hcju/isabelleps huggingface dataset:
tobd_F22_inj: fixes x :: "'d"   and xa :: "('e, 'd, 'f) F2"   and y :: "'d"   and r1 :: "(bd_type_F1 \<times> bd_type_F1) set" assumes "x \<in> F2set2 xa"   and "y \<in> F2set2 xa" shows "(toCard (F2set2 xa) (dir_image (r1 +c bd_F2) Abs_bd_type_F) x = toCard (F2set2 xa) (dir_image (r1 +c bd_F2) Abs_bd_type_F) y) = (x = y)"
distrib: fixes g :: "'c \<Rightarrow> 'b"   and h :: "'c \<Rightarrow> 'b"   and A :: "'c multiset" shows "(\<Sum>x\<in>#A. g x + h x) = \<Sum>\<^sub># (image_mset g A) + \<Sum>\<^sub># (image_mset h A)"
polyn_exprs_eq: fixes c :: "nat \<times> (nat \<Rightarrow> 'a)"   and d :: "nat \<times> (nat \<Rightarrow> 'a)"   and k :: "nat" assumes "pol_coeff S c"   and "pol_coeff S d"   and "k \<le> min (fst c) (fst d)"   and "\<forall>j\<le>k. snd c j = snd d j" shows "polyn_expr R X k c = polyn_expr R X k d"
less_trans: fixes D :: "('e, 'f) Order_scheme"   and a :: "'e"   and b :: "'e"   and c :: "'e" assumes "Order D"   and "a \<in> carrier D"   and "b \<in> carrier D"   and "c \<in> carrier D"   and "a \<prec>\<^bsub>D\<^esub> b"   and "b \<prec>\<^bsub>D\<^esub> c" shows "a \<prec>\<^bsub>D\<^esub> c"
FII: fixes v :: "'a"   and t :: "'a ifex"   and e :: "'a ifex"   and ti :: "'ni"   and ei :: "'ni" assumes "n = IF v t e"   and "(ti, t) \<in> R s"   and "(ei, e) \<in> R s" shows "ospec (fii v ti ei s) (\<lambda>(r, s'). (r, fi v t e) \<in> Q s \<and> I' s')"
i_f_join_def: fixes f :: "nat \<Rightarrow> 'b"   and I :: "nat set" shows "f \<Join>\<^sub> I \<equiv> f \<Down> Suc (Max I) \<Join>\<^sub> I"
f2: shows "rk {A, B, C, P, A'} \<le> 3"
V_4: fixes j :: "nat" shows "U j \<subseteq> V j"
bitlen_mantissa_Float: fixes m :: "int"   and e :: "int" shows "bitlen \<bar>mantissa (Float m e)\<bar> = (if m = 0 then 0 else bitlen \<bar>m\<bar> + e) - Float.exponent (Float m e)"
direct_path_return': fixes \<pi> :: "nat \<Rightarrow> 'n"   and x :: "'n"   and n :: "nat" assumes "is_path \<pi>"   and "\<pi> 0 = x"   and "x \<noteq> local.return"   and "\<pi> n = local.return" obtains \<pi>' :: "nat \<Rightarrow> 'n" and n' :: "nat" where "is_path \<pi>'" and "\<pi>' 0 = x" and "\<pi>' n' = local.return" and "\<forall>i>0. \<pi>' i \<noteq> x"
subgroups_of_size_def: fixes p :: "nat" shows "conjP.subgroups_of_size p = {H. subgroup H (G\<lparr>carrier := P\<rparr>) \<and> card H = p}"
Some_in_map_option_set: fixes y :: "'a"   and f :: "'b \<Rightarrow> 'a option set"   and x :: "'b" assumes "Some y \<in> f x" shows "Some y \<in> map_option_set f (Some x)"
inf_p: fixes x :: "'a" shows "x \<sqinter> - x = bot"
aasc_seq_mem: fixes a :: "ant"   and A :: "ant set"   and n :: "nat" assumes "a \<in> A"   and "\<nexists>m. m \<in> A \<and> (\<forall>x\<in>A. x \<le> m)" shows "aasc_seq A a n \<in> A"
isometry_on_def: fixes X :: "'d set"   and f :: "'d \<Rightarrow> 'e" shows "isometry_on X f = (\<forall>x\<in>X. \<forall>y\<in>X. dist (f x) (f y) = dist x y)"
rel_pmf_bind_reflI: fixes p :: "'a pmf"   and R :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'b pmf"   and g :: "'a \<Rightarrow> 'c pmf" assumes "\<And>x. x \<in> set_pmf p \<Longrightarrow> rel_pmf R (f x) (g x)" shows "rel_pmf R (bind_pmf p f) (bind_pmf p g)"
f1: fixes a :: "'a" shows "ad (ad (ad a)) = ad a"
unit_simps_4: shows "src (local.inv \<epsilon>\<^sub>1) = src d\<^sub>1"
elims_3: fixes x :: "'at set"   and xa :: "'at Literal" assumes "\<not> x \<Turnstile> xa" obtains I :: "'at set" and A :: "'at" where "x = Ia__" and "xa = Pos A__" and "A__ \<notin> Ia__"   | I :: "'at set" and A :: "'at" where "x = I" and "xa = Neg A" and "\<not> A \<notin> I"
arg_min_list_in: fixes xs :: "'a list"   and f :: "'a \<Rightarrow> 'b" assumes "xs \<noteq> []" shows "arg_min_list f xs \<in> set xs"
nonnull: shows "\<not> lnull (lfilter P (ltake (enat (Suc (lfind P lst))) lst))"
wf_monitor: fixes i :: "nat"   and v :: "'a option list"   and \<pi> :: "(char list \<times> 'a list) prefix" assumes "(i, v) \<in> joiner (\<lambda>k. M (splitter \<pi> k))" shows "wf_tuple (MFOTL.nfv \<phi>) (fv \<phi>) v"
kD_nsqn: fixes dip :: "nat"   and rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option" assumes "dip \<in> kD rt" shows "nsqn rt dip = nsqn\<^sub>r (the (rt dip))"
simple_match_to_of_matchI: fixes r :: "32 simple_match"   and p :: "(32, 'b) simple_packet_ext_scheme"   and ifs :: "char list list" assumes "simple_match_valid r"   and "simple_matches r p"   and "p_iiface p \<in> set ifs"   and "p_l2type p = 2048" shows "\<exists>gr\<in>set (simple_match_to_of_match r ifs). OF_match_fields gr p = Some True"
not_less_iMin: fixes k :: "'a"   and I :: "'a set" assumes "k < iMin I" shows "k \<notin> I"
mset_rev: fixes xs :: "'c list" shows "mset (rev xs) = mset xs"
finish_simps_9: fixes u :: "'d"   and s :: "('d, 'c) state_scheme" shows "state.more (param_DFS_defs.finish u s) = state.more s"
dbm_entry_val_add_3: fixes u :: "'c \<Rightarrow> 'd"   and c :: "'c"   and d :: "'c"   and a :: "'d DBMEntry"   and e :: "'c"   and b :: "'d DBMEntry" assumes "dbm_entry_val u (Some c) (Some d) a"   and "dbm_entry_val u (Some d) (Some e) b" shows "dbm_entry_val u (Some c) (Some e) (dbm_add a b)"
is_faithful: fixes f :: "'a"   and f' :: "'a" assumes "par f f'"   and "\<I> \<otimes> f = \<I> \<otimes> f'" shows "f = f'"
left_le_right: fixes i :: "int"   and j :: "int" assumes "opt_bst2_dom (i, j)" shows "(i = j \<longrightarrow> root (opt_bst2 i j) = i) \<and> (i < j \<longrightarrow> root (opt_bst2 i j) \<in> {root (opt_bst2 i (j - 1))..root (opt_bst2 (i + 1) j)})"
const_charn: fixes X :: "'d \<Rightarrow> 'e"   and \<tau> :: "'d"   and \<tau>' :: "'d" assumes "const X" shows "X \<tau> = X \<tau>'"
compat_right: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<sqsubseteq> y"   and "y \<sqsubset> z"   and "x \<in> A"   and "y \<in> A"   and "z \<in> A" shows "x \<sqsubset> z"
sp_compile_fundef: fixes get_arity :: "'fun \<Rightarrow> nat option"   and load_oracle :: "nat \<Rightarrow> Unboxed.type option"   and fd1 :: "('b, 'n, 'fun, 'd, 'e) Inca.instr fundef"   and fd2 :: "('b, 'n, 'fun, 'd, 'e, 'g, 'h, 'i) Ubx.instr fundef"   and F :: "'fun \<Rightarrow> 'o fundef option" assumes "compile_fundef get_arity load_oracle fd1 = Ok fd2"   and "\<And>x. rel_option (\<lambda>ar fd. arity fd = ar) (get_arity x) (F x)" shows "sp_fundef F fd2 (body fd2) = Ok [None]"
ee: shows "essentially_equal G (bs[0 := bs ! 0 \<otimes> u]) bs"
psimps: shows "parse_nodes s = (if s = [] \<or> take 2 s = ''</'' then Parser_Monad.return [] s else if hd s \<noteq> CHR ''<'' then (parse_text \<bind> (\<lambda>t. parse_nodes \<bind> (\<lambda>ns. Parser_Monad.return (XML_text (the t) # ns)))) s else (exactly ''<'' \<bind> (\<lambda>_. parse_name \<bind> (\<lambda>n. parse_attributes \<bind> (\<lambda>atts. oneof [''/>'', ''>''] \<bind> (\<lambda>e ts'. if e = ''/>'' then (parse_nodes \<bind> (\<lambda>cs. Parser_Monad.return (XML n atts [] # cs))) ts' else (parse_nodes \<bind> (\<lambda>cs. exactly ''</'' \<bind> (\<lambda>_. exactly n \<bind> (\<lambda>_. exactly ''>'' \<bind> (\<lambda>_. parse_nodes \<bind> (\<lambda>ns. Parser_Monad.return (XML n atts cs # ns))))))) ts'))))) s)"
simps_2: fixes x21 :: "'a tree"   and x22 :: "'a"   and x23 :: "'a tree" shows "\<langle>\<rangle> \<noteq> \<langle>x21, x22, x23\<rangle>"
IH: fixes n :: "nat" assumes "distinct xs"   and "n < length xs" shows "insert_spec xs (oid, Some (xs ! n)) ! Suc n = oid"
success_effectE: fixes c :: "'b Heap"   and h :: "heap" assumes "success c h" obtains r :: "'b" and h' :: "heap" where "effect c h h' r"
assms_2: fixes x :: "'a" assumes "xs = [x]" shows "P"
intro: fixes point_set :: "'b set"   and block_collection :: "'b set multiset"   and design_rep_number :: "int" assumes "\<And>x. x \<in> point_set \<Longrightarrow> block_collection rep x = design_rep_number" shows "constant_rep_design_axioms point_set block_collection design_rep_number"
assms_1: shows "\<guillemotleft>f : \<I> \<rightarrow> \<I>\<^sub>1\<guillemotright>"
apply_wiring_def: shows "apply_wiring = (\<lambda>(f, g). map_fun id (map_fun f (map_spmf (map_prod g id))))"
less_eq_prod_def: fixes p1 :: "'c \<times> 'd"   and p2 :: "'c \<times> 'd" shows "(p1 \<le> p2) = (fst p1 \<le> fst p2 \<and> snd p1 \<le> snd p2)"
assms_2: shows "e = Binop_i t iop \<or> e = Binop_f t fop \<or> e = Relop_i t irop \<or> e = Relop_f t frop"
M_in_subprob: fixes M :: "'a measure" assumes "prob_space M" shows "M \<in> space (subprob_algebra M)"
ball_eq_cball_iff: fixes x :: "'a"   and d :: "real"   and y :: "'a"   and e :: "real" shows "(ball x d = cball y e) = (d \<le> 0 \<and> e < 0)"
content_cball_conv_ball: fixes c :: "'b"   and r :: "real" shows "content (cball c r) = content (ball c r)"
hyps_1: shows "\<Gamma> \<turnstile> p \<leftrightarrow> q : T\<^sub>1\<rightarrow>T\<^sub>2"
h_plus_sum_finite: fixes i :: "nat" shows "integral\<^sup>N (count_space UNIV) (h_plus i) \<noteq> \<top>"
simps_26: fixes vn :: "'q \<Rightarrow> 'r fundef option"   and v :: "nat"   and vp :: "Unboxed.type option list" shows "sp_instr vn (ICJump v) vp = Error ()"
simps_1: fixes s :: "'a \<Rightarrow> bool" shows "val_ifex Trueif s = True"
mult_1_right: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a" assumes "class.monoid_mult one times" shows "times a one = a"
visits_via_trivial: fixes G :: "('c, 'd) ParityGame_scheme"   and P :: "'c llist"   and v0 :: "'c"   and p :: "Player"   and \<sigma> :: "'c \<Rightarrow> 'c"   and W :: "'c set"   and A :: "'c set" assumes "vmc_path G P v0 p \<sigma>"   and "v0 \<in> W" shows "visits_via P A W"
strict_order_iff: fixes b :: "'b multiset"   and a :: "'b multiset" shows "(b \<subset># a) = (a = a \<union># b \<and> a \<noteq> b)"
state_of_def: fixes s :: "'g Heap" shows "state_of s \<equiv> State (\<lambda>heap. the (execute s heap))"
subtopology_def: fixes U :: "'a topology"   and V :: "'a set" shows "subtopology U V = topology (\<lambda>T. \<exists>S. T = S \<inter> V \<and> openin U S)"
lvl_update_aux: fixes x :: "'c"   and y :: "'d"   and t :: "(('c \<times> 'd) \<times> nat) tree" shows "lvl (update x y t) = lvl t \<or> lvl (update x y t) = lvl t + 1 \<and> sngl (update x y t)"
maxR_code: fixes d :: "'d"   and x :: "'x"   and y :: "'x" shows "MaxR.maxR (Pd d) x y = (if (x, y) \<in> Pd d then y else x)"
is_run_def: fixes r :: "nat \<Rightarrow> 'a" shows "a.is_run r \<equiv> r 0 \<in> a.V0 \<and> ipath a.E r"
sum_bin: fixes x :: "'b"   and y :: "'b" shows "\<Sum> {x, y} = x + y"
weak_unit_cancel_left: fixes a :: "'a"   and f :: "'a"   and g :: "'a" assumes "weak_unit a"   and "ide f"   and "ide g"   and "a \<star> f \<cong> a \<star> g" shows "f \<cong> g"
mat_max_def: fixes A :: "'a mat"   and B :: "'a mat" shows "max\<^sub>m A B = mat (dim_row A) (dim_col A) (\<lambda>ij. max (A $$ ij) (B $$ ij))"
cdf_lim_at_top_prob: shows "(C \<longlongrightarrow> 1) at_top"
newton_interpolation_poly_int_def: fixes x_fs :: "(int \<times> int) list" shows "newton_interpolation_poly_int x_fs = (let xs = map fst x_fs in Let (map snd x_fs) (newton_poly_impl_int xs))"
indicator_mult_ennreal: fixes A :: "'c set"   and x :: "'c"   and r :: "real" shows "indicator A x * ennreal r = ennreal (indicat_real A x * r)"
add_decreasing2: fixes c :: "'a multiset"   and a :: "'a multiset"   and b :: "'a multiset" assumes "c \<subseteq># {#}"   and "a \<subseteq># b" shows "a + c \<subseteq># b"
sfilter_cong: fixes xs :: "[:'r:]"   and xs' :: "[:'r:]"   and p :: "'r \<rightarrow> tr"   and p' :: "'r \<rightarrow> tr" assumes "xs = xs'"   and "\<And>x. x \<in> sset xs \<Longrightarrow> p\<cdot>x = p'\<cdot>x" shows "sfilter\<cdot>p\<cdot>xs = sfilter\<cdot>p'\<cdot>xs'"
set_map: fixes f :: "'c \<Rightarrow> 'b"   and v :: "'c list" shows "set (map f v) = f ` set v"
boxInitD: fixes F :: "'a \<Rightarrow> bool" shows "\<box>Init F \<equiv> \<box>F"
prime_sum_upto_def: fixes f :: "nat \<Rightarrow> 'a"   and x :: "real" shows "prime_sum_upto f x = (\<Sum>p | prime p \<and> real p \<le> x. f p)"
Copied_1: shows "((ps, a) # branch) !. v = Some (qs, a')"
prems_3: shows "set (s # ss) \<subseteq> R"
bintrunc_bintrunc_ge: fixes n :: "nat"   and m :: "nat"   and w :: "int" assumes "n \<le> m" shows "take_bit n (take_bit m w) = take_bit n w"
assms_4: shows "(n'', nid'') \<in> his t n nid"
HasMethodDef_def: 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 C :: "char list"   and M :: "char list"   and mthd :: "ty list \<times> ty \<times> char list list \<times> expr"   and Cs :: "char list list" shows "P \<turnstile> C has M = mthd via Cs \<equiv> (Cs, mthd) \<in> MethodDefs P C M"
trivial_RModuleHom: fixes smult' :: "'f \<Rightarrow> 'b \<Rightarrow> 'b" assumes "\<forall>r\<in>UNIV. smult' r (0::'b) = (0::'b)" shows "RModuleHom UNIV (\<cdot>) V smult' 0"
code: fixes p :: "'b \<Rightarrow> bool"   and g21 :: "'b \<Rightarrow> 'a"   and g22 :: "'b \<Rightarrow> 'b"   and a :: "'b" shows "unfold_llist p g21 g22 a = (if p a then LNil else LCons (g21 a) (unfold_llist p g21 g22 (g22 a)))"
vbrelation_Limit_in_VsetI: fixes \<alpha> :: "V" assumes "Limit \<alpha>"   and "\<AA>\<lparr>Obj\<rparr> \<in>\<^sub>\<circ> Vset \<alpha>"   and "\<R>\<^sub>\<circ> (\<FF>\<lparr>ObjMap\<rparr>) \<in>\<^sub>\<circ> Vset \<alpha>" shows "\<FF>\<lparr>ObjMap\<rparr> \<in>\<^sub>\<circ> Vset \<alpha>"
lift_character_unchanged: fixes x :: "'a"   and \<chi>z :: "('a \<Rightarrow> complex) \<times> complex" assumes "x \<in> {\<one>}" shows "H.lift_character \<chi>z x = fst \<chi>z x"
bounded_iff: fixes b :: "'b word"   and c :: "'b word"   and a :: "'b word" shows "(signed.max b c \<le>s a) = (b \<le>s a \<and> c \<le>s a)"
simps_4: fixes e :: "(name, hf) finfun"   and A :: "fm" shows "eval_fm e (Neg A) = (\<not> eval_fm e A)"
d: fixes p :: "nat"   and x :: "int" shows "root_int_ceiling p x = (if 0 \<le> x then root_int_ceiling_pos p x else - root_int_floor_pos p (- x))"
simps: fixes a1 :: "('fenv_inca, 'c, 'fun) prog"   and a2 :: "('fenv_inca, 'c, ('fun, 'd) frame) state" shows "Sinca.load a1 a2 = (\<exists>F main fd H. a1 = Prog F H main \<and> a2 = State F H [Frame main 0 []] \<and> Finca_get F main = Some fd \<and> arity fd = 0)"
IH: fixes y :: "'a" assumes "set xs \<subseteq> A"   and "y \<in> A" shows "xs \<Squnion>\<^bsub>f\<^esub> y \<in> A"
continuous_on_eq_continuous_within: fixes s :: "'b set"   and f :: "'b \<Rightarrow> 'c" shows "continuous_on s f = (\<forall>x\<in>s. continuous (at x within s) f)"
to_tree_order_child2: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and nodes :: "(_) object_ptr list"   and child :: "(_) node_ptr" assumes "Shadow_DOM.heap_is_wellformed h"   and "ShadowRootClass.type_wf h"   and "ShadowRootClass.known_ptrs h"   and "h \<turnstile> Shadow_DOM.to_tree_order ptr \<rightarrow>\<^sub>r nodes"   and "cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r child \<noteq> ptr"   and "cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r child \<in> set nodes" obtains parent :: "(_) object_ptr" where "h \<turnstile> Shadow_DOM.get_parent child \<rightarrow>\<^sub>r Some parent" and "parent \<in> set nodes"
finite_tableaus_aux: fixes L :: "nat set"   and V :: "nat set"   and t0 :: "(nat \<times> linear_poly) list" shows "finite {t. lvars t = L \<and> rvars t = V - L \<and> \<triangle> t \<and> (\<forall>v. v \<Turnstile>\<^sub>t t = v \<Turnstile>\<^sub>t t0)}"
da_elim_cases_9: fixes Env :: "env"   and B :: "lname set"   and e :: "expr"   and c1 :: "stmt"   and c2 :: "stmt"   and A :: "assigned" assumes "Env\<turnstile> B \<guillemotright>\<langle>If(e) c1 Else c2\<rangle>\<guillemotright> A" obtains E :: "assigned" and C1 :: "assigned" and C2 :: "assigned" where "Env\<turnstile> B \<guillemotright>\<langle>e\<rangle>\<guillemotright> E" and "Env\<turnstile> B \<union> assigns_if True e \<guillemotright>\<langle>c1\<rangle>\<guillemotright> C1" and "Env\<turnstile> B \<union> assigns_if False e \<guillemotright>\<langle>c2\<rangle>\<guillemotright> C2" and "nrm A = nrm C1 \<inter> nrm C2" and "brk A = brk C1 \<Rightarrow>\<inter>  brk C2"
times_ow_def: fixes U :: "'b set"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" shows "times_ow U times \<equiv> \<forall>a b. a \<in> U \<longrightarrow> b \<in> U \<longrightarrow> times a b \<in> U"
exists_postrecording_violation_if_count_greater_0: fixes channel :: "nat \<Rightarrow> ('d \<times> 'd) option"   and trans :: "'d \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> bool"   and send :: "nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and recv :: "nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and init :: "('d, 'e, 'f) configuration"   and final :: "('d, 'e, 'f) configuration"   and t :: "('d, 'e, 'f) event list" assumes "computation channel trans send recv init final"   and "0 < computation.count_violations channel trans send recv init t" shows "\<exists>i. computation.postrecording_event channel trans send recv init t i \<and> 0 < card {j \<in> {i + 1..<length t}. computation.prerecording_event channel trans send recv init t j}"
binary_endofunctor_def: fixes C :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and T :: "'d \<times> 'd \<Rightarrow> 'd" shows "binary_endofunctor C T \<equiv> (category C \<and> product_category C C) \<and> product_category C (product_category.comp C C) \<and> binary_functor C C C T"
inputChainAlpha: fixes Xs :: "name set"   and xvec :: "name list"   and N :: "'a"   and P :: "('a, 'b, 'c) psi"   and Ys :: "name set"   and p :: "(name \<times> name) list" assumes "Xs \<sharp>* inputChain xvec N P"   and "Ys \<sharp>* N"   and "Ys \<sharp>* P"   and "set p \<subseteq> Xs \<times> Ys" shows "inputChain xvec N P = inputChain (p \<bullet> xvec) (p \<bullet> N) (p \<bullet> P)"
execute_parallel_operator_no_effect_if: fixes ops :: "'a strips_operator list"   and v :: "'a"   and s :: "'a \<Rightarrow> bool option" assumes "\<forall>op\<in>set ops. v \<notin> set (add_effects_of op) \<and> v \<notin> set (delete_effects_of op)" shows "execute_parallel_operator s ops v = s v"
IH: fixes i :: "nat"   and a :: "nat"   and j :: "nat"   and b :: "nat" assumes "i \<le> a"   and "a < 2 ^ n + i"   and "j \<le> b"   and "b < 2 ^ n + j" shows "square2 n i j - {(a, b)} \<in> tiling Ls"
inter_union_distrib_left: fixes A :: "'c multiset"   and B :: "'c multiset"   and C :: "'c multiset" shows "A \<inter># B + C = (A + C) \<inter># (B + C)"
nuc_prop1: fixes f :: "'b \<Rightarrow> 'b"   and x :: "'b"   and y :: "'b" assumes "nucleus f" shows "f (x \<cdot> y) = f (x \<cdot> f y)"
stable_on_def: fixes ds :: "D2 set"   and X :: "X4 set" shows "StableNoDecomp.stable_on ds X = (StableNoDecomp.individually_rational_on ds X \<and> StableNoDecomp.stable_no_blocking_on ds X)"
lesspoll_def: fixes A :: "'c set"   and B :: "'d set" shows "A \<prec> B \<equiv> A \<lesssim> B \<and> \<not> A \<approx> B"
countable_bipartite_web_def: fixes \<Gamma> :: "('k, 'l) web_scheme" shows "countable_bipartite_web \<Gamma> \<equiv> (\<^bold>V\<^bsub>\<Gamma>\<^esub> \<subseteq> A \<Gamma> \<union> B \<Gamma> \<and> A \<Gamma> \<subseteq> \<^bold>V\<^bsub>\<Gamma>\<^esub> \<and> (\<forall>x y. edge \<Gamma> x y \<longrightarrow> x \<in> A \<Gamma> \<and> y \<in> B \<Gamma>)) \<and> (A \<Gamma> \<inter> B \<Gamma> = {} \<and> (\<forall>x. x \<notin> \<^bold>V\<^bsub>\<Gamma>\<^esub> \<longrightarrow> weight \<Gamma> x = 0)) \<and> (\<forall>x. weight \<Gamma> x \<noteq> \<top>) \<and> countable \<^bold>E\<^bsub>\<Gamma>\<^esub>"
a1: shows "j < dim_col (Matrix.mat (2 ^ (n + 1)) 1 (\<lambda>x. complex_of_real (case x of (i, j) \<Rightarrow> if even i then 1 / sqrt 2 ^ (n + 1) else - 1 / sqrt 2 ^ (n + 1))))"
oInv_ex: fixes F :: "ordinal \<Rightarrow> ordinal"   and a :: "ordinal" assumes "normal F"   and "F 0 \<le> a" shows "\<exists>q. F q \<le> a \<and> a < F (oSuc q)"
compare_LT_trans: fixes eq :: "'b \<rightarrow> 'b \<rightarrow> tr"   and compare :: "'b \<rightarrow> 'b \<rightarrow> Ordering"   and x :: "'b"   and y :: "'b"   and z :: "'b" assumes "class.Ord_linear eq compare"   and "compare\<cdot>x\<cdot>y = LT"   and "compare\<cdot>y\<cdot>z = LT" shows "compare\<cdot>x\<cdot>z = LT"
\<omega>_simps_4: shows "local.dom \<omega> = w \<star>\<^sub>B w"
tendsto_real_sqrt: fixes f :: "'c \<Rightarrow> real"   and x :: "real"   and F :: "'c filter" assumes "(f \<longlongrightarrow> x) F" shows "((\<lambda>x. sqrt (f x)) \<longlongrightarrow> sqrt x) F"
upper_interval_of: fixes a :: "'b" shows "upper (interval_of a) = a"
"1_1": shows "(T, S) \<in> \<langle>A \<times>\<^sub>r B\<rangle>Rs1"
phi_equiv_mapping_refl: fixes ph :: "'g \<Rightarrow> ('node \<times> 'val, 'val list) mapping"   and g :: "'g"   and m :: "('val, 'a set) mapping" shows "uninst_code.phi_equiv_mapping ph g m m"
while_def: fixes b :: "'a"   and x :: "'a" shows "while b do x od = (t b \<cdot> x)\<^sup>\<star> \<cdot> !b"
eq: shows "Ps' = map (extend ( \<Phi>1 + \<Gamma>' \<Rightarrow>* \<Psi> + \<Delta>')) ps"
size1_if_complete: fixes t :: "'b tree" assumes "Tree.complete t" shows "size1 t = 2 ^ height t"
prefixes_length: fixes z :: "nat"   and i :: "nat"   and j :: "nat"   and x :: "nat" shows "length (local.prefixes z i j x) = Suc x"
append_minus: fixes xs :: "'b list"   and ys :: "'b list" shows "xs @ ys - xs = ys"
emeasure_map_pmf: fixes f :: "'d \<Rightarrow> 'c"   and M :: "'d pmf"   and X :: "'c set" shows "emeasure (measure_pmf (map_pmf f M)) X = emeasure (measure_pmf M) (f -` X)"
less_mat_def: fixes A :: "'a mat"   and B :: "'a mat" shows "(A < B) = (A \<le> B \<and> \<not> B \<le> A)"
ineq_all_m: fixes m :: "nat" assumes "n \<le> m" shows "u m / real m \<le> u n / real n + 3 * C / real m"
trail_Cons_iff: fixes u :: "'a"   and e :: "'b"   and es :: "'b list"   and w :: "'a" shows "trail u (e # es) w = (e \<in> arcs G \<and> u = tail G e \<and> e \<notin> set es \<and> trail (head G e) es w)"
natural_transformation_\<T>e: shows "natural_transformation (\<cdot>\<^sup>o\<^sup>p) (\<cdot>\<^sub>S) (Y a) F (\<T> e)"
ocnet_bigger_than_next: fixes n :: "net_tree" shows "oclosed (opnet (\<lambda>i. optoy i \<langle>\<langle>\<^bsub>i\<^esub> qmsg) n) \<Turnstile> (\<lambda>_ _ _. True, other nos_inc (net_tree_ips n) \<rightarrow>) global (\<lambda>\<sigma>. \<forall>i\<in>net_tree_ips n. no (\<sigma> i) \<le> no (\<sigma> (nhid (\<sigma> i))))"
axioms: fixes A :: "V"   and f :: "V" assumes "unop A f" shows "nop A (1\<^sub>\<nat>) f"
len_of_dflt_size_def: fixes uu :: "dflt_size itself" shows "len_of uu \<equiv> dflt_size_aux"
iteration_mono_eq: fixes x :: "int"   and n :: "int" assumes "x ^ p = n" shows "(n div x ^ pm + x * int pm) div int p = x"
Equality: fixes arity :: "'e \<Rightarrow> nat"   and Ana :: "('e, 'f) Term.term \<Rightarrow> ('e, 'f) Term.term list \<times> ('e, 'f) Term.term list"   and S :: "('e, 'f) strand_step list"   and \<delta> :: "'f \<Rightarrow> ('e, 'f) Term.term"   and t :: "('e, 'f) Term.term"   and t' :: "('e, 'f) Term.term"   and uu :: "poscheckvariant"   and S' :: "('e, 'f) strand_step list"   and \<theta> :: "'f \<Rightarrow> ('e, 'f) Term.term"   and public :: "'e \<Rightarrow> bool" assumes "intruder_model arity Ana"   and "simple S"   and "Some \<delta> = mgu t t'" shows "((S @ \<langle>uu: t \<doteq> t'\<rangle>\<^sub>s\<^sub>t # S', \<theta>), S @ S' \<cdot>\<^sub>s\<^sub>t \<delta>, \<theta> \<circ>\<^sub>s \<delta>) \<in> intruder_model.LI_rel arity public"
moebius_circline_comp: fixes M1 :: "moebius"   and M2 :: "moebius"   and H :: "circline" shows "moebius_circline (moebius_comp M1 M2) H = moebius_circline M1 (moebius_circline M2 H)"
preserves_comp: fixes g :: "'a"   and f :: "'a" assumes "S.seq g f" shows "L (g \<cdot>\<^sub>S f) = L g \<cdot>\<^sub>S L f"