hcju/isabelleps huggingface dataset:
l_get_owner_document\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def: fixes known_ptr\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) object_ptr \<Rightarrow> bool"   and get_parent :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr option) prog"   and type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) heap \<Rightarrow> bool"   and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_root_node :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr) prog"   and get_owner_document\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) document_ptr) prog"   and get_host :: "(_) shadow_root_ptr \<Rightarrow> ((_) heap, exception, (_) element_ptr) prog"   and get_host_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_owner_document :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) document_ptr) prog" shows "l_get_owner_document\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M known_ptr\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_parent type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_disconnected_nodes get_disconnected_nodes_locs get_root_node get_owner_document\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_host get_host_locs get_owner_document \<equiv> l_get_host get_host get_host_locs \<and> l_get_owner_document\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_parent known_ptr\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_disconnected_nodes get_disconnected_nodes_locs get_root_node get_owner_document\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M \<and> l_get_owner_document\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms get_disconnected_nodes get_root_node get_owner_document"
assms_1: shows "convert_atom_list var L (xs' @ x # xs) = Some L'"
assms_3: fixes x :: "'a"   and y :: "'a" shows "P x y = ((x, y) \<in> R)"
simps: fixes a1 :: "'opSym"   and a2 :: "'index \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set option" shows "wlsInp a1 a2 = (\<exists>delta inp. a1 = delta \<and> a2 = inp \<and> wlsOpS delta \<and> (\<forall>x. arOf delta x = None \<and> inp x = None \<or> (\<exists>s X. arOf delta x = Some s \<and> inp x = Some X \<and> wls s X)))"
simps_2: fixes s :: "'d \<Rightarrow> bool" shows "val_ifex Falseif s = False"
rec_comb: fixes rs :: "(name \<times> value) list"   and \<Gamma> :: "(name, value) fmap"   and t :: "sterm"   and css :: "(name, (Term_Class.term \<times> sterm) list) fmap"   and name :: "name"   and \<Gamma>' :: "(name, value) fmap"   and cs :: "(Term_Class.term \<times> sterm) list"   and u :: "sterm"   and u' :: "value"   and env :: "(name, value) fmap"   and uv :: "Term_Class.term"   and rhs :: "sterm"   and val :: "value" assumes "rs, \<Gamma> \<turnstile>\<^sub>v t \<down> Vrecabs css name \<Gamma>'"   and "fmlookup css name = Some cs"   and "rs, \<Gamma> \<turnstile>\<^sub>v u \<down> u'"   and "vfind_match cs u' = Some (env, uv, rhs)"   and "rs, \<Gamma>' ++\<^sub>f env \<turnstile>\<^sub>v rhs \<down> val" shows "rs, \<Gamma> \<turnstile>\<^sub>v t $\<^sub>s u \<down> val"
subst_open: fixes u :: "expr"   and xT :: "char list \<times> Unary_PCF.type"   and i :: "nat"   and t :: "expr"   and v :: "expr" assumes "lc u" shows "subst xT u (open i t v) = open i (subst xT u t) (subst xT u v)"
suc_least_AboveS: fixes a :: "nat + 'a Bonk_Schramm_extension_unfolded set"   and B :: "(nat + 'a Bonk_Schramm_extension_unfolded set) set" assumes "a \<in> AboveS wo_aux B" shows "(suc wo_aux B, a) \<in> wo_aux"
path_conforms_with_strategy_2: shows "\<forall>v. v \<in> lset P \<longrightarrow> \<sigma> v = \<sigma>' v"
wf_ciD3': fixes ci :: "'a instr \<Rightarrow> (nat \<times> nat \<times> 'a instr list \<times> (nat \<times> nat \<times> String.literal option \<times> nat \<times> nat) list) prog \<Rightarrow> 'b \<Rightarrow> 'a val list \<Rightarrow> 'a val list \<Rightarrow> String.literal \<Rightarrow> String.literal \<Rightarrow> nat \<Rightarrow> 'a frame list \<Rightarrow> bool"   and ins :: "'a instr"   and P :: "(nat \<times> nat \<times> 'a instr list \<times> (nat \<times> nat \<times> String.literal option \<times> nat \<times> nat) list) prog"   and h :: "'b"   and stk :: "'a val list"   and loc :: "'a val list"   and C :: "String.literal"   and M :: "String.literal"   and pc :: "nat"   and frs :: "'a frame list"   and pc' :: "nat" assumes "wf_ci ci"   and "ci ins P h stk loc C M pc frs"   and "pc \<le> pc'" shows "ci ins P h stk loc C M pc' frs"
balanced_alt_def_all: fixes point_set :: "'b set"   and block_collection :: "'b set multiset"   and grouping :: "int"   and index :: "int" assumes "t_wise_balance point_set block_collection grouping index" shows "\<forall>ps\<subseteq>point_set. int (card ps) = grouping \<longrightarrow> int (block_collection index ps) = index"
integral_measure_pmf: fixes A :: "'a set"   and M :: "'a pmf"   and f :: "'a \<Rightarrow> 'b" assumes "finite A"   and "\<And>a. \<lbrakk>a \<in> set_pmf M; f a \<noteq> (0::'b)\<rbrakk> \<Longrightarrow> a \<in> A" shows "measure_pmf.expectation M f = (\<Sum>a\<in>A. pmf M a *\<^sub>R f a)"
times_in_intervalI: fixes x :: "'a"   and X :: "'a interval"   and y :: "'a"   and Y :: "'a interval" assumes "x \<in>\<^sub>i X"   and "y \<in>\<^sub>i Y" shows "x * y \<in>\<^sub>i X * Y"
induct: fixes x1 :: "Core.term"   and x2 :: "Core.term"   and P :: "Core.term \<Rightarrow> Core.term \<Rightarrow> bool" assumes "x1 \<rightarrow>\<^sub>\<beta> x2"   and "\<And>T s t. P (Abs T s $ t) (subst_bv2 s 0 t)"   and "\<And>s t u. \<lbrakk>s \<rightarrow>\<^sub>\<beta> t; P s t\<rbrakk> \<Longrightarrow> P (s $ u) (t $ u)"   and "\<And>s t u. \<lbrakk>s \<rightarrow>\<^sub>\<beta> t; P s t\<rbrakk> \<Longrightarrow> P (u $ s) (u $ t)"   and "\<And>s t T. \<lbrakk>s \<rightarrow>\<^sub>\<beta> t; P s t\<rbrakk> \<Longrightarrow> P (Abs T s) (Abs T t)" shows "P x1 x2"
iLastStrong_def: fixes t0 :: "nat"   and I :: "nat set"   and P :: "nat \<Rightarrow> bool" shows "iLastStrong t0 I P \<equiv> \<diamond> t {iprev t0 I} \<down>< t0. P t"
lnull_ltlI: fixes xs :: "'c llist" assumes "lnull xs" shows "lnull (ltl xs)"
mem_samp_primes_gt_2: fixes prime_set :: "nat set"   and x :: "nat" assumes "rsa_base prime_set"   and "x \<in> set_spmf (rsa_base.sample_primes prime_set)" shows "2 < x"
prems_2: shows "nonzero xs (Cond (FDiv e1 e2))"
of_nat_0_less_iff: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and n :: "nat" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero" shows "less zero (semiring_1.of_nat one plus zero n) = (0 < n)"
Bnd'_def: shows "Bnd' = of_nat n * DBnd"
finfun_update_def: shows "finfun_update \<equiv> map_fun ($) (map_fun id (map_fun id Abs_finfun)) fun_upd"
cap'_simps: fixes x :: "'v"   and y :: "'v" shows "cap' (x, y) = (if edge \<Gamma> x y then g (x, y) else if edge \<Gamma> y x then 1 else 0)"
has_measure_limit_iff: fixes S :: "'c set"   and m :: "real" shows "(S \<in> lmeasurable \<and> m = Sigma_Algebra.measure lebesgue S) = (\<forall>e>0. \<exists>B>0. \<forall>a b. ball (0::'c) B \<subseteq> cbox a b \<longrightarrow> S \<inter> cbox a b \<in> lmeasurable \<and> \<bar>Sigma_Algebra.measure lebesgue (S \<inter> cbox a b) - m\<bar> < e)"
reachable_refl: fixes x :: "(recf \<times> nat list \<times> nat list) list \<times> nat option" shows "reachable x x"
index_Cons: fixes ps :: "'b list"   and a :: "'c"   and branch :: "('b list \<times> 'c) list"   and v :: "nat"   and qs :: "'b list"   and b :: "'c"   and v' :: "nat"   and q :: "'b"   and p :: "'b" assumes "((ps, a) # branch) !. v = Some (qs, b)"   and "qs !. v' = Some q" shows "\<exists>qs'. ((p # ps, a) # branch) !. v = Some (qs', b) \<and> qs' !. v' = Some q"
RETURN_rel_conv1: fixes f :: "'c \<Rightarrow> 'd"   and g :: "'e \<Rightarrow> 'f"   and A :: "('c \<times> 'e) set"   and B :: "('d \<times> 'f) set" shows "((RETURN \<circ> f, RETURN \<circ> g) \<in> A \<rightarrow> \<langle>B\<rangle>nres_rel) = ((f, g) \<in> A \<rightarrow> B)"
mult_pos_neg2: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.linordered_semiring_strict plus zero minus less_eq less times"   and "less zero a"   and "less b zero" shows "less (times b a) zero"
perm_append1_eq: fixes zs :: "'a list"   and xs :: "'a list"   and ys :: "'a list" shows "(zs @ xs <~~> zs @ ys) = (xs <~~> ys)"
ss_adversary_def: fixes x :: "'a"   and c1 :: "'a \<times> nat \<times> nat"   and c2 :: "'a \<times> nat \<times> nat" shows "ss_adversary x c1 c2 = (let (a, e, z) = c1; (a', e', z') = c2 in return_spmf (if e' < e then nat ((int z - int z') * Number_Theory_Aux.inverse (e - e') (order \<G>) mod int (order \<G>)) else nat ((int z' - int z) * Number_Theory_Aux.inverse (e' - e) (order \<G>) mod int (order \<G>))))"
Sup_aform'_le: fixes e'' :: "nat \<Rightarrow> real"   and r :: "'a \<times> 'a pdevs"   and p :: "nat" assumes "\<And>i. e'' i \<le> 1"   and "\<And>i. - 1 \<le> e'' i" shows "aform_val e'' r \<le> Sup_aform' p r"
reltos_into_relto_fun: fixes a :: "'a"   and b :: "'a"   and n :: "nat"   and B :: "('a \<times> 'a) set"   and A :: "('a \<times> 'a) set" assumes "(a, b) \<in> (B\<^sup>* O A O B\<^sup>*) ^^ n" shows "\<exists>as sel m. relto_fun A B n as sel m (a, b)"
merge_def: fixes I :: "'d set"   and J :: "'d set" shows "merge I J = (\<lambda>(x, y) i. if i \<in> I then x i else if i \<in> J then y i else undefined)"
rBang: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and P' :: "('a, 'b, 'c) psi"   and A\<^sub>P :: "name list"   and \<Psi>\<^sub>P :: "'b"   and C :: "'d" assumes "\<Psi> \<rhd> P \<parallel> !P \<longmapsto> \<tau> \<prec> P'"   and "guarded P"   and "extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>"   and "distinct A\<^sub>P"   and "\<And>C. Prop C \<Psi> (P \<parallel> !P) P' A\<^sub>P (\<Psi>\<^sub>P \<otimes> \<one>)"   and "\<Psi>\<^sub>P \<simeq> \<one>"   and "supp \<Psi>\<^sub>P = {}"   and "A\<^sub>P \<sharp>* \<Psi>"   and "A\<^sub>P \<sharp>* P"   and "A\<^sub>P \<sharp>* P'"   and "A\<^sub>P \<sharp>* C" shows "Prop C \<Psi> (!P) P' [] (\<one>)"
lex_comp'_pp_def: shows "lex_comp' = comp_of_ord lex_pp"
I: shows "\<lparr>PGMs = gc_coms, INIT = gc_system_init, FAIR = \<lambda>s. True\<rparr> \<Turnstile>\<^bsub>pre\<^esub> I"
conc_trans: fixes C :: "'b nres"   and R :: "('b \<times> 'c) set"   and B :: "'c nres"   and R' :: "('c \<times> 'd) set"   and A :: "'d nres" assumes "C \<le> \<Down> R B"   and "B \<le> \<Down> R' A" shows "C \<le> \<Down> R (\<Down> R' A)"
hom_var_pow_finsum: fixes K :: "'a set"   and p :: "'a list"   and A :: "('e, 'f) ring_scheme"   and h :: "'a list \<Rightarrow> 'e" assumes "subring K R"   and "p \<in> carrier (K [X])"   and "ring_hom_ring (K [X]) A h" shows "h p = (\<Oplus>\<^bsub>A\<^esub>t\<in>set (dense_repr p). h [fst t] \<otimes>\<^bsub>A\<^esub> h X [^]\<^bsub>A\<^esub> snd t)"
bl2wc_simps_1: fixes c :: "cell list" shows "bl2wc (Oc # tl c) = Suc (bl2wc c) - bl2wc c mod 2"
allocationRightUniqueRangeDomain: fixes a :: "('c set \<times> 'd) set"   and G :: "'c set"   and N :: "'d set" assumes "a \<in> possible_allocations_rel G N" shows "runiq a \<and> runiq (a\<inverse>) \<and> Domain a partitions G \<and> Range a \<subseteq> N"
integ_exis_vert: shows "\<forall>(k, \<gamma>)\<in>two_chain_vertical_boundary two_chain. line_integral_exists F {j} \<gamma>"
expectation_gpv_le_inline: fixes \<I> :: "('b, 'c) \<I>"   and \<I>' :: "('d, 'e) \<I>"   and callee :: "'a \<Rightarrow> 'b \<Rightarrow> ('c \<times> 'a, 'd, 'e) gpv"   and gpv :: "('f, 'b, 'c) gpv"   and f :: "'f \<Rightarrow> ennreal"   and s :: "'a" assumes "\<And>s x. x \<in> outs_\<I> \<I> \<Longrightarrow> plossless_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"   and "\<I> \<turnstile>g gpv \<surd>"   and "\<And>s x. x \<in> outs_\<I> \<I> \<Longrightarrow> \<I>' \<turnstile>g callee s x \<surd>" shows "expectation_gpv 0 \<I> f gpv \<le> expectation_gpv 0 \<I>' (\<lambda>(x, s). f x) (inline callee gpv s)"
primroot_shorter_root: fixes u :: "'b list"   and q :: "'b list" assumes "u \<noteq> \<epsilon>"   and "u \<in> q*" shows "\<^bold>|\<rho> u\<^bold>| \<le> \<^bold>|q\<^bold>|"
groundVar_def: fixes v :: "Term.var" shows "groundVar v = (case v of LVar ln \<Rightarrow> True | {accC,statDeclC,stat}e..fn \<Rightarrow> \<exists>a. e = Lit a | e1.[e2] \<Rightarrow> \<exists>a i. e1 = Lit a \<and> e2 = Lit i | InsInitV c v \<Rightarrow> False)"
cyk_def: fixes G :: "('c \<times> ('c, 'd) RHS) list"   and S :: "'c"   and w :: "'d list" shows "cyk G S w = (case init G w 0 of None \<Rightarrow> False | Some T \<Rightarrow> let len = length w in if len = 1 then mem S (T (0, 1)) else let T' = main G T len 0 2 in mem S (T' (0, len)))"
intro: fixes T :: "'a \<Rightarrow> 'b"   and V :: "'a set"   and W :: "'b set" assumes "bij_betw T V W" shows "FGModuleIso_axioms V T W"
igWlsInp_termMOD_wlsInp: fixes delta :: "'opSym"   and inp :: "'index \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set option" shows "igWlsInp termMOD delta inp = wlsInp delta inp"
is_Done_resumption_lub: fixes R :: "('f, 'g, 'h) Resumption.resumption set" shows "Resumption.resumption.is_Done (resumption_lub R) = (\<forall>r\<in>R. Resumption.resumption.is_Done r)"
prod_assn_pure_conv: fixes R1 :: "('f \<times> 'd) set"   and R2 :: "('g \<times> 'e) set" shows "pure R1 \<times>\<^sub>a pure R2 = pure (R1 \<times>\<^sub>r R2)"
curry0_def: fixes f :: "unit \<Rightarrow> 'e" shows "curry0 f = f ()"
small_setint: fixes k :: "nat"   and U :: "'a set" assumes "k \<le> K"   and "U \<in> PS.events"   and "emeasure P U < ennreal epsilon" shows "set_nn_integral P U ((nn_transfer_operator ^^ k) h) \<le> ennreal delta"
exhaust: fixes y :: "num" obtains "y = num.One"   | x2 :: "num" where "y = num.Bit0 x2"   | x3 :: "num" where "y = num.Bit1 x3"
mangoldt_split: fixes d :: "nat" shows "mangoldt d = mangoldt_1 d + mangoldt_even d + mangoldt_odd d"
diff_empty: fixes M :: "'a multiset" shows "M - {#} = M \<and> {#} - M = {#}"
compact_eq_bounded_closed: fixes S :: "'e set" shows "compact S = (bounded S \<and> closed S)"
fact_bounds_1: fixes n :: "nat" assumes "0 < n" shows "sqrt (2 * pi * real n) * (real n / exp 1) ^ n \<le> fact n"
assms_1: shows "\<forall>N\<in>set Ns. RSubmodule N"
1: shows "sqrt (real n) * cmod (sum \<chi> {1..x}) < real n * ln (real n)"
aborting_switch_independent: shows "\<forall>n s. raborting (rcswitch n s) = raborting s"
altc_sample_writer_def: fixes altc_sample :: "'g \<Rightarrow> ('h \<Rightarrow> 'f) \<Rightarrow> 'f"   and p :: "'g"   and f :: "'h \<Rightarrow> ('d, 'e, 'f) writerT" shows "altc_sample_writer altc_sample p f = WriterT (altc_sample p (\<lambda>p. run_writer (f p)))"
s_top_assoc: fixes H :: "'a set"   and K :: "'a set"   and L :: "'a set" assumes "G \<guillemotright> H "   and "G \<guillemotright> K "   and "G \<guillemotright> L " shows "H \<diamondop> K \<diamondop> L = H \<diamondop> (K \<diamondop> L)"
irbd_rbd_inv2_inv: fixes \<phi> :: "'c set \<Rightarrow> 'd set" assumes "(bd\<^sub>\<R> \<circ> bd\<^sup>-\<^sub>\<R>) \<phi> = \<phi>" shows "Sup_pres \<phi>"
s_Tr_3: shows "\<lless>\<pi>\<^sub>2, e\<^sub>2\<ggreater> I\<rightarrow> \<lless>[], e\<^sub>3\<ggreater>"
simps_5: fixes \<sigma> :: "nat \<Rightarrow> val"   and b :: "cexpr"   and e1 :: "cexpr"   and e2 :: "cexpr" shows "cexpr_sem \<sigma> (IF\<^sub>c b THEN e1 ELSE e2) = (if cexpr_sem \<sigma> b = TRUE then cexpr_sem \<sigma> e1 else cexpr_sem \<sigma> e2)"
coherent_Hcomp: fixes t :: "'c local.term"   and u :: "'c local.term" assumes "Arr t"   and "Arr u"   and "Src t = Trg u"   and "coherent t"   and "coherent u" shows "coherent (t \<^bold>\<star> u)"
extf_compat_append_left: fixes ground_heads_var :: "'a \<Rightarrow> 'b set"   and gt_sym :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and arity_sym :: "'b \<Rightarrow> enat"   and arity_var :: "'a \<Rightarrow> enat"   and extf :: "'b \<Rightarrow> (('b, 'a) tm \<Rightarrow> ('b, 'a) tm \<Rightarrow> bool) \<Rightarrow> ('b, 'a) tm list \<Rightarrow> ('b, 'a) tm list \<Rightarrow> bool"   and f :: "'b"   and gt :: "('b, 'a) tm \<Rightarrow> ('b, 'a) tm \<Rightarrow> bool"   and ys :: "('b, 'a) tm list"   and xs :: "('b, 'a) tm list"   and zs :: "('b, 'a) tm list" assumes "rpo_basis ground_heads_var gt_sym arity_sym arity_var extf"   and "extf f gt ys xs" shows "extf f gt (zs @ ys) (zs @ xs)"
integral_mono_AE: fixes M :: "'c measure"   and f :: "'c \<Rightarrow> real"   and g :: "'c \<Rightarrow> real" assumes "integrable M f"   and "integrable M g"   and "AE x in M. f x \<le> g x" shows "integral\<^sup>L M f \<le> integral\<^sup>L M g"
Setcompr_eq_image: fixes f :: "'b::type \<Rightarrow> 'a::type"   and A :: "'b::type set" shows "{(f::'b::type \<Rightarrow> 'a::type) x |x::'b::type. x \<in> (A::'b::type set)} = f ` A"
rep_eq: fixes x :: "'b poly \<times> 'b poly" shows "Rep_ratfps (quot_to_ratfps x) = (case x of (x, y) \<Rightarrow> let (x', y') = normalize_quot (x, y) in if constant_term y' = (0::'b) then 0 else quot_to_fract (x', y'))"
Ostrowski_base_mem: fixes n :: "nat"   and vv :: "nat \<Rightarrow> 'a \<Rightarrow> ant" assumes "vals_nonequiv K (Suc n) vv" shows "\<forall>j\<le>Suc n. (\<Omega>\<^bsub>K vv Suc n\<^esub>) j \<in> carrier K"
greatest_lower_bound: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "(- x \<le> - y \<and> - x \<le> - z) = (- x \<le> - y \<sqinter> - z)"
ipresAbs_def: fixes h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> 'a"   and hA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> 'b"   and MOD :: "('c, 'd, 'varSort, 'e, 'f, 'var, 'a, 'b, 'g) model_scheme" shows "ipresAbs h hA MOD \<equiv> \<forall>xs x s X. isInBar (xs, s) \<and> wls s X \<longrightarrow> hA (Abs xs x X) = igAbs MOD xs x (h X)"
msgrel_imp_eqv_freeright: fixes U :: "freemsg"   and V :: "freemsg" assumes "U \<sim> V" shows "freeright U \<sim> freeright V"
axioms_2: fixes A :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and B :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and F :: "'d \<Rightarrow> 'e"   and G :: "'d \<Rightarrow> 'e"   and \<tau> :: "'d \<Rightarrow> 'e" assumes "inverse_transformation A B F G \<tau>" shows "category B"
vrestriction_vconverse: fixes r :: "V"   and A :: "V" shows "r\<inverse>\<^sub>\<circ> \<restriction>\<^sub>\<circ> A = (r \<restriction>\<^sub>\<circ> A)\<inverse>\<^sub>\<circ>"
no_smallest_element_skipped_def: fixes carrier :: "'c set"   and indep :: "'c set \<Rightarrow> bool"   and weight :: "'c \<Rightarrow> 'd"   and E :: "'c set"   and F :: "'c set" assumes "weighted_matroid carrier indep" shows "weighted_matroid.no_smallest_element_skipped carrier indep weight E F = (\<forall>e\<in>carrier - E. \<forall>g>weight e. indep (insert e (weighted_matroid.limi weight F g)) \<longrightarrow> e \<in> weighted_matroid.limi weight F g)"
\<epsilon>: fixes \<eta> :: "real" assumes "0 \<le> \<eta>"   and "\<eta> < \<epsilon>" shows "a + \<eta> *\<^sub>R l \<in> rel_interior S"
lt_p: shows "sig_of_pair p = lt (poly_of_pair p)"
assoc_naturality: fixes f0 :: "'d"   and f1 :: "'d"   and f2 :: "'d" assumes "D.arr f0"   and "D.arr f1"   and "D.arr f2" shows "\<a>\<^sub>D[D.cod f0, D.cod f1, D.cod f2] \<cdot>\<^sub>D ((f0 \<otimes>\<^sub>D f1) \<otimes>\<^sub>D f2) = (f0 \<otimes>\<^sub>D f1 \<otimes>\<^sub>D f2) \<cdot>\<^sub>D \<a>\<^sub>D[D.dom f0, D.dom f1, D.dom f2]"
metric_CauchyI: fixes dist :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and uniformity :: "('c \<times> 'c) filter"   and "open" :: "'c set \<Rightarrow> bool"   and X :: "nat \<Rightarrow> 'c" assumes "class.metric_space dist uniformity open"   and "\<And>e. 0 < e \<Longrightarrow> \<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. dist (X m) (X n) < e" shows "uniform_space.Cauchy uniformity X"
Derivation_implies_leftderives_gen: fixes a :: "symbol list"   and D :: "(nat \<times> symbol \<times> symbol list) list"   and u :: "symbol list"   and v :: "symbol list" assumes "Derivation a D (u @ v)"   and "is_word u" shows "\<exists>w. leftderives a (u @ w) \<and> (v = [] \<longrightarrow> w = []) \<and> (\<forall>X. is_first X v \<longrightarrow> is_first X w)"
itrm_abstraction_axioms: shows "itrm_abstraction extra_rule itrm_bracket"
not_Iic_eq_empty: fixes h :: "'a multiset" shows "subset_mset.atMost h \<noteq> {}"
n2_type: fixes t1 :: "'c bro"   and h :: "nat"   and t2 :: "'c bro"   and a :: "'c" shows "(t1 \<in> Bp h \<and> t2 \<in> T h \<longrightarrow> n2 t1 a t2 \<in> Bp (Suc h)) \<and> (t1 \<in> T h \<and> t2 \<in> Bp h \<longrightarrow> n2 t1 a t2 \<in> Bp (Suc h))"
assms_2: shows "J \<subseteq> carrier (R Quot I)"
Bseq_iff2: fixes X :: "nat \<Rightarrow> 'a" shows "Bseq X = (\<exists>k>0. \<exists>x. \<forall>n. norm (X n + - x) \<le> k)"
diagonal_to_Smith_PQ: fixes A :: "(('b, 'c) vec, 'd) vec"   and bezout :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<times> 'b \<times> 'b \<times> 'b \<times> 'b"   and P :: "(('b, 'd) vec, 'd) vec"   and B :: "(('b, 'c) vec, 'd) vec"   and Q :: "(('b, 'c) vec, 'c) vec" assumes "isDiagonal A"   and "is_bezout_ext bezout"   and "(P, B, Q) = diagonal_to_Smith_PQ A bezout" shows "B = P ** A ** Q \<and> invertible P \<and> invertible Q \<and> B = diagonal_to_Smith A bezout"
not_conga_not_ang: fixes a :: "'p \<Rightarrow> 'p \<Rightarrow> 'p \<Rightarrow> bool"   and A :: "'p"   and B :: "'p"   and C :: "'p"   and A' :: "'p"   and B' :: "'p"   and C' :: "'p" assumes "QCongA a"   and "\<not> A B C CongA A' B' C'"   and "a A B C" shows "\<not> a A' B' C'"
le_\<I>_def: fixes \<I> :: "('c, 'd) \<I>"   and \<I>' :: "('c, 'd) \<I>" shows "(\<I> \<le> \<I>') = (outs_\<I> \<I> \<subseteq> outs_\<I> \<I>' \<and> (\<forall>x\<in>outs_\<I> \<I>. responses_\<I> \<I>' x \<subseteq> responses_\<I> \<I> x))"
i_Exec_Comp_Stream_Acc_Output__eq_Msg_State_Idle_conv': fixes k :: "nat"   and localState :: "'a \<Rightarrow> 'b"   and output_fun :: "'a \<Rightarrow> 'c message_af"   and trans_fun :: "'d message_af \<Rightarrow> 'a \<Rightarrow> 'a"   and input :: "nat \<Rightarrow> 'd message_af"   and c :: "'a"   and t :: "nat"   and m :: "'c message_af"   and t0 :: "nat"   and s :: "nat \<Rightarrow> 'a" assumes "Suc 0 < k"   and "State_Idle localState output_fun trans_fun (i_Exec_Comp_Stream_Acc_LocalState k localState trans_fun input c t)"   and "m \<noteq> NoMsg"   and "t0 = t * k"   and "s = i_Exec_Comp_Stream trans_fun (input \<odot> k) c" shows "(i_Exec_Comp_Stream_Acc_Output k output_fun trans_fun input c t = m) = ((\<not> State_Idle localState output_fun trans_fun (localState (s t2)). t2 \<U> t1 [0\<dots>] \<oplus> t0. output_fun (s t1) = m \<and> State_Idle localState output_fun trans_fun (localState (s t1))) \<or> (\<not> State_Idle localState output_fun trans_fun (localState (s t2)). t2 \<U> t1 [0\<dots>] \<oplus> t0. output_fun (s t1) = m \<and> (\<circle> t3 t1 [0\<dots>]. output_fun (s t4) = NoMsg. t4 \<U> t5 [0\<dots>] \<oplus> t3. output_fun (s t5) = NoMsg \<and> State_Idle localState output_fun trans_fun (localState (s t5)))))"
scf_alts: fixes agents :: "'a set"   and alts :: "'b set"   and scf :: "('a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'b set"   and R :: "'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "social_choice_function agents alts scf"   and "pref_profile_wf agents alts R" shows "scf R \<subseteq> alts"