hcju/isabelleps huggingface dataset:
single_valuedD: fixes r :: "('h \<times> 'i) set"   and x :: "'h"   and y :: "'i"   and z :: "'i" assumes "single_valued r"   and "(x, y) \<in> r"   and "(x, z) \<in> r" shows "y = z"
comp_null_2: fixes f :: "'c local.term set" shows "f \<cdot> \<F>C.null = \<F>C.null"
perm_swap_1: fixes a :: "name"   and b :: "name"   and x :: "'b" shows "[(a, b)] \<bullet> [(a, b)] \<bullet> x = x"
assms_1: shows "s\<^sub>2 = s\<^sub>1(r \<mapsto> x)"
tendsto_artanh: fixes f :: "'a \<Rightarrow> real"   and a :: "real"   and F :: "'a filter" assumes "(f \<longlongrightarrow> a) F"   and "- 1 < a"   and "a < 1" shows "((\<lambda>x. artanh (f x)) \<longlongrightarrow> artanh a) F"
While: fixes Q :: "(char list \<Rightarrow> int) \<Rightarrow> bool" shows "\<turnstile>\<^sub>t {wp\<^sub>t c Q} c {Q}"
agree2: shows "Vagree (sol 0, b) (mk_v I (OVar vid1) (sol 0, b) (sol 0)) (FVT (f1 fid1 vid1))"
w\<^sub>1: shows "if a \<in> alphabet t\<^sub>1 then if b \<in> alphabet t\<^sub>1 then weight (swapLeaves t\<^sub>1 w\<^sub>a a w\<^sub>b b) + freq t\<^sub>1 a + freq t\<^sub>1 b = weight t\<^sub>1 + w\<^sub>a + w\<^sub>b else weight (swapLeaves t\<^sub>1 w\<^sub>a a w\<^sub>b b) + freq t\<^sub>1 a = weight t\<^sub>1 + w\<^sub>b else if b \<in> alphabet t\<^sub>1 then weight (swapLeaves t\<^sub>1 w\<^sub>a a w\<^sub>b b) + freq t\<^sub>1 b = weight t\<^sub>1 + w\<^sub>a else weight (swapLeaves t\<^sub>1 w\<^sub>a a w\<^sub>b b) = weight t\<^sub>1"
STL1: fixes F :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool" assumes "\<turnstile> F" shows "\<turnstile> \<box>F"
exhaust: fixes y :: "'b \<times> 'c" obtains x1 :: "'b" and x2 :: "'c" where "y = (x1, x2)"
ddi_value: fixes l :: "nat"   and \<sigma> :: "'var \<Rightarrow> 'val"   and v :: "'var"   and k :: "nat" assumes "l dd\<^bsup>path \<sigma>,v\<^esup>\<rightarrow> k" shows "(\<sigma>\<^bsup>l\<^esup>) v = (\<sigma>\<^bsup>Suc k\<^esup>) v"
cont_hiding1: fixes Y :: "nat \<Rightarrow> 'b process"   and A :: "'b set" assumes "chain Y" shows "chain (\<lambda>i. Y i \ A)"
getGMUserCom_def: fixes s :: "('f, 'd, 'e) state" shows "getGMUserCom s = (getGMUser s, getGMCom s)"
lepoll_restricted_funspace: fixes A :: "'b set"   and B :: "'c set"   and k :: "'b \<Rightarrow> 'c" shows "{f. f ` A \<subseteq> B \<and> {x. f x \<noteq> k x} \<subseteq> A \<and> finite {x. f x \<noteq> k x}} \<lesssim> Fpow (A \<times> B)"
cases: fixes a :: "'d set" assumes "finite a" obtains "a = {}"   | A :: "'d set" and aa :: "'d" where "a = insert aa A" and "finite A"
sorted_many_eq: fixes x :: "'a"   and y :: "'a"   and zs :: "'a list" shows "Specialisation_Examples.sorted (x \<Colon> y \<Colon> zs) = (x \<le> y \<and> Specialisation_Examples.sorted (y \<Colon> zs))"
find_first_prefix: fixes x :: "'b"   and xs :: "'b list"   and i :: "nat"   and n :: "nat" assumes "find_first x xs = Some i"   and "i < n" shows "find_first x (take n xs) = Some i"
\<Phi>_simp: fixes yx :: "'d \<times> 'c" assumes "DopxC.ide yx" shows "\<Phi>.map yx = S.mkArr (HomC.set (F (fst yx), snd yx)) (HomD.set (fst yx, G (snd yx))) (inD \<circ> \<phi> (fst yx) \<circ> HomC.\<psi> (F (fst yx), snd yx))"
nocommonPP: shows "set (PPc c) \<inter> set (PPV V) = {}"
assms_2: fixes u :: "'a" assumes "u \<in> basis" shows "f *\<^sub>V u = (0::'b)"
equalOnD: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "equalOn A f g" shows "\<forall>x\<in>A. f x = g x"
prefix_fininf_prefix: fixes k :: "nat"   and w :: "'a stream" shows "stake k w \<le>\<^sub>F\<^sub>I w"
cyz: fixes z :: "'a"   and y :: "'a" assumes "B < norm z"   and "B < norm y" shows "connected_component (- S) y z"
val_by_cf: shows "val = (\<Sum>(u, v)\<in>outgoing s. cf (v, u))"
assms_4: shows "proj2_incident q (apply_cltn2_line l C)"
inject_1: fixes x1 :: "'g"   and y1 :: "'g" shows "(Inl x1 = Inl y1) = (x1 = y1)"
wary_arg: fixes arity_sym :: "'a \<Rightarrow> enat"   and arity_var :: "'b \<Rightarrow> enat"   and t :: "('a, 'b) tm" assumes "arity.wary arity_sym arity_var t" shows "arity.wary arity_sym arity_var (arg t)"
wls_swapAbs_involutive2: fixes us :: "'varSort"   and s :: "'sort"   and A :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set"   and zs :: "'varSort"   and y :: "'var"   and x :: "'var" assumes "wlsAbs (us, s) A" shows "A $[x \<and> y]_zs $[y \<and> x]_zs = A"
HSem_subst_expr_below: fixes ESem :: "'f \<Rightarrow> ('e \<Rightarrow> 'd) \<rightarrow> 'd"   and e1 :: "'f"   and x :: "'e"   and e2 :: "'f"   and \<Gamma> :: "('e \<times> 'f) list"   and \<rho> :: "'e \<Rightarrow> 'd" assumes "ESem e1\<cdot>(has_ESem.HSem ESem ((x, e2) # \<Gamma>)\<cdot>\<rho>) \<sqsubseteq> ESem e2\<cdot>(has_ESem.HSem ESem ((x, e2) # \<Gamma>)\<cdot>\<rho>)" shows "has_ESem.HSem ESem ((x, e1) # \<Gamma>)\<cdot>\<rho> \<sqsubseteq> has_ESem.HSem ESem ((x, e2) # \<Gamma>)\<cdot>\<rho>"
is_path_undir_append: fixes G :: "('a, 'b) graph"   and v :: "'a"   and p1 :: "('a \<times> 'b \<times> 'a) list"   and u :: "'a"   and p2 :: "('a \<times> 'b \<times> 'a) list"   and w :: "'a" assumes "valid_graph G"   and "is_path_undir G v p1 u"   and "is_path_undir G u p2 w" shows "is_path_undir G v (p1 @ p2) w"
lookup3: fixes a :: "coname"   and \<Delta> :: "(coname \<times> ty) list"   and y :: "name"   and x :: "name" assumes "a \<sharp> idc \<Delta> y" shows "lookupa x a (idc \<Delta> y) = Ax x a"
x: shows "x \<notin> TER (current_of_bipartite f)"
restrict_SomeD: fixes m :: "'e \<Rightarrow> 'd option"   and A :: "'e set"   and x :: "'e"   and y :: "'d" assumes "(m|\<^bsub>A\<^esub>) x = Some y" shows "m x = Some y"
id_fresh: fixes a :: "name"   and \<Gamma> :: "(name \<times> ty) list" assumes "a \<sharp> \<Gamma>" shows "a \<sharp> SN.id \<Gamma>"
times_X_pow_coeff: fixes g :: "nat \<Rightarrow> 'a"   and k :: "nat"   and n :: "nat" assumes "g \<in> carrier P" shows "(monom P \<one> k \<otimes>\<^bsub>P\<^esub> g) (n + k) = g n"
subset_pairs_def: fixes A :: "('e \<times> 'e) set"   and B :: "('e \<times> 'e) set" shows "(A \<subseteq>\<^sub>p B) = (\<forall>x. x \<in>\<^sub>p A \<longrightarrow> x \<in>\<^sub>p B)"
simps_5: fixes bs :: "'a list"   and a :: "poly"   and b :: "poly" shows "\<lparr>Sub a b\<rparr>\<^sub>p\<^bsup>bs\<^esup> = \<lparr>a\<rparr>\<^sub>p\<^bsup>bs\<^esup> - \<lparr>b\<rparr>\<^sub>p\<^bsup>bs\<^esup>"
intros_3: fixes fg :: "('a, 'b, 'c, 'd, 'e) flowgraph_rec_scheme"   and p :: "'b"   and r :: "'a list"   and c :: "'a list multiset" shows "((return fg p # r, c), LRet, r, c) \<in> trss fg"
subseqs_def: fixes P :: "nat \<Rightarrow> (nat \<Rightarrow> nat) \<Rightarrow> bool" shows "Diagonal_Subsequence.subseqs P \<equiv> \<forall>n s. strict_mono s \<longrightarrow> (\<exists>r'. strict_mono r' \<and> P n (s \<circ> r'))"
seq_char: fixes g :: "'c \<times> 'c"   and f :: "'c \<times> 'c" shows "C.VV.seq g f = (C.VV.arr f \<and> C.VV.arr g \<and> C.VxV.seq g f)"
rev_hsubsetD: fixes c :: "hf"   and A :: "hf"   and B :: "hf" assumes "c \<^bold>\<in> A"   and "A \<le> B" shows "c \<^bold>\<in> B"
lm015: fixes B :: "'d set"   and A :: "'c set" shows "possible_allocations_rel' A B = possible_allocations_rel A B"
f_seq_lim: fixes x :: "'a" shows "(\<lambda>i. f (seq x i)) \<longlonglongrightarrow> g x"
sup_lsubgroup: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and A :: "'b set"   and a :: "'b"   and b :: "'b" assumes "class.lgroup minus plus zero uminus inf less_eq less sup"   and "A \<in> lgroup.lsubgroup plus zero uminus inf sup"   and "a \<in> A"   and "b \<in> A" shows "sup a b \<in> A"
Suc_lessD: fixes m :: "nat"   and n :: "nat" assumes "Suc m < n" shows "m < n"
assms_2: shows "round_robin n0 \<sigma> s = \<lfloor>(t, \<lfloor>(ta, x', m')\<rfloor>, \<sigma>')\<rfloor>"
leftderives_pad: fixes \<alpha> :: "symbol list"   and \<beta> :: "symbol list"   and u :: "symbol list"   and v :: "symbol list" assumes "leftderives \<alpha> \<beta>"   and "is_word u"   and "is_sentence v" shows "leftderives (u @ \<alpha> @ v) (u @ \<beta> @ v)"
simps: fixes ires :: "cell list"   and rs :: "nat"   and l :: "cell list"   and r :: "cell list" shows "wcode_on_left_moving_2_O ires rs (l, r) = (\<exists>ln rn. l = Bk # Oc # ires \<and> r = Oc # Bk \<up> ln @ Bk # Bk # Oc \<up> Suc rs @ Bk \<up> rn)"
Gromov_product_add: fixes e :: "'b"   and x :: "'b"   and y :: "'b" shows "Gromov_product_at e x y + Gromov_product_at x e y = dist e x"
selection_axioms: fixes Sts :: "('a literal multiset set \<times> 'a literal multiset set \<times> 'a literal multiset set) llist" shows "selection (S_Q Sts)"
word_and_full_mask_simp: shows "x && mask LENGTH(32) = x"
new_shadow_root_get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t: fixes h :: "(_) heap"   and h' :: "(_) heap"   and new_shadow_root_ptr :: "(_) shadow_root_ptr"   and ptr :: "(_) object_ptr"   and getter :: "(_) RObject_ext \<Rightarrow> 'bc" assumes "h \<turnstile> new\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_M \<rightarrow>\<^sub>h h'"   and "h \<turnstile> new\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_M \<rightarrow>\<^sub>r new_shadow_root_ptr"   and "ptr \<noteq> cast\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r new_shadow_root_ptr" shows "preserved (get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t ptr getter) h h'"
simps_1: fixes uu :: "char list \<Rightarrow> bool"   and uv :: "'a pfm \<Rightarrow> bool" shows "peval uu uv \<^bold>\<bottom>\<^sub>! = False"
a_coset_add_zero: fixes M :: "'c set" assumes "M \<subseteq> carrier S" shows "M +>\<^bsub>S\<^esub> \<zero>\<^bsub>S\<^esub> = M"
mnhelper2: shows "of_bl (to_bl b) < 2 ^ (m - n)"
pathstart_reversepath: fixes g :: "real \<Rightarrow> 'a" shows "pathstart (reversepath g) = pathfinish g"
im_jointfun1: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'e"   and m :: "nat"   and g :: "nat \<Rightarrow> 'e" shows "jointfun n f m g ` {j. j \<le> Suc (n + m)} = f ` {j. j \<le> n} \<union> g ` {j. j \<le> m}"
ord_equiv_trans: fixes E :: "('g, 'h) Order_scheme"   and F :: "('i, 'j) Order_scheme" assumes "Order E"   and "Order F"   and "ord_equiv D E"   and "ord_equiv E F" shows "ord_equiv D F"
assms_2: shows "(\<lambda>n. LINT x|M. norm (u n x)) \<longlonglongrightarrow> 0"
c'_le_c: fixes A :: "'a" shows "C_of A \<le> CA_of A"
A1: shows "\<forall>S. (SourceTerm S, TargetTerm (\<lbrakk>S\<rbrakk>)) \<in> Rel \<and> (TargetTerm (\<lbrakk>S\<rbrakk>), SourceTerm S) \<in> Rel"
converse_Id_on: fixes A :: "'d set" shows "(Id_on A)\<inverse> = Id_on A"
thetaIfZO_def: shows "thetaIfZO \<equiv> {(if tst then c1 else c2, if tst then d1 else d2) |tst c1 c2 d1 d2. compatTst tst \<and> c1 \<approx>01 d1 \<and> c2 \<approx>01 d2}"
transfer_operations_rl_8: shows "(RETURN o2 appr_sup_inner ops optns, Sup_inners) \<in> appr_rell \<rightarrow> rl_rel \<rightarrow> \<langle>rnv_rel\<rangle>nres_rel"
dom_simp_1: shows "partial_magma.dom (concrete_category.COMP {False, True} (\<lambda>x y. Collect (graph.path {False, True} {False, True} (\<lambda>_. False) (\<lambda>_. True) x y)) (\<lambda>_ _ _ g f. f @ g)) quiver.Zero = quiver.Zero"
comp_simp: fixes g :: "'a"   and f :: "'a" assumes "S.seq g f" shows "S.comp g f = g \<cdot> f"
external_from_internal_invariant_vimage: fixes S :: "('a, 'b) spec"   and P :: "'a set"   and Q :: "'b set" assumes "reach S \<subseteq> P"   and "P \<subseteq> obs S -` Q" shows "oreach S \<subseteq> Q"
is_lub_bin: fixes below :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "class.po below"   and "below x y" shows "po.is_lub below {x, y} y"
\<psi>\<^sub>1_is_state: fixes n :: "nat" assumes "1 \<le> n" shows "state (n + 1) (Matrix.mat (2 ^ (n + 1)) 1 (\<lambda>x. complex_of_real (case x of (i, j) \<Rightarrow> if even i then 1 / sqrt 2 ^ (n + 1) else - 1 / sqrt 2 ^ (n + 1))))"
absorb_def: fixes d :: "('g, 'h) distinguisher"   and conv :: "('g, 'h, 'e, 'f) converter" shows "absorb d conv = map_gpv fst id (inline run_converter d conv)"
assms_2: shows "\<forall>a\<in>carrier R. \<forall>b\<in>carrier R. a \<otimes> b = \<zero> \<longrightarrow> a = \<zero> \<or> b = \<zero>"
hyps: fixes p :: "(nat \<times> int) list" assumes "dm \<le> length b"   and "length p = length b"   and "\<And>d. d < dm \<Longrightarrow> odd (ix p d) \<and> lv b d \<le> lv p d \<and> ix p d < (ix b d + 1) * 2 ^ (lv p d - lv b d) \<and> (ix b d - 1) * 2 ^ (lv p d - lv b d) < ix p d"   and "\<And>d. \<lbrakk>dm \<le> d; d < length b\<rbrakk> \<Longrightarrow> p ! d = b ! d" shows "p \<in> grid b {0..<dm}"
phytagorean_theorem_norm: fixes x :: "'a"   and y :: "'a" assumes "orthogonal x y" shows "(norm (x + y))\<^sup>2 = (norm x)\<^sup>2 + (norm y)\<^sup>2"
partitioning1_empty1: fixes ts :: "'b set list"   and ss :: "'b set list" assumes "{} \<notin> set ts" shows "set (partitioning1 ss ts) - {{}} = set (partitioning1 ss ts)"
pmf_cond: fixes p :: "'e pmf"   and s :: "'e set"   and x :: "'e" assumes "set_pmf p \<inter> s \<noteq> {}" shows "pmf (cond_pmf p s) x = (if x \<in> s then pmf p x / measure_pmf.prob p s else 0)"
fillAndUpdate_def: fixes len :: "nat"   and f :: "nat \<Rightarrow> 'b"   and i :: "nat"   and "value" :: "'b"   and fillValue :: "'b"   and j :: "nat" shows "fillAndUpdate len f i value fillValue j \<equiv> if j = i then value else if len \<le> j \<and> j < i then fillValue else f j"
take_prefix: fixes f :: "nat \<Rightarrow> nat option"   and k :: "nat"   and n :: "nat" assumes "f \<in> \<R>"   and "k \<le> n" shows "prefix f k = take (Suc k) (prefix f n)"
commit_correct: shows "okamoto_\<Sigma>_commit.abstract_com.correct"
conventional: fixes g :: "('a, 'b, 'c) gen_cfg_wf"   and n :: "'a"   and ns :: "'a list"   and m :: "'a"   and v :: "'b \<times> 'a \<times> Def"   and x :: "'a"   and v' :: "'b \<times> 'a \<times> Def" assumes "gen_wf.path2 g n ns m"   and "n \<notin> set (tl ns)"   and "v \<in> gen_wf.braun_ssa.allDefs g n"   and "v \<in> gen_wf.braun_ssa.allUses g m"   and "x \<in> set (tl ns)"   and "v' \<in> gen_wf.braun_ssa.allDefs g x" shows "gen_wf_var g v' \<noteq> gen_wf_var g v"
condA: shows "\<forall>a b c. (a, b) \<in> Rel \<and> (c, b) \<in> Rel \<and> (a, c) \<notin> Rel \<longrightarrow> (b, a) \<in> Rel \<or> (b, c) \<in> Rel"
steps_decomp: fixes xs :: "'a list"   and ys :: "'a list" assumes "steps (xs @ ys)"   and "xs \<noteq> []"   and "ys \<noteq> []" shows "steps xs \<and> steps ys \<and> last xs \<rightarrow> hd ys"
cis_inverse: fixes a :: "real" shows "inverse (cis a) = cis (- a)"
discI: fixes option :: "'f option"   and x2 :: "'f" assumes "option = \<lfloor>x2\<rfloor>" shows "option \<noteq> None"
root_int_floor_pos_upper: fixes p :: "nat"   and x :: "int" assumes "p \<noteq> 0"   and "0 \<le> x" shows "x < (root_int_floor_pos p x + 1) ^ p"
c_next_events: fixes xp :: "'e \<times> 'f"   and step :: "'g \<Rightarrow> 'e \<Rightarrow> 'g"   and out :: "'g \<Rightarrow> 'e \<Rightarrow> 'f"   and s\<^sub>0 :: "'g"   and xps :: "('e \<times> 'f) list" shows "(xp \<in> next_events (c_process step out s\<^sub>0) xps) = (\<exists>X. (xps @ [xp], X) \<in> c_failures step out s\<^sub>0)"
run_stake: fixes A :: "('a, 'b) nba"   and w :: "'a stream"   and V :: "(nat \<times> 'b) set"   and r :: "(nat \<times> 'b) stream"   and p :: "nat \<times> 'b"   and k :: "nat" assumes "grun A w V r p" shows "gpath A w V (stake k r) p"
simps_6: fixes v1 :: "'a"   and l1 :: "value" shows "local.less_aexp_aux (V v1) (L l1) = False"
\<psi>_in_hom: fixes x :: "'a"   and g :: "'b"   and y :: "'b" assumes "Adj.C.ide x"   and "\<guillemotleft>g : y \<rightarrow>\<^sub>D G x\<guillemotright>" shows "\<guillemotleft>\<psi> x g : F y \<rightarrow>\<^sub>C x\<guillemotright>"
that: fixes s :: "nat \<Rightarrow> real" assumes "filterlim s at_top sequentially"   and "(f \<circ> s) \<longlonglongrightarrow> p" shows "thesis"
assms_1: shows "\<exists>p\<^sub>1\<in>set ps. dist (case p\<^sub>0 of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) (case p\<^sub>1 of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) < \<delta>"
RETURN_refine_iff: fixes x :: "'f"   and R :: "('f \<times> 'g) set"   and y :: "'g" shows "(RETURN x \<le> \<Down> R (RETURN y)) = ((x, y) \<in> R)"
homI: fixes f :: "'e"   and A :: "'d"   and B :: "'d" assumes "f \<in> Ar\<^bsub>BB\<^esub>"   and "Dom\<^bsub>BB\<^esub> f = A"   and "Cod\<^bsub>BB\<^esub> f = B" shows "f \<in> Hom\<^bsub>BB\<^esub> A B"
rreq_oip: shows "paodv i \<TTurnstile> onl \<Gamma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V (\<lambda>(\<xi>, l). l \<in> {PRreq-:3..PRreq-:9} \<union> {PRreq-:15, PRreq-:24, PRreq-:26} \<longrightarrow> oip \<xi> \<in> kD (rt \<xi>) \<and> (osn \<xi> < sqn (rt \<xi>) (oip \<xi>) \<or> sqn (rt \<xi>) (oip \<xi>) = osn \<xi> \<and> the (dhops (rt \<xi>) (oip \<xi>)) \<le> Suc (hops \<xi>) \<and> the (flag (rt \<xi>) (oip \<xi>)) = val))"
1: shows "bp (P\<^sub>1 \<union> wrap B\<^sub>1 \<union> wrap {} \<union> (P\<^sub>2 \<union> wrap B\<^sub>2) \<union> wrap {u} \<union> {{v} |v. v \<in> V - {u}})"
add_neg_nonpos: fixes a :: "'c multiset"   and b :: "'c multiset" assumes "a \<subset># {#}"   and "b \<subseteq># {#}" shows "a + b \<subset># {#}"
inside_Un_outside: fixes S :: "'b set" shows "inside S \<union> outside S = - S"
intro: fixes r :: "('d \<times> 'd) set" assumes "Linear_order r" shows "MaxR r"
l6_13_1: fixes P :: "'p"   and A :: "'p"   and B :: "'p" assumes "P Out A B"   and "P A Le P B" shows "Bet P A B"
assms: shows "max_Var_floatariths fas \<le> DIM('a)"
one_reducible_to_def: shows "one_reducible_to = (\<lambda>A B. \<exists>f. one_reducible_to_via A B f)"
nbt_nest: fixes y :: "'a"   and x :: "'a" shows "no_better_than y carrier relation \<subseteq> no_better_than x carrier relation \<or> no_better_than x carrier relation \<subseteq> no_better_than y carrier relation"
cod_comp: fixes g :: "'a \<times> 'a"   and f :: "'a \<times> 'a" assumes "VV.seq g f" shows "VV.cod (VV.comp g f) = VV.cod g"
vempty_nin: fixes a :: "V" shows "a \<notin>\<^sub>\<circ> []\<^sub>\<circ>"
added_files_Deliver_Append_diff_collapse: fixes e :: "'b"   and e' :: "'b"   and i :: "'c"   and j :: "'c" assumes "e \<noteq> e'" shows "added_files [Deliver (i, Append j e)] e' = []"
single_valuedI: fixes r :: "('b \<times> 'c) set" assumes "\<And>x y z. \<lbrakk>(x, y) \<in> r; (x, z) \<in> r\<rbrakk> \<Longrightarrow> y = z" shows "single_valued r"
braun_iff_braun_indices: fixes t :: "'b tree" shows "braun t = (braun_indices t = {1..size t})"
ModelIkTpart_def: fixes wtFsym :: "'a \<Rightarrow> bool"   and wtPsym :: "'b \<Rightarrow> bool"   and arOf :: "'a \<Rightarrow> 'c list"   and resOf :: "'a \<Rightarrow> 'c"   and parOf :: "'b \<Rightarrow> 'c list"   and \<Phi> :: "('a, 'b) lit list set"   and infTp :: "'c \<Rightarrow> bool"   and prot :: "'c \<Rightarrow> bool"   and protFw :: "'c \<Rightarrow> bool"   and intT :: "'c \<Rightarrow> univ \<Rightarrow> bool"   and intF :: "'a \<Rightarrow> univ list \<Rightarrow> univ"   and intP :: "'b \<Rightarrow> univ list \<Rightarrow> bool" shows "ModelIkTpart wtFsym wtPsym arOf resOf parOf \<Phi> infTp prot protFw intT intF intP \<equiv> ProblemIkTpart wtFsym wtPsym arOf resOf parOf \<Phi> infTp prot protFw \<and> CM.Model wtFsym wtPsym arOf resOf parOf \<Phi> intT intF intP"
char_approx1: fixes M :: "real measure"   and n :: "nat"   and t :: "real" assumes "real_distribution M"   and "\<And>k. k \<le> n \<Longrightarrow> integrable M (\<lambda>x. x ^ k)" shows "cmod (char M t - (\<Sum>k\<le>n. (\<i> * complex_of_real t) ^ k / fact k * complex_of_real (LINT x|M. x ^ k))) \<le> 2 * \<bar>t\<bar> ^ n / fact n * (LINT x|M. \<bar>x\<bar> ^ n)"
t_ex: shows "\<exists>t. degree (b * f - t * #v) < degree v"
pq_order_subgrp_normal: fixes q :: "nat"   and p :: "nat" assumes "finite (carrier (G\<lparr>carrier := H\<rparr>))"   and "order (G\<lparr>carrier := H\<rparr>) = q * p"   and "prime p"   and "prime q"   and "p < q" shows "(THE Q. Q \<in> Hgroup.subgroups_of_size q) \<lhd> G\<lparr>carrier := H\<rparr>"
intros_1: fixes ns :: "('a \<times> 'a) set" shows "({#}, {#}) \<in> multpw ns"
null_Sequence\<^sub>b\<^sub>a\<^sub>s\<^sub>e_def: shows "null \<equiv> Abs_Sequence\<^sub>b\<^sub>a\<^sub>s\<^sub>e \<lfloor>\<bottom>\<rfloor>"
observe_execution_no_possible_step: fixes e :: "((nat \<times> nat) \<times> transition) fset"   and s :: "nat"   and r :: "nat \<Rightarrow>f value option"   and h :: "String.literal \<times> value list"   and es :: "(String.literal \<times> value list) list" assumes "possible_steps e s r (fst h) (snd h) = {||}" shows "observe_execution e s r (h # es) = []"
multiplicity_prime: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and p :: "'b" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize"   and "comm_semiring_1.prime_elem times one zero p" shows "factorial_semiring.multiplicity times one p p = 1"
bind_eq_return_pmf: fixes p :: "'d pmf"   and f :: "'d \<Rightarrow> 'c pmf"   and x :: "'c" shows "(p \<bind> f = return_pmf x) = (\<forall>y\<in>set_pmf p. f y = return_pmf x)"
is_shadow_root_ptr_kind_obtains: fixes document_ptr :: "(_) document_ptr" assumes "is_shadow_root_ptr_kind document_ptr" obtains shadow_root_ptr :: "(_) shadow_root_ptr" where "document_ptr = cast shadow_root_ptr"
null_cons_def: fixes g :: "('e, 'f) generator"   and s :: "'f" shows "null_cons g s = List.null (unstream g s)"
induct: fixes P :: "vname aexp list \<Rightarrow> vname aexp list \<Rightarrow> bool"   and a0 :: "vname aexp list"   and a1 :: "vname aexp list" assumes "P [] []"   and "\<And>a uu_. P [] (a # uu_)"   and "\<And>a uv_. P (a # uv_) []"   and "\<And>v t v' t'. (v = v' \<Longrightarrow> P t t') \<Longrightarrow> P (L v # t) (L v' # t')"   and "\<And>v t h' t'. P t t' \<Longrightarrow> P (V v # t) (h' # t')"   and "\<And>v va t h' t'. P t t' \<Longrightarrow> P (Plus v va # t) (h' # t')"   and "\<And>v va t h' t'. P t t' \<Longrightarrow> P (Minus v va # t) (h' # t')"   and "\<And>v va t h' t'. P t t' \<Longrightarrow> P (Times v va # t) (h' # t')"   and "\<And>h t v t'. P t t' \<Longrightarrow> P (h # t) (V v # t')"   and "\<And>h t v va t'. P t t' \<Longrightarrow> P (h # t) (Plus v va # t')"   and "\<And>h t v va t'. P t t' \<Longrightarrow> P (h # t) (Minus v va # t')"   and "\<And>h t v va t'. P t t' \<Longrightarrow> P (h # t) (Times v va # t')" shows "P a0 a1"
cfg_on_div_repcs_strong: fixes cfg :: "('s \<times> ('c \<Rightarrow> real) set) cfg"   and st :: "'s \<times> ('c \<Rightarrow> real)" assumes "cfg \<in> R_G_cfg_on_div (abss st)"   and "st \<in> S"   and "alternating cfg" shows "repcs st cfg \<in> cfg_on_div st"
absorb_iff2: fixes b :: "'c"   and a :: "'c" shows "(b \<le> a) = (min a b = b)"
USdBsym: fixes d :: "'d" shows "sym (\<approx>\<^bsub>d\<^esub>)"
sub_blocks: fixes bl :: "'a set" assumes "bl \<in># \<A>" shows "bl \<in># \<B>"
order_iff: fixes b :: "'c \<times> 'c \<Rightarrow> 'd"   and a :: "'c \<times> 'c \<Rightarrow> 'd" shows "(b \<preceq> a) = (a = a \<oplus> b)"
strip_step': fixes S :: "ivl Abs_State.st option"   and c :: "ivl Abs_State.st option acom" shows "strip (step_ivl S c) = strip c"
right_inverse: fixes x :: "'a" shows "x \<odot> inverse x = one"
3: shows "{} \<turnstile> SyntaxN.Ex j (Var j EQ \<guillemotleft>\<beta>(i::=\<guillemotleft>\<beta>\<guillemotright>)\<guillemotright> AND \<alpha>(i::=Var j)) IFF \<alpha>(i::=\<guillemotleft>\<beta>(i::=\<guillemotleft>\<beta>\<guillemotright>)\<guillemotright>)"
ladder_shift_j_length: fixes d :: "nat"   and L :: "(nat \<times> nat \<times> nat) list" shows "length (ladder_shift_j d L) = length L"
l2_inv8I: fixes x :: "l2_state" assumes "secret x \<subseteq> {NonceF (R $ sk) |R. R = test \<or> R \<in> partners}" shows "x \<in> l2_inv8"