hcju/isabelleps huggingface dataset:
list_all2_update_cong: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'a list"   and ys :: "'b list"   and x :: "'a"   and y :: "'b"   and i :: "nat" assumes "list_all2 P xs ys"   and "P x y" shows "list_all2 P (xs[i := x]) (ys[i := y])"
sorted_map_same: fixes f :: "'c \<Rightarrow> 'a"   and g :: "'c list \<Rightarrow> 'a"   and xs :: "'c list" shows "sorted (map f (filter (\<lambda>x. f x = g xs) xs))"
simps_1: fixes w :: "nat"   and a :: "'a" shows "alphabet (Leaf w a) = {a}"
euclid: fixes x :: "nat"   and y :: "nat" shows "rdom \<lceil>\<lambda>s. s ''x'' = x \<and> s ''y'' = y\<rceil> \<subseteq> wp (WHILE (\<lambda>s. s ''y'' \<noteq> zero_class.zero) INV (\<lambda>s. gcd (s ''x'') (s ''y'') = gcd x y) DO (''z'' ::= (\<lambda>s. s ''y'')) ; (''y'' ::= (\<lambda>s. s ''x'' mod s ''y'')) ; (''x'' ::= (\<lambda>s. s ''z'')) OD) \<lceil>\<lambda>s. s ''x'' = gcd x y\<rceil>"
simps_5: fixes dims :: "nat list"   and M :: "nat \<Rightarrow> complex mat"   and S :: "com"   and P :: "complex mat" shows "state_sig.wlp dims (While M S) P = state_sig.wlp_while dims (M 0) (M 1) (state_sig.wlp dims S) P"
hyps_3: shows "tgtOf (last tr_) = srcOf trn_"
positive_Qk: fixes l :: "nat" shows "positive (tensor_P (proj_psi_l l) (proj_k l))"
goal_of_def: shows "goal_of \<equiv> id \<circ> Record.iso_tuple_fst Record.tuple_iso_tuple \<circ> Record.iso_tuple_snd Record.tuple_iso_tuple \<circ> Record.iso_tuple_snd Record.tuple_iso_tuple \<circ> Record.iso_tuple_fst sas_plus_problem_ext_Tuple_Iso"
simps_15: fixes x11 :: "char list"   and x12 :: "'a list" shows "set_literal (Pos x11 x12) = set x12"
gfun_mem_iff: fixes f :: "(val \<times> val) fset"   and x :: "nat"   and e :: "exp"   and \<rho> :: "(nat \<times> bval) list" shows "good_fun f x e \<rho> = (\<forall>v1 v2. (v1, v2) \<in> fset f \<longrightarrow> (\<forall>v\<in>good v1. \<exists>v'. (x, v) # \<rho> \<turnstile> e \<Down> v' \<and> v' \<in> good v2))"
secure_def: fixes P :: "'b process"   and I :: "('c \<times> 'c) set"   and D :: "'b \<Rightarrow> 'c" shows "secure P I D \<equiv> \<forall>xs y ys Y zs Z. (y # ys, Y) \<in> futures P xs \<and> (zs, Z) \<in> futures P xs \<longrightarrow> (ipurge_tr I D (D y) ys, ipurge_ref I D (D y) ys Y) \<in> futures P xs \<and> (y # ipurge_tr I D (D y) zs, ipurge_ref I D (D y) zs Z) \<in> futures P xs"
simps_3: fixes m :: "nat"   and n :: "nat" shows "wellf (recf.Id m n) = (n < m)"
strict_coboundedI2: fixes b :: "'b multiset"   and c :: "'b multiset"   and a :: "'b multiset" assumes "b \<subset># c" shows "a \<inter># b \<subset># c"
select_convs_2: fixes Obj :: "'e set"   and Mor :: "'d set"   and Dom :: "'d \<Rightarrow> 'e"   and Cod :: "'d \<Rightarrow> 'e"   and Id :: "'e \<Rightarrow> 'd"   and Comp :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and more :: "'f" shows "mor\<^bsub>\<lparr>Obj = Obj, Mor = Mor, Dom = Dom, Cod = Cod, Id = Id, Comp = Comp, \<dots> = more\<rparr>\<^esub> = Mor"
simps_3: fixes t :: "nat"   and sop :: "nat set \<times> ((nat \<Rightarrow> nat option) \<Rightarrow> nat)" shows "issue_expr t (Tmp sop) = []"
Synchronized1Red1: fixes uf :: "bool"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and o' :: "'addr expr1"   and s :: "'heap \<times> 'addr val list"   and ta :: "('addr, 'thread_id, 'heap) external_thread_action"   and o'' :: "'addr expr1"   and s' :: "'heap \<times> 'addr val list"   and V :: "nat"   and e :: "'addr expr1" assumes "uf,P,t \<turnstile>1 \<langle>o',s\<rangle> -ta\<rightarrow> \<langle>o'',s'\<rangle>" shows "uf,P,t \<turnstile>1 \<langle>sync\<^bsub>V\<^esub> (o') e,s\<rangle> -ta\<rightarrow> \<langle>sync\<^bsub>V\<^esub> (o'') e,s'\<rangle>"
parAssocRight: fixes Rel :: "(ccs \<times> ccs) set"   and P :: "ccs"   and Q :: "ccs"   and R :: "ccs" assumes "\<And>S T U. (S \<parallel> (T \<parallel> U), S \<parallel> T \<parallel> U) \<in> Rel" shows "P \<parallel> (Q \<parallel> R) \<leadsto>[Rel] P \<parallel> Q \<parallel> R"
left_neutral: fixes a :: "'e" shows "(1::'e) \<cdot> a = a"
jviewIncr: shows "\<forall>a t s. jview a (t \<leadsto> s) = jviewIncr a (envObs a s) (jview a t)"
f3: shows "Deliver (i, Append i e) \<notin> set pre"
cet_minus_int: fixes \<Gamma> :: "nat \<Rightarrow> pdf_type"   and e :: "cexpr" assumes "\<Gamma> \<turnstile>\<^sub>c e : INTEG" shows "\<Gamma> \<turnstile>\<^sub>c -\<^sub>c e : INTEG"
maximal_ideal_ideal: fixes R :: "('e, 'f) Ring_scheme"   and mx :: "'e set" assumes "Ring R"   and "maximal_ideal R mx" shows "ideal R mx"
u_in_Tokens_at: shows "u \<in> TokensAt k (Gen P)"
V_i_def: fixes i :: "nat"   and V :: "'b set"   and E :: "('b \<times> 'b) set"   and M :: "('b \<times> 'b) set"   and L :: "'b \<Rightarrow> nat" shows "V_i i V E M L = \<Union> (edge_as_set ` matching_i i V E M L)"
lookup_def: fixes rep_key_order :: "'a \<Rightarrow> 'b key_order"   and list_of_oalist :: "'c \<Rightarrow> ('b \<times> 'd) list \<times> 'a"   and oalist_of_list :: "('b \<times> 'd) list \<times> 'a \<Rightarrow> 'c"   and xs :: "'c" assumes "oalist_abstract rep_key_order list_of_oalist oalist_of_list" shows "oalist_abstract.lookup rep_key_order list_of_oalist xs = oalist_raw.lookup_raw rep_key_order (list_of_oalist xs)"
multiplicity_same_power': 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 unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and p :: "'b"   and n :: "nat" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize" shows "factorial_semiring.multiplicity times one p (power.power one times p n) = (if p = zero \<or> dvd.dvd times p one then 0 else n)"
thetaParWL2_Wbis: shows "thetaParWL2 \<subseteq> Wbis"
leftderives_\<alpha>: shows "leftderives \<alpha> (terminals (drop r p @ [X]))"
FOREACH_body_def: fixes f :: "'c \<Rightarrow> 'd \<Rightarrow> 'e nres" shows "FOREACH_body f \<equiv> \<lambda>(xs, \<sigma>). let x = hd xs in f x \<sigma> \<bind> (\<lambda>\<sigma>'. RETURN (tl xs, \<sigma>'))"
lists_eq_set: fixes A :: "'b set" shows "lists A = {xs. order.greater_eq A (set xs)}"
SeqSubstFormP_Neg: fixes s :: "name"   and k :: "name"   and s1 :: "tm"   and k1 :: "tm"   and x :: "tm"   and x' :: "tm"   and v :: "tm"   and i :: "tm" assumes "atom s \<sharp> (k, s1, k1, x, x', v, i)"   and "atom k \<sharp> (s1, k1, x, x', v, i)" shows "{SeqSubstFormP v i x x' s1 k1, TermP i, VarP v} \<turnstile> SyntaxN.Ex s (SyntaxN.Ex k (SeqSubstFormP v i (Q_Neg x) (Q_Neg x') (Var s) (Var k)))"
v_def: shows "v = (\<lambda>t. np f t + np g t)"
assms_1: shows "\<forall>i xs. i \<le> n \<and> set xs \<subseteq> {0..n} \<longrightarrow> (0::'a) \<le> len m i i xs"
elims_2: fixes x :: "cmd"   and xa :: "w_node"   and xb :: "cmd" assumes "labels_nodes x xa xb" obtains prog :: "cmd" and l :: "nat" and c :: "cmd" where "x = prog" and "xa = (_ l _)" and "xb = c" and "labels prog l c"
swap_simps_3: fixes a :: "NominalSequents.var"   and c :: "NominalSequents.var"   and b :: "NominalSequents.var" assumes "a \<noteq> c"   and "b \<noteq> c" shows "[(a, b)] \<bullet> c = c"
l_adopt_node\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms: shows "l_adopt_node\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_owner_document get_parent get_parent_locs remove_child remove_child_locs get_disconnected_nodes get_disconnected_nodes_locs set_disconnected_nodes set_disconnected_nodes_locs get_ancestors_di get_ancestors_di_locs adopt_node adopt_node_locs adopt_node\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M adopt_node_locs\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M known_ptr type_wf get_child_nodes get_child_nodes_locs known_ptrs set_child_nodes set_child_nodes_locs get_host get_host_locs get_disconnected_document get_disconnected_document_locs remove"
cmp_pol_coeff_e: fixes A :: "('g, 'h) Ring_scheme"   and B :: "('g, 'i) Ring_scheme"   and Y :: "'g"   and f :: "'a \<Rightarrow> 'g"   and n :: "nat"   and c :: "nat \<Rightarrow> 'a" assumes "PolynRg A B Y"   and "f \<in> pHom R S X, A B Y"   and "pol_coeff S (n, c)" shows "pol_coeff B (n, cmp f c)"
conditionally_complete_imp_nonempty_imp_ex_extreme_bound_iff_ex_bound: fixes A :: "'b set"   and r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and X :: "'b set" assumes "conditionally_complete A r"   and "X \<subseteq> A"   and "X \<noteq> {}" shows "Ex (extreme_bound A r X) = (\<exists>b\<in>A. bound X r b)"
\<G>_Fset_def: fixes CC :: "'a literal multiset set" shows "\<G>_Fset CC = \<Union> (\<G>_F ` CC)"
Rep_pfun_inverse: fixes x :: "('a, 'b) pfun" shows "Abs_pfun (Rep_pfun x) = x"
distinct_list_update: fixes xs :: "'a list"   and a :: "'a"   and i :: "nat" assumes "distinct xs"   and "a \<notin> set xs - {xs ! i}" shows "distinct (xs[i := a])"
validE_NF_def: fixes P :: "'c \<Rightarrow> bool"   and f :: "'c \<Rightarrow> (('d + 'e) \<times> 'c) set \<times> bool"   and Q :: "'e \<Rightarrow> 'c \<Rightarrow> bool"   and E :: "'d \<Rightarrow> 'c \<Rightarrow> bool" shows "\<lbrace>P\<rbrace> f \<lbrace>Q\<rbrace>, \<lbrace>E\<rbrace>! \<equiv> \<lbrace>P\<rbrace> f \<lbrace>Q\<rbrace>, \<lbrace>E\<rbrace> \<and> no_fail P f"
VgeqBV: shows "BVG (the (snd (usubstappp \<sigma> (fst (usubstappp \<sigma> U \<alpha>)) \<alpha>))) \<subseteq> fst (usubstappp \<sigma> U \<alpha>)"
complete_direct_successors: fixes \<Gamma> :: "'c constr list"   and n :: "nat"   and \<Psi> :: "'c TESL_atomic list"   and \<Phi> :: "'c TESL_atomic list" shows "\<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<subseteq> \<Union> (HeronConf_interpretation ` \<C>\<^sub>n\<^sub>e\<^sub>x\<^sub>t \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi>)"
hom_sum: fixes f :: "'c \<Rightarrow> 'a poly poly"   and X :: "'c set" shows "map_poly (map_poly hom) (sum f X) = (\<Sum>x\<in>X. map_poly (map_poly hom) (f x))"
SKIP_def: shows "SKIP = [: id (\<lambda>x. id ((=) x)) :]"
simps: fixes a :: "'c ltl_prop_equiv_quotient" shows "sat_models a = {A. A \<Turnstile>\<^sub>P Rep a}"
filter_min_aux_distinct: fixes ys :: "'a list"   and xs :: "'a list" assumes "reflp rel"   and "distinct ys" shows "distinct (local.filter_min_aux xs ys)"
i_par_i: shows "(i::'a::refinement_lattice) \<parallel> i \<sqsubseteq> i"
\<eta>_in_terms_of_\<phi>: fixes y :: "'b" assumes "Adj.D.ide y" shows "Adj.\<eta> y = \<phi> y (F y)"
d\<^sub>y: shows "profile_single d\<^sub>y"
emeasure_restrict_space: fixes \<Omega> :: "'d set"   and M :: "'d measure"   and A :: "'d set" assumes "\<Omega> \<inter> space M \<in> sets M"   and "A \<subseteq> \<Omega>" shows "emeasure (restrict_space M \<Omega>) A = emeasure M A"
ordLeq_ordLess_trans: fixes r :: "('c \<times> 'c) set"   and r' :: "('d \<times> 'd) set"   and r'' :: "('e \<times> 'e) set" assumes "r \<le>o r'"   and "r' <o r''" shows "r <o r''"
eventually_nhds_in_open: fixes "open" :: "'b set \<Rightarrow> bool"   and s :: "'b set"   and x :: "'b" assumes "class.topological_space open"   and "open s"   and "x \<in> s" shows "\<forall>\<^sub>F y in topological_space.nhds open x. y \<in> s"
right_cancel: fixes b :: "'b perm"   and a :: "'b perm"   and c :: "'b perm" shows "(b * a = c * a) = (b = c)"
is_sup_join: fixes x :: "'c"   and y :: "'c" shows "is_sup x y (x \<squnion> y)"
suffix_ConsD: fixes x :: "'a"   and xs :: "'a list"   and ys :: "'a list" assumes "suffix (x # xs) ys" shows "suffix xs ys"
type_definition_coreflexive: shows "type_definition Rep_coreflexive Abs_coreflexive (\<down>(1::'b))"
x0: shows "real_of_1 (p, l, r) \<noteq> 0"
vectorize_mult_scalar: fixes p :: "'a \<Rightarrow>\<^sub>0 'c"   and q :: "'t \<Rightarrow>\<^sub>0 'c" shows "vectorize_poly (p \<odot> q) = MPoly_Type_Class.punit.monom_mult p 0 (vectorize_poly q)"
rows_to_keep_def: fixes A :: "'a mat" shows "rows_to_keep A = map snd (pivot_positions (gauss_jordan_single A\<^sup>T))"
green_typeII_chain_axioms: shows "green_typeII_chain i j F two_chain_typeII s"
inverse_equality: fixes u :: "'a"   and v :: "'a" assumes "u \<cdot> v = \<one>"   and "v \<cdot> u = \<one>"   and "u \<in> N"   and "v \<in> N" shows "sub.inverse u = v"
weakCongSimStatEq: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Rel :: "('b \<times> ('a, 'b, 'c) psi \<times> ('a, 'b, 'c) psi) set"   and Q :: "('a, 'b, 'c) psi"   and \<Psi>' :: "'b"   and Rel' :: "('b \<times> ('a, 'b, 'c) psi \<times> ('a, 'b, 'c) psi) set" assumes "\<Psi> \<rhd> P \<leadsto>\<guillemotleft>Rel\<guillemotright> Q"   and "\<Psi> \<simeq> \<Psi>'"   and "\<And>\<Psi> P Q \<Psi>'. \<lbrakk>(\<Psi>, P, Q) \<in> Rel; \<Psi> \<simeq> \<Psi>'\<rbrakk> \<Longrightarrow> (\<Psi>', P, Q) \<in> Rel'" shows "\<Psi>' \<rhd> P \<leadsto>\<guillemotleft>Rel'\<guillemotright> Q"
transT_MtransT: fixes tval :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and aval :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and cf :: "('a, 'c) com \<times> 'b"   and s' :: "'b" assumes "PL.transT tval aval cf s'" shows "PL.MtransT tval aval cf s'"
intros_3: fixes t1 :: "lam"   and t2 :: "lam"   and t3 :: "lam" assumes " t1 \<longrightarrow>\<^sub>\<beta>\<^sup>* t2"   and " t2 \<longrightarrow>\<^sub>\<beta>\<^sup>* t3" shows " t1 \<longrightarrow>\<^sub>\<beta>\<^sup>* t3"
ord_Zp_not_equal_ord_plus_minus: fixes x :: "nat \<Rightarrow> int"   and y :: "nat \<Rightarrow> int" assumes "x \<in> nonzero Zp"   and "y \<in> nonzero Zp"   and "ord_Zp x \<noteq> ord_Zp y" shows "ord_Zp (x \<ominus> y) = ord_Zp (x \<oplus> y)"
f1: shows "(r \<otimes> t \<otimes> v |\<^bsub>rel\<^esub> s \<otimes> u \<otimes> w) = (r \<otimes> (t \<otimes> v) |\<^bsub>rel\<^esub> s \<otimes> (u \<otimes> w))"
is_nan_divide: fixes x :: "('a, 'b) IEEE.float"   and y :: "('a, 'b) IEEE.float" assumes "IEEE.is_nan x \<or> IEEE.is_nan y" shows "IEEE.is_nan (x / y)"
assms_3: shows "c' \<equiv> complex_of_real (root n (cmod c)) * cis (arg c / real n)"
below_eq_trans: fixes below :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c"   and b :: "'c"   and c :: "'c" assumes "below a b"   and "b = c" shows "below a c"
double_dual_iso_def: fixes x :: "('a \<Rightarrow> complex) \<Rightarrow> complex" shows "G''.double_dual_iso x = (\<lambda>\<chi>. if character (Characters (Characters G)) \<chi> then \<chi> x else 0)"
fMax_finsert: fixes x :: "'m"   and A :: "'m fset" shows "fMax (finsert x A) = (if A = {||} then x else max x (fMax A))"
HPhase0Read_HInv4d_p: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and d :: "Disk" assumes "HPhase0Read s s' p d"   and "HInv4d s p" shows "HInv4d s' p"
dg_Rel_components_2: fixes \<alpha> :: "V" shows "dg_Rel \<alpha>\<lparr>Arr\<rparr> = ZFC_in_HOL.set {T. arr_Rel \<alpha> T}"
frechet_derivative_compose_eucl: fixes g :: "'b \<Rightarrow> 'c"   and x :: "'b"   and f :: "'c \<Rightarrow> 'd" assumes "g differentiable at x"   and "f differentiable at (g x)" shows "frechet_derivative (f \<circ> g) (at x) = (\<lambda>v. \<Sum>i\<in>Basis. (frechet_derivative g (at x) v \<bullet> i) *\<^sub>R frechet_derivative f (at (g x)) i)"
V_Least: fixes l :: "nat" shows "V (LEAST la. V la = V l) = V l"
edom_env: fixes S :: "'d set"   and m :: "'d \<Rightarrow> 'e" shows "edom (m f|` S) = edom m \<inter> S"
truncate_up_nonneg: fixes x :: "real"   and p :: "nat" assumes "0 \<le> x" shows "0 \<le> truncate_up p x"
a: shows "\<not> (\<exists>e\<in>lotteries_on out. \<exists>x\<in>out. pmf e x = 1)"
NI_altc_def: shows "NI_altc \<equiv> cset_nondetM_base.altc_nondet merge_id"
reduce_def: fixes f :: "nat \<Rightarrow> real"   and P :: "nat set" shows "reduce f P = f ` (P - (kmax f P \<union> kmin f P))"
valid_selector_def: fixes Rs :: "(('c, 'd) labeled_graph \<times> ('c, 'd) labeled_graph) set"   and selector :: "nat \<times> ('c \<times> nat \<times> nat) set \<Rightarrow> ((('c, 'd) labeled_graph \<times> ('c, 'd) labeled_graph) \<times> ('d \<times> nat) set) option" shows "valid_selector Rs selector \<equiv> \<forall>G. (worklist G Rs \<noteq> {} \<longrightarrow> (\<exists>(R, f)\<in>UNIV. selector G = Some (R, f) \<and> valid_selection Rs G R f)) \<and> (worklist G Rs = {} \<longrightarrow> selector G = None)"
tauLaw1SimLeft: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Rel :: "('b \<times> ('a, 'b, 'c) psi \<times> ('a, 'b, 'c) psi) set"   and Q :: "('a, 'b, 'c) psi" assumes "\<Psi> \<rhd> P \<leadsto><Rel> Q"   and "eqvt Rel"   and "\<And>\<Psi> P Q R. \<lbrakk>\<Psi> \<rhd> P \<sim> Q; (\<Psi>, Q, R) \<in> Rel\<rbrakk> \<Longrightarrow> (\<Psi>, P, R) \<in> Rel" shows "\<Psi> \<rhd> \<tau>.P \<leadsto><Rel> Q"
finite_lookup: fixes f :: "'a \<Rightarrow>\<^sub>0 'b" shows "finite {k. poly_mapping.lookup f k \<noteq> (0::'b)}"
truncate_up_shift_int: fixes p :: "nat"   and x :: "real"   and k :: "int" shows "truncate_up p (x * 2 powr real_of_int k) = truncate_up p x * 2 powr real_of_int k"
start_points_end_points_conv: fixes x :: "'a" shows "start_points x = end_points (x\<^sup>T)"
order_trans: fixes x :: "'b Word.word"   and y :: "'b Word.word"   and z :: "'b Word.word" assumes "x \<le>s y"   and "y \<le>s z" shows "x \<le>s z"
lossless_WT_gpv_induct: fixes \<I> :: "('d, 'e) \<I>"   and gpv :: "('f, 'd, 'e) gpv"   and P :: "('f, 'd, 'e) gpv \<Rightarrow> bool" assumes "lossless_gpv \<I> gpv"   and "\<I> \<turnstile>g gpv \<surd>"   and "\<And>p. \<lbrakk>lossless_spmf p; \<And>out c. IO out c \<in> set_spmf p \<Longrightarrow> out \<in> outs_\<I> \<I>; \<And>out c input. \<lbrakk>IO out c \<in> set_spmf p; out \<in> outs_\<I> \<I> \<Longrightarrow> input \<in> responses_\<I> \<I> out\<rbrakk> \<Longrightarrow> lossless_gpv \<I> (c input); \<And>out c input. \<lbrakk>IO out c \<in> set_spmf p; out \<in> outs_\<I> \<I> \<Longrightarrow> input \<in> responses_\<I> \<I> out\<rbrakk> \<Longrightarrow> \<I> \<turnstile>g c input \<surd>; \<And>out c input. \<lbrakk>IO out c \<in> set_spmf p; out \<in> outs_\<I> \<I> \<Longrightarrow> input \<in> responses_\<I> \<I> out\<rbrakk> \<Longrightarrow> P (c input)\<rbrakk> \<Longrightarrow> P (GPV p)" shows "P gpv"
aval_afold_N: fixes t :: "char list \<Rightarrow> int option"   and s :: "char list \<Rightarrow> int"   and a :: "aexp"   and n :: "int" assumes "approx t s"   and "afold a t = N n" shows "aval a s = n"
assms_2: shows "\<forall>x\<in>S. f x \<le> b"
assms: fixes x :: "'a"   and r :: "real" shows "compact (cball x r \<inter> S)"
limit_nonempty: fixes x :: "nat \<Rightarrow> 'e" assumes "finite (range x)" shows "\<exists>a. a \<in> limit x"
intro: fixes p :: "nat" assumes "prime p"   and "[p = 1] (mod 4)" shows "noninert_gauss_int_prime p"
P: shows "P = W * four_block_mat (1\<^sub>m 1) (0\<^sub>m 1 (n - 1)) (0\<^sub>m (n - 1) 1) P'"
n_distrib2_opp: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "!x \<cdot> !y \<oplus> !z = (!x \<oplus> !z) \<cdot> (!y \<oplus> !z)"
OclAnd_def: fixes X :: "'a state \<times> 'a state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>"   and Y :: "'a state \<times> 'a state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>" shows "X and Y \<equiv> \<lambda>\<tau>. case X \<tau> of \<bottom> \<Rightarrow> case Y \<tau> of \<bottom> \<Rightarrow> \<bottom> | \<lfloor>\<bottom>\<rfloor> \<Rightarrow> \<bottom> | \<lfloor>\<lfloor>True\<rfloor>\<rfloor> \<Rightarrow> \<bottom> | \<lfloor>\<lfloor>False\<rfloor>\<rfloor> \<Rightarrow> \<lfloor>\<lfloor>False\<rfloor>\<rfloor> | \<lfloor>\<bottom>\<rfloor> \<Rightarrow> case Y \<tau> of \<bottom> \<Rightarrow> \<bottom> | \<lfloor>\<bottom>\<rfloor> \<Rightarrow> \<lfloor>\<bottom>\<rfloor> | \<lfloor>\<lfloor>True\<rfloor>\<rfloor> \<Rightarrow> \<lfloor>\<bottom>\<rfloor> | \<lfloor>\<lfloor>False\<rfloor>\<rfloor> \<Rightarrow> \<lfloor>\<lfloor>False\<rfloor>\<rfloor> | \<lfloor>\<lfloor>True\<rfloor>\<rfloor> \<Rightarrow> Y \<tau> | \<lfloor>\<lfloor>False\<rfloor>\<rfloor> \<Rightarrow> \<lfloor>\<lfloor>False\<rfloor>\<rfloor>"
1: shows "F \<a>\<^sub>C[f, g, h] = \<Phi> (f, g \<star>\<^sub>C h) \<cdot>\<^sub>D (F f \<star>\<^sub>D \<Phi> (g, h)) \<cdot>\<^sub>D \<a>\<^sub>D[F f, F g, F h] \<cdot>\<^sub>D (D.inv (\<Phi> (f, g)) \<star>\<^sub>D F h) \<cdot>\<^sub>D D.inv (\<Phi> (f \<star>\<^sub>C g, h))"
add_le: fixes m :: "nat"   and ns :: "nat list" shows "m \<le> foldl (+) m ns"
that: fixes n :: "nat"   and m :: "nat" assumes "n \<noteq> 0"   and "\<bar>x\<bar> = real m / real n"   and "coprime m n" shows "thesis"
induces_adjunction: shows "adjunction (\<cdot>\<^sub>C) (\<cdot>\<^sub>D) replete_setcat.comp Adj'.\<phi>C Adj'.\<phi>D F G \<phi> \<psi> Adj.\<eta> Adj.\<epsilon> Adj.\<Phi> Adj.\<Psi>"
gd_nat_term_id: fixes cmp_term :: "('a \<times> 'b) nat_term_order" shows "gd_nat_term (\<lambda>x. x) (\<lambda>x. x) cmp_term"
gcd_proj1_iff: fixes one :: "'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and gcd :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and lcm :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and m :: "'a"   and n :: "'a" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize" shows "(gcd m n = normalize m) = dvd.dvd times m n"
Quotient_uint8: shows "Quotient (=) Abs_uint8 Rep_uint8 cr_uint8"
\<R>\<^sub>2_elim: fixes c\<^sub>1 :: "('Var, 'AExp, 'BExp) Stmt"   and mds :: "Mode \<Rightarrow> 'Var set"   and mem\<^sub>1 :: "'Var \<Rightarrow> 'Val"   and c\<^sub>2 :: "('Var, 'AExp, 'BExp) Stmt"   and mem\<^sub>2 :: "'Var \<Rightarrow> 'Val"   and \<Gamma> :: "'Var \<Rightarrow> Sec option" assumes "\<langle>c\<^sub>1, mds, mem\<^sub>1\<rangle> \<R>\<^sup>2\<^bsub>\<Gamma>\<^esub> \<langle>c\<^sub>2, mds, mem\<^sub>2\<rangle>" obtains \<Gamma>\<^sub>1 :: "'Var \<Rightarrow> Sec option" and \<Gamma>\<^sub>2 :: "'Var \<Rightarrow> Sec option" where "\<langle>c\<^sub>1, mds, mem\<^sub>1\<rangle> \<approx> \<langle>c\<^sub>2, mds, mem\<^sub>2\<rangle>" and "\<forall>x\<in>dom \<Gamma>. \<Gamma> x = Some High" and "\<turnstile> \<Gamma>\<^sub>1 {c\<^sub>1} \<Gamma>" and "\<turnstile> \<Gamma>\<^sub>2 {c\<^sub>2} \<Gamma>" and "mds_consistent mds \<Gamma>\<^sub>1" and "mds_consistent mds \<Gamma>\<^sub>2"
circ_left_slide: fixes x :: "'a"   and y :: "'a" shows "(x * y)\<^sup>\<circ> * x \<le> x * (y * x)\<^sup>\<circ>"
Rl: shows "rel_set \<chi> (Inl -` A1) (Inl -` A2)"
has_disc_negated_positiv_has_disc: fixes disc :: "'c \<Rightarrow> bool"   and neg :: "bool"   and m :: "'c match_expr" shows "(has_disc_negated disc neg m \<or> has_disc_negated disc (\<not> neg) m) = has_disc disc m"
uf: shows "{ofe_action x |x. \<exists>f. x = f \<and> f \<in> set (a # ft) \<and> \<gamma> (ofe_fields f) p \<and> (\<forall>fo\<in>set (a # ft). ofe_prio f < ofe_prio fo \<longrightarrow> \<not> \<gamma> (ofe_fields fo) p)} = {ofe_action x |x. x \<in> set ft \<and> \<gamma> (ofe_fields x) p \<and> (\<forall>fo\<in>set ft. ofe_prio x < ofe_prio fo \<longrightarrow> \<not> \<gamma> (ofe_fields fo) p)}"
IH: shows "ik\<^sub>e\<^sub>s\<^sub>t D \<subseteq> subterms\<^sub>s\<^sub>e\<^sub>t (M \<union> N)"