hcju/isabelleps huggingface dataset:
mod_tree_lemma1: fixes y :: "nat tree"   and x :: "nat tree" assumes "\<forall>i\<in>set_tree y. 0 < i" shows "x mod (x + y) = x"
two_composable_arrows_of_spans_axioms: shows "two_composable_arrows_of_spans (\<cdot>) prj0 prj1 f (src f)"
new_privs_preserved: fixes c :: "'Com\<^sub>C"   and mds :: "Mode \<Rightarrow> 'Var\<^sub>C set"   and mem :: "'Var\<^sub>C \<Rightarrow> 'Val"   and c' :: "'Com\<^sub>C"   and mds' :: "Mode \<Rightarrow> 'Var\<^sub>C set"   and mem' :: "'Var\<^sub>C \<Rightarrow> 'Val"   and x :: "'Var\<^sub>C"   and m :: "Mode" assumes "\<langle>c, mds, mem\<rangle>\<^sub>C \<leadsto>\<^sub>C \<langle>c', mds', mem'\<rangle>\<^sub>C"   and "x \<notin> range var\<^sub>C_of" shows "(x \<in> mds m) = (x \<in> mds' m)"
distinct_3: fixes x1 :: "real mpoly"   and x3 :: "real mpoly" shows "Less x1 \<noteq> Leq x3"
interv_sum_list_conv_sum_set_nat: fixes f :: "nat \<Rightarrow> 'c"   and m :: "nat"   and n :: "nat" shows "sum_list (map f [m..<n]) = sum f (set [m..<n])"
R19_4: shows "pmf (sds R19) d = 1 / 2"
card_eq_length: fixes dxs :: "'b set_dlist" assumes "ID CEQ('b) \<noteq> None" shows "card (Collect (DList_Set.member dxs)) = DList_Set.length dxs"
tree_unique: fixes x :: "'r tree"   and f :: "'r tree \<rightarrow> 'r tree" assumes "x = f\<cdot>x"   and "\<And>i t u. tree_take i\<cdot>t = tree_take i\<cdot>u \<Longrightarrow> tree_take (Suc i)\<cdot>(f\<cdot>t) = tree_take (Suc i)\<cdot>(f\<cdot>u)" shows "x = fix\<cdot>f"
arr_inv: fixes f :: "'b" assumes "D.iso f" shows "D.arr (D.inv f)"
refine_using_invariants: fixes R :: "('a \<times> 'b) set"   and I :: "'a set"   and J :: "'b set"   and Ta :: "('a, 'c) TS_scheme"   and Tc :: "('b, 'd) TS_scheme" assumes "{R \<inter> I \<times> J} TS.trans Ta, TS.trans Tc {> R}"   and "{I \<inter> Domain (R \<inter> UNIV \<times> J)} TS.trans Ta {> I}"   and "{J \<inter> Range (R \<inter> I \<times> UNIV)} TS.trans Tc {> J}"   and "init Tc \<subseteq> R `` init Ta"   and "init Ta \<subseteq> I"   and "init Tc \<subseteq> J" shows "PO_refines (R \<inter> I \<times> J) Ta Tc"
length_zip: fixes xs :: "'e list"   and ys :: "'f list" shows "length (xs || ys) = min (length xs) (length ys)"
trie_reverse_key_def: fixes ksv :: "'c list \<times> 'd" shows "trie_reverse_key ksv = (rev (fst ksv), snd ksv)"
psimps: fixes g :: "'g"   and v :: "'var"   and n :: "'node"   and phis :: "('node \<times> 'var, ('var \<times> 'node \<times> Def) list) mapping" assumes "readVariableRecursive_readArgs_dom (Inl (g, v, n, phis))" shows "readVariableRecursive g v n phis = (if v \<in> defs g n then ((v, n, SimpleDef), phis) else case predecessors g n of [] \<Rightarrow> ((v, n, PhiDef), Mapping.update (n, v) [] phis) | [m] \<Rightarrow> readVariableRecursive g v m phis | m # ab # lista \<Rightarrow> case Mapping.lookup phis (n, v) of None \<Rightarrow> let phis = Mapping.update (n, v) [] phis; (phis, args) = readArgs g v n phis (m # ab # lista) in ((v, n, PhiDef), Mapping.update (n, v) args phis) | Some x \<Rightarrow> ((v, n, PhiDef), phis))"
Card_order_wo_rel: fixes r :: "('c \<times> 'c) set" assumes "Card_order r" shows "wo_rel r"
succ_in_G: fixes n :: "nat" shows "\<forall>v\<in>succs n. v \<in> set (g_V G)"
distinct_23: fixes x3 :: "nat"   and x4 :: "nat" shows "Nonce x3 \<noteq> Key x4"
star_left_unfold_equal: fixes star :: "'c \<Rightarrow> 'c"   and one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and bot :: "'c"   and x :: "'c" assumes "class.left_kleene_algebra star one times sup less_eq less bot" shows "sup one (times x (star x)) = star x"
induct: fixes P :: "primfun \<Rightarrow> primfun \<Rightarrow> bool"   and a0 :: "primfun"   and a1 :: "primfun" assumes "\<And>x y. P (LnChain x) (LnChain y)" shows "P a0 a1"
Interp_imp_Interp: fixes C :: "'a literal multiset"   and D :: "'a literal multiset"   and D' :: "'a literal multiset" assumes "C \<le> D"   and "D \<le> D'"   and "Interp D \<Turnstile> C" shows "Interp D' \<Turnstile> C"
sets_pair_measure: fixes A :: "'d measure"   and B :: "'e measure" shows "sets (A \<Otimes>\<^sub>M B) = sigma_sets (space A \<times> space B) {a \<times> b |a b. a \<in> sets A \<and> b \<in> sets B}"
regular_event: fixes ev :: "('a, 'b, 'c) event" shows "regular_event ev \<equiv> isTrans ev \<or> isSend ev \<or> isRecv ev"
comp_cod_arr: fixes f :: "'a1"   and b :: "'a1" assumes "C1.arr f"   and "C1.cod f = b" shows "b \<cdot>\<^sub>1 f = f"
Abs_finfun_induct: fixes P :: "('a, 'b) finfun \<Rightarrow> bool"   and x :: "('a, 'b) finfun" assumes "\<And>y. y \<in> finfun \<Longrightarrow> P (Abs_finfun y)" shows "P x"
invertible_right_inverse: fixes u :: "'a" assumes "comm.additive.invertible u"   and "u \<in> R" shows "u - u = \<zero>"
set_empty2: fixes xs :: "'a list" shows "({} = list.set xs) = (xs = [])"
typeof_NoClassDefFoundError: fixes h :: "heap" assumes "preallocated h" shows "typeof\<^bsub>h\<^esub> (Addr (addr_of_sys_xcpt NoClassDefFoundError)) = \<lfloor>Class NoClassDefFoundError\<rfloor>"
lookup_fresh': fixes z :: "name"   and \<theta> :: "(name \<times> lam) list" assumes "z \<sharp> \<theta>" shows "\<theta> maps z to Var z"
zcount_eq_zero_iff: fixes M :: "'a zmultiset"   and x :: "'a" shows "(zcount M x = 0) = (x \<notin>#\<^sub>z M)"
maintain_invar2_1: fixes Q :: "'v \<Rightarrow> enat"   and \<pi> :: "'v \<Rightarrow> 'v option"   and u :: "'v"   and d :: "nat" assumes "prim_invar2 Q \<pi>"   and "Q u = enat d"   and "\<forall>v. enat d \<le> Q v" shows "prim_invar2 (Q' Q \<pi> u) (\<pi>' Q \<pi> u)"
simps_2: fixes OP :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and N :: "'a"   and r :: "'a" shows "rexp_of_list OP N [r] = r"
K_elem_repr_inj: fixes \<mu> :: "('s \<times> ('c \<Rightarrow> real) set) pmf"   and l :: "'s"   and R :: "('c \<Rightarrow> real) set" assumes "\<mu> \<in> \<K> (l, R)" shows "inj_on (\<lambda>(l, R). (l, SOME u. u \<in> R)) (set_pmf \<mu>)"
eq: shows "s.enum ` {i..j} = t.enum ` {i + d..j + d}"
some_H'h': fixes M :: "('b \<times> real) set set"   and E :: "'b set"   and p :: "'b \<Rightarrow> real"   and F :: "'b set"   and f :: "'b \<Rightarrow> real"   and c :: "('b \<times> real) set set"   and H :: "'b set"   and h :: "'b \<Rightarrow> real"   and x :: "'b" assumes "M = norm_pres_extensions E p F f"   and "c \<in> chains M"   and "graph H h = \<Union> c"   and "x \<in> H" shows "\<exists>H' h'. x \<in> H' \<and> graph H' h' \<subseteq> graph H h \<and> linearform H' h' \<and> Subspace.subspace H' E \<and> Subspace.subspace F H' \<and> graph F f \<subseteq> graph H' h' \<and> (\<forall>x\<in>H'. h' x \<le> p x)"
perm_restrict_perm_restrict: fixes f :: "'b \<Rightarrow> 'b"   and A :: "'b set"   and B :: "'b set" shows "perm_restrict (perm_restrict f A) B = perm_restrict f (A \<inter> B)"
sup_inf_absorb: fixes x :: "'a"   and y :: "'a" shows "x + x \<sqinter> y = x"
mcp_prefix_of_suffix: fixes vs :: "'e list"   and xs :: "'e list"   and V :: "'e list set"   and xs' :: "'e list" assumes "mcp (vs @ xs) V vs"   and "prefix xs' xs" shows "mcp (vs @ xs') V vs"
coeffs_nth: fixes p :: "'b poly"   and n :: "nat" assumes "p \<noteq> 0"   and "n \<le> degree p" shows "coeffs p ! n = coeff p n"
braun_del_hi: fixes t :: "'b tree"   and st :: "nat" assumes "braun t"   and "size t = st" shows "braun (del_hi st t)"
DFS_axioms: shows "DFS succs isNode invariant ins memb empt nodeAbs"
h: shows "((\<Sum>i = 0..<d + 1. f i * C ^ (d - i)) * C ^ (d mod 2) < 0) = ((\<Sum>i = 0..<d + 1. f i / C ^ i) < 0)"
cases: fixes a1 :: "'c set list"   and a2 :: "'c"   and a3 :: "'c" assumes "of_weak_ranking a1 a2 a3" obtains i :: "nat" and j :: "nat" and xs :: "'c set list" and x :: "'c" and y :: "'c" where "a1 = xs" and "a2 = y" and "a3 = x" and "i \<le> j" and "i < length xs" and "j < length xs" and "x \<in> xs ! i" and "y \<in> xs ! j"
Quotient_alt_def: fixes R :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and Abs :: "'c \<Rightarrow> 'd"   and Rep :: "'d \<Rightarrow> 'c"   and T :: "'c \<Rightarrow> 'd \<Rightarrow> bool" shows "Quotient R Abs Rep T = ((\<forall>a b. T a b \<longrightarrow> Abs a = b) \<and> (\<forall>b. T (Rep b) b) \<and> (\<forall>x y. R x y = (T x (Abs x) \<and> T y (Abs y) \<and> Abs x = Abs y)))"
conj_disj_distrib: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x \<^bold>\<sqinter> (y \<^bold>\<squnion> z) = x \<^bold>\<sqinter> y \<^bold>\<squnion> x \<^bold>\<sqinter> z"
simps: fixes random :: "natural \<times> natural \<Rightarrow> ('a \<times> (unit \<Rightarrow> term)) \<times> natural \<times> natural"   and nrandom :: "natural"   and seed :: "natural \<times> natural" shows "Random_Sequence.iter random nrandom seed = (if nrandom = 0 then Lazy_Sequence.empty else Lazy_Sequence (\<lambda>u. let ((x, uu_), seed') = random seed in Some (x, Random_Sequence.iter random (nrandom - 1) seed')))"
opt: shows "T\<^sub>p init qs Strat = T\<^sub>p_opt init qs"
map_mem_spmf_of_set: fixes B :: "'b set"   and A :: "'b set" assumes "finite B"   and "B \<noteq> {}" shows "map_spmf (\<lambda>x. x \<in> A) (spmf_of_set B) = spmf_of_pmf (bernoulli_pmf (real (card (A \<inter> B)) / real (card B)))"
sep_disj_addI3: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x + y ## z"   and "x ## y" shows "x ## y + z"
monomial_power: fixes t :: "'a"   and c :: "'b"   and n :: "nat" shows "monomial c t ^ n = monomial (c ^ n) (\<Sum>i = 0..<n. t)"
Pre_segment: fixes D :: "('c, 'd) Order_scheme"   and a :: "'c"   and b :: "'c" assumes "Worder D"   and "a \<in> segment D b"   and "ExPre (Iod D (segment D b)) a" shows "ExPre D a \<and> Pre D a = Pre (Iod D (segment D b)) a"
ih_5: shows "InvariantWatchListsCharacterization (getWatchList state') (getWatch1 state') (getWatch2 state')"
LIVE_def: shows "LIVE = [: x \<leadsto> (u, x') . u 0 = 0 \<and> x = x' :] \<circ> {.u, x. prec_st prec_ex rel_ex u x.} \<circ> [: (u, x) \<leadsto> y . (\<box> (\<lambda>u x y. rel_ex (u 0) (u 1) (x 0) (y (0::'a)))) u x y \<and> (\<box> \<diamond> (\<lambda>y. y (0::'a))) y :]"
continuous_on_mult_left: fixes s :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and c :: "'b" assumes "continuous_on s f" shows "continuous_on s (\<lambda>x. c * f x)"
entails_concl_compact_union: fixes EE :: "'f set"   and CC :: "'f set"   and DD :: "'f set" assumes "finite EE"   and "CC \<union> DD \<Turnstile> EE" shows "\<exists>CC'\<subseteq>CC. finite CC' \<and> CC' \<union> DD \<Turnstile> EE"
fimage_strict_vsubset: fixes f :: "V"   and A :: "V" shows "f `\<^sub>\<bullet> A \<subseteq>\<^sub>\<circ> f `\<^sub>\<bullet> \<D>\<^sub>\<bullet> f"
prod_lub_def: fixes Sup_a :: "'c set \<Rightarrow> 'c"   and Sup_b :: "'d set \<Rightarrow> 'd"   and Y :: "('c \<times> 'd) set" shows "prod_lub Sup_a Sup_b Y = (Sup_a (fst ` Y), Sup_b (snd ` Y))"
interval_eq_set_of_iff: fixes X :: "'b interval"   and Y :: "'b interval" shows "(X = Y) = (set_of X = set_of Y)"
add_1: fixes M :: "'b multiset" assumes "N \<subset># M" shows "(\<Sum>x\<in>#N. Suc (f x)) < (\<Sum>x\<in>#M. Suc (f x))"
simps_3: fixes t :: "'c local.term"   and u :: "'c local.term" shows "Can (t \<^bold>\<star> u) = (Can t \<and> Can u \<and> Src t = Trg u)"
authK_authentic: fixes A :: "agent"   and authK :: "nat"   and Peer :: "msg"   and Ta :: "msg"   and authTicket :: "msg"   and evs :: "event list" assumes "Crypt (shrK A) \<lbrace>Key authK, Peer, Ta, authTicket\<rbrace> \<in> parts (knows Spy evs)"   and "A \<notin> bad"   and "evs \<in> kerbIV_gets" shows "Says Kas A (Crypt (shrK A) \<lbrace>Key authK, Peer, Ta, authTicket\<rbrace>) \<in> set evs"
g_meas: shows "g \<in> N \<rightarrow>\<^sub>M P"
simps_11: fixes e :: "'c exp"   and M :: "char list"   and es :: "'c exp list" shows "ss_exp (e\<bullet>M(es)) = (case val_of e of None \<Rightarrow> ss_exp e | \<lfloor>v\<rfloor> \<Rightarrow> case map_vals_of es of None \<Rightarrow> the (ss_exps es) | \<lfloor>t\<rfloor> \<Rightarrow> e\<bullet>M(es))"
deadlockI: fixes s :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set" assumes "\<And>t x. \<lbrakk>thr s t = \<lfloor>(x, no_wait_locks)\<rfloor>; \<not> final x; wset s t = None\<rbrakk> \<Longrightarrow> t \<turnstile> \<langle>x, shr s\<rangle> \<wrong> \<and> (\<forall>LT. t \<turnstile> \<langle>x, shr s\<rangle> LT \<wrong> \<longrightarrow> (\<exists>lt\<in>LT. must_wait s t lt (dom (thr s))))"   and "\<And>t x ln l. \<lbrakk>thr s t = \<lfloor>(x, ln)\<rfloor>; 0 < ln $ l; \<not> waiting (wset s t)\<rbrakk> \<Longrightarrow> \<exists>l t'. 0 < ln $ l \<and> t \<noteq> t' \<and> thr s t' \<noteq> None \<and> has_lock (locks s $ l) t'"   and "\<And>t x w. thr s t = \<lfloor>(x, no_wait_locks)\<rfloor> \<Longrightarrow> wset s t \<noteq> \<lfloor>PostWS w\<rfloor>" shows "deadlock s"
avl_split_max: fixes t :: "('b \<times> nat) tree" assumes "avl t"   and "t \<noteq> \<langle>\<rangle>" shows "avl (fst (split_max t)) \<and> height t \<in> {height (fst (split_max t)), height (fst (split_max t)) + 1}"
ordinal_div_0: fixes x :: "ordinal" shows "x div 0 = 0"
tau_converg': fixes i :: "nat"   and x :: "nat"   and t :: "nat" assumes "dettime i x \<down>= t" shows "\<tau> i x \<down>= e_nth (block i t) x"
lSup_insert_LNil: fixes Y :: "'c llist set" shows "lSup (insert LNil Y) = lSup Y"
assms: shows "A \<subseteq>\<^sub>\<circ> B \<inter>\<^sub>\<circ> C"
assms_2: shows "get_coeffs var p = (A, B, C)"
coinduction: fixes rel :: "('b \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> bool"   and eval :: "'c \<Rightarrow> 'b"   and retr :: "('b \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and R :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "cong rel eval retr"   and "\<forall>R. R \<le> retr R \<longrightarrow> R \<le> (=)"   and "R \<le> retr (cong.gen_cong rel eval R)" shows "R \<le> (=)"
SN_partp_SN_part_eq: fixes r :: "('b \<times> 'b) set" shows "SN_partp (\<lambda>x xa. (x, xa) \<in> r) = (\<lambda>x. x \<in> SN_part r)"
observable_moves_preds_transfers_1: fixes S :: "'node SDG_node set"   and f :: "'edge \<Rightarrow> ('var, 'val, 'ret, 'pname) edge_kind"   and ms :: "'node list"   and s :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list"   and as :: "'edge list"   and ms' :: "'node list"   and s' :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list" assumes "S,f \<turnstile> (ms,s) =as\<Rightarrow> (ms',s')" shows "preds (map f as) s"
assms: shows "mopup_jump_over1 (Suc (2 * n), l, Bk # xs) lm n ires"
rel_spmf_of_set_bij: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and B :: "'d set"   and R :: "'c \<Rightarrow> 'd \<Rightarrow> bool" assumes "bij_betw f A B"   and "\<And>x. x \<in> A \<Longrightarrow> R x (f x)" shows "rel_spmf R (spmf_of_set A) (spmf_of_set B)"
length_pfd1: fixes x :: "'b"   and ys :: "('b \<times> nat) list" shows "length (snd (partial_fraction_decomposition x ys)) = length ys"
assms: shows "\<forall>i\<ge>n. \<exists>j\<ge>i. P (S i) (S j)"
assms: shows "s' = snd (fst (load_sub2 addr asi r win w s)) \<and> get_S (cpu_reg_val PSR s) = 0"
return_valid: fixes fg :: "('a, 'b, 'c, 'd, 'e) flowgraph_rec_scheme"   and p :: "'b" assumes "flowgraph fg" shows "proc_of fg (return fg p) = p"
Says_serverK_unique: fixes B' :: "agent"   and A' :: "agent"   and Na :: "nat"   and Nb :: "nat"   and PMS :: "nat"   and NA :: "nat"   and NB :: "nat"   and Y :: "msg"   and evs :: "event list"   and A :: "agent"   and B :: "agent" assumes "Says B' A' (Crypt (serverK (Na, Nb, PRF (PMS, NA, NB))) Y) \<in> set evs"   and "Notes A \<lbrace>Agent B, Nonce PMS\<rbrace> \<in> set evs"   and "evs \<in> tls"   and "A \<notin> bad"   and "B \<notin> bad"   and "B' \<noteq> Spy" shows "B = B'"
hom_zero: shows "h \<zero> = \<zero>\<^bsub>H\<^esub>"
Cauchy_theorem_triangle_interior: fixes a :: "complex"   and b :: "complex"   and c :: "complex"   and f :: "complex \<Rightarrow> complex" assumes "continuous_on (convex hull {a, b, c}) f"   and "f holomorphic_on interior (convex hull {a, b, c})" shows "(f has_contour_integral 0) (linepath a b +++ linepath b c +++ linepath c a)"
ACI_nPlus_PLUS: fixes xs1 :: "'c rexp list"   and xs2 :: "'c rexp list" assumes "xs1 \<noteq> []"   and "xs2 \<noteq> []"   and "\<forall>x\<in>set (xs1 @ xs2). \<nexists>r s. x = Plus r s"   and "sorted xs2"   and "distinct xs2" shows "ACI_nPlus (PLUS xs1) (PLUS xs2) = PLUS (sorted_list_of_set (set (xs1 @ xs2)))"
simps_5: fixes acc2 :: "'a list list"   and l1 :: "'a list"   and l2 :: "'a list"   and ls :: "'a list list" shows "merge_list acc2 (l1 # l2 # ls) = merge_list (merge l1 l2 # acc2) ls"
length_butlast: fixes xs :: "'c list" shows "|butlast xs| = |xs| - 1"
wt_ge_num_args_if_\<delta>_eq_0: fixes s :: "('s, 'v) tm" assumes "\<delta> = 0" shows "num_args s \<le> wt s"
map_option_eq_Some: fixes f :: "'f \<Rightarrow> 'e"   and xo :: "'f option"   and y :: "'e" shows "(map_option f xo = Some y) = (\<exists>z. xo = Some z \<and> f z = y)"
all_rename: fixes \<phi> :: "'fmla"   and x :: "'var"   and y :: "'var" assumes "\<phi> \<in> fmla"   and "x \<in> var"   and "y \<in> var"   and "y \<notin> Fvars \<phi>" shows "all x \<phi> = all y (subst \<phi> (Var y) x)"
intros: fixes A :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and B :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and S :: "('b, 'd) dds"   and S' :: "('c, 'e) dds" assumes "(A ===> rel_prod B (rel_dds A B)) (run S) (run S')" shows "rel_dds A B S S'"
covsi: shows "ov O s\<inverse> \<subseteq> ov \<union> f\<inverse> \<union> d\<inverse>"
tl_append2: fixes xs :: "'b list"   and ys :: "'b list" assumes "xs \<noteq> []" shows "tl (xs @ ys) = tl xs @ ys"
vsv_valneq_v11I: fixes A :: "V"   and f :: "V \<Rightarrow> V" assumes "\<And>x y. \<lbrakk>x \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> (VLambda A f); y \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> (VLambda A f); x \<noteq> y\<rbrakk> \<Longrightarrow> VLambda A f\<lparr>x\<rparr> \<noteq> VLambda A f\<lparr>y\<rparr>" shows "v11 (VLambda A f)"
monoid_axioms: shows "Group_Theory.monoid stfx.carrier_direct_lim stfx.add_rel (stfx.class_of V (zero_str\<^sub>Y V))"
lexs_imp_lex: fixes x :: "real \<times> real"   and y :: "real \<times> real" assumes "lexs x y" shows "Counterclockwise_2D_Arbitrary.lex x y"
some_in_eq: fixes A :: "'c set" shows "((SOME x. x \<in> A) \<in> A) = (A \<noteq> {})"
wf\<^sub>l\<^sub>s\<^sub>s\<^sub>t\<^sub>s_def: fixes \<S> :: "('lbl strand_label \<times> ('fun, 'var) stateful_strand_step) list set" shows "wf\<^sub>l\<^sub>s\<^sub>s\<^sub>t\<^sub>s \<S> \<equiv> (\<forall>\<A>\<in>\<S>. wf'\<^sub>s\<^sub>s\<^sub>t {} (unlabel \<A>)) \<and> (\<forall>\<A>\<in>\<S>. \<forall>\<A>'\<in>\<S>. fv\<^sub>l\<^sub>s\<^sub>s\<^sub>t \<A> \<inter> bvars\<^sub>l\<^sub>s\<^sub>s\<^sub>t \<A>' = {})"
execute_3: shows "fst agQ \<in> alphabet (complement A) \<and> snd agQ \<in> transition (complement A) (fst agQ) fP"
uconnectedV_insert_selfloop: fixes x :: "'a"   and F :: "'a uprod set" assumes "x \<in> verts" shows "Restr (uconnected (insert (Upair x x) F)) verts = Restr (uconnected F) verts"
dup_in_hom: fixes f :: "'a"   and a :: "'a"   and b :: "'a" assumes "\<guillemotleft>f : a \<rightarrow> b\<guillemotright>" shows "\<guillemotleft>\<d>[f] : a \<rightarrow> b \<otimes> b\<guillemotright>"
beta: fixes x :: "V"   and A :: "V"   and b :: "V \<Rightarrow> V" assumes "x \<in>\<^sub>\<circ> A" shows "VLambda A b\<lparr>x\<rparr> = b x"
var_rename_fv_set_disjoint: fixes M :: "('d, 'e \<times> nat) Term.term set"   and s :: "('d, 'e \<times> nat) Term.term"   and t :: "('f, 'e \<times> nat) Term.term" assumes "finite M"   and "s \<in> M" shows "fv s \<inter> fv (t \<cdot> var_rename (max_var_set (fv\<^sub>s\<^sub>e\<^sub>t M))) = {}"
pmaxsimpchainD_maxsimp: fixes xs :: "'a set list"   and x :: "'a set" assumes "pmaxsimpchain xs"   and "x \<in> set xs" shows "maxsimp x"
strict_lunit: fixes a :: "'a" assumes "ide a" shows "\<l>[a] = a"
group_right_one: fixes x :: "'a" shows "x * (1::'a) = x"
no_excess_imp_maxflow: assumes "\<forall>u\<in>V - {s, t}. excess f u = (0::'a)" shows "isMaxFlow f"
suffix_imp_sublist: fixes xs :: "'c list"   and ys :: "'c list" assumes "suffix xs ys" shows "sublist xs ys"
simps_3: fixes ODE1 :: "('b, 'a) ODE"   and ODE2 :: "('b, 'a) ODE" shows "BVO (OProd ODE1 ODE2) = BVO ODE1 \<union> BVO ODE2"
a: assumes "R (p \<bullet> x) y" shows "R (- p \<bullet> y) x"
tendsto: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'e \<Rightarrow> 'a"   and a :: "'a"   and F :: "'e filter" assumes "bounded_linear f"   and "(g \<longlongrightarrow> a) F" shows "((\<lambda>x. f (g x)) \<longlongrightarrow> f a) F"
x_as_monom: shows "[:0::'a, 1::'a:] = monom (1::'a) 1"
assms_2: fixes w :: "(real, 'n) vec" assumes "w \<in> S" shows "orthogonal v w"
times_tree_def: fixes x :: "'a tree"   and y :: "'a tree" shows "x * y = pure (*) \<diamondop> x \<diamondop> y"
simps_2: fixes p :: "int"   and i :: "'a"   and l :: "'a factor_tree"   and r :: "'a factor_tree" shows "product_factor_tree p (Factor_Node i l r) = (let L = product_factor_tree p l; R = product_factor_tree p r; f = factor_node_info L; g = factor_node_info R; fg = poly_mod.Mp p (f * g) in Factor_Node fg L R)"
wt: shows "P,T,mxs,length ins,xt \<turnstile> ins ! pc,pc :: \<Phi> C M"
cball_eq_cball_iff: fixes x :: "'b"   and d :: "real"   and y :: "'b"   and e :: "real" shows "(cball x d = cball y e) = (d < 0 \<and> e < 0 \<or> x = y \<and> d = e)"
\<iota>_in_hom: shows "\<F>C.in_hom \<F>C.\<iota>\<^sub>F\<^sub>M\<^sub>C (\<F>C.tensor \<F>C.unity \<F>C.unity) \<F>C.unity"
fun_sum_list_commute_canc: fixes f :: "'a \<Rightarrow> 'b"   and xs :: "'a list" assumes "\<And>x y. f (x + y) = f x + f y" shows "f (sum_list xs) = sum_list (map f xs)"
pred_inject: fixes P :: "'c \<Rightarrow> bool"   and a :: "'c"   and aa :: "'c stream" shows "pred_stream P (a ## aa) = (P a \<and> pred_stream P aa)"
commutes_with_composition: fixes x :: "'b"   and y :: "'b" assumes "x \<in> A"   and "y \<in> A" shows "f (multA x y) = multB (f x) (f y)"
flit_of_hlit_hlit_of_flit: fixes l :: "fterm literal" assumes "ground\<^sub>l l" shows "flit_of_hlit (hlit_of_flit l) = l"
abs_err: shows "\<bar>pdevs_val e (snd (fst X)) * pdevs_val e (snd (fst Y)) + ex * (fst (fst Y) + pdevs_val e (snd (fst Y))) + ey * (fst (fst X) + pdevs_val e (snd (fst X))) + ex * ey\<bar> \<le> snd (mult_aform X Y)"
succ_in_G: fixes G :: "nat graph_rec"   and n :: "nat" assumes "cfg_doms G" shows "\<forall>v\<in>cfg_doms.succs G n. v \<in> set (g_V G)"
induced_ghom_def: fixes F :: "('e, 'g) Group_scheme"   and G :: "('f, 'h) Group_scheme"   and f :: "'e \<Rightarrow> 'f" shows "f\<dieresis>\<^bsub>F,G\<^esub> = (\<lambda>X\<in>set_rcs F (gker\<^bsub>F,G\<^esub> f). f (SOME x. x \<in> X))"
IH: assumes "i + length xs \<le> length ys"   and "\<forall>jj<length xs. ys ! (i + jj) = xs ! jj" shows "sublist_at xs ys i"
secgwext_hosts_def: shows "secgwext_hosts \<equiv> [''hypervisor'', ''securevm1'', ''securevm2'', ''publicvm1'', ''publicvm2'', ''INET'']"
bin_cat_assoc_sym: fixes m :: "nat"   and n :: "nat"   and z :: "int"   and y :: "int"   and x :: "int" shows "concat_bit m (concat_bit n z y) x = concat_bit (min m n) z (concat_bit (m - n) y x)"
rvI: fixes n :: "'node SDG_node"   and as :: "'edge list"   and n' :: "'node SDG_node"   and S :: "'node SDG_node set"   and V :: "'var" assumes "parent_node n -as\<rightarrow>\<^sub>\<iota>* parent_node n'"   and "n' \<in> HRB_slice S"   and "V \<in> Use\<^bsub>SDG\<^esub> n'"   and "\<forall>n''. valid_SDG_node n'' \<and> parent_node n'' \<in> set (sourcenodes as) \<longrightarrow> V \<notin> Def\<^bsub>SDG\<^esub> n''" shows "V \<in> rv S n"
assert_all_def: fixes t :: "(nat \<times> linear_poly) list"   and as :: "('i \<times> 'a atom) list" shows "assert_all t as \<equiv> let s = assert_all_state t as in if \<U> s then Inl (the (\<U>\<^sub>c s)) else Inr (\<V> s)"
seqI': fixes f :: "'a"   and a :: "'a"   and b :: "'a"   and g :: "'a"   and c :: "'a" assumes "S.in_hom f a b"   and "S.in_hom g b c" shows "S.seq g f"
last_ConsL: fixes xs :: "'e list"   and x :: "'e" assumes "xs = []" shows "last (x # xs) = x"
map_fm_binders_negation_free: fixes \<phi> :: "atom fm"   and f :: "atom \<Rightarrow> nat \<Rightarrow> atom"   and n :: "nat" assumes "negation_free \<phi>" shows "negation_free (map_fm_binders f \<phi> n)"