hcju/isabelleps huggingface dataset:
inj: shows "inj_on (\<lambda>i. i mod k) {d..<d + k}"
assms: shows "(\<Phi>\<^sub>S\<inverse> \<Pi> \<A> k) v \<noteq> None"
refl: fixes x :: "'a" shows "(x, x) \<in> Rel"
Step2_2a: fixes p :: "PrIds"   and l :: "Locs"   and rmhist :: "state \<Rightarrow> PrIds \<Rightarrow> histState" shows "\<turnstile> Write rmCh mm ires p l \<and> ImpNext p \<and> [HNext rmhist p]_(c p, r p, m p, rmhist!p) \<and> $ImpInv rmhist p \<longrightarrow> S4 rmhist p$ \<and> unchanged (e p, c p, r p, rmhist!p)"
Sbis_mC_C: fixes c :: "('test, 'atom, 'choice) cmd"   and d :: "('test, 'atom, 'choice) cmd"   and s :: "'state"   and t :: "'state" assumes "c \<approx>s d"   and "s \<approx> t" shows "\<exists>P F. mC_C Sbis c d s t P F"
ceiling_less_cancel: fixes x :: "'b"   and y :: "'b" assumes "\<lceil>x\<rceil> < \<lceil>y\<rceil>" shows "x < y"
initial: fixes p :: "'state" assumes "init p" shows "p \<in> nodes"
l_zero: fixes A :: "('c, 'd) aGroup_scheme"   and a :: "'c" assumes "aGroup A"   and "a \<in> carrier A" shows "\<zero>\<^bsub>A\<^esub> \<plusminus>\<^bsub>A\<^esub> a = a"
semiconfluentpI: fixes r :: "'d \<Rightarrow> 'd \<Rightarrow> bool" assumes "\<And>x y z. \<lbrakk>r x y; r\<^sup>*\<^sup>* x z\<rbrakk> \<Longrightarrow> \<exists>u. r\<^sup>*\<^sup>* y u \<and> r\<^sup>*\<^sup>* z u" shows "semiconfluentp r"
sp_append_one: fixes g :: "('d \<times> nat, 'e) rgraph_scheme"   and v1 :: "'d \<times> nat"   and es :: "('d \<times> nat) edge list"   and e :: "('d \<times> nat) edge"   and v2 :: "'d \<times> nat"   and subs :: "(('d \<times> nat) \<times> 'd \<times> nat) set" shows "SubRel.subpath g v1 (es @ [e]) v2 subs = (SubRel.subpath g v1 es (src e) subs \<and> e \<in> edges g \<and> (tgt e = v2 \<or> (tgt e, v2) \<in> subs\<^sup>+))"
wqo_on_def: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set" shows "wqo_on P A = (transp_on P A \<and> almost_full_on P A)"
Least_def: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and P :: "'b \<Rightarrow> bool" shows "ord.Least less_eq P = (THE x. P x \<and> (\<forall>y. P y \<longrightarrow> less_eq x y))"
finite_fun_symbols: fixes t :: "('a, 'b) Term.term" shows "finite (funs_term t)"
downsetE: fixes xss :: "'b list set"   and xs :: "'b list" assumes "downset xss"   and "xs \<in> xss" shows "butlast xs \<in> xss"
BIT_ge0: fixes x :: "int"   and b :: "bool" shows "(0 \<le> x BIT b) = (0 \<le> x)"
simps: fixes prec :: "nat"   and I :: "float interval list"   and a :: "float interval list"   and t1 :: "taylor_model"   and t2 :: "taylor_model" shows "tm_min prec I a t1 t2 = tm_union prec I a t1 t2"
dual: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less_eq' :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less' :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and P :: "'b set"   and f :: "'b \<Rightarrow> 'c" assumes "OrderingSetMap less_eq less less_eq' less' P f" shows "OrderingSetMap (\<lambda>a b. less_eq b a) (\<lambda>a b. less b a) (\<lambda>a b. less_eq' b a) (\<lambda>a b. less' b a) P f"
is_minimal_basisD3: fixes B :: "('t \<Rightarrow>\<^sub>0 'c) set"   and p :: "'t \<Rightarrow>\<^sub>0 'c"   and q :: "'t \<Rightarrow>\<^sub>0 'c" assumes "is_minimal_basis B"   and "p \<in> B"   and "q \<in> B"   and "p \<noteq> q" shows "\<not> lt q adds\<^sub>t lt p"
code: fixes t :: "'a tree" shows "mirror t = Node (root t) (mirror (right t)) (mirror (left t))"
assms_2: shows "p\<^sub>0\<^sub>R \<noteq> p\<^sub>1\<^sub>R"
domD: fixes a :: "'d"   and m :: "'d \<Rightarrow> 'e option" assumes "a \<in> dom m" shows "\<exists>b. m a = Some b"
conseq: fixes \<Gamma> :: "'e \<Rightarrow> ('f, 'e, 'g) com option"   and \<Theta> :: "('f set \<times> 'e \<times> 'f set \<times> 'f set) set"   and F :: "'g set"   and P' :: "'d \<Rightarrow> 'f set"   and c :: "('f, 'e, 'g) com"   and Q' :: "'d \<Rightarrow> 'f set"   and A' :: "'d \<Rightarrow> 'f set"   and P :: "'f set"   and Q :: "'f set"   and A :: "'f set" assumes "\<forall>Z. \<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>(P' Z) c (Q' Z),(A' Z)"   and "\<forall>s. s \<in> P \<longrightarrow> (\<exists>Z. s \<in> P' Z \<and> Q' Z \<subseteq> Q \<and> A' Z \<subseteq> A)" shows "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P c Q,A"
rep_eq: fixes x :: "'b pmf" shows "pmf x = (\<lambda>x. measure_pmf.prob x {x})"
\<Phi>_n_def: fixes x :: "real" shows "\<Phi>_n x = x\<^sup>2 / 2 * ((real_of_int i' - 1) * 2 ^ (l + 1) + (real_of_int i + 1) * 2 ^ (l' + 1)) - x ^ 3 / 3 * 2 ^ (l' + 1) * 2 ^ (l + 1) - x * (real_of_int i + 1) * (real_of_int i' - 1)"
assms_1: shows "\<forall>(uu_, p)\<in>set \<mu>. 0 \<le> p"
resource_of_oracle_rprodl: fixes "oracle" :: "'c \<times> 'd \<times> 'e \<Rightarrow> 'a \<Rightarrow> ('b \<times> 'c \<times> 'd \<times> 'e) spmf"   and s1 :: "'c"   and s2 :: "'d"   and s3 :: "'e" shows "RES (map_fun rprodl (map_fun id (map_spmf (map_prod id lprodr))) oracle) ((s1, s2), s3) = RES oracle (s1, s2, s3)"
lookup_of_nat: fixes n :: "nat"   and k :: "'b" shows "lookup (of_nat n) k = (of_nat n when k = (0::'b))"
is_REL_state_aux1: fixes c :: "nat"   and n :: "nat"   and ts :: "nat \<Rightarrow> thread"   and t :: "nat"   and tk :: "nat" assumes "B.reachable (c, n, ts)"   and "is_REL_state ts"   and "t < N"   and "ts t = WAIT tk" shows "tk \<noteq> c"
hd_prog_append_Prog\<^sub>s\<^sub>b: fixes x :: "'b"   and sb :: "'b memref list"   and p :: "'b"   and p' :: "'b"   and mis :: "instr list" shows "hd_prog x (sb @ [Prog\<^sub>s\<^sub>b p p' mis]) = hd_prog p sb"
that: fixes s :: "real" assumes "0 < s"   and "\<And>z. z \<in> cball \<xi> s - {\<xi>} \<Longrightarrow> f z \<noteq> (0::'a)" shows "thesis"
subgraph_complete: fixes G :: "nat set \<times> nat set set" assumes "uwellformed G" shows "subgraph G (Ugraph_Lemmas.complete (uverts G))"
simps_2: fixes x :: "'a"   and ys :: "'a list" shows "linorder_class.sorted (x \<Colon> ys) = (Ball (set ys) ((\<le>) x) \<and> linorder_class.sorted ys)"
finsum_singleton: fixes f :: "'c \<Rightarrow> 'a"   and i :: "'c" assumes "f i \<in> R" shows "finsum {i} f = f i"
chine_assoc_props_3: shows "cods.Prj\<^sub>1\<^sub>0 \<cdot> cods.chine_assoc = cods.Prj\<^sub>0\<^sub>1"
prj_joint_monic: fixes f :: "'a"   and g :: "'a"   and h :: "'a"   and h' :: "'a" assumes "cospan f g"   and "seq (prj1 f g) h"   and "seq (prj1 f g) h'"   and "prj0 f g \<cdot> h = prj0 f g \<cdot> h'"   and "prj1 f g \<cdot> h = prj1 f g \<cdot> h'" shows "h = h'"
EncE: fixes v :: "i"   and x :: "\<kappa>"   and F :: "\<Pi>\<^sub>1" assumes "[\<lbrace>x,F\<rbrace> in v]" shows "\<exists>r o\<^sub>1. Some r = d\<^sub>1 F \<and> Some o\<^sub>1 = d\<^sub>\<kappa> x \<and> o\<^sub>1 \<in> en r"
induct: fixes P :: "Cooper.num \<Rightarrow> Cooper.num \<Rightarrow> bool"   and a0 :: "Cooper.num"   and a1 :: "Cooper.num" assumes "\<And>t c. P t (C c)"   and "\<And>t n. P t (Bound n)"   and "\<And>t i a. P t a \<Longrightarrow> P t (CN 0 i a)"   and "\<And>t v i a. P t a \<Longrightarrow> P t (CN (Suc v) i a)"   and "\<And>t a. P t a \<Longrightarrow> P t (Neg a)"   and "\<And>t a b. \<lbrakk>P t a; P t b\<rbrakk> \<Longrightarrow> P t (Add a b)"   and "\<And>t a b. \<lbrakk>P t a; P t b\<rbrakk> \<Longrightarrow> P t (Sub a b)"   and "\<And>t i a. P t a \<Longrightarrow> P t (Mul i a)" shows "P a0 a1"
norm_exp: fixes x :: "'a" shows "norm (exp x) \<le> exp (norm x)"
enumerate_in_set: fixes S :: "'b set"   and n :: "nat" assumes "infinite S" shows "wellorder_class.enumerate S n \<in> S"
F1map_congL: fixes x :: "('e, 'd, 'f) F1"   and f :: "'d \<Rightarrow> 'd"   and g :: "'f \<Rightarrow> 'f" assumes "\<forall>a\<in>F1set2 x. f a = a"   and "\<forall>a\<in>F1set3 x. g a = a" shows "F1map id f g x = x"
equiv_chain_2b: fixes f :: "nat \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a"   and c :: "'a"   and X :: "'a set" assumes "[f[a .. b ..  c]X]" shows "[[a .. b .. c]X]"
cosafety_fo_spec_axioms_def: fixes sat :: "'c trace \<Rightarrow> 'd list \<Rightarrow> nat \<Rightarrow> bool" shows "cosafety_fo_spec_axioms sat \<equiv> \<forall>\<sigma> v i. sat \<sigma> v i \<longrightarrow> (\<exists>\<pi>. prefix_of \<pi> \<sigma> \<and> (\<forall>\<sigma>'. prefix_of \<pi> \<sigma>' \<longrightarrow> sat \<sigma>' v i))"
simps_1: fixes A :: "(('b, 'c) vec, 'd) vec"   and i :: "'d"   and j :: "'c"   and bezout :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<times> 'b \<times> 'b \<times> 'b \<times> 'b" shows "bezout_iterate A 0 i j bezout = A"
simps: fixes p :: "char list"   and ts :: "fterm list" shows "hatom_of_fatom (p, ts) = (p, hterms_of_fterms ts)"
cases_scheme: fixes r :: "'a state_scheme" obtains id :: "nat" and no :: "nat" and nhid :: "nat" and msg :: "msg" and num :: "nat" and sid :: "nat" and more :: "'a" where "r = \<lparr>state.id = id, no = no, nhid = nhid, msg = msg, num = num, sid = sid, \<dots> = more\<rparr>"
tuple_pr_arr: fixes a :: "'a"   and b :: "'a"   and h :: "'a" assumes "ide a"   and "ide b"   and "seq (pr0 a b) h" shows "tuple (pr1 a b \<cdot> h) (pr0 a b \<cdot> h) = h"
scaleC_add_left: fixes scaleR :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and scaleC :: "complex \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and a :: "complex"   and b :: "complex"   and x :: "'b" assumes "class.complex_vector scaleR scaleC plus zero minus uminus" shows "scaleC (a + b) x = plus (scaleC a x) (scaleC b x)"
select_convs_3: fixes Obj :: "'e set"   and Mor :: "'d set"   and Dom :: "'d \<Rightarrow> 'e"   and Cod :: "'d \<Rightarrow> 'e"   and Id :: "'e \<Rightarrow> 'd"   and Comp :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and more :: "'f" shows "Dom \<lparr>Obj = Obj, Mor = Mor, Dom = Dom, Cod = Cod, Id = Id, Comp = Comp, \<dots> = more\<rparr> = Dom"
span_eq_dim: fixes S :: "('b, 'c) vec set"   and T :: "('b, 'c) vec set" assumes "vec.span S = vec.span T" shows "vec.dim S = vec.dim T"
left_invertible_transpose: fixes A :: "(('c, 'd) vec, 'e) vec" shows "(\<exists>B. B ** Finite_Cartesian_Product.transpose A = mat (1::'c)) = (\<exists>B. A ** B = mat (1::'c))"
tendsto_nhds_continuousI: fixes f :: "'b \<Rightarrow> 'c"   and l :: "'c"   and x :: "'b" assumes "f \<midarrow>x\<rightarrow> l"   and "f x = l" shows "(f \<longlongrightarrow> l) (nhds x)"
vector_all_zero_from_index_eq: fixes i :: "'a"   and A :: "('b, 'a) vec" shows "(\<forall>m\<ge>i. A $ m = (0::'b)) = vector_all_zero_from_index (mod_type_class.to_nat i, vec_to_iarray A)"
intros_2: fixes F :: "'effect fs_set"   and P :: "'state"   and Q :: "'state"   and f :: "'effect" assumes "P \<sim>\<cdot>[F] Q"   and "f \<in>\<^sub>f\<^sub>s F" shows "L_bisimilar (AC (f, F, \<langle>f\<rangle>P)) (AC (f, F, \<langle>f\<rangle>Q))"
lm036: fixes a :: "('c \<times> 'd set) set"   and G :: "'d set"   and N :: "'c set" assumes "a \<in> allAllocations N G" shows "a \<in> injections (Domain a) (Range a) \<and> Range a \<in> all_partitions G \<and> Domain a \<subseteq> N"
emeasure_eq_1_AE: fixes S :: "'s stream set"   and s :: "'s" assumes "S \<in> K.T.events s"   and "AE x in K.T s. x \<in> S" shows "emeasure (K.T s) S = 1"
ta_reduce_def: fixes TA :: "('a, 'b) tree_automaton_rec"   and P :: "'a set" shows "ta_reduce TA P \<equiv> \<lparr>ta_initial = ta_initial TA \<inter> P, ta_rules = reduce_rules (ta_rules TA) P\<rparr>"
is_ff_dghmE: fixes \<alpha> :: "V"   and \<AA> :: "V"   and \<BB> :: "V"   and \<FF> :: "V" assumes "\<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^sub>.\<^sub>f\<^sub>f\<^bsub>\<alpha>\<^esub> \<BB>" obtains "\<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^sub>.\<^sub>f\<^sub>a\<^sub>i\<^sub>t\<^sub>h\<^sub>f\<^sub>u\<^sub>l\<^bsub>\<alpha>\<^esub> \<BB>" and "\<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^sub>.\<^sub>f\<^sub>u\<^sub>l\<^sub>l\<^bsub>\<alpha>\<^esub> \<BB>"
sign_changes_number_rootat: fixes ps :: "rat poly list" shows "sign_changes_inf (map real_of_rat_poly ps) = sign_changes_number_rootat ps"
scalar_prod_int_rat_def: fixes x :: "int vec"   and y :: "rat vec" shows "x \<bullet>i y = y \<bullet> of_int_hom.vec_hom x"
fls_left_inverse_base_factor: fixes x :: "'a"   and f :: "'a fls" assumes "x \<noteq> (0::'a)" shows "fls_left_inverse (fls_base_factor f) x = fls_base_factor (fls_left_inverse f x)"
MakeCatCod: fixes f :: "'a"   and C :: "('b, 'a, 'c) Category_scheme" assumes "f \<in> mor\<^bsub>C\<^esub>" shows "cod\<^bsub>C\<^esub> f = cod\<^bsub>MakeCat C\<^esub> f"
matrix_inv_2: fixes M :: "(('a, 'b) vec, 'c) vec" assumes "invertible M" shows "M ** matrix_inv M = mat (1::'a)"
distinct_concat_map_E_1: fixes f :: "'b \<Rightarrow> 'a list"   and xs :: "'b list"   and x :: "'b"   and y :: "'b" assumes "distinct (concat (map f xs))"   and "x \<in> set xs"   and "y \<in> set xs"   and "x \<noteq> y" shows "set (f x) \<inter> set (f y) = {}"
simps_2: fixes a :: "'c"   and t1 :: "'c bt"   and t2 :: "'c bt" shows "inorder (Br a t1 t2) = inorder t1 @ [a] @ inorder t2"
ge_game_trans: fixes x :: "game"   and y :: "game"   and z :: "game" assumes "ge_game (x, y)"   and "ge_game (y, z)" shows "ge_game (x, z)"
is_empty_inj: fixes e :: "'b" assumes "is_empty e" shows "e = empty"
Dugundji: fixes C :: "'c set"   and U :: "'d set"   and S :: "'d set"   and f :: "'d \<Rightarrow> 'c" assumes "convex C"   and "C \<noteq> {}"   and "closedin (top_of_set U) S"   and "continuous_on S f"   and "f ` S \<subseteq> C" obtains g :: "'d \<Rightarrow> 'c" where "continuous_on U g" and "g ` U \<subseteq> C" and "\<And>x. x \<in> S \<Longrightarrow> g x = f x"
le: shows "0\<^sub>m n n \<le>\<^sub>L A"
add_rel_frac: fixes r :: "'a"   and s :: "'a"   and r' :: "'a"   and s' :: "'a" assumes "(r, s) \<in> R \<times> (R\<setminus>I)"   and "(r', s') \<in> R \<times> (R\<setminus>I)" shows "cq.add_rel (cq.frac r s) (cq.frac r' s') = cq.frac (r \<cdot> s' + r' \<cdot> s) (s \<cdot> s')"
UN: fixes K :: "'c set set"   and S :: "'c set set" assumes "\<And>k. k \<in> K \<Longrightarrow> generate_topology_on S k" shows "generate_topology_on S (\<Union> K)"
odd_pow_cong: fixes n :: "nat"   and m :: "nat" assumes "1 < n"   and "odd m" shows "[(n - 1) ^ m = n - 1] (mod n)"
inject_2: fixes x2 :: "(val \<times> val) fset"   and y2 :: "(val \<times> val) fset" shows "(VFun x2 = VFun y2) = (x2 = y2)"
tendsto_upperbound: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and "open" :: "'a set \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'a"   and x :: "'a"   and F :: "'b filter"   and a :: "'a" assumes "class.linorder_topology less_eq less open"   and "filterlim f (topological_space.nhds open x) F"   and "\<forall>\<^sub>F i in F. less_eq (f i) a"   and "F \<noteq> \<bottom>" shows "less_eq x a"
inj_xM: shows "inj_on f (set_mset (add_mset x M))"
False_imp_not_eventually: fixes P :: "'b \<Rightarrow> bool"   and net :: "'b filter" assumes "\<forall>x. \<not> P x"   and "net \<noteq> bot" shows "\<not> eventually P net"
not_elim: fixes P :: "'b \<Rightarrow> bool"   and x :: "'b"   and C :: "bool" assumes "(\<not> P) x"   and "P x" shows "C"
Can_red_2: fixes u :: "'a local.term" assumes "Ide u" shows "u\<^bold>\<down> \<in> VHom u \<^bold>\<lfloor>u\<^bold>\<rfloor>"
coeff_inject: fixes x :: "'a poly"   and y :: "'a poly" shows "(poly.coeff x = poly.coeff y) = (x = y)"
iT_Plus_iUntil_conv: fixes I :: "nat set"   and k :: "nat"   and P :: "nat \<Rightarrow> bool"   and Q :: "nat \<Rightarrow> bool" shows "(P t1. t1 \<U> t2 I \<oplus> k. Q t2) = (P (t1 + k). t1 \<U> t2 I. Q (t2 + k))"
U_int: fixes i :: "nat" shows "(U i has_integral p i) UNIV"
multi_member_skip: fixes x :: "'c"   and XS :: "'c multiset"   and y :: "'c" assumes "x \<in># XS" shows "x \<in># {#y#} + XS"
compact_closure: fixes S :: "'c set" shows "compact (closure S) = bounded S"
surjective: fixes r :: "('a, 'b) graph_scheme" shows "r = \<lparr>nodes = nodes r, edges = edges r, \<dots> = graph.more r\<rparr>"
BinOp_\<tau>red1t_xt2: fixes uf :: "bool"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and h :: "'heap"   and e2 :: "'addr expr1"   and xs :: "'addr val list"   and e2' :: "'addr expr1"   and xs' :: "'addr val list"   and v :: "'addr val"   and bop :: "bop" assumes "\<tau>red1gt uf P t h (e2, xs) (e2', xs')" shows "\<tau>red1gt uf P t h (Val v \<guillemotleft>bop\<guillemotright> e2, xs) (Val v \<guillemotleft>bop\<guillemotright> e2', xs')"
nnfis: shows "Measure.measure M A \<in> nnfis \<chi> A M"
induct: fixes P :: "(fid_t \<Rightarrow> (role_t \<times> agent list \<times> atom list) option) \<Rightarrow> (key \<times> nat \<times> nat) signal \<Rightarrow> bool"   and a0 :: "fid_t \<Rightarrow> (role_t \<times> agent list \<times> atom list) option"   and a1 :: "(key \<times> nat \<times> nat) signal" assumes "\<And>runz B A Kab Ts Ta. P runz (Running [B, A] (Kab, Ts, Ta))"   and "\<And>runz B A Kab Ts Ta. P runz (Commit [B, A] (Kab, Ts, Ta))"   and "\<And>runz va. P runz (Running [] va)"   and "\<And>runz vb va. P runz (Running [vb] va)"   and "\<And>runz vb vd vf vg va. P runz (Running (vb # vd # vf # vg) va)"   and "\<And>runz va. P runz (Commit [] va)"   and "\<And>runz vb va. P runz (Commit [vb] va)"   and "\<And>runz vb vd vf vg va. P runz (Commit (vb # vd # vf # vg) va)" shows "P a0 a1"
clkp_set_def: fixes A :: "('f \<times> ('d, 'e) cconstraint \<times> 'g \<times> 'd list \<times> 'f) set \<times> ('f \<Rightarrow> ('d, 'e) cconstraint)" shows "clkp_set A = collect_clki (inv_of A) \<union> collect_clkt (trans_of A)"
NotActualActualD: fixes v :: "i"   and \<phi> :: "\<o>" assumes "\<not> [\<^bold>\<A>\<^bold>\<A>\<phi> in v]" shows "\<not> [\<^bold>\<A>\<phi> in v]"
cop_F_rejected_inv_def: fixes Xd :: "'d \<Rightarrow> 'e"   and Xh :: "'d \<Rightarrow> 'f"   and Pd :: "'e \<Rightarrow> ('d \<times> 'd) set"   and Ch :: "'f \<Rightarrow> 'd set \<Rightarrow> 'd set"   and ds :: "'e set"   and fp :: "'d set" assumes "ContractsWithUnilateralSubstitutesAndIRC Xd Xh Pd Ch" shows "ContractsWithUnilateralSubstitutesAndIRC.cop_F_rejected_inv Xd Pd Ch ds fp = (\<forall>x\<in>Rf (Contracts.CH Ch) fp. \<exists>fp'\<subseteq>fp. x \<in> fp' \<and> above (Pd (Xd x)) x \<subseteq> fp' \<and> Xd x \<notin> Xd ` Contracts.CH Ch fp')"
i_take_take: fixes m :: "nat"   and n :: "nat"   and f :: "nat \<Rightarrow> 'b" shows "f \<Down> n \<down> m = f \<Down> min n m"
deriv_linear: fixes c :: "'a" shows "deriv ((*) c) = (\<lambda>z. c)"
assms_1: shows "sem\<^sub>e\<^sub>s\<^sub>t_c {} \<I> A"
mult_bounds_enclose_zero1_1: fixes la :: "'b"   and ua :: "'b"   and lb :: "'b"   and ub :: "'b" assumes "la \<le> (0::'b)"   and "(0::'b) \<le> ua" shows "min (la * lb) (min (la * ub) (min (lb * ua) (ua * ub))) \<le> (0::'b)"
Receive_def: fixes recv :: "nat \<Rightarrow> 'a Frame list"   and store :: "nat \<Rightarrow> 'a Frame list"   and activation :: "nat \<Rightarrow> nat list" shows "Receive recv store activation \<equiv> \<forall>t. if activation t = [] then store t = recv t else store t = []"
union_right_cancel: fixes M :: "'b multiset"   and K :: "'b multiset"   and N :: "'b multiset" shows "(M + K = N + K) = (M = N)"
segment_inc1: fixes b :: "'a"   and a :: "'a" assumes "b \<in> carrier D" shows "(a \<prec> b \<and> a \<in> carrier D) = (a \<in> segment D b)"
cInf_le_iff: fixes A :: "'a set"   and x :: "'a" assumes "A \<noteq> {}"   and "bdd_below A" shows "(Inf A \<le> x) = (\<forall>y>x. \<exists>a\<in>A. a < y)"
pivot_tableau_equiv: fixes s :: "('i, 'a) state"   and x\<^sub>i :: "nat"   and x\<^sub>j :: "nat"   and v :: "nat \<Rightarrow> 'a" assumes "\<triangle> (\<T> s)"   and "x\<^sub>i \<in> lvars (\<T> s)"   and "x\<^sub>j \<in> rvars_eq (eq_for_lvar (\<T> s) x\<^sub>i)" shows "v \<Turnstile>\<^sub>t \<T> s = v \<Turnstile>\<^sub>t \<T> (pivot x\<^sub>i x\<^sub>j s)"
a2': shows "\<exists>F\<subseteq>insert e E. is_offending_flows_min_set F \<lparr>nodes = V, edges = insert e E\<rparr> nP"
assms: shows "\<I> \<oplus>\<^sub>\<I> \<I>_common \<turnstile>res res \<surd>"
simps_1: fixes i :: "nat" shows "remove_nth i [] = []"
lastoccy: shows "i + 1 + length bs + 1 \<le> index (rev (take i cs) @ x # rev bs @ x # rev as @ h) (cs ! i)"
commute: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "ac_operator f" shows "f a b = f b a"
GiSi: shows "\<GG> ! i \<lhd> G\<lparr>carrier := \<GG> ! (i + 1)\<rparr>"
ex_step1I: fixes x :: "'b"   and xs :: "'b list"   and r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and y :: "'b" assumes "x \<in> set xs"   and "r y x" shows "\<exists>ys. step1 r ys xs \<and> y \<in> set ys"
1: shows "fmrestrict_set vs (exec_plan s (as_proj as vs)) = exec_plan (fmrestrict_set vs s) (as_proj as vs)"
Basic_Sound: fixes C :: "Class"   and m :: "Method"   and l :: "int"   and ll :: "int"   and code :: "(int \<times> Instr) list"   and T :: "(Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<times> Val \<Rightarrow> bool"   and MI :: "(Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool"   and Anno :: "(int \<times> ((Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool)) list"   and l1 :: "int"   and l2 :: "int"   and ins :: "Instr"   and G :: "((Class \<times> Method \<times> int) \<times> ((Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool) \<times> ((Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<times> Val \<Rightarrow> bool) \<times> ((Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool)) list"   and n :: "nat" assumes "Segment C m l ll code"   and "MST\<down>(C, m) = Some (T, MI, Anno)"   and "l \<le> l1"   and "l1 < ll"   and "l2 = l1 + 1"   and "code\<down>l1 = Some ins"   and "basic ins"   and "derivAssum G C m l2 (Cachera n)" shows "deriv G C m l1 (Cachera n)"
wfp_on_restrict_to_tranclp_wfp_on_conv: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set" shows "wfp_on (restrict_to P A)\<^sup>+\<^sup>+ A = wfp_on P A"
exhaust: fixes y :: "msg" obtains x11 :: "nat" and x12 :: "nat" and x13 :: "nat" and x14 :: "nat" and x15 :: "k" and x16 :: "nat" and x17 :: "nat" and x18 :: "nat" where "y = Rreq x11 x12 x13 x14 x15 x16 x17 x18"   | x21 :: "nat" and x22 :: "nat" and x23 :: "nat" and x24 :: "nat" and x25 :: "nat" where "y = Rrep x21 x22 x23 x24 x25"   | x31 :: "nat \<Rightarrow> nat option" and x32 :: "nat" where "y = Rerr x31 x32"   | x41 :: "nat" and x42 :: "nat" where "y = Newpkt x41 x42"   | x51 :: "nat" and x52 :: "nat" and x53 :: "nat" where "y = Pkt x51 x52 x53"
iT_Plus_inext: fixes n :: "nat"   and k :: "nat"   and I :: "nat set" shows "inext (n + k) (I \<oplus> k) = inext n I + k"
forall_finite_1: fixes P :: "nat \<Rightarrow> bool" shows "(\<forall>i<0. P i) = True"
span_scale: fixes x :: "'c"   and S :: "'c set"   and c :: "'a" assumes "x \<in> vs2.span S" shows "c *b x \<in> vs2.span S"
sint_of_int_eq: fixes x :: "int" assumes "- (2 ^ (LENGTH('b) - 1)) \<le> x"   and "x < 2 ^ (LENGTH('b) - 1)" shows "sint (word_of_int x) = x"
cobounded1: fixes a :: "'b multiset"   and b :: "'b multiset" shows "a \<inter># b \<subseteq># a"
bb: shows "anyinternal (cutoff red (ds @ [Right]) T\<^sub>2) (\<lambda>p. \<not> red ((ds @ [Right]) @ p))"