hcju/isabelleps huggingface dataset:
theI': assumes "\<exists>!x. \<exists>a. P a x" shows "\<exists>a. P a (THE x. \<exists>a. P a x)"
disjoint_notin1: fixes A :: "'b set"   and B :: "'b set"   and x :: "'b" assumes "A \<inter> B = {}"   and "x \<in> B" shows "x \<notin> A"
OFCLASS_Hermite_ring_def: shows "OFCLASS('b, Hermite_ring_class) \<equiv> (\<And>A. admits_triangular_reduction A)"
compose_eq': fixes A :: "'ag set"   and G :: "'ai \<Rightarrow> 'ah"   and F :: "'ag \<Rightarrow> 'ai" shows "compose A G F = restrict (G \<circ> F) A"
a_def: shows "a_conditions (insert_messages (M \<union> {Delete m})) a"
closed_segment_subset_domainI: fixes T :: "real set"   and t0 :: "real"   and t :: "real"   and s :: "real" assumes "interval T"   and "t0 \<in> T"   and "t \<in> T"   and "s \<in> {t0--t}" shows "s \<in> T"
\<nu>: shows "\<guillemotleft>\<nu> : L f \<Rightarrow> L f'\<guillemotright>"
point_mult_closed: fixes a :: "'a"   and b :: "'a"   and p :: "'a point"   and n :: "nat" assumes "a \<in> carrier R"   and "b \<in> carrier R"   and "on_curve a b p" shows "on_curve a b (point_mult a n p)"
that: fixes a :: "'a" assumes "a \<noteq> (0::'a)"   and "span S = {x. a \<bullet> x = 0}" shows "thesis"
prems: shows "wf\<^sub>e\<^sub>s\<^sub>t V A"
positions_in_row_unique: fixes s :: "('c \<times> bool list) stream"   and i :: "nat" assumes "\<exists>!p. snd (s !! p) ! i" shows "the_elem (positions_in_row s i) = (THE p. snd (s !! p) ! i)"
Span_exp_base: fixes K :: "'a set"   and x :: "'a" assumes "subfield K R"   and "x \<in> carrier R"   and "(algebraic over K) x" shows "Span K (exp_base x (degree (Irr K x))) = simple_extension K x"
Suc_Max_notin_set: fixes X :: "nat set"   and v :: "nat" assumes "finite X" shows "Suc (v + Max (insert 0 X)) \<notin> X"
X4_pow: fixes X :: "X4 set" shows "X \<in> set ` set (subseqs [X4.Xd1, X4.Xd1', X4.Xd2, Xd2'])"
no_winning_strategy_on_deadends: fixes G :: "('c, 'd) ParityGame_scheme"   and v :: "'c"   and p :: "Player"   and \<sigma> :: "'c \<Rightarrow> 'c" assumes "ParityGame G"   and "v \<in> (if p = Even then V0\<^bsub>G\<^esub> else V\<^bsub>G\<^esub> - V0\<^bsub>G\<^esub>)"   and "\<not> (\<exists>w\<in>V\<^bsub>G\<^esub>. v \<rightarrow>\<^bsub>G\<^esub> w)"   and "ParityGame.strategy G p \<sigma>" shows "\<not> ParityGame.winning_strategy G p \<sigma> v"
v11_vconverse_app_in_vdomain: fixes y :: "V" assumes "y \<in>\<^sub>\<circ> \<R>\<^sub>\<circ> vint_uminus" shows "vint_uminus\<inverse>\<^sub>\<circ>\<lparr>y\<rparr> \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> vint_uminus"
fps_divide_unit': fixes g :: "'a fps"   and f :: "'a fps" assumes "subdegree g = 0" shows "f / g = f * inverse g"
map_graph_fn_graphI: fixes G :: "('e, 'g) labeled_graph"   and f :: "'g \<Rightarrow> 'f" shows "graph (map_graph_fn G f)"
ReprF_combineWith_CapN: fixes \<phi> :: "'fmla" assumes "\<phi> \<in> fmla"   and "Fvars \<phi> = {}" shows "prveqlPT (instInp (encF N) \<langle>\<phi>\<rangle>) \<langle>neg \<phi>\<rangle>"
Out_def: fixes P :: "'p"   and A :: "'p"   and B :: "'p" shows "P Out A B \<equiv> A \<noteq> P \<and> B \<noteq> P \<and> (Bet P A B \<or> Bet P B A)"
R2_def: shows "R2 = (\<lambda>x. sum_upto (\<lambda>d. 1 / real (d\<^sup>2)) x - (\<zeta> - 1 / x))"
simple_refinement_safe_def: fixes \<R>\<^sub>A :: "((((Eg1.var, aexp, bexp) Stmt \<times> (Mode \<Rightarrow> Eg1.var set)) \<times> (Eg1.var \<Rightarrow> nat)) \<times> ((Eg1.var, aexp, bexp) Stmt \<times> (Mode \<Rightarrow> Eg1.var set)) \<times> (Eg1.var \<Rightarrow> nat)) set"   and \<R> :: "((((Eg1.var, aexp, bexp) Stmt \<times> (Mode \<Rightarrow> Eg1.var set)) \<times> (Eg1.var \<Rightarrow> nat)) \<times> ((var\<^sub>C, aexp\<^sub>C, bexp\<^sub>C) Stmt \<times> (Mode \<Rightarrow> var\<^sub>C set)) \<times> (var\<^sub>C \<Rightarrow> nat)) set"   and abs_steps :: "((Eg1.var, aexp, bexp) Stmt \<times> (Mode \<Rightarrow> Eg1.var set)) \<times> (Eg1.var \<Rightarrow> nat) \<Rightarrow> ((var\<^sub>C, aexp\<^sub>C, bexp\<^sub>C) Stmt \<times> (Mode \<Rightarrow> var\<^sub>C set)) \<times> (var\<^sub>C \<Rightarrow> nat) \<Rightarrow> 'a" shows "simple_refinement_safe \<R>\<^sub>A \<R> abs_steps \<equiv> \<forall>c\<^sub>A mds\<^sub>A mem\<^sub>1\<^sub>A mem\<^sub>2\<^sub>A c\<^sub>C mds\<^sub>C mem\<^sub>1\<^sub>C mem\<^sub>2\<^sub>C. (\<langle>c\<^sub>A, mds\<^sub>A, mem\<^sub>1\<^sub>A\<rangle>\<^sub>A, \<langle>c\<^sub>A, mds\<^sub>A, mem\<^sub>2\<^sub>A\<rangle>\<^sub>A) \<in> \<R>\<^sub>A \<and> (\<langle>c\<^sub>A, mds\<^sub>A, mem\<^sub>1\<^sub>A\<rangle>\<^sub>A, \<langle>c\<^sub>C, mds\<^sub>C, mem\<^sub>1\<^sub>C\<rangle>\<^sub>C) \<in> \<R> \<and> (\<langle>c\<^sub>A, mds\<^sub>A, mem\<^sub>2\<^sub>A\<rangle>\<^sub>A, \<langle>c\<^sub>C, mds\<^sub>C, mem\<^sub>2\<^sub>C\<rangle>\<^sub>C) \<in> \<R> \<longrightarrow> stops\<^sub>C \<langle>c\<^sub>C, mds\<^sub>C, mem\<^sub>1\<^sub>C\<rangle>\<^sub>C = stops\<^sub>C \<langle>c\<^sub>C, mds\<^sub>C, mem\<^sub>2\<^sub>C\<rangle>\<^sub>C \<and> abs_steps \<langle>c\<^sub>A, mds\<^sub>A, mem\<^sub>1\<^sub>A\<rangle>\<^sub>A \<langle>c\<^sub>C, mds\<^sub>C, mem\<^sub>1\<^sub>C\<rangle>\<^sub>C = abs_steps \<langle>c\<^sub>A, mds\<^sub>A, mem\<^sub>2\<^sub>A\<rangle>\<^sub>A \<langle>c\<^sub>C, mds\<^sub>C, mem\<^sub>2\<^sub>C\<rangle>\<^sub>C \<and> (\<forall>mds\<^sub>1\<^sub>C' mds\<^sub>2\<^sub>C' mem\<^sub>1\<^sub>C' mem\<^sub>2\<^sub>C' c\<^sub>1\<^sub>C' c\<^sub>2\<^sub>C'. eval_abv\<^sub>C \<langle>c\<^sub>C, mds\<^sub>C, mem\<^sub>1\<^sub>C\<rangle>\<^sub>C \<langle>c\<^sub>1\<^sub>C', mds\<^sub>1\<^sub>C', mem\<^sub>1\<^sub>C'\<rangle>\<^sub>C \<and> eval_abv\<^sub>C \<langle>c\<^sub>C, mds\<^sub>C, mem\<^sub>2\<^sub>C\<rangle>\<^sub>C \<langle>c\<^sub>2\<^sub>C', mds\<^sub>2\<^sub>C', mem\<^sub>2\<^sub>C'\<rangle>\<^sub>C \<longrightarrow> c\<^sub>1\<^sub>C' = c\<^sub>2\<^sub>C' \<and> mds\<^sub>1\<^sub>C' = mds\<^sub>2\<^sub>C')"
scaleC_cancel_right: fixes a :: "complex"   and x :: "'b"   and b :: "complex" shows "(a *\<^sub>C x = b *\<^sub>C x) = (a = b \<or> x = (0::'b))"
coclop_iso: fixes f :: "'b \<Rightarrow> 'b" assumes "coclop f" shows "mono f"
bdd_above_Icc: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a" assumes "class.preorder less_eq less" shows "preorder.bdd_above less_eq (ord.atLeastAtMost less_eq a b)"
assms_4: shows "interpret_floatariths ds qs = interpret_floatariths es rs"
add_mono: fixes a :: "'c DBMEntry"   and b :: "'c DBMEntry"   and c :: "'c DBMEntry"   and d :: "'c DBMEntry" assumes "a \<preceq> b"   and "c \<preceq> d" shows "dbm_add a c \<preceq> dbm_add b d"
assms: shows "\<exists>t1. Some t1 = SEQ n s1 \<and> (\<exists>t2. Some t2 = SEQ n s2 \<and> NIA t1 t2)"
countable_insert: fixes A :: "'k set"   and a :: "'k" assumes "countable A" shows "countable (insert a A)"
irel: shows "G\<turnstile>I\<preceq>I J"
iso_Field: fixes r :: "('a \<times> 'a) set"   and r' :: "('b \<times> 'b) set"   and f :: "'a \<Rightarrow> 'b" assumes "iso r r' f" shows "f ` Field r = Field r'"
has_locks_release_all_conv: fixes l :: "('b \<times> nat) option"   and t :: "'b" shows "has_locks (release_all l t) t = 0"
natural_quotient_hom_image: fixes H :: "'g set" shows "(\<lambda>a. if a \<in> G then BinOpSetGroup.Abs_G_perm (G // LCoset_rel H) (+) (LCoset_rel H `` {a}) else 0) ` G = quotient_group H"
size_regex_estimation: fixes x :: "'b"   and r :: "'b regex"   and y :: "nat"   and f :: "'b \<Rightarrow> nat" assumes "x \<in> atms r"   and "y < f x" shows "y < size_regex f r"
Pd_singletons_for_ds_def: fixes X :: "'x set"   and ds :: "'d set" shows "Pd_singletons_for_ds X ds \<equiv> override_on Pd (\<lambda>d. dX X d \<times> dX X d) ds"
le_0: fixes x :: "V" shows "0 \<subseteq>\<^sub>\<circ> x"
getFr: fixes xs :: "'var list"   and ts :: "'trm list"   and \<phi>s :: "'fmla list" assumes "set xs \<subseteq> var"   and "set ts \<subseteq> trm"   and "set \<phi>s \<subseteq> fmla" shows "getFr xs ts \<phi>s \<in> var"
ell2_ket_2: fixes a :: "'c" shows "ell2_norm (\<lambda>i. if a = i then 1 else 0) = 1"
drop_minus: fixes n :: "nat"   and xs :: "'b list"   and n' :: "nat" assumes "n < length xs"   and "n' = length xs" shows "drop (n' - Suc n) xs = rev xs ! n # drop (n' - n) xs"
countable_converseI: fixes A :: "('b \<times> 'c) set" assumes "countable A" shows "countable (A\<inverse>)"
preserves_ide: fixes a :: "'a arrow_of_spans_data" assumes "ide a" shows "ide (src a)"
iso_\<nu>'\<nu>_r: shows "local.iso ((\<a>[local.dom \<rho>, w', w] \<cdot> (\<nu>' \<star> w) \<cdot> \<nu>) \<cdot> \<r>[g])"
assms_2: shows "g piecewise_C1_differentiable_on {c..b}"
subset_map_restrict_sub_add: fixes T :: "'c set"   and S :: "'c set"   and m :: "'c \<Rightarrow> 'd option" assumes "T \<subseteq> S" shows "m |` (S - T) ++ m |` T = m |` S"
A: shows "(*) (0::'a) = (\<lambda>_. 0::'a)"
synthetic_div_unique_lemma: fixes c :: "'a"   and p :: "'a poly"   and a :: "'a" assumes "Polynomial.smult c p = pCons a p" shows "p = 0"
list_succ_in_conv1: fixes A :: "'b set"   and xs :: "'b list"   and x :: "'b" assumes "A \<inter> set xs = {}" shows "(list_succ xs x \<in> A) = (x \<in> A)"
arrivemsg_def: fixes i :: "nat"   and P :: "'d \<Rightarrow> bool"   and a :: "'d node_action" shows "arrivemsg i P a \<equiv> case a of ii\<not>ni:arrive(m) \<Rightarrow> ii = {i} \<longrightarrow> P m | _ \<Rightarrow> True"
assms: fixes v :: "'a vec" assumes "v \<in> set ls" shows "dim_vec v = n"
cod_comp: fixes g :: "'c"   and f :: "'c" assumes "C.seq g f" shows "C.cod (C g f) = C.cod g"
in_cong_bigtheta: fixes f :: "'b \<Rightarrow> 'c"   and F :: "'b filter"   and g :: "'b \<Rightarrow> 'c"   and h :: "'b \<Rightarrow> 'c" assumes "f \<in> \<Theta>[F](g)" shows "(f \<in> o[F](h)) = (g \<in> o[F](h))"
assms: shows "A \<turnstile>\<^sub>C callee1(s) \<approx> callee2(s')"
eventually_iff_iff: fixes F :: "'b filter"   and P :: "'b \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> bool" assumes "ultrafilter F" shows "(\<forall>\<^sub>F x in F. P x = Q x) = (eventually P F = eventually Q F)"
nat_pow_pow: fixes G :: "('b, 'c) monoid_scheme"   and x :: "'b"   and n :: "nat"   and m :: "nat" assumes "Group.monoid G"   and "x \<in> carrier G" shows "(x [^]\<^bsub>G\<^esub> n) [^]\<^bsub>G\<^esub> m = x [^]\<^bsub>G\<^esub> (n * m)"
brownHueristic_less_than: fixes n :: "nat"   and L :: "atom list"   and F :: "atom fm list"   and var :: "nat" assumes "brownsHeuristic n L F = var" shows "var \<le> n"
vcons_in_vcpowerI: fixes n :: "V"   and xs :: "V"   and A :: "V"   and x :: "V" assumes "n \<in>\<^sub>\<circ> \<omega>"   and "xs \<in>\<^sub>\<circ> A ^\<^sub>\<times> vcard xs"   and "x \<in>\<^sub>\<circ> A"   and "n = vcard (xs #\<^sub>\<circ> x)" shows "xs #\<^sub>\<circ> x \<in>\<^sub>\<circ> A ^\<^sub>\<times> n"
SUP_mult_left_ennreal: fixes c :: "ennreal"   and f :: "'c \<Rightarrow> ennreal"   and I :: "'c set" shows "c * Sup (f ` I) = (SUP i\<in>I. c * f i)"
Fun_def: fixes f :: "'s" shows "Fun f = restrict (img \<circ> (\<cdot>) f \<circ> inv_into (hom unity (local.dom f)) img) (local.set (local.dom f))"
elims: fixes x :: "('b, 'a) Term.term set"   and y :: "'a set" assumes "fv\<^sub>s\<^sub>e\<^sub>t x = y" obtains M :: "('b, 'a) Term.term set" where "x = M" and "y = \<Union> (fv ` M)"
dioph_invariant_def: fixes ks :: "int list"   and l :: "int" shows "dioph_invariant ks l = (\<lambda>(is, js). length is = nat (2 * max \<bar>l\<bar> (sum_list (map abs ks)) + 1))"
reds_det: fixes c :: "com"   and s :: "char list \<Rightarrow> val option"   and s\<^sub>1 :: "char list \<Rightarrow> val option"   and s\<^sub>2 :: "char list \<Rightarrow> val option" assumes "\<langle>c,s\<rangle> \<rightarrow>* \<langle>Skip,s\<^sub>1\<rangle>"   and "\<langle>c,s\<rangle> \<rightarrow>* \<langle>Skip,s\<^sub>2\<rangle>" shows "s\<^sub>1 = s\<^sub>2"
connected_def: fixes u :: "nat"   and v :: "nat" shows "connected u v \<equiv> \<exists>p. isPath u p v"
natural_transformation_axioms: shows "natural_transformation D C Adj.FGF.map F (\<epsilon>' \<circ> F)"
eq_divide_eq_numeral1_2: fixes a :: "'a"   and b :: "'a"   and w :: "num" shows "(a = b / - numeral w) = (if - numeral w \<noteq> (0::'a) then a * - numeral w = b else a = (0::'a))"
lexext_rev_irrefl: fixes xs :: "'b list"   and gt :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "\<forall>x\<in>set xs. \<not> gt x x" shows "\<not> lexext_rev gt xs xs"
resr3: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x \<cdot> y \<rightarrow> z = y \<rightarrow> x \<rightarrow> z"
tickFree_tl: fixes s :: "'a event list" assumes "s \<noteq> []"   and "tickFree s" shows "tickFree (tl s)"
simps: fixes a :: "'c"   and r :: "('d \<times> 'c) set" shows "(a \<in> Range r) = (\<exists>aa b. a = b \<and> (aa, b) \<in> r)"
cases: fixes a1 :: "'b"   and a2 :: "'b"   and a3 :: "'b \<Rightarrow> real"   and a4 :: "intv'" assumes "intv'_elem a1 a2 a3 a4" obtains u :: "'b \<Rightarrow> real" and x :: "'b" and y :: "'b" and c :: "int" where "a1 = xa__" and "a2 = y__" and "a3 = u__" and "a4 = Smaller' c__" and "u__ xa__ - u__ y__ < real_of_int c__"   | u :: "'b \<Rightarrow> real" and x :: "'b" and y :: "'b" and d :: "int" where "a1 = xb__" and "a2 = ya__" and "a3 = ua__" and "a4 = Const' d__" and "ua__ xb__ - ua__ ya__ = real_of_int d__"   | d :: "int" and u :: "'b \<Rightarrow> real" and x :: "'b" and y :: "'b" where "a1 = xc__" and "a2 = yb__" and "a3 = ub__" and "a4 = Intv' d" and "real_of_int d < ub__ xc__ - ub__ yb__" and "ub__ xc__ - ub__ yb__ < real_of_int (d + 1)"   | c :: "int" and u :: "'b \<Rightarrow> real" and x :: "'b" and y :: "'b" where "a1 = x" and "a2 = y" and "a3 = u" and "a4 = Greater' c" and "real_of_int c < u x - u y"
sum_nonneg_leq_bound: fixes s :: "'c set"   and f :: "'c \<Rightarrow> 'a"   and B :: "'a"   and i :: "'c" assumes "finite s"   and "\<And>i. i \<in> s \<Longrightarrow> bot \<le> f i"   and "comm_monoid_add.sum (\<squnion>) bot f s = B"   and "i \<in> s" shows "f i \<le> B"
dim_basis: fixes A :: "'a vec set" assumes "finite A"   and "KB.basis A" shows "KB.dim = card A"
app1: fixes A :: "'a trm"   and A' :: "'a trm"   and B :: "'a trm" assumes "A \<rightarrow>\<beta> A'" shows "App A B \<rightarrow>\<beta> App A' B"
explosion_counterexample: shows "\<not> valid_in {1} (Imp' (Con' (Pro ''p'') (Neg' (Pro ''p''))) (Pro ''q''))"
dom_distrib: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "ad (ad x) + ad (ad y) \<cdot> ad (ad z) = (ad (ad x) + ad (ad y)) \<cdot> (ad (ad x) + ad (ad z))"
sup_subset_mset_def: fixes A :: "'a multiset"   and B :: "'a multiset" shows "A \<union># B = A + (B - A)"
intuitionistic_sep_conj_sep_true_simp: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and sep_disj :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and P :: "'c \<Rightarrow> bool" assumes "class.sep_algebra plus zero sep_disj"   and "sep_algebra.intuitionistic plus sep_disj P" shows "sep_algebra.sep_conj plus sep_disj P (\<lambda>s. True) = P"
assms_3: assumes "\<not> b" shows "B \<subseteq> A"
NCut_axioms_def: fixes s :: "nat"   and t :: "nat"   and k :: "nat set" shows "NCut_axioms s t k \<equiv> s \<in> k \<and> t \<notin> k"
zle_add1_eq_le: fixes w :: "int"   and z :: "int" shows "(w < z + 1) = (w \<le> z)"
lens_indep_left_ext: fixes y :: "'d \<Longrightarrow> 'e"   and z :: "'f \<Longrightarrow> 'e"   and x :: "'g \<Longrightarrow> 'd" assumes "y \<bowtie> z" shows "x ;\<^sub>L y \<bowtie> z"
eventually_at_right_field_le: fixes P :: "'b \<Rightarrow> bool"   and x :: "'b" shows "eventually P (at_right x) = (\<exists>b>x. \<forall>y>x. y \<le> b \<longrightarrow> P y)"
is_tangle_diagram_length_rowlength: fixes w :: "brick list"   and ws :: "wall" assumes "is_tangle_diagram (w * ws)" shows "length (blockmat w) = rat_poly.row_length (kauff_mat ws)"
mon_e_simps_1: fixes fg :: "('b, 'c, 'd, 'a, 'e) flowgraph_rec_scheme"   and a :: "'f" shows "mon_e fg (LBase a) = {}"
orthogonal_self: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and norm :: "'a \<Rightarrow> real"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and sgn :: "'a \<Rightarrow> 'a"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and inner :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and x :: "'a" assumes "class.real_inner minus dist norm plus zero uminus scaleR sgn uniformity open inner" shows "real_inner.orthogonal inner x x = (x = zero)"
is_document_ptr_ref: fixes n :: "nat" shows "is_document_ptr (document_ptr.Ref n)"
pderivs_Atom: fixes s :: "'a list"   and c :: "'a" shows "pderivs s (Atom c) \<subseteq> {Atom c, One}"
Kbase_Kronecker: fixes n :: "nat"   and P :: "nat \<Rightarrow> ('a \<Rightarrow> ant) set" assumes "distinct_pds K n P" shows "\<forall>j\<le>n. \<forall>l\<le>n. (\<nu>\<^bsub>K P j\<^esub>) ((Kb\<^bsub>K n P\<^esub>) l) = \<delta>\<^bsub>j l\<^esub>"
simps_4: fixes prec :: "nat"   and a :: "floatarith"   and b :: "floatarith"   and bs :: "float interval option list"   and ss :: "nat list" shows "approx_form prec (LessEqual a b) bs ss = (case (approx prec a bs, approx prec b bs) of (None, b) \<Rightarrow> False | (Some ivl, None) \<Rightarrow> False | (Some ivl, Some ivl') \<Rightarrow> float_plus_up prec (upper ivl) (- lower ivl') \<le> 0)"
cases: fixes x :: "('b \<times> 'b) list \<times> 'b" obtains xa :: "'b" where "x = ([], xa__)"   | y :: "'b" and x' :: "'b" and xs :: "('b \<times> 'b) list" and xa :: "'b" where "x = ((y, x') # xs, xa)"
functions_of: fixes A :: "'c set"   and M :: "'d multiset"   and B :: "'d set" assumes "finite A"   and "set_mset M \<subseteq> B"   and "size M = card A" shows "functions_of A M \<in> (A \<rightarrow>\<^sub>E B) // domain_permutation A B"
seq_plus_closed: fixes s :: "nat \<Rightarrow> nat \<Rightarrow> int"   and s' :: "nat \<Rightarrow> nat \<Rightarrow> int" assumes "s \<in> carrier Zp\<^bsup>\<omega>\<^esup>"   and "s' \<in> carrier Zp\<^bsup>\<omega>\<^esup>" shows "s \<oplus>\<^bsub>Zp\<^bsup>\<omega>\<^esup>\<^esub> s' \<in> carrier Zp\<^bsup>\<omega>\<^esup>"
ChiRel_ChiPlus_trans: fixes U :: "'d"   and S :: "'d"   and A :: "('d, 'e, 'f) hierauto"   and T :: "'d" assumes "(U, S) \<in> ChiPlus A"   and "(S, T) \<in> ChiRel A" shows "(U, T) \<in> ChiPlus A"
tSplice_tObsC: fixes t :: "('a, 'ePubAct, 'es, 'pPubAct, 'ps) BEState Trace"   and t' :: "('a, 'ePubAct, 'es, 'pPubAct, 'ps) BEState Trace"   and a :: "'a" assumes "tObsC t = tObsC t'" shows "tObsC (t \<^bsub>\<^esub>\<bowtie>\<^bsub>a\<^esub> t') = tObsC t"
spec_imp_SafeStickyGlobVacantUpto_aux: fixes s :: "('c, 'd) configuration stream" assumes "alw (holds InvGlobVacantUptoImpliesNrec) s"   and "alw (holds InvGlobalRecordCount) s"   and "alw (holds InvGlobalIncomingInfoUpright) s"   and "alw (holds InvGlobalInfoAtBetaUpright) s"   and "alw next s" shows "alw SafeStickyGlobVacantUpto s"
r_one_unique: fixes G :: "('c, 'd) Group_scheme"   and a :: "'c"   and x :: "'c" assumes "Group G"   and "a \<in> carrier G"   and "x \<in> carrier G"   and "a \<cdot>\<^bsub>G\<^esub> x = a" shows "x = \<one>\<^bsub>G\<^esub>"
twiddleVal_IVal: fixes i1 :: "int"   and i2 :: "int"   and \<beta> :: "(Location \<times> Location) set" assumes "i1 = i2" shows "(\<beta>, IVal i1, IVal i2) \<in> twiddleVal"
part_not_discrete_card: fixes X :: "'a set" assumes "part X"   and "\<not> discrete X" shows "card {f. CK_nf_pos f} = 3"
linord_of_list_singleton: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "((x, y) \<in> linord_of_list [z]) = (x = z \<and> y = z)"
meta_impE: fixes P :: "prop"   and V :: "prop" assumes "PROP P \<Longrightarrow> PROP V"   and "PROP P" obtains "PROP V"
prems_1: shows "set (v # Vs) \<subseteq> set Ws"
right: fixes i :: "nat"   and j :: "nat" assumes "i < nr"   and "j < nc" shows "[:row A i \<bullet> col B j:] * p = row (map_mat (\<lambda>a. [:a:]) A) i \<bullet> col (map_mat (\<lambda>a. [:a:] * p) B) j"
TComp_var_instance_wt_subst_exists: fixes g :: "'fun"   and T :: "('fun, ('fun, 'atom) Term.term \<times> nat) Term.term list"   and U :: "('fun, ('fun, 'atom) Term.term \<times> nat) Term.term list" assumes "\<Gamma> (Fun g T) = Fun g (map \<Gamma> U)"   and "wf\<^sub>t\<^sub>r\<^sub>m (Fun g T)"   and "\<forall>u\<in>set U. \<exists>y. u = Var y"   and "distinct U" shows "\<exists>\<theta>. wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<theta> \<and> wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (subst_range \<theta>) \<and> Fun g T = Fun g U \<cdot> \<theta>"
norm_lt_square: fixes x :: "'b"   and a :: "real" shows "(norm x < a) = (0 < a \<and> x \<bullet> x < a\<^sup>2)"
l_set_val\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and set_val :: "(_) character_data_ptr \<Rightarrow> char list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_val_locs :: "(_) character_data_ptr \<Rightarrow> ((_) heap, exception, unit) prog set" shows "l_set_val\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf set_val set_val_locs \<equiv> type_wf = DocumentClass.type_wf \<and> set_val = l_set_val\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_set_val \<and> set_val_locs = l_set_val\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_set_val_locs"
sqmatrix_add_commute: fixes A :: "('c, 'd) sqmatrix"   and B :: "('c, 'd) sqmatrix" shows "A + B = B + A"
g_def: fixes n :: "nat" shows "g n = complex_of_real (sin_coeff (n + 1))"
4: shows "v ||| l ## sscan (count cs) v l = sdrop (length u) (stl w ||| count cs (shd w) k ## sscan (count cs) (stl w) (count cs (shd w) k))"
rep_eq: fixes x :: "'a fset"   and xa :: "'a fset" shows "fset (x |\<union>| xa) = fset x \<union> fset xa"
get_host_ok: fixes h :: "(_) heap"   and shadow_root_ptr :: "(_) shadow_root_ptr" assumes "Shadow_DOM.heap_is_wellformed h"   and "ShadowRootClass.type_wf h"   and "ShadowRootClass.known_ptrs h"   and "shadow_root_ptr |\<in>| shadow_root_ptr_kinds h" shows "h \<turnstile> ok get_host shadow_root_ptr"
fold_congs_2: fixes r :: "('c, 'd) rgraph_scheme"   and r' :: "('c, 'd) rgraph_scheme"   and v' :: "'c edge set"   and f :: "'c edge set \<Rightarrow> 'c edge set"   and f' :: "'c edge set \<Rightarrow> 'c edge set" assumes "r = r'"   and "edges r' = v'"   and "\<And>v. v' = v \<Longrightarrow> f v = f' v" shows "edges_update f r = edges_update f' r'"
assms_3: shows "\<exists>!x. x \<in> set xs \<and> h \<turnstile> P x \<rightarrow>\<^sub>r True"
eval_ps': shows "P \<turnstile> \<langle>blocks (ps', Ts', vs', e),(h, l(p \<mapsto> v))\<rangle> \<Rightarrow> \<langle>e',(h', l''')\<rangle>"
simps_2: fixes id :: "'a"   and f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and n :: "'a"   and y :: "'a"   and ys :: "'a list" assumes "Matrix_Tensor.mult id f" shows "mult.times f n (y # ys) = f n y # mult.times f n ys"
list_greater_non_memb: fixes xs :: "'c list"   and oid :: "'c" assumes "\<And>x. x \<in> set xs \<Longrightarrow> x < oid"   and "oid \<in> set xs" shows "False"
nat_pow_closed: fixes G :: "('a, 'b) monoid_scheme"   and x :: "'a"   and n :: "nat" assumes "Group.monoid G"   and "x \<in> carrier G" shows "x [^]\<^bsub>G\<^esub> n \<in> carrier G"
ESem_considers_fv: fixes e :: "'a"   and \<rho> :: "'b \<Rightarrow> 'c" shows "\<lbrakk> e \<rbrakk>\<^bsub>\<rho>\<^esub> = \<lbrakk> e \<rbrakk>\<^bsub>\<rho> f|` fv e\<^esub>"
exec_bound: fixes pc :: "nat"   and bp :: "instr array"   and s :: "integer"   and s0 :: "integer" assumes "pc < array_length bp"   and "s \<in> state_bound bp s0" shows "set (exec (array_get bp pc) (pc, s)) \<subseteq> config_bound bp s0"
Nil\<^sub>1: fixes P :: "nat exp prog"   and s :: "heap \<times> val list \<times> sheap" shows "P \<turnstile>\<^sub>1 \<langle>[],s\<rangle> [\<Rightarrow>] \<langle>[],s\<rangle>"
jointfun_mem: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'c"   and A :: "'c set"   and m :: "nat"   and g :: "nat \<Rightarrow> 'c"   and B :: "'c set"   and l :: "nat" assumes "\<forall>j\<le>n. f j \<in> A"   and "\<forall>j\<le>m. g j \<in> B"   and "l \<le> Suc (n + m)" shows "jointfun n f m g l \<in> A \<union> B"