hcju/isabelleps huggingface dataset:
dominating_chain_imp_dominating_chain': fixes Ftr :: "'a filter"   and gs :: "('a \<Rightarrow> real) list" assumes "Ftr \<noteq> bot"   and "\<And>g. g \<in> set gs \<Longrightarrow> filterlim g at_top Ftr"   and "landau_dominating_chain Ftr gs" shows "landau_dominating_chain' Ftr gs"
sep_disj_addI1: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x ## y + z"   and "y ## z" shows "x + y ## z"
compute_\<ff>s_SPEC_def: fixes s :: "'b list" shows "compute_\<ff>s_SPEC s \<equiv> SPEC (\<lambda>\<ff>s. length \<ff>s = length s + 1 \<and> (\<forall>j\<le>length s. \<ff>s ! j = \<ff> s j))"
base_out: fixes i :: "nat" assumes "n \<le> i" shows "base i = p"
map_image_filter_correct_aux1: fixes \<alpha>1 :: "'a \<Rightarrow> 'b \<Rightarrow> 'c option"   and invar1 :: "'a \<Rightarrow> bool"   and \<alpha>2 :: "'d \<Rightarrow> 'e \<Rightarrow> 'f option"   and invar2 :: "'d \<Rightarrow> bool"   and map_image_filter :: "('b \<times> 'c \<Rightarrow> ('e \<times> 'f) option) \<Rightarrow> 'a \<Rightarrow> 'd"   and m :: "'a"   and f :: "'b \<times> 'c \<Rightarrow> ('e \<times> 'f) option"   and k' :: "'e"   and v' :: "'f" assumes "map_image_filter \<alpha>1 invar1 \<alpha>2 invar2 map_image_filter"   and "invar1 m"   and "transforms_to_unique_keys (\<alpha>1 m) f" shows "invar2 (map_image_filter f m) \<and> (\<alpha>2 (map_image_filter f m) k' = Some v') = (\<exists>k v. \<alpha>1 m k = Some v \<and> f (k, v) = Some (k', v'))"
dom_map_of_zip: fixes xs :: "'a list"   and ys :: "'b list" assumes "length xs = length ys" shows "dom (map_of (zip xs ys)) = set xs"
\<Phi>_Fun_mapsto: fixes y :: "'d"   and f :: "'c"   and x :: "'c" assumes "D.ide y"   and "\<guillemotleft>f : F y \<rightarrow>\<^sub>C x\<guillemotright>" shows "\<Phi>.FUN (y, x) \<in> HomC.set (F y, x) \<rightarrow> HomD.set (y, G x)"
take_eq_Nil: fixes n :: "nat"   and xs :: "'c list" shows "(take n xs = []) = (n = 0 \<or> xs = [])"
binop_bij_def: fixes A :: "V"   and f :: "V" shows "binop_bij A f \<equiv> nop_bij A (2\<^sub>\<nat>) f"
linorder_linear: fixes x :: "'c"   and y :: "'c" shows "x \<le> y \<or> y \<le> x"
add_increasing: fixes a :: "'b"   and b :: "'b"   and c :: "'b" assumes "(0::'b) \<le> a"   and "b \<le> c" shows "b \<le> a + c"
id_funpow: fixes n :: "nat" shows "id ^^ n = id"
A: fixes a :: "'a"   and x :: "'b" shows "f a x \<le> f' a x"
assms_2: fixes x :: "'b"   and \<sigma> :: "'a" shows "nres_of (fi x \<sigma>) \<le> f x \<sigma>"
measurable_envelope_eq2: fixes A :: "'b set"   and E :: "'b set"   and M :: "'b measure" assumes "A \<subseteq> E"   and "E \<in> sets M"   and "emeasure M E < \<infinity>" shows "measurable_envelope M A E = (emeasure M E = outer_measure_of M A)"
prognosis_upd: fixes ae :: "Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and as :: "Arity list"   and u :: "Arity"   and \<Gamma> :: "(Vars.var \<times> exp) list"   and e :: "exp"   and S :: "stack_elem list"   and x :: "Vars.var" shows "prognosis ae as u (\<Gamma>, e, S) \<sqsubseteq> prognosis ae as u (\<Gamma>, e, Upd x # S)"
RangeE: fixes b :: "'d"   and r :: "('e \<times> 'd) set" assumes "b \<in> Range r" obtains a :: "'e" where "(a, b) \<in> r"
program_equalityI: fixes F :: "'d program"   and G :: "'d program" assumes "Init F = Init G"   and "Acts F = Acts G"   and "AllowedActs F = AllowedActs G" shows "F = G"
eq_fin_le_fin_transp: fixes w\<^sub>1 :: "'action list"   and w\<^sub>2 :: "'action list"   and w\<^sub>3 :: "'action list" assumes "w\<^sub>1 =\<^sub>F w\<^sub>2"   and "w\<^sub>2 \<preceq>\<^sub>F w\<^sub>3" shows "w\<^sub>1 \<preceq>\<^sub>F w\<^sub>3"
qGoodEnv_def: fixes rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qTerm option" shows "qGoodEnv rho \<equiv> (\<forall>xs. liftAll qGood (rho xs)) \<and> (\<forall>ys. |{y. rho ys y \<noteq> None}| <o |UNIV|)"
rtrancl_Un_subset: fixes R :: "('b \<times> 'b) set"   and S :: "('b \<times> 'b) set" shows "R\<^sup>* \<union> S\<^sup>* \<subseteq> (R \<union> S)\<^sup>*"
compactin_PiE: fixes X :: "'a \<Rightarrow> 'b topology"   and I :: "'a set"   and S :: "'a \<Rightarrow> 'b set" shows "compactin (product_topology X I) (Pi\<^sub>E I S) = (Pi\<^sub>E I S = {} \<or> (\<forall>i\<in>I. compactin (X i) (S i)))"
fmember_implies_member: fixes e :: "'a"   and f :: "'a fset" assumes "e |\<in>| f" shows "e \<in> fset f"
sum_distrib_right: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and f :: "'e \<Rightarrow> 'd"   and A :: "'e set"   and r :: "'d" assumes "class.semiring_0 plus zero times" shows "times (comm_monoid_add.sum plus zero f A) r = comm_monoid_add.sum plus zero (\<lambda>n. times (f n) r) A"
strand_leaks\<^sub>l\<^sub>s\<^sub>t_def: fixes \<A> :: "('lbl strand_label \<times> ('fun, 'var) strand_step) list"   and Sec :: "('fun, 'var) Term.term set"   and \<I> :: "'var \<Rightarrow> ('fun, 'var) Term.term" shows "strand_leaks\<^sub>l\<^sub>s\<^sub>t \<A> Sec \<I> \<equiv> \<exists>t\<in>Sec - declassified\<^sub>l\<^sub>s\<^sub>t \<A> \<I>. \<exists>l. \<I> \<Turnstile> \<langle>proj_unl l \<A> @ [send\<langle>t\<rangle>\<^sub>s\<^sub>t]\<rangle>"
param_empty: fixes R :: "('a \<times> 'b) set" shows "({}, {}) \<in> \<langle>R\<rangle>set_rel"
cut_ge_mem_iff: fixes x :: "'a"   and I :: "'a set"   and t :: "'a" shows "(x \<in> I \<down>\<ge> t) = (x \<in> I \<and> t \<le> x)"
Fix_2: fixes j :: "nat" assumes "hi < j"   and "j < length xs" shows "xs' ! j = xs ! j"
sup_inf_distrib2: fixes y :: "'d multiset"   and z :: "'d multiset"   and x :: "'d multiset" shows "y \<inter># z \<union># x = y \<union># x \<inter># (z \<union># x)"
l1_init_def: shows "l1_init \<equiv> {\<lparr>ik = {}, secret = {}, progress = Map.empty, signals = \<lambda>x. 0\<rparr>}"
SPEC_trans: fixes x :: "'c nres"   and y :: "'c nres"   and Postcond :: "'c \<Rightarrow> bool" assumes "x \<le> y"   and "y \<le> SPEC Postcond" shows "x \<le> SPEC Postcond"
rem_implicit_pres_ops_execute: fixes prob :: "(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"   and s :: "nat \<Rightarrow> nat option"   and name :: "char list" assumes "\<And>op. op \<in> set (ast_problem.ast\<delta> prob) \<Longrightarrow> is_standard_operator op"   and "s \<in> ast_problem.valid_states prob" shows "ast_problem.execute (rem_implicit_pres_ops prob) name s = ast_problem.execute prob name s"
asymp_equiv_imp_eventually_same_sign: fixes f :: "real \<Rightarrow> real"   and F :: "real filter"   and g :: "real \<Rightarrow> real" assumes "f \<sim>[F] g" shows "\<forall>\<^sub>F x in F. sgn (f x) = sgn (g x)"
add_inv_def: shows "add_inv = (\<lambda>blues reds onstack. \<not> (\<exists>v\<in>(blues - onstack) \<inter> F. (v, v) \<in> E\<^sup>+) \<and> reds \<subseteq> blues \<and> reds \<inter> onstack = {} \<and> red_dfs_inv E reachable reds onstack)"
fflip_app: fixes a :: "V"   and b :: "V"   and f :: "V" assumes "[a, b]\<^sub>\<circ> \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> f" shows "fflip f \<lparr>b, a\<rparr>\<^sub>\<bullet> = f \<lparr>a, b\<rparr>\<^sub>\<bullet>"
dsr2: fixes x :: "'a"   and y :: "'a" shows "domain_bdlo_monoid.d (1::'a) (\<cdot>) (\<sqinter>) \<top> (x \<cdot> domain_bdlo_monoid.d (1::'a) (\<cdot>) (\<sqinter>) \<top> y) = domain_bdlo_monoid.d (1::'a) (\<cdot>) (\<sqinter>) \<top> (x \<cdot> y)"
frontier_of_components_closed_complement: fixes S :: "'a set"   and C :: "'a set" assumes "closed S"   and "C \<in> components (- S)" shows "frontier C \<subseteq> S"
lems_2: fixes t :: "'b tree"   and a :: "'b"   and b :: "'b" shows "minima t a b = (a \<in> alphabet t \<and> b \<in> alphabet t \<and> a \<noteq> b \<and> (\<forall>c\<in>alphabet t. c \<noteq> a \<longrightarrow> c \<noteq> b \<longrightarrow> freq t a \<le> freq t c \<and> freq t b \<le> freq t c))"
listI: fixes xs :: "'a list"   and n :: "nat"   and A :: "'a set" assumes "length xs = n"   and "set xs \<subseteq> A" shows "xs \<in> list n A"
simps_1: fixes \<sigma> :: "(char list \<times> 'a list) trace"   and v :: "'a list"   and i :: "nat"   and r :: "char list"   and ts :: "'a MFOTL.trm list" shows "MFOTL.sat \<sigma> v i (formula.Pred r ts) = ((r, map (MFOTL.eval_trm v) ts) \<in> \<Gamma> \<sigma> i)"
lossless_inline: fixes \<I> :: "('b, 'c) \<I>"   and gpv :: "('d, 'b, 'c) gpv"   and \<I>' :: "('f, 'g) \<I>"   and callee :: "'e \<Rightarrow> 'b \<Rightarrow> ('c \<times> 'e, 'f, 'g) gpv"   and s :: "'e" assumes "lossless_gpv \<I> gpv"   and "\<I> \<turnstile>g gpv \<surd>"   and "\<And>s x. x \<in> outs_\<I> \<I> \<Longrightarrow> lossless_gpv \<I>' (callee s x)"   and "\<And>s x. x \<in> outs_\<I> \<I> \<Longrightarrow> results_gpv \<I>' (callee s x) \<subseteq> responses_\<I> \<I> x \<times> UNIV" shows "lossless_gpv \<I>' (inline callee gpv s)"
Chains_into_chain: fixes M :: "'a set"   and R :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "M \<in> Chains {(x, y). R x y}" shows "Complete_Partial_Order.chain R M"
myss2_axioms_def: fixes x :: "int \<Longrightarrow> 'a"   and y :: "int \<Longrightarrow> 'a"   and a :: "char list \<Longrightarrow> 'a"   and z :: "char list \<Longrightarrow> 'a" shows "myss2_axioms x y a z \<equiv> (vwb_lens z \<and> z \<bowtie> x \<and> x \<bowtie> z) \<and> (z \<bowtie> y \<and> y \<bowtie> z) \<and> z \<bowtie> a \<and> a \<bowtie> z"
intro: fixes T :: "'b set set" assumes "\<And>x y. \<lbrakk>x open\<^bsub>T\<^esub>; y open\<^bsub>T\<^esub>\<rbrakk> \<Longrightarrow> x \<inter> y open\<^bsub>T\<^esub>"   and "\<And>M. \<forall>m\<in>M. m open\<^bsub>T\<^esub> \<Longrightarrow> \<Union> M open\<^bsub>T\<^esub>" shows "topology T"
trms\<^sub>s\<^sub>s\<^sub>t\<^sub>p_subst: fixes a :: "('d, 'c) stateful_strand_step"   and \<theta> :: "'c \<Rightarrow> ('d, 'c) Term.term" assumes "set (bvars\<^sub>s\<^sub>s\<^sub>t\<^sub>p a) \<inter> subst_domain \<theta> = {}" shows "trms\<^sub>s\<^sub>s\<^sub>t\<^sub>p (a \<cdot>\<^sub>s\<^sub>s\<^sub>t\<^sub>p \<theta>) = trms\<^sub>s\<^sub>s\<^sub>t\<^sub>p a \<cdot>\<^sub>s\<^sub>e\<^sub>t \<theta>"
image_domain_partition_eq: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and B :: "'d set"   and p\<^sub>A :: "'c \<Rightarrow> 'c" assumes "f \<in> A \<rightarrow>\<^sub>E B"   and "p\<^sub>A permutes A" shows "(`) p\<^sub>A ` (\<lambda>b. {x \<in> A. f x = b}) ` B = (\<lambda>b. {x \<in> A. f (inv p\<^sub>A x) = b}) ` B"
2: shows "(g k \<cdot>\<^sub>v gs ! k) \<bullet> M.sumlist (map (\<lambda>i. h i \<cdot>\<^sub>v gs ! i) [0..<k]) = (0::'a)"
list_Previous_nth_if: fixes xs :: "'d list"   and init :: "'d"   and n :: "nat" shows "xs\<^bsup>\<leftarrow>' init\<^esup> n = (if n = 0 then init else xs ! (n - Suc 0))"
simps: fixes n :: "nat"   and xs :: "'b list" shows "Balance.bal n xs = (if n = 0 then (\<langle>\<rangle>, xs) else let m = n div 2; (l, ys) = Balance.bal m xs; (r, y) = Balance.bal (n - 1 - m) (tl ys) in (\<langle>l, hd ys, r\<rangle>, y))"
vcard_vempty: fixes A :: "V" shows "(vcard A = 0) = (A = 0)"
finite_dom_lookup: fixes t :: "('c, 'd) Trie.trie" shows "finite (dom (lookup_trie t))"
cond_conj_distr: fixes P :: "'d upred"   and b :: "'d upred"   and Q :: "'d upred"   and S :: "'d upred" shows "(P \<and> Q \<triangleleft> b \<triangleright> S) = (P \<and> Q) \<triangleleft> b \<triangleright> (P \<and> S)"
static_quantities_trading_strat: fixes p :: "'c \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> real" assumes "static_quantities p"   and "finite (support_set p)" shows "trading_strategy p"
cond_d_def: fixes a :: "nat list"   and b :: "nat list"   and xs :: "nat list"   and ys :: "nat list" shows "cond_d a b xs ys = (\<forall>l\<le>length b. take l b \<bullet> take l ys \<le> a \<bullet> xs)"
no_fail_def: fixes P :: "'c \<Rightarrow> bool"   and m :: "'c \<Rightarrow> ('d \<times> 'c) set \<times> bool" shows "no_fail P m \<equiv> \<forall>s. P s \<longrightarrow> \<not> snd (m s)"
vw: shows "v = (\<chi>i. complex_of_real (w $h i))"
sup_V_0_right: fixes a :: "V" shows "a \<squnion> 0 = a"
gbg_impl_rel_eext_def: fixes Rm :: "('f \<times> 'h) set"   and Rv :: "('e \<times> 'g) set" shows "\<langle>Rm, Rv\<rangle>gbg_impl_rel_eext \<equiv> \<langle>Rm, \<langle>\<langle>Rv\<rangle>list_set_rel\<rangle>list_set_rel\<rangle>gen_gbg_impl_rel_eext"
null_measure_has_subprob_density: fixes M :: "'a measure" assumes "space M \<noteq> {}" shows "has_subprob_density (null_measure M) M (\<lambda>_. 0)"
preallocated_new: fixes h :: "heap"   and a :: "nat"   and x :: "obj" assumes "preallocated h"   and "h a = None" shows "preallocated (h(a \<mapsto> x))"
homeomorphic_affine_sets: fixes S :: "'a set"   and T :: "'b set" assumes "affine S"   and "affine T"   and "aff_dim S = aff_dim T" shows "S homeomorphic T"
bij_Var_subst_compose_Var: fixes g :: "'a \<Rightarrow> 'b" assumes "bij g" shows "(Var \<circ> g) \<circ>\<^sub>s (Var \<circ> inv g) = Var"
convergentD: fixes X :: "nat \<Rightarrow> 'a" assumes "convergent X" shows "\<exists>L. X \<longlonglongrightarrow> L"
execute_abs: fixes s :: "nat \<Rightarrow> nat option"   and ast_op :: "char list \<times> (nat \<times> nat) list \<times> ((nat \<times> nat) list \<times> nat \<times> nat option \<times> nat) list \<times> nat" assumes "s \<in> valid_states"   and "ast_op \<in> set ast\<delta>"   and "is_operator_applicable_in s (abs_ast_operator ast_op)" shows "execute ((fst \<circ> the \<circ> lookup_action \<circ> abs_ast_operator) ast_op) s = s \<then>\<^sub>+ abs_ast_operator ast_op"
P1: shows "P1 \<in> carrier_mat (dim_row A - 1) (dim_row A - 1)"
equality: fixes r :: "('m, 'n, 'o, 'p) RDocument_scheme"   and r' :: "('m, 'n, 'o, 'p) RDocument_scheme" assumes "RObject.nothing r = RObject.nothing r'"   and "RDocument.nothing r = RDocument.nothing r'"   and "doctype r = doctype r'"   and "document_element r = document_element r'"   and "disconnected_nodes r = disconnected_nodes r'"   and "RDocument.more r = RDocument.more r'" shows "r = r'"
Nil: fixes A :: "'a \<Rightarrow> bool" shows "listsp A []"
add_smult_distrib_vec: fixes a :: "'b"   and b :: "'b"   and v :: "'b vec" shows "(a + b) \<cdot>\<^sub>v v = a \<cdot>\<^sub>v v + b \<cdot>\<^sub>v v"
sets_measure_of: fixes A :: "'a set set"   and \<Omega> :: "'a set"   and \<mu> :: "'a set \<Rightarrow> ennreal" assumes "A \<subseteq> Pow \<Omega>" shows "sets (measure_of \<Omega> A \<mu>) = sigma_sets \<Omega> A"
le_approx_def: fixes P :: "'c process"   and Q :: "'c process" shows "P \<sqsubseteq> Q \<equiv> D Q \<subseteq> D P \<and> (\<forall>s. s \<notin> D P \<longrightarrow> Ra P s = Ra Q s) \<and> min_elems (D P) \<subseteq> T Q"
Parallel_Strong_Soundness_Seq_aux: fixes Ts :: "('b ann_com option \<times> 'b set) list"   and i :: "nat"   and c0 :: "'b ann_com"   and c1 :: "'b ann_com" assumes "interfree Ts"   and "i < length Ts"   and "OG_Hoare.com (Ts ! i) = Some (AnnSeq c0 c1)" shows "interfree (Ts[i := (Some c0, pre c1)])"
V_BCLIM_not_in_Lim: shows "V\<^bsub>BC-LIM\<^esub> \<notin> LIM"
aux_lemmas_1: shows "subfield (rupture_surj A P ` poly_of_const ` K) (Rupt A P)"
induct: fixes P :: "('a \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c) \<Rightarrow> ('a \<times> 'c) list \<Rightarrow> ('a \<times> 'c) list \<Rightarrow> bool"   and a0 :: "'a \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c"   and a1 :: "('a \<times> 'c) list"   and a2 :: "('a \<times> 'c) list" assumes "\<And>f k v as k' v' bs. \<lbrakk>k' < k \<Longrightarrow> P f ((k, v) # as) bs; \<lbrakk>\<not> k' < k; k < k'\<rbrakk> \<Longrightarrow> P f as ((k', v') # bs); \<lbrakk>\<not> k' < k; \<not> k < k'\<rbrakk> \<Longrightarrow> P f as bs\<rbrakk> \<Longrightarrow> P f ((k, v) # as) ((k', v') # bs)"   and "\<And>f uu_. P f [] uu_"   and "\<And>f uv_. P f uv_ []" shows "P a0 a1 a2"
make_mat_works: fixes A :: "'a mat"   and m :: "nat"   and n :: "nat"   and i :: "nat"   and Q_step :: "'a mat" assumes "A \<in> carrier_mat m n"   and "Suc i < n"   and "Q_step = make_mat n (Suc i) (snd (Smith_1x2 (Matrix.mat 1 2 (\<lambda>(a, b). if b = 0 then A $$ (0, 0) else A $$ (0, Suc i)))))" shows "A $$ (0, 0) * Q_step $$ (0, Suc i) + A $$ (0, Suc i) * Q_step $$ (Suc i, Suc i) = (0::'a)"
pmf_embed_spmf_None: fixes f :: "'c \<Rightarrow> real" assumes "\<integral>\<^sup>+ x. ennreal (f x) \<partial>count_space UNIV \<le> 1" shows "pmf (embed_spmf f) None = 1 - enn2real (\<integral>\<^sup>+ x. ennreal (f x) \<partial>count_space UNIV)"
transfer_rule_of_nat: fixes R :: "'b \<Rightarrow> 'c \<Rightarrow> bool" assumes "R (0::'b) (0::'c)"   and "R (1::'b) (1::'c)"   and "(R ===> R ===> R) (+) (+)" shows "((=) ===> R) of_nat of_nat"
ipresWls_def: fixes h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> 'd"   and MOD :: "('e, 'f, 'g, 'sort, 'h, 'i, 'd, 'j, 'k) model_scheme" shows "ipresWls h MOD \<equiv> \<forall>s X. wls s X \<longrightarrow> igWls MOD s (h X)"
simps_2: fixes U :: "'f set"   and u :: "'f"   and r :: "'f list" shows "atU_s U (u # r) = (u \<in> U)"
isSend_def: fixes event :: "('a, 'b, 'c) event" shows "isSend event = (\<exists>x21 x22 x23 x24 x25 x26. event = Send x21 x22 x23 x24 x25 x26)"
case_2: fixes f1 :: "guest \<Rightarrow> room \<Rightarrow> key \<times> key \<Rightarrow> 'a"   and f2 :: "guest \<Rightarrow> room \<Rightarrow> key \<times> key \<Rightarrow> 'a"   and f3 :: "guest \<Rightarrow> room \<Rightarrow> 'a"   and x21 :: "guest"   and x22 :: "room"   and x23 :: "key \<times> key" shows "(case Enter x21 x22 x23 of Check_in x xa xb \<Rightarrow> f1 x xa xb | Enter x xa xb \<Rightarrow> f2 x xa xb | Exit x xa \<Rightarrow> f3 x xa) = f2 x21 x22 x23"
unsigned_or_eq: 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 v :: "'c word"   and w :: "'c word" assumes "class.semiring_bit_operations plus minus zero times one divide modulo bit push_bit drop_bit take_bit and or xor mask" shows "semiring_1.unsigned one plus zero (v OR w) = or (semiring_1.unsigned one plus zero v) (semiring_1.unsigned one plus zero w)"
tendsto_Im3: fixes g :: "'a \<Rightarrow> octo"   and a :: "octo"   and F :: "'a filter" assumes "(g \<longlongrightarrow> a) F" shows "((\<lambda>x. Im3 (g x)) \<longlongrightarrow> Im3 a) F"
orderI: fixes a :: "'c"   and b :: "'c" assumes "a = sup a b" shows "b \<le> a"
right_idem: fixes a :: "'c multiset"   and b :: "'c multiset" shows "a \<inter># b \<inter># b = a \<inter># b"
4: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x \<squnion> y \<squnion> z = x \<squnion> (y \<squnion> z)"
iso_in_hom_def: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b"   and a :: "'b"   and b :: "'b" assumes "category C" shows "category.iso_in_hom C f a b \<equiv> partial_magma.in_hom C f a b \<and> category.iso C f"
IH: fixes s0 :: "('a, 'b) fmap"   and PROB :: "(('a, 'b) fmap \<times> ('a, 'b) fmap) set"   and h :: "('a, 'b) fmap" assumes "s0 \<in> valid_states PROB"   and "as \<in> valid_plans PROB"   and "ListMem h (state_list s0 as)" shows "fmdom' h = fmdom' s0"
iT_Plus_neg_iEx_conv: fixes I :: "nat set"   and k :: "nat"   and P :: "nat \<Rightarrow> bool" shows "(\<diamond> t I \<oplus>- k. P t) = (\<diamond> t I \<down>\<ge> k. P (t - k))"
abs_eq: fixes xa :: "'a"   and x :: "'a \<Rightarrow> 'b option" shows "Mapping.delete xa (mapping.Mapping x) = mapping.Mapping (x(xa := None))"
am5: fixes x :: "'a"   and y :: "'a" shows "ad (x \<cdot> y) \<cdot> x \<cdot> ad y = ad (x \<cdot> y) \<cdot> x"
lambda_predicates_2_2: fixes \<phi> :: "\<kappa> \<Rightarrow> \<kappa> \<Rightarrow> \<o>"   and x :: "\<nu>"   and y :: "\<nu>" assumes "IsProperInXY \<phi>" shows "[[\<lparr>\<^bold>\<lambda>\<^sup>2 (\<lambda>x y. \<phi> (x\<^sup>P) (y\<^sup>P)),x\<^sup>P,y\<^sup>P\<rparr> \<^bold>\<equiv> \<phi> (x\<^sup>P) (y\<^sup>P)]]"
cobounded1: fixes a :: "'x \<Rightarrow>\<^sub>0 nat"   and b :: "'x \<Rightarrow>\<^sub>0 nat" shows "a \<preceq> ordered_powerprod_lin.max a b"
id_eq_2: fixes v :: "i"   and x :: "'b"   and y :: "'b" shows "[x \<^bold>= y \<^bold>\<rightarrow> y \<^bold>= x in v]"
r_mul_def: shows "r_mul \<equiv> Pr 1 Z (Cn 3 r_add [recf.Id 3 1, recf.Id 3 2])"
pickFT_length: fixes cl :: "('d, 'e, 'f) cmd list" assumes "WtFT cl = 1" shows "pickFT cl < length cl"
assms_2: fixes x :: "'a" assumes "x \<notin> S" shows "f x = x"
in_dom: fixes i :: "'a" assumes "i \<in> agents'" shows "Order_Predicates.preorder_on alts' (R' i)"
naturality: fixes \<mu> :: "'c" assumes "C.arr \<mu>" shows "\<tau>\<^sub>1 (C.cod \<mu>) \<cdot>\<^sub>D (G \<mu> \<star>\<^sub>D \<tau>\<^sub>0 (src\<^sub>C \<mu>)) = (\<tau>\<^sub>0 (trg\<^sub>C \<mu>) \<star>\<^sub>D F \<mu>) \<cdot>\<^sub>D \<tau>\<^sub>1 (C.dom \<mu>)"
satPB_def: fixes \<xi> :: "Preliminaries.var \<Rightarrow> univ"   and \<Phi> :: "(('fsym, 'tp) efsym, ('psym, 'tp) epsym) lit list set" shows "satPB \<xi> \<Phi> \<equiv> \<forall>c\<in>\<Phi>. satC \<xi> c"
len\<^sub>A: shows "length cms\<^sub>C = length cms"
set_intros_3: fixes ya :: "'a"   and x23 :: "'a tree"   and x21 :: "'a tree"   and x22 :: "'a" assumes "ya \<in> set_tree x23" shows "ya \<in> set_tree \<langle>x21, x22, x23\<rangle>"
rel_F_neg_distr_cond_eq: fixes tytok :: "('t \<times> 'u \<times> 'v \<times> 'w \<times> 'x \<times> 'y \<times> 'z \<times> 'aa \<times> 'ab \<times> 'ac) itself" shows "rel_F_neg_distr_cond (=) (=) (=) (=) (=) (=) (=) (=) (=) (=) (=) (=) tytok"
convert_obs_initial_inject: fixes ta :: "('a, 'b, 'c, 'd, 'e, 'f) thread_action"   and ta' :: "('a, 'b, 'c, 'd, 'e, 'f) thread_action" shows "(convert_obs_initial ta = convert_obs_initial ta') = (ta = ta')"
circ_sup_sub_sup_one: fixes x :: "'a"   and y :: "'a" shows "x * x\<^sup>\<circ> * (x \<squnion> y) \<le> x * x\<^sup>\<circ> * ((1::'a) \<squnion> y)"
cod_eqI: fixes b :: "'a \<times> 'a"   and f :: "'a \<times> 'a" assumes "CC.ide b"   and "CC.seq b f" shows "CC.cod f = b"
equivalence_relation_def: fixes x :: "'b trm \<Rightarrow> 'b trm \<Rightarrow> bool" shows "equivalence_relation x = (reflexive x \<and> symmetric x \<and> transitive x)"
of_hypreal_eq_0_iff: fixes x :: "real star" shows "((*f* of_real) x = 0) = (x = 0)"
compl_eq_compl_iff: fixes x :: "'a"   and y :: "'a" shows "(NOT x = NOT y) = (x = y)"
iter_tensor_of_state_is_state: fixes m :: "nat"   and A :: "complex Matrix.mat"   and n :: "nat" assumes "state m A"   and "1 \<le> n" shows "state (m * n) (A \<otimes>\<^bsup>n\<^esup>)"
l_adopt_node_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms: shows "l_adopt_node_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^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 adopt_node adopt_node_locs DocumentClass.known_ptr DocumentClass.type_wf get_child_nodes get_child_nodes_locs DocumentClass.known_ptrs set_child_nodes set_child_nodes_locs remove heap_is_wellformed parent_child_rel"
Vinv3_def: shows "Vinv3 = {s. ran (decisions s) \<subseteq> locked s}"
leading_monomial_tail: fixes p :: "'t \<Rightarrow>\<^sub>0 'c" shows "p = monomial (lc p) (lt p) + tail p"
bounded_linear_mult_right: fixes x :: "'b" shows "bounded_linear ((*) x)"
bound_deg_sum: fixes f :: "nat \<Rightarrow> 'a"   and g :: "nat \<Rightarrow> 'a"   and n :: "nat" assumes "f \<in> carrier P"   and "g \<in> carrier P"   and "deg R f \<le> n"   and "deg R g \<le> n" shows "deg R (f \<oplus>\<^bsub>P\<^esub> g) \<le> n"
gen_inv: fixes x :: "'e"   and G :: "('e, 'f) monoid_scheme"   and gens :: "'e set" assumes "x \<in> \<langle>gens\<rangle>\<^bsub>G\<^esub>" shows "inv\<^bsub>G\<^esub> x \<in> \<langle>gens\<rangle>\<^bsub>G\<^esub>"