hcju/isabelleps huggingface dataset:
return_def: fixes x :: "'h" shows "return x = sum_bot.Right x"
hcomplex_hcnj_add: fixes w :: "complex star"   and z :: "complex star" shows "hcnj (w + z) = hcnj w + hcnj z"
env_restr_perm: fixes p :: "perm"   and S :: "'c set"   and \<rho> :: "'c \<Rightarrow> 'd" assumes "supp p \<sharp>* S"   and "finite S" shows "(p \<bullet> \<rho>) f|` S = \<rho> f|` S"
VLambdaE: fixes x :: "V"   and A :: "V"   and f :: "V \<Rightarrow> V" assumes "x \<in>\<^sub>\<circ> VLambda A f" obtains a :: "V" where "a \<in>\<^sub>\<circ> A" and "x = \<langle>a, f a\<rangle>"
basic_type_less_right_simps_6: fixes \<tau> :: "'a basic_type" shows "(\<tau> < UnlimitedNatural) = (\<tau> = OclVoid)"
f_borel_Suc_expl_cond_expect: fixes p :: "real"   and M :: "bool stream measure"   and F :: "nat \<Rightarrow> bool stream measure"   and f :: "bool stream \<Rightarrow> real"   and n :: "nat"   and g :: "bool stream \<Rightarrow> 'b"   and N :: "'b measure"   and w :: "bool stream" assumes "infinite_cts_filtration p M F"   and "f \<in> borel_measurable (F (Suc n))"   and "g \<in> F n \<rightarrow>\<^sub>M N"   and "set_discriminating n g N"   and "g -` {g w} \<in> sets (F n)"   and "\<forall>y z. g y = g z \<and> y !! n = z !! n \<longrightarrow> f y = f z"   and "0 < p"   and "p < 1" shows "expl_cond_expect M g f w = p * f (pseudo_proj_True n w) + (1 - p) * f (pseudo_proj_False n w)"
NotAllD: fixes v :: "i"   and \<phi> :: "'a \<Rightarrow> \<o>" assumes "\<not> [\<^bold>\<forall>x. \<phi> x in v]" shows "\<exists>x. \<not> [\<phi> x in v]"
well_founded_extend: fixes A :: "'c set"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and B :: "'c set" assumes "well_founded A less"   and "well_founded B less"   and "\<forall>a\<in>A. \<forall>b\<in>B. \<not> less b a" shows "well_founded (A \<union> B) less"
O_def: fixes \<gamma> :: "('a, 'b, 'c) trans \<Rightarrow> bool"   and g :: "('a, 'b, 'c) trans \<Rightarrow> 'd" shows "BD_Security.O \<gamma> g \<equiv> BD_Security.filtermap \<gamma> g"
relto_fun: fixes as :: "nat \<Rightarrow> 'a"   and a :: "'a"   and m :: "nat"   and b :: "'a"   and sel :: "nat \<Rightarrow> bool"   and A :: "('a \<times> 'a) set"   and B :: "('a \<times> 'a) set"   and n :: "nat" assumes "as 0 = a"   and "as m = b"   and "\<And>i. i < m \<Longrightarrow> (sel i \<longrightarrow> (as i, as (Suc i)) \<in> A) \<and> (\<not> sel i \<longrightarrow> (as i, as (Suc i)) \<in> B)"   and "n = card {i. i < m \<and> sel i}"   and "(n = 0) = (m = 0)" shows "relto_fun A B n as sel m (a, b)"
Mp_coeff: fixes m :: "int"   and f :: "int poly"   and i :: "nat" shows "coeff (poly_mod.Mp m f) i = poly_mod.M m (coeff f i)"
atom_step3: fixes r :: "'a"   and l :: "'a"   and a :: "'a"   and b :: "'a"   and q :: "'a" assumes "r \<cdot> l \<le> l \<cdot> r"   and "a \<cdot> l \<le> l \<cdot> a"   and "b \<cdot> l \<le> l \<cdot> b"   and "q \<cdot> l \<le> l \<cdot> q"   and "b\<^sup>\<infinity> = b\<^sup>\<star>" shows "l\<^sup>\<infinity> \<cdot> r\<^sup>\<infinity> \<cdot> (a \<cdot> b\<^sup>\<infinity> \<cdot> q \<cdot> r\<^sup>\<infinity>)\<^sup>\<infinity> = (a \<cdot> b\<^sup>\<infinity> \<cdot> q + l + r)\<^sup>\<infinity>"
po_sw_compP: fixes f :: "String.literal \<Rightarrow> String.literal \<Rightarrow> ty list \<Rightarrow> ty \<Rightarrow> 'd \<Rightarrow> 'c"   and P :: "'d prog" shows "po_sw (compP f P) = po_sw P"
map_simp: fixes f :: "'a" assumes "L.arr f" shows "L.map f = f"
simps_2: fixes x :: "nat \<Rightarrow> bool"   and n :: "nat" shows "count x (Suc n) = (if x (Suc n) then count x n + 1 else count x n)"
stepC_app: fixes n :: "C"   and c :: "nat"   and ns :: "C list" assumes "n \<noteq> C []" shows "stepC c (C (unC n @ ns)) = C (unC (stepC c n) @ ns)"
F2set3_JF2set_JF2set: fixes a' :: "'d JF2"   and a :: "'d JF2"   and c :: "'d" assumes "a' \<in> F2set3 (dtor2 a)"   and "c \<in> JF2set a'" shows "c \<in> JF2set a"
pr_id3_1': shows "pr_conv_3_to_1 (\<lambda>x y z. x) \<in> PrimRec1'"
mbisim_dom_eq: fixes bisim :: "'a \<Rightarrow> 'b \<times> 'c \<Rightarrow> 'd \<times> 'e \<Rightarrow> bool"   and bisim_wait :: "'b \<Rightarrow> 'd \<Rightarrow> bool"   and s1 :: "('f, 'a) locks \<times> (('a \<Rightarrow> ('b \<times> 'f \<Rightarrow>f nat) option) \<times> 'c) \<times> ('a \<Rightarrow> 'g wait_set_status option) \<times> 'a set"   and s2 :: "('f, 'a) locks \<times> (('a \<Rightarrow> ('d \<times> 'f \<Rightarrow>f nat) option) \<times> 'e) \<times> ('a \<Rightarrow> 'g wait_set_status option) \<times> 'a set" assumes "FWbisimulation_base.mbisim bisim bisim_wait s1 s2" shows "dom (thr s1) = dom (thr s2)"
Beta_App_cong_aux_1: fixes t1 :: "lam"   and t2 :: "lam"   and s :: "lam" assumes " t1 \<longrightarrow>\<^sub>\<beta>\<^sup>* t2" shows " App t1 s \<longrightarrow>\<^sub>\<beta>\<^sup>* App t2 s"
add_succ_right: fixes m :: "'a"   and n :: "'a" shows "add m (succ n) = succ (add m n)"
power_inductr: fixes z :: "('b \<times> 'b) set"   and y :: "('b \<times> 'b) set"   and x :: "('b \<times> 'b) set"   and n :: "nat" assumes "z \<union> y O x \<subseteq> y" shows "z O rel_dioid.power x n \<subseteq> y"
lists_eq_set: fixes A :: "'a set" shows "lists A = {xs. set xs \<subseteq> A}"
FunctorCompMapsTo2: fixes f :: "'a"   and F :: "('b, 'g, 'a, 'h, 'c, 'i) Functor"   and G :: "('g, 'd, 'h, 'e, 'i, 'f) Functor" assumes "f \<in> mor\<^bsub>CatDom (F ;;: G)\<^esub>"   and "F \<approx>>;;; G"   and "PreFunctor (F ;;: G)" shows "F ;;: G ## f maps\<^bsub>CatCod (F ;;: G)\<^esub> F ;;: G @@ dom\<^bsub>CatDom (F ;;: G)\<^esub> f to F ;;: G @@ cod\<^bsub>CatDom (F ;;: G)\<^esub> f"
PAC_Format_induct: fixes \<V> :: "nat set"   and A :: "int mpoly multiset"   and \<V>' :: "nat set"   and A' :: "int mpoly multiset"   and P :: "nat set \<Rightarrow> int mpoly multiset \<Rightarrow> nat set \<Rightarrow> int mpoly multiset \<Rightarrow> bool" assumes "PAC_Format (\<V>, A) (\<V>', A')"   and "\<And>p q p' A \<V>. \<lbrakk>p \<in># A; q \<in># A; p + q - p' \<in> More_Modules.ideal polynomial_bool; vars p' \<subseteq> \<V>\<rbrakk> \<Longrightarrow> P \<V> A \<V> (add_mset p' A)"   and "\<And>p q p' A \<V>. \<lbrakk>p \<in># A; p * q - p' \<in> More_Modules.ideal polynomial_bool; vars p' \<subseteq> \<V>; vars q \<subseteq> \<V>\<rbrakk> \<Longrightarrow> P \<V> A \<V> (add_mset p' A)"   and "\<And>p A \<V>. p \<in># A \<Longrightarrow> P \<V> A \<V> (A - {#p#})"   and "\<And>p' x r. \<lbrakk>p'\<^sup>2 - p' \<in> More_Modules.ideal polynomial_bool; vars p' \<subseteq> \<V>; x \<notin> \<V>\<rbrakk> \<Longrightarrow> P \<V> A (\<V> \<union> {x' \<in> vars (p' - Var x). x' \<notin> \<V>}) (add_mset (p' - Var x) A)" shows "P \<V> A \<V>' A'"
asym: fixes b :: "'a"   and a :: "'a" assumes "b \<prec> a"   and "a \<prec> b" shows "False"
intros_3: fixes n :: "nat"   and d :: "nat" shows "i_set_indp [n\<dots>,d]"
spec_imp_SafeStickyGlobVacantUpto: fixes s :: "('c, 'd) configuration stream" assumes "spec s" shows "alw SafeStickyGlobVacantUpto s"
I: shows "\<Phi>.FUN a' (\<Psi>.FUN a' (\<iota> \<chi>')) = \<iota> \<chi>'"
disjoint_fusion: fixes x :: "i" assumes "\<exists>w. \<not> O w x" shows "\<forall>w. P w (\<sigma>z. \<not> O z x) = (\<not> O w x)"
tauChain_def: fixes P :: "ccs"   and P' :: "ccs" shows "P \<Longrightarrow>\<^sub>\<tau> P' \<equiv> (P, P') \<in> {(P, P') |P P'. P \<longmapsto>\<tau> \<prec> P'}\<^sup>*"
simps_2: fixes x :: "'d"   and xs :: "'d list" shows "rev (x # xs) = rev xs @ [x]"
AE_emeasure_singleton: fixes M :: "'b measure"   and x :: "'b"   and P :: "'b \<Rightarrow> bool" assumes "emeasure M {x} \<noteq> 0"   and "almost_everywhere M P" shows "P x"
Entry_unreachable: assumes "True" shows "gen_wf.inEdges gen_cfg_wf (gen_wf_Entry gen_cfg_wf) = []"
n'_shd: fixes N :: "'a stream measure"   and M :: "'a stream measure"   and y :: "'a" assumes "R N M" shows "distr (n' N M y) (count_space UNIV) shd = measure_pmf (return_pmf y)"
is_hom_GB_bound_def: fixes F :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set"   and b :: "nat" shows "is_hom_GB_bound F b = ((\<forall>f\<in>F. homogeneous f) \<longrightarrow> (\<forall>g\<in>punit.reduced_GB F. poly_deg g \<le> b))"
assms_3: shows "\<forall>C\<in>H. atom i \<sharp> C"
shift_snth: fixes xs :: "'a list"   and s :: "'a stream"   and n :: "nat" shows "(xs @- s) !! n = (if n < length xs then xs ! n else s !! (n - length xs))"
method_def: fixes P :: "'a prog"   and C :: "String.literal"   and M :: "String.literal" shows "method P C M \<equiv> THE (D, Ts, T, m). P \<turnstile> C sees M: Ts\<rightarrow>T = m in D"
simps_2: fixes I :: "('d \<times> 'd) set"   and D :: "'c \<Rightarrow> 'd"   and u :: "'d"   and xs :: "'c list"   and x :: "'c" shows "ipurge_tr I D u (xs @ [x]) = (if D x \<in> sinks I D u (xs @ [x]) then ipurge_tr I D u xs else ipurge_tr I D u xs @ [x])"
postSetpostList: fixes xs :: "('a \<times> ('b \<Rightarrow> 'c)) list"   and l :: "'a \<Rightarrow> 'c"   and s :: "'b" assumes "postList xs l s" shows "postSet (set xs) l s"
abs_eq: fixes xa :: "complex"   and x :: "'d" shows "xa *\<^sub>C to_conjugate_space x = to_conjugate_space (cnj xa *\<^sub>C x)"
A2': fixes a :: "'a"   and b :: "'a"   and p :: "'a"   and q :: "'a"   and r :: "'a"   and s :: "'a" assumes "a b \<equiv> p q"   and "a b \<equiv> r s" shows "p q \<equiv> r s"
encode_transitions_def: fixes \<Pi> :: "'b strips_problem"   and t :: "nat" shows "\<Phi>\<^sub>T \<Pi> t \<equiv> encode_operators \<Pi> t \<^bold>\<and> encode_all_frame_axioms \<Pi> t"
cases: fixes a1 :: "'node"   and a2 :: "'edge list"   and a3 :: "'node" assumes "a1 -a2\<rightarrow>* a3" obtains n :: "'node" where "a1 = n__" and "a2 = []" and "a3 = n__" and "valid_node n__"   | n'' :: "'node" and as :: "'edge list" and n' :: "'node" and a :: "'edge" and n :: "'node" where "a1 = n" and "a2 = a # as" and "a3 = n'" and "n'' -as\<rightarrow>* n'" and "valid_edge a" and "sourcenode a = n" and "targetnode a = n''"
project_constrains_project_set: fixes G :: "'c program"   and C :: "'c set"   and B :: "'c set" assumes "G \<in> C co B" shows "project h C G \<in> project_set h C co project_set h B"
horner_eval'_interval: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'b"   and g :: "nat \<Rightarrow> 'b interval"   and x :: "'b"   and I :: "'b interval"   and v :: "'b"   and V :: "'b interval" assumes "\<And>i. i < n \<Longrightarrow> f i \<in>\<^sub>i g i"   and "x \<in>\<^sub>i I"   and "v \<in>\<^sub>i V" shows "horner_eval' f x v n \<in>\<^sub>i horner_eval' g I V n"
abs_conv_abscissa_mult_const_left: fixes c :: "'b"   and f :: "'b fds" shows "abs_conv_abscissa (fds_const c * f) \<le> abs_conv_abscissa f"
DomainE: fixes a :: "'e"   and r :: "('e \<times> 'f) set" assumes "a \<in> Domain r" obtains b :: "'f" where "(a, b) \<in> r"
nofail_SPEC_iff: fixes m :: "'a nres" shows "nofail m = (m \<le> SPEC (\<lambda>_. True))"
F_ndet: fixes P :: "'a process"   and Q :: "'a process" shows "F (P |-| Q) = F P \<union> F Q"
BOX2_def: shows "BOX2 \<equiv> \<lambda>h x. cbox (x - h *\<^sub>R One) (x + h *\<^sub>R One)"
tautology_Rf: fixes A :: "'a"   and C :: "'a literal multiset"   and N :: "'a literal multiset set" assumes "Pos A \<in># C"   and "Neg A \<in># C" shows "C \<in> sr.Rf N"
of_int_eq_0_iff: fixes one :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and z :: "int" assumes "class.ring_char_0 one times plus zero minus uminus" shows "(ring_1.of_int one plus zero minus z = zero) = (z = 0)"
set_cond_bind_pmf: shows "set_pmf local.cond_bind_pmf = {x \<in> set_pmf p. set_pmf (f x) \<inter> A \<noteq> {}}"
stocks_portfolio_risk_neutral: fixes N :: "'a measure"   and pf :: "'b \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> real"   and n :: "nat" assumes "risk_neutral_prob N"   and "trading_strategy pf"   and "subalgebra N M"   and "support_set pf \<subseteq> stocks Mkt"   and "\<forall>n. \<forall>asset\<in>support_set pf. integrable N (\<lambda>w. prices Mkt asset (Suc n) w * pf asset (Suc n) w)" shows "\<forall>x\<in>support_set pf. AEeq N (real_cond_exp N (F n) (discounted_value r (\<lambda>m y. prices Mkt x m y * pf x m y) (Suc n))) (discounted_value r (\<lambda>m y. prices Mkt x m y * pf x (Suc m) y) n)"
l_get_owner_document\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def: fixes get_parent :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr option) prog"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and type_wf :: "(_) 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 :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) document_ptr) prog" shows "l_get_owner_document\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_parent known_ptr type_wf get_disconnected_nodes get_disconnected_nodes_locs get_root_node get_owner_document \<equiv> l_get_disconnected_nodes type_wf get_disconnected_nodes get_disconnected_nodes_locs \<and> l_get_root_node get_root_node get_parent \<and> l_get_owner_document\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms known_ptr get_disconnected_nodes get_root_node get_owner_document"
zero_notin_list_of_oalist: fixes xs :: "('c, 'b) oalist_ntm" shows "(0::'b) \<notin> snd ` set (fst (list_of_oalist_ntm xs))"
\<section>: fixes x :: "'a" assumes "x \<in> (\<Union>x\<in>S. \<Union>y\<in>T. {x - y})" shows "b < a \<bullet> x"
simps: fixes ko :: "'o"   and xs :: "('a \<times> 'c) list"   and ox :: "'o" shows "min_key_val_raw ko (xs, ox) = (if ko = ox then hd else min_list_param (\<lambda>x y. le ko (fst x) (fst y))) xs"
uniform_real_density_def: shows "uniform_real_density \<equiv> \<lambda>(a, b) x. ennreal (if a < b \<and> x \<in> {a..b} then inverse (b - a) else 0)"
eigen: shows "A *\<^sub>v v = e \<cdot>\<^sub>v v"
ife_FAIL_to_ASSERT_cnv: fixes \<Phi> :: "bool"   and m :: "'b nres" shows "(if \<Phi> then m else FAIL) = op_nres_ASSERT_bnd \<Phi> m"
fresh_type: fixes a :: "name"   and T :: "ty" shows "a \<sharp> T"
rep_eq: fixes x :: "('c, 'b) tllist"   and xa :: "('c, 'b) tllist" shows "local.tllist_ord x xa = (case (llist_of_tllist x, terminal x) of (xs1, b1) \<Rightarrow> \<lambda>(xs2, b2). if lfinite xs1 then b1 = b \<and> lprefix xs1 xs2 \<or> xs1 = xs2 \<and> flat_ord b b1 b2 else xs1 = xs2) (llist_of_tllist xa, terminal xa)"
FAccThrow: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and E :: "char list \<Rightarrow> ty option"   and e :: "expr"   and s\<^sub>0 :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and e' :: "expr"   and s\<^sub>1 :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and F :: "char list"   and Cs :: "char list list" assumes "P,E \<turnstile> \<langle>e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>throw e',s\<^sub>1\<rangle>" shows "P,E \<turnstile> \<langle>e\<bullet>F{Cs},s\<^sub>0\<rangle> \<Rightarrow> \<langle>throw e',s\<^sub>1\<rangle>"
add_vert_simps_4: fixes u :: "'a" shows "head (add_vert u) = head G"
commutes_with_composition: fixes x :: "'a"   and y :: "'a" assumes "x \<in> R"   and "y \<in> R" shows "additive.Class (x \<cdot> y) = additive.Class x [\<cdot>] additive.Class y"
lrev_induct: fixes xs :: "'b llist"   and A :: "'b set"   and P :: "'b llist \<Rightarrow> bool" assumes "xs \<in> A\<^sup>\<star>"   and "P LNil"   and "\<And>x xs. \<lbrakk>xs \<in> A\<^sup>\<star>; P xs; x \<in> A\<rbrakk> \<Longrightarrow> P (xs @@ x ## LNil)" shows "P xs"
dvd_0_right: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and a :: "'b" assumes "class.comm_semiring_1 times one plus zero" shows "dvd.dvd times a zero"
of_hypnat_0_less_iff: fixes n :: "nat star" shows "(0 < of_hypnat n) = (0 < n)"
\<phi>_in_hom_2: shows "\<guillemotleft>r\<^sub>0s\<^sub>1.\<phi> : r\<^sub>0 \<star> p\<^sub>1 \<Rightarrow> s\<^sub>1 \<star> p\<^sub>0\<guillemotright>"
normalize_quot_aux_4: fixes b :: "'b"   and a :: "'b" assumes "b \<noteq> (0::'b)" shows "gcd a b * unit_factor b dvd b"
add_lt_neutral: fixes a :: "'a"   and b :: "'a" assumes "a + b < b" shows "a < \<one>"
inject: fixes x1 :: "'b"   and x2 :: "'c"   and x3 :: "nat"   and x4 :: "('b, 'c) SkewBinomialTree list"   and y1 :: "'b"   and y2 :: "'c"   and y3 :: "nat"   and y4 :: "('b, 'c) SkewBinomialTree list" shows "(Node x1 x2 x3 x4 = Node y1 y2 y3 y4) = (x1 = y1 \<and> x2 = y2 \<and> x3 = y3 \<and> x4 = y4)"
del_left_size: fixes t :: "'b tree"   and x :: "'b"   and t' :: "'b tree" assumes "del_left t = (x, t')"   and "t \<noteq> \<langle>\<rangle>" shows "size t = size t' + 1"
extend_distance_fp_def: shows "extend_distance_fp = worec Bonk_Schramm_extension_unfolded_wo extend_distance"
eval_conj_atom: fixes L :: "atomUni list"   and x :: "real" shows "evalUni (list_conj_Uni (map AtomUni L)) x = (\<forall>l\<in>set L. aEvalUni l x)"
lt2_per2__lt: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and A' :: "'a"   and B' :: "'a"   and C' :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.Per Bet Cong A B C"   and "Tarski_neutral_dimensionless.Per Bet Cong A' B' C'"   and "Tarski_neutral_dimensionless.Lt Bet Cong A B A' B'"   and "Tarski_neutral_dimensionless.Lt Bet Cong B C B' C'" shows "Tarski_neutral_dimensionless.Lt Bet Cong A C A' C'"
insert_queue_invar: fixes q :: "('c, 'd) SkewBinomialTree list"   and e :: "'c"   and a :: "'d" assumes "queue_invar q" shows "queue_invar (local.insert e a q)"
pairwise_subset: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and S :: "'b set"   and T :: "'b set" assumes "pairwise P S"   and "T \<subseteq> S" shows "pairwise P T"
has_integral_sum: fixes T :: "'c set"   and f :: "'c \<Rightarrow> 'd \<Rightarrow> 'e"   and i :: "'c \<Rightarrow> 'e"   and S :: "'d set" assumes "finite T"   and "\<And>a. a \<in> T \<Longrightarrow> (f a has_integral i a) S" shows "((\<lambda>x. \<Sum>a\<in>T. f a x) has_integral sum i T) S"
setmonotone_def: fixes f :: "'b set \<Rightarrow> 'b set" shows "setmonotone f = (\<forall>X. X \<subseteq> f X)"
NP_def: shows "NP = Neg ''P'' [Fun ''c'' []]"
Hash_synth: fixes X :: "msg"   and H :: "msg set" assumes "Hash X \<in> synth H" obtains "Hash X \<in> H"   | "X \<in> synth H"
has_finite_products_if_has_finite_limits: fixes I :: "'b set" assumes "\<And>J. finite (Collect (partial_magma.arr J)) \<Longrightarrow> has_limits_of_shape J"   and "finite I"   and "I \<noteq> UNIV" shows "has_products I"
cases: fixes x :: "'a list" obtains "xa__ = []"   | x :: "'a" and ys :: "'a list" where "xa__ = x # ys"
hyps_3: fixes Q :: "pi" assumes "Q \<in> S - {Q'}"   and "Q \<longmapsto> Rs" shows "P \<longmapsto> Rs"
coprime_common_divisor_int: fixes a :: "int"   and b :: "int"   and x :: "int" assumes "coprime a b"   and "x dvd a"   and "x dvd b" shows "\<bar>x\<bar> = 1"
delta_pf_portfolio: fixes N :: "bool stream measure"   and der :: "bool stream \<Rightarrow> real"   and matur :: "nat" shows "portfolio (delta_pf N der matur)"
scopeExtSum: fixes x :: "name"   and P :: "pi"   and Q :: "pi" assumes "x \<sharp> P" shows "<\<nu>x>(P \<oplus> Q) \<sim>\<^sup>s P \<oplus> <\<nu>x>Q"
prems_4: shows "basis_reduction_add_rows_loop i fs (Suc j) = fs'"
stop_no_eval: fixes mds :: "Mode \<Rightarrow> 'Var set"   and mem :: "'Var \<Rightarrow> 'Val"   and c' :: "('Var, 'AExp, 'BExp) Stmt"   and mds' :: "Mode \<Rightarrow> 'Var set"   and mem' :: "'Var \<Rightarrow> 'Val" shows "(\<langle>Stop, mds, mem\<rangle>, \<langle>c', mds', mem'\<rangle>) \<notin> eval\<^sub>w"
convergesE: fixes F :: "'a set set"   and x :: "'a" assumes "F \<longlongrightarrow> x" obtains "nhds x \<subseteq> F"
Diff_Int: fixes A :: "'d set"   and B :: "'d set"   and C :: "'d set" shows "A - B \<inter> C = A - B \<union> (A - C)"
NegChecks: shows "x = \<forall>X\<langle>\<or>\<noteq>: F \<or>\<notin>: G\<rangle>"
ground_subst: fixes t :: "('a, 'c) Term.term"   and \<sigma> :: "'c \<Rightarrow> ('a, 'b) Term.term" shows "ground (t \<cdot> \<sigma>) = (\<forall>x\<in>vars_term t. ground (\<sigma> x))"
indicator_simps_2: fixes x :: "'e"   and S :: "'e set" assumes "x \<notin> S" shows "indicator S x = (0::'f)"
split_list: fixes x :: "'k"   and xs :: "'k list" assumes "x \<in> set xs" shows "\<exists>ys zs. xs = ys @ x # zs"
index_minus_vec_1: fixes i :: "nat"   and v\<^sub>2 :: "'b Matrix.vec"   and v\<^sub>1 :: "'b Matrix.vec" assumes "i < dim_vec v\<^sub>2" shows "(v\<^sub>1 - v\<^sub>2) $v i = v\<^sub>1 $v i - v\<^sub>2 $v i"
sphere_translation: fixes a :: "'a"   and c :: "'a"   and r :: "real" shows "sphere (a + c) r = (+) a ` sphere c r"
ci_stk_offerD: 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 stk' :: "'a val list" assumes "ci_stk_offer ci"   and "ci ins P h stk loc C M pc frs" shows "ci ins P h (stk @ stk') loc C M pc frs"
monad_reader_state_stateT: fixes return :: "'a \<times> 'b \<Rightarrow> 'c"   and bind :: "'c \<Rightarrow> ('a \<times> 'b \<Rightarrow> 'c) \<Rightarrow> 'c"   and ask :: "('d \<Rightarrow> 'c) \<Rightarrow> 'c" assumes "monad_reader return bind ask" shows "monad_reader_state (return_state return) (bind_state bind) (ask_state ask) get_state put_state"
mod_exp_eq: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bit :: "'a \<Rightarrow> nat \<Rightarrow> bool"   and a :: "'a"   and m :: "nat"   and n :: "nat" assumes "class.semiring_bits plus minus zero times one divide modulo bit" shows "modulo (modulo a (power.power one times (numeral.numeral one plus (num.Bit0 num.One)) m)) (power.power one times (numeral.numeral one plus (num.Bit0 num.One)) n) = modulo a (power.power one times (numeral.numeral one plus (num.Bit0 num.One)) (min m n))"
m_part: shows "M = {#y \<in># M. y = x#} + {#y \<in># M. y \<noteq> x#}"
E_wfD_2: fixes G :: "'b graph"   and v :: "'b"   and v' :: "'b" assumes "wf_graph G"   and "(v, v') \<in> edges G" shows "v' \<in> nodes G"
assms_2: fixes c :: "real set" assumes "c \<in> C" shows "c \<in> sets borel"
wfvbind: fixes \<rho> :: "val list"   and \<eta> :: "val set list"   and \<Gamma> :: "(ty \<times> nat) list \<times> nat"   and v :: "val"   and \<tau> :: "ty" assumes "\<turnstile> \<rho>,\<eta> : \<Gamma>"   and "v \<in> T \<tau> \<eta>" shows "\<turnstile> v # \<rho>,\<eta> : push_ty \<tau> \<Gamma>"
is_reducible_imp_is_reducible_lit: fixes Ds :: "'a literal list list"   and C :: "'a literal list"   and C' :: "'a literal list" assumes "reduce Ds C C' \<noteq> C'" shows "\<exists>L\<in>set C'. is_reducible_lit Ds (C @ remove1 L C') L"
Sup_upper: fixes x :: "'a"   and A :: "'a set" assumes "x \<in> A" shows "\<Sqinter> A \<le> x"