hcju/isabelleps huggingface dataset:
dr_Seq: fixes \<phi> :: "'c \<Rightarrow> 'd"   and P :: "'c \<Rightarrow> bool"   and A :: "bool \<Rightarrow> ('d \<Rightarrow> real) \<Rightarrow> 'd \<Rightarrow> real"   and B :: "bool \<Rightarrow> ('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real"   and Q :: "'c \<Rightarrow> bool"   and C :: "bool \<Rightarrow> ('d \<Rightarrow> real) \<Rightarrow> 'd \<Rightarrow> real"   and D :: "bool \<Rightarrow> ('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real" assumes "drefines \<phi> P A B"   and "drefines \<phi> Q C D"   and "\<guillemotleft> P \<guillemotright> \<tturnstile> wp B \<guillemotleft> Q \<guillemotright>"   and "well_def B"   and "well_def C"   and "well_def D" shows "drefines \<phi> P (A ;; C) (B ;; D)"
assms_1: shows "run (Heap_Monad.heap f) \<sigma> \<sigma>' res"
P3: shows "Coplanar X Y A P \<and> Coplanar X Y A Q \<and> Coplanar X Y B P \<and> Coplanar X Y B Q"
simps_15: fixes bs :: "real list"   and p :: "fm" shows "Ifm bs (A p) = (\<forall>x. Ifm (x # bs) p)"
jointly_exhaustive: fixes meets :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and \<I> :: "'b \<Rightarrow> bool"   and e :: "('b \<times> 'b) set"   and m :: "('b \<times> 'b) set"   and b :: "('b \<times> 'b) set"   and ov :: "('b \<times> 'b) set"   and d :: "('b \<times> 'b) set"   and s :: "('b \<times> 'b) set"   and f :: "('b \<times> 'b) set"   and p :: "'b"   and q :: "'b" assumes "class.arelations meets \<I> e m b ov d s f"   and "\<I> p"   and "\<I> q" shows "(p, q) \<in> b \<or> (p, q) \<in> m \<or> (p, q) \<in> ov \<or> (p, q) \<in> s \<or> (p, q) \<in> d \<or> (p, q) \<in> f\<inverse> \<or> (p, q) \<in> e \<or> (p, q) \<in> f \<or> (p, q) \<in> s\<inverse> \<or> (p, q) \<in> d\<inverse> \<or> (p, q) \<in> ov\<inverse> \<or> (p, q) \<in> m\<inverse> \<or> (p, q) \<in> b\<inverse>"
delete_store_commute: fixes i :: "'c"   and "is" :: "'c set"   and e1 :: "'d"   and e2 :: "'d"   and mo :: "'c" assumes "i \<notin> is" shows "\<langle>Delete is e1\<rangle> \<rhd> \<langle>Store e2 mo i\<rangle> = \<langle>Store e2 mo i\<rangle> \<rhd> \<langle>Delete is e1\<rangle>"
rel_spmf_mono_strong: fixes A :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and f :: "'d spmf"   and g :: "'e spmf"   and B :: "'d \<Rightarrow> 'e \<Rightarrow> bool" assumes "rel_spmf A f g"   and "\<And>x y. \<lbrakk>A x y; x \<in> set_spmf f; y \<in> set_spmf g\<rbrakk> \<Longrightarrow> B x y" shows "rel_spmf B f g"
deltafun_apply_eq: fixes b :: "'a"   and a :: "'b" shows "(b \<delta> a) a = b"
closer_to_chamber2_not_in_fold_im_chamber_system: fixes F :: "'a set" assumes "order.greater_eq (chamber_distance C F) (chamber_distance D F)" shows "F \<notin> ChamberComplex.chamber_system (fold_A \<turnstile> A)"
dom_dom: fixes f :: "'c" assumes "C.arr f" shows "C.dom (C.dom f) = C.dom f"
swaps_sort_sws: fixes k :: "'b \<Rightarrow> 'c"   and xs :: "'b list" shows "swaps (sort_sws k xs) xs = sort_key k xs"
sc_mem: fixes M :: "('e, 'f, 'g) Module_scheme"   and R :: "('f, 'h) Ring_scheme"   and a :: "'f"   and m :: "'e" assumes "R module M"   and "a \<in> carrier R"   and "m \<in> carrier M" shows "a \<cdot>\<^sub>s\<^bsub>M\<^esub> m \<in> carrier M"
abs_conv_abscissa_ln: fixes l :: "'a"   and f :: "'a fds" shows "abs_conv_abscissa (fds_ln l f) = abs_conv_abscissa (fds_deriv f / f)"
get_tag_name_pure: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and get_tag_name :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, char list) prog"   and get_tag_name_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and element_ptr :: "(_) element_ptr"   and h :: "(_) heap" assumes "l_get_tag_name type_wf get_tag_name get_tag_name_locs" shows "pure (get_tag_name element_ptr) h"
reflp_rtranclp: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool" shows "reflp R\<^sup>*\<^sup>*"
upd: shows "upd = snd (fst (avar G i a s'))"
nth_map: fixes n :: "nat"   and xs :: "'e list"   and f :: "'e \<Rightarrow> 'f" assumes "n < length xs" shows "map f xs ! n = f (xs ! n)"
hyps: fixes s :: "'a"   and ss2 :: "'a \<times> (bool \<times> 'b \<times> 'a) list" assumes "(s, xs) \<in> conv ars"   and "ss2 \<in> conv ars"   and "lst_conv (s, xs) = fst ss2" shows "(s, xs @ snd ss2) \<in> conv ars \<and> lst_conv (s, xs @ snd ss2) = lst_conv ss2"
fun_left_idem: fixes z :: "'a\<^sub>m option"   and x :: "'a\<^sub>m" shows "z \<bind> m x \<bind> m x = z \<bind> m x"
assms_2: shows "{y, z} \<subseteq> orbit f x"
isUb_def: fixes R :: "'b set"   and S :: "'b set"   and x :: "'b" shows "isUb R S x = (S *<= x \<and> x \<in> R)"
order_slim_node: fixes k :: "nat"   and a :: "'b"   and h :: "nat" shows "order k (slim_node k a h)"
erH_minus: fixes A :: "('e, 'f) Ring_scheme"   and B :: "('e, 'g) Ring_scheme"   and Y :: "'e"   and h :: "'a \<Rightarrow> 'e"   and p :: "'a" assumes "Idomain S"   and "PolynRg A B Y"   and "h \<in> rHom S B"   and "p \<in> carrier R" shows "erH R S X A B Y h (-\<^sub>a p) = -\<^sub>a\<^bsub>A\<^esub> erH R S X A B Y h p"
C: shows "IntAgrK (S (User m, n, run)) = Some c"
tq_bound3: shows "te q (i + 1) < te p (i + 1)"
lvl_0_iff: fixes t :: "('b \<times> nat) tree" assumes "invar t" shows "(lvl t = 0) = (t = \<langle>\<rangle>)"
all_envirD: fixes p :: "'c \<Rightarrow> bool"   and x :: "'c" assumes "all_envir p" shows "p x"
simps: fixes a1 :: "call \<times> (nat \<Rightarrow> 'a option) \<times> ((nat \<times> char list) \<times> 'a \<Rightarrow> 'a \<aproc> set) \<times> 'a"   and a2 :: "call \<times> (nat \<Rightarrow> 'a option) \<times> ((nat \<times> char list) \<times> 'a \<Rightarrow> 'a \<aproc> set) \<times> 'a" shows "a1 \<lessapprox> a2 = (\<exists>ve ve' c \<beta> b. a1 = (c, \<beta>, ve, b) \<and> a2 = (c, \<beta>, ve', b) \<and> ve \<lessapprox> ve')"
closure_idem_strong: fixes A :: "'a set" assumes "A \<subseteq> carrier S" shows "closure_of (closure_of A) = closure_of A"
assms: fixes z :: "'a"   and X :: "'a set" assumes "z \<in> Rf CH fp"   and "z \<in> X"   and "stable_on ds X" shows "False"
wf_same_fst: fixes P :: "'a \<Rightarrow> bool"   and R :: "'a \<Rightarrow> ('b \<times> 'b) set" assumes "\<And>x. P x \<Longrightarrow> wf (R x)" shows "wf (same_fst P R)"
length_zero_iff_borders_eq: fixes r :: "real_int" shows "(\<parallel>r\<parallel> = 0) = (left r = right r)"
mat_plus_right_mono: fixes B :: "'b mat"   and C :: "'b mat"   and A :: "'b mat"   and nr :: "nat"   and nc :: "nat" assumes "B \<ge>\<^sub>m C"   and "A \<in> carrier_mat nr nc"   and "B \<in> carrier_mat nr nc"   and "C \<in> carrier_mat nr nc" shows "A + B \<ge>\<^sub>m A + C"
polys_to_mat_index: fixes i :: "nat"   and ps :: "('d \<Rightarrow>\<^sub>0 'e) list"   and j :: "nat"   and ts :: "'d list" assumes "i < length ps"   and "j < length ts" shows "polys_to_mat ts ps $$ (i, j) = lookup (ps ! i) (ts ! j)"
game_single_def: fixes \<A> :: "unit \<Rightarrow> ('x \<times> 'hash, 'x, 'hash) gpv" shows "game_single \<A> = map_spmf (\<lambda>key. (key, (), {})) key_gen \<bind> (\<lambda>(c_o, c_a, s). exec_gpv (oracle_hash c_o) (\<A> c_a) s \<bind> (\<lambda>(guess, s'). case guess of (x, h) \<Rightarrow> return_spmf (h = hash c_o x \<and> x \<notin> s')))"
log2_lm1: fixes u :: "nat" assumes "0 < u" shows "u div 2 ^ log2 u = 1"
is_hashtable_def: fixes l :: "('a \<times> 'b) list list"   and ht :: "('a, 'b) hashtable" shows "is_hashtable l ht = the_array ht \<mapsto>\<^sub>a l * \<up> (ht_size l (the_size ht) \<and> ht_hash l \<and> ht_distinct l \<and> 1 < length l)"
Y'_def: fixes \<omega> :: "real" shows "Y' \<omega> = (if \<omega> \<in> {\<omega> \<in> {0<..<1}. \<not> isCont M.I \<omega>} then 0 else M.I \<omega>)"
full_rank_implies_invertible: fixes A :: "(('b, 'c) vec, 'c) vec" assumes "rank A = nrows A" shows "invertible A"
word_bw_comms_2: fixes x :: "'a word"   and y :: "'a word" shows "x || y = y || x"
h: shows "(\<Sum>x\<in>{monom (1::'a) x |x. x \<le> p}. smult (f x) x) = 0"
points_neq: shows "a \<noteq> b \<and> b \<noteq> c \<and> a \<noteq> c"
elims: fixes x :: "'c + 'a \<times> 'b"   and y :: "'a \<times> 'b" assumes "|x|\<^sub>r\<^sub>h = y" obtains r :: "'a" and h :: "'b" where "x = Inr (r, h)" and "y = (r, h)"   | uu_ :: "'c" where "x = Inl uu_" and "y = undefined"
add1: fixes A :: "'d"   and k :: "'d"   and a :: "'d"   and b :: "'d" assumes "A \<equiv> k + a" shows "A + b \<equiv> k + (a + b)"
Open: fixes P :: "pi"   and a :: "name"   and b :: "name"   and P' :: "pi" assumes "P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'"   and "a \<noteq> b" shows "<\<nu>b>P \<longmapsto>\<^sub>e a<\<nu>b> \<prec>\<^sub>e P'"
permutation_inject: fixes x :: "'b permutation"   and y :: "'b permutation" shows "((\<rightarrow>) x = (\<rightarrow>) y) = (x = y)"
eql: fixes A :: "fmla"   and H :: "fmla set" assumes "A \<in> equality_axioms" shows "H \<turnstile> A"
listgcd_greatest_mult: fixes xs :: "'b list"   and y :: "'b"   and z :: "'b" assumes "\<And>x. x \<in> set xs \<Longrightarrow> y dvd x * z" shows "y dvd some_gcd.listgcd xs * z"
deg_pm_minus_le: fixes t :: "'a \<Rightarrow>\<^sub>0 nat"   and s :: "'a \<Rightarrow>\<^sub>0 nat" shows "deg_pm (t - s) \<le> deg_pm t"
simps_2: fixes x :: "'a"   and e :: "(char list \<Rightarrow> int) \<Rightarrow> nat"   and xs :: "('a \<times> ((char list \<Rightarrow> int) \<Rightarrow> nat)) list"   and C :: "acom"   and l :: "'a \<Rightarrow> nat"   and s :: "char list \<Rightarrow> int" shows "preList ((x, e) # xs) C l s = (l x = preT C e s \<and> preList xs C l s)"
d_IN_monotone_convergence_SUP: fixes f :: "nat \<Rightarrow> 'a \<times> 'a \<Rightarrow> ennreal"   and y :: "'a" assumes "incseq (\<lambda>n x. f n (x, y))" shows "d_IN (\<lambda>e. \<Squnion>n. f n e) y = (\<Squnion>n. d_IN (f n) y)"
intros_13: fixes as :: "(('d, 'e, 'f) ann \<times> 'd set \<times> 'd set) list"   and cs :: "('d, 'e, 'f) com list"   and \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and \<Theta> :: "'e \<Rightarrow> ('d, 'e, 'f) ann list option"   and F :: "'f set"   and Q :: "'d set"   and A :: "'d set" assumes "length as = length cs"   and "\<forall>i<length cs. \<Gamma>, \<Theta> \<turnstile>\<^bsub>/F\<^esub> (fst (as ! i)) (cs ! i) (fst (snd (as ! i))), (snd (snd (as ! i)))"   and "interfree \<Gamma> \<Theta> F as cs"   and "\<Inter> (set (map (\<lambda>a. fst (snd a)) as)) \<subseteq> Q"   and "\<Union> (set (map (\<lambda>a. snd (snd a)) as)) \<subseteq> A" shows "\<Gamma>, \<Theta> \<turnstile>\<^bsub>/F\<^esub> (AnnPar as) (Parallel cs) Q, A"
dom_cod: fixes f :: "'a" assumes "arr f" shows "local.dom (cod f) = cod f"
h'_g: fixes y :: "nat" shows "(\<lambda>x. h' x y) sums g' y"
non_special_solutions_def: fixes a :: "nat list"   and b :: "nat list" shows "non_special_solutions a b = (let A = Max (set b); B = Max (set a) in minimize (check' a b (generate' A B a b)))"
q_def: shows "q \<equiv> if D \<le> - pi then 1 else if pi < D then - 1 else 0"
iface_is_wildcard_def: fixes ifce :: "iface" shows "iface_is_wildcard ifce \<equiv> iface_name_is_wildcard (iface_sel ifce)"
u: shows "take i u \<cdot> [u ! i] \<cdot> drop (Suc i) u = u"
maximal_independent_subset: fixes V :: "'a set" obtains B :: "'a set" where "B \<subseteq> V" and "cindependent B" and "V \<subseteq> cspan B"
mat_entry_explicit: fixes M :: "'b mat"   and m :: "nat"   and n :: "nat"   and i :: "nat"   and j :: "nat" assumes "M \<in> carrier_mat m n"   and "i < m"   and "j < n" shows "(M *\<^sub>v unit_vec n j) $ i = M $$ (i, j)"
graph_nodes_it_axioms_def: fixes \<alpha> :: "'e \<Rightarrow> ('f, 'g) graph"   and invar :: "'e \<Rightarrow> bool"   and nodes_list_it :: "'e \<Rightarrow> ('f list \<Rightarrow> bool) \<Rightarrow> ('f \<Rightarrow> 'f list \<Rightarrow> 'f list) \<Rightarrow> 'f list \<Rightarrow> 'f list" shows "graph_nodes_it_axioms \<alpha> invar nodes_list_it \<equiv> \<forall>g. invar g \<longrightarrow> set_iterator (nodes_list_it g) (nodes (\<alpha> g))"
orient_lit_inst_def: 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 L :: "'b literal"   and u :: "'b trm"   and v :: "'b trm"   and s :: "sign"   and \<sigma> :: "('b \<times> 'b trm) list" assumes "basic_superposition trm_ord sel pos_ord vars filter_trms" shows "basic_superposition.orient_lit_inst trm_ord L u v s \<sigma> = ((L = Pos (Eq u v) \<or> L = Pos (Eq v u)) \<and> (u \<lhd> \<sigma>, v \<lhd> \<sigma>) \<notin> trm_ord \<and> s = pos \<or> (L = Neg (Eq u v) \<or> L = Neg (Eq v u)) \<and> (u \<lhd> \<sigma>, v \<lhd> \<sigma>) \<notin> trm_ord \<and> s = neg)"
Lxx1: fixes xs :: "nat list"   and x :: "nat"   and y :: "nat" assumes "xs \<in> Lxx x y" shows "2 \<le> length xs"
closure_of_subset_subtopology: fixes X :: "'a topology"   and S :: "'a set"   and T :: "'a set" shows "subtopology X S closure_of T \<subseteq> S"
fun_relI: fixes A :: "('a \<times> 'b) set"   and f :: "'a \<Rightarrow> 'c"   and f' :: "'b \<Rightarrow> 'f"   and B :: "('c \<times> 'f) set" assumes "\<And>a a'. (a, a') \<in> A \<Longrightarrow> (f a, f' a') \<in> B" shows "(f, f') \<in> A \<rightarrow> B"
lift_envs'_def: fixes b :: "nat"   and R :: "event_data list set" shows "lift_envs' b R = (\<lambda>(xs, ys). xs @ ys) ` ({xs. length xs = b} \<times> R)"
word_and_or_mask_aligned2: fixes b :: "'b word"   and n :: "nat"   and a :: "'b word" assumes "is_aligned b n"   and "a \<le> mask n" shows "a + b = a OR b"
le_add_right_mono: fixes a :: "'b"   and b :: "'b"   and c :: "'b"   and d :: "'b" assumes "a \<le> b + c"   and "c \<le> d" shows "a \<le> b + d"
m_s_widen_rep: fixes X :: "'a set"   and S1 :: "'a \<Rightarrow> 'av"   and S2 :: "'a \<Rightarrow> 'av"   and x :: "'a" assumes "finite X"   and "S1 = S2 on - X"   and "\<not> S2 x \<le> S1 x" shows "(\<Sum>x\<in>X. m (S1 x \<nabla> S2 x)) < (\<Sum>x\<in>X. m (S1 x))"
zero_in_Field: assumes "Field wo_aux \<noteq> {}" shows "zero wo_aux \<in> Field wo_aux"
ring_axioms_def: fixes R :: "'d set"   and addition :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and multiplication :: "'d \<Rightarrow> 'd \<Rightarrow> 'd" shows "ring_axioms R addition multiplication \<equiv> (\<forall>a b c. a \<in> R \<longrightarrow> b \<in> R \<longrightarrow> c \<in> R \<longrightarrow> multiplication a (addition b c) = addition (multiplication a b) (multiplication a c)) \<and> (\<forall>a b c. a \<in> R \<longrightarrow> b \<in> R \<longrightarrow> c \<in> R \<longrightarrow> multiplication (addition b c) a = addition (multiplication b a) (multiplication c a))"
add_right_cancel: fixes b :: "'a"   and a :: "'a"   and c :: "'a" shows "(b + a = c + a) = (b = c)"
snd_app_args: fixes f :: "'e \<Rightarrow> 'c"   and args :: "('e list \<times> 'd \<times> ('e \<times> 'e + nat) list) \<times> nat" shows "snd (app_args f args) = snd args"
map_mirror_o_snd_polychain_of_eq: fixes x0 :: "'a"   and y :: "'a"   and xs :: "'a list" shows "map (mirror_point x0 \<circ> snd) (polychain_of y xs) = map snd (polychain_of (mirror_point x0 y) (map uminus xs))"
atLeastatMost_empty: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and b :: "'c"   and a :: "'c" assumes "class.order less_eq less"   and "less b a" shows "ord.atLeastAtMost less_eq a b = {}"
Arr_AB: shows "\<BB>\<lparr>Arr\<rparr> \<inter>\<^sub>\<circ> Hom \<BB> A B = Hom \<BB> A B"
h_R: fixes n :: "nat"   and x :: "nat"   and y :: "nat" assumes "0 < h n x y"   and "x \<le> n + 1"   and "y \<le> n + 1" shows "(x, y) \<in> R' n"
normalize_prod_mset: fixes A :: "'b poly multiset" shows "smult (inverse (lead_coeff (\<Prod>\<^sub># A))) (\<Prod>\<^sub># A) = (\<Prod>p\<in>#A. smult (inverse (lead_coeff p)) p)"
add_mset: fixes x :: "'c"   and N :: "'c multiset" shows "\<Sum>\<^sub># (add_mset x N) = x + \<Sum>\<^sub># N"
freshEnv_updEnv_I: fixes ys :: "'varSort"   and xs :: "'varSort"   and y :: "'var"   and x :: "'var"   and X :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('a, 'b, 'varSort, 'var, 'c) qTerm set option" assumes "ys \<noteq> xs \<or> y \<noteq> x"   and "fresh ys y X"   and "rho ys y = None"   and "\<And>zs z Z. \<lbrakk>zs \<noteq> xs \<or> z \<noteq> x; rho zs z = Some Z\<rbrakk> \<Longrightarrow> fresh ys y Z" shows "freshEnv ys y (rho [x \<leftarrow> X]_xs)"
permutation_mat_def: fixes n :: "nat"   and p :: "nat \<Rightarrow> nat" shows "permutation_mat n p = Matrix.mat n n (\<lambda>(i, j). if i = p j then 1::'d else (0::'d))"
nsqn_update_changed_kno_val: fixes rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option"   and ip :: "nat"   and dsn :: "nat"   and hops :: "nat"   and nhip :: "nat" assumes "rt \<noteq> update rt ip (dsn, kno, val, hops, nhip, {})" shows "nsqn (update rt ip (dsn, kno, val, hops, nhip, {})) ip = dsn"
rel_option_Some1: fixes R :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and x :: "'g"   and y :: "'h option" shows "rel_option R \<lfloor>x\<rfloor> y = (\<exists>y'. y = \<lfloor>y'\<rfloor> \<and> R x y')"
splitAt_distinct_ab_aux: fixes vs :: "'c list"   and a :: "'c list"   and b :: "'c list"   and ram :: "'c" assumes "distinct vs"   and "(a, b) = splitAt ram vs" shows "distinct a \<and> distinct b"
inf_mono: fixes inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and c :: "'b"   and b :: "'b"   and d :: "'b" assumes "class.semilattice_inf inf less_eq less"   and "less_eq a c"   and "less_eq b d" shows "less_eq (inf a b) (inf c d)"
join_def: fixes A :: "'b option list set"   and pos :: "bool"   and B :: "'b option list set" shows "join A pos B = (if pos then Option.these (join1 ` (A \<times> B)) else A - Option.these (join1 ` (A \<times> B)))"
IH: shows "xs \<in> shuffles (filter P xs) (filter (\<lambda>x. \<not> P x) xs)"
lunit_simps_2: fixes f :: "'a" assumes "B.ide f" shows "src\<^sub>B (B.lunit f) = src\<^sub>B f"
Inp: fixes delta :: "'opSym"   and inp :: "'index \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set option" assumes "wlsOpS delta"   and "\<And>i. arOf delta i = None \<and> inp i = None \<or> (\<exists>s X. arOf delta i = Some s \<and> inp i = Some X \<and> wls s X)" shows "wlsInp delta inp"
parts_crypt_mult_start: fixes s :: "(agent \<times> msg) set"   and s' :: "(agent \<times> msg) set"   and SK :: "nat option \<times> nat set \<times> nat set"   and A :: "nat"   and B :: "nat" assumes "s \<turnstile> s'"   and "Crypt (SesK SK) (A \<otimes> B) \<in> parts (used s')"   and "Crypt (SesK SK) (A \<otimes> B) \<notin> parts (used s)" shows "B \<in> fst (snd SK) \<and> (\<exists>n C. (Asset n, Token n (Auth_PriK n) B C SK) \<in> s') \<or> {A \<otimes> B, SesKey SK} \<subseteq> spied s"
b_2: shows "\<forall>a'\<ge>b. a' \<in> \<nat> \<longrightarrow> f \<in> \<Omega>(\<lambda>x. real x powr p * (1 + integral (\<lambda>u. g' u / u powr (p + 1)) a' (real x)))"
Figure_7: fixes i :: "nat"   and js :: "nat list"   and xs :: "nat list"   and ys :: "nat list" assumes "foldl1 f2 (map (h2 i) js) = Two"   and "js = xs @ ys"   and "xs \<noteq> []"   and "i = last xs" shows "i + 1 \<in> set ys"
fixp_induct_strong2_uc: fixes U :: "'f \<Rightarrow> 'd \<Rightarrow> 'e spmf"   and F :: "'f \<Rightarrow> 'f"   and C :: "('d \<Rightarrow> 'e spmf) \<Rightarrow> 'f"   and f :: "'f"   and P :: "('d \<Rightarrow> 'e spmf) \<Rightarrow> bool" assumes "\<And>x. spmf.mono_body (\<lambda>f. U (F (C f)) x)"   and "f \<equiv> C (spmf.fixp_fun (\<lambda>f. U (F (C f))))"   and "\<And>f. U (C f) = f"   and "spmf.admissible P"   and "P (\<lambda>_. return_pmf None)"   and "\<And>f'. \<lbrakk>spmf.le_fun (U f') (U f); spmf.le_fun (U f') (U (F f')); P (U f')\<rbrakk> \<Longrightarrow> P (U (F f'))" shows "P (U f)"
T_A'_nneg: shows "0 \<le> T_A' n"
causal_program_history: fixes sb\<^sub>1 :: "'a memref list"   and sb\<^sub>2 :: "'a memref list" assumes "sb = sb\<^sub>1 @ sb\<^sub>2" shows "\<exists>is. instrs sb\<^sub>2 @ Read volatile a t # is\<^sub>s\<^sub>b = is @ prog_instrs sb\<^sub>2"
mkarr_memb_ARR: fixes f :: "'c local.term set"   and t :: "'c local.term" assumes "ARR f"   and "t \<in> f" shows "mkarr t = f"
add_assoc: fixes a :: "'b"   and b :: "'b"   and c :: "'b" shows "a \<sqinter> b \<sqinter> c = a \<sqinter> (b \<sqinter> c)"
must_syncI: fixes P :: "'addr J_prog"   and t :: "'thread_id"   and x :: "'addr expr \<times> (String.literal \<Rightarrow> 'addr val option)"   and m :: "'heap" assumes "\<exists>taa x' m' s. red_mthr.r_syntax P t x m taa x' m' \<and> red_mthr.actions_ok s t taa" shows "red_mthr.must_sync P t x m"
Field_Restr_ofilter: fixes r :: "('c \<times> 'c) set"   and A :: "'c set" assumes "Well_order r"   and "ofilter r A" shows "Field (Restr r A) = A"
setdist_def: fixes s :: "'b set"   and t :: "'b set" shows "setdist s t \<equiv> if s = {} \<or> t = {} then 0 else Inf {dist x y |x y. x \<in> s \<and> y \<in> t}"
fds_zeta_def: shows "fds_zeta = fds (\<lambda>_. 1::'b)"
iso_class_memb_is_ide: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and F :: "'b set"   and f :: "'b" assumes "category C"   and "category.is_iso_class C F"   and "f \<in> F" shows "partial_magma.ide C f"
cases: fixes x :: "nat" obtains "x = 0"   | n :: "nat" where "x = Suc n"
R1: fixes a :: "letter"   and vs :: "letter list list"   and ws :: "letter list list"   and w :: "letter list" assumes "R a vs ws" shows "R a (w # vs) ((a # w) # ws)"
restrict_indets_ideal: fixes F :: "(('b option \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c) set" shows "restrict_indets ` ideal F = ideal (restrict_indets ` F)"
f_def: fixes x :: "nat" shows "f x = pr_index_enumerator n x"
homeomorphic_affine_sets: fixes S :: "'c set"   and T :: "'d set" assumes "affine S"   and "affine T"   and "aff_dim S = aff_dim T" shows "S homeomorphic T"
neg_equal_iff_equal: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.group_add minus plus zero uminus" shows "(uminus a = uminus b) = (a = b)"
rtp2: shows "positionVector e = \<alpha> ** positionVector x \<oplus> \<beta> ** positionVector y \<oplus> \<gamma> ** positionVector z"
inf_left_idem: fixes x :: "'a multiset"   and y :: "'a multiset" shows "x \<inter># (x \<inter># y) = x \<inter># y"
image_vimage_adjoints: fixes ob :: "'b \<Rightarrow> 'a"   and P :: "'b set"   and Q :: "'a set" shows "(ob ` P \<subseteq> Q) = (P \<subseteq> ob -` Q)"
abs_inner: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and norm :: "'a \<Rightarrow> real"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and sgn :: "'a \<Rightarrow> 'a"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and inner :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and Basis :: "'a set"   and abs :: "'a \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and i :: "'a"   and x :: "'a" assumes "class.ordered_euclidean_space Inf Sup minus dist norm plus zero uminus scaleR sgn uniformity open inner Basis abs inf sup less_eq less"   and "i \<in> Basis" shows "inner (abs x) i = \<bar>inner x i\<bar>"
extremeE: fixes X :: "'b set"   and r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and e :: "'b" assumes "extreme X r e" obtains "e \<in> X" and "\<And>x. x \<in> X \<Longrightarrow> r x e"
map_comp: fixes g1 :: "'g \<Rightarrow> 'd"   and g2 :: "'h \<Rightarrow> 'e"   and g3 :: "'i \<Rightarrow> 'f"   and f1 :: "'j \<Rightarrow> 'g"   and f2 :: "'k \<Rightarrow> 'h"   and f3 :: "'l \<Rightarrow> 'i"   and v :: "('j, 'k, 'l) expr" shows "map_expr g1 g2 g3 (map_expr f1 f2 f3 v) = map_expr (g1 \<circ> f1) (g2 \<circ> f2) (g3 \<circ> f3) v"