hcju/isabelleps huggingface dataset:
is_extensional: fixes f :: "('a \<Rightarrow> 'b, 'a \<Rightarrow> 'b) concrete_category.arr \<times> 'a" assumes "\<not> A_BxA.arr f" shows "E.map f = B.null"
abc_append_frist_steps_halt_eq': fixes lm :: "nat list"   and A :: "abc_inst list"   and n :: "nat"   and lm' :: "nat list"   and B :: "abc_inst list" assumes "abc_steps_l (0, lm) A n = (length A, lm')"   and "A \<noteq> []" shows "\<exists>n'. abc_steps_l (0, lm) (A @ B) n' = (length A, lm')"
dim_R_def: fixes M :: "'v set" shows "dim_R M = (if \<exists>ms. set ms \<subseteq> M \<and> RSpan ms = M then LEAST n. \<exists>ms. length ms = n \<and> set ms \<subseteq> M \<and> RSpan ms = M else 0)"
b: shows "V.basis (set (find_base_vectors (berlekamp_resulting_mat u)))"
r'_r: fixes x :: "(real, 'n) Finite_Cartesian_Product.vec" assumes "x \<in> Y" shows "r' x = r x"
div_plus_div_distrib_dvd_right: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and 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 modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and euclidean_size :: "'b \<Rightarrow> nat"   and c :: "'b"   and b :: "'b"   and a :: "'b" assumes "class.euclidean_semiring_cancel divide plus minus zero times one modulo euclidean_size"   and "dvd.dvd times c b" shows "divide (plus a b) c = plus (divide a c) (divide b c)"
sigma_sets_subset: fixes a :: "'d set set"   and M :: "'d measure" assumes "a \<subseteq> sets M" shows "sigma_sets (space M) a \<subseteq> sets M"
rel_sel: fixes R :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and a :: "('f, 'd) envT"   and b :: "('f, 'e) envT" shows "envT.rel_envT R a b = ((=) ===> R) (run_env a) (run_env b)"
power_even_abs: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sgn :: "'b \<Rightarrow> 'b"   and abs :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and n :: "nat"   and a :: "'b" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one"   and "even n" shows "power.power one times (abs a) n = power.power one times a n"
x0_1: shows "a11 * fst x0 + a12 * snd x0 = b1"
ncop__ncols: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p" assumes "\<not> Coplanar A B C D" shows "\<not> Col A B C \<and> \<not> Col A B D \<and> \<not> Col A C D \<and> \<not> Col B C D"
T: fixes i :: "'b" assumes "i \<in> I" shows "T i \<subseteq> topspace X"
cnf_of_encode_problem_structure_1: fixes \<Pi> :: "'b strips_problem"   and t :: "nat" shows "cnf (\<Phi>\<^sub>I \<Pi>) \<subseteq> cnf (\<Phi> \<Pi> t)"
HcompNml_Nml_Src: fixes t :: "'a local.term" assumes "Nml t" shows "t \<^bold>\<lfloor>\<^bold>\<star>\<^bold>\<rfloor> Src t = t"
scaleR2_rel_internal: fixes A :: "('a \<times> ('b \<times> 'c) set) set" shows "scaleR2_rel A = ((ereal_rel \<times>\<^sub>r ereal_rel) \<times>\<^sub>r A) O br (\<lambda>((l, u), X). scaleR2 l u X) (\<lambda>((l, u), uu_). ereal -` {l..u} \<noteq> {})"
Ball_def: fixes A :: "'b set"   and P :: "'b \<Rightarrow> bool" shows "Ball (A::'b set) (P::'b \<Rightarrow> bool) = (\<forall>x::'b. x \<in> A \<longrightarrow> P x)"
max_eq: shows "max (IArray.length 0) (IArray.length a) = IArray.length a"
simps_1: fixes uu :: "nat" shows "no_illegal_updates_code [] uu = True"
gotoD: fixes \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and r :: "'c rule list"   and s :: "state"   and t :: "state"   and m :: "'c match_expr"   and chain :: "char list" assumes "\<Gamma>,\<gamma>,p\<turnstile>\<^sub>g \<langle>r, s\<rangle> \<Rightarrow> t"   and "r = [Rule m (Goto chain)]"   and "s = Undecided"   and "matches \<gamma> m p" shows "\<exists>rs. \<Gamma> chain = Some rs \<and> \<Gamma>,\<gamma>,p\<turnstile>\<^sub>g \<langle>rs, s\<rangle> \<Rightarrow> t"
jpTraces_step_length_inv: fixes jp :: "'a \<Rightarrow> ('obs, 'aAct, 'c) Protocol"   and n :: "nat" shows "{t \<in> jpTraces jp. tLength t = Suc n} = {uu_. \<exists>eact aact t s. uu_ = t \<leadsto> s \<and> t \<in> {t \<in> jpTraces jp. tLength t = n} \<and> eact \<in> set (envAction (tLast t)) \<and> (\<forall>a. aact a \<in> set (actJP jp t a)) \<and> s = envTrans eact aact (tLast t)}"
HSem_add_fresh': fixes x :: "'b"   and \<Gamma> :: "('b \<times> 'a) list"   and \<rho> :: "'b \<Rightarrow> 'c"   and e :: "'a" assumes "atom x \<sharp> \<Gamma>"   and "x \<notin> edom \<rho>"   and "\<And>e \<rho>'. e \<in> snd ` set \<Gamma> \<Longrightarrow> \<lbrakk> e \<rbrakk>\<^bsub>\<rho>'\<^esub> = \<lbrakk> e \<rbrakk>\<^bsub>env_delete x \<rho>'\<^esub>" shows "env_delete x (\<lbrace> (x, e) # \<Gamma> \<rbrace>\<rho>) = \<lbrace> \<Gamma> \<rbrace>\<rho>"
gf_G_eq_gf_U_1: fixes K :: "'a \<Rightarrow> 'a pmf"   and x :: "'a"   and z :: "'b" assumes "MC_syntax.convergence_G K x x z" shows "MC_syntax.gf_G K x x z = (1::'b) / ((1::'b) - MC_syntax.gf_U K x x z)"
swap_simp: fixes x :: "'f"   and y :: "'e" shows "prod.swap (x, y) = (y, x)"
coinduct: fixes X :: "('b, 'c) dds \<Rightarrow> ('d, 'e) dds \<Rightarrow> bool"   and x1 :: "('b, 'c) dds"   and x2 :: "('d, 'e) dds"   and A :: "'b \<Rightarrow> 'd \<Rightarrow> bool"   and B :: "'c \<Rightarrow> 'e \<Rightarrow> bool" assumes "X x1 x2"   and "\<And>x1 x2. X x1 x2 \<Longrightarrow> \<exists>S S'. x1 = S \<and> x2 = S' \<and> (A ===> rel_prod B (\<lambda>uu_ uua_. X uu_ uua_ \<or> rel_dds A B uu_ uua_)) (run S) (run S')" shows "rel_dds A B x1 x2"
intro: fixes known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and type_wf :: "(_) heap \<Rightarrow> bool"   and known_ptrs :: "(_) heap \<Rightarrow> bool"   and heap_is_wellformed :: "(_) heap \<Rightarrow> bool"   and parent_child_rel :: "(_) heap \<Rightarrow> ((_) object_ptr \<times> (_) object_ptr) set"   and get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_parent :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr option) prog"   and get_parent_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_ancestors :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr list) prog"   and get_ancestors_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_root_node :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr) prog"   and get_root_node_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and to_tree_order :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr list) prog" assumes "l_get_root_node_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M known_ptr type_wf known_ptrs heap_is_wellformed parent_child_rel get_child_nodes get_child_nodes_locs get_disconnected_nodes get_parent get_parent_locs get_ancestors get_ancestors_locs get_root_node get_root_node_locs"   and "l_to_tree_order_wf heap_is_wellformed parent_child_rel type_wf known_ptr known_ptrs to_tree_order get_parent get_child_nodes" shows "l_to_tree_order_wf_get_root_node_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M known_ptr type_wf known_ptrs heap_is_wellformed parent_child_rel get_child_nodes get_child_nodes_locs get_disconnected_nodes get_parent get_parent_locs get_ancestors get_ancestors_locs get_root_node get_root_node_locs to_tree_order"
unowned_shared_nth_update: fixes i :: "nat"   and p :: "'a"   and "is" :: "instr list"   and xs :: "nat \<Rightarrow> nat option"   and sb :: "'a memref list"   and \<D> :: "bool"   and \<O> :: "nat set"   and \<R> :: "nat \<Rightarrow> bool option"   and \<O>' :: "nat set"   and p' :: "'a"   and is' :: "instr list"   and xs' :: "nat \<Rightarrow> nat option"   and sb' :: "'a memref list"   and \<D>' :: "bool"   and \<R>' :: "nat \<Rightarrow> bool option" assumes "i < length ts\<^sub>s\<^sub>b"   and "ts\<^sub>s\<^sub>b ! i = (p, is, xs, sb, \<D>, \<O>, \<R>)"   and "\<O> \<subseteq> \<O>'" shows "unowned_shared \<S>\<^sub>s\<^sub>b (ts\<^sub>s\<^sub>b[i := (p', is', xs', sb', \<D>', \<O>', \<R>')])"
ax_Normal_cases: fixes G :: "prog"   and A :: "'b triple set"   and P :: "'b assn"   and t :: "Term.term"   and Q :: "'b assn" assumes "G,A\<turnstile>{Normal P} t\<succ> {Q}"   and "G,A\<turnstile>{P \<and>. Not \<circ> normal} t\<succ> {Q}" shows "G,A\<turnstile>{P} t\<succ> {Q}"
sum_mod_poly: fixes k :: "nat"   and q :: "nat" assumes "0 < k" shows "(k - 1) * sum ((^) k) {..<q} < k ^ q"
finite_lists_length_le: fixes A :: "'c set"   and n :: "nat" assumes "finite A" shows "finite {xs. set xs \<subseteq> A \<and> length xs \<le> n}"
simplex_retraction1: fixes X :: "'b set set"   and f :: "'b \<Rightarrow> 'b"   and x :: "'b set" assumes "ChamberComplexRetraction X f"   and "x \<in> f \<turnstile> X" shows "fixespointwise f x"
isLeaf_pt_def: fixes p :: "ref"   and low :: "ref \<Rightarrow> ref"   and high :: "ref \<Rightarrow> ref" shows "isLeaf_pt p low high = (low p = Null \<and> high p = Null)"
rel_pmf_bind_reflI: fixes p :: "'c pmf"   and R :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'd pmf"   and g :: "'c \<Rightarrow> 'e pmf" assumes "\<And>x. x \<in> set_pmf p \<Longrightarrow> rel_pmf R (f x) (g x)" shows "rel_pmf R (p \<bind> f) (p \<bind> g)"
def_hashmap_size_iarray_def: shows "def_hashmap_size = (\<lambda>_. 10)"
assms_2: shows "m\<^sub>2 $$ k = None"
finsum_insert: fixes F :: "'b set"   and a :: "'b"   and f :: "'b \<Rightarrow> 'a vec" assumes "finite F"   and "a \<notin> F"   and "f \<in> F \<rightarrow> carrier_vec n"   and "f a \<in> carrier_vec n" shows "finsum V f (insert a F) = f a + finsum V f F"
ball_Un: fixes A :: "'d set"   and B :: "'d set"   and P :: "'d \<Rightarrow> bool" shows "(\<forall>x\<in>A \<union> B. P x) = ((\<forall>x\<in>A. P x) \<and> (\<forall>x\<in>B. P x))"
budan_fourier_le: fixes p :: "real poly"   and b :: "real" assumes "p \<noteq> 0" shows "int (proots_count p {x. x \<le> b}) \<le> changes_le_der b p \<and> even (changes_le_der b p - int (proots_count p {x. x \<le> b}))"
s2: shows "bounded (range (snd \<circ> f \<circ> r1))"
mod_mult_right_eq: fixes divide :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and times :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and one :: "'e"   and modulo :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and euclidean_size :: "'e \<Rightarrow> nat"   and a :: "'e"   and b :: "'e"   and c :: "'e" assumes "class.euclidean_semiring_cancel divide plus minus zero times one modulo euclidean_size" shows "modulo (times a (modulo b c)) c = modulo (times a b) c"
fmsubset_restrict_fset: fixes S :: "'a fset"   and m :: "('a, 'b) fmap" shows "fmrestrict_fset S m \<subseteq>\<^sub>f m"
split_block_1: fixes A :: "'a mat"   and sr1 :: "nat"   and sc1 :: "nat"   and A1 :: "'a mat"   and A2 :: "'a mat"   and A3 :: "'a mat"   and A4 :: "'a mat"   and sr2 :: "nat"   and sc2 :: "nat" assumes "split_block A sr1 sc1 = (A1, A2, A3, A4)"   and "dim_row A = sr1 + sr2"   and "dim_col A = sc1 + sc2" shows "A1 \<in> carrier_mat sr1 sc1"
new_root_def: fixes v0 :: "'v"   and s :: "('v, 'es) state_scheme" shows "new_root v0 s \<equiv> let c = Suc (counter s); d = discovered s(v0 \<mapsto> counter s); p = {v0} \<times> E `` {v0}; st = [v0] in s\<lparr>counter := c, discovered := d, pending := p, stack := st\<rparr>"
write_reg_low_equal: fixes s1 :: "'b sparc_state"   and s2 :: "'b sparc_state"   and t1 :: "'b sparc_state"   and w :: "word32"   and curr_win :: "'b word"   and rd :: "word5"   and t2 :: "'b sparc_state" assumes "low_equal s1 s2 \<and> t1 = snd (fst (write_reg w curr_win rd s1)) \<and> t2 = snd (fst (write_reg w curr_win rd s2))" shows "low_equal t1 t2"
restrict_def: fixes ord :: "WS1S_Prelim.order"   and P :: "'a fset" shows "restrict ord P = (case ord of FO \<Rightarrow> \<exists>x. P = {|x|} | SO \<Rightarrow> True)"
InvariantQCharacterization_def: fixes conflictFlag :: "bool"   and Q :: "Literal list"   and F :: "Literal list list"   and M :: "(Literal \<times> bool) list" shows "InvariantQCharacterization conflictFlag Q F M \<equiv> \<not> conflictFlag \<longrightarrow> (\<forall>l. l el Q = (\<exists>c. c el F \<and> isUnitClause c l (elements M)))"
take_bit_eq_mask: 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 push_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and drop_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and take_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and "and" :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and or :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and xor :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and mask :: "nat \<Rightarrow> 'b"   and n :: "nat"   and a :: "'b" assumes "class.semiring_bit_operations plus minus zero times one divide modulo bit push_bit drop_bit take_bit and or xor mask" shows "take_bit n a = and a (mask n)"
simps_2: fixes ab :: "nat"   and bb :: "nat"   and n :: "nat" shows "mv_boxes ab bb (Suc n) = mv_boxes ab bb n [+] mv_box (ab + n) (bb + n)"
cont2monofunE: fixes f :: "'e \<Rightarrow> 'f"   and x :: "'e"   and y :: "'e" assumes "cont f"   and "x \<sqsubseteq> y" shows "f x \<sqsubseteq> f y"
ex_post_efficient_sds_def: fixes agents :: "'a set"   and alts :: "'b set"   and sds :: "('a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'b pmf" shows "ex_post_efficient_sds agents alts sds \<equiv> social_decision_scheme agents alts sds \<and> ex_post_efficient_sds_axioms agents alts sds"
concat_conv_foldr: fixes xss :: "'b list list" shows "concat xss = foldr (\<cdot>) xss \<epsilon>"
preservation: shows "rel_preserves_binary_pred Rel (\<lambda>a. hasBarb a (STCalWB SWB TWB))"
preserves_arr: fixes f :: "'a" assumes "\<phi>.A.arr f" shows "\<phi>.B.arr (G f)"
L4_10: fixes \<phi> :: "'b ltlr"   and qs :: "'b node set"   and \<xi> :: "nat \<Rightarrow> 'b set" assumes "create_gba_from_nodes_precond \<phi> qs"   and "\<xi> \<Turnstile>\<^sub>r \<phi>" shows "gba.accept (create_gba_from_nodes \<phi> qs) \<xi>"
wf\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t_def: fixes \<theta> :: "'c \<Rightarrow> ('d, 'c) Term.term" shows "wf\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<theta> \<equiv> subst_domain \<theta> \<inter> range_vars \<theta> = {} \<and> finite (subst_domain \<theta>)"
type_wf_impl: shows "type_wf = DocumentClass.type_wf"
encode2_alter: fixes dims :: "nat list"   and A :: "nat set"   and vars :: "nat set"   and i :: "nat" assumes "{0..<length dims} \<subseteq> A" shows "partial_state.encode2 dims vars i = digit_decode (partial_state.dims2 dims vars) (nths (digit_encode dims i) (A - vars))"
seq_app_tail: fixes x :: "'a"   and s :: "nat \<Rightarrow> 'a" shows "tail (x ## s) = s"
ext_inject: fixes break_status :: "bool"   and return_status :: "bool"   and more :: "'b"   and break_status' :: "bool"   and return_status' :: "bool"   and more' :: "'b" shows "(\<lparr>break_status = break_status, return_status = return_status, \<dots> = more\<rparr> = \<lparr>break_status = break_status', return_status = return_status', \<dots> = more'\<rparr>) = (break_status = break_status' \<and> return_status = return_status' \<and> more = more')"
max_enat_minus1: fixes n :: "enat"   and m :: "enat"   and k :: "enat" shows "max n m - k = max (n - k) (m - k)"
find_works_not_None: fixes a :: "'c"   and xs :: "('c, 'd) bintree option list" assumes "Some a \<in> set (map (map_option priority) xs)" shows "Binomial_Queue.find a xs \<noteq> None"
norm_CBasis: fixes u :: "'a" assumes "u \<in> CBasis" shows "norm u = 1"
convex_on_dist: fixes S :: "'b set"   and a :: "'b" shows "convex_on S (dist a)"
z: shows "\<forall>y. O y z = (\<exists>x. (x = a \<or> x = b) \<and> O y x)"
that: assumes "\<And>w. \<lbrakk>u = v \<or> w \<noteq> u \<and> w \<noteq> v; w \<in> verts G\<rbrakk> \<Longrightarrow> arc_balance w p = 0"   and "u \<noteq> v \<Longrightarrow> arc_balance u p = - 1"   and "u \<noteq> v \<Longrightarrow> arc_balance v p = 1" shows "thesis"
Q_\<alpha>_def: fixes Qi :: "'q" shows "Q_\<alpha> Qi \<equiv> enat_of_option \<circ> Q_lookup Qi"
f_Exec_Stream_Init_Cons: fixes trans_fun :: "'e \<Rightarrow> 'd \<Rightarrow> 'd"   and x :: "'e"   and xs :: "'e list"   and c :: "'d" shows "f_Exec_Comp_Stream_Init trans_fun (x # xs) c = c # f_Exec_Comp_Stream_Init trans_fun xs (trans_fun x c)"
bet__lt2313: fixes A :: "'p"   and B :: "'p"   and C :: "'p" assumes "A \<noteq> B"   and "Bet A B C" shows "B C Lt A C"
permutes_univ: fixes p :: "'d \<Rightarrow> 'd" shows "(p permutes UNIV) = (\<forall>y. \<exists>!x. p x = y)"
closed_orbit_period_nonneg: fixes x :: "'a" assumes "closed_orbit x" shows "0 \<le> period x"
homomorphism: fixes f :: "'e \<Rightarrow> 'd"   and x :: "'e" shows "Applicative.ap (return_spmf f) (return_spmf x) = return_spmf (f x)"
pick_def: fixes X :: "'b set" shows "pick X \<equiv> SOME x. x \<in> X"
lexordp_eq_simps_1: fixes less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and ys :: "'d list" shows "ord.lexordp_eq less [] ys = True"
homomorphism: fixes f :: "'e \<Rightarrow> 'd"   and x :: "'e" shows "pure f <*> pure x = pure (f x)"
eq: shows "Set.filter (\<lambda>(x, y). x\<^sup>2 + y\<^sup>2 = p) (SIGMA x:{0..p}. {x..p}) = {(x, y). x \<le> y \<and> x\<^sup>2 + y\<^sup>2 = p}"
induct: fixes \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and x1 :: "'c rule list"   and x2 :: "state"   and P :: "'c rule list \<Rightarrow> state \<Rightarrow> bool" assumes "\<Gamma>,\<gamma>,p\<turnstile> x1 \<Rightarrow>\<^sub>z x2"   and "P [] Undecided"   and "\<And>m rs. matches \<gamma> m p \<Longrightarrow> P (Rule m Accept # rs) (Decision FinalAllow)"   and "\<And>m rs. matches \<gamma> m p \<Longrightarrow> P (Rule m Drop # rs) (Decision FinalDeny)"   and "\<And>m rs. matches \<gamma> m p \<Longrightarrow> P (Rule m Reject # rs) (Decision FinalDeny)"   and "\<And>m rs t. \<lbrakk>matches \<gamma> m p; \<Gamma>,\<gamma>,p\<turnstile> rs \<Rightarrow>\<^sub>z t; P rs t\<rbrakk> \<Longrightarrow> P (Rule m Log # rs) t"   and "\<And>m rs t. \<lbrakk>matches \<gamma> m p; \<Gamma>,\<gamma>,p\<turnstile> rs \<Rightarrow>\<^sub>z t; P rs t\<rbrakk> \<Longrightarrow> P (Rule m Empty # rs) t"   and "\<And>m rs t a. \<lbrakk>\<not> matches \<gamma> m p; \<Gamma>,\<gamma>,p\<turnstile> rs \<Rightarrow>\<^sub>z t; P rs t\<rbrakk> \<Longrightarrow> P (Rule m a # rs) t"   and "\<And>m chain rs\<^sub>1 m' rs\<^sub>2 rrs t. \<lbrakk>matches \<gamma> m p; \<Gamma> chain = Some (rs\<^sub>1 @ Rule m' Return # rs\<^sub>2); matches \<gamma> m' p; \<Gamma>,\<gamma>,p\<turnstile> rs\<^sub>1 \<Rightarrow>\<^sub>z Undecided; P rs\<^sub>1 Undecided; \<Gamma>,\<gamma>,p\<turnstile> rrs \<Rightarrow>\<^sub>z t; P rrs t\<rbrakk> \<Longrightarrow> P (Rule m (Call chain) # rrs) t"   and "\<And>m chain rs X rrs. \<lbrakk>matches \<gamma> m p; \<Gamma> chain = Some rs; \<Gamma>,\<gamma>,p\<turnstile> rs \<Rightarrow>\<^sub>z Decision X; P rs (Decision X)\<rbrakk> \<Longrightarrow> P (Rule m (Call chain) # rrs) (Decision X)"   and "\<And>m chain rs rrs t. \<lbrakk>matches \<gamma> m p; \<Gamma> chain = Some rs; \<Gamma>,\<gamma>,p\<turnstile> rs \<Rightarrow>\<^sub>z Undecided; P rs Undecided; \<Gamma>,\<gamma>,p\<turnstile> rrs \<Rightarrow>\<^sub>z t; P rrs t\<rbrakk> \<Longrightarrow> P (Rule m (Call chain) # rrs) t" shows "P x1 x2"
inject: fixes x2 :: "final_decision"   and y2 :: "final_decision" shows "(Decision x2 = Decision y2) = (x2 = y2)"
2: shows "[\<^bold>\<diamond>\<^bold>\<not>(\<^bold>\<forall>\<alpha>. \<^bold>\<not>\<phi> \<alpha>) \<^bold>\<rightarrow> \<^bold>\<not>(\<^bold>\<forall>\<alpha>. \<^bold>\<box>\<^bold>\<not>\<phi> \<alpha>) in v]"
pmf_cond_bind_pmf: fixes x :: "'a" shows "pmf local.cond_bind_pmf x = F x"
g_def: shows "g \<equiv> \<lambda>w. f w / w ^ Suc n"
add_zeror: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and zero :: "'c"   and x :: "'c" assumes "class.ab_near_semiring_one_zerol plus times one zero" shows "plus x zero = x"
advantage_def: fixes \<A> :: "(bool, bool list \<times> bool list + bool list \<times> bool list \<times> 'hash, (bool list \<times> bool list \<times> 'hash) option + bool list option) gpv" shows "local.ind_cca.advantage \<A> = \<bar>spmf (local.ind_cca.game \<A>) True - 1 / 2\<bar>"
i: shows "i < dim_row (submatrix A I UNIV * submatrix B UNIV J)"
d1: assumes "prod_list (map ((!) sign) xa) = - 1" shows "poly (prod_list (retrieve_polys qs xa)) x < 0"
UINF_as_Sup_collect': fixes f :: "'d \<Rightarrow> ('b, 'c) uexpr" shows "(\<Sqinter> P \<bullet> f P) = \<Sqinter>range f"
eq: shows "filter_mset ((\<subseteq>) ps) {#bl - {p}. bl \<in># \<B>#} = {#b - {p}. b \<in># {#b \<in># \<B>. ps \<subseteq> b - {p}#}#}"
list_of_vec_0: fixes n :: "nat" shows "list_of_vec (0\<^sub>v n) = replicate n (0::'a)"
one_aezfun_transfer: shows "(1::'a) \<delta>\<delta> (0::'b) = 1"
measure_Un3: fixes A :: "'c set"   and M :: "'c measure"   and B :: "'c set" assumes "A \<in> fmeasurable M"   and "B \<in> fmeasurable M" shows "Sigma_Algebra.measure M (A \<union> B) = Sigma_Algebra.measure M A + Sigma_Algebra.measure M B - Sigma_Algebra.measure M (A \<inter> B)"
homf_into_set: fixes A :: "'o" shows "Functor Hom(A,_) : AA \<longrightarrow> Set"
measurable_PiM_single': fixes I :: "'b set"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> 'd"   and N :: "'c measure"   and M :: "'b \<Rightarrow> 'd measure" assumes "\<And>i. i \<in> I \<Longrightarrow> f i \<in> N \<rightarrow>\<^sub>M M i"   and "(\<lambda>\<omega> i. f i \<omega>) \<in> space N \<rightarrow> (\<Pi>\<^sub>E i\<in>I. space (M i))" shows "(\<lambda>\<omega> i. f i \<omega>) \<in> N \<rightarrow>\<^sub>M Pi\<^sub>M I M"
Let_def: fixes s :: "'m"   and f :: "'m \<Rightarrow> 'l" shows "Let s f \<equiv> f s"
sifum_security_init_axioms: shows "sifum_security_init dma\<^sub>C \<C>_vars\<^sub>C eval\<^sub>C"
glbound_in_ofD_lbound: fixes P :: "'a set"   and x :: "'a"   and y :: "'a"   and b :: "'a" assumes "glbound_in_of P x y b" shows "lbound_of x y b"
prems: shows "set (x # Xs) \<subseteq> carrier_vec n"
emeasure_measure_pmf_finite: fixes S :: "'a set"   and M :: "'a pmf" assumes "finite S" shows "emeasure (measure_pmf M) S = ennreal (sum (pmf M) S)"
is_extensional: fixes f :: "'a \<times> 'a" assumes "\<not> B.VV.arr f" shows "cmp\<^sub>U\<^sub>P f = null"
E_bipartite_web: fixes \<Gamma> :: "('a, 'b) web_scheme" shows "\<^bold>E\<^bsub>bipartite_web_of \<Gamma>\<^esub> = (\<lambda>(x, y). (Inl x, Inr y)) ` \<^bold>E\<^bsub>\<Gamma>\<^esub> \<union> (\<lambda>x. (Inl x, Inr x)) ` (\<^bold>V\<^bsub>\<Gamma>\<^esub> - A \<Gamma> - B \<Gamma>)"
targets_dom: fixes \<mu> :: "'a" assumes "arr \<mu>" shows "targets (local.dom \<mu>) = targets \<mu>"
bangPres: fixes P :: "pi"   and Q :: "pi"   and Rel :: "(pi \<times> pi) set"   and Rel' :: "(pi \<times> pi) set" assumes "(P, Q) \<in> Rel"   and "\<And>R S. (R, S) \<in> Rel \<Longrightarrow> R \<leadsto><Rel> S"   and "\<And>R S T U. \<lbrakk>(R, S) \<in> Rel; (T, U) \<in> Rel'\<rbrakk> \<Longrightarrow> (R \<parallel> T, S \<parallel> U) \<in> Rel'"   and "\<And>R S x. (R, S) \<in> Rel' \<Longrightarrow> (<\<nu>x>R, <\<nu>x>S) \<in> Rel'"   and "\<And>R S. (R \<parallel> !R, S) \<in> Rel' \<Longrightarrow> (!R, S) \<in> Rel'"   and "bangRel Rel \<subseteq> Rel'"   and "eqvt Rel'" shows "!P \<leadsto><Rel'> !Q"