hcju/isabelleps huggingface dataset:
f_nxt_mono1: fixes f :: "'b \<Rightarrow> 'b ttree"   and f' :: "'b \<Rightarrow> 'b ttree"   and T :: "'b set"   and x :: "'b"   and x' :: "'b" assumes "\<And>x. paths (f x) \<subseteq> paths (f' x)" shows "paths (f_nxt f T x x') \<subseteq> paths (f_nxt f' T x x')"
seqI': fixes f :: "'a"   and a :: "'a"   and b :: "'a"   and g :: "'a"   and c :: "'a" assumes "\<guillemotleft>f : a \<rightarrow>\<^sub>A b\<guillemotright>"   and "\<guillemotleft>g : b \<rightarrow>\<^sub>A c\<guillemotright>" shows "A.seq g f"
is_unit_hom_iff: fixes x :: "'a poly poly" shows "(poly_y_x x dvd 1) = (x dvd 1)"
IFSC_def: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p"   and A' :: "'p"   and B' :: "'p"   and C' :: "'p"   and D' :: "'p" shows "A B C D IFSC A' B' C' D' \<equiv> Bet A B C \<and> Bet A' B' C' \<and> Cong A C A' C' \<and> Cong B C B' C' \<and> Cong A D A' D' \<and> Cong C D C' D'"
transition_of_st_mat_def: fixes a :: "'a st_mat"   and i :: "'a" shows "transition_of_st_mat a i = pmf_of_st_vec (transition_vec_of_st_mat a i)"
assigned_def: fixes V :: "'c"   and e :: "'c exp" shows "assigned V e \<equiv> \<exists>v e'. e = V:=Val v;; e'"
union_ring_is_field: fixes C :: "'c ring set" assumes "\<And>R. R \<in> C \<Longrightarrow> field R"   and "\<And>R S. \<lbrakk>R \<in> C; S \<in> C\<rbrakk> \<Longrightarrow> R \<lesssim> S \<or> S \<lesssim> R"   and "C \<noteq> {}" shows "field (union_ring C)"
distinct_7: fixes x5 :: "nat \<times> char list list" shows "Unit \<noteq> Ref x5"
simps: fixes step :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<times> 'a"   and s :: "'a"   and a :: "'b"   and ou :: "'c"   and s' :: "'a" shows "IO_Automaton.validTrans step (Trans s a ou s') = (step s a = (ou, s'))"
wf_samequot_exec: fixes n :: "nat"   and r :: "'a atom rexp"   and as :: "'a \<times> bool list" assumes "local.wf n r"   and "as \<in> (set \<circ> \<sigma> \<Sigma>) n" shows "local.wf n (samequot_exec as r)"
square_free_imp_separable: fixes f :: "'c poly" assumes "square_free f" shows "separable f"
apartment_faces: fixes X :: "'a set set"   and x :: "'a set"   and y :: "'a set" assumes "X \<in> \<A>"   and "x \<in> X"   and "order.greater_eq x y" shows "y \<in> X"
inner_minus_left: fixes x :: "'a"   and y :: "'a" shows "inner (- x) y = - inner x y"
covector_vector_comp: fixes v :: "'a" assumes "v \<sqinter> top = v" shows "- id v \<sqinter> v = bot"
is_new_action_simps_3: fixes a :: "'a"   and al :: "addr_loc"   and v :: "'a val" shows "is_new_action (NormalAction (ReadMem a al v)) = False"
assms_1: shows "dis \<phi> \<psi> \<pi>l"
partially_localize_CR: fixes r :: "('b \<times> 'b) set" shows "CR r = (\<forall>x y z. (x, y) \<in> r \<and> (x, z) \<in> r\<^sup>* \<longrightarrow> (y, z) \<in> r\<^sup>\<down>)"
innermost_intvl: fixes \<phi> :: "atom fm"   and xs :: "'b list"   and l :: "'b"   and x :: "'b"   and y :: "'b" assumes "nqfree \<phi>"   and "nolb \<phi> xs l x"   and "l < x"   and "x \<notin> EQ \<phi> xs"   and "DLO.I \<phi> (x # xs)"   and "l < y"   and "y \<le> x" shows "DLO.I \<phi> (y # xs)"
as1: fixes x :: "'a" shows "ad x \<cdot> x = (0::'a)"
transp_rel_pmf: fixes R :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "transp R" shows "transp (rel_pmf R)"
finite_UNIV_list_def: shows "finite_UNIV = Phantom('a list) False"
Fun_lam_pr1: shows "Fun \<Lambda>_pr1 = restrict (Fun (\<Lambda> a b c g) \<circ> (\<lambda>x\<^bold>\<in>local.set (local.prod a b). UP (hfst (DOWN x)))) (local.set (local.prod a b))"
lconcatMap_def: shows "lconcatMap \<equiv> \<Lambda> f. lconcat oo lmap\<cdot>f"
path_component_of_maximal: fixes X :: "'b topology"   and s :: "'b set"   and x :: "'b" assumes "path_connectedin X s"   and "x \<in> s" shows "s \<subseteq> path_component_of_set X x"
beta_coerce_cfun: fixes f :: "'g \<rightarrow> 'h"   and x :: "'f" shows "COERCE('g \<rightarrow> 'h, 'f \<rightarrow> 'e)\<cdot>f\<cdot>x = COERCE('h, 'e)\<cdot>(f\<cdot>(COERCE('f, 'g)\<cdot>x))"
cinner_ge_zero: fixes scaleR :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and scaleC :: "complex \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and norm :: "'b \<Rightarrow> real"   and sgn :: "'b \<Rightarrow> 'b"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool"   and cinner :: "'b \<Rightarrow> 'b \<Rightarrow> complex"   and x :: "'b" assumes "class.complex_inner scaleR scaleC plus zero minus uminus dist norm sgn uniformity open cinner" shows "0 \<le> cinner x x"
Cd_Above: fixes d :: "'b"   and X :: "'a set" shows "Cd d X = Above (Pd d) (X \<inter> Field (Pd d)) \<inter> X"
simps_1: fixes x :: "'c" shows "weight (Var x) = w0"
dim_eq_card: fixes B :: "('b \<Rightarrow>\<^sub>0 'c) set"   and V :: "('b \<Rightarrow>\<^sub>0 'c) set" assumes "phull B = phull V"   and "phull.independent B" shows "phull.dim V = card B"
carr_dsum_prod1: fixes I :: "'h set"   and M :: "'h \<Rightarrow> ('i, 'j) aGroup_scheme" shows "\<forall>x. x \<in> carr_dsumag I M \<longrightarrow> x \<in> carr_prodag I M"
union_mset_add_mset_left: fixes a :: "'a"   and A :: "'a multiset"   and B :: "'a multiset" shows "add_mset a A + B = add_mset a (A + B)"
rel_prod_inject: fixes R1 :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and R2 :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "'a"   and b :: "'c"   and c :: "'b"   and d :: "'d" shows "rel_prod R1 R2 (a, b) (c, d) = (R1 a c \<and> R2 b d)"
proj_mono_prefix: fixes t :: "cnf llist"   and t' :: "cnf llist"   and c :: "'id" assumes "lprefix t t'" shows "lprefix (\<pi>\<^bsub>c\<^esub>t) (\<pi>\<^bsub>c\<^esub>t')"
degree_sum_le_Max: fixes F :: "'b set"   and f :: "'b \<Rightarrow> 'c poly" assumes "finite F" shows "degree (sum f F) \<le> (MAX x\<in>F. degree (f x))"
Eq_Trans_E: fixes H :: "fm set"   and x :: "tm"   and u :: "tm"   and t :: "tm"   and A :: "fm" assumes "H \<turnstile> x EQ u"   and "insert (t EQ u) H \<turnstile> A" shows "insert (x EQ t) H \<turnstile> A"
raborting_consistent: shows "\<forall>s t u. rvpeq u s t \<longrightarrow> raborting s u = raborting t u"
sup_bot_left: fixes sup :: "'x \<Rightarrow> 'x \<Rightarrow> 'x"   and less_eq :: "'x \<Rightarrow> 'x \<Rightarrow> bool"   and less :: "'x \<Rightarrow> 'x \<Rightarrow> bool"   and bot :: "'x"   and x :: "'x" assumes "class.bounded_semilattice_sup_bot sup less_eq less bot" shows "sup bot x = x"
simps: fixes n :: "nat"   and l :: "cell list"   and r :: "cell list" shows "inv_loop1_exit n (l, r) = (0 < n \<and> (l, r) = ([], Bk # Oc # Bk \<up> n @ Oc \<up> n))"
eclose_facts_1: fixes X :: "hf" shows "Transset (eclose X)"
eventually_finite: fixes x :: "ereal"   and f :: "'b \<Rightarrow> ereal"   and F :: "'b filter" assumes "\<bar>x\<bar> \<noteq> \<infinity>"   and "(f \<longlongrightarrow> x) F" shows "\<forall>\<^sub>F x in F. \<bar>f x\<bar> \<noteq> \<infinity>"
currentLevelZeroTrailEqualsItsPrefixToLevelZero: fixes M :: "('a \<times> bool) list" assumes "currentLevel M = 0" shows "M = prefixToLevel 0 M"
fv\<^sub>s\<^sub>t\<^sub>p_is_subterm_trms\<^sub>s\<^sub>t\<^sub>p: fixes x :: "'c"   and a :: "('d, 'c) strand_step" assumes "x \<in> fv\<^sub>s\<^sub>t\<^sub>p a" shows "Var x \<sqsubseteq>\<^sub>s\<^sub>e\<^sub>t trms\<^sub>s\<^sub>t\<^sub>p a"
star_inductr_var_equiv: fixes y :: "'a"   and x :: "'a" shows "(y \<cdot> x \<le> y) = (y \<cdot> qstar x \<le> y)"
holomorphic_on_imp_continuous_on: fixes f :: "complex \<Rightarrow> complex"   and s :: "complex set" assumes "f holomorphic_on s" shows "continuous_on s f"
dsj: shows "\<bar>d fs (Suc j)\<bar> \<le> int N ^ (m - 1)"
lex_ord_pair_EqD: fixes x :: "'o"   and xs :: "('a \<times> 'd) list"   and ys :: "('a \<times> 'e) list"   and f :: "'a \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> order option"   and k :: "'a" assumes "oalist_inv_raw x xs"   and "oalist_inv_raw x ys"   and "lex_ord_pair x f xs ys = Some Eq"   and "k \<in> fst ` set xs \<union> fst ` set ys" shows "f k (lookup_pair x xs k) (lookup_pair x ys k) = Some Eq"
iMOD_add: fixes r :: "nat"   and m :: "nat"   and k :: "nat" shows "[ r, mod m ] \<oplus> k = [ r + k, mod m ]"
axioms_1: fixes wtFsym :: "'a \<Rightarrow> bool"   and wtPsym :: "'b \<Rightarrow> bool"   and arOf :: "'a \<Rightarrow> nat"   and parOf :: "'b \<Rightarrow> nat"   and \<Phi> :: "('a, 'b) lit list set"   and D :: "'c \<Rightarrow> bool"   and intF :: "'a \<Rightarrow> 'c list \<Rightarrow> 'c"   and intP :: "'b \<Rightarrow> 'c list \<Rightarrow> bool" assumes "U.Model wtFsym wtPsym arOf parOf \<Phi> D intF intP" shows "U.Problem wtFsym wtPsym arOf parOf \<Phi>"
tangent_space_def: fixes p :: "'c" shows "manifold_eucl.dest.tangent_space p = {X. manifold_eucl.dest.is_derivation X p \<and> extensional0 manifold_eucl.dest.diff_fun_space X}"
qFresh_imp_ex_qAFresh: fixes V :: "'var set"   and X :: "('d, 'e, 'varSort, 'var, 'f) qTerm"   and fZs :: "'var \<Rightarrow> 'varSort set" assumes "finite V"   and "qGood X"   and "\<forall>z\<in>V. \<forall>zs\<in>fZs z. qFresh zs z X" shows "\<exists>X'. qGood X' \<and> X #= X' \<and> (\<forall>z\<in>V. \<forall>zs\<in>fZs z. qAFresh zs z X')"
S_2: fixes n :: "nat" shows "AE x in M. X n x \<in> S"
cases: fixes a :: "('Var, 'AExp, 'BExp) Stmt" assumes "no_await a" obtains x :: "'Var" and e :: "'AExp" where "a = x \<leftarrow> e"   | c1 :: "('Var, 'AExp, 'BExp) Stmt" and c2 :: "('Var, 'AExp, 'BExp) Stmt" where "a = c1__ ;; c2__" and "no_await c1__" and "no_await c2__"   | c1 :: "('Var, 'AExp, 'BExp) Stmt" and c2 :: "('Var, 'AExp, 'BExp) Stmt" and b :: "'BExp" where "a = Stmt.If ba__ c1 c2" and "no_await c1" and "no_await c2"   | c :: "('Var, 'AExp, 'BExp) Stmt" and b :: "'BExp" where "a = While b ca__" and "no_await ca__"   | "a = Skip"   | "a = Stop"   | c :: "('Var, 'AExp, 'BExp) Stmt" and m :: "'Var ModeUpd" where "a = c@[m]" and "no_await c"
exi_rename: fixes \<phi> :: "'fmla"   and x :: "'var"   and y :: "'var" assumes "\<phi> \<in> fmla"   and "x \<in> var"   and "y \<in> var"   and "y \<notin> Fvars \<phi>" shows "exi x \<phi> = exi y (subst \<phi> (Var y) x)"
coinduct: fixes X :: "('a, 'b) resource \<Rightarrow> ('a, 'b) resource \<Rightarrow> bool"   and x1 :: "('a, 'b) resource"   and x2 :: "('a, 'b) resource"   and A :: "'a set" assumes "X x1 x2"   and "\<And>x1 x2. X x1 x2 \<Longrightarrow> \<exists>res res'. x1 = res \<and> x2 = res' \<and> (\<forall>x. x \<in> A \<longrightarrow> rel_spmf (rel_prod (=) (\<lambda>uu_ uua_. X uu_ uua_ \<or> A \<turnstile>\<^sub>R uu_ \<sim> uua_)) (run_resource res x) (run_resource res' x))" shows "A \<turnstile>\<^sub>R x1 \<sim> x2"
t1t2: fixes t :: "real" assumes "t \<in> {t1<..<t2}" shows "flow0 x t \<notin> {a<--<b}"
to_nat_less_card: fixes j :: "'d" shows "mod_type_class.to_nat j < CARD('d)"
injections_def: fixes X :: "'c set"   and Y :: "'d set" shows "injections X Y = {R. Domain R = X \<and> Range R \<subseteq> Y \<and> runiq R \<and> runiq (R\<inverse>)}"
d_least: shows "\<forall>E\<in>{E \<in> N_of_state (Sup_state Sts) \<union> P_of_state (Sup_state Sts) \<union> Q_of_state (Sup_state Sts). subsumes E C}. \<not> strictly_subsumes E D"
igSubstCls_fromMOD: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) Recursion.model" assumes "gSubstCls MOD" shows "igSubstCls (fromMOD MOD)"
not_postdominate_source_not_postdominate_target: fixes n :: "'node"   and a :: "'edge" assumes "\<not> n postdominates sourcenode a"   and "valid_node n"   and "valid_edge a"   and "intra_kind (kind a)" obtains ax :: "'edge" where "sourcenode a = sourcenode ax" and "valid_edge ax" and "\<not> n postdominates targetnode ax"
poly\<M>: fixes C :: "'a set" assumes "C \<in> \<M>" shows "polytope C"
mult_assoc: fixes a :: "'b \<Rightarrow> 'b"   and b :: "'b \<Rightarrow> 'b"   and c :: "'b \<Rightarrow> 'b" shows "a \<circ> b \<circ> c = a \<circ> (b \<circ> c)"
pr_naturality_2: fixes g :: "'a"   and b :: "'a"   and d :: "'a"   and f :: "'a"   and a :: "'a"   and c :: "'a" assumes "arr g"   and "local.dom g = b"   and "cod g = d"   and "arr f"   and "local.dom f = a"   and "cod f = c" shows "pr1 c d \<cdot> local.prod f g = f \<cdot> pr1 a b"
drop_bit_rec: 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 n :: "nat"   and a :: "'a" assumes "class.semiring_bit_shifts plus minus zero times one divide modulo bit push_bit drop_bit take_bit" shows "drop_bit n a = (if n = 0 then a else drop_bit (n - 1) (divide a (numeral.numeral one plus (num.Bit0 num.One))))"
\<theta>: shows "\<guillemotleft>\<theta> : p\<^sub>0 \<star> w \<Rightarrow> u\<guillemotright>"
inf1D1: fixes A :: "'b \<Rightarrow> bool"   and B :: "'b \<Rightarrow> bool"   and x :: "'b" assumes "inf A B x" shows "A x"
Act_Basic_1: fixes v :: "i"   and \<phi> :: "\<o>" shows "[\<^bold>\<A>\<phi> \<^bold>\<or> \<^bold>\<A>\<^bold>\<not>\<phi> in v]"
card_eq_SucD: fixes A :: "'e set"   and k :: "nat" assumes "card A = Suc k" shows "\<exists>b B. A = Set.insert b B \<and> b \<notin> B \<and> card B = k \<and> (k = 0 \<longrightarrow> B = {})"
foldr_append: fixes f :: "'b \<Rightarrow> 'a \<Rightarrow> 'a"   and xs :: "'b list"   and ys :: "'b list"   and a :: "'a" shows "foldr f (xs @ ys) a = foldr f xs (foldr f ys a)"
s'A_m_le: fixes m :: "nat"   and n :: "nat" assumes "m \<le> length (paid_A ! n)" shows "swaps (drop (length (paid_A ! n) - m) (paid_A ! n)) (s_A n) = s'_A n m"
map_poly_pCons: fixes f :: "'c \<Rightarrow> 'b"   and c :: "'c"   and p :: "'c poly" assumes "f (0::'c) = (0::'b)" shows "map_poly f (pCons c p) = pCons (f c) (map_poly f p)"
l_cancel_one': fixes x :: "nat \<Rightarrow> int"   and a :: "nat \<Rightarrow> int" assumes "x \<in> carrier Zp"   and "a \<in> carrier Zp" shows "(x = x \<oplus> a) = (a = \<zero>)"
Dags_Nodes_cases: fixes P :: "dag \<Rightarrow> dag \<Rightarrow> 'd \<Rightarrow> bool"   and var :: "'d"   and fnct :: "ref \<Rightarrow> ref"   and n :: "nat"   and levellista :: "ref list list"   and low :: "ref \<Rightarrow> ref"   and high :: "ref \<Rightarrow> ref" assumes "\<And>t1 t2. P t1 t2 var = P t2 t1 var"   and "\<And>t1 t2. \<lbrakk>t1 \<in> Dags (fnct ` Nodes n levellista) low high; t2 \<in> Dags (fnct ` Nodes n levellista) low high\<rbrakk> \<Longrightarrow> P t1 t2 var"   and "\<And>t1 t2. \<lbrakk>t1 \<in> Dags (fnct ` Nodes n levellista) low high; t2 \<in> Dags (fnct ` Nodes (Suc n) levellista) low high; t2 \<notin> Dags (fnct ` Nodes n levellista) low high\<rbrakk> \<Longrightarrow> P t1 t2 var"   and "\<And>t1 t2. \<lbrakk>t1 \<in> Dags (fnct ` Nodes (Suc n) levellista) low high; t1 \<notin> Dags (fnct ` Nodes n levellista) low high; t2 \<in> Dags (fnct ` Nodes (Suc n) levellista) low high; t2 \<notin> Dags (fnct ` Nodes n levellista) low high\<rbrakk> \<Longrightarrow> P t1 t2 var" shows "\<forall>t1 t2. t1 \<in> Dags (fnct ` Nodes (Suc n) levellista) low high \<and> t2 \<in> Dags (fnct ` Nodes (Suc n) levellista) low high \<longrightarrow> P t1 t2 var"
intro: fixes sb :: "'b memref list"   and is\<^sub>s\<^sub>b :: "instr list" assumes "\<And>sb\<^sub>1 sb\<^sub>2. sb = sb\<^sub>1 @ sb\<^sub>2 \<Longrightarrow> \<exists>is. instrs sb\<^sub>2 @ is\<^sub>s\<^sub>b = is @ prog_instrs sb\<^sub>2" shows "causal_program_history is\<^sub>s\<^sub>b sb"
P_sup_eq_E_sup: fixes P :: "'s stream \<Rightarrow> bool"   and s :: "'s" assumes "Measurable.pred St P" shows "P_sup s P = E_sup s (indicator {x \<in> space St. P x})"
sinvar_valid_remove_SOME_offending_flows: fixes m :: "'b SecurityInvariant_configured"   and nodesG :: "'b set"   and edgesG :: "('b \<times> 'b) set" assumes "configured_SecurityInvariant m"   and "c_offending_flows m \<lparr>nodes = nodesG, edges = edgesG\<rparr> \<noteq> {}" shows "c_sinvar m \<lparr>nodes = nodesG, edges = edgesG - (SOME F. F \<in> c_offending_flows m \<lparr>nodes = nodesG, edges = edgesG\<rparr>)\<rparr>"
in_KeysI: fixes s :: "'c"   and f :: "'c \<Rightarrow>\<^sub>0 'd"   and F :: "('c \<Rightarrow>\<^sub>0 'd) set" assumes "s \<in> keys f"   and "f \<in> F" shows "s \<in> Keys F"
prems_2: fixes S :: "'a set" assumes "S \<in> insert X \<F>" shows "\<exists>a g. a \<notin> U \<and> continuous_on (S - {a}) g \<and> g ` (S - {a}) \<subseteq> T \<and> (\<forall>x\<in>S \<inter> K. g x = h x)"
A1_A2: shows "mat_of_rows (dim_col A) (map (Matrix.row A) [0..<k]) @\<^sub>r mat_of_rows (dim_col A) (map (Matrix.row A) [k..<dim_row A]) \<in> carrier_mat (dim_row A) (dim_col A)"
ideD_3: fixes a :: "'a" assumes "ide a" shows "cod a = a"
ide_right: shows "ide (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0)\<^sup>*"
trans_tm_forgets: fixes "is" :: "name list"   and t :: "tm" assumes "atom ` set is \<sharp>* t" shows "trans_tm is t = trans_tm [] t"
mHom_add: fixes N :: "('i, 'b, 'j) Module_scheme"   and f :: "'a \<Rightarrow> 'i"   and m :: "'a"   and n :: "'a" assumes "R module N"   and "f \<in> mHom R M N"   and "m \<in> carrier M"   and "n \<in> carrier M" shows "f (m \<plusminus> n) = f m \<plusminus>\<^bsub>N\<^esub> f n"
lock_actions_ok'E: fixes l :: "('c \<times> nat) option"   and t :: "'c"   and las :: "lock_action list" assumes "lock_actions_ok' l t las" obtains "lock_actions_ok l t las"   | xs :: "lock_action list" and ys :: "lock_action list" where "las = xs @ Lock # ys" and "lock_actions_ok l t xs" and "\<not> may_lock (upd_locks l t xs) t"
pis: fixes p :: "real poly"   and i :: "nat" assumes "(p, i) \<in> set pis" shows "degree p \<le> 2 \<or> (\<forall>x\<in>set (coeffs p). x \<in> \<rat>)"
usubstr_nth: fixes \<sigma> :: "(char \<Rightarrow> trm option) \<times> (char \<Rightarrow> trm option) \<times> (char \<Rightarrow> fml option) \<times> (char \<Rightarrow> game option)"   and R :: "fml list \<times> fml"   and k :: "nat" assumes "usubstr \<sigma> R \<noteq> None"   and "0 \<le> k"   and "k < length (fst R)" shows "fst (the (usubstr \<sigma> R)) ! k = the (usubstappf \<sigma> Denotational_Semantics.allvars (fst R ! k))"
project_set_def: fixes X :: "'c set"   and b :: "'c"   and y :: "real" shows "project_set X b y = SPEC (\<lambda>ivl. X \<inter> plane b y \<subseteq> ivl \<and> ivl \<subseteq> plane b y)"
list2FWpolicy2list: fixes p :: "((int \<times> int) set set, int) Combinators" shows "C (list2FWpolicy (policy2list p)) = C p"
cases: fixes r :: "a0n_state" obtains signals :: "signal \<Rightarrow> nat" where "r = \<lparr>signals = signals\<rparr>"
set_fold_aps: fixes aps :: "'g \<Rightarrow> 'e list \<Rightarrow> 'e list \<Rightarrow> 'f \<Rightarrow> ('g \<times> 'f \<times> 'e) list \<Rightarrow> ('g \<times> 'f \<times> 'e) list"   and flag :: "'g"   and gs :: "'e list"   and bs :: "'e list"   and hs :: "'f list"   and ps :: "('g \<times> 'f \<times> 'e) list" assumes "\<And>gs bs h ps. set (aps flag gs bs h ps) = set ps \<union> Pair flag ` ({h} \<times> (set gs \<union> set bs))" shows "set (fold (aps flag gs bs) hs ps) = Pair flag ` (set hs \<times> (set gs \<union> set bs)) \<union> set ps"
bij_combine: fixes A :: "'c set"   and B :: "'c set"   and C :: "'d set"   and D :: "'d set"   and f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" assumes "A \<subseteq> B"   and "C \<subseteq> D"   and "bij_betw f A C"   and "bij_betw g (B - A) (D - C)" obtains h :: "'c \<Rightarrow> 'd" where "bij_betw h B D" and "\<And>x. x \<in> A \<Longrightarrow> h x = f x" and "\<And>x. x \<in> B - A \<Longrightarrow> h x = g x"
f_def: shows "f \<equiv> \<lambda>(x, y). ordertype B s * ordermap A r x + ordermap B s y"
split_Nil_iff: fixes xs :: "'c list"   and ys :: "'c list" shows "(splice xs ys = []) = (xs = [] \<and> ys = [])"
rename_Stable: fixes h :: "'c \<Rightarrow> 'd"   and F :: "'c program"   and A :: "'c set" assumes "bij h" shows "(rename h F \<in> Stable (h ` A)) = (F \<in> Stable A)"
vinsert_vsingleton: fixes a :: "V"   and b :: "V" shows "vinsert a (set {b}) = set {a, b}"
senior_states_card_mono: fixes A :: "'b set"   and q :: "'b"   and n :: "nat" assumes "A \<subseteq> senior_states q n" shows "card A \<le> card (senior_states q n)"
join_def: shows "join \<equiv> map_fun id (map_fun alist.impl_of (map_fun alist.impl_of Alist)) join_raw"
TDomain: fixes T :: "('a, 'b) Axioms"   and A :: "'a"   and e :: "('a, 'b) Expression"   and B :: "'a" shows "TDomain (TermEquivCl' T A e B) = A"
simps_7: fixes v :: "String.literal"   and va :: "typ" shows "beta_reducible (Ct v va) = False"
assms: fixes x :: "'a" assumes "x \<in> A" shows "f x abs_summable_on B"
Sup_insert: fixes S :: "real set"   and x :: "real" assumes "bounded S" shows "Sup (insert x S) = (if S = {} then x else max x (Sup S))"
ctr_transfer_1: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool" shows "rel_option R None None"
ccSUP_const: fixes A :: "'c set"   and f :: "'a" assumes "A \<noteq> {}" shows "(SUP i\<in>A. f) = f"
bounded_clinear_def: fixes f :: "'d \<Rightarrow> 'e" shows "bounded_clinear f \<equiv> clinear f \<and> bounded_clinear_axioms f"
empty_Union_conv: fixes A :: "'c set set" shows "({} = \<Union> A) = (\<forall>x\<in>A. x = {})"
full_col_rank_imp_independent_columns: fixes A :: "(('a, 'b) vec, 'c) vec" assumes "rank A = ncols A" shows "vec.independent (columns A)"
ou_witness_not_ed_ou: fixes X :: "ou_witness set" shows "\<not> ed_ou X"
when_when: fixes a :: "'a"   and P :: "bool"   and Q :: "bool" shows "(a when P when Q) = (a when P \<and> Q)"
eventually_mono_sequentially: fixes P :: "nat \<Rightarrow> bool"   and k :: "nat"   and Q :: "nat \<Rightarrow> bool" assumes "eventually P sequentially"   and "\<And>x. P (x + k) \<Longrightarrow> Q (x + k)" shows "eventually Q sequentially"
length_append: fixes xs :: "'k list"   and ys :: "'k list" shows "length (xs @ ys) = length xs + length ys"
calc_x_axis_intersection_fun_mono: fixes x1 :: "real"   and x2 :: "real" assumes "1 < x1"   and "x1 < x2" shows "x2 - sqrt (x2\<^sup>2 - 1) < x1 - sqrt (x1\<^sup>2 - 1)"
compute_floorlog: fixes b :: "nat"   and x :: "nat" shows "floorlog b x = (if 0 < x \<and> 1 < b then floorlog b (x div b) + 1 else 0)"
ndepth_def: fixes n :: "('f, 'g) Old_Datatype.node" shows "ndepth n \<equiv> case Rep_Node n of (f, x) \<Rightarrow> LEAST k. f k = Inr 0"
Range: fixes y :: "ZF"   and f :: "ZF" shows "y |\<in>| HOLZF.Range f = (\<exists>x. Opair x y |\<in>| f)"
size_of_rhs: fixes p :: "real poly"   and qs :: "real poly list"   and subsets :: "nat list list" shows "dim_vec (v_vec p qs subsets) = length subsets"
intruder_deduct_hom_ComposeH: fixes X :: "('fun, 'var) Term.term list"   and f :: "'fun"   and M :: "('fun, 'var) Term.term set"   and \<A> :: "('lbl strand_label \<times> ('fun, 'var) strand_step) list"   and Sec :: "('fun, 'var) Term.term set" assumes "length X = arity f"   and "public f"   and "\<And>x. x \<in> set X \<Longrightarrow> \<langle>M;\<A>;Sec\<rangle> \<turnstile>\<^sub>h\<^sub>o\<^sub>m x"   and "homogeneous\<^sub>l\<^sub>s\<^sub>t (Fun f X) \<A> Sec"   and "Fun f X \<in> GSMP (trms\<^sub>l\<^sub>s\<^sub>t \<A>)" shows "\<langle>M;\<A>;Sec\<rangle> \<turnstile>\<^sub>h\<^sub>o\<^sub>m Fun f X"
disjoint_product_over_sum: fixes x :: "i"   and y :: "i"   and z :: "i" assumes "O x y"   and "\<not> O x z" shows "x \<otimes> (y \<oplus> z) = x \<otimes> y"
f_join_iTILL: fixes xs :: "'b list"   and n :: "nat" shows "xs \<Join>\<^sub> [\<dots>n] = xs \<down> Suc n"
subst_type: fixes \<Delta> :: "binding list"   and U :: "POPLmark.type"   and \<Gamma> :: "binding list"   and t :: "trm"   and T :: "POPLmark.type"   and u :: "trm" assumes "\<Delta> @ VarB U \<Colon> \<Gamma> \<turnstile> t : T"   and "\<Gamma> \<turnstile> u : U" shows "\<down>\<^sub>e 1 0 \<Delta> @ \<Gamma> \<turnstile> t[\<parallel>\<Delta>\<parallel> \<mapsto> u] : \<down>\<^sub>\<tau> 1 \<parallel>\<Delta>\<parallel> T"
glbound_in_of_def: fixes P :: "'a set"   and x :: "'a"   and y :: "'a"   and b :: "'a" shows "glbound_in_of P x y b \<equiv> b \<in> P \<and> lbound_of x y b \<and> (\<forall>a\<in>P. lbound_of x y a \<longrightarrow> b \<succeq> a)"