hcju/isabelleps huggingface dataset:
quad_next4_id: fixes f :: "face"   and v :: "nat" assumes "|vertices f| = 4"   and "distinct (vertices f)"   and "v \<in> \<V> f" shows "f \<bullet> (f \<bullet> (f \<bullet> (f \<bullet> v))) = v"
1: shows "{xs |xs. set xs \<in> set ` XS \<and> distinct xs} = \<Union> {{xs |xs. set xs = A \<and> distinct xs} |A. A \<in> set ` XS}"
simps_1: fixes arity_sym :: "'a \<Rightarrow> enat"   and arity_var :: "'b \<Rightarrow> enat"   and x :: "'b" shows "arity.arity_hd arity_sym arity_var (Var x) = arity_var x"
vwb_lens_wb: fixes x :: "'b \<Longrightarrow> 'c" assumes "vwb_lens x" shows "wb_lens x"
assms_2: shows "l2 = l2a @ [h] @ l2b"
ct_def: fixes s :: "'s" shows "ct s = (SOME D. D \<in> K s \<and> lfp (\<lambda>F s. \<Sqinter>D\<in>K s. \<integral>\<^sup>+ t. g t (F t) \<partial>measure_pmf D) s = \<integral>\<^sup>+ t. g t (lfp (\<lambda>F s. \<Sqinter>D\<in>K s. \<integral>\<^sup>+ t. g t (F t) \<partial>measure_pmf D) t) \<partial>measure_pmf D)"
d1_def: shows "st2.d1 = prod_list st2.dims1"
assms_4: shows "f \<cdot> h = g \<cdot> k"
assms: shows "FI_RESULT [(B, B), (A, A)] C emp F"
ta_bisim_def: fixes bisim :: "'a \<Rightarrow> 'b \<times> 'c \<Rightarrow> 'd \<times> 'e \<Rightarrow> bool"   and ta1 :: "('f, 'a, 'b, 'c, 'g, 'h) thread_action"   and ta2 :: "('f, 'a, 'd, 'e, 'g, 'h) thread_action" shows "ta_bisim bisim ta1 ta2 \<equiv> \<lbrace>ta1\<rbrace>\<^bsub>l\<^esub> = \<lbrace>ta2\<rbrace>\<^bsub>l\<^esub> \<and> \<lbrace>ta1\<rbrace>\<^bsub>w\<^esub> = \<lbrace>ta2\<rbrace>\<^bsub>w\<^esub> \<and> \<lbrace>ta1\<rbrace>\<^bsub>c\<^esub> = \<lbrace>ta2\<rbrace>\<^bsub>c\<^esub> \<and> \<lbrace>ta1\<rbrace>\<^bsub>o\<^esub> = \<lbrace>ta2\<rbrace>\<^bsub>o\<^esub> \<and> \<lbrace>ta1\<rbrace>\<^bsub>i\<^esub> = \<lbrace>ta2\<rbrace>\<^bsub>i\<^esub> \<and> list_all2 (nta_bisim bisim) \<lbrace>ta1\<rbrace>\<^bsub>t\<^esub> \<lbrace>ta2\<rbrace>\<^bsub>t\<^esub>"
order_pos_iff: fixes p :: "'b poly"   and a :: "'b" assumes "p \<noteq> 0" shows "(0 < order a p) = (poly p a = (0::'b))"
assms_5: shows "(x, y) \<in> relations (clock_repMC (toSet cec)) a"
vimage_vdomain_vsubset: fixes r :: "V"   and A :: "V" shows "r `\<^sub>\<circ> A \<subseteq>\<^sub>\<circ> r `\<^sub>\<circ> \<D>\<^sub>\<circ> r"
assms_2: shows "\<not> hb (r, Store e2 mo2 r) (i, Store e1 mo i)"
rat_to_normalized_int_poly_1: fixes p :: "rat poly"   and d :: "rat"   and q :: "int poly" assumes "rat_to_normalized_int_poly p = (d, q)" shows "p = Polynomial.smult d (of_int_poly q)"
rev_map: fixes f :: "'d \<Rightarrow> 'c"   and xs :: "'d list" shows "rev (map f xs) = map f (rev xs)"
hyps_21: shows "A\<^sub>Q \<sharp>* \<Psi>\<^sub>P"
rewrite_MultiportPorts_one: fixes \<beta> :: "'b common_primitive \<Rightarrow> ('b, 'c) tagged_packet_scheme \<Rightarrow> ternaryvalue"   and m :: "'b common_primitive match_expr"   and \<alpha> :: "action \<Rightarrow> ('b, 'c) tagged_packet_scheme \<Rightarrow> bool"   and a :: "action"   and p :: "('b, 'c) tagged_packet_scheme" assumes "primitive_matcher_generic \<beta>"   and "normalized_nnf_match m" shows "matches (\<beta>, \<alpha>) (replace_primitive_matchexpr (is_MultiportPorts, multiportports_sel) rewrite_MultiportPorts_one m) a p = matches (\<beta>, \<alpha>) m a p"
chine_hcomp_props_6: fixes \<mu> :: "'a arrow_of_spans_data"   and \<nu> :: "'a arrow_of_spans_data" assumes "arr \<mu>"   and "arr \<nu>"   and "src \<nu> = trg \<mu>" shows "\<p>\<^sub>1[Leg0 (Cod \<nu>), Leg1 (Cod \<mu>)] \<cdot> chine_hcomp \<nu> \<mu> = Chn \<nu> \<cdot> \<p>\<^sub>1[Leg0 (Dom \<nu>), Leg1 (Dom \<mu>)]"
iT_Minus_Minus: fixes a :: "nat"   and b :: "nat"   and I :: "nat set" shows "a \<ominus> (b \<ominus> I) = I \<down>\<le> b \<oplus> a \<oplus>- b"
mem_uminus: fixes x :: "'a" assumes "x \<in> S" shows "- x \<in> S"
skl1_refines_trans_l1: shows "{R11} TS.trans l1, TS.trans skl1 {> R11}"
atLeast0_lessThan_Suc_shift: fixes g :: "nat \<Rightarrow> 'b"   and n :: "nat" shows "prod g {0..<Suc n} = g 0 * prod (g \<circ> Suc) {0..<n}"
ucast_leq_mask: fixes n :: "nat"   and x :: "'b word" assumes "LENGTH('b) \<le> n" shows "UCAST('b \<rightarrow> 'c) x \<le> mask n"
carrier_vecD: fixes v :: "'a vec"   and n :: "nat" assumes "v \<in> carrier_vec n" shows "dim_vec v = n"
le_ASSUMEI: fixes \<Phi> :: "bool"   and M :: "'maa"   and M' :: "'maa" assumes "\<Phi>"   and "\<Phi> \<Longrightarrow> M \<le> M'" shows "M \<le> abind (aASSUME \<Phi>) (\<lambda>_. M')"
cl_altdef: fixes X :: "'a set" assumes "X \<subseteq> \<E>" shows "\<E>.cl X = \<Union> {Y \<in> Pow \<E>. X \<subseteq> Y \<and> \<E>.rank_of Y = \<E>.rank_of X}"
neg_le_pos_iff: fixes A :: "'a"   and B :: "'a" shows "(Neg A \<le> Pos B) = (A < B)"
dg_prod_ArrE: fixes f :: "V"   and I :: "V"   and \<AA> :: "V \<Rightarrow> V" assumes "f \<in>\<^sub>\<circ> dg_prod I \<AA>\<lparr>Arr\<rparr>" obtains "vsv f" and "\<D>\<^sub>\<circ> f = I" and "\<And>i. i \<in>\<^sub>\<circ> I \<Longrightarrow> f\<lparr>i\<rparr> \<in>\<^sub>\<circ> \<AA> i\<lparr>Arr\<rparr>"
disc_17: fixes x21 :: "nat"   and x22 :: "'a"   and x23 :: "'a"   and x24 :: "'b"   and x25 :: "'b"   and x26 :: "'c" shows "\<not> isSnapshot (Send x21 x22 x23 x24 x25 x26)"
O6I: fixes F :: "'fmla set"   and t :: "'trm"   and n :: "nat" assumes "nprv F (LLq t (Num n))"   and "t \<in> atrm"   and "F \<subseteq> fmla"   and "finite F" shows "nprv F (LLq t (suc (Num n)))"
parts_insert_MPair: fixes X :: "msg"   and Y :: "msg"   and H :: "msg set" shows "parts (insert \<lbrace>X, Y\<rbrace> H) = insert \<lbrace>X, Y\<rbrace> (parts (insert X (insert Y H)))"
assms: shows "qtable (Suc n) A (mem_restr (lift_envs R)) P X"
frange_vempty: shows "\<R>\<^sub>\<bullet> []\<^sub>\<circ> = []\<^sub>\<circ>"
Dlec_def: shows "Dlec \<equiv> {L \<in> (\<lambda>L. L \<inter> {x. x \<bullet> i \<le> c}) ` \<D>. content L \<noteq> 0}"
continuous_map_from_subtopology: fixes X :: "'b topology"   and X' :: "'c topology"   and f :: "'b \<Rightarrow> 'c"   and S :: "'b set" assumes "continuous_map X X' f" shows "continuous_map (subtopology X S) X' f"
eq_divide_imp: fixes c :: "'a"   and a :: "'a"   and b :: "'a" assumes "c \<noteq> (0::'a)"   and "a * c = b" shows "a = b / c"
rolling_rule_rtl: fixes g :: "'e \<rightarrow> 'd"   and f :: "'d \<rightarrow> 'e" shows "g\<cdot>(fix\<cdot>(f oo g)) \<sqsubseteq> fix\<cdot>(g oo f)"
supp: shows "emeasure M {x \<in> space M. f x \<noteq> 0} < \<infinity>"
matrix_vector_column: fixes A :: "(('b, 'd) vec, 'c) vec"   and x :: "('b, 'd) vec" shows "A *v x = (\<Sum>i\<in>UNIV. x $ i *s Finite_Cartesian_Product.transpose A $ i)"
alleqset: shows "\<forall>x. \<forall>(d, e, f)\<in>set a. d * x\<^sup>2 + e * x + f = 0"
IH: fixes i :: "nat" assumes "n < card S"   and "i \<le> pick S n" shows "card {a \<in> S. a < i} \<le> n"
mset_subset_eq_multiset_union_diff_commute: fixes B :: "'a multiset"   and A :: "'a multiset"   and C :: "'a multiset" assumes "B \<subseteq># A" shows "A - B + C = A + C - B"
set_cond_pmf: fixes p :: "'h pmf"   and s :: "'h set" assumes "set_pmf p \<inter> s \<noteq> {}" shows "set_pmf (cond_pmf p s) = set_pmf p \<inter> s"
preorder_order_ow_def: fixes U\<^sub>1 :: "'a set"   and le\<^sub>1 :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and ls\<^sub>1 :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and U\<^sub>2 :: "'b set"   and le\<^sub>2 :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and ls\<^sub>2 :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "preorder_order_ow U\<^sub>1 le\<^sub>1 ls\<^sub>1 U\<^sub>2 le\<^sub>2 ls\<^sub>2 \<equiv> preorder_ow U\<^sub>1 le\<^sub>1 ls\<^sub>1 \<and> order_ow U\<^sub>2 le\<^sub>2 ls\<^sub>2"
agents_eq: shows "agents' = {i. R' i \<noteq> (\<lambda>_ _. False)}"
borel_integrable_atLeastAtMost': fixes a :: "real"   and b :: "real"   and f :: "real \<Rightarrow> 'a" assumes "continuous_on {a..b} f" shows "set_integrable lborel {a..b} f"
simps: fixes t1 :: "'a option list"   and s :: "nat set"   and t2 :: "'a option list" shows "isSameIntersection t1 s t2 = (\<forall>i\<in>s. t1 ! i = t2 ! i)"
sp_assoc: fixes x :: "real"   and y :: "real"   and z :: "real" shows "sp (sp x y) z = sp x (sp y z)"
list_of_mset_def: fixes m :: "'b multiset" shows "list_of_mset m = (SOME l. m = mset l)"
of_complex_cblinfun_apply: fixes x :: "complex"   and y :: "'a" shows "of_complex x *\<^sub>V y = x *\<^sub>C y"
cbig_def2: fixes \<sigma>' :: "'state"   and coll :: "'coll"   and P :: "'prog"   and \<sigma> :: "'state" shows "((\<sigma>', coll) \<in> cbig P \<sigma>) = (\<exists>n. (\<sigma>', coll) \<in> csmall_nstep P \<sigma> n \<and> \<sigma>' \<in> endset)"
suc_inField: fixes r :: "('c \<times> 'c) set"   and B :: "'c set" assumes "wo_rel r"   and "B \<subseteq> Field r"   and "AboveS r B \<noteq> {}" shows "suc r B \<in> Field r"
weight_from_list_def: fixes l :: "(('a \<times> 'a) \<times> nat) list" shows "weight_from_list l \<equiv> foldr (\<lambda>((u, v), d) w. w({u, v} := d)) l (\<lambda>_. 0)"
monotone_on_def: fixes X :: "'c set"   and r :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and s :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'd" shows "monotone_on X r s f \<equiv> \<forall>x y. x \<in> X \<longrightarrow> y \<in> X \<longrightarrow> r x y \<longrightarrow> s (f x) (f y)"
continuous_on_divide: fixes s :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "continuous_on s f"   and "continuous_on s g"   and "\<forall>x\<in>s. g x \<noteq> (0::'b)" shows "continuous_on s (\<lambda>x. f x / g x)"
similar_mat_wit_four_block: fixes A1 :: "'a mat"   and B1 :: "'a mat"   and P1 :: "'a mat"   and Q1 :: "'a mat"   and A2 :: "'a mat"   and B2 :: "'a mat"   and P2 :: "'a mat"   and Q2 :: "'a mat"   and URA :: "'a mat"   and UR :: "'a mat"   and LLA :: "'a mat"   and LL :: "'a mat"   and n :: "nat"   and m :: "nat" assumes "similar_mat_wit A1 B1 P1 Q1"   and "similar_mat_wit A2 B2 P2 Q2"   and "URA = P1 * UR * Q2"   and "LLA = P2 * LL * Q1"   and "A1 \<in> carrier_mat n n"   and "A2 \<in> carrier_mat m m"   and "LL \<in> carrier_mat m n"   and "UR \<in> carrier_mat n m" shows "similar_mat_wit (four_block_mat A1 URA LLA A2) (four_block_mat B1 UR LL B2) (four_block_mat P1 (0\<^sub>m n m) (0\<^sub>m m n) P2) (four_block_mat Q1 (0\<^sub>m n m) (0\<^sub>m m n) Q2)"
simps_1: fixes s :: "'a \<Rightarrow> bool option" shows "trace_parallel_plan_strips s [] = [s]"
CauchyI': fixes dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool"   and X :: "nat \<Rightarrow> 'b" assumes "class.metric_space dist uniformity open"   and "\<And>e. 0 < e \<Longrightarrow> \<exists>M. \<forall>m\<ge>M. \<forall>n>m. dist (X m) (X n) < e" shows "uniform_space.Cauchy uniformity X"
jordan_nf_char_poly: fixes A :: "'b mat"   and n_as :: "(nat \<times> 'b) list" assumes "jordan_nf A n_as" shows "char_poly A = (\<Prod>(n, a)\<leftarrow>n_as. [:- a, 1::'b:] ^ n)"
\<theta>_neq_0: shows "\<theta> \<noteq> 0"
emeasure_space_1: fixes M :: "'a pmf" shows "emeasure (measure_pmf M) (space (measure_pmf M)) = 1"
P_upd: fixes dir :: "('i, 'a) Direction"   and s :: "('i, 'a) state"   and I :: "'i list" assumes "dir = Positive \<or> dir = Negative"   and "Q A (lt dir) (UBI dir) (LBI dir) (UB dir) (LB dir) (UBI_upd dir) (UI dir) (LI dir) (LE dir) (GE dir) s" shows "Q A (lt dir) (UBI dir) (LBI dir) (UB dir) (LB dir) (UBI_upd dir) (UI dir) (LI dir) (LE dir) (GE dir) (set_unsat I s)"
obtain_at_base: fixes X :: "atom set" assumes "finite X" obtains a :: "'a" where "atom a \<notin> X"
insert_subset_synth_analz: fixes X :: "msg"   and H :: "msg set" assumes "X \<in> synth (analz H)" shows "insert X H \<subseteq> synth (analz H)"
ipresIGFreshAll_termMOD: fixes Delta :: "('b, 'c, 'd, 'e, 'f) signature"   and h :: "('b, 'c, 'd, 'a, 'f) qTerm set \<Rightarrow> 'g"   and hA :: "('b, 'c, 'd, 'a, 'f) qAbs set \<Rightarrow> 'h"   and MOD :: "('i, 'j, 'd, 'k, 'l, 'a, 'g, 'h, 'm) model_scheme" assumes "FixSyn TYPE('a) Delta" shows "ipresIGFreshAll h hA (FixSyn.termMOD Delta) MOD = FixSyn.ipresFreshAll Delta h hA MOD"
zero_on_le: fixes f :: "'d \<Rightarrow> 'c"   and X :: "'d set"   and x :: "'d" shows "(zero_on f X x \<le> f x) = (x \<in> X \<longrightarrow> (0::'c) \<le> f x)"
lem_cont_e: fixes v :: "i"   and F :: "\<Pi>\<^sub>1" shows "[\<^bold>\<diamond>(\<^bold>\<exists>x. \<lparr>F,x\<^sup>P\<rparr> \<^bold>& \<^bold>\<diamond>\<^bold>\<not>\<lparr>F,x\<^sup>P\<rparr>) \<^bold>\<equiv> \<^bold>\<diamond>(\<^bold>\<exists>x. \<^bold>\<not>\<lparr>F,x\<^sup>P\<rparr> \<^bold>& \<^bold>\<diamond>\<lparr>F,x\<^sup>P\<rparr>) in v]"
inconsistentCharacterization: fixes valuation :: "Literal list" shows "inconsistent valuation = (\<exists>literal. literalTrue literal valuation \<and> literalFalse literal valuation)"
app_nonrestrict_Seq: fixes s :: "hf"   and k :: "hf"   and z :: "hf"   and X :: "hf" assumes "Seq s k"   and "z \<^bold>\<notin> X" shows "app (nonrestrict s X) z = app s z"
2: shows "step (\<gamma>\<^sub>o \<top>) (\<gamma>\<^sub>c C) \<le> \<gamma>\<^sub>c C"
hide_rules_4: fixes a :: "'b"   and B :: "'b set"   and P :: "'b process" assumes "a \<in> B" shows "(a \<rightarrow> P) \ B = P \ B"
one_idem: shows "(1::'a) + (1::'a) = (1::'a)"
coeff_power_closed: fixes A :: "'a set"   and p :: "'a mpoly"   and n :: "nat"   and m :: "nat \<Rightarrow>\<^sub>0 nat" assumes "ring_closed A"   and "\<And>m. MPoly_Type.coeff p m \<in> A" shows "MPoly_Type.coeff (p ^ n) m \<in> A"
le_diff_eq: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c"   and c :: "'c"   and b :: "'c" assumes "class.ordered_ab_group_add plus zero minus uminus less_eq less" shows "less_eq a (minus c b) = less_eq (plus a b) c"
mat_less_def: fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c"   and g :: "'a \<Rightarrow> 'b \<Rightarrow> 'c" shows "mat_less f g = (mat_less_eq f g \<and> f \<noteq> g)"
traverse_def: fixes E :: "('b \<times> 'b) fset" shows "traverse E = dds_of (\<lambda>visited A. (fImage E A |-| visited, visited |\<union>| A)) {||}"
step_9: shows "InvariantWatchesDiffer (getF state') (getWatch1 state') (getWatch2 state')"
when_apply_cong: fixes C :: "bool"   and C' :: "bool"   and s :: "'d"   and s' :: "'d"   and m :: "'d \<Rightarrow> (('e + unit) \<times> 'd) \<times> bool"   and m' :: "'d \<Rightarrow> (('e + unit) \<times> 'd) \<times> bool" assumes "C = C'"   and "s = s'"   and "C' \<Longrightarrow> m s' = m' s'" shows "whenE C m s = whenE C' m' s'"
ex_countable_basis: fixes "open" :: "'c set \<Rightarrow> bool" assumes "class.second_countable_topology open" shows "\<exists>B. countable B \<and> topological_space.topological_basis open B"
fixp_def: fixes f :: "'a \<Rightarrow> 'a" shows "a.fixp f = luba (a.iterates f)"
word_eq_unatI: fixes v :: "'b word"   and w :: "'b word" assumes "unat v = unat w" shows "v = w"
ipresWls_wlsBinp: fixes delta :: "'opSym"   and binp :: "'bindex \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set option"   and hA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> 'd"   and MOD :: "('e, 'f, 'varSort, 'sort, 'g, 'h, 'i, 'd, 'j) model_scheme" assumes "wlsBinp delta binp"   and "ipresWlsAbs hA MOD" shows "igWlsBinp MOD delta (lift hA binp)"
strip_pfp: fixes g :: "'b \<Rightarrow> 'c"   and f :: "'b \<Rightarrow> 'b"   and x0 :: "'b"   and x :: "'b" assumes "\<And>x. g (f x) = g x"   and "pfp f x0 = Some x" shows "g x = g x0"
drop_all: fixes xs :: "'a list"   and n :: "nat" assumes "length xs \<le> n" shows "drop n xs = []"
map_to_set_dom: fixes m :: "'b \<Rightarrow> 'c option" shows "dom m = fst ` map_to_set m"
sorted_list_of_set_length: fixes A :: "'b set" shows "length (sorted_list_of_set A) = card A"
upd_idemp: fixes b :: "'b"   and b' :: "'b"   and a :: "'a"   and b'' :: "'b" assumes "b \<noteq> b'" shows "upd a b'' (upd a b' (cnst b)) = upd a b'' (cnst b)"
map\<^sub>1_def: fixes V\<^sub>B :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and H\<^sub>B :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<a>\<^sub>B :: "'c \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<i>\<^sub>B :: "'c \<Rightarrow> 'c"   and src\<^sub>B :: "'c \<Rightarrow> 'c"   and trg\<^sub>B :: "'c \<Rightarrow> 'c"   and V\<^sub>C :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and H\<^sub>C :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and \<a>\<^sub>C :: "'d \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and \<i>\<^sub>C :: "'d \<Rightarrow> 'd"   and src\<^sub>C :: "'d \<Rightarrow> 'd"   and trg\<^sub>C :: "'d \<Rightarrow> 'd"   and V\<^sub>D :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and H\<^sub>D :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and \<a>\<^sub>D :: "'e \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'e"   and \<i>\<^sub>D :: "'e \<Rightarrow> 'e"   and src\<^sub>D :: "'e \<Rightarrow> 'e"   and trg\<^sub>D :: "'e \<Rightarrow> 'e"   and F :: "'d \<Rightarrow> 'e"   and \<Phi>\<^sub>F :: "'d \<times> 'd \<Rightarrow> 'e"   and G :: "'d \<Rightarrow> 'e"   and \<Phi>\<^sub>G :: "'d \<times> 'd \<Rightarrow> 'e"   and H :: "'c \<Rightarrow> 'd"   and \<Phi>\<^sub>H :: "'c \<times> 'c \<Rightarrow> 'd"   and \<tau>\<^sub>0 :: "'d \<Rightarrow> 'e"   and \<tau>\<^sub>1 :: "'d \<Rightarrow> 'e"   and f :: "'c" assumes "pseudonatural_transformation_whisker_right V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D F \<Phi>\<^sub>F G \<Phi>\<^sub>G H \<Phi>\<^sub>H \<tau>\<^sub>0 \<tau>\<^sub>1" shows "pseudonatural_transformation_whisker_right.map\<^sub>1 H \<tau>\<^sub>1 f = \<tau>\<^sub>1 (H f)"
poly_map_iteratei_axioms: shows "poly_map_iteratei \<alpha> invar list_it"
I_out_na_def: fixes s :: "'c list"   and t :: "'c list" shows "I_out_na s t \<equiv> \<lambda>(i, j, pos). (\<forall>ii<i. \<not> sublist_at s t ii) \<and> (case pos of None \<Rightarrow> j = (0::'b) | Some p \<Rightarrow> p = i \<and> sublist_at s t i)"
ord_less_eq_trans: fixes less :: "'m \<Rightarrow> 'm \<Rightarrow> bool"   and a :: "'m"   and b :: "'m"   and c :: "'m" assumes "less a b"   and "b = c" shows "less a c"
inj_on_id: fixes A :: "'c set" shows "inj_on id A"
restore_rs: fixes xs :: "nat list"   and ffp :: "nat"   and gs :: "recf list"   and f :: "recf"   and anything :: "nat list" shows "{\<lambda>nl. nl = 0 \<up> max (Suc (length xs)) (Max (insert ffp ((\<lambda>(aprog, p, n). n) ` rec_ci ` set gs))) @ rec_exec (Cn (length xs) f gs) xs # 0 \<up> length gs @ xs @ anything} mv_box (max (Suc (length xs)) (Max (insert ffp ((\<lambda>(aprog, p, n). n) ` rec_ci ` set gs)))) (length xs) {\<lambda>nl. nl = 0 \<up> length xs @ rec_exec (Cn (length xs) f gs) xs # 0 \<up> (max (Suc (length xs)) (Max (insert ffp ((\<lambda>(aprog, p, n). n) ` rec_ci ` set gs))) - length xs) @ 0 \<up> length gs @ xs @ anything}"
mono_onI: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "\<And>r s. \<lbrakk>r \<in> A; s \<in> A; r \<le> s\<rbrakk> \<Longrightarrow> f r \<le> f s" shows "mono_on f A"
choose_arc_arc: fixes x :: "'a" assumes "x \<noteq> bot" shows "arc (choose_arc x)"
flowD_capacity_IN: fixes \<Delta> :: "('a, 'b) network_scheme"   and f :: "'a \<times> 'a \<Rightarrow> ennreal"   and x :: "'a" assumes "flow \<Delta> f" shows "d_IN f x \<le> d_IN (capacity \<Delta>) x"
semiring_normalization_rules_18: fixes times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and lx :: "'d"   and rx :: "'d"   and ry :: "'d" assumes "class.comm_semiring_1 times one plus zero" shows "times lx (times rx ry) = times (times lx rx) ry"
StandardControlDependencePDG_def: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> 'c edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and Def :: "'b \<Rightarrow> 'd set"   and Use :: "'b \<Rightarrow> 'd set"   and state_val :: "'c \<Rightarrow> 'd \<Rightarrow> 'e"   and Exit :: "'b" shows "StandardControlDependencePDG sourcenode targetnode kind valid_edge Entry Def Use state_val Exit \<equiv> Postdomination sourcenode targetnode kind valid_edge Entry Exit \<and> CFGExit_wf sourcenode targetnode kind valid_edge Entry Def Use state_val Exit"
chine_simps_2: shows "C.dom \<nu>'.chine = \<nu>'.dom.apex"
tarski_absolute_space_def: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and B :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool" shows "tarski_absolute_space C B \<equiv> tarski_first5 C B \<and> tarski_absolute_space_axioms B"
distinct_1: fixes a :: "'e"   and b' :: "'f" shows "pure a \<noteq> Inr b'"
dvdI: fixes a :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and b :: "'a"   and k :: "'a" assumes "a = times b k" shows "dvd.dvd times b a"
analz_Un_implAuthSet: fixes G :: "msg set"   and H :: "msg set" assumes "Enc_keys_clean (G \<union> H)" shows "analz ({asym.implAuth A B M |A B M. M \<in> G} \<union> H) \<subseteq> synth (analz (G \<union> H)) \<union> - payload"
assms_2: shows "x <\<^sub>v (zeroes n)[i := b]"
is_parallel_solution_for_problem_def: fixes \<Pi> :: "('a, 'b) strips_problem_scheme"   and \<pi> :: "'a strips_operator list list" shows "is_parallel_solution_for_problem \<Pi> \<pi> \<equiv> \<Pi>\<^sub>G \<subseteq>\<^sub>m execute_parallel_plan (\<Pi>\<^sub>I) \<pi> \<and> list_all (list_all (\<lambda>op. ListMem op (\<Pi>\<^sub>\<O>))) \<pi>"
PiE_dflt_def: fixes A :: "'a set"   and dflt :: "'b"   and B :: "'a \<Rightarrow> 'b set" shows "PiE_dflt A dflt B = {f. \<forall>x. (x \<in> A \<longrightarrow> f x \<in> B x) \<and> (x \<notin> A \<longrightarrow> f x = dflt)}"
max_subset_3: fixes A :: "'b set"   and B :: "'b set"   and P :: "'b set \<Rightarrow> bool" assumes "finite A"   and "B \<subseteq> A"   and "P B" shows "card B \<le> card (max_subset A P)"
preserves_ide: fixes a :: "'d" assumes "D.ide a" shows "D.ide (D.R a)"
discI_2: fixes generat :: "('g, 'h, 'i) generat"   and x21 :: "'h"   and x22 :: "'i" assumes "generat = IO x21 x22" shows "\<not> is_Pure generat"
distinct_20: fixes x5 :: "char list" shows "Class x5 \<noteq> NT"
doms_append_1: fixes \<Gamma> :: "binding list"   and \<Delta> :: "binding list" shows "ty_dom (\<Gamma> @ \<Delta>) = ty_dom \<Gamma> \<union> ty_dom \<Delta>"
d2_5: fixes K1 :: "'a set"   and K2 :: "'a set" assumes "K1 \<in> p2"   and "K2 \<in> p2"   and "K1 \<noteq> K2" shows "interior K1 \<inter> interior K2 = {}"
regular_closed_p: fixes x :: "'c \<times> 'c \<Rightarrow> 'd" shows "matrix_p_algebra.regular (\<ominus> x)"
Poly_cCons_eq_pCons_Poly: fixes a :: "'b"   and p :: "'b list" shows "Poly (cCons a p) = pCons a (Poly p)"
c: shows "flip y zs ! y = (\<not> zs ! y)"
T_def: shows "T = integral {real_of_int a..real_of_int b} (\<lambda>t. (pbernpoly (Suc n) t / real (Suc n)) *\<^sub>R f' t)"
tends: shows "AE x in lebesgue_on {a..b}. (\<lambda>k. (if x \<le> a + 1 / real (Suc k) then real (Suc k) * d * (a + 1 / real (Suc k) - x) else 0)\<^sup>2) \<longlonglongrightarrow> (if x = a then d\<^sup>2 else 0)"
game\<^sub>2_def: fixes x :: "nat"   and y :: "nat" shows "game\<^sub>2 x y = coin_spmf \<bind> (\<lambda>b. exec_gpv oracle' (\<A>\<^sub>1 (\<^bold>g [^] x)) ({}, empty) \<bind> (\<lambda>(((m\<^sub>1, m\<^sub>2), \<sigma>), s, s_h). TRY assert_spmf (valid_plains m\<^sub>1 m\<^sub>2) \<bind> (\<lambda>_. spmf_of_set (nlists UNIV len_plain) \<bind> (\<lambda>h. let cipher = (\<^bold>g [^] y, h [\<oplus>] (if b then m\<^sub>1 else m\<^sub>2)) in exec_gpv oracle' (\<A>\<^sub>2 cipher \<sigma>) (s, s_h) \<bind> (\<lambda>(b', s', s_h'). return_spmf (b' = b, \<^bold>g [^] (x * y) \<in> s')))) ELSE (coin_spmf \<bind> (\<lambda>b. return_spmf (b, \<^bold>g [^] (x * y) \<in> s)))))"
trivial_IrrFinGroupRepI: fixes smult :: "('a, 'g) aezfun \<Rightarrow> 'b \<Rightarrow> 'b" assumes "of_nat (card G) \<noteq> (0::'a)"   and "\<forall>a\<in>group_ring. smult a (0::'b) = (0::'b)" shows "IrrFinGroupRepresentation G smult 0"
mono_minus_ivl: fixes iv1 :: "ivl"   and iv2 :: "ivl" assumes "iv1 \<le> iv2" shows "- iv1 \<le> - iv2"
isolated_verts_dv: shows "H.isolated_verts = isolated_verts - {u}"