hcju/isabelleps huggingface dataset:
assoc'_is_natural_1: fixes \<mu> :: "'a"   and \<nu> :: "'a"   and \<tau> :: "'a" assumes "arr \<mu>"   and "arr \<nu>"   and "arr \<tau>"   and "src \<mu> = trg \<nu>"   and "src \<nu> = trg \<tau>" shows "\<a>\<^sup>-\<^sup>1[\<mu>, \<nu>, \<tau>] = ((\<mu> \<star> \<nu>) \<star> \<tau>) \<cdot> \<a>\<^sup>-\<^sup>1[local.dom \<mu>, local.dom \<nu>, local.dom \<tau>]"
vint_plus_closed: fixes x :: "V"   and y :: "V" assumes "x \<in>\<^sub>\<circ> \<int>\<^sub>\<circ>"   and "y \<in>\<^sub>\<circ> \<int>\<^sub>\<circ>" shows "x +\<^sub>\<int> y \<in>\<^sub>\<circ> \<int>\<^sub>\<circ>"
sep_add_zero_sym: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and sep_disj :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c" assumes "class.sep_algebra plus zero sep_disj" shows "plus zero x = x"
simps_2: fixes x21 :: "'b"   and x22 :: "'b"   and x23 :: "'c" shows "Null \<noteq> MkArr x21 x22 x23"
set_update_distinct: fixes xs :: "'g list"   and n :: "nat"   and x :: "'g" assumes "distinct xs"   and "n < length xs" shows "set (xs[n := x]) = insert x (set xs - {xs ! n})"
fields_Object: fixes G :: "prog" assumes "wf_prog G" shows "DeclConcepts.fields G Object = []"
lub_upper: fixes A :: "'b llist set"   and x :: "'b llist" assumes "Complete_Partial_Order.chain lprefix A"   and "x \<in> A" shows "lprefix x (lSup A)"
simps_1: fixes l :: "'d tree"   and v :: "'d"   and r :: "'d tree" shows "left \<langle>l, v, r\<rangle> = l"
v11_vrrestriction: fixes A :: "V" shows "v11 (r \<restriction>\<^sup>r\<^sub>\<circ> A)"
a1: shows "s' = snd (fst (save_restore_instr instr s)) \<and> get_S (cpu_reg_val PSR s) = 0"
present_alloc: fixes xs :: "'a list"   and h :: "heap" shows "Array.present (snd (Array.alloc xs h)) (fst (Array.alloc xs h))"
cblinfun_extension_exists_def: fixes S :: "'a set"   and \<phi> :: "'a \<Rightarrow> 'b" shows "cblinfun_extension_exists S \<phi> = (\<exists>B. \<forall>x\<in>S. B *\<^sub>V x = \<phi> x)"
continuous_on_at_right: fixes a :: "'a"   and b :: "'a"   and f :: "'a \<Rightarrow> 'b" assumes "continuous_on {a..b} f"   and "a < b" shows "continuous (at_right a) f"
unfold: shows "kvs = take n kvs @ (k, v) # kvs'"
fresh_3: fixes a :: "name"   and x2 :: "'a"   and x1 :: "('a, 'b, 'c) input" shows "a \<sharp> x2\<lparr>x1 = (a \<sharp> x2 \<and> a \<sharp> x1)"
chamber_system_def: shows "domain.\<C> \<equiv> Collect domain.chamber"
select_convs_6: fixes Functions :: "'b \<Rightarrow> (real, 'a) vec \<Rightarrow> real"   and Predicates :: "'a \<Rightarrow> (real, 'a) vec \<Rightarrow> bool"   and Contexts :: "'c \<Rightarrow> ((real, 'a) vec \<times> (real, 'a) vec) set \<Rightarrow> ((real, 'a) vec \<times> (real, 'a) vec) set"   and Programs :: "'a \<Rightarrow> (((real, 'a) vec \<times> (real, 'a) vec) \<times> (real, 'a) vec \<times> (real, 'a) vec) set"   and ODEs :: "'a \<Rightarrow> (real, 'a) vec \<Rightarrow> (real, 'a) vec"   and ODEBV :: "'a \<Rightarrow> 'a set"   and more :: "'d" shows "ODEBV \<lparr>Functions = Functions, Predicates = Predicates, Contexts = Contexts, Programs = Programs, ODEs = ODEs, ODEBV = ODEBV, \<dots> = more\<rparr> = ODEBV"
usubstappp_choice: fixes \<sigma> :: "(char \<Rightarrow> trm option) \<times> (char \<Rightarrow> trm option) \<times> (char \<Rightarrow> fml option) \<times> (char \<Rightarrow> game option)"   and U :: "variable set"   and \<alpha> :: "game"   and \<beta> :: "game" shows "usubstappp \<sigma> U (\<alpha> \<union>\<union> \<beta>) = (fst (usubstappp \<sigma> U \<alpha>) \<union> fst (usubstappp \<sigma> U \<beta>), Choiceo (snd (usubstappp \<sigma> U \<alpha>)) (snd (usubstappp \<sigma> U \<beta>)))"
SPEC_BallI: fixes f :: "'a nres"   and X :: "'b set"   and P :: "'b \<Rightarrow> 'a \<Rightarrow> bool" assumes "nofail f"   and "\<And>x. x \<in> X \<Longrightarrow> f \<le> SPEC (P x)" shows "f \<le> SPEC (\<lambda>r. \<forall>x\<in>X. P x r)"
snoc: shows "(find_indices x Vs = []) = (x \<notin> set Vs)"
comp_fun_idem_sup: fixes sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "class.semilattice_sup sup less_eq less" shows "comp_fun_idem sup"
list_order_insert_between: fixes ops :: "('a \<times> 'a option) list"   and oid :: "'a"   and ref :: "'a"   and x :: "'a" assumes "insert_ops (ops @ [(oid, Some ref)])"   and "list_order ops ref x" shows "list_order (ops @ [(oid, Some ref)]) oid x"
Frame: fixes P :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool"   and C :: "com"   and Q :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool"   and F :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool" assumes "\<turnstile>\<^sub>3\<^sub>' {P} C { Q}" shows "\<turnstile>\<^sub>3\<^sub>' {P \<and>* F} C { Q \<and>* F}"
one_in_kernel: shows "\<one>\<^bsub>DirProds (\<lambda>i. G\<lparr>carrier := S i\<rparr>) I\<^esub> \<in> kernel (DirProds (\<lambda>i. G\<lparr>carrier := S i\<rparr>) I) (G\<lparr>carrier := J\<rparr>) (\<lambda>x. finprod G x I)"
simps_5: fixes x6 :: "'a OCL_Types.type"   and y6 :: "'a OCL_Types.type" shows "(OrderedSet x6 = OrderedSet y6) = (x6 = y6)"
Bound: fixes a :: "name"   and x :: "name"   and Q' :: "pi" assumes "Q \<longmapsto> a<\<nu>x> \<prec> Q'"   and "x \<sharp> P"   and "x \<sharp> Q"   and "x \<noteq> a"   and "x \<sharp> C" shows "\<exists>P'. P \<Longrightarrow>a<\<nu>x> \<prec> P' \<and> (P', Q') \<in> Rel"
mult_2_right: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and z :: "'a" assumes "class.semiring_numeral one times plus" shows "times z (numeral.numeral one plus (num.Bit0 num.One)) = plus z z"
natural_transformation_axioms: shows "natural_transformation (\<cdot>\<^sup>o\<^sup>p) (\<cdot>\<^sub>S) F (\<Phi>'.Y a) \<Psi>.map"
single_valuedp_single_valued_eq: fixes r :: "('b \<times> 'c) set" shows "single_valuedp (\<lambda>x y. (x, y) \<in> r) = single_valued r"
simps_22: fixes f1 :: "'a"   and f2 :: "'a"   and f3 :: "'a"   and f4 :: "'a"   and f5 :: "pdf_type \<Rightarrow> pdf_type \<Rightarrow> 'a" shows "(case UNIT of UNIT \<Rightarrow> f1 | BOOL \<Rightarrow> f2 | INTEG \<Rightarrow> f3 | REAL \<Rightarrow> f4 | PRODUCT x xa \<Rightarrow> f5 x xa) = f1"
assms_2: shows "I (dom (\<alpha> m)) \<sigma>0"
out546__sams: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and E :: "'p"   and D :: "'p"   and F :: "'p" assumes "A \<noteq> B"   and "B \<noteq> C"   and "E Out D F" shows "SAMS A B C D E F"
sets_Sup_in_sets: fixes M :: "'b measure set"   and N :: "'b measure" assumes "M \<noteq> {}"   and "\<And>m. m \<in> M \<Longrightarrow> space m = space N"   and "\<And>m. m \<in> M \<Longrightarrow> sets m \<subseteq> sets N" shows "sets (\<Squnion> M) \<subseteq> sets N"
delete_Some_x_set_of: fixes x :: "'b"   and t :: "'b tree"   and t' :: "'b tree" assumes "delete x t = Some t'" shows "x \<in> set_of t \<and> x \<notin> set_of t'"
dist_measure_has_density: fixes v :: "val"   and dst :: "pdf_dist" assumes "v \<in> type_universe (dist_param_type dst)" shows "has_density (dist_measure dst v) (stock_measure (dist_result_type dst)) (dist_dens dst v)"
igSwapIGAbs_def: fixes MOD :: "('c, 'd, 'varSort, 'sort, 'e, 'f, 'g, 'h, 'i) Iteration.model_scheme" shows "igSwapIGAbs MOD \<equiv> \<forall>zs z1 z2 xs x s X. isInBar (xs, s) \<and> igWls MOD s X \<longrightarrow> igSwapAbs MOD zs z1 z2 (igAbs MOD xs x X) = igAbs MOD xs (x @xs[z1 \<and> z2]_zs) (igSwap MOD zs z1 z2 X)"
is_ubD: fixes S :: "'a set"   and u :: "'a"   and x :: "'a" assumes "S <| u"   and "x \<in> S" shows "x \<sqsubseteq> u"
lm096: fixes x :: "'c"   and xs :: "'c list"   and Y :: "'d list" shows "set (injectionsAlg (x # xs) Y) = (\<Union>f\<in>set (injectionsAlg xs Y). {f \<union> {(x, y)} |y. y \<in> set Y - Range f})"
sel_rews_3: fixes u :: "'a\<^sub>\<bottom>" shows "unbox\<cdot>(Box\<cdot>u) = u"
fresh_swap_compose: fixes X :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and xs :: "'varSort"   and y :: "'var"   and z :: "'var"   and x :: "'var" assumes "good X"   and "fresh xs y X"   and "fresh xs z X" shows "X #[y \<and> x]_xs #[z \<and> y]_xs = X #[z \<and> x]_xs"
Int_mono: fixes A :: "'b set"   and C :: "'b set"   and B :: "'b set"   and D :: "'b set" assumes "order.greater_eq C A"   and "order.greater_eq D B" shows "order.greater_eq (C \<inter> D) (A \<inter> B)"
assms_1: fixes a :: "'a"   and b :: "'a"   and c :: "'a"   and d :: "'a" assumes "P a b c d" shows "P a b d c"
exp_ge_add_one_self: fixes x :: "real" shows "1 + x \<le> exp x"
eqdE: fixes x :: "'b list"   and y :: "'b list"   and u :: "'b list"   and v :: "'b list" assumes "x \<cdot> y = u \<cdot> v"   and "\<^bold>|x\<^bold>| \<le> \<^bold>|u\<^bold>|" obtains t :: "'b list" where "x \<cdot> t = u" and "t \<cdot> v = y"
finite_UNIV_surj_inj: fixes f :: "'c \<Rightarrow> 'c" assumes "finite UNIV"   and "surj f" shows "inj f"
basis_wf_insert_ln_3: fixes b :: "real \<Rightarrow> real" assumes "basis_wf [b]" shows "is_expansion (MS (MSLCons (1, 1) MSLNil) (\<lambda>x. ln (b x))) [\<lambda>x. ln (b x)]"
surjective: shows "e2.Class ` Sigma I \<FF>2 = e2.Partition"
transfer_bounded_sesquilinear_bounded_antilinearI: fixes g :: "'a \<Rightarrow> 'b \<Rightarrow> 'c"   and f :: "'a \<Rightarrow> 'b \<Rightarrow>\<^sub>C\<^sub>L 'c" assumes "g = (\<lambda>i. cblinfun_apply (f i))" shows "bounded_sesquilinear g = bounded_antilinear f"
distinct_insort: fixes x :: "('b, 'c) Combinators"   and xs :: "('b, 'c) Combinators list"   and l :: "'b set list" shows "distinct (FWNormalisationCore.insort x xs l) = (x \<notin> set xs \<and> distinct xs)"
funct: shows "funct_OT_14 msgs (c0, c1) \<bind> (\<lambda>(o1, o2). S1_14_OT msgs () \<bind> D) = S1_14_OT msgs () \<bind> D"
of_rat_3: fixes x :: "rat" shows "real_of_3 (of_rat_3 x) = real_of_rat x"
prob_Collect_eq_1: fixes s :: "'a cfg"   and P :: "'a cfg stream \<Rightarrow> bool" assumes "{x \<in> space (MC.T s). P x} \<in> MC.T.events s" shows "(MC.T.prob s {x \<in> space (MC.T s). P x} = 1) = almost_everywhere (MC.T s) P"
comp_inj_on: fixes f :: "'e \<Rightarrow> 'f"   and A :: "'e set"   and g :: "'f \<Rightarrow> 'g" assumes "inj_on f A"   and "inj_on g (f ` A)" shows "inj_on (g \<circ> f) A"
zero: fixes x :: "'d" shows "0 *\<^sub>C x = (0::'d)"
A: shows "const (P (\<lambda>_. OclAllInstances_generic pre_post Type (mk \<lparr>heap = \<sigma>'(oid \<mapsto> Object), assocs = A\<rparr>)))"
length_eq_distinct: fixes xs :: "'c list"   and ys :: "'c list" assumes "set xs = set ys"   and "distinct xs"   and "length xs = length ys" shows "distinct ys"
DSourcesA22_L1: shows "DSources level1 sA22 = {sA11, sA41}"
ge: fixes i :: "'a" assumes "i \<in> K" shows "g i \<le> f i"
assms: fixes i :: "'b" assumes "i \<in> I" shows "((\<lambda>x. f x i) \<longlongrightarrow> (0::'c)) F"
simps_2: fixes x :: "'d"   and xs :: "'d list" shows "distinct (x # xs) = (x \<notin> set xs \<and> distinct xs)"
alw_shift: fixes \<phi> :: "'a stream \<Rightarrow> bool"   and xl :: "'a list"   and xs :: "'a stream" assumes "alw \<phi> (xl @- xs)" shows "alw \<phi> xs"
assms: shows "\<exists>c\<^sub>1>0. \<exists>c\<^sub>2>0. \<exists>n. \<forall>m>n. f m \<le> c\<^sub>1 * g m \<and> g m \<le> c\<^sub>2 * f m"
face_of_Int_subface: fixes A :: "'b set"   and B :: "'b set"   and C :: "'b set"   and D :: "'b set" assumes "A \<inter> B face_of A"   and "A \<inter> B face_of B"   and "C face_of A"   and "D face_of B" shows "C \<inter> D face_of C \<and> C \<inter> D face_of D"
diamond_cube_gen_boundary_valid: fixes k :: "int"   and \<gamma> :: "real \<Rightarrow> real \<times> real" assumes "(k, \<gamma>) \<in> boundary diamond_cube_gen" shows "valid_path \<gamma>"
simps_1: fixes \<S> :: "nat \<Rightarrow> bool option" shows "non_volatile_writes_unshared \<S> [] = True"
left_treeE: fixes u :: "'a"   and s :: "'a"   and t :: "'a" assumes "u \<in> left_tree s t" shows "u \<in> V \<and> s \<in> set (u \<leadsto> t)"
inv_equality: fixes G :: "('e, 'f) monoid_scheme"   and y :: "'e"   and x :: "'e" assumes "Group.group G"   and "y \<otimes>\<^bsub>G\<^esub> x = \<one>\<^bsub>G\<^esub>"   and "x \<in> carrier G"   and "y \<in> carrier G" shows "inv\<^bsub>G\<^esub> x = y"
sel_4: fixes x21 :: "('c, 'd) tree"   and x22 :: "'c"   and x23 :: "'d"   and x24 :: "('c, 'd) tree" shows "rsub (Node x21 x22 x23 x24) = x24"
exhaust: fixes y :: "float10" obtains x1 :: "int" and x2 :: "int" where "y = x1 \<e> x2"
typ_of_eq_ax_1: shows "typ_of eq_reflexive_ax = Some propT"
hd_coeffs1_def: fixes \<phi> :: "atom fm" shows "hd_coeffs1 \<phi> = (let m = zlcms (map hd_coeff (atoms\<^sub>0 \<phi>)) in And (Atom (Dvd m 0 [1])) (map\<^sub>f\<^sub>m (hd_coeff1 m) \<phi>))"
fuse_converter: fixes core :: "('i, 'k, 'a, 'c, 'e, 'g) core"   and rest :: "('j, 'k, 'b, 'd, 'f, 'h, 'l) rest_scheme"   and s_core :: "'i"   and s_rest :: "'j" shows "RES (fused_resource.fuse core rest) (s_core, s_rest) = fuse_converter \<rhd> RES (callee_of_core core) s_core \<parallel> RES (callee_of_rest rest) s_rest"
assms: fixes y :: "'a" shows "mcont luba orda lub_spmf (ord_spmf (=)) (g y)"
len_lexext_iff_lenlex: fixes gt :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and ys :: "'b list"   and xs :: "'b list" shows "len_lexext gt ys xs = ((xs, ys) \<in> lenlex {(x, y). gt y x})"
CHECK_def: fixes f :: "unit \<Rightarrow> 'a"   and P :: "bool" shows "CHECK f P = (if P then RETURN () else let _ = f () in SUCCEED)"
unwrapS_unwrapS'_eq: shows "unwrapS = unwrapS'"
binop_v11_def: fixes A :: "V"   and f :: "V" shows "binop_v11 A f \<equiv> nop_v11 A (2\<^sub>\<nat>) f"
prems_2: shows "n \<notin> fst (pn_atoms (F \<^bold>\<or> G))"
l2: shows "u ^ 3 + 3 * u * v\<^sup>2 * s\<^sup>2 + a * u\<^sup>2 + a * v\<^sup>2 * s\<^sup>2 + b * u + c + (3 * u\<^sup>2 * v + v ^ 3 * s\<^sup>2 + 2 * a * u * v + b * v) * s = 0"
stabilizer_def: fixes m :: "'a set" shows "H'act.stabilizer m = {g \<in> carrier H'. \<phi> g m = m}"
simps: fixes a1 :: "'opSym"   and a2 :: "'bindex \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set option" shows "wlsBinp a1 a2 = (\<exists>delta binp. a1 = delta \<and> a2 = binp \<and> wlsOpS delta \<and> (\<forall>x. barOf delta x = None \<and> binp x = None \<or> (\<exists>us s A. barOf delta x = Some (us, s) \<and> binp x = Some A \<and> wlsAbs (us, s) A)))"
verit_eq_simplify_6: fixes x :: "'e"   and y :: "'e" assumes "x = y" shows "x \<le> y"
wf_net_tree_children: fixes p1 :: "net_tree"   and p2 :: "net_tree" assumes "wf_net_tree (p1 \<parallel> p2)" obtains "wf_net_tree p1" and "wf_net_tree p2"
bounded_linear_compose: fixes f :: "'b \<Rightarrow> 'c"   and g :: "'d \<Rightarrow> 'b" assumes "bounded_linear f"   and "bounded_linear g" shows "bounded_linear (\<lambda>x. f (g x))"
sc_linear_span: fixes A :: "'b set"   and H :: "'a set"   and a :: "'b"   and h :: "'a" assumes "ideal R A"   and "H \<subseteq> carrier M"   and "a \<in> A"   and "h \<in> H" shows "a \<cdot>\<^sub>s h \<in> linear_span R M A H"
not_eq_order_implies_strict: fixes a :: "'b zmultiset"   and b :: "'b zmultiset" assumes "a \<noteq> b"   and "b \<subseteq>#\<^sub>z a" shows "b \<subset>#\<^sub>z a"
sp_from_old_verts_imp_sp_in_old: fixes g :: "('d \<times> nat, 'e) rgraph_scheme"   and subs :: "(('d \<times> nat) \<times> 'd \<times> nat) set"   and e :: "('d \<times> nat) edge"   and g' :: "('d \<times> nat, 'e) rgraph_scheme"   and v1 :: "'d \<times> nat"   and v2 :: "'d \<times> nat"   and es :: "('d \<times> nat) edge list" assumes "sub_rel_of g subs"   and "ArcExt.extends g e g'"   and "v1 \<in> Graph.vertices g"   and "v2 \<in> Graph.vertices g"   and "SubRel.subpath g' v1 es v2 subs" shows "SubRel.subpath g v1 es v2 subs"
iprev_append: fixes A :: "nat set"   and B :: "nat set"   and n :: "nat" assumes "finite A"   and "A \<noteq> {}"   and "B \<noteq> {}"   and "Max A < iMin B" shows "iprev n (A \<union> B) = (if n \<in> A then iprev n A else if n = iMin B then Max A else iprev n B)"
IDirProds_eq_finprod_PiE: fixes I :: "'d set"   and S :: "'d \<Rightarrow> 'a set" assumes "finite I"   and "\<And>i. i \<in> I \<Longrightarrow> subgroup (S i) G" shows "IDirProds G S I = (\<lambda>x. finprod G x I) ` Pi\<^sub>E I S"
eq_onp_same_args: fixes P :: "'e \<Rightarrow> bool"   and x :: "'e" shows "eq_onp P x x = P x"
char_to_digit_simps_1: shows "char_to_digit CHR ''0'' = 0"
Gramian_determinant_ge1: fixes k :: "nat" assumes "k \<le> length fs" shows "(1::'a) \<le> Gramian_determinant fs k"
acc_relfunD: fixes s :: "'s"   and s' :: "'s" assumes "(s, s') \<in> acc" obtains n :: "nat" where "((\<lambda>a b. b \<in> set_pmf (K a)) ^^ n) s s'"
bt_val_augment_1: fixes t :: "('d, 'c) bintree"   and q :: "('c, 'd) pq" shows "set (bt_dfs val t) \<union> set |q| = set |bt_augment t q|"
colosless_gpv_catch_gpv_conv_map: fixes gpv :: "('f, 'd, 'e) gpv" assumes "colossless_gpv \<I>_full gpv" shows "catch_gpv gpv = map_gpv Some id gpv"
seq: shows "\<sigma> \<Turnstile> bind\<^sub>S\<^sub>E (mbind\<^sub>F\<^sub>a\<^sub>i\<^sub>l\<^sub>P\<^sub>u\<^sub>r\<^sub>g\<^sub>e (a # S) ioprog) P"
r_findr_prim: shows "prim_recfn 1 r_findr"
PROTECT2_def: fixes x :: "'d"   and y :: "prop" shows "PROTECT2 x PROP y \<equiv> x"
sq_mtx_inv_one: shows "1\<^sup>-\<^sup>1 = 1"
right_inverse: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and z :: "'b"   and inverse :: "'b \<Rightarrow> 'b"   and a :: "'b" assumes "group f z inverse" shows "f a (inverse a) = z"
heads_pos_iff: fixes xs :: "bool list" shows "(0 < heads xs) = (True \<in> set xs)"
simps_2: fixes s :: "nat \<Rightarrow> 'd"   and k :: "nat" shows "inf_truncate s (Suc k) = inf_truncate s k @ [s (Suc k)]"
notI_PL: fixes A :: "'a set"   and p :: "'a" assumes "A \<turnstile> p .\<longrightarrow>. \<bottom>" shows "A \<turnstile> .\<not> p"
R_respect: shows "(\<lambda>x. {R `` {x}}) respects R"
nn_integral_count_space: fixes A :: "'a set"   and f :: "'a \<Rightarrow> ennreal" assumes "finite {a \<in> A. 0 < f a}" shows "integral\<^sup>N (count_space A) f = (\<Sum>a | a \<in> A \<and> 0 < f a. f a)"
prod_A_is_p_B_bound: fixes n :: "nat" assumes "B n = 1" shows "p = (\<Prod>i = 0..<n. A i ^ Suc i)"
ap_spmf_homo: fixes f :: "'e \<Rightarrow> 'd"   and x :: "'e" shows "return_spmf f \<diamondop> return_spmf x = return_spmf (f x)"
fix_eq: shows "P\<^sub>n\<^sub>o\<^sub>r\<^sub>m\<lbrakk>Tr\<^sub>l\<^sub>p,Up\<^sub>l\<^sub>p\<rbrakk> = (\<Lambda> X. (\<lambda>s. Mprefix (Tr\<^sub>l\<^sub>p s) (\<lambda>e. X (Up\<^sub>l\<^sub>p s e))))\<cdot>(P\<^sub>n\<^sub>o\<^sub>r\<^sub>m\<lbrakk>Tr\<^sub>l\<^sub>p,Up\<^sub>l\<^sub>p\<rbrakk>)"
freshAbs_substAbs_ident: fixes X :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and A :: "('a, 'b, 'varSort, 'var, 'c) qAbs set"   and xs :: "'varSort"   and x :: "'var" assumes "good X"   and "goodAbs A"   and "freshAbs xs x A" shows "A $[X / x]_xs = A"
no_edges_imp_all_nodes_terminal: fixes V :: "node fset"   and \<Lambda> :: "node \<Rightarrow> assertion_gadget" shows "terminals (Graph V \<Lambda> []) = V"
a3: shows "start_points x \<cdot> end_points y = 0"
assms_2: assumes "b"   and "b'" shows "RETURN S1 \<le> \<Down> R S1'"
in_uedge_conv: fixes x :: "'b"   and u :: "'b"   and v :: "'b" shows "(x \<in> uedge (u, v)) = (x = u \<or> x = v)"
rep_eq: fixes x :: "nat"   and xa :: "nat"   and xb :: "nat \<Rightarrow>\<^sub>0 'a" shows "lookup (lowerPowers x xa xb) = (\<lambda>xc. if xc \<in> {x..<x + xa} then 0::'a else lookup xb (dec_above x xa xc))"
elims: fixes x :: "'b list"   and xa :: "'b list"   and y :: "'b list" assumes "remove_all_cycles x xa = y" obtains xs :: "'b list" where "x = []" and "xa = xsa__" and "y = xsa__"   | xb :: "'b" and xs :: "'b list" and ys :: "'b list" where "x = xb # xs" and "xa = ys" and "y = remove_all_cycles xs (start_remove ys xb [])"
id_fcomp: fixes g :: "'b \<Rightarrow> 'c" shows "fcomp id g = g"
oalist_abstract_axioms: shows "oalist_abstract key_order_of_nat_term_order_inv list_of_oalist_ntm OAlist_ntm"
Pair_Rep_def: fixes a :: "'a"   and b :: "'b" shows "Pair_Rep a b = (\<lambda>x y. x = a \<and> y = b)"
gen_one: fixes G :: "('d, 'e) monoid_scheme"   and gens :: "'d set" shows "\<one>\<^bsub>G\<^esub> \<in> \<langle>gens\<rangle>\<^bsub>G\<^esub>"
count_simps_1: shows "count_of [] = (\<lambda>_. 0)"
AReturn_def: fixes ch :: "state \<Rightarrow> PrIds \<Rightarrow> ('a, 'b) chan"   and p :: "PrIds"   and v :: "state \<Rightarrow> 'b" shows "Return ch p v \<equiv> $Calling ch p \<and> rbit<ch!p>$ = $cbit<ch!p> \<and> res<ch!p>$ = $v"
fun_coeff_def: fixes f :: "nat \<Rightarrow> 'a"   and var :: "nat" shows "fun_coeff f var = f var"
subst_mset_empty: fixes \<sigma> :: "'e \<Rightarrow> ('c, 'd) Term.term" shows "subst_mset \<sigma> {#} = {#}"
Astack_append_map_Dummy: fixes S' :: "stack_elem list"   and S :: "stack_elem list" assumes "Astack S' = 0" shows "Astack (S @ S') = Astack S"
graph_union_iff: fixes G\<^sub>1 :: "('c, 'd) labeled_graph"   and G\<^sub>2 :: "('c, 'd) labeled_graph" shows "(graph_union G\<^sub>1 G\<^sub>2 = G\<^sub>2) = (edges G\<^sub>1 \<subseteq> edges G\<^sub>2 \<and> vertices G\<^sub>1 \<subseteq> vertices G\<^sub>2)"
nonzero_component_def: fixes s :: "char list"   and X :: "'c set"   and n :: "'c" shows "nonzero_component s X n = Inf_inner X n \<bind> (\<lambda>I. Sup_inner X n \<bind> (\<lambda>S. CHECK (\<lambda>_. print_msg s) (0 < I \<or> S < 0)))"
lmeasurable_negligible_symdiff: fixes S :: "'a set"   and T :: "'a set" assumes "S \<in> lmeasurable"   and "negligible (S - T \<union> (T - S))" shows "T \<in> lmeasurable"
preserves_locally_sound_mode_use_def: fixes \<R> :: "((('Com\<^sub>A \<times> (Mode \<Rightarrow> 'Var\<^sub>A set)) \<times> ('Var\<^sub>A \<Rightarrow> 'Val)) \<times> ('Com\<^sub>C \<times> (Mode \<Rightarrow> 'Var\<^sub>C set)) \<times> ('Var\<^sub>C \<Rightarrow> 'Val)) set" shows "preserves_locally_sound_mode_use \<R> \<equiv> \<forall>lc\<^sub>A lc\<^sub>C. abs.locally_sound_mode_use lc\<^sub>A \<and> (lc\<^sub>A, lc\<^sub>C) \<in> \<R> \<longrightarrow> conc.locally_sound_mode_use lc\<^sub>C"
Integral_same_iff: fixes a :: "real"   and f :: "real \<Rightarrow> real"   and k :: "real" shows "Integral (a, a) f k = (k = 0)"
bl_word_rotl_dt: fixes n :: "nat"   and w :: "'b word" shows "to_bl (word_rotl n w) = drop (n mod LENGTH('b)) (to_bl w) @ take (n mod LENGTH('b)) (to_bl w)"
prems: shows "jm \<in> DeclConcepts.imethds G J sig"
elims_3: fixes x :: "('b, 'c, 'd, 'sort, 'e, 'f, 'g) semDom_scheme"   and xa :: "'varSort \<times> 'sort"   and xb :: "('varSort, 'f) sAbs" assumes "\<not> sWlsAbs x xa xb" obtains SEM :: "('b, 'c, 'd, 'sort, 'e, 'f, 'g) semDom_scheme" and xs :: "'varSort" and s :: "'sort" and xs' :: "'varSort" and sF :: "'f \<Rightarrow> 'f" where "x = SEM" and "xa = (xs, s)" and "xb = sAbs xs' sF" and "\<not> (isInBar (xs, s) \<and> xs = xs' \<and> (\<forall>sX. if sWls SEM (asSort xs) sX then sWls SEM s (sF sX) else sF sX = sDummy SEM s))"
permutes_orbit_subset: fixes f :: "'e \<Rightarrow> 'e"   and S :: "'e set"   and x :: "'e" assumes "f permutes S"   and "x \<in> S" shows "orbit f x \<subseteq> S"
C2: shows "(\<Union>n. (T ^^ n) --` C) = (\<Union>n\<in>{1..}. (T ^^ n) --` C)"
mark_object_axioms: fixes m :: "'a" shows "mark_object (mutator m) (\<lambda>s. ghost_hs_phase (s (mutator m)) \<noteq> hp_Idle)"
assms_2: fixes x :: "'a" assumes "RETURN x \<le> m" shows "f x \<le> m'"
trace_core_conv_trace_core_aux: fixes core :: "('f, 'a, 'b, 'd, 'c, 'e) core"   and p :: "'f spmf"   and tr :: "('a + 'b \<times> 'c + 'd \<times> 'e) list" shows "trace_core' core p tr = (\<lambda>e. weight_spmf (trace_core_aux core p tr \<bind> (\<lambda>s. cpoke core s e)), \<lambda>ia. trace_core_aux core p tr \<bind> (\<lambda>s. map_spmf fst (cfunc_adv core s ia)), \<lambda>iu. trace_core_aux core p tr \<bind> (\<lambda>s. map_spmf fst (cfunc_usr core s iu)))"
suminf_indicator: fixes A :: "nat \<Rightarrow> 'c set"   and x :: "'c" assumes "disjoint_family A" shows "(\<Sum>n. indicator (A n) x) = indicator (\<Union> (range A)) x"
Union_disjoint: fixes C :: "'b set set"   and g :: "'b \<Rightarrow> 'c" assumes "\<forall>A\<in>C. finite A"   and "\<forall>A\<in>C. \<forall>B\<in>C. A \<noteq> B \<longrightarrow> A \<inter> B = {}" shows "sum g (\<Union> C) = (sum \<circ> sum) g C"
Pow_not_empty: fixes A :: "'b set" shows "Pow A \<noteq> {}"