hcju/isabelleps huggingface dataset:
ide_char: fixes Obj :: "'a set"   and Hom :: "'a \<Rightarrow> 'a \<Rightarrow> 'b set"   and Id :: "'a \<Rightarrow> 'b"   and Comp :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "('a, 'b) concrete_category.arr" assumes "concrete_category Obj Hom Id Comp" shows "partial_magma.ide (concrete_category.COMP Obj Hom Comp) f = (f \<noteq> concrete_category.Null \<and> concrete_category.Dom f \<in> Obj \<and> concrete_category.Cod f = concrete_category.Dom f \<and> concrete_category.Map f = Id (concrete_category.Dom f))"
poly2_const: fixes a :: "'a"   and x :: "'a"   and y :: "'a" shows "poly2 [:[:a:]:] x y = a"
canonical_fun_def: fixes S :: "'c set"   and is_open :: "'c set \<Rightarrow> bool"   and \<FF> :: "'c set \<Rightarrow> 'd set"   and \<rho> :: "'c set \<Rightarrow> 'c set \<Rightarrow> 'd \<Rightarrow> 'd"   and b :: "'d"   and add_str :: "'c set \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and mult_str :: "'c set \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and zero_str :: "'c set \<Rightarrow> 'd"   and one_str :: "'c set \<Rightarrow> 'd"   and I :: "'c set set"   and U :: "'c set"   and x :: "'d" assumes "direct_lim S is_open \<FF> \<rho> b add_str mult_str zero_str one_str I" shows "direct_lim.canonical_fun \<FF> \<rho> I U x = direct_lim.class_of \<FF> \<rho> I U x"
keys_delete: fixes k :: "'b"   and m :: "('b, 'c) mapping" shows "keys (delete k m) = keys m - {k}"
assms: shows "h \<turnstile> ok insert_before ptr node reference_child"
someI_ex: fixes P :: "'h \<Rightarrow> bool" assumes "\<exists>x. P x" shows "P (SOME x. P x)"
WCFG_induct: fixes x1a :: "cmd"   and x2a :: "w_node"   and x3a :: "(char list \<Rightarrow> val option) edge_kind"   and x4a :: "w_node"   and P :: "cmd \<Rightarrow> w_node \<Rightarrow> (char list \<Rightarrow> val option) edge_kind \<Rightarrow> w_node \<Rightarrow> bool" assumes "x1a \<turnstile> x2a -x3a\<rightarrow> x4a"   and "\<And>prog. P prog (_Entry_) (\<lambda>s. False)\<^sub>\<surd> (_Exit_)"   and "\<And>prog. P prog (_Entry_) (\<lambda>s. True)\<^sub>\<surd> (_ 0 _)"   and "P Skip (_ 0 _) \<Up>id (_Exit_)"   and "\<And>V e. P (V:=e) (_ 0 _) \<Up>\<lambda>s. s(V := interpret e s) (_ 1 _)"   and "\<And>V e. P (V:=e) (_ 1 _) \<Up>id (_Exit_)"   and "\<And>c\<^sub>1 n et n' c\<^sub>2. \<lbrakk>c\<^sub>1 \<turnstile> n -et\<rightarrow> n'; P c\<^sub>1 n et n'; n' \<noteq> (_Exit_)\<rbrakk> \<Longrightarrow> P (c\<^sub>1;; c\<^sub>2) n et n'"   and "\<And>c\<^sub>1 n et c\<^sub>2. \<lbrakk>c\<^sub>1 \<turnstile> n -et\<rightarrow> (_Exit_); P c\<^sub>1 n et (_Exit_); n \<noteq> (_Entry_)\<rbrakk> \<Longrightarrow> P (c\<^sub>1;; c\<^sub>2) n et ((_ 0 _) \<oplus> #:c\<^sub>1)"   and "\<And>c\<^sub>2 n et n' c\<^sub>1. \<lbrakk>c\<^sub>2 \<turnstile> n -et\<rightarrow> n'; P c\<^sub>2 n et n'; n \<noteq> (_Entry_)\<rbrakk> \<Longrightarrow> P (c\<^sub>1;; c\<^sub>2) (n \<oplus> #:c\<^sub>1) et (n' \<oplus> #:c\<^sub>1)"   and "\<And>b c\<^sub>1 c\<^sub>2. P (if (b) c\<^sub>1 else c\<^sub>2) (_ 0 _) (\<lambda>s. interpret b s = Some true)\<^sub>\<surd> ((_ 0 _) \<oplus> 1)"   and "\<And>b c\<^sub>1 c\<^sub>2. P (if (b) c\<^sub>1 else c\<^sub>2) (_ 0 _) (\<lambda>s. interpret b s = Some false)\<^sub>\<surd> ((_ 0 _) \<oplus> #:c\<^sub>1 + 1)"   and "\<And>c\<^sub>1 n et n' b c\<^sub>2. \<lbrakk>c\<^sub>1 \<turnstile> n -et\<rightarrow> n'; P c\<^sub>1 n et n'; n \<noteq> (_Entry_)\<rbrakk> \<Longrightarrow> P (if (b) c\<^sub>1 else c\<^sub>2) (n \<oplus> 1) et (n' \<oplus> 1)"   and "\<And>c\<^sub>2 n et n' b c\<^sub>1. \<lbrakk>c\<^sub>2 \<turnstile> n -et\<rightarrow> n'; P c\<^sub>2 n et n'; n \<noteq> (_Entry_)\<rbrakk> \<Longrightarrow> P (if (b) c\<^sub>1 else c\<^sub>2) (n \<oplus> #:c\<^sub>1 + 1) et (n' \<oplus> #:c\<^sub>1 + 1)"   and "\<And>b c'. P (while (b) c') (_ 0 _) (\<lambda>s. interpret b s = Some true)\<^sub>\<surd> ((_ 0 _) \<oplus> 2)"   and "\<And>b c'. P (while (b) c') (_ 0 _) (\<lambda>s. interpret b s = Some false)\<^sub>\<surd> (_ 1 _)"   and "\<And>b c'. P (while (b) c') (_ 1 _) \<Up>id (_Exit_)"   and "\<And>c' n et n' b. \<lbrakk>c' \<turnstile> n -et\<rightarrow> n'; P c' n et n'; n \<noteq> (_Entry_); n' \<noteq> (_Exit_)\<rbrakk> \<Longrightarrow> P (while (b) c') (n \<oplus> 2) et (n' \<oplus> 2)"   and "\<And>c' n et b. \<lbrakk>c' \<turnstile> n -et\<rightarrow> (_Exit_); P c' n et (_Exit_); n \<noteq> (_Entry_)\<rbrakk> \<Longrightarrow> P (while (b) c') (n \<oplus> 2) et (_ 0 _)" shows "P x1a x2a x3a x4a"
bin_sign_sc: fixes n :: "nat"   and b :: "bool"   and w :: "int" shows "bin_sign (bin_sc n b w) = bin_sign w"
mono_mult: fixes r :: "('b \<times> 'b) set"   and r' :: "('b \<times> 'b) set" assumes "r \<subseteq> r'" shows "mult r \<subseteq> mult r'"
more_than_one_mset_mset_diff: fixes a :: "'b"   and M :: "'b multiset" assumes "a \<in># M - {#a#}" shows "set_mset (M - {#a#}) = set_mset M"
eq_sym: fixes eq :: "'b ltln \<Rightarrow> 'b ltln \<Rightarrow> bool"   and \<phi> :: "'b ltln"   and \<psi> :: "'b ltln" assumes "ltl_equivalence eq"   and "eq \<phi> \<psi>" shows "eq \<psi> \<phi>"
sublinear_intros_3: fixes f :: "'a \<Rightarrow> 'a" shows "sublinear (wp (Apply f))"
HSigmaD1: fixes a :: "hf"   and b :: "hf"   and A :: "hf"   and B :: "hf \<Rightarrow> hf" assumes "\<langle>a, b\<rangle> \<^bold>\<in> HSigma A B" shows "a \<^bold>\<in> A"
intros: fixes v :: "nat"   and ps :: "nat \<times> (form, nat) antecedent" assumes "v |\<in>| vertices"   and "\<And>pth v'. \<lbrakk>task.path v v' pth; task.terminal_vertex v'\<rbrakk> \<Longrightarrow> ps \<in> snd ` set pth" shows "v \<in> task.scope ps"
prems_1: shows "S,kind \<turnstile> (m # ms,s) =as\<Rightarrow>\<^sub>\<tau> (m' # mx # msx,s')"
eucl_rel_int: fixes k :: "int"   and l :: "int" shows "eucl_rel_int k l (k div l, k mod l)"
sep_conj_impl: fixes P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool"   and s :: "'a"   and P' :: "'a \<Rightarrow> bool"   and Q' :: "'a \<Rightarrow> bool" assumes "(P ** Q) s"   and "\<And>s. P s \<Longrightarrow> P' s"   and "\<And>s. Q s \<Longrightarrow> Q' s" shows "(P' ** Q') s"
u_in_hom: shows "\<guillemotleft>u : src u \<rightarrow> src r\<guillemotright>"
nvalid_def: fixes \<Gamma> :: "'e \<Rightarrow> ('f, 'e, 'g) com option"   and n :: "nat"   and F :: "'g set"   and P :: "'f set"   and c :: "('f, 'e, 'g) com"   and Q :: "'f set"   and A :: "'f set" shows "\<Gamma>\<Turnstile>n:\<^bsub>/F\<^esub> P c Q,A \<equiv> \<forall>s t. \<Gamma>\<turnstile> \<langle>c,s\<rangle> =n\<Rightarrow> t \<longrightarrow> s \<in> Normal ` P \<longrightarrow> t \<notin> Fault ` F \<longrightarrow> t \<in> Normal ` Q \<union> Abrupt ` A"
Z_strict_def: shows "local.Z_strict = {f \<in> {0..<n} \<rightarrow> {0..<m}. (\<forall>i. i \<notin> {0..<n} \<longrightarrow> f i = i) \<and> strict_mono_on f {0..<n}} \<times> {\<pi>. \<pi> permutes {0..<n}}"
meet_below1: fixes x :: "'a"   and z :: "'a"   and y :: "'a" assumes "x \<sqsubseteq> z" shows "x \<sqinter> y \<sqsubseteq> z"
known_ptr_defs: fixes ptr :: "(_) object_ptr" shows "DocumentClass.known_ptr ptr = (CharacterDataClass.known_ptr ptr \<or> (case cast ptr of None \<Rightarrow> False | Some document_ptr \<Rightarrow> is_document_ptr document_ptr))"
assms_1: shows "\<GG> : \<BB> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^sub>.\<^sub>t\<^sub>i\<^sub>n\<^sub>y\<^bsub>\<alpha>\<^esub> \<CC>"
pairself_image_prod: fixes f :: "'d \<Rightarrow> 'c"   and A :: "'d set"   and B :: "'d set" shows "pairself f ` (A \<times> B) = f ` A \<times> f ` B"
a_src: shows "a = a1 @ [fst p] @ a2"
nn_integral_dominated_convergence: fixes u :: "nat \<Rightarrow> 'a \<Rightarrow> ennreal"   and M :: "'a measure"   and u' :: "'a \<Rightarrow> ennreal"   and w :: "'a \<Rightarrow> ennreal" assumes "\<And>i. u i \<in> borel_measurable M"   and "u' \<in> borel_measurable M"   and "w \<in> borel_measurable M"   and "\<And>j. AE x in M. u j x \<le> w x"   and "integral\<^sup>N M w < \<infinity>"   and "AE x in M. (\<lambda>i. u i x) \<longlonglongrightarrow> u' x" shows "(\<lambda>i. integral\<^sup>N M (u i)) \<longlonglongrightarrow> integral\<^sup>N M u'"
bidi_2: fixes a :: "'b" assumes "a \<in> arcs G" shows "edge_rev M a \<noteq> a"
P: shows "P ** A = fst (foldl (echelon_form_of_column_k bezout) (A, 0) [0..<Suc k])"
simps_2: fixes v :: "int" shows "fromB (A v) = None"
edom_AEstack: fixes as :: "Arity list"   and S :: "stack_elem list" shows "edom (AEstack as S) \<subseteq> fv S"
TIMES_def: fixes xs :: "'a language list" shows "TIMES xs \<equiv> foldr Times xs One"
dep3_event_permute_4: fixes \<E> :: "'b set"   and \<P> :: "'b set set"   and Q :: "'b set"   and R :: "'b set"   and S :: "'b set"   and x :: "'b" assumes "MinkowskiPrimitive \<E> \<P>"   and "MinkowskiPrimitive.dep3_event \<P> Q R S x" shows "MinkowskiPrimitive.dep3_event \<P> S Q R x"
subgroup_action: fixes G :: "('c, 'd) monoid_scheme"   and \<phi> :: "'c \<Rightarrow> 'e \<Rightarrow> 'e"   and M :: "'e set"   and H :: "'c set" assumes "group_action G \<phi> M"   and "subgroup H G" shows "group_action (G\<lparr>carrier := H\<rparr>) \<phi> M"
mem_mod_low_equal: fixes s1 :: "'b sparc_state"   and s2 :: "'b sparc_state"   and t1 :: "'b sparc_state"   and addr :: "word36"   and val :: "word8"   and t2 :: "'b sparc_state" assumes "low_equal s1 s2 \<and> t1 = mem_mod 10 addr val s1 \<and> t2 = mem_mod 10 addr val s2" shows "low_equal t1 t2"
not_interp_to_Interp_imp_le: fixes A :: "'a"   and C :: "'a literal multiset"   and D :: "'a literal multiset" assumes "A \<notin> interp C"   and "A \<in> Interp D" shows "C \<le> D"
in_set_conv_decomp_last: fixes x :: "'a"   and xs :: "'a list" shows "(x \<in> set xs) = (\<exists>ys zs. xs = ys @ x # zs \<and> x \<notin> set zs)"
WbisT_transT: fixes indis :: "('a \<times> 'a) set"   and c :: "('b, 'c) com"   and d :: "('b, 'c) com"   and tval :: "'b \<Rightarrow> 'a \<Rightarrow> bool"   and aval :: "'c \<Rightarrow> 'a \<Rightarrow> 'a"   and s :: "'a"   and t :: "'a"   and s' :: "'a" assumes "PL_Indis indis"   and "(c, d) \<in> PL_Indis.WbisT tval aval indis"   and "(s, t) \<in> indis"   and "PL.transT tval aval (c, s) s'" obtains t' :: "'a" where "PL.MtransT tval aval (d, t) t'" and "(s', t') \<in> indis"
CHECK_rule: fixes c :: "bool"   and \<Psi> :: "unit \<Rightarrow> bool"   and \<Phi> :: "'c \<Rightarrow> bool"   and e :: "'c" assumes "c \<Longrightarrow> \<Psi> ()"   and "\<not> c \<Longrightarrow> \<Phi> e" shows "CHECK c e \<le> ESPEC \<Phi> \<Psi>"
hyps: fixes y :: "'a"   and b :: "'a"   and c :: "'a" assumes "R\<inverse>\<inverse> y x"   and "R\<^sup>*\<^sup>* y b"   and "R\<^sup>*\<^sup>* y c" shows "\<exists>d. R\<^sup>*\<^sup>* b d \<and> R\<^sup>*\<^sup>* c d"
inv_iso_def: fixes hom :: "('a, 'b, 'e, 'f) digraph_isomorphism" shows "inv_iso hom \<equiv> \<lparr>iso_verts = the_inv_into (verts G) (iso_verts hom), iso_arcs = the_inv_into (arcs G) (iso_arcs hom), iso_head = head G, iso_tail = tail G\<rparr>"
DF_RUN_refine_F: fixes A :: "'a set" shows "DF A \<sqsubseteq>\<^sub>F RUN A"
that: fixes i :: "nat" assumes "limit x = range (suffix i x)"   and "limit y = range (suffix i y)" shows "thesis"
Cons_1: fixes s :: "'a"   and \<alpha> :: "'a"   and t :: "'a" assumes "(s, (\<alpha>, t) # xs) \<in> seq lrs" shows "set (labels (t, xs)) \<subseteq> (ars\<^sup>+)\<inverse> \<down>s {\<alpha>}"
divides_mult_imp_divides: fixes R :: "('c, 'd) ring_scheme"   and a :: "'c"   and b :: "'c" assumes "a divides\<^bsub>Ring_Divisibility.mult_of R\<^esub> b" shows "a divides\<^bsub>R\<^esub> b"
inv_char: fixes x :: "'a"   and y :: "'a" assumes "x \<in> carrier G"   and "y \<in> carrier G"   and "x \<otimes> y = \<one>"   and "y \<otimes> x = \<one>" shows "inv x = y"
connected_distI: fixes c :: "nat \<times> nat \<Rightarrow> 'b"   and v :: "nat"   and d :: "nat"   and v' :: "nat" assumes "Graph.dist c v d v'" shows "Graph.connected c v v'"
U_function_ring_def: fixes R :: "('a, 'b) ring_scheme" shows "U_function_ring R \<equiv> ring R"
le_None: fixes x :: "'b option"   and r :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "(x \<sqsubseteq>\<^bsub>Opt.le r\<^esub> None) = (x = None)"
interior_open: fixes S :: "'b set" assumes "open S" shows "interior S = S"
pd_total: fixes x :: "'n"   and z :: "'n"   and y :: "'n" assumes "x pd\<rightarrow> z"   and "y pd\<rightarrow> z" shows "x pd\<rightarrow> y \<or> y pd\<rightarrow> x"
Rf_Sup_subset_Rf_Liminf: fixes Ns :: "'a literal multiset set llist" assumes "chain (\<triangleright>) Ns" shows "Rf (Sup_llist Ns) \<subseteq> Rf (Liminf_llist Ns)"
are_all_operator_effects_consistent_set: fixes ops :: "('b, 'c) strips_operator_scheme list" shows "are_all_operator_effects_consistent ops = (\<forall>op\<^sub>1\<in>set ops. \<forall>op\<^sub>2\<in>set ops. set (add_effects_of op\<^sub>1) \<inter> set (delete_effects_of op\<^sub>2) = {} \<and> set (delete_effects_of op\<^sub>1) \<inter> set (add_effects_of op\<^sub>2) = {})"
LatticeTransitivity: fixes G :: "'a \<Rightarrow> bool"   and H :: "'b \<Rightarrow> bool"   and F :: "'c \<Rightarrow> bool" shows "\<turnstile> (G \<leadsto> H) \<and> (F \<leadsto> G) \<longrightarrow> (F \<leadsto> H)"
cpu_reg_mod_y_privilege: fixes s' :: "'b sparc_state"   and w :: "word32"   and s :: "'b sparc_state" assumes "s' = cpu_reg_mod w Y s \<and> get_S (cpu_reg_val PSR s) = 0" shows "get_S (cpu_reg_val PSR s') = 0"
vcard_vemptyD: fixes A :: "V" assumes "vcard A = 0" shows "A = 0"
map_of_alist_to_aset: fixes xs :: "('b \<times> 'c) list" assumes "unique_keys_set (set xs)" shows "map_of_aset (set xs) = map_of_alist xs"
flipped_reflections_distinct_lconjseq_eq_lconjseq: fixes ss :: "'w list" assumes "ss \<in> lists S"   and "distinct (lconjseq ss)" shows "flipped_reflections (sum_list ss) = set (lconjseq ss)"
separated_by_disjoint: fixes A :: "'b set"   and B :: "'b set"   and x :: "'b"   and y :: "'b" assumes "separated_by {A, B} x y"   and "A \<inter> B = {}"   and "x \<in> A" shows "y \<in> B"
local_soundness: fixes r :: "'rule"   and s :: "'sequent"   and sl :: "'sequent fset" assumes "r \<in> R"   and "eff r s sl"   and "\<And>s'. s' |\<in>| sl \<Longrightarrow> ssat s'" shows "ssat s"
Scan_apply_setmonotone: fixes x :: "item"   and I :: "item set"   and T :: "(symbol \<times> character list) set"   and k :: "nat" assumes "x \<in> I" shows "x \<in> Scan T k I"
elims: fixes x :: "nat"   and y :: "nat" assumes "len x = y" obtains "x = 0" and "y = 0"   | "x = Suc 0" and "y = 1"   | va :: "nat" where "x = Suc (Suc va)" and "y = Suc (len (Suc (Suc va) div 2))"
isSimplePath_append: fixes s :: "nat"   and p1 :: "(nat \<times> nat) list"   and p2 :: "(nat \<times> nat) list"   and t :: "nat" shows "isSimplePath s (p1 @ p2) t = (\<exists>u. isSimplePath s p1 u \<and> isSimplePath u p2 t \<and> set (pathVertices_fwd s p1) \<inter> set (pathVertices_fwd u p2) = {u})"
dist_triangle5: fixes x :: "'b"   and u :: "'b"   and y :: "'b"   and z :: "'b"   and t :: "'b" shows "dist x u \<le> dist x y + dist y z + dist z t + dist t u"
nb\<^sub>1: shows "\<forall>op\<in>set ops. SAS_Plus_Representation.is_operator_applicable_in I op"
ideal_prod_distr_2: fixes I :: "'a set"   and J :: "'a set"   and K :: "'a set" assumes "ideal I R"   and "ideal J R"   and "ideal K R" shows "(J <+> K) \<cdot> I = J \<cdot> I <+> K \<cdot> I"
uni: shows "{l. length l = m \<and> sum_list l = N} = {l. length l = m \<and> sum_list l = N \<and> hd l = 0} \<union> {l. length l = m \<and> sum_list l = N \<and> hd l \<noteq> 0}"
s\<^sub>1s\<^sub>2: shows "s\<^sub>1 \<leadsto>\<^sup>* s\<^sub>2"
integrable_ln_Gamma_aux: fixes n :: "nat"   and N :: "nat" shows "(\<lambda>x. complex_of_real (pbernpoly n x) / (complex_of_real x + s) ^ n) integrable_on {0..real N}"
test_dist_var1: fixes p :: "'a"   and q :: "'a"   and r :: "'a" assumes "test p"   and "test q"   and "test r" shows "(p + q) \<cdot> (!p + r) = !p \<cdot> q + p \<cdot> r"
strongAppend: 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 Rel' :: "('b \<times> ('a, 'b, 'c) psi \<times> ('a, 'b, 'c) psi) set"   and R :: "('a, 'b, 'c) psi"   and Rel'' :: "('b \<times> ('a, 'b, 'c) psi \<times> ('a, 'b, 'c) psi) set" assumes "\<Psi> \<rhd> P \<leadsto><Rel> Q"   and "\<Psi> \<rhd> Q \<leadsto>[Rel'] R"   and "eqvt Rel''"   and "insertAssertion (extractFrame R) \<Psi> \<hookrightarrow>\<^sub>F insertAssertion (extractFrame Q) \<Psi>"   and "{(\<Psi>, P, R) |\<Psi> P R. \<exists>Q. (\<Psi>, P, Q) \<in> Rel \<and> (\<Psi>, Q, R) \<in> Rel'} \<subseteq> Rel''"   and "\<And>\<Psi> P Q \<Psi>'. (\<Psi>, P, Q) \<in> Rel' \<Longrightarrow> (\<Psi> \<otimes> \<Psi>', P, Q) \<in> Rel'" shows "\<Psi> \<rhd> P \<leadsto><Rel''> R"
map_pair_cong: fixes xs :: "('d \<times> 'e) list"   and f :: "'d \<times> 'e \<Rightarrow> 'd \<times> 'f"   and g :: "'d \<times> 'e \<Rightarrow> 'd \<times> 'f" assumes "\<And>kv. kv \<in> set xs \<Longrightarrow> f kv = g kv" shows "map_pair f xs = map_pair g xs"
sup_continuous_lfp: fixes F :: "'c \<Rightarrow> 'c" assumes "sup_continuous F" shows "lfp F = (SUP i. (F ^^ i) bot)"
CanonicalCat'MapsTo: fixes T :: "(ZF, ZF) Axioms"   and f :: "ZF"   and X :: "ZF"   and Y :: "ZF" assumes "f maps\<^bsub>CanonicalCat' T\<^esub> X to Y"   and "ZFAxioms T" shows "\<exists>ef. f = [X,ef,Y]\<^bsub>T\<^esub> \<and> Sig aS\<^bsub>T\<^esub> \<triangleright> Vx : X \<turnstile> ef : Y"
isomorphic_implies_hpar_4: fixes f :: "'a"   and f' :: "'a" assumes "f \<cong> f'" shows "trg f = trg f'"
LeadsTo_weaken: fixes F :: "'b program"   and A :: "'b set"   and A' :: "'b set"   and B :: "'b set"   and B' :: "'b set" assumes "F \<in> A \<longmapsto>w A'"   and "B \<subseteq> A"   and "A' \<subseteq> B'" shows "F \<in> B \<longmapsto>w B'"
3: shows "\<forall>a. ListMem a as \<longrightarrow> agree (snd a) s"
accI: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b" assumes "\<And>y. r y x \<Longrightarrow> Wellfounded.accp r y" shows "Wellfounded.accp r x"
at_lst_asgd_not_ge: fixes x :: "'a"   and y :: "'a" assumes "carrier \<noteq> {}"   and "x \<in> carrier"   and "y \<in> carrier"   and "x \<notin> at_least_as_good y carrier relation" shows "(x, y) \<notin> relation"
INDEP_def: fixes v :: "'a" shows "INDEP v \<equiv> True"
elims_2: fixes x :: "value"   and xa :: "value" assumes "x < xa" obtains n :: "int" and s :: "String.literal" where "x = Num n" and "xa = Str s"   | s1 :: "String.literal" and s2 :: "String.literal" where "x = Str s1" and "xa = Str s2" and "s1 < s2"   | n1 :: "int" and n2 :: "int" where "x = Num n1" and "xa = Num n2" and "n1 < n2"
has_integral_add: fixes f :: "'c \<Rightarrow> 'd"   and k :: "'d"   and S :: "'c set"   and g :: "'c \<Rightarrow> 'd"   and l :: "'d" assumes "(f has_integral k) S"   and "(g has_integral l) S" shows "((\<lambda>x. f x + g x) has_integral k + l) S"
tab_in_vhom': shows "\<guillemotleft>\<rho> : r\<^sub>1 \<Rightarrow> r \<star> r\<^sub>0\<guillemotright>"
inv_locate_a_Bk: fixes as :: "nat"   and am :: "nat list"   and q :: "nat"   and aaa :: "cell list"   and ires :: "cell list" assumes "inv_locate_a (as, am) (q, aaa, []) ires" shows "inv_locate_a (as, am) (q, aaa, [Bk]) ires"
strongSimWeakSim: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi"   and Rel :: "('b \<times> ('a, 'b, 'c) psi \<times> ('a, 'b, 'c) psi) set" assumes "(\<Psi>, P, Q) \<in> Rel"   and "\<And>\<Psi>' R S. (\<Psi>', R, S) \<in> Rel \<Longrightarrow> insertAssertion (extractFrame S) \<Psi>' \<hookrightarrow>\<^sub>F insertAssertion (extractFrame R) \<Psi>'"   and "\<And>\<Psi>' R S. (\<Psi>', R, S) \<in> Rel \<Longrightarrow> \<Psi>' \<rhd> R \<leadsto>[Rel] S"   and "\<And>\<Psi>' R S \<Psi>''. (\<Psi>', R, S) \<in> Rel \<Longrightarrow> (\<Psi>' \<otimes> \<Psi>'', R, S) \<in> Rel" shows "\<Psi> \<rhd> P \<leadsto><Rel> Q"
has_derivative: fixes g :: "'c \<Rightarrow> 'a"   and g' :: "'c \<Rightarrow> 'a"   and F :: "'c filter" assumes "(g has_derivative g') F" shows "((\<lambda>x. f (g x)) has_derivative (\<lambda>x. f (g' x))) F"
ret_path_none_pd: fixes prob :: "('a, 'b, 'c, 'd) ifc_problem"   and x :: "'a" assumes "IFC prob"   and "x \<in> IFC_def.nodes prob"   and "x \<noteq> IFC_def.return prob" obtains \<pi> :: "nat \<Rightarrow> 'a" and n :: "nat" where "IFC_def.is_path prob \<pi>" and "\<pi> 0 = x" and "\<pi> n = IFC_def.return prob" and "\<forall>i>0. \<not> IFC_def.is_pd prob x (\<pi> i)"
diff_enat_def: fixes a :: "enat"   and b :: "enat" shows "a - b = (case a of enat x \<Rightarrow> case b of enat y \<Rightarrow> enat (x - y) | \<infinity> \<Rightarrow> 0 | \<infinity> \<Rightarrow> \<infinity>)"
append_is_polynomial: fixes K :: "'a set"   and p :: "'a list"   and n :: "nat" assumes "subring K R"   and "polynomial K p"   and "p \<noteq> []" shows "polynomial K (p @ replicate n \<zero>)"
c_x_prop: fixes x :: "('a \<times> 'a set) set" shows "1\<^sub>\<pi> \<cdot> x = 1\<^sub>\<pi>"
sbelows_rel_internal: fixes A :: "('c \<times> 'd) set" shows "sbelows_rel A = \<langle>\<langle>A\<rangle>sctn_rel\<rangle>list_set_rel O br sbelow_halfspaces top"
vfsequence_vremove_vcons_vfsequence: fixes xs' :: "V"   and x :: "V" assumes "dg_Rel \<alpha> = xs' #\<^sub>\<circ> x" shows "vfsequence xs'"
FG_advice_minus_subformulas: fixes \<psi> :: "'b ltln"   and \<phi> :: "'b ltln"   and Y :: "'b ltln set" assumes "\<psi> \<notin> subformulas\<^sub>\<nu> \<phi>" shows "\<phi>[Y - {\<psi>}]\<^sub>\<mu> = \<phi>[Y]\<^sub>\<mu>"
R: fixes B :: "'a set"   and U :: "'a set" assumes "A \<subseteq> B"   and "B open"   and "U \<in> nhds x"   and "B \<inter> U = {}" shows "R"
exist_path_length_def: fixes v :: "'v"   and l :: "nat" shows "exist_path_length v l \<equiv> \<exists>v' ps. is_trail v' ps v \<and> length ps = l"
perp_exists: fixes A :: "'p"   and B :: "'p"   and PO :: "'p" assumes "A \<noteq> B" shows "\<exists>X. PO X Perp A B"
cong_equivp: fixes rel :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and eval :: "'b \<Rightarrow> 'a"   and retr :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and R :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "cong rel eval retr"   and "cong.cong rel eval R" shows "equivp R"
Supq_distl: fixes x :: "'a"   and Y :: "'a set" shows "x \<cdot> \<Squnion> Y = \<Squnion> ((\<cdot>) x ` Y)"
alphaAbs_qAbs_iff_all_equal_or_qAFresh: fixes X :: "('d, 'e, 'varSort, 'var, 'f) qTerm"   and X' :: "('d, 'e, 'varSort, 'var, 'f) qTerm"   and xs :: "'varSort"   and x :: "'var"   and xs' :: "'varSort"   and x' :: "'var" assumes "qGood X"   and "qGood X'" shows "(qAbs xs x X $= qAbs xs' x' X') = (xs = xs' \<and> (\<forall>y. (y = x \<or> qAFresh xs y X) \<and> (y = x' \<or> qAFresh xs y X') \<longrightarrow> X #[[y \<and> x]]_xs #= X' #[[y \<and> x']]_xs))"
uminus_mat: fixes A :: "complex mat"   and n :: "nat" assumes "A \<in> carrier_mat n n" shows "- A = - 1 \<cdot>\<^sub>m A"
U: shows "UNIV = (\<lambda>((a, b), c). \<lparr>es = a, ps = b, \<dots> = c\<rparr>) ` ((UNIV \<times> UNIV) \<times> UNIV)"
reads_singleton: fixes f :: "('x, 'y, 'z) prog"   and h :: "'x"   and h' :: "'x" shows "reads {preserved f} f h h'"
single_row_inv: fixes j' :: "nat"   and j :: "nat"   and n :: "nat"   and i' :: "nat"   and m :: "nat \<Rightarrow> nat \<Rightarrow> 'a"   and k :: "nat" assumes "j' < j"   and "j \<le> n"   and "i' \<le> n" shows "fw m n k i' j i' j' = fw m n k i' j' i' j'"
insert_Diff_if: fixes x :: "'d"   and A :: "'d set"   and B :: "'d set" shows "insert x A - B = (if x \<in> B then A - B else insert x (A - B))"
simps_2: fixes lc :: "'b"   and q :: "'b poly"   and r :: "'b poly"   and d :: "'b poly"   and dr :: "nat" shows "divide_poly_main lc q r d dr 0 = q"
infsetsum_altdef: fixes f :: "'e \<Rightarrow> 'd"   and A :: "'e set" shows "infsetsum f A = set_lebesgue_integral (count_space UNIV) A f"
HcompNml_Nml: fixes t :: "'a local.term"   and u :: "'a local.term" assumes "Nml (t \<^bold>\<star> u)" shows "t \<^bold>\<lfloor>\<^bold>\<star>\<^bold>\<rfloor> u = t \<^bold>\<star> u"
inf_commute: fixes inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "class.semilattice_inf inf less_eq less" shows "inf x y = inf y x"