hcju/isabelleps huggingface dataset:
le_less_linear: fixes less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and x :: "'e"   and y :: "'e" assumes "class.linorder less_eq less" shows "less_eq x y \<or> less y x"
assms_1: shows "\<GG>' = op_dghm \<GG>"
not_image_stut: fixes f :: "nat \<Rightarrow> nat"   and sub :: "'b run"   and r :: "'b run"   and n\<^sub>0 :: "nat"   and n :: "nat"   and k :: "nat" assumes "dilating f sub r"   and "n\<^sub>0 = Max {i. f i \<le> n}"   and "f n\<^sub>0 < k \<and> k \<le> n" shows "\<nexists>k\<^sub>0. f k\<^sub>0 = k"
af_imp_almost_full_on: fixes A :: "'a set"   and P :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "af A P" shows "almost_full_on P A"
numbers_of_marks_simps_1: fixes n :: "nat" shows "numbers_of_marks n [] = {}"
drop_bit_word_beyond: fixes n :: "nat"   and w :: "'b word" assumes "LENGTH('b) \<le> n" shows "drop_bit n w = 0"
simps_1: fixes less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and k :: "'b" shows "ord.rbt_lookup less Empty k = None"
linord_helper_less_eq1_def: fixes a :: "('c, 'd) linord_helper"   and b :: "('c, 'd) linord_helper" shows "linord_helper_less_eq1 a b \<equiv> case a of LinordHelper a1 a2 \<Rightarrow> case b of LinordHelper b1 b2 \<Rightarrow> a1 < b1 \<or> a1 = b1 \<and> a2 \<le> b2"
6: shows "project visible s\<^sub>2 = project visible s\<^sub>3"
ext: fixes g :: "'t"   and f :: "'t" assumes "g \<cdot>\<acute> f \<noteq> S'.null" shows "S'.seq g f"
in_pair_alt: fixes p :: "'e \<times> 'e"   and A :: "('e \<times> 'e) set" shows "(p \<in>\<^sub>p A) = (p \<in> A \<or> prod.swap p \<in> A)"
prems_2: shows "history_consistent (\<theta> |` (dom \<theta> - read_tmps ys)) p (x # xs)"
jordan_block_pow_dim_2: fixes n :: "nat"   and a :: "'b"   and r :: "nat" shows "dim_col (jordan_block n a ^\<^sub>m r) = n"
indE: fixes x :: "'a"   and y :: "'a"   and S :: "'a set" assumes "(x, y) \<in> induced S r"   and "S \<subseteq> A" shows "(x, y) \<in> r"
openin_prod_Times_iff: fixes X :: "'c topology"   and Y :: "'d topology"   and S :: "'c set"   and T :: "'d set" shows "openin (prod_topology X Y) (S \<times> T) = (S = {} \<or> T = {} \<or> openin X S \<and> openin Y T)"
mult_matrix_n: fixes A :: "'b matrix"   and n :: "nat"   and B :: "'c matrix"   and fadd :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and fmul :: "'b \<Rightarrow> 'c \<Rightarrow> 'd" assumes "ncols A \<le> n"   and "nrows B \<le> n"   and "fadd (0::'d) (0::'d) = (0::'d)"   and "fmul (0::'b) (0::'c) = (0::'d)" shows "mult_matrix fmul fadd A B = mult_matrix_n n fmul fadd A B"
not_eq_newpkt_pkt: fixes no :: "nat"   and sid :: "nat" shows "\<not> eq_newpkt (Pkt no sid)"
precedesFilter: fixes a :: "'b"   and b :: "'b"   and list :: "'b list"   and f :: "'b \<Rightarrow> bool" assumes "precedes a b list"   and "f a"   and "f b" shows "precedes a b (filter f list)"
inter_line_ext_1: fixes l :: "Lines"   and m :: "Lines"   and n :: "Lines" assumes "Pascal_Property.inter l m \<in> line_ext n"   and "l \<noteq> m"   and "l \<noteq> n" shows "Pascal_Property.inter l m = Pascal_Property.inter l n"
f1i_sum: shows "map of_int_hom.vec_hom fs ! i = gs.M.sumlist (map (\<lambda>j. fs.gs.\<mu> i j \<cdot>\<^sub>v fs.gs.gso j) [0..<i]) + fs.gs.gso i"
strict_from_inj_F_strict: fixes xa :: "nat \<Rightarrow> nat" assumes "xa \<in> local.F_inj" shows "strict_from_inj n xa \<in> local.F_strict"
sel_2: fixes x1 :: "'b"   and x2 :: "'a"   and x3 :: "'c list" shows "state.heap (State x1 x2 x3) = x2"
finite_entropy_def: fixes S :: "'d measure"   and X :: "'a \<Rightarrow> 'd"   and f :: "'d \<Rightarrow> real" shows "finite_entropy S X f = (distributed M S X (\<lambda>x. ennreal (f x)) \<and> integrable S (\<lambda>x. f x * log b (f x)) \<and> (\<forall>x\<in>space S. 0 \<le> f x))"
iiws: fixes loc :: "'loc" shows "c_imp c loc + c_work c loc = zmset_of (mset_set (set_antichain (frontier (c_pts c loc)))) + (\<Sum>loc'\<in>UNIV. after_summary (zmset_of (mset_set (set_antichain (frontier (c_imp c loc'))))) (summary loc' loc))"
sum: fixes g :: "'h \<Rightarrow> 'a"   and S :: "'h set"   and y :: "'b" shows "h (sum g S) y = (\<Sum>a\<in>S. h (g a) y)"
mod_integer_ge: fixes b :: "integer"   and a :: "integer"   and x :: "integer" assumes "b \<le> 0"   and "0 < a" shows "b \<le> x mod (a + 1)"
assms: fixes i :: "nat" assumes "i < a" shows "length (f i) = d"
new_character_data_is_l_new_character_data: shows "l_new_character_data DocumentClass.type_wf"
rep_eq: fixes x :: "'b"   and xa :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and xb :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and xc :: "integer \<Rightarrow> 'b"   and xd :: "'b mat_impl"   and xe :: "integer" shows "Rep_mat_impl (eliminate_entries_i2 x xa xb xc xd xe) = (case Rep_mat_impl xd of (nr, nc, a) \<Rightarrow> \<lambda>i'. (nr, nc, let ai' = IArray.sub' (a, i') in IArray.tabulate (integer_of_nat nr, \<lambda>i. let ai = IArray.sub' (a, i) in if i = i' then ai else let vi'j = xc i in if vi'j = x then ai else IArray.tabulate (integer_of_nat nc, \<lambda>j. xa (IArray.sub' (ai, j)) (xb vi'j (IArray.sub' (ai', j))))))) xe"
anonymous_profile_def: shows "anonymous_profile R1 = image_mset (weak_ranking \<circ> R1) (mset_set agents)"
scopeFresh: fixes x :: "name"   and P :: "pi" assumes "x \<sharp> P" shows "<\<nu>x>P \<sim>\<^sup>s\<^sub>l P"
canon_ang_def: fixes \<alpha> :: "real" shows "\<downharpoonright>\<alpha>\<downharpoonleft> = (THE \<alpha>'. canon_ang_P \<alpha> \<alpha>')"
continuous_on_cos: fixes s :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "continuous_on s f" shows "continuous_on s (\<lambda>x. cos (f x))"
BSI_implies_SI_for_modified_view: fixes ES :: "'b ES_rec"   and \<V> :: "'b V_rec"   and \<V>' :: "'b V_rec" assumes "BSPTaxonomyDifferentCorrections ES \<V>"   and "BSI \<V>' Tr\<^bsub>ES\<^esub>"   and "\<V>' = \<lparr>V = V\<^bsub>\<V>\<^esub> \<union> N\<^bsub>\<V>\<^esub>, N = {}, C = C\<^bsub>\<V>\<^esub>\<rparr>" shows "SI \<V> Tr\<^bsub>ES\<^esub>"
ParPres': fixes \<Psi>' :: "'b"   and S :: "('a, 'b, 'c) psi"   and T :: "('a, 'b, 'c) psi"   and U :: "('a, 'b, 'c) psi" assumes "(\<Psi>', S, T) \<in> Rel'" shows "(\<Psi>', U \<parallel> S, U \<parallel> T) \<in> Rel'"
divergent_\<R>_divergent: fixes xs :: "('c \<Rightarrow> real) stream" assumes "pred_stream (\<lambda>u. u \<in> V) xs"   and "divergent xs"   and "stream_trans xs" shows "\<R>_div (smap (\<lambda>u. [u]\<^sub>\<R>) xs)"
if_then_Some_else_None_eq_1: fixes a :: "bool"   and b :: "'a"   and c :: "'a" shows "((if a then Some b else None) = Some c) = (a \<and> b = c)"
Mid_perm: fixes A :: "'p"   and B :: "'p"   and C :: "'p" assumes "A Midpoint B C" shows "A Midpoint B C \<and> A Midpoint C B"
2: assumes "ide f \<and> C.iso (Leg0 (Dom f))" shows "adjunction_in_bicategory (\<bullet>) (\<star>) assoc unit src trg f \<lparr>Chn = Chn f, Dom = \<lparr>Leg0 = Leg1 (Dom f), Leg1 = Leg0 (Dom f)\<rparr>, Cod = \<lparr>Leg0 = Leg1 (Dom f), Leg1 = Leg0 (Dom f)\<rparr>\<rparr> (unit_for_map f) (counit_for_map f)"
evalOP_Num: fixes n :: "nat" shows "DBNum n \<Down> DBNum n"
subst_ex_same: fixes x :: "'a \<Longrightarrow> 'b"   and \<sigma> :: "'c \<Rightarrow> 'b"   and v :: "('a, 'c) uexpr"   and P :: "'b upred" assumes "mwb_lens x" shows "\<sigma>(x \<mapsto>\<^sub>s v) \<dagger> (\<exists> x \<bullet> P) = \<sigma> \<dagger> (\<exists> x \<bullet> P)"
y_eq_v1_solves: assumes "y = v1" shows "\<exists>paths. DisjointPaths G v0 v1 paths \<and> card paths = Suc sep_size"
HPhase1or2Write_maxBalInp: fixes s :: "state"   and s' :: "state"   and q :: "Proc"   and d :: "Disk"   and b :: "nat"   and v :: "InputsOrNi" assumes "HPhase1or2Write s s' q d"   and "maxBalInp s b v" shows "maxBalInp s' b v"
ipresWlsAbs_def: fixes hA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> 'h"   and MOD :: "('i, 'j, 'varSort, 'sort, 'k, 'l, 'm, 'h, 'n) model_scheme" shows "ipresWlsAbs hA MOD \<equiv> \<forall>us s A. wlsAbs (us, s) A \<longrightarrow> igWlsAbs MOD (us, s) (hA A)"
deriv_left_inverse: fixes f :: "complex \<Rightarrow> complex"   and S :: "complex set"   and g :: "complex \<Rightarrow> complex"   and T :: "complex set"   and w :: "complex" assumes "f holomorphic_on S"   and "g holomorphic_on T"   and "open S"   and "open T"   and "f ` S \<subseteq> T"   and "\<And>z. z \<in> S \<Longrightarrow> g (f z) = z"   and "w \<in> S" shows "deriv f w * deriv g (f w) = 1"
plossless_obsf_distinguisher: fixes \<I> :: "('m, 'n) \<I>"   and \<A> :: "('m, 'n) distinguisher" assumes "plossless_gpv \<I> \<A>"   and "\<I> \<turnstile>g \<A> \<surd>" shows "plossless_gpv (exception_\<I> \<I>) (obsf_distinguisher \<A>)"
fps_of_poly_pCons: fixes c :: "'b"   and p :: "'b poly" shows "fps_of_poly (pCons c p) = fps_const c + fps_of_poly p * fps_X"
abs_eq_iff': fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and abs :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.linordered_ring plus zero minus uminus less_eq less abs times" shows "(abs a = b) = (less_eq zero b \<and> (a = b \<or> a = uminus b))"
LCons_def: shows "LCons \<equiv> \<Lambda> a app. llist_abs\<cdot>(sinr\<cdot>(:up\<cdot>a, up\<cdot>app:))"
open_induct_on: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set"   and Q :: "'b \<Rightarrow> bool"   and x :: "'b" assumes "qo_on P A"   and "dc_on P A"   and "open_on P Q A"   and "x \<in> A"   and "\<And>x. \<lbrakk>x \<in> A; \<And>y. \<lbrakk>y \<in> A; strict P y x\<rbrakk> \<Longrightarrow> Q y\<rbrakk> \<Longrightarrow> Q x" shows "Q x"
orthogonal_mat_of_cols: fixes ws :: "'b vec list"   and n :: "nat" assumes "set ws \<subseteq> carrier_vec n"   and "corthogonal ws"   and "length ws = n" shows "corthogonal_mat (mat_of_cols n ws)"
one_sheaf_spec_def: fixes U :: "'a set set" shows "one_sheaf_spec U \<equiv> \<lambda>\<pp>\<in>U. quotient_ring.one_rel (R\<setminus>\<pp>) R (+) (\<cdot>) \<zero> \<one>"
neumann_3B_b_non_strict: fixes u :: "'a pmf"   and v :: "'a pmf"   and \<alpha> :: "real" assumes "u \<succeq>[\<R>] v"   and "\<alpha> \<in> {0..1}" shows "u \<succeq>[\<R>] mix_pmf \<alpha> u v"
iso_char': fixes F :: "('a, 'a set) Maps.arr" shows "Maps.iso F = (Maps.arr F \<and> (\<exists>f. f \<in> Maps.Map F \<and> equivalence_map f))"
subst_neq: fixes x :: "name"   and y :: "name"   and u :: "lam" assumes "x \<noteq> y" shows "Var x[y::=u] = Var x"
eq_reflection: fixes x :: "'a"   and y :: "'a" assumes "x = y" shows "x \<equiv> y"
act_act: fixes g :: "cltn2"   and h :: "cltn2"   and b :: "proj2" assumes "g \<in> carrier \<lparr>carrier = UNIV, monoid.mult = cltn2_compose, one = cltn2_id\<rparr>"   and "h \<in> carrier \<lparr>carrier = UNIV, monoid.mult = cltn2_compose, one = cltn2_id\<rparr>" shows "apply_cltn2 (apply_cltn2 b g) h = apply_cltn2 b (g \<otimes>\<^bsub>\<lparr>carrier = UNIV, monoid.mult = cltn2_compose, one = cltn2_id\<rparr>\<^esub> h)"
inv_solve_left: fixes a :: "'c"   and b :: "'c"   and c :: "'c" assumes "a \<in> carrier H"   and "b \<in> carrier H"   and "c \<in> carrier H" shows "(a = inv\<^bsub>H\<^esub> b \<otimes>\<^bsub>H\<^esub> c) = (c = b \<otimes>\<^bsub>H\<^esub> a)"
image_affinity_atLeastAtMost_diff: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and 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 m :: "'b"   and c :: "'b"   and a :: "'b"   and b :: "'b" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs" shows "(\<lambda>x. minus (times m x) c) ` ord.atLeastAtMost less_eq a b = (if ord.atLeastAtMost less_eq a b = {} then {} else if less_eq zero m then ord.atLeastAtMost less_eq (minus (times m a) c) (minus (times m b) c) else ord.atLeastAtMost less_eq (minus (times m b) c) (minus (times m a) c))"
PrfP_graph_aux_def: shows "PrfP_graph_aux \<equiv> lfp (\<lambda>p x1 x2. \<exists>PrfP_aux l s sl m n sm sn k t. x1 = (s, k, t) \<and> x2 = LstSeqP s k t AND All2 n (SUCC k) (SyntaxN.Ex sn (HPair (Var n) (Var sn) IN s AND (AxiomP (Var sn) OR SyntaxN.Ex m (SyntaxN.Ex l (SyntaxN.Ex sm (SyntaxN.Ex sl (Var m IN Var n AND Var l IN Var n AND HPair (Var m) (Var sm) IN s AND HPair (Var l) (Var sl) IN s AND (ModPonP (Var sm) (Var sl) (Var sn) OR ExistsP (Var sm) (Var sn) OR SubstP (Var sm) (Var sn))))))))) \<and> atom l \<sharp> (s, sl, m, n, sm, sn) \<and> atom sl \<sharp> (s, m, n, sm, sn) \<and> atom m \<sharp> (s, n, sm, sn) \<and> atom n \<sharp> (s, k, sm, sn) \<and> atom sm \<sharp> (s, sn) \<and> atom sn \<sharp> s)"
finfun_fst_def: fixes f :: "('a, 'b \<times> 'c) finfun" shows "finfun_fst f = finfun_comp fst f"
ltl_const_equiv_and_const_1: fixes \<phi>\<^sub>1 :: "'a ltln"   and \<phi>\<^sub>2 :: "'a ltln" shows "\<phi>\<^sub>1 and\<^sub>n \<phi>\<^sub>2 \<sim>\<^sub>C true\<^sub>n = (\<phi>\<^sub>1 \<sim>\<^sub>C true\<^sub>n \<and> \<phi>\<^sub>2 \<sim>\<^sub>C true\<^sub>n)"
one_circle: fixes l :: "real"   and r :: "real"   and P :: "real poly"   and p :: "nat" assumes "l < r"   and "proots_count (map_poly complex_of_real P) (circle_diam l r) = 0"   and "P \<noteq> 0"   and "degree P \<le> p" shows "Bernstein_changes p l r P = 0"
c3: shows "c3 = map (\<lambda>d. \<langle>check: pair (t, s) \<doteq> pair d\<rangle>\<^sub>s\<^sub>t) (unlabel Di)"
\<Z>_subset_Suc: fixes k :: "nat"   and n :: "nat" shows "\<Z> k n \<subseteq> \<Z> k (Suc n)"
Theorem4: fixes p :: "L"   and G :: "Var \<Rightarrow> L"   and c :: "IMP"   and H :: "Var \<Rightarrow> L"   and X :: "((Var \<Rightarrow> Val) \<Rightarrow> (Var \<Rightarrow> Val) \<Rightarrow> bool) set"   and q :: "L" assumes "(p, G, c, H) \<in> HS" shows "\<exists>\<Phi>. X \<rhd> c : Sec (Q p H) (EQ G q) (EQ H q) \<Phi>"
start_nth_unstable: fixes p :: "nat" assumes "p \<in> {1..<length (g_V G)}"   and "succs p \<noteq> {}" shows "\<not> stable r step start p"
llist_all2_same: fixes P :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and xs :: "'e llist" shows "llist_all2 P xs xs = (\<forall>x\<in>lset xs. P x x)"
rsp_on_def: fixes A :: "'b set"   and f :: "'b set \<Rightarrow> 'b set" shows "rsp_on A f = {(x, y). \<exists>B\<subseteq>A. x \<in> Rf f B \<and> y \<in> f B}"
distinct_93: fixes x6 :: "arith_type"   and x8 :: "shift_type" shows "arith_type x6 \<noteq> shift_type x8"
ZObisL_ZObis: fixes cval :: "'a \<Rightarrow> 'b \<Rightarrow> real"   and indis :: "('b \<times> 'b) set"   and aval :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and tval :: "'d \<Rightarrow> 'b \<Rightarrow> bool"   and cl :: "('d, 'c, 'a) cmd list"   and dl :: "('d, 'c, 'a) cmd list"   and n :: "nat" assumes "PL_Indis cval indis"   and "PL_Indis.ZObisL aval tval cval indis cl dl"   and "n < length cl \<or> n < length dl" shows "(cl ! n, dl ! n) \<in> PL_Indis.ZObis aval tval cval indis"
Some_mem_default_rel: fixes x :: "'a"   and y :: "'b"   and d :: "'b"   and X :: "('a \<times> 'b) set" shows "((Some x, y) \<in> \<langle>X\<rangle>default_rel d) = ((x, y) \<in> X)"
Tree_wf\<^sub>pE: fixes a\<^sub>p :: "('e, 'f, 'g) Tree\<^sub>p"   and b\<^sub>p :: "('e, 'f, 'g) Tree\<^sub>p" assumes "(a\<^sub>p, b\<^sub>p) \<in> Tree_wf\<^sub>p" obtains a :: "('e, 'f, 'g) Tree" and b :: "('e, 'f, 'g) Tree" where "a\<^sub>p = abs_Tree\<^sub>p a" and "b\<^sub>p = abs_Tree\<^sub>p b" and "(a, b) \<in> Tree_wf"
power2_less_sqrt_iff: fixes x :: "real"   and y :: "real" shows "(x\<^sup>2 < y) = (0 \<le> y \<and> \<bar>x\<bar> < sqrt y)"
Fun_mapsto: fixes f :: "'s" assumes "S.arr f" shows "S.Fun f \<in> extensional (S.set (S.dom f)) \<inter> (S.set (S.dom f) \<rightarrow> S.set (S.cod f))"
sat_slice_iff: fixes v :: "event_data list"   and S :: "event_data list set"   and \<sigma> :: "(String.literal \<times> event_data list) trace"   and V :: "String.literal \<Rightarrow> (nat \<Rightarrow> event_data list set) option"   and i :: "nat"   and \<phi> :: "Formula.formula" assumes "v \<in> S" shows "Formula.sat \<sigma> V v i \<phi> = Formula.sat (map_\<Gamma> (\<lambda>D. D \<inter> relevant_events \<phi> S) \<sigma>) V v i \<phi>"
ipd_1: shows "\<pi> l = ipd (\<pi> k)"
assms_2: shows "B \<subseteq> dgrad_p_set d m"
commute: fixes a :: "'b"   and b :: "'b" shows "a AND b = b AND a"
i_less_k: shows "mod_type_class.to_nat i < k"
normalize_lead_coeff: fixes R :: "('c, 'd) ring_scheme"   and p :: "'c list" assumes "ring R"   and "length (ring.normalize R p) < length p" shows "lead_coeff p = \<zero>\<^bsub>R\<^esub>"
intros: fixes \<omega> :: "(('test, 'atom, 'choice) cmd \<times> 'state) stream"   and s :: "('test, 'atom, 'choice) cmd \<times> 'state" assumes "enabled (shd \<omega>) (stl \<omega>)"   and "shd \<omega> \<in> G s" shows "enabled s \<omega>"
is_liftingD_1: fixes f :: "'a \<Rightarrow> 'b"   and basis :: "(real \<Rightarrow> real) list"   and basis' :: "(real \<Rightarrow> real) list"   and C :: "'a" assumes "is_lifting f basis basis'" shows "eval (f C) = eval C"
B: shows "abs_conv_abscissa fds_zeta \<le> 1"
B: fixes f :: "'a set \<Rightarrow> 'a"   and b :: "'a" assumes "\<forall>Y\<in>F. f Y \<in> Y"   and "b \<in> x" shows "fa b f ` ({x} \<union> F) \<in> {insert (f x) (f ` F) |f. f x \<in> x \<and> (\<forall>Y\<in>F. f Y \<in> Y)}"
assms_2: shows "(\<lambda>s. P s \<and> \<not> T s) \<le> |Y] Q"
D_1: shows "D \<in> decomps\<^sub>e\<^sub>s\<^sub>t A N \<I>"
group_subset_point_subset: fixes G :: "'a set"   and G' :: "'a set"   and ps :: "'a set" assumes "G \<in> \<G>"   and "G' \<subseteq> G"   and "ps \<subseteq> G'" shows "ps \<subseteq> \<V>"
D: shows "xs @ [x] \<in> snd (Rep_process P)"
f_Exec_Stream_nth2: fixes n :: "nat"   and xs :: "'d list"   and c :: "'e"   and trans_fun :: "'d \<Rightarrow> 'e \<Rightarrow> 'e" assumes "n \<le> length xs" shows "(c # f_Exec_Comp_Stream trans_fun xs c) ! n = f_Exec_Comp trans_fun (xs \<down> n) c"
is_lub_def: fixes below :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and S :: "'d set"   and x :: "'d" assumes "class.po below" shows "po.is_lub below S x = (po.is_ub below S x \<and> (\<forall>u. po.is_ub below S u \<longrightarrow> below x u))"
pre_chainmap: fixes f :: "'c \<Rightarrow> 'a"   and g :: "'d \<Rightarrow> 'b"   and \<Pi> :: "('c, 'd) chain" shows "pre (chainmap f g \<Pi>) = f (pre \<Pi>)"
residual_network_sel_2: fixes f :: "'v \<times> 'v \<Rightarrow> ennreal"   and x :: "'v"   and y :: "'v" shows "capacity (RES.residual_network f) (x, y) = (if edge (residual_network f) x y then capacity (residual_network f) (x, y) - f (x, y) else if y = source (residual_network f) then 0 else f (y, x))"
fall2_def: fixes F :: "('state \<times> 'aprop set) stream \<Rightarrow> ('state \<times> 'aprop set) stream \<Rightarrow> ('state \<times> 'aprop set) stream list \<Rightarrow> bool" shows "fall2 F \<equiv> fall (\<lambda>\<pi>. fall (F \<pi>))"
lift_is_cptn: fixes c :: "('b com option \<times> 'b) list"   and P :: "'b com" assumes "c \<in> cptn" shows "map (lift P) c \<in> cptn"
rule0: fixes a :: "'d" shows "a \<equiv> a + (0::'d)"
assms_1: fixes A :: "'a set" assumes "A \<in> sets F" shows "set_nn_integral M A f = set_nn_integral M A g"
compat_def: fixes I :: "'a set"   and theta :: "('b \<times> 'b) set"   and f :: "'a \<Rightarrow> 'b" shows "Resumption_Based.compat I theta f \<equiv> \<forall>i j. {i, j} \<subseteq> I \<and> i \<noteq> j \<longrightarrow> (f i, f j) \<in> theta"
fold_append: fixes f :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and xs :: "'c list"   and ys :: "'c list" shows "fold f (xs @ ys) = fold f ys \<circ> fold f xs"
sp_through_de_decomp: fixes e :: "('c \<times> nat) edge"   and subs :: "(('c \<times> nat) \<times> 'c \<times> nat) set"   and g :: "('c \<times> nat, 'd) rgraph_scheme"   and v1 :: "'c \<times> nat"   and es :: "('c \<times> nat) edge list"   and v2 :: "'c \<times> nat" assumes "tgt e \<notin> subsumees subs"   and "out_edges g (tgt e) = {}"   and "SubRel.subpath g v1 es v2 subs"   and "e \<in> set es" shows "\<exists>es'. es = es' @ [e] \<and> e \<notin> set es'"
join_fmap_return: fixes xs :: "'d\<cdot>'e" shows "join\<cdot>(fmap\<cdot>return\<cdot>xs) = xs"
F1: fixes u :: "complex" assumes "u \<in> S"   and "u \<noteq> a" shows "(\<lambda>z. if z = a then deriv f a else (f z - f a) / (z - a)) field_differentiable at u within S"
eq_equivp: shows "equivp (\<sim>\<^sub>P)"
assms_3: shows "ContractsWithBilateralSubstitutesAndIRC Xd Xh Pd' (Ch'(h := Ch''))"
op_igb_graph_degeneralize_ext_def: fixes ecnv :: "('a, 'c) igb_graph_rec_scheme \<Rightarrow> 'b"   and G :: "('a, 'c) igb_graph_rec_scheme" shows "op_igb_graph_degeneralize_ext ecnv G \<equiv> igb_graph.degeneralize_ext G ecnv"
one_diff_power_eq: fixes x :: "'a"   and n :: "nat" shows "(1::'a) - x ^ n = ((1::'a) - x) * sum ((^) x) {..<n}"
ltrm_of_X_minus: fixes R :: "('c, 'd) ring_scheme"   and a :: "'c" assumes "UP_cring R"   and "a \<in> carrier R"   and "\<one>\<^bsub>R\<^esub> \<noteq> \<zero>\<^bsub>R\<^esub>" shows "monom (UP R) (X_poly_minus R a (deg R (X_poly_minus R a))) (deg R (X_poly_minus R a)) = X_poly R"
one_strongly_connected_iff_1_eq: fixes x :: "'a" shows "one_strongly_connected x = (x\<^sup>T ; 1 ; x\<^sup>T = x\<^sup>+)"
star_left_unfold: fixes y :: "'a" shows "(1::'a) \<squnion> y\<^sup>+ \<le> y\<^sup>\<star>"
pdevs_val_const_pdevs_of_list: fixes c :: "real"   and xs :: "'b list" shows "pdevs_val (\<lambda>_. c) (pdevs_of_list xs) = c *\<^sub>R sum_list xs"
disjoint: shows "A \<Gamma>' \<inter> B \<Gamma>' = {}"
hyps_3: assumes "\<And>i. i \<in> J \<Longrightarrow> f i \<noteq> 0" shows "degree (prod f J) = (\<Sum>i\<in>J. degree (f i))"
Bot_delR_simp: fixes \<Gamma> :: "'a formula multiset"   and \<Delta> :: "'a formula multiset" shows "\<Gamma> \<Rightarrow> \<bottom>, \<Delta> = \<Gamma> \<Rightarrow> \<Delta>"
linear_indep_image_lemma: fixes f :: "'b \<Rightarrow> 'c"   and B :: "'b set"   and x :: "'b" assumes "linear (*a) (*b) f"   and "finite B"   and "vs2.independent (f ` B)"   and "inj_on f B"   and "x \<in> vs1.span B"   and "f x = (0::'c)" shows "x = (0::'b)"
is_ground_subst_def: fixes subst_atm :: "'a \<Rightarrow> 'b \<Rightarrow> 'a"   and \<sigma> :: "'b" shows "substitution_ops.is_ground_subst subst_atm \<sigma> = (\<forall>A. substitution_ops.is_ground_atm subst_atm (subst_atm A \<sigma>))"
m_gt_0: fixes m :: "nat" assumes "LoadBalancing m" shows "0 < m"
dropWhile_queue_rep: fixes f :: "'a \<Rightarrow> bool"   and q :: "'a queue" shows "linearize (dropWhile_queue f q) = dropWhile f (linearize q)"
simps_4: fixes \<phi> :: "'b formula" shows "SOV (FNot \<phi>) = SOV \<phi>"
ivl_disj_un_9: fixes l :: "'e"   and u :: "'e" assumes "l \<le> u" shows "{..l} \<union> {l<..u} = {..u}"
valid_W_invD0_2: fixes r :: "'d"   and s :: "'f process_name \<Rightarrow> ('e, 'f, 'g, 'd) Model.local_state"   and p :: "'f process_name"   and q :: "'f process_name" assumes "r \<in> W (s p)"   and "valid_W_inv s" shows "r \<notin> ghost_honorary_grey (s q)"
integral_mult_left: fixes c :: "'a"   and M :: "'b measure"   and f :: "'b \<Rightarrow> 'a" assumes "c \<noteq> (0::'a) \<Longrightarrow> integrable M f" shows "LINT x|M. f x * c = integral\<^sup>L M f * c"
simps_2: fixes f :: "'e \<Rightarrow> 'f \<Rightarrow> 'd"   and x1 :: "'e"   and x2 :: "'f" shows "(case (x1, x2) of (x, xa) \<Rightarrow> f x xa) = f x1 x2"
right_iff_left: fixes \<sigma> :: "'a"   and \<tau> :: "'a" shows "right \<sigma> \<tau> = left \<tau> \<sigma>"
univalentD: fixes R :: "('e \<times> 'f) set"   and x :: "'e"   and y :: "'f"   and z :: "'f" assumes "univalent R"   and "(x, y) \<in> R"   and "(x, z) \<in> R" shows "z = y"
assms: shows "coincide_on \<sigma> \<sigma>' (vars_of_cl (cl_ecl C))"
top_on_acom_def: fixes C :: "'a st option acom"   and X :: "char list set" shows "top_on\<^sub>c C X = (\<forall>a\<in>set (annos C). top_on\<^sub>o a X)"
simps_16: fixes x4 :: "int"   and x3 :: "int" shows "Smaller' x4 \<noteq> Greater' x3"