hcju/isabelleps huggingface dataset:
seq_mult_closed: fixes s :: "nat \<Rightarrow> nat \<Rightarrow> int"   and s' :: "nat \<Rightarrow> nat \<Rightarrow> int" assumes "s \<in> carrier Zp\<^bsup>\<omega>\<^esup>"   and "s' \<in> carrier Zp\<^bsup>\<omega>\<^esup>" shows "s \<otimes>\<^bsub>Zp\<^bsup>\<omega>\<^esup>\<^esub> s' \<in> carrier Zp\<^bsup>\<omega>\<^esup>"
assms_1: shows "\<lparr>\<nu>x\<rparr>F = \<lparr>\<nu>y\<rparr>G"
not_ordLess_iff_ordLeq: fixes r :: "('a \<times> 'a) set"   and r' :: "('b \<times> 'b) set" assumes "Well_order r"   and "Well_order r'" shows "((r', r) \<notin> ordLess) = (r \<le>o r')"
a: fixes x :: "'a"   and y :: "'a" shows "R x y = (R x = R y)"
sgn_divide: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and sgn :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.field_abs_sgn divide inverse times one plus zero minus uminus abs sgn" shows "sgn (divide a b) = divide (sgn a) (sgn b)"
simps: fixes r :: "real"   and c :: "real"   and cs :: "real list"   and a :: "atom" shows "subst\<^sub>0 (Eq r (c # cs)) a = (case a of Less s (d # ds) \<Rightarrow> Less (s - r * d / c) (ds - (d / c) *\<^sub>s cs) | Eq s (d # ds) \<Rightarrow> Eq (s - r * d / c) (ds - (d / c) *\<^sub>s cs))"
simps_1: fixes edge :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and more :: "'b" shows "edge \<lparr>edge = edge, \<dots> = more\<rparr> = edge"
inject_1: fixes x1 :: "'a"   and y1 :: "'a" shows "(Rval x1 = Rval y1) = (x1 = y1)"
list_all3_mono_strong: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> bool"   and xs :: "'a list"   and ys :: "'b list"   and zs :: "'c list"   and Q :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> bool" assumes "list_all3 P xs ys zs"   and "\<And>x y z. \<lbrakk>x \<in> set xs; y \<in> set ys; z \<in> set zs; P x y z\<rbrakk> \<Longrightarrow> Q x y z" shows "list_all3 Q xs ys zs"
addr_loc_type_conv_addr_loc_type: fixes typeof_addr :: "'a \<Rightarrow> htype option"   and P :: "'c prog"   and h :: "'b" shows "heap_base.addr_loc_type (\<lambda>_. typeof_addr) P h = heap_base'.addr_loc_type TYPE('b) typeof_addr P"
FIN_not_subseteq_NUM: shows "\<not> FIN \<subseteq> NUM"
transp: fixes A :: "'a set"   and h :: "'a \<Rightarrow> 'b"   and bo :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and m :: "'a \<Rightarrow> 'a \<Rightarrow> 'a option" assumes "merge_on A h bo m" shows "transp bo"
min_height_size1_if_incomplete: fixes t :: "'b tree" assumes "\<not> Tree.complete t" shows "2 ^ min_height t < size1 t"
wt_e: shows "\<lparr>prg = G, cls = accC, lcl = L\<rparr>\<turnstile>e\<Colon>-RefT statT"
assms: shows "\<guillemotleft>t : local.dom f \<rightarrow> \<one>\<guillemotright>"
cases: fixes x :: "tm" obtains y :: "name" and xa :: "tm" and z :: "name" where "atom y \<sharp> (xa, z)" and "atom z \<sharp> xa" and "x = xa"
monad_fail_def: fixes return :: "'d \<Rightarrow> 'e"   and bind :: "'e \<Rightarrow> ('d \<Rightarrow> 'e) \<Rightarrow> 'e"   and fail :: "'e" shows "monad_fail return bind fail \<equiv> monad return bind \<and> monad_fail_axioms bind fail"
arrI: fixes f :: "'a" assumes "left a f" shows "Left_a.arr f"
length_remdups_card_conv: fixes xs :: "'b list" shows "length (remdups xs) = card (set xs)"
covector_restrict_comp_conv: fixes x :: "'a"   and y :: "'a" assumes "top \<sqinter> x = x" shows "y \<sqinter> x \<le> y \<sqinter> id x"
expandTauFrame: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and P' :: "('a, 'b, 'c) psi"   and A\<^sub>P :: "name list"   and \<Psi>\<^sub>P :: "'b"   and C :: "'e" assumes "\<Psi> \<rhd> P \<longmapsto> \<tau> \<prec> P'"   and "extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>"   and "distinct A\<^sub>P"   and "A\<^sub>P \<sharp>* P"   and "A\<^sub>P \<sharp>* C" obtains \<Psi>' :: "'b" and A\<^sub>P' :: "name list" and \<Psi>\<^sub>P' :: "'b" where "extractFrame P' = \<langle>A\<^sub>P', \<Psi>\<^sub>P'\<rangle>" and "\<Psi>\<^sub>P \<otimes> \<Psi>' \<simeq> \<Psi>\<^sub>P'" and "A\<^sub>P' \<sharp>* C" and "A\<^sub>P' \<sharp>* P'" and "distinct A\<^sub>P'"
discriminant_nonneg: fixes a :: "real"   and b :: "real"   and c :: "real"   and x :: "real" assumes "a \<noteq> 0"   and "0 \<le> discrim a b c" shows "(a * x\<^sup>2 + b * x + c = 0) = (x = (- b + sqrt (discrim a b c)) / (2 * a) \<or> x = (- b - sqrt (discrim a b c)) / (2 * a))"
"1_2": fixes x :: "'a poly" assumes "k \<noteq> 0"   and "x = root_unity k"   and "\<not> x dvd p" shows "decompose_prod_root_unity_main (map_poly hom p) (k - 1) = map_prod id (map_poly hom) (decompose_prod_root_unity_main p (k - 1))"
AI_wn_def: fixes c :: "com" shows "AI_wn c = pfp_wn (local.step' \<top>) (bot c)"
zero_ratfps_code: shows "quot_of_ratfps 0 = (0, 1)"
next_candidates_def: fixes n :: "nat" shows "next_candidates n = (if n = 0 then (1001, primes_1000) else (n + 30, [n, n + 2, n + 6, n + 8, n + 12, n + 18, n + 20, n + 26]))"
prj_in_homs_1: shows "\<guillemotleft>\<p>\<^sub>0[\<mu>.leg0, HH\<nu>\<pi>\<rho>.leg1] : H\<mu>HH\<nu>\<pi>\<rho>.chine \<rightarrow>\<^sub>C HH\<nu>\<pi>\<rho>.chine\<guillemotright>"
goodI: fixes i :: "nat"   and j :: "nat"   and P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "nat \<Rightarrow> 'b" assumes "i < j"   and "P (f i) (f j)" shows "good P f"
distinct_5: fixes x2 :: "name"   and x31 :: "tm"   and x32 :: "tm" shows "Var x2 \<noteq> Eats x31 x32"
den_gt_0: fixes x :: "nat" assumes "x \<in> set_tree den" shows "0 < x"
glb_lower: fixes cl :: "'b potype"   and S :: "'b set"   and x :: "'b" assumes "CL cl"   and "S \<subseteq> pset cl"   and "x \<in> S" shows "(glb S cl, x) \<in> order cl"
fconst_on_vinsert: fixes a :: "V"   and c :: "V"   and A :: "V" shows "vinsert [a, c]\<^sub>\<circ> (fconst_on A c) = fconst_on (vinsert a A) c"
lprefix_ltake: fixes u :: "'b llist"   and v :: "'b llist" assumes "u \<le> v" shows "u = ltake (llength u) v"
simps_5: fixes \<mu> :: "'a ltln"   and \<mu>' :: "'a ltln" shows "pure_eventual (\<mu> U\<^sub>n \<mu>') = (\<mu> = true\<^sub>n \<or> pure_eventual \<mu>')"
F_def: fixes a :: "'a"   and b :: "'b" shows "F a b = CBlinfun (f a b)"
eucl_truncate_down_le: fixes x :: "'a"   and y :: "'a"   and w :: "nat" assumes "x \<le> y" shows "eucl_truncate_down w x \<le> y"
5: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x \<sqinter> y \<squnion> x \<sqinter> z \<le> x \<sqinter> (y \<squnion> z)"
seqI: fixes f :: "('e set, 'e \<Rightarrow> 'e) concrete_category.arr"   and g :: "('e set, 'e \<Rightarrow> 'e) concrete_category.arr" assumes "arr f"   and "arr g"   and "local.dom g = cod f" shows "seq g f"
f_bounded: fixes x :: "'b" assumes "x \<in> space K" shows "\<bar>f x\<bar> \<le> B"
cross3_simps_12: fixes b :: "'a"   and a :: "'a"   and c :: "'a" shows "b \<cdot> (a \<cdot> c) = a \<cdot> (b \<cdot> c)"
assms_1: fixes i :: "nat" assumes "i < length T" shows "S ! i \<in> set \<langle>a --\<guillemotright> b\<rangle>\<langle>T ! i\<rangle>"
LCons_llist_of_stream: fixes x :: "'c"   and xs :: "'c stream" shows "LCons x (llist_of_stream xs) = llist_of_stream (x ## xs)"
affine_parallel_def: fixes S :: "'b set"   and T :: "'b set" shows "affine_parallel S T = (\<exists>a. T = (+) a ` S)"
csk: shows "cs\<^bsup>\<pi>\<^esup> k' = xs"
simps_2: fixes n :: "nat"   and c :: "int"   and t :: "MIR.num" shows "numgcdh (CN n c t) = (\<lambda>g. gcd c (numgcdh t g))"
lw_2: fixes n :: "nat" assumes "0 < n" shows "w n \<le> real n * lw"
abs_eq: fixes x :: "'c \<Rightarrow> 'd option"   and xb :: "'c"   and xa :: "'d" assumes "eq_onp (\<lambda>m. finite (dom m)) x x" shows "fmupd xb xa (Abs_fmap x) = Abs_fmap (map_upd xb xa x)"
triangle_circumscribes_triangle_def: fixes A' :: "Points"   and B' :: "Points"   and C' :: "Points"   and A :: "Points"   and B :: "Points"   and C :: "Points" shows "triangle_circumscribes_triangle A' B' C' A B C \<equiv> incid A (line B' C') \<and> incid C (line A' B') \<and> incid B (line A' C')"
inverts_components: fixes a :: "'c" assumes "Cop.ide a" shows "S.inverse_arrows (\<Phi>' a) (\<Psi>'.map a)"
get_tag_name_new_shadow_root: fixes h :: "(_) heap"   and new_shadow_root_ptr :: "(_) shadow_root_ptr"   and h' :: "(_) heap"   and r :: "(_) heap \<Rightarrow> (_) heap \<Rightarrow> bool"   and ptr' :: "(_) element_ptr" assumes "h \<turnstile> new\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_M \<rightarrow>\<^sub>r new_shadow_root_ptr"   and "h \<turnstile> new\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_M \<rightarrow>\<^sub>h h'"   and "r \<in> get_tag_name_locs ptr'" shows "r h h'"
ord_eq_le_trans: fixes a :: "'g"   and b :: "'g"   and less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and c :: "'g" assumes "a = b"   and "less_eq b c" shows "less_eq a c"
that: fixes M' :: "real" assumes "exponential_order M' c I" shows "thesis"
nat_eq_diff_eq: fixes c :: "nat"   and b :: "nat"   and a :: "nat" assumes "c \<le> b" shows "(a = b - c) = (a + c = b)"
PiM_component: fixes i :: "nat" assumes "i \<in> UNIV" shows "distr (Pi\<^sub>M UNIV (\<lambda>n. exponential (R n))) (exponential (R i)) (\<lambda>\<omega>. \<omega> i) = exponential (R i)"
FWbisimulation_base_aux_def: fixes final1 :: "'a \<Rightarrow> bool"   and r1 :: "('d, 'b, 'a, 'c, 'e, 'f) semantics"   and final2 :: "'g \<Rightarrow> bool"   and r2 :: "('d, 'b, 'g, 'h, 'e, 'f) semantics" shows "FWbisimulation_base_aux final1 r1 final2 r2 \<equiv> multithreaded final1 r1 \<and> multithreaded final2 r2"
zero_im_sheaf_def: shows "im0.zero_im_sheaf \<equiv> \<lambda>V. 0::'e"
top_apply: fixes x :: "'c" shows "top_class.top x = top_class.top"
CConcStart: fixes xs :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators list"   and a :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators"   and x :: "int \<times> (int \<times> int \<times> protocol) \<times> (int \<times> int \<times> protocol) \<times> DummyContent" assumes "xs \<noteq> []"   and "Cp a x = \<bottom>" shows "Cp (list2FWpolicy (xs @ [a])) x = Cp (list2FWpolicy xs) x"
bv_leqs_eq: fixes xs :: "bool list"   and ys :: "bool list" shows "xs \<preceq>\<^sub>b ys = ((\<forall>i<length xs. xs ! i \<longrightarrow> ys ! i) \<and> length xs = length ys)"
a_r_def: shows "a_r = (\<lambda>r. f (\<lambda>i. if i < r then 0 else l))"
invertible_commutes_with_inverse: fixes a :: "'a" assumes "invertible a"   and "a \<in> G" shows "\<eta> (inverse a) = inverse' \<eta> a"
codomains_null: shows "C1.codomains C1.null = {}"
\<theta>_simps_4: shows "local.dom \<theta>' = f \<star> w"
nadds: fixes q :: "'t \<Rightarrow>\<^sub>0 'b" assumes "q \<in> B" shows "\<not> lt q adds\<^sub>t lt p"
step: fixes s :: "'a" assumes "fst g s" shows "(snd (snd g s), s) \<in> R"
fempty_iff: fixes c :: "'a" shows "(c |\<in>| {||}) = False"
bet_cong3: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and A' :: "'p"   and B' :: "'p" assumes "Bet A B C"   and "Cong A B A' B'" shows "\<exists>C'. A B C Cong3 A' B' C'"
set_conv_list: fixes g :: "'c \<Rightarrow> 'b"   and xs :: "'c list" shows "sum g (set xs) = sum_list (map g (remdups xs))"
sel_2: fixes x1 :: "'d set"   and x2 :: "'c set"   and x3 :: "'d \<Rightarrow> 'c \<Rightarrow> 'c set"   and x4 :: "'c \<Rightarrow> bool" shows "initial (nba x1 x2 x3 x4) = x2"
hseqI': fixes \<mu> :: "('a, 'a set) Maps.arr arrow_of_spans_data"   and \<nu> :: "('a, 'a set) Maps.arr arrow_of_spans_data" assumes "Span.arr \<mu>"   and "Span.arr \<nu>"   and "Span.src \<nu> = Span.trg \<mu>" shows "Span.hseq \<nu> \<mu>"
set_tag_name_types_preserved: fixes w :: "((_) heap, exception, unit) prog"   and element_ptr :: "(_) element_ptr"   and h :: "(_) heap"   and h' :: "(_) heap" assumes "w \<in> set_tag_name_locs element_ptr"   and "h \<turnstile> w \<rightarrow>\<^sub>h h'" shows "type_wf h = type_wf h'"
Rep_formula': fixes x :: "('a, 'b, 'c) formula" shows "hereditarily_fs (Rep_formula x)"
line_extension_def: fixes K :: "'a set"   and a :: "'a"   and E :: "'a set" shows "line_extension K a E = K #> a <+> E"
abs_eq_iff: fixes x :: "'a ltln"   and y :: "'a ltln" shows "(abs_ltln\<^sub>C x = abs_ltln\<^sub>C y) = x \<sim>\<^sub>C y"
balanced: fixes ps :: "'a set" assumes "ps \<subseteq> \<V>"   and "int (card ps) = \<t>" shows "int (\<B> index ps) = \<Lambda>\<^sub>t"
assms: shows "({#}, A) \<in> ns_mul_ext ns s"
rreq_oip_kD: shows "paodv i \<TTurnstile> onl \<Gamma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V (\<lambda>(\<xi>, l). l \<in> {PRreq-:3..PRreq-:22} \<longrightarrow> oip \<xi> \<in> kD (rt \<xi>))"
elims: fixes x :: "val"   and xa :: "val"   and y :: "val option" assumes "x \<squnion> xa = y" obtains n :: "nat" and n' :: "nat" where "x = VNat n" and "xa = VNat n'" and "y = (if n = n' then Some (VNat n) else None)"   | f :: "(val \<times> val) fset" and f' :: "(val \<times> val) fset" where "x = VFun f" and "xa = VFun f'" and "y = Some (VFun (f |\<union>| f'))"   | va :: "(val \<times> val) fset" and v :: "nat" where "x = VFun va__" and "xa = VNat v" and "y = None"   | vb :: "nat" and va :: "(val \<times> val) fset" where "x = VNat vb" and "xa = VFun va" and "y = None"
ldrop_0: fixes xs :: "'b llist" shows "ldrop 0 xs = xs"
iprev_less_mono_if_neq_iMin: fixes a :: "nat"   and b :: "nat"   and I :: "nat set" assumes "a < b"   and "a \<in> I"   and "b \<in> I"   and "iMin I \<noteq> a" shows "iprev a I < iprev b I"
netFlow_def: shows "netFlow \<equiv> sum f (outgoing' k) - sum f (incoming' k)"
wadjust_backto_standard_pos_B_Bk_Oc: fixes m :: "nat"   and rs :: "nat"   and list :: "cell list" assumes "wadjust_backto_standard_pos_O m rs ([], Oc # list)" shows "wadjust_backto_standard_pos_B m rs ([], Bk # Oc # list)"
pos_add_strict: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "class.strict_ordered_comm_monoid_add plus zero less_eq less"   and "less zero a"   and "less b c" shows "less b (plus a c)"
\<Delta>''_sel_2: shows "capacity \<Delta>'' = split (capacity \<Delta>)"
basis_null_space_eq_basis_left_null_space_transpose: fixes A :: "(('b, 'c) vec, 'd) vec" shows "basis_null_space A = basis_left_null_space (Finite_Cartesian_Product.transpose A)"
vimage_Collect_eq: fixes f :: "'d \<Rightarrow> 'e"   and P :: "'e \<Rightarrow> bool" shows "f -` Collect P = {y. P (f y)}"
all_in_\<pi>': shows "\<forall>l j. is_kth_obs \<pi> l j \<longrightarrow> (\<exists>j'. cs\<^bsup>\<pi>\<^esup> j = cs\<^bsup>\<pi>'\<^esup> j')"
is_natural_2: fixes f :: "'a" assumes "A.arr f" shows "\<tau> (A.cod f) \<cdot>\<^sub>B F f = \<tau> f"
fun_Compl_def: fixes A :: "'c \<Rightarrow> 'd" shows "- A = (\<lambda>x. - A x)"
bf_if_def: fixes v :: "'c"   and t :: "('c \<Rightarrow> bool) \<Rightarrow> 'd"   and e :: "('c \<Rightarrow> bool) \<Rightarrow> 'd" shows "bf_if v t e \<equiv> bf_ite (bf_lit v) t e"
has_derivative_continuous: fixes f :: "'c \<Rightarrow> 'd"   and f' :: "'c \<Rightarrow> 'd"   and x :: "'c"   and s :: "'c set" assumes "(f has_derivative f') (at x within s)" shows "continuous (at x within s) f"
Seq\<^sub>1: fixes P :: "nat exp prog"   and e\<^sub>0 :: "nat exp"   and s\<^sub>0 :: "heap \<times> val list \<times> sheap"   and v :: "val"   and s\<^sub>1 :: "heap \<times> val list \<times> sheap"   and e\<^sub>1 :: "nat exp"   and e\<^sub>2 :: "nat exp"   and s\<^sub>2 :: "heap \<times> val list \<times> sheap" assumes "P \<turnstile>\<^sub>1 \<langle>e\<^sub>0,s\<^sub>0\<rangle> \<Rightarrow> \<langle>Val v,s\<^sub>1\<rangle>"   and "P \<turnstile>\<^sub>1 \<langle>e\<^sub>1,s\<^sub>1\<rangle> \<Rightarrow> \<langle>e\<^sub>2,s\<^sub>2\<rangle>" shows "P \<turnstile>\<^sub>1 \<langle>e\<^sub>0;; e\<^sub>1,s\<^sub>0\<rangle> \<Rightarrow> \<langle>e\<^sub>2,s\<^sub>2\<rangle>"
cong_reverse_identity: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and C :: "'a"   and D :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Cong A A C D" shows "C = D"
inv_in_hom: fixes f :: "'b"   and a :: "'b"   and b :: "'b" assumes "B.iso f"   and "\<guillemotleft>f : a \<rightarrow>\<^sub>B b\<guillemotright>" shows "\<guillemotleft>B.inv f : b \<rightarrow>\<^sub>B a\<guillemotright>"
mono: fixes a :: "'a"   and b :: "'a" assumes "a \<in> I"   and "b \<in> I"   and "x < a"   and "a \<le> b" shows "f a \<le> f b"
complement_blocks_def: shows "\<B>\<^sup>C \<equiv> image_mset des.block_complement \<B>"
bounded_bdd_above: fixes P :: "'c \<Rightarrow> real" assumes "Expectations.bounded P" shows "bdd_above (range P)"
routing_ipassmt_wi_subsetted: fixes a1 :: "char list"   and b1 :: "'b wordinterval"   and tbl :: "'b routing_rule list"   and b2 :: "'b wordinterval" assumes "(a1, b1) \<in> set (routing_ipassmt_wi tbl)"   and "(a1, b2) \<in> set (routing_port_ranges tbl wordinterval_UNIV)" shows "wordinterval_to_set b2 \<subseteq> wordinterval_to_set b1"
orbit_sim_step: fixes s :: "'b"   and f :: "'b \<Rightarrow> 'b"   and t :: "'b" assumes "s \<in> orbit f t" shows "f s \<in> orbit f (f t)"
between_norm: fixes a :: "'a"   and b :: "'a"   and x :: "'a" shows "between (a, b) x = (norm (x - a) *\<^sub>R (b - x) = norm (b - x) *\<^sub>R (x - a))"
P6: shows "Col T B C \<longrightarrow> Coplanar A B C D"
residual_assoc: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x \<rightarrow> y \<leftarrow> z = x \<rightarrow> y \<leftarrow> z"
intros_2: fixes y :: "'a"   and x :: "'a"   and l :: "'a list" shows "y # x # l <~~> x # y # l"
accepting_pair\<^sub>G\<^sub>R_LTS_def: fixes \<delta> :: "('c \<times> 'd \<times> 'c) set"   and q\<^sub>0 :: "'c"   and P :: "('c \<times> 'd \<times> 'c) set \<times> ('c \<times> 'd \<times> 'c) set set"   and w :: "nat \<Rightarrow> 'd" shows "accepting_pair\<^sub>G\<^sub>R_LTS \<delta> q\<^sub>0 P w \<equiv> \<exists>r. LTS_is_inf_run \<delta> w r \<and> r 0 = q\<^sub>0 \<and> limit (\<lambda>i. (r i, w i, r (Suc i))) \<inter> fst P = {} \<and> (\<forall>I\<in>snd P. limit (\<lambda>i. (r i, w i, r (Suc i))) \<inter> I \<noteq> {})"
subset_natUnion: fixes X :: "'b set"   and f :: "nat \<Rightarrow> 'b set" assumes "\<forall>n. X \<subseteq> f n" shows "X \<subseteq> natUnion f"
assms: shows "\<forall>i. \<exists>j. f (i, j) \<noteq> bot"
stable_insort_not_equiv: fixes cmp :: "'b comparator"   and y :: "'b"   and x :: "'b"   and xs :: "'b list" assumes "compare cmp y x \<noteq> Equiv" shows "filter (\<lambda>y. compare cmp y y = Equiv) (Sorting_Algorithms.insort cmp x xs) = filter (\<lambda>y. compare cmp y y = Equiv) xs"
normalize_polynomial: fixes R :: "('c, 'd) ring_scheme"   and K :: "'c set"   and p :: "'c list" assumes "ring R"   and "polynomial\<^bsub>R\<^esub> K p" shows "ring.normalize R p = p"
cases: fixes \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and a1 :: "'c rule list"   and a2 :: "state" assumes "\<Gamma>,\<gamma>,p\<turnstile> a1 \<Rightarrow>\<^sub>r a2" obtains "a1 = []" and "a2 = Undecided"   | m :: "'c match_expr" and rs :: "'c rule list" where "a1 = Rule m__ Accept # rs__" and "a2 = Decision FinalAllow" and "matches \<gamma> m__ p"   | m :: "'c match_expr" and rs :: "'c rule list" where "a1 = Rule ma__ Drop # rsa__" and "a2 = Decision FinalDeny" and "matches \<gamma> ma__ p"   | m :: "'c match_expr" and rs :: "'c rule list" where "a1 = Rule mb__ Reject # rsb__" and "a2 = Decision FinalDeny" and "matches \<gamma> mb__ p"   | m :: "'c match_expr" and rs :: "'c rule list" where "a1 = Rule mc__ Return # rsc__" and "a2 = Undecided" and "matches \<gamma> mc__ p"   | rs :: "'c rule list" and t :: "state" and m :: "'c match_expr" where "a1 = Rule md__ Log # rsd__" and "a2 = ta__" and "\<Gamma>,\<gamma>,p\<turnstile> rsd__ \<Rightarrow>\<^sub>r ta__"   | rs :: "'c rule list" and t :: "state" and m :: "'c match_expr" where "a1 = Rule me__ Empty # rse__" and "a2 = tb__" and "\<Gamma>,\<gamma>,p\<turnstile> rse__ \<Rightarrow>\<^sub>r tb__"   | m :: "'c match_expr" and rs :: "'c rule list" and t :: "state" and a :: "action" where "a1 = Rule mf__ a # rsf__" and "a2 = tc__" and "\<not> matches \<gamma> mf__ p" and "\<Gamma>,\<gamma>,p\<turnstile> rsf__ \<Rightarrow>\<^sub>r tc__"   | m :: "'c match_expr" and chain :: "char list" and rs :: "'c rule list" and X :: "final_decision" and rrs :: "'c rule list" where "a1 = Rule mg__ (Call chain__) # rrs__" and "a2 = Decision X" and "matches \<gamma> mg__ p" and "\<Gamma> chain__ = Some rsg__" and "\<Gamma>,\<gamma>,p\<turnstile> rsg__ \<Rightarrow>\<^sub>r Decision X"   | chain :: "char list" and rs :: "'c rule list" and rrs :: "'c rule list" and t :: "state" and m :: "'c match_expr" where "a1 = Rule m (Call chain) # rrs" and "a2 = t" and "\<Gamma> chain = Some rs" and "\<Gamma>,\<gamma>,p\<turnstile> rs \<Rightarrow>\<^sub>r Undecided" and "\<Gamma>,\<gamma>,p\<turnstile> rrs \<Rightarrow>\<^sub>r t"
one_stalk_def: fixes S :: "'d set"   and is_open :: "'d set \<Rightarrow> bool"   and \<FF> :: "'d set \<Rightarrow> 'e set"   and \<rho> :: "'d set \<Rightarrow> 'd set \<Rightarrow> 'e \<Rightarrow> 'e"   and b :: "'e"   and add_str :: "'d set \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'e"   and mult_str :: "'d set \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero_str :: "'d set \<Rightarrow> 'e"   and one_str :: "'d set \<Rightarrow> 'e"   and I :: "'d set set"   and x :: "'d"   and V :: "'d set" assumes "stalk S is_open \<FF> \<rho> b add_str mult_str zero_str one_str I x" shows "stalk.one_stalk \<FF> \<rho> one_str I V \<equiv> direct_lim.class_of \<FF> \<rho> I V (one_str V)"
lconfD: fixes G :: "prog"   and s :: "st"   and vs :: "'a \<Rightarrow> val option"   and Ts :: "'a \<Rightarrow> ty option"   and n :: "'a"   and T :: "ty" assumes "G,s\<turnstile>vs[\<Colon>\<preceq>]Ts"   and "Ts n = Some T" shows "G,s\<turnstile>the (vs n)\<Colon>\<preceq>T"
CRotate_eq_rotateC: fixes p :: "((int \<times> int) set set, int) Combinators" shows "CRotate p = C (rotatePolicy p)"
Close1: fixes P :: "pi"   and y :: "name"   and P'' :: "pi"   and a :: "name"   and x :: "name"   and P' :: "pi"   and Q :: "pi"   and Q' :: "pi" assumes "P \<Longrightarrow>\<^sub>ly in P''\<rightarrow>a<x> \<prec> P'"   and "Q \<Longrightarrow>\<^sub>l a<\<nu>y> \<prec> Q'"   and "y \<sharp> P"   and "y \<sharp> Q" shows "P \<parallel> Q \<Longrightarrow>\<^sub>l \<tau> \<prec> <\<nu>y>(P' \<parallel> Q')"
encode_problem_with_operator_interference_exclusion_def: fixes \<Pi> :: "'b strips_problem"   and t :: "nat" shows "\<Phi>\<^sub>\<forall> \<Pi> t \<equiv> \<Phi>\<^sub>I \<Pi> \<^bold>\<and> (encode_operators \<Pi> t \<^bold>\<and> (encode_all_frame_axioms \<Pi> t \<^bold>\<and> (encode_interfering_operator_exclusion \<Pi> t \<^bold>\<and> (\<Phi>\<^sub>G \<Pi>) t)))"
perm_elem_finite: fixes M2 :: "('e, 'f, 'g) FSM"   and V :: "'e list set"   and V'' :: "('e \<times> 'h) list set"   and M1 :: "('e, 'h, 'i) FSM" assumes "is_det_state_cover M2 V"   and "well_formed M2"   and "V'' \<in> Perm V M1" shows "finite V''"
indRelLSTPO_impl_SRel_and_TRel_weakly_preserve_barbs_2: fixes SRel :: "('procS \<times> 'procS) set"   and TRel :: "('procT \<times> 'procT) set" assumes "rel_weakly_preserves_barbs (indRelLSTPO SRel TRel) (STCalWB SWB TWB)" shows "rel_weakly_preserves_barbs TRel TWB"
comp_inv_arr': fixes f :: "'a" assumes "C.iso f" shows "C.inv f \<cdot>\<^sub>C f = C.dom f"
lossless_S2_xor: fixes A :: "bool \<times> bool"   and out :: "bool" shows "lossless_spmf (S2_xor A out)"
open_ou_witness_def: fixes X :: "ou_witness set" shows "open X = (ou_witness.a \<notin> X \<or> X = UNIV)"