hcju/isabelleps huggingface dataset:
starset_n_starset: fixes As :: "nat \<Rightarrow> 'a set"   and A :: "'a set" assumes "\<forall>n. As n = A" shows "*sn* As = *s* A"
Abs_atoms_inject: fixes x :: "'a set"   and y :: "'a set" assumes "x \<in> range atom_map"   and "y \<in> range atom_map" shows "(Abs_atoms x = Abs_atoms y) = (x = y)"
hom_boundary_carrier: fixes p :: "int"   and X :: "'b topology"   and S :: "'b set"   and c :: "(((nat \<Rightarrow> real) \<Rightarrow> 'b) \<Rightarrow>\<^sub>0 int) set" shows "hom_boundary p X S c \<in> carrier (homology_group (p - 1) (subtopology X S))"
wf_J_prog'_def: shows "wf_J_prog' = wf_prog wf_J_mdecl'"
Cond: fixes E :: "env"   and dt :: "loc \<Rightarrow> ty option"   and e0 :: "expr"   and e1 :: "expr"   and T1 :: "ty"   and e2 :: "expr"   and T2 :: "ty"   and T :: "ty" assumes "E,dt\<Turnstile>e0\<Colon>-PrimT Boolean"   and "E,dt\<Turnstile>e1\<Colon>-T1"   and "E,dt\<Turnstile>e2\<Colon>-T2"   and "prg E\<turnstile>T1\<preceq>T2 \<and> T = T2 \<or> prg E\<turnstile>T2\<preceq>T1 \<and> T = T1" shows "E,dt\<Turnstile>e0 ? e1 : e2\<Colon>-T"
INT_extend_simps_1: fixes A :: "'b \<Rightarrow> 'a set"   and C :: "'b set"   and B :: "'a set" shows "\<Inter> (A ` C) \<inter> B = (if C = {} then B else \<Inter>x\<in>C. A x \<inter> B)"
rel_fun_mono_strong: fixes A :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and B :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'e"   and g :: "'d \<Rightarrow> 'f"   and A' :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and B' :: "'e \<Rightarrow> 'f \<Rightarrow> bool" assumes "rel_fun A B f g"   and "A' \<le> A"   and "\<And>x y. \<lbrakk>x \<in> f ` {x. Domainp A' x}; y \<in> g ` {x. Rangep A' x}; B x y\<rbrakk> \<Longrightarrow> B' x y" shows "rel_fun A' B' f g"
inf_chain_bad: fixes f :: "nat \<Rightarrow> 'a"   and i :: "nat" assumes "local.inf_chain f" shows "local.bad (f i)"
cont_W: fixes b :: "(char list \<Rightarrow> int) \<Rightarrow> bool"   and r :: "((char list \<Rightarrow> int) \<times> (char list \<Rightarrow> int)) set" shows "cont (W b r)"
get_heap_def: shows "get_heap = Prog (\<lambda>h. h \<turnstile> return h)"
span_add_eq: fixes x :: "'a"   and S :: "'a set"   and y :: "'a" assumes "x \<in> ideal S" shows "(x + y \<in> ideal S) = (y \<in> ideal S)"
Rep_atoms_inverse: fixes x :: "'a atoms" shows "Abs_atoms (Rep_atoms x) = x"
finite_ball_include: fixes S :: "'a set"   and a :: "'a" assumes "finite S" shows "\<exists>e>0. S \<subseteq> ball a e"
simps: fixes \<phi> :: "'b stream \<Rightarrow> bool"   and a :: "'b stream" shows "alw \<phi> a = (\<exists>xs. a = xs \<and> \<phi> xs \<and> alw \<phi> (stl xs))"
cltn2_inverse_is_K2_isometry: fixes J :: "cltn2" assumes "is_K2_isometry J" shows "is_K2_isometry (cltn2_inverse J)"
degree_m_mult_eq: fixes p :: "int poly"   and q :: "int poly" assumes "Mp p \<noteq> Mp 0"   and "Mp q \<noteq> Mp 0" shows "degree_m (p * q) = degree_m p + degree_m q"
finite_carrier: fixes carrier :: "'b set"   and le :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "finite_total_preorder_on carrier le" shows "finite carrier"
snoc: shows "set ats' \<Turnstile>\<^sub>i \<B>\<I> (assert_bound_loop ats' (init t))"
frequently_eq_eventually: fixes P :: "'a \<Rightarrow> bool" shows "frequently P U = eventually P U"
next_subset_dfs: fixes S :: "('a \<times> 'b \<times> 'a) list"   and xs :: "('a \<times> 'b \<times> 'a) list"   and x :: "'a \<times> 'b \<times> 'a" assumes "set S \<subseteq> reach\<^sub>t (set \<Sigma>) \<delta> q\<^sub>0"   and "list_all (\<lambda>t. t \<in> reach\<^sub>t (set \<Sigma>) \<delta> q\<^sub>0) xs"   and "x \<in> set xs" shows "x \<in> set (dfs S xs)"
r_list_encode: fixes xs :: "nat list"   and n :: "nat" assumes "length xs = Suc n" shows "eval (r_list_encode n) xs \<down>= Partial_Recursive.list_encode xs"
AE_distr_iff: fixes f :: "'b \<Rightarrow> 'c"   and M :: "'b measure"   and N :: "'c measure"   and P :: "'c \<Rightarrow> bool" assumes "f \<in> M \<rightarrow>\<^sub>M N"   and "{x \<in> space N. P x} \<in> sets N" shows "almost_everywhere (distr M N f) P = (AE x in M. P (f x))"
sel_3: fixes x1 :: "'b literal multiset multiset"   and x2 :: "'b literal multiset"   and x3 :: "'b literal multiset" shows "concl_of (Infer x1 x2 x3) = x3"
sortedP_right_of_last: fixes le :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and zs :: "'a list"   and y :: "'a" assumes "linorder_list0.sortedP le zs"   and "y \<in> set zs"   and "y \<noteq> last zs" shows "le y (last zs)"
abs_eq: shows "Inf = (\<lambda>X. Abs_atoms (Collect atom \<inter> \<Inter> (Rep_atoms ` X)))"
null_sets_completion_iff2: fixes A :: "'b set"   and M :: "'b measure" shows "(A \<in> null_sets (completion M)) = (\<exists>N'\<in>null_sets M. A \<subseteq> N')"
vconcat_vempty: shows "vconcat []\<^sub>\<circ> = []\<^sub>\<circ>"
execn_blockAbrupt: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and bdy :: "('e, 'd, 'f) com"   and init :: "'e \<Rightarrow> 'e"   and s :: "'e"   and n :: "nat"   and t :: "'e"   and return :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and c :: "'e \<Rightarrow> 'e \<Rightarrow> ('e, 'd, 'f) com" assumes "\<Gamma>\<turnstile> \<langle>bdy,Normal (init s)\<rangle> =n\<Rightarrow> Abrupt t" shows "\<Gamma>\<turnstile> \<langle>block init bdy return c,Normal s\<rangle> =n\<Rightarrow> Abrupt (return s t)"
store: fixes \<C> :: "t_context"   and n :: "nat"   and a :: "nat"   and tp :: "tp option"   and t :: "t"   and off :: "nat" assumes "memory \<C> = Some n"   and "load_store_t_bounds a tp t" shows "\<C> \<turnstile> [Store t tp a off] : [T_i32, t] _> []"
L: shows "(\<lambda>xa. ereal (Gromov_product_at e x (v xa))) \<longlonglongrightarrow> ereal (Gromov_product_at e x y)"
eval\<Pi>\<^sub>1_cases: fixes y :: "\<upsilon> \<Rightarrow> j \<Rightarrow> i \<Rightarrow> bool" assumes "y \<in> UNIV" obtains x :: "\<Pi>\<^sub>1" where "y = eval\<Pi>\<^sub>1 x"
insert_if: fixes A :: "'d set"   and g :: "'d \<Rightarrow> 'e"   and x :: "'d" assumes "finite A" shows "prod g (insert x A) = (if x \<in> A then prod g A else g x * prod g A)"
cycles_def: fixes execute :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and enabled :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and p :: "'b" shows "transition_system_initial.cycles execute enabled p \<equiv> {w. transition_system.path execute enabled w p \<and> fold execute w p = p}"
Nil: fixes w :: "int" shows "bl_to_bin_aux [] w = w"
fmdrop_fset_twice: fixes S :: "'c fset"   and T :: "'c fset"   and m :: "('c, 'd) fmap" shows "fmdrop_fset S (fmdrop_fset T m) = fmdrop_fset (S |\<union>| T) m"
pdom_plus: fixes f :: "('a, 'b) pfun"   and g :: "('a, 'b) pfun" shows "pdom (f + g) = pdom f \<union> pdom g"
take_bit_not_take_bit: fixes 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 divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bit :: "'c \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and drop_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and take_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and "and" :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and or :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and xor :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and mask :: "nat \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and not :: "'c \<Rightarrow> 'c"   and n :: "nat"   and a :: "'c" assumes "class.ring_bit_operations plus minus zero times one divide modulo bit push_bit drop_bit take_bit and or xor mask uminus not" shows "take_bit n (not (take_bit n a)) = take_bit n (not a)"
sum_distrib_left: fixes r :: "'c"   and f :: "'d \<Rightarrow> 'c"   and A :: "'d set" shows "r * sum f A = (\<Sum>n\<in>A. r * f n)"
listrel_Cons: fixes r :: "('b \<times> 'a) set"   and x :: "'b"   and xs :: "'b list" shows "listrel r `` {x # xs} = set_Cons (r `` {x}) (listrel r `` {xs})"
prems: shows "[:- a, 1:] * p \<noteq> 0"
finite_sup_least: fixes A :: "'a set" assumes "finite A"   and "A \<subseteq> carrier L"   and "A \<noteq> {}" shows "is_lub L (\<Squnion>A) A"
TS_x': fixes h0 :: "nat list"   and x :: "nat"   and y :: "nat"   and h :: "nat list" shows "Partial_Cost_Model.T_on' (rTS h0) ([x, y], h) [x] = 0 \<and> Partial_Cost_Model.config' (rTS h0) ([x, y], h) [x] = ([x, y], rev [x] @ h)"
sel: fixes x :: "fol fm" shows "concl (Thm x) = x"
cast_shadow_root_ptr_not_document_ptr_2: fixes document_ptr :: "(_) document_ptr"   and shadow_root_ptr :: "(_) shadow_root_ptr" shows "cast document_ptr \<noteq> cast shadow_root_ptr"
to_bl_eq_rev: fixes w :: "'b word" shows "to_bl w = map (bit w) (rev [0..<LENGTH('b)])"
f1: shows "Deliver (i, Create i e) \<in> set pre \<or> Deliver (i, Append i e) \<in> set pre \<or> (\<exists>mo. Deliver (i, Expunge e mo i) \<in> set pre) \<or> (\<exists>mo. Deliver (i, Store e mo i) \<in> set pre)"
Impl_sound_lemma: fixes A :: "triple set"   and B :: "triple set"   and f :: "'a \<Rightarrow> 'b \<Rightarrow> triple"   and Ms :: "'b set"   and Cm :: "'b"   and na :: "nat"   and z :: "'a" assumes "\<forall>z n. Ball (A \<union> B) (nvalid n) \<longrightarrow> Ball (f z ` Ms) (nvalid n)"   and "Cm \<in> Ms"   and "Ball A (nvalid na)"   and "Ball B (nvalid na)" shows "\<Turnstile>na: f z Cm"
diag_mat_transpose: fixes A :: "'b mat" assumes "dim_row A = dim_col A" shows "diag_mat A\<^sup>T = diag_mat A"
cbv_star_implies_eval: fixes t :: "lam"   and v :: "lam" assumes " t \<longrightarrow>cbv* v"   and "val v" shows "t \<Down> v"
set_less_eq_aux'_refl: fixes A :: "'a set" shows "(A \<sqsubseteq>'' A) = (A \<in> infinite_complement_partition)"
adj_iso1: fixes f :: "'b \<Rightarrow> 'c"   and g :: "'c \<Rightarrow> 'b" assumes "f \<stileturn> g" shows "mono f"
simps_1: fixes states :: "hf set"   and init :: "hf"   and final :: "hf set"   and nxt :: "hf \<Rightarrow> 'b \<Rightarrow> hf"   and more :: "'c" shows "dfa.states \<lparr>dfa.states = states, init = init, final = final, nxt = nxt, \<dots> = more\<rparr> = states"
traces_CHAOS_sub: fixes A :: "'b set" shows "T (CHAOS A) \<subseteq> {s. set s \<subseteq> ev ` A}"
ideD_1: fixes a :: "'a \<times> 'a \<times> 'a" assumes "CCC.ide a" shows "CCC.arr a"
h2: shows "t2 \<noteq> Tip \<and> sortedTree h t2 \<longrightarrow> sortedTree h (wrm h t2)"
ord_embed_inj: fixes f :: "'b \<Rightarrow> 'c" assumes "ord_embed f" shows "inj f"
chain_le: fixes I :: "nat \<Rightarrow> 'a set"   and m :: "nat" assumes "\<forall>n. I n \<subseteq> I (n + 1)" shows "\<forall>n\<le>m. I n \<subseteq> I m"
TryCatchRedsFinal: fixes e\<^sub>1 :: "char list exp"   and s\<^sub>0 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and a :: "nat"   and h\<^sub>1 :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and l\<^sub>1 :: "char list \<Rightarrow> val option"   and P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list"   and D :: "char list"   and fs :: "char list \<times> char list \<Rightarrow> val option"   and C :: "char list"   and e\<^sub>2 :: "char list exp"   and V :: "char list"   and e\<^sub>2' :: "char list exp"   and h\<^sub>2 :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and l\<^sub>2 :: "char list \<Rightarrow> val option" assumes "P \<turnstile> \<langle>e\<^sub>1,s\<^sub>0\<rangle> \<rightarrow>* \<langle>Throw a,(h\<^sub>1, l\<^sub>1)\<rangle>"   and "h\<^sub>1 a = \<lfloor>(D, fs)\<rfloor>"   and "P \<turnstile> D \<preceq>\<^sup>* C"   and "P \<turnstile> \<langle>e\<^sub>2,(h\<^sub>1, l\<^sub>1(V \<mapsto> Addr a))\<rangle> \<rightarrow>* \<langle>e\<^sub>2',(h\<^sub>2, l\<^sub>2)\<rangle>"   and "final e\<^sub>2'" shows "P \<turnstile> \<langle>try e\<^sub>1 catch(C V) e\<^sub>2,s\<^sub>0\<rangle> \<rightarrow>* \<langle>e\<^sub>2',(h\<^sub>2, l\<^sub>2(V := l\<^sub>1 V))\<rangle>"
coset_eq: shows "\<forall>x\<in>carrier G. A #> x = x <# A"
ss422iv: fixes y :: "'a"   and x :: "'a" assumes "is_p_fun y"   and "x \<le> y"   and "y ; 1 \<le> x ; 1" shows "x = y"
Sign_def: shows "Sign \<equiv> map_fun Re (map_fun Re Ab) cSign"
csv_hyp: shows "\<forall>r. consistent_sign_vec_copr (qs1 @ qs2) r = consistent_sign_vec_copr qs1 r @ consistent_sign_vec_copr qs2 r"
IH: fixes gpv :: "('d, 'e, 'f) gpv" shows "exp1 (map_gpv' g h k gpv) = exp2 gpv"
sndI: fixes x :: "'g \<times> 'h"   and y :: "'g"   and z :: "'h" assumes "x = (y, z)" shows "snd x = z"
strategyproofness_1: shows "(pmf (sds R26) a < pmf (sds R8) d \<or> pmf (sds R26) a + (pmf (sds R26) c + pmf (sds R26) d) < pmf (sds R8) d + (pmf (sds R8) b + pmf (sds R8) a)) \<or> pmf (sds R26) a = pmf (sds R8) d \<and> pmf (sds R26) a + (pmf (sds R26) c + pmf (sds R26) d) = pmf (sds R8) d + (pmf (sds R8) b + pmf (sds R8) a)"
cblinfun_apply_cases: fixes y :: "'c \<Rightarrow> 'd" assumes "y \<in> {f. bounded_clinear f}" obtains x :: "'c \<Rightarrow>\<^sub>C\<^sub>L 'd" where "y = (*\<^sub>V) x"
scale: fixes s1 :: "'c \<Rightarrow> 'd \<Rightarrow> 'd"   and s2 :: "'c \<Rightarrow> 'e \<Rightarrow> 'e"   and f :: "'d \<Rightarrow> 'e"   and r :: "'c"   and b :: "'d" assumes "module_hom s1 s2 f" shows "f (s1 r b) = s2 r (f b)"
sorted_list_prop: fixes l :: "real list"   and x :: "real" assumes "sorted l"   and "0 < length l"   and "l ! 0 < x"   and "x \<le> l ! (length l - 1)" shows "\<exists>n. n + 1 < length l \<and> l ! n \<le> x \<and> x \<le> l ! (n + 1)"
fm_gen_by_prod_def: fixes R :: "('e, 'f) Ring_scheme"   and P :: "('g \<times> 'h, 'e, 'i) Module_scheme"   and M :: "('g, 'e, 'i) Module_scheme"   and N :: "('h, 'e, 'i) Module_scheme" shows "FM\<^bsub>R\<^esub> P M N = (R module P \<and> free_generator R P (M \<times>\<^sub>c N))"
ij_le2_1: shows "i < prod_list (nths (dims A) I)"
IH_1: shows "2 ^ min_height l \<le> KD_Tree.size_kdt l"
dbm_le_not_inf: fixes a :: "'c DBMEntry"   and b :: "'c DBMEntry" assumes "a \<preceq> b"   and "b \<noteq> \<infinity>" shows "a \<noteq> \<infinity>"
pick_insert_index: fixes I :: "nat set"   and k :: "nat"   and a :: "nat"   and i :: "nat"   and a' :: "nat" assumes "card I = k"   and "a \<notin> I"   and "i < k"   and "pick (insert a I) a' = a"   and "a' < card I + 1" shows "pick (insert a I) (insert_index a' i) = pick I i"
eliminate_entries_gen_def: fixes minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and v :: "nat \<Rightarrow> 'c"   and A :: "'c mat"   and I :: "nat"   and J :: "nat" shows "eliminate_entries_gen minus times v A I J = mat (dim_row A) (dim_col A) (\<lambda>(i, j). if i \<noteq> I then minus (A $$ (i, j)) (times (v i) (A $$ (I, j))) else A $$ (i, j))"
path_weightp_def: fixes weights :: "'a \<Rightarrow> 'a \<Rightarrow> 'b antichain"   and l1 :: "'a"   and l2 :: "'a"   and s :: "'b" assumes "graph weights" shows "graph.path_weightp weights l1 l2 s \<equiv> \<exists>xs. graph.path weights l1 l2 xs \<and> s = foldr (+) (map (\<lambda>(s, l, t). l) xs) (0::'b)"
trms_subsumes_and_red_inf: fixes D :: "'a eclause"   and C :: "'a eclause"   and \<eta> :: "('a \<times> 'a trm) list"   and S :: "'a eclause set"   and P :: "'a eclause set"   and \<sigma> :: "('a \<times> 'a trm) list"   and \<theta> :: "('a \<times> 'a trm) list" assumes "trms_subsumes D C \<eta>"   and "redundant_inference (subst_ecl D \<eta>) S P \<sigma>"   and "\<sigma> \<doteq> \<theta> \<lozenge> \<eta>" shows "redundant_inference C S P \<sigma>"
internal_case_prod_def: shows "??.Product_Type.internal_case_prod \<equiv> case_prod"
conj_com: fixes Cn :: "'b set \<Rightarrow> 'b set"   and true_PL :: "'b"   and false_PL :: "'b"   and imp_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and not_PL :: "'b \<Rightarrow> 'b"   and conj_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and disj_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and equiv_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and p :: "'b"   and q :: "'b"   and A :: "'b set" assumes "Supraclassical_logic Cn true_PL false_PL imp_PL not_PL conj_PL disj_PL equiv_PL" shows "(conj_PL p q \<in> Cn A) = (conj_PL q p \<in> Cn A)"
assms_2: shows "\<exists>j. suffix (n + j) w \<Turnstile>\<^sub>n af \<psi> (w [n \<rightarrow> j + n])[X]\<^sub>\<nu>"
fin_2: shows "\<forall>AA\<in>AAA. finite AA"
store_preserved: fixes s :: "s"   and vs :: "v list"   and es :: "e list"   and i :: "nat"   and s' :: "s"   and vs' :: "v list"   and es' :: "e list"   and \<S> :: "s_context"   and ts :: "t list" assumes "\<lparr>s;vs;es\<rparr> \<leadsto>_ i \<lparr>s';vs';es'\<rparr>"   and "store_typing s \<S>"   and "\<S>\<bullet>None \<tturnstile>_ i vs;es : ts" shows "store_typing s' \<S>"
count_empty: fixes a :: "'e" shows "count {#} a = 0"
simps_16: fixes prec :: "nat"   and ord :: "nat"   and I :: "float interval list"   and a :: "float interval list"   and l :: "floatarith"   and r :: "floatarith"   and env :: "taylor_model list" shows "approx_tm prec ord I a (Max l r) env = (case (approx_tm prec ord I a l env, approx_tm prec ord I a r env) of (None, b) \<Rightarrow> None | (Some t1, None) \<Rightarrow> None | (Some t1, Some t2) \<Rightarrow> Some (tm_max prec I a t1 t2))"
tendsto_set_lebesgue_integral_at_top: fixes a :: "real"   and M :: "real measure"   and f :: "real \<Rightarrow> 'a" assumes "\<And>b. a \<le> b \<Longrightarrow> {a..b} \<in> sets M"   and "set_integrable M {a..} f" shows "((\<lambda>b. set_lebesgue_integral M {a..b} f) \<longlongrightarrow> set_lebesgue_integral M {a..} f) at_top"
less_mem_of_Nset: fixes x :: "nat"   and n :: "nat" assumes "x < n" shows "x \<in> {i. i \<le> n}"
extend_8: shows "[a] @- b\<^sub>1 @- u' =\<^sub>I u"
external_from_internal_invariant_vimage: fixes S :: "('e, 'f) spec"   and P :: "'e set"   and Q :: "'f set" assumes "reach S \<subseteq> P"   and "P \<subseteq> obs S -` Q" shows "oreach S \<subseteq> Q"
assms: shows "\<guillemotleft>f : b \<rightarrow> a\<guillemotright>"
simps_5: fixes prec :: "nat"   and x :: "floatarith"   and a :: "floatarith"   and b :: "floatarith"   and bs :: "float interval option list"   and ss :: "nat list" shows "approx_form prec (AtLeastAtMost x a b) bs ss = (case (approx prec x bs, approx prec a bs, approx prec b bs) of (None, b) \<Rightarrow> False | (Some ivlx, None, ba) \<Rightarrow> False | (Some ivlx, Some ivl, None) \<Rightarrow> False | (Some ivlx, Some ivl, Some ivl') \<Rightarrow> float_plus_up prec (upper ivl) (- lower ivlx) \<le> 0 \<and> float_plus_up prec (upper ivlx) (- lower ivl') \<le> 0)"
not_irreducible_factor_yields_prime_factors: fixes u :: "'b poly"   and f :: "'b poly"   and P :: "'b poly set" assumes "u dvd f"   and "finite P"   and "f = \<Prod>P"   and "P \<subseteq> {q. irreducible q \<and> monic q}"   and "0 < degree u"   and "\<not> irreducible u" shows "\<exists>pi pj. pi \<in> P \<and> pj \<in> P \<and> pi \<noteq> pj \<and> pi dvd u \<and> pj dvd u"
finite_compl: fixes A :: "'c set" assumes "finite A" shows "finite (- A) = finite UNIV"
mono_INF: fixes f :: "'b \<times> 'c \<Rightarrow> 'd"   and A :: "'e \<Rightarrow> 'b \<times> 'c"   and I :: "'e set" assumes "prod.mono f" shows "f (Inf_prod (A ` I)) \<le> (\<Sqinter>x\<in>I. f (A x))"
image_le_pred: fixes f :: "'c \<Rightarrow> 'b"   and A :: "'c set"   and G :: "'b \<Rightarrow> bool" shows "(\<P> f A \<subseteq> {s. G s}) = (\<forall>x\<in>A. G (f x))"
Fun: fixes x2a :: "('f, 'b) Term.term" assumes "x2a \<in> set ts" shows "weight (x2a \<cdot> \<sigma>) = weight x2a + (\<Sum>x\<in>#vars_term_ms (SCF x2a). weight (\<sigma> x) - w0)"
is_ground_subst_atm: fixes A :: "'a"   and \<sigma> :: "'s" assumes "is_ground_atm A" shows "A \<cdot>a \<sigma> = A"
not_hd_in_tl: fixes x :: "'c"   and xs :: "'c list" assumes "x \<noteq> hd xs"   and "x \<in> set xs" shows "x \<in> set (tl xs)"
DiffI: fixes c :: "'e"   and A :: "'e set"   and B :: "'e set" assumes "c \<in> A"   and "c \<notin> B" shows "c \<in> A - B"
bdd_above_bot: fixes bot :: "'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a set" assumes "class.order_bot bot less_eq less" shows "preorder.bdd_below less_eq A"
span_insert_idI: fixes r :: "'a"   and B :: "'a set" assumes "r \<in> More_Modules.ideal B" shows "More_Modules.ideal (insert r B) = More_Modules.ideal B"
assms_2: fixes x :: "complex" assumes "x \<in> A" shows "f x \<noteq> 1"
monom_mult: fixes a :: "'a"   and b :: "'a"   and n :: "nat"   and m :: "nat" assumes "a \<in> carrier R"   and "b \<in> carrier R" shows "monom P (a \<otimes> b) (n + m) = monom P a n \<otimes>\<^bsub>P\<^esub> monom P b m"
conv_callee_parallel: fixes callee1 :: "'z \<Rightarrow> 'r \<Rightarrow> ('t \<times> 'z, 'v, 'x) gpv"   and callee2 :: "'aa \<Rightarrow> 's \<Rightarrow> ('u \<times> 'aa, 'w, 'y) gpv"   and s :: "'z"   and s' :: "'aa" shows "CNV (callee1 \<ddagger>\<^sub>I callee2) (s, s') = CNV callee1 s |\<^sub>= CNV callee2 s'"
3: fixes xs :: "('d \<times> unit) list"   and a :: "'d list" shows "map_iterator_dom (foldli xs) top (#) a = rev (map_iterator_dom (foldli xs) (\<lambda>_. True) (\<lambda>x l. l @ [x]) (rev a))"
simps_5: fixes b :: "bool" shows "ESem (Bool b) = (\<Lambda> \<rho>. (\<Lambda> x. x)\<cdot>(B\<cdot>(Discr b)))"
prob_T': fixes P :: "'s stream \<Rightarrow> bool"   and I :: "'s pmf" assumes "Measurable.pred (stream_space (count_space UNIV)) P" shows "\<P>(x in T' I. P x) = measure_pmf.expectation I (\<lambda>s. T.prob s {x \<in> space (T s). P (s ## x)})"
L_def: shows "L = (1::'a)\<^sup>\<circ> * bot"
psubst_subst_compose_freshEnv: fixes rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('a, 'b, 'varSort, 'var, 'c) qTerm set option"   and X :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and Y :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and ys :: "'varSort"   and y :: "'var" assumes "goodEnv rho"   and "good X"   and "good Y"   and "freshEnv ys y rho" shows "X #[Y / y]_ys #[rho] = X #[rho] #[Y #[rho] / y]_ys"
heap_is_wellformed_is_l_heap_is_wellformed: shows "l_heap_is_wellformed ShadowRootClass.type_wf ShadowRootClass.known_ptr Shadow_DOM.heap_is_wellformed Shadow_DOM.parent_child_rel Shadow_DOM.get_child_nodes get_disconnected_nodes"
homI: fixes G :: "('b, 'c) monoid_scheme"   and h :: "'b \<Rightarrow> 'd"   and H :: "('d, 'e) monoid_scheme" assumes "\<And>x. x \<in> carrier G \<Longrightarrow> h x \<in> carrier H"   and "\<And>x y. \<lbrakk>x \<in> carrier G; y \<in> carrier G\<rbrakk> \<Longrightarrow> h (x \<otimes>\<^bsub>G\<^esub> y) = h x \<otimes>\<^bsub>H\<^esub> h y" shows "h \<in> hom G H"
square_def: fixes A :: "state \<times> state \<Rightarrow> bool"   and v :: "state \<Rightarrow> 'b" shows "[A]_v \<equiv> A \<or> unchanged v"
expr_list_inj_term_simp: fixes es :: "expr list" shows "\<langle>es\<rangle> = In3 es"
ircD: fixes f :: "'d set \<Rightarrow> 'd set"   and a :: "'d"   and B :: "'d set" assumes "irc f"   and "a \<notin> f (insert a B)" shows "f (insert a B) = f B"
meet_interchange: fixes w :: "'a"   and x :: "'a"   and y :: "'a"   and z :: "'a" shows "(w \<cdot> x) ; (y \<cdot> z) \<le> w ; y \<cdot> x ; z"
senior_senior: fixes \<Sigma> :: "'c set"   and \<delta> :: "'d \<Rightarrow> 'c \<Rightarrow> 'd"   and q\<^sub>0 :: "'d"   and w :: "nat \<Rightarrow> 'c"   and x :: "nat"   and n :: "nat" assumes "semi_mojmir \<Sigma> \<delta> q\<^sub>0 w" shows "semi_mojmir_def.senior \<delta> q\<^sub>0 w (semi_mojmir_def.senior \<delta> q\<^sub>0 w x n) n = semi_mojmir_def.senior \<delta> q\<^sub>0 w x n"
split_sum: shows "int (card G + sum card (\<G> - {G})) = \<v>"
V_def: shows "V \<equiv> {v. \<forall>x\<in>\<X>. (0::'a) \<le> v x}"
get_lower_bound_def: fixes U :: "'a set"   and V :: "'a set" shows "get_lower_bound U V = (SOME W. W \<in> I \<and> W \<subseteq> U \<and> W \<subseteq> V)"
set_fold: fixes f :: "'d list \<Rightarrow> 'd list \<Rightarrow> 'd list"   and g :: "'c \<Rightarrow> 'd list"   and xs :: "'c list"   and ys :: "'d list" assumes "\<And>x ys. set (f (g x) ys) = set (g x) \<union> set ys" shows "set (fold (\<lambda>x. f (g x)) xs ys) = (\<Union>x\<in>set xs. set (g x)) \<union> set ys"