hcju/isabelleps huggingface dataset:
sort_of_atom_var: fixes n :: "nat"   and ty :: "ty" shows "sort_of (atom (Var n ty)) = sort_of_ty ty"
singleton_single: fixes default :: "unit \<Rightarrow> 'a"   and x :: "'a" shows "singleton default (single x) = x"
HCollectI: fixes a :: "hf"   and A :: "hf"   and P :: "hf \<Rightarrow> bool" assumes "a \<^bold>\<in> A"   and "P a" shows "a \<^bold>\<in> HCollect P A"
deriv_id: shows "deriv id = (\<lambda>z. 1::'a)"
mask_range_to_bl: fixes ptr :: "'b word"   and bits :: "nat" assumes "is_aligned ptr bits" shows "mask_range ptr bits = {x. take (LENGTH('b) - bits) (to_bl x) = take (LENGTH('b) - bits) (to_bl ptr)}"
r_const: fixes c :: "nat"   and x :: "nat" shows "eval (r_const c) [x] \<down>= c"
Sym: fixes \<Psi>' :: "'b"   and S :: "('a, 'b, 'c) psi"   and T :: "('a, 'b, 'c) psi" assumes "(\<Psi>', S, T) \<in> Rel" shows "(\<Psi>', T, S) \<in> Rel"
inf_continuous_def: fixes F :: "'c \<Rightarrow> 'd" shows "inf_continuous F = (\<forall>M. decseq M \<longrightarrow> F (\<Sqinter> range M) = (\<Sqinter>i. F (M i)))"
takeWhile_nth: fixes j :: "nat"   and P :: "'c \<Rightarrow> bool"   and xs :: "'c list" assumes "j < length (takeWhile P xs)" shows "takeWhile P xs ! j = xs ! j"
steps'_length: fixes qs :: "'c list"   and as :: "(nat \<times> nat list) list"   and n :: "nat"   and s :: "'c list" assumes "length qs = length as"   and "n \<le> length as" shows "length (steps' s qs as n) = length s"
zmax_ge_r: fixes y :: "int"   and x :: "int" shows "y \<le> zmax x y"
hyps: fixes n :: "nat" assumes "rotate m' xs = rotate n xs" shows "rotate (m' - n) xs = xs"
case_2: fixes f1 :: "'d \<Rightarrow> 'c"   and f2 :: "'e \<Rightarrow> 'c"   and f3 :: "('d \<Rightarrow> 'd \<Rightarrow> 'd) \<Rightarrow> 'c"   and x2 :: "'e" shows "(case Load x2 of Const x \<Rightarrow> f1 x | Load x \<Rightarrow> f2 x | Apply x \<Rightarrow> f3 x) = f2 x2"
prems_2: shows "\<forall>c. c \<in> set (clause # Wl') \<longrightarrow> 0 \<le> c \<and> c < length (getF stateA)"
mult_commute: fixes a :: "('a \<times> 'a) set"   and b :: "('a \<times> 'a) set" assumes "valid_frac a"   and "valid_frac b" shows "mult_rel a b = mult_rel b a"
assms_1: fixes n :: "nat" assumes "m \<le> n" shows "(\<Sum>k\<le>N. c k * f (n + k)) = g (n + N)"
sel_8: fixes x21 :: "nat"   and x22 :: "'d"   and x23 :: "'d"   and x24 :: "'e"   and x25 :: "'e"   and x26 :: "'f" shows "getId (Send x21 x22 x23 x24 x25 x26) = x21"
strict_sorted_imp_append_less: fixes xs :: "'a list"   and ys :: "'a list" assumes "strict_sorted (xs @ ys)" shows "xs < ys"
hoare_kcomp: fixes P :: "'b \<Rightarrow> bool"   and G :: "'b \<Rightarrow> 'c set"   and R :: "'c \<Rightarrow> bool"   and F :: "'c \<Rightarrow> 'd set"   and Q :: "'d \<Rightarrow> bool" assumes "P \<le> |G] R"   and "R \<le> |F] Q" shows "P \<le> |G ; F] Q"
h: shows "h \<circ> (\<lambda>t. [fst t] \<otimes>\<^bsub>K [X]\<^esub> X [^]\<^bsub>K [X]\<^esub> snd t) \<in> set (dense_repr p) \<rightarrow> carrier A"
frac_less_eq: fixes y :: "'a"   and z :: "'a"   and x :: "'a"   and w :: "'a" assumes "y \<noteq> (0::'a)"   and "z \<noteq> (0::'a)" shows "(x / y < w / z) = ((x * z - w * y) / (y * z) < (0::'a))"
prems_2: shows "(l # ts) ! i = (p, is, \<theta>, Ghost\<^sub>s\<^sub>b A L R W # sb, \<D>, \<O>, \<R>)"
assms_6: shows "\<forall>x\<in>{0..<n}. f x = g (\<phi> x)"
i_take_drop_map: fixes m :: "nat"   and n :: "nat"   and f :: "nat \<Rightarrow> 'b" shows "f \<Down> n \<up> m = map f [m..<n]"
atLeastatMost_subset_iff: fixes a :: "'e word"   and b :: "'e word"   and c :: "'e word"   and d :: "'e word" shows "(signed.atLeastAtMost a b \<subseteq> signed.atLeastAtMost c d) = (\<not> a \<le>s b \<or> c \<le>s a \<and> b \<le>s d)"
tpOfV_def: shows "tpOfV \<equiv> Eps tpOfV_pred"
i: shows "i \<in> I \<and> i \<in> II"
closed_cbox: fixes a :: "'b"   and b :: "'b" shows "closed (cbox a b)"
nlists_def: shows "nlists \<equiv> \<lambda>A n. {x. nlistsp A n x}"
in_dom: fixes i :: "'agent" assumes "i \<in> agents" shows "Order_Predicates.preorder_on alts (R i)"
is_aligned_shift: fixes k :: "'a word"   and m :: "nat" shows "is_aligned (k << m) m"
alpha_tConj: fixes tset1 :: "('a, 'b, 'c, 'd) Tree set['a]"   and tset2 :: "('a, 'b, 'c, 'd) Tree set['a]" shows "(tConj tset1 =\<^sub>\<alpha> tConj tset2) = rel_bset (=\<^sub>\<alpha>) tset1 tset2"
cindex_poly_rec: fixes a :: "real"   and b :: "real"   and p :: "real poly"   and q :: "real poly" assumes "a < b"   and "poly (p * q) a \<noteq> 0"   and "poly (p * q) b \<noteq> 0" shows "cindex_poly a b q p = cross (p * q) a b + cindex_poly a b (- (p mod q)) q"
List_app: fixes p :: "ref"   and h :: "ref \<Rightarrow> ref"   and as :: "ref list"   and bs :: "ref list" shows "List p h (as @ bs) = (\<exists>y. Path p h y as \<and> List y h bs)"
BC: shows "execution trans sends start [cfg] [] \<and> nonUniform (last [cfg]) \<and> (\<exists>cfgList' msgList'. nonUniform (last cfgList') \<and> prefixList [cfg] cfgList' \<and> prefixList [] msgList' \<and> execution trans sends start cfgList' msgList' \<and> (\<exists>msg'. execution.minimalEnabled [cfg] [] msg' \<and> msg' \<in> set msgList'))"
cnf_of_operator_encoding_structure: fixes \<Pi> :: "'b strips_problem"   and t :: "nat" shows "cnf (encode_operators \<Pi> t) = cnf (encode_all_operator_preconditions \<Pi> (\<Pi>\<^sub>\<O>) t) \<union> cnf (encode_all_operator_effects \<Pi> (\<Pi>\<^sub>\<O>) t)"
lift_subst_lt: fixes i :: "nat"   and j :: "nat"   and t :: "dB"   and s :: "dB" assumes "i < j + 1" shows "lift (t[s/j]) i = lift t i[lift s i/j + 1]"
to_fun_mult: fixes g :: "nat \<Rightarrow> 'a"   and f :: "nat \<Rightarrow> 'a"   and x :: "'a" assumes "g \<in> carrier P"   and "f \<in> carrier P"   and "x \<in> carrier R" shows "to_fun (f \<otimes>\<^bsub>P\<^esub> g) x = to_fun f x \<otimes> to_fun g x"
c4: fixes x :: "('a \<times> 'a set) set"   and y :: "('a \<times> 'a set) set" shows "x \<cdot> y \<cdot> 1\<^sub>\<pi> = x \<cdot> (y \<cdot> 1\<^sub>\<pi>)"
MGT_lemma: fixes C :: "((state \<Rightarrow> state \<Rightarrow> bool) \<times> com \<times> (state \<Rightarrow> state \<Rightarrow> bool)) set"   and c :: "com" assumes "\<forall>p. C \<tturnstile> {MGT (CALL p)}" shows "C \<tturnstile> {MGT c}"
singular_relboundary_alt: fixes p :: "nat"   and X :: "'c topology"   and S :: "'c set"   and c :: "((nat \<Rightarrow> real) \<Rightarrow> 'c) \<Rightarrow>\<^sub>0 int" shows "singular_relboundary p X S c = (\<exists>d e. singular_chain (Suc p) X d \<and> singular_chain p (subtopology X S) e \<and> chain_boundary (Suc p) d = c + e)"
closed_nonneg_Reals_complex: shows "closed \<real>\<^sub>\<ge>\<^sub>0"
path_decomp': fixes xs :: "'a list"   and x :: "'a"   and ys :: "'a list" assumes "H.path (xs @ x # ys)" shows "H.path (xs @ [x])"
simps: fixes a :: "'a"   and r :: "('b \<times> 'a) set" shows "(a \<in> Range r) = (\<exists>aa b. a = b \<and> (aa, b) \<in> r)"
stutnempty: fixes s :: "nat \<Rightarrow> 'b"   and n :: "nat" assumes "\<not> stutstep s n" shows "notemptyseq (s |\<^sub>s n)"
point_indices_def: fixes t :: "int" shows "point_indices t \<equiv> {int (\<B> index ps) |ps. int (card ps) = t \<and> ps \<subseteq> \<V>}"
cl_in_subset_carrier: fixes \<E> :: "'a set"   and X :: "'a set" assumes "\<E> \<subseteq> \<E>" shows "\<E>.cl_in \<E> X \<subseteq> \<E>"
c: shows "(\<lambda>z. if z = 1 then c else (z - 1) * deriv zeta z - deriv zeta z / zeta z) analytic_on {s. 1 \<le> Re s}"
single_inv: fixes f :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and z :: "'e"   and i :: "'e \<Rightarrow> 'e"   and a :: "'e"   and m :: "int" assumes "commute_bpp f (aug_pm_set z i {a})"   and "assoc_bpp (aug_pm_set z i {a}) f"   and "ipp_cond1 {a} i"   and "ipp_cond2 z {a} i f"   and "ipp_cond3 z i"   and "inv_ipp z i f {a}"   and "commute_bpp f (aug_pm_set z i {a})"   and "zeroA z i f {a} z" shows "\<^sub>i- m\<Odot>a\<^bsub>f,i,z\<^esub> = (- m)\<Odot>a\<^bsub>f,i,z\<^esub>"
dtermsl_add_stermsl_beforeD: fixes q :: "('a, 'b, 'c, 'd) seqp"   and p :: "('a, 'b, 'c, 'd) seqp" assumes "q \<in> dtermsl p" shows "\<exists>ps\<in>stermsl p. q \<in> dtermsl ps"
S_list_respects_labels: fixes B :: "'c set"   and \<phi> :: "'a \<Rightarrow> 'c"   and ss :: "'a permutation list"   and v :: "'a" assumes "label_wrt B \<phi>"   and "ss \<in> lists (\<Union>(f, g)\<in>fundfoldpairs. {Abs_induced_automorph f g})"   and "v \<in> \<Union> X" shows "\<phi> (sum_list ss \<rightarrow> v) = \<phi> v"
ct_suffix_ts_conv_suffix: fixes ts :: "t list"   and ts' :: "t list" shows "ct_suffix (to_ct_list ts) (to_ct_list ts') = suffix ts ts'"
cases: fixes a1 :: "(ctx \<times> cld) list"   and a2 :: "char list list" assumes "fields_in_path a1 a2" obtains "a1 = []" and "a2 = []"   | cld :: "cld" and cl_f_list :: "(cl \<times> char list) list" and ctxcld_list :: "(ctx \<times> cld) list" and fs :: "char list list" and fs' :: "char list list" and ctx :: "ctx" where "a1 = [(ctx, cld)] @ ctxcld_list" and "a2 = fs'" and "class_fields cld (map (\<lambda>(cl_XXX, f_XXX). fd_def cl_XXX f_XXX) cl_f_list)" and "fields_in_path ctxcld_list fs" and "fs' = map (\<lambda>(cl_XXX, f_XXX). f_XXX) cl_f_list @ fs"
xs_decomp: shows "xs = xs_pre @ v # w # xs_post"
r_prepend: fixes r :: "recf"   and vs :: "nat list"   and x :: "nat" assumes "recfn 1 r" shows "eval (r_prepend vs r) [x] = (if x < length vs then Some (vs ! x) else eval r [x - length vs])"
induct: fixes P :: "'a list \<Rightarrow> bool"   and a0 :: "'a list" assumes "P []"   and "\<And>x. P [x]"   and "\<And>v vb vc. \<lbrakk>P (take (length (v # vb # vc) div 2) (v # vb # vc)); P (drop (length (v # vb # vc) div 2) (v # vb # vc))\<rbrakk> \<Longrightarrow> P (v # vb # vc)" shows "P a0"
map\<^sub>0_simp: fixes a :: "'b" assumes "B.obj a" shows "FoH.map\<^sub>0 a = \<tau>.F.map\<^sub>0 (H.map\<^sub>0 a)"
row_add_row_def: fixes A :: "(('d, 'e) vec, 'f) vec"   and a :: "'f"   and b :: "'f"   and q :: "'d"   and i :: "'f" shows "row_add_row A a b q i = (\<chi>j. if i = a then A $ a $ j + q * A $ b $ j else A $ i $ j)"
deg_n_def: fixes R :: "('d, 'e) Ring_scheme"   and S :: "('d, 'f) Ring_scheme"   and X :: "'d"   and p :: "'d" shows "deg_n R S X p = c_max S (SOME c. cf_sol R S X p c)"
crename_AndL2: fixes R :: "trm"   and a :: "coname"   and b :: "coname"   and x :: "name"   and N :: "trm"   and y :: "name" assumes "R[a\<turnstile>c>b] = AndL2 x.N y"   and "x \<sharp> R" shows "\<exists>N'. R = AndL2 x.N' y \<and> N'[a\<turnstile>c>b] = N"
iWHILEI_def: fixes bind :: "'c \<Rightarrow> ('d \<Rightarrow> 'b) \<Rightarrow> 'b"   and return :: "'d \<Rightarrow> 'b"   and I :: "'d \<Rightarrow> bool"   and b :: "'d \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'c"   and s0 :: "'d" shows "iWHILEI bind return I b f s0 \<equiv> REC (WHILEI_body bind return I b f) s0"
assms_1: fixes x :: "real"   and y :: "real" assumes "b \<le> x"   and "x \<le> y" shows "f x \<le> f y"
B_trusts_YM4_newK: fixes K :: "nat"   and NB :: "nat"   and evs :: "event list" assumes "Crypt K (Nonce NB) \<in> parts (knows Spy evs)"   and "Nonce NB \<notin> analz (knows Spy evs)"   and "evs \<in> yahalom" shows "\<exists>A B NA. Says Server A \<lbrace>Crypt (shrK A) \<lbrace>Agent B, Key K, Nonce NA, Nonce NB\<rbrace>, Crypt (shrK B) \<lbrace>Agent A, Key K\<rbrace>\<rbrace> \<in> set evs"
SFAssInitThrow\<^sub>1: fixes P :: "nat exp prog"   and e\<^sub>2 :: "nat exp"   and s\<^sub>0 :: "heap \<times> val list \<times> sheap"   and v :: "val"   and h\<^sub>1 :: "heap"   and l\<^sub>1 :: "val list"   and sh\<^sub>1 :: "sheap"   and C :: "char list"   and F :: "char list"   and t :: "ty"   and D :: "char list"   and a :: "nat exp"   and s' :: "heap \<times> val list \<times> sheap" assumes "P \<turnstile>\<^sub>1 \<langle>e\<^sub>2,s\<^sub>0\<rangle> \<Rightarrow> \<langle>Val v,(h\<^sub>1, l\<^sub>1, sh\<^sub>1)\<rangle>"   and "P \<turnstile> C has F,Static:t in D"   and "\<nexists>sfs. sh\<^sub>1 D = \<lfloor>(sfs, Done)\<rfloor>"   and "P \<turnstile>\<^sub>1 \<langle>INIT D ([D],False) \<leftarrow> unit,(h\<^sub>1, l\<^sub>1, sh\<^sub>1)\<rangle> \<Rightarrow> \<langle>throw a,s'\<rangle>" shows "P \<turnstile>\<^sub>1 \<langle>C\<bullet>\<^sub>sF{D} := e\<^sub>2,s\<^sub>0\<rangle> \<Rightarrow> \<langle>throw a,s'\<rangle>"
simulation1: fixes s1 :: "'s1"   and s2 :: "'s2"   and tl1 :: "'tl1"   and s1' :: "'s1" assumes "s1 \<approx> s2"   and "s1 -1-tl1\<rightarrow> s1'"   and "\<not> \<tau>move1 s1 tl1 s1'" shows "\<exists>s2' s2'' tl2. s2 -\<tau>2\<rightarrow>* s2' \<and> s2' -2-tl2\<rightarrow> s2'' \<and> \<not> \<tau>move2 s2' tl2 s2'' \<and> s1' \<approx> s2'' \<and> tl1 \<sim> tl2"
add_0_iff: fixes b :: "'a"   and a :: "'a" shows "(b = b + a) = (a = (0::'a))"
id_stutter_sampler: fixes \<sigma> :: "nat \<Rightarrow> 'a" shows "stutter_sampler id \<sigma>"
\<tau>exec_1_def: fixes addr2thread_id :: "'a \<Rightarrow> 'b"   and thread_id2addr :: "'b \<Rightarrow> 'a"   and spurious_wakeups :: "bool"   and empty_heap :: "'c"   and allocate :: "'c \<Rightarrow> htype \<Rightarrow> ('c \<times> 'a) set"   and typeof_addr :: "'c \<Rightarrow> 'a \<Rightarrow> htype option"   and heap_read :: "'c \<Rightarrow> 'a \<Rightarrow> addr_loc \<Rightarrow> 'a val \<Rightarrow> bool"   and heap_write :: "'c \<Rightarrow> 'a \<Rightarrow> addr_loc \<Rightarrow> 'a val \<Rightarrow> 'c \<Rightarrow> bool"   and P :: "(nat \<times> nat \<times> 'a instr list \<times> (nat \<times> nat \<times> String.literal option \<times> nat \<times> nat) list) prog"   and t :: "'b"   and \<sigma> :: "'a option \<times> 'c \<times> 'a frame list"   and \<sigma>' :: "'a option \<times> 'c \<times> 'a frame list" shows "JVM_heap_base.\<tau>exec_1 addr2thread_id thread_id2addr spurious_wakeups empty_heap allocate typeof_addr heap_read heap_write P t \<sigma> \<sigma>' = (JVM_heap_base.exec_1 addr2thread_id thread_id2addr spurious_wakeups empty_heap allocate typeof_addr heap_read heap_write P t \<sigma> \<lbrace>\<rbrace> \<sigma>' \<and> JVM_heap_base.\<tau>Move2 typeof_addr P \<sigma>)"
Hoare_While: fixes G :: "((Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool) set"   and c :: "OBJ"   and b :: "BExpr"   and I :: "(Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> 'a \<Rightarrow> bool" assumes "G \<rhd> c : \<lambda>s t. \<forall>r. evalB b (fst s) \<longrightarrow> I s r \<longrightarrow> I t r" shows "G \<rhd> While b c : \<lambda>s t. \<forall>r. I s r \<longrightarrow> I t r \<and> \<not> evalB b (fst t)"
handle3: fixes ck :: "bool"   and env :: "v sem_env"   and s1 :: "'b state"   and e :: "exp0"   and s2 :: "'b state"   and a :: "abort"   and pes :: "(pat \<times> exp0) list" assumes "evaluate ck env s1 e (s2, Rerr (Rabort a))" shows "evaluate ck env s1 (Handle e pes) (s2, Rerr (Rabort a))"
span_explicit: fixes b :: "'b set" shows "cspan b = {\<Sum>a\<in>t. r a *\<^sub>C a |t r. finite t \<and> t \<subseteq> b}"
cases: fixes x :: "'b gexp" obtains uu_ :: "bool" where "x = Bc uu_"   | a1 :: "'b aexp" and a2 :: "'b aexp" where "x = Eq a1__ a2__"   | a1 :: "'b aexp" and a2 :: "'b aexp" where "x = Gt a1 a2"   | v :: "'b" and l :: "value list" where "x = In v l"   | g1 :: "'b gexp" and g2 :: "'b gexp" where "x = Nor g1 g2"
FF_def: shows "FF \<equiv> \<lambda>\<mu>\<nu>. if C.VV.arr \<mu>\<nu> then (F (fst \<mu>\<nu>), F (snd \<mu>\<nu>)) else D.VV.null"
inv_rule_from_INV_rule: fixes T :: "('a, 'b) TS_scheme"   and I :: "'a set"   and P :: "'a set" assumes "init T \<subseteq> I"   and "I \<subseteq> P"   and "{I} TS.trans T {> I}" shows "reach T \<subseteq> P"
Compl_in_sets_lebesgue: fixes A :: "'b set" shows "(- A \<in> sets lebesgue) = (A \<in> sets lebesgue)"
list_set_rel_append: fixes x :: "'a"   and s :: "'b"   and a :: "'a \<Rightarrow> 'b"   and I :: "'a \<Rightarrow> bool"   and xs :: "'a list"   and S :: "'b set" assumes "(x, s) \<in> br a I"   and "(xs, S) \<in> \<langle>br a I\<rangle>list_set_rel"   and "s \<notin> S" shows "(xs @ [x], insert s S) \<in> \<langle>br a I\<rangle>list_set_rel"
unfold: fixes s :: "'d"   and h :: "'d \<rightarrow> ('e, 'd) Step" assumes "s \<noteq> \<bottom>" shows "unfold\<cdot>h\<cdot>s = unfold2\<cdot>(unfold\<cdot>h)\<cdot>(h\<cdot>s)"
st_proj2_inv_inv: fixes x :: "'b" shows "st_proj2 (st_proj2_inv x) = x"
P1_game_alt_def: fixes m1 :: "'msg1"   and m2 :: "'msg2"   and D :: "'view1 \<Rightarrow> bool spmf" shows "P1_game_alt m1 m2 D = coin_spmf \<bind> (\<lambda>b. funct m1 m2 \<bind> (\<lambda>(out1, out2). R1 m1 m2 \<bind> (\<lambda>rview. S1 m1 out1 \<bind> (\<lambda>sview. D (if b then rview else sview) \<bind> (\<lambda>b'. return_spmf (b = b'))))))"
adds_def: fixes b :: "'a"   and a :: "'a" shows "(b adds a) = (\<exists>k. a = b + k)"
h: fixes B :: "'c set" assumes "B \<subseteq> A \<and> B \<subseteq> carrier V \<and> gen_set B" shows "B = A"
asym: fixes a :: "'a \<times> 'b"   and b :: "'a \<times> 'b" assumes "less_prod a b"   and "less_prod b a" shows "False"
Sup_empty: fixes Inf :: "'c set \<Rightarrow> 'c"   and Sup :: "'c set \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "Sup {} = bot"
spr_simObs_def: fixes envObsC :: "'c \<Rightarrow> 'e" shows "spr_simObs envObsC \<equiv> \<lambda>a. (\<lambda>s. (envObsC (es s), ODList.lookup (ps s) a)) \<circ> snd \<circ> ODList.hd \<circ> snd"
preserves_arr: fixes f :: "'c" assumes "J'.arr f" shows "J.arr (\<psi> f)"
comm1Aux: fixes \<Psi> :: "'b"   and \<Psi>\<^sub>Q :: "'b"   and R :: "('a, 'b, 'c) psi"   and K :: "'a"   and xvec :: "name list"   and N :: "'a"   and R' :: "('a, 'b, 'c) psi"   and A\<^sub>R :: "name list"   and \<Psi>\<^sub>R :: "'b"   and P :: "('a, 'b, 'c) psi"   and M :: "'a"   and L :: "'a"   and P' :: "('a, 'b, 'c) psi"   and A\<^sub>Q :: "name list"   and A\<^sub>P :: "name list"   and \<Psi>\<^sub>P :: "'b"   and Q :: "('a, 'b, 'c) psi" assumes "\<Psi> \<otimes> \<Psi>\<^sub>Q \<rhd> R \<longmapsto> K\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> R'"   and "extractFrame R = \<langle>A\<^sub>R, \<Psi>\<^sub>R\<rangle>"   and "\<Psi> \<otimes> \<Psi>\<^sub>R \<rhd> P \<longmapsto> M\<lparr>L\<rparr> \<prec> P'"   and "\<Psi> \<otimes> \<Psi>\<^sub>Q \<otimes> \<Psi>\<^sub>R \<turnstile> M \<leftrightarrow> K"   and "\<langle>A\<^sub>Q, (\<Psi> \<otimes> \<Psi>\<^sub>Q) \<otimes> \<Psi>\<^sub>R\<rangle> \<hookrightarrow>\<^sub>F \<langle>A\<^sub>P, (\<Psi> \<otimes> \<Psi>\<^sub>P) \<otimes> \<Psi>\<^sub>R\<rangle>"   and "extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>"   and "extractFrame Q = \<langle>A\<^sub>Q, \<Psi>\<^sub>Q\<rangle>"   and "distinct A\<^sub>P"   and "distinct A\<^sub>R"   and "A\<^sub>R \<sharp>* A\<^sub>P"   and "A\<^sub>R \<sharp>* A\<^sub>Q"   and "A\<^sub>R \<sharp>* \<Psi>"   and "A\<^sub>R \<sharp>* P"   and "A\<^sub>R \<sharp>* Q"   and "A\<^sub>R \<sharp>* R"   and "A\<^sub>R \<sharp>* K"   and "A\<^sub>P \<sharp>* \<Psi>"   and "A\<^sub>P \<sharp>* R"   and "A\<^sub>P \<sharp>* P"   and "A\<^sub>P \<sharp>* M"   and "A\<^sub>Q \<sharp>* R"   and "A\<^sub>Q \<sharp>* M" obtains K' :: "'a" where "\<Psi> \<otimes> \<Psi>\<^sub>P \<rhd> R \<longmapsto> K'\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> R'" and "\<Psi> \<otimes> \<Psi>\<^sub>P \<otimes> \<Psi>\<^sub>R \<turnstile> M \<leftrightarrow> K'" and "A\<^sub>R \<sharp>* K'"
open_connected_component: fixes S :: "'a set"   and x :: "'a" assumes "open S" shows "open (connected_component_set S x)"
observable_transitionI: fixes P :: "'state"   and Q :: "'state"   and \<alpha> :: "'act"   and Q' :: "'state"   and P' :: "'state" assumes "P \<Rightarrow> Q"   and "Q \<rightarrow> \<langle>\<alpha>,Q'\<rangle>"   and "Q' \<Rightarrow> P'" shows "P \<Rightarrow>{\<alpha>} P'"
compl_le_compl_iff: fixes x :: "'a"   and y :: "'a" shows "(- x \<le> - y) = (y \<le> x)"
seq_char: fixes g :: "'a \<times> 'a"   and f :: "'a \<times> 'a" shows "H.VxV.seq g f = (V.seq (fst g) (fst f) \<and> V.seq (snd g) (snd f))"
map_sq_matrix_ident: fixes M :: "('c, 'd) sq_matrix" shows "map_sq_matrix (\<lambda>x. x) M = M"
while_dual_star: fixes p :: "'b Assertion"   and x :: "'b" shows "(While p do x) = ({\<cdot> p } * x) ^ \<otimes> * {\<cdot> - p }"
bij_is_surj: fixes f :: "'a \<Rightarrow> 'b" assumes "bij f" shows "surj f"
nat_to_set_inj: fixes u :: "nat"   and v :: "nat" assumes "nat_to_set u = nat_to_set v" shows "u = v"
ide_char': fixes a :: "'a1" shows "C1.ide a = (C1.arr a \<and> (C1.dom a = a \<or> C1.cod a = a))"
dvd_trans: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and a :: "'a"   and b :: "'a"   and c :: "'a" assumes "class.comm_monoid_mult times one"   and "dvd.dvd times a b"   and "dvd.dvd times b c" shows "dvd.dvd times a c"
multiset_eq_iff: fixes M :: "'c multiset"   and N :: "'c multiset" shows "(M = N) = (\<forall>a. count M a = count N a)"
optimize_matches_option_simple_ruleset: fixes rs :: "'b rule list"   and f :: "'b match_expr \<Rightarrow> 'b match_expr option" assumes "simple_ruleset rs" shows "simple_ruleset (optimize_matches_option f rs)"
tm_inject_apps: fixes \<zeta> :: "('c, 'd) hd"   and ss :: "('c, 'd) tm list"   and \<xi> :: "('c, 'd) hd"   and ts :: "('c, 'd) tm list" shows "(apps (Hd \<zeta>) ss = apps (Hd \<xi>) ts) = (\<zeta> = \<xi> \<and> ss = ts)"
inject_3: fixes x31 :: "char list"   and x32 :: "expr"   and y31 :: "char list"   and y32 :: "expr" shows "(\<lparr>x31\<rparr>x32 = \<lparr>y31\<rparr>y32) = (x31 = y31 \<and> x32 = y32)"
finite_pdigraph_axioms: shows "finite_pdigraph \<alpha> I \<AA>"
elims_2: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and x :: "'a local.term" assumes "monoidal_language C"   and "monoidal_language.Diag C x" obtains "x = \<^bold>\<I>"   | f :: "'a" where "x = \<^bold>\<langle>f__\<^bold>\<rangle>" and "partial_magma.arr C f__"   | f :: "'a" and u :: "'a local.term" where "x = \<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<otimes> u" and "partial_magma.arr C f \<and> monoidal_language.Diag C u \<and> u \<noteq> \<^bold>\<I>"
fake_Inj: fixes M :: "chan"   and chan :: "chan set"   and bad :: "agent set"   and IK :: "msg set" assumes "M \<in> chan" shows "M \<in> fake bad IK chan"
nth_sorted_list_of_set_greaterThanLessThan: fixes n :: "nat"   and j :: "nat"   and i :: "nat" assumes "n < j - Suc i" shows "sorted_list_of_set {i<..<j} ! n = Suc (i + n)"
SUP_L_P_upper: fixes v :: "'a"   and i :: "'b"   and u :: "'c"   and P :: "'a \<Rightarrow> 'b \<Rightarrow> 'e" assumes "pair v i \<sqsubset> u" shows "P v i \<sqsubseteq> SUP_L_P P u i"
alg_dvd_of_int_iff: fixes x :: "int"   and y :: "int" shows "(of_int x alg_dvd of_int y) = (x dvd y)"
ma_smult: shows "\<ominus> a \<odot>\<^bsub>M\<^esub> x \<in> carrier M"
distinct_replacefacesAt21: fixes i :: "nat"   and Fss :: "face list list"   and "is" :: "nat list"   and newFs :: "face list"   and olfF :: "face" assumes "i < |Fss|"   and "i \<in> set is"   and "distinct is"   and "distinct (Fss ! i)"   and "distinct newFs"   and "set (Fss ! i) \<inter> set newFs \<subseteq> {olfF}" shows "distinct (replacefacesAt2 is olfF newFs Fss ! i)"
prime_primitive_root_exists: fixes n :: "nat" assumes "1 < n"   and "prime n" shows "\<exists>x. residue_primroot n x"
supp_Conj\<^sub>\<alpha>: fixes tset\<^sub>\<alpha> :: "('a, 'b, 'c, 'd) Tree\<^sub>\<alpha> set['a]" shows "supp (Conj\<^sub>\<alpha> tset\<^sub>\<alpha>) = supp tset\<^sub>\<alpha>"
pointermap_p_valid_def: fixes p :: "nat"   and m :: "('a, 'b) pointermap_scheme" shows "pointermap_p_valid p m \<equiv> p < length (entries m)"