hcju/isabelleps huggingface dataset:
exec_plan_fdom_subset: fixes as :: "(('a, 'b) fmap \<times> ('a, 'b) fmap) list"   and PROB :: "(('a, 'b) fmap \<times> ('a, 'b) fmap) set"   and s :: "('a, 'b) fmap" assumes "as \<in> valid_plans PROB" shows "fmdom' (exec_plan s as) \<subseteq> fmdom' s \<union> prob_dom PROB"
elims_3: fixes x :: "('b \<Rightarrow> 'b) \<times> 'b hrel"   and xa :: "('b \<Rightarrow> 'b) \<times> 'b hrel" assumes "\<not> x \<rightarrow>\<^sub>u xa" obtains \<sigma> :: "'b \<Rightarrow> 'b" and P :: "'b hrel" and \<rho> :: "'b \<Rightarrow> 'b" and Q :: "'b hrel" where "x = (\<sigma>, P)" and "xa = (\<rho>, Q)" and "\<not> \<langle>\<sigma>\<rangle>\<^sub>a ;; P \<sqsubseteq> \<langle>\<rho>\<rangle>\<^sub>a ;; Q"
ext_inject: fixes om :: "'d \<Rightarrow> 'f"   and am :: "'e \<Rightarrow> 'g"   and more :: "'h"   and om' :: "'d \<Rightarrow> 'f"   and am' :: "'e \<Rightarrow> 'g"   and more' :: "'h" shows "(\<lparr>om = om, am = am, \<dots> = more\<rparr> = \<lparr>om = om', am = am', \<dots> = more'\<rparr>) = (om = om' \<and> am = am' \<and> more = more')"
prems: fixes a :: "'a" assumes "a \<in> insert a__ A" shows "f a < \<top>"
zero_notin_S: shows "0 \<notin> (\<Union>(f, g)\<in>fundfoldpairs. {Abs_induced_automorph f g})"
assms_2: shows "t \<in> GSMP (trms\<^sub>l\<^sub>s\<^sub>t (proj l \<A>)) \<inter> GSMP (trms\<^sub>l\<^sub>s\<^sub>t (proj l' \<A>))"
Image_singleton: fixes r :: "('c \<times> 'b) set"   and a :: "'c" shows "r `` {a} = {b. (a, b) \<in> r}"
rp: shows "rat_to_int_poly (of_int_poly p) = (c, q)"
proj_sing1: fixes \<theta> :: "('a, 'sz) trm" shows "singleton \<theta> vid1 = \<theta>"
a_rcos_self: fixes H :: "'a set"   and G :: "('a, 'b) ring_scheme"   and x :: "'a" assumes "abelian_subgroup H G"   and "x \<in> carrier G" shows "x \<in> H +>\<^bsub>G\<^esub> x"
mult_four_block_mat: fixes A1 :: "'a mat"   and nr1 :: "nat"   and n1 :: "nat"   and B1 :: "'a mat"   and n2 :: "nat"   and C1 :: "'a mat"   and nr2 :: "nat"   and D1 :: "'a mat"   and A2 :: "'a mat"   and nc1 :: "nat"   and B2 :: "'a mat"   and nc2 :: "nat"   and C2 :: "'a mat"   and D2 :: "'a mat" assumes "A1 \<in> carrier_mat nr1 n1"   and "B1 \<in> carrier_mat nr1 n2"   and "C1 \<in> carrier_mat nr2 n1"   and "D1 \<in> carrier_mat nr2 n2"   and "A2 \<in> carrier_mat n1 nc1"   and "B2 \<in> carrier_mat n1 nc2"   and "C2 \<in> carrier_mat n2 nc1"   and "D2 \<in> carrier_mat n2 nc2" shows "four_block_mat A1 B1 C1 D1 * four_block_mat A2 B2 C2 D2 = four_block_mat (A1 * A2 + B1 * C2) (A1 * B2 + B1 * D2) (C1 * A2 + D1 * C2) (C1 * B2 + D1 * D2)"
fun_left_comm: fixes y :: "'b fset set"   and x :: "'b fset set"   and z :: "'b fset set" shows "y \<otimes>\<^sub>m x \<otimes>\<^sub>m z = x \<otimes>\<^sub>m y \<otimes>\<^sub>m z"
row_id_matrix: fixes i :: "nat"   and nr :: "nat"   and nc :: "nat" assumes "i < nr" shows "row (id_matrix nr nc) i = unit_vec nc i"
bounded: fixes n :: "nat" shows "\<bar>f (Suc n) - f n\<bar> \<le> c / x ^ n"
has_density_embed_measure: fixes f :: "'c \<Rightarrow> 'd"   and N :: "'c measure"   and f' :: "'d \<Rightarrow> 'c"   and M :: "'c measure"   and \<delta> :: "'c \<Rightarrow> ennreal" assumes "inj f"   and "\<And>x. x \<in> space N \<Longrightarrow> f' (f x) = x" shows "has_density (embed_measure M f) (embed_measure N f) (\<delta> \<circ> f') = has_density M N \<delta>"
inferences_from_def: fixes CC :: "'a literal multiset set" shows "sr_ext.inferences_from CC = {\<gamma> \<in> ground_sound_\<Gamma>. infer_from CC \<gamma>}"
DAList_Multiset_fold: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and fn :: "'b \<Rightarrow> nat \<Rightarrow> 'c \<Rightarrow> 'c"   and e :: "'c"   and al :: "('b, nat) alist" assumes "comp_fun_commute f"   and "\<And>a n x. fn a n x = (f a ^^ n) x" shows "fold_mset f e (Bag al) = DAList_Multiset.fold fn e al"
cases_scheme: fixes r :: "('b, 'c) graph_scheme" obtains nodes :: "'b set" and edges :: "('b \<times> 'b) set" and more :: "'c" where "r = \<lparr>nodes = nodes, edges = edges, \<dots> = more\<rparr>"
trivial_empty_or_singleton: fixes X :: "'a set" shows "trivial X = (X = {} \<or> X = {the_elem X})"
directed_eq_rtrancl: fixes x :: "'b"   and A :: "'b set"   and r :: "('b \<times> 'b) set" shows "(x \<in> directed_towards A r) = (\<exists>a\<in>A. (x, a) \<in> r\<^sup>*)"
elims: fixes x :: "trm option"   and y :: "trm option" assumes "Differentialo x = y" obtains \<theta> :: "trm" where "x = Aterm \<theta>" and "y = Aterm (Differential \<theta>)"   | "x = undeft" and "y = undeft"
collapse_1: fixes literal :: "'a literal" assumes "is_pos literal" shows "Pos (atm_of literal) = literal"
inj_semiring_hom_axioms: shows "inj_semiring_hom to_fract"
sup_assoc: fixes sup :: "'x \<Rightarrow> 'x \<Rightarrow> 'x"   and less_eq :: "'x \<Rightarrow> 'x \<Rightarrow> bool"   and less :: "'x \<Rightarrow> 'x \<Rightarrow> bool"   and x :: "'x"   and y :: "'x"   and z :: "'x" assumes "class.semilattice_sup sup less_eq less" shows "sup (sup x y) z = sup x (sup y z)"
axioms_2: fixes R :: "('h \<times> 'i) set"   and A :: "('h, 'j) graph_rec_scheme"   and B :: "('i, 'k) graph_rec_scheme" assumes "simulation R A B" shows "graph B"
prj_simps_10: shows "C.arr f_src_src.Prj\<^sub>1"
N_decomp: shows "\<exists>I J K. N = I + J \<and> z = I + K \<and> J \<noteq> {#} \<and> (\<forall>k\<in>#K. \<exists>j\<in>#J. (k, j) \<in> r)"
has_locks_acquire_locks_conv: fixes l :: "('b \<times> nat) option"   and t :: "'b"   and n :: "nat" assumes "may_lock l t" shows "has_locks (acquire_locks l t n) t = has_locks l t + n"
sWlsAbsDisj_def: fixes SEM :: "('g, 'h, 'i, 'sort, 'j, 'k, 'l) semDom_scheme" shows "sWlsAbsDisj SEM \<equiv> \<forall>us s us' s' sA. isInBar (us, s) \<and> isInBar (us', s') \<and> sWlsAbs SEM (us, s) sA \<and> sWlsAbs SEM (us', s') sA \<longrightarrow> us = us' \<and> s = s'"
interleave_rev1: fixes w :: "'b list"   and w1 :: "'b list"   and w2 :: "'b list" assumes "w \<in> w1 \<otimes> w2" shows "rev w \<in> rev w1 \<otimes> rev w2"
nn_integral_cmult: fixes f :: "'b \<Rightarrow> ennreal"   and M :: "'b measure"   and c :: "ennreal" assumes "f \<in> borel_measurable M" shows "\<integral>\<^sup>+ x. c * f x \<partial>M = c * integral\<^sup>N M f"
intros_14: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c :: "('e, 'd, 'f) com" shows "\<Gamma>\<turnstile>c \<down> Stuck"
face_of_Int: fixes t1 :: "'b set"   and S :: "'b set"   and t2 :: "'b set" assumes "t1 face_of S"   and "t2 face_of S" shows "t1 \<inter> t2 face_of S"
var_order_rel'': fixes x :: "char list"   and y :: "char list" shows "((x, y) \<in> var_order_rel) = (x < y)"
simps_1: fixes i :: "int"   and j :: "int" shows "local.wpl i j \<langle>\<rangle> = 0"
simplex_empty: fixes n :: "int" shows "(n simplex {}) = (n = - 1)"
frac_ge_0: fixes x :: "'a" shows "(0::'a) \<le> frac x"
no_deficient_nodes: shows "\<forall>v\<in>V - {s, t}. sum f (outgoing v) \<le> sum f (incoming v)"
clinear_continuous_at: fixes f :: "'b \<Rightarrow> 'c"   and x :: "'b" assumes "bounded_clinear f" shows "isCont f x"
subset_empty: fixes A :: "'g set" shows "(A \<subseteq> {}) = (A = {})"
Par1F: fixes P :: "pi"   and \<alpha> :: "freeRes"   and P' :: "pi"   and Q :: "pi" assumes "P \<Longrightarrow>\<alpha> \<prec> P'" shows "P \<parallel> Q \<Longrightarrow>\<alpha> \<prec> P' \<parallel> Q"
constant_function_closed: fixes R :: "('c, 'd) ring_scheme"   and a :: "'c"   and S :: "'e set" assumes "ring_functions R"   and "a \<in> carrier R" shows "constant_function S a \<in> carrier (function_ring S R)"
excess_non_negative: shows "\<forall>v\<in>V - {s, t}. (0::'capacity) \<le> excess f v"
remove_child_pointers_preserved: fixes w :: "((_) heap, exception, unit) prog"   and ptr :: "(_) object_ptr"   and owner_document :: "(_) document_ptr"   and h :: "(_) heap"   and h' :: "(_) heap" assumes "w \<in> remove_child_locs ptr owner_document"   and "h \<turnstile> w \<rightarrow>\<^sub>h h'" shows "object_ptr_kinds h = object_ptr_kinds h'"
drop_bit_eq_div: fixes 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 divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bit :: "'b \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and drop_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and take_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and n :: "nat"   and a :: "'b" assumes "class.semiring_bit_shifts plus minus zero times one divide modulo bit push_bit drop_bit take_bit" shows "drop_bit n a = divide a (power.power one times (numeral.numeral one plus (num.Bit0 num.One)) n)"
reflexive: fixes p :: "'d"   and A :: "('e, 'f, 'd, 'g) FSM_scheme" shows "p \<in> reachable A p"
eq_iff: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" assumes "class.order less_eq less" shows "(x = y) = (less_eq x y \<and> less_eq y x)"
left_commute: fixes b :: "'b"   and a :: "'b"   and c :: "'b" shows "b XOR a XOR c = a XOR b XOR c"
bis_Comp: fixes B1 :: "'d set"   and B2 :: "'e set"   and s1 :: "'d \<Rightarrow> ('f, 'd, 'e) F1"   and s2 :: "'e \<Rightarrow> ('g, 'd, 'e) F2"   and B1' :: "'h set"   and B2' :: "'i set"   and s1' :: "'h \<Rightarrow> ('f, 'h, 'i) F1"   and s2' :: "'i \<Rightarrow> ('g, 'h, 'i) F2"   and P1 :: "('d \<times> 'h) set"   and P2 :: "('e \<times> 'i) set"   and B1'' :: "'j set"   and B2'' :: "'k set"   and s1'' :: "'j \<Rightarrow> ('f, 'j, 'k) F1"   and s2'' :: "'k \<Rightarrow> ('g, 'j, 'k) F2"   and Q1 :: "('h \<times> 'j) set"   and Q2 :: "('i \<times> 'k) set" assumes "bis B1 B2 s1 s2 B1' B2' s1' s2' P1 P2"   and "bis B1' B2' s1' s2' B1'' B2'' s1'' s2'' Q1 Q2" shows "bis B1 B2 s1 s2 B1'' B2'' s1'' s2'' (P1 O Q1) (P2 O Q2)"
preserves_reflects_arr: fixes f :: "'a" shows "arr (f \<otimes> cod \<iota>) = arr f"
simps: fixes ts :: "'f \<Rightarrow> ('g \<times> 'h \<Rightarrow>f nat) option"   and t :: "'f" shows "free_thread_id ts t = (ts t = None)"
no_grey_refs_imp_valid_W_inv: fixes s :: "'e process_name \<Rightarrow> ('f, 'e, 'g, 'h) Model.local_state" assumes "no_grey_refs s"   and "\<forall>p. sys_mem_store_buffers p s = []" shows "valid_W_inv s"
semiring_of_sets_axioms: shows "semiring_of_sets \<Omega> (sigma_sets \<Omega> M)"
clinearI: fixes f :: "'a \<Rightarrow> 'b" assumes "\<And>b1 b2. f (b1 + b2) = f b1 + f b2"   and "\<And>r b. f (r *\<^sub>C b) = r *\<^sub>C f b" shows "clinear f"
wf_rderiv: fixes idx :: "'n"   and \<phi> :: "('a, 'k) aformula"   and x :: "'x" assumes "local.wf idx \<phi>" shows "local.wf idx (rderiv x \<phi>)"
remove_child_preserves_known_ptrs: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and child :: "(_) node_ptr"   and h' :: "(_) heap" assumes "heap_is_wellformed h"   and "type_wf h"   and "known_ptrs h"   and "h \<turnstile> remove_child ptr child \<rightarrow>\<^sub>h h'" shows "known_ptrs h'"
connectedin_eq_not_separated_subset: fixes X :: "'c topology"   and S :: "'c set" shows "connectedin X S = (S \<subseteq> topspace X \<and> (\<nexists>C1 C2. S \<subseteq> C1 \<union> C2 \<and> S \<inter> C1 \<noteq> {} \<and> S \<inter> C2 \<noteq> {} \<and> separatedin X C1 C2))"
split_list: fixes x :: "'f"   and xs :: "'f list" assumes "x \<in> set xs" shows "\<exists>ys zs. xs = ys @ x # zs"
isint_Mul: fixes e :: "MIR.num"   and bs :: "real list"   and c :: "int" assumes "isint e bs" shows "isint (Mul c e) bs"
3: shows "(\<Sum>i<n. if i = a then g i else (0::'b)) = (if a = a then g a else (0::'b))"
noops_sound: fixes \<Psi> :: "('a, 'b) sas_plus_problem"   and \<pi>s :: "('a, 'b) sas_plus_operator list" assumes "SAS_Plus_Semantics.is_serial_solution_for_problem (prob_with_noop \<Psi>) \<pi>s" shows "SAS_Plus_Semantics.is_serial_solution_for_problem \<Psi> (rem_noops \<pi>s)"
narrow_PrimT2: fixes G :: "prog"   and S :: "ty"   and pt :: "prim_ty" assumes "G\<turnstile>S\<succ>PrimT pt" shows "\<exists>t. S = PrimT t \<and> G\<turnstile>PrimT t\<preceq>PrimT pt"
word_eqI: fixes u :: "'c word"   and v :: "'c word" assumes "\<And>n. n < size u \<longrightarrow> u !! n = v !! n" shows "u = v"
sgn_neg: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sgn :: "'b \<Rightarrow> 'b"   and abs :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and a :: "'b" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one"   and "less a zero" shows "sgn a = uminus one"
irc_onI: fixes A :: "'a set"   and f :: "'a set \<Rightarrow> 'a set" assumes "\<And>B a. \<lbrakk>B \<subseteq> A; a \<in> A; a \<notin> f (insert a B)\<rbrakk> \<Longrightarrow> f (insert a B) = f B" shows "irc_on A f"
enat_def: fixes n :: "nat" shows "enat n = Abs_enat (Some n)"
nb: shows "\<forall>x\<in>set xs. bound0 (f x)"
Sup_inf: fixes B :: "'a set"   and a :: "'a" shows "\<Squnion> B \<sqinter> a = (\<Squnion>b\<in>B. b \<sqinter> a)"
le_lt_comparator_of_2: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "class.linorder less_eq less" shows "lt_of_comp (linorder.comparator_of less) = less"
fresh_3: fixes a :: "name" shows "a \<sharp> \<tau> = True"
vector_smult_component: fixes c :: "'d"   and y :: "('d, 'e) vec"   and i :: "'e" shows "(c *s y) $ i = c * y $ i"
check_extension_l_side_cond_err_def: fixes v :: "char list"   and p :: "(char list list \<times> int) list"   and p' :: "(char list list \<times> int) list"   and q :: "(char list list \<times> int) list" shows "check_extension_l_side_cond_err v p p' q = SPEC (\<lambda>_. True)"
OUTfromChCorrect_data12: shows "OUTfromChCorrect data12"
Rep_Arity_inc: fixes a' :: "Arity" shows "Rep_Arity (inc\<cdot>a') = Suc (Rep_Arity a')"
HS_E_eval: fixes G :: "Var \<Rightarrow> L"   and e :: "Expr"   and t :: "L"   and q :: "L"   and r :: "Var \<Rightarrow> Val"   and s :: "Var \<Rightarrow> Val" assumes "(G, e, t) \<in> HS_E"   and "EQ G q r s"   and "LEQ t q" shows "evalE e r = evalE e s"
prv_exu_exi: fixes x :: "'var"   and \<phi> :: "'fmla" assumes "x \<in> var"   and "\<phi> \<in> fmla"   and "prv (exu x \<phi>)" shows "prv (exi x \<phi>)"
ineq: fixes x :: "'a" assumes "x \<in> V" shows "\<bar>f x\<bar> \<le> c * \<parallel>x\<parallel>"
that: fixes u :: "real" assumes "0 \<le> u"   and "u \<le> 1"   and "\<And>x. 0 \<le> x \<and> x < u \<Longrightarrow> g x \<in> interior S"   and "g u \<notin> interior S"   and "u = 0 \<or> g u \<in> closure S" shows "thesis"
assms_3: shows "s \<equiv> if 0 \<le> p then 1 else - 1"
VPow_vempty: shows "VPow 0 = set {0}"
body_body': shows "unwrap oo body oo wrap = body' oo unwrap oo wrap"
assms: shows "a \<in>\<^sub>\<circ> r\<inverse>\<^sub>\<circ> `\<^sub>\<circ> A"
set_tree_delete: fixes join :: "('b \<times> 'c) tree \<Rightarrow> 'b \<Rightarrow> ('b \<times> 'c) tree \<Rightarrow> ('b \<times> 'c) tree"   and inv :: "('b \<times> 'c) tree \<Rightarrow> bool"   and t :: "('b \<times> 'c) tree"   and x :: "'b" assumes "Set2_Join join inv"   and "Tree2.bst t" shows "Tree2.set_tree (Set2_Join.delete join x t) = Tree2.set_tree t - {x}"
simps_2: fixes e :: "(name, nat) finfun"   and A :: "fmla"   and B :: "fmla" shows "eval_fmla e (A OR B) = (eval_fmla e A \<or> eval_fmla e B)"
mult_1: fixes V :: "'b set"   and x :: "'b" assumes "vectorspace V"   and "x \<in> V" shows "1 \<cdot> x = x"
consistent_def: fixes A :: "(('b, 'c) vec, 'd) vec"   and b :: "('b, 'd) vec" shows "consistent A b = (\<exists>x. is_solution x A b)"
fps_X_plus_const_power: fixes c :: "'a"   and n :: "nat" assumes "c \<noteq> (0::'a)" shows "(fps_X + fps_const c) ^ n = fps_const (c ^ n) * (fps_binomial (of_nat n) oo fps_const (inverse c) * fps_X)"
case_eq_if: fixes f1 :: "'b \<Rightarrow> 'a"   and f2 :: "'b \<Rightarrow> 'a"   and literal :: "'b literal" shows "(case literal of Pos x \<Rightarrow> f1 x | Neg x \<Rightarrow> f2 x) = (if is_pos literal then f1 (atm_of literal) else f2 (atm_of literal))"
strict_prefixE: fixes xs :: "'a list"   and ys :: "'a list" assumes "strict_prefix xs ys" obtains "prefix xs ys" and "xs \<noteq> ys"
size_bst_of_list_distinct: fixes xs :: "'a list" assumes "distinct xs" shows "size (bst_of_list xs) = length xs"
simps: fixes R :: "'a sequent list \<times> 'a sequent \<Rightarrow> bool"   and a1 :: "'a sequent"   and a2 :: "nat" shows "derivablep R a1 a2 = ((\<exists>C. a1 = C \<and> a2 = 0 \<and> R ([], C)) \<or> (\<exists>r m. a1 = snd r \<and> a2 = m + 1 \<and> R r \<and> fst r \<noteq> [] \<and> (\<forall>p\<in>set (fst r). \<exists>n\<le>m. derivablep R p n)))"
lunit'_simps_3: fixes f :: "'d" assumes "D.ide f" shows "trg\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[f] = trg\<^sub>D f"
divide_fds_def: fixes f :: "'a fds"   and g :: "'a fds" shows "f / g = f * inverse g"
not_vertex: fixes \<Gamma> :: "('a, 'b) web_scheme"   and x :: "'a" assumes "countable_bipartite_web \<Gamma>"   and "x \<notin> A \<Gamma>"   and "x \<notin> B \<Gamma>" shows "\<not> vertex \<Gamma> x"
one_D_integral_measurable: shows "(\<lambda>x. LBINT y. f (x, y)) \<in> borel_measurable lborel"
count_of_join_raw: fixes ys :: "('b \<times> nat) list"   and xs :: "('b \<times> nat) list"   and x :: "'b" assumes "distinct (map fst ys)" shows "count_of xs x + count_of ys x = count_of (join_raw (\<lambda>x (x, y). x + y) xs ys) x"
length_nat_non0: fixes n :: "nat" assumes "n \<noteq> 0" shows "length_nat n = Suc (length_nat (n div 2))"
induct: fixes P :: "'s \<Rightarrow> bool"   and a0 :: "'s" assumes "\<And>s. (\<And>x2. generator g s = Skip x2 \<Longrightarrow> P x2) \<Longrightarrow> P s" shows "P a0"
prems: fixes x :: "'a" assumes "x \<in> insert x__ A" shows "g x \<noteq> 0"
NotR: fixes F :: "'a formula"   and \<Gamma> :: "'a formula multiset"   and \<Delta> :: "'a formula multiset" assumes "F, \<Gamma> \<Rightarrow> \<Delta>" shows "\<Gamma> \<Rightarrow> \<^bold>\<not> F, \<Delta>"
longest_common_prefix_prefix2: fixes xs :: "'b list"   and ys :: "'b list" shows "xs \<and>\<^sub>p ys \<le>p ys"
zero_carrier_vec: fixes n :: "nat" shows "0\<^sub>v n \<in> carrier_vec n"
exec: shows "\<Gamma>\<turnstile> \<langle>c,Normal s\<rangle> =n\<Rightarrow> t"
ras_get_def: fixes s :: "'c array \<times> nat"   and i :: "nat" shows "ras_get s i \<equiv> let (a, n) = s in array_get a (n - (i + 1))"
basic_trans_rules_31: fixes A :: "'b set"   and B :: "'b set"   and c :: "'b" assumes "order.greater_eq B A"   and "c \<in> A" shows "c \<in> B"
Inv_Ide: fixes V :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and src :: "'a \<Rightarrow> 'a"   and trg :: "'a \<Rightarrow> 'a"   and a :: "'a local.term" assumes "bicategorical_language V src trg"   and "bicategorical_language.Ide V src trg a" shows "bicategorical_language.Inv V a = a"
cntE: fixes f :: "'b \<Rightarrow> 'c"   and S :: "'b set set"   and T :: "'c set set" assumes "f \<in> cnt S T" obtains "f \<in> carrier\<^bsub>S\<^esub> \<rightarrow> carrier\<^bsub>T\<^esub>" and "\<forall>m. m open\<^bsub>T\<^esub> \<longrightarrow> carrier\<^bsub>S\<^esub> \<inter> f -` m open\<^bsub>S\<^esub>"
smallest_accepting_rank_def: shows "smallest_accepting_rank\<^sub>R \<equiv> if accept\<^sub>R' then Some (LEAST i. finite fail \<and> finite (local.merge i) \<and> infinite (succeed i)) else None"
I: fixes xa :: "'a" shows "\<bottom>\<bottom> = - (- xa \<squnion> - (- xa))"
acnc: shows "all_chains (no_call_to c) \<Gamma> rs"
split_asm: fixes P :: "'b \<Rightarrow> bool"   and f1 :: "'b"   and f2 :: "'c \<Rightarrow> 'c list \<Rightarrow> 'b"   and list :: "'c list" shows "P (case list of [] \<Rightarrow> f1 | x # xa \<Rightarrow> f2 x xa) = (\<not> (list = [] \<and> \<not> P f1 \<or> (\<exists>x21 x22. list = x21 # x22 \<and> \<not> P (f2 x21 x22))))"
nn_integral_snd: fixes f :: "val \<times> val \<Rightarrow> ennreal" assumes "f \<in> borel_measurable (stock_measure t1 \<Otimes>\<^sub>M N')" shows "\<integral>\<^sup>+ y. \<integral>\<^sup>+ x. f (x, y) \<partial>stock_measure t1 \<partial>N' = integral\<^sup>N (stock_measure t1 \<Otimes>\<^sub>M N') f"
intro: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<mu> :: "'b arrow_of_spans_data" assumes "arrow_of_spans C \<mu>"   and "identity_arrow_of_spans_axioms C \<mu>" shows "identity_arrow_of_spans C \<mu>"
add_strict_increasing2: fixes a :: "'c multiset"   and b :: "'c multiset"   and c :: "'c multiset" assumes "{#} \<subseteq># a"   and "b \<subset># c" shows "b \<subset># a + c"
AE_integrable_fst': fixes f :: "'b \<times> 'c \<Rightarrow> 'd" assumes "integrable (lborel \<Otimes>\<^sub>M lborel) f" shows "AE x in lborel. integrable lborel (\<lambda>y. f (x, y))"
topspace_subtopology: fixes U :: "'c topology"   and V :: "'c set" shows "topspace (subtopology U V) = topspace U \<inter> V"
that: fixes K :: "'a set"   and C :: "nat \<Rightarrow> 'a set" assumes "negligible K"   and "\<And>n. compact (C n)"   and "S = \<Union> (range C) \<union> K" shows "thesis"
bind_is_OK_E: fixes h :: "'a"   and f :: "('a, 'b, 'd) prog"   and g :: "'d \<Rightarrow> ('a, 'b, 'c) prog" assumes "h \<turnstile> ok Heap_Error_Monad.bind f g" obtains x :: "'d" and h' :: "'a" where "h \<turnstile> f \<rightarrow>\<^sub>r x" and "h \<turnstile> f \<rightarrow>\<^sub>h h'" and "h' \<turnstile> ok g x"
subset_divisors_adds: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and a :: "'a"   and b :: "'a" assumes "class.comm_monoid_add plus zero" shows "({c. plus.adds plus c a} \<subseteq> {c. plus.adds plus c b}) = plus.adds plus a b"
word_rep_drop: fixes bl :: "bool list" shows "to_bl (of_bl bl) = replicate (LENGTH('a) - length bl) False @ drop (length bl - LENGTH('a)) bl"
state_measure_def: fixes V :: "nat set"   and \<Gamma> :: "nat \<Rightarrow> pdf_type" shows "state_measure V \<Gamma> \<equiv> Pi\<^sub>M V (\<lambda>x. stock_measure (\<Gamma> x))"
in_components_connected: fixes c :: "'b set"   and s :: "'b set" assumes "c \<in> components s" shows "connected c"
sub_num_simps_4: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and one :: "'a"   and k :: "num" assumes "class.neg_numeral minus plus zero uminus" shows "neg_numeral.sub minus plus one (num.Bit0 k) num.One = numeral.numeral one plus (Num.BitM k)"
nres_monad1: fixes x :: "'d"   and f :: "'d \<Rightarrow> 'c nres" shows "RETURN x \<bind> f = f x"
vector_transpose_matrix: fixes x :: "('b, 'd) vec"   and A :: "(('b, 'd) vec, 'c) vec" shows "x v* Finite_Cartesian_Product.transpose A = A *v x"
less_fun_def: fixes f :: "'q \<Rightarrow> 'r"   and g :: "'q \<Rightarrow> 'r" shows "(f < g) = (f \<le> g \<and> \<not> g \<le> f)"
real_of_int_float: fixes x :: "int" shows "real_of_int x \<in> float"
simps_3: fixes va :: "nat" shows "nat_to_sch (Suc (Suc va)) = (let u = mod7 (c_fst (Suc (Suc va))); v = c_snd (Suc (Suc va)); v1 = c_fst v; v2 = c_snd v; sch1 = nat_to_sch v1 in Let (nat_to_sch v2) (loc_f u sch1))"
abs_eq: fixes xa :: "int"   and x :: "int" shows "Fract xa x = Abs_Rat (if x = 0 then (0, 1) else (xa, x))"
intros: fixes A :: "('f \<times> ('d, 'e) cconstraint \<times> 'g \<times> 'd list \<times> 'f) set \<times> ('f \<Rightarrow> ('d, 'e) cconstraint)"   and k :: "'d \<Rightarrow> 'e"   and X :: "'d set" assumes "\<forall>(x, m)\<in>clkp_set A. m \<le> k x \<and> x \<in> X \<and> m \<in> \<nat>"   and "collect_clkvt (trans_of A) \<subseteq> X"   and "finite X" shows "valid_abstraction A X k"
lowEquivalence_relevant_nodes_Entry: fixes s :: "'state"   and s' :: "'state"   and S :: "'node set" assumes "s \<approx>\<^sub>L s'"   and "(_High_) \<notin> backward_slice S" shows "\<forall>V\<in>rv S (_Entry_). state_val s V = state_val s' V"
irreflexiv_Elem: fixes a :: "ZF" shows "\<not> Elem a a"
ceq: shows "equal_base.list_distinct ceq' xs \<or> ID CEQ('a) = None"
succ_imp_graph_outside: fixes l :: "lit"   and cnf :: "lit uprod set" assumes "var l \<notin> vars_of_cnf cnf" shows "succ_imp_graph cnf l = {}"
at_left_ereal: fixes r :: "real" shows "at_left (ereal r) = filtermap ereal (at_left r)"