hcju/isabelleps huggingface dataset:
elims: fixes \<Delta> :: "('a, 'b) network_scheme"   and g :: "'a \<times> 'a \<Rightarrow> ennreal"   and x :: "nat"   and xa :: "'a \<times> 'a"   and y :: "ennreal" assumes "flow_network \<Delta> g"   and "flow_network.h_plus \<Delta> g x xa = y" obtains xb :: "'a" and ya :: "'a" where "x = 0" and "xa = (xb__, ya__)" and "y = (if xb__ = source \<Delta> then g (xb__, ya__) else 0)"   | i :: "nat" and xb :: "'a" and ya :: "'a" where "x = Suc i" and "xa = (xb, ya)" and "y = (if flow_network.enum_v \<Delta> (Suc i) = xb \<and> d_OUT (flow_network.h_plus \<Delta> g i) xb < d_IN (flow_network.h_plus \<Delta> g i) xb then let total = d_IN (flow_network.h_plus \<Delta> g i) xb - d_OUT (flow_network.h_plus \<Delta> g i) xb; share = g (xb, ya) - flow_network.h_plus \<Delta> g i (xb, ya); shares = d_OUT g xb - d_OUT (flow_network.h_plus \<Delta> g i) xb in flow_network.h_plus \<Delta> g i (xb, ya) + share * total / shares else flow_network.h_plus \<Delta> g i (xb, ya))"
recvmsg_simps_4: fixes P :: "'e \<Rightarrow> bool"   and ip :: "nat" shows "recvmsg P (\<not>unicast ip) = True"
\<epsilon>_def: shows "F'G'.\<epsilon> \<equiv> transformation_by_components.map (\<cdot>\<^sub>B) (\<cdot>\<^sub>B) F'G'.FG.map F'G'.\<epsilon>o"
him_of_real: fixes a :: "real" shows "HIm (octo_of_real a) = 0"
ennreal_mult_numeral: fixes x :: "real"   and n :: "num" shows "ennreal x * numeral n = ennreal (x * numeral n)"
seqr_post_transfer: fixes q :: "('c, 'b) urel"   and P :: "('b, 'c) urel"   and R :: "('c, 'b) urel" assumes "out\<alpha> \<sharp> q" shows "P ;; (q \<and> R) = (P \<and> q\<^sup>-) ;; R"
limit_jordan_block: fixes k :: "nat"   and la :: "complex"   and i :: "nat"   and j :: "nat"   and off :: "nat" assumes "(k, la) \<in> set n_as"   and "i < k"   and "j < k" shows "(\<lambda>N. (jordan_block k la ^\<^sub>m K off N) $$ (i, j) * complex_of_real (C off N)) \<longlonglongrightarrow> (if i = 0 \<and> j = k - 1 \<and> cmod la = 1 \<and> k = m then la ^ off else 0)"
IH_3: fixes x :: "'a list list"   and xa :: "'a list"   and xb :: "nat"   and xc :: "'a"   and xd :: "'a list \<times> 'a list \<times> 'a list"   and xe :: "'a list"   and y :: "'a list \<times> 'a list"   and xda :: "'a list"   and ya :: "'a list"   and xdb :: "nat"   and xdc :: "nat" assumes "\<not> length xs \<le> 20"   and "x = chop 5 xs"   and "xa = map slow_median x"   and "xb = ((length xs + 4) div 5 - 1) div 2"   and "xc = mom_select xb xa"   and "xd = partition3 xc xs"   and "(xe, y) = xd"   and "(xda, ya) = y"   and "xdb = length xe"   and "xdc = length xda" shows "T_mom_select xb xa \<le> T'_mom_select (length xa)"
prems_2: shows "P \<turnstile> C_ \<prec>\<^sup>1 D"
le_def: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and e\<^sub>1 :: "'a err"   and e\<^sub>2 :: "'a err" shows "Err.le r e\<^sub>1 e\<^sub>2 = (case e\<^sub>2 of Err \<Rightarrow> True | OK y \<Rightarrow> case e\<^sub>1 of Err \<Rightarrow> False | OK x \<Rightarrow> x \<sqsubseteq>\<^bsub>r\<^esub> y)"
match_1: fixes \<nu> :: "'a"   and \<mu> :: "'a"   and \<tau> :: "'a" assumes "\<nu> \<star> \<mu> \<noteq> null"   and "(\<nu> \<star> \<mu>) \<star> \<tau> \<noteq> null" shows "\<mu> \<star> \<tau> \<noteq> null"
trace_a_def: fixes X :: "('b \<times> ('c \<times> 'b) list) set" shows "trace_a X = {(p, []) |p. \<nexists>x. x \<in> X \<and> p = first x}"
enumchain_mono: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and N :: "'b"   and i :: "nat" assumes "infinitely_many2 P N" shows "N \<le> infinitely_many2.enumchain P N i \<and> infinitely_many2.enumchain P N i < infinitely_many2.enumchain P N (Suc i)"
le_infI: fixes x :: "'c multiset"   and a :: "'c multiset"   and b :: "'c multiset" assumes "x \<subseteq># a"   and "x \<subseteq># b" shows "x \<subseteq># a \<inter># b"
Gauss_Jordan_in_ij_0: fixes A :: "(('c, 'd) vec, 'b) vec"   and j :: "'d"   and i :: "'b"   and a :: "'b" assumes "\<exists>n. A $ n $ j \<noteq> (0::'c) \<and> i \<le> n"   and "a \<noteq> i" shows "Gauss_Jordan_in_ij A i j $ a $ j = (0::'c)"
delay_bisimulation_measure_flip: fixes trsys1 :: "'g \<Rightarrow> 'h \<Rightarrow> 'g \<Rightarrow> bool"   and trsys2 :: "'i \<Rightarrow> 'j \<Rightarrow> 'i \<Rightarrow> bool"   and bisim :: "'g \<Rightarrow> 'i \<Rightarrow> bool"   and tlsim :: "'h \<Rightarrow> 'j \<Rightarrow> bool"   and \<tau>move1 :: "'g \<Rightarrow> 'h \<Rightarrow> 'g \<Rightarrow> bool"   and \<tau>move2 :: "'i \<Rightarrow> 'j \<Rightarrow> 'i \<Rightarrow> bool"   and \<mu>1 :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and \<mu>2 :: "'i \<Rightarrow> 'i \<Rightarrow> bool" assumes "delay_bisimulation_measure trsys1 trsys2 bisim tlsim \<tau>move1 \<tau>move2 \<mu>1 \<mu>2" shows "delay_bisimulation_measure trsys2 trsys1 (flip bisim) (flip tlsim) \<tau>move2 \<tau>move1 \<mu>2 \<mu>1"
mono_upclosed: fixes f :: "'c \<Rightarrow> 'd" assumes "mono f" shows "\<forall>Y. upclosed_set Y \<longrightarrow> upclosed_set (f -` Y)"
eq: shows "GEN_OP node_eq_impl (=) (node_rel \<rightarrow> node_rel \<rightarrow> bool_rel)"
marked_insertions_def: fixes m :: "'e" shows "mut_m.marked_insertions m = (\<lambda>s. \<forall>x. x \<in> set (sys_mem_store_buffers (mutator m) s) \<longrightarrow> marked_insertion x s)"
sin_of_real_pi_half: shows "sin (of_real pi / (2::'a)) = (1::'a)"
conseq1: fixes G :: "'b triple set"   and P' :: "'b assn"   and c :: "com"   and Q :: "'b assn"   and P :: "'b assn" assumes "G|-{P'}. c .{Q}"   and "\<forall>Z s. P Z s \<longrightarrow> P' Z s" shows "G|-{P}. c .{Q}"
simps_7: fixes x :: "nat"   and a :: "floatarith"   and b :: "floatarith"   and vs :: "real list" shows "isDERIV x (Min a b) vs = False"
ae_C: shows "AE x in measure_pmf N. x \<in> C"
mkM_def: fixes T :: "'s Trace set" shows "mkM T \<equiv> \<lparr>worlds = T, relations = \<lambda>a. {(t, t'). {t, t'} \<subseteq> T \<and> jview a t = jview a t'}, valuation = envVal \<circ> tLast\<rparr>"
bidirected_digraph_del_arc: fixes G :: "('c, 'd) pre_digraph"   and arev :: "'d \<Rightarrow> 'd"   and a :: "'d" assumes "bidirected_digraph G arev" shows "bidirected_digraph (pre_digraph.del_arc (pre_digraph.del_arc G (arev a)) a) (perm_restrict arev (arcs G - {a, arev a}))"
perm_bij: fixes pi :: "(NominalSequents.var \<times> NominalSequents.var) list"   and x :: "'a"   and y :: "'a" shows "(pi \<bullet> x = pi \<bullet> y) = (x = y)"
assms_2: assumes "p \<noteq> 0"   and "ipoly p x = (0::'a)" shows "thesis"
assms_3: shows "\<forall>s\<in>{t<..<t'}. flow0 x s \<noteq> flow0 x t"
sort_key_by_quicksort: fixes f :: "'c \<Rightarrow> 'a"   and xs :: "'c list" shows "sort_key f xs = sort_key f (filter (\<lambda>x. f x < f (xs ! (length xs div 2))) xs) @ filter (\<lambda>x. f x = f (xs ! (length xs div 2))) xs @ sort_key f (filter (\<lambda>x. f (xs ! (length xs div 2)) < f x) xs)"
echelon_form_of_column_k_iarrays_def: fixes A' :: "'a iarray iarray \<times> nat \<times> ('a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<times> 'a \<times> 'a \<times> 'a)"   and k :: "nat" shows "echelon_form_of_column_k_iarrays A' k = (let (A, i, bezout) = A'; nrows_A = nrows_iarray A; column_Ak = column_iarray k A; all_zero_below_i = vector_all_zero_from_index (i + 1, column_Ak) in if i = nrows_A \<or> A !! i !! k = (0::'a) \<and> all_zero_below_i then (A, i, bezout) else if all_zero_below_i then (A, i + 1, bezout) else let n = least_non_zero_position_of_vector_from_index column_Ak i; interchange_A = interchange_rows_iarray A i n in (bezout_iterate_iarrays interchange_A (nrows_A - 1) i k bezout, i + 1, bezout))"
csdi: shows "s O d\<inverse> \<subseteq> b \<union> m \<union> ov \<union> f\<inverse> \<union> d\<inverse>"
vars\<^sub>s\<^sub>s\<^sub>t\<^sub>p_cases_7: fixes X :: "'c list"   and F :: "(('d, 'c) Term.term \<times> ('d, 'c) Term.term) list"   and G :: "(('d, 'c) Term.term \<times> ('d, 'c) Term.term) list" shows "vars\<^sub>s\<^sub>s\<^sub>t\<^sub>p \<forall>X\<langle>\<or>\<noteq>: F \<or>\<notin>: G\<rangle> = fv\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s F \<union> fv\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s G \<union> set X"
int2: shows "integrable M (\<lambda>x. G x * (X x - real_cond_exp M F X x))"
seq_compact_def: fixes S :: "'b set" shows "seq_compact S = (\<forall>f. (\<forall>n. f n \<in> S) \<longrightarrow> (\<exists>l\<in>S. \<exists>r. strict_mono r \<and> (f \<circ> r) \<longlonglongrightarrow> l))"
no_leading_Cons: fixes P :: "'b \<Rightarrow> bool"   and x :: "'b"   and xs :: "'b list" shows "no_leading P (x # xs) = (\<not> P x)"
actors_graph_def: fixes g :: "igraph" shows "actors_graph g \<equiv> {x. \<exists>y. y \<in> nodes g \<and> x \<in> agra g y}"
trss_xchange_context_s: fixes s :: "'f list"   and c :: "'f list multiset"   and e :: "('g, 'h) label"   and s' :: "'f list"   and csp :: "'f list multiset"   and fg :: "('f, 'g, 'h, 'i, 'j) flowgraph_rec_scheme"   and cn :: "'f list multiset" assumes "((s, c), e, s', csp + c) \<in> trss fg"   and "mon_c fg cn \<subseteq> mon_c fg c" shows "((s, cn), e, s', csp + cn) \<in> trss fg"
nat_add_max_left: fixes m :: "nat"   and n :: "nat"   and q :: "nat" shows "max m n + q = max (m + q) (n + q)"
deny_all_def: fixes p :: "'a" shows "D\<^sub>U p = \<lfloor>deny ()\<rfloor>"
test_bit_2p: fixes n :: "nat"   and m :: "nat" shows "word_of_int (2 ^ n) !! m = (m = n \<and> m < LENGTH('b))"
diagram_axioms: shows "diagram (\<cdot>\<^sub>J) (\<cdot>) (J_C.Map d)"
is_processT4: fixes P :: "'c process" shows "\<forall>s X Y. (s, Y) \<in> F P \<and> X \<subseteq> Y \<longrightarrow> (s, X) \<in> F P"
c1_c1': shows "c1 \<subseteq>\<^sub>g c1'"
r_not_def: shows "r_not \<equiv> Cn 1 r_sub [r_const 1, r_sign]"
simps_1: fixes ts :: "'a \<Rightarrow> ('b \<times> 'c \<Rightarrow>f nat) option" shows "thread_oks ts [] = True"
smash_systems_def: fixes p :: "real poly"   and qs1 :: "real poly list"   and qs2 :: "real poly list"   and subsets1 :: "nat list list"   and subsets2 :: "nat list list"   and signs1 :: "rat list list"   and signs2 :: "rat list list"   and mat1 :: "rat mat"   and mat2 :: "rat mat" shows "smash_systems p qs1 qs2 subsets1 subsets2 signs1 signs2 mat1 mat2 = (qs1 @ qs2, kronecker_product mat1 mat2, subsets_smash (length qs1) subsets1 subsets2, signs_smash signs1 signs2)"
rel_cases: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "'c stream"   and b :: "'d stream" assumes "stream_all2 R a b" obtains x1a :: "'c" and x2a :: "'c stream" and y1a :: "'d" and y2a :: "'d stream" where "a = x1a ## x2a" and "b = y1a ## y2a" and "R x1a y1a" and "stream_all2 R x2a y2a"
leD: fixes y :: "'b list"   and x :: "'b list" assumes "y \<le>p x" shows "\<not> x <p y"
assms_10: shows "InvariantCllCharacterization (getCl state) (getCll state) (getC state) (getM state)"
arr_lunit: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and T :: "'a \<times> 'a \<Rightarrow> 'a"   and \<alpha> :: "'a \<times> 'a \<times> 'a \<Rightarrow> 'a"   and \<iota> :: "'a"   and a :: "'a" assumes "monoidal_category C T \<alpha> \<iota>"   and "partial_magma.ide C a" shows "partial_magma.arr C (monoidal_category.lunit C T \<alpha> \<iota> a)"
fresh_Op_simp: fixes inp :: "'b \<Rightarrow> ('b, 'c, 'varSort, 'var, 'd) qTerm set option"   and binp :: "'c \<Rightarrow> ('b, 'c, 'varSort, 'var, 'd) qAbs set option"   and xs :: "'varSort"   and x :: "'var"   and delta :: "'d" assumes "goodInp inp"   and "goodBinp binp" shows "fresh xs x (Op delta inp binp) = (freshInp xs x inp \<and> freshBinp xs x binp)"
simps: fixes w1l :: "32 Word.word"   and w1u :: "32 Word.word"   and w2l :: "32 Word.word"   and w2u :: "32 Word.word" shows "tu w1l w1u w2l w2u = wmax (wmax (wtimes w1l w2l) (wtimes w1u w2l)) (wmax (wtimes w1l w2u) (wtimes w1u w2u))"
p1d_def: fixes x :: "'a \<times> 'a"   and xa :: "'a \<times> 'a" shows "p1d x xa = (38::'a) * (fst xa * fst x) / (15::'a) - (69::'a) * fst xa / (38::'a) - (27::'a) * (fst xa * (fst x)\<^sup>2) / (28::'a) - (24::'a) * (fst xa * fst x ^ 3) / (43::'a) + (14::'a) * snd xa / (29::'a) + ((651::'a) * (fst x * snd xa) + (651::'a) * (fst xa * snd x)) / (441::'a) + ((8554::'a) * ((fst x)\<^sup>2 * snd xa) + (17108::'a) * (fst xa * (fst x * snd x))) / (2209::'a) - ((560::'a) * (fst x ^ 3 * snd xa) + (1680::'a) * (fst xa * ((fst x)\<^sup>2 * snd x))) / (256::'a) - (6::'a) * (snd xa * snd x) / (17::'a) - ((36::'a) * (fst x * (snd xa * snd x)) + (18::'a) * (fst xa * (snd x)\<^sup>2)) / (81::'a) - ((1240::'a) * ((fst x)\<^sup>2 * (snd xa * snd x)) + (1240::'a) * (fst xa * (fst x * (snd x)\<^sup>2))) / (400::'a) + snd xa * (snd x)\<^sup>2 / (34::'a) + ((177::'a) * (fst x * (snd xa * (snd x)\<^sup>2)) + fst xa * snd x ^ 3 * (59::'a)) / (3481::'a)"
ImpR_inv': fixes \<Gamma> :: "'b formula multiset"   and F :: "'b formula"   and G :: "'b formula"   and \<Delta> :: "'b formula multiset"   and n :: "nat" assumes "\<Gamma> \<Rightarrow> F \<^bold>\<rightarrow> G, \<Delta> \<down> n" shows "F, \<Gamma> \<Rightarrow> G, \<Delta> \<down> n"
simps: fixes KP :: "('c \<times> 'd) set" shows "treap_of KP = (if infinite KP \<or> KP = {} then \<langle>\<rangle> else let m = arg_min_on snd KP; L = {p \<in> KP. fst p < fst m}; R = {p \<in> KP. fst m < fst p} in \<langle>treap_of L, m, treap_of R\<rangle>)"
r_update_aux_def: shows "r_update_aux \<equiv> let f = r_constn 2 0 in Let (Cn 5 r_snoc [recf.Id 5 1, Cn 5 r_ifeq [recf.Id 5 0, recf.Id 5 3, recf.Id 5 4, Cn 5 r_nth [recf.Id 5 2, recf.Id 5 0]]]) (Pr 3 f)"
alwD: fixes P :: "'a stream \<Rightarrow> bool"   and x :: "'a stream" assumes "alw P x" shows "P x"
surj_fin_dim: fixes K :: "('b, 'c) ring_scheme"   and V :: "('b, 'd, 'e) module_scheme"   and W :: "('b, 'f, 'g) module_scheme"   and T :: "'d \<Rightarrow> 'f" assumes "linear_map K V W T"   and "vectorspace.fin_dim K V"   and "T ` carrier V = carrier W" shows "vectorspace.fin_dim K W"
binop_wf_dfa: fixes A :: "nat bdd list \<times> bool list"   and B :: "nat bdd list \<times> bool list"   and n :: "nat"   and f :: "bool \<Rightarrow> bool \<Rightarrow> bool" assumes "prod_DFS A B n" shows "wf_dfa (binop_dfa f A B) n"
div_mult_self1: fixes divide :: "'m \<Rightarrow> 'm \<Rightarrow> 'm"   and plus :: "'m \<Rightarrow> 'm \<Rightarrow> 'm"   and minus :: "'m \<Rightarrow> 'm \<Rightarrow> 'm"   and zero :: "'m"   and times :: "'m \<Rightarrow> 'm \<Rightarrow> 'm"   and one :: "'m"   and modulo :: "'m \<Rightarrow> 'm \<Rightarrow> 'm"   and euclidean_size :: "'m \<Rightarrow> nat"   and b :: "'m"   and a :: "'m"   and c :: "'m" assumes "class.euclidean_semiring_cancel divide plus minus zero times one modulo euclidean_size"   and "b \<noteq> zero" shows "divide (plus a (times c b)) b = plus c (divide a b)"
Cons_def: fixes x :: "'b"   and xs :: "'b SList.list" shows "x # xs = Abs_List (CONS (Leaf x) (Rep_List xs))"
equality_axioms_hold: fixes A :: "fmla"   and e :: "(name, nat) finfun" assumes "A \<in> equality_axioms" shows "eval_fmla e A"
iTILL_E: fixes x :: "nat"   and n :: "nat" assumes "x \<in> [\<dots>n]" obtains "x \<le> n"
DgrDataRefinement1_def: fixes D :: "'e \<times> 'e \<Rightarrow> 'f \<Rightarrow> 'f"   and R :: "'e \<Rightarrow> 'f \<Rightarrow> 'g"   and D' :: "'e \<times> 'e \<Rightarrow> 'g \<Rightarrow> 'g" shows "DgrDataRefinement1 D R D' = (\<forall>i j. DataRefinement (D (i, j)) (R i) (R j) (D' (i, j)))"
simps: fixes a :: "msg"   and H :: "msg set" shows "(a \<in> analz H) = ((\<exists>X. a = X \<and> X \<in> H) \<or> (\<exists>X Y. a = X \<and> \<lbrace>X, Y\<rbrace> \<in> analz H) \<or> (\<exists>X Y. a = Y \<and> \<lbrace>X, Y\<rbrace> \<in> analz H) \<or> (\<exists>K X. a = X \<and> Crypt K X \<in> analz H \<and> Key (invKey K) \<in> analz H))"
poly_weak_mono_all_E: fixes p :: "('c monom \<times> 'd) list"   and f :: "'c \<Rightarrow> ('e monom \<times> 'd) list"   and g :: "'c \<Rightarrow> ('e monom \<times> 'd) list" assumes "poly_weak_mono_all p"   and "\<And>x. f x \<ge>p g x \<and> g x \<ge>p zero_poly" shows "poly_subst f p \<ge>p poly_subst g p"
h_def: shows "h = (\<lambda>x. b_least2 f x (g x))"
gallery_Cons_reduce: fixes x :: "'a set"   and xs :: "'a set list" assumes "gallery (x # xs)" shows "gallery xs"
poly_map_poly: fixes hom :: "'b \<Rightarrow> 'c"   and p :: "'b poly"   and x :: "'b" assumes "comm_semiring_hom hom" shows "poly (map_poly hom p) (hom x) = hom (poly p x)"
interpret_floatarith_FDERIV_floatarith_append: fixes fa :: "floatarith"   and x :: "'b"   and params :: "real list" assumes "\<And>i j. i < DIM('b) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params)"   and "max_Var_floatarith fa \<le> DIM('b) + length params" shows "((\<lambda>x. interpret_floatarith fa (list_of_eucl x @ params)) has_derivative (\<lambda>d. interpret_floatarith (FDERIV_floatarith fa [0..<DIM('b)] (map Var [length params + DIM('b)..<length params + 2 * DIM('b)])) (list_of_eucl x @ params @ list_of_eucl d))) (at x)"
Continuous_Monotonic: fixes H :: "'a upred \<Rightarrow> 'a upred" assumes "Continuous H" shows "Monotonic H"
coboundedI1: fixes c :: "'d multiset"   and a :: "'d multiset"   and b :: "'d multiset" assumes "c \<subseteq># a" shows "c \<subseteq># a \<union># b"
code: fixes xs :: "('f, 'g) tllist" shows "llist_of_tllist xs = (case xs of TNil x \<Rightarrow> LNil | TCons x xs' \<Rightarrow> LCons x (llist_of_tllist xs'))"
rd_is_CSP1: fixes r :: "('c, 'd) alpha_rp_scheme \<times> ('c, 'd) alpha_rp_scheme \<Rightarrow> bool"   and p :: "('c, 'd) alpha_rp_scheme \<times> ('c, 'd) alpha_rp_scheme \<Rightarrow> bool" shows "R (r \<turnstile> p) is CSP1 healthy"
simps_1: fixes s :: "('c, 'd) tm" shows "apps s [] = s"
bNaux: fixes x :: "('b, 'c) Combinators"   and y :: "('b, 'd) Combinators" assumes "\<not> member DenyAll x"   and "\<not> member DenyAll y"   and "first_bothNet x = first_bothNet y" shows "{first_srcNet x, first_destNet x} = {first_srcNet y, first_destNet y}"
major: shows "partn_lst \<beta> (n # \<alpha>) \<gamma>"
hyps: fixes i :: "nat" assumes "e \<in> set (elements M')"   and "elementLevel e M' + i \<le> level"   and "i \<le> level" shows "e \<in> set (elements (prefixToLevel_aux M' level i))"
defAss_def: fixes \<alpha>n :: "'a \<Rightarrow> 'b list"   and invar :: "'a \<Rightarrow> bool"   and inEdges' :: "'a \<Rightarrow> 'b \<Rightarrow> ('b \<times> 'c) list"   and Entry :: "'a \<Rightarrow> 'b"   and defs :: "'a \<Rightarrow> 'b \<Rightarrow> 'd set"   and phis :: "'a \<Rightarrow> 'b \<times> 'd \<Rightarrow> 'd list option"   and g :: "'a"   and m :: "'b"   and v :: "'d" shows "CFG_SSA_base.defAss \<alpha>n invar inEdges' Entry defs phis g m v = (\<forall>ns. graph_path_base.path2 \<alpha>n invar inEdges' g (Entry g) ns m \<longrightarrow> (\<exists>n\<in>set ns. v \<in> CFG_SSA_base.allDefs defs phis g n))"
linorder: fixes comp :: "'b \<Rightarrow> 'b \<Rightarrow> order" assumes "comparator comp" shows "class.linorder (le_of_comp comp) (lt_of_comp comp)"
comp_assoc: fixes h :: "'a \<times> 'a"   and g :: "'a \<times> 'a"   and f :: "'a \<times> 'a" shows "VV.comp (VV.comp h g) f = VV.comp h (VV.comp g f)"
assms_2: shows "wf\<^sub>s\<^sub>s\<^sub>t A"
marked_imp_black_or_grey_1: fixes s :: "'d process_name \<Rightarrow> ('a, 'd, 'b, 'c) Model.local_state"   and r :: "'c" assumes "marked r s" shows "black r s \<or> grey r s"
vconverse_atI: fixes r :: "V"   and a :: "V"   and b :: "V" assumes "vsv r"   and "a \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> r"   and "r\<lparr>a\<rparr> = b" shows "\<langle>b, a\<rangle> \<in>\<^sub>\<circ> r\<inverse>\<^sub>\<circ>"
intros_1: fixes T :: "typ"   and s :: "Core.term"   and t :: "Core.term" shows "Abs T s $ t \<rightarrow>\<^sub>\<beta> subst_bv2 s 0 t"
cname_of_def: shows "cname_of = h.cname_of undefined"
G_def: shows "G = fps_poly_swap2 hensel_fpxs1"
that_3: assumes "\<not> discrete X \<and> ed_ou X" shows "thesis"
Quotient3_rel: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and Abs :: "'a \<Rightarrow> 'b"   and Rep :: "'b \<Rightarrow> 'a"   and r :: "'a"   and s :: "'a" assumes "Quotient3 R Abs Rep" shows "(R r r \<and> R s s \<and> Abs r = Abs s) = R r s"
cdf_lim_at_bot: shows "(cdf M2 \<longlongrightarrow> 0) at_bot"
indexed_nominal_ts_def: fixes satisfies :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and transition :: "'b \<Rightarrow> ('d, 'b) residual \<Rightarrow> bool" shows "indexed_nominal_ts TYPE('a) satisfies transition \<equiv> nominal_ts satisfies transition \<and> indexed_nominal_ts_axioms TYPE('a) TYPE('b)"
neg_equal_0_iff_equal: fixes a :: "'a" shows "(- a = (0::'a)) = (a = (0::'a))"
ideal_field_disj: fixes B :: "'c set" shows "ideal B = {0::'c} \<or> ideal B = UNIV"
hcomp_in_vhom: fixes \<mu> :: "'b"   and f :: "'b"   and g :: "'b"   and \<nu> :: "'b"   and h :: "'b"   and k :: "'b" assumes "\<guillemotleft>\<mu> : f \<Rightarrow>\<^sub>D g\<guillemotright>"   and "\<guillemotleft>\<nu> : h \<Rightarrow>\<^sub>D k\<guillemotright>"   and "src\<^sub>D h = trg\<^sub>D f" shows "\<guillemotleft>\<nu> \<star>\<^sub>D \<mu> : h \<star>\<^sub>D f \<Rightarrow>\<^sub>D k \<star>\<^sub>D g\<guillemotright>"
red_crit_lifting_family: fixes q :: "'q" assumes "q \<in> Q" shows "calculus Bot_F Inf_F (\<lambda>N1 N2. entails_q q (\<G>_Fset_q q N1) (\<G>_Fset_q q N2)) (Red_I_\<G>_q q) (Red_F_\<G>_q q)"
Abs_multiset_inject: fixes x :: "'b \<Rightarrow> nat"   and y :: "'b \<Rightarrow> nat" assumes "x \<in> multiset"   and "y \<in> multiset" shows "(Abs_multiset x = Abs_multiset y) = (x = y)"
assms_1: shows "\<GG> : op_dg \<BB> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<CC>"
atLeastAtMost_singleton: fixes a :: "'e word" shows "signed.atLeastAtMost a a = {a}"
Throw: fixes e :: "char list exp"   and xs :: "char list exp list" shows "map_val2 (throw e # xs) [] (throw e # xs)"
ba_mp_2: fixes uminus :: "'b \<Rightarrow> 'b"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and x :: "'b"   and y :: "'b"   and z :: "'b" assumes "class.boolean_algebra_mp uminus sup" shows "sup (uminus (sup (uminus x) y)) (sup z y) = sup y (sup z x)"
update_since_2: fixes \<sigma> :: "(String.literal \<times> event_data list) trace"   and V :: "String.literal \<Rightarrow> (nat \<Rightarrow> event_data list set) option"   and R :: "event_data list set"   and args :: "Monitor.args"   and \<phi> :: "Formula.formula"   and \<psi> :: "Formula.formula"   and aux :: "'msaux"   and ne :: "nat"   and n :: "nat"   and rel1 :: "event_data option list set"   and rel2 :: "event_data option list set"   and rel :: "event_data option list set"   and aux' :: "'msaux"   and I :: "\<I>"   and pos :: "bool" assumes "wf_since_aux \<sigma> V R args \<phi> \<psi> aux ne"   and "qtable n (fv \<phi>) (mem_restr R) (\<lambda>v. Formula.sat \<sigma> V (map the v) ne \<phi>) rel1"   and "qtable n (fv \<psi>) (mem_restr R) (\<lambda>v. Formula.sat \<sigma> V (map the v) ne \<psi>) rel2"   and "(rel, aux') = update_since args rel1 rel2 (\<tau> \<sigma> ne) aux"   and "fv \<phi> \<subseteq> fv \<psi>"   and "args_ivl args = I"   and "args_n args = n"   and "args_L args = fv \<phi>"   and "args_R args = fv \<psi>"   and "args_pos args = pos" shows "qtable n (fv \<psi>) (mem_restr R) (\<lambda>v. Formula.sat \<sigma> V (map the v) ne (Sincep pos \<phi> I \<psi>)) rel"
I_dnf: fixes \<phi> :: "atom fm"   and xs :: "'b list" assumes "nqfree \<phi>" shows "(\<exists>as\<in>set (dnf \<phi>). \<forall>a\<in>set as. I\<^sub>d\<^sub>l\<^sub>o a xs) = DLO.I \<phi> xs"
notin_Dag_update_r: fixes q :: "ref"   and t :: "dag"   and p :: "ref"   and l :: "ref \<Rightarrow> ref"   and r :: "ref \<Rightarrow> ref"   and y :: "ref" assumes "q \<notin> set_of t" shows "Dag p l (r(q := y)) t = Dag p l r t"
netgmap_netgmap_rhs: fixes i :: "nat"   and s2 :: "'c net_state"   and sr :: "'c \<Rightarrow> 'd \<times> 'e"   and s1 :: "'c net_state" assumes "i \<in> net_ips s2" shows "(fst (netgmap sr s1) ++ fst (netgmap sr s2)) i = fst (netgmap sr s2) i"
INF1_E: fixes B :: "'b \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'b set"   and b :: "'a"   and a :: "'b" assumes "(\<Sqinter> (B ` A)) b" obtains "B a b"   | "a \<notin> A"
sum_gp_multiplied: fixes m :: "nat"   and n :: "nat"   and x :: "'b" assumes "m \<le> n" shows "((1::'b) - x) * sum ((^) x) {m..n} = x ^ m - x ^ Suc n"
prj0_in_hom': fixes f :: "'a"   and g :: "'a" assumes "C.cospan f g" shows "\<guillemotleft>\<p>\<^sub>0[f, g] : C.dom \<p>\<^sub>0[f, g] \<rightarrow>\<^sub>C C.dom g\<guillemotright>"
SKIPp: fixes s :: "char list \<Rightarrow> int" shows "\<down>\<^sub>t (SKIP, s) = Suc 0"