hcju/isabelleps huggingface dataset:
add_eq_0_iff: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.group_add minus plus zero uminus" shows "(plus a b = zero) = (b = uminus a)"
deriv_uminus: shows "deriv uminus = (\<lambda>z. - (1::'a))"
asciis_of_zero: shows "asciis_of_literal STR '''' = []"
s_r: shows "s r = Some (\<sigma>, \<tau>, \<E> [Ref (VE v)])"
simps_9: fixes t :: "'a local.term"   and u :: "'a local.term"   and v :: "'a local.term" shows "\<lbrace>\<^bold>\<a>\<^bold>[t, u, v\<^bold>]\<rbrace> = \<alpha> (\<lbrace>t\<rbrace>, \<lbrace>u\<rbrace>, \<lbrace>v\<rbrace>)"
simps_2: fixes \<phi> :: "'a ltln"   and \<psi> :: "'a ltln" shows "rewrite_syn_imp (\<phi> or\<^sub>n \<psi>) = (if \<phi> \<turnstile>\<^sub>s \<psi> then rewrite_syn_imp \<psi> else if \<psi> \<turnstile>\<^sub>s \<phi> then rewrite_syn_imp \<phi> else if not\<^sub>n \<phi> \<turnstile>\<^sub>s \<psi> \<or> not\<^sub>n \<psi> \<turnstile>\<^sub>s \<phi> then true\<^sub>n else mk_or (rewrite_syn_imp \<phi>) (rewrite_syn_imp \<psi>))"
ennreal_pmf_map: fixes f :: "'d \<Rightarrow> 'c"   and p :: "'d pmf"   and x :: "'c" shows "ennreal (pmf (map_pmf f p) x) = integral\<^sup>N (measure_pmf p) (indicator (f -` {x}))"
f3: shows "dims (tensor_from_vec (dims B) (vec_smult \<alpha> (vec A))) = dims B"
factor_def: fixes G :: "('b, 'c) monoid_scheme"   and a :: "'b"   and b :: "'b" shows "a divides\<^bsub>G\<^esub> b = (\<exists>c\<in>carrier G. b = a \<otimes>\<^bsub>G\<^esub> c)"
intros_2: fixes uv :: "'c" shows "Le uv \<prec> \<infinity>"
add_edge_def: shows "add_edge \<equiv> \<lambda>(u, v) d g. add_edge1 (v, u) d (add_edge1 (u, v) d g)"
normal_imp_commuting: fixes G :: "('e, 'f) monoid_scheme"   and A :: "'e set"   and B :: "'e set"   and x :: "'e"   and y :: "'e" assumes "Group.group G"   and "A \<lhd> G"   and "B \<lhd> G"   and "A \<inter> B \<subseteq> {\<one>\<^bsub>G\<^esub>}"   and "x \<in> A"   and "y \<in> B" shows "x \<otimes>\<^bsub>G\<^esub> y = y \<otimes>\<^bsub>G\<^esub> x"
sup_continuous_mult_left_ennreal: fixes f :: "'a \<Rightarrow> ennreal"   and c :: "ennreal" assumes "sup_continuous f" shows "sup_continuous (\<lambda>x. c * f x)"
vpa_trans_observable_moves: fixes cs :: "'edge list"   and as :: "'edge list"   and m :: "'node"   and m' :: "'node"   and s :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list"   and s' :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list"   and rs :: "'edge list"   and S :: "'node SDG_node set" assumes "valid_path_aux cs as"   and "m -as\<rightarrow>* m'"   and "preds (kinds as) s"   and "transfers (kinds as) s = s'"   and "valid_call_list cs m"   and "\<forall>i<length rs. rs ! i \<in> get_return_edges (cs ! i)"   and "valid_return_list rs m"   and "length rs = length cs"   and "length s = Suc (length cs)" obtains ms :: "'node list" and ms'' :: "'node list" and s'' :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list" and ms' :: "'node list" and as' :: "'edge list" and as'' :: "'edge list" where "S,kind \<turnstile> (m # ms,s) =slice_edges S cs as\<Rightarrow>* (ms'',s'')" and "S,kind \<turnstile> (ms'',s'') =as'\<Rightarrow>\<^sub>\<tau> (m' # ms',s')" and "ms = targetnodes rs" and "length ms = length cs" and "\<forall>i<length cs. call_of_return_node (ms ! i) (sourcenode (cs ! i))" and "slice_edges S cs as = slice_edges S cs as''" and "m -as'' @ as'\<rightarrow>* m'" and "valid_path_aux cs (as'' @ as')"
split_block_def: fixes A :: "'b mat"   and sr :: "nat"   and sc :: "nat" shows "split_block A sr sc = (let nr = dim_row A; nc = dim_col A; nr2 = nr - sr; nc2 = nc - sc; A1 = mat sr sc (($$) A); A2 = mat sr nc2 (\<lambda>(i, j). A $$ (i, j + sc)); A3 = mat nr2 sc (\<lambda>(i, j). A $$ (i + sr, j)); A4 = mat nr2 nc2 (\<lambda>(i, j). A $$ (i + sr, j + sc)) in (A1, A2, A3, A4))"
lookupa_fresh: fixes a :: "coname"   and \<theta>_c :: "(coname \<times> name \<times> trm) list"   and y :: "name" assumes "a \<sharp> \<theta>_c" shows "lookupa y a \<theta>_c = Ax y a"
hyps_4: assumes "\<And>a b. \<lbrakk>a \<in> A; b \<in> A; a \<noteq> b\<rbrakk> \<Longrightarrow> coprime a b" shows "Lcm A = normalize (\<Prod>A)"
DmdImpl: fixes F :: "'a \<Rightarrow> bool"   and G :: "'a \<Rightarrow> bool" assumes "\<turnstile> F \<longrightarrow> G" shows "\<turnstile> \<diamond>F \<longrightarrow> \<diamond>G"
wt'_wt: fixes P :: "(char list list \<times> char list exp) prog"   and h :: "heap"   and sh :: "sheap"   and E :: "char list \<Rightarrow> ty option"   and e :: "char list exp"   and T :: "ty" assumes "P,E,h,sh \<turnstile> e :' T" shows "P,E,h,sh \<turnstile> e : T"
two_face_Seed: fixes p :: "nat" shows "2 \<le> |faces Seed\<^bsub>p\<^esub>|"
all_acquired_takeWhile_subset: fixes P :: "'b memref \<Rightarrow> bool"   and sb :: "'b memref list" shows "all_acquired (takeWhile P sb) \<subseteq> all_acquired sb"
bit_minus_iff: fixes a :: "'a"   and n :: "nat" shows "bit (- a) n = ((2::'a) ^ n \<noteq> (0::'a) \<and> \<not> bit (a - (1::'a)) n)"
wf_wcode_double_case_le: shows "wf wcode_double_case_le"
X5_pow: fixes X :: "X5 set" shows "X \<in> set ` set (subseqs [X5.Xd1, X5.Xd1', X5.Xd2, X5.Xd2', Xd2''])"
Sig2Mor: fixes A :: "'t"   and e :: "('t, 'f) Expression"   and B :: "'t" assumes "Sig iS \<triangleright> Vx : A \<turnstile> e : B" shows "\<exists>g. L\<lbrakk>Vx : A \<turnstile> e : B\<rbrakk> \<rightarrow> IMor g"
model_to_dimacs_neg_nin: fixes n_vars :: "nat"   and x :: "nat"   and M :: "nat \<Rightarrow> bool"   and a :: "nat" assumes "n_vars \<le> x" shows "int x \<notin> set (model_to_dimacs_model M [a..<n_vars])"
FullSpec_def: fixes s :: "('b, 't, 'loc) Combined.configuration stream" shows "FullSpec s = (holds InitConfig s \<and> alw Next s)"
finite_inc_above_ne: fixes f :: "nat \<Rightarrow> 'b"   and c :: "'b"   and b :: "nat"   and i :: "nat" assumes "finite {x. f x \<noteq> c}" shows "finite {x. f (inc_above b i x) \<noteq> c}"
assms_3: shows "\<not> (\<exists>n'\<ge>n. n' < n'' \<and> \<parallel>id\<parallel>\<^bsub>t n'\<^esub>)"
cases: fixes G :: "('a, 'b) Graph_scheme"   and a :: "'a list" assumes "Digraph G"   and "Digraph.walk G a" obtains "a = []"   | v :: "'a" where "a = [v__]" and "v__ \<in> V\<^bsub>G\<^esub>"   | v :: "'a" and w :: "'a" and vs :: "'a list" where "a = v # w # vs" and "v \<rightarrow>\<^bsub>G\<^esub> w" and "Digraph.walk G (w # vs)"
fisher_yates_aux_correct: fixes i :: "nat"   and xs :: "'a list" shows "fisher_yates_aux i xs = map_pmf ((@) (take i xs)) (shuffle (drop i xs))"
frac_nat_add_id: fixes n :: "nat"   and r :: "real" shows "frac (real n + r) = frac r"
assms: shows "(as \<inter> supp x, x) \<approx>set (=) supp p (as' \<inter> supp x', x')"
assms_3: shows "fixespointwise s (C0 \<inter> D0)"
unique_box_desc_1: fixes v :: "i"   and \<phi> :: "\<nu> \<Rightarrow> \<o>" shows "[(\<^bold>\<exists>!x. \<^bold>\<box>\<phi> x) \<^bold>\<rightarrow> (\<^bold>\<forall>y. y\<^sup>P \<^bold>= (\<^bold>\<iota>x. \<phi> x) \<^bold>\<rightarrow> \<phi> y) in v]"
lem1: shows "\<forall>x\<in>cbox (0::'a) One. \<forall>y\<in>cbox (0::'a) One. \<forall>i\<in>Basis. label x i \<noteq> label y i \<longrightarrow> \<bar>f x \<bullet> i - x \<bullet> i\<bar> \<le> norm (f y - f x) + norm (y - x)"
frequently_elim1: fixes P :: "'b \<Rightarrow> bool"   and F :: "'b filter"   and Q :: "'b \<Rightarrow> bool" assumes "frequently P F"   and "\<And>i. P i \<Longrightarrow> Q i" shows "frequently Q F"
f_def: fixes y :: "nat"   and x :: "nat" shows "f y x = set_to_nat (nat_to_set (c_fst x) - {z \<in> nat_to_set (c_snd x). z < y})"
set_cond_pmf: fixes p :: "'c pmf"   and s :: "'c set" assumes "set_pmf p \<inter> s \<noteq> {}" shows "set_pmf (cond_pmf p s) = set_pmf p \<inter> s"
rcm_A: fixes a :: "'a"   and b :: "'a" shows "a * (a r\<rightarrow> b) = b * (b r\<rightarrow> a)"
sequentially_consistentI: fixes E :: "('a \<times> ('b, 'a) obs_event action) llist"   and P :: "'c prog"   and ws :: "nat \<Rightarrow> nat" assumes "\<And>r. r \<in> read_actions E \<Longrightarrow> P,E \<turnstile> r \<leadsto>mrw ws r" shows "sequentially_consistent P (E, ws)"
lucas_coprime_lemma: fixes m :: "nat"   and a :: "nat"   and n :: "nat" assumes "m \<noteq> 0"   and "[a ^ m = 1] (mod n)" shows "coprime a n"
extended_distance_set_middle: fixes dist :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and uniformity :: "('c \<times> 'c) filter"   and "open" :: "'c set \<Rightarrow> bool"   and a :: "'c Bonk_Schramm_extension_unfolded"   and b :: "'c Bonk_Schramm_extension_unfolded" assumes "class.metric_space dist uniformity open"   and "a \<in> metric_space.extended_distance_set dist"   and "b \<in> metric_space.extended_distance_set dist" shows "middle a b \<in> metric_space.extended_distance_set dist"
valid_app_mono: fixes \<Delta> :: "(name \<times> ty) list"   and \<Gamma>\<^sub>1 :: "(name \<times> ty) list"   and \<Gamma>\<^sub>2 :: "(name \<times> ty) list" assumes "valid (\<Delta> @ \<Gamma>\<^sub>1)"   and "supp \<Delta> \<sharp>* \<Gamma>\<^sub>2"   and "valid \<Gamma>\<^sub>2"   and "\<Gamma>\<^sub>1 \<sqsubseteq> \<Gamma>\<^sub>2" shows "valid (\<Delta> @ \<Gamma>\<^sub>2)"
infinite_finite_induct: fixes P :: "'q set \<Rightarrow> bool"   and A :: "'q set" assumes "\<And>A. infinite A \<Longrightarrow> P A"   and "P {}"   and "\<And>x F. \<lbrakk>finite F; x \<notin> F; P F\<rbrakk> \<Longrightarrow> P (insert x F)" shows "P A"
deg_le_sect_zero: fixes X :: "'b set" shows "deg_le_sect X 0 = {0}"
G1_bigo: shows "(\<lambda>x. G1 x - (x * (ln x)\<^sup>2 - x * ln x)) \<in> O(\<lambda>x. (ln x)\<^sup>2)"
dematricize_def: fixes rmodes :: "nat set"   and A :: "'m mat"   and ds :: "nat list" shows "dematricize rmodes A ds = tensor_from_lookup ds (\<lambda>is. A $$ (digit_decode (nths ds rmodes) (nths is rmodes), digit_decode (nths ds (- rmodes)) (nths is (- rmodes))))"
mmore_def: shows "mmore = Record.iso_tuple_snd Measurement_System_ext_Tuple_Iso"
distinct_1: fixes x1 :: "prim_ty"   and x2 :: "ref_ty" shows "PrimT x1 \<noteq> RefT x2"
imp_foreach_def: fixes tsl :: "'d \<Rightarrow> 'e list Heap"   and s :: "'d"   and c :: "'c \<Rightarrow> bool Heap"   and f :: "'e \<Rightarrow> 'c \<Rightarrow> 'c Heap"   and \<sigma> :: "'c" shows "imp_foreach tsl s c f \<sigma> \<equiv> tsl s \<bind> (\<lambda>l. imp_nfoldli l c f \<sigma>)"
ts_id_def_1: shows "ts_id @ [t, t] = ts''"
prj_simps_13: shows "C.dom \<mu>\<nu>\<pi>.Prj\<^sub>1 = H\<mu>H\<nu>\<pi>.chine"
assms_2: fixes R :: "pi"   and S :: "pi" assumes "R \<simeq> S" shows "F R \<leadsto>\<guillemotleft>weakBisim\<guillemotright> F S"
rtrancl_O_push: fixes S :: "('a \<times> 'a) set"   and R :: "('a \<times> 'a) set" assumes "S O R \<subseteq> R O S\<^sup>*" shows "S\<^sup>* O R \<subseteq> R O S\<^sup>*"
HomFtorId': fixes C :: "('d, 'e, 'f) LSCategory_scheme"   and X :: "'d"   and Y :: "'d" assumes "LSCategory C"   and "X \<in> obj\<^bsub>C\<^esub>"   and "Y \<in> obj\<^bsub>C\<^esub>" shows "Hom\<^bsub>C\<^esub>[X,id\<^bsub>C\<^esub> Y] = id\<^bsub>SET'\<^esub> (Hom\<^bsub>C\<^esub> X Y)"
set_the_range_of_is_range_of_sas_plus_if: fixes \<Psi> :: "('c, 'd) sas_plus_problem"   and v :: "'c" assumes "is_valid_problem_sas_plus \<Psi>"   and "v \<in> set (\<Psi>\<^sub>\<V>\<^sub>+)" shows "set (the (range_of \<Psi> v)) = \<R>\<^sub>+ \<Psi> v"
foldl_cong: fixes a :: "'a"   and b :: "'a"   and l :: "'b list"   and k :: "'b list"   and f :: "'a \<Rightarrow> 'b \<Rightarrow> 'a"   and g :: "'a \<Rightarrow> 'b \<Rightarrow> 'a" assumes "a = b"   and "l = k"   and "\<And>a x. x \<in> set l \<Longrightarrow> f a x = g a x" shows "foldl f a l = foldl g b k"
assms_1: fixes i :: "'a" assumes "i \<in> I" shows "space (E i) = space (F i)"
PiE_mem: fixes f :: "'b \<Rightarrow> 'c"   and S :: "'b set"   and T :: "'b \<Rightarrow> 'c set"   and x :: "'b" assumes "f \<in> Pi\<^sub>E S T"   and "x \<in> S" shows "f x \<in> T x"
suc_least_AboveS: fixes a :: "'a"   and B :: "'a set" assumes "a \<in> local.AboveS B" shows "(suc B, a) \<in> r"
r1: shows "r1 \<in> set (fst (splitAt r2 vs))"
toProcess_sidx: fixes sidx :: "nat"   and p :: "proc"   and ss :: "(integer \<times> edge list) iarray"   and a :: "nat"   and n :: "String.literal"   and l :: "(String.literal, nat) assoc_list"   and idx :: "nat"   and r :: "edgeIndex \<times> procArg list \<times> varDecl list" assumes "toProcess sidx p = (ss, a, n, l, idx, r)" shows "idx = sidx"
zero_le_double_add_iff_zero_le_single_add: 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 a :: "'a" assumes "class.linordered_ab_group_add plus zero minus uminus less_eq less" shows "less_eq zero (plus a a) = less_eq zero a"
dep_sos_imp_mem_dep: fixes PROB :: "(('b, 'c) fmap \<times> ('b, 'd) fmap) set"   and S :: "'b set set"   and vs :: "'b set" assumes "dep_var_set PROB (\<Union> S) vs" shows "\<exists>vs'. vs' \<in> S \<and> dep_var_set PROB vs' vs"
induced_map_iterates: fixes n :: "nat"   and A :: "'a set"   and x :: "'a" shows "(A.induced_map A ^^ n) x = (induced_map A ^^ (\<Sum>i<n. A.return_time_function A ((A.induced_map A ^^ i) x))) x"
op_inter_fst_def: fixes X :: "('a \<times> 'b) set"   and Y :: "'a set" shows "op_inter_fst X Y = X \<inter> Y \<times> UNIV"
rel_Grp: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'b" shows "rel_pmf (BNF_Def.Grp A f) = BNF_Def.Grp {x. set_pmf x \<subseteq> A} (map_pmf f)"
set_pmf_finite: fixes s :: "'s"   and D :: "'s pmf" assumes "s \<in> S"   and "D \<in> K s" shows "finite (set_pmf D)"
lemma_3_8_ii: fixes F :: "'a set"   and a :: "'a"   and b :: "'a" assumes "F \<in> filters" shows "cong_l F a b = (\<exists>x y. x \<in> F \<and> y \<in> F \<and> a * x = b * y)"
invariant_1E: fixes plr :: "int poly \<times> rat \<times> rat" assumes "invariant_1 plr" obtains "root_cond plr (real_of_1 plr)" and "sgn (rai_lb plr) = sgn (rai_ub plr)" and "sgn (real_of_1 plr) = real_of_rat (sgn (rai_ub plr))" and "\<exists>!x. root_cond plr x" and "poly_cond (poly_real_alg_1 plr)" and "0 < degree (poly_real_alg_1 plr)" and "primitive (poly_real_alg_1 plr)"
tendsto_add: fixes f :: "'d \<Rightarrow> 'e"   and a :: "'e"   and F :: "'d filter"   and g :: "'d \<Rightarrow> 'e"   and b :: "'e" assumes "(f \<longlongrightarrow> a) F"   and "(g \<longlongrightarrow> b) F" shows "((\<lambda>x. f x + g x) \<longlongrightarrow> a + b) F"
OK_extend_iff: fixes h :: "'c \<times> 'd \<Rightarrow> 'e"   and I :: "'f set"   and F :: "'f \<Rightarrow> 'c program" assumes "Extend h" shows "OK I (\<lambda>i. extend h (F i)) = OK I F"
bit1_def: shows "bit1 = (\<lambda>x. Suc (bit0 x))"
1: shows "(\<exists>x. linear_orderable_1 x) = (\<exists>x. linear_orderable_2 x)"
equivalence_maps_compose: fixes f :: "'a"   and f' :: "'a" assumes "equivalence_map f"   and "equivalence_map f'"   and "src f' = trg f" shows "equivalence_map (f' \<star> f)"
finite_dimensional_vector_space_axioms: shows "finite_dimensional_vector_space (*b) B2"
eqOn_def: fixes P :: "pvar set"   and env :: "pvar \<Rightarrow> nat"   and \<pi>l :: "('state \<times> 'aprop set) stream list"   and env1 :: "pvar \<Rightarrow> nat"   and \<pi>l1 :: "('state \<times> 'aprop set) stream list" shows "local.eqOn P env \<pi>l env1 \<pi>l1 \<equiv> \<forall>p. p \<in> P \<longrightarrow> \<pi>l ! env p = \<pi>l1 ! env1 p"
is_path_undir_split': fixes v :: "'v"   and p1 :: "('v \<times> 'w \<times> 'v) list"   and u :: "'v"   and w :: "'w"   and u' :: "'v"   and p2 :: "('v \<times> 'w \<times> 'v) list"   and v' :: "'v" shows "is_path_undir G v (p1 @ (u, w, u') # p2) v' = (is_path_undir G v p1 u \<and> ((u, w, u') \<in> E \<or> (u', w, u) \<in> E) \<and> is_path_undir G u' p2 v')"
ht_hash_in_bounded_hashcode_nat_not_i_not_in_dom_i: fixes l :: "('a \<times> 'b) list list"   and i :: "nat"   and k :: "'a" assumes "ht_hash l"   and "1 < length l"   and "i < length l"   and "i \<noteq> bounded_hashcode_nat (length l) k" shows "k \<notin> dom (map_of (l ! i))"
that: fixes p' :: "('a, 'b, 'c, 'd) seqp"   and pp' :: "('a, 'b, 'c, 'd) seqp" assumes "p = {l}unicast(fip, fmsg) . p' \<triangleright> pp'"   and "case a of unicast x xa \<Rightarrow> l' \<in> labels \<Gamma> q1 | _ \<Rightarrow> l' \<in> labels \<Gamma> q2" shows "thesis"
length_word_rsplit_exp_size: fixes n :: "nat"   and w :: "'d word" assumes "n = LENGTH('c)" shows "length (word_rsplit w) = (size w + n - 1) div n"
as: fixes i :: "nat" shows "(as i, as (Suc i)) \<in> R \<union> S"
simps_2: fixes j :: "nat"   and xs :: "nat list"   and i :: "nat"   and bezout :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<times> 'b \<times> 'b \<times> 'b \<times> 'b"   and P :: "(('b, 'c) Finite_Cartesian_Product.vec, 'c) Finite_Cartesian_Product.vec"   and A :: "(('b, 'd) Finite_Cartesian_Product.vec, 'c) Finite_Cartesian_Product.vec"   and Q :: "(('b, 'd) Finite_Cartesian_Product.vec, 'd) Finite_Cartesian_Product.vec" shows "diagonal_to_Smith_i_PQ (j # xs) i bezout (P, A, Q) = (if A $h from_nat i $h from_nat i dvd A $h from_nat j $h from_nat j then diagonal_to_Smith_i_PQ xs i bezout (P, A, Q) else let (p, q, u, v, d) = bezout (A $h from_nat i $h from_nat i) (A $h from_nat j $h from_nat j); A' = diagonal_step A i j d v; (P', Q') = diagonal_step_PQ A i j bezout in diagonal_to_Smith_i_PQ xs i bezout (P' ** P, A', Q ** Q'))"
wfPUNIVI: fixes r :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "\<And>P x. \<forall>x. (\<forall>y. r y x \<longrightarrow> P y) \<longrightarrow> P x \<Longrightarrow> P x" shows "wfP r"
HFail_HInv5_q1: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and q :: "Proc" assumes "HFail s s' p"   and "p \<noteq> q"   and "Inv2a_inner s' q"   and "maxBalInp s (bal (dblock s q)) (inp (dblock s q))" shows "maxBalInp s' (bal (dblock s' q)) (inp (dblock s' q))"
Sup_vec_def: fixes X :: "('a, 'b) vec set" shows "Sup X = (\<chi>i. SUP x\<in>X. x $ i)"
merge': fixes h1 :: "'b heap"   and h2 :: "'b heap" assumes "is_root' h1"   and "is_root' h2" shows "hom (merge' h1 h2) = Pairing_Heap_Tree.merge (hom h1) (hom h2)"
int_simps_11: fixes P :: "'a \<Rightarrow> bool" shows "\<turnstile> (#False \<longrightarrow> P) = #True"
convex_alt: fixes s :: "'b set" shows "convex s = (\<forall>x\<in>s. \<forall>y\<in>s. \<forall>u. 0 \<le> u \<and> u \<le> 1 \<longrightarrow> (1 - u) *\<^sub>R x + u *\<^sub>R y \<in> s)"
smap_empty_def: fixes k :: "'b" shows "{}. k = {}"
encode_initial_state_def: fixes \<Pi> :: "'b strips_problem" shows "\<Phi>\<^sub>I \<Pi> \<equiv> let I = \<Pi>\<^sub>I; vs = \<Pi>\<^sub>\<V> in \<^bold>\<And>map (\<lambda>v. encode_state_variable 0 (index vs v) (I v) \<^bold>\<or> \<bottom>) (filter (\<lambda>v. I v \<noteq> None) vs)"
Refinement_basic: fixes Sc :: "('a, 'b) spec"   and R :: "('c \<times> 'a) set"   and Sa :: "('c, 'd) spec"   and pi :: "'b \<Rightarrow> 'd" assumes "init Sc \<subseteq> R `` init Sa"   and "{R} TS.trans Sa, TS.trans Sc {> R}"   and "obs_consistent R pi Sa Sc" shows "refines R pi Sa Sc"
sum_relI_2: fixes r :: "'c"   and r' :: "'d"   and Rr :: "('c \<times> 'd) set"   and Rl :: "('e \<times> 'f) set" assumes "(r, r') \<in> Rr" shows "(Inr r, Inr r') \<in> \<langle>Rl, Rr\<rangle>sum_rel"
step: fixes Sup :: "'r set \<Rightarrow> 'r"   and less_eq :: "'r \<Rightarrow> 'r \<Rightarrow> bool"   and less :: "'r \<Rightarrow> 'r \<Rightarrow> bool"   and x :: "'r"   and f :: "'r \<Rightarrow> 'r" assumes "class.ccpo Sup less_eq less"   and "x \<in> ccpo.iterates Sup less_eq f" shows "f x \<in> ccpo.iterates Sup less_eq f"
norm_butterfly: fixes \<psi> :: "'b"   and \<phi> :: "'a" shows "norm (butterfly \<psi> \<phi>) = norm \<psi> * norm \<phi>"
TimesL_def: fixes r :: "'d regex"   and S :: "'d regex set" shows "TimesL r S = Times r ` S"
N_entails_B: shows "N \<Turnstile>\<inter>\<G> {B}"
disjoint_family_onI: fixes I :: "'a set"   and A :: "'a \<Rightarrow> 'b set" assumes "\<And>i j. \<lbrakk>i \<in> I \<and> j \<in> I; i \<noteq> j\<rbrakk> \<Longrightarrow> A i \<inter> A j = {}" shows "disjoint_family_on A I"
hom: shows "\<forall>x\<in>carrier G. \<forall>y\<in>carrier G. h (x \<otimes>\<^bsub>G\<^esub> y) = h x \<otimes>\<^bsub>H\<^esub> h y"
class_assoc_type_def: fixes \<A> :: "'a" shows "class_assoc_type \<A> \<equiv> Set \<langle>\<A>\<rangle>\<^sub>\<T>[1]"
cycle_free_up_to_diag: fixes m :: "nat \<Rightarrow> nat \<Rightarrow> 'a"   and k :: "nat"   and n :: "nat"   and i :: "nat" assumes "cycle_free_up_to m k n"   and "i \<le> n" shows "(0::'a) \<le> m i i"
fds_nth_mult: fixes f :: "'b fds"   and g :: "'b fds" shows "fds_nth (f * g) = dirichlet_prod (fds_nth f) (fds_nth g)"
match_def: fixes \<mu> :: "(M \<times> W) set" shows "Non_Strict.match \<mu> = (inj_on fst \<mu> \<and> inj_on snd \<mu> \<and> \<mu> \<subseteq> (\<Union>m. {m} \<times> Field (Pm m)) \<inter> (\<Union>w. Field (Pw w) \<times> {w}))"
vec_list: fixes v :: "'a vec" shows "vec_of_list (list_of_vec v) = v"
better_Cut_substn2: fixes x :: "name"   and P :: "trm"   and y :: "name"   and M :: "trm"   and a :: "coname"   and N :: "trm"   and b :: "coname" assumes "x \<sharp> (P, y)"   and "y \<sharp> M" shows "Cut <a>.M x.N{y:=<b>.P} = Cut <a>.M x.N{y:=<b>.P}"
rbt_delete: fixes t :: "('b \<times> color) tree"   and x :: "'b" assumes "rbt t" shows "rbt (RBT_Set.delete x t)"
d_transitive_PreLoebian_imp_Loebian: fixes x :: "'a" assumes "\<forall>y.  |x\<rangle>  |x\<rangle> y \<le>  |x\<rangle> y"   and "PreLoebian x" shows "Loebian x"
rPar1: fixes P' :: "('a, 'b, 'c) psi"   and C :: "'d" assumes "\<Psi> \<rhd> P \<longmapsto> \<tau> \<prec> P'" shows "Prop C \<Psi> (P \<parallel> !P) (P' \<parallel> !P)"
cases: fixes x :: "'a btree \<times> 'a" obtains y :: "'a" where "x = (Leaf, y__)"   | ts :: "('a btree \<times> 'a) list" and t :: "'a btree" and y :: "'a" where "x = (Node ts t, y)"
bd_FGcontra_def: shows "bd_FGcontra = bd_F"
invh_upd_1: fixes t :: "(('c \<times> 'd) \<times> tcolor \<times> 'c \<times> 'd) tree"   and x :: "'c"   and y :: "'d" assumes "invh t" shows "invh (upd x y t)"
compress_extra_correct_matchexpr: fixes \<beta> :: "'b common_primitive \<Rightarrow> ('b, 'c) tagged_packet_scheme \<Rightarrow> ternaryvalue"   and \<alpha> :: "action \<Rightarrow> ('b, 'c) tagged_packet_scheme \<Rightarrow> bool"   and m :: "'b common_primitive match_expr" assumes "primitive_matcher_generic \<beta>" shows "matches (\<beta>, \<alpha>) m = matches (\<beta>, \<alpha>) (compress_extra m)"
convex_hulls_are_convex: fixes A :: "'b vec set"   and n :: "nat" assumes "A \<subseteq> carrier_vec n" shows "gram_schmidt.convex n (gram_schmidt.convex_hull n A)"
nb\<^sub>8_1: shows "dom (map_of (map (\<lambda>v. (v, True)) (add_effects_of op'))) = set (effect_of op)"
induct: fixes P :: "fm \<Rightarrow> bool"   and a0 :: "fm" assumes "\<And>a. P (Lt a)"   and "\<And>a. P (Le a)"   and "\<And>a. P (Gt a)"   and "\<And>a. P (Ge a)"   and "\<And>a. P (Eq a)"   and "\<And>a. P (NEq a)"   and "\<And>p. P p \<Longrightarrow> P (NOT p)"   and "\<And>p q. \<lbrakk>P p; P q\<rbrakk> \<Longrightarrow> P (And p q)"   and "\<And>p q. \<lbrakk>P p; P q\<rbrakk> \<Longrightarrow> P (Or p q)"   and "\<And>p q. \<lbrakk>P p; P q\<rbrakk> \<Longrightarrow> P (Imp p q)"   and "\<And>p q. \<lbrakk>P p; P q\<rbrakk> \<Longrightarrow> P (Iff p q)"   and "P T"   and "P F"   and "\<And>v. P (E v)"   and "\<And>v. P (A v)" shows "P a0"
projection_zero: shows "projection {0::'a} = (\<lambda>_. 0::'a)"
not_nonzero_Zp: fixes x :: "nat \<Rightarrow> int" assumes "x \<in> carrier Zp"   and "x \<notin> nonzero Zp" shows "x = \<zero>"
lhd_ldrop: fixes n :: "enat"   and xs :: "'c llist" assumes "n < llength xs" shows "lhd (ldrop n xs) = lnth xs (the_enat n)"
set_sel_1: fixes a :: "('d, 'e, 'f) generat" assumes "is_Pure a" shows "generat.result a \<in> generat_pures a"
inj_on_empty: fixes f :: "'b \<Rightarrow> 'c" shows "inj_on f {}"
fref_ncI: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'c \<Rightarrow> 'd"   and R :: "('a \<times> 'c) set"   and S :: "('b \<times> 'd) set" assumes "(f, g) \<in> R \<rightarrow> S" shows "(f, g) \<in> R \<rightarrow>\<^sub>f S"
prems_4: fixes a :: "'a"   and b :: "'a" assumes "a \<in> set (x0 # xs) \<union> set ys"   and "b \<in> set ys"   and "rel a b" shows "a = b"
run_of_lasso_pre_1: fixes L :: "'Q lasso" assumes "is_lasso_pre L" shows "is_run (run_of_lasso L)"