hcju/isabelleps huggingface dataset:
assms: fixes x :: "'a" shows "g\<cdot>(f\<cdot>x) = x"
strict_mono_less_eq: fixes less_eq :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and less :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and f :: "'f \<Rightarrow> 'g"   and x :: "'f"   and y :: "'f" assumes "class.linorder less_eq less"   and "order.strict_mono less f" shows "(f x \<le> f y) = less_eq x y"
is_sig_red_cong': fixes sing_reg :: "'t \<Rightarrow> 't \<Rightarrow> bool"   and top_tail :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and F :: "('t \<Rightarrow>\<^sub>0 'b) set"   and p :: "'t \<Rightarrow>\<^sub>0 'b"   and q :: "'t \<Rightarrow>\<^sub>0 'b" assumes "is_sig_red sing_reg top_tail F p"   and "lt p = lt q"   and "rep_list p = rep_list q" shows "is_sig_red sing_reg top_tail F q"
subfield_Units: fixes K :: "'c set"   and R :: "('c, 'd) ring_scheme" assumes "subfield K R" shows "Units (R\<lparr>carrier := K\<rparr>) = K - {\<zero>\<^bsub>R\<^esub>}"
unch_defs: fixes v :: "'a \<Rightarrow> 'b"   and w :: "nat \<Rightarrow> 'a" shows "(Unchanged v) w = (v (second w) = v (first w))"
det_bind: fixes m :: "'c dres"   and M :: "'c nres"   and f :: "'c \<Rightarrow> 'd dres"   and F :: "'c \<Rightarrow> 'd nres" assumes "nres_of m \<le> M"   and "\<And>x. nres_of (f x) \<le> F x" shows "nres_of (m \<bind> f) \<le> M \<bind> F"
split_relation_length: fixes xs :: "'a list"   and ls :: "'a list"   and rs :: "'a list" shows "split_relation xs (ls, rs) (length xs) = (ls = xs \<and> rs = [])"
qle_trans: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<preceq> y"   and "y \<preceq> z" shows "x \<preceq> z"
delete_edge_set_nodes: fixes v :: "'b"   and v' :: "'b"   and G :: "'b list_graph" shows "set (nodesL (FiniteListGraph.delete_edge v v' G)) = set (nodesL G)"
SeqSum: fixes x :: "ZF"   and f :: "nat \<Rightarrow> ZF" shows "Elem x (SeqSum f) = (\<exists>n. Elem x (f n))"
func_pre: fixes f :: "nat \<Rightarrow> 'c"   and n :: "nat"   and A :: "'c set" assumes "f \<in> {j. j \<le> Suc n} \<rightarrow> A" shows "f \<in> {j. j \<le> n} \<rightarrow> A"
simps_5: fixes M :: "('fun, 'var) Term.term set"   and X :: "'var list"   and F :: "(('fun, 'var) Term.term \<times> ('fun, 'var) Term.term) list"   and S :: "('fun, 'var) strand_step list" shows "\<lbrakk>M; \<forall>X\<langle>\<or>\<noteq>: F\<rangle>\<^sub>s\<^sub>t # S\<rbrakk>\<^sub>d' = (\<lambda>\<I>. ineq_model \<I> X F \<and> \<lbrakk>M; S\<rbrakk>\<^sub>d' \<I>)"
for_in_RI: fixes x :: "'h"   and R :: "('h \<times> 'i) set" assumes "x \<in> Domain R" shows "(x, fun_of_rel R x) \<in> R"
non_terminating_implies_F: fixes P :: "'b process" assumes "non_terminating P" shows "CHAOS UNIV \<sqsubseteq>\<^sub>F P"
type_aexpr: fixes \<Gamma> :: "'Var \<Rightarrow> 'BExp set option"   and e :: "'AExp" shows "\<Gamma> \<turnstile>\<^sub>a e \<in> \<Union> (to_total \<Gamma> ` aexp_vars e)"
CatIdDomCod_2: fixes C :: "('d, 'e, 'f) Category_scheme"   and X :: "'d" assumes "Category C"   and "X \<in> obj\<^bsub>C\<^esub>" shows "cod\<^bsub>C\<^esub> (id\<^bsub>C\<^esub> X) = X"
word_minus_one_le_leq: fixes x :: "'a word"   and y :: "'a word" assumes "x - 1 < y" shows "x \<le> y"
bind_SB_def: fixes f :: "'c \<Rightarrow> ('d \<times> 'c) set"   and g :: "'d \<Rightarrow> 'c \<Rightarrow> ('b \<times> 'c) set"   and \<sigma> :: "'c" shows "bind\<^sub>S\<^sub>B f g \<sigma> = (\<Union>(out, \<sigma>)\<in>f \<sigma>. g out \<sigma>)"
homeomorphic_imp_continuous_map: fixes X :: "'c topology"   and Y :: "'d topology"   and f :: "'c \<Rightarrow> 'd" assumes "homeomorphic_map X Y f" shows "continuous_map X Y f"
rel_cases: fixes R :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'b literal"   and b :: "'c literal" assumes "rel_literal R a b" obtains x :: "'b equation" and y :: "'c equation" where "a = Pos x" and "b = Pos y" and "rel_equation R x y"   | xa :: "'b equation" and ya :: "'c equation" where "a = Neg xa" and "b = Neg ya" and "rel_equation R xa ya"
spm_mult_le_dual_prts: fixes A1 :: "(nat \<times> (nat \<times> 'a) list) list"   and A2 :: "(nat \<times> (nat \<times> 'a) list) list"   and c1 :: "(nat \<times> (nat \<times> 'a) list) list"   and c2 :: "(nat \<times> (nat \<times> 'a) list) list"   and y :: "(nat \<times> (nat \<times> 'a) list) list"   and r1 :: "(nat \<times> (nat \<times> 'a) list) list"   and r2 :: "(nat \<times> (nat \<times> 'a) list) list"   and b :: "(nat \<times> (nat \<times> 'a) list) list"   and A :: "'a matrix"   and c :: "'a matrix"   and x :: "'a matrix" assumes "sorted_sparse_matrix A1"   and "sorted_sparse_matrix A2"   and "sorted_sparse_matrix c1"   and "sorted_sparse_matrix c2"   and "sorted_sparse_matrix y"   and "sorted_sparse_matrix r1"   and "sorted_sparse_matrix r2"   and "sorted_spvec b"   and "le_spmat [] y"   and "sparse_row_matrix A1 \<le> A"   and "A \<le> sparse_row_matrix A2"   and "sparse_row_matrix c1 \<le> c"   and "c \<le> sparse_row_matrix c2"   and "sparse_row_matrix r1 \<le> x"   and "x \<le> sparse_row_matrix r2"   and "A * x \<le> sparse_row_matrix b" shows "c * x \<le> sparse_row_matrix (add_spmat (mult_spmat y b) (let s1 = diff_spmat c1 (mult_spmat y A2); s2 = diff_spmat c2 (mult_spmat y A1) in add_spmat (mult_spmat (pprt_spmat s2) (pprt_spmat r2)) (add_spmat (mult_spmat (pprt_spmat s1) (nprt_spmat r2)) (add_spmat (mult_spmat (nprt_spmat s2) (pprt_spmat r1)) (mult_spmat (nprt_spmat s1) (nprt_spmat r1))))))"
keys_eq_supp: fixes s :: "'a \<Rightarrow>\<^sub>0 'b" shows "keys s = supp_fun (lookup s)"
np_spec_def: fixes np :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> nat \<times> 'g \<Rightarrow> (bool \<times> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list" shows "np_spec np = (\<forall>gs bs hs data. snd ` set (np gs bs hs data) \<subseteq> set hs \<times> (set gs \<union> set bs \<union> set hs) \<and> set hs \<times> (set gs \<union> set bs) \<subseteq> snd ` set (np gs bs hs data) \<and> (\<forall>a b. a \<in> set hs \<longrightarrow> b \<in> set hs \<longrightarrow> a \<noteq> b \<longrightarrow> (a, b) \<in>\<^sub>p snd ` set (np gs bs hs data)) \<and> (\<forall>p q. (True, p, q) \<in> set (np gs bs hs data) \<longrightarrow> q \<in> set gs \<union> set bs))"
stabiliser_A_def: shows "stabiliser_A = {id, rotate_A, rotate_A \<circ> rotate_A}"
contour_integrable_lmul: fixes f :: "complex \<Rightarrow> complex"   and g :: "real \<Rightarrow> complex"   and c :: "complex" assumes "f contour_integrable_on g" shows "(\<lambda>x. c * f x) contour_integrable_on g"
length_chop_part_le: fixes ys :: "'a list"   and n :: "nat"   and xs :: "'a list" assumes "ys \<in> set (chop n xs)" shows "length ys \<le> n"
lmeasurable_closure: fixes S :: "'c set" assumes "bounded S" shows "closure S \<in> lmeasurable"
that: fixes point :: "((variable \<times> String.literal set) \<times> typ) list" assumes "assoc_match x y subs = Some point"   and "fold (\<lambda>(T, U) subs. Option.bind subs (assoc_match T U)) (zip xs ys) (Some point) = Some subs'" shows "thesis"
st: shows "s \<in> \<S>\<^sub>a \<and> t \<in> \<S>\<^sub>a \<and> t \<le>\<^sub>\<S> \<lambda> (s, B) \<and> t \<circ> C \<noteq> C"
prime_factors_dvd: 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 x :: "'b" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize"   and "x \<noteq> zero" shows "set_mset (factorial_semiring.prime_factorization zero times one normalize x) = {p. normalization_semidom.prime zero times one normalize p \<and> dvd.dvd times p x}"
transfer: shows "rel_fun pcr_word (rel_fun pcr_word pcr_word) (+) (+)"
pure_monoid_add_def: fixes zero :: "'b" shows "zero.pure_monoid_add zero = Monoid_ap zero"
1: fixes x :: "V"   and y :: "V" assumes "x \<in> elts \<alpha>"   and "y \<in> elts \<alpha>"   and "x < y" shows "(u + x, u + y) \<in> VWF"
cover_of_select_index: fixes X :: "'c set"   and U :: "'c set"   and index :: "real set"   and cover :: "real \<Rightarrow> 'c set"   and x :: "'c" assumes "cover_of_subset X U index cover"   and "x \<in> U" shows "x \<in> cover (cover_of_subset.select_index index cover x)"
transform_disjoint: fixes temp :: "char list"   and c :: "cmd"   and V :: "char list"   and e :: "expr" assumes "subcmd (transform_aux temp c) (V:=e)"   and "newname c temp" shows "V \<notin> rhs_aux e"
rep_eq: fixes x :: "transaction" shows "the_Transaction\<^sub>m (embed_transaction x) = Unblinded (map_prod (map_prod Unblinded Unblinded) (map embed_view) (the_Transaction x))"
Order_ext_def: fixes rel :: "('c \<times> 'c) set"   and more :: "'d" shows "\<lparr>rel = rel, \<dots> = more\<rparr> \<equiv> Record.iso_tuple_cons Order_ext_Tuple_Iso rel more"
fold_congs_8: fixes r :: "('b, 'c) sparc_state_scheme"   and r' :: "('b, 'c) sparc_state_scheme"   and v' :: "sparc_state_var"   and f :: "sparc_state_var \<Rightarrow> sparc_state_var"   and f' :: "sparc_state_var \<Rightarrow> sparc_state_var" assumes "r = r'"   and "state_var r' = v'"   and "\<And>v. v' = v \<Longrightarrow> f v = f' v" shows "state_var_update f r = state_var_update f' r'"
delay_bisimulation_diverge_locale: fixes t :: "'t" shows "delay_bisimulation_diverge (r1 t) (r2 t) (bisim t) ta_bisim_bisim_syntax \<tau>move1 \<tau>move2"
zip_Cons_Cons: fixes x :: "'b"   and xs :: "'b list"   and y :: "'c"   and ys :: "'c list" shows "x # xs || y # ys = (x, y) # (xs || ys)"
assms_3: fixes pred :: "'b" shows "P (Pred\<^sub>\<alpha> pred)"
rel_sel: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "'c sctn"   and b :: "'d sctn" shows "rel_sctn R a b = (R (normal a) (normal b) \<and> pstn a = pstn b)"
intro: fixes U :: "'a set"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and one :: "'a" assumes "comm_semiring_ow U plus times"   and "cancel_comm_monoid_add_ow U plus minus zero"   and "zero_neq_one_ow U one zero"   and "comm_monoid_mult_ow U times one"   and "comm_semiring_1_cancel_ow_axioms U minus times" shows "comm_semiring_1_cancel_ow U plus minus zero times one"
assms_2: shows "s2' \<equiv> interpret_floatarith s2 vs"
mersenne_mod2_less: fixes n :: "nat"   and k :: "int" assumes "2 < n"   and "k \<le> 2 ^ (2 * n + 2)" shows "mersenne_mod2 k n < 2 ^ n + 5"
top_in_filter: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and top :: "'b"   and F :: "'b set" assumes "class.order_top less_eq less top"   and "ord.filter less_eq F" shows "top \<in> F"
sorted_inorder_list_subtrees: fixes ts :: "('b btree \<times> 'b) list" assumes "sorted_less (inorder_list ts)" shows "\<forall>sub\<in>set (subtrees ts). sorted_less (inorder sub)"
s_mul_ext_local_mono: fixes xs :: "'a multiset"   and ys :: "'a multiset"   and ns :: "('a \<times> 'a) set"   and ns' :: "('a \<times> 'a) set"   and s :: "('a \<times> 'a) set"   and s' :: "('a \<times> 'a) set" assumes "set_mset xs \<times> set_mset ys \<inter> ns \<subseteq> ns'"   and "set_mset xs \<times> set_mset ys \<inter> s \<subseteq> s'"   and "(xs, ys) \<in> s_mul_ext ns s" shows "(xs, ys) \<in> s_mul_ext ns' s'"
ga_automaton_simps_3: shows "ga.V0 = {pid_init_gc}"
oFix_eq_self: fixes F :: "ordinal \<Rightarrow> ordinal"   and a :: "ordinal" assumes "F a = a" shows "oFix F a = a"
ih: shows "(prod_list (map ((!) sign) xa) = 1) = (0 < poly (prod_list (retrieve_polys qs xa)) x)"
inter_guards_Guard: fixes f :: "'f"   and g1 :: "'d set"   and bdy1 :: "('d, 'e, 'f) com"   and c2 :: "('d, 'e, 'f) com"   and c :: "('d, 'e, 'f) com" shows "((Guard f g1 bdy1 \<inter>\<^sub>g c2) = Some c) = (\<exists>g2 bdy2 bdy. c2 = Guard f g2 bdy2 \<and> (bdy1 \<inter>\<^sub>g bdy2) = Some bdy \<and> c = Guard f (g1 \<inter> g2) bdy)"
fun_eq_iff: fixes f :: "'a \<Rightarrow> 'c"   and g :: "'a \<Rightarrow> 'c" shows "(f = g) = (\<forall>x. f x = g x)"
take_bit_and: fixes n :: "nat"   and a :: "'a"   and b :: "'a" shows "take_bit n (a AND b) = take_bit n a AND take_bit n b"
trm_rep_is_lower: fixes trm_ord :: "('b trm \<times> 'b trm) set"   and sel :: "'b literal set \<Rightarrow> 'b literal set"   and pos_ord :: "'b eclause \<Rightarrow> 'b trm \<Rightarrow> (indices list \<times> indices list) set"   and vars :: "'b set"   and filter_trms :: "'b literal set \<Rightarrow> 'b trm set \<Rightarrow> 'b trm set"   and t :: "'b trm"   and S :: "'b eclause set" assumes "basic_superposition trm_ord sel pos_ord vars filter_trms" shows "t \<noteq> basic_superposition.trm_rep trm_ord sel t S \<longrightarrow> (basic_superposition.trm_rep trm_ord sel t S, t) \<in> trm_ord"
symetric: fixes P :: "pi"   and Q :: "pi" assumes "P \<simeq>\<^sup>s Q" shows "Q \<simeq>\<^sup>s P"
smult_conv_map_poly: fixes c :: "'a"   and p :: "'a poly" shows "Polynomial.smult c p = map_poly ((*) c) p"
induct: fixes x1 :: "'effect fs_set"   and x2 :: "('b, 'pred, 'act, 'effect) formula"   and P :: "'effect fs_set \<Rightarrow> ('b, 'pred, 'act, 'effect) formula \<Rightarrow> bool" assumes "x2 \<in> \<A>[x1]"   and "\<And>xset F. \<lbrakk>finite (supp xset); \<And>x. x \<in> set_bset xset \<Longrightarrow> x \<in> \<A>[F]; \<And>x. x \<in> set_bset xset \<Longrightarrow> P F x\<rbrakk> \<Longrightarrow> P F (Conj xset)"   and "\<And>F x. \<lbrakk>x \<in> \<A>[F]; P F x\<rbrakk> \<Longrightarrow> P F (FL_Formula.Not x)"   and "\<And>f F \<phi>. f \<in>\<^sub>f\<^sub>s F \<Longrightarrow> P F (Pred f \<phi>)"   and "\<And>f F \<alpha> x. \<lbrakk>f \<in>\<^sub>f\<^sub>s F; bn \<alpha> \<sharp>* (F, f); x \<in> \<A>[L (\<alpha>, F, f)]; P (L (\<alpha>, F, f)) x\<rbrakk> \<Longrightarrow> P F (Act f \<alpha> x)" shows "P x1 x2"
action_orderE: fixes E :: "('a \<times> ('b, 'a) obs_event action) llist"   and a :: "nat"   and a' :: "nat" assumes "E \<turnstile> a \<le>a a'" obtains "a \<in> actions E" and "a' \<in> actions E" and "is_new_action (action_obs E a)" and "is_new_action (action_obs E a') \<longrightarrow> a \<le> a'"   | "a \<in> actions E" and "a' \<in> actions E" and "\<not> is_new_action (action_obs E a)" and "\<not> is_new_action (action_obs E a')" and "a \<le> a'"
sorted_sublist_wrt_le: fixes hi :: "nat"   and lo :: "nat"   and xs :: "'c list"   and R :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "hi \<le> lo"   and "hi < length xs" shows "sorted_sublist_wrt R xs lo hi"
circlepath_add_half: fixes z :: "complex"   and r :: "real"   and x :: "real" shows "circlepath z r (x + 1 / 2) = circlepath z r (x - 1 / 2)"
preserves_quasi_inverses: fixes f :: "'c"   and g :: "'c" assumes "C.quasi_inverses f g" shows "D.quasi_inverses (F f) (F g)"
abs_add_abs: fixes abs :: "'a \<Rightarrow> '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 a :: "'a"   and b :: "'a" assumes "class.ordered_ab_group_add_abs abs plus zero minus uminus less_eq less" shows "abs (plus (abs a) (abs b)) = plus (abs a) (abs b)"
Compositionality_axioms: shows "Compositionality ES1 ES2 \<V> \<V>1 \<V>2"
jordan_matrix_carrier: fixes n_as :: "(nat \<times> 'b) list" shows "jordan_matrix n_as \<in> carrier_mat (sum_list (map fst n_as)) (sum_list (map fst n_as))"
map_add_dom_eq: fixes m :: "'c \<Rightarrow> 'd option"   and m' :: "'c \<Rightarrow> 'd option" assumes "dom m = dom m'" shows "m ++ m' = m'"
set_of_appr_takeD: fixes xs :: "real list"   and n :: "nat"   and XS :: "'b list" assumes "xs \<in> set_of_appr (take n XS)" shows "xs \<in> take n ` set_of_appr XS"
vars_clause_def: fixes C :: "('b, 'a) Term.term literal multiset" shows "vars_clause C = \<Union> (set_mset (image_mset vars_lit C))"
ipurge_tr_aux_insert: fixes I :: "('c \<times> 'c) set"   and D :: "'b \<Rightarrow> 'c"   and v :: "'c"   and U :: "'c set"   and xs :: "'b list" shows "ipurge_tr_aux I D (insert v U) xs = ipurge_tr_aux I D U (ipurge_tr I D v xs)"
HFun_Sigma_subst: fixes r :: "tm"   and i :: "name"   and t :: "tm" shows "(HFun_Sigma r)(i::=t) = HFun_Sigma (subst i t r)"
subdiv_distinct_verts0_2: shows "v \<noteq> w"
FS_FMT_1: shows "ee = LOC (LCall p # w)"
ODE_bound_effect: fixes s :: "real"   and t :: "real"   and sol :: "real \<Rightarrow> (real, 'a) vec"   and I :: "('b, 'c, 'a) interp"   and ODE :: "('b, 'a) ODE"   and X :: "(real, 'a) vec set"   and b :: "(real, 'a) vec" assumes "s \<in> {0..t}"   and "(sol solves_ode (\<lambda>_. ODE_sem I ODE)) {0..t} X" shows "Vagree (sol 0, b) (sol s, b) (- BVO ODE)"
S2_def: fixes rmhist :: "state \<Rightarrow> PrIds \<Rightarrow> histState"   and p :: "PrIds" shows "S2 rmhist p = S rmhist True False False clkA rpcA histA histA p"
card_of_empty1: fixes r :: "('a \<times> 'a) set" assumes "Well_order r \<or> Card_order r" shows "|{}| \<le>o r"
fg_init_config_def: fixes A :: "('f, 'g, 'h, 'i) CHOAlgorithm_scheme"   and config :: "('g, 'f, 'j) config"   and coord :: "'f \<Rightarrow> 'f" shows "fg_init_config A config coord \<equiv> round config = (\<lambda>p. 0) \<and> (\<forall>p. CinitState A p (state config p) (coord p)) \<and> sent config = (\<lambda>p. {}) \<and> rcvd config = (\<lambda>p. Map.empty) \<and> network config = {}"
rep_eq: fixes x :: "'b pmf" shows "set_pmf x = {x. measure_pmf.prob x {x} \<noteq> 0}"
pw_Inf_nofail: fixes C :: "'b nres set" shows "nofail (Inf C) = (\<exists>x\<in>C. nofail x)"
sup_eq_neg_inf: fixes a :: "'a"   and b :: "'a" shows "sup a b = - inf (- a) (- b)"
thetaZOCh_ZObis: fixes ch :: "'choice"   and c1 :: "('test, 'atom, 'choice) cmd"   and d1 :: "('test, 'atom, 'choice) cmd"   and c2 :: "('test, 'atom, 'choice) cmd"   and d2 :: "('test, 'atom, 'choice) cmd" assumes "compatCh ch"   and "c1 \<approx>01 d1"   and "c2 \<approx>01 d2" shows "thetaZOCh ch c1 c2 d1 d2 \<subseteq> ZObis"
HSem_bot_ind: fixes P :: "(Vars.var \<Rightarrow> C \<rightarrow> CValue) \<Rightarrow> bool"   and \<Gamma> :: "(Vars.var \<times> exp) list" assumes "adm P"   and "P \<bottom>"   and "\<And>\<rho>'. P \<rho>' \<Longrightarrow> P (\<^bold>\<N>\<lbrakk> \<Gamma> \<^bold>\<rbrakk>\<^bsub>\<rho>'\<^esub>)" shows "P (\<N>\<lbrace>\<Gamma>\<rbrace>)"
member_pred_def: fixes eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and P :: "'b Predicate.pred"   and x :: "'b" shows "member_pred eq P x = (\<exists>y. eq x y \<and> pred.eval P y)"
mod_minus_eq: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and euclidean_size :: "'c \<Rightarrow> nat"   and uminus :: "'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c" assumes "class.euclidean_ring_cancel divide plus minus zero times one modulo euclidean_size uminus" shows "modulo (uminus (modulo a b)) b = modulo (uminus a) b"
coeffs_eq_iff: fixes p :: "'a poly"   and q :: "'a poly" shows "(p = q) = (coeffs p = coeffs q)"
ring_record_simps_11: fixes carrier :: "'d set"   and mult :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and zero :: "'d"   and add :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and more :: "'e" shows "\<zero>\<^bsub>\<lparr>carrier = carrier, monoid.mult = mult, one = one, zero = zero, add = add, \<dots> = more\<rparr>\<^esub> = zero"
div_diff: fixes 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 divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and c :: "'a"   and a :: "'a"   and b :: "'a" assumes "class.idom_modulo times one plus zero minus uminus divide modulo"   and "dvd.dvd times c a"   and "dvd.dvd times c b" shows "divide (minus a b) c = minus (divide a c) (divide b c)"
circle_cube_boundary_explicit: fixes i :: "real \<times> real"   and j :: "real \<times> real"   and d :: "real" assumes "circle i j d" shows "boundary (circle.circle_cube d) = {circle.circle_left_edge d, circle.circle_right_edge d, circle.circle_bot_edge d, circle.circle_top_edge d}"
i_Exec_Stream_Pre_Post1: fixes c_n :: "'a"   and trans_fun :: "'b \<Rightarrow> 'a \<Rightarrow> 'a"   and input :: "nat \<Rightarrow> 'b"   and c :: "'a"   and n :: "nat"   and x_n :: "'b"   and P1 :: "'b \<Rightarrow> bool"   and P2 :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool" assumes "c_n = (i_Exec_Comp_Stream trans_fun input c)\<^bsup>\<leftarrow> c\<^esup> n"   and "x_n = input n" shows "(P1 x_n \<and> P2 c_n \<longrightarrow> Q (i_Exec_Comp_Stream trans_fun input c n)) = (P1 x_n \<and> P2 c_n \<longrightarrow> Q (trans_fun x_n c_n))"
stable_pair_on_Cd_match: fixes ds :: "'b set"   and XD_XH :: "'a set \<times> 'a set"   and d :: "'b" assumes "stable_pair_on ds XD_XH"   and "d \<in> ds" shows "Cd d (match XD_XH) = Cd d (fst XD_XH)"
iadd_SUP_le_iff: fixes n :: "enat"   and f :: "'b \<Rightarrow> enat"   and A :: "'b set"   and y :: "enat" shows "(n + \<Squnion> (f ` A) \<le> y) = (if A = {} then n \<le> y else \<forall>x\<in>A. n + f x \<le> y)"
F'_f: fixes s :: "complex" assumes "1 < Re s" shows "eval_fds F' s = f' s"
filterS_defined: fixes a :: "('b, 'c) Stream"   and p :: "'b \<rightarrow> tr" assumes "a \<noteq> \<bottom>" shows "filterS\<cdot>p\<cdot>a \<noteq> \<bottom>"
toplevel_summands_flatten_ACI_norm_image_Union: fixes r :: "'c rexp"   and s :: "'c rexp" shows "toplevel_summands (PLUS (sorted_list_of_set (ACI_norm ` toplevel_summands r \<union> ACI_norm ` toplevel_summands s))) = ACI_norm ` toplevel_summands r \<union> ACI_norm ` toplevel_summands s"
triple_valid_def: fixes G :: "prog"   and n :: "nat"   and t :: "'b triple" shows "G\<Turnstile>n:t = (case t of {P} t\<succ> {Q} \<Rightarrow> \<forall>Y s Z. P Y s Z \<longrightarrow> type_ok G t s \<longrightarrow> (\<forall>Y' s'. G\<turnstile>s \<midarrow>t\<succ>\<midarrow>n\<rightarrow> (Y', s') \<longrightarrow> Q Y' s' Z))"
WT_callee: fixes s :: "'s1" assumes "I1 s" shows "\<I> \<turnstile>c oracle1 s \<surd>"
HC_intros_8: fixes F :: "'b formula"   and G :: "'b formula" shows "AX10 \<turnstile>\<^sub>H ((F::'b formula) \<^bold>\<and> (G::'b formula)) \<^bold>\<rightarrow> F"
eq_on_def: fixes A :: "'a set" shows "eq_on A = (\<lambda>x y. x \<in> A \<and> x = y)"
avar_state_no_jump: fixes G :: "prog"   and i :: "val"   and a :: "val"   and s :: "state"   and j :: "jump" assumes "fst (snd (avar G i a s)) = Some (Jump j)" shows "fst s = Some (Jump j)"
terminates_Skip': fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'a, 'c) com option"   and s :: "('b, 'c) xstate" shows "\<Gamma>\<turnstile>Skip \<down> s"
first: fixes Mkt :: "('c, 'd) discrete_market"   and v :: "real"   and pf :: "'d \<Rightarrow> nat \<Rightarrow> 'c \<Rightarrow> real"   and asset :: "'d" shows "remaining_qty Mkt v pf asset (Suc 0) = (\<lambda>w. (v - val_process Mkt pf 0 w) / prices Mkt asset 0 w)"
homeomorphic_space_contractibility: fixes X :: "'a topology"   and Y :: "'b topology" assumes "X homeomorphic_space Y" shows "contractible_space X = contractible_space Y"
residue_cong: fixes f :: "complex \<Rightarrow> complex"   and g :: "complex \<Rightarrow> complex"   and z :: "complex"   and z' :: "complex" assumes "\<forall>\<^sub>F z in at z. f z = g z"   and "z = z'" shows "residue f z = residue g z'"
simps_2: fixes x :: "'a"   and uu :: "'a ppath" shows "pp_first (ppath.Cons x uu) = x"
bdia_fbox_galois: fixes R :: "('f \<times> 'd) set"   and f :: "'f \<Rightarrow> 'e"   and g :: "'d \<Rightarrow> 'e" shows "(\<forall>x.  \<langle>R| f x \<le> g x) = (\<forall>x. f x \<le> |R] g x)"