hcju/isabelleps huggingface dataset:
StarLR_Plus: fixes r :: "'a language"   and s :: "'a language"   and t :: "'a language" shows "StarLR (Plus r s) t = Plus (StarLR r t) (StarLR s t)"
sup_assoc: fixes x :: "'g multiset"   and y :: "'g multiset"   and z :: "'g multiset" shows "x \<union># y \<union># z = x \<union># (y \<union># z)"
p'_has_vector_derivative: fixes t :: "real"   and s :: "real set" shows "(p' has_vector_derivative p'' t) (at t within s)"
jkbpT_tLength_inv: fixes jkbp :: "'b \<Rightarrow> ('b, 'c, 'd) GC list"   and jview :: "'b \<Rightarrow> 'e Trace \<Rightarrow> 'f"   and t :: "'e Trace"   and envInit :: "'e list"   and envAction :: "'e \<Rightarrow> 'g list"   and envTrans :: "'g \<Rightarrow> ('b \<Rightarrow> 'd) \<Rightarrow> 'e \<Rightarrow> 'e"   and envVal :: "'e \<Rightarrow> 'c \<Rightarrow> bool"   and T :: "'e Trace set"   and n :: "nat" assumes "PreEnvironmentJView jkbp jview"   and "t \<in> PreEnvironmentJView.jkbpT jkbp envInit envAction envTrans envVal jview T"   and "tLength t = n" shows "t \<in> PreEnvironmentJView.jkbpTn jkbp envInit envAction envTrans envVal jview n T"
phi_def: fixes g :: "'g"   and v :: "'val" shows "phi g v = Mapping.lookup (phis g) (defNode g v, v)"
simps_2: fixes r :: "real"   and n :: "nat"   and w :: "'c" shows "disc_rfr_proc r (Suc n) w = (1 + r) * disc_rfr_proc r n w"
and_num_4194304_128: shows "4194304 AND 128 = 0"
vsubset_vsingleton_leftD: fixes a :: "V"   and A :: "V" assumes "set {a} \<subseteq>\<^sub>\<circ> A" shows "a \<in>\<^sub>\<circ> A"
length_acc_lengths: fixes acc :: "nat"   and ls :: "'a list list" shows "length (acc_lengths acc ls) = length ls"
sq_norm_zero_vec: fixes n :: "nat" shows "\<parallel>0\<^sub>v n\<parallel>\<^sup>2 = (0::'a)"
one_rel_carrier: fixes U :: "'a set" assumes "U \<in> I" shows "\<lfloor>U, \<one>\<^bsub>U\<^esub>\<rfloor> \<in> carrier_direct_lim"
space_PiF_singleton_eq_product: fixes I :: "'d set"   and M :: "'d \<Rightarrow> 'e measure" assumes "finite I" shows "space (Pi\<^sub>F {I} M) = (\<Pi>' i\<in>I. space (M i))"
overlap_monotonicity: fixes x :: "i"   and y :: "i"   and z :: "i" assumes "P x y"   and "O z x" shows "O z y"
is_span: shows "C.span \<mu>.dom.leg0 \<mu>.dom.leg1"
bind_lunionM: fixes f :: "'a list \<Rightarrow> 'm"   and m :: "'m"   and m' :: "'m" assumes "\<And>A B. f (A @ B) = bind (f A) (\<lambda>x. bind (f B) (\<lambda>y. return (x @ y)))" shows "bind (lunionM m m') f = lunionM (bind m f) (bind m' f)"
extremum_strict: fixes a :: "'a" shows "\<not> a < \<bottom>"
mult_subdistr_var: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "(x \<cdot> y) ; z \<le> x ; z \<cdot> y ; z"
dual_leq: fixes x :: "'c"   and y :: "'c" shows "(dual x \<sqsubseteq> dual y) = (y \<sqsubseteq> x)"
Lp_I_2: fixes p :: "real"   and f :: "'b \<Rightarrow> real"   and M :: "'b measure" assumes "0 < p"   and "f \<in> borel_measurable M"   and "integrable M (\<lambda>x. \<bar>f x\<bar> powr p)" shows "Norm (\<LL> (ennreal p) M) f = (LINT x|M. \<bar>f x\<bar> powr p) powr (1 / p)"
Nil: fixes y :: "'t \<Rightarrow>\<^sub>0 'c"   and ys :: "('t \<Rightarrow>\<^sub>0 'c) list" shows "local.lexordp [] (y # ys)"
AnalBinds_lookup: fixes exp :: "exp \<Rightarrow> 'd \<rightarrow> 'c"   and \<Gamma> :: "(Vars.var \<times> exp) list"   and ae :: "Vars.var \<Rightarrow> 'd\<^sub>\<bottom>"   and x :: "Vars.var" shows "(ExpAnalysis.AnalBinds exp \<Gamma>\<cdot>ae) x = (case map_of \<Gamma> x of None \<Rightarrow> \<bottom> | Some e \<Rightarrow> fup\<cdot>(exp e)\<cdot>(ae x))"
contains_considers: fixes n :: "nat"   and y :: "nat \<times> form" assumes "infinite (deriv s)"   and "contains f n y" shows "\<exists>m. considers f (n + m) y"
minus_mod_eq_mult_div: 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 a :: "'a"   and b :: "'a" assumes "class.semiring_modulo plus minus zero times one divide modulo" shows "minus a (modulo a b) = times b (divide a b)"
symbolic_run_interp_cons_morph: fixes \<gamma> :: "'a constr"   and \<Gamma> :: "'a constr list" shows "\<lbrakk> \<gamma> \<rbrakk>\<^sub>p\<^sub>r\<^sub>i\<^sub>m \<inter> \<lbrakk>\<lbrakk> \<Gamma> \<rbrakk>\<rbrakk>\<^sub>p\<^sub>r\<^sub>i\<^sub>m = \<lbrakk>\<lbrakk> \<gamma> # \<Gamma> \<rbrakk>\<rbrakk>\<^sub>p\<^sub>r\<^sub>i\<^sub>m"
xor_compl_left: fixes x :: "'c"   and y :: "'c" shows "NOT x XOR y = NOT (x XOR y)"
\<omega>_simps_1: shows "S.arr \<i>[a]"
simps_1: fixes s :: "('e, 'f) locks \<times> (('f \<Rightarrow> ('x \<times> 'e \<Rightarrow>f nat) option) \<times> 'g) \<times> ('f \<Rightarrow> 'h wait_set_status option) \<times> 'f set"   and t :: "'f"   and T :: "'f" shows "cond_action_ok s t (Join T) = (case thr s T of None \<Rightarrow> True | \<lfloor>(x, ln)\<rfloor> \<Rightarrow> t \<noteq> T \<and> final x \<and> ln = no_wait_locks \<and> wset s T = None)"
divide_cblinfun_def: fixes A :: "'a \<Rightarrow>\<^sub>C\<^sub>L 'b"   and B :: "'a \<Rightarrow>\<^sub>C\<^sub>L 'b" shows "A / B = A * inverse B"
continuous_on_arsinh: fixes A :: "real set" shows "continuous_on A arsinh"
Rel_abs: fixes A :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and B :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'e" assumes "\<And>x y. Transfer.Rel A x y \<Longrightarrow> Transfer.Rel B (f x) (g y)" shows "Transfer.Rel (rel_fun A B) f g"
sum_mono: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and K :: "'e set"   and f :: "'e \<Rightarrow> 'd"   and g :: "'e \<Rightarrow> 'd" assumes "class.ordered_comm_monoid_add plus zero less_eq less"   and "\<And>i. i \<in> K \<Longrightarrow> less_eq (f i) (g i)" shows "less_eq (comm_monoid_add.sum plus zero f K) (comm_monoid_add.sum plus zero g K)"
simplex_real_3: fixes cs :: "constraint list"   and v :: "(nat, rat) mapping" assumes "simplex cs = Inr v" shows "\<langle>v\<rangle> \<Turnstile>\<^sub>c\<^sub>s set cs"
limproc_Seq_F1: fixes Y :: "nat \<Rightarrow> 'c process"   and S :: "'c process" assumes "chain Y" shows "F (lim_proc (range Y) `;` S) = F (lim_proc (range (\<lambda>i. Y i `;` S)))"
emeasure_count_space_eq_0: fixes A :: "'c set"   and X :: "'c set" shows "(emeasure (count_space A) X = 0) = (X \<subseteq> A \<longrightarrow> X = {})"
ipurge_ref_aux_append: fixes I :: "('f \<times> 'f) set"   and D :: "'e \<Rightarrow> 'f"   and U :: "'f set"   and xs :: "'e list"   and ys :: "'e list"   and X :: "'e set" shows "ipurge_ref_aux I D U (xs @ ys) X = ipurge_ref_aux I D (sinks_aux I D U xs) ys X"
hausdorff_distance_sym: fixes A :: "'b set"   and B :: "'b set" shows "hausdorff_distance A B = hausdorff_distance B A"
linorder_not_less: fixes x :: "'g"   and y :: "'g" shows "(\<not> x < y) = (y \<le> x)"
minus_eq: fixes f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and gm :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and g1 :: "'b \<Rightarrow> 'b"   and gn1 :: "'b \<Rightarrow> 'b"   and x :: "'b"   and y :: "'b" assumes "periodic_fun f g gm g1 gn1" shows "g x (- y) = gm x y"
insert_not_empty: fixes a :: "'k"   and A :: "'k set" shows "insert a A \<noteq> {}"
iso_point_subset_block_iff: fixes p :: "'a set"   and bl :: "'a set" assumes "p \<subseteq> \<V>"   and "bl \<in># \<B>" shows "(p \<subseteq> bl) = (\<pi> ` p \<subseteq> \<pi> ` bl)"
get_dom_component_subset: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and c :: "(_) object_ptr list"   and ptr' :: "(_) object_ptr" assumes "heap_is_wellformed h"   and "type_wf h"   and "known_ptrs h"   and "h \<turnstile> get_dom_component ptr \<rightarrow>\<^sub>r c"   and "ptr' \<in> set c" shows "h \<turnstile> get_dom_component ptr' \<rightarrow>\<^sub>r c"
subdprodr_def: fixes y :: "nat list" shows "subdprodr y = (\<forall>l\<le>n. take l b \<bullet> take l y \<le> a \<bullet> map (max_x (take l y)) [0..<m])"
asym_greater: fixes cmp :: "'b comparator"   and a :: "'b"   and b :: "'b" assumes "compare cmp a b = Greater" shows "compare cmp b a \<noteq> Greater"
cases: fixes x :: "'b \<times> sort_dir" obtains xa :: "'b" where "x = (xa__, sort_dir.Left)"   | xa :: "'b" where "x = (xa, sort_dir.Right)"
span_add_eq2: fixes y :: "'t \<Rightarrow>\<^sub>0 'c"   and S :: "('t \<Rightarrow>\<^sub>0 'c) set"   and x :: "'t \<Rightarrow>\<^sub>0 'c" assumes "y \<in> pmdl S" shows "(x + y \<in> pmdl S) = (x \<in> pmdl S)"
map_matrix_def: fixes f :: "'e \<Rightarrow> 'b"   and x :: "(('e, 'c) vec, 'd) vec" shows "map_matrix f x = (\<chi>i j. f (x $ i $ j))"
fmdom_notD: fixes x :: "'d"   and m :: "'d \<rightharpoonup>\<^sub>f 'e" assumes "x |\<notin>| fmdom m" shows "fmlookup m x = None"
tuple_is_extensional: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and prj0 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and prj1 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b"   and g :: "'b"   and h :: "'b"   and k :: "'b" assumes "elementary_category_with_pullbacks C prj0 prj1"   and "\<not> category.commutative_square C f g h k" shows "elementary_category_with_pullbacks.tuple C prj0 prj1 h f g k = partial_magma.null C"
dghm_tdghm_comp_components_1: fixes \<HH> :: "V"   and \<NN> :: "V" shows "(\<HH> \<circ>\<^sub>D\<^sub>G\<^sub>H\<^sub>M\<^sub>-\<^sub>T\<^sub>D\<^sub>G\<^sub>H\<^sub>M \<NN>)\<lparr>NTMap\<rparr> = (\<lambda>a\<in>\<^sub>\<circ>\<NN>\<lparr>NTDGDom\<rparr>\<lparr>Obj\<rparr>. \<HH>\<lparr>ArrMap\<rparr>\<lparr>\<NN>\<lparr>NTMap\<rparr>\<lparr>a\<rparr>\<rparr>)"
cInf_greaterThanLessThan: fixes y :: "'a"   and x :: "'a" assumes "y < x" shows "Inf {y<..<x} = y"
snth_SCons_Suc: fixes x :: "'d"   and xs :: "'d stream"   and n :: "nat" shows "(x ## xs) !! Suc n = xs !! n"
sphere_def: fixes x :: "'b"   and e :: "real" shows "sphere x e = {y. dist x y = e}"
simps_3: fixes t :: "'c local.term"   and u :: "'c local.term" shows "Arr (t \<^bold>\<star> u) = HSeq t u"
comp_apply: fixes f :: "'i \<Rightarrow> 'h"   and g :: "'j \<Rightarrow> 'i"   and x :: "'j" shows "(f \<circ> g) x = f (g x)"
pick_def: fixes x :: "'a"   and y :: "'a" shows "local.pick x y = (SOME xs. chain R (llist_of (x # xs @ [y])))"
card_of_cong: fixes r :: "('e \<times> 'e) set"   and r' :: "('f \<times> 'f) set" assumes "r =o r'" shows "|Field r| =o |Field r'|"
borel_measurable_uminus_ereal: fixes f :: "'c \<Rightarrow> ereal"   and M :: "'c measure" assumes "f \<in> borel_measurable M" shows "(\<lambda>x. - f x) \<in> borel_measurable M"
healthy_wp_Seq: fixes t :: "bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and u :: "bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real" assumes "healthy (wp t)"   and "healthy (wp u)" shows "healthy (wp (t ;; u))"
safe_delayed_reach_n: shows "safe_reach_upto n safe_delayed c\<^sub>0"
subst_mono: fixes t :: "('a, 'b) Term.term"   and u :: "('a, 'b) Term.term"   and s :: "'b \<Rightarrow> ('a, 'c) Term.term" assumes "t \<sqsubseteq> u" shows "t \<cdot> s \<sqsubseteq> u \<cdot> s"
up_to_iff: fixes J :: "nat set"   and n :: "nat" assumes "finite J" shows "(up_to J \<le> n) = (\<forall>i\<in>J. i < n)"
D_front_tickFree_subset: fixes P :: "'a process" shows "D P \<subseteq> Collect front_tickFree"
tendsto_cos: fixes f :: "'a \<Rightarrow> 'b"   and a :: "'b"   and F :: "'a filter" assumes "(f \<longlongrightarrow> a) F" shows "((\<lambda>x. cos (f x)) \<longlongrightarrow> cos a) F"
cramer: fixes A :: "(('a, 'b) vec, 'b) vec"   and x :: "('a, 'b) vec"   and b :: "('a, 'b) vec" assumes "det A \<noteq> (0::'a)" shows "(A *v x = b) = (x = (\<chi>k. det (\<chi>i j. if j = k then b $ i else A $ i $ j) / det A))"
integral_minus: fixes M :: "'e measure"   and f :: "'e \<Rightarrow> 'd" shows "LINT x|M. - f x = - integral\<^sup>L M f"
py: shows "(p, y) \<in> \<langle>term_poly_list_rel \<times>\<^sub>r int_rel\<rangle>list_rel"
prod_diff_prod: fixes prod :: "'a \<Rightarrow> 'b \<Rightarrow> 'c"   and x :: "'a"   and y :: "'b"   and a :: "'a"   and b :: "'b" assumes "bounded_bilinear prod" shows "prod x y - prod a b = prod (x - a) (y - b) + prod (x - a) b + prod a (y - b)"
lfilter_def: fixes P :: "'c \<Rightarrow> bool" shows "lfilter P \<equiv> llist.fixp_fun (\<lambda>lfilter xs. case xs of LNil \<Rightarrow> LNil | LCons x xs' \<Rightarrow> if P x then LCons x (lfilter xs') else lfilter xs')"
assms_3: fixes oid :: "'a"   and oper :: "'b"   and ref :: "'a" assumes "(oid, oper) \<in> ops"   and "ref \<in> deps oper" shows "ref < oid"
incseqD: fixes f :: "nat \<Rightarrow> 'b"   and i :: "nat"   and j :: "nat" assumes "incseq f"   and "i \<le> j" shows "f i \<le> f j"
semiring_normalization_rules_2: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and a :: "'b"   and m :: "'b" assumes "class.comm_semiring_1 times one plus zero" shows "plus (times a m) m = times (plus a one) m"
bij_betw_inv_into_RIGHT: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set"   and A' :: "'b set"   and B' :: "'b set" assumes "bij_betw f A A'"   and "B' \<subseteq> A'" shows "f ` inv_into A f ` B' = B'"
inj_on_wrap_iso_arcs_f: fixes A :: "'b set"   and f :: "'b \<Rightarrow> 'b"   and B :: "'d set" assumes "A \<subseteq> arcs G"   and "f \<in> A \<rightarrow> A"   and "B = iso_arcs hom ` A"   and "inj_on f A" shows "inj_on (wrap_iso_arcs hom f) B"
IH: fixes h :: "'a"   and l1 :: "'a list" assumes "subseq (h # l1) l2" shows "\<exists>l2aa l2b. l2 = l2aa @ [h] @ l2b \<and> \<not> ListMem h l2aa"
listsum0: fixes As :: "'b tensor list"   and ds :: "nat list" assumes "\<And>A. A \<in> set As \<Longrightarrow> A = tensor0 ds" shows "listsum ds As = tensor0 ds"
write_actions_iff: fixes a :: "nat"   and E :: "('a \<times> ('b, 'a) obs_event action) llist" shows "(a \<in> write_actions E) = (a \<in> actions E \<and> is_write_action (action_obs E a))"
basisI: fixes X :: "'a set" assumes "indep X"   and "\<And>x. x \<in> carrier - X \<Longrightarrow> \<not> indep (insert x X)" shows "basis X"
Infinitesimal_add_hypreal_of_real_less: fixes x :: "real"   and y :: "real"   and u :: "real star" assumes "x < y"   and "u \<in> Infinitesimal" shows "hypreal_of_real x + u < hypreal_of_real y"
assms_2: fixes S :: "'a set" assumes "S \<in> \<F>" shows "closedin (top_of_set (\<Union> \<F>)) S"
DERIV_imp_deriv: fixes f :: "'c \<Rightarrow> 'c"   and f' :: "'c"   and x :: "'c" assumes "(f has_field_derivative f') (at x)" shows "deriv f x = f'"
project_def: fixes h :: "'d \<times> 'e \<Rightarrow> 'f"   and C :: "'f set"   and F :: "'f program" shows "project h C F = mk_program (project_set h (Init F), project_act h ` Restrict C ` Acts F, {act. Restrict (project_set h C) act \<in> project_act h ` Restrict C ` AllowedActs F})"
rpr_less_le_trans: fixes x :: "'a"   and r :: "('a \<times> 'a) set"   and y :: "'a"   and z :: "'a"   and A :: "'a set" assumes "x \<^bsub>r\<^esub>\<prec> y"   and "y \<^bsub>r\<^esub>\<preceq> z"   and "rpr A r" shows "x \<^bsub>r\<^esub>\<prec> z"
hermitian_inner_prod_real: fixes A :: "complex mat"   and n :: "nat"   and v :: "complex vec" assumes "A \<in> carrier_mat n n"   and "v \<in> carrier_vec n"   and "hermitian A" shows "inner_prod v (A *\<^sub>v v) \<in> \<real>"
urel32_power: fixes x :: "uint32"   and x' :: "int"   and y :: "uint32"   and y' :: "nat" assumes "urel32 x x'"   and "urel32 y (int y')" shows "urel32 (power_p32 pp x y) (power_p p x' y')"
executableI_step: fixes p :: "'state"   and a :: "'transition" assumes "p \<in> nodes"   and "enabled a p" shows "a \<in> executable"
Assigno_undef: fixes x :: "variable"   and \<theta> :: "trm option" shows "(Assigno x \<theta> = undefg) = (\<theta> = undeft)"
H: shows "H \<in> carrier_mat (dim_row A) (dim_col A)"
partner_test: fixes R :: "fid_t"   and R' :: "fid_t" assumes "R \<in> partners"   and "partner_runs R R'" shows "R' = test"
ext_inject: fixes arrays :: "typerep \<Rightarrow> nat \<Rightarrow> nat list"   and refs :: "typerep \<Rightarrow> nat \<Rightarrow> nat"   and lim :: "nat"   and more :: "'a"   and arrays' :: "typerep \<Rightarrow> nat \<Rightarrow> nat list"   and refs' :: "typerep \<Rightarrow> nat \<Rightarrow> nat"   and lim' :: "nat"   and more' :: "'a" shows "(\<lparr>arrays = arrays, refs = refs, lim = lim, \<dots> = more\<rparr> = \<lparr>arrays = arrays', refs = refs', lim = lim', \<dots> = more'\<rparr>) = (arrays = arrays' \<and> refs = refs' \<and> lim = lim' \<and> more = more')"
termFSwMorph_def: fixes Delta :: "('b, 'c, 'd, 'e, 'f) signature"   and h :: "('b, 'c, 'd, 'a, 'f) qTerm set \<Rightarrow> 'g"   and hA :: "('b, 'c, 'd, 'a, 'f) qAbs set \<Rightarrow> 'h"   and MOD :: "('b, 'c, 'd, 'e, 'f, 'a, 'g, 'h, 'i) Recursion.model_scheme" assumes "FixSyn TYPE('a) Delta" shows "FixSyn.termFSwMorph Delta h hA MOD \<equiv> FixSyn.presWlsAll Delta h hA MOD \<and> FixSyn.presCons Delta h hA MOD \<and> FixSyn.presFreshAll Delta h hA MOD \<and> FixSyn.presSwapAll Delta h hA MOD"
th: shows "a = Floor (snd (zsplit0 t)) \<and> n = fst (zsplit0 t)"
closedin_def: fixes U :: "'a topology"   and S :: "'a set" shows "closedin U S = (S \<subseteq> topspace U \<and> openin U (topspace U - S))"
comp_def: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and g :: "'b"   and f :: "'b" assumes "dual_category C" shows "dual_category.comp C g f \<equiv> C f g"
parTrace_snoc: fixes indis :: "('a \<times> 'a) set"   and tval :: "'b \<Rightarrow> 'a \<Rightarrow> bool"   and aval :: "'c \<Rightarrow> 'a \<Rightarrow> 'a"   and cfl :: "(('b, 'c) com \<times> 'a) list"   and cf :: "('b, 'c) com \<times> 'a" assumes "PL_Indis indis" shows "PL_Indis.parTrace tval aval (cfl @ [cf]) = (cfl \<noteq> [] \<longrightarrow> PL_Indis.parTrace tval aval cfl \<and> PL.transC tval aval (last cfl) cf)"
weakSimE_2: 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 Q' :: "('a, 'b, 'c) psi" assumes "\<Psi> \<rhd> P \<leadsto><Rel> Q"   and "\<Psi> \<rhd> Q \<longmapsto> \<tau> \<prec> Q'" shows "\<exists>P'. \<Psi> \<rhd> P \<Longrightarrow>\<^sup>^\<^sub>\<tau> P' \<and> (\<Psi>, P', Q') \<in> Rel"
ipresIGAbs_termMOD: fixes h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> 'e"   and hA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> 'f"   and MOD :: "('g, 'h, 'varSort, 'i, 'j, 'var, 'e, 'f, 'k) model_scheme" shows "ipresIGAbs h hA termMOD MOD = ipresAbs h hA MOD"
weak_ranking_altdef: shows "weak_ranking (map_relation (inv f) le) = R.weak_ranking_aux (inv f -` carrier)"
Schwarz2: fixes f :: "complex \<Rightarrow> complex"   and r :: "real"   and s :: "real"   and w :: "complex" assumes "f holomorphic_on ball 0 r"   and "0 < s"   and "ball w s \<subseteq> ball 0 r"   and "\<And>z. cmod (w - z) < s \<Longrightarrow> cmod (f z) \<le> cmod (f w)" shows "f constant_on ball 0 r"
simps_2: fixes x :: "'a"   and l :: "'a tree"   and y :: "'a"   and r :: "'a tree" shows "lookup_cost x \<langle>l, y, r\<rangle> = (if x = y then 0 else if x < y then Suc (lookup_cost x l) else Suc (lookup_cost x r))"
linorder_not_less: fixes x :: "'b"   and y :: "'b" shows "(\<not> x < y) = (y \<le> x)"
finsert_iff: fixes a :: "'a"   and b :: "'a"   and A :: "'a fset" shows "(a |\<in>| finsert b A) = (a = b \<or> a |\<in>| A)"
join_mask_def: fixes n :: "nat"   and X :: "nat set" shows "join_mask n X = map (\<lambda>i. i \<in> X) [0..<n]"
lcm_gcd: fixes a :: "'a"   and b :: "'a" shows "lcm a b = normalize (a * b div gcd a b)"
localSS'_aux: shows "(\<lambda>x. \<forall>a aa b. \<not> (case x of (x, u, v) \<Rightarrow> \<lambda>ab. u = 0 \<and> (case ab of (y, u', v') \<Rightarrow> u' = Suc 0 \<and> v = v')) (a, aa, b)) = (\<lambda>(x, u, v). 0 < u)"
inv: shows "TS_inv' (Partial_Cost_Model.config' (rTS h0) ([x, y], h) qs) (last qs) [x, y]"
In_I_max: shows "I n \<subseteq> I (max n m)"
\<ll>_is_natural_isomorphism: shows "natural_isomorphism (\<bullet>) (\<bullet>) L local.map \<ll>.map"
vrestriction_def: fixes D :: "V" shows "vrestriction D = (\<lambda>\<langle>r, A\<rangle>\<in>\<^sub>\<circ>D. set {\<langle>a, b\<rangle> |a b. a \<in>\<^sub>\<circ> A \<and> b \<in>\<^sub>\<circ> A \<and> \<langle>a, b\<rangle> \<in>\<^sub>\<circ> r})"
fmp: fixes F :: "'a \<Rightarrow> bool"   and G :: "'a \<Rightarrow> bool" assumes "\<turnstile> F"   and "\<turnstile> F \<longrightarrow> G" shows "\<turnstile> G"
H: shows "\<Delta> @ VarB U \<Colon> \<Gamma> \<turnstile> t : T"
rt_graph_not_dip_trans: fixes ip :: "nat"   and ip' :: "nat"   and \<sigma> :: "nat \<Rightarrow> state"   and dip :: "nat" assumes "(ip, ip') \<in> (rt_graph \<sigma> dip)\<^sup>+" shows "ip \<noteq> dip"
ge_tpoly_def: fixes q :: "('v pvar, hmultiset) tpoly"   and p :: "('v pvar, hmultiset) tpoly" shows "(q \<ge>\<^sub>p p) = (\<forall>A. legal_zpassign A \<longrightarrow> eval_tpoly A (map_tpoly (\<lambda>x. x) zhmset_of p) \<le> eval_tpoly A (map_tpoly (\<lambda>x. x) zhmset_of q))"
cfg_on_def: fixes K :: "'a \<Rightarrow> 'a pmf set"   and s :: "'a" assumes "Markov_Decision_Process K" shows "Markov_Decision_Process.cfg_on K s = {cfg. Markov_Decision_Process.cfg_onp K s cfg}"
S_fixes_fundchamber_image_int: fixes s :: "'a permutation" assumes "s \<in> (\<Union>(f, g)\<in>fundfoldpairs. {Abs_induced_automorph f g})" shows "s `\<rightarrow> (C0 \<inter> s `\<rightarrow> C0) = C0 \<inter> s `\<rightarrow> C0"
VPowD: fixes A :: "V"   and B :: "V" assumes "A \<in>\<^sub>\<circ> VPow B" shows "A \<subseteq>\<^sub>\<circ> B"
d_OUT_h_plus_not_infty': fixes \<Delta> :: "('a, 'b) network_scheme"   and g :: "'a \<times> 'a \<Rightarrow> ennreal"   and x :: "'a"   and i :: "nat" assumes "flow_network \<Delta> g"   and "x \<noteq> sink \<Delta>" shows "d_OUT (flow_network.h_plus \<Delta> g i) x \<noteq> \<top>"
IH: fixes m :: "nat" assumes "m < length xs" shows "rotate n xs ! m = xs ! ((m + n) mod length xs)"
concat_def: shows "concat \<equiv> rec_list [] (\<lambda>x xs. (@) x)"
fds_abs_summable_zeta: fixes s :: "'a" assumes "1 < s \<bullet> (1::'a)" shows "fds_abs_converges fds_zeta s"
Joints_mapE: fixes ys :: "'a list"   and YS :: "('a \<times> 'a pdevs) list" assumes "ys \<in> Joints YS" obtains e :: "nat \<Rightarrow> real" where "ys = map (aform_val e) YS" and "\<And>i. e i \<in> {- 1..1}"
wf_trm_TComp_subterm: fixes s :: "('fun, 'var) Term.term"   and t :: "('fun, 'var) Term.term" assumes "wf\<^sub>t\<^sub>r\<^sub>m s"   and "t \<sqsubset> s" obtains f :: "'fun" and T :: "('fun, 'atom) Term.term list" where "\<Gamma> s = Fun f T"
Rf_sat: fixes N :: "'a literal multiset set" assumes "satisfiable (N - Rf N)" shows "satisfiable N"
finals_app_map: fixes vs :: "val list"   and es :: "'a exp list" shows "finals (map Val vs @ es) = finals es"
conjugate_conjugate_sprod: fixes v :: "'b vec"   and n :: "nat"   and w :: "'b vec" assumes "v \<in> carrier_vec n"   and "w \<in> carrier_vec n" shows "conjugate (conjugate v \<bullet> w) = v \<bullet>c w"
with_proj_simps_5: fixes G :: "'a pair_pre_digraph" shows "head (with_proj G) = snd"
sorted_subseqs_sorted: fixes xs :: "'c list"   and ys :: "'c list" assumes "sorted xs"   and "ys \<in> set (subseqs xs)" shows "sorted ys"
null_eqI: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and n :: "'b" assumes "partial_magma C"   and "\<And>f. C n f = n \<and> C f n = n" shows "n = partial_magma.null C"
Collect_eqvt: fixes p :: "perm"   and P :: "'a \<Rightarrow> bool" shows "p \<bullet> {x. P x} = {x. (p \<bullet> P) x}"
simps_2: fixes key :: "'a \<Rightarrow> 'b"   and bs :: "'a list" shows "merge key [] bs = bs"
conj_one_right: fixes x :: "'a" shows "x \<^bold>\<sqinter> \<one> = x"
finite_Lim_measure_incseq: fixes A :: "nat \<Rightarrow> real set" assumes "range A \<subseteq> sets M"   and "incseq A" shows "(\<lambda>i. Sigma_Algebra.measure M (A i)) \<longlonglongrightarrow> Sigma_Algebra.measure M (\<Union> (range A))"
ok_Join_iff1: fixes F :: "'c program"   and G :: "'c program"   and H :: "'c program" shows "F ok (G \<squnion> H) = (F ok G \<and> F ok H)"
xj: shows "x\<^sub>j \<in> rvars_eq (eq_for_lvar (\<T> s) x\<^sub>i)"
leq_antisym: fixes y :: "'b"   and x :: "'b" assumes "y \<le> x"   and "x \<le> y" shows "x = y"