv4_addr}"
Rats_def: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b" assumes "class.field_char_0 divide inverse times one plus zero minus uminus" shows "field_char_0.Rats divide one plus zero minus = range (field_char_0.of_rat divide one plus zero minus)"
le_sum_list_less: fixes xs :: "nat list"   and ys :: "nat list" assumes "xs \<le>\<^sub>v ys"   and "sum_list xs < sum_list ys" shows "xs <\<^sub>v ys"
ext_compat_list_axioms_def: fixes ext :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> 'a list \<Rightarrow> bool" shows "ext_compat_list_axioms ext \<equiv> \<forall>y x gt xs xs'. y \<noteq> x \<longrightarrow> gt y x \<longrightarrow> ext gt (xs @ y # xs') (xs @ x # xs')"
mini_maxi_keys_le: fixes x :: "'c"   and xs :: "'c list"   and key :: "'c \<Rightarrow> 'd" assumes "x \<in> set xs" shows "key (xs ! mini xs key) \<le> key (xs ! maxi xs key)"
wellFormedCons: fixes xvec :: "'d list"   and Tvec :: "'e list"   and \<sigma> :: "('d list \<times> 'e list) list" shows "wellFormedSubst ((xvec, Tvec) # \<sigma>) = (length xvec = length Tvec \<and> distinct xvec \<and> wellFormedSubst \<sigma>)"
coregular_tcsigs_def: fixes sub :: "('a \<times> 'a) set"   and tcs :: "'b \<Rightarrow> ('a \<Rightarrow> 'a set list option) option" shows "coregular_tcsigs sub tcs \<equiv> \<forall>ars\<in>ran tcs. \<forall>c\<^sub>1\<in>dom ars. \<forall>c\<^sub>2\<in>dom ars. class_leq sub c\<^sub>1 c\<^sub>2 \<longrightarrow> list_all2 (sort_leq sub) (the (ars c\<^sub>1)) (the (ars c\<^sub>2))"
z2: shows "lincomb c X = 0\<^sub>v n"
remap_cta: fixes f :: "nat set option \<Rightarrow> 'a" assumes "inj_on f tac.Q" shows "complete_tree_automaton (ta_remap f TAC) A"
not_mem_less_INF: fixes f :: "'d \<Rightarrow> 'c"   and x :: "'d"   and S :: "'d set" assumes "f x < Inf (f ` S)"   and "x \<in> S" shows "False"
mem'_eq: shows "subst (g (to_partial mem |` X)) mem' = mem'"
rtrancl_on_mono: fixes s :: "('c \<times> 'c) set"   and r :: "('c \<times> 'c) set"   and F :: "'c set"   and G :: "'c set"   and a :: "'c"   and b :: "'c" assumes "s \<subseteq> r"   and "F \<subseteq> G"   and "(a, b) \<in> rtrancl_on F s" shows "(a, b) \<in> rtrancl_on G r"
edge_det: fixes a :: "'b LDCFG_node \<times> 'c edge_kind \<times> 'b LDCFG_node"   and a' :: "'b LDCFG_node \<times> 'c edge_kind \<times> 'b LDCFG_node" assumes "lift_valid_edge valid_edge sourcenode targetnode kind Entry Exit a"   and "lift_valid_edge valid_edge sourcenode targetnode kind Entry Exit a'"   and "src a = src a'"   and "trg a = trg a'" shows "a = a'"
remdups_rev_code_ps_fst: fixes p :: "'a"   and ps :: "'a list"   and rsa :: "'a list" shows "remdups_rev_code (p # ps) rsa = remdups_rev_code ps (removeAll p rsa)"
except_Compl: fixes p :: "'b \<Rightarrow>\<^sub>0 'c"   and U :: "'b set" shows "except p (- U) = p - except p U"
zip_assoc: fixes xs :: "'c list"   and ys :: "'d list"   and zs :: "'e list" shows "xs || ys || zs = map2 (\<lambda>(x, y) z. (x, y, z)) (xs || ys) zs"
assms: shows "Partitions.partn_lst r B [x, y] n"
Fix1: fixes \<phi> :: "((Var \<Rightarrow> Val) \<times> (Var \<Rightarrow> Val) \<Rightarrow> bool) \<Rightarrow> (Var \<Rightarrow> Val) \<times> (Var \<Rightarrow> Val) \<Rightarrow> bool"   and s :: "Var \<Rightarrow> Val"   and t :: "Var \<Rightarrow> Val" assumes "Monotone \<phi>"   and "FIX \<phi> (s, t)" shows "\<phi> (FIX \<phi>) (s, t)"
remove_def: fixes node_ptr :: "(_) node_ptr" shows "remove node_ptr = Heap_Error_Monad.bind (get_parent node_ptr) (\<lambda>parent_opt. case parent_opt of None \<Rightarrow> return () | Some parent \<Rightarrow> Heap_Error_Monad.bind (remove_child parent node_ptr) (\<lambda>_. return ()))"
arr_char: fixes f :: "'a \<times> 'a" shows "B.VV.arr f = (B.arr (fst f) \<and> B.arr (snd f) \<and> src\<^sub>B (fst f) = trg\<^sub>B (snd f))"
hext_trans: fixes h :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and h' :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and h'' :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option" assumes "h \<unlhd> h'"   and "h' \<unlhd> h''" shows "h \<unlhd> h''"
is_expansion_imp_smallomega: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and is_expansion :: "'a \<Rightarrow> (real \<Rightarrow> real) list \<Rightarrow> bool"   and expansion_level :: "'a itself \<Rightarrow> nat"   and eval :: "'a \<Rightarrow> real \<Rightarrow> real"   and zero_expansion :: "'a"   and const_expansion :: "real \<Rightarrow> 'a"   and powr_expansion :: "bool \<Rightarrow> 'a \<Rightarrow> real \<Rightarrow> 'a"   and power_expansion :: "bool \<Rightarrow> 'a \<Rightarrow> nat \<Rightarrow> 'a"   and trimmed :: "'a \<Rightarrow> bool"   and dominant_term :: "'a \<Rightarrow> real \<times> real list"   and basis :: "(real \<Rightarrow> real) list"   and F :: "'a"   and b :: "real \<Rightarrow> real"   and e :: "real" assumes "class.multiseries divide inverse minus plus times uminus is_expansion expansion_level eval zero_expansion const_expansion powr_expansion power_expansion trimmed dominant_term"   and "basis_wf basis"   and "is_expansion F basis"   and "filterlim b at_top at_top"   and "trimmed F"   and "\<forall>g\<in>set basis. (\<lambda>x. ln (g x)) \<in> o(\<lambda>x. ln (b x))"   and "e < 0" shows "eval F \<in> \<omega>(\<lambda>x. b x powr e)"
simps_1: fixes v :: "'v"   and v' :: "'v" shows "is_trail v [] v' = (v = v' \<and> v' \<in> V)"
ordLeq_iff_ordLess_or_ordIso: fixes r :: "('a \<times> 'a) set"   and r' :: "('b \<times> 'b) set" shows "(r \<le>o r') = (r <o r' \<or> r =o r')"
minus_plus: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and s :: "'b"   and t :: "'b"   and u :: "'b" assumes "class.comm_powerprod plus minus zero"   and "plus.adds plus s t" shows "plus (minus t s) u = minus (plus t u) s"
addfunsetD_range: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and M :: "'d set" assumes "f \<in> addfunset A M" shows "range f \<subseteq> M"
scale_left_diff_distrib: fixes a :: "real"   and b :: "real"   and x :: "'c" shows "(a - b) *\<^sub>R x = a *\<^sub>R x - b *\<^sub>R x"
unit_counit_G: fixes x :: "'a" assumes "Adj.C.ide x" shows "G x = G (Adj.\<epsilon>o x) \<cdot>\<^sub>D Adj.\<eta>o (G x)"
matrix_construction_is_kronecker_product_R: fixes subs1 :: "(nat list \<times> nat list) list"   and n1 :: "nat"   and signs1 :: "rat list list"   and signs2 :: "rat list list"   and subs2 :: "(nat list \<times> nat list) list" assumes "\<And>l i. \<lbrakk>l \<in> set subs1; i \<in> set (fst l) \<or> i \<in> set (snd l)\<rbrakk> \<Longrightarrow> i < n1"   and "\<And>j. j \<in> set signs1 \<Longrightarrow> length j = n1" shows "M_mat_R (signs_smash signs1 signs2) (subsets_smash_R n1 subs1 subs2) = kronecker_product (M_mat_R signs1 subs1) (M_mat_R signs2 subs2)"
prems: shows "\<Gamma>\<turnstile> \<langle>mark_guards f (Catch c1 c2),s\<rangle> =n\<Rightarrow> t"
intF_def: shows "intF \<equiv> ProblemIkTpart_TEModel.intF arOf resOf tpD tpFD (\<lambda>\<sigma>. D) eintF"
E: shows "E = (\<lambda>s. 1 + 2 * (4 - nat (s a)))"
l_def: fixes F :: "'a filter"   and g :: "'a \<Rightarrow> 'b" shows "l F g = {f. \<forall>c>0. \<forall>\<^sub>F x in F. R (norm (f x)) (c * norm (g x))}"
assms_1: shows "\<Psi> \<rhd> P \<leadsto>\<guillemotleft>Rel\<guillemotright> Q"
reachable1_awalk: fixes G :: "('c, 'd) pre_digraph"   and u :: "'c"   and v :: "'c" assumes "wf_digraph G" shows "(u \<rightarrow>\<^sup>+\<^bsub>G\<^esub> v) = (\<exists>p. pre_digraph.awalk G u p v \<and> p \<noteq> [])"
unit_counit_G: fixes x :: "'a" assumes "Adj.C.ide x" shows "G' x = G' (Adj'.\<epsilon>o x) \<cdot>\<^sub>D Adj'.\<eta>o (G' x)"
idx_inj: fixes bound :: "'a bound"   and x :: "'a"   and y :: "'a" assumes "in_bound less_eq0 less0 bound x"   and "in_bound less_eq0 less0 bound y"   and "idx0 bound x = idx0 bound y" shows "x = y"
measurable_trivial_measure: fixes f :: "'a \<Rightarrow> 'b"   and M :: "'a measure"   and X :: "'b set" assumes "f \<in> space M \<rightarrow> X"   and "f -` X \<inter> space M \<in> sets M" shows "f \<in> M \<rightarrow>\<^sub>M trivial_measure X"
inject_4: fixes x51 :: "bexp"   and x52 :: "com"   and y51 :: "bexp"   and y52 :: "com" shows "(WHILE x51 DO x52 = WHILE y51 DO y52) = (x51 = y51 \<and> x52 = y52)"
assms_2: shows "v \<in> V\<^bsub>subgame V'\<^esub>"
prems_2: shows "\<pi> =\<^bsub>k\<^esub> \<pi>'"
set_disconnected_nodes_get_child_nodes: fixes ptr :: "(_) document_ptr"   and h :: "(_) heap"   and h' :: "(_) heap"   and ptr' :: "(_) object_ptr" shows "\<forall>w\<in>set_disconnected_nodes_locs ptr. h \<turnstile> w \<rightarrow>\<^sub>h h' \<longrightarrow> (\<forall>r\<in>get_child_nodes_locs ptr'. r h h')"
ab_group_add_ow_axioms_def: fixes S :: "'b set"   and pls :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and z :: "'b"   and mns :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and um :: "'b \<Rightarrow> 'b" shows "ab_group_add_ow_axioms S pls z mns um \<equiv> (\<forall>a. a \<in> S \<longrightarrow> pls (um a) a = z) \<and> (\<forall>a b. a \<in> S \<longrightarrow> b \<in> S \<longrightarrow> mns a b = pls a (um b)) \<and> (\<forall>a. a \<in> S \<longrightarrow> um a \<in> S)"
set_rw1: shows "cols_upt_k A (Suc k) = insert (column (mod_type_class.from_nat (Suc k)) A) (cols_upt_k A k)"
LeftDerivationFix_cut_prefix: fixes \<delta> :: "symbol list"   and \<alpha> :: "symbol list"   and i :: "nat"   and D :: "(nat \<times> symbol \<times> symbol list) list"   and j :: "nat"   and \<gamma> :: "symbol list" assumes "LeftDerivationFix (\<delta> @ \<alpha>) i D j \<gamma>"   and "derivation_ge D (length \<delta>)"   and "length \<delta> \<le> i"   and "is_word \<delta>" shows "\<exists>\<gamma>'. \<gamma> = \<delta> @ \<gamma>' \<and> LeftDerivationFix \<alpha> (i - length \<delta>) (derivation_shift D (length \<delta>) 0) (j - length \<delta>) \<gamma>'"
subsetD: fixes A :: "'g set"   and B :: "'g set"   and c :: "'g" assumes "A \<subseteq> B"   and "c \<in> A" shows "c \<in> B"
isomorphicE: fixes a :: "'a"   and a' :: "'a" assumes "a \<cong> a'" obtains f :: "'a" where "\<guillemotleft>f : a \<Rightarrow> a'\<guillemotright> \<and> local.iso f"
trivial_phis_def: fixes p :: "('a, 'c, 'd, 'b) gen_ssa_cfg_wf \<Rightarrow> ('a \<times> 'b, 'b list) mapping"   and g :: "('a, 'c, 'd, 'b) gen_ssa_cfg_wf" shows "uninst_trivial_phis p g = Set.filter (\<lambda>(n, v). uninst_trivial v (the (Mapping.lookup (p g) (n, v)))) (Mapping.keys (p g))"
assms_1: shows "M \<subseteq> subformulas\<^sub>\<mu> \<phi>"
AL_delete5: fixes b :: "'a"   and a :: "'a"   and l :: "('a \<times> 'b) list"   and x :: "'b option" assumes "b \<noteq> a"   and "l\<down>b = x" shows "delete l a\<down>b = x"
sum_measure_le_measure_inter: fixes A :: "'a set"   and B :: "'a set" assumes "A \<in> events"   and "B \<in> events" shows "prob A + prob B \<le> 1 + prob (A \<inter> B)"
abs_induct: fixes P :: "'a zmultiset \<Rightarrow> bool"   and x :: "'a zmultiset" assumes "\<And>y. P (abs_zmultiset y)" shows "P x"
substAbs_vsubstAbs_trans: fixes A :: "('d, 'e, 'varSort, 'var, 'f) qAbs set"   and Y :: "('d, 'e, 'varSort, 'var, 'f) qTerm set"   and ys :: "'varSort"   and y1 :: "'var"   and y :: "'var" assumes "goodAbs A"   and "good Y"   and "freshAbs ys y1 A" shows "A $[y1 // y]_ys $[Y / y1]_ys = A $[Y / y]_ys"
igSwapIGAbs_fromMOD: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) Recursion.model" assumes "gSwapGAbs MOD" shows "igSwapIGAbs (fromMOD MOD)"
ordering_top_axioms: shows "ordering_top (\<le>) (<) \<top>"
mat_mult_index: fixes nr :: "nat"   and n :: "nat"   and m1 :: "'b list list"   and nc :: "nat"   and m2 :: "'b list list"   and i :: "nat"   and j :: "nat"   and ze :: "'b"   and pl :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and ti :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" assumes "mat nr n m1"   and "mat n nc m2"   and "i < nr"   and "j < nc" shows "mat_multI ze pl ti nr m1 m2 ! j ! i = scalar_prodI ze pl ti (row m1 i) (col m2 j)"
inserted_ids_last_ins: fixes xs :: "('c \<times> ('c, 'd) list_op) list"   and oid :: "'c"   and ref :: "'c option"   and val :: "'d" shows "inserted_ids (xs @ [(oid, Insert ref val)]) = inserted_ids xs @ [oid]"
heap_is_wellformed_children_in_heap: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and children :: "(_) node_ptr list"   and child :: "(_) node_ptr" assumes "heap_is_wellformed h"   and "h \<turnstile> get_child_nodes ptr \<rightarrow>\<^sub>r children"   and "child \<in> set children" shows "child |\<in>| node_ptr_kinds h"
unique_declaration: fixes G :: "prog"   and m :: "memberdecl"   and C :: "qtname"   and n :: "memberdecl" assumes "G\<turnstile> m declared_in C"   and "G\<turnstile> n declared_in C"   and "memberid m = memberid n" shows "m = n"
b_2: shows "t \<in> trms\<^sub>s\<^sub>s\<^sub>t\<^sub>p (snd b) \<union> pair ` setops\<^sub>s\<^sub>s\<^sub>t\<^sub>p (snd b)"
inverse_less_imp_less: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and 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 sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less (inverse a) (inverse b)"   and "less zero a" shows "less b a"
r_prod_encode_prim: shows "prim_recfn 2 r_prod_encode"
dvd_mod: 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 k :: "'b"   and m :: "'b"   and n :: "'b" assumes "class.euclidean_semiring_cancel divide plus minus zero times one modulo euclidean_size"   and "dvd.dvd times k m"   and "dvd.dvd times k n" shows "dvd.dvd times k (modulo m n)"
measure_closure: fixes S :: "'c set" assumes "bounded S"   and "negligible (frontier S)" shows "Sigma_Algebra.measure lebesgue (closure S) = Sigma_Algebra.measure lebesgue S"
abs_eq: fixes xa :: "'b"   and x :: "'b \<Rightarrow> 'c option" shows "delete xa (Mapping x) = Mapping (x(xa := None))"
effect_returnE: fixes x :: "'a"   and h :: "heap"   and h' :: "heap"   and r :: "'a" assumes "effect (return x) h h' r" obtains "r = x" and "h' = h"
evaln_evalc: fixes c :: "com"   and s :: "state"   and n :: "nat"   and t :: "state" assumes "<c,s> -n-> t" shows "<c,s> -c-> t"
simps_1: fixes v :: "'c option"   and m :: "('d \<times> ('d, 'c) trie) list" shows "lookup_trie (Trie v m) [] = v"
derivation_shift_def: fixes D :: "(nat \<times> symbol \<times> symbol list) list"   and left :: "nat"   and right :: "nat" shows "derivation_shift D left right = map (\<lambda>d. (fst d - left + right, snd d)) D"
set_mset_mset: fixes xs :: "'k list" shows "set_mset (mset xs) = set xs"
borel_measurable_vimage_open: fixes f :: "'a \<Rightarrow> 'b"   and S :: "'a set" shows "(f \<in> borel_measurable (lebesgue_on S)) = (\<forall>T. open T \<longrightarrow> {x \<in> S. f x \<in> T} \<in> sets (lebesgue_on S))"
of_complex_def: shows "of_complex \<equiv> map_fun id abs_complex_homo of_complex_hcoords"
some_sym_eq_trivial: fixes x :: "'b" shows "(SOME y. x = y) = x"
NonUniformExecutionStep: fixes cfgList :: "('p, 'v, 's) configuration list"   and msgList :: "('p, 'v) message list" assumes "initial (hd cfgList)"   and "nonUniform (last cfgList)"   and "execution trans sends start cfgList msgList" shows "\<exists>cfgList' msgList'. nonUniform (last cfgList') \<and> prefixList cfgList cfgList' \<and> prefixList msgList msgList' \<and> execution trans sends start cfgList' msgList' \<and> initial (hd cfgList') \<and> (\<exists>msg'. execution.minimalEnabled cfgList msgList msg' \<and> msg' \<in> set (drop (length msgList) msgList'))"
card_0_eq: fixes A :: "'b set" assumes "finite A" shows "(card A = 0) = (A = {})"
chain_imageI: fixes le_a :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and Y :: "'b set"   and le_b :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'c" assumes "Complete_Partial_Order.chain le_a Y"   and "\<And>x y. \<lbrakk>x \<in> Y; y \<in> Y; le_a x y\<rbrakk> \<Longrightarrow> le_b (f x) (f y)" shows "Complete_Partial_Order.chain le_b (f ` Y)"
null_rec_1: fixes x :: "'a"   and xs :: "'a list" shows "List.null (x # xs) = False"
num_eq_card_orbit: fixes P :: "'a set" assumes "P \<in> subgroups_of_size (p ^ a)" shows "subgroups_of_size (p ^ a) = conjG.orbit P"