hcju/isabelleps huggingface dataset:
in_listE: fixes xs :: "'a list"   and n :: "nat"   and A :: "'a set" assumes "xs \<in> list n A" obtains "length xs = n" and "set xs \<subseteq> A"
S_finite: shows "\<forall>i\<in>{1..m}. finite (S i)"
Lindelof_openin: fixes \<F> :: "'a set set"   and U :: "'a set" assumes "\<And>S. S \<in> \<F> \<Longrightarrow> openin (top_of_set U) S" obtains \<F>' :: "'a set set" where "\<F>' \<subseteq> \<F>" and "countable \<F>'" and "\<Union> \<F>' = \<Union> \<F>"
simps_2: fixes v :: "nat" shows "oddpart (Suc v) = (if odd (Suc v) then Suc v else oddpart (Suc v div 2))"
limsup_shift_k: fixes u :: "nat \<Rightarrow> 'a"   and k :: "nat" shows "limsup (\<lambda>n. u (n + k)) = limsup u"
unat_ucast_mask: fixes a :: "'c word" shows "unat (UCAST('c \<rightarrow> 'b) a) = unat (a AND mask LENGTH('b))"
is_starext_def: fixes F :: "'c star \<Rightarrow> 'c star"   and f :: "'c \<Rightarrow> 'c" shows "is_starext F f = (\<forall>x y. \<exists>X\<in>Rep_star x. \<exists>Y\<in>Rep_star y. (y = F x) = (\<forall>\<^sub>F n in \<U>. Y n = f (X n)))"
integrable_bounded_linear: fixes T :: "'d \<Rightarrow> 'e"   and M :: "'f measure"   and f :: "'f \<Rightarrow> 'd" assumes "bounded_linear T"   and "integrable M f" shows "integrable M (\<lambda>x. T (f x))"
sub_eq: shows "nths (verticesFrom f v) (set es) = removeNones (natToVertexList v f es)"
Un_UNIV_right: fixes A :: "'a set" shows "A \<union> UNIV = UNIV"
Sup_enat_def: fixes A :: "enat set" shows "Sup A = (if A = {} then 0 else if finite A then Max A else \<infinity>)"
a_inv: fixes a :: "ant"   and b :: "ant" assumes "a + b = 0" shows "a = - b"
awalk_decomp: fixes u :: "'a"   and p :: "'b list"   and v :: "'a"   and w :: "'a" assumes "H.awalk u p v"   and "w \<in> set (H.awalk_verts u p)" shows "\<exists>q r. p = q @ r \<and> H.awalk u q w \<and> H.awalk w r v"
transfer_not: fixes p :: "bool"   and P :: "nat \<Rightarrow> bool" assumes "p \<equiv> eventually P \<U>" shows "\<not> p \<equiv> \<forall>\<^sub>F n in \<U>. \<not> P n"
paid_part: shows "(\<Sum>i<length qs. \<Sum>(x, y)\<in>{(x, y). x \<in> set init \<and> y \<in> set init \<and> x < y}. ALG_P (snd (Strat ! i)) x y (steps' init qs Strat i)) = (\<Sum>(x, y)\<in>{(x, y). x \<in> set init \<and> y \<in> set init \<and> x < y}. ALG_Pxy Strat qs init x y)"
strip_while: fixes f :: "'a acom \<Rightarrow> 'a acom"   and P :: "'a acom \<Rightarrow> bool"   and c :: "'a acom"   and c' :: "'a acom" assumes "\<forall>c. strip (f c) = strip c"   and "while_option P f c = Some c'" shows "strip c' = strip c"
mem_carrier: fixes x :: "cltn2" assumes "x \<in> Collect is_K2_isometry" shows "x \<in> carrier \<lparr>carrier = UNIV, monoid.mult = cltn2_compose, one = cltn2_id\<rparr>"
numVars_def: fixes problem :: "(char list \<times> nat option \<times> char list list) list \<times> nat list \<times> (nat \<times> nat) list \<times> (char list \<times> (nat \<times> nat) list \<times> ((nat \<times> nat) list \<times> nat \<times> nat option \<times> nat) list \<times> nat) list" shows "ast_problem.numVars problem \<equiv> length (ast_problem.astDom problem)"
tail_invar_cons_down: fixes t :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree"   and bq :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree list" assumes "SkewBinomialHeapStruc.tail_invar (t # bq)" shows "SkewBinomialHeapStruc.tail_invar bq"
set_map: fixes f :: "'c \<Rightarrow> 'b"   and v :: "'c up\<^sub>i" shows "set_up\<^sub>i (map_up\<^sub>i f v) = f ` set_up\<^sub>i v"
Pan_image_eq: fixes x :: "nat"   and A :: "nat set" shows "(Pan x \<in> Pan ` A) = (x \<in> A)"
SeqSubstFormP_graph_aux_def: shows "SeqSubstFormP_graph_aux \<equiv> lfp (\<lambda>p x1 x2. \<exists>SeqSubstFormP_aux l s k v u sl sl' m n sm sm' sn sn' x x'. x1 = (v, u, x, x', s, k) \<and> x2 = LstSeqP s k (HPair x x') AND All2 l (SUCC k) (SyntaxN.Ex sl (SyntaxN.Ex sl' (HPair (Var l) (HPair (Var sl) (Var sl')) IN s AND (SubstAtomicP v u (Var sl) (Var sl') OR SyntaxN.Ex m (SyntaxN.Ex n (SyntaxN.Ex sm (SyntaxN.Ex sm' (SyntaxN.Ex sn (SyntaxN.Ex sn' (Var m IN Var l AND Var n IN Var l AND HPair (Var m) (HPair (Var sm) (Var sm')) IN s AND HPair (Var n) (HPair (Var sn) (Var sn')) IN s AND (Var sl EQ Q_Disj (Var sm) (Var sn) AND Var sl' EQ Q_Disj (Var sm') (Var sn') OR Var sl EQ Q_Neg (Var sm) AND Var sl' EQ Q_Neg (Var sm') OR Var sl EQ Q_Ex (Var sm) AND Var sl' EQ Q_Ex (Var sm')))))))))))) \<and> atom l \<sharp> (s, k, v, u, sl, sl', m, n, sm, sm', sn, sn') \<and> atom sl \<sharp> (s, v, u, sl', m, n, sm, sm', sn, sn') \<and> atom sl' \<sharp> (s, v, u, m, n, sm, sm', sn, sn') \<and> atom m \<sharp> (s, n, sm, sm', sn, sn') \<and> atom n \<sharp> (s, sm, sm', sn, sn') \<and> atom sm \<sharp> (s, sm', sn, sn') \<and> atom sm' \<sharp> (s, sn, sn') \<and> atom sn \<sharp> (s, sn') \<and> atom sn' \<sharp> s)"
simps_4: fixes var :: "nat"   and p :: "real mpoly" shows "substNegInfinity var (Neq p) = Neg (allZero p var)"
exhaust: fixes y :: "'b Predicate.seq" obtains "y = seq.Empty"   | x21 :: "'b" and x22 :: "'b Predicate.pred" where "y = seq.Insert x21 x22"   | x31 :: "'b Predicate.pred" and x32 :: "'b Predicate.seq" where "y = seq.Join x31 x32"
H'_2: shows "B \<in> g \<turnstile> \<C>"
locally_negligible_alt: fixes S :: "'b set" shows "negligible S = (\<forall>x\<in>S. \<exists>U. openin (top_of_set S) U \<and> x \<in> U \<and> negligible U)"
\<beta>: shows "\<beta> \<in> elts (ordertype M VWF)"
DisP': fixes a :: "'c"   and ps :: "('b, 'c) fm list"   and branch :: "(('b, 'c) fm list \<times> 'c) list"   and p :: "('b, 'c) fm"   and q :: "('b, 'c) fm"   and A :: "'c set"   and n :: "nat" assumes "(p \<^bold>\<or> q) at a in' set ((ps, a) # branch)"   and "A, n \<turnstile> (p # ps, a) # branch"   and "A, n \<turnstile> (q # ps, a) # branch" shows "A, n \<turnstile> (ps, a) # branch"
tmadd_blt: fixes n :: "nat"   and t :: "tm"   and s :: "tm" assumes "tmboundslt n t"   and "tmboundslt n s" shows "tmboundslt n (tmadd t s)"
set_in_finite_cone: fixes Vs :: "'b vec set"   and n :: "nat" assumes "Vs \<subseteq> carrier_vec n"   and "finite Vs" shows "Vs \<subseteq> gram_schmidt.finite_cone n Vs"
inject_1: fixes x2 :: "'a bro"   and y2 :: "'a bro" shows "(N1 x2 = N1 y2) = (x2 = y2)"
wf_cdecl_def: fixes wf_md :: "'b prog \<Rightarrow> char list \<Rightarrow> 'b mdecl \<Rightarrow> bool"   and P :: "'b prog" shows "wf_cdecl wf_md P \<equiv> \<lambda>(C, D, fs, ms). (\<forall>f\<in>set fs. wf_fdecl P f) \<and> distinct_fst fs \<and> (\<forall>m\<in>set ms. wf_mdecl wf_md P C m) \<and> distinct_fst ms \<and> (C \<noteq> Object \<longrightarrow> is_class P D \<and> (D, C) \<notin> (subcls1 P)\<^sup>* \<and> (\<forall>(M, b, Ts, T, m)\<in>set ms. \<forall>D' b' Ts' T' m'. P \<turnstile> D sees M, b' :  Ts'\<rightarrow>T' = m' in D' \<longrightarrow> b = b' \<and> P \<turnstile> Ts' [\<le>] Ts \<and> P \<turnstile> T \<le> T')) \<and> wf_clinit ms"
sep_disj_commute: fixes x :: "'a"   and y :: "'a" shows "x ## y = y ## x"
hom_injI: assumes "\<forall>x\<in>carrier G. h x = \<one>\<^bsub>H\<^esub> \<longrightarrow> x = \<one>" shows "inj_on h (carrier G)"
a_nonneg: fixes a :: "nat \<Rightarrow> real"   and n :: "nat" assumes "shapiro_tauberian a"   and "0 < n" shows "0 \<le> a n"
a_def: shows "a \<equiv> SOME a. a \<in> A"
bunch_of_lemmata_about_matches_2: fixes \<gamma> :: "('b \<Rightarrow> 'c \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'c \<Rightarrow> bool)"   and a :: "action"   and p :: "'c" shows "matches \<gamma> MatchAny a p"
eq_self_neq_FF: fixes eq :: "'d \<rightarrow> 'd \<rightarrow> tr"   and x :: "'d" assumes "class.Eq_equiv eq" shows "eq\<cdot>x\<cdot>x \<noteq> FF"
l6: shows "- (ordinate C - ordinate D) * abscissa X + (abscissa C - abscissa D) * ordinate X = - abscissa C * (ordinate C - ordinate D) + ordinate C * (abscissa C - abscissa D)"
trm_rep_is_lower: fixes t :: "'a trm"   and S :: "'a eclause set" shows "t \<noteq> trm_rep t S \<longrightarrow> (trm_rep t S, t) \<in> trm_ord"
GB_alt_1_dgrad_p_set: fixes d :: "'a \<Rightarrow> nat"   and G :: "('t \<Rightarrow>\<^sub>0 'c) set"   and m :: "nat" assumes "dickson_grading d"   and "G \<subseteq> dgrad_p_set d m" shows "is_Groebner_basis G = (\<forall>f\<in>pmdl G. f \<in> dgrad_p_set d m \<longrightarrow> (red G)\<^sup>*\<^sup>* f 0)"
ENF_offending_case1: fixes sinvar :: "'b graph \<Rightarrow> ('b \<Rightarrow> 'c) \<Rightarrow> bool"   and P :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and G :: "'b graph"   and nP :: "'b \<Rightarrow> 'c" assumes "SecurityInvariant_withOffendingFlows.sinvar_all_edges_normal_form sinvar P"   and "\<not> sinvar G nP" shows "{{(e1, e2). (e1, e2) \<in> edges G \<and> \<not> P (nP e1) (nP e2)}} = SecurityInvariant_withOffendingFlows.set_offending_flows sinvar G nP"
econc_fun_def: fixes E :: "('g \<times> 'e) set"   and R :: "('h \<times> 'f) set" shows "\<Down>\<^sub>E E R \<equiv> \<Down> (\<langle>E, R\<rangle>sum_rel)"
fds_zeta_def: shows "fds_zeta = fds (\<lambda>_. 1::'a)"
SUP_mono: fixes Inf :: "'h set \<Rightarrow> 'h"   and Sup :: "'h set \<Rightarrow> 'h"   and inf :: "'h \<Rightarrow> 'h \<Rightarrow> 'h"   and less_eq :: "'h \<Rightarrow> 'h \<Rightarrow> bool"   and less :: "'h \<Rightarrow> 'h \<Rightarrow> bool"   and sup :: "'h \<Rightarrow> 'h \<Rightarrow> 'h"   and bot :: "'h"   and top :: "'h"   and A :: "'i set"   and B :: "'j set"   and f :: "'i \<Rightarrow> 'h"   and g :: "'j \<Rightarrow> 'h" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "\<And>n. n \<in> A \<Longrightarrow> \<exists>m\<in>B. less_eq (f n) (g m)" shows "less_eq (Sup (f ` A)) (Sup (g ` B))"
ru_4: shows "root_cond (p, l, r) (real_of_1 (p, l, r))"
simps_10: fixes f :: "'f" shows "mark_guards f Throw = Throw"
sup_idemp: fixes a :: "'a" shows "a \<squnion>1 a = a"
intro: fixes D :: "('c, 'd) Order_scheme" assumes "Torder D"   and "Worder_axioms D" shows "Worder D"
init_segment_insert: fixes S :: "nat set"   and T :: "nat set"   and n :: "nat" assumes "init_segment S T"   and "T \<lless> {n}" shows "init_segment S (insert n T)"
h_nn: shows "AE x in M. 0 \<le> h x"
map_fun_comp: fixes f :: "'f \<Rightarrow> 'h"   and g :: "'i \<Rightarrow> 'g"   and f' :: "'h \<Rightarrow> 'd"   and g' :: "'e \<Rightarrow> 'i" shows "f ---> g \<circ> (f' ---> g') = (f' \<circ> f) ---> g \<circ> g'"
get_return_edges_call_of_return_nodes: fixes cs :: "'edge list"   and m :: "'node"   and rs :: "'edge list" assumes "valid_call_list cs m"   and "valid_return_list rs m"   and "\<forall>i<length rs. rs ! i \<in> get_return_edges (cs ! i)"   and "length rs = length cs" shows "\<forall>i<length cs. call_of_return_node (targetnodes rs ! i) (sourcenode (cs ! i))"
solution_def: fixes D :: "nat" assumes "pell D" shows "pell.solution D = (\<lambda>(a, b). a\<^sup>2 = (1::'a) + of_nat D * b\<^sup>2)"
word_oa_dist2: fixes x :: "'a word"   and y :: "'a word"   and z :: "'a word" shows "x OR y AND z = (x OR y) AND (x OR z)"
parent_child_rel_parent: fixes h :: "(_) heap"   and child_node :: "(_) node_ptr"   and parent :: "(_) object_ptr" assumes "heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M h"   and "h \<turnstile> Shadow_DOM.get_parent child_node \<rightarrow>\<^sub>r Some parent" shows "(parent, cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r child_node) \<in> Shadow_DOM.parent_child_rel h"
div_const_unit_poly: fixes c :: "'b"   and p :: "'b poly" assumes "is_unit c" shows "p div [:c:] = smult ((1::'b) div c) p"
take_Cons': fixes n :: "nat"   and x :: "'c"   and xs :: "'c list" shows "take n (x # xs) = (if n = 0 then [] else x # take (n - 1) xs)"
append_expunge_commute: fixes i1 :: "'c"   and mo :: "'c"   and e1 :: "'d"   and e2 :: "'d"   and i2 :: "'c" assumes "i1 \<noteq> mo" shows "\<langle>Append i1 e1\<rangle> \<rhd> \<langle>Expunge e2 mo i2\<rangle> = \<langle>Expunge e2 mo i2\<rangle> \<rhd> \<langle>Append i1 e1\<rangle>"
add_instr_cache_privilege: fixes s' :: "'b sparc_state"   and s :: "'b sparc_state"   and addr :: "word32"   and y :: "word32"   and m :: "word4" assumes "s' = add_instr_cache s addr y m"   and "get_S (cpu_reg_val PSR s) = 0" shows "get_S (cpu_reg_val PSR s') = 0"
Entry_cdep_SDG_path: fixes as :: "'edge list"   and n' :: "'node" assumes "(_Entry_) -as\<rightarrow>\<^sub>\<iota>* n'"   and "inner_node n'"   and "\<not> method_exit n'" obtains ns :: "'node SDG_node list" where "CFG_node (_Entry_) cd-ns\<rightarrow>\<^sub>d* CFG_node n'" and "ns \<noteq> []" and "\<forall>n''\<in>set ns. parent_node n'' \<in> set (sourcenodes as)"
alpha_Tree_eqvt: fixes t1 :: "('b, 'c, 'd) Tree"   and t2 :: "('b, 'c, 'd) Tree"   and p :: "perm" assumes "t1 =\<^sub>\<alpha> t2" shows "p \<bullet> t1 =\<^sub>\<alpha> p \<bullet> t2"
exhaust_sel: fixes vertex :: "'b vertex" obtains "vertex = SOURCE"   | "vertex = vertex.SINK"   | "vertex = Inner (vertex.inner vertex)"
next_symbol_def: fixes x :: "item" shows "next_symbol x = (if is_complete x then None else Some (item_rhs x ! item_dot x))"
Unf_idem: fixes \<phi> :: "'b ltln" shows "Unf (Unf \<phi>) \<sim>\<^sub>P Unf \<phi>"
inject_2: fixes x2 :: "'b"   and y2 :: "'b" shows "(TargetTerm x2 = TargetTerm y2) = (x2 = y2)"
Utrans_P_def: fixes vars :: "nat set"   and U :: "complex mat" shows "Utrans_P vars U = Utrans (mat_extension dims vars U)"
set_card_diff_ge_zero: fixes A :: "'a set"   and B :: "'a set" assumes "finite A"   and "finite B"   and "A \<noteq> B"   and "card A = card B" shows "0 < card (A - B)"
sim_submodel_aux: fixes s :: "'s set \<times> 's"   and n :: "nat" assumes "s \<in> worlds (Clock.MCSn n)" shows "gen_model clock_simMC s = gen_model (Clock.MCSn n) s"
inv_one: shows "inv\<^bsub>DirProds Hs J\<^esub> \<one>\<^bsub>DirProds Hs J\<^esub> = \<one>\<^bsub>DirProds Hs J\<^esub>"
dnf_ltln_of_dnf_min_dnf: fixes \<phi> :: "'b ltln" shows "dnf (ltln_of_dnf (min_dnf \<phi>)) = min_dnf \<phi>"
Abs_perm_inject: fixes x :: "atom \<Rightarrow> atom"   and y :: "atom \<Rightarrow> atom" assumes "x \<in> perm"   and "y \<in> perm" shows "(Abs_perm x = Abs_perm y) = (x = y)"
to_polynomial_def: fixes R :: "('a, 'b) ring_scheme" shows "to_polynomial R = (\<lambda>a. monom (UP R) a 0)"
subset: shows "{t..t + h} \<subseteq> {t..u}"
assms: shows "unifier\<^sub>l\<^sub>s \<sigma> (L\<^sub>1 \<union> L\<^sub>2)"
bit_eq_iff: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bit :: "'b \<Rightarrow> nat \<Rightarrow> bool"   and a :: "'b"   and b :: "'b" assumes "class.semiring_bits plus minus zero times one divide modulo bit" shows "(a = b) = (\<forall>n. bit a n = bit b n)"
divide_1: fixes R :: "('b, 'c) ring_scheme"   and x :: "'b" assumes "field R"   and "x \<in> carrier R" shows "x \<oslash>\<^bsub>R\<^esub> \<one>\<^bsub>R\<^esub> = x"
assms: fixes x :: "V"   and y :: "V" assumes "x \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> r"   and "y \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> r"   and "x \<noteq> y"   and "r\<lparr>x\<rparr> \<noteq> r\<lparr>y\<rparr>" shows "P (r\<lparr>x\<rparr>) (r\<lparr>y\<rparr>)"
finsum_vec_empty: fixes n :: "nat"   and f :: "'b \<Rightarrow> 'a vec" shows "finsum_vec TYPE('a) n f {} = 0\<^sub>v n"
conn_head: shows "head G a \<notin> H.isolated_verts"
edka5_run_def: fixes cf :: "nat \<times> nat \<Rightarrow> int"   and am :: "nat \<Rightarrow> nat list" shows "edka5_run cf am \<equiv> while\<^sub>T (\<lambda>(cf, brk). \<not> brk) (\<lambda>(cf, uu_). assert (RGraph c s t cf) \<bind> (\<lambda>_. bfs2_op am cf \<bind> (\<lambda>p. case p of None \<Rightarrow> return (cf, True) | Some p \<Rightarrow> assert (p \<noteq> []) \<bind> (\<lambda>_. assert (Graph.isShortestPath cf s p t) \<bind> (\<lambda>_. resCap_cf_impl cf p \<bind> (\<lambda>bn. augment_cf_impl cf p bn \<bind> (\<lambda>cf. assert (RGraph c s t cf) \<bind> (\<lambda>_. return (cf, False))))))))) (cf, False) \<bind> (\<lambda>(cf, uu_). compute_rflow cf \<bind> return)"
hyp_port_for_spec': fixes f :: "'form"   and t :: "('form, 'rule, 'subst, 'var) itnode rose_tree"   and "is" :: "nat list" assumes "f \<in> hyps_along t is" shows "case hyp_port_for' t is f of (is', i, h) \<Rightarrow> prefix (is' @ [i]) is \<and> i < length (inPorts' (iNodeOf (tree_at t is'))) \<and> hyps (iNodeOf (tree_at t is')) h = Some (inPorts' (iNodeOf (tree_at t is')) ! i) \<and> f = subst (iSubst (tree_at t is')) (freshen (iAnnot (tree_at t is')) (labelsOut (iNodeOf (tree_at t is')) h))"
expectation_gpv_cong: fixes fail :: "ennreal"   and fail' :: "ennreal"   and \<I> :: "('a, 'b) \<I>"   and \<I>' :: "('a, 'b) \<I>"   and gpv :: "('c, 'a, 'b) gpv"   and gpv' :: "('c, 'a, 'b) gpv"   and f :: "'c \<Rightarrow> ennreal"   and g :: "'c \<Rightarrow> ennreal" assumes "fail = fail'"   and "\<I> = \<I>'"   and "gpv = gpv'"   and "\<And>x. x \<in> results_gpv \<I>' gpv' \<Longrightarrow> f x = g x" shows "expectation_gpv fail \<I> f gpv = expectation_gpv fail' \<I>' g gpv'"
is_chainD: fixes f :: "nat \<Rightarrow> 'a set"   and x :: "'a"   and m :: "nat"   and n :: "nat" assumes "is_chain f"   and "x \<in> f m" shows "x \<in> f (m + n)"
POR_simps_2: fixes x :: "ValD" shows "(x por ValTT) = ValTT"
proj_psi_minus_1_Q2: shows "adjoint (2 \<cdot>\<^sub>m tensor_P proj_psi (1\<^sub>m K) - 1\<^sub>m d) * Q2 * (2 \<cdot>\<^sub>m tensor_P proj_psi (1\<^sub>m K) - 1\<^sub>m d) = Q1"
x: shows "(x, T) \<notin> fv (mk_eq (Abs T (subst_bvs1' s 1 (map (\<lambda>(x, y). Fv x y) vs))) (Abs T (subst_bvs1' t 1 (map (\<lambda>(x, y). Fv x y) vs)))) \<union> FV \<Gamma> \<union> fv s"
rel_option_None2: fixes P :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'c option" shows "rel_option P x None = (x = None)"
mkArr_def: fixes S :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and img :: "'b \<Rightarrow> 'b"   and \<AA> :: "('c \<times> 'c) set"   and A :: "'b set"   and B :: "'b set"   and F :: "'b \<Rightarrow> 'b" assumes "set_category_given_img S img \<AA>" shows "set_category_given_img.mkArr S img \<AA> A B F = (if A \<subseteq> Collect (category.terminal S) \<and> |A| <o \<AA> \<and> B \<subseteq> Collect (category.terminal S) \<and> |B| <o \<AA> \<and> F \<in> A \<rightarrow> B then THE f. f \<in> {f. partial_magma.in_hom S f (set_category_given_img.mkIde S img \<AA> A) (set_category_given_img.mkIde S img \<AA> B)} \<and> set_category_given_img.Fun S img f = restrict F A else partial_magma.null S)"
sur: shows "f ` ({..<q} - {0}) = {..<q} - {0}"
H: shows "\<Gamma> \<turnstile> (\<lambda>:T\<^sub>1\<^sub>1. t\<^sub>1\<^sub>2) \<bullet> t\<^sub>2 : T"
abs_Tree\<^sub>p_eq: fixes p :: "perm"   and t :: "('e, 'f, 'g) Tree" shows "abs_Tree\<^sub>p (p \<bullet> t) = abs_Tree\<^sub>p t"
simps_2: fixes x3 :: "char list list"   and y3 :: "char list list" shows "(Called x3 = Called y3) = (x3 = y3)"
mono_def: fixes Retr :: "'a \<Rightarrow> 'b" shows "Bisim.mono Retr \<equiv> \<forall>theta theta'. theta \<le> theta' \<longrightarrow> Retr theta \<le> Retr theta'"
simps: fixes a1 :: "s"   and a2 :: "s_context" shows "store_typing a1 a2 = (\<exists>\<S> \<C>s tfs ns ms tgs insts fs tclss bss gs. a1 = \<lparr>inst = insts, funcs = fs, tab = tclss, mem = bss, globs = gs\<rparr> \<and> a2 = \<S> \<and> \<S> = \<lparr>s_inst = \<C>s, s_funcs = tfs, s_tab = ns, s_mem = ms, s_globs = tgs\<rparr> \<and> list_all2 (inst_typing \<S>) insts \<C>s \<and> list_all2 (cl_typing \<S>) fs tfs \<and> list_all (tab_agree \<S>) (concat tclss) \<and> list_all2 (\<lambda>tcls n. n \<le> length tcls) tclss ns \<and> list_all2 mem_agree bss ms \<and> list_all2 glob_agree gs tgs)"
equpto: shows "(\<pi> @\<^bsup>n\<^esup> \<pi>') =\<^bsub>n\<^esub> \<pi>"
parallel_composition: fixes V :: "'com list"   and V' :: "'com list"   and d :: "'d" assumes "length V = length V'"   and "\<forall>i<length V. [V ! i] \<approx>\<^bsub>d\<^esub> [V' ! i]" shows "V \<approx>\<^bsub>d\<^esub> V'"
exhaust: fixes y :: "H2" obtains "y = H1"   | "y = H2"
skip: fixes \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d" shows "\<Gamma>,\<gamma>,p\<turnstile> [] \<Rightarrow>\<^sub>z Undecided"
assms_2: shows "inverse_transformations A B G H \<tau> \<tau>'"
strict_trans2: fixes a :: "'c word"   and b :: "'c word"   and c :: "'c word" assumes "a <s b"   and "b \<le>s c" shows "a <s c"
trans: fixes a :: "'b list"   and b :: "'b list"   and c :: "'b list" assumes "prefix a b"   and "prefix b c" shows "prefix a c"
gu: shows "graph G\<^sub>1 \<and> graph G\<^sub>2 \<and> graph_union G\<^sub>1 G\<^sub>2 = G\<^sub>2"
eucl_of_list_append_zeroes: fixes xs :: "real list"   and n :: "nat" shows "eucl_of_list (xs @ replicate n 0) = eucl_of_list xs"
prems: fixes n :: "nat" assumes "n < length (ia # ias)" shows "interrupt_action_ok' (redT_updIs is (take n (ia # ias))) ((ia # ias) ! n)"
rm_vars_singleton: fixes v :: "'a"   and s :: "'a \<Rightarrow> ('b, 'a) Term.term" shows "rm_vars {v} s = rm_var v s"
f13: shows "snd x' \<otimes> r' \<in> carrier R"
most_recent_write_for_write_actionsD: fixes P :: "'a prog"   and E :: "('b \<times> ('c, 'b) obs_event action) llist"   and ra :: "nat"   and wa :: "nat" assumes "P,E \<turnstile> ra \<leadsto>mrw wa" shows "wa \<in> write_actions E"
borel_measurable_lebesgue_integral: fixes f :: "'c \<Rightarrow> 'b \<Rightarrow> 'd"   and N :: "'c measure" assumes "(\<lambda>(x, y). f x y) \<in> borel_measurable (N \<Otimes>\<^sub>M M2)" shows "(\<lambda>x. integral\<^sup>L M2 (f x)) \<in> borel_measurable N"
unset_bit_def: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bit :: "'a \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and drop_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and take_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and "and" :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and or :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and xor :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and mask :: "nat \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and not :: "'a \<Rightarrow> 'a"   and n :: "nat"   and a :: "'a" assumes "class.ring_bit_operations plus minus zero times one divide modulo bit push_bit drop_bit take_bit and or xor mask uminus not" shows "ring_bit_operations.unset_bit one push_bit and not n a = and a (not (push_bit n one))"
count_image_mset_eq_card_vimage: fixes A :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and b :: "'c" assumes "finite A" shows "count (image_mset f (mset_set A)) b = card {a \<in> A. f a = b}"
sums_diff: fixes f :: "nat \<Rightarrow> 'a"   and a :: "'a"   and g :: "nat \<Rightarrow> 'a"   and b :: "'a" assumes "f sums a"   and "g sums b" shows "(\<lambda>n. f n - g n) sums (a - b)"
mult_1: fixes a :: "'b" shows "(1::'b) * a = a"
disjoint_union: fixes C :: "'b set set"   and B :: "'b set set" assumes "disjoint C"   and "disjoint B"   and "\<Union> C \<inter> \<Union> B = {}" shows "disjoint (C \<union> B)"
stab_unfold: fixes s :: "s"   and i :: "nat"   and j :: "nat"   and cl :: "cl" assumes "stab s i j = Some cl" shows "\<exists>k. inst.tab (inst s ! i) = Some k \<and> j < length (s.tab s ! k) \<and> s.tab s ! k ! j = Some cl"
Zp_int_inc_closed: fixes n :: "int" shows "[n] \<cdot> \<one> \<in> carrier Zp"
KL_density: fixes b :: "real"   and f :: "'a \<Rightarrow> real" assumes "1 < b"   and "random_variable borel f"   and "AE x in M. 0 \<le> f x" shows "KL_divergence b M (density M (\<lambda>x. ennreal (f x))) = expectation (\<lambda>x. f x * log b (f x))"
that: fixes x :: "'a" assumes "x \<in> cball a e"   and "f x = x" shows "thesis"
simps_8: fixes \<phi> :: "atom fm"   and \<Gamma> :: "real list" shows "eval (AllQ \<phi>) \<Gamma> = (\<forall>x. eval \<phi> (x # \<Gamma>))"
distinct_53: fixes x61 :: "'c"   and x62 :: "('a, 'b, 'c) Stmt" shows "While x61 x62 \<noteq> Stop"
inf_closed: fixes A :: "'a set" assumes "A \<subseteq> carrier L" shows "\<Sqinter>\<^bsub>L\<^esub>A \<in> carrier L"