hcju/isabelleps huggingface dataset:
length2_def: fixes l :: "tm list" shows "length2 l \<equiv> case l of [] \<Rightarrow> False | [x] \<Rightarrow> False | [x, xa] \<Rightarrow> True | x # xa # ab # list \<Rightarrow> False"
word_of_int_uint: fixes w :: "'b word" shows "word_of_int (uint w) = w"
sees_methods_rec: fixes P :: "'a prog"   and C :: "char list"   and D :: "char list"   and fs :: "fdecl list"   and ms :: "'a mdecl list"   and Mm :: "char list \<Rightarrow> ((staticb \<times> ty list \<times> ty \<times> 'a) \<times> char list) option"   and Mm' :: "char list \<Rightarrow> ((staticb \<times> ty list \<times> ty \<times> 'a) \<times> char list) option" assumes "class P C = \<lfloor>(D, fs, ms)\<rfloor>"   and "C \<noteq> Object"   and "P \<turnstile> D sees_methods Mm"   and "Mm' = Mm ++ (map_option (\<lambda>m. (m, C)) \<circ> map_of ms)" shows "P \<turnstile> C sees_methods Mm'"
wordinterval_eq_def: fixes r1 :: "'b wordinterval"   and r2 :: "'b wordinterval" shows "wordinterval_eq r1 r2 = (wordinterval_subset r1 r2 \<and> wordinterval_subset r2 r1)"
pdevs_apply_inner_scaleR_pdevs: fixes a :: "'b"   and x :: "'b pdevs"   and i :: "nat" shows "pdevs_apply (inner_scaleR_pdevs a x) i = (a \<bullet> pdevs_apply x i) *\<^sub>R pdevs_apply x i"
f_3: shows "prob_space (density M (\<lambda>x. ennreal (f x)))"
val_normif: fixes s :: "'a \<Rightarrow> bool"   and env :: "('a, bool) mapping"   and t :: "'a ifex"   and t1 :: "'a ifex"   and t2 :: "'a ifex" assumes "agree s env" shows "val_ifex (normif env t t1 t2) s = val_ifex (if val_ifex t s then t1 else t2) s"
pure_dual_def: shows "pure_dual \<equiv> map_fun id (map_fun id B) (\<lambda>pure. pure)"
elims: fixes x :: "'b \<Rightarrow> 'c"   and xa :: "('d \<times> ('c \<Rightarrow> 'a) \<times> 'e) \<times> 'b"   and y :: "'a" assumes "solution_nsc x xa = y" obtains solution_s :: "'b \<Rightarrow> 'c" and asi :: "'d" and tv :: "'c \<Rightarrow> 'a" and ui :: "'e" and s :: "'b" where "x = solution_s" and "xa = ((asi, tv, ui), s)" and "y = tv (solution_s s)"
P3_a: fixes x :: "'a" shows "(x l\<rightarrow> (1::'a)) r\<rightarrow> (1::'a) = (1::'a)"
IH_2: assumes "isin x3_ x" shows "size x3_ = Suc (size (delete x x3_))"
rQuot_star: fixes c :: "'b"   and A :: "'b list set" shows "rQuot c (star A) = star A @@ rQuot c A"
subset: shows "set (characterize_consistent_signs_at_roots_copr p qs) \<subseteq> set (reduction_signs p qs signs subsets (M_mat signs subsets))"
suminf_le_const: fixes f :: "nat \<Rightarrow> 'b"   and x :: "'b" assumes "summable f"   and "\<And>n. sum f {..<n} \<le> x" shows "suminf f \<le> x"
clear_subsets: shows "subsets_smash (length qs1) (get_subsets (calculate_data p qs1)) (get_subsets (calculate_data p qs2)) = get_subsets (snd (combine_systems p (qs1, calculate_data p qs1) (qs2, calculate_data p qs2)))"
scale_cancel_left: fixes scale :: "'d \<Rightarrow> 'e \<Rightarrow> 'e"   and a :: "'d"   and x :: "'e"   and y :: "'e" assumes "vector_space scale" shows "(scale a x = scale a y) = (x = y \<or> a = (0::'d))"
widens_Cons: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and x :: "ty"   and xs :: "ty list"   and ys :: "ty list" shows "P \<turnstile> (x # xs) [\<le>] ys = (\<exists>z zs. ys = z # zs \<and> P \<turnstile> x \<le> z \<and> P \<turnstile> xs [\<le>] zs)"
scaleR_left_diff_distrib: fixes a :: "real"   and b :: "real"   and x :: "'b" shows "(a - b) *\<^sub>R x = a *\<^sub>R x - b *\<^sub>R x"
drop_bit_of_0: 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 n :: "nat" assumes "class.semiring_bit_shifts plus minus zero times one divide modulo bit push_bit drop_bit take_bit" shows "drop_bit n zero = zero"
content_eq_zero_iff: fixes p :: "'d poly" shows "(content p = (0::'d)) = (p = 0)"
abs_eq: fixes xa :: "nat"   and x :: "nat \<Rightarrow> 'c" shows "vec xa x = Abs_vec (xa, mk_vec xa x)"
get_component_def: shows "Core_DOM_Components.get_component \<equiv> l_get_component\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_component Core_DOM_Functions.get_root_node Core_DOM_Functions.to_tree_order"
minit_def: fixes \<phi> :: "Formula.formula" shows "minit \<phi> = (let n = Formula.nfv \<phi> in \<lparr>mstate_i = 0, mstate_m = minit0 n \<phi>, mstate_n = n\<rparr>)"
AE_uniform_measure: fixes M :: "'a measure"   and A :: "'a set"   and P :: "'a \<Rightarrow> bool" assumes "emeasure M A \<noteq> 0"   and "emeasure M A < \<infinity>" shows "almost_everywhere (uniform_measure M A) P = set_almost_everywhere A M P"
rep_eq: fixes x :: "('a, 'b) mapping" shows "Mapping.lookup x = mapping.rep x"
tail_is_vert: fixes u :: "nat"   and v :: "nat" assumes "(u, v) \<in> g_E G" shows "u \<in> set (g_V G)"
lcm_complex_def: shows "lcm = normalization_euclidean_semiring_class.lcm"
subcategory_axioms: shows "subcategory (\<cdot>) (left a)"
path_split_snoc: fixes prog :: "cmd"   and n :: "w_node"   and as :: "(w_node \<times> (char list \<Rightarrow> val option) edge_kind \<times> w_node) list"   and n' :: "w_node" assumes "While_CFG.path prog n as n'"   and "as \<noteq> []" obtains a' :: "w_node \<times> (char list \<Rightarrow> val option) edge_kind \<times> w_node" and as' :: "(w_node \<times> (char list \<Rightarrow> val option) edge_kind \<times> w_node) list" where "as = as' @ [a']" and "While_CFG.path prog n as' (sourcenode a')" and "valid_edge prog a'" and "n' = targetnode a'"
usubst_upd_pr_var_id: fixes x :: "'e \<Longrightarrow> 'f" assumes "vwb_lens x" shows "[x \<mapsto>\<^sub>s &x] = id"
simps: fixes a1 :: "nat"   and a2 :: "'b tensor" shows "cprank_max a1 a2 = ((\<exists>ds. a1 = 0 \<and> a2 = tensor0 ds) \<or> (\<exists>A B j. a1 = Suc j \<and> a2 = A + B \<and> dims A = dims B \<and> cprank_max1 A \<and> cprank_max j B))"
closure_halfspace_gt: fixes a :: "'c"   and b :: "real" assumes "a \<noteq> (0::'c)" shows "closure {x. b < a \<bullet> x} = {x. b \<le> a \<bullet> x}"
A\<^sub>A_updates_x: shows "mem\<^sub>A_of mem\<^sub>1\<^sub>C x = v \<longrightarrow> mem\<^sub>A_of mem\<^sub>2\<^sub>C x \<noteq> v'"
unit_triangle_1: fixes a :: "'a" assumes "B.obj a" shows "\<i>\<^sub>B[a] \<star>\<^sub>B a = (a \<star>\<^sub>B \<i>\<^sub>B[a]) \<cdot>\<^sub>B \<a>\<^sub>B[a, a, a]"
subgroup_self: shows "subgroup (carrier (Z 1)) (Z 1)"
sup_ge1: fixes x :: "'b"   and y :: "'b" shows "x \<le> sup x y"
rep_eq: fixes x :: "ennreal" shows "enn2ereal (inverse x) = inverse (enn2ereal x)"
UniqueSucState_Target_ChiRel_StepConf: fixes TS :: "('d \<times> (('d, 'e) expr \<times> ('f data \<Rightarrow> bool) \<times> 'e set \<times> 'f pupdate) \<times> 'd) set"   and ST :: "('d, 'e, 'f) status"   and S :: "'d"   and A :: "('d, 'e, 'f) seqauto"   and T :: "'d"   and U :: "'d"   and V :: "'d" assumes "TS \<in> HPT ST"   and "S \<in> StepConf (HA ST) (Conf ST) TS"   and "A \<in> SAs (HA ST)"   and "A \<in> the (CompFun (HA ST) S)"   and "T \<in> States A"   and "U \<in> States A"   and "T \<in> StepConf (HA ST) (Conf ST) TS"   and "T \<noteq> U"   and "(V, U) \<in> ChiRel (HA ST)"   and "U \<in> HAInitStates (HA ST)" shows "V \<notin> Target TS"
assms: shows "\<S>\<bullet>\<C> \<turnstile> [Callcl cl] : t1s' _> t2s'"
simps_1: fixes e :: "'a"   and a :: "'b"   and ts :: "('a, 'b) BinomialHeapStruc.BinomialTree list" shows "BinomialHeapStruc.tree_invar (BinomialHeapStruc.Node e a 0 ts) = (ts = [])"
measurable_sfirst2: fixes P :: "'c \<Rightarrow> 'd stream \<Rightarrow> bool"   and N :: "'c measure"   and M :: "'d measure" assumes "Measurable.pred (N \<Otimes>\<^sub>M stream_space M) (\<lambda>(x, \<omega>). P x \<omega>)" shows "(\<lambda>(x, \<omega>). sfirst (P x) \<omega>) \<in> N \<Otimes>\<^sub>M stream_space M \<rightarrow>\<^sub>M count_space UNIV"
residue_mult_comm: fixes k :: "nat"   and x :: "int"   and y :: "int" shows "x \<otimes>\<^bsub>residue_ring (p ^ k)\<^esub> y = y \<otimes>\<^bsub>residue_ring (p ^ k)\<^esub> x"
Od_carrier: fixes R :: "('a \<times> 'a) set"   and S :: "'a set" shows "carrier (D\<lparr>carrier := S, rel := R\<rparr>) = S"
simps_41: fixes f1 :: "nat \<Rightarrow> nat list \<Rightarrow> 'a"   and f2 :: "nat \<Rightarrow> nat list \<Rightarrow> 'a"   and f3 :: "form \<Rightarrow> form \<Rightarrow> 'a"   and f4 :: "form \<Rightarrow> form \<Rightarrow> 'a"   and f5 :: "form \<Rightarrow> 'a"   and f6 :: "form \<Rightarrow> 'a"   and x5 :: "form" shows "(case FAll x5 of PAtom x xa \<Rightarrow> f1 x xa | NAtom x xa \<Rightarrow> f2 x xa | FConj x xa \<Rightarrow> f3 x xa | FDisj x xa \<Rightarrow> f4 x xa | FAll x \<Rightarrow> f5 x | FEx x \<Rightarrow> f6 x) = f5 x5"
meet_assoc: fixes x :: "'c"   and y :: "'c"   and z :: "'c" shows "(x \<sqinter> y) \<sqinter> z = x \<sqinter> (y \<sqinter> z)"
prems_3: shows "f permutes \<Union> (sset (xs # xss))"
the_adj_chamber_neq: fixes C :: "'a set"   and z :: "'a set" assumes "folding.chamber C"   and "z \<lhd> C" shows "the_adj_chamber C z \<noteq> C"
same_prefix_prefix: fixes xs :: "'c list"   and ys :: "'c list"   and zs :: "'c list" shows "xs @ ys \<le> xs @ zs = ys \<le> zs"
SUP_constant: fixes Inf :: "'c set \<Rightarrow> 'c"   and Sup :: "'c set \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and c :: "'c"   and A :: "'d set" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "Sup ((\<lambda>y. c) ` A) = (if A = {} then bot else c)"
valid_single: fixes fg :: "('a, 'b, 'c, 'd, 'e) flowgraph_rec_scheme"   and s :: "'a list" shows "valid fg {#s#}"
tm_defined_2: shows "{s \<in> {t1..<t2}. flow0 x s \<in> {a<--<b}} \<noteq> {}"
fderiv: shows "((\<lambda>t. fst (mk_v I ODE (sol 0, b) (sol t))) has_derivative (\<lambda>t'. t' *\<^sub>R (\<chi>i. if i \<in> ODE_vars I ODE then ODE_sem I ODE (sol s) $ i else 0))) (at s within {0..t})"
Method_i_i_i_o_o_o_oI: fixes x :: "'a prog"   and xa :: "String.literal"   and xb :: "String.literal"   and xc :: "ty list"   and xd :: "ty"   and xe :: "'a option"   and xf :: "String.literal" assumes "x \<turnstile> xa sees xb: xc\<rightarrow>xd = xe in xf" shows "pred.eval (Method_i_i_i_o_o_o_o x xa xb) (xc, xd, xe, xf)"
subgraph_preserves_hom: fixes A :: "('e, 'f) labeled_graph"   and B :: "('e, 'f) labeled_graph"   and X :: "('e, 'g) labeled_graph"   and h :: "('g \<times> 'f) set" assumes "subgraph A B"   and "graph_homomorphism X A h" shows "graph_homomorphism X B h"
iso_runit: fixes a :: "'a" assumes "S.ide a" shows "S.iso (M.runit a)"
om_nu_tau: fixes x :: "'a" shows "\<nu> x\<^sup>\<omega> + \<nu> x\<^sup>\<star> \<cdot> \<tau> x \<le> x\<^sup>\<omega>"
af_\<mu>LTL: fixes \<phi> :: "'a ltln"   and w :: "nat \<Rightarrow> 'a set" assumes "\<phi> \<in> \<mu>LTL" shows "w \<Turnstile>\<^sub>n \<phi> = (\<exists>i. af \<phi> (prefix i w) \<sim> true\<^sub>n)"
irreducible\<^sub>d_mE: fixes m :: "int"   and f :: "int poly" assumes "poly_mod.irreducible\<^sub>d_m m f" obtains "0 < degree (poly_mod.Mp m f)" and "\<And>g h. \<lbrakk>degree (poly_mod.Mp m g) < degree (poly_mod.Mp m f); degree (poly_mod.Mp m h) < degree (poly_mod.Mp m f)\<rbrakk> \<Longrightarrow> poly_mod.Mp m f \<noteq> poly_mod.Mp m (g * h)"
assms_1: shows "Pair \<notin> \<Union> (funs_term ` SMP (trms\<^sub>s\<^sub>s\<^sub>t S))"
gen_power_0: fixes G :: "('c, 'd) cyclic_group_scheme"   and r :: "nat" assumes "cyclic_group G"   and "\<^bold>g\<^bsub>G\<^esub> [^]\<^bsub>G\<^esub> r = \<one>\<^bsub>G\<^esub>"   and "r < order G" shows "r = 0"
cinner_mult_right': fixes scaleR :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and scaleC :: "complex \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and norm :: "'b \<Rightarrow> real"   and sgn :: "'b \<Rightarrow> 'b"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool"   and cinner :: "'b \<Rightarrow> 'b \<Rightarrow> complex"   and a :: "'c"   and b :: "'c"   and m :: "complex" assumes "class.complex_inner scaleR scaleC plus zero minus uminus dist norm sgn uniformity open cinner" shows "\<langle>a, b * of_complex m\<rangle> = \<langle>a, b\<rangle> * m"
ord_Zp_def: shows "ord_Zp = padic_val p"
polysub_norm: fixes p :: "'b poly"   and q :: "'b poly" assumes "isnpoly p"   and "isnpoly q" shows "isnpoly (p -\<^sub>p q)"
append_simpl3: fixes y :: "'b list"   and i :: "nat"   and z :: "'c"   and zs :: "'c list" assumes "length y < i" shows "i < length (z # zs) * length y \<longrightarrow> i - length y < length zs * length y"
expand_ord_def: fixes \<phi> :: "'d ltlr" shows "expand_ord \<phi> \<equiv> inv_image (finite_psupset (old_next_limit \<phi>) <*lex*> less_than) (\<lambda>(n, ns). (old_next_pair ` ns, size_set (new n)))"
simps_1: shows "SkewBinomialHeapStruc.rank_skew_invar [] = True"
delete_add_edge: fixes a :: "'c"   and H :: "('c, 'd) graph"   and c :: "'c"   and w :: "'d" assumes "a \<in> nodes H"   and "c \<in> nodes H"   and "(a, w, c) \<notin> edges H" shows "delete_edge a w c (add_edge a w c H) = H"
cond_spmf_pair_spmf: fixes p :: "'h spmf"   and q :: "'i spmf"   and A :: "'h set"   and B :: "'i set" shows "cond_spmf (pair_spmf p q) (A \<times> B) = pair_spmf (cond_spmf p A) (cond_spmf q B)"
UnE: fixes c :: "'f"   and A :: "'f set"   and B :: "'f set" assumes "c \<in> A \<union> B" obtains "c \<in> A"   | "c \<in> B"
EndPhase1_HInv3_t: fixes s :: "state"   and s' :: "state"   and t :: "Proc"   and p :: "Proc"   and q :: "Proc"   and d :: "Disk" assumes "EndPhase1 s s' t"   and "HInv3_inner s p q d"   and "t \<noteq> p"   and "t \<noteq> q" shows "HInv3_inner s' p q d"
simps: fixes a :: "'a"   and f :: "'a \<Rightarrow> 'a"   and x :: "'a" shows "(a \<in> orbit f x) = (a = f x \<or> (\<exists>y. a = f y \<and> y \<in> orbit f x))"
f3: shows "(r'' |\<^bsub>rel\<^esub> s'') \<otimes>\<^bsub>rec_rng_of_frac\<^esub> (r' |\<^bsub>rel\<^esub> s') = (r'' \<otimes> r' |\<^bsub>rel\<^esub> s'' \<otimes> s')"
assms_3: fixes i :: "nat"   and j :: "nat" assumes "i < nr"   and "j < nc" shows "f (i, j) = f' (i, j)"
line_extension_is_subgroup: fixes E :: "'a set"   and a :: "'a" assumes "subgroup E (add_monoid R)"   and "a \<in> carrier R" shows "subgroup (line_extension K a E) (add_monoid R)"
simps_2: fixes delta :: "'e"   and inp :: "'a \<Rightarrow> ('a, 'b, 'c, 'd, 'e) qTerm option"   and binp :: "'b \<Rightarrow> ('a, 'b, 'c, 'd, 'e) qAbs option"   and delta' :: "'e"   and inp' :: "'a \<Rightarrow> ('a, 'b, 'c, 'd, 'e) qTerm option"   and binp' :: "'b \<Rightarrow> ('a, 'b, 'c, 'd, 'e) qAbs option" shows "(qOp delta inp binp #= qOp delta' inp' binp') = (delta = delta' \<and> sameDom inp inp' \<and> sameDom binp binp' \<and> liftAll2 (#=) inp inp' \<and> liftAll2 ($=) binp binp')"
Goal: fixes P :: "pi"   and a :: "name"   and x :: "name"   and P' :: "pi" assumes "P \<longmapsto> a<x> \<prec> P'"   and "x \<sharp> P"   and "valid P" shows "valid P'"
bind_runit: fixes x :: "'b option" shows "Option.bind x Some = x"
REL_INTF_def: fixes R :: "'a"   and I :: "'b" shows "REL_INTF R I \<equiv> True"
complete_join_all: fixes ts :: "'a tree23s"   and n :: "nat" assumes "\<forall>t\<in>trees ts. complete t \<and> height t = n" shows "complete (join_all ts)"
del_vert_def: fixes G :: "('c, 'd) pre_digraph"   and v :: "'c" shows "pre_digraph.del_vert G v = \<lparr>verts = verts G - {v}, arcs = {a \<in> arcs G. tail G a \<noteq> v \<and> head G a \<noteq> v}, tail = tail G, head = head G\<rparr>"
sat_eqI: fixes m :: "'b sat"   and n :: "'b sat" assumes "nat_of m = nat_of n" shows "m = n"
append_cols_def: fixes A :: "'a mat"   and B :: "'a mat" shows "A @\<^sub>c B = four_block_mat A B (0\<^sub>m 0 (dim_col A)) (0\<^sub>m 0 (dim_col B))"
inv_inv_eq: fixes f :: "'a \<Rightarrow> 'b" assumes "bij f" shows "inv (inv f) = f"
composite_simps_2: shows "src ((\<mu> \<star> \<nu>) \<star> \<pi>) = src \<pi>"
sets_Collect_conj: fixes P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool" assumes "{x \<in> \<Omega>. P x} \<in> M"   and "{x \<in> \<Omega>. Q x} \<in> M" shows "{x \<in> \<Omega>. Q x \<and> P x} \<in> M"
ivp_solsD_4: fixes X :: "real \<Rightarrow> 'b"   and f :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and U :: "'b \<Rightarrow> real set"   and S :: "'b set"   and t\<^sub>0 :: "real"   and s :: "'b" assumes "X \<in> Sols f U S t\<^sub>0 s" shows "t\<^sub>0 \<in> U s"
empty: shows "Lcm\<^sub>f\<^sub>i\<^sub>n {} = (1::'a)"
RDirProd_list_iso1: fixes R :: "('a, 'b) ring_scheme"   and Rs :: "('a, 'b) ring_scheme list" shows "(\<lambda>(a, as). a # as) \<in> ring_iso (RDirProd R (RDirProd_list Rs)) (RDirProd_list (R # Rs))"
finter_lower2: fixes A :: "'b fset"   and B :: "'b fset" shows "A |\<inter>| B |\<subseteq>| B"
hom_decomp_concat: fixes pss :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list list" assumes "\<And>ps. ps \<in> set pss \<Longrightarrow> hom_decomp ps" shows "hom_decomp (concat pss)"
may_lock_upd_lock_conv: fixes l :: "('b \<times> nat) option"   and t :: "'b"   and L :: "lock_action" assumes "lock_action_ok l t L" shows "may_lock (upd_lock l t L) t = may_lock l t"
If_le: fixes b :: "bexp"   and c1 :: "'b acom"   and c2 :: "'b acom"   and S :: "'b"   and c :: "'b acom" shows "(IF b THEN c1 ELSE c2 {S} \<le> c) = (\<exists>c1' c2' S'. c = IF b THEN c1' ELSE c2' {S'} \<and> c1 \<le> c1' \<and> c2 \<le> c2' \<and> S \<le> S')"
unit_vec_def: fixes n :: "nat"   and i :: "nat" shows "unit_vec n i = vec n (\<lambda>j. if j = i then 1::'b else (0::'b))"
unit_factor: fixes G :: "('c, 'd) monoid_scheme"   and a :: "'c"   and b :: "'c" assumes "Group.comm_monoid G"   and "a \<otimes>\<^bsub>G\<^esub> b \<in> Units G"   and "a \<in> carrier G"   and "b \<in> carrier G" shows "a \<in> Units G"
Abs_dataspace_inverse: fixes y :: "'a set list" assumes "y \<in> dataspace" shows "Rep_dataspace (Abs_dataspace y) = y"
apply_rsp': fixes R1 :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and R2 :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'e"   and x :: "'b"   and y :: "'c" assumes "rel_fun R1 R2 f g"   and "R1 x y" shows "R2 (f x) (g y)"
integral_eq_nn_integral: fixes f :: "'c \<Rightarrow> real"   and M :: "'c measure" assumes "f \<in> borel_measurable M"   and "AE x in M. 0 \<le> f x" shows "integral\<^sup>L M f = enn2real (\<integral>\<^sup>+ x. ennreal (f x) \<partial>M)"
hyps_3: assumes "list_all stateful_strand_step.is_Equality (unlabel A)" shows "list_all stateful_strand_step.is_Equality (unlabel (dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t A))"
set: fixes x1 :: "'e"   and x2 :: "'e stream" shows "sset (x1 ## x2) = insert x1 (sset x2)"
cond_def: fixes P :: "'b \<times> 'b \<Rightarrow> bool"   and b :: "'b \<times> 'b \<Rightarrow> bool"   and Q :: "'b \<times> 'b \<Rightarrow> bool" shows "P \<triangleleft> b \<triangleright>  Q \<equiv> b \<and> P \<or> \<not> b \<and> Q"
rep_eq: fixes x :: "'a \<Rightarrow>\<^sub>C\<^sub>L 'b" shows "cblinfun_apply (- x) = (\<lambda>xa. - cblinfun_apply x xa)"
m_s_h: fixes X :: "char list set"   and S :: "'a st" assumes "finite X" shows "m\<^sub>s S X \<le> h * card X"
variable_disjoint_def: fixes C :: "'b eclause"   and D :: "'b eclause" shows "variable_disjoint C D = (vars_of_cl (cl_ecl C) \<inter> vars_of_cl (cl_ecl D) = {})"
prj_tuple_2: fixes f :: "'a"   and g :: "'a"   and h :: "'a"   and k :: "'a" assumes "commutative_square f g h k" shows "\<p>\<^sub>1[f, g] \<cdot> \<langle>h \<lbrakk>f, g\<rbrakk> k\<rangle> = h"
RSpan_contains_spanset: fixes ms :: "'m list" assumes "set ms \<subseteq> M" shows "set ms \<subseteq> RSpan ms"
H_seq: fixes r :: "('a \<times> 'a) set"   and y :: "('a \<times> 'a) set"   and q :: "('a \<times> 'a) set"   and p :: "('a \<times> 'a) set"   and x :: "('a \<times> 'a) set" assumes "rel_kat.H r y q"   and "rel_kat.H p x r" shows "rel_kat.H p (x ; y) q"
mod_diff1_eq1: fixes a :: "nat"   and b :: "nat"   and m :: "nat" assumes "a \<le> b"   and "a mod m \<le> b mod m" shows "(b - a) mod m = b mod m - a mod m"
nth_list_update_alt: fixes xs :: "'a list"   and i :: "nat"   and x :: "'a"   and j :: "nat" shows "xs[i := x] ! j = (if i < length xs \<and> i = j then x else xs ! j)"
eq_iff: fixes a :: "'b"   and b :: "'b" shows "(a = b) = (b \<le> a \<and> a \<le> b)"
FAccNull\<^sub>1: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> nat exp) list) list"   and e :: "nat exp"   and s\<^sub>0 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> val list"   and s\<^sub>1 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> val list"   and F :: "char list"   and D :: "char list" assumes "P \<turnstile>\<^sub>1 \<langle>e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>null,s\<^sub>1\<rangle>" shows "P \<turnstile>\<^sub>1 \<langle>e\<bullet>F{D},s\<^sub>0\<rangle> \<Rightarrow> \<langle>THROW NullPointer,s\<^sub>1\<rangle>"
exhaust: fixes y :: "('b, 'c) result" obtains x1 :: "'b" where "y = Rval x1"   | x2 :: "'c error_result" where "y = Rerr x2"
has_contour_integral_eq: fixes f :: "complex \<Rightarrow> complex"   and y :: "complex"   and p :: "real \<Rightarrow> complex"   and g :: "complex \<Rightarrow> complex" assumes "(f has_contour_integral y) p"   and "\<And>x. x \<in> path_image p \<Longrightarrow> f x = g x" shows "(g has_contour_integral y) p"
simps_2: fixes j :: "'a"   and J :: "'a list"   and s :: "'a simplex_state" shows "assert_all_simplex (j # J) s = (case assert_simplex j s of Inl x \<Rightarrow> Inl x | Inr x \<Rightarrow> assert_all_simplex J x)"
AxWVT: fixes m :: "Body"   and k :: "Body"   and b :: "Body"   and x :: "'a Point" assumes "IOb m"   and "IOb k" shows "k sees b at x = m sees b at wvt m k x"
thm_relation_negation_5_1: fixes v :: "i"   and F :: "\<Pi>\<^sub>1" shows "[F \<^bold>\<noteq> F\<^sup>- in v]"
bit0_def: fixes n :: "'a" shows "bit0 n = (2::'a) * n"
sum_Suc_diff': fixes m :: "nat"   and n :: "nat"   and f :: "nat \<Rightarrow> 'a" assumes "m \<le> n" shows "(\<Sum>i = m..<n. f (Suc i) - f i) = f n - f m"
croots2: fixes p :: "complex poly" assumes "degree p = 2" shows "{x. poly p x = 0} = set (croots2 p)"
copyp0: fixes x :: "'a"   and y :: "'a"   and z :: "'a"   and k :: "nat" assumes "- (x \<squnion> - y) = z" shows "- (x \<squnion> - (y \<squnion> k \<otimes> (x \<squnion> z))) = z"
coprime_dvd_mult_right_iff: fixes one :: "'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and gcd :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and lcm :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and a :: "'a"   and c :: "'a"   and b :: "'a" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize"   and "algebraic_semidom.coprime times one a c" shows "dvd.dvd times a (times c b) = dvd.dvd times a b"
422: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "- - (x \<sqinter> y) \<sqinter> (- x \<sqinter> z) = bot"
weak_hb_def: fixes history :: "nat \<Rightarrow> 'c event list"   and msg_id :: "'c \<Rightarrow> 'd"   and m1 :: "'c"   and m2 :: "'c" assumes "network history msg_id" shows "network.weak_hb history m1 m2 \<equiv> network.hb history m1 m2 \<or> m1 = m2"
mult_le_cancel_iff1: fixes z :: "'a"   and x :: "'a"   and y :: "'a" assumes "(0::'a) < z" shows "(x * z \<le> y * z) = (x \<le> y)"