hcju/isabelleps huggingface dataset:
simps_2: fixes i :: "int"   and l :: "int tree"   and k :: "int"   and r :: "int tree" shows "leaves i \<langle>l, k, r\<rangle> = incr2 ` (leaves i l \<union> leaves (k + 1) r)"
mu: shows "\<bar>gs1.\<mu> i (i - 1)\<bar> \<le> 1 / 2"
finite_total_preorder_of_weak_ranking: fixes xs :: "'c set list"   and A :: "'c set" assumes "\<Union> (set xs) = A"   and "is_finite_weak_ranking xs" shows "finite_total_preorder_on A (of_weak_ranking xs)"
gt_ex: fixes less_eq :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and less :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and x :: "'f" assumes "class.no_top less_eq less" shows "\<exists>y. less x y"
psubst_new_free'_1: fixes c :: "'a"   and n :: "'a"   and t :: "'a FOL_Fitting.term" assumes "c \<noteq> n" shows "new_term n (psubstt (id(n := c)) t)"
zero_less_norm_iff: 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 x :: "'a" assumes "class.real_normed_vector minus dist norm plus zero uminus scaleR sgn uniformity open" shows "(0 < norm x) = (x \<noteq> zero)"
assms_5: shows "\<forall>f\<in>Collect P. |vertices f| \<le> 4"
get_disconnected_nodes_ptr_in_heap: fixes h :: "(_) heap"   and document_ptr :: "(_) document_ptr" assumes "h \<turnstile> ok get_disconnected_nodes document_ptr" shows "document_ptr |\<in>| document_ptr_kinds h"
num_induct: fixes P :: "num \<Rightarrow> bool"   and x :: "num" assumes "P num.One"   and "\<And>x. P x \<Longrightarrow> P (Num.inc x)" shows "P x"
smc_Cod_vdomain: shows "\<D>\<^sub>\<circ> (\<BB>\<lparr>Cod\<rparr>) = \<BB>\<lparr>Arr\<rparr>"
scalar_prod_smult_right: fixes w :: "'a Matrix.vec"   and v :: "'a Matrix.vec"   and k :: "'a" assumes "dim_vec w = dim_vec v" shows "w \<bullet> (k \<cdot>\<^sub>v v) = k * (w \<bullet> v)"
that: fixes f :: "V" assumes "v11 f"   and "\<D>\<^sub>\<circ> f = A"   and "\<R>\<^sub>\<circ> f \<subseteq>\<^sub>\<circ> B" shows "thesis"
lsub: shows "{l (- dr)--l dr} \<subseteq> l ` {- dr..dr}"
2201: fixes x :: "'a"   and y :: "'a" shows "- x \<squnion> - - (y \<squnion> x) = top"
swapEnv_vsubstEnv: fixes rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('a, 'b, 'varSort, 'var, 'c) qTerm set option"   and zs :: "'varSort"   and z1 :: "'var"   and z2 :: "'var"   and ys :: "'varSort"   and y1 :: "'var"   and y :: "'var" assumes "goodEnv rho" shows "rho &[y1 // y]_ys &[z1 \<and> z2]_zs = rho &[z1 \<and> z2]_zs &[y1 @ys[z1 \<and> z2]_zs // y @ys[z1 \<and> z2]_zs]_ys"
set_filter: fixes P :: "'b \<Rightarrow> bool"   and xs :: "'b list" shows "set (filter P xs) = {x \<in> set xs. P x}"
assms_3: fixes i :: "'a" assumes "i \<in> I" shows "Group.group ((f \<circ> Gs) i)"
that: fixes r :: "real" assumes "0 < r"   and "ball a r \<subseteq> s"   and "inj_on f (ball a r)" shows "thesis"
inv_def: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b" assumes "category C" shows "category.inv C f = (SOME g. category.inverse_arrows C f g)"
simps_2: fixes X :: "freemsg"   and Y :: "freemsg" shows "freenonces (MPAIR X Y) = freenonces X \<union> freenonces Y"
norm_signed_idem: fixes w :: "bit list" shows "norm_signed (norm_signed w) = norm_signed w"
simps_2: fixes \<gamma> :: "('b \<Rightarrow> 'c \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'c \<Rightarrow> bool)"   and m :: "'b match_expr"   and a :: "action"   and rs :: "'b rule list"   and P :: "'c set" shows "rmshadow \<gamma> (Rule m a # rs) P = (if \<forall>p\<in>P. \<not> matches \<gamma> m a p then rmshadow \<gamma> rs P else Rule m a # rmshadow \<gamma> rs {p \<in> P. \<not> matches \<gamma> m a p})"
induct: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x1 :: "'a"   and x2 :: "'a"   and P :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "rtranclpR r x1 x2"   and "\<And>a. P a a"   and "\<And>a b c. \<lbrakk>rtranclpR r a b; P a b; r b c\<rbrakk> \<Longrightarrow> P a c" shows "P x1 x2"
Union_eq_wens_single: fixes W :: "'b set set"   and act :: "('b \<times> 'b) set"   and B :: "'b set" assumes "\<forall>k. \<not> W \<subseteq> wens_single_finite act B ` {..k}"   and "W \<subseteq> insert (wens_single act B) (range (wens_single_finite act B))" shows "\<Union> W = wens_single act B"
prime_imp_strong_fermat_witness: fixes p :: "nat"   and a :: "nat" assumes "prime p"   and "2 < p"   and "a \<in> {1..<p}" shows "strong_fermat_liar a p"
bisimilar_is_bisimulation: shows "L_transform.is_bisimulation (\<sim>\<cdot>\<^sub>L)"
map_def: fixes f :: "'a" shows "\<rho>.map f = (if arr f then \<r>[cod f] \<cdot> T (f, \<I>) else null)"
\<F>\<G>_implies_FG: fixes \<phi> :: "'b ltln"   and w :: "nat \<Rightarrow> 'b set" shows "\<forall>\<psi>\<in>\<F>\<G> \<phi> w. w \<Turnstile>\<^sub>n F\<^sub>n (G\<^sub>n \<psi>[\<G>\<F> \<phi> w]\<^sub>\<Pi>\<^sub>1)"
orthogonal_matrix_id: shows "orthogonal_matrix (mat (1::'d))"
single_valued_def: fixes r :: "('c \<times> 'd) set" shows "single_valued r = (\<forall>x y. (x, y) \<in> r \<longrightarrow> (\<forall>z. (x, z) \<in> r \<longrightarrow> y = z))"
Int_space_eq2: fixes \<Omega> :: "'c set"   and M :: "'c set set"   and x :: "'c set" assumes "semiring_of_sets \<Omega> M"   and "x \<in> M" shows "x \<inter> \<Omega> = x"
bij_betw_iff_card: fixes A :: "'c set"   and B :: "'d set" assumes "finite A"   and "finite B" shows "(\<exists>f. bij_betw f A B) = (card A = card B)"
convex_lsc_improper: fixes f :: "'b \<Rightarrow> ereal"   and x :: "'b" assumes "Lower_Semicontinuous.convex_on UNIV f"   and "\<not> proper f"   and "lsc f" shows "f x = \<infinity> \<or> f x = - \<infinity>"
set_3: fixes x1 :: "'a" shows "fv (Var x1) = {x1}"
semidecisiveI: fixes C :: "'a set"   and Is :: "'a set"   and A :: "'b set"   and x :: "'b"   and y :: "'b"   and scf :: "('a \<Rightarrow> ('b \<times> 'b) set) \<Rightarrow> ('b \<times> 'b) set" assumes "C \<subseteq> Is"   and "\<And>P. \<lbrakk>profile A Is P; \<And>i. i \<in> C \<Longrightarrow> x \<^bsub>(P i)\<^esub>\<prec> y; \<And>i. i \<in> Is - C \<Longrightarrow> y \<^bsub>(P i)\<^esub>\<prec> x\<rbrakk> \<Longrightarrow> x \<^bsub>(scf P)\<^esub>\<prec> y" shows "semidecisive scf A Is C x y"
negligible_iff_measure: fixes S :: "'a set" shows "negligible S = (S \<in> lmeasurable \<and> Sigma_Algebra.measure lebesgue S = 0)"
items_def: shows "items \<equiv> \<lambda>H. {x. itemsp (\<lambda>x. x \<in> H) x}"
cnf_of_operator_effect_encoding_contains_add_effect_clause_if: fixes \<Pi> :: "'b strips_problem"   and k :: "nat"   and t :: "nat"   and op :: "'b strips_operator"   and v :: "'b" assumes "is_valid_problem_strips \<Pi>"   and "k < t"   and "op \<in> set (\<Pi>\<^sub>\<O>)"   and "v \<in> set (add_effects_of op)" shows "{(Operator k (index (\<Pi>\<^sub>\<O>) op))\<inverse>, (State (Suc k) (index (\<Pi>\<^sub>\<V>) v))\<^sup>+} \<in> cnf (encode_all_operator_effects \<Pi> (\<Pi>\<^sub>\<O>) t)"
carrier_not_empty: fixes G :: "('f, 'g) monoid_scheme" assumes "Group.monoid G" shows "carrier G \<noteq> {}"
hlp1: fixes i :: "'a ifex"   and "is" :: "'a ifex list"   and a :: "'a"   and val :: "bool" assumes "i \<in> set is"   and "lowest_tops is = Some a"   and "ifex_ordered i" shows "a \<notin> ifex_var_set (restrict_top i a val)"
flatten_while: fixes gs :: "('f \<times> 'd set) list"   and b :: "'d set"   and c :: "('d, 'e, 'f) com" shows "flatten (while gs b c) = [while gs b c]"
safe: shows "\<O>s,\<R>s,i\<turnstile> (RMW a t (D, f) cond ret A L R W # is, \<theta>, m, \<D>, \<O>, \<S>) \<surd>"
assms: shows "[x \<^bold>=\<^sub>E y in v]"
ACC: shows "\<forall>A\<in>F. limit r \<inter> A \<noteq> {}"
bij_mono_map_set_to_nat_def: fixes A :: "'a set" shows "bij_mono_map_set_to_nat A = (\<lambda>x. if x \<in> A then the_inv_into {..<card A} ((!) (sorted_list_of_set A)) x else card A)"
simps: fixes ws :: "'a list"   and uu :: "'a set" shows "local.rtrancl_while_test (ws, uu) = (ws \<noteq> [] \<and> p (hd ws))"
nat_info_filtration: shows "init_triv_filt M nat_filtration"
lin_tss_aux: shows "lin_ts_mmuaux aux = linearize tss"
induced_subgraph_def: fixes V :: "nat set"   and G :: "nat set \<times> nat set set" shows "induced_subgraph V G = (V, uedges G \<inter> all_edges V)"
mem_of_Nset: fixes x :: "nat"   and n :: "nat" assumes "x \<le> n" shows "x \<in> {i. i \<le> n}"
loop: fixes A :: "('a \<times> ('b, 'd) cconstraint \<times> ('b set \<times> 'a) pmf) set \<times> ('a \<Rightarrow> ('b, 'd) cconstraint)"   and st :: "'a \<times> ('b \<Rightarrow> 'd)" assumes "Probabilistic_Timed_Automaton A" shows "return_pmf st \<in> Probabilistic_Timed_Automaton.K A st"
lspasl_e_eq: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and sep_disj :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and h1 :: "'a"   and h2 :: "'a"   and h0 :: "'a" assumes "class.heap_sep_algebra plus zero sep_disj" shows "heap_sep_algebra.tern_rel plus sep_disj h1 h2 h0 = (heap_sep_algebra.tern_rel plus sep_disj h1 h2 h0 \<and> heap_sep_algebra.tern_rel plus sep_disj h2 h1 h0)"
simps_17: fixes x61 :: "msg"   and x62 :: "msg"   and x1 :: "agent" shows "\<lbrace>x61, x62\<rbrace> \<noteq> msg.Agent x1"
listt_update_in_list: fixes xs :: "'a list"   and n :: "nat"   and A :: "'a set"   and x :: "'a"   and i :: "nat" assumes "xs \<in> list n A"   and "x \<in> A" shows "xs[i := x] \<in> list n A"
insert_if: fixes A :: "'b set"   and g :: "'b \<Rightarrow> 'c"   and x :: "'b" assumes "finite A" shows "sum g (insert x A) = (if x \<in> A then sum g A else g x + sum g A)"
iTILL_finite: fixes n :: "nat" shows "finite [\<dots>n]"
dom_char: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b local.term set" assumes "free_monoidal_category C" shows "partial_magma.dom (free_monoidal_category.comp C) f = (if partial_magma.arr (free_monoidal_category.comp C) f then free_monoidal_category.mkarr C (monoidal_language.Dom C (free_monoidal_category.rep C f)) else partial_magma.null (free_monoidal_category.comp C))"
inter_plus_ideal_eq_carrier: fixes n :: "nat"   and I :: "nat \<Rightarrow> 'a set" assumes "\<And>i. i \<le> Suc n \<Longrightarrow> ideal (I i) R"   and "\<And>i j. \<lbrakk>i \<le> Suc n; j \<le> Suc n; i \<noteq> j\<rbrakk> \<Longrightarrow> I i <+> I j = carrier R" shows "\<Inter> (I ` {..n}) <+> I (Suc n) = carrier R"
elims: fixes x :: "'a list"   and xa :: "'a list"   and y :: "'a list" assumes "longest_common_prefix x xa = y" obtains xb :: "'a" and xs :: "'a list" and ya :: "'a" and ys :: "'a list" where "x = xb # xs" and "xa = ya # ys" and "y = (if xb = ya then xb # longest_common_prefix xs ys else [])"   | uv_ :: "'a list" where "x = []" and "xa = uv_" and "y = []"   | uu_ :: "'a list" where "x = uu_" and "xa = []" and "y = []"
comp_arr_ide: fixes a :: "'s"   and f :: "'s" assumes "ide a"   and "seq f a" shows "f \<cdot> a = f"
NoSporadic_stable: fixes \<Phi> :: "'b TESL_atomic list" shows "\<lbrakk>\<lbrakk> \<Phi> \<rbrakk>\<rbrakk>\<^sub>T\<^sub>E\<^sub>S\<^sub>L \<subseteq> \<lbrakk>\<lbrakk> NoSporadic \<Phi> \<rbrakk>\<rbrakk>\<^sub>T\<^sub>E\<^sub>S\<^sub>L"
run_pause_env: fixes pause :: "'c \<Rightarrow> ('d \<Rightarrow> 'a) \<Rightarrow> 'a"   and out :: "'c"   and c :: "'d \<Rightarrow> ('b, 'a) envT"   and r :: "'b" shows "run_env (pause_env pause out c) r = pause out (\<lambda>i. run_env (c i) r)"
zero_HOM: fixes M :: "('e, 'f, 'g) Module_scheme"   and R :: "('f, 'h) Ring_scheme"   and N :: "('i, 'f, 'j) Module_scheme" assumes "R module M"   and "R module N" shows "mzeromap M N = \<zero>\<^bsub>HOM\<^bsub>R\<^esub> M N\<^esub>"
closed_Union: fixes S :: "'a set set" assumes "finite S"   and "\<forall>T\<in>S. closed T" shows "closed (\<Union> S)"
intro: fixes known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and get_owner_document :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) document_ptr) prog"   and get_root_node :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr) prog"   and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog" assumes "known_ptr = DocumentClass.known_ptr"   and "get_owner_document = l_get_owner_document\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_owner_document get_root_node get_disconnected_nodes" shows "l_get_owner_document\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms known_ptr get_disconnected_nodes get_root_node get_owner_document"
intro: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and D :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and F :: "'b \<Rightarrow> 'a"   and G :: "'a \<Rightarrow> 'b"   and \<eta> :: "'b \<Rightarrow> 'b"   and \<epsilon> :: "'a \<Rightarrow> 'a" assumes "category C"   and "category D"   and "functor D C F"   and "functor C D G"   and "natural_isomorphism D D (identity_functor.map D) (G \<circ> F) \<eta>"   and "natural_isomorphism C C (F \<circ> G) (identity_functor.map C) \<epsilon>" shows "equivalence_of_categories C D F G \<eta> \<epsilon>"
name_fresh_left: fixes a :: "name"   and pi :: "(name \<times> name) list"   and x :: "'b" shows "a \<sharp> pi \<bullet> x = rev pi \<bullet> a \<sharp> x"
HORun_def: fixes A :: "('a, 'b, 'c, 'd) CHOAlgorithm_scheme"   and rho :: "nat \<Rightarrow> 'a \<Rightarrow> 'b"   and HOs :: "nat \<Rightarrow> 'a \<Rightarrow> 'a set" shows "HORun A rho HOs \<equiv> SHORun A rho HOs HOs"
Cons_1: fixes k :: "nat"   and rcs :: "recf list" assumes "length rgs = k"   and "length rcs = k"   and "0 < k"   and "\<forall>j<k. rec_exec (rcs ! j) xs = 0" shows "Embranch (zip (map rec_exec rgs) (map (\<lambda>r args. 0 < rec_exec r args) rcs)) xs = 0"
cases: fixes x :: "('c \<times> nat) tree" obtains l :: "('c \<times> nat) tree" and a :: "'c" and lv :: "nat" and r :: "('c \<times> nat) tree" where "x = \<langle>l, (a, lv), r\<rangle>"   | "x = \<langle>\<rangle>"
commute: fixes f :: "'a::type \<Rightarrow> 'a::type \<Rightarrow> 'a::type"   and a :: "'a::type"   and b :: "'a::type" assumes "abel_semigroup (f::'a::type \<Rightarrow> 'a::type \<Rightarrow> 'a::type)" shows "(f::'a::type \<Rightarrow> 'a::type \<Rightarrow> 'a::type) (a::'a::type) (b::'a::type) = f b a"
enumerate_Suc': fixes S :: "'b set"   and n :: "nat" shows "wellorder_class.enumerate S (Suc n) = wellorder_class.enumerate (S - {wellorder_class.enumerate S 0}) n"
nempty_dom: fixes e :: "'a \<Rightarrow> 'b option" assumes "e \<noteq> Map.empty" obtains m :: "'a" where "m \<in> dom e"
circline_reciprocal_def: shows "circline_reciprocal = conjugate_circline \<circ> circline_inversion"
a_type_wf_def: fixes h :: "(_) heap" shows "ObjectClass.type_wf h = True"
case_prodE2: fixes Q :: "'b \<Rightarrow> bool"   and P :: "'c \<Rightarrow> 'd \<Rightarrow> 'b"   and z :: "'c \<times> 'd" assumes "Q (case z of (a, b) \<Rightarrow> P a b)" obtains x :: "'c" and y :: "'d" where "z = (x, y)" and "Q (P x y)"
exhaust: fixes y :: "'a ms" obtains x1 :: "('a \<times> real) msllist" and x2 :: "real \<Rightarrow> real" where "y = MS x1 x2"
finite_list': fixes A :: "'a set" assumes "finite A" obtains xs :: "'a list" where "A = set xs" and "distinct xs" and "length xs = card A"
Collect_cong: fixes P :: "'c \<Rightarrow> bool"   and Q :: "'c \<Rightarrow> bool" assumes "\<And>x. P x = Q x" shows "{x. P x} = {x. Q x}"
times_matrix_assoc: fixes x :: "(nat \<times> nat) \<times> nat \<times> nat"   and y :: "(nat \<times> nat) \<times> nat \<times> nat"   and z :: "(nat \<times> nat) \<times> nat \<times> nat" shows "x \<otimes> y \<otimes> z = x \<otimes> (y \<otimes> z)"
assms_3: shows "dip \<in> kD (rt (\<sigma> nhip))"
distinct_25: fixes x31 :: "('a, 'b, 'c) Stmt"   and x32 :: "'a ModeUpd"   and x51 :: "'c"   and x52 :: "('a, 'b, 'c) Stmt"   and x53 :: "('a, 'b, 'c) Stmt" shows "x31@[x32] \<noteq> Stmt.If x51 x52 x53"
hyps_1: fixes x :: "Vars.var"   and \<rho> :: "Vars.var \<Rightarrow> 'Value" assumes "x \<in> domA as" shows "\<lbrakk> the (map_of as x) \<rbrakk>\<^bsub>\<rho>\<^esub> = \<lbrakk> the (map_of as x) \<rbrakk>\<^bsub>\<rho> f|` fv (the (map_of as x))\<^esub>"
x1_def: shows "x1 = Max (insert (x - 1) ((\<lambda>X. x + (Inf (Re ` X) - x) / 2) ` T))"
bidirected_digraphI_pair_sd: fixes G :: "'b pair_pre_digraph"   and H :: "'b pair_pre_digraph" assumes "pair_sd G H" shows "pair_bidirected_digraph H"
assms_2: fixes x :: "V" assumes "x \<in>\<^sub>\<circ> B" shows "P x = Q x"
a3: shows " \<theta><(x, close \<Gamma> T1) # \<Gamma>> \<turnstile> t2 : \<theta><T2> "
h1: shows "Sup {norm (f x) / norm x |x. x \<noteq> (0::'a)} = Sup {norm (f x) / norm x |x. True}"
arrE: fixes f :: "'a \<times> 'a" assumes "CC.arr f" obtains "arr (fst f) \<and> arr (snd f)"
hyps: fixes x2aa :: "('sf + 'a, 'sz) trm" assumes "x2aa \<in> range x2a"   and "Vagree \<nu> \<omega> (\<Union>i\<in>{i. Inr i \<in> SIGT x2aa}. FVT (\<sigma> i))" shows "sterm_sem (adjointFO I \<sigma> \<nu>) x2aa = sterm_sem (adjointFO I \<sigma> \<omega>) x2aa"
fmrel_upd: fixes P :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and m :: "'g \<rightharpoonup>\<^sub>f 'e"   and n :: "'g \<rightharpoonup>\<^sub>f 'f"   and x :: "'e"   and y :: "'f"   and k :: "'g" assumes "fmrel P m n"   and "P x y" shows "fmrel P (m(k \<mapsto>\<^sub>f x)) (n(k \<mapsto>\<^sub>f y))"
has_integral_substitution_strong: fixes s :: "real set"   and a :: "real"   and b :: "real"   and g :: "real \<Rightarrow> real"   and c :: "real"   and d :: "real"   and f :: "real \<Rightarrow> 'a"   and g' :: "real \<Rightarrow> real" assumes "finite s"   and "a \<le> b"   and "g a \<le> g b"   and "g ` {a..b} \<subseteq> {c..d}"   and "continuous_on {c..d} f"   and "continuous_on {a..b} g"   and "\<And>x. x \<in> {a..b} - s \<Longrightarrow> (g has_real_derivative g' x) (at x within {a..b})" shows "((\<lambda>x. g' x *\<^sub>R f (g x)) has_integral integral {g a..g b} f) {a..b}"
order_refl: fixes x :: "'a" shows "x \<preceq> x"
get_document_ptr_simp6: fixes document_ptr :: "(_) document_ptr"   and character_data_ptr :: "(_) character_data_ptr"   and f :: "'i option RCharacterData_scheme"   and h :: "(_) heap" shows "get document_ptr (put character_data_ptr f h) = get document_ptr h"
unitary_gen_def: fixes M :: "complex \<times> complex \<times> complex \<times> complex" shows "unitary_gen M = (\<exists>k. k \<noteq> 0 \<and> mat_adj M *\<^sub>m\<^sub>m M = k *\<^sub>s\<^sub>m eye)"
pr_key_set_unused: fixes evs :: "event list"   and S :: "agent \<times> nat \<times> nat \<Rightarrow> session"   and A :: "msg set"   and U :: "msg set"   and H :: "msg set" assumes "(evs, S, A, U) \<in> protocol"   and "H \<subseteq> range Key \<union> range Pri_AgrK - U" shows "analz (H \<union> A \<union> spies evs) = H \<union> analz (A \<union> spies evs)"
wf_tuple_empty: fixes n :: "nat"   and v :: "'a option list" shows "wf_tuple n {} v = (v = replicate n None)"
nth_shiftl: fixes w :: "'a word"   and m :: "nat"   and n :: "nat" shows "(w << m) !! n = (n < LENGTH('a) \<and> m \<le> n \<and> w !! (n - m))"
0: shows "a \<in> maximalStrictAllocations N (set \<Omega>) b \<and> Outside' {seller} (the_elem (argmax (sum (summedBidVector (pseudoAllocation (randomEl (takeAll (\<lambda>x. winningAllocationRel (N \<union> {seller}) (set \<Omega>) ((\<in>) x) b) (allAllocationsAlg (N \<union> {seller}) \<Omega>)) r) <| ((N \<union> {seller}) \<times> finestpart (set \<Omega>))) (N \<union> {seller}) (set \<Omega>))) (maximalStrictAllocations N (set \<Omega>) b))) = Outside' {seller} a \<and> a \<in> maximalStrictAllocations N (set \<Omega>) b"
rep_eq: fixes x :: "'b \<Rightarrow>\<^sub>L 'd"   and xa :: "'c \<Rightarrow>\<^sub>L 'd" shows "blinfun_apply (comp12 x xa) = (\<lambda>(a, b). blinfun_apply x a + blinfun_apply xa b)"
has_field_derivative_Gen_Shleg: fixes n :: "nat"   and k :: "nat"   and x :: "real" shows "(Gen_Shleg n k has_real_derivative Gen_Shleg n (Suc k) x) (at x)"
integrable_on_1_iff: fixes f :: "'c \<Rightarrow> (real, 1) vec"   and S :: "'c set" shows "(f integrable_on S) = ((\<lambda>x. f x $ 1) integrable_on S)"
G_is_essentially_surjective: shows "essentially_surjective_functor (\<cdot>\<^sub>D) (\<cdot>\<^sub>C) F"
ur_closure': fixes A :: "'b set"   and H :: "(('c, 'b) fm list \<times> 'b) set"   and i :: "'b"   and p :: "('c, 'b) fm" assumes "Hintikka A H"   and "p at i in' H"   and "\<forall>a. p = Nom a \<or> p = (\<^bold>\<diamond> Nom a) \<longrightarrow> a \<in> A" shows "\<exists>a\<in>assign A H i. p at a in' H"
distinct_278: fixes x14 :: "Syntax.term"   and x13 :: "Syntax.term" shows "Unroll x14 \<noteq> Roll x13"
to_nat_from_nat_id: fixes times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and uminus :: "'c \<Rightarrow> 'c"   and one :: "'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and Rep :: "'c \<Rightarrow> int"   and Abs :: "int \<Rightarrow> 'c"   and x :: "nat" assumes "class.mod_type times minus plus zero uminus one less_eq less Rep Abs"   and "x < CARD('c)" shows "mod_type.to_nat Rep (mod_type.from_nat Abs x) = x"
simps: fixes a1 :: "nat \<Rightarrow> Syntax.type"   and a2 :: "nat \<Rightarrow> Syntax.type"   and a3 :: "ctxt"   and a4 :: "Syntax.type"   and a5 :: "Syntax.type" shows "(a1 , a2 \<turnstile>\<^sub>c\<^sub>t\<^sub>x\<^sub>t a3 : a4 \<Leftarrow> a5) = ((\<exists>\<Gamma> \<Delta> T. a1 = \<Gamma> \<and> a2 = \<Delta> \<and> a3 = \<diamond> \<and> a4 = T \<and> a5 = T) \<or> (\<exists>\<Gamma> \<Delta> E T1 T2 U t. a1 = \<Gamma> \<and> a2 = \<Delta> \<and> a3 = E \<^sup>\<bullet> t \<and> a4 = T2 \<and> a5 = U \<and> \<Gamma> , \<Delta> \<turnstile>\<^sub>c\<^sub>t\<^sub>x\<^sub>t E : T1 \<rightarrow> T2 \<Leftarrow> U \<and> \<Gamma> , \<Delta> \<turnstile>\<^sub>T t : T1))"
vreal_fsingleton_in_fproduct_vreal: fixes a :: "real" shows "[a\<^sub>\<real>]\<^sub>\<circ> \<in>\<^sub>\<circ> \<real>\<^sub>\<circ> ^\<^sub>\<times> 1\<^sub>\<nat>"
hyps_2: shows "n dd\<^bsup>path \<sigma>,v\<^esup>\<rightarrow> l"
wadjust_loop_erase_nonempty_6: fixes m :: "nat"   and rs :: "nat"   and c :: "cell list"   and list :: "cell list" shows "wadjust_on_left_moving_O m rs (c, Bk # list) = False"
sigma_primepower: fixes p :: "nat"   and e :: "nat" assumes "prime p" shows "(p - 1) * sigma (p ^ e) = p ^ (e + 1) - 1"
pf: shows "pivot_fun (row_echelon A) f (dim_col (row_echelon A))"
vertexI2: fixes \<Gamma> :: "('a, 'b) graph_scheme"   and x :: "'a"   and y :: "'a" assumes "edge \<Gamma> x y" shows "vertex \<Gamma> y"
l_neg: fixes x :: "'a" assumes "x \<in> carrier G" shows "\<ominus> x \<oplus> x = \<zero>"
ex_inf: fixes x :: "'c"   and y :: "'c" assumes "Lattice.class.lattice TYPE('c)" shows "\<exists>inf. is_inf x y inf"
smC_def: fixes a :: "'b"   and b :: "'b"   and c :: "'b"   and d :: "'b" shows "norm_metric.smC a b c d \<equiv> norm_dist a b = norm_dist c d"
inner_lv_rel_eq: fixes xs :: "real list"   and xa :: "real list"   and x'a :: "'b" assumes "length xs = DIM('b)"   and "(xa, x'a) \<in> lv_rel" shows "inner_lv_rel xs xa = eucl_of_list xs \<bullet> x'a"
Rep_Abs_Seq: fixes P :: "'a process"   and Q :: "'a process" shows "Rep_process (Abs_process ({(t, X). (t, X \<union> {tick}) \<in> F P \<and> tickFree t} \<union> {(t, X). \<exists>t1 t2. t = t1 @ t2 \<and> t1 @ [tick] \<in> T P \<and> (t2, X) \<in> F Q} \<union> {(t, X). t \<in> div_seq P Q}, div_seq P Q)) = ({(t, X). (t, X \<union> {tick}) \<in> F P \<and> tickFree t} \<union> {(t, X). \<exists>t1 t2. t = t1 @ t2 \<and> t1 @ [tick] \<in> T P \<and> (t2, X) \<in> F Q} \<union> {(t, X). t \<in> div_seq P Q}, div_seq P Q)"
simps_11: fixes v :: "'a rexp" shows "local.le_rexp One (Star v) = True"
sdim_UNIV: shows "UNIV = {Length, Mass, Time, Current, Temperature, Amount, Intensity}"
finite_set_zmset: fixes M :: "'a zmultiset" shows "finite (set_zmset M)"
ipresIGFreshAbs_termMOD: fixes hA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> 'h"   and MOD :: "('i, 'j, 'varSort, 'k, 'l, 'var, 'm, 'h, 'n) model_scheme" shows "ipresIGFreshAbs hA termMOD MOD = ipresFreshAbs hA MOD"
diag_mat_const_eq: fixes c :: "'d" shows "diag_mat (\<lambda>i. c) = mat c"